588 
 
 Very 
 
 and Terrestrial 
 
 Albedoes
 
 I 
 
 THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 OF CALIFORNIA 
 
 LOS ANGELES
 
 SCIENTIFIC PAPERS 
 
 THE 
 
 WESTWOOD ASTSOPHYSICAL OBSERVATORY 
 
 UMKR I. 
 
 LUNAR AND TERRESTRIAL 
 ALBEDOES 
 
 By 
 FRANK W. VERY 
 
 - THE FOUR MPANY
 
 
 OCCASIONAL SCIENTIFIC PAPERS 
 
 OF THE 
 WESTWOOD ASTROPHYSICAL OBSERVATORY 
 
 NUMBER I. 
 
 LUNAR AND TERRESTRIAL ALBEDOES 
 
 By 
 FRANK W. VERY 
 
 BOSTON 
 
 THE FOUR SEAS COMPANY 
 1917
 
 HISTORICAL NOTE 
 
 FOUNDED in 1906, the Westw'ood Astrophysical Observatory owes 
 its inception to aid from Percival Lowell. In beginning a special 
 series of its publications, the writer wishes to place on record his 
 indebtedness to the warm sympathy and encouragement of a faith- 
 ful friend. Himself an ardent lover of freedom, Dr. Lowell never 
 interfered with the writer's free and independent ordering of the 
 researches conducted at Westwood, but with a rare disinterested- 
 ness he placed at his disposal numerous spectrograms taken at the 
 Lowell Observatory by the skilful hands of Dr. V. M. Slipher for 
 measurement with apparatus of the writer's. design. Nevertheless, 
 the gain was mutual, for the results throw unexpected light on 
 some of Lowell's own researches and demonstrate that complete 
 independence in respect to control and motives of action is not in- 
 compatible with a consistent working together for a common end. 
 
 Lowell had been greatly interested in the research which 
 forms the subject of the present communication, with its obvious 
 bearing on the problem of planetary temperature. In his "Tem- 
 perature of Mars," he had adopted 0.75 for the albedo of a half 
 clouded earth, and I, in my "Greenhouse Theory and Planetary 
 Temperature/' had taken 0.70 for the same datum, differing but 
 little from the value now found for the geometrical albedo of the 
 earth, which is 0.72. 
 
 Let me also place on record as a result of my intimate associa- 
 tion with him, my recognition of the fact that his theories were 
 based on an elaborate accumulation of unsurpassed evidence, that 
 he was always open-minded to new evidence, and that, while pre- 
 senting some revolutionary new conceptions, he did not hesitate to 
 modify his own ideas when convinced that they could be im- 
 proved. It is this Willingness to revise that constitutes the true 
 man of science. That there was very little for him to change, as 
 his researches progressed, is a testimony to Lowell's thoroughness 
 and to his deep insight into nature's mysteries. 
 
 With gratitude to God for the gift of a friend generous, 
 thoughtful for others, and noble in his ideals, keenly critical, but 
 kindly appreciative, learned, but modest 
 
 I dedicate these researches 
 an the Jflrmnrij of 
 
 Jlernual 
 
 9176: 

 
 THE WESTWOOD ASTROPHYSICAL OBSERVATORY 
 
 THE WESTWOOD ASTROPHYSICAL OBSERVATORY is situated in 
 Westwood, Massachusetts. Its approximate position and altitude 
 (derived from the topographical map of the United States Geolog- 
 ical Survey) are 
 
 Latitude = 42 12' 58" North. 
 Longitude = 71 1 1' 58" West. 
 Altitude = 190 feet above sea level. 
 
 Its publications hitherto have been in current scientific per- 
 iodicals, especially, Lowell Observatory Bulletin, American Jour- 
 nal of Science, Astro physical Journal, Science, Astronomische 
 Nachrichten, and Bulletin Astronomique. 
 
 The Observatory possesses special instruments for the study 
 of solar radiation and atmospheric transmission, for delicate heat 
 measurements, the utilization of solar radiation and study of the 
 "greenhouse" effect, photometry, spectral line and band com- 
 parator, etc. Eor several years it had the use of a fine silver-on- 
 glass concave mirror of 12 inches aperture and 10 feet focal 
 length, which was loaned by its maker, Dr. J. A. Brashear. The 
 mirror was used in researches on the transmission of terrestrial 
 radiation by the aqueous vapor of the atmosphere. 
 
 Special researches are being actively prosecuted on at- 
 mospheric transmission and the solar constant, quantitative mea- 
 surements of the intensity of spectral lines, planetary atmospheres 
 and temperatures, greenhouse theory, contributions to the theory 
 of nebulae and novae, measurements' of the earth's albedo and of 
 that of the moon for all parts of the visible spectrum. The latter 
 researches form the subject of the present communication.
 
 LUNAR AND TERRESTRIAL ALBEDOES 
 
 Introduction. 
 
 THE word albedo (derived from the Latin albus, white) has 
 been used by astronomers to designate the fraction of the sun's 
 luminous rays reflected by a planet at full phase, allowance being 
 made for the distances of the planet from sun and earth and for 
 the dimensions of the reflecting body. If the planet were a smooth 
 sphere with perfect specular reflection, it would be itself invis- 
 ible, but would present w r ithin the diminutive limits of its disk a 
 complete picture of the surrounding heavens, distorted by spher- 
 ical aberration, but otherwise exact ; and within this image the 
 reflection of the sun would surpass in brilliancy all other objects, 
 shining like a star at a point on the planet's disk distant from the 
 center by the radius of the disk multiplied by the cosine of half 
 the elongation of the planet from the sun. But whatever specular 
 surfaces there may be on the planets of our solar system, they 
 are of too limited extent to be recognized as such ; and the plane- 
 tary reflection of light is to be classed under the head of a gen- 
 erally diffusive one, though not necessarily an equable one in all 
 directions; and in fact there are diversities in the distribution of 
 the reflected light to different parts of the sphere which must be 
 considered in getting the phase-curve of the illumination, and 
 which are not entirely without influence even if we confine our 
 attention to the reflection sent earthward at full phase, while they 
 are vital to the determination of the complete reflection to the 
 sphere. 
 
 Since all of the planets, except possibly some of the smaller 
 asteroids, are spheroidal bodies, it is not necessary for purposes 
 of intercomparison to refer their albedoes to the standard specific 
 reflectivity of a flat surface; but it is desirable to distinguish 
 clearly between the only thing which is certainly measurable in 
 most cases, which is (i) the geometrical albedo at full phase, 
 or the amount of light sent earthwards at the planet's full phase, 
 
 5
 
 6 LUNAR AND TERRESTRIAL ALBEDOES 
 
 compared with that which would be sent by a sphere of the same 
 size and at the same distance, which possesses "perfect diffusive 
 reflectivity; and (2) that integration of the reflection to the 
 entire sphere, or the spherical albedo, whose determination 
 requires a knowledge of the phase-law. This law is very imper- 
 fectly known, except in the case of the ipoon, and hence there are 
 rival hypotheses which give more than one kind of "spherical" 
 albedo. There is even a diversity of usage in regard to what shall 
 be called the "geometrical" albedo, although there need be no 
 discrepancies in the facts of observation on which it is based. 
 A very few words will suffice to make the fundamental distinctions 
 plain as to their general principles ; but the remoter consequences 
 of the acceptance of the diverse points of view lead to discussions 
 of some complexity whose complete unfolding can not be ex- 
 hibited in the limits of this paper, but enough will be presented to 
 give an intelligible conception of the subject. 
 
 If we measure the .amount of light received by the eye from 
 the full moon, that is to say, if we find the reflection bf sunlight 
 by a spheroidal surface to a point (since the pupil of the eye is 
 virtually a point), we shall get the same value whether the moon 
 is near the horizon or in the zenith (after correcting for the 
 absorption by the earth's atmosphere) ; and it seems natural to 
 take this constant light-quantity as the basis of the geometrical 
 albedo referred to a definite point in space, comparing it with the 
 quantity of light which would be given if the whole sky were filled 
 with moons of perfectly diffusive reflecting quality, and viewed by 
 turning the eye progressively to all parts of the sky and summing 
 the successive impressions. This geometrical ratio of the reflec- 
 tion to a point compared with the perfectly diffuse reflection at 
 that point from an ideal body of the same size and in the same 
 situation, is the one considered in this paper and is what is meant 
 by the geometrical albedo. 
 
 But if, instead of this, we take the illumination of an extended 
 surface by the hypothetical sky full of moons, it is necessary to 
 take into account the diminution of superficial illumination from 
 those rays which are at low angles to the surface, and even sup- 
 posing an absence of atmosphere, the surface illumination pro- 
 duced by a sky full of moons will only be half as great as the 
 sum of the illuminations supposing each moon to be successively
 
 LUNAR AND TERRESTRIAL ALB EDO ES 7 
 
 transported to the zenith. Thus the "surface illumination" is one 
 half of the geometrical albedo. 
 
 FIGURE i 
 
 Lambert showed in his "Photometria" (cap. II.) that if we 
 seek the illuminating power (L) of a circular luminary of radius 
 SR = r (Fig. i), whose center is at any point (S) of zenith- 
 distance SZ = "C,, upon a surface at C, we may obtain L by sum- 
 ming a series of annuli concentric with S and of radius SX = x, 
 where, if an element (dx, dq>) of the annulus is at the angle 
 ZSX = q>. from the vertical through 5", the area of the element 
 is dx dy sin x. Hence 
 
 dx, 
 
 = I I sin x cos 
 
 since the illumination of the surface at C varies in proportion 
 to cos . 
 
 By spherical trigonometry, if s is the zenith-distance of any 
 point X on the annulus, 
 
 cos z = cos cos x -f- sin sin x. cos cp, 
 and
 
 8 LUNAR AND TERRESTRIAL ALB EDO ES 
 
 L = \ \ dx d<p sin x [cos cos x -)- sin sin x cos qp] 
 
 -' 
 
 A first integration relatively to qp between cp = o and 
 tp = 360, or 2ji, gives 
 
 L = I </.* sin x 2n cos cos #. 
 
 Integrating this with respect to .r from x = o to .r = r, 
 L = 2Jt cos t, I sin ;r cos .r t/^r 
 
 I - COS 2.Y 
 = 2n COS ^ (- -) =r Jt COS 
 
 4 
 
 This gives for the illuminating power fZJ of the moon at 
 the zenith to that of a sky full of moons (L') upon an extended 
 surface at C, as a first approximation, 
 
 L :L' = n cos o (sin 2 15' 33") : ji cos o (sin 2 90) 
 = i : 48,875. 
 
 But if we consider the luminous effect upon a point, such as 
 the eye, or the heating effect upon the bulb of a thermometer 
 which may 'likewise be taken as a point, instead of that commun- 
 icated to an extended surface, then, neglecting atmospheric 
 absorption, it is necessary to find the ratio of illuminations by 
 taking the ratio of the area of the apparent lunar disk to the 
 hemispherical sky area, a ratio which is half as great as the 
 one just given. For a disk as small as that of a planet, the area 
 may be taken = Jt sin 2 Q r 2 , where o is the angular value of 
 the radius of the disk and r is the distance of the planet. Com- 
 paring this with the area of the hemisphere, 2jir 2 , the latter ex- 
 ceeds the iormer in the ratio, 2 : sin 2 Q, which, for the moon's 
 semi-diameter, Q = I$' 33", gives for the ratio of the light re- 
 flected to a point from the two sources, 
 
 97750 : i 
 with a similar degree of approximation to the preceding value.
 
 LUNAR AND TERRESTRIAL ALB EDO ES g 
 
 The integration of the total light reflected by the illuminated 
 hemisphere of the planet in all directions requires the introduction 
 of hypotheses. The first is Lambert's hypothesis of uniform 
 diffuse reflection, of which the following account is substantially 
 that of Mullen 
 
 FIGURE 2 
 
 Consider a diffusely reflecting surface-element, ds (Fig. 2) 
 illuminated under any angle of incidence, i. If L is the quantity 
 of light which falls normally on the unit of surface, then ds 
 receives L ds cos i, of which a certain fractbn cL ds cos i is re- 
 flected normally, and in any other direction, such as that of the 
 emanation angle E, the light-quantity dq = cL cos i ds cos E is 
 reflected, provided the surface reflects as well at one angle as 
 at another. 
 
 Construct a hemisphere with radius i about ds of which an 
 element dw receives the fractional light-quantity 
 
 dQ = dq dw = cL ds cos i cos E d(a. 
 
 Then since the element rfca has the width dv sin E and the height 
 </E, or the angular area </ sin E dv, the total light-quantity has the 
 value 
 
 Q = cL ds cos i 
 
 Jir/2 
 
 COS E sin E (/ 
 
 f 
 
 V 
 
 dv,
 
 io LUNAR AND TERRESTRIAL ALBEDOES 
 
 JTT 72 /?2JT 
 
 cos 8 sin E dt= l /2, and I dv = 2it, 
 t/ 
 
 Q = L ds cos i A, 
 
 where the factor A is a fraction which tells how much of the 
 incoming light is reflected to a hemisphere of radius i. A, which 
 is always smaller than i, is simply called the albedo of the sub- 
 stance by Lambert. 
 
 From the two equations for Q, it follows that c = A/n, and 
 thence we have Lambert's law of illumination by diffusely re- 
 flecting substances in the well known form : 
 
 dq= L ds cos i cos e, 
 
 Ji 
 
 or 
 
 dq F! ds cos i cos e 
 
 if r = 
 
 Calculation of the quantity of light sent to the earth at dif- 
 ferent phases of a reflecting planet requires some further slight 
 modifications. 
 
 Let a plane be drawn through the middle point of the planet 
 at right angles to the earth-planet line ; and let its intersection 
 with the planet's surface be represented by the circle ABCD 
 (Fig- 3)- The perpendicular to this plane in the direction of the 
 earth is shown diagrammatically by ME. MS is drawn in the 
 directon of the sun. The arc of a great circle ES is the phase- 
 angle a, taken from full phase. An element ds of the visible 
 hemisphere of the planet is connected with E and S by great 
 circles of the sphere. Arc S ds = i, arc E ds=E. Latitude 
 of <ly = \|j = F ds. Longitude fro'm E = EF = (d. 
 
 From the right-angled spherical triangles FSds and FEds, we 
 have the relations : 
 
 cos i = cos i|> cos (co a), 
 cos E== cos ty cos (0. 
 
 If the semi-diameter of the planet is Q, the linear dimensions 
 of ds are Q dty in the direction of the meridian and 9 rfto cos ip 
 along a parallel. Hence 
 
 surface of ds = Q 2 cos ip <fa> d\\).
 
 LUNAR AND TERRESTRIAL ALBEDOES n 
 
 J 
 
 We now introduce two rival hypotheses, or laws of reflection, 
 (i) that reflection is uniform in all directions, (2) that it varies 
 according to a definite law, and get 
 
 (0 
 
 Lambert's Law : 
 
 i a. dq^ = r\ o 2 cos 3 \p dty cos (co a) cos co dco, 
 
 Lommel-Seeliger Law : 
 
 cos co cos (co a) 
 
 ib. d. = 
 
 o cos 
 
 cos (co a) -}-?. cos co 
 
 a'co, 
 
 where P., = 
 
 L 
 
 k 
 
 and X = 
 
 , k being the coefficient of 
 
 absorption of the rays which enter into the interior of the sub- 
 stance of the planet's surface, k l the coefficient of interior ab- 
 sorption of the returning rays on the way to emission, and p. 
 the diffusive power of the body. In general t <, because the 
 outgoing rays have lost their more absorbable ingredients. If the 
 material is strongly colored, k may be very much larger than k r
 
 12 LUNAR AND TERRESTRIAL ALBEDOES 
 
 The derivation of the Lommel-Seeliger equation which takes 
 account of interior reflection, and diffusion is very complicated. 
 The final equation is 
 
 cos i cos 6 
 
 ds 
 
 cos i -(- A cos E 
 
 Confining attention here to the Lambert equation, the formula 
 
 must be integrated over that part of the illuminated surface 
 
 visible from the earth. The integration limits for ip are 
 
 - Jt/2 and -f- jt/2, and those for to arc 11/2 -f a and 4- Jt/2, 
 
 pr/2 pr/2 
 
 whence <7 1 = r i o 2 I cos 3 ip chp I cos (w a) cos co dto. 
 
 t/ - IT /- t/a-TT/2 
 
 But 
 
 J1T / 2 A*TT / I! 
 
 cos 3 \p (/x))= I cos- ip fl?(sin 
 -7T/2 e / -7T/2 
 
 = r 
 
 ip] c/(sin \p) = - 
 And 
 
 J 
 
 7T/2 
 
 cos (to a) cos to 
 
 a- TT /2 
 
 a) afco 
 
 J7T / 2 S*TT / 2 
 
 cos v. dw -}- l /2 I cos (20) a 
 -7T/2 t/a-TT/2 
 
 r: = /^2 [ ( Jt a) cos a -f- sin a] . 
 Therefore 
 
 (2) q 1 = r i Q 2 2/3 [sin a -f- (n a) cos a]. 
 
 For the full phase, when sun, earth and planet stand in a straight 
 line, a o and the reflected light is g 1 (o) = I^o 2 2/3:1. Hence 
 we have for the ratio 
 
 Light at phase-angle a : Light at full phase, 
 
 (3) 9iA?i (o) = [sin a -f (it a) cos a].
 
 LUNAR AND TERRESTRIAL ALB EDO ES 13 
 
 A similar integration for the Lommel-Seeliger law gives 
 
 |~i o 
 
 (4) q-= [i sin a/2 tan a/2 log cot a/4]. 
 
 But for a= o, ^., (0) = (T 2 o 2 :fi)/2, and 
 
 (5) ^2/ < /2 (0) : T g i n a / 2 tan a / 2 lg cot a/4. 
 
 The distribution of light over the apparent disk of the planet 
 varies according to the adopted law. Euler's law would demand 
 uniform light, except for a narrow strip of sudden diminution at 
 the terminator and an excessively narrow, but exceedingly bright 
 rim at the illuminated limb. Nothing of the sort is observed, and 
 this law may be dismissed at once. Moreover, Euler considered 
 nothing but superficial reflection, just as Bouguer did, whereas 
 the penetration of the light into a thin surfa'ce layer, even in the 
 most opaque substances, is of great importance. 
 
 Lambert's law appears to work fairly well where the reflec- 
 ting medium is of the nature of cloud with internal diffusion and 
 multiple reflection from innumerable widely dispersed and finely 
 divided particles, such as ice crystals, or dust, or the liquid water 
 particles of ordinary cloud. The Lommel-Seeliger Law is more 
 appropriate for extended solid surfaces at various inclinations to 
 the incident light. A composite inter-mingling of solid surface 
 and cloud requires a mixture of the two laws. 
 
 In the following Table are given the computed values of the 
 functions of the phase-angle, <p (a), for intervals of 5 according 
 to several theories. These quantities are then multiplied by 
 others proportional to the areas of the corresponding zones, 
 sin a Aa = A(i cos a), to give the values in the last three 
 columns which, being summed, produce the proportional factors 
 for the spherical albedo. If the intervals had been taken small 
 enough, the sum of the differences of versine a,2 [A (i cos a)], 
 would have been exactly 2 which, multiplied by 2 JD, gives the 
 area of the sphere of unit radius. The average reflection by a 
 planet to the sphere has an intensity l / if the reflection is perfect 
 (normal specific reflectivity = i), this being the mean between 
 the quantities ( l / 2 and o) sent in the directions of source and 
 antipodal point. Calling S [A (i cos a) q> (a)] the spherical 
 factor, its values given in the last line of the table, are
 
 i 4 LUNAR AND TERRESTRIAL ALBEDOES 
 
 0.75 by Lambert's Law. 
 
 1.50 by the Lommel-Seeliger Law. 
 
 0.35 by the lunar phase-curve. 
 
 While the spherical albedo can not exceed unity, there may be 
 various distributions of light to the sphere. Thus, for perfect 
 reflection, the diverse spherical factors obtained from the sum- 
 mations in the table are consistent with geometrical albedoes of 
 0.50, 1.33, 0.67, and 2.86, the last being for the phase-law of the 
 moon where the reflection at full phase is extraordinarily large. 
 
 Limits of 
 
 4 
 
 (a) 
 
 <P (a) 
 
 en (a) 
 
 (p(a) X A(i cos a) 
 
 phase-angle 
 
 ! <X 2 
 
 Ad-cosa) 
 
 Lambert 
 
 Lommel- 
 Seeliger 
 
 i \ / 
 
 Moon 
 
 Lambert 
 
 Lommel- 
 Seeliger 
 
 Moon 
 
 5 
 
 .0038 
 
 .996 
 
 .999 
 
 .957 
 
 .0038 
 
 .0038 
 
 .0036 
 
 5 10 
 
 .0114 
 
 .986 
 
 .994 
 
 .869 
 
 .0112 
 
 .0113 
 
 .0099 
 
 10 15 
 
 .0189 
 
 .972 
 
 .985 
 
 .783 
 
 .0184 
 
 .0186 
 
 .0148 
 
 15 20 
 
 .0262 
 
 .953 
 
 .974 
 
 .703 
 
 .0250 
 
 .0255 
 
 .0184 
 
 20 25 
 
 . 0334 
 
 .928 
 
 .961 
 
 .628 
 
 .0310 
 
 .0321 
 
 .0210 
 
 25 30 
 
 .0403 
 
 .898 
 
 .947 
 
 . 557 
 
 . 0362 
 
 .0382 
 
 .0224 
 
 30 35 
 
 .0469 
 
 .863 
 
 .931 
 
 .494 
 
 .0405 
 
 .0437 
 
 .0232 
 
 35 40 
 
 .0531 
 
 .824 
 
 .915 
 
 .440 
 
 .0438 
 
 .0486 
 
 .0234 
 
 40 45 
 
 .0589 
 
 .781 
 
 .898 
 
 .392 
 
 .0460 
 
 . 0529 
 
 .0231 
 
 45 50 
 
 .0643 
 
 .734 
 
 .880 
 
 .348 
 
 .0472 
 
 .0566 
 
 .0224 
 
 50 55 
 
 .0692 
 
 .684 
 
 .862 
 
 .310 
 
 .0473 
 
 . 0597 
 
 .0215 
 
 55 60 
 
 .0736 
 
 .633 
 
 .844 
 
 .275 
 
 .0466 
 
 .0621 
 
 .0202 
 
 60 65 
 
 .0774 
 
 .581 
 
 .826 
 
 .243 
 
 .0450 
 
 .0639 
 
 .0188 
 
 65 70 
 
 .0806 
 
 .531 
 
 .808 
 
 .213 
 
 .0428 
 
 .0651 
 
 .0172 
 
 70 75 
 
 .0832 
 
 .481 
 
 .790 
 
 .185 
 
 .0400 
 
 .0657 
 
 .0154 
 
 75 80 
 
 . 0852 
 
 .434 
 
 .772 
 
 .159 
 
 .0370 
 
 .0658 
 
 . 0135 
 
 80 85 
 
 .0865 
 
 .388 
 
 .755 
 
 .136 
 
 .0336 
 
 .0653 
 
 .0118 
 
 85 90 
 
 .0872 
 
 .342 
 
 .738 
 
 .'115 
 
 . 0298 
 
 .0644 
 
 .0100 
 
 90 95 
 
 .0872 
 
 .298 
 
 .721 
 
 .096 
 
 .0260 
 
 .0629 
 
 .0084 
 
 95 100 
 
 .0865 
 
 .256 
 
 .705 
 
 .080 
 
 .0221 
 
 .0610 
 
 .0069 
 
 100 105 
 
 .0852 
 
 .218 
 
 .690 
 
 .066 
 
 .0186 
 
 .0588 
 
 .0056 
 
 105 110 
 
 .0832 
 
 .183 
 
 .675 
 
 .054 
 
 .0152 
 
 . 0562 
 
 .0045 
 
 110 115 
 
 .0806 
 
 .150 
 
 .661 
 
 .044 
 
 .0121 
 
 .0533 
 
 .0035 
 
 115 120 
 
 .0774 
 
 .120 
 
 .648 
 
 .035 
 
 .0093 
 
 .0502 
 
 .0027 
 
 120 125 
 
 .0736 
 
 .094 
 
 .636 
 
 .028 
 
 .0069 
 
 .0468 
 
 .0021 
 
 125 130 
 
 .0692 
 
 .073 
 
 .625 
 
 .021 
 
 .0050 
 
 .0433 
 
 .0015 
 
 130 135 
 
 .0643 
 
 .055 
 
 .614 
 
 .016 
 
 .0035 
 
 .0395 
 
 .0010 
 
 135 140 
 
 .0589 
 
 .040 
 
 .605 
 
 .012 
 
 .0024 
 
 .0356 
 
 .0007 
 
 140 145 
 
 .0531 
 
 .028 
 
 .596 
 
 .010 
 
 .0015 
 
 .0316 
 
 .0005 
 
 145 150 
 
 .0469 
 
 .019 
 
 .589 
 
 .008 
 
 .0009 
 
 .0276 
 
 .0004 
 
 150 155 
 
 .0403 
 
 .015 
 
 .582 
 
 .006 
 
 .0006 
 
 .0235 
 
 .0002 
 
 155 160 
 
 .0334 
 
 .012 
 
 .577 
 
 .005 
 
 .0004 
 
 .0193 
 
 .0002 
 
 160 165 
 
 .0262 
 
 .010 
 
 .572 
 
 .005 
 
 .0003 
 
 .0150 
 
 .0001 
 
 165 170 
 
 .0189 
 
 .008 
 
 .569 
 
 .004 
 
 .0002 
 
 .0108 
 
 .0001 
 
 170 175 
 
 .0114 
 
 .006 
 
 .567 
 
 .004 
 
 .0001 
 
 . 0065 
 
 .0001 
 
 175 180 
 
 .0038 
 
 .004 
 
 .566 
 
 .004 
 
 .0000 
 
 .0022 
 
 .0000 
 
 Sums 
 
 2.0202 
 
 
 
 
 0.7503 
 
 1 . 4874 
 
 0.3491 

 
 LUNAR AND TERRESTRIAL ALBEDOES 
 
 DISCUSSION OF THE OBSERVATIONS 1 
 
 In the Astro physical Journal for April, 1916, Professor H. N. 
 Russell has discussed some recent observations of the writer on 
 the earth-shine, from which the earth's albedo had been obtained 
 indirectly. The observations are of two sorts (i) direct visual 
 comparisons of parts of the moon (lit by the sun's rays) and of 
 other parts of similar quality, lit by the earth-shine, with the light 
 of a flame seen through blue glass; and (2) comparisons of the 
 relative intensities of all the colors in the spectrum between 
 violet and red from their relative photographic effect on spectro- 
 grams. 
 
 The earth-shine spectrograms, 2 along with similar ones of the 
 moon and of the sky, were measured at the Westwood Observ- 
 atory by means of the comparator originally designed for quantita- 
 tive measures of the intensities of atmospheric bands on the 
 Lowell Observatory spectrograms of Mars and the Moon. With 
 this instrument I have already obtained approximate determina- 
 tions of the amounts of the water vapor and oxygen in the at- 
 mosphere of Mars, and I am now engaged in measuring the in- 
 tensities of the Fraunhofer lines in the solar spectrum with an 
 accuracy which has not been approached hitherto. These facts 
 show that on the score of precision, the comparator is capable of 
 excellent work, though, like all instruments dependent on photo- 
 graphy for the registration of intensities, it involves the complex- 
 ities of photographic laws. These complexities, however, I have 
 
 1 The essential features of this discussion were presented before 
 the American Astronomical Society at its Nineteenth Meeting at 
 Swarthmore, Pennsylvania, August 31, 1916. 
 
 2 The spectrograms were made for me through the kindness of 
 Dr. Percival Lowell by Dr. V. M. Slipher at Flagstaff. Several of 
 them were taken under exceptionally favorable atmospheric conditions. 
 See Frank W. Very "The Photographic Spectrography of the Earth- 
 Shine," Astronomische Nuchrichten, Nr. 4819-20, Bd. 201, s. 353-400, 
 November, 1915; also "Atmospheric Transmission," Science, N. S., 
 Vol. XLIV., No. 1127, pages 168-171, August 4, 1916. 
 
 15
 
 16 LUNAR AND TERRESTRIAL ALBEDOES 
 
 endeavored to minimize, and I have in large part succeeded in 
 eliminating them by an extensive study of the photographic prob- 
 lem for each wave-length and a wide range of exposures on more 
 than one kind of plate. Professor Russell's criticisms of my work 
 with the spectral line and band comparator are largely founded on 
 misapprehensions. He has taken unwarrantable liberties with my 
 figures and by so doing has rejected my work on the spectro- 
 grams on insufficient grounds. When correctly reduced, the two 
 methods give results which are in good agreement, but on the 
 whole, those from the spectrograms are the more reliable. 1 
 
 In his first article, 2 Russell adopts 465,000 : I for the ratio 
 of sunlight to full-moon light ; and on page 184 of the April Jour- 
 nal he expresses preference for the ratio 9,000 : 1 3 between sun- 
 light and full-earth light on the moon. According to this, the ratio 
 of full-earth light to full-moon light is 
 
 465,000 : 9,000 == 51.7 : i, 
 
 1 The visual photometric values of the earth-shine which are de- 
 scribed in my paper on "The Earth's Albedo" (Astronomische Nach- 
 richten, Nr. 4696, Bd. 196, s. 269-290, November, 1913) were obtained 
 with a special earth-shine photometer which might be improved in the 
 light of the experience gained with it. Although free from photo- 
 graphic difficulties, the method has difficulties of its own, as may be 
 recognized from the elaborate researches which were required in estab- 
 lishing the constants of the various absorbent pieces. Owing to the 
 faintness of the earth-shine, the low altitude of the crescent moon 
 when the measures have to be made, and the varying transparency of 
 the atmosphere, there are further difficulties which Professor Russell 
 generously allows for in his criticism, but he has not understood some 
 of the minor details. 
 
 It must be remembered that the spectrograms were made with an 
 analyzing spectroscope, and that the values obtained relate to the 
 intrinsic brightness of definite regions on the moon where the reflect- 
 ing quality of the surface is far from uniform, and the range of 
 luminous values with the phase is wide, so that small displacements on 
 the surface may give considerable alteration of light. The observed 
 differences are due to these unavoidable vicissitudes, rather than to 
 errors of observation; but such differences as these tend to average 
 out from the general mean. In the earth-shine exposures, the slit 
 was placed half on and half outside the dark limb of the moon to give 
 the sky spectrum needed in the reductions. 
 
 2 Astrophysical Journal for March, 1916, p. 125. 
 
 3 On page 194 (op. cit.) , this ratio is attributed to me, but I have 
 not given it, and prefer the ratio 10,000 : 1.
 
 LUNAR AND TERRESTRIAL ALB EDO ES 17 
 
 which, since the angular area of the earth as seen from the moon 
 is 13.4 times that of the moon seen from the earth, makes the 
 earth's albedo 51.7/13.4 = 3.86 times that of the moon. My 
 own determination of this ratio is considerably larger, namely, 
 4.8 : i. 
 
 The ratio of sunlight to moonlight is not easily measured 
 with precision on account of the wide range of intensities involved 
 and the uncertainties of atmospheric absorption. Whether we 
 take the ratio 465,000 : i, difference of magnitude = 14.17 
 (Russell), or 618,000 : i, difference of magnitude = 14.48 
 (Zollner), or even a value as large as Wollaston's (801,000 : i) 
 we shall still be inside the actual divergences of some very good 
 observers. A critical examination of sources of error will im- 
 prove this result greatly. 
 
 Zollner's explanation of the peculiar efficacy of the lunar 
 mountains in emphasizing the peak of the lunar phase-curve at 
 the full, with Searle's emendation which notes the contribution 
 to the same effect by crevices which retain the sunshine away 
 from observation until the short interval when the rays strike 
 their floors, suffices for the anomalies of this phase-curve. Wheth- 
 er the elevations are mountains, as is usually assumed, or innu- 
 merable crystalline facets, that is, whether the roughness is on 
 a large or a small scale, is a matter of indifference. Zollner's 
 diagrammatic figure is well characterized by Russell as "artifi- 
 cial," but the fact of a general excessive roughness of the lunar 
 surface is probable enough and natural enough, even though 
 it may not be as obvious as the mountains are to telescopic vision. 
 Nevertheless, although Zollner's hypothetical moon behaves in 
 some respects like the actual moon near the time of full, the anal- 
 ogy would fail if pushed to its limit, especially in the early 
 crescent phases, and the title "true" which was used by Zollner 
 to designate the "albedo" of this hypothetical body is a misnomer 
 when applied to the actual moon. Russell's criticism by -ques- 
 tioning the foundation of an almost unanimous acceptance of 
 Zollner's value and terminology, performs a much needed use. 
 In this he has followed Guthnick more or less. Miiller, also, had 
 previously characterized Zollner's value of the mean (52) slope 
 of the lunar upheavals as "illusory." 
 
 In getting the lunar albedo, Zollner allowed for the rotundity
 
 18 LUNAR AND TERRESTRIAL ALBEDOES 
 
 of the moon, and by an unfortunate misapplication of Lambert's 
 law of reflection from a uniformly diffusing sphere 1 found that 
 the full moon should reflect 
 
 P ( 3/2 )X 48,980 73,470 
 
 of sunlight, if it were such a sphere. The corresponding 
 value of the albedo he called the "apparent albedo" (scheinbare 
 Albedo) a term which is misleading, since it implies that 
 the moon's reflection which is actually observed is this quantity 
 given by computation, and that the reflection up to this point 
 follows Lambert's law, which is not true. 
 
 The total radiation of the moon, including both reflected and 
 emitted rays, does appear to follow pretty nearly a sequence 
 which can be derived from Lambert's law, multiplied by the 
 factor 2/3, a number which seems to turn up on every hand in 
 this research. This may be seen from the phase-curve given in 
 my "Prize Essay on the Distribution of the Moon's Heat and 
 its Variation with the Phase," where I found 
 At first quarter, total radiation = 17.7% of radiation from 
 
 full moon. 
 At last quarter, total radiation = 24.8% of radiation from 
 
 full moon. 
 Mean (quadrature) total radiation = 21.25% f radiation from 
 
 full moon. 
 Lambert's law, multiplied by 2/3 = 21.22% of radiation from 
 
 full moon. 
 
 Here the radiation which is measured, is made up of two 
 parts one which is wholly reflected, and the other an emission 
 from a heated surface whose temperature varies from a maxi- 
 mum at the center of the disk to a minimum at the limb, while 
 the surfaces of equal temperature are concentric zones. The 
 luminous reflection, at any rate, follows the opposite law and is 
 greater at the limb, and probably the non-luminous rays are 
 reflected in the same way. The smaller total radiation at first 
 quarter is of course due to the fact that the moon is getting hotter 
 and the energy is being expended in modifying the subsurface 
 
 1 The law was designed to give the spherical albedo. What was 
 needed here was simply the geometrical albedo.
 
 LUNAR AND TERRESTRIAL ALB EDGES 19 
 
 thermal gradient, while the heat thus retained is given out again 
 in the lunar afternoon, or at last quarter. The point which I 
 wish to make clear is that, as far as the reflection of the moon's 
 lit/ lit is concerned, the introduction of Lambert's law at this 
 or any other stage of the computation was a mistake. 
 
 The value "p = 0.1195," which Zollner calls "die scheinbare 
 Albedo des Mondes," is a hypothetical value which is not imme- 
 diately given by observation, but is obtained by restricting the 
 definition of albedo to diffuse reflection from a nonexistent smooth 
 sphere on the supposition that a factor 2/3 must be introduced. 
 I will return to this later, but will note here that the moon does 
 not reflect much after the approved fashion of a sphere, but acts, 
 to all appearance, more like a flat surface, or even like one a 
 little dished at the margin ; for whereas a diffusive sphere should 
 send out light from the marginal zones at full into a rear hem- 
 isphere, whereby these zones as seen from the front should be 
 considerably fainter than the center, it is found, on the contrary, 
 that the limb in the actual moon is in fact the brighter of the two. 
 Both front and rear reflections from the limb are exceptionally 
 large, so that these portions of the lunar surface are but little 
 heated by the sun's rays. 
 
 The arguments by which Zollner persuaded himself that he 
 had arrived at the "true" value of the moon's albedo from his 
 "apparent" albedo are somewhat involved. One of the first needs 
 is that our procedures and definitions may be -clarified and sim- 
 plified. Let us begin by considering the Lambert law. 
 
 On page 176 of his second article, Professor Russell says 
 correctly in speaking of the reflective function of the phase-angle, 
 cp(a) : "The whole amount of light reflected by the planet to the 
 celestial sphere will be proportional to 
 
 f 
 
 c/ t 
 
 cp(a) sin a da. 
 
 If it shone in all directions with the brightness of 
 the full phase, the emitted light would be 2.0 on 
 the same scale." But this being so, the factor q which 
 transforms from the geometrical albedo at opposition to the 
 spherical albedo should be the integral just given, and not twice
 
 20 LUNAR AND TERRESTRIAL ALBEDOES 
 
 that quantity, which is Russell's equation (7), because the value 
 of the integral alone without the coefficient 2 is 2.0, if <p(a) = i 
 everywhere. For Lambert's law, 
 
 f * 
 
 (6) q = I <p(a) sin a e?a 0.75, 
 
 = f 
 
 and this is the Lambert factor for spherical albedo, or 
 it is the spherical albedo if the geometrical albedo at 
 full phase is unity, instead of #=1.5 as given by Russell. 
 Similarly, for the lunar phase-curve, q=O-3$, which is 
 to be multiplied into the observed geometrical albedo from 
 the full moon to give the lunar spherical albedo. Since the values 
 of q have been taken two times too large, all of the numbers in 
 Russell's Table I., op. cit., page 179, should be divided by two. On 
 the other hand, in getting q for the Lommel-Seeliger law, Russell 
 has inconsistently dropped the factor 2, which would make his 
 value the same as mine, were it not that there is a further in- 
 accuracy in the integration by which he gets "(7=1.6366." His 
 equation (7) should give ^ = 3.0. 
 
 Lambert's formula for spherical albedo, 
 
 L = (I/JD) [sin a a cos a] -)- cos a, 
 
 where a is the moon's elongation, or phase-angle from conjunc- 
 tion, or as Russell prefers to put it, employing phase-angles from 
 opposition', (in which respect I shall follow his procedure) 
 
 <p(a) = (i/ft) [sin a-f- (:t a) cos a], 
 
 gives the light at quadrature, L 90 = i/jt = 0.318, when the light 
 at the full phase is unity. For the emission of its own radiation 
 combined with the reflection, the radiant observation already 
 quoted would seem to favor the value, R QO = i/( 1.511) = 0.212. 
 Here, however, we must note that, though the emission may be 
 independent of its direction, the distribution of temperature is 
 not uniform, and the result is a complex of two different 
 functions of a. 
 
 Under these circumstances we can attach little significance 
 to this special value. It is difficult to see how a particular numer- 
 ical factor can survive this double vicissitude. If the factor 2/3 
 occurs undisguised in both functions, it will become 4/9 
 in their product; if in opposite senses, it will cancel out; 

 
 LUNAR AND TERRESTRIAL ALBEDOES 21 
 
 and if the factor is found in one case, but a different one is 
 substituted for it in the other, it could not be so easily recognized. 
 By Lambert's law, at the full, L = jtX ^-90 3- T 4 -^90- 
 For the planet Venus, reflection = L = (17.283) L 90 = 3-53^90- 
 Lunar emitted and reflected radiation = R = (3/2) JtX -^90 
 
 90 . 
 
 Lunar reflected light = L = (i/.ios) L ao = 9-52L 9 
 
 The factors connecting the observed light at full phase 
 with the light at quadrature are evidently empirical. Those 
 connecting spherical and geometrical albedoes are probably 
 equally empirical. The appearance of the factor (3/2) Ji in my 
 lunar radiation observation, to which I have called attention, is 
 rather striking, yet it is probably no better than a coincidence. 
 Zollner must have been under a great misapprehension when he 
 attempted to introduce the factor 3/2 into the discussion of his 
 lunar observations, for it does not fit the facts. 
 
 There is, it is true, a universal usage for which the factor 
 3/2 is appropriate, namely : The reflection from a sphere of any 
 ordinarily diffusive material is 2/3 of that returned with per- 
 pendicular incidence and reflection from a plane surface of the 
 same substance, and in comparing the reflection of the entire 
 spherical body of a planet with that from a plane surface of 
 some terrestrial substance, it would be appropriate, provided we 
 could be sure that the light is diffusively reflected, to multiply 
 the reflection from the sphere by 3/2 to put it on terms of 
 equality with the recognized reflective power, or specific reflect- 
 ivity, of the given terrestrial material in the form of a flat surface 
 when this is viewed normally. If the flat surface reflects the 
 light at an angle of 45, its reflection must . be multiplied by 
 cos 45 = 0.707, and in this case the two reflectors are already on 
 terms of approximate equality. It does not appear that the factor 
 3/2 was introduced by Zollner with any such end as this in view, 
 but simply because it occurs in the Lambert formula. As applied 
 by Zollner in his lunar theory, this use of the factor 3/2 has been 
 a stumbling block in the path of subsequent research on the subject. 
 The whole of this cloudy lucubration should be swept aside. 
 Zollner's original lunar observations are among the best that 
 have ever been made, and they deserve to be rescued from the 
 scandalous treatment of his theoretical discussion.
 
 22 LUNAR AND TERRESTRIAL ALBEDOES 
 
 At the start, Zollner has evidently adopted the incorrect idea 
 that the fraction of light from the sun received upon the moon's 
 surface and which has to be considered, is the fraction of the 
 total luminous output of the sun to the entire sphere, and he thus 
 gets for the denominator the number 48,980. He sees that this 
 number is too small for his observations and makes the hypothesis 
 that it must be multiplied by 3/2, giving 73,470, to which he 
 assigned the symbol p. But this is in turn too small, and he 
 introduces the further hypothesis of the lunar mountains with 
 slopes of 52, getting a new factor x, and x/ = 107,300, with 
 which his value of the so-called "true" albedo, 0.174, was obtained 
 by comparing it with the observed ratio of sunlight to moonlight. 
 If he had started out with the correct conception that the moon 
 receives a certain fraction of the light emitted by the half sphere 
 of the sun which is turned toward the planet, he would have 
 obtained at once the number 97,960 (slightly different from the 
 one which I have used, because we have adopted slightly different 
 values of the moon's semidiameter) and he would have 
 obtained at once for the geometrical albedo of the moon, 
 97,960/618,000 = 0.1585. 
 
 In short, Zollner started with a wrong number, multiplied 
 this by 3/2 and then by nearly another 3/2, or in all by nearly 
 2%, instead of by 2 exactly, as he should have done, and thus 
 by a threefold error he reached a result which was nearly right, 
 but solely by accident. 
 
 Russell starts with the same erroneous conception that the 
 sun's complete spherical emission should be considered, but im- 
 mediately abandons it (though without noting the fact) for 
 another, not necessarily incorrect, but different from mine, since 
 he does not introduce 4 into the numerator of his expression for 
 p, nor yet the number 2 which would give what I call the geomet- 
 rical albedo, but multiplies by i, whence his p is one half of 
 the geometrical albedo, given by eye observation, and represents 
 surface illumination as I have shown in the Introduction. 
 He then makes the reverse change by introducing 2 into 
 his value of q, outside the integral, where it does 
 not belong if by q is meant the spherical factor, as in my 
 equation (6), so that his q is two times mine and through the 

 
 LUNAR AND TERRESTRIAL ALBEDOES 23 
 
 cancellation of these opposite transformations we finally reach 
 similar results for the spherical albedo. 
 
 The geometrical albedo, A v which, like Russell's p, "depends 
 only on the geometrical and photometric relations of the planet 
 as observed at the full phase" 1 is correctly given by equation (7), 
 
 2A/ (1 sin 2 S 
 
 (7) A, = ~^- . 2 , 
 
 sin ,y sin o 
 
 where S is the apparent semi-diameter of the sun as seen from 
 the earth, ,y and <T O are the semi-diameters of the sun as seen 
 from the planet and of the planet as seen from the earth at the 
 time of opposition, and M is the ratio of the light received from 
 the planet at mean opposition, to the light of the sun as observed 
 from the earth. This equation is the same as the middle one 
 of Miiller's (i4) 2 and is a special case derived from the Lommel- 
 Seeliger theory, which, although it is a theory of spherical 
 reflection, gives the geometrical albedo at this particular point, 3 
 which Russell's equation (5) does not do. 
 
 Professor Russell says: "Let r be the mean radius of the 
 planet's disk, and R its distance from the sun, and M be the ratio 
 of the apparent brightness of the planet at the full phase, and at 
 distance A from the earth, to that of the sun at unit distance. 
 The fraction of the sun's whole radiation which the planet in- 
 tercepts is r 2 /4R 2 ." * This is all true, but it is not what we want to 
 know. The fraction of the sun's light emitted by the solar hem- 
 isphere which is visible from the planet, and which the planet 
 intercepts, is nr 2 /2nR 2 = r z /2.R 2 ; and M being the ratio of the 
 observed planetary light at full phase to sunlight and A the 
 distance from the earth at the time of observation, so that if the 
 other things remain the same, Af A 2 = const., an alternative ex- 
 pression for the geometrical albedo is 
 
 Tt is not necessary in this problem of the reflection of the sun's 
 
 1 H. N. Russell, in Astrophysical Journal for April, 1916, p. 177. 
 
 2 Photometric der Gestirne, p. 65. 
 
 3 The geometrical albedo is in fact the same as the spherical 
 albedo on the assumption that the spherical factor is unity (q = 1). 
 
 4 Astrophysical Journal, April, 1916, p. 176.
 
 24 LUNAR AND TERRESTRIAL ALBEDOES 
 
 rays to consider the sun's invisible hemisphere. The reflection 
 would be the same if this were dark. For the particular problem 
 under discussion, the other side of the sun is as if it were non- 
 existent, and it has nothing to do with the question. That there 
 may be no misunderstanding, I repeat that Russell's final expression 
 for what he denotes under the symbol p, and which I call surface 
 illumination, must be multiplied by 2 in order to obtain the 
 quantity A 2 or the geometrical albedo; and therefore his factor 
 "q" which reduces to spherical albedo is to be divided by 2, so 
 that, apart from all other considerations, that is, granting the 
 reliability of the original data, there should be no difference be- 
 tween us in respect to the spherical albedoes ("A" of Russell's 
 Table V), provided we could agree in regard to the best phase- 
 law to be adopted. 1 
 
 The quantity "p" is denned in two ways in Russell's paper : 
 "The factor p may also be defined as the ratio of the actual 
 brightness of the planet at the full phase to that of a self-luminous 
 body of the same size and position, which radiates as much light 
 from each unit of its surface as the planet receives from the sun 
 under normal illumination" (Op. cit., pp. 177-178). 
 
 By this definition, as interpreted by Russell, the quantity p is 
 proportional to Miiller's Af , since all of the planetary values in 
 Russell's equation have been reduced to unit distance, and Af n 
 is a ratio to sunshine at unit distance. But from (7) 
 
 M = J-^oX const., 
 
 or Russell's p, like Miiller's A/ , is proportional to the half of the 
 geometrical albedo. 
 
 The writer would interpret the definition itself differently, 
 because a planet of radius r and distance from the sun R, will 
 receive from the sun, if L is the sun's total spherical emission of 
 light, the light-quantity L/4nR 2 on each unit of normally ex- 
 posed surface. But the planet "receives" light from the sun on 
 only one half of the planetary surface, and hence, if it radiates 
 from "each unit of its surface" self-luminously (or what amounts 
 . to the same thing, if the light falling normally on a plane surface 
 equal to the planet's section, is wholly emitted by a hemispherical 
 surface of that planet) the emitted light is given out through 
 
 1 This will be considered in a separate paper.
 
 LUNAR AND TERRESTRIAL ALB EDO ES 25 
 
 a surface twice as great as the receiving surface, and should be 
 on the average intrinsically half as intense as the received light. 
 If the intrinsic brightness of the emitted light is 
 
 the total light emitted by the sphere is 
 
 L r~ 
 
 L' = 4JL/V 2 = 
 
 But this does not represent the real conditions, because the defi- 
 nition itself is faulty, since the planet does not receive light 
 "under normal illumination" on every part of its exposed hem- 
 isphere. If, therefore, we make my parenthetical substitution, 
 and omit the stipulation that each unit of surface shall radiate 
 as much light as it receives "under normal illumination," the 
 total light emitted by the hemisphere is 
 
 I r 2 
 
 L' = znJr* = - -, 
 4 K- 
 
 and we have returned to the previous proposition relating 
 to the fraction of the sun's whole spherical emission which 
 the planet intercepts. This, as I have said, does not 
 concern us in the problem of reflection, where the planet 
 reflects light solely from the visible hemisphere of the sun, 
 and the invisible solar hemisphere with its emission need not be 
 considered, since none of its light reaches the planet. 
 
 \Ve have, therefore, for the reflection-ratio, if all of the 
 sunlight is reflected, 
 
 . L L' r- 
 
 L /~ = 2 T- = 
 
 2 L 2R 2 
 
 and the geometrical albedo is the ratio of the observed 
 reflection to this value ; whence Russell's first definition, 
 if modified so as to bring it into accord with the actual 
 conditions, agrees with my definition of the geometrical albedo. 
 
 On the other hand, Russell's second definition can not be 
 thus reconciled. It reads as follows : 
 
 "The factor p may be defined verbally as the ratio of the 
 observed brightness of the planet at full phase to that of a flat 
 disk of the same size and in the same position, illuminated and
 
 26 LUNAR AND TERRESTRIAL ALBEDOES 
 
 viewed normally, and reflecting all the incident light in accordance 
 with Lambert's law" (op. cit., p. 188). 
 
 If the sphere shines "in all directions with the brightness 
 of the full phase" (op. cit., p. 176), the quantity which I call q 
 (eq. 6) is 
 
 q fp(a) sin a da = 2.0. 
 
 y. 
 
 *J 
 
 But if the sphere shines according to Lambert's law, the value 
 of the integral is 3/4. If, however, the further proviso be made 
 that p is "the ratio of the observed brightness of the planet at 
 full phase to that of a flat disk," since the ratio of reflection from 
 a sphere illuminated and viewed from the front (or that reflection 
 corresponding to its geometrical albedo), is to the reflection from a 
 flat disk, illuminated and viewed normally, as 2/3 : i, the 3/4 
 must be multiplied by 2/3 giving 1/2 as the maximum "flat-disk" 
 value of the spherical albedo. The same fraction expresses the 
 relation between the intregral computed for cp(ct) as given by 
 Lambert's law (q) and by the Lommel-Seeliger law (q 2 ), 
 namely, 
 
 Since, however, the total reflection, or spherical albedo, must be 
 the same and equal to the incident light on either hypothesis 
 with complete and diffusive reflection, it follows that whatever 
 differences result from these hypotheses must fall upon q and p 
 equably and oppositely in order that their product, A = qp, may 
 remain the same for the given planet. That is to say, whatever 
 variations there may be in the distribution of light to different 
 zones by reflection according to the rival theories, the sum total, 
 or spherical albedo, must be the same for either if the reflection 
 is complete. Hence if q is obtained by Lambert's law, p must be 
 twice as large as it would be if q were given by the Lommel- 
 Seeliger law. There appears, therefore, to be an inconsistency in 
 Russell's second definition, and his "p" matches the condition 
 demanded by the Lommel-Seeliger law, while my doubled value, 
 though given as a particular case of the Lommel-Seeliger formula, 
 has to be combined with the Lambert value of q, if A = qp is to 
 be kept constant. We thus reach the curious dilemma that
 
 LUNAR AND TERRESTRIAL ALB EDO ES 27 
 
 neither of these two rival theories can dispense with the other. 
 As if to enforce this point, the actual phase-curve of Venus fol- 
 lows a mixture of the two laws. It becomes exceedingly difficult 
 to "mind your p's and q's," under these circumstances. 
 
 If we knew the mean temperature of a planet for all latitudes 
 from the equator to the poles, we should no doubt find some 
 relation between the thermal quantity corresponding to this tem- 
 perature and the spherical albedo of the*planet. In general, we 
 may anticipate that the greater the spherical albedo is, the less will 
 be the heat, but not necessarily in any exact proportionality, be- 
 cause the blanketing action of the planet's atmosphere is the 
 principal factor in the retention of any heat which the surface 
 may receive. Many geologists believe that a continually cloudy 
 atmosphere, which would certainly reflect most of the sun's rays, 
 but would retain terrestrial heat, was largely responsible for the 
 growth of a tropically luxuriant vegetation within the Arctic 
 circle in past ages. Morever, some highly important climatic 
 properties, such as the melting of snow in high latitudes and the 
 possibility or nonpossibility of a permanent fee-cap, depend on the 
 maximum summer temperature, rather than on the mean tem- 
 perature. The maximum temperature at the sub-solar point is 
 somewhat intimately related to the reflection of total radiation 
 spherically, and to some extent the latter follows the luminous 
 albedo. The winter snows of Mars reach nearly as low a latitude 
 as on earth, but no lower. The feebler sunshine of Mars is better 
 conserved because it suffers a smaller reflective loss in passing 
 through a clearer and rarer atmosphere. In fact, the albedo of 
 Mars is so much smaller than the earth's that Mars would have 
 the hotter climate of the two, in spite of greater distance from the 
 sun, were it not that a rare atmosphere permits an easier escape 
 of Martian surface radiation. On a planet with hardly any air, 
 but having a long period of insolation and approximation to a 
 steady state of thermal equilibrium, the sub-solar effect of the 
 sun's rays must be nearly equal to the solar constant multiplied by 
 one minus the spherical reflection of solar rays of every wave- 
 length (i ^! (t) ). Thus, for the moon, the reflection of total 
 radiation in connection with the temperature (both of which are 
 measurable) has a bearing on the problem of the solar constant,
 
 28 LUNAR AND TERRESTRIAL ALBEDOES 
 
 although it may not be possible to utilize the information fully for 
 lack of other data. 
 
 If a completely and uniformly diffusive reflecting sphere of 
 indefinitely great radius be drawn about the sun as a center and 
 including the earth, and if the central radius of the segment of 
 this sphere in view from the night side of the earth be in the 
 prolongation of the line from the sun to the earth, or what 
 amounts to the same thing, if we imagine an ideal night sky to be 
 "packed" with perfectly reflecting full moons, such a segment of 
 the sphere (embraced in a hemisphere as viewed from the earth), 
 or such a sky, should reproduce sunlight at the earth's distance. 
 The moon sends us 1/98,317 part of the light reflected from such 
 a hemisphere to a point, 1 and the full moon, which may be likened 
 to a circular disk of 15*32. "7 radius cut out from this surface, 
 should send us that fraction of sunlight, if it were not that it does 
 not reflect in that way, but absorbs all but a small part of the com- 
 bined luminous and nonluminous radiation received from the sun. 
 Since, however, when equilibrium is attained, the combined emis- 
 sion and diffuse reflection of rays of every wave-length must equal 
 the total of solar radiation received (unless there is some excep- 
 tional specular reflection in a particular direction, of which there 
 is no evidence, and except for a slight retention of heat to be 
 radiated away during the night) a measurement of the heat 
 received from the total radiations of sun and. moon, respectively, 
 should approximate to this ratio. Such a measurement was made 
 by Director Langley and myself at the Allegheny Observatory 
 from which the fraction 1/96,509 was obtained, which seems to 
 be in sufficiently close agreement with the theory. 
 
 Combining the above-named theoretical value with the observed 
 
 ratio of at full moon, we have the following 
 
 moonlight 
 
 Values of the geometrical lunar albedo : 
 By Zollner's observation, A 2 = - =0.159, 
 
 1 Namely, light from surface of hemisphere : light from lunar 
 
 i 
 
 Q Ot T^ ^ O 
 
 disk = -.-- - = __-__- x 1.00515 = 98,317 : 1. 
 
 o,r>2 rr gm J g gm .> Jg' 32". 7 A
 
 LUNAR AND TERRESTRIAL ALBEDOES 29 
 
 With Russell's adopted ratio, A 2 = 9 ' 3 * 7 =o.2ii, 1 
 
 08 
 
 With Miiller's adopted ratio, ^ 2 - =0.173. 
 
 5^9'Soc* 
 
 Mean A, = 0.181. 
 
 Miiller points out that numerical values will differ according 
 to the definitions of albedo of which there are several. Of three 
 theories given in his book with much detail, neither one is even 
 remotely applicable to the moon, except in so far as the particular 
 values by the Lommel-Seeliger and Euler theories for full phase 
 do coincide with the geometrical albedo, on account of the afore- 
 said identity of the equations for this special case. The albedoes 
 "by Seeliger's definition" which are set down by Miiller are obtain- 
 ed on the limiting assumption that the coefficients of absorption 
 of incoming and outgoing rays have the ratio X=i, which makes 
 the coefficient in the Seeliger formula for A 2 = 2, or the same as in 
 the formula for geometrical albedo. Otherwise, if X differs from 
 unity, we have 
 
 A = i, numerical coefficient = 2.0000 
 A = 2, numerical coefficient = 1.8924 
 >. = 3, numerical coefficient = 1.8679 
 X = 4, numerical coefficient = 1.8628 
 A, = 5, numerical coefficient 1.8639 
 A, = 6, numerical coefficient = 1.8673 
 A. = 10, numerical coefficient = 1.8846 
 
 The albedoes "by Lambert's definition" are spherical albedoes de- 
 rived from the geometrical albedoes by applying the factor 
 q = 0.75, which is obtained by integration of the Lambert phase- 
 curve. Muller leaves the reader to choose for himself between 
 these values of the moon's albedo: 
 
 "A 1 = 0.129 (by Lambert's definition), 
 ^2 = 0.172 (by Seeliger's definition)." 2 
 
 The use of the Lambert theory and of the constant factor 0.75 in 
 passing from A 2 to A lt prevents the values of A 1 in Miiller's book 
 from being regarded as spherical albedoes, except in those cases 
 where Lambert's law may possibly be followed approximately. 
 
 1 Russell himself, as already noted, divides this by 2 to get his 
 "p," obtaining p = 0.105, and for Zollner's value, p = 0.08. 
 
 2 Photometric der Gestirne, p. 343.
 
 30 LUNAR AND TERRESTRIAL ALBEDOES 
 
 With my value of the earth : moon ratio and Miiller's geomet- 
 rical albedo of the moon, the earth's geometrical albedo is 
 
 A C2 = 4.8 X o- 1 ? 2 = 0-826. 
 
 The ratio 4.8 : I applies only to the geometrical albedoes. The 
 spherical albedoes adopted by Russell, namely, 
 
 "A" = A mi = 0.073 f r tne moon, 
 "A".= A ei = 0.45 for the earth, 
 
 have the larger ratio A ei : A mi = 6.16 : i, and I shall show 
 presently that this ratio ought to be still further increased. On 
 the other hand Russell's values of p which are proportional to 
 geometrical albedoes have a ratio smaller than mine, namely : 
 ^(e) . ^(m) __ Aez . Am2 == 3.86 : i, 
 
 and one which does not agree with his adopted ratio of sunlight 
 to moonlight. A revision on this account is certainly required. 
 Russell's lunar value, "/> = 0.105," if ^ represents the 
 "reflecting power" of the lunar surface, 1 would require 
 that the moon should be composed of something almost as dark 
 as dark grey slate, or nearly like trachyte lava, o.io, according 
 to the figures which he quotes from Wilsing and Scheiner. But 
 excluding the very -brightest and darkest spots which are of rela- 
 tively small area, there are extensive dark regions on the moon 
 whose average total-radiation reflection (bolometrically deter- 
 mined by measuring the transmission of lunar radiation through 
 a glass plate which cuts off practically all of the emitted rays and 
 distinguishes between these and the reflected ones) is from 10 to 
 12 per cent., while that of correspondingly situated bright regions 
 (similarly determined) is from 20 to 25 per cent. Since the 
 moon's surface is about equally divided between such "dark" 
 and "bright" areas, a mean total-radiation reflection of 0.15 to 
 0.185 (average = 0.168) is indicated by my bolometric measures 
 which form a useful check on my photometric results. 2 
 
 1 Russell says (op. cit., p. 192) : "Wilsing and Scheiner have de- 
 termined the reflecting power of many ordinary rocks, using an ap- 
 proximately flat, rough, natural surface normal to the incident and 
 reflected rays. Their formula of reduction gives exactly the quantity 
 which has been designated by p." To the writer, it looks as if p, the 
 planetary illuminating power, should be multiplied by 3/2 before mak- 
 ing this comparison. 
 
 2 Some samples of these are to be found in my "Photometry of a 
 Lunar Eclipse," Astrophysical Journal, November, 1895, p. 299-300.
 
 LUNAR AND TERRESTRIAL ALBEDOES 31 
 
 If the sum total of reflected rays of every wave-length agrees 
 approximately with unaltered solar radiation, the preceding frac- 
 tion must be increased a little to represent the result as it would 
 be found outside the earth's atmosphere ; because the solar reflected 
 rays are of shorter wave-length than the rays emitted by the moon, 
 and they are differently modified in passing through the at- 
 mosphere, which alters the relative values of the terms of the 
 comparison. Except for certain bands of selective absorption, the 
 longer waves are more readily transmitted by the air. It becomes 
 increasingly evident that the solar constant is about 3.5 (C. G. 
 Min.), but this is reduced to 1.5 at sea-level, so that the real trans- 
 mission of solar rays by the atmosphere is 3/7. In a seasonally 
 comparable observation of the moon, 48 per cent, of its emitted 
 radiation entered through the air. Reducing to conditions outside 
 the atmosphere by these values, 
 
 radiation reflected by the moon !6.8X(7/3) I 
 
 radiation absorbed by the moon 83.2 X ( I( V4.8) ~ 4.42 
 and the true percentage of total solar radiation reflected from 
 the moon is 100/5.42 = 18.5%, which differs little from a mean of 
 the three results quoted for the reflected light of the moon, 
 A.,= 18.196- These, however, as I shall show, need to be dimin- 
 ished somewhat. 
 
 The question whether the invisible and longer solar waves of 
 radiation are better or worse reflected by the moon than the visible 
 ones has never been definitely settled, and indeed there is diversity 
 of opinion as to the relative reflection by the moon of different 
 colors in the visible spectrum. We need not consider the great 
 bands of "metallic" reflection by quartz near 9/x and the large 
 reflection by many common terrestrial substances between 8 and 
 lOfji, for there is very little solar radiation of these wave-lengths 
 to suffer reflection. 1 Metals have greater specular reflection for 
 infra-red radiation just beyond the visible spectrum than for lu- 
 minous rays ; but metals are not in question here. The lunar reflec- 
 tion is almost entirely diffusive, and we wish to know how sub- 
 stances which reflect diffusely behave to infra-red rays between 
 0.7 and 3-O/u.. Eighteen years ago, I published the value of 13.1 
 
 1 See my paper on "The Temperature Assigned by Langley to the 
 Moon," Science, N. S., Vol. XXXVII, No. 964, pp. 949-957, June 20, 
 1913.
 
 LUNAR AND TERRESTRIAL ALBEDOES 
 
 per cent, for the lunar reflection of total solar radiation, 1 but I 
 now think that this should be increased to the value given above, 
 (-18.5%), because in my former work I under-rated the absorp- 
 tion of solar radiation by the air. 
 
 On the other hand, on the strength of Zollner's oft quoted, but 
 little studied value of what he calls the "true" lunar albedo 
 (17.4%) I had formerly supposed that luminous rays are better 
 reflected than the visible ones from 0.7 to 3-O/x; but it now appears 
 probable from measures which are to follow, that this relation 
 must be reversed, and that the larger luminous reflection which 
 would result from the lunar-solar ratios adopted by Miiller and 
 Russell, can not be accepted. In fact, in place of Zollner's hitherto 
 accepted albedo must be substituted the smaller value A 2 = o.i$(), 
 which follows from his own observations (entirely apart from any 
 considerations whatsoever as to the shape of the moon, or as to its 
 surface quality, or the peculiarities of its phase law). 
 
 I will now give a series of ratios of sunlight to moonlight for 
 homogeneous radiations in the visible spectrum derived from my 
 spectro-photometric observations, published in Astronomische 
 Nachrichtcn, Nr. 4820 (s. 385 386) which, as there given, are 
 corrected for atmospheric absorption only. The original values 
 are all that is necessary for a comparison of the relative reflection 
 of different colors by the moon, but for our present purpose they 
 
 I 
 
 Sunlight 
 
 A 
 
 Sunlight 2 
 
 Moonlight 
 
 Moonlight 
 
 p- 
 
 
 /* 
 
 
 0.40 
 
 531,000 
 
 0.56 
 
 682,000 
 
 0.42 
 
 559,000 
 
 0.58 
 
 666,000 
 
 0.44 
 
 587,000 
 
 0.60 
 
 646,000 
 
 0.46 
 
 607,000 
 
 0.62 
 
 619,000 
 
 0.48 
 
 640,000 
 
 0.64 
 
 600,000 
 
 0.50 
 
 669,000 
 
 0.66 
 
 513,000 
 
 0.52 
 
 681,000 
 
 0.68 
 
 456,000 
 
 0.54 
 
 685,000 
 
 Mean = 
 
 609,000 
 
 1 Astrophysical Journal, Vol. VIII, p. 275, December, 1898. 
 
 2 The reduction factor to moon's mean distance from the earth, 
 and to full moon, rests on the following data : 
 
 Series 1 Series 2 Series 3 
 
 Mean phase-angle from full, a 30 a = 18 a = 6 
 
 Light (from lunar phase-curve), 0.53 0.69 0.90 
 
 Moon's parallax, 3374" 3415" 3455" 
 
 Reduction factors, 0.515 0.687 0.916
 
 LUNAR AND TERRESTRIAL ALBEDOES 33 
 
 require. further reduction for the distance of the moon and for the 
 interval to exact full moon. The original figures have been 
 multiplied by the factor 0.70 and are fully corrected. The 
 spectro-photometric method yields results which have one special 
 advantage. They are entirely free from the troublesome Purkinje 
 effect which has vitiated much of the previous measurement. 
 
 The mean ratio of sunlight to moonlight for light of every color 
 within the visible spectrum is 
 
 sunlight : moonlight = 609,000 : i, 
 
 but considering that the central region in the green affects the eye 
 most powerfully, a mean visual ratio of 681,000 : I is to be 
 preferred. 
 
 It is evident from inspection of the numbers in this table that, 
 while the reflection of blue and violet light by the moon is larger 
 than that in the green and yellow, there is also a large reflection 
 in the red which increases in the direction of the infra-red. 
 Unfortunately, there is a dearth of observations of the reflective 
 power of ordinary terrestrial materials in the region between 0.7 
 and 3-O/x, where there is a great block of solar radiant energy ; 
 but I think we may conclude that this region is probably more 
 reflected by the moon than the visible part of the spectrum. In 
 this case, Russell's ratio (i : 465,000) may answer well enough 
 for the reflection of solar infra-red radiation by the moon, but it 
 is much too large for the reflection of visible rays. 
 
 The earth can not have the same ratio of reflection for visible 
 and infra-red rays that the moon does, because the earth's reflec- 
 tion is mainly atmospheric, with visible rays somewhat better 
 reflected than the infra-red. If the moon's geometrical albedo 
 for visible rays is A mz = 0.15, that of the earth is 
 A e2 -- 4.8 X ai 5 = O-7 2 -' but the reflection of total radiation 
 by the earth, unlike that by the moon, is smaller than for visible 
 rays (because the infra-red rays are but little reflected by the 
 atmosphere). It can hardly exceed A e2 w = 0.70, and 
 may be as low as 0.50. For the present I shall adopt A e2 (l) =0.60. 
 Whatever values are finally adopted ought to be consistent among 
 themselves and with the general principles now under discussion. 
 
 In my visual photometric work on the earth-shine, the intrinsic 
 brightness of the moon at quadrature is taken = 0.16 times the 
 light at full moon, giving
 
 34 LUNAR AND TERRESTRIAL ALBEDOES 
 
 full-moon light: full-earth light = 1,600 : 0.16 = 10,000 : i, 
 since I found that full earth-shine at the time of new moon 
 must be about 1/1,600 of the light of a corresponding sun- 
 lit area of the moon near quadrature. 1 It was not neces- 
 sary for this purpose that the whole illuminated surface of 
 the moon should be measured, nor is it possible to make such a 
 measure of the earth-shine directly at new moon ; but it is essential 
 that the lunar surfaces to be compared shall be similar, and 
 Professor Russell's arbitrary change of my mean ratio is not ad- 
 missible, even after granting the large probable error of the 
 result. 2 
 
 I propose to give equal weights to the results of my own 
 measures and to those of Zollner as now correctly reduced. Com- 
 paring the lunar-solar ratio with the moonlight : earth-shine ratio 
 given above, we have (with Very's value) 
 
 sunlight : full-earth light = 681,000 : 10,000 68.1 : i, 
 (with Zollner's value) 
 
 sunlight : full-earth light = 618,000 : 10,000 = 61.8 : i. 
 Allowing for the greater area of the earth as seen from the moon, 
 these ratios become : 
 
 68.1/13.4 = 5.08, and 61.8/13.4 = 4.61, mean = 4.8. 
 Moon's geometrical albedo (for the visual effect) 
 
 According to Very, A m2 = 98,317/681,000 = 0.144 "] 
 
 A ,- ^..,, [mean=0. 15 
 
 According to Zollner, /4 m2 =98,3i7/6i8,ooo=:o.i59 J 
 
 Earth's geometrical albedo = 4.8 X o>1 5 0.72 
 Geometrical reflection of total radiation : 
 
 Moon = 0.185, Earth = 0.60 (?) 
 
 I take the reduction factor for spherical albedo, q = 0.35 for the 
 moon from the integration of its phase-curve, and twice this, or 
 9 = 0.70 for the earth, which is a little less than the Lambert 
 value. 
 
 Spherical albedo of moon, A mi = 0.35 X 0.15 = 0.053 
 Spherical albedo of earth, /4 ei = 0.70 X -7 2 = 0.504. 
 
 1 Astronomische Nachrichten, Nr. 4696, s. 286. 
 
 2 With the increased assurance given by the good agreement of 
 the photographic result, I do not believe that this error can amount to 
 as much as 10%. Some weighting of the observations is perhaps 
 desirable.
 
 LUNAR AND TERRESTRIAL ALBEDOES 35 
 
 Moon's Stellar Magnitude: 
 
 Difference of magnitude from sun = log 681,000/0.4 = -j- 14.58 
 Stellar magnitude of sun (Russell) 26.72 
 
 Stellar magnitude of moon (Very) = 12.14 
 
 Photographic magnitude of moon (King) - 11.37 
 
 (with Russell's phase-curve, etc.) 
 
 Moon's Color-Index (King- Very) =-{-0.77, 
 
 or a little less than that of the sun, 1 which agrees with Abney's 
 photographic observations, confirmed by my spectro-photometric 
 comparison of sun and moon, in showing that the moonlight is 
 bluer than sunlight. I have shown that the moonlight is redder 
 than sunlight in the extreme red, but these rays do not count for 
 much either photographically or visually, and therefore do not 
 effect the color-index as usually defined. 
 
 Earth's Stellar Magnitude (Very) : 
 
 As seen from moon, -12.14 4- 58 = 16.72 
 
 As seen from sun, 16.72-}- 12.95= 3-77 
 
 I make no further claim for my previously published value of 
 the earth's albedo 2 than that its ratio to the moon's albedo has 
 been fairly well determined. The previous figures were based 
 upon Zollner's published lunar albedo and must be diminished a 
 little according to what precedes ; but this will not effect the argu- 
 ment for a high value of the solar constant, because this rests on 
 wholly different grounds. If I could accept in principle Professor 
 Russell's argument that my measurement of the earth-shine, "far 
 from being inconsistent with Abbot's value of the solar constant 
 (1.93 calories) is actually in agreement with it," 3 since it has 
 now been shown that Russell's p must be doubled to give the 
 geometrical albedo, I might claim that Abbot's constant should 
 be doubled ! But unhappily this simple method of disposing of the 
 solar-constant problem will not work. A high value of terrestrial 
 reflection of total solar radiation is indeed inconsistent with a low 
 value of the solar-constant, but a low value of this reflection is not 
 necessarily inconsistent with a high value of solar radiation, because 
 the atmospheric depletion of the sun's rays is composed of several 
 
 1 Color-index of sun = between + 0.8 and + 0.9, that of Capella 
 being + 1.0. 
 
 2 Astronomische Nachrichten, Nr. 4820, s. 400. 
 
 3 Astrophysical Journal, April, 1916, p. 195.
 
 36 LUNAR AND TERRESTRIAL ALB E DOES 
 
 parts. If reflection is found to be less potent than has been sup- 
 posed, this simply puts a heavier burden on the other processes of 
 depletion. 
 
 I have elsewhere concluded 1 that only about 18 per cent, of 
 the sun's rays, received upon the entire sunward hemisphere of the 
 earth, are effective at the earth's surface in production of tem- 
 perature. Out of the 82 per cent, of solar radiation lost by the 
 sunward hemisphere of the earth in one way or another in pas- 
 sing through the earth's atmosphere, the measurement of the 
 earth's spherical albedo which has just been given indicates that 
 approximately 50 are reflected back to space by air, or by clouds, 
 including a reflection of a few per cent, by the solid or liquid sur- 
 face of the earth. The rest of the depletion is divided among 
 agencies which go under the general name of "absorption," but 
 this also is really a complex of several processes. 
 
 As was pointed out in my "Note on Atmospheric Radiation," 2 
 a portion of the incoming solar radiation of short wave-length is 
 used up in the upper air in ionization of atmospheric ingredients, or 
 in the production of ozone and other highly efficient absorbents; 
 and since there is at present no way of finding out how potent 
 this part of the atmospheric process may be (except possibly 
 through an interpretation of certain little understood facts made 
 known to us in the study of atmospheric thermodynamics) it is 
 possible, as Dr. Louis Bell has suggested to me, that more solar 
 energy than we imagine is lost in the ionization processes ; and in 
 this case quite a little of the remaining 32 per cent, of "absorp- 
 tion," so-called, may be ionization by solar radiation of very short 
 wave-length at great altitudes in the atmosphere, these rays being 
 wholly obliterated in the process. 
 
 I have alluded to the changes which Professor Russell has 
 introduced into my earth-shine measures as founded on mis- 
 apprehensions, and must now substantiate this claim. On page 
 185 of his second article we are told that "Table IVA contains data 
 derived from Very's paper" ; but the mode of derivation is not 
 consistent. The first three numbers in the last column have been 
 obtained by multiplying the ratio of exposure-durations for earth- 
 shine and for sunlit moon by the ratio of photographic intensities ; 
 
 1 Astrophysical Journal, Vol. XXXIV, p. 382, Dec., 1911. 
 
 2 American Journal of Science, Vol. XXXIV, p. 533, Dec., 1912.
 
 LUNAR AND TERRESTRIAL ALBEDOES 37 
 
 but the last two numbers have been found by dividing the first 
 quantities by the second. By this means, the two sets of numbers 
 (for January and August respectively) which, if they had been 
 correctly derived, would have been entirely different, because as 
 yet uncorrected for photographic peculiarities, are brought into 
 seeming approximate agreement, and the conclusion is reached 
 that the photographic correction which I have derived for the rel- 
 atively over-exposed lunar spectrograms was unnecessary, and that 
 my corrected values are wrong ! By this wholly erroneous argu- 
 ment, Russell supports his reduction of my ratio of the earth's 
 albedo to the moon's from the spectrograms, to a quantity about 
 half as great as mine. It is needless to say that the seeming agree- 
 ment of the numbers in the last column of Table IVA is wholly 
 accidental. 
 
 On page 184 (op. cit.} Professor Russell says that my "con- 
 clusions regarding the relative intensity of the light of these two 
 sources [the earth-lit and the sun-lit portions of the moon] depend 
 on assumptions regarding the photographic action of exposures to 
 light of different brightness." On the contrary, my results do not 
 rest on "assumptions," but on carefully executed quantitative 
 measurements of the photographic effects in question throughout 
 the entire visible spectrum. When properly reduced, there is no 
 difference between the results of the photographic and of the 
 visual observations. The statement (op. cit., p. 186) that "the 
 photographic observations therefore make the earth-shine only 
 half as bright as do the visual observations," is consequently en- 
 tirely wrong; and the conclusion that "this is just what might be 
 expected if the plates had followed the ordinary law for faint 
 illumination and long exposure, and been 'less sensitive than 
 i X t' >" is equally erroneous. The error has come from the in- 
 correct reduction of my observations by Russell in the aforesaid 
 Table IVA. 
 
 The discrepancy which Professor Russell thinks he finds 
 between my theory and my observations in connection with the 
 phase-curve of the moon (op. cit., p. 186 to 187) does not really 
 exist. The observations had first to be reduced to a constant unit 
 of comparison surface, and then to a selected lunar phase-angle. 
 As it happened, the two corrections in a particular case were of 
 equal numerical value, but opposite sign. Limited areas of the
 
 38 LUNAR AND TERRESTRIAL ALBEDOES 
 
 moon were necessarily observed, and the comparison in every case 
 in the visual measures was between "twin circular apertures in 
 black card, each subtending 7' of arc on the celestial sphere, one 
 above, and the other below the horizontal line of junction of the 
 last pair of reflecting prisms." 1 One of these apertures was illu- 
 minated by the standard light, and the other by a sample of the 
 lunar surface. In preparing the material of A. N. 4696 for the press, 
 I have omitted a remark which is needed to complete the sense. 
 After the first equation at the top of page 286 in that paper, there 
 should be inserted these words : "The ratio of intrinsic brightness 
 for the particular (limited) region of the moon under observation 
 is the inverse of that just given." A quotation from my note book 
 will clear up the matter fully: "At cp = 44, M = 0.090. 
 At (p = 87, M = 0.058. Ratio 1.55 : i.oo. For equal moon- 
 light E/M (for (p = 44) must be multiplied by 1.55." The phase- 
 reduction required division by the same number. This peculiarity 
 is due to an exceptionally large reflection from the lunar substance 
 when the angle of incidence is large and nearly equal to the angle 
 of reflection, as was the case for the particular lunar region 
 observed with the moon's elongation, qp=44 ; but the reflection 
 diminished as the angle of incidence of the solar rays on this spot 
 decreased. The Lommel-Seeliger law was formulated to deal with 
 just such peculiarities. My published result: 
 
 "E 44 : E 87 = (0.0003477 X i -55) : 0.0002210 
 
 = 0.0005389 : 0.0002210 
 -2.438 : i," 
 
 (op. cit., p. 286) still stands and is fairly comparable with the 
 ratio for Venus, computed by me from Miiller's result, namely, 
 
 V 44 : V 87 = 2.100 : i, 
 or as Russell gives it for elongations 30 and 90, 
 
 V 30 : V 90 = 2.74 : i.oo, 
 
 where Lambert's law would give 2.94 : i.oo. The agreement is 
 
 close enough to show that the phase-law for the earth resembles 
 
 that for Venus, approaching, however, a little more nearly to 
 
 the Lambert law, and is quite different from that for the moon. 
 
 On page 189, Professor Russell says : "Very's observations 
 
 of the earth-shine indicate that the mean full earth, as seen from 
 
 1 Astronomische Nachrichten, Nr. 4696, s. 269.
 
 LUNAR AND TERRESTRIAL ALBEDOES 39 
 
 the moon is forty times brighter than the full moon as seen from 
 the earth," where the forty should be sixty-eight. 
 
 Russell's adopted value of the moon's stellar magnitude 
 ( 12.55) ' s -4 I magnitude brighter than mine ( 12.14). 
 Zollner's result from a comparison with the sun gave the 
 intermediate value, 12.24, while that from his Capella com- 
 parison, 12. 18, approaches still more nearly to mine. 1 
 
 The last line of Russell's Table V. (op. cit., p. 190) which 
 purports to give "the earth (from Very's reductions of Slipher's 
 spectrograms)" is misleading, since he has substituted his own 
 reduction for mine. 
 
 A final word may be permitted on the vicissitudes of the solar- 
 lunar light-ratio. Bouguer, who obtained a ratio of 300,000 : I 
 for sunlight to moonlight (Traite d'Optique, p. 87, 1760) was 
 careful to observe when the full moon was near its mean distance 
 and when both bodies were at the same altitude ; but unfortunately, 
 he thought it necessary to use identical optical means in either 
 case, and therefore his candle had to be at a distance of 50 feet 
 for the moon and i^ feet for the sun, so that the illuminations 
 actually measured were in the ratio of 1407 : I, both moonlight 
 and sunlight having been much reduced. Under these cir- 
 cumstances, the bluer light of the heavenly bodies being compared 
 with reddish candle light, the moonlight, on account of its greater 
 faintness and of the relatively greater sensitiveness of rod-vision 
 for faint blue light as the general illumination diminished, had an 
 undue advantage, in the candle comparison, over sunlight, as will 
 be evident from a short table in my paper on "The Earth's 
 Albedo." 2 Bouguer's 300,000 must be at least doubled to correct 
 for this error. The spectrophotometric method entirely removes 
 this difficulty. 
 
 1 From Zollner (op. cit., p. 125 and p. 105). 
 
 Log ratio Sun : Capella = 10.7463 
 Log ratio Sun : Moon = 5.7910 
 
 Log ratio Moon : Capella = 4.9553 
 Log ratio divided by 0.4 - 12.39 
 
 Stellar magnitude of Capella = + 0.21 
 
 Stellar magnitude of Moon = 12.18 
 2 Astronomische Nachrichten, Nr. 4696, s. 276.
 
 40 LUNAR AND TERRESTRIAL ALBEDOES 
 
 Dr. W. H. Wollaston, who found 1 that the sun = 
 
 5,563 X ( 12) 2 = 801,072 moons, 
 
 used better observing conditions, but he made only two 
 readings on the moon. In one observation at full, his 
 candle was placed at 12 feet. In the other, made at a time 
 when, if the atmosphere had been equally transparent the light 
 should have been 0.84 of that at full moon, the same candle- 
 reading "12 feet," was recorded. One can not help surmising that 
 neither reading was better than a rough approximation. 
 
 Bond's solar-lunar ratio, 471,000 : I, is an underestimate for 
 the same reason that Bouguer's value is too small. Zollner's 
 criticism of Bond's fireworks as not accurate enough for standards 
 is also fully justified. 
 
 Zollner's own measurements of the ratio of sunlight to moon- 
 light appear to have been made with great care ; but in reducing 
 them he becomes lost in the mazes of an unnecessarily complex 
 argument. With the removal of this blemish, no fault can be 
 found with the new value deduced from the original measures. 
 The same can not be said of Zollner's isolated measurements of 
 the earth-shine- which require unknown corrections for skylight. 
 The earth-shine observations of Arago and Laugier have been 
 utilized by me in conjunction with my own with which they are 
 in good agreement. 
 
 Various other more or less aberrant values of the moon's 
 albedo usually err from inadequate correction for changes in 
 atmospheric transparency. As an instance of a great name attached 
 to an extraordinarily small value which is simply impossible, may 
 be cited that of William Thomson (Lord Kelvin) : "70,000 : i" 
 for the ratio of sunlight to full-moon light. 2 
 
 Whatever faults may still remain in the values which are 
 given here, they at least have this merit, that they are consistent 
 among themselves, which is very far from being the case with 
 the results which have been published hitherto. 
 WESTWOOD ASTROPHYSICAL OBSERVATORY, 
 August, /o/(5. 
 
 1 Philosophical Transactions of the Royal Society of London, Vol. 
 CXIX, p. 19-27, 1829. 
 
 2 Poggendorff's Jubelband, p. 624, 1874. 
 
 3 Nature, Vol. XXVII, p. 279, January 18, 1883. 
 
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