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Vol. 4. Part I.
SEPTEMBER, 1922
Number 19
.
BULLETIN
OF THE
:
NATIONAL RESEARCH
COUNCIL
CELESTIAL MECHANICS
REPORT OF THE COMMITTEE ON CELESTIAL MECHANICS OF THE
NATIONAL RESEARCH COUNCIL
By
E. W. BROWN, Professor of Mathematics, Yale University, Chairman; G. D. BIRKHOFF
Professor of Mathematics, Harvard University; A. O. LEUSCHNER,
Professor of Astronomy, University of California;
H. N. RUSSELL, Professor of Astronomy,
Princeton University
.
PUBLISHED BY THE NATIONAL RESEARCH COUNCIL
OF
THE NATIONAL ACADEMY OF SCIENCES
WASHINGTON, D. C.
1922
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math Sin
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BULLETIN
OF THE
NATIONAL RESEARCH COUNCIL
Vol. 4. Part 1
SEPTEMBER, 1922
Number 19
CELESTIAL MECHANICS
REPORT OF THE COMMITTEE ON CELESTIAL MECHANICS OF THE
NATIONAL RESEARCH COUNCIL*
BY
E. W. BROWN, Professor of Mathematics, Yale University, Chairman;
G. D. BIRKHOFF, Professor of Mathematics, Harvard University;
A. 0. LEUSCHNER, Professor of Astronomy, University of
California; H. N. RUSSELL, Professor of Astronomy,
Princeton University
CONTENTS
Part I.
Part II.
Part III.
2
9
The Solar System.
Celestial Mechanics as Applied to the Stars...
The Theory of the Problem of Three or More
Bodies....
17
REPORT OF THE COMMITTEE ON CELESTIAL MECHANICS
Celestial mechanics, broadly interpreted, is involved in practi-
cally all the astronomy of the present time. The limited meaning
of the term now usually adopted refers only to those problems
in which the law of gravitation plays the chief or only part, and more
particularly to those which deal with the motions of bodies about
one another and with their rotations. This limitation will, in any
case, be adopted in this report since surveys dealing with other
aspects of astronomy have been written or are contemplated. It
is, however, necessary not to be too rigid about the border lines,
especially in considering questions where the gravitational action
does not fully account for the observed phenomena.
The report has three general divisions: I, the solar system; II,
the stellar system; III, the theoretical aspects of the general prob-
lem of three or more bodies. It is not intended to contain a com-
* The membership of the committee is: E. W. Brown, chairman, G. D. Birkhoff,
A. 0. Leuschner, H. N. Russell.
1
2
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
QB
plete account of the present status of the subject. More empha-
sis has been laid on those portions which are under discussion at
the present time and on problems, at present unsolved, which need
discussion and solution.
܀
PART I. THE SOLAR SYSTEM
The order of treatment is as follows: The moon, the eight major
planets, their satellites other than the moon, the asteroids or minor
planets, comets. The general view of the subject now as in the
past, has been to consider the consequences of the law of gravita-
tion, the extent to which it accounts for the observed motions
leading to the discovery of other possible influences—and predic-
tion for future observation and comparison with theory.
The Moon. The gravitational motion has been worked out
sufficiently to satisfy all observational needs of the past and prob-
ably of some centuries in the future, and the results are fully em-
bodied in tables constructed to furnish the moon's position without
excessive labor. The observational data are the daily Green-
wich observations (weather permitting) since 1750, isolated series
of observations, eclipses, occultations sipce the beginning of the
sixteenth century, and occasional ancient records of eclipses and
occultations during the past forty centuries. These have led to
the establishment of the following differences from a purely gravi-
tational theory :
(a) An apparent secular acceleration of the moon's mean mo-
tion of about 4"5* per century, per century, combined with an ac-
celeration of the earth's mean motion about the sun (“'acceleration
of the sun”) of a little over 1" with probable errors, according to
Fotheringham, to whom the latest figures are due, of about #0".5.
The former has frequently been attributed to a slowing down of the
earth's rate of rotation due to tidal friction: the new work of
G.I. Taylora and H. Jeffreys has rendered this explanation very prob-
able both qualitatively and quantitatively, especially as it also
accounts for most of the sun's acceleration.
(b) A long-period term of some 275 years period and 13" ampli-
tude in the mean longitude, obtained from observations extending
* This is the coefficient of t2 in the expression for the longitude generally mis-
named the acceleration. The true acceleration is therefore twice this amount.
1 M. N. R. A. S., 80, p. 581.
2 Phil. Trans. R. S. , 220, p. 1.
3 Ibid., 221, p. 239; M. N. R. A. S., 80, 309.
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
3
+
over about the same time. Numerous hypotheses have been ad-
vanced to account for this deviation, but none of them rest on any
secure physical basis nor have they received independent testimony.1
(c) Fluctuations which are evident in the observations of the
past 170 years and well defined during the last 70 years. In the
former time their principal period seems to have decreased from
some 70 years to about 40 years, with an amplitude of some 3" or
4". In 1914 E. W. Brownpointed out that similar fluctuations of
much smaller amplitude could be traced in the motions of the earth
and of Mercury: these fluctuations were confirmed by Glauert3
who found them also in the longitude of Venus. The latter also
showed that they could all be moderately well accounted for by
changes in the rate of rotation of the earth. No cause is assigned
for these changes and their magnitude, amounting sometimes to
as much as a loss or gain of 08.07 in one year in the rotation of the
earth considered as a clock, makes the acceptance of the hypoth-
esis difficult. It hass been uggested that this hypothesis might be
tested by observations of the eclipses of Jupiter's satellites, which
at present seem to furnish the only possible means for the purpose.
The Major Planets. The theories and tables by Newcomb and
Hill seem to satisfy present needs, except perhaps those of Jupiter
and Saturn, into which some small errors have crept but are now
in process of examination by Innes. The sufficiency of the adopted
theories is well shown by the theoretical and observed secular
changes of the perihelia, nodes, eccentricities and inclination. The
only large outstanding difference, that of the perihelion of Mercury,
is fully accounted for by Einstein's addition to the Newtonian law,
although one or two others need to be kept in mind as being per-
haps in excess of their actual errors. It may be mentioned that the
Einstein addition causes an increase of about 2" in the centennial
motions of the lunar perigee and node* but this is just at the limit
of accuracy of Brown's theory and probably much beyond detec-
tion by observation for many decades to come.
Attempts made to discover a supposed trans-Neptunian planet
by its perturbations on Neptune or Uranus have been unsuccessful.
The errors of the latter though considerable when the tables of Lever-
1 See E. W. Brown, Amer. Jour. Sc., Ser. 4, 29, p. 529.
2 Brit. Assoc. Report, 1914, p. 320.
3 M. N. R. A. S., 75, p. 489.
4 de Sitter, M. N. R. A. S., 77, p. 172.
5 P. Lowell, Lowell Obs. Trans., 1.
4
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
rier or Gaillot are used, have become very small with Newcomb's
tables, and the observations of Neptune are not sufficient for the
purpose.
From time to time, numerical relations of the masses, distances,
etc., like those contained in Bode's law, appear but have not so far
given theoretical results. A curious fact concerning the distribu-
tion of the poles of the planetary orbits, noted by H. C. Plummer, 1
deserves mention.
Satellites. Neither the Uranian nor Neptunian systems present
many points of interest to the theoretical astronomer, on account
of their distances from the sun and Earth. The four inner satellites
of Jupiter, partly on account of the librational relation between
three of them and partly because of the possibility of testing the
constancy of the rate of rotation of the Earth by observations of
their eclipses, have been again considered by R. A. Sampson,
who has worked out their theory and has published tables. The
outer satellites present several features of mathematical and
physical interest. In Hyperion and Titan, satellites of Saturn,
there is another case of libration worked out to a limited extent by
Newcomb. Since the issue of Vol. IV of Tisserand's Mécanique
Celeste in 1896, which contains a full account of the work to that
date, the orbit of Phoebe (Saturnian system), and of the sixth
satellite of Jupiter have been worked out by Ross.2
Asteroids. Nearly one thousand of these bodies are now known.
From the gravitational point of view, they possess the greatest
mathematical interest, on account of the large perturbations pro-
duced in their orbits by Jupiter.
Long period inequalities constitute the chief difficulty in all the
gravitational problems of the solar system, on account of the further
approximations needed to obtain the required degree of accuracy
of the numerical values of the comparatively large coefficients.
The older methods went ahead without reference to them and car-
ried out the approximations as they were needed, as for instance in
Hansen's method used by G. W. Hill for Jupiter and Saturn in which
a very long period term with a large coefficient causes most of the
trouble. In the newer methods initiated by Gyldén and his follow-
ers, among them Backlund and Brendel, an attempt is made to
introduce such terms as early as possible so as to diminish the
ultimate work of computation.
1 M. N. R A. S., 76, p. 378.
2 Harvard Annals, 53, VI.
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
5
There are two types of oscillation which differ in their mathe-
matical treatment and in their physical results. The ordinary
long-period type is that in which a forced oscillation has a period
near that of a free oscillation. But when the two periods become
almost exactly the same, the free oscillation is compelled to take
the period of the forced oscillation, and there is then a new oscilla-
tion of finite period about this position: the latter is called a libra-
tion. It is well known that though the periods of the asteroids have
a considerable range, there are none certainly known whose periods
are exactly 1/2, 1/3, ²/5 that of Jupiter, while there are considerable
numbers with periods a little different from these fractions. It
is obvious that the resonance has some relation to the distribution,
but so far all mathematical investigation has failed to show any
reason for the gaps: there is no evidence of instability in the deduc-
tions from the equations of motion. Attempts to search for the
cause in cosmogonic speculations or in a resisting medium have been
made, but a more complete investigation of the gravitational effects
is needed. The problem is similar to that of the divisions in Saturn's
ring, connected with perturbations produced by the satellites. The
question is complicated by the fact that several cases of libration
exist without apparent instability, e. g., amongst the satellites of
Jupiter, of Saturn, in the Trojan group of asteroids whose mean
period is the same as that of Jupiter and best known of all, in the
rotation of the moon which has the same period as that of its revolu-
tion around the earth.
Numerous statistical investigations have been carried out, but
little has been deduced from them, except in the way of confirming
known perturbative effects.
Closely related in importance to the foregoing purely theoretical
considerations is the practical problem of suitable numerical meth-
ods for the representation of the motion of the asteroids. Hansen's
and the Gylden-Brendel methods have been referred to. Of funda-
mental importance for practical purposes are also the methods in-
augurated by Bohlin for the group determination of the perturba-
tions of planets which have a mean motion nearly commensurable
with that of Jupiter. Bohlin's method rests on Hansen and has
been followed by Von Zeipel, Leuschner, and D. F. Wilson, by the
former in application to the group 1/2, by the latter to the group
2/5. Bohlin's developments are general for all groups, with special
application to the group 1/3. A feature of these methods is the use
of elements which are similar to the elements ordinarily known as
6
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
mean elements. In all of these methods the mean motion, eccen-
tricity, and inclination may lead to complications. The success of
any method depends upon the possibility of meeting the complica-
tions arising from critical values of these three elements separately
and jointly. This has not been attempted in practice to a degree
of precision which would reveal any departures from the motion
under the Newtonian law of the type of the motion of the perihelion
of Mercury, although for planets with moderate inclination and
eccentricity and a mean motion not nearly commensurable with
that of Jupiter, Hansen's method appears entirely suitable.
The principal aim of astronomers at the present time is to repre-
sent the motions with sufficient approximation to serve purposes of
identification and observation. Even with this limitation of
accuracy the difficulties are considerable. Leuschner published
preliminary results of his experience with the Watson asteroids. I
Unpublished later results of the planets 10 Hygiea and 175 Andro-
mache verify his conclusions that the revised tables of von Zeipel
for the group 72 will give the most satisfactory results for all known
planets of this group. It would seem therefore extremely advisable
to have tables computed on Bohlin's or similar plans for other than
the three groups for which they are available. The Trojan group,
however, does not appear to lend itself to treatment by Bohlin's
method without certain modifications. The perturbations of this
group are being successfully dealt with by E. W. Brown, and Wil-
kins” has represented the observations of 884 Priamus by his own
treatment to within 10" of arc from one opposition to the next by
including second order perturbations of the first degree in the
eccentricity, inclination, and deviation from the center of libration.
This method will more than answer practical requirements from
one opposition to the next, while Brown's developments are in-
tended to represent positions at any time. .
Brendel classifies the planets after Gyldén as ordinary, character-
istic and critical planets according to the ratio of their mean mo-
tions to that of Jupiter, the critical planets being those of close com-
mensurability. Application of his method has been made by Bren-
del to one hundred planets with mean motions from 800 to 852.
The elements are instantaneous but not osculating and perturba-
tions greater than 3'.4 within fifty years are included so as to re-
produce the geocentric places within 20' for one hundred years.
1 Proceedings of the National Academy of Sciences, 5, pp. 67-76, March, 1919.
2 A. N. 208, 233.
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
7
The secular perturbations in the longitude of the perihelion and of
the node are the same for all these planets. The periodic perturba-
tions of the various elements adopted are computed with the aid of
five to nine constants for each planet and four arguments, linear
with the time. Later Brendel' has published improved elements
for sixty of these planets and added three to the list.
Among other applications of Brendel's method are those by
Labitzkel who has published mean elements and approximate
perturbations by Jupiter for nineteen of the planets with mean
motions between 780" and 857" with a somewhat lower degree of
approximation than that aimed at by Brendel for the previous list.
Bodahas published approximate perturbations for one hundred and
eight planets of the group 1/3, Hestia group, with mean motions
ranging from 845" to 958", in two groups according to whether the
mean motion is greater or less than 897". The perturbations are
not very large in these cases because the longitudes of perihelia are
near those of Jupiter. These approximations by Brendel's method
serve a very useful purpose.
Among other investigations which may serve for comparison of
the suitability of various methods are those by Notebauma and
Osten.3 The former has developed the perturbations of 433
Eros after Leverrier. Commensurability with the mean motions
of the three perturbing planets is not involved. The latter has
developed the perturbations of the third order due to two major
planets for 447 Valentine by Hansen's method, reproducing the
normal places within = 1" to 2" over 19 years. A preliminary
review of the present status of the determination of the per-
turbations of minor planets is in progress under the direction of
Leuschner.
Of interest in connection with the question of stability for com-
mensurable planets are certain numerical results. Brown's studies
and von Zeipel's developments do not indicate instability. Miss
Levy's unpublished developments by the von Zeipel method place
the mean mean motion of Andromache at 619.5" and according to
Berberich's computations by the method of special perturbations the
osculating value of the mean motion has diminished with slight peri-
odic variations from 617.7" in 1877 to 607.8" in 1921, so that the
1 Brendel, A. N. 195, 417; A. N., 200, 1; Labitzke, A. N., 212, 217; Boda, A. N.
212, 219.
2 A. N., 214, 153.
3 Astr. Abh., 15; A. N., 210, 130.
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8
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
hitherto accepted gap has been well invaded in this case. On the
other hand Krassowski comes to the conclusion on the basis of Bren-
del's theory that for planets of mean motion near 400" or 4/3
times Jupiter's mean motion, the motion of an asteroid would
become unstable between the limits 391.5" and 401.7". (Travaux
de la Société des Sciences de Varsovie III. Classe des sciences
math. et. nat. Nr. 12.)
Comets. The principal question of interest regarding comets is
that of their origin, whether they are members of the solar system
with elliptic orbits or enter it from without in parabolic or hyper-
bolic orbits. The possibility of the “capture” of a comet by diver-
sion from a parabolic into an elliptic orbit has been recognized since
the days of Laplace and discussed in much detail by H. A. New-
ton. 1 Leuschner? has shown that the numerous “parabolic”
orbits which appear in catalogues of comets, represent cases in which
the observations are insufficient to detect the deviations from a
parabola which almost always appear when the observations are
sufficiently numerous and cover a long enough interval. In the
latter cases the orbit is usually definitely elliptic. The osculating
orbit near perihelion is occasionally hyperbolic, but Fayets and
Strömgren“ have shown that in every case of this sort the approach
has been in an elliptic orbit which was later converted to a hyper-
bolic by planetary perturbations.
All comets so far recorded appear, therefore, to be members of
the solar system. The origin of the cometary orbits of shorter
period, especially of the numerous group having a period between
four and eight years, has been attributed to capture of comets of
longer period by encounters with the major planets. All investi-
gators agree in confirming this explanation in the case of the “Jupi-
ter family described above. Russel15 has shown that there is
very little evidence of such capture among the comets of periods
between 10 and 2,000 years, and that the supposed “families” of
Saturn, Uranus, and Neptune have little or no foundation. Their
large number is probably due to the disruption of an originally
smaller number of comets by close approach to Jupiter as suggested
1 Memoirs Nat. Acad. Sciences, 6, 7–23, 1893.
2 Pub. A. S. P., 19, 67–71, 1907.
3 Annales de l'observatoire de Paris, Mémories, 26, 1910.
4 Pub. fra Kobenhavn's Observatorium, 19, 61, 1914.
5 A. J., 33, 49–61, 1920. (This paper contains a number of references to earlier
works.)
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
9
by Callandreau' and several groups of comets of probably common
origin have been indicated by Fayet.2
These comets may have had substantially their present periods
for an indefinite period or have attained them by the slow summa-
tion of small perturbations of the ordinary sort.
The motion of the matter in the tails of comets, which observa-
tions show to be repelled by the Sun, and probably also by the
comet's head,3 also presents problems of considerable interest,
though the unavoidable lack of precision in the observations makes
exact computations difficult.
PART II. CELESTIAL MECHANICS AS APPLIED TO THE STARS
The incidence of the application of mathematical analysis to
sidereal problems is quite different from that which is found within
the solar system. The determination of orbits and disturbed mo-
tion here take a subordinate place, and are largely supplanted by
questions relating to the statistical equilibrium of systems of great
numbers of stars, to the internal constitution, rotation, and vibra-
tions of gaseous masses, and to theories of the evolution of the
Universe and its separate portions. Throughout this field the
dynamical discussion must keep in close touch with statistical
methods and, above all, with physics, especially atomic physics.
1. The problem of the determination of orbits, whether of visual
double stars or of spectroscopic or eclipsing binaries, differs radically
from that presented by the orbit of a planet, comet or satellite,
because of the great difference in the percentage accuracy of the
observations. Errors amounting to ten per cent of the observed
quantity are habitually met with, and it is, therefore, hopeless to
attempt to derive reliable elements from the theoretical minimum
number of data.
Only when numerous observations, covering at least a consider-
able fraction of the period, are available, is it worth while to com-
pute; and graphical methods, based upon curves drawn to repre-
sent the whole course of the observations, are universally employed
—though the later treatment is often in part analytical. The num-
ber of existing methods of solution is considerable, and their practi-
cal application is an art, fully as much as a science. Preference
should be given to those procedures which enable the computer to
1 Annales de l'Obs. de Paris, 22, p. 1-47, 1902.
2 Bull. A st., 28, 170, 1911.
Eddington M. M., 70, 442-458, 1910.
3
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REPORT OF COMMITTEE ON CELESTIAL MECHANICS
keep closest to the original observations, rather than to the empiri-
cal curves which have been drawn to represent them.
An excellent detailed discussion of all three sorts of binary stars
will be found in Aitken's recent volume. 1
2. In view of these facts, it is not surprising that relatively little
work has been done upon perturbations in multiple stellar systems,
where orbital motion is shown. Such systems, so far as is known,
always exhibit a close pair, attended at a relatively considerable
distance by a companion (sometimes itself double) revolving in an
orbit of much longer period. In a few cases the system thus formed
has itself a remote attendant, presumably in very slow orbital mo-
tion.
The masses of all the components, so far as they are ascertain-
able, are of the same order of magnitude. The problems thus pre-
sented are analogous to those of the Lunar, rather than the Plane-
tary Theory, but are complicated by the great eccentricities of the
orbits, which sometimes exceed 0.7.
There is, however, not as yet a single system in which the orbital
elements of both the close and the wide pair are fully known. When
the close pair is telescopically separable, the period of the distant
companion is so long that it will be several centuries before a re-
liable orbit can be computed. Seeliger has discussed the per-
turbations in such systems. The cases in which one or both com-
ponents of a visual binary are themselves spectroscopic binaries of
short period are more promising, but certain elements (notably the
inclination of the orbit plane) cannot be found from the spectro-
scopic observations. Two such systems 4,5 have been observed long
enough to show definite evidences of perturbations of the close sys-
tem (advance of the line of apsides and changes in period) and it
is very desirable that they should be studied analytically.
3. The distribution and motions of the stars in space, and in
particular the distribution of velocities which is known as "Star
Streaming' afford an attractive field for dynamical study. Here
we are concerned with the statistical distribution of the coordinates
2
1 R. G. Aitken, The Binary Stars (New York, 1918) Chapters 4, 6 and 7.
a Geminorum and € Hydrae.
3 H. Seeliger, Abhandlungen de Münchener Akad., II KI, 17, 1011 (1888) (>Cancri)
Astronomische Nachrichten, 173, 327 (1907) (€Hydrae).
k. Pegasi; F. Henroteau, Lick Observatory Bulletin, 9, 120 (1918). Period of close
pair 5.97 days; of wide pair 11.35 years.
5 13 Ceti; J. S. Paraskevopoulos, Astrophysical Journal, 52, 110 (1920). Short period
2.08 days; long period 6.88 years.
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
11
1
and velocities of an enormous number of bodies under their mutual:
gravitation, and the analysis resembles in many ways that employed
in the kinetic theory of gases.
The stars are, however, so small in comparison with the distances
between them that the influence of collisions, or even of encounters
such that their mutual attraction changes their directions of motion
by a degree, may be neglected, unless enormously long intervals of
time are involved. The "time of relaxation" which would be re-
quired to produce extensive alterations in the velocities of the stars
by such encounters is estimated by Jeans? as 1014 years and by
Charlier as 1016 years.
The motions of the stars under the general attraction of the
whole mass of stars have been discussed by Eddington“ and Jeans: 5
They find that a steady state in which star-streaming occurs is
possible with a spherical “universe" and the direction of streaming
radial, or with a "universe” shaped like a figure of revolution, and
with streaming taking place along circles coaxial with the axis of
symmetry. The first of these models is unlike the actual universe
of stars, while the applicability of the second is uncertain.
It is,
however, very doubtful whether our universe is in a steady state-
especially in view of the enormous time which would be required to
reach one.
Jeans has shown that a similar type of streaming might
be produced as the result of successive encounters of our "universe"
with other star clusters, but in a later discussion? he reverts tenta-
tively to the previous hypothesis.
In the globular clusters, the motions are unknown, but the
distribution of the stars in space is remarkably similar from cluster
to cluster, and follows closely the law p = Poll + r2/a2) / (where
p is the density of distribution, r the distance from the centre, and
po and a are constants). Jeans: and Eddington' have discussed
this question. They find suggestions that this distribution may
represent the nearest approach to a state of equipartition of energy
1 Summarized by Jeans, "Problems of Cosmogony" (Cambridge University Press,
1919) Chapter X.
J. H. Jeans, M. N. R. A. S., 74, 112 (1913).
3 C. V. L. Charlier, Meddelanden från Lunds Astron. Observatorium, Series II, No.
16, 84 (1917).
4 A. S. Eddington, M. N. R. A. S., 74, 5 (1913); 75, 366 (1915), and 76, 37 (1915).
5 J. H. Jeans, M. N. R. A. S., 76, 70 (1915).
6 J. H. Jeans, M. N. R. A. S., 76, 552 (1916).
7 "Problems of Cosmogony," pp. 236-242.
8 M. N. R. A. S., 76, 567 (1916); “Problems of Cosmogony,!' p. 245.
9 M. N. R. A. S., 76, 572 (1916).
2
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REPORT OF COMMITTEE ON CELESTIAL MECHANICS
which is possible without a scattering of the outer stars to infinity;
but the problem is by no means yet solved.
4. The modern theory of the internal constitution of the stars
begins with the work of Schwarzschild' who called attention to the
importance of “radiative equilibrium." The transfer of heat out-
ward from the interior takes place almost entirely by the emission
of radiation and its absorption in overlying layers, and the tem-
perature gradient is determined by the outgoing flux of heat and
the opacity of the material to the radiation. Schwarzschild dealt
only with the atmosphere of the Sun, but Eddington extended the
analysis to the interior of the stars, both those of low density, with-
in which the simple gas laws are obeyed at all points, and later:
to those of higher density. He was the first to point out the impor-
tance of radiation pressure, which at the very high temperatures
that prevail inside the stars becomes great enough to counteract
a large part of the gravitational force, and of ionization, which
breaks up most of the atoms into nuclei (or small nuclear groups)
and free electrons, and greatly reduces the molecular weight. Upon
certain plausible assumptions, he concluded that the ratio of the
radiation pressure to the total pressure is constant throughout the
star. Hence, the gas pressure is proportional to the fourth power
of the temperature--a relation which suffices to define the whole
internal constitution of the star, when combined with the law con-
necting temperature, pressure, and density, for which in general,
Eddington adopts a simplified form of Van der Waals' Law.
If ß is the ratio of the gas pressure to the total pressure it follows
that (1 - B) / 34 = CM²m+ where M is the mass of the star, m the
mean molecular weight of the material composing it, and C is a
constant, which is the same for all stars of low density, and depends
only on fundamental physical constants of gravitation, radiation
and gas-theory.
For bodies of mass less than about 3 X 1032 grams, (1 - B)/B4
is small, and the radiation pressure is almost negligible. For
masses greater than 3 X 1034 grams (1 – B)/84 is large, and the
radiation pressure almost neutralizes gravitation, and is the domi-
nant influence in the internal equilibrium. The interval within which
the change takes place is exactly that in which all known stellar
masses lie. The masses of the stars appear, therefore, to be deter-
1 Göttingen Nachrichten, 1906, 141.
2 M. N. R. A. S., 77, 16 (1916).
3 M. N. R. A. S., 77, 596 (1917).
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
13
ๆ
mined by the fundamental properties of atoms. Bodies of smaller
mass do not radiate enough to be visible as stars, while those of
greater mass are in such a delicate state of equilibrium that they tend
to break up into smaller masses.
When the mean density increases, and departures from the
simple laws must be considered, the constant C decreases rapidly.
Eddington has determined it by quadratures.
The total radiation from a star's surface is proportional to
M1 B)/K where K is the coefficient of opacity of the material.
As the star contracts, the radiation will be relatively great, and
nearly constant, till the density becomes considerable, and will
then fall steadily.
Certain of Eddington's assumptions have been severely criticized
by Jeans,' who proposes a modified theory. The principal point
at issue is the constancy of the ratio of radiation pressure to gas
pressure throughout the star. This will be constant, if, and only if,
the product Kn is constant, where K is the opacity of the material,
and the mean rate of generation of heat, per unit time, per unit
mass, in the portion of the star nearer the center than the region con-
sidered. It is very probable that n increases toward the center and
that K decreases. Eddington, in his original discussion, took them
as separately constant; but this is unnecessary. He does not pre-
sent his results as rigorous solutions of the physical problem, but
as solutions of a simplified problem analogous to the more com-
plicated reality. The manner in which his "model" agrees with
the known properties of the stars is so striking as fully to justify
his contention that the assumptions on which it is based are probably
not far from the truth.
Jeans has further shown that, if the energy of the stars is derived
from gravitational contraction, those of moderate mass would
develop more rapidly than those of either small or great mass.
5. The oscillations of a gaseous star about its normal equilibrium
have been discussed by Moulton and Eddington* with a view to
the explanation of the variation of stars of the short period or Ce-
pheid type. Moulton's conclusion that a very small oscillation from
a prolate to an oblate spheroidal form would account for great
1 M. N. R. A. S., 78, 28. Reply by Eddington, Ibid., 78, 113 (1917).
2 M. N. R. A. S., 78, 36 (1917) and 79, 319 (1919) "Problems of Cosmogony,"
Chapter VIII.
3 Astrophysical Journal, 29, 261 (1909).
4 M. N. R. A. S., 79, 2, 1918 and 79, 177 (1919).
--
14
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
variations in brightness, does not appear to correspond to the
phenomena.
The type of pulsation considered by Eddington (which appears
far more promising as an explanation of the facts) is a bodily expan-
sion and contraction during which the radius changes by ten or
twenty per cent, the temperature rising as the star contracts and
falling again as it expands. The changes in temperature account
for the variability in light and color, while the outward and inward
motions of the surface explain the observed changes in radial ve-
locity; certain important details of the observed variation, especially
the wide departure of the oscillations from simple harmonic motion,
remain incompletely explained. The conclusion of the most general
interest is perhaps that if the energy supply of the star o
Cephei were derived from gravitational contraction alone, its period
should be shortening several hundred times as fast as the actually
observed rate of change.
6. The problem of the configurations of equilibrium of a rotating
incompressible mass of fluid is an old one. Successive stages in its
development are marked by the spheroid of Maclaurin, the ellipsoid
of Jacobi, and the pear-shaped figure of Poincaré. The stability
of this latter figure had been investigated by G. H. Darwin and
Liapounoff, with contradictory results, and the problem was only
resolved in 1916 by Jeans, who showed that an expansion to the
third order of small quantities was required to arrive at decisive
results, and that the pear-shaped figure was unstable.
It follows that a slowly contracting and rotating mass of incom-
pressible fluid, after following the series of spheroids and ellipsoids,
would become physically unstable, and go through a period of
rapid change, and probably break up into two separate masses.
The more difficult question of the behavior of a rotating compressi-
ble mass is of greater astrophysical importance. This too has been
attacked by Jeans. He finds that there are two possible mechan-
isms of breaking up. The mass may divide into two (or perhaps
more) isolated parts by fission, or the centrifugal force at some point
or line upon its surface may become equal to the gravitational
attraction, and a stream or sheet of matter may be thrown off into
1 "On the Potential of Ellipsoidal Bodies,” Philosophical Transaction A, 215, 27
(1914). “On the Instability of the Pear-Shaped Figure, “Phil. Trans. A., 217,1 (1916).
"Problems of Cosmogony," Chapters IV, V.
? "The Configurations of Rotating Compressible Masses,” PhilosophicalTransac-
tions, A. 218, 157 (1917). “Problems of Cosmogony," Chapter VII.
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
15
1
space. When the mass is of uniform density the first of these proc-
esses happens: when it is greatly concentrated at the center, the
second.
Intermediate situations may be represented by assuming that the
compressible material obeys the equation of state p = Kpr (where
y = co gives homogeneity while y = 6/5 is found to lead to an
infinitely great central condensation). It is found that fission
takes place if y exceeds 2.2, while if y is less than this limit, the
spheroidal surface is distorted, develops a sharp edge, and matter is
thrown off in a sheet in the equatorial plane. This limiting value
of y corresponds to a surprisingly low degree of central condensa-
tion in a spherical mass—the central density being only about three
times the mean density.
7. The origin and evolution of binary stars has been consider-
ably discussed. Direct observational evidence, showing pairs of
stars revolving almost in contact, makes it almost certain that such
systems have been formed by the fission of a single rotating mass,
and this explanation may be accepted for the spectroscopic binaries
of short period. The densities of these stars are, however, so low
that it is doubtful whether the value of y of the gas of which they
are formed can be as great as 2, but Jeans has shown that the influ-
ences of ionization and radiation pressure may remove this difficulty.
The wider visual binaries, with periods from five years to many
centuries, present a more difficult problem. Moulton” and others
have proved that their periods can never have been very much
shorter than at present, and that, if they were formed by fission,
the density at the time of separation must have been exceedingly
small. Moreover, their orbits are often very eccentric, and Nölkes
has shown that tidal friction is incompetent to produce such high
eccentricities from the nearly circular initial orbits. Triple and
multiple systems (as Russell4 has shown) exhibit a grouping into
close pairs, with widely distant companions (cf. 2 above) which is
a necessary consequence of the theory that they have been formed
by repeated fission, but may also be a consequence of other modes
of origin.
Jeans' has pointed out that the effect of encounters between bi-
1 Phil. Trans. A., 218, 208 (1917).
2 F. R. Moulton, Astrophysical Journal, 29, 12–13, 1909.
3 Fr. Nölke, Abh. Nat. Ver. Bremen, 20, Teil. 2, 1911.
4 H. N. Russell, Astrophysical Journal, 31, 185 (1910).
5 M. N. R. A. S., 79, 100 (1918), and 79, 408 (1919); "Problems of Cosmogony,"
Chapter XI.
16
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
2
nary systems and other stars which happen to pass near them tend
in the long run, to an equipartition of energy between the radial
and transverse components of motion, and to an average orbital
eccentricity of 0.64—a little greater than the observed mean value.
He supposes that most of this action has taken place in the remote
past, when the stars were closer together than now (see below) and
that the majority of visual binaries probably started as neigh-
boring nuclei when the stars were originally formed (in agreement
with a previous conclusion of Moulton“).
8. The theory of rotating masses has been employed by Jeans
in a bold and brilliant hypothesis of the origin of spiral nebulae,
and even of our universe of stars. A huge rotating mass of rare-
fied gas (or cloud of dust), would, upon contraction, assume a spher-
oidal form, then as this grew more flattened, become lens-shaped
till, in time, matter was thrown off by centrifugal force in the equa-
torial plane. The tidal forces due to the attraction of the rest of
the universe would localize the regions of ejection near two opposite
points on the equator, and the gas, streaming out from these, would
form two spiral arms enclosing the nucleus.
This is a remarkably good model of a spiral nebula. Moreover
Jeans shows that, if the rate of ejection is rapid enough, the out-
going stream of gas will become unstable, and tend to break up into
condensations or nuclei under its own gravitational attraction.
Condensations of this sort are conspicuous on the photographs of
many spiral nebulae. If the nebula is big enough, they may be so
massive that they ultimately condense into stars.
The masses calculated for spiral nebulae on this hypothesis are
very great-5000 to 500,000,000 times the mass of the Sun-but
the mean density is excessively low (4 X 10-17 grams per cubic
centimetre). These values are roughly of the order of magnitude
indicated by other observational data.
More tentatively, Jeans postulates that a similar huge nebula,
in the course of hundreds of millions of years, may have completely
dispersed its substance into star-forming condensations, which,
circulating in the general plane of the nebula, and spreading out to
some sixty times its original diameter may have given rise to the
existing galactic system of stars.
1 See note 2, p. 15.
2 M. N. R. A. S., 77, 186 (1917); “Problems of Cosmogony," Chapter IX.
..
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
17
PART III. THE THEORY OF THE PROBLEM OF THREE OR
MORE BODIES1
When three or more bodies (taken as particles) move according
to the Newtonian law of gravitation, the mathematical determina-
tion of their motion presents great difficulty. The so-called re-
stricted problem of three bodies in the plane is that special case in
which two of the bodies move in circles about their center of gravity,
while the third body is of negligible mass and moves in their plane
attracted by them. We turn first to this case, which has particular
importance for the reason that many of the fundamental character-
istics of the more general problem appear in simple form.
THE RESTRICTED PROBLEM OF THREE BODIES IN THE PLANE
(a) Periodic orbits.
Hillwas the first to realize the importance of certain periodic
orbits in the restricted problem of three bodies for the lunar theory.
Later G. H. Darwin3 undertook extensive calculations based on
mechanical quadrature and found other classes of such orbits not
obtainable by Hill's methods. Moulton and his students have
applied the method of analytic continuation of Poincaré to the
treatment of various periodic orbits. Brown, Strömgren and
others have concerned themselves with types of periodic orbits of
particular astronomical significance. In general only a beginning
has been made with the determination of all the types of periodic
orbits. Rigorously proven qualitative results are rare.?
It should be noted that the periodic orbits referred to, form
closed curves in the plane rotating with the finite bodies. Orbits
of ejection in which the small body collides periodically with one
of the finite bodies are included. The singularity of collision can
be completely disposed of by mathematical transformation.8
As in most dynamical problems there are many types of non-
1 For a report on recent literature see E. O. Lovett, Quarterly Journ. Math., 42,252–315
(1911).
2 G. W. Hill, Am. Journ. Math., 1, 5–26, 129–147, 245–260 (1878).
3 G. H. Darwin, Acta Math., 21, 99-242 (1897).
* F. R. Moulton, Proc. Math. Cong., Cambridge, England, 2, 182–187 (1913); Also
see Periodic Orbits, Carnegie Inst., Washington, 1920.
5 E. W. Brown, M. N. R. A. S., 1911.
6 E. Strömgren, A. N., 168, 105-108 (1905); 174, 33-46 (1907).
? G. D. Birkhoff, Rend. Circ. Mat. Palermo., 39, 265–337 (1915). Other references
are given in this paper.
8 T. N. Thiele, A. N., 138, 1-10 (1895); T. Levi-Civita, Acta Math., 30, 305-327
(1906).
18
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
2
periodic orbits. Poincaré was the first to recognize sufficiently the
central importance of the periodic orbits and conjectured that every
orbit may be approximated to for an arbitrary length of time by a
periodic orbit. This conjecture has neither been proved nor dis-
proved, but it has been shown that for very general classes of
dynamical problems, including the restricted problem of three
bodies, every stable orbit has either certain properties of recurrence,
or asymptotically approaches and recedes from orbits with such
properties.
(b) Integrability.
The concept of integrability is one which admits of various in-
terpretations. Thus, if the differential equations of the problem
under consideration are such that the coördinates may be expressed
in terms of "known functions" of the time, the dynamical problem
may be called integrable. Unfortunately a function which is not
regarded as known at one time may be admitted later to the class
of known functions. For instance, Painlevé in his Stockholm
Lectures: admits all functions defined by infinite series which con-
verge uniformly. If this be accepted, it follows immediately that
the restricted problem of three bodies is integrable. For, as was
stated above, it is possible by change of variables to eliminate en-
tirely the singularities of collision, and then find series of the type
required. Unfortunately this type of integrability is of doubtful
importance.
Sundman* in his epoch-making work on the general problem
of three bodies established that this problem also is integrable in the
sense of Painlevé. But he was not able to draw any conclusions
therefrom, and his series were useless for purposes of computation.
With so many more or less justifiable concepts of integrability,
the question arises whether any one is to be preferred before the
others. The answer seems to be that the differential equations of
a dynamical problem should be called integrable in the vicinity of
a particular periodic solution if the formal trigonometric series for
this solution and nearby solutions converge for all values real or
complex of the variables involved. In this sense it has not yet been
demonstrated that the restricted problem of three bodies is not
1 H. Poincaré, Les méthodes nouvelles de la Mécanique Céleste, Paris, 1892–1899.
2 G. D. Birkhoff, Bull. Soc. Math. France, 40, 305-323 (1912); Acta Math., 43,
1-119 (1920).
3 Painlevé, Lecons sur la Théorie Analytique des Équations Différentielles, professées à
Stockholm, Paris, 1897.
4 C. F. Sundman, Acta Math., 36, 105–192 (1912).
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
19
1
integrable although Poincaré has shown that these series do not
converge uniformly for all values of one of the parameters of the
problem.
(c) Reducibility.
In order to obtain a comprehension of the restricted problem of
three bodies, it is necessary to deal with the totality of orbits for a
given value of the energy constant. A partial explanation is the
following: For periodic orbits the motion may be followed in-
definitely. But a non-periodic orbit may have such complexity as
to approach and recede from periodic orbits and more generally to
be so related to other orbits that it is not possible to isolate the
orbit completely.
Now there are essentially three arbitrary constants involved in
the restricted problem of three bodies, namely the two relative
coördinates of the particle and the angular coördinate giving the
direction of motion. If these three constants be interpreted as
rectangular coördinates in space, the totality of orbits may be
represented as the stream lines of an incompressible fluid in steady
motion in this space. To a periodic motion will correspond a
closed stream line. Hence if we imagine the closed stream line to
be cut by a stationary surface S, there will correspond to the suc-
cessive intersections of this surface by a stream line a sequence of
points of the surface. The transformation T of the surface which
takes each point into the next following one on the same stream
line, and in particular takes the point of intersection of the closed
stream line with the surface into itself, has a very intimate connec-
tion with the dynamical problem. In fact, by this means, first
introduced by Poincaré,1 we are enabled to reduce the restricted
problem to the transformation of a surface into itself.
Another way of seeing this reducibility is the following: Consider
the direct variational periodic orbit of Hill in the restricted problem
of three bodies. Orbits (for the same energy constant) which cross
the line of apsides with nearly the same direction and abscissa as
this periodic orbit are determined by the abscissa r of crossing and
the direction given by an angular variable o. When the small
body projected in this manner crosses a second time after a com-
plete circuit we have a new pair of variables r', o'. Now r, o may
be regarded as the rectangular coördinates of a point in the plane.
That transformation of the plane which carries r, o into r', o consti-
1 See note 1, p. 18.
20
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
tutes a transformation T. It is to be observed that the variational
orbit corresponds to an invariant point of the transformation.
It will be found that the important qualitative properties of the
orbit are mirrored in corresponding properties of the transforma-
tion T. Thus if the problem were integrable in the specific sense
referred to above, the transformation near the invariant point
would be exactly of the nature of a rotation in which the angle of
rotation varies with the distance from the invariant point.
As Poincaré showed, not only the orbits near a particular periodic
orbit but the totality of orbits can be treated by means of a trans-
formation T. The transformation may be regarded as a trans-
formation of the surface of a sphere into itself, approximately like
a rotation in which the angle of rotation varies with latitude. The
two fixed points correspond to the fundamental direct and retro-
grade periodic orbits. 1,2
(d) Stability.
The outstanding problem of stability for the fundamental direct
periodic orbit is this: will all slightly disturbed orbits remain in-
definitely in the vicinity of this periodic orbit? In terms of the
transformation T of a surface S, such stability would imply the
existence of an infinite set of invariant closed curves as near as
desired to the invariant point corresponding to this periodic orbit.
In the interpretation by means of fluid motion, such stability would
therefore mean the presence of infinitely many torus-shaped canals
of stream lines enclosing the closed stream line which corresponds
to the periodic orbit.
Levi-Civitas has shown that when the periodic orbit is of such
a type that the mean motion of the small body is commensurable
with the mean motion of the two other bodies, there will not be
stability in this sense. In fact, there will then be orbits approach-
ing and receding from the given periodic orbit. However, there
remains the possibility that the degree of instability is limited
In the case of incommensurable mean motions the formal series
are available, and this implies stability in the usual astronomical
However, there is a distinction between the type of mathe-
matical stability referred to above and astronomical stability.
The astronomical type can be treated by direct computational
1 See note 1, p. 18.
2 G. D. Birkhoff, Rend. Circ. Mat., Palermo, 39, 265–337 (1915). Other references
are given in this paper.
3 T. Levi-Civita, Ann. di Mat. ser. 3, 5, 221–309. 1991.
sense.
1
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
21
2
methods, but the mathematical type presents the utmost diffi-
culty.
From the practical standpoint one may ask the following two
interesting questions. Suppose that the body of small mass is
subject to arbitrary slight disturbance, although initially in the
fundamental direct periodic orbit. What is the least time in which
the particle can deviate by a stated amount from that orbit? What
is the probable length of time that will elapse before it deviates by
this amount? Neither of these questions appear to have received
consideration despite their interest for the lunar theory.
THE PROBLEM OF THREE OR MORE BODIES
1
Most of the facts outlined above have their analogues in the
more general problem.
The great increase in complexity precludes any attempt, at an
enumeration of the types of periodic orbits. Nevertheless such
orbits must be basic in any attempted treatment of the questions
which arise.
The question of ideal collision has been partially disposed of in the
fundamental paper of Sundman referred to above and in later papers
of Levi-Civita. Sundman established that when the three bodies
do not move in one plane a triple collision is impossible and that at
double collision the behavior of the colliding bodies is essentially
the same as in the ordinary two-body problem.
Sundman went further and proved that the sum of the three mu-
tual distances will exceed always a specified positive constant.
Birkhoff has announced that the work of Sundman can be ex-
tended to apply not only to the usual problem of three bodies
attracting each other under the Newtonian law, but to n bodies
under similar laws, and that furthermore Sundman's methods may
be applied to give the conclusion that if the area integral constants
be assigned and if the mutual distances are small enough initially,
the sum of the mutual distances increases indefinitely with lapse of
time.
For periodic motions the sum of these distances remains fipite of
course, and the same is true of motions asymptotic to these periodic
motions. It appears possible, however, that in all other cases the
sum of the three distances increases indefinitely.
Thus in an idealized earth, sun, moon problem it is likely that
1 T. Levi-Civita, Acta Math., 42, 99-144 (1919).
22
REPORT OF COMMITTEE ON CELESTIAL MECHANICS
the earth and moon will recede from the sun while the moon ap-
proaches the earth.
In the case of n nearby bodies we may anticipate that one or
more will recede gradually from the others with lapse of time until
finally the bodies are all remote from each other or moving in nearby
pairs as in the two body problem. The source of the potential
energy required will of course lie either in the high initial velocities
or in the near approach of the paired bodies. From time to time,
then, these bodies may approach so near as to collide. These
conjectures seem in harmony with the facts of stellar distribution.
The interest of the pure mathematician in the problem of three
or more bodies has been stimulated by its importance for an un-
derstanding of the past and future of the stellar universe. The
entrance upon the field of the theory of relativity of Einstein has
altered this situation considerably. If the relativistic point of
view prevails there can be little doubt that new factors of the ut-
most importance will be introduced in astronomical speculation
concerning great lapses of time, although for limited intervals of
time the classical problem of three or more bodies will maintain its
importance. Only the very simplest features of the modifications
required by the theory of relativity have as yet been determined,
mainly those for a very small body in the presence of a central
body.
1 K. Schwarzschild, Sitz. Preuss. Akad. Wiss., 35, 189–196(1916). See also W.
De Sitter, M. N. R. A. S., 76, (1916).
1
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