ASTRONOMY LIBRARY A TREATISE ON THE SUN'S RADIATION AND OTHER SOLAR PHENOMENA'' IN CONTINUATION OF THE METEOROLOGICAL TREATISE ON ATMOSPHERIC CIRCULATION AND RADIATION, 1915 BY FRANK H. BIGELOW, M.A., L.H.D PROFESSOR OF METEOROLOGY IN THE U. S. WEATHER BUREAU, iSpI-IQIO AND IN THE ARGENTINE METEOROLOGICAL OFFICE SINCE IQIO FIRST EDITION NEW YORK JOHN WILEY & SONS, INC. LONDON: CHAPMAN & HALL, LIMITED 1918 * # * Copyright, 1918, by FRANK H. BIGELOW PUBLISHERS PRINTING COMPANY 207-217' West Twenty-fifth Street, New York PREFACE THERE are two fundamental problems in solar physics that need an immediate solution: (l) the nature of the solar radiation, whether it is black or gray, that is, whether it has full or imper- fect efficiency, together with its amount at the sun, and hi the different layers of the earth's atmosphere; (2) the physical con- ditions at which this radiation generates, and the pressures, temperatures, densities, and gas efficiencies that are concerned with the observed phenomena in the solar spectra. It has been found that the system of non-adiabatic meteorology, which was explained in the Author's Treatise on Atmospheric Circula- tion and Radiation becomes equally applicable in the solar atmospheres by means of a simple transformation. These terms have been computed for several monatomic gases, by the method of trials, and the results are not only instructive in many directions, but they are in complete harmony with the observations made with the spectro-heliograph and the spectro- bolometer. The radiation is black at 5.85 gram calories per square centimeter per minute, when reduced to the equivalent at the distance of the earth; it loses 1.87 calories in the hemi- spherical non-adiabatic shell of the sun, and is effective on the outside of the earth's atmosphere at 3.98 calories; the course of its depletion is measured by the bolometer to 2.47 calories at the sea level. This conforms with the thermodynamic con- ditions of the two atmospheres. The pyrheliometer fails to record three important depletions, and it, therefore, cannot account for more than 1.94 calories by the Bouguer formula. The following Treatise is a continuation of the earlier one on Meteorology. FRANK H. BIGELOW. SOLAR AND MAGNETIC OBSERVATORY, PlLAR-CORDOBA ARGENTINA. December, 1917. in 380098 TABLE OF CONTENTS PAGE PREFACE .- 1 . iii CHAPTER I THE THERMODYNAMIC PROCESSES IN THE SOLAR ATMOSPHERE . . . 1 Introduction * . . . . . . . . ....... 1 Historical Remarks ..... -w. , '. 9 Derivation of the Non-Adiabatic System of Equations .... 10 Preliminary Summary of Conditions 13 The Working Equations 15 Adiabatic and Non-Adiabatic . . 16 The Thermodynamic Equation for the Conservation of Energy . . 18 The Initial Constants and Coefficients for the Computations in the Solar Atmosphere 19 Notation and Fundamental Formulas for the Electromagnetic Field . 28 Poynting's Equation for the Flux of Radiation 30 Pressure of Radiation 32 The Mean Flux of Energy . . . .'...... . . 33 The Mean Pressure of Radiation ...... . . . . . 34 Emission and Temperature, Stefan's Law 34 Summary of the Formulas of Radiation 35 Tables of the Constituents of Radiation in the Volume .... 37 Definition of Terms and Dimensions 38 Table of Astronomical Constants 41 CHAPTER II COMPUTATION OF THE THERMODYNAMIC TERMS 43 General Remarks . 43 The Atomic and Molecular Weights . . 56 The Distribution of the Temperatures .58 The Heights at Which the Same Temperature Occurs for the Differ- ent Elements by the Hyperbolic Law . 62 The Constant Temperature Near the Photosphere and the Variations of the Temperature Outside the Isothermal Region .... 66 The Distribution of the Pressures 68 The Distribution of the Densities 74 The Gas Efficiency of the Different Elements . . 78 The Cause of the Sharp Limb of the Sun 79 v VI TABLE OF CONTENTS CHAPTER III PAGE THE DETERMINATION OF THE "SOLAR CONSTANT" OF RADIATION IN THE ISOTHERMAL LAYER OF THE SUN 83 Statement of the Problem 83 The Distribution of the Free Heat (Qi Qo) 85 The Distribution of the Entropy (Si So) 88 The Distribution of the Work of Expansion (Wi Wo) .... 91 The Distribution of the Inner Energy (/i-t/o) 92 The Distribution of the Radiation Potential KIQ 95 Method of Computing the Coefficients and Exponents in the Formula of Radiation . . . . , . .... ... . . . . 96 The Distribution of the Exponent (a) in K w = c T a 100 The Distribution of the Coefficient (log c) .101 The Solar Constant of Radiation 103 The Mean Values of the Coefficient and the Exponent of Radiation in the Stefan Law 104 The Logarithmic Spiral Formulas 109 The Logarithmic Spiral for a =35 10' Ill The Solar True Radiation at the Distance of the Earth . . . .115 The Average Physical Conditions in the Solar Strata Where the Radi- ation Originates . .' . . . " . . . . . . . . . 116 The Evaluation of J a = Ci TV-Co TV in gr. cal./cm. 2 min. . . .117 CHAPTER IV THE COEFFICIENTS IN THE STEFAN AND THE WIEN-PLANCK FORMULAS FOR BLACK-BODY RADIATION ,. . 120 The Conversion from the (M.K.S.) to the (C.G.S.) System . . . 120 The Coefficients in the Wien-Planck Formula of Spectrum Radiation 127 The Formulas of Computation 129 The Thermodynamics of Radiation in the Solar Atmospheres . . 134 Tables for the Mean Square Velocity 135 Tables for the Number of Molecules per Cu. Cm .136 Tables for the Thermal Coefficient in P V=K T . . . . . . 138 Tables for the Number of Molecules in One Gram ..... 139 Tables for the Mean Kinetic Energy of Motion . . . . . .141 . Tables for the Boltzmann's Entropy Coefficient 142 Tables for the Planck's Wirkungsquantum 146 Tables for the Wien-Planck Coefficients ci, c 2 . .... 149,151 Tables for the Volume Intensity Coefficient . 153 Tables for the Kinetic Energy per Unit Volume . . . . . . 155 Check on the Computations 160 Second Computation of the Wien-Planck Coefficients . . . .161 CHAPTER V THE ELEMENTS OF BLACK RADIATION IN THE ATMOSPHERES OF THE SUN AND THE EARTH. . '*. ~ -^8 The Concentrations of Black Radiation in Gaseous Media . . . 168 TABLE OF CONTENTS Vll THE ELEMENTS OF BLACK RADIATION IN THE ATMOSPHERES OF THE SUN AND THE EARTH Continued PAGE Tables for the Specific Volume Density . . . . ... . 171 Tables for the Total Pressure per Unit Volume . . . . . . 175 Tables for the Energy of Black Radiation . . , : . .'- . . . 177 Tables for the Mechanical Force 179 Tables for the Specific Entropy . 181 Tables for the Specific Intensity of the Entropy Radiation . . . 183 Tables for the Unpolarized Specific Intensity 185 Tables for the Total Volume Inner Kinetic Energy . . .. , . . 186 Certain Data in the Radiation Terms . ...... . . . 187 Tables for the Thermodynamic Pressure of Kinetic Energy . . . 188 The Discontinuity at the Levels Where Radiation Is Generated . 191 CHAPTER VI RECONCILIATION OF THE DATA DERIVED FROM THE PYRHELIOMETER AND THE BOLOMETER 192 The Poynting Surface-Flux and Volume-Density Equation . . . 192 The Cause of the Change from the Total Solar Radiation at 5.85 Calories to the Effective Radiation 3.98 Calories Received on the Outer Strata of the Earth's Atmosphere 195 Summary of the Terrestrial Thermodynamic Data 200 The Thermodynamic Data in the 1000-Meter Levels . . ; . . 206 Resume of the Preceding Results 210 The Potential Energy of the Solar Radiation in the Sun's Atmosphere . 211 The Potential Energy of the Solar Radiation in the Earth's Atmosphere 214 The Third Thermodynamic Depletion . . . .' . ' _. . ,. . 216 The Scattered and the Absorbed Radiation ....... 216 The Pyrheliometer and the Bolometer Observations 217 The Depletion of the Solar Radiation from 5.85 Calories in the Iso- thermal Layers of the Sun to 1.50 Calories at the Sea Level of the Earth 218 The Line and Band Absorption and the Scattering 222 Direct Readings of the Pyrheliometers at Great Heights . . . . 225 Summary of the Pyrheliometer Results as Reduced by Bigelow's Method 226 Computation of the Pyrheliometer Data Leading to 3.98 gr. cal./cm. 2 min. as the Intensity of the Solar Radiation at the Earth . . . 227 Synchronism of the Solar and the Terrestrial Variations During the Interval 1900-1915 in Argentina . 231 Short- and Long-Range Forecasts !/ . .. . . 236 CHAPTER VII OTHER SOLAR PHENOMENA . . . . . . 240 Restatement of the General Line of Argument 240 The Effect of the Solar Isothermal Shell upon the Visible Surface Phenomena . 242 Vlll TABLE OF CONTENTS OTHER SOLAR PHENOMENA Continued P AGE The Granulations, Faculse, Flocculi, and Prominences .... 243 Granulations 249 Faculse, Flocculi, Prominences 250 The Origin of the Sun-Spots 251 The Circulation in a Solar Vortex or Sun-spot 257 The Inward and Outward Velocities in the Solar Vortices . . .261 The Invisible Deep-seated Thermodynamic Processes .... 262 The General Circulation of the Sun in Latitude 263 The General Circulation of the Sun in Longitude 267 The Chromosphere and the Inner Corona . . - 268 The Outer Solar Corona . . ....... .... > . .269 Solar Magnetism . . . . . 271 The Polar Rays of the Solar Coronas as Evidence of a Magnetized Sphere 273 The Zeeman Effect in the Sun's Atmosphere ....... 275 The Distribution of the Solar Magnetism as Determined by the Zee- man Effect 277 The Solar Spectra 280 The General Solar Spectrum 281 Atmospheric Refraction and Scattering . . . " . . . . . . 285 The Formulas of Refraction . . ... 286 The Density p as Computed by the Non-adiabatic, the Adiabatic, and the Bessel Formulas 289 The Atmospheric Transmission for Different Wave-lengths p k , Col- lected According to the Pyrheliometer p w 290 Results for Washington, Mt. Wilson, and Mt. Whitney . 291,- 292, 293 The Terrestrial Values of the Index of Refraction . . . . ' . . 293 The Solar Values of the Index of Refraction 294 General Remarks 295 Variation of the Intensity of the Sun's Radiation in Longitude . . 300 CHAPTER VIII THREE THEORIES OF RADIATION 304 The Derivation of the Wien-Planck Formula for Black Radia- tion in a Spectrum ... . . 305 The Derivation of Bigelow's Functions for the Potential h . . 307 Additional Formulas ..... . . 310 The Variable k in all Atmospheres . . ' 312 The Atmospheric Pressure . 313 The Variable Potential Coefficient h 313 The Solar Volume Density of Radiation 314 The Terrestrial Volume Density of Radiation 315 Evaluation of lr-7;;) and 1 7-^1 318 \k TJB \k i/ p The Variations in the Wien Displacement Law .'... . . . . 319 Another Formula for the Quantity hs ......... 320 TABLE OF CONTENTS IX THREE THEORIES OF RADIATION Continued PAGE The Kinetic and the Potential Energies in Radiation, Determined from Thermodynamics . , .321 The Molecular Potential Energy in the Earth's Atmosphere . . 325 The Molecular Potential Energy in the Sun 326 Summary . . . . .'..... 328 Derivation of Certain Formulas for Radiation, lonization, and Atmospheric Electricity 329 lonization, Potential Energy, and Frequency . ...... . 330 Atmospheric Electricity . . . ... 331 The Terrestrial Atmospheric Electricity . . . ' . . . . . 333 The Solar Atmospheric Electricity . . 339 Atmospheric Electricity and the Diurnal Convection . . . . 342 The Fundamental Quantities in Meteorology and in Astrophysics . 345 Practical Series of Thermodynamic Terms 348 The Relations Between the Kinetic and the Potential Energies in Orbital Oscillations . ... ... 349 Bohr's Theory of Non-Radiating Orbits in Atoms . . . . . 350 The Series of Spectral Lines t . . . . . . . 351 Derivation of the Orbital Formulas ... . ' . . ' . . . 352 Moseley's Law, Bigelow Form and Millikan Form 355 Evaluation of I, 357 \Tl 2 T 2 2 / The Electronic Orbits in the K and L Series of Radiation Lines for /*- Variable . .../'.' '. ^. . . . 358 K a Radiation . . . . . . . . . . . '. . , . . 359 L a Radiation ,.*.... 361 The Interpenetration of the Electron Orbits at the Contact of CoUision ... . . . ,.. \ 362 The Electromagnetic Waves Due to the Sudden Motion and Stoppage of an Electric Charge in Collisions 363 The Variable Intensity of the Sun's Radiation in the 26.68- Day Period of the Synodic Rotation , 365 International Character Numbers . .. . ^. . . . .. . . 366 Synchronism Between the Solar Radiation Intensity and the Terrestrial Magnetic Variations in the 365-Day and the 26.68- Day Periods 368 The Effect of Dust in the Lower Strata . . . ..... . 369 The Systems of Units Employed in Meteorology . 371 The Bar as a Practical Unit for Absolute Pressure 373 Long Range Forecasts of Weather Conditions . . . . . . 374 Appendix. The Pyrheliometer and the Poynting Equation . . 375 A TREATISE ON THE SUN'S RADIATION AND OTHER SOLAR PHENOMENA CHAPTER I The Thermodynamic Processes in the Solar Atmosphere Introduction THIS Treatise on Solar Physics is the direct continuation of the author's Meteorological Treatise on the Circulation and Radia- tion in the Atmospheres of the Earth and the Sun, 1915. In the earlier work the fundamental principles and the formulas were suf- ficiently explained, so that it is unnecessary to repeat them here. In this research there are several important points essential to the discussion that should be kept in mind by the reader. (1) In the Boyle-Gay Lussac Law, P = p R T, aU the terms, including the gas efficiency R, are variable. This is the general, non-adiabatic case, which alone is applicable to free atmospheres, the adiabatic case being only occasionally developed. This change of R a = constant to R = variable is very revolutionary, because it carries with it the so-called constants, k, h, Ci, c 2 , in the Wien-Planck spectrum formula, and a, cr, in the general Stefan formula for black radiation. (2) The transition from terrestrial values to initial solar values of Pv = R T is effected by, G^ gravity acceleration at the sun # gravity acceleration at the earth so that y P . yv = y R .y T, at one point on the sun. From this point, by the method of trials with the dynamic and the gravity equations, the complete thermodynamic system has been worked out for several gases, from hydrogen, monatomic, Hi = 1.00 and diatomic, H 2 = 2.00, to mercury Hg = 198. Each gas develops independently of all the others, and possesses 1 ylLH A ^EATISE ON THE SUN'S RADIATION a low level adiabatic base below the solar photosphere, a middle isothermal region containing the photosphere, and a non-adia- batic region extending to the vanishing planes at different heights, H 2 at 25,000 kilometers above the photosphere, He at 11,000, carbon (assumed monatomic) at 4,000, calcium at 1,000, zinc at 675, cadmium at 380, mercury at 210 kilometers. The sharp disk of the sun is not an optical effect, but the vanishing level of the heavy metallic vapors under the operation of the prevailing thermodynamic conditions. The computed system is in ad- mirable harmony with that which is indicated by the spectro- heliographic observations. (3) The equation of condition for radiation is derived directly from the first law of thermodynamics, dQ = dW + d U, by substituting d W = P dv, *~- T) _ >. r l 1 . * 10 -tVlO L J. . fll VQ Vi VQ The dimension of each term is easily reduced to ^- . T~jT 2 J > tne mean kinetic energy per unit volume, and it is so described by Boltzmann, Heaviside, Lorentz, W. Wien, Planck, A. L. Day and C. E. van Ostrand, Richardson, and others generally. When the volume energy is transmitted with the velocity c the result is radiant energy or flux, r M L M-\ \_Y~T2' ~T ~ T3 r conversion factor of radiation-den- sity from (M. K. S.) to (C. G. S.) is 10; that of the radiant- Dplyin Joule flux is 1,000. Applying to a = - j , the dimensions give: K S-- ' - m. sec. 2 deg. 4 m. 3 deg. 4 ' ' cm. sec. 2 deg. 4 cm. 3 deg. 4 ' (C. G. S.) Apply ing. to a = a. , the dimensions give: kilog. J ule (M K S)- gr ' sec. 3 deg. 4 m. 2 sec. deg. 4 ' ' sec. 3 deg. 4 erg. s < cm. 2 sec. deg. THERMODYNAMIC PROCESSES IN SOLAR ATMOSPHERE 3 The earlier treatise was concerned exclusively with the volume density equations of ponderable gases, and not with the flux equation. This treatise passes from the volume density of the kinetic energy of gases to the electromagnetic flux of radiation in the pure aether, and to its source near the bottom of the isothermal layers in the solar envelope. In this process the Poynting Equation for surface-flux and volume density is employed. (4) In computing the Boltzmann entropy coefficient, k = /("" wi 7? -jy. = ~yr~, where N = the number of #-atoms per unit mass, it is evident that while in the adiabatic system k, K, N, are all constants, k, K, are certainly variable in the non-adiabatic system. If H is the inner molecular kinetic energy per unit 3 volume, H = P = U, in monatomic gases, and EQ is the in- trinsic kinetic energy of each molecule, assumed in the usual kinetic theory of gases to be a universal constant, we have for TJ op the number of molecules per unit volume, n = TT = TTTT, and JbQ z LQ thence N = n , where m H is the mass of the standard P WH hydrogen molecule. It appears from the computations that N does not remain constant, but diminishes with the height, especially in the very rarefied high-level strata. We, therefore, encounter a dilemma, because either N or EQ is variable, and they may both be variables. If it is thought that N = nifj is necessarily constant, it assumes a certain view regarding mass, which is not yet settled in the analytic discussions. If E is made a variable it modifies the foundations of the kinetic theory of gases. Finally, if E Q is a variable, diminishing with the height, the law is unknown, and k, h, Ci, c*, become different variables from those herewith computed in the first case. We have, ac- cordingly, retained E = constant and allowed N to become variable, in the preliminary discussion of gaseous radiation. The system has, also, been given for N = constant and E = variable. (5) The computations have led to the following important result regarding the value of the solar constant of radiation. In each gas, at its special depth below the photosphere, there is a sudden change in the value of all the variable coefficients entering into the process of radiation. The mean temperature in these layers of radiation is 7655 A. absolute; the mean logarithmic value of a in the Stefan law, JQ = a T a is log a = 5.74000. and the exponent a = 4.00; so that J Q at the distance of the srr cal earth is equivalent to 5.854 '- r-. This agrees with the cm. 2 nun. predicted value, Fig. 54, preceding volume; the terrestrial thermodynamics established the effective radiation reaching the earth as. 3.980 calories, equivalent to about 6950 A.; the ob- served depletion by the bolometer continues from station to station down to Washington, 2.47 calories. The scattering process is apparently equivalent to 1.87 calories in the solar atmosphere, 0.76 calorie in the high levels of the earth's atmo- sphere, and 0.25 to 0.54 calorie in the low levels; the absorption process in the earth's atmosphere increases to about 0.120 calorie; the free heat at sea level is 1.50 calories by the pyrheliometer. Abbot's extrapolated value of the solar constant, 1.95 calories, is about one-third of the true solar constant. He has, therefore, made an error in supposing that the sun, though having high temperatures, radiates with very low efficiency. On the con- trary, the data of the computations on solar thermodynamics, and those on the terrestrial thermodynamics, both agree with the results of the bolometer in making the solar radiation of full efficiency. (6) More specifically, the process of depletion of the solar intensity of radiation from 5.85 gr. cal./cm. 2 min. to 1.30-1.50 calories at the sea level of the earth's atmosphere proceeds by the following steps. a. Temperature of origin for all the gases = 7655 absolute A. in a deep isothermal layer. b. Exponent in the Stefan Law of black radiation = 4.00. THERMODYNAMJC PROCESSES IN SOLAR ATMOSPHERE c. Equivalent intensity at the mean distance of the earth = 5.85 calories. d. The depletion in the hemispherical shell of the sun's atmosphere, including the isothermal and the non-adiabatic layers, is measured by the decrease in brightness of the solar disk between the center and the limb. Abbot's Table 55, Vol. III., Ann. S. I., gives the bolometer intensities for several spectrum lines, in parts of the intensity at the center, for certain points of the radial distance. Factors are computed for each line showing its contribution when referred to an undepleted intensity from the center to the limb. The following summary gives the ordinates computed for black radiation at the tem- peratures 7655, 6950, 5810, 5456, and Abbot's observed ordinates as extrapolated by him to the outside of the earth's atmosphere, marked Total, at Mt. Whitney, Mt. Wilson, and Washington. The corresponding spectra are in relative num- bers, but they can be reduced to calories by the general factor 20.9: COURSE OF DEPLETION OF THE SOLAR RADIATION DOWN TO THE SEA LEVEL SOLAR BOLOMETER PYRHELI- THERMODYNAMICS OBSERVATIONS OMETER Wave Lengths T 7655 Black Factor of Deple- tion Effec- tive of Sun to Earth T 6950 Black Total as Extra- polated Mt. Whit- ney Mt. Wilson Wash- ington Total as Extra- polated 5810 Sea Level Gener- ally 5450 0.323M 9.790 0.580 5.678 5.406 1.500 0.828 0.764 1.578 0.932 .386 10.552 .589 6.215 6.348 3.690 2.959 2.733 1.728 2.218 1.453 .433 10.294 .633 6.516 6.514 5.472 4.413 4.265 3.370 2.586 .790 .456 9.884 .661 6.533 6.454 6.051 4.992 4.883 3.918 2.618 .824 .481 9.411 .679 6.390 6.268 6.056 5.254 5.063 4.118 2.656 .885 .501 9.031 .690 6.231 6.116 6.052 5.445 5.195 4.268 2.684 .931 .534 8.337 .709 5.911 5.946 5.768 5.267 5.017 4.202 2.648 .942 .604 6.893 .738 5.091 4.959 4.994 4.670 4.454 3.833 2.480 .884 .670 5.721 .762 4.359 4.217 4.070 3.856 3.758 3.294 2.229 .732 .699 5.188 .770 3.995 3.664 3.664 3.495 3.444 3.063 2.116 .663 .866 3.191 .806 2.572 2.511 2.403 2.471 2.451 2.208 1.602 1.302 1.031 1.994 .830 1.655 1.603 1.536 1.748 1.744 1.574 1.162 0.970 1.225 1.199 .845 1.013 0.986 0.985 1.065 1.055 1.000 0.732 .626 1.655 0.459 .884 0.406 .389 .466 0.456 0.410 0.440 .256 .226 2.097 .204 .899 .183 .176 .211 .201 .191 .181 .130 .117 Calories 5.85 3.98 3.98 3.22 2.96 2.87 2.47 1.94 1.50 6 A TREATISE ON THE SUN'S RADIATION The solar radiation arrives at the earth as black and of an equivalent temperature 6950. e. The terrestrial thermodynamics reproduces the same co- efficient and exponent in the Stefan Law as correspond with black body radiation at 6950 on the levels 60000 to 65000 meters. /. The summary of the several thermodynamic terms in the earth's atmosphere, from the sea level to the vanishing plane, derived directly from the prevailing temperatures, which is the precise equivalent of the effective radiation, whatever may be the details of the paths of the radiant energy, gives the following results: 1. Summation of the free heat 4.08 gr. cal./cm 2 min. 2. Summation of the hydrostatic pres- sure 4.08 " 3. Summation of the inner energy and work 4.08 " 4. Summation of the black radiation. . . 3.94 5. The free heat in the stratum 3.92 " 6. The hydrostatic pressure in the stratum 4.00 " * 7. The inner energy in the stratum 3.92 " 8. The external work and gas efficiency. 4.02 " 9. The kinetic and potential energies with the absorbed radiation 3.90 " g. The energy of the solar radiation in the atmosphere separates itself into the following terms: I. The kinetic energy for temperature effects. II. The potential energy = 0.641 kinetic energy. III. The specific heat change from constant volume to con- stant pressure, or from the inner energy of the Boyle- Gay Lussac Law to the external energy due to the impressed gravitation. IV. The energy lost by non-selective scattering. V. The energy lost by selective absorption. VI. The energy transformed into ionization products, free electric charges, and magnetic fields. THERMODYNAMIC PROCESSES IN SOLAR ATMOSPHERE 7 h. The effective intensity 3.98 calories minus (III + IV + V + VI) gives the bolometer ordinates approximately. The further subtraction of the potential energy II gives the data observed by the pyrheliometer in bulk, as the kinetic energy which produces its temperature. Hence, the proper reduction of the pyrheliometer observations follows: H = the kinetic energy, observed by the pyrheli- ometer in the zenith. / = the potential energy = 0.641 H. k (H + /) = (100 - p) (H + J) = the proportional part lost by scattering. A U e = 0.012 (e e Q ) j a small correction for vapor pressure. a the nearly constant line and band absorption in the spectrum. R = the nearly constant specific heat term at the station. S = the effective solar constant outside the earth's atmosphere. S= (H + 7) + k (H + J) + A U e + (a + R) = 3.98 calories. These reductions of the pyrheliometer observations agree at many stations in producing 3.98 calories. This refers to La Confianza (4485 m.), Mt. Whitney (4420), La Quiaca (3465), Mt. Wilson (1780), Bassour (1160), Mt. Weather (526), Cordoba (438), Pilar (340), Washington, D. C. (34). The Bouguer for- mula of depletion is used to compute p and /io = H in the zenith, but it does not employ the objectionable Langley-Abbot method of extrapolation beyond its natural limits. Each station produces 3.98 calories with variations on its own level. (7) The most interesting and striking result is that the solar radiation originates in a per saltum process, resembling a radio- active discharge, due to a readjustment of the electrons and to thermal collisions. In the case of hydrogen it was necessary to treat it as a monatomic element, HI = 1.00, below the radiation level, but as a diatomic element, HZ = 2.00, above that level, in order to terminate on the observed vanishing plane, namely, 8 A TREATISE ON THE SUN'S RADIATION the top of the inner corona at 25000 kilometers. This is direct evidence of dissociation below, and association above the plane where the radiation originates. That plane is just above the true adiabatic strata where the isothermal layer begins, and at the levels where the temperature, the pressure, the density, and the gas coefficient, all suffer very quick changes of gradient. The factor of change in the coefficients of radiation is of the order 10~ 6 , dropping from a very high tension to a very low tension in a short vertical distance. The effect of this phenom- enon, as well as the general variability of the coefficients of radiation, upon the theories of radiation now under discussion, cannot possibly be indicated. It has been our purpose to bring forward the main thermodynamic conditions at the sun, under which the radiation originates, and no attempt is made here to interpret them fully. (8) The sun's atmosphere consists of a gaseous envelope containing a spherical isothermal shell, at the mean temperature of about 7687 C. This phenomenon serves to explain several of the common solar problems. Above the isothermal layer the temperatures decrease rapidly to C., and below it the tem- peratures increase to very great values. Hence, low- tempera- ture spectra are projected upon a uniform continuous spectrum at a definite temperature of 7687 for a background; the flash spectrum, at 500 kilometers, develops for the light gases above the heavy metallic strata o.f the photosphere; the granulation is an optical effect of segregation within the isothermal layer; the sun spots have two independent parts: (l) The lower on the under side of the isothermal layer, generating a convectional vortex like the terrestrial hurricanes, and forming the dark umbra; (2) the upper, located above the isothermal layer, and falling into the umbral vortex, as into the depression at the center of a hurricane, thus producing the penumbra; the faculae, flocculi, and prominences, are convectional effects of rising gases originally at very high pressures and temperatures, discharging through the isothermal layer, and seeking a new equilibrium above it; the general circulation of the sun must conform to the con- tinuous production of this isothermal layer at every point of THERMODYNAMIC PROCESSES IN SOLAR ATMOSPHERE 9 the photosphere; the radioactive discharge within the isothermal layer sets free vast numbers of electrons, which make up the surface electric charge of the sun, with coronal dispersion to space under the pressure of solar radiation; the movements of free electrons produce the Zeeman effect in the sun-spot vortices ; they cause the coronal polar magnetic fields, and the general polarization throughout the sun. Historical Remarks The problems of the physical constitution and the processes in the Sun have engaged the attention of scientists for three centuries. Since the invention of the telescope the statistical data regarding the sun-spots, and the solar rotation in different latitudes, have been studied ; with the application of the spectro- scope the data regarding the prominences, or hydrogen flames on the disk, have been classified, since 1871; with the develop- ment of the spectroheliograph the constitution of the visible spots, faculae and flocculi, has been closely analyzed, and the pressures in certain levels of a few gases have been approxi- mately determined; with the discovery of the ionization of molecules, and the existence of free electrons, together with the radioactive processes in atoms and molecules, and the struc- tural arrangement of the positive nucleus with the circulating negative electrons in the atoms, including their abrupt redis- tribution by the association of atoms into molecules, or the dissociation of molecules into atoms, under certain pressures, densities, thermal efficiencies, the physical field has been en- larged to an extraordinary degree; -the attack upon the thermal spectrum by means of the bolometer, and the measurement of the amount of radiation received at stations in the lower atmo- sphere of the earth by the pyrheliometer, have afforded much information regarding the coefficients of emission and absorption of the solar radiant energy in the earth's atmosphere; some progress has been made in regard to the synchronism of the solar variations as observed and the corresponding effects upon the meteorological or climatic conditions prevailing as tern- 10 A TREATISE ON THE SUN ? S RADIATION peratures, pressures, vapor pressures, circulation, electric and magnetic variations in different terrestrial latitudes and longitudes. In spite of the extensive literature regarding the material just mentioned, it must be admitted that the scientific knowl- edge available is really superficial, because there has been no thoroughly comprehensive analysis of the many physical con- ditions which must exist in order to produce the very complex phenomena that are observed on the sun. All the above methods depend upon visual phenomena, but these are of limited scope in respect of the fundamental conditions in the solar gases which produce them. It has seemed to the writer that thermodynamics constitutes the royal road between the earth and the sun, and that the system of equations which can repro- duce the general conditions in the earth's atmosphere should be able to do the same in the sun's atmospheres. With this purpose in view, the terrestrial dynamic and thermodynamic relations have been studied, in order to develop such a system of equa- tions as is effective and at the same time simple enough for practical computations. In certain papers published in the U. S. Monthly Weather Review, 1905, 1906, and in* Bulletins No. 3, No. 4, of the Argentine Meteorological Office, 1912, 1914, and in my Meteorological Treatise, 1915, such equations and their applications have been explained for the earth's atmo- sphere. The present treatise contains their modification so that they become equally valid in the sun's envelope, thereby prov- ing that this thermodynamic system is of universal application, as to any star where the surface gravitation is known from the astronomical masses of binary stellar systems. While the labor of developing this method of computation has been very con- siderable, it has now become relatively simple in consequence of the discovery of certain laws which regulate the distribution of the fundamental quantities. Derivation of the Non-adiabatic System of Equations The change from the adiabatic to the non-adiabatic system of equations consists in making the gas coefficient of thermal THERMODYNAMIC PROCESSES IN SOLAR ATMOSPHERE 11 efficiency R a variable in the Boyle- Gay Lussac Law, P = p R T. In the adiabatic system Ra = constant for each gas, and it is generally appropriate to the conditions of laboratory practise, but these do not conform to those prevailing in free atmospheres. If the conditions in a free atmosphere were adiabatic, then the tern- rr\ rr\ perature gradients would be at the fixed rate, = 777-; #1 ZQ Lp a where Cp a is the specific heat at a constant pressure, Cp a = R a , and K = Cp a /Cv a , the ratio of the two specific heats. But, as a matter of fact, the observed temperature gradients T* T* ft are different, such that = -Jh , so that Cp = *i - ZQ Cp 1 if R - , and R must be a variable coefficient. The most compact way to arrive at the general equation of condition, Treatise (27), is as follows: Take all the terms variable in Pv = R T, and integrate between two levels z\ . 26, within which stratum the temperature gradient remains constant. (1) Pdv + vdP = RdT+TdR, (2) P 10 Oi - w ) + *io (Pi - Po) = -Rio fa - To) + T w (Ri - jRo), We have the term for the external work, (3) P 10 (v 1 -v Q ) = (W l -W Q )=R 10 (T a -T )-v 10 (P l -P Q ), (333), so that by substitution, R IQ (T a - To) = 10 (Z\ - To) + Tio (Ri - Ro). The equations (2), (3) give the relations between the work, the hydrostatic pressure per unit density, and the gas efficiency in the stratum, but the gravitation acceleration g (21 z ) does not yet appear in the equation. In order to introduce any ex- ternal impressed forces, such as gravitation or circulation, we proceed as follows, by adding and subtracting Cp a (T a TO) = g (zi 20). Hence, (4) OFx-PFo) + ^~ = - g(zi - 86) ~ (Cp a -Ru) (T a -T ). PlO 12 A TREATISE ON THE SUN'S RADIATION We employ the following forms in the first equation of thermodynamics, Treatise, pp. 81, 82, (5) (Gi - Co) = (JFi - Wo) + (U, - U ), (6) (ft -ft) = (CP.- cp 10 ) (r.- r.), (7) (Wi - TFo) = (R* - Cp m ) (T a - To), (8) (tf, - tf.) = (Cp a - #,) (r.- To) = Cv m (T- a - To), so that by substitution, (9) + *(*- so) = - ^ ^ - (W, - W ) - (U, - U ), PlO (10) + g&- so) = - ~^ - (Q l - Co). PlO If the kinetic energy of circulation | (q^ q Q 2 ) is obtained by observing the velocities (qi . q ) on the levels (z t . ZQ) respec- tively, a secondary value of (Qi Qo) = (Qi Qo) 1 2 (?i 2 ~~ Hydrostatic pressure, . . = D E B, Pio Work of expansion, (Wi - JFo) . . . = A B E = A C E, Total inner energy, (Ui - U ) = A E B C, Free heat, (ft - Q ) = EEC = ABC = DEC, Gas efficiency, R 10 (T a -T ) = A C D = A B D, Pyrheliometer radiation, equivalent to EFGB, Bolometer radiation, equivalent to E A C B, Kinetic energy of the Poynting flux = E F G B, Potential energy of the Poynting flux = FACG, Dissipated energy of the Poynting flux = ADC, Kinetic energy of solar radiation . . = B G, Potential energy of solar radiation = G C. By keeping these conditions in mind the reader will more readily follow the argument of the following pages. The Working Equations There are only two important steps which are required in order to transform the adiabatic system of equations into the non-adiabatic system actually applicable at the sun. The first of these is the change of R a constant into R = variable in the Boyle-Gay Lussac Law, P = p R T; and the second is the multiplication of all the terms in Pv = R T by 7 = Go/ go, the ratio of the acceleration of gravitation at the sun relatively to that at the earth, so that, p " T ' T If R a = constant the result is that the temperature gradient 16 A TREATISE ON THE SUN'S RADIATION . T a To go T a TO Go is = -prr on the earth, and = 77 On Zi ZQ . Lp a Zi ZQ Lp a the sun, these being fixed gradients for each gas. They exist in the lower levels of the earth's atmosphere in the tropics, and in certain strata below the photosphere of the sun, but above these strata there are in every atmosphere isothermal strata, and above these non-adiabatic strata, where no such temperature gradients can occur. For this reason the adiabatic formulas, which have been universally employed, that is, wherever R a has been treated as a constant, are inapplicable in two deep strata of every atmosphere. In order to pass from the adiabatic to the non-adiabatic conditions, it is only necessary to put the T a To adiabatic gradient ... factor n = ^ TFT = r 1 T- r> ln to the coefficients TI TO observed gradient of the formulas. Adiabatic and Non-adiabatic (14) Pressure. log P, - log P = ^-j- (log Z\ - log To) ; n K log P, - log Po = ^ (log Ti - log To). (15) Density. log pi log p = ~T (log TI log n log Pl - log po = (log Ti - log TO). K J. (16) Thermal Efficiency. log 7?i - log ^ = 0; log R, - log^o =(-!) (log 7\- log To). The importance of this step is seen in the fact that the entire thermodynamic and radiation systems, instead of depend- ing upon a set of universal constants, become involved in a series of variable coefficients. The constants apply only within the adiabatic strata, while the variable coefficients are neces- sary in every non-adiabatic stratum, that is, wherever the natural temperature gradients differ from the fixed adiabatic rate. It is this fact, namely, the adoption of the usual laboratory adia- batic equations for free atmospheres, that has prevented meteor- ology from entering the region of the exact sciences. Change THERMODYNAMIC PROCESSES IN SOLAR ATMOSPHERE 17 R a from a constant into a variable, and the physics of atmo- spheres becomes a perfectly simple branch of thermodynamics. Examining the dimensions of the terms in the Boyle- Gay Lussac Law, we have for the absolute temperature, P M L s T 2 T = = *''* degree = l degree > so that the absolute temperature in degrees has no dimensions, being only the ratio between the pressure and the product of the density with the thermal efficiency. The last is merely the velocity of the molecules per degree, which pertain to the num- ber in the mass p when producing a pressure P. It is evident that in the laboratory conditions, the gas enclosed in a vessel of impermeable walls is very different from the same gas in the lower strata of a free atmosphere, which absorbs or emits radiant energy to space. As a matter of fact, the R begins at full adia- batic value in some low-lying strata, and thence diminishes to on the vanishing plane of the gas. If R is variable, so are all the following quantities also variables. (18) Specific heats, Cp = --: R } C v = - - R. K JL K JL Thermal efficiency, K = m R, where m = the molecular weight. & Number of molecules per unit mass, N = m v n = -r. Number of molecules per unit volume, n = N/m v = -^-j, ~ ]T~- Boltzmann's entropy coefficient, k = K/N. where Planck's Wirkungsquan turn, h = ( -J , c ^ a ' c = the velocity of light, a = 1.0823 (Planck), a = variable. Wien-Planck coefficients, Ci = 8 TT c h, c 2 = c h/k. 6 a. c\ Stefan coefficients, a = - -, a = in JQ = CO OO O5 1-1 i i I-H CD COC^i ttOto^Oi O CO O5 TjH CO CO W W CO i-J tO TH CO (NCOCOOOOO (M s s x 2 CO (N ^3 I s * i I C cd co 06 co CO C^ 8 II s^ 00 I; OO odd 10 o CO CO i-J CO (N odd *O O 1 1 I O 8 8 8 o o o irH 58S8 ^ CO (M iO O 00 O CO CO CD i^ ^O OS CD CO ^^ i i CO 1O OS I I I I I I- I I 6 8 S 8 OS OS O f^co ^ Si cqoqW'^^Os r-5i-5ddi-5i 5i-5i-5 I I OOSi i^Ot^-COiOO iOCOJCOi ^Osl>Cl O C^ CO C^ *O ddrHioosdco' 1 * rH CO >O OS I I I I I I I I I I rt cd 8. a |^| 5 *3 O 33 O o rt ca 3 -M C 3 C ? .?3 o a o c c ^r"i, ^1^3131 &3 CJ z ^ 5 a 3 'fl I H fc w 1 3 B 1 4J A TREATISE ON THE SUN'S RADIATION : = 8 00 CO i-< i i CO O2 CO to O CO 00 O O4 dOOTHr-IrH^IN OOOOOOOto^? CO CO 00 CO i 1 rH CO co co *"H co os co to r-(C^O5(M'-Hl^i> CO CO t^ CO fH ^^ O^ COCOOCOOOOd OOOOOOOiO^ Assumed monatom Diatomic Monatomic Assumed monatom Monatomic 8^5 ^^ T-H CO ^^ t> T~^ O O O O rH rH CO ddddcoddd 1 1 1 1 Dia Mo Assu Mo C C rH O5 l>; TjJ (N rH rH O O O 00 00 Oi iO erH 1C rH lO Oi r- 1 lO *O rH CO rH rH 00 *0 CO O t> CO t>- O >O 00 _ _- _- t^ 00 OS O CO i I LO T-H IQ C^ s C< rH rH IO rH O O O O a I 1 .y .a o i ago 1 ^2 g 5 g en aj o W3 O Islls COCOOOOOIOOO CO CO CO ^H CO ^^ C^ 00 OtO-^t^-^r^OSGO i IT IT ICOr tOO' ' O5COCOOOCOT-IOOO fNcDOO i i O CD TJH CO rH ca .a g. Isi II . a 26 A TREATISE ON THE SUN S RADIATION HO U 3 I 2 W Number W 3 M | a i c 3 1 to 4 s- 1 OS OS CO ^ ICOCOOOCO oo '"^ oo i> oo os *o co CO O O OO iO Tt< i ICO !> CO i-H *O i 1 OS !> ^ t^O l>- O IO TjH co C O O OS 00 !> t^OSTt^TtHOS l> CO O 00 O 1C coi>ppcqi>osco f> IO OS CO CO OS i~^ CO CO T I O O CO i-H H w ) (r. - T ) - Pvs (t>i o), (36) Inner energy (U, - Uo) = (Cp, - RIO) (T a - T ) = (ft - ft) - (Wi ~ Wo), and for the second law of thermodynamics, (37) Entropy (S, - So) = &-=-& = (Cp a - Q 10 ) ( T ~ T ] . J- 10 V -t 10 ' From these the potential of radiation is /OON K o o p rT * (38; AIO = - = -- /io = c L . V 1 VQ Vi V The coefficient c and exponent a are computed by the methods of Bigelow's Meteorological Treatise. From the trial equation (39) log C = log K 1Q - A log r 10 , 28 A TREATISE ON THE SUN'S RADIATION we have, by using successive values of K iQ and T 10 , log (log K!- log K ) lo * A = log (log Ti - log To)' which gives certain values of A that can be combined with the corresponding log C in Equation (39). These values of (-4, log C) are plotted on a diagram, and the mean line is drawn, from which a table for (a, log c) is made. Trial values of i are assumed, and log c is computed from (41) log c = log K 10 a log TIQ. In this table the corresponding value of a is interpolated, and if i = a the trial value is correct. Two trial values (i, a^) are generally enough to produce the pair values (a, log c) on the same horizontal line of the table of the gas. Examples of the results of these computations will be given, and they are un- usually interesting in determining the source of the solar radiation in the different gases and their relation to a in the Stefan Law, (42) /o = a T*. Notation and Fundan (43) Field strength . Impressed forces Cental Formulas for the Electric Field EI (E x . E y . E z )i e e E = D - = K Ei + eo + e ^- = b(b x .b y .b z ) ot 47rC = 4-ir.cE 4?r pv **j 47T/0 = D + 4lT (C+pv+j] Electromagnetic Field Magnetic Field Hi (H x .H Z .Hy)l ho h H=*- = tf Hi + //o + h - B (B x . By. B z ) oi Motion of the medium Total force.... (44) Displacement current Conduction current Convection cur- rent Motion of the medium Total current. . 47T g 47T<7 = B + iirg THERMODYNAMIC PROCESSES IN SOLAR ATMOSPHERE 29 The impressed forces and the motion of the medium will not be required in the problems concerning the radiation. (45) Maxwell's Laws of Circuitation I 47r/o = > + 47T (C + PV+J) = curl (H - fc) 47r / = ) + 47r (c_|_ pv ) = curl# 1 47rr 1 = Z)-f47rC = curl H (46) II 47rG = B + 47rg = - curl (E - *>) 47rG = B = - curl t 47rr 2 = B = - curlE (47) Potential Energy J = .(E x D x + EyD y + E Z D Z ) d 6r Sir (48) Kinetic Energy K = ^ fff(H* B x + H,B, + H z B,)d r = (49) Joule's Heat Q = fff (E x C x + E y C y + E Z C Z ) d i (50) Activity (51) Scalar product (A B) = A B cos A B = A x B x + A y B y + A x B z . (52) Vector product [A B] = (A y B z - A z B y ) + (A Z B X - A x B z ) + (A x B y - A y B x ) = V.AB. (53) Divergence div A = * + + . (54) Curl-rotation curl 4, = + ; curl ^ y = " 30 A TREATISE ON THE SUN'S RADIATION The five fundamental equations are Usual Units Lorentz* Units (55) 1. Divergence div D = p div D = p (56) 2. div B = o div B = o (57) 3. Curl curl D= -H curl D = - ~ H c (58)4. " curl# = >-t-47rpz; curlF = i (Z> ~ (59) 5. M r h c a e nical /? = E + [v.H] F = D + i fo.ff] Poynting's Equation for the Flux of Radiation The following demonstration of the Poynting equation is according to Richardson's exposition, Electron Theory of Matter, page 202. This theorem proves that the rate at which work is done on the electric charges within a given volume d r has two com- ponents, first, the rate at which energy flows into the volume d T through the surface, and second, the rate of loss of the energy by the electromagnetic field within the volume. Take the mechanical force acting upon p electrons in d r. s (60) Fp = p(E + [v.ff])dr, E and H represent the total electric and magnetic forces. The activity, or rate of working of the forces in the volume, is, (61) A = /// vp(E+[c.H])d r. c = the velocity of propagation in the medium. The second term is always zero, so that the activity is (62) A - Expand by the formulas for curl, THERMODYNAMIC PROCESSES IN SOLAR ATMOSPHERE 31 Collect the terms and integrate by parts, rearranging them, (64) A = -^ff(E y H z -E z Hy)dydz+(E z H x - E x H z )dzdx + (E x H y - EyHjdxdy Substitute dydz = ldS, dzdx = mdS, dxdy = ndS, surface integral. From H* = H x * + H y * + H,*, we hzve2H~ = 2 Ql O * Since rot E = , we can employ the components; sub- 8 * stitute them, i3--ffrtfaM.+ IrJ J J ( \ Qt U --[dr. A = r, ff w ds -r* /// 9 T ( * ff2 + * 2) dT - Change the general E . H to the specific K E , M H Q . -L V.EH-H-j. E and H are perpendicular to each other, and the propaga- tion is perpendicular to both. The internal energies are equal, 32 A TREATISE ON THE SUN ? S RADIATION In a beam of parallel plane polarized light the electric and magnetic vectors move in equal periodic waves, E = Eo cos ( 16T , (80) Entropy. 5 = = a r 3 = -y : = ^-pj K T' LL 36 A TREATISE ON THE SUN'S RADIATION (81) Entropy Radiation. L = -j- a T* = - s = K/T = 47T 167T __ JL 47r m T' LPdeg _c_^_u_ r M -i ' LPdegJ* (82) Radiation Energy. 7 = -p = - - K = - -K = o o C o C c_F r M 2' (83) Mechanical Force. F = F e + F m = - /. (84) Radiation Intensity. E A = - = --- u v = ^r-r 7, A O 7T A 6Zi IT" A (85) Monochromatic Spectrum lines. = -p f * V = >^ H in c 1 THERMODYNAMIC PROCESSES IN SOLAR ATMOSPHERE 30 1, > > ^ ^ v V"V__l!_>: ** s CJ \ A ! S 1 i tD bC 0) u *O Q^ bi ^ ^ U v- u C en b/j bfi O w . u flj 1-t 1-1 -rt" '8 s s s s ^ 3 U U (O ^T 1 *T c :> C 5 C 5 C 5 C 5 C D C D 6 b b C "gl V 1 g > . ' i 1 0) 5t 1 II S ^ M # s g , -S . 1 ^^ (1) ', x J &u *; # $ w b^ "V$ 8!$ Hi ri ^M g S O i 1 i l '2 * & ^ S^^ ^15 13 S ** ^ "' I j ^ \ CO 1 C^ - ^ s II j^ fX^ | t* tn g S4 ^ i ft CO 1 1 b C 3 |j 8 3 - ^ 3 N ol CT rr ^ 1 & fc R C K w p 1 O T-l TH 40 A TREATISE ON THE SUN S RADIATION ;- *V >- V " - " 1* Ti bfl bfl cu ? 0) ! _ r 1 s c/5 O '"1 u s? "s &? a S* s i ^ s fc 8 * u bb cu bfl o 0) || II II cu 1 U "O -o bfl CJ CJ O (J >-< 8" . 35 sj y d^ i 6 6 d 6 . C 5 O O O C 5 C > % o CU 0) CJ bfl cu H 1 -o 3 9 CO j *4U *]* 4 J2 M * _ 1 ^ A " S J, " S ^ 8 ' 3 o 8 ll C .2 I * 1 fe 1 i .2 ; I \ ( 1 ^ : >> a fc c G -in rj C O CU 3 2 B . o TS ^ S -5 .a .2 .a i 3 i in > p4 ^ en black radi Mechanical 1 1 I M bfl S vo|, II b j2 _c c s = 1 ^ o ^ u C CJ I g w THERMODYNAMIC PROCESSES IN SOLAR ATMOSPHERE TABLE 4 TABLE OF ASTRONOMICAL CONSTANTS Meter-Kilogram-Second (M. K. S.) System 41 Number Logarithm r = Mean radius of the earth (Bessel's spheroid) 6370191 m. 6 8041525 R R/r D = Mean radius of the sun, (Auwer'sdiam.31'59.26") = Ratio of the radii R/r. . . = Distance of the sun to the earth 694800800 m. 109.071 149 340 870 000 m 8.8418603 2.0377078 11 1741786 RID = Ratio R/D 0046525 7.6676817 10 (R/DY /A 2 = Ratio I 1 . 000021645 5.3353634 10 P \D) = Parallax of the sun, p = tan (r/D) . 8" 80756 5 6299739 10 P = Parallax of the sun (Newcomb) 8 7965 S/Si = Ratio of the surfaces, S/Si = (109. 071) 2 11896 4 4 0754156 V/Yi = Ratio of the volumes, V/Vi = (109. 071) 3 1297548 6 1131234 Pa Ps m = Average density of the earth (Harkness) = Average density of the M fry SUn = m(R) Pa '-' = Mass of the earth in kilo- grams . 5.576kil./m.3 1. 43287 kil./m. 3 6.0377Xl0 24 kil. 0.7463230 0.1562056 24.780872 M = Mass of the sun in kilo- grams 2.0132X10 30 kil. 30.303878 M/m M/m G Go 7 k = Ratio, mass of the sun to the earth, M/m = Ratio, mass of the sun to the earth (Newcomb) . = Acceleration per second at surface of the sun = Acceleration per second at surface of the earth. . . . = Ratio of the accelera- M r 2 tions, G/go = m K The gravitation constant, k = g r z /m 333431 333432 274.843 m./sec. 9.8060 m./sec. 28.028 6.5906X10~ U 5.523006 5.523008 2.4390844 0.9914920 1.4475904 9.818925-20 V = Velocity of the earth in its orbit 29806 6 m./sec. 4.474313 18. 52 12 miles/sec. 1.267670 42 A TREATISE ON THE SUN S RADIATION TABLE 4^-Continued TABLE OF ASTRONOMICAL CONSTANTS Meter-Kilogram-Second (M. K. S.) System Number Logarithm / = Acceleration at the dis- tance of the earth = - . 0.0059491 m./sec. 7.774446-10 Check,/ = -(^Y go. m \D J 7.774446-10 S = Rate at which the earth falls toward the sun = %f 0.0029746 m./sec. 7.473416-10 1" (second of arc) = radius of sun in kilometers radius of sun in seconds of arc 724.030 kil. 2 . 8597565 The astronomical constants given in Table 4 will be found useful in reference to the general relations between the earth and the sun, and these data will be employed in this Treatise. CHAPTER II Computation of the Thermodynamic Terms General Remarks HAVING obtained the data for one point of thermal equi- librium in the gas under consideration, it is necessary to proceed by the method of trials from level to level, both above and below that point. In order to accomplish this purpose, the formulas (l) to (5) are worked through in the non-adiabatic form; then the terms in the gravitation formula are computed: (43) G. (z. - Z ) = - Z=Z - (Cp a - CpJ (T. - T,)_+ A G\. p "> fe - *) Take a series of values of T, as 7\, T 2 , T 3 , . . . T w , and compute the residuals A Go (zi z ) . There is one definite value of T which will make the residual a minimum. As T n approaches this value the residuals diminish; as T n passes the minimum residua] these increase. After some practise with the curve of the residuals the value of T may be readily found approximately for each new level, and as the curves for P, p, R, T, develop it is easy to project T quite accurately. The labor of working through the first four elements, hydrogen (diatomic), calcium, cadmium, mercury, was very great, but the curves soon re- vealed the law of the temperature distribution with the height above the photosphere, so that the other elements, hydrogen (monatomic), helium, carbon, zinc, were computed with very little need of extensive trials. The number of trial computations was from six to ten for hydrogen (diatomic), but it diminished to less than two in the last elements that were tried. It is now easy to assign the temperatures, pressures, densities, and thermal efficiencies for any gas quite closely without the use of the complete set of formulas. An examination of the curves on 43 44 A TREATISE ON THE SUN S RADIATION COMPUTATION OF THE THERMODYNAMIC TERMS 46 A TREATISE ON THE SUN'S RADIATION COMPUTATION OF THE THERMODYNAMIC TERMS 47 a 48 A TREATISE ON THE SUN'S RADIATION COMPUTATION OF THE THERMODYNAMIC TERMS 49 50 A TREATISE ON THE SUN S RADIATION COMPUTATION OF THE THERMODYNAMIC TERMS 51 52 A TREATISE ON THE SUN J S RADIATION Figs. 2-9 shows that they are all constructed on the same model, while Figs. 10-13 indicate that they form families of curves which are interrelated by hyperbolic conditions with the parameter m. The constitution of the atmospheres of the several gases on Z 25000 20000 15000 10000 5000 -5000 -10000 He Top of the nnercorom 4n Top of the Jhromosphe 2000 4000 6000 8000 10000 13000 FIG. 10. T. Temperature in Absolute Centigrade Degrees. Z = height in kilometers from the photosphere. the sun is exactly the same in principle as was found in the atmosphere of the earth. This is seen by comparing the curves of Fig. 1, which summarizes the terrestrial data for the balloon ascension, Uccle, September 13, 1911. Compare Bigelow's Treatise, Tables 96, 97; In the case of the earth's atmosphere, the lower adiabatic section is not very clearly developed at the latitude of Belgium, because the temperature gradients in the COMPUTATION OF THE THERMODYNAMIC TERMS 53 lower levels do not agree with the adiabatic rate. This would, however, be found generally for the balloon ascensions made in the Tropics. The isothermal layers, 12000 to 37000 meters, and the non-adiabatic layers, 38000 to 90000 meters, are clearly determined. The peculiar secondary curvature in the gas 20,000 H 15,000 10,000 5000 -5000 -10,000 20 40 60 80 100 120 FIG. 11. A. Pressure in Terrestrial Atmosphe efficiency R, which occurs in the levels where the temperature gradient is at its greatest value, is found in all the atmospheres. In the case of the sun there were from 50 to 70 points deter- mined on each of the curves of P, p, R, T, from deep within the adiabatic region to the vanishing plane of the gas. An inspection of Figs. 2-9 shows clearly that the gases Hi, H 2 , He, C, Ca, Zn, Cd, 54 A TREATISE ON THE SUN'S RADIATION Eg differ from one another chiefly in the value of the vertical scale, or depth, within which the gas develops its distribution. The original computations in logarithms check very closely throughout; these are reserved for another publication. In order to make clear the series of fundamental laws of distribution, 15,000 10,000 5000 -5000 -10,000 0.100 0.500 o.uoo 0.200 0.300 0.400 FIG. 12. p. Density. kil grams ; (M.K.S.) meter 2 L 3 the values of T, P, A, p, R have been interpolated, where neces- sary, so that the values are here given on the same selected levels for each of the gases. As all of the terms will be studied in detail, it is not now necessary to explain further the Figs. 2-9. It will be convenient to give at this point Table 5 of the atomic and molecular weights of all the chemical elements, together with the approximate order of brightness of occurrence of those identified upon the sun generally, but in spots and faculae especially. COMPUTATION OF THE THERMODYNAMIC TERMS 55 a = the atomic weight referred to hydrogen, H = 1.00, (0 = 15.88). m = the corresponding molecular weight for some ele- ments. ai = the atomic weight referred to H = 1.008 (O = 16.00). Z 20000 19000 He 10000 5000 C c Zn CdQ Hg He -5000 25000 \ He Photosphere 50000 75000 100000 FIG. 13. R. Gas-Coefficient. velocity 2 136000 150000 degree ' jPdeg. (M. K. S.) m-i = the corresponding molecular weight. O B = the order of brightness in the solar spectrum (l to 36). O L = the order of the number of lines in the solar spec- trum (1 to 36). O S = the number of lines of the element in sun-spots. 56 A TREATISE ON THE SUN 7 S RADIATION TABLE 5 THE ATOMIC AND MOLECULAR WEIGHTS The Occurrence of the Chemical Elements on the Sun Element 5 H = 1.00 m H= 1.008 mi OB Order OL Order OS No. OF No. Aluminum. . . Antimony . . . Argon . . Al Sb A As Ba Bi B Br Cd Cs Ca C Ce Cl Cr Co Cb Cu Dy Er Eu F Gd Ga Ge Gl Au He Ho H In I Ir Fe Kr La Pb Li Lu Mg Mn Hg Mo Nd 26.88 119.24 39.56 74.37 136.28 206.35 10.91 79.29 111.51 131.57 39.75 11.91 139.14 35.18 51.59 58.50 92.76 63.07 161.21 166.37 150.79 18.85 156.05 69.35 71.92 9.03 195.64 3.97 162.20 1.000 113.89 125.91 191.57 55.40 82.26 137.90 205.56 6.88 173.61 24.13 54.49 198.47 95.24 143.16 27.10 120.20 39.88 74.96 137.37 208.00 11.00 79.92 112.40 132.81 40.07 12.00 140.25 35.46 52.00 58.97 93.50 63.57 162.50 167.70 152.00 19.00 157.30 69.90 72.50 9.10 197.20 4.00 163.50 1.008 114.80 126.92 193.10 55.84 82.92 139.00 207.20 6.94 175.00 24.32 54.93* 200.06 96.00 144.30 39.56 297.48 39.88 299.84 9 25 Arsenic Barium 15 D 24 D 2 Bismuth Boron . . Bromine 158.58 111.51 131.57 39.75 159.84 112.40 '132.81 40.07 27 1 16 28 11 7 22 25 26 11 7 10 5 6 16 30 Cadmium.. . . Caesium Calcium 60 1 2 386 118 33 105 47 *78 66 Carbon Chlorine 70.36 70.92 Chromium. . . Cobalt Columbium. . Copper Dysprosium. . Erbium 35 28 . . . Europium . . . Fluorine 38.00 37.70 Gadolinium.. Gallium Germanium. . Glucinum. . . . Gold 30 29 33 32 4 2 Helium . . 3.97 4.00 3 Holmium. . . . Hydrogen. . . . Indium. 2.000 2.016 253.84 3 D D 2 21 34 22 D D 1 14 35 Iodine . . . 251.82 1108 327 24 Iridium Iron Krypton Lanthanum . . Lead Lithium Lutecium. . . . Magnesium. . Manganese. . Mercury Molybdenum Neodymium . 82.26 82.92 6.88 6.94 6 13 D 20 24 19 4 D 17 12 8 167 3 5 21 '42 198.47 200.06 COMPUTATION OF THE THERMODYNAMIC TERMS TABLE 5 (Continued} THE ATOMIC AND MOLECULAR WEIGHTS The Occurrence of the Chemical Elements on the Sun 57 Element 5 a H = 1.00 m ai H - 1.008 mi OB Order OL Order 05 No. OF No. Neon . . Ne Ni Nb M N Os Pd P Pt K Pr Ra Rh Rb Ru Sa Sc Se Si Ag Na Sr S Ta Te Tb Tl Th Tm Su Ti W U V Xe Yb Yt Zn Zr 19.86 58.21 92.76 220.63 13.90 189.39 15.88 105.85 30.79 193.65 38.79 139.78 224.21 102.08 84.77 100.89 149.21 43.75 78.57 28.08 107.02 22.82 86.93 31.81 180.06 126.49 157.94 202.38 230.56 167.16 117.76 47.72 183.00 236.31 50.59 129.71 172.12 88.00 64.85 89.88 19.86 20.02 58.68 93.50 220.40 14.01 190.90 16.00 106.70 31.04 195.20 39.10 140.90 226.00 102.90 85.45 101.70 150.40 44.10 79.20 28.30 107.88 23.00 87.63 32.06 181.50 127.50 159.20 204.00 232.40 168.50 118.70 48.10 184.00 238.20 51.00 130.20 173.50 88.70 65.37 90.60 20.02 5 2 251 71 Nickel Niobium .... Niton(radium emanation) Nitrogen .... Osmium. . . . 27.80 31.76 28.02 32.00 D D 23 D 36 D D 18 D 36 . . . . . . Oxygen Palladium . . . Phosphorus. . Platinum. . . . Potassium. . . Praseodym- ium. 123.16 124.16 39. io ... ... 38.79 Radium Rhodium. . . . Rubidium. . . Ruthenium . . Samarium . . . Scandium. . . . Selenium .... Silicon 31 D 27 D 84.77 85.45 17 13 45 34 4 6 2 157.14 158.40 Silver .... 32 4 12 D. 31 20 23 D 8 2 Sodium Strontium . . . Sulfur 22.82 23.00 64.12 63.62 Tantalum. . . . Tellurium. . . Terbium Thallium Thorium.. . . 252.98 255.00 D D D D ... Thulium Tin. . 33 10 D D 14 34 3 D D 8 Titanium. . . . Tungsten.. . . Uranium .... Vanadium. . . Xenon Ytterbium. . . Yttrium Zinc Zirconium . . . 432 131 176 42 130.20 129.71 64*85 18 26 19 15 29 9 3 3 7 19 2 65.37 58 F = the number of lines of the element in the flash spectrum. D = doubtful identification in the solar spectrum. The order of brightness is taken from Abbot's Sun, page 206. The number in sun-spots, Astro physical Journal, June, 1913, St. John. The number in the flash spectrum, Astro physical Journal. March, 1915, Mitchell. 1 am indebted to Professor F. W. Clarke for the revised values of the atomic weights, which are those of the Inter- national Committee for 1916. *: The Distribution of the Temperatures The computations were conducted in height by taking as 0, or the reference plane, the point of equilibrium that was derived from the terrestrial data through the factor p, but as this point has no special significance in solar physics it has been found convenient to define two other planes of reference, (1) the surface of the photosphere, and (2) the bottom of the isothermal layer. By means of the spectroheliographic observations made at Mt. Wilson, and by other spectroscopic determinations, the pressure on the photosphere lies between 5.80 and 6.00 standard terrestrial atmospheres, and this has been accepted as the defini- tion of the level of the photosphere. Compare Fig. 11, where all the pressure curves pass through this point, no matter to what heights and depths the several pressures extend in their distribution. This is confirmed as correct by the general fact that the diatomic hydrogen vanishes at 25000 kilometers above the photosphere, at the top of the inner corona as observed in eclipses, that the top of the chromosphere, 5000 kilometers, is at the level where the hydrogen pressure and density become very small, as in Figs. 11, 12. The light gases extend beyond the level of the reversing layer which is at 400-500 kilometers, but the heavy gases do not reach it normally, only in great disturbances, and so do not generally appear in the flash spec- trum of the Fraunhofer lines. The vanishing monatomic molec- COMPUTATION OF THE THERMODYNAMIC TERMS 59 ular weights range for the flash spectrum from 60 to 80, so that helium (4), carbon (as 12), calcium (40), zinc (65), appear readily, while cadmium (112), mercury (198), are seen with difficulty. Thus, the flash spectrum by Adams* gives some- what the following distribution of lines by numbers as counted: Carbon (12), 105; calcium (40), 33; cerium (139), 47; cobalt (59), 66; chromium (52), 78; iron (55), 327; glucinum (9), 2; helium (4), 3; lanthanum (138), 24; magnesium (24), 5; man- ganese (54), 21; sodium (23), 6; neodymium (143), 42; nickel (58), 71; scandium (44), 34; silicon (28), 4; strontium (87), 2; titanium (48), 131; vanadium (51), 42; yttrium (89), 19; zinc (65), 2; as contained in the tables. The exceptions are cerium (139), lanthanum (138). Table 6 contains the temperatures, and it shows clearly the distribution of the isothermal temperature, deep for hydrogen and shallow for mercury, with non-adiabatic layers above and adiabatic layers below it. It is seen, from Fig. 11, that the pressures have one point in common in passing the plane of the photosphere at 6 atmospheres, and from Fig. 10, that the tem- peratures have one nearly vertical line in passing the photosphere, namely, the isothermal layer, having a temperature for each gas of about 7650 at the bottom, and of about 7710 at the top of this characteristic layer. The means refer to the tempera- tures in the isothermal region to the left of the broken line of separation. The non-adiabatic region is above this line and the adiabatic region is below it. The mean value of the temperature of the isothermal layer is about 7687 absolute Centigrade. Compare the Table 6 and Figs. 2 to 9. These figures were constructed on different convenient vertical scales of heights for ordinates, and with various abscissas for P, p, R, T, respectively, in order to show the interrelation of these quantities for each gas; the Figs. 10 to 13 bring together on the same vertical scale all the elementary gases, for temperature, pressure, density, gas co- efficient, in order to determine the general laws of their dis- tribution. It is easily found that the equilateral hyperbola, x y = c, is fundamental and since this passes over into the * Astroph. Journ,, March, 1915. 60 A TREATISE ON THE SUN'S RADIATION TABLE 6 THE TEMPERATURES OF THE DIFFERENT ELEMENTS ON THE SAME LEVELS z Kilometers Hi 1.00 H-2 2.00 He 4 12 Co 40 Zu 65 Cd 112 Hg 198 Means 25000.... 20000.... 15000... 10000.... 5000.... 1000... 900.... 800.... 700.... 600.... 500.... 400.... 300.... 200.... 100.... 90.... 80.... 70.... 60.... 50.... 40.... 30.... 20.... 10.... Photos. 0. . - 10.... - 20.... - 30.... - 40.... - 50... - 60.... - 70.... - 80.... - 90.... - 100... - 200.... 300 3320 4300 5370 6600 7550 955 2055 3495 5890 Topo f the Inn er Coron a 740 3780 '6 .. . 300 900 1500 2200 2950 3750 4700 5750 6800 7500 7600 Top o Chromos phere 7688 7680 7685 7691 7695 7698 7702 7705 7698 7697 7696 7681 7694 7701 7705 7696 7696 7699 7698 7693 7691 7690 7688 7686 7684 7687 7685 7683 7687 7690 7686 7684 7684 7682 7684 7677 7674 7672 7669 7670 7681 7661 7661 7668 7678 7686.7 7696 7710 7708 7706 7705 7704 7703 7702 7702 7701 7701 7701 7701 7701 7701 7701 7701 7701 7701 7701 7701 7701 7701 7701 7701 7701 7701 7701 7701 7700 7699 7698 7697 7696 7695 7694 7694 7693 7692 7691 7686 7681 7676 7671 7666 7680 7679 7678 7677 7676 7675 7674 7673 7672 7671 7671 7671 7671 7671 7671 7670 7670 7670 7670 7670 7670 7670 7670 7670 7669 7669 7669 7669 7669 7669 7668 7667 7666 7665 7664 7663 7662 7661 7660 7662 7676 7640 7651 7665 7690 7600 7650 7670 7690 7705 7715 7714 7713 7712 7711 7710 7710 7709 7709 7708 7708 7707 7707 7706 7705 7705 7704 7704 7703 7703 7702 7702 7701 7700 7699 7698 7697 7696 7695 7693 7691 7689 7687 7685 7665 | 7680 | 9451 11354 13255 15157 17059 18961 20863 22765 24667 6450 6700 7000 7250 7500 7615 640 1790 3120 4480 5940 7500 7600 Revers 1200 2900 5700 6000 ing layer 7715 7710 7705 7704 7703 7702 7701 7700 7699 7698 7698 7697 7696 7695 7694 7693 7692 7691 7691 7690 7689 7688 7687 7686 7685 7680 2500 3950 4450 4950 5450 5900 6350 6750 7100 7400 7600 7650 7690 7705 7715 7713 7711 7709 7707 7705 7703 7701 7699 7697 7695 7693 7691 7689 7687 7685 7665 7676 7650 7690 7720 7735 7730 7725 7720 7718 7715 7711 7707 7703 7699 7695 7691 7687 7683 7679 7675 7700 6600 6950 7300 7475 7650 7670 7690 7687 7684 7680 7677 7673 7669 7665 7670 7667 7663 7659 , 7655 7651 7660 7656 7652 7686 7720 7800 8550 9494 10436 11381 20816 30251 39686 49121 58556 67991 77426 86861 96296 190648 285000 379352 473704 568056 662408 756760 851112 945470 1039822 1134174 12742 18045 23348 28651 33954 39257 44560 49863 55166 108193 161220 214247 267274 320201 373228 426255 479282 532309 585336 638363 - 400.... - 500.... - 600... - 700.... - 800... - 900.... - 1000.... - 2000.... - 3000.... - 4000.... - 5000.... - 6000.... - 7000.... - 8000.... - 9000.... -10000.... -12000.... -14000.... 7675 7670 7665 7658 7651 7646 7640 11974 17680 23387 29094 34801 40508 46215 51922 57629 63336 69043 8651 10553 12455 14357 16259 18161 20063 39083 58103 77123 96143 115163 134183 153203 172223 191243 229283 267323 13797 16881 19965 23049 26133 29217 32301 63140 93979 124818 155657 186496 217335 248174 279013 309852 340691 371530 7661 7656 7651 7690 8000 8500 9187 10561 11934 8476 -47.554-686.90-1902.17-5706.5-19021.7 -30839.3 -53027.5 -94352.0 Adiabatic gradient per 1000 kilometers. COMPUTATION OF THE THERMODYNAMIC TERMS 61 logarithmic law of decrease or depletion, this distribution con- forms to the natural law of physical conditions. The bottom of the isothermal layer is nearly the same as the top of the adiabatic strata, making some allowance for the transition from the adiabatic to the isothermal conditions; the top of the isothermal layer is nearly coincident with the bottom of the non-adiabatic strata. The adiabatic lines of pressure are nearly parallel to the plane of the photosphere, especially for the heavy gases; the non-adiabatic lines of pressure are nearly vertical to the plane of the photosphere; the isothermal layer of temperature occurs within the layers where the pressure is changing rapidly from very large to very small gradients. The density line for the light gases in the adiabatic layers is nearly vertical, as hydrogen, but horizontal for the heavy gases; the gas coefficient has a very great range for hydrogen, but contracts to a very short range for mercury. It follows that the solar atmospheres constitute a very perfect thermodynamic engine in the adiabatic and isothermal layers, but one of less efficiency in the non-adiabatic region, which can be reduced to correspond- ing secondary adiabatic and isothermal components. The relations between the height z, the atomic weight m, and the temperature T, are very interesting and instructive, as can be seen in the following tables: TABLE 7 BOTTOM, TOP, AND DEPTH OF THE ISOTHERMAL LAYER Take m = the atomic weight, or molecular weight (monatomic). ZA the bottom of the isothermal layer zi = the top of the isothermal layer zi ZA = the depth of the isothermal layer Element m ZA zi ZI-ZA Hydrogen Hi 1.00 2.00 4.00 12.00 40.00 65.00 112.00 198.00 -12000 - 6000 - 3000 - 1000 - 300 - 185 - 107 - 60 +3600 + 1800 + 900 + 300 + 90 + 55 + 32 + 18 15600 7800 3900 1300 390 240 139 78 Hydrogen Hz . ... Helium He Carbon C Calcium Ca Zinc Zn Cadmium C'd Mercury Hg 62 A TREATISE ON THE SUN'S RADIATION Hence, the following general hyperbolic laws prevail: Bottom of the isothermal layer, m . Z A = 12000 kilometers Top of the isothermal layer, m . z 1 = + 3600 " Depth of the isothermal layer, m.(z I ZA) = 15600 " The corresponding depths for any other gas can be found on the assumption that it is monatomic, but since other molecular combinations may prevail above a certain level within the isothermal layer, the conditions on the sun become so com- plicated by the association of the atoms into molecules of different configurences and valences that it is necessary to reserve this portion of the subject. The Heights at which the Same Temperature Occurs for the Different Elements, by the Hyperbolic Law, m z = C T . An inspection of Fig. 10, which gives the temperature lines of the different elements, on the same scales of ordinates and abscissas, suggests that the curves form a family whose par- ameter is m, connected by the law of the equilateral hyperbola, m z = C T , where C T is a constant for a selected temperature T. To test this theorem, the heights z at which the same tempera- ture T occurs for the different elements m, was interpolated from the original tables, and the results are collected in Table 8 for certain temperatures, T = 000, 1000, 2000, . . . 12000. The first column contains the element and its atomic weight m, the second the height z at which the temperature T was computed by the thermodynamic formulas, the third the product m z. It is easily seen that the products tend toward a constant value for the same T, and the mean is the constant C T , referred to the photosphere. Hence, we obtain the hyperbolic constants from which the height may be computed where the given tem- perature occurs in the different elements. T = occurs at the he ght 1000 2000 3000 4000 5000 6000 7000 45400/m in kilometers 37792 30246 24304 19611 15422 11713 7826 COMPUTATION OF THE THERMODYNAMIC TERMS . 63 Isothermal Layer 8000 occurs at the height 14036/m in kilometers 9000 " " " " -16594 " " 10000 " " " " -18972 " " 11000 " " " " -21223 " " 12000 " " " " -23522 " " The non-adiabatic group can be referred to the bottom of the isothermal layers by adding 12000 to the respective values C T , as 45400 + 12000 = 57400 for the temperature T = 0. The first group for T = gives the height of the vanishing plane of the several elements. Hydrogen does not exist in the monatomic state, m = 1.00 above the photosphere, but only in the diatomic form of the molecule for m = 2.00. It is, how- ever, very convenient to have H = 1.00 fully computed, be- cause it is the implied standard of reference for all the chemical elements and the entire scheme is more readily understood by reference to this H = 1.00, as if it actually existed. It will be shown that the molecule H z = 2.00 does not exist below a certain plane in the isothermal layer, but only in the atomic form Hi = 1.00. It was necessary to use HI = 1.00 in certain levels, and H z = 2.00 in other levels, and this constitutes an important piece of evidence that at certain values of P, p, R, T, there is dissociation of H 2 into two atoms of fli, or conversely that at this level there is association of two atoms of HI into one molecule H z . An examination of the products m z of Table 8 seems to indicate that the variations are quite accidental in their general arrangement. It should be remembered that the data have been obtained by the method of trials applied in succession to eight elements, so that they are entirely in- dependent of one another. The general process of the compu- tations, therefore, resulted in a family of equilateral hyperbolas, which can be readily developed for the fundamental theory of the distribution of the solar elements. Since the Naperian logarithms are based upon the law of the equilateral hyperbola, it follows that an exponential law of decay or depletion must be placed at the foundations of solar gaseous distributions in height under the force of the gravity acceleration there existing. As already stated, it was found that the first four elements 64 A TREATISE ON THE SUN S RADIATION TABLE 8 DISTRIBUTION OF THE TEMPERATURE HEIGHTS BY THE HYPERBOLIC LAW, mz= (CONST.) r Temp. r= 000 T= 1000 r = 2000 T = 3000 T = 4000 m 3 ms z mz z mz z mz z mz Hi 1.. 48000 48000 42000 42000 32900 32900 26800 26800 21500 21500 H^ 2.. 25500 51000 19740 39480 14710 29200 11460 22920 8720 17440 He 4.. 11500 46000 9500 38000 7680 30720 6060 24240 4720 18880 C 12 4000 48000 3280 39360 2690 32280 2040 24480 1660 19920 Ca 40.. 1050 42000 880 35200 727 29080 594 23760 472 18880 Zn 65.. 675 43900 565 36725 483 31395 408 26520 335 21775 Cd 112.. 380 42600 312 34944 247 27664 196 21952 151 16912 Hg 198. . 210 41500 185 36630 144 28512 120 23760 109 21582 To the Phot osphere 45400 37792 30246 24304 19611 12000 12000 12000 12000 12000 Bottom of Isother- 57400 49792 42246 36304 31611 mal layer Temp. r = 5000 T = 6000 r= 7000 m z mz z mz z mz The temperature of the top of the isothermal layer is 7700; of the bottom 7620; of the photo- Hi I.. 16600 16600 12400 12400 8400 8400 sphere 7687; of the radiation layer H 2 2.. 6560 13120 4800 9600 2910 5820 7655. He 4.. 3670 14680 2720 10080 1770 7080 C 12.. 1500 18000 1160 13920 800 9600 The constant *for the photo- Ca 40. . 370 14800 277 11080 226 9040 sphere + 12000 becomes the Zn 65. . 263 17095 196 12740 131 8515 constant for the bottom of the Cd 112.. 120 13440 93 10416 68 7616 isothermal layer. Hg 198.. 79 15642 68 13464 33 6534 To the Phot osphere 15422 11713 7826 Bottom of Isother- 12000 12000 12000 mal layer 27422 23713 19826 Temp. T = 8000 r = 9000 T = 10000 T= 11000 T= 12000 m z mz z mz z mz z mz z mz Hi 1 . . -13000 -13000 -15062 -15062 -17200 -17200 -19300 -19300 -21450 -21450 #2 2.. - 8000 -16000 - 9740 -19580 -11207 -22414 -12630 -25260 -14080 -28160 He 4.. - 3220 -12880 - 3772 -15088 - 4200 -16800 - 4815 -19260 - 5345 -21380 C 12.. - 1267 -15204 - 1482 -17784 - 1660 -19820 - 1827 -21924 - 2000 -24000 Ca 40.. - 366 -14640 - 418 -16720 - 472 -18880 - 523 -20920 - 576 -23040 Zn 65.. - 211 -13715 - 245 -15925 - 275 -17875 - 309 -20085 - 342 -22230 Cd 112. . - 107 -11984 - 130 -14560 - 149 -16680 - 167 -18704 - 189 -21168 Hg 198. . - 53 -10494 - 75 -14850 - 86 -17028 - 96 -19008 - 107 -21186 To the Phot osphere -13490 -16196 -18337 -20558 -22827 COMPUTATION OF THE THERMODYNAMIC TERMS 65 pointed to the existence of this elementary law, so that it became easy to locate the temperature curves of He, C, Zn, Cd, -from which fact the later computations were greatly facilitated. On Fig. 14, the values of C T have been plotted, showing that GOOOO 40000 30000 20000 10000 -10000 -2000(7 ~30000 2000 T Temperature 4000 6000 8000 10000 12000 FIG. 14. Distribution of the Temperature Heights by the Hyperbolic Law, mz = (Const) r- there, is a convex curvature downward in the non-adiabatic region, a nearly vertical line in the isothermal region, and a straight line in the adiabatic region. An auxiliary adiabatic line has been drawn in the non-adiabatic region, and by resolu- tion the non-adiabatic curve can be broken up into its adiabatic 66 A TREATISE ON THE SUN'S RADIATION and isothermal components. The non-adiabatic branch ter- minates on the ordinate T = 0, but the adiabatic branch can be extended as far downward as the law of gases prevails. The Constant Temperature Near the Photosphere and the Variations of Temperature Outside of the Isothermal Region The temperatures of the different gases were computed at convenient intervals, differing from one gas to the other, but by interpolation the temperatures for the eight selected gases can be found on any level. Table 6 contains such a compilation of solar temperatures from 25,000 kilometers above the plane of the photosphere to 14000 kilometers below that plane. It is seen that the table divides itself by a broken line into three parts, the middle containing nearly constant or isothermal temperatures; the upper part in which the temperatures on the same level decrease from a maximum for HI = 1.00 to a minimum or to for the heavier elements; and the lower part in which the temperature on the same level increases from a minimum for HI = 1.00 to a maximum for the heaviest element Hg 198. Taking the mean values in the isothermal region, accord- ing to the number of elements on the left of the broken line of division, it is seen that the temperature is nearly a constant throughout this region, and that the general mean temperature of the isothermal region is 7686.7. It is noted that the depth of the isothermal region is great for the light gases, and small for the heavy gases, as 15600 kilometers for HI = 1.00 and 78 kilometers for Hg = 198, according to the hyperbolic laws. On the other hand, outside of the isothermal region, on the same level there are enormous changes in temperature from the light to the heavy gases, as 7696 to on the 1000-kilometer level, or 7651 to 945470 on the - 10000-kilometer level. In the adiabatic region the temperatures have been extended below the actual levels of the computations by merely adding the adiabatic gradients. These can be studied for what they may be worth, but it is thought that at such high temperatures, pressures, densities as are implied, the simple gaseous law, COMPUTATION OF THE THERMODYNAMIC TERMS 67 P = p R T, must become greatly modified in the passing of the gas into its liquid, viscous, or solid state. It is perhaps superfluous to remark that in the solar en- velope there is no constant temperature except in the isothermal layers near the photosphere, and that the popular idea of think- ing of the sun as having fixed temperatures on the same level is erroneous. By equation (6) it is seen that temperature has no dimensions, such as pressure, density, and gas efficiency possess, but that it merely represents a sort of thermal equilibrium -I g 'J T being merely a ratio, the same value occurs at different heights for the different elements, or else different values of the ratio T occur on the same level for the different elements. Indeed, the temperature ratios are so very complicated in all atmos- pheres composed of different gases as to become intelligible only after special analyses. The energy of radiation originates as a special effect within the isothermal layers of the different gases, so that it is proper to speak of a constant temperature of radiation, but this ignores the complex phenomena that exist in the non-adiabatic and the adiabatic regions. Similarly, the temperature of a star as determined in the spectrum refers only to the source of radiation in its isothermal layer, and not to the other temperatures above or below that region. It should be noted that on Fig. 10, and in Table 6, the temperature of the black body solar radiation is somewhat less than 7687 Centigrade Absolute, and it will be shown to be 7655. There is nothing characteristic of the solar radiation at the temperature 5810, as deduced from the pyrheliometer data by Mr. Abbot and- other ob- servers. Indeed, it only remains to determine the value of the coefficient a in, (92) / to obtain the solar constant of radiation at the distance of the earth. We shall show that a at the sun's isothermal layer agrees 68 with the Kurlbaum coefficient, and hence that J Q = 5.85 gr. cal./cm. 2 min. before any depletions occur in the solar and in the terrestrial atmospheres. The Distribution of the Pressures The values of 'P, the pressure in kilograms per meter 2 , and the values of A, the pressure in standard terrestrial atmospheres, have been collected in Tables 9 and 10 respectively, for the same values of z, for the eight elements. The original computations contain many more points between z = 1000 and z = 25000, or the vanishing plane of the gas. It is necessary to use P in all thermodynamic discussions, but A is a practical unit that may be conveniently employed in descriptive explanations, and its meaning is easily understood in the different levels. It is to be noted that P has almost exactly the same value on the plane of the photosphere for each gas, the mean value being, P = 616190 kil./m. 2 , A = 6.0814 standard atmospheres. There is no other plane on which this relation holds, all the pressures diminishing in horizontal lines above the photosphere, from HI to the heavy gases, and increasing below the photo- sphere from HI to the heavy gases. In the adiabatic region, below the photosphere, these computations can be extended to enormous pressures corresponding with the great temperatures. On the other hand, in vertical directions, the hyperbolic law m z = (Const)p for a selected pressure holds true as shown by Table 11. The height of the vanishing plane is seen under the first section for A = 0, where the product approximates the mean value m z = 45002, and similarly in the following sections. COMPUTATION OF THE THEEMODYNAMIC TERMS 69 TABLE 9 THE PRESSURE OF THE DIFFERENT ELEMENTS ON THE SAME LEVELS (Kilograms /meter 2 ), P (M. K. S.) z Kilom- eters Hi 1.00 H2 2.00 He 4 C 12 Ca 40 Zn 65 Cd 112 Hg 198 Means 25000 20000 15000 10000 5000 1000 900 800 700 600 500 400 300 200 100 90 80 70 60 50 40 30 20 10 Photos. -10 -20 -30 -40 -50 60 2648.2 12877 44710 120450 276700 515610 523640 531780 540000 548440 556960 565630 574430 583350 592410 593350 594250 595170 596090 597010 597930 598860 599780 600700 601630 602570 603500 604450 605390 606330 607270 608210 609160 610110 611060 620670 630430 640360 650430 660670 671060 681620 692340 703230 821980 958160 1116920 1305550 1526000 1778800 2073500 2426700 2839900 3348700 3858900 n -9000 2426700 2 . 727x10 0.03778 110.52 8165.6 108265 440880 454850 468670 483210 498220 513688 529640 546110 563090 580600 592390 584170 585960 587760 589560 591370 593190 595000 596830 598657 600550 602460 604390 606300 608230 610170 612110 614060 616010 617970 637910 658490 679730 701667 723480 746020 769250 793220 817920 1115480 1522400 2078830 2839900 3874000 5287000 7158670 9555800 12498900 20247000 30917100 -4500 2459365 Top of the Inner Corona 4 0.24958 10770.3 328086 348870 371020 394560 419610 446250 474760 505060 537290 571590 575100 578670 582260 585890 589530 593190 596870 600580 604310 608063 611790 615510 619290 623060 626870 630700 634560 638430 642330 646250 686820 729930 775750 824440 877240 933400 993160 1056760 1124390 2082760 3873360 6776430 10718300 15786300 22074500 Top of Chromo sphere 92408 114260 141280 172020 209435 252660 304807 367570 443250 534560 544300 554210 564320 574000 585070 595740 606600 617620 628860 640370 652700 665240 678050 691080 704380 717930 731750 745820 760170 774780 931240 1119300 1345370 1622400 1956450 2357200 2839900 342350Q 4127100 19705500 52206000 105061000 -5 7.755x10 0.46894 26.881 362.57 2350.4 9785.8 30407 75783 148280 287830 306410 326180 347230 369640 393510 418850 445830 474540 505110 537620 571630 607760 646200 687070 730520 777570 827660 880960 937720 998070 2064900 3871580 6955330 11488400 17435200 24918300 34050000 44937800 57678750 306112700 0.771869 83.238 2110.5 16411 73052 231670 256110 283120 312970 345990 382510 422470 466620 515380 569230 628814 695150 768500 849580 939200 1038290 1147800 1268900 1402800 1550800 1714440 4700900 10972600 Reversi ng Layer 3.7724 4185.1 110313 140109 169904 209405 248906 301903 354900 428378 501856 606612 711367 856834 1002300 1211515 1420730 1713140 2005550 2422725 2839900 3425725 4011550 15310000 3.150x10 11380 19967 32999 51958 78746 115760 165960 233350 323060 442456 603000 816820 1121320 1526210 2090450 2839900 3853640 5144000 6682670 8468400 10514000 616190 2593212 70 80 90 -100 -200 -300 -400 -500 -600 -700 -800 -900 -1000 -2000 -3000 -4000 -5000 -6000 -7000 -8000 -9000 -10000 -12000 -14000 Radiatio layer ZR Pressure -2250 2530410 -750 2598550 -225 2516570 -140 2909024 -80 2839900 -45 2465175 2593212 70 A TREATISE ON THE SUN S RADIATION i i co 600 &: 1 13 rH t- 00 CM "tf CO CO rH CO CO IO os eo rH rH CM CO ^ IO S3 a Reversing Layer 3.723X10-6 .04130 1 . 0887 CO CO O tf OS iH r-i CM CM CM IO rH OS 00 co * **" 10' t-' ,5s i 1 ,1 fcH O S Q ~ bs^ 06 'CM t- CO rH OS rH t- * OS - TJ CM ' 9 co o t- eo o So <* co Tf CO t> rj< 00 'i-i CM CM rH CM IO OO * co os -* ^ rH CM rl< CO 00 rH CM CO CO 00 rH CO t* t* t* 00 nil i CM o co eo co eo * T* <* T* 10 10 1010101010 10 10 10 10 CO 'k O5NOOOCO lOOOCMCOrH cct-ooo eo^cot-os Nil! CO Tj< rH OS CO ^SS2 t- t- t- 00 00 SS8 eo 10 t- os 00 OO OO 00 i 10 10 10 10 10 10 10 10 10 IO 1O 1O IO IO kO *8 co CO CM O COrHCOOOrH OOOOOOOSCM OO^rHtS OrHCMCO^ Hill O5 OS O O rH IO <* T}I CO CM IO CO t- 00 OS 00 OO OO OO 00 rH O OS 00 os os os os t- OOOrHCM 1O1O1O1O1O IO IO IO IO IO 10 10 10 10 10 IO IO IO 1O 10 oo oo os os o oo oo co CM jo eo eo g ^ TH 1-1 CM eo * 10 o eo t^.^4^00 iOt^-CO^CM XOOO OSi-liOtH Oi-HiOi-HO ^H0t>00 CMOSrjtOOOO N COCOOOiH ^*t~O^fOO CMOST^t i ( O <^ CM 00 COOOCM Tfiocooo OSOCMCO^* eor-iocM oeoe^ot- t-Tfuj eoeoeoeo eot-ti>t- t-osi-ieo OOOOCO OSCDOSOSTl< i-l-^OCM OOOOCMO oo-^coos eo-^CMOOs oooo-^eo ecc-NCMO ioocoosr-i cot THT}< OOCMCDOCO t t- o 10 eocOi-HOco ooi^CMt oo oco O O i * iH i-HCMC^COCO CO t . COOSN t>OCM^H OOOift iHOOSOS OSOr-lTjtf- OOO io 10 10 io eo co eo eo eo eo eo' co CD' eo t-' t-' t-' t-' oo TH jo o oo oo CM O TJ j eo rjlO eOOO CMrH 10 10 10 10 eo eo eo r-l CO O* OS 00 eoeoeo eoeoeocoeo CDOOOST-ICM lot-oeooo osoeot-c I X S| CD CO (M OS i-H CO CO CO 5< 5 co CD co m O CO OO CD ^r o co co I-H t- co o m m CD t- t- oo CO CO O O o 1 in 2 T T u O OS Tt TH t- CD 1-1 m -^ co os OO O CO TP CO co co oo m UO e in *o 1 X X oo co oo -^ co t- m IH ^ CD t- OS i-H CO S t^ 0? N NJ CO o. oo m 1-1 ^ i-i co 00 TH CO CO CO CO CO CO CO CO CO CO H * ' T T T c o o o o co ^ w G} 00 lO ^ O CO t- t- t> t- t> o t> co oo co m o m ^ co TH os m co t~ t- S oo * H 85 T T os i-l t- o os CD t~ OO O r)< CD 00 OS CO Tj< t- o co oo ^1 *" I XX os in TH CD oo m co m oo in eg os CD co t- oo oo os o pili CO CO i-l i-t O CD C~ OO OS O os os oo t~ m in m m CO CO CO CO . 002547 ~ 1 1 1 CO O CD 00 CO t- CO -^ CO IN i-l i-l iH m c~ os TH co oo oo oo os os OO 1> t- i-H Tf rH CO CO CO 00 in t- os I-H co os os os o o t> o co m oo o co m t- os TH ^J< t- llli 09 *> t- in -* o o o o co o o o o o o O 9 t- 00 CO gg n< o I-H TH o * CO CO i-H OS co co co <* t- TH CO T* oo c O o o o O O o o o o o o o o e> o o o o o o O O O O O OS 00 t- CO 10 3iSS w o o o o o 10 O' in o in CO CO TH iH O OS 00 t- CD m TI< co co r-t S COMPUTATION OF THE THERMODYNAMIC TERMS 77 O IO Tf Tf CO Tf OS CD CO TfTfO iO O OS CO IO CD COOJIO OOCOTfCOCO OO 00 rH CO rH 00 IO CO rH O> rHCOCO eo co Tf 10 co co eo rH o co O oo eo t~ loeoo OOTfrH O CO CO O 10 CO OS Tf IO CO 00 rH IO rH O OO O rH Tf OSOrH COTfCOt^OS COO>CO rHrH rHrHrHrHrH CO^fOO li III O * 00 CD CD CO CO Tf ft -^ lO CD OO 00 O5 O i-H N (M CO CO CO IO CD rH Tf co oo co o> co co co eo co o 00 O CD 10 ^f C7> CO 00 CO i-l rH CO CO CO Ssl O OJOOOU5 OOCOOOCDCO co rHcocoo cooeocoio t- OOOOOCO t-Tff^CO'O 00 OOOSrHCO COlOt^OSrH -f O O O> CO CO kO O i-H TH O 00 CD ^O IO IO CO t- i IO U5 IO CO CO CO rH Tf CO 00 Tf rH . Tf eo co co co co O5 O rH CO CO Tf IO CD CO CO CO CO CO CO CO CO CO CO CO CO O O> l- t- CO Tf OO OO IO t> OS rH Tf OS O rH CO i co eo eo eo eo eo eo w co t- rH 10 o co oo co Sco oo o co 10 oo CO CO t- t- t- t- o o o o o o o VOCOt-CO rHCOrHTfTf OrHOrH lOOlOrH OOTfrHOOCO lOCOrHN OOrHCOCO OOrHTfCOOS CO^OCOt- OrHrHrH rHCOCOCOCO COCDOU5 Tf Oi O CO CO 10 co t- o o 05 rH CD CO O Tf CO t- 04 CO lOrH t-eOO>lOi-l t>t-OOO Tfio lococot-oo ooTfoco coco cococococo coTfioio COCO COCOCOCDCO COCDCOCO COTfOJIO C-JO>OO rncocoo CDOTf OOOlOrHrH U3COOO TfcOCDt-t- S2 SS2SS8 *K* I " 8"-| rHCOCOTf lOCDt-OOOS Mil I I II I I I I I I I I I I llll ^ t- 00 rH iii o 3 ko oo 3 S!5 gj .2 8.8 I 1 s I 78 A TREATISE ON THE SUN ? S RADIATION The Gas Efficiency of the Different Elements The gas efficiency R is the velocity-square per degree of temperature, and hence is the kinetic energy of the gas mole- cules in their movements of translation, oscillation, and collision, corresponding to a change of one degree of temperature. This is the thermodynamic coefficient and does not refer to the electromagnetic oscillations within the molecules and atoms, or the forces concerned with association, dissociation, and ionization. The distribution of R is given in Table 13 for the same co- ordinates. We find that there are two places in which the hyperbolic law applies. Along the photosphere, R = 127702/w, or R m = 127702, and along the line dividing the adiabatic layers from the iso- thermal layers, the relation is R a = 281808/w, or R a m = 281808, that is, approximately along the broken line, though the transi- tion is not so abrupt in the computations. The relations for R are shown clearly on Fig. 13, with a very large range for hydro- gen, contracting to a small range for mercury. Hence, one degree of temperature signifies a large change in the velocity- square of the hydrogen molecule, and only a small change in the kinetic energy of the heavy mercury molecule, in fact, in ac- cordance with Clausius' Law, that kinetic energies are constant for the different elements. The secondary deflection in the curves for R occurs at the height of the greatest gradient in the temperatures, and is the compensation for it. In the lower levels of each gas R becomes a vertical line, and these lines are connected together by the constant R a m = 281808. In the case of calcium the values in the adiabatic region seem to require a slight readjustment by a second computation. This example is left in order to illustrate how by another trial such divergences can be removed, that is, by selecting somewhat lower values of the temperature T in this case. At the levels of the source of the radiation layers the value COMPUTATION OF THE THERMODYNAMIC TERMS 79 The Cause of the Sharp Limb of the Sun The cause of the sharp visible edge of the sun, that is, the configuration of the disk in the midst of a gaseous envelope, has been a problem of much difficulty to explain. Perhaps the favorite theory is that this layer is composed of a cloudy con- densation, as evidenced by the granulation, the appearance of the spots, faculae, and similar forms. Another view is that of Schmidt, who advocates the application of circular refraction, by which a ray of light originating at great depths in an at- mosphere of densities diminishing from the center outward, will at a proper height pursue a circular path. Visible surface phenomena may in this way actually exist at much lower depths than the surface of the disk, so that sun-spots and the other phenomena represent deep-seated optical effects. Julius has applied the principles of anomalous dispersion, or irregular refraction, among neighboring masses which differ greatly in density from one another, to account for the prominences, the Fraunhofer lines, as absorption lines enveloped in dispersion bands, reversals of lines in the chromosphere, some of the sun- spot appearances, and similar effects. Abbot intimates that the effect of scattering upon the outgoing solar radiation is such as to "prevent us from seeing toward the center of the sun, when looking directly at the middle of the solar disk, to more than 5000 miles (8047) kilometers, below the reversing layer." "At the edge of the disk, owing to the oblique line of sight, gaseous scattering will probably extinguish almost all yellow light start- ing from more than 500 miles below the chromosphere, while an even less thickness suffices for blue or violet light." It may be remarked that all these hypothetical suggestions make of the sun's disk an optical effect of some sort, rather than a material distribution under the thermodynamic laws prevailing in the solar envelope. An examination of Tables 6 for T, 9 for P, 12 for p, and the corresponding figures 10 for T,ll for A or P, 12 for p, makes it evident that the sharp edge of the solar disk is due to the rapid diminution in height of the heavier gases under the solar 80 A TREATISE ON THE SUN S RADIATION rH 00 10 a CM i as S S S t- 00 OS U5 OS N O CO CO l CMCOeOCOCO COCO g|S Ot-NO O o t- t- NCMN I O OOSOOt" oooo o oooo fCOCMiH . i-INCO** 8 I I I I o JS a* COMPUTATION OF THE THERMODYNAMIC TERMS 81 OS t- "- N 10 c> 10 to N O 00 O t- 00 <* t- CO t- t> OO O eo Tji 10 CD t> o> N N M 10 C< ^ 00 -^ O ^H O * 5O i-H O Oi O i-H N CO 03 CO CO CO CO >-! 00 00 t- t- O5 1C IN C5 > oo os o o i SC- 00 100 f rj< IO t- O t- r)< r-l 00 c- t- t- t- t- t- t- CJ CO 00 OS N CO T-! (M t^ 10 CJ CO oo co w ko co t- oo o 1-1 01 co "* <* -t t- t- t- t~ * "* S3 O t- U3 COCOTfTtlT}" T}lOl6 r-i TJ< T-iocnoooo OiHcocpt-i o I I I I I I I I r-l CSI CO 1 I I I I 0000 0000 t- 00 OS O I I I I T I I 82 gravitation. Thus, by Table 8, calcium (40) vanishes at 1000 kilometers above the photosphere, that is, 1".4 arc; zinc (65) at 675 kilometers = 0".9; cadmium (112) at 385 kilometers = 0".5; mercury (198) at 210 kilometers = 0".3. Since the radius of the sun is 31' 59".26 = 1919".26, it follows that all the gases whose atomic weight is greater than 40 are so distributed as to vanish physically and materially at one two- thousandth of the radius of the sun from the plane of the photosphere, as defined by the common pressure of six atmospheres for all the elements. That is to say, these heavy gases pass from enormous values of tem- perature (Table 6), and from enormous values of pressure (Tables 9, 10), to vanishing values in a very narrow vertical stratum. The density changes (Table 12) are by no means so surprising, except that all the densities for the heavy gases are really very small. The sharp disk of the sun is, therefore, a thermodynamic and gravitational effect upon gases of which the monatomic weights are somewhat greater than the 40 for cal- cium. This element just floats comfortably upon the layers of still heavier gases, and it is easily observed in the spectro- heliograph as a surface phenomenon, having many configurations, as in the faculae and flocculi. The lighter gases, * especially helium and hydrogen, rise to great heights, the hydrogen to the top of the inner corona. The gases between helium and calcium, up to bromine, columbium, germanium, krypton, molybdenum, rubidium, selenium, strontium, yttrium, zirconium, ought to just reach the level of the reversing layer, if they exist in the solar envelope. These distributions conform well with the re- sults of the spectroheliograph observations upon the height of the chemical elements in the sun. CHAPTER III The Determination of the "Solar Constant" of Radiation in the Isothermal Layer of the Sun Statement of the Problem IN Bulletins No. 3, 1912; No. 4, 1914, of the Oficina Meteoro- logica Argentina, and in the Treatise on Circulation and Radiation, 1915, an account was given of an effort to solve the dis- crepancies in the data of radiation as observed by the pyrheli- ometer and the bolometer. The criticisms regarding the former consisted in showing that in using the Bouguer formula for pure radiation in the forms (96) I = I Q e~ km , or 7 = To/*, for absorption and transmission respectively, it is not proper to interpret the original formula, (97) 7 = 7 8ec2 , as if sec z = were sec z = m, for m^ = 1, the depth of the Wo atmosphere. The graph (log I . sec z) of observation serves only up to sec z = 1 in the zenith, but it cannot be extended to the non-mathematical sec z = o without unwarranted assump- tions. These are, that p is constant from the sea level to the vanishing plane of the earth's atmosphere, whereas in fact p is very variable; no account is taken of the potential, specific heat, and ionization energies which are absorbed to balance the kinetic energy, so that the extrapolated value of 7 = 1.950 gr. cal./cm. 2 min. depends upon a very incomplete treatment of the observations made at stations having different elevations above the sea level; the thermodynamics of the radiation throughout the earth's atmosphere shows that the total amount of black body radiation falling on the outermost strata is about 83 84 A TREATISE ON THE SUN'S RADIATION 3.980 gr. cal./cm. 2 min., and that this suffers a series of deple- tions which reduce it to about 1.500 gr. cal./cm. 2 min. at the sea level. The data of the bolometer spectrum indicate that the sur- viving long- wave ordinates, X = 1.50^ to X = 2.50 /x, seem to originate in a radiating solar temperature of 7700 Centigrade absolute, while the middle waves, X = 0.45 M to X = 1.50 M, appear to represent a radiation at about 6900, having lost a large amount from the original source, and the short waves, X = 0.00 M to X = 0.045 /z, have evidently been excluded en- tirely by reflection or absorption within the high levels, 40,000 to 70,000 meters, of the earth's atmosphere. The middle or- dinates, X = 0.45 n to X = 1.50 fjL, suffer progressive depletion from one high station to another, and arrive at sea level with a value of about 2.47 gr. cal./cm. 2 min., when fully corrected for band and minor losses in the thermal spectrum. These two types of observations afford such divergent data that it is im- possible to reconcile them on the hypothesis that the solar radiation originates in gases whose temperature is about 6900, but whose efficiency corresponds only to a temperature of about 5800. This is, of course, equivalent to assuming that the Kurlbaum coefficient of radiation a in the Stefan Law, at the distance of the earth D, m /-. does not hold true at the sun. Otherwise, (98) 7. = , (6900) 4 (|) 2 = a (5800) 4 (|) 2 , so that th^re results, If we take as a mean value of the Kurlbaum coefficient log o- = 5.74103, then we should have for the defective log 0-1 = - 5.44000, so that while a = 5.509 X 10~ 5 , the defective or inefficient value would be ai = 2.754 X 10~ 5 , half as much as a. SOLAR CONSTANT" OF RADIATION 85 We have already proved that the mean temperature of the sun's isothermal layer is about 7687, as in Table 6. It only remains to determine the temperature within the isothermal layer at which the emitted radiation is generated, as well as the value of = J (Cp l + Cpo) = J C^i + Ro) r, the mean of the non-adiabatic values at the K 1 top and the bottom of the stratum. In the computations lead- ing to the adiabatic value R a there is one fact of interest. At the initial point, determined from the terrestrial values, we have for calcium vapor on the sun, as an example, 7 X P (earth) = P (sun) = 28.028 X 101323.5 = 2839900 (M. K. S.) T x T " = T " = 28.028 X 273 = 7651.6 (M. K. S.) y~ l x p " = p " = 1.7999/28.028 = 0.064219 (M. K. S.) 7 X R " = R " = 28.028 X 206.205 = 5779.5 (M. K. S.) This value of R is not the adiabatic R a , but only one point in the gas at some distance above the adiabatic strata. In computing the terms of the gravity equation, go X 28.028 (, - so) = - Pl ~ P - (Cp a - CAo) (r. - r ), Pio this value of (Cp a - Cp lQ ) (7\ - T ) = (Qi - Qo) 1 is not the same as (Qi Qb) in (34), because it is not referred to the true adiabatic Cp a = R a _ . The initial value of R serves as a preliminary in computing R a . Thus, (Qi Qo) 1 is a positive 86 A TREATISE ON THE SUN'S RADIATION term, that is, it has positive values which must be added to the p p positive value of - to produce G (zi z ) ; below this PlO initial R we find that (Qi Qo) 1 is a negative quantity to be p p subtracted from -, which has now become greater than PlO G (zi ZQ) . Proceeding in this way, R enters at last into a stratum where it becomes a constant, R a , which has a true adiabatic value. Now the course of the development of the J^-term depends upon the trial temperatures T from level to level, and if these are not exactly correct there will be a corresponding error in 'Cp a = % R a 7, and all the subsequent terms derived from it. This K ~~~ L , explanation is needed to account for the minor irregularities that occur in the tables for eight elements. In Table 14 the free heat has been reduced to 20-kilometer strata, whatever may have been the intervals in the computation, for the sake of comparison at different heights. Only a small portion of the data in hand is here quoted, but this is full in the isothermal layer and near the level of the photosphere, where it is likely to be of most practical usefulness. A similar arrange- ment is made for the other tables of this Chapter. We note that along the lower boundary of the table the value f (Qi ~~ (?o) is a vanishing quantity in the adiabatic region, and that along the upper diagonal a common maximum value of about 6500000 is reached for all the elements whatever the height may be between the adiabatic level and the vanishing plane. The irregularities in this diagonal line depend upon the facts that the vanishing plane does not occur on the exact height quoted in the Table, and that the adiabatic value Cp a may not have been found with precision by the method of trials in this computation. In Table 14 the mean value of (Qi Q ) on the photosphere is 3443372 for 20 kilometers, or 172.1686 for one meter, this being the mean kinetic energy or velocity square per degree. It appears that all the chemical elements on the sun have the same kinetic energy at the surface of the photosphere. Above that level 87 >oooo o ! iis i t-o >*co ow II iooi-Hco 0> NOS( I II I I I I I I I I I I I ; g! o = ^ < lilit^ScS CJ^tSJnS < SScOCN^H O OoSrfcoS 00< ^^^^^ ^ rn< 0000' T}T}l WO t- -^ ( [ I TTTTT 7 ^TTT 7 i i 1 7 i' So o o< COO(M I I I I I co eo co co eo I I I II g3 5 SS^^o 6-oco^.o ^^o^o cScSclcS^S cS cSclcScSS co'SSco'S S^^S : I I I I I I I I I I I I I I I I I I I I j>J . I I I I I 5 COCONNCSJ COCOCOCOCO I I I I I OOOOO O OOOOO OOOOO 0000 OOOOO O OOOOO OOCOCOO O-^Ol^OOO COOOOOOT)< C OS OOOOt-t-D Tj Cfl t-OOOOO ^ g moo cMcSco.-iS o cocot^c^S III | Ml s T7 i i i i i i i i i i i 1 1 ** IO CO IO -H ^< IO t~ i ( CD OO IO t- CM i-l CO ft osoi-H ot-ji,-io> co .-it-coot- 1 1 1 1 II 1 o tooo toioioiOTi< -* ^cocoeow 1 1 1 1 II 1 T77 11111 i 11111 i 1 QO CDCO^t-lO OOiOt-^ t-OOOCM^. t- 1 1 8O t- IO i-H i-H r-(O5t-lOCO CM OSt-incOiH 00 i i | I || i 83;^ ^^^^^ ^ ^.^^^^ ca | | | | || | 7 1 i i i 11111 i 11111 i 02 Oi * T!< 10 00 OCM1000CM CD Tji <* TJI U5 t> OS t><3> IO CO 1 1 1 81OOIOIOIO OOrJ IO CM OJ IO Oi Tf Ct-CD IOCMOOOCD O 1 1 THCO rji^-^^-^ coeoeococo co coeocococo cocoeoiN-i^-!dodc'coio^j ^J^^^^ ^Soi * 1 rj< CD N i-t T i i i i 11111 11111 i 11111 11111 ii i .1 O CO *O IO (M COCOCQC^rH i-HrHi-Hr-ti-H iH T-HrHi-Hr-Ht-^ OOO^C^OO ^OOCO^ t- T^ IOOO 00 1 77 i i i i i i i i 11111 i 11111 11111 i i i i z Kilometers 0000 ooooo ooooo o ooooo ooooo oooo OOOO OOOOO OOCDTl on the vanishing plane. These data are useful in testing the theories of radiation which have been founded upon various hypotheses and as- sumptions. Table 16 gives the distribution of the work of expansion (Wi W Q ) in 20-kilometer strata, and it is of an exactly inverse model to that of (Qi Q Q ) and (Ui U ). The mean value on the photosphere is 2009412, and on the levels above the photo- sphere there is decrease from hydrogen toward mercury, from a maximum to vanishing values, while below the photosphere there is increase toward a maximum. The mean value of this maximum of work is 4122458, and it is the common value of all the elements in the adiabatic strata. It diminishes to on the vanishing planes of the several gases. It should be noted that the mean value on the photosphere is about the mean of the common maximum 4122458, which is, also, the same as that for hydrogen, and the common minimum on the vanishing planes. The distribution of the inner energy is like that of (Qi Q Q ). It has the mean value 5467133 on the photosphere; on the levels above the photosphere it increases from a minimum value, namely, that of monatomic hydrogen, H\ = 1.00, at 4122458 to a maximum value which is 6656690 for all the elements; below the photosphere it decreases to the same minimum of hydrogen 4122458. It should be noted that the mean value on the photosphere is about the mean of the maximum 6656690 and the minimum 4122458. Mean values are applied to individual gases. The Equation of Conservation of Energy has three cases: The First Law of Thermodynamics, (d ~ Co) = (Wi - Wo) + (*7i - Uo). (1) On the vanishing planes, - 6656690 = 0000000 - 6656690. (2) On the photosphere, (approximate) , - 3443372 = 2009412 - 5467133. SOLAR CONSTANT" or RADIATION 91 o oooo< tO in>OO^J<0! i ^ t i * iH ^ < I co i-Hcococo- 11 3IIS' CO CO CO CO S 55 oooo oom t- vo oo t~ 1000 O Ntt^Wt- 00 O . .5g! COOr)I-HCO^ OrHCOCO'* O5i-(CO 5DC-OOOOO5 OOOOO O i-l i-l i-H i-l i-l r-(COrl< g gg^gS || CJ (N (N N M Tj00 cococococo cococococo coco o ooooo ooooo oooooo< OiOOCOO OOOOO OOOOCOOO< SOOCOCO CO O CO ** t- U3 < 1-HOrH O Tj< IO j-4 OS 05 > o co H co o co oo < nO -H Tjl t- i-H CO O t -ICO CO CO CO CO CO CO t ooooo ooooo I I I I*? 92 A TREATISE ON THE SUN S RADIATION o ooooo ' C<1 t CD 1C CO i-H o in kom inio I I I I I I oooo co eo OS <* CO O t- N oo t- 1- Tftf eoeo I I I I N 5iooS?i S: U3 Tj< rjt T}< Tjt CO CO I I I I I I I OOO OCOCO IO5OO IO OOi-HCO CC-^O CD CO tO I I I ooici ooco os oo t I I I I I oo rl< T* II :ss.s s 3 8S3S8 8S OOJOSOIOO OO t-tOCOiO-'J 1 O> *-l coioicioio 10 m in m m m T}"-^ I II I I I I I II I II O OOOOO OOOOO 00 oooc 05 O rj< O 00 CD jHCOCDOi ^OOt-CD0^05 SOO OOOOO O >00 OOOCOrflM (8J JL 777 "SOLAR CONSTANT" OF RADIATION 93 (3) On the adiabatic levels, 0000000 = 4122458 - 4122458. The Second Law of Thermodynamics, T w (Si So) = (Qi go), on the photosphere is 7689.5 X - 447.8 = - 3443372. The results of Tables 14, 16, 17 have been plotted on Fig. 16 in order to exhibit the general mutual relations of the free heat (Qi QQ), work of expansion (Wi WQ), and the inner energy (Ui Uo)j of the different elements in their distribution with the distance in a vertical direction from the level of the photo- sphere. Allowance is made for imperfections in the results of the computations by the method of trials, the essential facts being that the lines pass through the points indicated in the mean values under Table 15 at the respective heights of the top of the adiabatic layers, the plane of the photosphere, and the vanishing planes. The first law of thermodynamics is con- served on every level, and the heights as ordinates over any value, taking in the three terms, are distributed by the law of the equilateral hyperbola, x y = c, already given. The corresponding constants can easily be computed by collecting the values from the tables, but they will be omitted in this place. The necessary conclusion is that the thermodynamic system (Q. W. /.), as well as the dynamic system (P. p. R. T.), are distributed according to the exponential law of decay, counting from the adiabatic region upward. These data can, therefore, be utilized in constructing the general laws of solar physics in their relations to molecular and atomic conditions. We shall proceed to develop their consequences in relation to the "solar constant of radiation." It will be convenient to compute the radiation potential, KIQ = - = c T a , exponent (a), and the (log c), in the Vi VQ law of radiation, K w = c T a , in order to study their connection with the Stefan Law J = a T 4 . The specific volume v = l/p is computed from the values collected in Table 12, and thence the values, _ tr. - u. Kv> - ~ cT ' 94 A TREATISE ON THE SUN S RADIATION "SOLAR CONSTANT" OF RADIATION 95 I CDOCOCDCD O t- 00 O rH Ol I I I I iH rH CMCO-'tt-rH IIII I Mil t- ooooo t- COlO 00OOCOt-Tl< t-COCOOCO 00 COt-COt-Ol i-IOOTjcD T 1 CO CD t^ CM >O 00 T}< CO O rHcot>cM t>t-t^6do6 dbdboooooo 60 6606600000 asasasoo CM o as -4000. . 2.3857 -3.34790 2.4270 -3.18210 -8.33854 E -4500. . 2.4080 -3.26224 2.4276 -3.18348 -8.33992 .a -5000.. 2.3938 -3.32427 2.4280 -3.18440 -8.34084 jj % -5500. . 2.4178 -3.22897 2.4285 -3.18555 -8.34199 ,Q .2 -6000. . 2.4151 -3.24294 2.4288 -3.18624 -8.34268 < -6500.. 2.4252 -3.20316 2.4292 -3.18716 -8.34360 *s -7000.. 2.4291 -3.18916 2.4295 -3.18885 -8.34529 1 ! I . ! 1 I Can be extended downward in definitely 98 A TREATISE ON THE SUN'S RADIATION it extended from the same point (LOO, 0.00000), though there may be a question whether it should not be parallel to the non- adiabatic line, the shift being parallel rather than triangular. The practical effect is the same because (a, log c) extends for so short a distance in the square that no important change could come from a slightly different slope. The practical range of (a, log c) in the non-adiabatic region is in the small rectangle. Table 20A contains the mean values adopted in computing A 6.00 4.00 200 0.00 LogC -10.00000 -20.00000 FIG. 17. Helium. The (A\ log C) Lines in the Adiabatic and the Non- adiabatic Regions. the pair values (a, log c) for helium. Each of the elements HI, H 2 , He, C, Ca, Zn, Cd, Eg, has been computed in the same way, and these auxiliary tables differ from one another only by minor variations, dependent upon the uncertainties due to the method of trials. Thus, collecting together the ini- tial values for a = 4.00, we have the following results for log c. See Table 19. log A = Formula (40). log (log 7\ - log To)' log C = log KIQ - A log TV Formula (39). Plot (A, log C) and construct the auxiliary table for (a, log c). OF RADIATION 99 Take trial values of a (a lt a 2 . . . ) and compute log c = log KIQ a log TIQ] then take the corresponding pair values (a, log c) from the table. If i (trial) = a (computed) the pair (a, log c) is correct. If a\ < a, take #2 = \ (<*i + ) and proceed to a second trial. When a u = a, this is the adopted (a, log c). These mean values of Table 20B might be used to construct a common table for all the chemical elements, to serve in a re- computation based upon the series of mean values of the differ- ent hyperbolic constants that have already been obtained. At present, the results given have all been computed for each ele- ment, quite independent of one another. Using the auxiliary Table 20A, the pair values (a, log c) TABLE 20A THE WORKING VALUES (, LOG c) Adiabatic (a Log c) Non-adiabatic (ex Log c) a log c a log c 4.00 -6.80000 4.08 -10.78000 4.07 -10.74500 3.50 -4.50000 . 4.06 -10.71000 g "So 4.05 -10.67500 as 3 2.50 -3.35000 c a 4.04 -10.64000 r 2.49 -3.32700 g 4.03 -10.60500 53 2.48 -3.30400 !_ 4.02 -10.57000 aj 2.47 -3.28100 1 4.01 -10.53500 a 0) 2.46 -3.25800 cn 4.00 -10.50000 J3 2.45 -3.23500 M 3.99 -10.46500 O 2.44 -3.21200 -M 3.98 -10.43000 4-> H 2.43 -3.18900 3.97 -10.39500 1 2.42 -3.16600 *T3 0) 3.96 -10.36000 1 2.41 -3.14300 3.95 -10.32500 j> 2.40 -3.12000 3.94 -10.29000 2.39 -3.09700 J^ 3.93 -10.25500 Jl 2.38 -3.07400 15 3.92 -10.22000 2.37 -3.05100 .y 3.91 -10.18500 1 2.36 -3.02800 2 3.90 -10.15000 2.34 -4.98200 en 3.80 -11.80000 2.33 -4.95900 & _ _ 2.32 -4.93700 3.70 -11.45000 2.31 -4.91300 _ _ 2.30 -4.89000 3.60 -11.10000 Slope -0.100 -0.23000 Slope -0.100 -0.35000 100 A TREATISE ON THE SUN S RADIATION TABLE 21 BUTION OF THE EXPONENT (a) IN K i0 = cT a ormation between the adiabatic and the non-adiabatic strata DISTR The s TJI 10 CDOO -foo 10 t- ooo eoiot- > O rH NN NN CO N N CO NNN IN N NN NN N N NN NNN Non- diabatic Means 3 as Z om CO O5 O5 -t ( t~ ( to t-o o t- oo oo oo< 0> t-00 0505 05 05 05O 05NU3010 i TH CO OO 32 ! S I I CO COCO COCO CO CO CO-* I . CO cot>ooa OOCDCDO5CO COCOCOCO-^' O5OSO5OSOJ oocft OOO OO O5 OS O5 coccococo co coco N oo5O5Or-i N eo-t 10 O 0050500 O 00 O CO-* * rj< Tjl Tjl <* T* T* -** OiO ^f t~ 00 OOO NCO IO 3 COCO CO NN NN N 3 -# O lOr-l t- lOOJCOt-O 5 rf t- r-liO OO OOi-Hr-IN q r-t r-i NN N eoeoeococo CO-* -^ -* rjlTf TJ i-l CON t-< CD O5 CO D < CO CO -4" ^ o oo o < .8 N NCO TH N O5 O5 O} O5 O5 O5 O5 O5 O5 O5 O5 O5 O5 O5 O5 C5 O5 O5 O5 O5 O5 O5 O5 O> O coco co'co' co co' coco" co co' co co coco" co co' co co coco" co co' co coco" co" -*' O N< r-l COI 10 101 0?05 0> 3 CO CO CO COCO CO CO CO O 00t> U3 Oi-!< t- t-oo 05 eiO( oo oo oo oo 5 O5O5O5O5 O5 CO COCO COCO CO CO COCO CO COCOCOCOCO CO COCO CO COCO CO COCO COCO <* * Tjf* NNN II I I""* CO-i'tOOOO OOOO I I I I I I I I r-l N -* CDOO I 1411, i i i i "SOLAR CONSTANT" OF RADIATIQN t> COO) CO OCO COTjt 00 O **CO 001O"3 OS OSCO OS COO "*00 "5j< i-H CO^ COCOCO 1C -*t- OS OO OO t- CO COrH -<}IOOCO to CDt- t- OOOS i-HCO CO 00 OOO1 t-t-OO ICO COCO CO rH rHrH rHrHTH 1 1 sis COCOCO CO COCO COCO CO CO COCO CO CO CO 1C i-HIO O CO ^ COCO 1C O - oo T}< co 10 co ic O CO rH T}" t- t- coco o' d o'o o d 7 7 77 7 7 ) CO ( o' o' o o o' 7777 i TH THrH CO -*CO CDt=- r-i t- t-t- t- t-t- t-t- CO o'o o do d o'd do d d o'o' 77777 77777 7777 1 1 1 t- t- co.co CO COCO I I u^ lCCOOSCDiO ic ic ic co CO ooi-i ic icco co COCO CO SCO iH Oil OS CO CD< os co oo co i 00 O O OOOOO O H i-H i-H TH THrHi-HTHrH i-t I I I I Mill I 00 O 00 rHiH rH i-HiH I I I M o oo rH iH i-H I I I CO1O C^ Tt< O t> 50^! CO CO CO co'co co* I I I O 10 O OS t- * 1C i-H O co t- 10 os ic 10 O 1COS -*f OOCO r}< CDOO i-H CO CD JOJOCO CO T}< 1C t- OOOS CO 00 So" 1 1 1 00 O O OOOOO O 00 O OO O 00 OO O HTH TH TH ^^^^^ 7 77 7 77 7 77 77 7 CO COCO I I I CO 1C CO OSOSOS0300 CO t-OS O i-HCO O COCO 00-* CDCOOt-^f i-H OOlCCOOt- COr! 0> 0) 0> 1 1 1 1 1 1 1 os 00 os 1 - OS rH O IO * OS CM IO CO* ** Tj< TJ< rji Tj< CO CO W 10 10 i-l CO o co co co * eg 05 co 10 10 o 10 ib t- O O5 O5 O5 OS Ci Oi 1 1 1 1 1 1 1 co co s co OS 1 CM 00 cOOCOOOO(M OJOOOOOiS i o Ol O O O IO i-l CO oo 10 co co co 10 05 t^ CO CO O t OS O O tO lO lO O U3 00 IO CO CO OJ 11 1 1 1 1 1 OS 1 1 U en I t- a> Tf oo co o N o> OJ ^ ^ 00 O5 00 *& CO * *' TjJ Tj<' CO CO CO / s "n? Tj <0 N N CO 'S* lO ** CO OO N O] W CO O iO OSOOOOOOi CO Tjt i* Tjt - S i-< f os' os' oi o> o> os' os' 1 1 1 1 1 1 1 ^J< IO s CO OS 1 2 X 4 o *4- s IO 4 SI S o 8 !s 10 4-> j3 lllllsi o o o I rf O O lO VO pa t- T* Tf lO Tj< C- lO CO U3 CO OS OO lO O 00 CO iH CM CS1 lO lO t- t- CO CO CO CO CO 00 i oj 1 _c CO N H OQ 3 -d a 0) 1 CO -^ -^ "* * "t CO ^ 1 OS OS OS OS OS OS O> 1 1 1 I 1 1 1 OS 1 4-> d ~ JSB^ 2 ' VJ T3 T* CD ,C *o ss^^sss SSSScigg TjJ Tjl Tf Tjl Tj< -^ CO* S S Tf CG 1 S NOOOO^Ht- lO C- CM O CO OO CO tM O kO Tf i-l O OS o ** co >o oo os os 00 CO CO CO CO CO 00 os' os' os os' os os' os 1 1 1 1 1 1 1 t- os c- os 1 s .^ en V 3 00 % -f ifl oo T-H Irt CO 1C CO ^1" 00 CO ^" O3 O O to lO kO O 6- os io os -^ co o CO O ^f O C il CO CO CO CO CO CO CO 3 co co co co' co' co co 1 1 1 1 1 1 1 CO 1 eo'os 1 1 * co o eo 10 co t- rH IO CO 00 t~ IO OS CO CO CO CO rH O OS CO t- co co 10 os m T}< t- COOS O O CO O OS St- Tf U5 O O5 U5 t- TJ< oo co t- co rH rH CO CO CO CO CO IO CO t* CO CO CO CO CO CO CO co co co eo co co co 1 1 1 1 1 1 1 eo 1 coos I coco 1 . 10 10 o t eo co rH 1 rH rH CO CO CO CO CO 5 i CD IO t- s os oo co rH CO eooo "" CO CO CO CO CO CO CO 1 1 1 1 1 1 1 CO 1 coos' 1 coco 1 8S3S8SB ! t- CO OS rH OO rH O t- CO IO 00 rH rH CO CO CO CO CO to i CO CO lOrH IO rH OS 00 SCO 03 *i M NN ^__^ CO CO CO CO CO CO CO 1 1 1 1 1 1 1 CO 1 O CO CO OS 00 CO CO t- rH CO t- rH oo oo co os t- oo t- rH rH CO CO CO CO CO co S o IOIO 8 II CO CO CO CO CO CO CO CO 6 v^S to CO CO CO CO CO CO CO i 1 1 1 1 1 1 eo 1 a s y co os 1 coco 1 IO O CO OS CO OO 00 CO CO CO CO CO O O fl jf w IO O t- t- CD rf Tf co TJI os eo eo us 10 oo oo os c^ oo c^ 1 a c f 4U g w >< ON H coco KNNNNNN M O) g bE CO CO CO CO CO CO CO 1 1 1 1 1 1 1 CO 1 .0 O _1 H) 3 coos' 1 y coco 1 t- IO CO 00 >* T* O t- oo 10 co co o co CO CO CO CO CO -H O o CO I g rH 10 co * co eo t- eo 10 co *o co o co 1 2 ^ k 5THERM II i s 1 CO CO CO CO CO CO CO CO 4-> o CO CO CO CO CO CO CO 1 1 1 1 1 1 1 eo 1 03 oJ a coos' 1 5 coco 1 O 00 IO IO OO CO t~ oo oo co co eo rH -f CO CO CO CO CO rH O 1 13 > co -o 0) cfl 53 t- t eo co oo NNNNNWN > co co co co eo co eo 1 1 1 1 1 1 1 CO 1 c^ coos 1 coco 1 * CO 000000 00 OS iO -^ rfi O TJ< co co rf eo co rH o 3 CO rH CD CO CO O O IO t- t> OO IO O CD rH rH CO CO CO CO CO 1 M > t- eo eo oo wp52 " co co eo co eo eo co 1 1 1 1 1 1 1 CO 1 co'os 1 coco I lOlOOt-rHOO CD O> O Tj< Tf O Tj< IOOO rH COO O os ooo oo Tji o o O 00 TJ< OJ IO O CO oo oo oo 01 1- os c- rH rH CO CO CO CO CO 1 t- co O 00 oos cow eo'eo'NNeo'eo'co' co co co co eo co co 1 1 1 1 1 1 1 eo 1 1 coco 1 ::;:;;; '.'.'.'.'.'.'. I-H ^t co o m co oo U &3 U U N O &3 1 s rH TP CO 10 CO 00 rHT)COrHOS s ; ! vi M 108 A TREATISE ON THE SUN^S RADIATION among them at various depths. Section I contains the mean values of the exponent a in the isothermal layer, and it ranges from 4.0526 to 3.9937, approximately 4.00 on the average; Section II contains the coefficient log c, which ranges from - 9.81409 down to - 9.60739 (C. G. S.); Section III contains the values of the exponent a in the adiabatic layer, and it ranges from 2.4241 down to 2.4110; Section IV contains the values of the coefficient log c, and it ranges from 2.27072 down to 2.24019 (C. G. S.) ; Section V contains the corresponding values for the diatomic hydrogens H 2 = 2.00, and it shows that the system (a, log c) is the same in the isothermal layer, but that it is very different in the adiabatic strata, since the exponent a ranges from 3.3522 down to 3.3213 and the coefficient log c from 6.84068 down to - 6.73767 (C. G. S.). The minus sign affects only the characteristic of the logarithm. An examination of Table 24 indicates that, although there are minor differences among the elements, there prevails a gen- eral harmony in all respects. The same range of the mean values is found for each element. Such differences as exist depend upon the inaccuracies with which the adiabatic Ra and Cp a were computed by the method of trials. The most im- portant data are the adiabatic values just before (2.1) entering the radiation layer, and those values which occur just within (1.2) the isothermal layer. In the values of log c of the adia- batic strata, the hydrogen HI and helium He are apparently a little too small in comparison with the other elements. It is be- lieved that an adjusted computation, following the hyperbolic constants throughout, will produce a mean value under (l) about like that now under (2). The mean 2.25450 has, therefore, been adopted for the following discussions regarding the value of the solar constant of radiation. The general problem is to interpret the following data: Isothermal Layer, a = 4.00, log c = - 9.81400 (C. G. S.). Adiabatic Strata, a = 2.4172, log c = - 2.25450 " The sudden transition from the adiabatic to the isothermal conditions seems physically like the release of a powerful spring, the discharge of a cannon, or a radioactive process, in which the 109 compressed state of the adiabatic strata is per saltum transformed into the weak state of the isothermal layer, the coefficient c being one three-millionth less in value. It is assumed that this represents the process of transformation in the configuration of the electrons, atoms, molecules, the change of state being accompanied by a power- Jul radiation into space. This is the source of the solar constant of radiation, whatever its history of depletion may become. We have therefore to evaluate the mean value of the coefficient log c, of which the limits have been determined, supposing this mean value to be the a of the Stefan Law, /o = 10 ) (T. - r ), L 2 it must have the dimensions of the specific heat, ^7 , the dimension of (vi VQ) = -rjrJ therefore, has the di- JVL Vi ^o . D M M fa-m mension ^-3 . -77- = r ^ 9 , (M. K. S.). T 2 deg. U L T 2 deg. kilog. meter 2 ML 2 ... . For 1 Joule = = -^ , ^ 2 , this is equivalent to kilogram Joule Joule meter second 2 degree meter 3 deg. volume deg.' ST. cm. 2 Similarly, in the (C. G. S.) system, for 1 erg = p, (?i Qo , gram erg dyne - becomes - 2 9 = ~ = - 11 ;. PI VQ cm. sec. 2 cm. 3 cm. 2 M Since -j-^- = 10, we have, (M. K. S.) X 10 = (C. G. S.) and the conversion factor is 10 from (M. K. S.) to (C. G. S.). PIO is the pressure as work per unit volume, and not force per unit area, though the dimensions are the same, Pressure = work per unit volume = M L 2 T~ . L =M L~ T~ . Pressure = force per unit area = M L T~*. L~ 2 = M IT 1 T*. Its conversion factor from (M. K. S.) to (C. G. S.) is 10. (Ui UQ) is the inner energy per unit volume, due again to the kinetic energy of the molecular motions. Since (Ui Uo) = (Cp a - Rio) (T a - T ), it foUows that the same factor of con- version applies as for (Qi Q ). Kio is the energy of the electromagnetic radiation per unit volume, and it is so denned by all authorities. Since TIQ and the exponent a are without dimensions, it follows that the coefficient c is the energy per (degree)** in the unit volume. 122 A TREATISE ON THE SUN'S RADIATION It may be noted in passing that Mr. Frank W. Very, of the Westwood Astrophysical Observatory, insists upon interpreting this equation of condition as if it were, Joules . , Joules , - -- instead of ~, -- (M. K. S.). area volume Very> JouIes = Joule ?= kilog 1 ^ J [Ml, Factor=1000 . area m 2 sec. 2 m 2 ' LpU> Joules Joules kilog. m 2 1 [~ M ~| Bigelow, - iL - i -- =*- - = ^ . -, ^ - , Factor=10. 'volume m 3 sec. 2 m 3 L L PJ He has published an erroneous criticism regarding my com- putations in asserting that the conversion factor is 1000 in- stead of 10. This is evidently contrary to the common law of pressure in meteorology, which converts the pressure P = 101323 . 5 2i^? into 1013235 . ess*. and not into meter 2 cm. 2 ' 101323500 - . That would destroy the entire meteorological system in practice. It may be proper in this connection to make a few quotations as to the meaning of u a T* when referring to the Stefan formula; KIQ = cr T 4 evidently has identical dimensions. Lorentz, "The Theory of Electrons," page 74. " :S the Boltzmann entropy coefficient. (137) k = *L i, - r, erg sec., the Planck Wir- kungsquantum. a = 1.0823 as developed by Planck, " Die Theorie der Warmestrahlung," page 165, - =l+^+y 4 +^ 4 +.. . = 1.0823. (138) a = - - as determined from K^ = a T a . c Since k is a variable coefficient in non-adiabatic atmospheres, it follows that h is, also, a variable coefficient. COEFFICIENTS IN STEFAN AND WIEN-PLANCK FORMULAS 133 Wien-Planck coefficients. (139) a = Sir c h, L-JT-. J> erg X cm. (140) c 2 = -T-, [L . deg.], cm. deg. (141) a = ^H -1FT ' Ts ' ^il' .JfL^ Stefan coeffi- cients. c rJf L 2 1 L 1 M "1 gram (142) ^-^[- r .-.-^-.^^^, 1 ^- t deg." erg cm. 2 sec. deg. 4 (143) A = 4.1851 X 10 7 , "^r' ergs, mechanical equivalent of heat in small calories per sec. 2 60 ff gram calories , r = - - ; : - 9 the measure of the kinetic energy oi A cm. 2 mm. molecular motion in calories of heat. Since all the terms of the equations of condition represent kinetic energy of molecular motion, [_ I > in one form or another, and since A repre- sents a definite amount of this kinetic energy, or work done per unit volume, it follows that they can all be reduced to the corresponding number of calories. The computations prove r*f\ that -r- X 10 derived directly from the (log c, a) set of pairs A. (M. K. S.) through the factor 0.000014336, is exactly the same in value as when derived through the general formula, Indeed, all the resulting values of a (M. K. S.) have been checked by this independent computation in the (C. G. S.) to verify the clerical work of the computations. The following tables contain the data of the several elements, as they are collected together for intercomparisons, and for the derivation of the general laws which underlie the solar thermo- 134 A TREATISE ON THE SUN ? S RADIATION dynamics. This fragmentary collection, apart from the con- tinuity of the computations, necessarily deprives the data of much of its original force. The number of points found in the tables is about one-third of the number for which the computa- tions were made. They are quite full near the photosphere, but in the high and the low levels the data have been very much diminished in the amount to be included in the tables. The Thermodynamics of Radiation in the Solar Atmospheres Q The table for H = P has been omitted because it can be LI immediately computed from Table 9; similarly, the table for U = CvpT = His omitted because our compilations are for monatomic gases, except hydrogen, H2 = 2.00, as has been / 8 \i stated; also, the table for 7 = (^J 2 q = 0.92132 q [9.96441] is omitted. The computations for the Maxwell mean free path length are not computed on account of the difficulty of deter- 77 mining the coefficient v\ in l max = Q no^7 ' ' even ^ tne coen ^- U.oUyoi p 7 cient 0.30967 is still available. Consequently, the computation 7 for the number of collisions per second, v = 1 , must be de- tmax ferred. Table 31 contains the square root of the mean square- velocity q= V 2 2 . An examination of the results along the common plane of the photosphere for the different elements gives the general value, q =,54289 / V~^". Hence, \ m f = 1.4736 X 10 9 [9.16838]. The universal mean kinetic energy is equivalent to, (145) |m^ = |p wz ,= | w ^r=|pF=|^r=1.4736Xl0 9 , which is the same as for monatomic hydrogen. This conforms to the Clausius law of the constant mean kinetic energy at the photosphere. COEFFICIENTS IN STEFAN AND WIEN-PLANCK FORMULAS 135 i-i ooooo t-co 10 eo TP T-ITJIOOO eo O Ti < loojcoioc- eo ooeoi OO IOOSOJ1OOO CXI ?OOTt<< rH eoeoTjTj<-^ 10 io5oo< U5MCO OOO^*COT-I rH eOCONOUJ eoi-io oscoiot-Oi H wot-oi> uso t- * W t-OCOt-CXI OS OlOTj OU3 en 1-1 OONt-COt- Ot-t-lON O t-OOC CCT5U500r-l t-T-tO^J^uDio 10 i-iT)< ot-meoco t- i-Heo oot-uioso eo OO5 OCO>OOi-( Oi T)< iH CO i-l Tj< OJt>OlfllO rHO>?OCOO t- COOt-Tjli-( COVOOO(MCO I0p>i-lt- U30p t- t- O5CO O500O O OOrH COCDrH 00 O t- N CO is!!! CO I* rH C SIO COOOCOt-U5 rHO 10 COrHCOOt- rHOO ^HTJIC- rH CO CO OJ 00 00 COO N* CO CO Tjl 10 rH l> sill- - S- xxxx" " x" " t-NOOOCD Nt-mO< OSCJ^C^rH T^t^CO^OC COWOOO WrHN^C O t-CO< U50JO( X X U5 COrHOS t> O CO CO CD CO CO 05 rH 05 X X ) OSO rH rH t- cowl 1 -^ OS' |o o, . . XX X" OrH WOOOJ< -I- 1 " x" x t- CO O U5 t N om >rHOco * mt-oooos comooin'^' looeooo co os>ocN3rH 00 si I t- si ?2 XXXX X OOt~iO-^ IO cj co o co oj oo 00 00 (N i-t t- CO t- N CO 10 t-' OJ rH rH .h ' oo 0000 OOCOTf (N 1 1 1 i OOOOO OO o o o oo o o O O OOOOO eg Tj CD oo o cxi <* 1 1 1 1777 COEFFICIENTS IN STEFAN AND WIEN-PLANCK FORMULAS 137 The data for the planes of the radiation, Z R , give a similar result, but with a different constant, q = 71241/ ^Jm. Hence, J m q 2 = 2.5376 X 10 9 , [9.40443]. Finally, on the vanishing planes q = and the mean kinetic energy vanishes. There must, therefore, be a complex gradual diminution of the general mean kinetic energy from level to level, counting from the inner layers to the vanishing planes. The values in the lower levels can be computed as long as the gaseous law P = p R T, or P V = K T, continues to represent the natural conditions, that is, before viscous coefficients become necessary. The value of q is important in several formulas. The general kinetic energy, PV = KT==~mq 2 . o The specific kinetic energy, Pv = RT = -q 2 . o The hydrostatic pressure, P = p RT = pq 2 . o 3 1 The molecular kinetic energy, H = - P = - p q 2 . 2 2 By Table 9 the mean pressure along the layers where the black radiation originates is P = 2593212 (M. K. S.). 3 Since H = P = E n, we have in M. K. S. for n = 8.8990 X 10 23 2 H = 3889800, and E n = 3895000, which checks the procedure approximately, especially the re- sults by the method of trials upon which the entire series of data depends. The average number of molecules per cubic centimeter on the plane of the photosphere is 2.1187 X 10 17 , about one-fourth the number on the radiation levels. The num- ber diminishes to zero on the vanishing planes, very rapidly in the topmost strata. The number of molecules per cu. cm. on the plane of the photosphere is about one-hundredth the amount in the earth's atmosphere near the sea level, 2.1187 X 10 17 against 2.7278 X 10 19 . 138 A TREATISE ON THE SUN S RADIATION 03 i . A k i ft! ^ kJk IN 00 1 woo Woo, - . -. O5 OSO TH COO5 N -*CDl>000 (M TH 10 o mcD THNMIN / X C/5 O iHCO^t O UO W THCDTHOO O TH odaiO 10 O5 OOCOU300 O O5 t-rH N O5 W ^ * 0^ tf OOOOOSOiH 1 N IO TH N T)< CO* - - - IO t-O CO CO t- i-(THOOO5O TH TH C<1 (M C<1 CD CMCOCOCOCO Tjcoco w , ^ t> COT** CDNOCD-^ WCD-^CDN M T}t> lOt>O^THOO "* W O O 00 OO OO ia t- Ti< o c3 l O'^GC'^'^^t < CO TH W 00 W) *O IO 7 THWNNWWCC, H ^ rH OOCOt-NCO OOOOW5 COTHOO : : : : H v ! '. '. ' ' i : : : : : : : o' d o' o c> ooooo d o d d o' OOOOO OOOOO OOOCD^d IOOIOO "3 T-I N d 1 I 1 | rH W "3 1 SO 00 O 1 1 1 1 1 if c^coooo^ COEFFICIENTS IN STEFAN AND WIEN-PLANCK FORMULAS 139 CO |OQ V) W Ei a "I I s* X XX "* COCMt-t-rH (O r-ICReO US ^O^COi-H T-H OINCJ5 i-H CO CM " rjl OS -* OS T < r-lTHCMCMCM OS CO CO OS CO eooscoosos eo-*uscot> II I: = , -. ,,,.'. = = = = = = = = = = = = = 1 = XX X lr-1 USOOSOOOO CM CO i-l 00 OS rH CMOO-^OCO CM OO^i-Ht-CO oo o c *3< eo co 1-1 CM coeoeoosco (M oo in cvios OS OS COCO CO rfCOCOCOCO COCOCMUSi-l OSTj-*i-H COOO^HUirH t> OS FH t> rH ( OSOOOUSOS CMCOeOrHOl OSOOrHCM COU3CPOseO< 0000000005 eo' os eo' co co' eo *' TJ TH' n< US (0 t> Os iH r- 1:1: x" x* SrH COO< OS 05rH( eot-i-HrHCM eocoeoeoco CM CM CM lO COO (N t~ Oi t* t> CO Ci ^io 1 d 1 X X oS o ^ m COCXN^CO CO OO lOOJ kO t*OCO O O o ^ 00 CO r^ Sift 22: xx" - 2- - - " x" " : : : : : 5 oseo * og t-co to (MOO CO cooi -^ o co t-o I-H oico ^ Jow^Soo 10 CSCOOOINCO Tf CO 1% " 000 S3 xxx oo8!2g X CJ (M (N Csl CO CO co w coco co oo 10 ss n . **. . .^ . 1 ci 1 CO 02 X X CO COO CO M iiill CO CO <* T!< >* CO UiiftiOlOlO CD Oi ff> CO CO OO > oS IOCO O t-10 rH tfi 22 2 xx x" " :, i : 8 ^ OOioot-co co oj m o C 0>O> OOO -* co oo o I-H in ^ c\i m 1-1 oi oo COIOOOINIO W IM O CO t> t- 00 Tf 10 o ' N ' >-5 X X OOU5NOJCO CO OOO -H CO lO lOOSCOt-rH O> 00 CO t- CO W O ^*CO CO O5 O4 CO TJ" (M ** CO 00 C5 i-( r-( !M CO fO in O 00 Td ^ N 0000 ooco-tM OOOOO Ol ^ CO OOO 1 1 1 1^ ! IliiiS! 1 J I * ^ V Nj N." ts^[ s 142 A TREATISE ON THE SUN S RADIATION I. CO05 TfOOO tHO COH i-l O t-lOO tOtO 00 TjtlOlOCO T-I N ^J* t-*O>O i i O CM O5 05 C3 i"* COO O 05 05 05 OO O5 O5 O 05 O5 05 O O OO O rH 05 coco eo'eo'eo* coco co'eo co co co co'eo coco' eo cocoeoeo co' coco' co'eo co t-rjt O005 CO 00 Tt< OS>0 00 OOOO t- CO CO CO CO CO < Is 00 t- l> 000000 00t> ss- XX* 10 CO 05 ^-(O05 0000 too co t- I-H 005 t-io eo N ClrHCO OO OO Gi C5 coco coco eo co coeoco coco 05 1- to 5 Tf OS U3 COt>Tl< >^ \n o 10010 oio o 050>ioeo t^ t- t~l>t~ t^t- l> 00000000 coco coco co co eococo coco co cocoeoeo 5S5 ggS ii XX Is co't> x x i-l05 to O coco eococo coco coco eo eo cocoeo coco eo eocoeoco eo ooo o O N CO CON rH VOIO IO1 os o o< t>00 OOt coco eococo coco coco co eo eoeoeo coco co cocoeoeo eo coco lOCOr-l r-l t> to eo 10 -t t- tO <* to iH t- <* ooeoggg 05 05 050500 NCOCO Tj TH CM I-H 1 CM CM N CM CM CM CM II 1 CM CO N coos os t- co co t- OOi-H t- t-C- CO I I! X i 1 1 f 0. x x" CO rH T- rH CDO OOt- CO t-C us coe* 1 ^ . . , . 00 X XX 4 OS t- ^H OS t- OO 1 00 O rH 00 CO Sco TH o o oooo ii I rH i 1 CO (M I 03 X CM WC co os Tj< rH i- H NOS ON-* COO 3 ^ CO i-l CO rj< CMOO 3 CO C t- co 11 .. 1 1 o $ XX t-oo-^< os t- COO^ CDU OOOOCM mo OS t O ^ T IOMOS CD t-OSi-H I os CD co o oo co T}< 1* COCO CO CO CM CMCM X IS 8.0490X1 1 CO* te XX X i-H CM iH OOOS t- t- o co 03 w TJ< TH c mt i in I! 1 X ' 1 .7790X10-26 .1383 " .8862 " .1736 " .8926X10- 27 tMiH TJIN^H r4j. H CD** CMCM 00 1 a 0. us lull li t t co co cDinm 1 3 COCO CO CO CO COCO coco CO NCM CMCM CM CD' rH t- 111. . 1 1. I.. % XXX COCDOi-ltM COOS (M>000 OU lOOSOrfCO O5CO OONCD TJ< o OSCMC-COCO inin rfrJ 0912 " 0738 X 10-23 1 6264X10-26) 5400 " 4750 ' 1 ^NtMrH^H ^rH *** r-Hr '.'.'.'.'. '.'.'.'.'. '. 1 ddddd do odd dc OOOOO OO OOO OO OOOOO OOO CO^CM rH CMNr-1,-1 N 3 OO OOO OO COTl< CM CM TfCD 1 1 1 i 2 0H "17 i 11 1 '7 T o o o 7 O O OOO ""777 1 COEFFICIENTS IN STEFAN AND WIEN-PLANCK FORMULAS 147 We have here a sudden transition in the value of h, similar to that discussed for CO sss oo co t CD OS* 05 CON OSOS 05 S-* rHOO kO UJrl* 05 OS OSOS kO ^ 05 NOS r) rH -rH i-H O IO OO r-t iH HO O O O O 00 O O O O * OOCO "# 00( 3 ^ "^ ^ in t rH C >O5 00 t-O rH COC 05CO U5 OOi-H O3O5 Tj< O5 IO THO O 0501 * 00 CO 5 O -H O3O 00 rH t- COO COOO i-l i-(OO O5 O5 O5 O5 O5 O5 O5 O5 O5 O5 05 050505 00 00 0000 00 00 00 0000 00 0500 OOt-t- 00 O5OCO -^ IO COCO O 00 50 O5 CO D O TH O3 rH O5 OO CO-^f CO O5 t- -^ CO O5 t- mo3rH oo t- t- 1- t- (O to oo in COeOOSfOCO CO CO" CO CO CO CO COCOCOCOCO COCOrHrHrH O5 O5OSO5 O5 O5 O5O5 O5 O5 O5 OS O5 O5 O5 O5 O5 OS O5 O5 O5 N ^O rj< 00 D (O rj< O rH U5 O Tj( OO rH IO U3 00 rjl rHO5 t-' So co t-t- CO s t- t-' 00 O t-kO n os kocM > X X rH OS 00 CO. CMO CO . COO COOSCM CCM CO .CM OOO i-H OS CMCO iH rH CO >00 00 OSOS Oi-l . t^CM OSOO t- t- t-t- t- t-t- t- t-t- OOOO 00 OS OSO i-HCMCM ft! ajaj N NN 1! N 00 CO. s 88 o o o oo CM CM. 1 1 i i 7 7 * f f s^ 154 A TREATISE ON THE SUN S RADIATION d Sit 40000 meters reduced to the sun 6.1657 X 10" 13 13.81702. The value of Ci = 8 TT c h increases upward from the low levels to the vanishing plane. In the solar atmosphere it passes through a* sudden change in the radiation layer, and its mean value, derived from the logarithmic law of depletion, agrees closely with the value of c\ = 8 TT c h, as derived from computations of the terrestrial thermodynamics, at the height of the top of the earth's isothermal layer. ~ 15 The solar value (ci) 8.2393 X 10~ is the same as in the earth's atmosphere at the height 42000 meters above the sea level. The value of c z , computed from terrestrial balloon ascen- sions, agrees at the top of the isothermal layer, with the laboratory value 1.3702 and 1.4430, respectively. Reducing the balloon value to the sun by the factor 7^ = 3.0376, it becomes 4.1621, while the logarithmic mean value by the factor 0.461 is 40.787 at the sun, about ten times as great. The same abrupt change occurs in c 2 as in Ci on passing the radiation layer, 0.62.96 below and 88.496 above Z R for all the gases computed. At- tention is called to diatomic hydrogen, which agrees with all the gases above this layer, but is very different below it, while monatomic hydrogen agrees above and below with the other gases. It is supposed that this indicates dissociation of the hydrogen diatomic molecule below the plane of radioactivity. TABLE 45 COMPUTED VALUES OF a = - FROM THREE BALLOON ASCENSIONS 1911 Uccle, Uccle, Uccle, Mean Values June 9 Sept. 13 Nov. 9 up to 60000 Meters z- 90000... 1 6403 X10~ 19 80000 2 2546 X10- 17 2 3765 X10~ 16 70000. .7310X10- 21 * 1.0422X10- 15 60000. .3995X10- 15 * 4.1111 " 4.1970 " 4.1541X10-1 5 7. 134 X 10-" 50000. .4519X10-"* 1. 0137X10-" 1. 0091X10-" 10.1140 40000. : .4091 1.4221 1.4287 14.203 g 3 rt >- Jj 30000. .7945 1.8195 1.8153 18.098 |1|| | 20000. 3.0686 3.0476 3.0058 30.407 10000. 4.6766 4.5073 4.7964 47.268 a) >28 o"" 1 0000. 5.0810 4.9880 5 . 4694 51 . 795 HS.al-rfi * Omitted in the means. a Decreases with the height in the atmospheres of the earth and the sun, especially in the rarefied gases of the uppermost strata. COEFFICIENTS IN STEFAN AND WIEN-PLANCK FORMULAS 155 i a> SS5 s .; SooLcOCO OS N S 5 S 3 3 5 5 00 rH CO 00 TjfTjt t- C- lO OO OS COrH OS CO rH ^ rH ^ t 00 OS 00 O CO OrH CM TJ< T)< 10 10 m m m lOCO kO OS ^ kOCO 00 CMCM S COCO SS CO in m in in m m m in 6.9846X10-* 1.7970X10-2 5.9479X10- 8 5.3028X10- 9 8 5 o , , X . . , . ,00 X X : : 1000 10 CD-* CM CO CO rH in CM CM Oi CO Tj< CO OO lO CO CM -r* Tjt rH CM CO kO COOS 00 CD CO CO t- OOOS rH Oi O CD 00 | OSOO 3 ^ *s x X X og Tj< t- OS OO 00 CO co i-i os *' Tj CO* CO CO CM COOS t- CM CC t- O 00- OO O CM -co t> * co t- rH CO |0 00 rH ^ CMCM ^ OO OSrJi t-o to co 10 cooo t-o t- osos osos kO rH T . . . .11. . r-,N X X X 11 CO OOrHrH rH CO OS rH CO COCM CO Tj< t- CO CM OS kO kOTf CO rH kO 00 CM t-CM tr- lO kO kO CO CO t- t- CMrH CO Tj Tf OOO OS 0000 rH IOO rHCM CO COtM ii dd.. ^ XX os co eo coco' eo co" ^' rj< X X rHrH CM CO^ C0t2 . CO kO SkO O i-l O le^g 1 ko'rn' 000 dd CM XXX OSlOOOSrH OS OOO O kO CO co o t- oo 10 oo OrHCM co co eo tfOSCOkOCM O rHCMCO * kO kO t> 00 OS O CMU5 00 eo co ^< ko co t> b ko 10 ko ko 10 10 xx I-IT)< oo kOtM osco t- TI OSCM eoeoN OCM t- OOO rHCM kO CO ^ OS CMt-CM lii| CM CO CM -^ lO CO COCOCO CO CO CO to co co to coco co oco o oo t-t- t- os koco eococo c o d dd.. 'ZJ 2 .2 v ^ X O OS CM CO rH t~ CM rH rH CM CM CO CO ** 00 00 00 00 OO 00 OO X X CMCO t- lOCO rHOi i-( ifi OOOOO O OOO O O O N 77 T 1 I 1 1 rH CM <* COOO OCM^ 1 1 1 1 1 1 1 1 rHrHrH 1 1 1 1 : 156 A TREATISE ON THE SUN'S RADIATION SUMMARY OF THE RESULTS FOR a = l Number Logarithm General mean in the isothermal layer 6 8397 XlO- 19 19 83504 02 Mean just #bove the radiation layer . 9 3 127 X10~ 19 19 96907 a Logarithmic mean, for the factor 0.461 a,\ Mean just below the radiation layer 7. 5738 XlO- 15 2 3960 XlO- 12 -15.87932* 12 37949 Balloon ascensions at 55000 meters 7 390 XlO- 15 15 86864* reduced to the sun Laboratory value reduced to the sun 7. 390 XlO- 15 7 3900 XlO- 15 -15.86864 15 86864 Laboratory value at the earth 7 3900 XlO- 15 _ie; 86864* Compare the * values. The y factor is 1.00, since gives v 13 / y' 3 = 1.00, so that for the coefficient a the reduction from the values at the earth to those corresponding with them at the sun disappears. The value of a changes with the height in each atmosphere. TABLE 47 COMPUTED VALUES OF a = ^ FROM THREE BALLOON ASCENSIONS 4 1911 Uccle, June 9 Uccle, Sept. 13 Uccle, Nov. 9 Mean Values up to 60000 Meters 2 = 90000.... 80000.... 1.2302X10-9 1.7824X10-6 1.0790X10- 5 3.1478 " 7.5883 " 1.0716 XlO- 4 1.3613 " 2.2543 " 3.5973 " 4.1020 " 3.1156X10-5 7.5957 " 1.0650 XlO- 1 1.3572 " 2.2805 " 3.4934 " 3.5510 " 5. 3557 XIO- 8 1 0) -U OJ 1) 2 => fcjS PlM 2.8183X10-* 7. 81 63 XlO- 6 3.0834X10-5 7.6030 " 1. 0666X10-* 1.3646 " 2.2856 " 3.3805 " 3.7410 " 70000. . 60000. . 50000. . 40000. . 30000. . 20000. . 10000. . 0000... 3 . 5482 XlG-u* 1.0496X10-5* 1.0889 XlO- 4 * 1.0568 1.3458 ' 2.3015 ' 3.5075 ' 3.8101 ' * Omitted from the means. tO \r\ rtS kA CO h CO CA-vK *# \ft ~-4 --H O II t- OOrJtCO TH CO O lO oo Oicoeo O o TH CM t- COCM^ to CO t- CM CO COtOCO CO Tj OrH X 10 * OCM CD Oi Oi Oi CO O Oi oi oooo o CM 00 00 oo' oi X O t- Tj< C- O t- O coco ) 00 Oi Oi 1 10 to 00 Jl X O 00 CM t- CM CO 00 TH CO Tf O CO t- t-00 00 X S3,THg8 Oi *tl CO 00 Oi TH CO t- O CO CO Oi CO CO kO CO COCO CO t~t> X X 00 CMCMTH SS?c$S TH OOCMCO CO W OOO CO CO CO CO CO t- OO >Tf OiOi IS S3 t- Tj< OOf-l OO O (M IO O t- CMt- iH CO ** ^J CO CO CO CO t-t- coco OiOi CO Oit- COCO b- t-00 x OTH 11 rH Oi iOrH t-eo t- t- Oir-l CM^ t- CO Tf CO i-lOi iOi O CM^* COt- TH to to 10 IO IO to tO tOtO IO IO IO IO lO IO to IO CO COCO COCO t- 111. xxx" CO 10 CO CO t- O to CO COO CO CO CD CO t> THOOO CO O TjfOi ^ Oi r}< tot THCOO to 00 CMCO TH IO O COC COt-CM CD CO IOCO 00 Oi TH CM THCO CM OiC lOrH CM COI t- Oi t- eo TfTjl kO TH X t-Oi Oi O 00 rH O TH 00 OO 00 Oi Oi Oi Oi OiO> Oi OiOi Oi OiOi THiH TH TH 00 Oi "a O OrH CM_ CM CO COCO CO II CO CO oooo TITH oo oeo tot- OOOi CM O t- ^TH coco -^ to to cot- 0000 00 OOOO OOOO XX O CO tOCO oo to t- T}< t- ^"O CO OTH If III HI || is 5 % %% s i|| || || I lOOtOOtO TH ^ ''I'|||||THCMT}< CMCMi-lTH III N 0134X 2546X 5270X 6329X s ooo I ooto Sco to TfO CM OOO CM t-TH I * * ; i 1 g Jl 2' Above Z ...... Below. .7 ...... Diatomic hydrog Mean value in th 158 A TREATISE ON THE SUN'S RADIATION The excessive changes in a above 55000 meters involves problems of difficulty in the kinetic theory of gases. From the level 55000 meter, where a = 7.390 X 10 ~ 15 , to the sea level, the value of a steadily increases to 51.795 X 10 ~ 15 , where it is about seven times as large as at 55000 meters. This is, however, compensated by a change of the exponent from a = 4.00 to a = 3.82 on the sea level. SUMMARY OF THE RESULTS FOR 0014336 [_ 3 . 15644] . 4.1851 X 10 7 His factor violates the law of dimensions, and leads to im- possible conditions, incompatible with solar and terrestrial atmospheric physics. He has transferred the factor of the surface integral in the Poynting equation to the volume in- tegral, which is erroneous. Second Computation of the Wien-Planck Coefficients, Taking N = constant (6.062 X 10 23 ) and E Q = variable. The preceding Tables, 36 to 50, have been computed on the assumption that the kinetic energy of one molecule is a constant, E = 4.3769 X 10~ n , so that N and the other quan- tities dependent upon it become variables. The difficulty raised by this dilemma has been stated on page 140. We have in Tables 51 and 53 compiled the data by using N = constant (6.062 X 10 23 ) in the formulas for k, h, c\, c 2 , E , n, instead of the E Q = constant, as in Table 34. Table 52 is for the solar elements, and Table 53 is for atmospheric air in three balloon ascensions. The heights z for each element are in part the same as those in the Tables 36-50, and by comparing the re- spective tables it can easily be seen what has been the effect of making this change. The values of a = are identical 2 ' in both computations, so that the Stefan Law JQ = a T* re- mains unaltered in its values. The distribution in the spectrum becomes somewhat different, due to the changes in c\ and c 2 . It is remarkable that while c\ is larger below the plane of the source of the solar radiation z s , and smaller above it, they nearly agree in value on that plane. That is to say, the N = constant system is about equal to the E = constant system for the critical strata in the solar atmosphere where the black radiation is generated. Similarly, c 2 is larger below and smaller above 162 A TREATISE ON THE SUN'S RADIATION the planes Z R , but nearly equal on that plane for the N = con- stant and the E = constant systems. This is seen in Table 51, where the final values are as follows: STT c h, N = constant - 13.83601 EQ = constant. , 13.84623 k N = constant EQ = constant Means Laboratory value Means - 13.84112 Laboratory value ^. . . < - 13.62338 . ch log c 2 = -r, 0.89025 0.87570 0.88298 0.64181 TABLE 51 COMPARISON OF LOG ci, LOG c 2 , AT THE RADIATION LEVELS, AS COMPUTED WITH N = CONSTANT (HEAVY TYPE) AND WITH E = CONSTANT (LIGHT TYPE) LOG Ci = 8* ch Means ch LOG c z = -r K Element Below Above Below Above Means Hi =1.00 -14.81319 -12.83545 -13.82432 -1.83162 1 . 90906 0.87034 -14.81319 -12.90829 -13.86074 -1.83162 1 . 92727 0.87945 # 2 =2.00 -13.99743 -12.82635 -12.41189* 0.97706 1.88327 1.43017* -13.94548 -12.87856 -12.41202* 0.96467 1.89632 1.43050* H =4 -14.88971 -12.88834 -13.88903 -1.84879 1 . 96140 0.90510 -14.68828 -12.96015 -13.82422 -1.79843 1.97935 0.88889 C =12....... -14.76247 -12.85613 -13.80930 -1.78090 1 . 94005 0.86047 -14.76247 -12.94243 -13.85245 -1.78090 1.96162 0.87126 Ca =40. -14.87619 -12.72973 -13.80296 -1.82750 1 . 95076 0.88913 -14.70853 -12.95001 -13.82927 -1.77764 1 . 96922 0.87343 Zw=65 -14.89982 12.84379 -13 87181 1.82960 1.95123 0.89042 -14.78310 -12.96068 -13.87189 -1.80042 1.98045 . 89044 Cd=112 -14.78527 -12.86050 -13.82289 -1.80370 1 . 93985 0.87178 -14.78527 -12.94107 -13.86317 -1.80370 1.95999 0.88185 Hg=198 14.84221 12.80124 -13 82173 1.81595 2.07311 0.94453 -14.77151 -12.87220 -13.82186 -1.79828 1 . 89085 0.84457 General mean for N =constant. . . 13 . 83601 General mean for 2V = constant . 89025 " Eo=constant. . 13.84623 " Eo=constant 0.87570 Average for both 13 841 1 2 Average for hnth 88298 Laboratory vali ie 1 Q QQQQ . . . lo . bZooo Laboratory value 0.64181 COEFFICIENTS IN STEFAN AND WIEN-PLANCK FORMULAS 163 The general result is that the Wien-Planck formula, STJ-EA Sirch 1 ch X 5 has reproduced the usual laboratory values of Ci, c 2 , approxi- mately, at the critical places of the generation of the black radiation in the solar atmospheres of the monatomic gases. For the diatomic hydrogen the values of log c\ and log Cz are TABLE 52 SECOND COMPUTATION OF THE WIEN-PLANCK COEFFICIENTS, TAKING N= CONSTANT (6.062 XlO 23 ) AND E Q = VARIABLE Hydrogen (Monatomic, .Hi = 1.00) 2 k h c\ C2 Eo n 26000 . . . 4.9256X10- 16 8.6208X10-25 6.5000X10-13 52.506 2. 3272 XlO- 12 1.1933X10 16 24000 . . . 5.9027 " 1.0819 XlO- 24 8.1578 " 54.990 3.0988 1 . 8335 20000 . . . 7.8382 " 1.5520 " 1.1701 XlO- 12 59.399 5.0554 3 . 8207 16000... 9.8098 " 2.0810 " 1.5691 " 63 . 641 7.5780 7.0117 14000... 1.0769 XlO- 15 2. 3579 i ' 1.7778 " 65 . 687 9 . 0456 9.1445 10000... 1.2489 " 2.8762 ' 2.1686 " 69.087 1.2364 XlO- 11 1.4613X10 17 6000... 1.4490 " 3.4737 ' 2.6191 " 71 . 920 1.6192 2.1900 4000... 1.6005 " 3.8934 ' 2.9355 " 72.980 1 . 8365 2.6454 2000... 1.8028 " 4.4488 ' 3.3543 " 74.028 2.0796 3.1874 0... 2.0368 " 5.1160 ' 3.8574 " 75.355 2.3528 3.8356 -2000. . . 2.3108 ' 5.8831 " 4.4358 " 76.383 2 . 6656 4.6255 -4000. . . 2.6155 ' 6.7940 " 5.1226 " 77 . 930 3 . 0134 5 . 5597 -6000. . . 2.9671 ' 7.8345 " 5.9070 " 79.214 3.4141 6.7046 -8000. .. 3.3588 ' 9.0800X10-24 6. 8462 XlO- 12 81 . 108 3.8596 8.0586 -10000. .. 3.8135 ' 8.6264X10-26 6. 5134X10-" 0.67861 4.3769 9.7324 -12000... 4.2966 ' 1.0079X10-25 7.5990 " . 70370 4.9562 1.1727X10 18 -14000. . . 4.3882 ' 1.0331 " 7.7890 " 0.70623 5.5789 1.4006 " Hydrogen (Diatomic, #2 =2.00) z k h c\ C2 E n 25000. .5898 XlO- 1 ' 1.0563X10- 27 7. 9645X10-" 199.33 1.9073 XlO- 17 2.1450X102 20000. .6208 XlO- 1 * 4.2837X10-25 3. 2292X10-" 79.287 2.3221X10-" 2.4403Xioi 2 15000. .5557 " 1.8554X10-24 1.3989X10- 12 73.670 2.3291X10-12 7. 1175X10" 10000. .5397X10- 1 * 3.8151 " 2.8764 74.333 8.0910 " 1.4794XlO'6 5000. .9279 " 4.9243 " 3.7129 76.627 1.7033X10-" 9.5363 " 1000. .2183 " 5.6586 " 4.2665 76.527 2.5555 2.5879X10 17 0. 2.4254 " 6.2094 " 4.6818 76.772 2.7915 3 . 1220 -1000. 2.6588 " 6.8247 " 5.1456 77.003 3.0549 4.0161 -2000. 2.9075 " 7.4135 " 5.5896 76.493 3.3414 5.0075 -4000. 3.4901 " 8.8918 " 6. 7043 XlO- 12 76.432 3.9997 6.1927 -6000 . 4.1642 " 1.3185 " 9.9410X10- 1 ' 9.4986 4.7884 1.1859 -8000. 4 . 7642 " 1 . 5241 " 1.1491 XlO- 12 9.5973 5.7169 1.8783 -10000. 4 . 8734 " 1.5609 " 1.1769 " 9.6086 6.7155 2.7928 -12000. 4 . 8734 " 1.5573 " 1 . 1742 " 9 . 5865 7.7197 3.9342 -14000. 4 . 8734 " 1.5454 " 1.1652 " 9 . 5130 8.7238 5.3160 164 A TREATISE ON THE SUN ? S RADIATION TABLE 52 Continued Helium, w=4 z k h Cl C2 So M 10000... 2.3627X10-17 2.4814X10-26 1.8710X10-" 31 . 509 2.6225X10-" 1.4275X1013 5000... 8.3012XlO-i 1.7937X10-2* 1.3524X10-12 64.923 4.7070X10-12 3.4323X10" 2000 . . . 1.4058X10-16 3.7343 " 2.8156 " 79.692 1.4234X10-11 1.8050X10 17 1000 . . . 1.6192 " 4.4438 " 3.4286 " 82.332 1 . 8459 " 2.6656 " 0... 2.0444 " 5.7778 " 4.3563 " 84.786 2.3628 " 3.8603 " -1000... 2.6210 " 7.6713 " 5.7840 " 87.806 3.0215 " 5.5820 " -2000. . . 3.3628 " 1.0256X10-23 7.7328 " 91.496 3.8665 " 8.0802 " -4000 . . . 4.3720 " 1.0288X10-25 7.7573 " 0.70598 6.1979 " 1 . 6400X10 18 -6000. .. 4.3720 " 1.0256 " 7.7327 " 0.70373 8.6926 " 2.7241 " -7000. . . 4.3720 " 1.0135 " 7.7172 " 0.70233 9.9405 " 3.3310 " Carbon, m = 12 z k ft Cl C2 Eo n 1000 1.1494X10-15 2.8513X10-2* 2.1499X10-12 74.418 1.1121X10-11 1.2464X10 17 800.... 1.2552 " 3.1986 " 2.4116 " 76.448 1.3179 " 1 . 6080 " 600.... 1.3713 " 3.5568 " 2.6817 " 77.650 1 . 5427 " 2.0364 " 400.... 1 . 5490 " 4.0875 " 3.0819 " 79.164 1 . 7925 " 2.5506 " 200.... 1.8016 " 4.8216 " 3.6354 " 80.288 2.0822 " 3 . 1932 " 0.... 2.0899 " 5.7181 " 4.3113 " 82.080 2.4122 " 3.9819 " -200.... 2.4308 " 6.7633 " 5.0994 " 83.470 2.8021 " 4.9850 " -400.... 2.8198 " 8.0402 " 6.0621 " 85.540 3.2463 " 6.2164 " -600 3.2798 " 9.5230X10- 24 7.1802X10-12 87.106 3.7708 " 7.7825 " -800 3.8135 " 7.6756X10- 26 5.7873X10-14 0.60381 4.3769 " 9.7324X" -1000. ... 4.4354 " 9.2620 " 6.9833 " 0.62646 5.0829 " 1.2179X10 18 -2000.... 5.2888 " 1.1456X10-25 8.6378 " 0.64984 9.4995 " 2.9039 " Calcium, w=40 z k h Cl C2 Eo n 1000 5.7870X10-1" 4.6618X10- 27 3.5148X10-15 24 . 167 2.6041X10-15 4.4671X1011 800.... 1.9029XlO-i 2.7974X10-25 2.1092X10-13 44.102 4.2815X10-13 9.4174X10" 600.... 5.7680 " 1.1172X10-24 8.4232 " 57.924 2.5602X10-12 1.3771X10" 400. ... 1.0112X10-15 2.3329 " 1.7589X10-12 69.212 7.1287 " 6.3981 " 200.... 1.3173 " 3.4023 " 2.5652 " 77.482 1.3436X10-" 1 . 6553X10" 100.... 1 . 5573 " 4.1731 " 3.1464 " 80.393 1 . 7519 " 2.4644 " 1.9462 " 5.3485 " 4.0326 " 82.446 2.2493 " 3 . 5853 " -100 2.4990 " 7.1182 " 5.3670X10-12 85.452 2.8809 " 5.1966 " -200.... 3.3512 " .9730X10-24 7.5195X10-W 89.283 3 . 8531 " 8.0497 " -400 4.8558 " .0880X10-25 8.2035 " 0.67220 6.2630 " 1 . 6658X10 18 -600.... 4.8558 " .0843 " 8.1754 " 0.66990 9.0452 " 2.8913 " -800.... 4.8558 " .0807 " 8.1480 " 0.66765 1.1822XlO-io 4.3203 " -1000 4.8558 " .0780 ." 8.1280 " 0.66601 1.4580 " 5.9273 " -2000 4 . 8558 " 1 . 0742 " 8.0992 " 0.66365 2.8459 " 1.6134X10" COEFFICIENTS IN STEFAN AND WIEN-PLANCK FORMULAS 165 TABLE 52 Continued Zinc, w=65 z k h c\ C1 Eo n 600.. . 4.1704X10-1 7 4.4253X10-26 3.3365X10-14 31 . 833 4.0037X10-M 2.6926X10" 400. . . 5.2404XlO-i 1.0250X10-2" 7.7280X10-13 58.677 2.4523X10-12 1.2909XlQi 200. . . 1.1361X10-15 2.8737 " 2.1667X10-12 75.882 1.0122X10-" 1.0825X1017 100.. . 1.4277 " 3.8275 " 2.8859 " 80.427 1.6061 " 2.1636 " 0.. . 2.0693 " 5.8203 " 4.3883 " 84.378 2.3947 " 3.9386 " -100.. .. 3.1069 " 9.2562X10-24 6.9790X10-12 89.378 3.5768 " 7.1897 " -200.. .. 4.6771 " 1.0531X10-25 7.9400X10-1 4 0.67546 5.3544 " 1.3169X1018 Cadmium, m = 112 z k h Cl c-i Eo n 200. . . 7.4130X10-16 1.4297X10-24 1.0780X10-12 57.859 3.2243X10-12 1.9470XlQi 100.. . 1.3961X10-15 3.4522 " 2.6029 " 74.177 7.9580 " 1.3862X1Q1 7 80. . 1.4332 " 3.6483 " 2.7507 " 76.366 .4188X10-11 1.7963 " 60. . 1 . 5095 " 3.9150 " 2.9518 " 77.803 .1020 " 2.2536 " 40.. 1.6601 " 4.3665 " 3.2922 " 78.906 .2700 " 2.7946 20.. 1.8970 " 5.0824 " 3.8320 " 80.375 .4588 " 3.4403 " 0.. 2.1827 " 5.9411 " 4.4795 " 81 . 658 .6772 " 4.2413 " -20.. 2.5116 " 6.9680 " 5.2537 " 83.420 1 . 9237 " 5.2101 " -40. . 2 . 8843 " 8.1738 " 6.1629 " 85.016 2.2119 " 6.4231 " -60. . 3.3145 " 9.6192X10-24 7.2527X10-12 87.066 2.5981 " 7.7193 " -80. . 3.8135 " 8.0893X10-26 6.0991X10-14 0.63636 4.3769 " 9.7324 " -100. . 4.3397 " 9.4570 " 7.1303 " 0.65375 5.0254 " 1.1974X1018 -200. . 4.4926 " 9.8138 " 7.3995 " 0.65533 8.5872 " 2.6744 " Mercury, w = 198 z k h Cl cz So n 200 .. . 2.6278X10- 17 2.1953X10-26 1.6552X10-14 25.062 1.8161X10-15 2.6013X10" 100.. . 8.1208X10-16 1.4672X10-24 1.1063X10-12 54.203 4.8116X10-12 3.5475X1Q18 80. . . 9.9204 " 1.9254 " 1.-4517 " 58.225 7.3663 " 6.7196 " 60.. . 1.1787X10-15 2.4217 " 1.8259 " 61.637 1.0431X10-11 1.1323X10 17 40. . . 1.3882 " 2.9922 " 2.2561 " 64 . 662 1.4056 1.7711 " 20. . . 1.6529 " 3.7052 " 2.7936 " 67.248 1 . 8347 2.6473 " 0. . . 2.0469 " 4.7212 " 3 . 5597 " 69.196 2.3549 3.8409 " -20.. . 2.6258 " 6.2974 " 4.7481 " 71.948 3.0181 5.5729 " -40.. . 3.3720 " 8.3922X10-24 6.3276X10-12 118.33 3.8721 8.0980 " -60.. . 4.2269 " 9.2225X10-26 6.9537X10-14 0.65456 4.9454 1.1688X10 18 -80.. . 4.3282 " 9.4758 " 7.1438 " 0.65680 6.1636 1.6263 " -100. . . 4.3282 " 9.4488 " 7.1243 " 0.65493 7.3885 2.1345 " quite different from the means just computed. A study of Tables 52 and 53, above and below the radiation level Z R , and an intercomparison of the entire N = constant system with the EQ = constant system, make it difficult to decide whether one is better than the other, since the divergence is about the .same in opposite directions. 166 A TREATISE ON THE SUN S RADIATION TABLE 53 SECOND COMPUTATION OF THE WIEN- PLANCK COEFFICIENTS TAKING N = CONSTANT (6.062 XlO 23 ) AND E = VARIABLE Balloon Ascension, Uccle, June 9, 1911 z k h Cl C2 Eo n 70000. ... 2.2849XlO-a> 7.0356X10-30 5.3049X10- 18 9.23760 2.4145X10-21 1.6448X10 60000. ... 6. 8192X10- 18 2. 1060 XlO- 28 1.5879X10-16 0.92648 2.6598X10-16 4.2395X10" 50000. ... 3. 6806 XlO- 17 9.1420 " 6.8928 " 0.74515 3 . 1470X10-1 4 1.4747X1015 40000. ... 4.1715 " 1.0911 XlO- 27 8.2268 " 0.78470 1.0199 " 3.3560X1017 30000 . ... 4.9893 " 1.2780 " 9.6358 " 0.76844 1.6389 " 1. 0788 X 10" 20000. ... 7.9440 " 1.9870 " 1.4982X10-15 0.75040 2.5665 " 3.2545 " 10000. ... 1.1963 XlO- 16 2 . 9803 " 2.2471 " 0.74738 4.0538 " 1.0027X10 1 9 000. ... 1.3598 " 3.4419 " 2.5955 " 0.75890 3.9417 " 2.5393 " Balloon Ascension, Uccle, September 13, 1911 z k h Cl cz Eo n 80000 70000 60000 . . . 1.9762X10-19 5. 4412 XlO- 18 1.4529 XlO- 17 2.9574 " 4.2809 " 5.0749 " 8.0032 " 1.1778X10-16 1.3606 " 2.9057X10- 29 1.7195X10- 28 4.0310 " 7.6977 " 1.1260X10-25 1.3013 " 2.0114 " 2.9542 " 3.4632 " 2. 1909 XlO- 17 1.2965X10-16 3 . 0393 " 5 . 8039 " 8.4898 " 9.8118 " 1.5165X10-" 2.2274 " 2.6111 " 4.41110 0.94805 0.83240 0.78085 0.82626 0.76927 0.75397 0.75265 0.76357 2.9648X1Q- 19 3.6735X10- 16 1.7435 XlO- 15 4.8800 " 1.0661 X10~ 14 1 . 6976 " 2.6380 " 4.1275 " 5.9699 " 2.2120X106 9.0932X1013 4.2047X1015 5.2979X1016 3. 6261 XlO 17 1.1399 " 3.3742X10 18 1.0157X10 19 2.5191 " 50000 40000 30000 20000 10000 000 Balloon Ascension, Uccle, November 9, 1911 z k h Cl C2 Eo n 90000 7.0927X10-20 9.7702X10-30 7. 3665 XlO- 18 4.1325 4.2489X10-19 6.4684X106 80000 1.7830X10- 18 6.3581X10-2* 4.7939X10- 17 1 . 0698 1.0701X10-16 5.2278X1012 70000 6.1459 V 1.8166X10- 28 1.3696X10-16 0.88672 6.4536 " 4.3616X10 14 60000 1.3574 XlO- 17 3 . 6568 " 2.7571 " 0.80816 2. 0368 XlO- 15 7.3798X1015 50000 2 . 7384 " 6.9577 " 5.2459 " 0.76223 4.9301 " 7.2954X1016 40000 3.9570 " 1.0123 XlO- 27 7.6325 " . 76747 1. 0033X10-" 3. 7383 XlO 17 30000 4.7053 " 1 . 1774 " 8.8774 " 0.75609 1 . 5904 " 1.1619X10 18 20000 7.3330 " 1 . 7982 " 1.3559X10- 15 0.73568 2.4552 " 3.3843 " 10000 1.1528X10-16 2.8129 " 2.1209 " 0.73202 3.8242 " 1.0098X10 19 000 1.3606 " 3.3585 " 2.5322 " 0.74048 5.6494 " 2.6326 " Table 52 summarizes the following results for the N constant system: The Boltzmann entropy coefficient dimin- ishes with the height, without any per saltum change at the level of radiation ZR; the same is true of the kinetic energy of one molecule E , and for the number of molecules per unit volume n. On the other hand, the Planck Wirkungsquantum h, COEFFICIENTS IN STEFAN AND WIEN-PLANCK FORMULAS 167 and the Wien-Planck coefficients all change abruptly at the radiation levels Z R . In making up the mean change for any of the chemical elements it is sufficient to add to the lower term 0.461 (upper . . . lower) instead of taking the direct arithmetical mean. Table 53, for the earth's atmosphere, shows that there is a gradual diminution in the values of h, c\, c 2 , as well as in those for k, Eo, n. The intercomparison between the solar and the terrestrial elements can be readily studied, and applied in any theoretical discussions regarding the laws that govern the processes of radiation. The foregoing data have been computed directly from the non-adiabatic thermodynamics of the atmospheres of the sun and the earth, and these represent the accomplished effects of radiation upon the gases per unit volume, as summarized in the temperature which is the general parameter. We shall now proceed to give the data relating to the electromagnetic energy more directly, in order to be able to examine the distribution of the radiant energy that is, its concentration, or its depar- ture from the normal value in the aether, outside the atmos- pheric gaseous media. It has been explained that the nearly perfect solar black radiation seems to arrive at the levels about 50000 meters above the sea level without much change in its conditions, and that there is a progressive concentration from that height down to the sea level. NOTE. The coefficient h is to be regarded as indicating the potential energy controlling the movements in h v, where v is the frequency, while k r is a general constant in a given atmosphere, and represents the corresponding kinetic energy of a single atom or electron. CHAPTER V The Elements of Black Radiation in the Atmospheres of the Sun and the Earth The Concentrations of Black Radiation in Gaseous Media WE must distinguish clearly between the three stages of the physical conversion of the black body radiation as electro- magnetic energy in the aether, outside the gaseous media, and the final effect as temperature of the gaseous media through the kinetic energy of the molecules. In any given volume of the gaseous media, (1) there is absorbed a certain amount of radiant energy, and this is not a variable amount so long as the temperature is steady; (2) this constant amount of energy per fM L? 1 ~j ^ YS \ * s maintained by a flux of energy entering and leaving the volume in a constant flow per unit time, with rM . L 2 i L-\ rMn f * the velocity of light, [ ^ 2 j- z y I = [^ J , from the non- gaseous volumes; (3) there is a marked difference in the volume density of the constant aether-flux through it, and of the volume kinetic energy at different levels in the atmosphere as developed by the computations, having the nature of concentration, or convergence and divergence of the energy, away from the simple black radiation of the aether itself. These changes in concen- tration are expressions of the operation of many physical proc- esses, measured in the bulk, and although these processes are not understood, it may be of importance to give the correspond- ing quantities which come immediately from the general for- mulas of radiation, as summarized by Planck in Die Theorie der Warmestrahlung, Leipzig, 1913. The following computations depend entirely upon the volume density, u = a T*, where T is the temperature of the volume, and the volume energy per 168 ELEMENTS OF BLACK RADIATION IN THE ATMOSPHERES 169 deg. 4 is taken from the thermodynamic computations, as in Table 22. The coefficient (c) of that table is identical with a- per unit volume; and a = , this c being velocity .of light. This trans- c ition from my thermodynamic notation to the Planck notation should cause no confusion. The solar radiation on entering the earth's atmosphere cannot be treated in the simple manner just indicated, because all of the electromagnetic radiant energy is not converted into molecular kinetic energy per unit volume, as determined by the steady temperature that is observed. Rather, the incoming radiation separates into three branches, (l) the absorbed part making temperature; (2) the scattered or reflected part that does not make tem- perature; (3) the free radiation of high-frequency that pene- trates to the sea level, and is there converted into a lower fre- quency, to return with a further complex absorption and scatter- ing to space. These processes have been indicated in the Treatise, also in Bulletin No. 4, 0. M. A. The reader should bear the line of our argument in mind. Having computed KW = c T a from the thermodynamic temperature conditions of the atmo- sphere, the temperature representing the final absorptions of all kinds, where (c) differs from the black body o-, and (a) is different from 4.00, we next passed from (a) to 4.00, retaining 4 the same value of (c), as a variable O CO oi si c^i oo oi -J 63X10-8 54X10-* X t-oo t-coi OS OS OS OS XX* OS TfOS rl t-' 00 00 OS 0> 05 O> eoeoTi' ' ' ' Tf * t- to m n co o oo esi * as to o Tt oo o] co 10 co oo 05 t- H Tf N Tf N rH r-i * t> T-i CO XXXXXX > CO <* C< H t- -" CO xxxxx . , 1 xxxxx x ELEMENTS OF BLACK RADIATION IN THE ATMOSPHERES 173 The balloon volume density is about 4.63 times that of the laboratory (a). The former concentrates by this factor, but it is compensated by lowering the exponent from 4.00 to 3.80. The computations for u = a T* = . a T* are as follows: From Table 6 take the temperature T of the element at the height indicated, and compute log T 4 ; from Table 22 take log c in the (M. K. S.) system, and convert it into log a in the (C. G. S.) system by the factor 10; multiply a by -|- = 1.3333 X 10- 10 , [log -j- - 10.12494], to produce (a); finally, compute the volume density u. There are several remarks regarding the values of u in the solar atmosphere, Table 54, and in the ter- restrial atmosphere, Table 55. Above the transition level Z R the exponent in u = a T 4 resulted at about the Stefan black body value, 4.00, as in Tables 18-22, so that the low-efficiency values of Table 54 are those which actually exist. The ob- served thermal efficiency of the inner corona during eclipses has been singularly low. Below Z R the exponent averages about 2.420, Table 24, so that the actual efficiency passes through the radiation levels with a simple gradient. The po- tential efficiency is, however, very great below Z R , as is seen from Table 54 by using the exponent 4.00. The imprisoned potential radiation changes abruptly by a factor whose mean value is, Factor of radiation = 3.2442 X 10~ 7 (monatomic) " " = 1.5880 X 10~ 3 (diatomic) This is not quite correct, because the values of u should be extrapolated to the exact heights Z R , and this it is difficult to do practically without additional computations. The mean value of u on the photosphere is u p = 2.4512 X io- 3 . In the earth's atmosphere the first section of Table 55 gives a few values computed from balloon ascensions and the mean values up to 60000 meters; in the second section the values are computed from the mean temperatures T t and log a = 174 A TREATISE ON THE SUN'S RADIATION - 15.86494 from Table 26. It is noted that the balloon and the laboratory values coincide at about 50000 meters, but that the former concentrate sevenfold at the sea level. The incoming solar radiation, therefore, undergoes a process of concentration in the successive volumes from 50000 meters to the sea level, until it becomes 4.63 times as dense per unit volume as the density computed directly from the laboratory value, from log a = 15.86494 and the exponent 4.00. This result should not be misinterpreted. For with the concen- tration of the coefficient there is a diminution of the exponent, 4(T from 4.00 to 3.80, so that the effective u a = T 3 ' 80 remains c about the same as if there had been no concentration of the coefficient. Take an example from the isothermal region, for T = 220 and the exponent 3.72, computing for both cases: Black radiation, u = [- 15.86497] X 220 4 - 00 = 1.716 X 10 ~ 5 ; Concentrated coefficient) i 5 .86497] X 4.63 X 220^ = with smaller exponent, J 1.754 X 10 ~ 5 . A very slight adjustment would equalize those results. Similar conditions prevail on the sun. Take the data from. Table 24, for T = 7655. Nearly black radiation, u = [- 9.81409] X - X 7655 4 - 053 = c 4.795 X 10 -*; Concentrated coefficientl ^ = { _ 22mQ] x 4 with smaller exponent, J c 5.362 X 10 ~ 3 . Here, again, small adjustments would produce an equality. This concentration at the sun is [6.42610], or conversely, the dim- inution of the coefficient is [- 7.57390] = 3.749 X 10~ 7 , while that given on Table 54, which is not quite accurate, is 3.2442 X 10~ 7 . This remarkable interplay between coefficient and ex- ponent we have interpreted as connected with radiation in the solar atmosphere at the levels Z R , with further scattering in the upper levels, amounting to an equivalent 1.87 gr. cal./cm. 2 min. ELEMENTS OF BLACK RADIATION IN THE ATMOSPHERES 175 a I "'' , . X XX~ " X 00 OCOrHOlO CO at Tt rHooosio co O 1O^J< rHOS t- CO CO t-COOO t-00 rH 00 COrHCOWOS rH XX X rHrHCQCOrH X X O COOSt-OCM iH rHCO-^COCO t- ig, 5 XX" X o oo o t in 10 rH 00 * ^ O N oo' od o N" eo' oi X10- 8 X10-5 X10- 4 OJCOCO CSIOOOCO-^ CJ CO CO t- t- rH Tt i 00 N CO OO CD f NCO >t>0>rH ON cot>t>t>ao NCO isiLi xxx" x CD OS CD t- ^ t- O5 i-H CO IO OOC4OiOi t> OO^CNCO! rHcoeoeoN lOrHiocnco t> COOSOt-T}< rHlOOprHlO 00 cocoooco rHCOW5tOi O- rHCO IO t- OJ rH ooioogos co eo< OJCOT}TjtU5 OS I XX 00 X XX iHOO 3 g{ ill XXX rHCO U5U5kOlOCD Jiiil xxxxx X X O OJ OS OS OS rH rH* , rHOO^fCOt- kONOOTfO t- 00 CM O OS CO t- sssss sssss s?s3?a i=i=: X X O5 lO (M O t- CN OS t-OSO CM CO rHCO t- CO Tj< O CO N ^COt^rH 00 IO CM OS IO CJ lO CM 00 IO CM 00 t-WOSOSO OOOOrH rH X XX 00 O rH PH N CO ** 0000000004 i i i g: CMTtCOOOO O< I I I I r-t N <}< CO 00 O CM rt I I I ' I I I I I rHrHrH I I I I I XX ^* 00 ^J* 00 d iO 176 A TREATISE ON THE SUN S RADIATION at the distance of the earth. Similarly, in the earth's atmos- phere, we retained the coefficient unchanged, but by making the exponent 4.00 in place of that found in the pair value (log c. a) we passed from the selective absorption J a to the black / , which added the amount scattered and the free heat, since J a = 1.46 and J = 3.95 calories. The pressure of the radiation energy per unit volume is taken p = f u, so that all the conditions that were described under u can be transferred to p. The factor of transition at Z R is the same; p (photosphere) = 8.1707 X 10" 4 . TABLE 57 VALUES OF p IN THE EARTH'S ATMOSPHERE 1911 June 9 Sept. 13 Nov. 9 Means z 90000. . . _ _ 1. 3998X10-" _ 80000... 9.3418X10- 13 2.0280X10- 10 _ 70000... *2.5231X10- 20 1. 4245X10-* 1.1514X10- 8 60000... *2.1316X10- 10 5.6128X10- 8 1.3990X10- 7 9.8014X10- 8 50000... *5. 1086X10" 4.9474X10- 7 6.9750 " 5.9613X10- 7 40000... 3.3158X10- 6 3. 5993X10-" 3.8849X10- 6 3.7423X10- 6 30000... 1.3759X10- 5 1.4997X10- 5 1.5590X10- 5 1.4782X10- 5 20000. . . 2.2020 " 2.3711 " 2.4864 " 2.3532 " 10000. . . 4.0597 " 4.4819 " 3.8207 " 4.12fO " 100. . . 1. 1929X10-* 1. 2171X10-* 1.0703X10- 4 1.1601X10- 4 * Omitted from the means. This pressure probably is due to the kinetic movements of the electrons in the volume as distinguished from P, the hydro- static pressure in the gas due to the kinetic energy of the atoms and molecules. The formulas give numerous interrelations between the electronic and the molecular energies. Nichols and Hull give the radiation pressure 7.01 X 10" 5 erg. dynes cm. 3 cm. 2 ' If A = the coefficient of absorption, and / = constant emissive black energy, for all bodies by KirchhorTs Law, then (Tf \ -j Ej = I (1 A) = ELEMENTS OF BLACK RADIATION IN THE ATMOSPHERES 177 1 11 o iS o T-H rHi-i:: : r-i : : FNI-C 1-1 v XX X XX v COOOCOCOOJ 00 OMOOl C-COO5C-O i-l CMNCOI OOCOCCOOrJt CO COOrHI tfOrHCMCO -^ COCncO- o ^jOrHCOco co*oo- rH OOCO^t-r-t iH iHlHI i I. x x' 0-2 O- 2 0* X XX X 10 T*lOTjt-t- 00 00050>05rH COCO xxxx ^OO-^ U3 OC-iH t- t ._ ,_!!< COCOt>C~t- CO 00 00 Ol O5 O5 1.1112X10- 5.2515X10* 2.5904X105 7.7150 " 1.6684X10* cocccoioeo t- coTtcdt>od ododododco" oi xxx co t- mo Oi-lOCO XX X ^COOiCOlO CO COOOC ^SSSS? S3 S5J ::::: 3S2 XXX COCOOOOOOO OCOCO T-C^t>OCO t- T^ co^ot-oco ot^co t-t-t-0000 00 0000000000 OOOOOOOJOS >-(COCO iiili 1. xxxxx x" SSS SSco^, oooc* coc^^^ 1 ^^ NCOCOCOCO ir500^JOt*< :ss 1. 1. u IH co co 1-10 t-o-<*ooco ^eo< OO rH CO CO CO CMr-(O5t-CO O i-l ( 00 t- rH O5 00 rH rH i ( rH rH rH rH i XX CO iH O CO CO rH O o 10 eo eo O5 1- 10 rf r-l rH 00 1C T)< O5 co co * * oo 05 eo ,-J rH* ^-i ^ CO* CM *<" ooooo co^coooo ogoogoo o o o o o o o CO T}< CO OO O CO -^< 1 1 1 1777 178 A TREATISE ON THE SUN'S RADIATION the radiation transmitted. Hence, (148) I = E + I -AI and the emission equals the absorption for any other gaseous medium. Since / can be computed from u in all thermodynamic atmospheres, it becomes useful in discussing the side of the general problem that pertains to the electromagnetic and the electron terms. TABLE 59 VALUES OF / IN THE EARTH'S ATMOSPHERE 1911 June 9 Sept. 13 Nov. 9 Means 2 90000... _ 1. 7590X10-" __ 80000... _ 1.1739X10- 13 2.5484X10- 9 _ 70000... *3.1706X10' 19 1.7901X10- 8 1.4469X10- 7 60000... *2.6786X10- 9 7.0533X10- 7 1.7580X10-' 1.2633X10- 6 50000... *6.4196X10- 7 6.2171X10- 6 8.7650 " 7.4911 " -40000... 4.1667X10- 5 4.5234X10- 5 4.8819X10- 5 4.3240X10- 5 30000... 1.7290X10' 4 1.8847X10- 4 1.9592X10- 4 1.8576X10- 4 20000... 2.7671 " 2.9795 " 3.1246 " 2.9571 " 10000... 5.1016 " 5.6321 " 4.8017 " 5.1785 " 100... 1.4991X10- 3 1.5295X10- 3 1.3450X10- 8 1.4579X10- 3 * Omitted from the means. The energy of black radiation on the photosphere is about the same as in the earth's atmosphere at 40000 m. The energy at Z R goes through an abrupt change. p = the dynamical pressure of the radiation enclosed in a perfectly reflecting vessel; F = the pressure exerted mechani- cally by the electromagnetic wave front. This is the Maxwell pressure of light, verified by Lebedew, Nichols, and Hull. These STT are related, F p. The equivalent light pressure in the sun to that in the earth's atmosphere is near the top of the respective gases. This pres- sure diminishes very rapidly on approaching the vanishing planes. ELEMENTS OF BLACK RADIATION IN THE ATMOSPHERES 179 X10- X10-1 " 5. 6 1.3 2.7 4.8 8.2 XXX X o inico 00 CM U5 t- ^f IO C- iH M kO^ r-l S3 i 4 X oo eoto"* t- tocooOTJico o ^ OOTf t- CM t- 0 i. XX" >co x eo to o 9356 XIO- 17 1202X10-" 4754 XIO- 13 tOOiCOt-T-H O OtOOlOO CO ^ <* Tj< IO IO IO^ODD 16X10-6 00X10-5 XXX X co oo t-eoo XX ! 5 i Tf^ to to IO 1 X IS t CO T-H IO rH TH coos t-ioeo kOlOiOlO to XX !2 5SSS2 >tDt*0904 iH XX X 03 to Tj< t-oeoeo ii. xxf CMCOCOCOCO CO TfTjTjiTliTjt lOOOOOJr-l CitOt- to' to to to to 10 >ototototo xxxxx 88S S?5^ CM IO IO< Tf > t-05000000 0000000000 00 0000000000 OOOOOOOJCft II x x" ii i Tt 05 to i-H c-eooio XX to co CM 90 o c iHUSrHCO U3 t>t>t-t- t-t-t-t-t- t- ?o o t-ooe t-' t-' t-' C- t-' t-' 00 00 <3J OS i-i i ooooo CMTlOOOO 1 1 II 1 1 180 A TREATISE ON THE SUN'S RADIATION TABLE 61 VALUES OF F IN THE EARTH'S ATMOSPHERE 1911 June 9 Sept. 13 Nov. 9 Means * 90000... _ _ 1. 1727X10-" 80000... _ 7.8260X10- 24 1. 6990X10-" 70000... *2.1138X10- 29 1.1934X10- 18 9.6458X10- 18 60000. . . *1.7858X10- 19 4. 7022X10-" 1.1720X10- 16 8.2111X10- 17 50000... *4.2798X10- 17 4.1448X10- 16 5.8433 " 4.9941X10- 16 40000. . . 2.7778X10- 15 3.0157X10- 15 3.2546X10- 15 3.0160X10' 15 30000... 1.1527X10- 14 1.2564X10- 14 1.3061X10- 14 1.2617X10- 14 20000... 1.8448 " 1.9864 " 2.0830 " 1.9714 " 10000... 3.4011 " 3.7548 " 3.2011 " 3.4523 " 100... 9.9940 " 1.0197X10' 13 8.9664 " 9.7191 " * Omitted from the means. The Nichols and Hull value of p gives F = X 7.01 X 10~ 5 c = 5.8727 X 10~ 14 and this occurs near the top of the isothermal layer in the earth's atmosphere. (dS\ 4 u 4 aT* From the differential equation, \~Ty)T = ~^ ~T = ~% ' ~T~~~ 4 a T 3 , it follows that the specific entropy per unit volume per o degree is 5 = a T 3 . The ratio at Z R is 3.3012 X 10~ 7 , and this o is equivalent to using a temperature 7690.2. It should be noted ds 1 that the structure of the spectrum depending upon == j^ (Planck, 135) refers to the pure aether enclosed with matter in an adiabatic volume, and it is not directly applicable to gaseous atmospheres open to space, wherein the term for work d W cannot be omitted, and d W = P d v assumed negligible. 8L ds Planck has developed two cases for the relations - = -: (149) I. (135), = = ; II. (81), L = s. P, and (76) ELEMENTS OF BLACK RADIATION IN THE ATMOSPHERES 181 i |i, 5 . x xx" " *J< COiOlTj<00 CO O5COO3 CD t- rji CON * COCO o OOCO rH CO O COOr-NCO 04 X > OCOr-l O I23S XXX O> t- lOCOi-H < X -l O t>CO 03 CO CO CO CO eo coco co co co o o XX X oo t- * oocoooeo HOOO t- OOO5 T-I M coko t-ooos 1-101 CD CO CS1 IN M CO-CO x 00 1ONO5COCO 0000000000 P5lOrJ 00 OOOOt-t-t- IONO5COCO NW^< CO ^iOCOt-OO CO CO CSI N N OOOiO CO CO CO CO CO CO ^ IO CO t~ 00 W N CO i-i IO O (M L eg co-t co 00 OO iH ( xxxxx CO-*tiOt-OS i-tCqiOOOi cocoooNio Oicocoo< COCOCOOOO t-lOCO,-l< CM i-l IO <* OJ OOOOiHi NCOCOCOi-l TfrJtlOiOl iH t>COO5U3 IO- tlNrHOiOO rHI .11. . I. ' xx" " x" iN^H iH 00 -^ CO N 00 00 10SIO 00 U) iH TJI OS t- CO CO US t- OW IOIO IfllOlO i. o, x" x" X iO CO 00 O> kO Oi ( C3 03 CO 00 OS 05 < NtO^CNlCO "^TjCM rHCOCOOCM >O IOI )-( lOOOi-tTfOO -l TJM CM ^TMOtQlO CO C0< XX X OOOi-lO t- icoco cMt-eoooeo oo I NT* Oi-ITfCOO j-< iCMOCO t~O5O 0000 S- 1 x" x eo tf o !OH CM i-IVOOOOOO lOOiOOl Ot-t-lOtM TfiOiOCOC O t ^ rH 00 kO t- 000000 CO CO CO CO CO t si XX t-c~ t- 1- 1- lOCN O OT* TH T* 0000010 OOCOCOOb- 'IOTJIO5OS Tf O5 T!< Oi CO >C-000>10 OiONCOlO 000. O O F^ T-( T-HJ TH ^H XXX X X OOeO-^Tt r)< "* o inosoo' oo b-OTj<( b- 0>OO< 00 O 3 : : : : : ; ; FNTHS IH XX X 10 kOlo'ujuilO 10 co' CON CM rH 23 2= : = : XX X ON CO T-I >-l >H -l COCOOb-' * Tf HOOiONO5 CMO5COC<1< NN b-COOb-CO CMCOiOb-C 000 XXX IH Ncorftioco iHtMe CO O5NOOOi-l t>OO( o >-! co TJ< 10 b- coco< CO tOCOCOCOCO t-00< 12% SS! !SS S xxxxx CDOTJ<00 OCOOrJ t> b-' b-^ i> t> i 22-1 xx" x" OJ b-COO5iOH CM b- TJ< T-H -oseo ION Ib-b- >-IN OOOJOOiH ^ -^ us us us co co' co' co co' o co co co co : -j -^< 1-1 ^ t>o ( >IOO b-OCO b-< )TJ< 10 b- cooo co< JiOlO IOCDCO b-l - - : 22: = xx" oo o N n oo b- oo b~ 10 co 00 O5 OS TT COCOCOCOCO COCOCOCOCO OO OOCO OOOOO (N^COOOO M M OO - - 186 A TREATISE ON THE SUN'S RADIATION F-3 II I , s I 00| t- -# CO Tl< CO OO' rH rH rH M' CO CO Tf !=: X Sco m CM CO >rH CO IO eo tu3 rH OOCCt < i-H O IO rH t>OOrHrHrH rH rH CO N CO CO IO 272X10" 7X101 7 " CO 00 NlOOOrH^ rHO CO Tj< mt-OOOrH COO TH 1-4 rHrHrHCON CC i IO si,i a...-,, = ,.,r, = = = = = s xxx x x COW5OU3CO OCOCOOSCO W OCOCOOO^JI OOOOCOCOOO CO OO"5COrHUi COCOCOCOCO CO OCDCOO5CO WkOCOt~00 IO rHCOrH-tod rH rH rH rH rH rH rH rH rH rH rH N 00 U5 CO* 00* rH S. , S x" " x* ^ QO iO 00 CO CO 00 O C^l ^* CO 00 O CO ^O 00 Ci CO 00 *C 00 1C t-OiCOOW COCO^^Tj* Tj* TfUSlClCO COOiWCOO t II I ; , . XX X* " o||, I XXX* X T-I^H^HrfrH rHrHrHrHrH rH rH rH rH rH rH rH rH rH rH r-l rH rH rH rH rH CO N CO CO Tji Tf in O O rn: ::rH : : : : : : : : : : : : : : : : ;:::; 5 3 2 ; 3 5 S X X CD Tf O5 to t- COrHOOiC- COCOCO-^lO CO OOOrHCO^O US-^COCOrH lOCOOSOSCO- TJ< CD rH os ^ ooosoorH cococococo co coeocoeoco TCCOOOOCO rHcocoio< CO O O ^* O CO CO ^ ^ ^ ^ Tj< -^ -^ -^ ^J< -^ ^* ^ ^ ^* ^ ^* ^ kO IO CO 00 O W CO < 0000 o ooooo ooooo ooooooo OOCO-^CO CO-tfCOOOO OOOOO OOOOOOO ICOrHrH ' ' ' ' If ^ T T T f f f ELEMENTS OF BLACK RADIATION IN THE ATMOSPHERES 187 of this research. The data here presented, from the thermody- namic side, must be taken into the account, especially at the solar source of radiation. TABLE 69 VALUES OF (UiUo) IN THE EARTH'S ATMOSPHERE 1911 June 9 Sept. 13 Nov. 9 Means 90000 2. 5798 XlO 5 80000 3 1797 XlO 6 6 4850 XlO 7 70000 . *4 1555 X10 4 2 2265 XlO 8 3.9119X10 8 60000 *1.6122X10 8 1.0568 XlO 9 1.2344 XlO 9 1.1456 XlO 9 50000 *1. 6014X10 2.9581 " 2.9880 ' 2.9731 " 40000 6.1829 " 6.4619 " 6.0909 ' 6.2452 " 30000 9.9355 " 1.0291 XlO 10 9.6396 ' 9.9554 " 20000 1.5560X10 9 1.5992 " 1.4883 XlO 10 1.5478 XlO 10 10000 2.4574 " 2.5023 " 2.3182 ' 2.4593 " 100 3.5843 " 3.6189 " 3.4247 ' 3.5426 " * Omitted from the means. P is the thermodynamic hydrostatic pressure due to the kinetic energy of the molecules per unit volume, and p is the rM U 1 -, kinetic energy of the electrons per unit volume, ^ - . , so that their ratio is a simple number. Within the solar isothermal layer the ratio is not far from the velocity of light, as if p X c = P, so that electrons having the velocity of light have the effect of molecules in producing volume pressure. This value of the ratio P/p is found near the bottom of the earth's atmosphere. Below the plane of solar radiation the ratio drops, and p ap- proaches P in value. Certain Data in the Radiation Terms Computed Directly from the Formula u a T* and Its Dependents for Black Radiation It will be convenient to give some examples of the values of these radiation terms as derived from the standard value of log a = 15.86494. We have selected certain temperatures for the computation in the atmospheres of the sun and the earth. 188 A TREATISE ON THE SUN'S RADIATION - 7.8722X109 1.6899X10* . . 7.7793X10 & o o. ... x x" " * O5CMCM OS * TH TH CM CS1 (NCO 10 - Ill- xxx" ooFH^-!ast> r-i : & 1 = = = = = X O asot-ot- d IO^CD t- as *J< TH iH C<1 COiH lO co OS S S x" xx" oo OTH uj ooooMOico X CO ?! Nco 00 XX~ lOlOr-l OSOOt-COlO THiHTJ* -#IOCOC>00 1 L. 1768XlOio L.4044 " L.6319 " L.8595 " J. 0870X10io II XX 2 gg t ^ : .. S3 ooS. . ..... NCOOSt-lO IOOU5OIO *#*$ i-t CQ *!* K3COCOt-C~ i 00 9.0862X109 1.0098X101 1.1110 " 1.2121 " 1.3133 ' III- - xxx" " co TJ< TJ* as 10 co CO 00 lO O 1O t> COTfrltCOO * N TH 00 10 -^ TH 10 ^ : H ' - 2 | x" " OOOTHiHCO CO-^COOOO ooooujiH ojcoo-<*as THU300COTj< COOOrHCOlO COCO'T)*'OCO t^t>oooooo 00 s , ^ asasas THTH 1- III 1 X~ XXX X CMOWNCO ^gSS 5 THU5Ot>CM TH T-! TH ri oo TH as as ; o oo : 10 co . 1 M : * II : : : : : XX 1 s " X i-4U5CiCO t- III T ^ i CJco ou5t-oooo asasasasas OS Oi Oi Oi Oi rH M xxx" 0} CO (M 00 U5 O *O Oi "^ 00 O CO ^O 00 O ICCKOOO t>OiTH^<> OOOOOOOOOi I O W CO iO CO oo, . . , xx" T : CO 04 O C^ CO -^ T} 1 1C ^O W M3 W 1C UD lO 1O U3WCOCOCO ococot-t- as T-I TJ< us 10 w Tf -l-ll- - $ X THOOWOOO ooor-icoT)* inuiiomio in o oeo t- oeo TH t-eo oco CM CO kO t-OO X XX l|j THrHWCOCO lOCOCOCOCO COCOCOCOCO co co co co CD COCOt-t-t- OOTHrH-,TH^lTH * ' i IIP p. 1 OOOOO d-^* co ooo 11117 ra-^fCOOOO OOOOOOO I I I I iH Kinetic energy, (Ui - Uo) Ratio of pressures, P P Pressure, P . . . 25247850 0.0000014889 2207720000 6161900 0.000000629 1277040000 998337 0.0012152 2870300 56808 0.00015823 1636900 24531 0.00001696 872613 Density, p tti Efficiency, m R. . . . Mean values of the entire system were taken for P 9 p, R, T, and the standard log a = 15.86494 in Table 26. These values should be compared with those given on Tables 54-71, 190 A TREATISE ON 1HE SUN S RADIATION in order to understand the departures that occur in the thermo- dynamic computed values. It should be noted that it is the variation in the coefficient and exponent of u = a T a that is primarily responsible for these changes. 10 11 12 13 14 15 16 17 18 -10 -9 -8 -7 -6 -5 -4 -3 T 2 -1 FIG. 21. The Great Discontinuity in the Radiation Data Occurring at the Z R Levels. Calcium Vapor, P and ( Ui C/ ) have no discontinuity at the ZR level ; w, p, K, /, s, L, all suffer a per saltum change at that level where the radiation is generated. ELEMENTS OF BLACK RADIATION IN THE ATMOSPHERES 191 The Discontinuity at the Levels Where the Solar Radiation is Generated In order to illustrate the discontinuity in several terms on the levels where the solar radiation originates, a diagram, Fig. 21, has been constructed for Calcium Vapor on a composite scale, such that the ordinates are in kilometers from the photo- sphere and the abscissas are the logarithm's characteristics. The hydrostatic pressure P and the inner kinetic energy (Ui UQ) run out smoothly across this level Z R , but the density u, pressure />, intensity K, energy 7, entropy s, and entropy radiation L, all suffer the marked discontinuity which has been computed and illustrated in the tables. It is seen that while the thermo- dynamic terms P, p, R do not go through such a discontinuity, all the terms involved in the generation of the radiation have been affected by this sudden variation, so that the physical processes must be different. The thermodynamic terms are probably confined to the kinetic and the potential energies of the molecules, while the radiation terms apparently are to be referred to some change in the structural motions of the elec- trons as free electric charges. CHAPTER VI Reconciliation of the Data Derived from the Pyrheliometer and the Bolometer The Poynting Equation THE resolution of the problem of the intensity of the solar radiation outside of the earth's atmosphere depends upon the interpretation that ought to be placed upon the two sides of the Poynting surface-flux, volume-density equation, (66) j- The surface flux is equal to the volume changes plus the waste. The observations with pyrheliometers and bolometers are commonly recorded in terms of the surface-flux, density ~\JT T 2 per aether-volume times the velocity of propagation, ~ 2 Ti * T- = r-3' so that conversions from the (M. K. S.) to the JL/ I. -L (C. G. S.) system require the factor 1000. On the other hand, the thermodynamic computations are properly conducted in ML 2 I M the volume-density form, with dimension ^ 2 j^ = y- so that the conversion factor is 10. As already stated, Mr. Very has made the erroneous criticism that the conversion factor for the volume-density data of thermodynamics should be 1000 instead of 10, thus transferring the surface-flux factor 1000 to the volume-density, and so violating the law of dimen- sions. While the data of the two sides of the equation must be equivalent, no confusion can be permitted regarding the con- version of computations from one system of units to another. The fundamental difficulty with this problem in practise is that the solar radiation in the electromagnetic form of energy 192 RECONCILIATION OF DATA 193 is transformed in the earth's atmosphere into equivalent thermo- dynamic forms, but these are so different in their manifestations that it is not easy fully to identify them. The pyrheliometer produces data which are very unlike the data derived from the bolometer, and each of these differs in form from the thermo- dynamic effects as recorded by the atmospheric conditions. In fact, it is still impractical to trace these transformations in detail, and this cannot be done until the physics of the trans- formation of energy in gases has been fully established. At the present time it is necessary to determine the total effects, first as a series of summations, and then proceed to the details so far as it is possible to do so. It should be noted that Mr. Abbot, and many other ob- servers, are approaching the general problem from too narrow a point of view, and are trying to reconcile these conflicting results, as heretofore indicated, by an overemphasis upon the complete sufficiency of the pyrheliometer to solve the problem. They are obliged to assume that the sun does not radiate like a black body at its high temperature, 7655, in order to account for the bolometer discrepancy, thus debasing its 4.00 calories down to about 2.00 calories. Nor do they take into account the positive thermodynamic evidence that 4.00 calories are expended in the earth's atmosphere. Mr. Abbot's first as- sumption that the sun does not radiate at black body efficiency, at a temperature of 7655, is disproved by the solar thermo- dynamics, summarized in the preceding chapters; his second assumption, that the terrestrial thermodynamics, as computed, does not require consideration is based only upon Mr. Very's erroneous statements regarding the conversion factor from (M. K. S.) to (C. G. S.), thus placing himself in conflict with the law of dimensions, as well as with the uniform practice of meteorologists and mathematical physicists. Mr. Abbot summarizes his evidence regarding the Intensity of the Solar Radiation outside the atmosphere, Smithsonian Miscellaneous Collections, Volume 65, Number 4, 1915, and repeats his opinion that this intensity is 1.93 calories per sq. cm. per minute. He arrives at this conclusion by the original 194 A TREATISE ON THE SUN'S RADIATION Langley Method of discussing the Bouguer Formula of deple- tion, as if the extrapolation of the graph (sec z, log 7) could be extended from sec z = 1.00, in the zenith, to sec z = 0.00, on the outside of the earth's atmosphere. The last step is non- mathematical, and as a physical process it is entirely unlike those preceding it (sec z\ . /i), (sec z 2 . 7 2 ), etc., so that it is in- competent to integrate from the station to the top of the at- mosphere through the numerous unknown physical processes involved. The formula 7 = 7 p se< z is competent to determine p at the station, but not to trace out the varying values of p in the higher strata; it is competent to determine the ratio 7// at the station, but not to discover the constituent physical terms that make up 7o as a thermodynamic effect, from the parameter T to the other terms of the general law, P p R T in its non- adiabatic free atmosphere application. He supposes that by repeating this imperfect extrapolation at many stations, and under many surface conditions, the resulting agreement among them becomes proof that the physical result is complete. This is accompanied by a general evasion of many other facts which are clearly in contradiction to his pyrheliometer result. On the other hand, we shall be able to point out that there are great physical tracts in thermodynamics which are never recorded by the pyrheliometer, as compared with the full effect of the solar radiation, and thus establish our thesis that this apparatus registers only a portion of the incoming solar energy. His " New Evidence" consists of pyrheliometer data, regis- tered automatically up to the height of about 22000 meters in balloon ascensions, and this is a welcome and important addition to our available data. These data will be accepted as substan- tially correct, and adopted into our reconstructed system. Mr. Fowle has published the results of his investigation into the percentage of absorption and scattering by dry air and aqueous vapor, as determined from the bolometer graphs at Washington, D. C., Mt. Wilson, and Mt. Whitney. These will be incorporated into the system without any discussion. We shall show that there are three decisive thermodynamic factors which must still be taken into the account, in order to reconcile the pyrheliometer RECONCILIATION OF DATA 195 data with the bolometer data, and both of these with the ther- modynamic data of the terrestrial and the solar radiation. It is probable that the scheme here explained is capable of great refinement regarding the physical processes in atmospheres, and that these will lead to important contributions to the mathe- matical analysis of the theory of radiation and the underlying constitution of matter. It should, also, be noted that the new high-level pyrheliometer data fit in with the curve which had already been adopted to represent this term from the sea level to the top of the atmosphere. The new data, therefore, serve in no respect to modify the original problem, which is to explain the broad discrepancy between the result by the pyrheliometer, 2.00 calories, and the result by the bolometer 4.00 calories, without denying the approximate efficiency of the sun as a black radiator at the temperature of 7655 and 5.85 calories. The Cause of the Change from the Total Solar Radiation 5.85 Calories to the Effective Radiation 3.98 Calories Received on the Outer Strata of the Earths Atmosphere It has been shown from solar thermodynamics that the radiation originates as black, and at an intensity equivalent to 5.85 calories at the distance of the earth. Of this amount it is found from the terrestrial thermodynamics that 3.98 calories are received on the outer strata of the earth's atmosphere and that about 1.40 calories reach the sea level. The former of these depletions is solar and the latter is terrestrial. In order to explain the change from the true solar to the effective solar radiation, we shall assume the data of Table 55, Vol. Ill, Smith- sonian Institution, which contain the relative intensity of the radiation for several different wave lengths, from X = 0.323 /* to X = 2.097 ju, as measured by the bolometer along a solar radius in terms of 1.000 at the center of the disk and 0.000 at the limb. The slit of the bolometer is set to a given wave length, and the disk of the sun in drifting across it produces a relative intensity curve with maximum at the center. Fig. 22 contains the set of relative curves, the dotted line corresponding with 0.470 /*> which 196 A TREATISE ON THE SUN S RADIATION is the common maximum ordinate of the system. The large dots show the limits of the cosine-ordinates, which express approximately the law of extinction for the shortest wave length. The coefficients of intensity are taken as the ordinates and the 1.000 .900 .800 .700 .10 .200 .100 .10 .20 .30 .40 .50 .55 .60 .65 .70 .75 .80 | g .90 { Sjl.OO * FIG. 22. The Loss of Radiation Between the Center and the Limb of the Disk of the Sun. (Abbot's Table 53, Vol. III). parts of the radius as the abscissas. The phenomenon of the diminution of the brightness of the sun's disk from the center to the limb is commonly attributed to scattering, and the figure shows how this is to be distributed in the spectrum. The short waves are depleted more than the long waves, and they both are increasingly absorbed in some function of the path-length from points below the surface to points of escape in the direc- tion of the earth. This complex subject will be resumed in Chapter VII. RECONCILIATION OF DATA 197 S3AJno aq^ Xq '3uip3O3Jd asoq^ UIQJJ spBtu 'UIBJ3 ip B UIOJJ pa^BOS SBAV UUinjOD o o o o o o o t- o * o Tf eo oo 00 N O CO t- 05 O CO 50 t- ^ o eo t- t- t- t- 200 rH t- o i-H t- Tf OOOOOOiOO5O500N eo t- in > 10 o eo * o oo os eo eo 03 eo co eo eo 03 soeococococococoeoeococoeoeoco o eo co <-H -i T)< oo eo in oo o eo eo -^ -^ji ^< 10 uj 198 A TREATISE ON THE SUN'S RADIATION In order to determine the effective intensity of the solar radiation towards the earth, it is necessary to integrate the parts contributed by each wave length over one hemisphere of the sun. To do this the data of Table 73, Section II, are com- puted as follows: Take the mean value of the intensity for two successive points, and multiply this by 2?r X Pi2, where p i2 is the mean radius of the two points. Example: Line 0.323 /*, Pl = 0.650 I I = 0.775 P 2 = 0.750 7 2 = 0.690 Pi 2 = 0.700 7 12 = 0.733 27T pi 2 7 12 = 6.283 X 0.700 X 0.733 = 3.224. Section II contains the products for each group, so that the horizontal sums give relative numbers of the effective intensity of the several spectrum lines over the hemisphere. Had there been no depletion, the same hemisphere would have given an intensity 1.000 in each zone, with the total 41.585, while the line 0.323 /* gives 24.112, so that the factor of efficiency is 0.580 for this line. The last column in II gives the efficiency factor for each of the selected lines in the spectrum. Whatever may be the original solar radiation, the hemispherical action of the sun's atmosphere is to reduce the lines in these proportions. We have shown that the solar radiation originates at T = 7655, as black radiation, and the coordinates as computed by the Wien-Planck Formula appear in Table 79. They can be compared there with the energy spectrum curve for 6950, 5810, 5450, as well as with the bolometer ordinates at Washington, Mt. Wilson, Mt. Whitney, and the total as extrapolated by Abbot to the outer limit of the atmosphere. The ordinates are given in relative numbers, but for intercomparison they are homogeneous, and for reduction to calories the factor 1/20.9 is sufficient. We will interpolate for the curves 7655, 6950, total bol- ometer, the ordinates corresponding with the wave lengths herein adopted. Those for the 7655-curve will be multiplied by the factors of efficiency, and the product is to be compared with the ordinates for the 6950 and the total curves. RECONCILIATION OF DATA 199 TABLE 74 COMPARISON OF THE BLACK SOLAR INTENSITY OF RADIATION AT 7655 WITH THE EFFECTIVE, THE THERMODYNAMIC INTENSITY AT 6950, AND THE EXTRAPOLATED BOLOMETER TOTAL. A T 7655 Factor Effective at Sun T 6950 Total Bolometer 0.323 M 9.790 0.580 5 678 5 406 1 500 386 . ... 10 552 589 6 215 6 348 3 690 0.433 10 294 633 6 516 6 514 5 472 456 . 9 884 661 6 533 6 454 6 051 0.481 9.411 679 6 390 6 268 6 056 501 . 9 031 690 6 231 6 116 6 052 0.534 8.337 709 5 911 5 946 5 768 604 . 6 898 738 5 091 4 959 4 994 0.670 5.721 762 4 359 4 217 4 070 699 . 5 188 770 3 995 3 664 3 664 0.866 3 191 806 2 572* 2 511 2 403 1 031 1 994 830 1 655 1 603 1 536 1.225 1 199 845 1 013 986 985 1.655 0.459 0.884 406 389 466 2.097 .. 204 899 183 176 211 A comparison of the effective radiation at the sun, as com- puted from black radiation at 7655, with the black radiation at 6950, shows that they are practically identical throughout the spectrum. This means that the solar radiation, which is equiva- lent to 5.85 calories at its strata of origin below the photosphere, is effective at the earth to the amount 3.98 calories, and that about 1.87 calories have been lost by the hemispherical scattering from the disk and by absorption in the superincumbent strata. By comparison of these curves with the Abbot ordinates as determined bv bolometer observations, and extrapolated to the 200 limit of the earth's atmosphere, it appears that there is sub- stantial agreement from X = 0.500 /* to the end of the spectrum in the long waves, but that there is a great depletion at the end of the spectrum in the short waves. This is in accordance with the data in Table 83 and Fig. 26. The depletion in the earth's atmosphere can be discussed under the thermodynamic terms, which agree with the bolometer in requiring 3.98 calories to fall upon the earth's atmosphere. Of this the pyrheliometer measures only 1.40 calories at the sea level, extrapolated to 1.940 calories on the vanishing plane. This is only half the effective solar constant and it is only one-third the true solar constant. The question as to whether the sun radiates as a black body seems to be settled in the affirmative. The physical laws regarding the processes of absorption and transmission in each atmosphere remain to be studied, but it is evident that we have obtained the P, p, R, T, in all the strata concerned in the sun, and through- out those at the earth, so that the laws of transmission, refraction, and numerous other functions can be approached with suitable material for their discussion. It should also be remembered that radiation from an iso- thermal layer, if sufficiently thick, must be black radiation. It has been found that the solar gases all pass through such deep iso- thermal strata, and we should, therefore, expect the solar radia- tion to be black. Furthermore, it is found from the terrestrial thermodynamics that about 3.98 calories of black radiation are actually consumed in the earth's atmosphere. Summary of the Terrestrial Thermodynamic Data It will be convenient at this point to summarize the results of the thermodynamic data as computed, and reported in Bulletin No. 4, 0. M. A., 1914, or in the Meteorological Treatise, 1915. These summaries are compiled for the balloon ascensions, Uccle, June 9, September 13, and November 9, 1911, from Tables 12-26 of the bulletin. The sums were taken for each ascension through the 5000-meter intervals indicated in Table 75 under z\ So; the means of these sums in each stratum for the three ascensions were taken in the (M. K. S.) system of the computations, and RECONCILIATION OF DATA 201 these were converted into the equivalent gr. cal./cm. 2 min. for the following discussions; finally, the successive sums 2 were computed from the vanishing plane to the bottom of the re- spective strata. The first and second columns under each element refer in the first to the strata-means, and in the second to the general sum from the top of the atmosphere to the bottom of the several strata. The third column under 2 (Qi Q ) , 2 (Ui Uo), 2 g (zi ZQ), has been computed by subtracting 1.5804 from the second column, for reasons that will be stated. In order to understand the relations of the thermodynamic terms to one another, a resume of the non-adiabatic formulas is given: Resume of the Thermodynamic Formulas (isi) _ A^Z = + (w 1 - w,} - R IO (r. - TO), Pio - Cpw (T a - TQ) = (R w - Cpio - Rio) (T a - T ). Pi- Po _ Pio - Cpw (T a - To) = (Cpa- Cpio - Cpa) (T a ~ T ). (152) - R 10 (T a - To) = - Pl "" P - (Wi - Wo), Pio R 10 (T a TO) = ( Cpio RIO ~t~ Cpio) (T a TQ). - RlO (Ta~ T Q )= + g (Zi - Zo) + (Ui - Uo), ~ RlO (T a - To) = ( ~ Cpa + Cpa ~ Rio) (T a ~ TO). (153) + (Wi - Wo) = - Pl " P + *io (T a - To), Pio (Rio Cpio) (T a To) = ( Cpio + RIO) (T a TO). + (Wi - Wo) = - Pl ~ P - (U, - Uo)- g (*i - zo), Pio Ta-To) = (- Cpjo - Cp a + RlO + Cpa) (T a ~ TQ). (154) - (Ui - Uo) = + #10 (T a -T )-\-g (i - zo), -(Cpa-Rw) (Ta-To) = (Ru - Cpa) (T a - TO). - (Ui - Uo) = + Pl ~ P + (Wi - Wo) + g (zi - zo), Pio - (Cpa- Rio) (Ta-To) = (Cpv> + .#10 - Cp m - Cpa) (Ta-To). (155) - (ft - Co) = - (Wi - Wo) - (Ui - Uo), -To) = (- Rio + Cp w - Cp a + RIO) (T a -T ). 202 A TREATISE ON THE SUN'S RADIATION - (ft - Co) = g & - So) + PlO -(Cpa-Cpu) (Ta-To) = (- Cp a + Cpu) (T a ~ T ). - (Qi - ft) = g fa - 2b) + Pio (T fl - To) - (Wi - Wo) , - (Cpa-Cp w ) (T -T ) = (- C^+ Pio - Pio + Q> 10 ) (T a - T Q ). (156) g (Zi - Zo) = - (ft - ft) + (Wi - Wo)-RlO (T a - To), Pi- Po g (zi - Zo) = - (ft - ft) - PlO - b) = - (IFi - TFo) - (tfi - Z7 ) - Pio g (21 - Zb) = ~ (^1 - ?7o) - ^10 (T a ~ To), - Q>a (Ta~ To) = (- Q> a + RlO ~ RU) (T a ~ TO). These formulas are easily verified from Fig. 23. Special attention may be directed to the following facts: 1. The boundary 8 A = 2 g (z t ZQ) limits the adiabatic p -p system, while 4 B A = S -- limits the non-adiabatic PlO system, and the difference between them is S (ft ft) = 04C^4=4J5^48, counted in the proper algebraic directions. 2. The work 2 + (Wi - PF ) is the area 03^=4C^7. 3. The efficiency S ^10 (T a T ) = 3 ^1 5 4 = 7 ^L 8. 4. The 2 - ^^L = S - (ft - ft) = i Sg (* - 20). PlO 5. Hence, the adiabatic area is formed from the non-adiabatic area by simply adding and subtracting Cp a (T a To), as in formula (4). It follows that the differences for the strata, (Wi Wo), (Ui Uo), (Q i ft), are the same in both systems, and that we can transfer the area 4 D A 8 to coincide with the area A D 4, and this is convenient for several reasons. p _ p 6. The total hydrostatic energy 2 = - - - measures the Pio effect of the solar radiation, after it has been transformed from the electromagnetic form into the thermodynamic form, and it RECONCILIATION OF DATA 203 can be measured by the sum of the free heat as well, S (Qi Q ). The sum for Abbot's value of the solar constant 1.94 calories is found on the hydrostatic curve at about 18000 meters above the sea level, on the free heat curve at 40000 meters, but on the black radiation curve at 7000 meters. Mr. Abbot has taken my TABLE 75 SUMMARY OF THE TERRESTRIAL THERMODYNAMICS Ucde, June 9, Sept. 13, Nov. 9, 1911, Data from Bulletin No. 4 Pi-Po f 2ff\ r\ \ z\ zo 2 (Wi Wo) PlO (tfi (Jo) 70-66 0.0184 0.0184 0.0254 0.0254 0.6623 0.6623 -0.9181 65-61 0.0321 0.0505 0.0439 0.0693 0.6441 1.3064 -0.2740 60-56 0.0572 0.1077 0.0780 0.1473 0:6118 1.9182 0.3278 55-51 0.0925 0.2002 0.1279 0.2752 0.5625 2.4807 0.9003 50-46 0.1386 0.3388 0.1926 0.4678 0.4990 2.9797 1.3993 45-41 0.1584 0.4972 0.2521 0.7199 0.4694 3.4491 1.8687 40-36 0.1410 0.6382 0.1988 0.9187 0.4955 3.9446 2.3642 35-31 0.1623 0.8005 0.2283 1.1470 0.4669 4.4115 2.8311 30-26 0.1993 0.9998 0.2804 1.4274 0.4157 4.8272 3.2468 25-21 0.2521 1.2519 0.3546 1.7820 0.3426 5.1698 3.5894 20-16 0.3183 1.5702 0.4496 2.2314 0,2467 5.4165 3.8361 15-11 0.3985 1.9687 0.5657 2.7971 0.1374 5.5539 3.9735 10- 6 0.4563 2.4250 0.6259 3.4230 0.0784 5.6323 4.0519 5- 1 0.4691 2.8941 0.6591 4.0821 0.0302 5.6625 4.0821 2.8941 4.0821 5.6625 31 ZO 2-(tfi-Z7o) 2 g (a -a) 70-66 0.6807 0.6807 -0.8997 0.6879 0.6879 -0.8925 65-61 0.6762 1.3569 -0.2235 0.6890 1.3769 -0.2035 60-56 0.6679 2.0248 0.4444 0.6901 2.0670 0.4866 55-51 0.6551 2.6799 1.0995 0.6912 2.7582 1.1778 50-46 0.6362 3.3161 1.7357 0.6924 3.4506 1.8702 45-41 0.6278 3.9439 2.3635 0.6934 4.1440 2.5636 40-36 0.6364 4.5803 2.9999 0.6945 4.8385 3.2581 35-31 0.6292 5.2095 3.6291 0.6956 5.5341 3.9537 30-26 0.6152 5.8247 4.2443 0.6967 6.2308 4.6504 25-21 0.5991 6.4238 4.8434 0.6978 6.9286 5.3482 20-16 0.5647 6.9885 5.4081 0.6989 7.6275 6.0371 15-11 0.5379 7.5264 5.9460 0.7000 8.3275 6.7471 10- 6 0.5243 8.0507 6.4703 0.7011 9.0286 7.4482 5- 1 0.4996 8.5503 6.9699 0.6882 9.7168 8.1364 8.5503 9.7168 204 A TREATISE ON THE SUN'S RADIATION TABLE 75 Continued ZlZO 2 Rio (Ta -To) 2 A J 2 A7a S-Ja 70-66 0.0070 0.0072 0.00039 0.00039 0.00002 0.00002 65-61 0.0188 0.0200 0.00070 0.00109 0.00008 0.00010 3.990 60-56 0.0396 0.0422 0.00211 0.00320 0.00029 0.00039 3.990 55-51 0.0750 0.0783 0.00577 0.00897 0.00098 0.00137 3.989 50-46 0.1290 0.1345 0.01282 0.02179 0.00531 0.00668 3.983 45-41 0.2227 0.2001 0.08993 0.11172 0.00922 0.01590 3.974 40-36 0.2805 0.2582 0.22285 0.33457 0.02343 0.03933 3.951 35-31 0.3465 0.3246 0.12940 0.46397 0.02665 0.06598 3.934 30-26 0.4276 0.4061 0.13468 0.59865 0.04446 0.11044 3.880 25-21 0.5301 0.5048 0.13975 0.73840 0.06792 0.17836 3.812 20-16 0.6612 0.6390 0.16625 0.90465 0.10836 0.28672 3.703 15-11 0.8284 0.8011 0.50154 1.40619 0.19539 0.48211 3.508 10- 6 0.9980 0.9779 1 . 15080 2.55699 0.40720 0.88931 3.101 5- 1 1.1880 1.1665 1.38090 3.93789 0.56672 1.45603 2.534 3.93789 1 . 45603 statements regarding the summation along the 2 (Qi Q ) curve as admission that the pyrheliometer measures the true solar constant, because this curve has its half-sum at that height, where the atmosphere has already a very low density. On the other hand, the half-sum for the hydrostatic curve is at 18000 meters, and for the black radiation it is at 8000 meters. These complex summations must be carefully studied before their final interpretation is permitted. The transformation between the electromagnetic radiation and the thermodynamic volume-effects involves some processes and terms that the pyrheliometer does not register, and it is on this account that Mr. Abbot's value of the solar constant is erroneous. The purpose of the foregoing summary is to indicate that we can pass from a non-adiabatic system, whose boundary curve is Cpw (T a jT ), into a strictly adiabatic system, whose boundary is Cp a (T a To) g (zi z ). The Boyle-Gay Lussac Law in free atmospheres refers to the non-adiabatic (P. p. R. 7\), which become adiabatic when such impressed forces as gravitation are superposed upon them. The mirror-enclosure, within which the radiation formulas are usually developed, does not exist in free atmospheres, but in these gases the enclosing walls are removed and these thermody- RECONCILIATION OF DATA 205 TABLE 76 SUMMARY OF THE ABSORBED AND THE BLACK BODY RADIATIONS AS COMPUTED WITHIN THE TERRESTRIAL ATMOSPHERE gr. cal. I. S Ja = the total radiation absorbed = 2 cm. 2 min. z Meters Uccle 3 Omahar 3 Europe 6 U.S. 7 Tropics 6 Means S-Ja 90000 3.980 Data from 80 _ _ _ 3.980 Tables 32-36, 70 _ _ _ _ _ 3.980 Bulletin No. 4 60 .... 3.980 50000.... 0.001 . 0.001 3.979 45 .... 0.007 0.007 3.973 40 .... 0.016 0.016 3.964 35 0.039 0.039 3.941 30000 0.065 _ _ _ 0.065 3.915 ft '"'...-'. 0.110 _ _ 0.110 _ 0.110 3.870 20 0.177 0.178 _ 0.175 _ 0.177 3.803 15 0.286 0.293 0.274 0.278 0.280 0.282 3.698 10000 0.481 0.488 0.460 0.483 0.516 0.486 3.494 8 0.623 0.615 0.609 0.615 0.676 0.628 3.352 6 0.784 0.820 0.788 0.785 0.852 0.806 3.174 4 0.982 1.029 0.974 0.999 1.023 1.001 2.979 Mt. Whitney 2 1.217 1.325 1.196 1.225 1.220 1.237 2.743 Mt. Wilson 000 .... 1.455 1.624 1.407 1.429 1.465 1.476 2.504 Washington, D. C. II. 2 Jo = the total black body radiation = 2 (ciT^c TV) gr. cal. cm. 2 min. z Meters Uccle 3 Omaha 3 Europe 6 U.S. 7 Tropics 6 Means 90000 QA - - - - - - oU .... 70 60 .... 0.003 0.003 50000 0.014 _ _ _ _ 0.014 45 0.032 _ _ _ _ O.S32 40 0.117 0.117 As the result of all the avail- 35 .... 0.345 0.345 able thermodynamic data the 30000 0.478 0.478 value 3.98 has been adopted 25 .... 0.618 _ 0.618 0.618 for the surface summations of 20 .... 0.760 0.760 _ 0.766 _ 0.762 the intensity of the solar radia- 15 .... 0.937 0.962 0.915 0.887 0.940 0.928 tion. 10000.... 1.352 1.272 1.259 1.331 1.635 1.370 8 1.714 1.498 1.673 1.704 2.101 1.738 6 2.178 2.010 2.164 2.200 2,599 2.230 4 2.675 2.408 2.655 2.780 3.062 2.716 2 3.274 2.967 3.210 3.391 3.647 3.298 000 3.808 3.609 3.743 3.946 4.338 3.909 namic curves take their place. The mirror- wall condition is reached only near the bottom of the earth's atmosphere, or in the adiabatic strata of the solar gases, and it is on this account 206 A TREATISE ON THE SUN'S RADIATION that the practical transformation problem has been so difficult. The main result that we wish to note follows: THE SOLAR RADIATION ORIGINATES AS BLACK RADIATION AT AN EQUIVALENT OF 5.85 CALORIES; IT HAS BEEN FOUND TO BE EQUIVALENT TO ABOUT 3.98 CALORIES IN THE EARTH'S ATMOSPHERE; THE EVIDENCE IS THAT IT IS STILL BLACK RADIATION AT THE LEVELS 50000-60000 METERS, AS INDICATED IN THE TABLES 54~70. These data are taken from Tables 32-36, Bui. No. 4, 0. M. A., and they summarize the amount of the absorbed radiation S J a) and the black body radiation 2 /o, down to the levels indicated on the column of z in succession. From a consideration of all the available data, taking account of the season of the year of the 25 balloon ascensions as computed, and the latitude, it has been thought proper to assume S / = 3.980 as the effec- tive solar radiation at the distance of the earth on the vanishing plane of the atmosphere. The column (S J a ) shows the progress of the depletion to the sea level, and it is closely in agreement with the bolometer results at the stations occupied. Compare Meteorological Treatise, pages 380, 389. The Thermodynamic Data in the lOOO-Meter Levels In Table 75 and Fig. 23 the thermodynamic data were sum- marized in the 5000-meter strata, and in the deep strata from the top of the atmosphere down to the levels indicated. In Table 77 and Fig. 24 Jhe data are retained in the 1000-meter strata for + (Wi- TF ), - Pl "" P , -(Qi - Qo) and -(Ui-U Q ). Pio The summations for the black radiation S A / and for selective radiation S A J a are repeated on Fig. 24. There are two points to be indicated in the construction of Figs. 23 and 24. In the balloon ascensions, as extended to 70000, 80000, 90000 meters, it is evident that the summations may be carried too high above the effective strata, into an asymptotic region, to represent properly the true solar radiation. The incoming column of the electromagnetic radiation may be represented as contained be- tween parallel planes on a fixed scale; at some levels this begins RECONCILIATION OF DATA 207 to transform when in contact with the gaseous media, and it is this effective height that is to be determined. On Fig. 23 the area 4 C A = 2 (Qi Qo), and this is also equal to the area 4 B A 8, whenever the curves 4 B A and 4 C A coincide at the 70000 60000 50000 .40000 30000 20000 10000 0123 A 56 78 FIG. 23. Summary of the Terrestrial Thermodynamic Data. points 4 and A. If the area 4 C A 8 is extended too high the points 4 and A diverge along the two axes, and this is due to the arbitrary addition of + Cp a and Cp a to the non-adiabatic formula, which is required to produce the adiabatic formula. Hence, by making the curves 4 B A and 4 C A coincide at 4 and A, the effective height of the atmosphere as a thermo- dynamic medium is determined. In this case it is about 60000 meters above the sea level. My first computations for Huron, September 1, 1910, assumed this height to be 50000 meters; but the second computations extend it to 60000 meters, with a very rarefied coronal strata of an asymptotic character continuing up to 90000 meters. The thermal contents of this coronal 208 A TREATISE ON THE SUN S RADIATION stratum are insignificant, although it can be clearly developed in the computations. Having thus shown that the base 04 is equal to the base 48, the first being strictly non-adiabatic, and the second the 70000 60000 50000 .40000 30000 20000 10000 "0 0.50 1.00 1.50 2.00 2.50 3.00 3.50 FIG. 24. The Thermodynamic Data in the Several Strata Having a Depth of 1000 Meters. addition (Qi Q ) required to transform it into the adiabatic system g (zi z ) existing in the earth's atmosphere, it becomes proper to superpose the two systems upon each other when dealing with the individual strata. Furthermore, since the in- coming solar radiation has one fixed value from the top to the bottom of the earth's atmosphere, 3.98 calories, it follows that this must be the value in every stratum, in whatever manner it may become subdivided by the physical processes of the transformation. Since, by Poynting's Law, the summation in every stratum must be the same for the radiation-flux and for the thermodynamic volume-density of the energy, we shall adopt the same vertical RECONCILIATION OF DATA 209 boundary for the two systems, namely, an ordinate at 3.98 calories, the effective value of the intensity of the solar radiation at the vanishing plane of the earth's atmosphere. Since the gravitation energy in the 1000-meter strata is 9806, (M. K. S.) system, this is equivalent to 0.1405 gr. cal./cm. 2 min., and it would be a proper scale for use with the thermodynamic data. But it is more instructive to adopt the full summation value 3.98, to which the factor of reduction is 28.74 that is, m, the molecular weight. At the same time the small asymptotic region above 60000 meters has been omitted. It may become possible to resume the computations on these high levels, adapt- ing them more closely to a definite normal height for the atmo- sphere at its coronal base levels. At present we are concerned with detecting the several elements into which the solar radiation is transformed within the earth's atmosphere. TABLE 77 THE THERMODYNAMIC DATA IN THE 1000-METER LEVELS EXTERN. IL WORK HYDROS. PRESSURE FREE HEAT INNER ENERGY Height, in Meters z /TIT- TT/ \ m(Wi- Pi-Po , Pl - Po -h- -m (Qi- -O/i- -m (Ui- Wo) pio PIO Qo) Qo) Z/o) Uo) 70000. . . 0.0030 0.0849 0.0041 0.1478 0.1334 3.9233 0.1363 3.9180 65000. . . 0.0050 0.1441 0.0056 0.1580 0.1307 3.7554 0.1357 3.8997 60000 . . . 0.0090 0.2577 0.0124 0.3557 0.1207 3.4669 0.1344 3.8619 55000. . . 0.0156 0.4486 0.0197 0.5656 0.1169 3.3594 0.1321 3.7970 50000. . . 0.0238 0.6848 0.0378 1.0861 0.1052 3.0232 0.1293 3.7166 45000 . . . 0.0323 0.9294 0.0453 1.3006 0.0932 2.6786 0.1255 3.6071 40000 . . . 0.0286 0.8231 0.0408 1 . 1734 0.0979 2.8128 0.1265 3.6360 35000. . . 0.0300 0.8628 0.0422 1.2134 0.0968 2.7815 0.1268 3.6443 30000. . . 0.0367 1.0534 0.0515 1.4804 0.0858 2.4663 0.1245 3.5772 25000. . . 0.0459 1.3177 0.0645 1.8531 0.0750 2.1552 0.1208 3.4726 20000. .. 0.0579 1.6630 0.0815 2.3408 0.0581 1.6686 0.1158 3.3275 15000. .. 0.0730 2.0975 0.1031 2.9624 0.0357 1.0270 0.1087 3.1246 10000. . . 0.0870 2.4992 0.1219 3.5030 0.0192 0.5525 0.1062 3.0518 5000. .. 0.0901 2.5879 0.1304 3.7476 0.0099 0.2869 0.1027 2.9523 000. . . 0.0991 2.8476 0.1392 4.0013 0.0000 0.0000 0.1007 2.8953 The first column under the respective thermodynamic terms is the mean value for the three balloon ascensions reduced to gr. cal./cm. 2 min. This column is multiplied by m = 28.74, the adopted molecular weight of the atmosphere. There is a margin of inaccuracy in the lowest stratum, between the sea level and the height of the station Uccle, which is 100 meters, because it 210 A TREATISE ON THE SUN'S RADIATION is not exactly known how well the adiabatic assumption which has been adopted for this stratum actually holds true. In our final result we have taken m (Wi W ) = m (Ui Z7 ) = 2.82. When the gas is in contact with the surface of the earth it is probable that the thermodynamic terms undergo a small modifi- cation, which should be more carefully examined. An especial research will be required to determine the mean values of all these terms in the first 1000 meters above the surface of the stations, where they are to be applied. Resume of the Preceding Results When the results of the preceding investigations have been collected together, it appears that there are twelve lines of com- putation and observation, more or less independent of one an- other, which converge upon 3.98 gr. cal./cm. 2 min. as the equiva- lent of the effective intensity of the solar radiation on the vanish- ing plane of the earth's atmosphere. 1. Solar thermodynamics, Table 28, Table 73 = 3.90 2. Free heat, 2 - (Q 1 - Co), Table 75 = 4.08 p p 3. Hydrostatic pressure, 2 , Table 75 = 4.08 4. Inner energy and external work, 2 [ (Ui U Q ) (Wi - TFo)], Table 73 ,:. = 4.08 5. Black radiation, 2 A 7 , Table 75 = 3.94 6. Free heat in the stratum, - m (Qi - Co), Table 77. .= 3.92 p p 7. Hydrostatic pressure in stratum, m , Table 77 = 4.00 8. Inner energy in stratum, -m(Ui- J7 ), Table 77 . . = 3.92 9. External work and gas efficiency ? m (Wi Wo), + RIO (T a - To), Table 77 = 4.02 10. Kinetic + potential/energy + absorbed radiation (7 + II + 2 A Jo) = 3 - 90 11. Total radiation by Table 79, I + II + III + IV + V + VI =3.98 12. The data of the pyrheliometer, Table 83, Fig. 26. ... = 3.98 Adopted mean of the twelve processes = 3.98 RECONCILIATION OF DATA 211 The concurrence of so extensive a series of methods of dis- cussion in fixing the solar radiation at the flux-intensity of 3.98 gr. cal./cm. 2 min. at the distance of the earth justifies the rejection of Abbot's imperfect method of extrapolation by the Bou- guer formula that is, those results from the pyrheliometer which reach only one-half the solar constant reduced to the distance of the earth, and only one-third of the true solar constant at its strata of origin below the photosphere. 1. True solar intensity of radiation 5.85 calories 2. Effective solar intensity at the distance of the earth 3.98 " 3. Effective intensity by the bolometer (indicated) 3.98 " 4. Effective intensity by thermodynamics 3.98 " 5. Extrapolated intensity by the pyrheliometer 1.95 " 6. Intensity at the sea level by the pyrheliometer 1.50 " The Potential Energy of the Solar Radiation in the Sun's Atmosphere The primary fact regarding the electromagnetic solar radia- tion is that half of its energy is kinetic or magnetic, and half of its energy is potential or electric. This holds true in the aether, in the space between the atmosphere of the sun and the atmosphere of the earth, but it is not true in the gaseous media of either of these atmospheres. The reason that the kinetic and the potential energies are not equally divided in gaseous media, as they are in the pure aether, is that gases have three degrees of freedom, while the )- Cp a (T a - To). Since Cp a = Cvw + RW, this is equivalent to the addition of the term .Rio, whose values may be studied in the formula, (165) + R 10 (T a - To) = + ^^ + OFi - Wo) = PlO -(Ci-eo)-g(*i-*>). The region - R 1Q (T a - T Q ) is indicated as III on Fig. 26, in conformity with the conditions already explained. Compare Table 75. The Scattered and the Absorbed Radiation The incoming radiation suffers two other important deple- tions, (l) by scattering or reflection on the molecules, and dust contents of the atmosphere, whereby no influence is effected upon the temperature, the energy being lost to space; and (2) by absorption, which is generally a potential effect rather than a kinetic effect upon the translation of the molecules. The region of scattered radiation is numbered IV on Figs. 25, 26, and one should distinguish between the low-level scattering and the high-level scattering, the former probably pertaining to the wave lengths 0.35 /-i to 0.80 M, and the latter to the shorter wave lengths 0.00 n to 0.35 M- There are two regions of ab- sorption, the low-level marked V, and the high-level included in III, the latter above the isothermal levels wherein there is no absorption, and the former below the isothermal strata, extending through the convection region to the sea level. The foregoing analysis admits that, RECONCILIATION OF DATA 217 (1) The pyrheliometer measures the kinetic energy as de- pleted by the scattering and the absorption in the lower levels, that is, I, IV, V; but it denies that, (2) The pyrheliometer measures the potential energy of region II, the thermodynamic term R of region III, the high- level scattering of region VI, and the high-level absorption of region VI, as well as the ionization of the same region VI. The boundaries within VI are still a subject of research. The Pyrheliometer and Bolometer Observations We shall now collect together the accepted data of the observations derived from the pyrheliometer and the bolometer, in order to learn to what extent they conform to the foregoing analysis of the thermodynamic data. It will be remembered that there are two distinct methods of discussing the pyrheli- ometer observations heretofore employed: (1) The Langley-Abbot method, in which only the kinetic energy, as it is effective at the instrument, is accounted for; and, (2) The Bigelow method, which is competent to include the potential energy with the kinetic energy, at least to some extent. The former method of extrapolation results in a series of values that are inconsistent at the several stations when taken in sufficient detail. The latter harmonizes them in a form which is simple for intercomparisons. Neither of these pyrheliom- eter methods succeeds in registering all the terms which are demanded by the bolometer and by the thermodynamics of the earth's atmosphere. This is in the Meteorological Treatise. (3) A third method of reducing the pyrheliometer observa- tions to about 3.98 calories is illustrated in Table 83, and all our data have finally been recomputed by this method, since it comprises the terms I, II, III, IV, V, VI, and requires no extrapolation beyond sec z = 1, thus being complete on the level of observation. The general view is that while the electromagnetic radiation has the kinetic and the potential energies in equal parts, and the atmosphere has 1.00 part kinetic and 0.641 part potential, the complex silver disk of the pyrheliometer turns 1.00 part 218 A TREATISE ON THE SUN'S RADIATION kinetic energy into temperature, but conceals 1.00 part poten- tial energy within the atoms and molecules. Similarly, the bolometer thread of metal conceals 1.00 part potential energy, but gives relative ordinates of kinetic energy in correct ratio to the surviving spectrum lines. The Depletion of the Solar Radiation from 5.5$ Calories in the Isothermal Layers of the Sun to 1.50 Calories at the Sea Level of the Earth We are now in possession of the data which are necessary in order to describe with considerable clearness the course of depletion of the original solar black body radiation, or the solar constant, which amounts to about 5.85 gram calories per square centimeter per minute, down to about the 1.50 calories that are ordinarily received at the sea level of the earth's atmosphere. This assumes that the quantities of the radiation at the sev- eral levels in the sun's and the earth's atmospheres have been corrected for instrumental and observational defects, and all reduced to the scale of the earth's mean distance from the sun. It will not be necessary to recapitulate in this place the results of the research in the earth's atmosphere, as these may be found in the Treatise on Atmospheric Circulation and -Radiation 1915, and in Bulletins No. 3, 1912, No. 4, 1914, of the Argentine Meteorological Office. These are sufficiently summarized in Tables 75, 76, and 77, and Figures 23 and 24, and they will be briefly explained. In Table 79 the wave lengths in the spectrum are taken for the differences 0.05 /*, up to 2.50 v, and wherever the bolometer ordinates of the four central columns are lacking in Abbot's observations the necessary interpolations have been added to fill out the columns uniformly, so that the general sums may become homogeneous and comparable. The columns under 7655, 6950, 5810, 5450, are computed from the Wien- Planck Formula, using the constants as follows, Ci = 5.575 X 10~ 15 , c 2 = 1.4455: 5.575 X 10" 15 5.575 X 10 5 / 1.4455 \ / 6277.4 *(e kT - RECONCILIATION OF DATA 219 Since X = 10V, Ci (C. G. S.) is multiplied by 10 20 for /*, and c z (C. G. S.) is multiplied by 10 4 for /*. The column 7655 corresponds with the solar constant, 5.85 calories reduced to its equivalent at the distance of the TABLE 79 SUMMARY OF THE DATA OF THE SOLAR RADIATION, SHOWING THE DEPLETION FROM THE SOLAR CONSTANT AT 5.85 CALORIES TO 1.50 CALORIES AT THE SEA LEVEL Wave Length THERMODYNAMICS BOLOMETER PYRHELIOMETER 7655 6950 Total Mt. Whitney Mt. Wilson Wash- ington 5810 5450 0.00/u.... 05 0.000 0.300 0.775 1.380 2.998 6.490 9.194 10.490 10.594 9.992 9.052 8.000 6.969 6.078 5.186 4.512 3.838 3.348 2.858 2.505 2.152 1.897 1.641 1.455 1.268 1.131 0.993 0.890 0.786 0.709 0.631 0.571 0.511 0.464 0.417 0.381 0.344 0.315 0.286 0.263 0.240 0.222 0.203 0.188 0.172 0.160 0.148 0.138 0.127 0.118 0.109 0.000 0.100 0.375 0.700 1.154 3.018 4.854 6.054 6.544 6.499 6.127 5.594 5.004 4.442 3.879 3.419 2.958 2.604 2.250 1.987 1.724 1.529 1.333 1.187 1.041 0.932 0.823 0.740 0.657 0.594 0.530 0.480 0.431 0.393 0.355 0.325 0.294 0.270 0.245 0.226 0.207 0.191 0.175 0.162 0.148 0.158 0.127 0.119 0.110 0.103 0.095 0.440 2.700 4.350 6.050 6.060 5.630 5.050 4.350 3.650 3.160 2.670 2.470 2.260 1.960 1.660 1.460 1.260 1.130 1.030 0.940 0.900 0.780 0.710 0.670 0.620 0.570 0.530 0.470 0.430 0.380 0.350 0.320 0.290 0.260 0.240 0.230 0.210 0.200 0.180 0.160 0.140 0.120 0.100 0.070 0.040 1.800 3.410 4.930 5.450 5.180 4.720 4.100 3.490 3.050 2.600 2.410 2.220 1.930 1.630 1.440 1.240 1.120 1.020 0.930 0.880 0.770 0.700 0.660 0.610 0.560 0.510 0.460 0.420 0.370 0.340 0.310 0.280 0.260 0.230 0.220 0.200 0.190 0.170 0.150 0.130 0.110 0.090 0.060 0.030 1.660 3.150 4.840 5.200 4.930 4.500 3.970 3.440 3.000 2.570 2.390 2.200 1.910 1.620 1.420 1.220 1.110 1.010 0.920 0.870 0.750 0.690 0.650 0.600 0.560 0.510 0.450 0.410 0.360 0.330 0.300 0.270 0.250 0.230 0.210 0.190 0.180 0.160 0.140 0.120 0.100 0.080 0.050 0.030 2.400 3.870 4.270 4.170 3.840 3.450 3.060 2.680 2.310 2.160 2.010 1.760 1.500 1.380 1.200 1.100 1.000 1.000 0.910 0.820 0.740 0.680 0.620 0.570 0.530 0.480 0.440 0.400 0.350 0.320 0.290 0.260 0.240 0.220 0.180 0.170 0.150 0.130 0.110 0.090 0.070 0.040 0.020 0.000 0.020 0.050 0.090 0.150 0.590 1.319 1.883 2.348 2.609 2.685 2.631 2.495 2.305 2.114 1.917 1.720 1.547 1.374 1.234 1.094 0.984 0.873 0.786 0.698 0.630 0.562 0.510 0.457 0.415 0.373 0.341 0.308 0.232 0.256 0.236 0.215 0.198 0.180 0.166 0.152 0.141 0.130 0.121 0.111 0.104 0.096 0.090 0.083 0.078 0.072 0.000 0.010 0.020 0.040 0.067 0.307 0.721 1.179 1.559 1.809 1.931 1.947 1.893 1.778 1.662 1.526 1.389 1.261 1.133 1.024 0.915 0.827 0.739 0.668 0.598 0.542 0.486 0.442 0.397 0.362 0.327 0.299 0.271 0.249 0.226 0.208 0.190 0.176 0.161 0.149 0.137 0.127 0.117 0.109 0.101 0.094 0.087 0.081 0.075 0.070 0.065 0.10 15 0.20 25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 . 80 0.85 0.90 0.95 1.00 1.05 1.10 . 1.15 1 20 1.25 1 30 1.35 1 40 1.45 1 . 50 1 . 55 1 . 60 1.65 1 70 1.75 1 . 80 1.85 1.90 1.95 2.00 2 05 2.10 2 15 2.20 2 25 2.30 . 2.35 2.40 . . 2.45 2 . 50 Sums 123.474 83.266 67.250 61.380 59.550 51.990 39.773 30.551 220 A TREATISE ON THE SUN S RADIATION TABLE 79 Continued Factor to reduce to gr. cal./cm. 2 min. = 20.9 Reduced Thermo- dynamics. . . . 5.91 5.85 3.98 3.98 3.22 3.23 2.94 2.96 2.85 2.87 2.49 2.47 1.90 1.94 1.46 1.50 Log T Log T* Log ( R V - Log \D) T ' ' Log Jo Jo 3.88395 15.53580 -15.23180 0.76760 5.856 3.84198 15.36792 -15.23180 0.59970 3.978 - - - - 3.76418 15.05672 -15.23180 0.28852 1 943 3.73640 14.94560 -15.23186 0.17740 1 505 earth, and it is derived from thermodynamics; the column 6950 is the effective radiation on the outermost stratum of the earth's atmosphere, and it is derived also from thermody- namics, as indicated; the column under 5810 corresponds with Abbot's extrapolated value of the solar constant 1.94 calories (compare Abbot's " Sun," p. 298); the column 5450 is equiva- lent to about 1.50 calories, the full amount measured by the pyrheliometer on the sea level. The four columns marked Total, Mt. Whitney, Mt. Wilson, Washington, are derived from Abbot's bolometer ordinates, Vol. II, Annals Smithsonian Institution, or the Astrophysical Journal, October, 1911, the method of reducing the original arbitrary units to the probable calories being given in Bulletin No. 3, p. 83. The values at the several wave lengths have been made as homogeneous as practicable in Table 79 and the corresponding curves are plotted on Fig. 25, where their mutual relations can be examined. The sums of the several columns are taken, and as they are com- parable there should be a common factor to reduce the areas which are equivalent to the curves to the corresponding calories. Such a factor has been found to be 20.90. The factor used in Bui. No. 4, page 84, is 20.72, where the same table in part may be found. The lower portion of Table 79 contains the reductions in calories at the distance of the earth. Under " reduced " values the sums of the columns divided by the factor 20.9 are given; under " thermodynamics " are found the corresponding values, as deduced from the several independent discussions by thermodynamics, from bolometer observations, and from pyrheliometer reductions. It is readily seen that the RECONCILIATION OF DATA 221 Gr. cal./cm. 2 min cal.=S )lar Constant (Bigelow). 3.22 cal.=0utside atmosphere (Abbot) 1.94 cal.= Solar Constant (Abbot) 0.50 1.00 1.50 2.00 2.50/f FIG. 25. Summary of the Data Derived from Thermodynamics, Bolometers, and Pyrheliometers. 222 A TREATISE ON THE SUN S RADIATION data of Table 79 constitute a nearly homogeneous scheme, from the solar constant in the isothermal layers of the sun to the bottom of the earth's atmosphere at the sea level. The reduced values of Table 79 are to be compared with the computed values derived from thermodynamics and the observations with bolometers and pyrheliometers. The Line and Band Absorption and the Scattering In addition to the amount of the scattering and the absorp- tion already indicated, there is a small additional amount re- corded as depleted areas of a triangular shape within the smooth spectrum curves heretofore assumed. Scattering and absorp- tion modify the original effective solar spectrum, 3.98 calories on the earth's vanishing plane, by depressing many ordinates more or less, and by selective depletion at certain lines and bands. Mr. F. E. Fowle has studied these spectrum regions with the following results. The lines and bands, together with their absorbent, oxygen or aqueous vapor, are here given. THE LINE AND BAND ABSORBENTS Line n Absorbent Line M Absorbent B ? 0.69 Oxygen $ 1.13 Water Vapor a 72 Water Vapor i/'.. . 1.42 Water Vapor A 76 Oxygen J2 1.89 Water Vapor 0.81 Water Vapor OJi 2.01 1 93 Water Vapor ti>2 2.05 ? Astrophysical Journal, December, 1915. By referring to Fig. 25 the position of these lines can be located, and their relative ordinates for the stations, Mt. Whit- ney, Mt. Wilson, and Washington, can be calculated from Table 79, the factor of reduction to calories being 20.9. Mr. Fowle gives the percentages under certain conditions, and the corre- sponding calories referred to 1.930 calories for the temperature spectrum corresponding with 5800. It is believed that this ref- erence curve is unconfirmed in thermodynamics, and while adopt- ing Mr. Fowle's relative ordinates as percentages, we reduce them RECONCILIATION OF DATA 223 to the ordinates of Fig. 25, at least approximately, according to the bolometer ordinates of the station, Mt. Whitney 2.96 calories, Mt. Wilson 2.87, Washington 2.47, as given in Table 80. TABLE 80 MEAN DEPLETIONS DUE TO SCATTERING AND ABSORPTIONS IN PERCENTAGES AND IN CALORIES I. Reduced to Calories Observed by the Bolometer Station Precipi- table Water in Cm. DRY AIR AQUEOUS VAPOR Scatt. Absd. Scatt. Absd. Mount Whitney 4420 meters 0.00 0.11 0.25 0.50 7.3% 0.5% -% 0.5 0.5 1.0 -% 4.1 5.2 6.2 7.3 0.5 0.7 5.2 Mount Wilson 1720 meters 0.00 0.33 0.50 1.00 2.00 7.8 0.5 1.0 1.6 2.1 4.7 5.7 6.2 7.8 9.3 7.8 0.5 2.4 7.3 Washington Sea Level 0.00 0.50 1.80 2.40 9.3 0.5 4.1 13.5 19.7 6.2 7.8 8.3 9.3 0.5 12.4 7.4 Reduced to Calories gr. cal. DRY AIR AQUEOUS VAPOR TOTAL cm. 2 min. Scatt. Absd. Scatt. Absd. Scatt. Absd. Mt. Whitney Mt. Wilson Washington 2.96 2.87 2.47 0.216 0.224 0.230 0.015 0.014 0.012 0.021 0.069 0.306 0.154 0.209 0.182 0.237 0.169 0.293 0.223 0.536 0.194 II. Reduced to the Calories Observed by the Pyrheliometer Mt. Whitney Mt. Wilson 1.72 1.64 0.126 0.128 0.009 0.008 0.012 0.039 0.089 0.120 0.138 0.167 0.098 0.128 Washington 1.50 0.140 0.007 0.186 0.111 0.326 0.118 224 In Section I the percentages of the scattering and the ab- sorption, as measured from the bolometer ordinates, have been reduced to gr. cal./cm. 2 min.; in Section II the same percent- ages have been applied to the mean values of the radiation intensity observed by the pyrheliometer. These mean values give a general idea of the values to be expected, and they must be applied to the observations by the bolometer and the pyr- heliometer, respectively, in such a way that both series shall give the same value of the effective solar radiation, which should be alike in all strata from the sea level to the vanishing plane of the earth's atmosphere. We may now confirm the foregoing analysis of the thermo- dynamic data and the observed data of radiation as given, by comparing these with the data of the bolometer as contained in Table 79 and Figs. 25, 26. The several regions from I to VII have been marked on them respectively. I. Kinetic energy up to the curve 5450, equal to 1.50 calories. II. Potential energy between the curve 5450 and the bolometer curve for Washington, 2.47 1.50 = 0.97 calorie as compared with 0.95 calorie on Table 79. III. The specific heat term, (Cp Cv) = R, which is made up from the absorption of the short wave ordinates. This amounts on the sea level to 1.19 0.19 = 1.00 calorie. The summation of the short wave ordinates from 0.00 /* to 0.38 /z is about 20.231; if this is divided by the factor 20.9 the result is 0.98 calorie. The absorption of the short wave ordinates is necessary in order to change the specific heats of the atmosphere from that of the inner energy Cv, U u = Cv p T, into that of the gravitation energy U p , g (zi z ) = Cp (T a T ). IV. The reflected or scattered energy. V. The absorbed energy used in molecular and atomic forms. The detailed distribution of these energies in the lowest strata will require more study. On the Washington level U = H + / + s = 2.62 and on Table 79, 2.49; on the Mt. Wilson level U = 2.82 and on Table 79, 2.85; on the Mt. Whitney level U = 2.98 and on Table 79, 2.94. These points are indi- RECONCILIATION OF DATA 225 cated on Fig. 26. Compare the results from thermodynamics. VI. The transition region from the electromagnetic energy to the thermodynamic energy of ionization may provisionally 70000 60000 50000 40000 30000 20000 10000 FIG. 26. The Seven Components Which Make up the Intensity of the Solar Radiation at the Earth, 3.98 gr. cal./cm. 2 min. be assigned to the high-level absorptions along the slope of the ordinates from the wave length 0.30 /* to 0.50 /*. VII. The great depletion from the total solar intensity, 5.85 calories, to the effective intensity at the earth, 3.98 calories, making the difference 1.87 calories, is contained in the area between the 7655 and the 6950 black body curves. Direct Readings of the Pyrheliometers at Great Heights In a paper, " New Evidence on the Intensity of Solar Radia- tion Outside the Atmosphere," 1^15, Mr. Abbot summarizes the results of his direct readings of pyrheliometers on balloons 226 A TREATISE ON THE SUN'S RADIATION up to about 22000 meters. Taking these in connection with the pyrheliometer readings in the zenith at Mt. Whitney, Mt. Wilson, Washington, D. C., we have the following data: TABLE 81 SUMMARY OF THE DIRECT READINGS OF THE PYRHELIOMETERS WHEN REDUCED TO THE ZENITH, AND WITHOUT ANY EXTRAPOLATION TO THE SUPPOSED SOLAR CONSTANT. Location of the Pyrheliometer Height in Meters Zenith reading gr. cal./cm. 2 min. Abbot balloon ascension, July 11, 1914 22000 .84 Peppier balloon ascension, Oct. 19, 1913. . 7500 .76 Mt. Whitney, maximum 4420 72 Mt. Wilson, maximum 1780 64 (modified) Washington, D.C., or sea level . .58 (modified) The data derived by the self-registering pyrheliometer car- ried on balloons serve to fix new points along the graph that has heretofore been assigned to the pyrheliometer curves, as in Fig. 26, so that the main features of the problem have not been in the least modified by their addition. The problem is the recon- ciliation of the pyrheliometer and the bolometer observations without making the assumption that the sun does not radiate energy as a black body. The mean value of I at Mt. Wilson is 1.53, the maximum being 1.64, and the mean value at Wash- ington is 1.33, the maximum being about 1.58 calories. Summary of the Pyrheliometer Results as Reduced by Bigelow's Method The result of the discussion of the pyrheliometer observations at different stations, by Bigelow's method of reduction, is col- lected in Table 37, Bulletin No. 4, 0. M. A., of which the following is an extract. The height z is in meters, and the free heat received by the pyrheliometer is 7 , after the depletions, due to aqueous vapor, dust, and other matter near the station, have been eliminated. It may be noted that by the Bigelow method of reduction the pyrheliometer is made to record the kinetic energy of region RECONCILIATION OF DATA 227 TABLE 82 SUMMARY OF THE AMOUNTS OF THE SOLAR RADIATION AS DETERMINED BY PYRHELIOMETER OBSERVATIONS Station z la La Confianza 4483 2.142 gr. cal Mt. Whitney 4420 2.138 cm. 2 min. La Quiaca Humahuaca . 3465 2939 2.005 1.930 Maimara 2384 1.852 These are plotted on Fig Mt. Wilson 1780 1.771 26 as crosses x x x x x Tuiuv 1302 1.705 while the Abbot data ap- Bassour Mt. Weather 1160 526 1.675 1 601 pear on the graph which is marked " Pyrheliom- Cordoba 438 1.592 eter." Filar 340 1.572 Potsdam 89 1.538 Washington 34 1.529 Sea level o 1 525 I and in part the potential energy of region II. It is apparent that the course of the line of crosses on Fig. 26 is seeking the line (H + /) which bounds the potential region. It is to be hoped that suitable observations can be made in balloon as- censions, which will enable this idea to be verified. Abbot's observation is along the line H, while Bigelow's reduction has a very different vertical gradient from Abbot's, that is, along the dotted line. Computation of the Pyrheliometer Data, Leading to 3.98 gr. cal./ cm. 2 min. as the Intensity of the Solar Radiation at the Earth The results of our computation are contained in Table 83 and Fig. 26. The data of Tables 60-69, Bulletin No. 4, Oficina Meteorologica Argentina, have been adopted, except for the stations Cordoba and La Quiaca, where the individual obser- vations were computed throughout the years 1912, 1913, 1914, and 1915, and the general means of the entire series are placed in Table 83. This table, therefore, represents the average con- dition at each station after the local and periodic action has been eliminated. The method applied to the daily observations is equally valid. 228 A TREATISE ON THE SUN S RADIATION We have exhibited the following components, into which the solar radiation divides itself in the earth's atmosphere: 1. H = the kinetic energy of the translation of the mole- cules, measured by the pyrheliometer as temperature. It is evident that this instrument is a means of registering only kinetic energy, and that the various forms of the potential energy in the atmosphere do not come into its competency. The bol- ometer, on the other hand, by means of its relative ordinates, does indicate some portions of this potential energy in its results, though in a confused fashion. The thermodynamics has the power to classify clearly the distribution of the various forms of the potential energy in all the strata of the atmosphere. TABLE 83 COMPUTATION OF THE PYRHELIOMETER DATA TO INCLUDE THE POTENTIAL ENERGY, THE SPECIFIC HEAT, AND THE IONIZATION 2 * e k H J H+J 5 a .V, R 5 (a+R)' Name i "*"* W) gl of the . 1 0) *O & |i 3 "S la .0 2 . *J Station .fl ^ 3 ' M C3 ^2 gJO = the latitude of the station. e = the vapor pressure in millimeters. Unless this is reduced to a mean value for the station there will be an annual variation. The successive mean eo are 2, 2, 4, 5, 7, 8, 10, 5 mm. for these stations. The coefficient . 012 was determined as the equivalent in calories for 1 mm. of e. Hence A = 0.012 (e-e ). k = the coefficient of scattering, s = k (H + J). H = 7i, the kinetic energy. This is the value I\ of the zenith intensity of the radiation, observed by the pyrheliometer, and reduced by the Bouguer Formula / = /o p 8ecz . No extra- polation is made beyond sec 2 = 1. J = the potential energy, J = 0.641 H. H + J = U the total inner energy after its losses by scattering and absorption. (a +R) = the specific heat effect plus the absorption. R is computed from thermodynamic data as in Table 75, and elsewhere, and an approximate value has been assigned for each station. If the mean solar constant at the earth is 3.980 calories, then we have a = S-(H+J+s + Ae+R). The absorption in Table 83 has been obtained in this manner and it is in harmony with Fowle's results. RECONCILIATION OF DATA 229 2. / = the potential energy in the atmosphere, correspond- ing with the kinetic energy through the formula, / = 0.641 H. 3. s = the scattering, or reflection of the radiation on the air contents, without affecting the temperature as such on the levels below that of the reflection. If the coefficient is k as obtained from k = (1.00 p), where p is computed from the pyrheliometer observations, then s k (H + J) the part which is not transmitted. a = the true absorption, as shown by the band and line depletions of the ordinates of the several stations. It is com- puted indirectly from the adopted mean intensity of the solar radiation, S = 3.980 calories, by the formula, (167) a = S - (H + J + k (H + f) + A e + R). A e = the vapor pressure correction to a mean value of <>, with a coefficient 0.012, (168) Ae = 0.012 (e - e ). R the specific heat energy, (169) Ru (T a - To) = (Cp a - Cv 1Q ) (T a - To), which is necessary to change the inner energy at the specific heat of constant volume U v into the inner energy at the specific heat of constant pressure U P . The Boyle- Gay Lussac Law in the free gas leads to U v , where there is no gravitation, but to Up where there is gravitation, as by formulas, (l) to (10). Thus, we have, generally, (170) U v = Cv. P T= (Cp -R} P T (171) U p = Cp P T= (Cv + R) P T = -gpz It is U v which is depleted by scattering, absorption, and ionization, so that we have, (172) U v = H + J + s + a + &e + E (ionization.) To obtain the stratum-intensity of the radiation. (173) S=U V + R P T, in conformity with final equilibrium against the impressed force of gravitation. It should be noted that we have arrived at the solar radia- tion S, and by Table 83 find it to be 3.980 calories on each stratum. In this process there has been no extrapolation by the Bouguer 230 A TREATISE ON THE SUN'S RADIATION formula beyond its legitimate mathematical function in the zenith, so that all difficulties on that score are fully obviated. E = the ionization, in the great region marked VI, which lies between (H + 7) + (s + a) and U v . There are two points to be noted in respect of the ionization: (1) Since the potential energy of the electromagnetic radiation is 1.00 on a scale of 2.00, and the atmospheric potential energy is 0.641, it follows that there is a change of 0.359 on the same scale. Hence, we have, E = 0.359 X^ X 3.980 = 0.714 - gr ' a ' the amount of the solar radiation which is probably transformed into electric energy of the free ions in the upper levels, as pro- duced by the action of the short waves upon the atmospheric atoms and molecules. (2) In Fig. 26, the transition between the electromagnetic potential energy and the thermodynamic potential energy is represented as abrupt or discontinuous. This is probably not the case in nature, and it is likely that a gradual change occurs in the coronal region, between 90000 and 50000 meters, while the process is intensified along the upper branch of the isothermal layer, 50000 to 37000 meters, where the temperature* is rapidly changing its value. // the production of ions is to be associated with rapid changes of temperature in atmospheres, we have five conspicuous regions where ionization occurs: 1. At the bottom of the solar isothermal layer. 2. At the top " " 3. At the top " the terrestrial isothermal layer. 4. At the bottom " " 5. At the bottom of the atmosphere and in the earth. These are the regions where electric charges are known to accumulate and to manifest themselves, as solar static electric fields, as cur- rents -of electricity inducing variations in the normal magnetic field, as auroras in the high level electricity in the earth's at- mosphere, thunderstorm energy in the convectional region, and at the surface of the earth itself. RECONCILIATION OF DATA 231 Synchronism of the Solar and the Terrestrial Variations during the Interval 1900-1915 in Argentina It is of interest to extend the computation of the intensity of the radiation to the annual values, in order to compare the variations with the other series of annual variations derived from various solar and terrestrial sets of data. This can be done immediately for the years 1903-1916, and the record can be carried back to 1883 by means of the observations at Mont- pellier, Geneva, and Warsaw. Table 84 contains the summary of the annual values. It may be recalled that the computa- tion here employed is the third attempt to make the pyrheli- ometer data conform to the requirements of solar and terrestrial thermodynamics, which sustain the bolometer data in requir- ing about 4.00 calories for the effective solar radiation at the earth, in place of the 2.00 calories derived from the pyrheli- ometer as interpreted by the Langley-Abbot method of dis- cussing the Bouguer formula of depletion. The first method introduced a functional correction to 7 due to the vapor pressure effect, (174) / = (I c - 0.0214 e) ^ sec2 , (I = I c - 0.0214 e). The second method was an attempt to reconcile the very dis- cordant results derived from stations at different altitudes, and consisted in making p w a function of the vapor pressure, (175) 7 = / (#.-/)"- where each station has an equation for/ (e), Bulletin No. 4, 0. M. A., Chapter 5; Treatise, Chapter VII. Both of these are methods of extrapolation, but the third method, described above, com- pletely rejects every attempt to extrapolate beyond sec z = 1, and it succeeds in reducing each station on its. own level to the solar intensity independently of other stations, and without making any use of certain bolometer factors, which are excessively complex, and impractical of attainment under ordinary conditions of pyr- heliometer observations. In transferring these results to Fig. 27, the value 3.914 for Mt. Whitney, 1910, seems to be rather too small, due to the 232 A TREATISE ON THE SUN'S RADIATION TABLE 84 MEAN ANNUAL VALUES AS COM 001>CO l> 1 1 1 1 O5OO CO ^^ CO CO CO O5 O5 CO CO i CO CO CO CO CO CO CO CO j li flj ^3 S-S &rC ri "S -M 3 T3 T3 C ^'2 M IH tj rt 0) CO 'Si ^ S i cti I" 8 5 rt 1-s fi- ll w en gr tx '5 QJ ^ 4-J 3 rt cx 2 i 3 fe rt _T! .2 S g s 15 II fl ! RECONCILIATION OF DATA 233 limited number of the observations, but it is retained; the value for Mt. Weather, 1912, 3.790, seems to have been unduly depressed by the dust from the Volcano Katmai. There are local conditions of this sort which tend to distort the annual means, and as they cannot be eliminated from the normal in any simple manner, it becomes necessary to utilize a larger number of stations for pyrheliometer records in various parts of the world. These should be made and reduced on an ac- cepted uniform plan, which conforms to the requirements of thermodynamics as well as of homogeneous statistics. The annual values in Table 84 produce a curve which is similar in its general features to the one in Table 93 of the Treatise, though it here presents an additional minor maximum for the year 1911. We shall next proceed to compare these results with other easily accessible data, as given in Table 85 and Fig. 27. Similar tables of comparison have been prepared for Argentina, from 1870 to 1916, and they present quite the same general facts of synchronism throughout this interval. It is necessary to receive prompt reports of the frequencies of the sun-spots and the prominences in order to utilize them in any forecast service for the use of the public. There are several facts which appear quite distinctly. (1) The sun-spot frequencies, the prominence numbers, the solar radiation, the amplitudes of the horizontal magnetic force, the precipitation, temperature, vapor pressure, and barometric pressure all agree in conforming to the long period of 11 years. (2) They all conform with certain local irregularities in respect of maxima and minima in the subordinate 3.75-year period. (3) The sun-spots, prominences, magnetic field, and baro- metric pressure all move together in a direct synchronism, while the radiation, the precipitation, the temperature in part, and the vapor pressure move in an inverse synchronism. (4) Taking the entire set of curves as attempting to express a fundamental law of solar activity, it is thought that a minor crest occurs at the maximum of the 11 -year period, that a 234 A TREATISE ON THE SUN'S RADIATION TABLE 85 SUMMARY OF THE SOLAR AND TERRESTRIAL OBSERVATIONS SHOWING A SYNCHRONISM IN THE VARIATIONS IN THE H-YEAR AND THE 3.75-YEAR PERIODS Year of Observation Sun- spots Promi- nences Radia- tion Hori- zontal Mag. Ampl. Precipi- tation Temper- ature Vapor Pressure Baro- metric Pressure 1900.. 114 128 1589 855 17 81 10 50 749 28 1901 33 88 1374 573 17 87 9 88 49 43 1902 60 47 1464 716 17 68 10 35 48 59 1903 293 118 3 989 1926 775 17 49 10 52 49 22 1904 1905 503 762 291 302 3.979 3 955 1787 2194 914 719 17.17 16 75 10.30 10 22 49.12 49 24 1906 646 261 3 975 2015 613 17 55 10 07 49 33 1907 1908 745 583 427 346 3.971 3 940 2299 2584 648 615 16.83 17 11 9.86 9 55 49.52 49 97 1909 527 365 3 960 2334 566 17 08 9 34 49 90 1910 223 259 3 952 2354 651 16 96 9 31 49 59 1911.. 68 163 3 960 1923 751 16 40 9 26 49 75 1912 43 126 3 978 1520 811 17 51 10 09 49 72 1913 17 135 4 003 1564 704 18 19 10 53 49 44 1914 . 115 198 4 002 1672 928- 17 28 10 78 48 77 1915 350 250 3 990 2635 773 17 32 10 06 48 92 1916 540 400 3.920 2676 486 17.65 8.87 49.00 Sun-spots. Data from Wolfer's Frequency Numbers, Meteorologische Zeitschrift, May, 1902, May, 1915. Prominences. Data from Ricco's Distribuzione delle protuberanze, Mem. della Soc. degli Spettro. Italiani, March, 1914. Radiation. Bigelow's reduction of the pyrheliometer data. Horizontal Magnetic Amplitudes. Bigelow's compilation from European observatories, and Pilar, Argentina. Precipitation, Temperature, Vapor Pressure, and Barometric Pressure, from the Argentina stations, Tucuman, Andalgala, Goya, Concordia, Cordoba, Pilar, Buenos Aires, Victorica, Bahia Blanca, Patagones. second crest occurs near the top of the ascending branch, that a third crest occurs near the top of the descending branch, and a fourth crest near the bottom of the descending branch. Some- times one or the other of these minor crests fails in the record, or it is distorted by the local conditions, or the sun itself does not always x work upon the normal plan, so that the result is complex. Each Argentine station clearly resembles this model under the influence of minor distortions, and they support this proposition. (5) In the case of the temperature, it is debatable whether the minor crests are not inverse on the long 11 -year period, so RECONCILIATION OF DATA 235 1900 1905 1910 1915 1920 FIG. 27. Synchronism of the Solar and Terrestrial Variations. 236 A TREATISE ON THE SUN.'s RADIATION as to become practically direct. This must be due to the in- fluence of precipitation and clouds, making the temperature at the surface beneath them opposite to similar temperature data above the cloud sheet, if it could be systematically recorded. (6) If the rule here proposed conforms to natural solar and terrestrial variations, it follows that one can readily construct a new solar curve in advance of the current year, giving it the mean 11-year and the mean 3.75-year periods, so that a long- range forecast of general annual conditions becomes practical. In this way the 11-year maximum is in 1918 and the four minor maxima occur in 1916, 1918-19, 1920-21, and 1923. These should be the years of minimum rainfall in Argentina, relative temperature maxima, low vapor pressure and high barometric pressure. // sufficient observations are maintained to keep in touch with any eccentric behavior on the part of the sun, they will enable us to modify this general statement from year to year, and it is generally true that a forecast for a year or two is without doubt possible. Local peculiarities will require a special study. The effect of the ocean upon coast cities, and the effect of mountain ranges upon cities located near them, make such stations less valuable, as they tend to disturb the smoothness of the result. This is easily seen at Buenos Aires and Bahia Blanca on the one hand, and at Mendoza and San Juan, near the Cordilleras, on the other hand. The inland stations are less liable to dis- turbances, and these respond to the solar impulses with greater regularity. It is noted that the long period comes out much better in Patagonia than in northern Argentina, in the case of the barometric pressure; while the latitude of Buenos Aires is more favorable for the periods of the vapor pressure. It is believed that with suitable experience such a system of long- range forecasts is easily available for Argentina and many other countries. Short- and Long-Range Forecasts The practical application of solar physics in meteorology consists in developing a system of interpretation of solar phe- nomena, such that the variations of the solar activity may be- RECONCILIATION OF DATA 237 come intelligible in terms of climatic effects. The complexity of the physical mechanism is so great that the progress toward the solution is slow, although there is no reasonable doubt that an effective synchronism exists. The efforts to justify this procedure have been entirely statistical, whereby a succession of maxima and minima has been determined in a general way, coordinating the two parts of the process. Many irregularities are encountered, and in a country of violent cyclonic aption, like the United States, the symptoms are much confused^ in Europe they are nearly extinct, while in Argentina they are comparatively simple and distinct. Criticism is so mixed up with an incomplete knowledge of the science that the growth of the research is unduly retarded; the enormous accumulation of the statistical data makes its treatment very oppressive; the data are not sufficiently homogeneous to bring out clearly trie small true solar residuals; the observations are made in the interests of short-range forecasts to the entire neglect of the long-range forecasts of solar physics; the several branches of the subject are so scattered as to be unmanageable, the solar physics, the bolometric spectra, the magnetic field, the electric ionization, the climatic meteorology, all being separated among different administrative offices ; the published reports from these different sources are so retarded that they are useless in any study of current forecasts whatsoever; there exists no systematic machinery for handling this kind of world-wide material. It is inevitable that several international institutes shall be established at least four, one in the United States, one in Ar- gentina, one in Europe or Asia, and one in Africa or Australia which shall cooperate upon a fixed general plan, utilize selected data, and treat the subject from the world-wide point of view. The long-range system will begin with annual forecasts, already easy in Argentina, and advance to details in accordance with the progress in solar physics. Such international institutes must be equipped with the highest type of apparatus, continu- ally improved as conditions permit. The progress in solar physics has been retarded, as far as concerns meteorology, by the insistence upon immediate fore- 238 A TREATISE ON THE SUN'S RADIATION cast results, before the fundamental laws have been classified. This demand has already made it unnecessarily difficult to obtain the facilities requisite to work gut the problems. It should be remembered how small is the margin of success in short-range forecasts, with all the extensive coordination that the national services actually possess. In the United States the annual amount of fair weather averages about seventy-five per cent and the rain twenty-five per cent. Of this twenty- five per cent of rain, for predictions at fixed 12-hour intervals in advance, 12 to 24, 24 to 36, 36 to 48 hours, the maximum suc- cess is about 33 per cent, making 83% the average actual fore- cast; similarly with the temperature. All the percentages above that amount, that is, a gain of only eight per cent, are due to elastic rules of verification. These are changed from time to time, but, in fact, there has been no real improvement in the past thirty years. In Argentina the fair weather percentage is higher, and the corresponding average, also, higher. The real utility of weather services is in the general clima- tological records of rainfall and temperature, and in an occa- sional important forecast of storm conditions, frost conditions, and, especially, in the educational propaganda. The scientific development has been entirely sporadic, depending on some in- dividual initiative. As matters stand, quite aside from the idea of international cooperation by central institutes, there is still a very wide margin for gain in the theory and practice of long-range forecasts. (1) Up to the present time meteorology has had no central unifying analysis. The investigations have been based upon the adiabatic formulas, which are wholly in- applicable in the non-adiabatic atmospheres of the earth and sun. It cannot be doubted that a sound central theory will have the same good influence upon atmospheric thermodynamics, that Newton's gravitation and Kepler's laws of motion had upon the primitive astronomy. (2) In the matter of solar radia- tion the amount available is not 1.94 calories, as has been claimed, but 5.85 calories, a margin of 200% to the good for future developments. (3) The method of distributing meteorology, radiation, magnetism, and solar physics, among distinct and RECONCILIATION OF DATA 239 independent institutions, is on the same level of efficiency as an army would be to repel invasion with the infantry in New York, the artillery in New Orleans, the cavalry in San Francisco, and the commissariat in Chicago. (4) The value of long- range forecasts is very great economically, and the practical ability to gain 8 or 10% on fair weather forecasts is not so forbidding, as may be supposed. Until the possibilities of science have been exhausted, there is not the least ground for withholding men and money from such an enterprise. Unless the government services approach this subject in an efficient way, there can be no doubt that the demands of commerce and agriculture will seek for other channels of development, which will extend the range of the forecasts far beyond their present limits. It should be noted that the small range of the solar and terrestrial synchronisms of the radiation, and its various manifestations, culminates in the case of the precipitation in Argentina by registering about 40% of the total as an annual variation. This is enough to distinguish good and bad agri- cultural years, and fully justifies all the cost of the research. When this is perfected it will be of prime value to the national interests. In the United States the registration approaches 30% of the total in some localities. CHAPTER VII Other Solar Phenomena Restatement of the General Line of Argument BEFORE entering upon a brief discussion of the other solar phenomena, following from the results of the thermodynamic v relations which have been developed on the sun, it is proper to restate the general line of argument that has been explained regarding the solar radiation. It has been shown that all gases of atmospheres open to space on one side, but resting upon a solid globe like the earth, or a viscous core like the sun, possess the same thermodynamic configuration for the elements of the Boyle- Gay Lussac law, P = p R T. These are illustrated in Figures 2-13. Near the bottom of the iso- thermal layer, where the temperature gradient suddenly changes from the adiabatic rate, there is a conversion of the chemical elements from their electron-ion-atomic state into their elec- tron-ion-molecular configuration. During this process there occurs a powerful readjustment of the electric charges, accom- panied by numerous rapid changes in the velocities of the elec- trons, the electromagnetic oscillations within the gas, and release of plane electromagnetic waves into space. The thermo- dynamic elements fix the temperature of the radiation for all elements at about 7655, and the evidence is clear that such radiation is primarily of black energy, including all the wave lengths in the spectrum. This energy passes through deep layers of superincumbent gas, and the electromagnetic energy suffers its first depletion in passing over from electric oscilla- tions into thermodynamic collisions and free paths, whereby the equivalent value in gram calories per cm. 2 per minute at the distance of the earth changes from 5.85 calories to 3.98 calories. The latter amount is determined as that which is 240 OTHER SOLAR PHENOMENA 241 received at the earth, since the computation of it within the solar gases involves many physical terms and coefficients that are still difficult of assignment. The loss of brightness between the center and the limb conforms to this 1.87 calories. The determination of the terrestrial intensity, 3.98 calories, depends upon three different lines of research. I. The observed ordinates of the thermal spectrum by means of the bolometer are best satisfied by several different effective temperatures, such that the wave lengths 1.50 ju- 2.50 /z originate at about 7655, while the wave lengths of 0.00 ju to 1.50 M belong to a complex spectrum energy of 6950 with certain depletions. II. Such a radiant energy in the space through which the earth passes in its orbit is competent to generate certain tem- peratures in the earth's atmosphere, in equilibrium with the kinetic energy, potential energy, inner energy at the specific heat of constant volume, and the inner energy at the specific heat of constant pressure as required by the force of gravita- tion. Such conditions of equilibria are built up with the lapse of time; they are independent of the direction of the radiant ray, whether reflected at the earth's surface or direct from space; and they are accompanied by such pres- sures, densities, thermal coefficients as are determined by ob- servations between the sea level and the vanishing plane of the atmosphere. The several lines of thermodynamic computations all converge upon the value 3.98 calories for the effective in- tensity of the solar radiation at the distance of the earth, thus agreeing with the bolometer ordinates between 0.00 M and IJJO'ji. III. The pyrheliometer measures the kinetic energy of radia- tion in temperature-degrees at the distance of the sun from the zenith, and by the Bouguer formula its value can be found in the zenith, together with the coefficient of absorption near the station. In the computations the corresponding air-poten tial energy, the corrections for the specific heat, the depletion by scattering and by absorption, can be supplied, and these again lead to 3.98 calories at stations in the northern and 242 A TREATISE ON THE UN's RADIATION the southern Americas. The Langley-Abbot method of reduc- ing the pyrheliometer observations involves a number of assump- tions and omissions which invalidate it. 1. Extrapolation to the non-mathematical sec 2 = 0. 2. Assumption that sec z 1 necessarily is identical with the depth of the earth's atmosphere. 3. Assumption that the line and band depletions used in making up the bolometer factor should be referred to the spec- trum of 6000, instead of 6950, which is purely arbitrary. 4. Omission of the potential energy term. 5. Omission of the specific heat term. 6. Omission of the ionization energy term. Removing these defects the pyrheliometer data become accordant with the bolometer data, and the thermodynamics of the atmospheres of the earth and the sun. The terrestrial meteorological data are much better satisfied practicably by reference to a radiation energy of about 4.00 calories than they are by that of 2.00 calories. In fact, it is impracticable to build up the prevailing temperatures, pressures, and densities, together with their climatological changes, by employing Abbot's solar intensity of 1.930 calories. We shall now proceed to make further use of the solar thermo- dynamic data in reference to the origin of the sun-spots, faculae, prominences, sharp disk of the sun, apparent diminution of brightness between the center and the edge of the sun, the corona, the solar magnetic field, the solar electrostatic field, the general circulation in the sun's mass. Each of these sub- jects requires a prolonged research for developing its details, so that only a short descriptive statement will be practicable in this chapter. The Effect of the Solar Isothermal Shell upon the Visible Surface Phenomena The most prominent fact regarding the physical constitu- tion of the sun is that there exists an isothermal shell of the mean temperature of about 7680 located in the midst of the solar gases at a fixed distance from the center of the mass. OTHER SOLAR PHENOMENA 243 This shell is maintained by the thermodynamic processes in this position by the interrelated actions of gravitation and radiation, as has been explained. The outer surface is called the photosphere, and it appears as a sharp circle on the disk, because the heavy gases vanish near those levels, and on ac- count of their large number they produce the observed optical brilliancy of the sun. The lighter gases extend upward through the levels of the flash spectrum, the chromosphere, and the inner corona, the latter being most conspicuous in hydrogen. Beneath the isothermal layer the gases recede at adiabatic rates to great depths, which finally coalesce in the quasi-solid nucleus. Compare Section I, Fig. 28. In the midst of the isothermal layer the gases go through their thermodynamic development in strata having a special depth for each gas, and yet they are entirely independent of one another. The isother- mal layer is a complex of numerous gases, each one agreeing with all the others as to temperature, while differing as to the vertical scale of their strata of equal temperatures. We may, therefore, having recognized the very complicated structure of the isothermal shell, proceed with further explanations, sub- stituting the simple idea of a single shell of uniform tempera- ture, at the mean distance R = 694800800 meters from the center of the sun. The sun-spots, faculae, and spectra, all testify to the complex internal structure of the solar gases in the neighborhood of the photosphere, and some progress has been made heretofore in classifying these phenomena, though this was much impeded by the lack of a definite knowledge of the pressures, densities, and gas efficiencies which are asso- ciated with them. The Granulations, Facula, Flocculi, and Prominences The literature regarding the appearance and origin of these solar phenomena is very extensive, and it consists of numerous explanations of the probable causes which produced them, in accordance with some observed physical conditions on the sun. We shall have much advantage in this respect from an accurate 244 A TREATISE ON THE SUN^S RADIATION knowledge of the temperatures, pressures, densities, and gas efficiencies that prevail, and it is not too much to say that approximate computations can easily be made, such as would reproduce these appearances in the neighborhood of the photo- FIG. 28. I. Formation of the Solar Photosphere. II. Formation of the Granulation, Prominences, and Sun-spots. III. Elements of the General Circulation within the Sun. IV. Refraction in the Solar Atmosphere. sphere. I shall avail myself freely of the results of the Mt. Wilson Observatory, and of the numerous papers from other authors in the Astrophysical Journal, in stating the facts per- taining to the several topics. The granulations, faculae, flocculi, and prominences are OTHER SOLAR PHENOMENA 245 correlative phenomena, all depending upon thermodynamic processes in several levels. The granulation is a convective result on the top of the photosphere; the faculae lie in the midst of or just above the photosphere, while the flocculi are in the higher levels of the convective columns; and the promi- nences may extend by eruption far beyond the normal levels of the hydrogen atmosphere. The forces of segregation de- pend upon the intrusion of the temperatures, and all the ele- ments associated with them, into levels which are inconsistent with the equilibria of gravitation. If there is an equilibrium, then the equation of condition holds. (176) G fe ~ So) = - Pl ~ P - * ( > \\ | / /^fr^LSTr** Cold ^s FIG. 29. General Scheme of the Structure of a Sun-spot Vortex, and Its Relations to the Reversing Layer. (Bigelow.) of explosive outbursts of imprisoned gases retained within a quasi-liquid restraining envelope. Secchi, Lockyer, Oppolzer, and others, consider the sun-spots the center of downpouring gases, or meteoric material, bringing cooler temperatures from the higher strata. - Young sees in the spot an area of sinking of the photosphere and elevation of a ring surrounding it. Fox, and others, observe that spot-birth is always preceded by an eruption of faculae in a ring outside of the penumbra, most pronounced on the following side. Schmidt applies circular refraction to make the spot a mirage effect, while the phenomenon really occurs at considerable depths below the photosphere. Julius applies the optical effects of anomalous dispersion and complex refraction to the entire series of spot and attendant phenomena. Schaeberle finds mechanical forces enough to account for the conditions. OTHER SOLAR PHENOMENA 253 Brester applies chemical luminescence and moving flashes, among other phenomena of that kind. Sidgreaves utilizes the chill of expansion to account for the low temperature in the central areas of spots. Hale, Adams, and others, find that the spectra indicate an almost universal weakening of the enhanced lines in sun-spots, and consider this as evidence of the low temperature in the reversing layer over the spots themselves. Abbot determines lower temperatures in the spots than in the surrounding photosphere by bolometer observations. The observers of the Mt. Wilson Observatory have measured in the displacement of the spectrum lines that there are out- ward velocities in the lower part of the sun-spot region and in- ward velocities in the upper part. There are evidences of a slow spiral inward movement from the photosphere at consid- erable distances into the penumbra, and thence downward to the umbra. Langley pointed out that the solar surface is striated in vertical columns, like sheaves of wheat, that the quiescent tops of the segregations appear as granules, and that when bent over they form the filaments in the penumbra. Faye tried to account for a solar vortex as a whirlpool in a stream having relatively different eastward velocities. Bigelow wrote as follows in respect of the connection between solar vortices and the dumb-bell-shaped vortex, Monthly Weather Review, October, 1908: "The sun-spots occur on the outer surface of the photo- sphere and extend inward toward the center of the sun. They consist visibly of a nucleus which is practically structureless, and a penumbra which is striated radially with much regularity. The observed movements of the material composing the penum- bra are from the outer edge of the disturbed area in the photo- sphere toward the umbra, and the radial striae usually termi- nate in ends which are bent downward toward the interior of the sun. The motion of a particle starting on the outer edge of the penumbra is primarily inward, and then rather suddenly downward. This corresponds so closely to the motion in the 254 A TREATISE ON THE SUN'S RADIATION upper levels of a dumb-bell-shaped vortex where the circulation is downward that it seems proper to suggest this explanation of the origin and structure of the sun-spots. The sun-spots would correspond to the layers between the sections a z = 180 and az = 170 (M. W. R., Oct. 1907, p. 475, fig. 3), if the cir- culation is downward. In this limited region there is practically little rotary velocity v, the vertical velo'city w becomes important only when approaching the abrupt curvature, which is here assumed to be on the outer edge of the umbra, but in the pe- numbra the radial velocity is conspicuous. The sun-spot may be caused by layers of matter inside the sun's photosphere oper- ating to draw material downward, warm layers being super- posed upon cold layers at the section which corresponds with the lower plane of the sun-spot vortex." Comparable ideas are quoted by C. E. St. John, from Moore's " Meteorology," in his paper on the radial motion in sun-spots, Astrophysical Journal, Vol. XXXVII, 1913. This paper pre- sents certain values of the outward and the inward radial veloci- ties u in different levels, as derived from the displacement spec- trum lines. Generally, the sun-spot is now considered to be a complex vortex, and we shall proceed to utilize our solar data in order to examine more closely its probable origin, and struc- ture. It will be seen that my views of 1908 have been modified in details in consequence of this new data. It should be remembered, as explained in my "Meteorological Treatise," that Nature abhors a discontinuity in the curves of pressure and density, and that temporary abnormalities in these curves, caused by imperfect thermodynamic adjustments, are compensated by horizontal currents in general. Since a hori- zontal current transfers its energy into a vertical force, by the general equations of motion it follows that these currents occur at the boundary of cool and warm strata, whenever the tem- perature gradient becomes abrupt and abnormal. In the solar atmosphere a small convectional lift of the adiabatic strata, at the lower boundary of the isothermal stratum, produces at once a horizontal outflow from the center or point of greatest dis- continuity in T. The lower planes of the several isothermal OTHER SOLAR PHENOMENA 255 strata are therefore the planes of vortical outflow, so long as the internal vertical convection from the interior of the sun persists. Since all gases must pass through the transformation from adiabatic to isothermal temperatures, that is from large negative gradients to small positive gradients, it is evident that the mass of gas containing (Qi Q Q ) can not dispossess itself of this heat content with sufficient rapidity to conform to the change in temperature gradient without other special processes. One of these has been described, namely, the generation of the solar radiation, and the other is the production of the sun-spot vortices. Hence, concentrations of radiation and of sun-spot phenomena happen wherever the internal thermodynamics within the enormous mass of the sun, operating by its own laws, may determine its great currents of general circulation. These concentrations may occur in latitude, as on the maxima of the sun-spot belts, or in longitude, as indicated by the suc- cessive maxima and minima along the equator, as heretofore found to be the case by several lines of evidence. The general result is to produce a series of spherical harmonics, and there is little doubt that their characteristics can be worked out from our materials. Since the same sun-spot endures for months on occasions, it is necessary to determine not only what are the forces which originate them, but what physical process sustains them during such long intervals of time. This, in connection with the minor features enumerated above, is the object of our research. The discovery of the permanent existence of a spherical isothermal shell around the sun, of persistent thermodynamic origin, greatly facilitates our effort. It is seen that all of the gases which by vertical convection, either from below upward or from above downward, attempt to pass through this layer, are necessarily reduced to this isothermal temperature, between 7650 and 7700, whatever may be their temperature on approaching this shell. For brevity, we shall continue to speak of the parameter T as implying the long train of thermodynamic processes that have already been described, including associa- tion and dissociation, radiation and absorption, dynamic move- 256 A TREATISE ON THE SUN'S RADIATION ments and thermodynamic transformations. It is especially to be remarked that in the adiabatic strata, below the isothermal layer, a short vertical movement of a mass of gas by convec- tion changes its level of temperature through many degrees, and, similarly, after passing the isothermal layer, there is a fur- ther great change of temperature with the height. This is en- tirely true of the heavy gases, and progressively less true of the light gases. TABLE 87 VERTICAL DISTANCE REQUIRED FOR A CHANGE OF 1000 Element Above the Isothermal Layer Below the Isothermal Layer Hydrogen, HI 1 Hydrogen, H 2 2 Helium, He 4 Carbon, C 12 6630 (' 4200 1700 620 145 85 56 30 ? Hi 4360 (, 1940 980 340 82 68 28 20 ft 3 CO 2100 K 1450 500 180 53 33 21 11 m. (4) 1 to s ll Calcium, Ca 40 Zinc, Zn 65 Cadmium, Cd 112. Mercury, Hg 198 Table 87 collects these vertical temperature gradients into three classes, as from to 4000 marked (2), 5000 to 7000 marked (3) above the isothermal layer, and 8000 to 12000 marked (4) below it in the adiabatic region. The change of gradient is progressive so that the table is merely illustrative. . 6330 3600 2150 , , , The ratios , , , show how to proceed from one m m m element to another through the atomic weights. Evidently short vertical changes for the heavy gases are matched by long vertical changes for the light gases. In order to understand the significance of a vertical convec- tive movement, we require the general equation of condition, including the kinetic energy of circulation, G (Zi - So) = - Pio - -A) - (Qi - Co). On a level of equilibrium without circulation, the term i (q\ q z o) disappears, but if by convection the mass containing OTHER SOLAR PHENOMENA 257 (Qi Qo) free heat is moved to a higher level, the equilibrium is destroyed temporarily, and in the process of adjustment the term J (q\ q z o) is set in operation, and it will be maintained as long as continuous vertical convection renews the (Qi Q Q ) Sit the abnormal level. Computing the available velocity, we have for different heights that is, for different discontinuities of the thermodynamic levels: TABLE 88 HORIZONTAL VELOCITY FOR VERTICAL CONVECTION TO DIFFERENT HEIGHTS Change in Height H 2 He t c Ca Zn Cd Hg Means 50 Kilometers 5 " 1484 469 1581 500 1703 539 1673 529 1483 469 1658 524 1597 505 1597 505 1 " 210 223 241 237 210 234 226 226 If the mass containing the free heat, as computed at the top of the adiabatic levels, or in the lower part of the isothermal stratum, is raised 50 kilometers*, it can be compensated on the average by a horizontal velocity of 1597 meters per second; for 5 kilometers by 505, and for 1 kilometer by 226 meters per second by the formula, (177) Kinetic Energy = - \ (q\ - to the true solar vortex. Professor Hale's preliminary descrip- tion of the formation of solar vortices is in harmony with the above account, though it lacks the fundamental isothermal action. The spectrum shows a series of line and band changes which can readily be understood by applying these principles. Indeed, a very complete solution of the distribution of P. p . R. T. at all levels can be drawn out from the Formulas of my Treatise, pages 172, 174, 199-202. We shall adopt these formulas, and compute a few vortex tubes, in order to bring out the dimen- sions of the vortices on the sun and their attendant phenomena. Very small spots have a radius of the umbra varying from 400 to 800 kilometers, while in very large spots the umbra has a radius of 40000 to 50000 kilometers, the radius of the pe- OTHER SOLAR PHENOMENA 259 numbra being 150000 kilometers. For a common spot the out- side radii may be taken, for the umbra, 10000 kilometers, " " penumbra, 25000 " " faculae, 500000 We shall compute tubes between 60474 kilometers (l) and 956 kilometers (6), on a level 500 kilometers below the reference plane, and extending downward to 50000 kilometers. Within these adopted data nearly all sun-spots will be readily contained. Th.e horizontal velocities on the 500 kilometer plane will range from 0.212 to 13.39 kilometers per second. To find log p, we take, i (log 60474 - log 956.25)= \ (4.78157-2.98057)= + 0.36020. By a little trial computing, it results that the current function $ = v w = 6 400 000 (6.80618), and the tube constant, C = = 0.0000035 ( 6.54407). Then the successive values of w, C, u, v, w, i, 17, are computed on the. 500 kilometer level, by the for- mulas indicated. They are next extended down to the 50000 kilometer level. An examination of the v, or tangential velo- cities, makes it probable that the actual vortex does not extend much beyond the horizontal lines where the velocity is 2500 kilometers per second. It is likely that, with sufficient study, the velocities indicated by the spectroscope may be made to determine the theoretical size and power of a given vortex. TABLE 89 TYPICAL SOLAR VORTEX CONFORMS TO KALE'S DATA Radii in kilometers, w 2 = -. Log 6.80618 Lines LogC Kilometers (1) -6.54407 Outside (2) -5.26447 (3) -5.98487 (4) -4.70527 (5) -3.42567 (6) -2.14607 Center 21ogp +0.72040 z= 500 1000 2000 5000 . . CO 60474. 42762. 30232. 19124. CO 26287. 18658. 13193. 8344. 00 11513. 8140.7 5756.3 3640.6 CO 5023.1 3551.9 2511.5 1588.5 CO 2191.7 1549.8 1095.8 693.06 oo 956.25 676.17 478.12 302.39 10000: 20000 30000 40000 50000 13523. 9562. 7807. 6761. 6047. 5900.1 4172.0 3406.5 2950.1 2638.6 2574.3 1820.3 1486.3 1287.2 1151.2 1123.2 794 . 22 648.49 561.60 502.31 490.08 346 . 53 282.94 245.04 219.17 213.82 151.20 123.45 106.91 95.63 260 A TREATISE ON THE SUN'S RADIATION TABLE 89 Continued Radial velocity in kilometers/second, u = Cw z = 500 1000 00 0.2117 1497 00 0.4851 3430 00 1.1119 7862 00 2.5482 1 8019 00 5.8404 4 1298 00 13.3860 9 4652 2000 1058 2426 5559 1 2741 2 9202 6 6929 5000 10000. . . . 0.0669 0473 0.1534 1085 0.3516 2486 0.8062 5698 1 . 8469 1 3060 4.2329 2 9931 20000 30000 . . 0.0335 0273 0.0767 0626 0.1758 1435 0.4029 3290 0.9234 7540 2.1164 1 7281 40000 0237 0542 1243 2849 6530 1 4696 50000 0212 0485 1112 2548 5840 1 3386 Tangential velocity in kilometers/second, v = 2 = 500 1000 105.83 149 66 242.56 343 02 555.91 785 02 1274.1 1801 9 2920.2 4129 7 6692.8 9465 2000 . . 211 66 485 11 1111 8 2548 2 5840 4 13386 5000 334 66 767 02 1758 4029 1 9234 4 21164. 10000. . . . 473 28 1084 7 2486 1 5698 13059 29931 20000 669.33 1534.0 3515 8 8057 8 18468. 42327. 30000 . . . 819 75 1878 8 4306 1 9869 2 22619 51842 40000 50000 946.56 1058 30 2169.4 2425 5 4972.2 5559 1 11396. 12741 26119. 29202 59861. 66928 Vertical velocity in kilometers/second, w = 2 Cz z => 0. . o o o o o 500 0.0035 0.0184 0.0966 0.5073 2.6648 13.998 1000 . . 0070 0368 1932 1 0146 5 3211 27 996 2000 0.0140 0.0735 3863 2 0292 10 659 55.994 5000. . 10000 20000 30000 0.0350 0.0700 0.1400 2100 0.1839 0.3677 0.7354 1 1031 0.9658 1.9312 3.8631 5 7946 5.0731 10.146 20.292 30 439 26.648 53 . 296 106.59 159 89 139.98 279.96 559.93 839 88 40000 50000 0.2800 3500 1.4708 1 8385 7.7262 9 6576 40.585 50 731 213.19 266 48 1119.8 1399 8 Tangential angle, tan i , _ o 90 0' 0" 90 0' 0" 500 1000 . . 2000.. .. 5000. 6 53 3 27 1 43 41 6 53 3 27 1 43 41 10000.. .. 20000. . 30000.. .. 40000.. .. 50000.. .. 21 10 007 005 004 ; 21 10 007 005 004 Vertical angle, tan 17 = z = o 500 1000 2000 00' 0" 007 10 14 22 31 43 53 Oil 018 0' 0" 16 22 31 49 1 10 1 39 021 2 20 2 36 0' 0" 36 51 1 12 1 53 2 40 3 46 4 37 056 5 58 0' 0" 1 22 1 56 2 44 4 20 067 8 39 10 36 12 15 14 1 0' 0" 038 4 26 6 16 9 55 14 2 19 50 24 18 28 3 31 22 0' 0" 7 11 10 10 14 23 22 44 32 10 45 28 55 42 1 4 19 1 11 53 i 5000 10000 .... 20000 30000. . 40000 50000 Formulas. Meteorological Treatise, p. 172 OTHER SOLAR PHENOMENA 261 Inward and Outward Velocities in Solar Vortices The velocities of inward and outward motions in the solar vortex, made at the Mt. Wilson observatory by C. E. St. John, and summarized in the Annual Report for 1913, are interesting in this connection. Generally the light gases have an inward radial velocity and the heavy gases an outward velocity. TABLE 90 SUMMARY OF THE INWARD AND OUTWARD RADIAL VELOCITIES Element At. Wt. Line Height Kilometers Vleters/sec. Direction Hydrogen . . 2 Ha 20000 1500 Inward Calcium 40 H 7 HO'^COO CO i-H T-H 1-1 COOOOOi irJHiOCOCOCOO5(N t>CO T ii II-HI 1 i ( 00 Oi 1> OS.t^ rH i 1 (M COCOOi i-H C<1 i 1 f-H i 1 COiO(MI^^OO(MCOQ OiCOOGOCOlMi-HCOO lOrHOCDC5CiCOrH1^-*IO>OOOCDCOOOOOO rH rH rH rH (M lOC^COlOOiCOCOC^C^COrH I rHrH ^Z -44 I ^T 1 T~H ^rHCOOOOrHrHrHOOlNOO OOOrHrHrHOOlN 00 m TJI 1 1 1 1 s I I 00 TH O^ t^ ^D rH (^ 00 00 rH rH rH rH 1C O '*'*OO OOCO iOl>(NCOCQ CO CO W rH rH ; 7 i I rH | | | | | 10 CD CO I>COCO(M 1 iT^rHCOOO CD ss 1 1 00 7 O^oO COOOO IO Tt^Ot^O5l>iO(NCOiOCOOO <>GOCOrHrH 1-HCDO5O5 T^lOCOrHrH I (MOrHCOl> I c |^ t/ u I~l (t t dv m -T7 = k z y eu . H a t FIG. 31. Elements of the Zee- man Effect. These equations are solved by the following terms: (180) x = x e st , provided, ms z x = k 2 x + e H . s y , (181) y = y Q e st , provided, m. s 2 y Q = k 2 y Q e H . s x , ,. _x , . k .2 IT , ^,27T Vw . ., . , (182) where 5 = ^ F =. = ^ =, and T - is the period. Vw T k This occurs when H = 0, and there is no magnetic field, so that T represents the undisturbed periodic motion of the line- constituents. If H has a value, 27TVW 276 A TREATISE ON THE SUN'S RADIATION The sign + signifies a greater periodic motion for the posi- tive rotation, and the sign for a smaller period. (1) When viewed along the magnetic field the two circular components accelerate or retard the normal rotation, change the wave length and the position of two resultants in the spec- trum. The original spectral line becomes two circularly polar- ized lines of equal intensity rotating in opposite directions, and symmetrically displaced in respect of the undisturbed position. (2) When the ion is viewed across the magnetic field, at right angles to H, the component along z is unaltered in period and position; the two circular components, seen in the xy plane, appear to be plane polarized, the direction of vibration being at right angles to the central s-component. The spectral line is broken up into three distinct plane polarized lines, the central vibrating parallel to the magnetic field and the two outer components vibrating at right angles to it. Since the polarization may be circular, elliptical, or com- plex, it follows that there may be developed several component of these two principal types, and numerous lines have been deter- mined by experiment in strong magnetic fields. Such division of a simple spectral line into components is, therefore, proof that a magnetic field exists of sufficient strength to be detected and measured. This method was successfully applied by Dr. G. E. Hale, in 1908, to the sun-spots, wherein the spectrum lines were broadened and subdivided in conformity with these tests. This constitutes proof that free ions or electrons (e.m) exist in the sun-spots, and that the vortical motion is suffi- ciently rapid to produce strong magnetic fields. Similarly, the Zeeman Effect has been generally found in the sun's atmos- phere, and its corresponding magnetic field has been deter- mined, so that the subject of solar magnetization of the entire mass, or local magnetic fields accompanied by electric currents, becomes a very important subject of research. The literature is already extensive, so that only a few conclusions can be mentioned. This simple theory of accounting for the Zeeman Effect was first proposed by Lorentz, but at has been extended to comprise very complex motions, including the rotation of OTHER SOLAR PHENOMENA 277 the axes themselves. The ions in high temperature gases may by their motions produce the most varied kinds of optical spectral effects. The Distribution of the Solar Magnetism as Determined by the Zeeman Effect If a positive charge rotates anti-clockwise and a negative charge rotates clockwise, a positive magnetic field is generated along the z-axis. Hence, by means of the Zeeman Effect, it is possible to determine the polarity in the sun-spots and in other localities on the sun. The observations on the sun- spots show that there are in each hemisphere about as many of one polarity as the other; that as the sun-spots are com- monly generated in pairs they are likely to be of opposite polarities; that the intensity of the magnetic field in the sun- spots increases radially toward the axis and downward from the plane of the photosphere; that the effect is distributed in many anomalous ways among the lines of the spectrum. Dr. Hale has properly associated these phenomena with a true vortex motion, wherein the inner tubes rotate faster than the outer and more rapidly with the depth below the reference plane. It is thought that the introduction of the isothermal shell, the depths of the several gases, the computed P, p, R, T, are all in complete harmony with the requirements of the solar spectrum. If the ions are chiefly generated near the bottom of the isothermal layer of each gas, the intrusion of abnormal temperatures from below, as well as from above, will induce segregation of the ions, the negative going to the cooler and the positive to the warmer strata, in a general way. Since the deflecting force of rotation near the equator is small on the sun, it follows that the underflowing hot sheet will rotate indif- ferently in forming the dependent vortex, so that it is probable that a group of sun-spots consists of independent axes of funnel- shaped tubes, rather than of curved horseshoe vortices termi- nating on the free surface of a stratum, since no such definite layer exists for the light gases, though by complex interaction between light and heavy gases some such complex layer may 278 A TREATISE ON THE SUN'S RADIATION be formed as a photospheric stratum. The structure of this region is so very complicated that the tangled record of the spectrum is to be expected. There are many statements re- garding the height of the lines of flow which will need to be modified. The method of double reversal, or that of the long and short arcs, while they give indications of the height, should be interpreted in terms of the general thermodynamic require- ments that must prevail on the sun. The general magnetic field of the sun has been determined with considerable precision, as well as have some of its character- istics. Generally, the magnetic field has opposite signs in the two hemispheres, such that the true positive magnetic pole is on the south side of the plane of the ecliptic on the southern hemisphere of the sun and the negative magnetic pole is on the north side, exactly as is the case with the distribution of the earth's magnetism. It has been suggested that the nega- tive ions as a whole are more distant from the axis of rotation than are the positive ions, and that the rotation of the sun produces its magnetism. It has been thought that the internal magnetization is produced by ampere electric circuital currents rotating about lines which are somewhat parallel to the sun's axis of rotation, as the inner field of a spherical magnet is located. The maximum production of ions is near the bottom of the isothermal layer, and the maximum of disturbed circulation is also near that level. Hence, magnetic field increases from a minimum in the levels above the photosphere to this level. Whether that isothermal value is a true maximum, or only a step toward a higher value in the lower levels, is not now known. The average value of the solar magnetic field is ap- proximately as follows: Near the coronal poles, 50 gausses In small sun-spots, 1000 " In largest sun-spots, 5000 " The sun's polar magnetism is about 80 times as much as the earth's polar vertical field, which is 0.66000 C. G. S. units. From such data the entire magnetic system of the sun can be readily computed by the well-known formulas. OTHER SOLAR PHENOMENA 279 It has been found that the Zeeman Effect indicates that there is a maximum in each hemisphere in the latitudes 45, and that a sine curve expresses the distribution, so that there is zero-value at the poles and at the equator. By referring to Fig. 30 of the general solar circulation, it is seen that the maxi- mum vertical circulation has been placed in about latitudes =*= 45 in each hemisphere, with minimum at the poles and at the equator. Hence, it may be inferred that there is a sec- ondary magnetic solar field due to vertical circulation in latitude, as if the ions were transported upward and downward in these paths. Very similar convectional circuits in the earth's atmos- phere are found to account quite fully for the existing diurnal magnetic variations, and there are other circuits concerned with the general disturbances in latitude. If circulation in zonal sheets in latitude is sufficient to cause a persistent maxima in certain latitudes, it follows that a similar segregated circulation in longitude is sufficient to account for the maxima of the magnetic field and the maxima of the radia- tion itself, arranged in longitude, as is suggested by the maxima of Fig. 30. There are persistent and unquestioned impulses from the sun operating on the earth's atmosphere in numerous manifestations, which indicate that there is a general periodic action in 26.68 days, upon which are superposed eight distinct maxima and minima, arranged more or less permanently in longitude. These are marked most clearly in the variations of the terrestrial magnetic field, and less emphatically in the meteorological field. It is stated that these maxima in longi- tude are not found in the spectrum changes. This can be readily understood from the fact that the spectrum observed belongs generally to the strata lying above the level of the iso- thermal layer, while the maximum impulses themselves originate in convectional operations which are rigorously confined below that stratum. The sun-spots penetrate this layer, but they have never been found to synchronize perfectly in their fre- quency with the system of magnetic pulses that are observed. The minor variations in the spectral lines, due to changes in pressure, density, and motion, are too superficial to take 280 account of the deep sources of the solar radiant energy and its allied forces. The radiation energy of the sun and the mag- netic field of the earth constitute a sensitive mechanism which registers these maximal fluxes of energy, and it would be re- quiring too much of the spectrum to attempt to see in it a full record of such solar actions. The interpretation of all the symptoms of these fields will doubtless be greatly improved by experience, and it may happen that some telltale signs can be discovered which will serve to indicate the presence of maxima in the radiation. At present the negative results from the spectroheliograph should not be interpreted as conclusive evi- dence that the problem of maxima has been exhausted as to their synchronisms. The Solar Spectra It is not our purpose to study the characteristics of the solar spectra themselves so much as to point out the physical conditions under which they are formed. A good description of them may be found in Abbot's "Sun," and the technical articles are collected in the Astro physical Journal from the Mt. Wilson and other observatories. The following summary of the facts of observation is derived from these sources, and the important matter is to compare them with the data of the preceding tables of solar physical conditions. It will be seen that a 'very large addition has been made to the knowledge of such condi- tions, and that while the general harmony between the ob- served and computed data is excellent, there will be needed many modifications in the inferences that have been made from the behavior of the spectral lines. The present knowledge of the prevailing temperatures, pressures, densities, and gas coefficients, in all strata for gases ranging from hydrogen to mercurial vapor, and the heights at which they occur, will greatly improve the interpretation of the reversals, the shifts, the broadening of the lines, the values of the long and short lines in respect of height, and similar problems. It will be pointed out that we have the material for computing the coeffi- cients of transmission and absorption of the different lines of OTHER SOLAR PHENOMENA 281 the spectrum, and the indices of refraction in all layers of the atmospheres of the earth and the sun. From these data, and those in the tables, studies can be made in atomic and molecular physics, under the conditions of solar temperature and gravi- tation, and thus escape from many laboratory limitations. The General Solar Spectrum The solar spectrum consists of a continuous background, crossed by dark lines which are commonly identified with those of the terrestrial elements. The spectra of all gases become continuous at high temperatures and high pressures. By Table 6 the temperatures in the isothermal layer range from 7650 to 7700, and in the adiabatic. layer they increase rapidly to enormous values; by Table 10 the pressures in terrestrial at- mospheres are about 6.08 at the photosphere, about 20 atmos- pheres at the bottom of the isothermal layer, and they soon become enormous in the adiabatic strata; by Table 12 the density for hydrogen is ^ that of the earth's normal atmos- phere at sea level, and mercury vapor has Vio that density, while at the bottom of the isothermal layer its density is % that of air, so that terrestrial sea-level densities occur only in the adiabatic layer, beyond the vision of the spectroscope; the gas efficiencies are indicated in Table 13. These are the conditions for the continuous spectrum. In the reversing layer the tem- peratures of the light gases, 2 to 60 atomic weight, have been lowered by 2000 to 3000 degrees, while the heavy gases and vapors do not extend to the altitude of 400 to 500 kilometers. These dark lines are formed by interposing a cooler absorbing layer of the same gas between the continuous spectrum and the spectroscope. Most of the dark lines are solar, as iron (55), nickel (58), calcium (40), titanium (48), cobalt (59), chromium (52), magnesium (24), carbon (12), vanadium (51), sodium (23), magnesium (24), hydrogen (2), these being the lines just before reaching the axis of the curves of Fig. 18. The heavy vapors and gases are seen with difficulty, only by intrusion above their normal levels. In the earth's atmosphere there are dark lines 282 A TREATISE ON THE SUN'S RADIATION due to absorption at very low temperatures by oxygen, car- bonic acid, and aqueous vapor. Tests are used to determine whether a line is solar or ter- restrial: (1) at high and low sun, the terrestrial lines are stronger at low sun; (2) the east and west limbs of the sun directed simultaneously on the slit give the Doppler Effect of shift for solar lines, while the terrestrial lines remain unchanged. The principal lines of the spectrum are marked, in Ang- strom units, namely, tenth-meter = 0.000 000 000 1 meter. K 3933.68 calcium, solar. H 3968.49 calcium, H y = G 4340.47 hydrogen i( Hp = F 4861.35 " " b 5183.62 magnesium " E 5269.55 iron " Z> 2 5889.98 sodium " H a = C 6562.84 hydrogen " B 6869.97 oxygen terrestrial a 7184.57 aqueous vapor " A 7593.83 oxygen The intensities of the lines are classified: ^ 0000 = the most difficult to see. 1 = just clearly visible on Rowland's spectrum map. 1000 = the strong calcium lines, H. K. In the solar spectrum the intensities decrease with the increase in the atomic weight, till radium (224) and uranium (236) may exist in the sun without being seen in the lines; there are only a few non-metallic elements; oxygen and helium exist; a very little absorbing gas produces a dark line; lines are dark only by contrast, so that the flash spectrum of the reversing layer consists of bright lines in the places of dark lines, as it is seen in eclipses at the moment the photosphere disappears; short waves are more absorbed than long waves, but the lines from 1.50 n to 2.50 p pass through the intervening layers of the solar and terrestrial atmospheres with only selec- tive band absorptions, while the short waves from 0.00 /z to OTHER SOLAR PHENOMENA 283 0.35 /x generally have disappeared; the intermediate lines 0.35 ju to 1.50 M are depleted in an irregular manner. Whatever changes the temperature, pressure, and density, relatively to certain normal values on the different levels, also causes variations in the position and shape of the lines. Thus, convection by temporary transportation changes these normal relations in vertical directions, while circulation changes them in horizontal directions, so that on the sun these varia- tions are numerous and very complex. Increase of pressure shifts the wave lengths toward one end or other of the spectrum, and it broadens certain lines. On the whole, the shifts increase with the wave length, but there are many arbitrary conditions. On the other hand, change in velocity shifts all lines proportional to their wave lengths. Due to the rotation of the sun, all solar lines have a little greater wave length than the corresponding terrestrial lines by a few thousandths of an Angstrom. The general pressure is from 4 to 6 atmospheres, as determined by the spectrum. There is unsymmetric broadening of some lines, and greater shift on the side of the long waves than on the side of the short waves. There is a common vertical circulation upward in general of 0.1 to 0.3 kilometers per second; there are large vertical move- ments in the granulations and pores, in the faculae arid sun- spots, and they become at times excessive in the solar promi- nences. These can all be discussed by the formulas of this Treatise. Lines are also classified by their temperatures: Enhanced lines = high temperatures = spark conditions, Average lines = low temperatures = arc conditions. The enhanced lines generally indicate vertical convection, bringing high-temperature conditions upward as in the umbra of spots, faculae, and granulations; low-temperature conditions occur over the center of sun-spots in the reversing layer, in the penumbra and in the pores between the filaments and the granules wherever there is downward circulation; irregular mixtures occur in all levels due to horizontal movements. The chromosphere, especially in the reversing layer, has a 284 A TREATISE ON THE SUN'S RADIATION spectrum opposite to that of the photosphere; there are double reversals in some lines, as 3933.667, wherein KI is probably stationary, while K%, K% has a vertical velocity upward of 1.97 kilometers, and K 3 a downward velocity of 1.14 kilometers per second. These represent layers of different levels, KI lowest, K 2 higher and rising, K 9 highest and falling. In the sun- spots the vortex circulations bring enhanced lines from below upward into the vortex of the penumbra; the high level gases of the chromosphere descend with their own lower temperatures, as in the penumbra and the reversing layer; there are lines from layers still somewhat normal, so that the sun-spot spectra are very complex and differ from that of the surrounding photo- sphere as indicated. Seen in superposition, there are many phenomena, such as increase in the shadings and wings; en- hanced lines apparently weaker in spots through lowering of temperatures from higher strata; H a weaker in spots; maxi- mum ordinate of radiation shifts from the penumbra to umbra with the change in the temperature; the short waves are rela- tively much weaker in the umbra than in the penumbra and the ratio approaches 1.000 in the long waves; temperature over sun-spots less than that of the surrounding photosphere; more scattering and absorption above spots than above the photo- sphere; many Fraunhofer lines are strengthened and many are weakened in the sun-spots as compared with the photo- sphere; sun-spot vapors are too cool to produce strong absorp- tion of the enhanced lines; great abundance of flu tings in the sun-spots; high temperatures produce complete dissociation in the lower part of the sun-spot vortex, but tendency to associa- tion in the central part of the isothermal layer and an abun- dance of associated compounds above the photosphere itself; reduction of the continuous background in spots is greatest for the short waves. As between the spectra at the center of the disk and the limb of the sun, the lines are generally displaced toward the red on account of an increase in total pressure under hemi- spherical curvature. Hydrogen (2), sodium (23), calcium (40), magnesium (24) show no displacement; titanium (48), vana- OTHER SOLAR PHENOMENA 285 dium (51), scandium (44) show moderate displacement; iron (55), nickel (58) show considerable up to 0.007 Angstroms. That is, high atomic weights are but little displaced; enhanced lines show maximum displacement. The violet edges of the lines do not shift. The limb spectra are weaker than at the center, and the violet lines need much more exposure at the limb; the Fraunhofer lines are much changed, especially in the violet; the strong lines lose their shading or wings; the enhanced, high temperature lines are weakened at the limb; the lines which are strong in spots are strong at the limb; the hydrogen H a is widened at the limb. Atmospheric Refraction and Scattering The general relations between the refraction or change of direction of a ray of light passing through an atmosphere and the loss of the energy by non-selective scattering upon the molecules are expressed by Rayleigh's Formula: , 32T(tf-l)fc 327r3(M-l) 2 . J*_ 3 X 4 n . Bo 3 X 4 go Wo Po in the following notation, k = coefficient of scattering, /* = index of refraction, X = the wave length in centimeters, lo = the height of the homogeneous atmosphere at the stratum whose pressure is B , B = the barometric pressure at the level under discussion, n = the number of molecules per cu. cm., System = (C. G. S.) /IOP\ r i loB. p m BQ B p m B (185) We have, -=- = -- = -- = JJQ PO -t>0 PO Hence, by transformations, (186) ( M _ 1)2= ^ P 1 -- , Table 3, Treatise. go Po On the sea level, from which the homogeneous height may be computed, or on the photosphere of the sun, the pressure P = PQ. The formula was devised to compute k at any height above the plane of reference, but since we have computed 286 A TREATISE ON THE SUN'S RADIATION w, p, P, at numerous points in each atmosphere, we can at once take them at the point in any stratum of any gas for which they have been prepared. The conditions are, however, so complex for the different wave lengths that it is possible merely to give some approximate results in the atmospheres of the earth and the sun as examples of a method that can be utilized in discussing these important subjects. It will be necessary to obtain the values of k from the observations with the pyrheliom- eter and the bolometer, and it will be convenient to summarize the formulas of the theory of refraction. THE FORMULAS OF REFRACTION, FIG. 28, IV Terms Polar Coordinates Angle Incidence Angle Refraction Refraction Index of Refraction Upper point (R + z), (0+<#) ili iz /i, /i. S,, 5 2 . . . Mi /*2 Lower point R z

fr R sin , atmospheric refraction II. /i (12 + z) = /IQ ^ sin r, circular refraction III. A(^ + 2) < ^o^sinr, imaginary refraction. 288 A TREATISE ON THE SUN'S RADIATION The index of refraction is connected with the density of the stratum by the general equation, / onoN M 2 1 = 4 K p ..dn 2 K . d p , (202) . n and = , . , , for K = a constant. /cp' Several hypotheses (Chauvenet) have been employed to determine p, the density in the strata above the sea-level value, p . r> n 7? T* (203) = p ^ , by the Boyle-Gay Lussac Law. FQ po AO 1 I. It is throughout these discussions assumed that R = RQ = constant, and consequently it is erroneously supposed that the atmosphere is stratified along adiabatic gradients. This being the case, the discussions need complete revision throughout the formulas. This is an exceedingly complicated problem, and it may prove that the empirical formula of Bessel cannot be improved for practical purposes. The first supposi- tion regarding the temperature is that T = T , and is isothermal. Hence, (204) L = P- = e ~l FQ po This does not lead to satisfactory results. II. Assume the following relations, which are easily found: ~ R' This finally leads to the equation, (206) F ' - = F = x - 2T- ro p lo 2 to It can be easily shown that the temperatures do not con- form to this formula in the higher strata. III. Assume, with Bessel, the following fundamental formula, (207) = e-f = l-- + ii 2 -...= Hence, PO OTHER SOLAR PHENOMENA 289 Take h = 7991.04, and h = 227775.7 meters. In common logarithms, (209) log P = log po - M (| - -J-). TABLE 92 THE DENSITY p AS COMPUTED BY THE NON-ADIABATIC, THE ADIABATIC, AND THE BESSEL FORMULAS Balloon Ascension, Uccle, June 9, 1911 (210) Non-adiabatic. Log pi = log po + ~~T (log Ti - log T ) i\ ~ J. (211) Adiabatic. (212) Bessel. Log pi = log Po Log pi = log Po - M \ - Height in Meters. Non-adiabatic (Bigelow) Adiabatic Bessel 2 = 50000 . . . 0.0009 0003 0.009 45000 0.0054 0.0022 0.053 40000 0160 0049 097 35000. . . 0293 0.0088 0.177 30000 0.0511 0.0188 0.0324 25000 0885 0407 0593 20000... ... 0. 1543 0.0901 0.1084 15000 2708 2012 1983 10000. . . 4753 . 4179 0.3627 5000. 0.7830 0.7229 0.6633 000. . 1 2132 1 2132 1 2132 Taking the same value of po = 1.2132 at the sea level, it is seen that the adiabatic is lower in amount than the non-adia- batic value up to 50000 meters; the Bessel value is lower than the non-adiabatic throughout; it is less than the adiabatic from the sea level to 15000 meters, and then remains interme- diate between the adiabatic and the non-adiabatic values up to 50000 meters. It will be exceedingly difficult to develop any formula that can supply the place of the non-adiabatic temperature gradient, while depending solely upon the surface conditions. In the case of the Bessel formula for refraction, upon which the working tables are constructed, it is probable 290 A TREATISE ON THE SUN'S RADIATION that the exponents A and X make some compensation for the actual inaccuracy in the density p. (213) B =o 0* T A tan f = Bessel's Formula. r/?e Atmospheric Transmission for Different Wave Lengths p K Collected According to the Value of p w by the Pyrheliometer Volumes II and III of the Annals of the Astrophysical Obser- vatory of the Smithsonian Institution contain the coefficients of transmission as measured by the pyrheliometer p w , together with the coefficients for several wave lengths as determined by the bolometer p^. These have been collected in groups with p w Sit convenient intervals, 0.900 0.890, 0.890 0.880, 0.880 0.870, etc., and the mean values were obtained for the several years. These were united in general means, and they were plotted on large diagrams, with p^ for ordinates and X the wave length for abscissas. Each group under the pyrheli- ometer p w results in a curve of transmission coefficients, and the series of p w make a family of curves. These were adjusted to probable values, and from them were scaled the final values which appear in Tables 93, 94, 95, for Washington, Mt. Wilson, and Mt. Whitney, respectively. Resuming the black spectrum for 6950, corresponding with 3.98 gr. cal./cm. 2 min., the spectrum values for the different wave lengths were multiplied by the coefficients of transmis- sion under the several values of p w , resulting in the groups of pi and / A . The latter represent the transmitted values of the radiation, and as they are homogeneous throughout they give relative values. Take the sums for the selected wave lengths along the spectrum; the factor 16.7 suffices to reduce these sums to calories, and it is seen that for each value of p w there are corresponding calories, such as, 2.79, 2.71, . . . 2.20, 2.13 for Washington, " 2.95, 2.92, . . . 2.72, 2.63 for Mt. Wilson, " 3.04, 2.99, . . . 2.78, 2.72 for Mt. Whitney. The pyrheliometric transmission at different values has its counterpart in the bolometric transmission at different wave OTHER SOLAR PHENOMENA 291 PH H w S a g< 2 bo p| < Q J> fc W go Q K^ I I S | S ii J3 -Hc5OOSCOC-COOT}"iOOOOOieOt-O3O>t--^NOO500 IO O T-H O 00 i-( IO O> U5 d O5 t- D 't CO CO C<1 N <-H rH i-( rH iH O O t- 1- 1- oo oo oo o> oj o> cj cs Oi os T-li-IC^COI oj o> cj cs Sliiigiiiig oo oo oo os os os os T} os os os os os os os TfeoeoocccDOC^M-^icioioiommiflioiflkOioioiomious < Ui O t> OO 00 00 OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OSU5CTj oo as o N co * 10 w t- oo os < ^5, ' ^t 10 c- oo oo oo Oi 05 as a> o> os as o as os os as as Oi os a> as as < T)O^-^'OSDO5O5TlOOO3O . OSTt^0^05t2coSt-2S(NO>t--Si3^NwS! iH CO* Tj< Tj< Tj< Tj Tji CO* N N r-i iH m>io ' COC^t^ eCeOO- ' ^*COt JINN(NCt--^NTi'cooo-^O-HOOlOCOi-lOOS 00 Oi Oi O> OJ OJ O5 O5 O5 OS O5 05 OS O3 OS Oi OS OJ OS O> OS OJ cD-iO5oom^Ha5Oix>oiooi-ic5iOiO5a5aiOi ^ OOOOOOOOOOOi I * N CS1 N W N M d OTHER SOLAR PHENOMENA 293 lengths. This group of results conforms in their average values to those which have been heretofore employed, Table 79, but they express in detail the effects of the atmospheric conditions upon the spectrum processes of scattering. TABLE 95 THE COEFFICIENTS OF TRANSMISSION AND THE ORDINATES IN THE SPECTRUM FOR DIFFERENT VALUES OF THE PYRHELIOMETER p w Mt. Whitney, 4420 meters Pyrheliometer t-w 0.938 0.918 0.898 0.878 0.858 0.838 r = 6950 A P* 'A *A 'A P* A *A J \ *A 'A X A OftO 9^iL 7 3^ . uu u . ZD/J- . . . . 0.30 . . I . oO 4.85 .510 2.47 .508 2.46 .506 2.45 .504 2.44 .502 2.43 .500 2.43 0.35 .... 6.05 .670 4.05 .631 3.82 .595 3.60 .574 3.47 .556 3.36 .545 3.30 . 40 . . 6.54 .810 5.30 .772 5.05 .743 4.86 .700 4.58 .665 4.35 .642 4.20 0.45 6.50 .877 5.70 .854 5.55 .826 5.37 .792 5.15 .772 5.02 .750 4.88 0.50 .... 6.13 .913 5.60 .898 5.50 .878 5.38 .852 5.22 .827 5.07 .804 4.93 0.55 .. 5.59 .930 5.20 .919 5.14 .904 5.05 .883 4.94 .861 4.81 .837 4.68 0.60 .. . 5.00 .942 4.71 .932 4.66 .921 4.61 .907 4.54 .884 4.42 .859 4.30 0.70 .. . 3.88 .961 3.74 .950 3.70 .941 3.66 .929 3.61 .911 3.54 .890 3.46 . 80 2.96 .971 2.87 .962 2.85 .952 2.82 .941 2.79 .927 2.74 .909 2.69 0.90 . 2.25 .976 2.20 .967 2.18 .958 2.16 .947 2.13 .934 2.10 .921 2.07 1.00 . 1.72 .978 1.68 .970 1.67 .962 1.65 .952 1.64 .939 1.62 .926 1.59 .10 . 1.33 .979 1.30 .972 1.29 .963 1.28 .953 1.27 .943 1.25 .930 1.24 .20 . 1.04 .980 1.03 .972 1.02 .964 1.01 .955 1.00 .946 .99 .931 .98 .30 . 0.82 .980 .80 .972 .80 .964 .79 .956 .78 .947 .78 .932 .76 .40 . 0.66 .980 .65 .973 .64 .965 .64 .956 .63 .947 .63 .932 .62 .50 . 0.53 .980 .52 .973 .52 .965 .51 .956 .51 .947 .50 .933 .49 .60 . 0.43 .980 .42 .973 .42 .966 .42 .957 .41 .947 .41 .934 .40 .70 . 0.36 .980 .35 .973 .35 .966 .35 .957 .34 .948 .34 .935 .34 .80 . 0.29 .980 .28 .973 .28 .966 .28 .957 .28 .948 .27 .936 .27 .90 . 0.25 .980 .25 .973 .24 .966 .24 .957 .24 .948 .24 .937 .23 2.00 . 0.21 .980 .21 .973 .20 .966 .20 .958 .20 .948 .20 .938 .20 2.10 . 0.18 .980 .18 .973 .18 .966 .17 .958 .17 .948 .17 .938 .17 2.20 . 0.15 .980 .15 .973 .15 .966 .14 .958 .14 .949 .14 .939 .14 2.30 . 0.13 .980 .13 .973 .13 .966 .13 .958 .12 .950 .12 .940 .12 2 . 40 0.11 .980 .11 .973 .11 .967 .11 .958 .11 .950 .10 .941 .10 2 . 50 . . . 0.10 .981 .10 .973 .10 .967 .10 .958 .10 .951 .10 .942 .09 2 . 60 . . . 0.09 .981 .09 .973 .09 .967 .09 .959 .09 .951 .09 .942 .08 65.50 50.09 49.10 48.07 46.90 45.79 44.76 Factor 16.7 765 750 734 716 .699 .683 3.98 3.04 2.99 2.92 2.85 2.78 2.72 The Terrestrial Values of the Index of Refraction (n 1) We shall utilize these coefficients of transmission p^ first to compute the coefficients of absorption, (214) 294 A TREATISE ON THE SUN'S RADIATION and then proceed by the Rayleigh Formula (184) to compute the values of (/* 1) for the index of refraction, From Tables 93, 94, 95, interpolate for four selected wave lengths 0.323 /i, 0.481 /*, 0.670 /i, 1.225 /i, as examples, the values of p\ corresponding to p w . Compute the equivalent coeffi- cients of scattering. In the Rayleigh Formula we have for 0.323 /i, X 4 = (0.0000323 cm.) 4 ; g is the acceleration of gravita- tion 980.6; n is the number of molecules per cubic centimeter, Tables 15, 21, 27, note page 55, Bui. No. 4, 0. M. A.; p is the density from Tables 9, 17, 23; P is the pressure from Tables 9, 17, 23. The mean values for these three balloon ascensions were used in this formula. The resulting values of (/* 1) for Washington, Mt. Wilson, and Mt. Whitney are given for four wave lengths, in order to illustrate the variability in the value of (/* l). It increases from large values of p w , the pyrheliometric coefficient of transmission, to low values of p w \ it increases always with the increase in the wave length; it diminishes generally with the altitude of the station. The Solar Values of the Index of Refraction The data are taken for the computations from Abbot's Table 55, Vol. III., which gives the relative brightness for dif- ferent lines, from the center to the limb at several radial dis- tances. These distances are a = r sin f , from which the zenith distance and the sec are computed. In the sun's atmos- phere we can take the path length of the ray, as in Section IV, Fig. 28, proportional to the sec , so that by the Bouguer Formula, sec sec In order to compute 7 , /i, /2, . the relative intensity, we proceed as follows: Take the relative brightness from Table 55; interpolate in the black spectrum of 7655 for the given wave length, and divide by 20.9, in this distribution of the OTHER SOLAR PHENOMENA 295 spectrum line distances, to reduce to gr. cal./cm. 2 min., and this is the assumed value of 7 at the center of the disk; mul- tiply 7 by the observed brightness at the other radial distances, so that we have the successive values, 7 , 7i, 7 2 , . . . ; these were plotted in pairs on a diagram whose abscissas are sec f and whose ordinates are log 7, as in the usual pyrheliometric reductions; the resulting values of p K appear in the Section I, Table 97. From these values of p^ compute k^ in Section II. From k^ compute (/* 1) in Section III, for four wave-lengths of hydrogen, carbon, calcium, mercury. The gravity accelera- tion G = 27484.3 cm./sec.; n = the number of molecules per cu. cm., Table 32; p is the density, Table 12; P is the pressure, Table 9. It is seen that (/* 1) decreases from the center to the limb of the sun; it increases with the wave length; and it increases with the molecular weight. The terrestrial values of (/z 1) in the lower strata of the atmosphere are about ten times as large as they are for the solar gases in the neighborhood of the photosphere. The values of (M 1) decrease gradually to vanishing values in the outermost strata of both atmospheres. General Remarks There can be no greater hardship than to present the results of an extensive logarithmic computation by means of a few selected numerical values at the end. The entire working of the formulas in their details is lost upon the reader, and the relations are in themselves so complex that it is quite im- possible to follow them intelligently apart from the actual processes that are involved. In this case the fundamental for- mulas of thermodynamics are employed in a new field for their application, and, as a matter of fact, the entire series checks within one or two units in the fifth decimal of logarithms. In the solar data the numbers are unusually large, but even the considerable values here quoted in the tables are not sufficient to reproduce the checks. The assumed value of the tempera- ture at any point implies the succeeding terms as computed. 296 A TREATISE ON THE SUN*S RADIATION f o 00 OS CO -* CO O 00 CD coSooos s d coS^oo oSeo o d ^t t-ooos 9 O r-i ^ OS ION 1 d coioc-oo os m co I-H o d d d ioos t- os t- c-oo -^< d |9

Tt< T)< 00 N G" 2 i | "So c 3 d ct moo ^os ^ M 10 S??kOH (4 -M M o i u " 1 o s O ** m T-IOOO rH -, VH COOS COOO g O oososeo^ s 1 c 1 '3 ^* t 00 OS 43 t*W^O InO |d--- & d--- v d IT- 1 c/5 i rt COrH OO JDOCO ?3! d OSi-4tH< I-l t- rH C moo t-< oo > o> to gs32 .2 O* O O CJOOt-N NOOt-COOO CO -^ -^ (M (M - CO CO CO 00 **H 00 *O 1>- CO O O Tfl ^ Tf Tj< 00 I>- CO ^O >O O C^ T-I rt< GO W O O5 O (M CO CO CO I: CO > o CO CO i i i i co *o oo co r^ ^ T^ 10 OTHER SOLAR PHENOMENA 299 " O OS 3 TH M t^> 03 OS I*** CO rH TjH ^J O CO ^^ t^* to cO CO CO l> CO . *O CO 1O TH CO OS 00 O O 00 *O J>-t>.lO *O l> IO CO Tfl rH O 1>- (NOOrH rH cq t^. (N^fl>O "* 00 CO iO O 00 O O Oi-t >OOS i ( T-H ^ i-H i i CO i i C^ CO O5 CO CO CO ^jJJ !>OOrHCO (MOOTJHOO COO5COI> O O C^OrHrHTtli IC^COO CO rH O5 (M CO 00 rH COO5 TtHCOCPOO OSQOCOrH COrH 00 CO O O (MT-I I^COC^^ r\ O rHCO Oi-HCl^O CO 00 OOO5C3CO COC3COOO O5O O O t^ 00 O CO ^O CO "3 1 00 ^t 1 CO l> 00 T^ to (N O 00 -^ O CO (M t^ rH Tfi 10 CO 00 O5 co >o -^ 00 rH t^ C^ *O O^l CO C^ t^* CO OS to t^ C^ C^J CO CO C^l TjH OrH(Mt>. (NWCOCO 3.3.3.3. 3.3.3.3. 3.3.3,3. 3.3.3.3. CO-tO>O CO i I O lO COT-iOtO CO i I O O C^OOt^fM (NOOt^(N (NOOt^lN T* (219) = 1 = -: = p . u v proportional to the f\ rj h v radiation density at the v frequency. * Planck, " Warmestrahlung," p. 139. t Richardson, " Electron Theory of Matter," p. 353. 306 A TREATISE ON THE SUN'S RADIATION _K + !._*_! + A SL !.-*T,l hv^~ 2 " hv " 2' A, 2 " hv " 2* (222) From the Boltzmann Entropy Law, S = k log w, Planck deduces the probability formula, (223) NS = k\ogW=-Nk o Take P = , and give m successive values, "n (224) NS = -Nk [-^log + (| - l) log (~ - l)]. 1 U I Substitute = 7 -- h IT, and differentiate for dS, d U. t] n v Zi U J_ _T _3_ J O 1 /, 7 l O Z. /. I O (io 1 /? tl V R It V T~" o" "TT + "o" n V ft V . / ^^\ .^. 1 //z^V . X ( hv Y rv \og (I + j-j, ) omitting j ^ +-3 V^rJ ""' ' (227) Hence, . = log (l + ^), and H = i + ^. ^ T 1 (228) The radiation p u v = T~ = -*r - , for the ^ frequency. Integrating for the total spectrum, STT^ /Vrfv 487r/f PLANCK'S THEORY OF RADIATION 307 (230) u = - ~ T* = a T 4 , Stefan Law. ' c n These formulas are first approximations to natural conditions of radiation, and since k, h, a, are not universal constants, they are of only formal value, though they can be used in succession wherever a certain set of values (k, h) are adopted. They, therefore, lead to families of spectra, which are superposed on the same level in atmospheres consisting of several independent gases. The formula of integration (227) T-~, = log ( 1 + r K J. ^ K is tautological when ryris a small term; generally it is not decisive. The following computations show how far the adiabatic case, upon which Planck's discussion is founded, is from satisfying the natural conditions of atmospheric radiation. Planck's function is a form of the probability curve, which gives the distribution of the ordinates of intensity according to their relative frequency of occurrence in seeking a maximum parameter, the Temperature. It seems, therefore, that the thermodynamic conditions in a given volume of an atmosphere should be capable of reaching the same result from the col- lisions of n molecules. We shall proceed to deduce two formulas, one for the non-adiabatic branch and one for the adiabatic branch of the curve for h, into which the Planck" formula divides itself at the bottom of the solar isothermal layers, as heretofore explained. Compare Fig. 32 for these three curves of the potential h of the vibrating electrons. Derivation of Bigelow's functions for the potential h. The first of the formulas is for the non-adiabatic branch, and it is derived from the terms P.p.R.T, taking them in the bulk, as an integrated effect. The second of the formulas is for the adiabatic branch, and it is deduced directly from the constituents of P.p.R.T, by the derivatives in terms of n. N. T, which are much more fundamental. Indeed, it will be proved that (n. N. T) is a better system for thermodynamics than the usual (P. p. R. T). 308 A TREATISE ON THE SUN'S RADIATION The General Equations of Condition d P n K d T K -J- = 7=l--T + ^ i)^-"^ log (). pO/ K - \1 O/ dp n dT 1 /T\ = -- 7 -Tfr + - r log ( _ ) d n. p K IT K I \./o/ iog The Boyle-Gay Lussac Law, with all terms variable. (234) Pv = RT. Differentiate and make the substitutions. (235) Pdv + vdP = RdT + TdR. J K 1 K (238) RdT ........................ = RdT. r ^7 T^ T^ (239) T d R = T # (n - 1) -r + ^ log ( - 1) R d T + R T log dn (240) This is the non-adiabatic form. For n = 1, we have, (241) Pdv + vdP = R a dT, the adiabatic form. BIGELOW'S FIRST THEORY OF RADIATION 309 7 rrt Since n = ~r> and g d z = Cp a d T a , we have by sub- stitution and addition, (242) Pdv + vdP + gdz = -(Cp a - R)dT a + RT log T This is the equation of condition in non-adiabatic atmos- pheres. Hence, by integration, and with P d v = d W, (243) gio (*i - 20) = - Pl ~ P - (Cp a - R w ) (T a - To) - Pio (Wi - W ) + #10 r w log (Y\ (n^ - no). (244) gio (zi 2o) = PlO R w r, log In my computations the last term in (HI n Q ) has been omitted, but more exactly it must be included. My equations for radiation depend upon , = K, while Planck's depend upon T7T~\ = J- It is interesting to note that they have such close relations as have been already pointed out, besides others, as in the examples on the following pages. It is evident that the Wien displacement law X m T = 0.2891 constant applies only to the adiabatic case, while X m T is a variable in all non-adiabatic atmospheres. It is thought that these values are too small, according to certain bolometric data, and that the X m T values for X OT in centimeters may have to be decreased in the logarithm by about 0.30000 to make the X m somewhat larger. This amounts to moving the Planck plotted curve a little to the left of its position. This discrepancy may be due to imperfections in the computation, such as the rr\ omission of the term R T log (jr J d n in the equation of con- dition. Until further progress has been made in these details the subject must be left open for study. 310 A TREATISE ON THE SUN'S RADIATION Additional Formulas It is seen that the derivatives are only for P. p . R . T, which treat of the gas from the integrated or the bulk conditions, rather than from their individual molecular constituents. The full significance of the change of the adiabatic constants to the non-adiabatic variable coefficients in atmospheres cannot be completely anticipated, especially in molecular-atomic-elec- tronic physics. However, there emerge from the preceding computations certain general formulas, of which a few may be mentioned. Cp 5 Kinetic energy in monatomic gases, K = -^- = . (245) - o NkT = - Cp . 5 Kinetic energy in gases whose K = g- is not (246) . 13 K 1 K 1 K N k T = r -kTV = HV = Cv P TV K Specific heat at constant volume Cv, 1 PV 1 NkT HV 1 11 (247) Cv = - - -- = - - -- Tfr- = = - T T- q 2 . K 1 mT K 1 mT m T K 1 3 * T The specific heat in terms of k, h, v (Wien-Planck), h V _h 2 v 2 e" T e" T - (Einstein). The inner energy in the kinetic theory of gases is related to the pressure as follows: (249) (U, - U ) K = -j (Pi - P.) = rf (A - P.) ' by the kinetic theory. (250) If (Ui - U ) Th = C w (T a - T ) Th = (Cp a - R 1Q ) (T a - T ) BIGELOW'S FIRST THEORY OF RADIATION 311 is the inner energy as derived from the non-adiabatic thermo- dynamics; the two are connected by the following formula: (251) (Ui - U ) K = P10 -- T g * - Pressure, number of molecules, and the kinetic energy. (252) P = nk T = 4- P ? 2 - PV = NkT = -^-mq 2 . o o (253) -| * T = -i- y P ? 2 = i y m $, for three degrees of freedom. (255) Po - Pi = Wio k T -rf log - = c (zi - 2) ^ Pio T w h. JV1 HI & It follows that Planck's Wirkungsquantum h can be expressed in thermodynamic terms. (256) h= - So) _w Tio Pio 2 c is the velocity of light, (zi zb) the depth of the column of gas where P is the pfessure at ZQ, and PI that at z\\ T w is the mean temperature of the column in which the gradient has the wi same value through the stratum; is half the atomic weight in Zi monatomic gases, and it, therefore, is the number of the electrons in each molecule of this kind. Hence, h represents the potential energy which exists between pairs of electrons per degree in the simplest form of the gaseous structure. This seems to be fundamentally true in solar gases, but to need an additional term in terrestrial gases. It was proved by trial computations that must be used 312 A TREATISE ON THE SUN'S RADIATION and not m, as if about one-half of the number of electrons form- ing the atom were concerned in this potential action. Since the adiabatic branch has a different function it seems to be implied that the electrons may be in another form of configura- tion, perhaps wholly dissociated under the very high prevailing temperature, more than 8000. It should be noted that in the Planck formula, wherein c . h . k are assumed to be universal constants in the exponent, , /r , such that for Cz = r = con- K \ J. K stant. For the adiabatic case, the distribution is different in the non-adiabatic case. Here L . ^ becomes T-^F and k T is a cou- rt \ 1 k 1 stant for each atmosphere under its gravitation, while h and v are both variables. It is entirely probable that the potential h between electrons of the same gas should vary in value from one level to the other while the kinetic energy is a constant, 3 1 k T = mu 2 = constant for each atom. The Variable k in All Atmospheres It has been verified throughout the computations that (k T) E has one constant value in the earth's atmosphere, and that (k T) s has a constant value in the sun's atmosphere, such that, (257) (k T) E X 7 2 = ( T) s . log T 2 = 2.89514. C. G. S. M. K. S. (258) Earth, (kT) E = 3.7145 X 10- 14 -14.56990 -21.56990 (259) Sun, (kT) s = 2.9179 X10- 11 -11.46508 -18.46508 The kinetic energy of each atom is one constant in the earth's atmosphere and another constant in the sun's atmosphere. They are so related that (260) r = T*, where T = = 28 - 028 > (*T)B g which is the ratio of the surface gravity accelerations. 2 This is correct since we assumed k T = E = constant. o BIGELOW'S FIRST THEORY OF RADIATION 313 The Atmospheric Pressure Since P = n k T, it follows that the pressure in gases depends upon the number of molecules present in the unit volume. Furthermore, (261) P = n k T pressure, (262) p = n -^ density, N (263) R = k the gas efficiency, and (264) Boyle-Gay Lussac Law,P = P RT = nkT =~ . ^ .k T. These propositions greatly simplify the theory of gases, especially in view of the probable numerical number of the m electrons = . Zi The Variable Potential Coefficient h It remains to illustrate the relation of the variable potential coefficient to the computed volume density of the radiation by the two formulas: * 4 1 IIP P (265) Thermodynamic, h B = T- r . - . =- . -^, - - > c (Zi ZQ) m 1 10 r 10 2 Bigelow. /48 TT R \ 1/8 k' 13 (266) Radiation, h P = ( -} , Planck. ^ a ' c In tables 98-102 is collected the summary of the computed values of u, using these two values of h for the sun and the earth, respectively. They are collected by temperatures, instead of by heights, since the several gases give nearly the same values of k, h, at the same temperatures, as computed. This applies to the monatomic elements and hydrogen in the Bigelow formula, but hydrogen is given separately under the Planck formula. In the Bigelow computation the exponent a is taken 4.00, as in the Stefan formula, while in the Planck formula both h and a vary in such a way as to produce very nearly the same value of the volume density of the radiation u, as can be seen 314 A TREATISE ON THE SUN'S RADIATION by comparing Tables 98 and 99. This is an additional proof that the solar radiation is black, and it has been hereby computed directly from the thermodynamic non-adiabatic data. In the case of the earth's atmosphere, there is brought out a large discrepancy in the Bigelow formula, as in Tables 100 and 101. The h B (Bigelow) is much smaller than h p (Planck), and the resulting u is impossible. On computing the correction AA, which is required to change h into hi = h -\- Ah, such that u\ = u in Tables 100 and 101, the values of Ah are seen in Table 101. In order to exhibit these relations, they are plotted in Fig. 32. The h B formula extends from -24.60000 to -20.00000 in a smooth curve; the h p extends from -26.60000 to -26.00000, thence by a sudden sal turn to 23.10000, whence it follows nearly along the h B curve. The saltum occurs, as already indicated, near the bottom of the sun's isothermal layer, and it is apparently related to the change of configuration from free electrons into atoms or monatomic molecules. The dia- tomic H 2 is intermediate. TABLE 98 THE SOLAR VOLUME DENSITY OF RADIATION *i*-a7v. Using 7lfc i i i-o - -n rB - , w) ' lB ~ ^ ( \ ' ~ T * P \.>lgClC C \Z\ ZQ) 7n JL 10 *!() ~2~ T k h a u 679.. Log. -14.65022 -14.34924 -14.10323 -15.94234 -15.82945 -15.74361 -15.66212 -15.58445 -15.59363 -15.56757 -15.51222 -15.46630 -15.42037 -15.38579 -15.34803 Log. -22.94069 -22.46783 -22.07325 -23.79901 -23.60079 -23.46848 -23.25572 -23.13332 -23.12281 -23.01921 -24.94727 -24.85549 -24.78105 -24.69859 -24.63058 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 Log. -11.88767 - 7.46057 - 6.55587 - 6.43543 - 5.93421 - 5.32002 - 4.94194 - 3.24510 - 3.33051 - 3.68471 - 3.88839 - 2.29867 - 2.37027 - 2.62641 - 2.79080 1484 2485 3719 4564 5526 6604 7629 7687 8369 9443 10511 11552 12572 13405 BIGELOW'S FIRST THEORY OF RADIATION TABLE 99 /487r/3\M & 4 /3 ,_. Using &p = ( I (Planck) For the monatomic elements 315 r k h a u 679.. 1484 2485 . -22.83036 -22.23233 23 83910 3.7556 3.9103 3 9701 -9.52655 -7.88278 5.15670 3719 4564 -23.60358 -23.45863 3.9860 3 . 9923 -5.97200 -4.33229 5526 6604. -23.34590 -23 22775 3.9975 3 9836 -4.67842 4 97341 7629. . Same 23 09578 4 0179 3.42749 7687.. 8369 9443 values of k -26.92865 -26.88846 -26.81290 2.4100 2.4185 2.4182 -3.73345 -3.87328 -3.96063 10511.. 11552 12572.. -26.75073 -26.66880 26 63955 2.4222 2.4189 2 4217 -2.26759 -2.27328 -2.33339 13405 26 59442 2 4191 -2.36464 For the diatomic hydrogen T k h a u 7687.. 8369. Same -24.06865 24 00975 3.3316 3 3468 -3.89554 2 15091 9443 10511 values of k -25.94549 -25 88354 3.3483 3 3490 -2.04349 -2.59640 11552 25 82674 3 . 3487 -2.59076 TABLE 100 THE TERRESTRIAL VOLUME DENSITY OF RADIATION 487T/3 fr , c* ' h 3 ' 1 J_ J_ Po-Pi L - zo) ' m_ * Tio ' 2 u = a T-. Using h B (Bigelow) z T k h a u Meters Log. Log. Log. 66000... 52.0 -15.03573 -20.99536 4.000 -26.80222 56000... 86.7 -16.71084 -19.14377 4.000 -27.94551 46000. . . 109.3 -16.55127 -20.83233 4.000 -26.64395 36000... 216.3 -16.23881 -20.22718 4.000 -24.37532 26000... 222.5 -16.22207 -20.19276 4.000 -24.48070 16000... 217.3 -16.23216 -20.22384 4.000 -24.38674 6000... 257.0 -16.16597 -20.06449 4.000 -24.89151 2000. . . 279.8 -16.12525 -21.99349 4.000 -23.08931 500. . . 285.5 -16.11579 -21.98890 4.000 -23.10028 316 A TREATISE ON THE SUN'S RADIATION TABLE 101 /48x\M 4 /3 Using h P =( 1 ._ (Planck). z T k h a u 66000... 52.0 -25.47579 2.737 -11.19360 56000... 86.7 -26.62427 3.399 - 8.33910 46000... 109.3 -26.26206 3.601 - 7.54148 36000... 216.3 Same -27.86323 3.577 - 6.49953 26000. . . 222.5 values -27.77386 3.668 - 6.95808 16000. . . 217.3 of* -27.71347 3.768 - 5.36961 6000. . . 257.0 -27.58766 3.819 - 5.88588 2000. . . 279.8 -27.52770 3.826 - 4.06088 500... 285.5 -27.51076 3.829 - 4.11486 TABLE 102 Correcting Bigelow's Formula, hi = h + A k z r Ah hi a Ml 66000... 52.0 -6.20287 -24,19823 4.000 -11.19361 56000... 86.7 -7.86880 -25.01257 4.000 - 8.33911 46000... 109.3 -7.70082 -26.53315 4.000 - 7.54149 36000... 216.3 -7.96526 -26.19244 4.000 - 6.49954 26000... 222.5 -7.84087 -26.03363 4.000 - 6.95809 16000... 217.3 -7.67271 -27.89655 4.000 - 5.36861 6000... 257.0 -7.66854 -27.73303 4.000 - 5.88589 2000... 279.8 -7.67614 -27.66963 4.000 - 4.06089 500. . . 285.5 -7.66175 -27.65071 4.000 - 4.11485 Extend the branch of the h p curve to 24.30000. Since the transformation of energy is accompanied by an emission of radiation, and an accession to the potential energy, we may regard the area between h B and h p as a measure of this energy of stored potential in the structure of the atoms. In the earth's atmosphere, the h p curve appears at the top of the diagram, 27.50000 to 25.00000, while the other curve k B is near the vanishing point of the solar curves at 21.00000. The differences between h B and h p are given in Table 102. It will be desirable to investigate further the relations between these formulas. BIGELOW'S FIRST THEORY OF RADIATION THREE THEORIES OF RADIATION. 317 log. h -20,00000 -21.00000 -22.00000 -23.00000 -24.00000 -25.0COOO -26.00000 -27.00000 The variable potential energy of radiation h. 1 1 1 Po-Pi I, Non-adiabatic, hy = (Zi So) W C 2 Hrtl i i /48 7T /3\ ' 3 w ' , Planck, hp = ( . \ a J c III, Adiabatic, H A = - ^^"^ - - (Vi Vo) i0 n v m FIG. 32. The Non-Adiabatic and the Adiabatic Branches of the Curve of the Functions of Solar Radiation h. 318 A TREATISE ON THE SUN'S RADIATION Certain Relations Between the Adiabatic and the Non-Adiabatic hv Values of y-y . It is proper to study the factors which are required for reducing the non-adiabatic values of h v/k T in the solar and terrestrial atmospheres to those required to conform to the Planck formula, as developed for the variables (k . a). Table 103 gives the factor of reduction for h B , which is A c . 7 3 . TABLE 103 EVALUATION OF ( -^ I. Terrestrial Conditions. T hB hB*m log h v 1 ( kV \ Ac.y 3 og m ag Ac.y* l g \kT) B 52 -26.51601 12.73207 -13.24808 -13.69795 0.67818 1.12805 86.7 -26.66442 12.95409 -13.61851 -13.45265 1.04861 0.88275 109.3 26.35298 13 05469 13 40767 13 34496 83777 77506 216 3 ... 27 74783 13 35113 13 09896 13 20430 52906 63440 222.5 -27.71341 13.36340 -13.07681 -13.17228 0.50691 0.60238 217.3 -27.74449 13.35313 -13.09762 -13.09123 0.52772 0.52133 257.0 -27.58514 13.42600 -13.01114 -14.99801 0.44124 0.42811 279.8 -27.51414 13.46292 -14.97706 -14.99233 0.40716 0.42243 285.5 -27.50955 13.47168 -14.98123 -14.98743 0.41133 0.41753 II. Solar Conditions. 273 -21.50777 13.45223 -8.96000 -8.75200 3.49492 3.28629 679 -22.94069 13.84794 -8.78863 -8.59493 3.32355 3.12985 1484 -22.46783 14.18750 -8.63533 -8.33040 3.17025 2 . 86532 2485 -22.07325 14.41140 8.48465 -8.20021 3.01957 2.73513 3719 23.79901 14.58650 8.38551 8.12100 2 . 92043 2 . 65592 4564 -23.60079 14 . 67542 -8.27621 -8.09364 2.81113 2 . 62856 5526 -23.46848 14.75848 -8.22696 -8.05470 2.76188 2.58962 6604 -23.25572 14.83588 -8.09160 -8.04626 2 . 63652 2.58118 7629 -23.13332 14 . 89854 -8.03186 -8.01561 2.56678 2.55053 Below the Isothermal Layer. 7687.. -23.12281 14.90183 -11.88806 -11.80980 0.42298 . 34472 8369 -23.01921 14.93874 -11.82137 -11.78979 0.35629 0.31471 9443 -24.94727 14.99118 -11.80187 -11.77086 0.33679 0.30578 10511 -24.85529 15.03771 11.75662 11.75561 0.29154 0.29053 11552 -24.78105 15.07848 -11.72295 -11.73234 0.25687 0.26726 12572 -24.69859 15.11548 -11.67749 -11.72891 0.21241 0.26383 13405 -24 . 63058 15.14335 -11.63735 -11.71841 0.17227 0.25333 The entire system is reduced to the equivalent of the ex- ponent 4.00, as in the Stefan Law. THE WIEN DISPLACEMENT LAW 319 In the terrestrial section I the divisor A c . y 3 consists of the mean factor Ac = 2.13658 required to reduce from the coeffi- cients corresponding with the exponent 2.42 to that for 4.00 and 7 3 . Since A c.y 3 is equal to a little more than y 4 = y 4/3 x 3 for h z in the equation, it is possible that y 4 is the fundamental reduc- tion for h from the solar adiabatic layer to the terrestrial non- the Bigelow (~TJ\ , and the Planck \~rf\ , as derived from adiabatic layer. The values of 7-= are thus nearly the same in , and the Planck \~rf their respective formulas for each. The Planck formula is some- what indeterminate, because it contains the unknown coefficient (48 TT a k*\ 1/3 --- i) y v m refers to the frequencies (J, Cr ' 2891 for the maximum wave lengths, X m = . The Variations in the Wien Displacement Law The Wien Displacement Law is reduced to the form \ m T = . -j-j and since (h, k) are variables it follows that \ m T must p K vary with them. For the selected temperatures T, compute -r and \ m T in succession, and finally the corresponding maximum fv wave length ^ m in terms of (/A). On multiplying \ m T, in \(cm), the displacement constant is about 0.1465 in the sun's non- adiabatic strata; about 0.1185 in the sun's adiabatic strata, about 0.2891 in the higher levels of the earth's atmosphere, but gradually diminishing to about 0.1500 near the sea level. This suggests that the Wien Displacement Law for (h . k) variables is only relatively a constant under special thermodynamic con- ditions, and that these differ from one body to another. The law was deduced as a function of (X.T), but it must be made a more complex function of (h . k . X . T). This difficult subject will require further investigations. 320 A TREATISE ON THE SUN'S RADIATION TABLE 104 THE WIEN DISPLACEMENT LAW, \ m T = - .-T-. r h k Am T A (M) T h k Am T Am (M) Sun's Atmosphere Earth's Atmosphere 220. . -11.40081 3.18137 6.912 52.0 -10.44006 4.22125 320.00 283. . -11.39881 3.18000 5.348 86.7 -11.91343 3.69462 45.35 679. -11.38978 3.17097 2.183 109.3 -11.71079 3.49198 28.40 1484. -11.39576 3.17695 1.013 216.3 -11.62442 3.40561 11.76 2485. -11.37677 3.16796 0.592 222.5 -11.55179 3.33298 9.68 3719. -11.38766 3.16885 0.388 217.3 -11.48131 3.26250 8.42 4564. -11.39555 3.17674 0.329 257.0 -11.42169 3.20288 6.21 5526. -11.38639 3.16758 0.266 279.8 -11.40245 3.18364 5.46 6604. -11.36788 3.14907 0.213 285.5 -11.39497 3.17616 5.26 7629. -11.34555 3.12674 0.176 7687. -11.33502 3.11621 0.170 8369. -11.32089 3.10208 0.151 9443. -11.30074 3.08193 0.128 10511. -11.28443 3.06562 0.111 11552. -11.25189 3.03308 0.094 12572. -11.25376 3.03495 0.086 13405. -11.24639 3.02758 0.080 c = 3 X 10 10 10.47712; = 4.9651 0.69593; , 13.78119 (M) Another Formula for the Quantity h B , Computed from Thermody- namic Data We will proceed to develop a series of formulas which will connect together definitely the radiation function that has been used, - , and the thermodynamic potential and kinetic energies, together with the radiation terms which enter the Wien-Planck formula. Furthermore, it will be possible to derive the terms of the ionization which make the free electricity of atmospheres, and its relation to the variable potential energy h, and the wave length or the wave frequency related thereto. It may be stated that there is a complete confirmation of the results of the discussion of the radiation as developed in the preceding chapters, so that the data are thoroughly self-con- sistent, and can be derived from at least three independent methods of computation. They throw much light upon the problems of atomic physics, which will be briefly mentioned. BIGELOW'S SECOND THEORY OF RADIATION 321 The Planck formula for the Wirkungsquantum h p was de- rived from a statistical analysis of the probable distribution of the elementary oscillators, coupled with the thermodynamic data. We have attempted to produce similar results by re- ferring to the pressure distribution of the electrons in the atoms of different weights. There has evidently been a partial success, since one branch of the solar curve has been obtained. There was, however, a difference in the extension to the adiabatic strata in the sun, and there was a wide discrepancy in applying the formula to the earth's atmosphere. Another formula has been derived from the fundamental equations of thermodynamics, in which it is the molecules that are considered as the units, rather than the atoms or elec- trons. The purpose has been to compute the amount of the potential energy which is expended in one oscillation of a molecule during the passage of one wave. Hence, the rate of the change of the potential energy per unit variation in the volume, the number of molecules in the unit volume, and the wave frequency are the significant terms. A large number of primary formulas are developed in terms of n, the number of molecules per unit volume, and N y the nurhber of molecules per unit mass. It is possible to develop the Boyle-Gay Lussac Law and the First Law of Thermodynamics in terms of n and N, SQ that these two fundamental quantities are the proper bases for all thermody- namic discussions of the problems of radiation. The Kinetic and the Potential Energies in Radiation, Determined from Thermodynamics It is evident that the complete relations between the radia- tion and the thermodynamics of atmospheres must be expressed in the terms of the kinetic and the potential energies. Specifically, it is necessary to compute the values of k, h, in the Wien-Planck formulas directly from the primary atmospheric quantities. The following development shows how the Boyle-Gay Lussac Law and the First Law of Thermodynamics can be expressed in terms of n, the number of molecules in the unit volume, and N, the number of molecules in the unit mass, in place of 322 A TREATISE ON THE SUN'S RADIATION the usual P, p, R, T. We have the following thermodynamic definitions : (267) Boyle-Gay Lussac Law. P = pRT. (268) Pressure, P = nkT. dP = k T.dn, for kT= constant. n (dn n dN\ (269) Density, p = m -j. d p = m (-^r - -^ . ) . 1 N 1 fdN N dn\ (270) Volume, v = -- . dv = ( ----- ). m n m \ n n n ' (271) Efficiency, R = k -. dR = ~ (kdN + Ndk). (272) Kinetic Energy, PV=mRT = N . k T = K T. (273) Specific Kinetic Energy, Pv = RT = N . ^ = ^ T. Differentiate and substitute in succession : (274) Pdv + vdP = RdT + TdR = R d T a . m n m n n m m n (277) RdT=^-NdT=^N^f. Ill ill J. (278) TdR= (kdN + Ndk) = (dN + N^ m v m V k /- \ kT ( J *r dn , *r dn \ kT j (279) Pdv+vdP = [d N N h ^ ) = d N. m V n n / m ^ ' m \ T k ' m The last step comes from k T = Constant, so that, This checks the differentiated equation. Integrate each term for the First Law of Thermodynamics: (282) Work, (Wi - Wo) = P w (f, - n,) = ( - No) - -l J BIGELOW'S SECOND THEORY OF RADIATION 323 (283) Hydrostatic) P^Po = = k_T ,-*, pressure, J p 10 m w (284) Efficiency I, R 1Q (T, - T ) = N 10 Ti ~ TQ . m IIQ (285) Efficiency II, T w (& - R ) = F(#, - # ) + #10 m i (286) Efficiency III, i?io (Z\ - T ) + Ti (A - #>) = ^10 . (T a -T Q )=^(N,-N Q \ (287) Free Heat, Q l - Q = Cp a (T a - T ) - Cp 1Q (T a - T ). -To ao K 1 io k T Ni- No nio = - g fa - 2 ) + K n w k T (vi - V ). It is easily proved that, dn K dN I I dN /oon\ l~O i o (289) - = - - 5^ - .! %= . n w K 1 Nio K 1 m which were used in the last transformations. (290) Work, Wi-W Q = R lo (T a - T ) - Cp 10 (T a - T ). n w kT K 1 m HIQ = HIQ k T (Vi Z> ). Total Inner energy (291) Ui-U, = Cp a (T a - To) - ft, (r. - To). = - g ( Zl - z} - ~ (N, - No). . Ni - No -r (*-*)- - -;-. = g(zi z ) + HIQ k T (K 1) fa 324 A TREATISE ON THE SUN*S RADIATION From the last equation the radiation function is found: Radiation function rr f \ i f i\ L T> = AW = g (i - Zo) K . m -TT- . - - + (K - 1) n w k T -tVio HI HQ = g Pio (21 Zo) K - - + (K 1) ttio & r. Wi WQ The density of the radiation is the potential energy together with the kinetic energy for the mass in the column (zi z ), g PIO (21 Zo) = Mioj which contains HIQ molecules. We may assume that this potential energy is also expressed by n 10 h v, where v = the frequency, v = -, and h is the potential energy A per oscillation in the path of the ray. Hence, we have, using the first form, (293) n 10 h v = + g , ",, so that, (fli VQ) (294) ^ = + ,fe^).A.l. 5 (Vi - VQ) n w v If we assume the Wien displacement law, (295) vm T m = 0.2891 (Planck), we have, rr\ (296) "" = i = ' HenC6 ' (297) , = +g 6 where the temperature is that of T m = Va c m 2891 (AQ-r^xa J? a ^ -) -^ ........... ...... Planck's formula. (299) h B = g / 2l ~ ^ .. . .Bigelow's formula. 6 (fi - Po)w n v m h B is the change in the potential energy per unit volume, per molecule, per wave vibration, and it therefore is the mean am- plitude of the potential energy of each molecule in one oscillation. BIGELOW'S SECOND THEORY OF RADIATION 325 Planck's Wirkungsquantum should be interpreted in this manner. The values of h increase with decreasing temperatures, and h is a wide variable. v m can be computed for each T m . The diagram of h B and h p , Fig. 32, shows that the curves are similar except that the amplitude of h p is considerably greater than that of h B at the isothermal layer. TABLE 105 THE MOLECULAR POTENTIAL ENERGY IN THE EARTH'S ATMOSPHERE Balloon Ascension, Uccle, September 13, 1911. C. G. S. g (zi - o) r . . for n v m C.T m 0.2891 ATMOSPHERIC AIR, m = 28.736 ATMOSPHERIC AIR, m = 28.736 r Logh B Logh p T LogA s Log h p 6... 11 16 21 25 29 33 37 41 45 49... 53 57 61 65... 68 -23.26522 -24.54772 -24.36712 -24.18255 -24.02646 -25.88035 -25.75124 -25.63559 -25.53110 -25.43576 -25.34825 -25.26741 -25.19222 -25.12184 -25.10459 -26.99386 -26.93365 -26.87620 -26.82071 -26.76907 -26.71771 -26.66646 -26.61849 -26.57217 -26.52723 -26.48460 -26.44156 -26.40049 -26.36057 -26.32076 -26.28411 -26.24687 -26.20997 -26.17416 -26.14021 -26.10516 -26.07145 -26.04097 -26.01089 -27.98406 -27.95397 -27.93361 -27.91444 -24.87100 -24.38936 -24.06853 -25.83527 -25.67431 -25.53936 -25.42188 -25.31896 -25.22653 -26.14329 -25.06905 -25.00024 -26.93678 -26.87784 -26.82108 -26.77527 -26.73127 -26.68964 -26.64962 -26.61150 -26.57385 -26.53964 -26.50511 -26.47190 -26.44065 -26.40964 -26.38119 -26.35157 -26.32412 -26.29646 -26.27335 -26.25144 -26.22934 -26.20745 -26.18839 -26.16340 -26.13304 -26.09912 -26.05750 -26.00915 -27.94784 -27.90550 -27.87792 215 . -27.89619 -27.87693 -27.85846 -27.83852 -27.81995 -27.80147 -27.78081 -27.75996 -27.74117 -27.72689 -27.70765 -27.68779 -27.66777 -27.64765 -27.62869 -27.61086 -27.59003 -27.57062 -27.55197 -27.53594 -27.51127 -27.49137 -27.47081 -27.45316 -27.43505 -27.41406 -27.39562 -27.37797 -27.36947 -27.34308 -27.32628 -27.30934 -27.29594 -27.28524 -27.27666 -27.26710 -27.25455 -27.24332 -27.19248 -27.17990 -27.86071 -27.85048 -27.84036 -27.83007 -27.82077 -27.81150 -27.80257 -27.79344 -27.78630 -27.77903 -27.77231 -27.76609 -27.76096 -27.75494 -27.74904 -27.74384 -27.73645 -27.73111 -27.72509 -27.72083 -27.71711 -27.70810 -27.70350 -27.69099 -27.67917 -27.66168 -27.64418 -27.62738 -27.60787 -27.59043 -27.57564 -27.56082 -27.55130 -27.53482 -27.52762 -27.52060 -27.51321 -27.50851 -27.50438 -27.49951 216 7 218.5 220.0 221 222.0 223 0. . 223 5 224.0 223 6 223 . 1 . . 222.8 221.7 221.2 220 4 220.1 219.8 219.0. . 218.0 217.1. . 71 74 77 80 83 86 216 216.5 217 . 1 218.3 220.5 227.0 233.7 240.3 248.1 255.1 261.7. . . 267.4 270.9 278.4. . . 281.4 284.5 287.2 289.0 290.4 292.5 89... 92 95 98 101 104 107 110.. . 112 114 116 118 120 124 130 138.. . 150 166 187. 202 211 326 A TREATISE ON THE SUN'S RADIATION TABLE 106 THE MOLECULAR POTENTIAL ENERGY IN THE SUN. r r c i, g (i go) 11, c c . T m C. G. S. KB -, r . . for v m = rtor ,, (vi-vo)io n v m \ m 0.2891 HYDROGEN 1.00 HYDROGEN 2.00 HELIUM 4.00 CARBON 12.00 r Log* a T Logh B r Logfe B r Log/ B 160 -20 . 10922 250 -21.94535 245 -21.86380 240 -21.99583 330 -21.52664 415 -21.21621 400 -21.90457 500 -21.39246 500 -22.94440 490 -21.29088 585 -22.70645 610 -21.18744 675 -22.49559 750 -22.94654 765 -22.35991 740 -22.86200 860 -22.14042 850 -22.72569 955 -23.99073 1025 -22 . 54442 1055 -23.85296 1000 -22.46554 1155 -23.73776 1120 -22.25270 1300 -22.21134 1260 -23.65236 1270 -22.14394 1365 -23.50624 1475 -23.40681 1430 -23.89265 1575 -23.90338 1585 -23.31500 1550 -23.87751 1700 -23 . 15858 1850 -23.63666 1815 -23.15270 1830 -23.64833 1790 -23.62562 1935 -23.07209 2055 -23.00079 2150 -23.51698 2180 -24.93299 2110 -23.45087 2190 -23.38746 2305 -24.86936 2475 -23.34117 2435 -24.80839 2400 -23.27727 2565 -24.72080 2630 -23.19183 .2 2695 -24.69632 2710 -23.12253 2 2800 -23.18871 2835 -24.64426 35 2985 -24.59380 3040 -24.98431 3150 -23.05115 3145 -24.52125 3110 -23.02073 'is 3315 -24.50106 3500 -24.92993 3495 -24.45832 3390 -24.86087 3620 -24.97693 3 3685 -24.41730 3780 -24.73894 ^J 3900 -24.81793 3885 -24.37848 g 4095 -24.34152 4200 -24.64921 4160 -24.75180 & 4300 -24.71832 4315 -24.30639 4545 -24.27298 4700 -24.62845 4785 -24.24133 4680 -24.55788 4720 -24.64203 5045 -24.21100 . 5150 -24.54440 5325 -24.18455 5180 -24.47627 5300 -24.54450 5600 -24.46466 5610 -24.15442 5890 -24.12830 5700 -24.40198 5900 -24 . 45604 6100 -24.39754 6160 -24.10343 6230 -24.33423 6420 -24.07941 6450 -24 . 37770 6600 -24.33859 6660 -24.04641 6750 -24.27258 6880 -24.02639 7000 -24.31002 7100 -24.27664 7218 -24.00748 7300 -24.21116 ^ 7450 -24.21528 7500 -25.98655 7500 -24.23340 c 7650 -24.16071 7685 -25.96720 7600 -24.15831 rt 7690 -24.10892 7680 -25.95259 7715 -24 . 10382 7715 -24.16797 >-* 7701 -24.05411 7675 -25.93409 7705 -24.05359 7705 -24.10742 13 7691 -24 . 00129 7670 -25.90632 7695 -24.00160 7695 -24.04491 7681 -25.94928 7665 -25.88558 7685 -25.94759 7685 -25.97958 fc 7671 -25.89508 7660 -25.82240 7675 -25.89577 7675 -25.91712 -*j 7661 -25.84133 7652 -25.90050 7665 -25.83907 7665 -25.84929 1 7652 -25.72315 7662 -25.99194 7652 -25.78725 7652 -25.78587 7665 -25.75970 7676 -25.78850 7640 -25.74790 7640 -25.70234 7652 -25.71170 7665 -25.67275 7750 -25.66468 7690 -25.73580 7690 -25.63693 7680 -25.73808 v S 8000 -25.59774 35 8476 -25.67732 8500 -25.56087 8500 -25.68037 8550 -25.59687 8951 -25.65733 9187 -25.52482 9451 -25.63315 9691 -25.53836 9427 -25.63427 9874 -25.49587 1 9902 -25.62351 10561 -25.46506 10402 -25.59186 10833 -25.49080 10378 -25.59313 11248 -25.43779 11353 -25.55426 3 10853 -25.57395 11935 -25.42004 12304 -25.51977 11974 -25.44933 * 13255 -25.48769 13115 -25.40844 14207 -25.45774 14257 -25.41550 15158 -25.42963 15398 -25.39486 BIGELOW'S SECOND THEORY OF RADIATION 327 TABLE 106 Continued THE MOLECULAR POTENTIAL ENERGY IN THE SUN. CALCIUM 40. 00 ZINC 64.85 CADMIUM 111.51 MERCURY 198.41 T Logh B T Log/ B r Log/ B T Logh B 230 -21.55292 300 -21.40258 300 -21.51639 420 -21.14156 550 -22.72159 600 -22.76097 600 -22.86002 640 -22.68030 900 -22.37932 890 -22.30522 900 -22.44028 850 -22.35401 1200 -22.03800 1180 -22.00821 1200 -22 . 06172 1150 -22.01314 1500 -22.75701 1480 -23 . 77339 1500 -23.74720 1450 -23.72689 1850 -23.51871 1790 -23.57788 1800 -23.48390 1800 -23.48438 2200 -23.32328 2110 -23.41177 2150 -23.25608 2150 -23.28636 2440 -23.26827 2550 -23.15603 2500 -23.07288 2550 -23.11613 2780 -23.14226 2950 -23.00686 2900 -24.97003 3000 -24.97029 3120 -23.03067 3350 -24.87851 5 3460 -24.93021 3400 -24 . 76029 3450 -24 . 84759 B 3750 -24.76517 C C/3 3800 -24 . 83917 3900 -24.68863 3950 -24.73723 u 4200 -24.66204 4140 -24.75554 ' 4480 -24.67869 JS 4400 -24.52947 4450 -24.64143 S 4700 -24.56977 4830 -24.60720 1 5000 -24.43197 4950 -24.55546 5225 -24.48668 5190 -24.54078 5560 -24.47676 5450 -24.48681 z 5750 -24.39883 5940 -24.42049 5700 -24.34787 5900 -24.40644 6300 -24.33804 6320 -24.36625 6350 -24.33924 6700 -24.31533 6600 -24.26731 6800 -24.29567 6750 -24.27724 7200 -24.23752 7100 -24.26657 7300 -24.20161 7100 -24.21855 u 7500 -24.14207 7500 -24 . 22694 7400 -24.16189 >l 7715 -24.12398 7745 -24.17468 7650 -24.14205 7600 -24.11135 3 7705 -24 . 07731 7735 -24.13613 7670 -24.06122 7695 -24.02532 7725 -24 . 09650 7690 -24.08396 7667 -24.00043 13 7685 -25.96788 7715 -24.04370 7684 -24.02540 7663 -25.94067 & 7675 -25.90878 7705 -24.00437 7677 -25.96701 1 7665 -25.81193 7695 -25.96132 7669 -25.90505 7659 -25.88993 7652 -25.94886 7685 -25.91737 7660 -25.84675 7656 -25.83591 S 7675 -25.87538 7652 -25.79100 7652 -25.78992 11 7665 -25.82680 7652 -25.78635 7640 -25.74322 7700 -25.69642 7700 -25.70688 7720 -25.73446 7800 -25.74041 d 8400 -25.65650 8500 -25.67005 8550 -25.68319 2 9171 -25.61771 S 9600 -25.57183 9942 -25.58258 9561 -25.61561 9494 -25.63361 .0 10713 -25.55075 10621 -25.57053 10437 -26.59501 11504 -25.49726 11484 -25.52071 11682 -25.52968 11381 -25.55770 1 12255 -25.49272 12742 -25.49247 12324 -25.52798 3 13406 -25.43299 13026 -25.46650 13803 -25.45793 13268 -25.49152 2 15308 -25.37649 14864 -25.42503 14211 -25.46186 ^ 17210 -25.32660 15924 -25.40965 15155 -25.43731 19112 -25.28173 21014 -25.19050 328 A TREATISE ON THE SUN'S RADIATION It should be remembered that in computing, the value of a in the denominator is the same as the coefficient c in the quasi-Stefan Law (301) K M = cT b and that the coefficient c and the exponent b were obtained by the method of trials. Since these vary together, and b is about 4 . 00 in the non-adiabatic layers, and about 2 . 42 in the adiabatic layers, it follows that the mutual adjustment of c, b, will produce the same values for h by the Planck and the Bigelow formulas, for the same value of - .It will be necessary to study more vi - v fully the relations between them before reaching any opinion as to their meaning. But it is clear that they have the same hor- izontal branch in the isothermal layer, and that they can be re- duced to the same values in the adiabatic and non-adiabatic layers. It is seen, furthermore, that if the line radiation in the spectrum of any element is due to a definite value of h, which represents a given pressure and temperature, then the series of spectral lines will be spaced as commonly observed, by giving h certain fixed increments in steps. In this case the series of lines of an element is due to its independent radiations at such levels as produce the given P. T., and if the scale can be fixed these lines can be located in the levels 26, Zi, z In the earth's atmosphere the agreement between h p and h B is very close, and these small variations can be adjusted through the c coefficient and b exponent. Summary It should be noted that the h B equation in its expanded form, /orkoN 7 K kT TO. TQ , . HIQ (302) h B = 7 . N a f . (K - 1) m n TT . K l m l a Q iv i iV o J_ 0.2891 * x- T* ' 10 C 1. rn T TV. A7 H OQQ1 (303) h B = K.kT T a -T N a 0.2891 AY- .# ' ^ r m ' DERIVATION OF CERTAIN FORMULAS 329 reduces finally to terms in (T, N), and includes the Wien Displace- ment Law. A new system of physics can be founded upon (T, n, N) since these include the potential energy h, and the kinetic energy k for each oscillation. Whether the molecule and the atom are constructed as electric oscillators (Planck), or as a large volume nucleus of positive charges (Thomson), or as a small condensed positive nucleus (Rutherford), the h, k, are not coefficients of the internal structure, but of the outer forces of col- lision between independent molecules. The secondary effects of such collisions in producing electromagnetic radiation, and vibra- tory adjustments of structures between the positive and the negative charges, lead to the fundamental problems in physics, chemistry, and the constitution of matter. It is hoped that some further progress can be made in this direction by the study of the spectral series, the atomic numbers and allied problems, from the point of view that h, k, n, N, are all primary variable quantities in the standard equations of thermodynamics and electromagnetics. Derivation of Certain Formulas for Radiation, lonization, and Atmospheric Electricity It seems that the fundamental formula for the radiation, (304) 4r is competent to produce the Wien-Planck formula for radiation, though the terms are differently interpreted; the fundamental relation between ionization, potential energy, and frequency of the vibration; and the quantity of the electric charge in the unit volume. Generally, we have for the single #-atom, of which there are n w in the unit volume, the potential energy = hv = g(zi ZQ) (K 1) m -^ '~^r, and the kinetic energy = IV 1 l\o (K 1) k T, so that, for single F-atoms, (305) Ul ~ U = h v + ( K - 1) k T, Vi VQ as the primitive form, whereas Planck assumed, (306) u = Ul ~ U " = h v + kT. 330 A TREATISE ON THE SUN'S RADIATION This will lead to the Planck form of the function with the change of 1 to K 1, that is, 0.666 for the monatomic gases and approximately 0.410 for the diatomic gases. Thus, (K 1) In this formula it is k T which is the variable constant in any given atmosphere, while h v is the variable potential energy. It has become evident that this potential energy is external to the atoms and molecules, such as occurs in the thermal colli- sions between atoms or molecules consisting of m unit charges e. Hence the theory of the generation of radiation is transferred from the saltum changes within the circulating orbits of an atom into the mutual repulsions of the atoms and molecules in the col- lisions which integrate into the existing pressure under the force of gravitation. Evidently the potential energy changes from level to level. lonization, Potential Energy, and Frequency It will be convenient to make the following transformations: 1 N From the equation for the specific volume, v = -- , we m n ' have, , N 1 fdN N dn\ (308) d v = I ---- ) . Integrate, m \ n n n / /onn x / x 1 Ni N (309) (Vi fl ) = m = J_ m m I 1 A T io ni n \ nw 10 /' n w K - 1 N W tf i - No\ K i J j No K 1 ' m nw Substitute in the general equation of radiation (304) , (310) = g -7-^ r- + n w k T, for nw charges of e. fli ^o \y\ Vo)w Hence we shall assume that the potential energy of one H- atom in thermal collisions is, IONIZATION.. POTENTIAL ENERGY. AND FREQUENCY . 331 (311) hv=- g ( Zl -z Q ).~.or, (312) h v = + g (zi - z ) (K - 1) m that is, in terms of the number of ^-charges per unit volume n, or per unit mass N. We can separate this equation into the terms for the ioni- zation, the potential energy, and the frequency changes, as follows : Take the second differences of (Ni No), fhdv 4- v dh\ (313) d (Ni No) = g (zi Z ) (K 1) m ( ^ 9 ) . v n v ' Integrate, (314) A (Ni - No) = - g (zi - z ) (* 1) f JT~ Since for one //-charged atom, (315) (Ni No) = + g (zi Zo) (K 1) m 7- - on the average, A (Ni -No) hi- ho vi - PQ N^No h 1Q v lQ This equation is valid in all atmospheres. Compare the Tables 107, 108, and 109. Atmospheric Electricity The potential equation is immediately applicable to the problem of the origin of atmospheric electricity and the quantity of electric charges in the unit volume by adopting the Einstein formula, (317) h v = Ve. The fundamental electrostatic equations are, (318) Potential = V = . (319) Electric force = E = - ^ = ~. d r r (320) Work of potential = W = Ve = = hv. 332 A TREATISE ON THE SUN ? S RADIATION The preceding development assumes that the fundamental charges (e . e) are alike, bemg in repulsion during the collisions and along the free path, so that we have only to transform the expressions for h v into volts per unit distance in order to com- pute the accumulated charge in the stratum (z 2 Zi). W (321) Q = -^- = (F 2 - FO (*, - Zl ). We have 1 E. S. U. = 300 volts. 1 Volt = 0.00333 E. S. U. = 0.00333 . Hence, cm. 3 , , Volts Volts (322) - = -- meter 100 cm. Preliminary trials on the data of atmospheric electricity indicate that m must be identified with the molecular weight in the earth's atmosphere, m = 28.74, approximately. Thence the formula for Q in successive strata gives the quantities which when summed together produce the electric charges on each level above the plane of reference, as the earth's surface or the top of the sun's adiabatic strata. The results are indicated in Tables 110 and 111. Tables 107, 108, and 109 give the results for the terms, T -- , the rate of change of the potential in the stratum, "10 , the rate of change of the frequency in the stratum, A ^ _ , the rate of change of the ionization in the stratum, for the balloon ascension, Uccle, September 13, 1911, in the terrestrial atmosphere, and for helium and zinc in the sun's atmosphere, the other elements being similar in their develop- ment. Figs. 33 and 34 show the relative values for the earth's atmosphere, m = 28, and for the solar atmosphere of helium, ATMOSPHERIC ELECTRICITY 333 m = 4.00. The potential is the value which occurs in each wave length and its oscillation so that it can be analyzed accord- ing to the formulas of Chapter I. The ionization term marks the disintegration of the mass according to the changes in the density per unit mass. These data, therefore, become of primary importance in studies of the atomic physics. In the earth's atmosphere the convectional, isothermal, and non-adiabatic regions are well defined. At the top of the isothermal strata there is a rapid change in the frequency and ionization, while the potential increases quite steadily in value. In the solar atmosphere there is no ionization in the adia- batic strata; there are sudden large changes in the frequency and ionization at the isothermal layer, including the photosphere, where the radiation is especially generated; in the non-adiabatic region they increase together to very large values. Figs. 33, 34. It should be remarked that Abbot's value of the solar constant of radiation, corresponding with the temperature 5810, finds no support in these results. They are in harmony with the view that the solar radiation originates at the temperature of 7655. The Terrestrial Atmospheric Electricity There are many hypotheses in the literature of atmospheric electricity regarding the origin and the distribution of the power- ful electric charges which are a characteristic and permanent constituent of all atmospheres, but they are generally deficient and unsatisfactory. We shall employ the Einstein relation, Ve = h v, as the electrostatic form of the prevailing potential energy of radiation, reducing it to the form, Volts 300 = h v. 100 m, for voltage, and meter e (Qi ~ Co) = (21 - z ) (Fi - Fo), for quantity. Table 110 contains the results for (Fi- F ), (Qi - Co) arranged by the arguments z and T. The values of (Fi Fo) have three distinct stages: (Fi Fo) varies between 1.00 and 2.00 in the convection strata; it increases in the isothermal region to about 10.00 at 334 A TREATISE ON THE SUN S RADIATION TABLE 107 UCCLE, SEPTEMBER 13, 1911 -A (Ni - N ) hi -h i No z hi -ho V\ VQ Cum -A(ATi-ATo) Meters hio V10 oil in Nt-No 80000 o.o 78000 11.0 0.40997 -.37017 .02980 .34468 76000..... 21.0 .35561 - . 17392 . 18169 .25819 74000 29.0 .30862 - . 12903 . 17959 . 18751 72000 37.0 .23946 - . 10255 . 13691 . 15143 70000 45.0 .20080 -.08517 .11563 . 12360 68000 53.0 .17266 -.07271 .09995 . 10518 66000 61.0 .15453 -.06348 .09105 .10040 64000 68.0 . 14302 -.04318 .09984 .09899 62000 74.0 .12761 -.03972 .08789 .09122 60000 80.0 .11813 -.03682 .08191 .08354 58000 86.0 .11035 -.03428 .07607 .07876 56000..... 92.0 . 10339 -.03210 .07129 .07437 54000 98.0 .09032 -.03012 .06020 .06923 52000 104.0 .09166 -.02841 .06325 .06514 50000 110.0 .08437 -.01806 .06631 .07018 48000 114.0 .08491 -.01735 .00756 .06761 46000 118.0 .07813 -.01701 .06712 .05769 44000 124.0 .07763 -.04727 .03036 .02031 42000 138.0 .06920 -.08318 -.01398 -.02941 40000 166.0 .06925 -.11897 -.04972 -.04545 38000 202.0 .04413 -.04359 .00054 .00177 36000 215.0 .04433 -.01017 .03416 .03823 34000 218.5 .04590 -.00681 .03909 .03830 32000 221.0 .04254 -.00452 .03802 .04068 30000 223.0 .04800 -.00458 .04392 .03799 28000 224.0 .03289 .00181 .03470 .03050 26000 223.1 .04572 .00149 .04721 .04747 24000 221.7 .04633 .00296 .04929 .03775 22000 220.4 .04103 .00140 .04243 .03978 20000 219.8 .04467 .00360 .04827 .04097 18000 218.0 .04230 .00416 .04646 .05027 16000 216.0 .04586 -.00192 .04394 .03674 14000 217.1 .04065 -.00553 .03502 .04092 12000 220.5 .04829 -.02907 .01922 .01405 10000 233.7 .04062 -.02781 .01281 .01156 8000 248.1 .03772 -.02788 .00984 .01234 6000 261.7 .03900 -.02156 .01744 .01901 4000 270.9 .02463 -.02730 -.00267 -.00497 2000 284.5 .02869 .00937 .01932 .01986 1000 289.0 .02690 -.00486 .02204 .01975 500 290.4 i 100 292.5 IONIZATION, POTENTIAL ENERGY AND FREQUENCY 335 TABLE 108 HELIUM, m = 4.00 -A (ft - No) hi-h* Ni- No &io z hi -ho VI VQ Cum -A(Ni-No) Kilometers km via ollltl Ni-No 11000 240 1.34090 -.94892 0.39198 1.46400 10500 490 .91438 -.40648 .50790 2.75790 10000 740 .85434 -.29885 . 55549 . 91804 9500 1000 .70840 -.23779 .47061 .66130 9000 1270 .59495 - . 19963 .39532 .57256 8500 1550 .51579 - . 16564 .35015 .42196 8000 1830 . 44697 -.14211 .30486 .35483 7500 2110 .39452 - . 12863 .26589 .29792 7000 2400 .35252 - . 12130 .23122 .25232 6500 2710 .31562 -.11484 .20078 .21132 6000 3040 .28231 -.10883 .17348 .18633 5500 3390 .27827 -.10878 .16949 .15458 5000 3780 .20587 - . 10526 .10061 .11610 4500 4200 .20952 -.10886 .10066 .10376 4000 4680 .19040 -.10141 .08899 .08683 3500 5180 . 17294 .09560 .07734 .07686 3000 5700 .15337 -.08885 .06452 .06935 2500 6230 .J4159 -.08009 .06150 .06373 2000 6750 . 14122 -.07850 ' 06272 .07444 1500 7300 . 12146 .04025 .08021 . 10524 1000 7600 . 12532 .01503 .11028 . 13702 500 7715 .11500 .00215 .11715 .11000 0.... 7705 .11953 .00045 .11998 .11217 -500. . . . 7695 . 12139 .00133 . 12272 .11282 -1000.... 7685 .11918 .00129 . 12047 .15327 -1500... . 7675 . 13037 .00132 . 13169 . 12795 -2000.... 7665 .11918 .00174 . 12092 . 13338 -2500... . 7652 .11311 -.00371 .10948 .06938 -3000.... 7680 . 13272 - . 10136 .03136 .00805 -3500.... 8500 .10862 - . 10595 .00267 -4000.... 9452 .09498 -.09579 -.00081 -4500... . 10402 .08654 -.08744 -.00090 -5000.... 11353 .07939 -.08041 -.00102 -5500.... 12304 .07382 -.07437 -.00056 -6000.... 13255 .06892 -.06927 -.00035 -6500.... 14206 .06473 -.06478 -.00005 -7000.... 15158 The curves for the other solar elements are similar. 336 A TREATISE ON THE SUN S RADIATION TABLE 109 ZINC, m =64.85 -A (Ni - No) = hi -ho Ni No h 10 "*" T hi -ho vi vo Cum -A (Ni - No) hio VlO Ni-No 650 625 230 420 0.88216 . 97240 -0.58461 - 41512 0.29755 55728 0.35806 1 28730 600 575... . 640 890 .81372 65837 - .32679 28017 .48693 37820 .74920 49726 550 525 1180 1480 .52790 44272 - .22564 18960 .30236 25312 .37027 29049 500 475 450 1790 2110 2440 .37788 .32768 28813 - .16408 - .14508 13022 .21380 . 18260 15791 .23516 . 19575 16789 425 400 2780 3120 .25552 23009 - .11528 10338 . 14024 12671 . 14861 13217 375 3460 . 20909 - 09363 11546 11993 350 325... 3800 4140 . 19197 17649 - .08566 07887 . 10631 09762 . 10972 09979 300 275 250 225 4480 4830 5190 5560 . 16424 . 15272 . 14706 12940 - .07519 - .07369 - .06698 06612 .08905 .07903 .08108 06328 .09004 .08076 .07448 06818 200 175 150 125 100 5940 6320 6700 7100 7500 .12470 .11710 .11246 .09117 12020 - .06199 - .05835 - .05799 - .03176 05517 .06271 .05875 .05447 .05941 06503 .06330 .05903 .05350 .06189 08657 75 50... 7745 7735 .08871 .09117 .00130 00130 .09001 09247 . 10455 10450 25 7725 . 12147 .00130 12277 12488 - 25 - 50 75 7715 7705 7695 7685 .08956 .09907 .10111 09661 .00130 .00128 .00133 00129 .09086 .10035 . 10244 09790 .08666 . 12484 .08619 12522 -100 -125 -150 175... 7675 7665 7652 7640 .11174 .09308 .09922 .08363 .00129 .00176 .00151 00782 .11303 .09484 . 10073 07581 . 10237 . 10645 . 10274 05130 -200 -225 -250 -275 -300 -325... . . -350..... -375 7700 8400 9171 9942 10713 11484 12255 13026 .11588 .08925 .08084 .07324 .06916 .06442 .06036 - .08697 - .08775 - .08067 - .07465 - .06946 - .05893 - .06047 .02891 .00150 .00017 -.00141 -.00030 + .00549 -.00011 .00429 IONIZATION, POTENTIAL ENERGY, FREQUENCY 337 I 80,000 70,000 , -A(Ni-No) _j- _( KI _TX O ) ~^^ ->- 60,000 h r h n ^ \ \ \ \ s \ A h 10 50,000 Xon .adiabatic R egion \ \ \ \ \ \ \ 40,000 S ^^ . -*^^ 30,000 $ " (y "n" T\ IA (NI - NO) ~^r NrN 20,000 IE othermal R egion j i 10,000 V / 000 Convectk nal Region *-e \ \,'\ .30000 .25000 .20000 .15000 .10000 .05000 .00000 Values of fhe Computed Ratios (The frequuency is plotted with the minus sign) FIG. 33. lonization, Potential Energy and Frequency in the Terrestrial Atmosphere. 338 A TREATISE ON THE SUN'S RADIATION // t-l < 1 ATMOSPHERIC ELECTRICITY 339 36,000 meters, but falls to 15.00 at 40,000 meters, then rises steadily to very large values as 1000.00 on the vanishing plane. The values of the computed quantity by summation range from 80 volts on the surface upwards to about 200,000 volts at the top of the isothermal layer; this value is nearly the same through the stratum 34,000 45,000 meters; above that level (Qi Q ) increases to very large values. Comparing with the average charges observed in balloon ascensions, Drexel, Nebraska, November and December, 1915, the increases in vertical value are in close accord up to 4,000 meters. This distribution is in agreement with the known conditions, generally, with increasing values in the convectional and iso- thermal region, a layer of steady values at about 200,000 volts, and a supercharged non-adiabatic region to the vanishing plane. The curve of voltage follows closely that of the free heat (Qi Qo). When the charge overpasses the insulating capacity of the gaseous medium there is a discharge as in accumulations in the lower strata during temporary thunderstorms, or permanently in the strata above the isothermal layers. This latter is, then, the fundamental cause of the free electric charges which go over into the polar auroras and the disturbances of the normal magnetic field as was explained in the Meteorological Treatise, Chapter VI. The cause of atmospheric electricity appears to be a ther- modynamic-radiation effect which is due to the deficiencies of the non-adiabatic physical conditions relative to the adiabatic conditions. Adiabatic strata have no free electricity, but, on diverging into the non-adiabatic temperature gradients, there is set up a process of transformation of energy which is of the electrostatic type, though it is really an integrated electro- magnetic form of potential energy. The Solar Atmospheric Electricity On Table 111 are collected the values of (Fi F ) computed for helium and zinc by the same formula. It is seen that the potential (Fi F ) vanishes in the adiabatic strata; in the 340 A TREATISE ON THE SUN S RADIATION TABLE 110 TERRESTRIAL ATMOSPHERIC ELECTRICITY 300 meter h v. 100 m. z T (Fi - Fo) z T o,-n, S (Qi - Qo) Observed at Drexel, Neb. Nov., Dec.. 1915 1915 79000... 6.0 36000 215 .0 10.66 222307 Elevation o9o JMeters 78000.. . 11 .0 293.3 35000 216 .7 11.01 211647 77000.. . 16 .0 990.9 34000 218 .5 11.32 200637 76000.. . 21 .0 1012.1 33000 220 .0 10.66 189317 75000... 25 .0 723.4 32000 221 .0 10.21 178657 03 b 1 74000... 29 .0 638.3 31000 222 .0 11.11 168447 73000... 33 .0 491.4 30000 223 .0 11.28 157337 J2 c 72000.. . 37 .0 386.0 29000 223 .5 8.69 146057 03 71000. .. 41 .0 310.0 28000 224 .6 8.38 137367 II 70000... 45 .0 252.6 27000 223 .6 10.64 128987 a 69000... 49 .0 209.0 26000 223 .1 9.82 117347 03 ^ "** 68000... 53 .0 175.5 25000 222 .8 10.13 107527 'B h 67000 . . . 57 .0 149.6 24000 221 .7 9.30 97397 (j C 66000. .. 61 .0 132.0 23000 221 .2 8.37 88097 *" 65000. . . 65 .0 124.9 22000 220 .4 7.30 79727 * x 64000... 68 .0 114.4 21000 220 .1 8.09 72427 I 63000 . . . 71 .0 99.2 20000 219 .8 7.57 64337 c 62000. . . 74 .0 87.84 19000 219 .0 7.09 56767 rt S 61000. . . 77 .0 74.02 18000 218 .0 5.84 49677 . c 60000 . . . 80 .0 68.89 17000 217 .1 8.40 43837 O j 59000. . . 83 .0 64.30 16000 216 .0 5.65 35437 a rt 58000.. . 86 .0 54.83 15000 216 .5 5.40 29787 ctf 57000... 89 .0 49.09 14000 217 .1 4.18 24387 CJ . i 56000.. . 92 .0 44.37 13000 218 .3 3.62 20207 v nn 55000.. . 95 .0 38.87 12000 220 .5 2.15 16587 &.s 54000.. . 98 .0 37.35 11000 227 .0 1.46 14437 :> 53000. . . 101 .0 33.00 10000 233 .7 1.39 12977 J u 52000 . . . 104 .0 30.27 9000 240 .3 1.14 11587 H & o 51000.. . 107 .0 28.37 8000 248 .1 1.05 10447 o "> 50000.. . 110 .0 28.72 7000 255 .1 1.37 9397 49000. . . 112 .0 25.55 6000 261 .7 1.80 8027 48000. . . 114 .0 24.72 5000 267 .4 0.82 6227 47000... 116 .0 22.31 4000 270 .9 0.73 5407 6052 7232 46000. . . 118 .0 19.65 3000 278 .4 0.91 4677 4451 5226 45000.. . 120 .0 14.53 2500 281 .4 1.09 4227 4149 4815 44000. . . 124 .0 8.84 2000 284 .5 1.69 3682 3519 4219 43000. . . 130 .0 2.99 1500 287 .2 2.09 2837 2071 2970 42000 . . . 138 .0 - 4.13 1000 289 .0 1.96 1792 946 1422 41000. . . 150 .0 -11.64 500 290 .4 1.83 812 127 240 40000. . . 166 .0 -15.32 100 292 .5 80 39000... 187 .0 - 9.69 38000... 202 .0 0.18 37000 . . . 211 .0 7.45 ATMOSPHERIC ELECTRICITY 341 TABLE 111 SOLAR ATMOSPHERIC ELECTRICITY V 300 meter h v. 100 m. I HELIUM, m = 4 ZINC m = 65 z T (Vi - Fo) 2 T (Vi - Fo) 11000 10500 240 490 3.703X10 7 1.093X10 7 650 625 230 420 1.014X10 8 1.1632 XlO 8 10000 9500 740 1000 6.428X10 6 3.005X10 6 600 575 640 890 5.371X10 7 2. 507 XlO 7 9000 1270 1 . 565 X 10 6 550 1180 1.371 XlO 7 8500 8000 1550 1830 9.254X10 6 5 . 697 X 10 5 525 500 1480 1790 8.438X10 6 5 . 676 X 10 6 7500 7000 2110 2400 3.689X10 5 2 481 XlO 5 ^ 475 450 2110 2440 3.892X10 6 2 824 XlO 6 6500 6000 2710 3040 1.635X10 5 1.229 XlO 5 o 3 P 425 400 2780 3120 2. 154 XlO 6 1.696 XlO 6 2 5500 5000 4500 3390 3780 4200 1.014 XlO 5 5. 204 XlO 4 4 746 XlO 4 Q. P" 1* 375 350 325 3460 3800 4140 1.367 XlO 6 1.125 XlO 6 9 311X10 5 P c- p- 4000 3500 4680 5180 3.668X10 4 2 922 XlO 4 300 275 4480 4830 7.730X10 6 6 436 XlO 5 I n 3000 2500 2000 1500 1000 . 5700 6230 6750 7300 7600 2.433X10 4 2. 103 XlO 4 2. 016 XlO 4 2. 421 XlO 4 2 985 XlO 4 250 225 200 175 150 5190 5560 5940 6320 6700 5. 761 XlO 5 4. 331 XlO 5 4.032X10 5 3.551X10 5 3 082 XlO 5 500 - 500 7715 7705 7695 2. 840 XlO 4 2.619X10 4 2 . 333 X 10 4 ? 125 100 75 50 7100 7500 7745 7735 3. 215 XlO 5 3.228TX10 5 4. 285 XlO 5 3 956 XlO 5 -1000 -1500 -2000 -2500 -3000 -3500 7685 7675 7665 7652 7680 8500 2. 019 XlO 4 1.945 XlO 4 1.594 XlO 4 1.248X10 4 thermal. 25 - 25 - 50 - 75' 100 7725 7715 7705 7695 7685 7675 4.717X10 5 3. 167 XlO 5 3. 140 XlO 5 2.900X10 5 2. 308 XlO 5 2 607 XlO 5 ! -4000 4500 . 9452 10402 - >* -125 150 7665 7652 3.161X10 5 706 XlO 5 P- -5000 -5500 -6000 -6500... 11353 12304 13255 14206 - Adiabatic. -175 -200 225 7640 7700 8400 1.306 XlO 5 0.473X10 5 -7000 15158 -250 -275 -300 -325 -350 -375 9171 9942 10713 11484 12255 13026 - Adiabatic. 342 A TREATISE ON THE SUN*S RADIATION isothermal region helium has a nearly constant value of about 20,000 volts per meter, and zinc 350,000 volts; in the non- adiabatic strata they increase to very large values. This is in harmony with the distribution of the terrestrial atmospheric electric charges. It is noted that in the adiabatic layers where the umbra of sun spots is developed, that is, in the low-level primary vortex, there are no free charges. The magnetic effects noted must be due to the charged atoms and molecules, rather than to the free electricity; in this region the Stark electrical effect is lack- ing. In the penumbra and the upper secondary vortex there is free electricity, but without rapid rotations and no Stark effect. The upper levels are highly charged, and this constitutes the source of the solar surface electrical density. Violent readjust- ments are continuously necessary in the high levels of all atmospheres. Atmospheric Electricity and the Diurnal Convection Similar computations were applied to the data of the diurnal distribution of the atmospheric electricity, using the same elements that were described on pages 98 to 112 of the Meteoro- logical Treatise, from the surface up to 3,000 meters. Table 112, and Figs. 35 and 36, briefly indicate these results. The figures are somewhat smoothed, because a single computation is not sufficient to produce accurate normal values, but the general distribution is clearly indicated. The ionization is shown on Fig. 35. It is seen that we have, 1. A maximum at the 8 A.M. and 8 P.M. hours at the surface, becoming 6 A.M. and 10 P.M. on the 3,000-meter level. 2. There is a minimum from 2 P.M. at the surface to 4 P.M. at 3,000 meters, and there is a morning minimum at about 2A.M. 3. At the 1,000-meter level, generally near the cumulus cloud base, there is a minimum, with negative ionization (ab- sorption). This leaves positive ionization above and below, so that this level is favorable for great electrical disturbances, ATMOSPHERIC ELECTRICITY 343 as in the production of thunderstorms, and lightning discharges either horizontally or to the earth. 4. The lowest curve represents the surface electric potential, with maxima at 8 A.M., 8 P.M., and minima at 4 A.M., 2 P.M. These must be closely associated with the thermodynamic ionization as computed. The system of arrows for the hor- izontal and vertical electric currents, H, V, which produce the observed diurnal components of the magnetic field in Argentina (Meteorological Treatise, pages 317-329), evidently has about the same distribution with abrupt changes at 8 A.M. and 8 P.M. Similarly, the 8 A.M., 8 P.M. maxima of the diurnal vapor pressure, coefficient of electrical dissipation, evaporation, and other surface phenomena, must depend upon this thermodynamic process. Further researches will be continued. TABLE 112 THE DIURNAL VARIATIONS OF ATMOSPHERIC ELECTRICITY Cordoba-Pilar Data Volts per Meter 2 2A.M. 6A.M. 10 A.M. 2 P.M. 6P.M. 10 P.M. 3000 98 07 *98 11 97 99 97 71 97 77 97 86 2500 .. 97 54 97 43 97 32 97 43 97 66 97 43 2000... 96 71 96 91 96 90 97 15 97 52 96 91 1500 96 80 96 29 96 15 96 92 97 21 96 29 1000 800 97.09 96 55 95.26 95 78 95.86 95 17 95.88 96 27 96.39 96 99 95.46 95 65 600... 95 97 95 57 94 87 96 69 97 09 95 57 400 95 09 95 01 94 96 96 83 96 89 95 34 200 94.23 93.82 95.11 96 36 96.42 94 85 000... 93.38 92 44 94 94 95 49 95 42 94 06 Total Voltage = S ( 21-20 ) (Fi-Fo) 3000 2500 2000 1325 1060 645 2081 1742 1481 1341 1006 796 1090 950 810 977 922 852 1574 1359 1099 1500 .. 690 1171 421 695 697 789 1000 800 . 835 727 656 760 276 138 175 251 287 407 374 412 600 400 611 435 718 606 78 96 335 363 427 389 396 350 200 000 263 93 368 92 126 92 269 95 295 95 252 94 344 A TREATISE ON THE SUN'S RADIATION 10P.M. 2A.M. 6A.M.. 10A.M. 2P.M. 6P.M. 10P.M. 2A.M. FlG. 35. lonization, Potential Energy and Frequency. ATMOSPHERIC ELECTRICITY 345 Table 112 and Fig. 36 contain the results of the compu- tations of the total voltage in the distribution caused by the diurnal period, that is, the solar radiation during the day and the obscuration during the night. It is seen that there exists on a given level z, generally from the surface to 3,000 meters, a minimum value at 2 P.M., with maxima averaging 8 A.M. and 8 P.M. The computed and the observed systems are prac- tically in accord throughout the twenty-four hours, and in all the levels of record. There is an isolated maximum extending along near the 1,000-meter level, from 10 P.M. to 6 A.M., at the top of the level of vapor convection. The midday minimum extends upward toward the right on the diagram, and it gradu- ally spreads out into a semi-diurnal curve in the high levels. This entire series of phenomena is, therefore, closely connected with the temperature convection, and they are all the by- products of it. There have been many speculative conjectures regarding the origin of these semi-diurnal meteorological periods, but they have been usually of a secondary character. The primary cause is clearly to be ascribed to the many complex processes which are due to the thermodynamics of radiation. It is thought that with sufficient experience the formulas that have been deduced here, and illustrated, can be made to yield other valuable data regarding the atomic and subatomic activities which are con- cerned in the variations of the fundamental terms expressed by T\ n\ NI hi v\ X and their very numerous derivatives. In Figs. 35 and 36 it is seen that, by interpolating the values of F/meter along any level, the well-known distribution is obtained, with maxima at 8 A.M. and 8 P.M., and with minima at 4 A.M. and 2 P.M., especially above 1,000 meters. The Fundamental Quantities in Meteorology and in Astrophysics It has become evident that the quantities which are to be most useful in the higher problems of Meteorology and Astro- physics are those which develop the fundamental terms of 346 A TREATISE ON THE SUN S RADIATION 10 P.M. 2 A 6 P.M. 10 P.M". 2 A.M. .Volts., 300.. hf . m 10 o m . = 3QQ_. g (.-.) i, 100m . Meter e e (y,-t? ), FIG. 36. Atmospheric Electricity. FUNDAMENTAL QUALITIES 347 our equations, namely, (T, n, N } h, k). The usual forms of the Boyle- Gay Lussac Law, (323) P = P RT, Pv = RT, P = .mR.T, PV = KT m may be written, (324) (n k T) m v = m (v n k) T. This involves the following definitions: Pressure. Since k T is constant = 3.7145 X 1(T 14 (C. G. S.) earth, and " " " = 2.9179 X KT 11 . sun, it represents the mean kinetic energy of one //-atom, and the pressure per unit volume is proportional to the .number of //-atoms in the specific volume, P = n k T. For any other gas whose molecular weight is m the pressure is (n k T) m per unit volume, and in the volume v the pressure or kinetic energy is, (325) P V = (n k T) m v. Gas Efficiency. Grouping the same quantities in the second form (v n k) , k is now a variable, and we have the gas efficiency of one //-atom for one* degree of temperature, (326) R= (vnk) so that the total kinetic energy for m atoms and the tem- perature T, is (327) K T = m (n v k) T. The corresponding dimensions are: =KT. Hence, the primitive equations become: (329) P = n k T for the kinetic energy of //-atoms per unit v. (330) P V = N k T for the kinetic energy of m and //"-atoms in v. The entire thermodynamic system, therefore, depends upon the variations in (n. N) and these are more fundamental than (P. p. R. T), since the latter are summation or integral results. 348 A TREATISE ON THE SUN S RADIATION Practical Series of, Thermodynamic Terms Since k T = constant, n is immediately found from (331) n = 7-^ = , , from barometric data. K J. K JL N must be computed from the density by the non-adiabatic formulas. This can be shortened somewhat, as follows: 1 The value of = = 0.711 [9.85194]. Assuming an initial pressure by the barometer BQ, and an initial air density PQ, it follows that the difference of the barometric logarithms multiplied by 0.711 is the difference of the density logarithms. For an example, Uccle, September 13, 1911, Height 2 in meters B (M.K.S.) 4000 4695 5000 4136 6000 meters 3636 log B 1 67164 1 61658 1 56058 (log BI log .BO) 0.05506 0.05600 Factor 0.711 0.711 doe: Di loe: DO) 03915 -0 03982 log; DO 1 93158 log pi -1.89243 . 1.85261 p density 0.8542 0.7806 0.7122 loe; v 06842 10757 14739 v volume 1 . 1706 1.2811 1.4041 N = n m v n m/p is now readily computed. (334) R = v n k can be found at once from R = =, = Tpr. pi pi We, therefore, have obtained, P, p, v, R, n, N, at the temperature T. k T constant Then k = =- = , the kinetic coefficient. (335) Then h = g (zi z ) . - - . the potential coefficient. Z>1 VQ HIQ This system is compact and simple, and it at once provides the data for physical studies in atmospheres. The entire future ENERGIES IN THE ORBIT 349 of meteorology and astrophysics depends upon obtaining the correct non-adiabatic density p and volume v. It is evident that but little progress has been made heretofore for lack of this fundamental quantity. The Relations Between the Kinetic and the Potential Energies in Orbital Oscillations Assume the following data: VQ = the velocity of the mass P in an orbit of amplitude a, with the angular velocity co = - = 20 p in the periodic time 0. / = the time elapsed from the epoch on the axis B. ti = the time elapsed from the epoch a on the axis O C. 1 co " = T = 2; = the fre - quency.

)] = | w a 2 co 2 cos 2 w (J )3) = J w co 2 * 2 . PS is a maximum when E z = 0; P x = when E x is a maximum. It is evident that the periodic oscillations along the free paths of the collisions will be sorted into groups according to the lengths of the free paths. If we identify E x with the thermodynamic h it is possible to compute the correlative quantities in terms of the electromagnetic data. The interpenetration of the orbits of the negative electrons which are carried along with the positive nucleus, whose sphere of action is the limit of the free path, will produce subatomic disturbances in the configurations and temporary instability. Bohr conceives that the negative electron charges e pass suddenly from one stable configuration to an- other in order to effect generation of radiation. This would render the atoms excessively unstable, inasmuch as the number of collisions and wave frequency is an enormous numerical quantity per second. The subatomic agitations conform to the X-ray series of spectra, and the* interatomic collisions, which are external, produce the visible series of spectra for a given element. This allows full scope for the development of both types of series in terms of the variables h, k. Bohr's Theory of Non-Radiating Orbits in Atoms Bohr's theory of successive non-radiating orbits in the structure of the atoms of the chemical elements, in order to account for the observed position of the lines of a series in the spectrum, has attained such success as to become the basis for further investigations. It assumes that the orbital structure of the electrons depends upon the internal constants, h = 6.548 X 10~ 27 , m = 8.845 X 10~ 28 , e = 4.774 X lO' 10 E. S. U, or 1.5913 X ID' 20 E. M. U. Thence, it follows that, BOHR'S THEORY OF RADIATION 351 (343) r being an integer characteristic for the orbits, 1, 2, 3, 4 . . . For r = 1, the inner orbit gives the convergence frequency. The Series of Spectral Lines The frequencies of the series of spectral lines can be expressed by B aimer's Formula, (344) r -V(l - ,) where m takes successive integral values, or by Bohr's Formula, 2 7T 2 w e 4 / 1 1 \ (345) ^_^_ (_-_), - where T2 and n are whole numbers. w = mass, e = charge, /? = potential of the electron, such that (346) = 3.235 X 10 15 , for E = e. Since the curves on Fig. 32 are functions of T and /?, both being variable, we may note the following fact: If the h ordinates be drawn for equal T intervals, as 7\ T =A degrees, they are spaced in positions very similar to those of the above formulas. It is supposed that this is in harmony with the computations which depend upon h and T, as they are related to the thermodynamic conditions of the gas. In this case the spacing depends upon the temperature of the live emissions in succession, and they become a method of determining the temperatures of the solar gases at different depths. Further- more, variations in the positions of the lines of a series, as hydrogen, indicate temporary changes in the local temperatures of solar emission. The positions of the lines, as measured in the spectrum, may be used inversely to establish the position and the correct curve of the (h . T) function. It is evident that h is a variable, and we shall show that the spectrum lines can be produced by the vibrations imposed upon the electronic orbits by reaction from the external collisions of the atoms and molecules. 352 A TREATISE ON THE SUN'S RADIATION Derivation of the Orbital Formulas Assume the following relations: a = the orbital radius. v = the orbital frequency = the vibration frequency. /> V = the potential at the distance r. 3 V e (347) =- = -- = the central force. (j V E e (348) W = = V e = the work done between charges E, e. v (349) co = - - = 2irv = 2ir =the angular velocity. 6 A (350) e = - - = = = the periodic time. CO V C 1 co c v mo 2 (351) , = T = = -= = . (352)

= a u = 2ir a . -^- = ^f- A r n E e e e (355) Kinetic Energy. %miP = hv=V e = = = rna ra g (zi ZQ) ,.. mv 2 2hv 2V e 2Ee (356) Acceleration to Center. = = = ; a a a rna 2 BIGELOW'S AND BOHR'S FREQUENCY 353 r h 2 (357) Radius of Orbit, a = . Log. 2 7r 2 m e 2 = & 7T ttt C - 45.59979. (358) Velocity. = T-. Log. 2 ire 2 = - 18.15594. 2 7T 2 fH 6* (359) Frequency. v = ^ ^ 2ir 2 me* = - 64.95755. 47T 3 w e 4 (360) Angular Velocity, co = r/J~- L S- 4 7r 3 w g 4 = - 63.75573. 9 2 AM /?3 (361) Potential. F= ^ 2 . Log. 2Tr 2 me* = - 54.27867. M7 7J" /7 77 T (362) Nuclear charge. = - 2 7T 2 m & (363) Kinetic Energy. J m ^ 2 = ^r^ . Log. 2 7r 2 m - 64.95755. N (364) Acceleration to Center. - = . d T ft Log. 8 r 4 w 2 e 6 = - 108.85837. These can be used in computing the elements in the jRT-series, Z-series, and the other constituents of the spectrum. From the theory of electronic orbits we have directly the centrifugal force = %mv 2 = hv = - , where E is the charge on the positive nucleus, and n the number of electrons; hydro- gen n = 2. For the velocity in the orbit, v = 2 TT a. v, where the angular frequency and the number of vibrations v are assumed to be the same. Hence, we have, (365) Frequency. = = (366) Radius. a mirv 354 A TREATISE ON THE SUN ? S RADIATION The central force of acceleration is, mv 2 2Ee Ee . (367) / = = - = , for hydrogen, n = 2. The charge on the central nucleus of positive electricity is, Charge (+), = m a mi) h = = e. Hence, e TT (368) m = -TT-. Thence, , N . (369) Frequency, * = -i- 2 ), hydrogen. We compute the two formulas for v and X = : v g (zi - zo) 1 1 V = 2 w* m e* 1 1 \ Lyman as observed, also Millikan Science April 6, 1917 0.00003650 (vi wo) 10 * wio * h Bigelow Bohr Log. g (ZI-ZG) Log. (ViVo)w Log. n w Log. h Log. , w Log. c Log. X Wave length (cm) X m 13.13805 5.15619 17.11295 -25.96720 Log. 7T 2 Log. m Log. e* Log. 2 Log. A' Log. j/ Log. c Log. X X 0.99430 -28.94670 -38.71552 -64.65652 0.30103 -79.44770 - 2.23634 14.90171 10.47712 -79.74873 14.90779 10.47712 - 5.57541 0.00003760 - 5.56933 0.00003710 The Bigelow formula seems to be equivalent to the Bohr formula for the case of the hydrogen series, since the convergence wave length is practically the same. It is evident that since in the Bigelow formula the h is an external potential, while in the Bohr formula h is a function of the kinetic energy, h = m MOSELEY'S LAW 355 it will be necessary to examine the formulas in their other relations. Our thermodynamic data refer in all cases to the maximum 2891 v m , Am, in the Wien Displacement Law, A m = -7^, so that m these results are practically identical. It should be noted that the Bigelow form uses the h m of the external potential, while Bohr uses // = the constant or adiabatic value. We may, therefore, conclude that the Bohr computation refers to the adiabatic case, generally, and the Bigelow computation to the nonadiabatic case near the bottom of the solar reversing layer. Moseley's equation for the relation between the frequency and the atomic numbers, (370) v N = A(N- b) 2 or V N = (a + b N) 2 or Uhler's more accurate equation, which is hyperbolic, (371) v N = A+BN + D 6 _ N (Physical Review, April, 1917), * shows that the ultimate relations between the atomic numbers in the K a , K^ L a , L& L y , types of the X-ray series are really very complicated. It remains to be seen what form further experiments will assign to these structural relations. We shall, therefore, reserve this subject for further examination, though it has seemed proper in this chapter to indicate some of the interesting developments which come from the non-adiabatic thermodynamics . Sanford applies the Bohr theory 'to the K and L series with much success in Physical Review, May, 1917. These subjects will be resumed in a further publication on the structure of matter. Moseley's Law Moseley found an important relation between the atomic numbers of the elements in certain' series, by assuming that the 356 A TREATISE ON THE SUN*S RADIATION addition of the b unit charges, b . e, to the nucleus suffices to pass from one element N to the next in order N+b. For r = 1 (convergence) . in *"'" 'in (372) / = ~ = ^- . 4 7r 2 a 2 2 = 4 7r 2 m a , 2 ^Q7cA / 2]g i j / E 2 e (373) /i = = - and/2 = r, for w = 2. hence. w i 2 ai 2 # 2 2 (374) /i ai 2 = Ej. e = 4 ?r 2 (375) / 2 a 2 2 = E 2 e = 4 Moseley finds that the frequency is proportional to the square of the nuclear charge. (376) = TTg = Moseley's Law. Hence, (377) f 1 = f^ 4 . ^ and i 3 ^ 3 = 2 3 a 2 3 , so that Lz Lz #2 (378) (Millikan) EI EI = a 2 E 2 . This is derived from the central acceleration. On the other hand, from equation (330), we have, by assuming that the kinetic energy in the orbit is a constant, a different relation. (379) (Bigelow) ai E 2 = a 2 EI There are, therefore, two distinct solutions by (346) and (347), the former depending upon assuming that the frequency in the orbits of the electrons is the same as the frequency in the spectral lines; the latter assumes that the kinetic energy in the orbit is a constant for all electrons, the radial distance conform- ing thereto. The former assumes that h is constant, and the latter that h is a variable; the former supposes that the funda- mental relations are internal to the atom, the latter that the primary relations are external to the atom, and that the varia- tions in the position of the lines of the spectrum are due to perturbations impressed by interpenetration of the active orbits of the electric charges during the collisions. The Bohr theory makes the variation of energy which causes radiation to depend EVALUATION OF SERIES FACTORS 357 TABLE 112 . Evaluation of (I 1\ (- 2 --, j l !v ^ - - 2 j J__JL cos 4> ' -p^^^ Tl 2 T2 2 """-V^^ Tl = 1 T2 = 2 0.750 0.750 41 25' X. 3 0.889 0.889 27 15 N. 4 0.938 0.938 20 17 5 0.960 0.960 16 16 M 6 0.972 0.972 13 36 jb 7 0.980 0.980 11 15 1 8 0.984 0.984 10 16 3' 9 0.988 0.988 8 52 3 10 0.990 0.990 8 8 11 0.992 0.992 7 15 Conver gence v 1.000 \ n-2 0.139 0.556 56 12' 11 7 5 - 132 *ij H Ix 4 0.188 0.752 41 13 X 5 0.210 0.840 32 51 bd \ 6 0.222 0.888 27 23 & \ 7 0.230 0.920 23 5 c \ 8 0.234 0.936 20 46 \ 9 0.238 0.952 17 50 n V 10 0.240 0.960 16 7 \ 11 0.242 0.968 14 31 \ Conver gence v 0.250 11 -^ 8651 3 "Tj O sr *SN. Tl = 3 T 2 = 4 0.049 0.441 65 50' jjljs \ 5 0.071 0.640 50 8 6 0.083 0.748 41 35 & 7 0.091 0.820 34 28 1 8 0.096 0.865 30 8 9 0.099 0.892 26 52 n k 10 0.101 0.910 24 30 V 11 0.103 0.928 21 52 \ Conver gence v 0.111 11 9 7 6 5 4 (6 FIG. 38. Orbital Distributions. 358 A TREATISE ON THE SUN'S RADIATION upon the electrons jumping from one stable non-radiating orbit to another, but this would make the structure of atoms wholly unstable and precarious. The flat Saturnian systems of orbits should probably be superseded by a series of orbits which are arranged upon the surface of a sphere, in order to produce the polarizations and magnetizations that exist in molecules. The evaluation of the term f -Jin the expression for the frequency v gives the relative position of the spectrum lines in different series. The function cos

is the angle from the equator. If the electrons should revolve on these planes, as indicated, the frequencies would correspond with an Amperean polarized sphere. If the several atoms occur in different environments of h, depths, densities, temperatures, the corresponding radiations conform to those observed in the spectrum. This research will be continued The Electronic Orbits in the K and L Series of Radiation Lines for h- Variable. We have computed the values of the several terms according to formulas (322) to (339) for the K a and L a series, quoting Uhler's and Sanford's wave length values X, as given in the American Physical Review for April and May, 1917. From these values of X the frequency v is computed, and with this v the corresponding variable h by the formula, (380) ' <<1M This variable h is then applied to a, v, V, % m v 2 , , E, in succession. Finally, we have, v> (381) A = = a constant by Moseley's Law. It is noted that in the K a radiation the value of A is somewhat smaller in the middle than at the ends of the series; the same is true of the L a series. In each case the divergence is small, and EVALUATION OF SERIES FACTORS 359 ^ 2 ^ < s < I & > rH 00 IO C-0000 rH I ooooo iocoeoc~rH oseocoorH t-ooi OS 00 O Tj< IO OJlOOrHCO O rH rH 1 t-tooooooo oooooooooo ooso< ooooo ooooo ooo< t-t-t-c-t- t-t-t-t-t- t- t- t- I o ososososos f Mill TjICOTf t- OS 1000 co 10 1- oooo os os os ososososos Mill t-oioc- )OS rHCOTjl NO] IOOO rH > OS N OS <> O3 rHIOOSCOrH IO OS IN OS CM OOOOO Oi T-H ioo] i oooo os loeocoioio t-iot-ioio t-os>OrH( CM c~ o 01 os COIOOSCON os^rHooi rH SSrHrH 7 77777 Mill ososososos ososososos ososososos Mill Mill Mill t-eo r-i t- t- ION i-H ( .OSOOO i r-i d d d OS^DOSrHO COOOrHCOO rHrHCMNCO CO CO CO 5< " TT ^J< -^ IO IO O US O fO CO oodo'd o'dodo' oo'do'd do odd Vfr"^ t-OSrHkO !>OSrHTj OS OS CO T-l iH rHOsoot-o iOTjcocoeo CM N CM CM r-J OOOIOOOS OS kO T}< rj< O 360 If ^ 3 r,i ^ A TREATISE ON THE SUN'S RADIATION CD rji t-l Hi t~ CSOOOOCS^ CJCOOOW5CO t-COt>Tii^-l IO t- Tfi lO O t- Ot-Ni-HOO t-COCOTfCM ^OOOCOCO CM -t- OO'O OTH ^Hi-Hi-IOlCO kfl-^i-KMCO I00t>t> 10 (N -si 8 tn OS-^OOC^ OCOOSiH (COCO CO ^oooooooooo oooooooooo oooooooooo oooooooooo oo I I I I I I I I I I I I I I I I I I I I I ICM CO t- M t- CO O i-< CO CO CO rH TjlOOOiHOS 00 1O rH CO "3 I ** COCOCMCOCO * CM 00 t- t- ICO ^WCOt-OO 0>OO^CO b-COOiiOCO t- CO (M 00 O C7i Ci O O O Ilii i-t CM CO ' iom m ie> 10 Mill I I I II I I I I I S^:! UOOOJOWO CDt^OJOi-H COTtHCOO> OiOCMCO'i* t- >COCOt^t-t- t-t~t-0000 OOOOOOOOOO OOOSO5OOJ O o'odo'o' o'o'o'o'd o'ddo'd do odd -! 3 00 C050S0505 OSOSOOOOOO OOOOOOOOt- t-t-C-t-t> CC OU5 Tj< r-l COU5O rH "500 s oo t- i-l i-lTjiOt~O U3rHt>COCO o < IflTjtTjicOCM CMiHiHOOJ OOOOt-t-CO O t-t-t-t-t- t-t-t-t-co cococococo co TTTTT ^TTTT ^TT'T T do odd do odd ddoo'd o'do'o'd d lOCOt-OOOi Ot-H Ot-HIMCOiO CO t- 00 OS O loiomioio U3iooioco EVALUATION OF SERIES FACTORS 361 '-teoNW us N CM t- oo m eo t- 1- t- 1- t- 1- 1- t> t- c- o IH N N O oooooooooo ooooooooos oo eo 1-1 ")" < Mr}< 00 rH 10 < Oo eo t- eo > ,_) IO IO IO<0< 10-H-i t-i.-iMeoeo eoeow*^ 1 ^J< ^c o < Q t* t- oo oo oo ^ / * ^^ ^^ . > t~ * gssr ( lOO >-l , - t-OSM^rH OO 00 OS OS O 'OOOOO 77777 ooooos 7777 ' ISSS! 05 OS OS < I I I I ' ' ' ' ' ' ' ' ' ' ' ' ' ' OOrHW< d o d d d t- co to com 1-1 N *-it~?oeooo tomtnco t- eo Ti< I-H o> oo^c0 OOrH< o o o o Illll I II I I Illll Illll Illll Illll ; ssii >NrHNrH >U 3?^ ^So3 SS! '^^^^^ I I I I 1 01 0404 ( I I I I .^coc, Nt-o^a S5 wo< -(NHeo i o 1-. w ^ io< >iovo-*feo eoMr-ioooo t-t^yssbio ibi IOSOSOSOS OSOSOSOOOO OOOOOOOOOO OOOOOOOOOO oooooooooo oooooooooo (NOJNCOtN N(N(NNM Illll Illll Illll Illll OOSOOW OrH t-rmot- ^t- Tf O -ios > t- rf t- i-H IrHl-HOSOJ -H i-! d d 362 A TREATISE ON THE SUN'S RADIATION it proves that Moseley's Law conforms to the variable h in the series. Since the potential energy h represents the resistance to the kinetic energy in the medium, because we have assumed the kinetic energy constant, and the central force variable from one element to another, it is seen that the theory of orbits takes on very much greater flexibility, and really represents the complex of the thermodynamic conditions which produce radiation. These vary with the depth in gases, temperature, density, pressure, Kinetic FlG. 39. Interpenetration of the Orbits of the Electrons in the Atoms During Collisions. gas efficiency, surrounding each atom as represented by n and N. Hence, different subordinate series of the same element originate in different environments on the outside, while the several chemical elements are derived from the different structural con- figurations of the electrons on the spherical surface of the atoms. The Interpenetration of the Electron Orbits at the Contact of Collision The complex systems of atoms in collision must cause inter- penetration of the individual orbits of the electrons, and this VARIABLE INTENSITY OF THE SOLAR RADIATION 363 must communicate a series of internal vibrations to the radiat- ing particles or electric charges. The potential energy // in the radiation function is related to the kinetic energy k in much the same way that has just been described. If k T is a constant then h v is a variable, and a very complex variable whose mean values alone can appear in the thermodynamic formulas. The important point to recognize is that the electrons may remain on their stationary orbits accord- ing to the structure of the chemical elements, and acquire the perturbations producing the radiation during the confusion of interpenetration. This avoids the difficulty in Bohr's theory which requires the electrons to pass from one orbit to another during radiation. The amperean spherical distribution of the interacting elementary charges, E at the center, and e in the several orbits, probably present the basis for the structural permanences which are inherent in the chemical elements. Fig. 39 represents the collision of two polarized atoms whose orbits are projected on the equatorial plane. The Electromagnetic Waves Due to the Sudden Motion and Stoppage of an Electric Charge in Collisions Following Heaviside's exposition of the effect of suddenly starting or stopping the motion of an electric charge, as in col- lisions, we have the distribution of the electric disturbance D and the magnetic induction B, in producing the plane elec- tromagnetic waves of Fig. 40. Let p take on suddenly the velocity v, so that in the time t it reaches the polar position of the sphere +p. The outside radial displacement changes into a current along the meridians from the positive to the negative pole, with its magnetic induction on the parallels in the vector sense, so that the polar field be- comes the plane wave [D . B], as the sphere enlarges. Let p be suddenly stopped, then the displacement reverses from the negative to the positive pole along the meridians, with induction on the parallels in the opposite direction, and renewed inner radial displacement to the center. This ; also, releases a plane 364 A TREATISE ON THE SUN'S RADIATION wave [D . B] at the positive pole. Similarly, during collisions of two orbit-atoms, there are sudden motions and stoppages, with reversed displacements and inductions, sending plane Sudden Stopping. Sudden Starting. FIG. 40. The Sudden Motion and Stoppage of an Electric Charge p. electromagnetic waves into space. When the orbits of atoms in collision interpenetrate, these waves are complicated in their frequencies, in accordance with the results seen in their charac- teristic spectrum lines. Furthermore, during successive collisions at the end of the free path C, there exist the potential and the kinetic energies, which may be analyzed as potential along the free path with h as the average value, and 2 h as the maximum value, the time of one period being second of time, while the kinetic energy v is expressed by the motion in the circle, such that k T = J m v 2 . (382) divW = - e<>J - hoG + Q + j + H (383) W = V (E - eo) (H - fc) = V E l H = v (j + H). All these forces, expressed and implied, are in action during the collisions of complex atoms and molecules. VARIABLE INTENSITY OF THE SOLAR RADIATION 365 The Variable Intensity of the Sun's Radiation in the 26.68-Day Period of the Synodic Rotation Besides showing that the intensity of the solar radiation is variable in the n-year period to the amount of i% to 2%, it appears that a similar variation occurs as the sun turns on its axis. Fig. 41 gives the normal direct curves of the solar varia- tion, as registered in the terrestrial magnetic field and the meteorological elements, according to the author's papers of 1893, 1895, 1898, and the Meteorological Treatise, 1915. The Cordoba-Pilar curve of the pyrheliometric mean intensities, 1912-1916, results in a nearly identical curve, and this proves that the solar radiation is variable in solar longitude and affects all the terrestrial elements in the 26.68-day period. Similarly, the La Quiaca pyrheliometric data produce the small curve in the 12345678 9 10 H 12 13 14 15 1617 18 19 20 21 22 23 2426 2627 FIG. 41. The variable intensity of the solar radiation in the 26.68-day synodic period. inverse for^n, the amplitudes being much larger. It seems that there are inversion and damping of the variations in the intensity of the radiation in the lower atmosphere, a subject of importance for further research. 366 A TREATISE ON THE SUN'S RADIATION International Character Numbers The character numbers of the amplitude of the disturbance of the magnetic field, as published by the International Com- mission for 35 stations during 1915, produces nearly the same inverse curve. This curve distributes its minor crests on four 10. 20 26 FlG. 42 Magnetic Character Numbers, 1915. International Commission, 35 Stations. axes, whose center is eccentric to the axis of rotation, as if the center of radiation revolves about the center of mass of the sun. This eccentricity of radiation is evidently sufficient to form the basis of weather forecasts, because these maximum points have been proved to persist through 70 years, 1840-1916. It will be necessary to abandon the practise of publishing magnetic and meteorological data on the calendar months, which has no scientific meaning, and substitute the 26.68-day period with epoch, June 13.72, 1887. Compare the Meteor- ological Treatise, p. 334. VARIABLE INTENSITY OF THE SOLAR RADIATION 367 365-day period Jan. Feb. Mar. April May Jane July Aug. Sept. Oct. Nov. Dec. Radiation Gr. Cal. Cm. 2 Min. 4.000 Clear days Hazy 3.950 days 3.900 Amplitude - \ Variations -70 50 \ Frequency Magnetic -12 Direct Type -16 FlG. 43. Synchronous variations of the radiation and the magnetic field in the 365-day period. FIG. 44. Synchronous variation of the radiation and the magnetic field in the 20 .68-day period. 368 A TREATISE ON THE SUN'S RADIATION Synchronism between the solar radiation intensity and the terrestrial magnetic variations in the jdj-day and the 26.68-day periods The 365-day period is for the annual revolution of the earth about the sun, and the 26.68-day period is for the synodic rota- tion of the sun on its axis. The pyrheliometer observations in Cordoba-Pilar have been divided into three classes, correspond- ing with the local weather conditions: Clear j when the sky is apparently free from haze and dust. Haze, when there is a milky haze due to vapor or ice. Dust, when the low level dust has been raised by wind. It was found practical to determine correcting ordinates which should reduce the haze and dust days to clear days. In order to eliminate the effect of the annual revolution, the clear, haze, and dust days have been reduced to the mean clear days, whose average value of the solar radiation is 3.980 gr. cal./cm. 2 min. Fig. 43 shows that the clear-day radiation, the hazy-day radiation, the amplitudes of irregularities, and the frequency of the magnetic direct type have maxima when the sun is on the earth's equator. Compare Bulletin No. 21, U. S. Weather Bureau, 1898, pages 100-108. Fig. 44 shows that the solar radiation and the magnetic field synchronize in the 26.68-day period, when the radiation is re- duced to the mean clear standard. It follows that the compilation of tables in the unscientific calendar months must be abandoned and collections in the 26.68-day period substituted. All observations of radiation must be freed from the imperfections due to haze and dust in the low levels before they are compiled. All the data of meteorology, terrestrial magnetism, atmospheric electricity, and solar physics generally, must conform to these criteria. In order to obtain the foregoing results it has been neces- sary to supplement the ordinary method of using the Bouguer formula by the table on page 370 of correction- ordinates com- piled at Cordoba-Pilar, Table 115. They are the result of separating the 5 -years' record, 19121916, into the three groups indicated. Other stations must prepare similar tables repre- senting their climatic conditions. EFFECT OF DUST 369 The Effect of Dust in the Lower Strata The results of observations with pyrheliometers, to determine the intensity of the solar radiation, and the amount of its varia- tions in different intervals, have been always unsatisfactory, because they were defective in two particulars: (1) The poten- tial energy of the radiation is omitted entirely in the computa- tions; (2) the effect of the dust and impurities in the lower strata has not been eliminated. Unfortunately, the records as usually published give insufficient data from which either of these terms can be fully computed. It is necessary to know the intensity at various zenith distances, and the coefficient of trans- mission; also, the temperature, vapor pressure and barometric pressure, for computing the potential terms; the state of the sky, clear, hazy, dusty, force of the wind, for determining the dust effect on the scattering. Neither the Smithsonian Astrophysical Observatory, nor the U. S. Weather Bureau, publish their in- tensities with enough auxiliary data to make it practical to include their numerous observations in a definitive reduction, such as Cordoba-Pttar now possesses. In Argentina the five-year records at Cordoba-Pilar have furnished the following system of monthly corrections, which, added to the zenith value of the intensity, produce results such as are illustrated in the preceding diagrams. The necessity for these corrections is evident from the fact that the normal in- tensity of radiation is always lessened by atmospheric impurities, the changes being in one direction. Hence, any mean value of a series depends upon the number of imperfect days which have been incorporated, so that the resulting mean values merely reflect the effects of the admixture of such defective conditions. The following tables indicate the corrections referred to, with examples of the results. It is certainly improper to ascribe to the radiation of the sun itself those terms which depend upon the local atmospheric conditions. 370 A TREATISE ON THE SUN'S RADIATION TABLE 115 CORDOBA-PlLAR, 1912-1917 Table of Reduction to the Mean Clear System Month Clear Haze Dust January. -0 010 +0 040 +0 170 February +0 . 030 +0 . 050 +0.220 March April . . -0.030 -0 030 +0.060 +0 060 +0.150 +0 150 May +0.020 +0 . 050 +0.200 June +0.030 +0.060 +0.210 July . . +0 020 +0 050 +0.200 August September +0.010 -0.010 +0.020 0.000 +0.190 +0.170 October 020 010 +0 160 November December -0.030 +0.030 0.000 +0.020 +0.150 +0.210 TABLE 116 DATA REDUCED TO THE MEAN CLEAR DAYS Months 1912 1913 1914 1915 1916 1917 Jan., Feb 4 059 3 946 4 001 4 012 3 888 3 960 Mar., April . 4.013 4.031 3.921 3 949 3.876 3 957 May, June Tulv, Auc 3.988 4 037 4.007 3 995 4.022 3 979 4.033 3 952 3.920 3 974 3.956 3 952 Sept., Oct Nov., Dec 3.988 3.934 4.003 4.076 4.030 4.029 3.959 4.023 3.886 3.973 Means 4 004 4 014 3 997 3 988 3 913 3 956 It is seen, among other things, that the year 1912 is near the maximum, while it was recorded and discussed in the Northern Hemisphere Mt. Wilson, Mt. Weather, Washington, Bassour as a minimum, the latter being due simply to the dust in the atmosphere from the volcano Katmai. In Pilar, during dry weather, a wind storm fills the lower strata with dust, and the observed intensity drops to 3.70, 3.60, or even 3.40 calories. Hence, the inclusion of such dust effects in any mean values simply vitiates them, and the results are useless. Until these defects can be fully eliminated from the observed radiation SYSTEMS OF UNITS EMPLOYED IN METEOROLOGY 371 intensities, the data can not be utilized in any of the solar radiation problems. The Systems of Units Employed in Meteorology In the United States the public use of the British system of measures made it necessary to employ some of them in the Government Weather Service. These are the pressure in inches of mercury, the temperature in degrees Fahrenheit, the wind velocity in miles per hour. The density has been always er- roneously computed on the adiabatic system, the gas efficiency has never been other than a constant, and all the dependent thermodynamic terms are consequently in error. These weather- map data are, therefore, inconsistent, and can be reduced to the M. K. S. or the C. G. S. systems only by laborious transforma- tions, which it is impractical to apply to so large a mass of records as has been compiled in the United States since 1871. These data are, therefore, only remotely accessible to scientific studies. In order to supplement this public system the Weather Bureau has employed a different but mixed system, compiled from several sources. Similarly, the International system is not self -consistent, and is likewise inconvenient in scientific researches. Furthermore, these imperfect data are assembled in such a way as further to embarrass their value. 1. The Washington 8 A.M. and 8 P.M. hours are made the times for simultaneous observations throughout the United States, but they become 7 A.M., 7 P.M. at Chicago, 6 A.M., 6 P.M. at Omaha, and 5 A.M., 5 P.M. at San Francisco, so that the daily means are always incomplete and generally erroneous. 2. These daily means of inconsistent data are compiled in monthly tables, according to the calendar months, but these have no proper connection with solar recurrent phenomena. It would be difficult to devise a program of work more com- petent than that one to obscure the fact of solar synchronism in the terrestrial weather conditions. Table 117 contains an example of these systems, based upon the balloon ascension at Drexel, Nebraska, October 31, 1915, which illustrates the divergent and inconsistent methods in use. 372 A TREATISE ON THE SUN S RADIATION 3 PQ i> i-H *< W 31 PQ 2 s sto wg cj . a; &> G o> 111 S ' 1 CO T-I CO l> OS i 1 O5 3 4J C3 O fn ^ O CO I-H a g CO i-l e c s I 8 : o x rt o i . 00 . ' T d^ 00 ^^ r7( ^5 ^D 00