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W. L. (William Larkin) Webb.

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development of Astronomical Photography, and partly by the
solution of the restricted problem of three bodies. This two-
fold advance led to a great increase in our knowledge of the
nebulae and of their mode of development into Cosmical Systems,
under laws which are demonstrated to be consistent with the
established principles of the mechanics of the heavens.

The present epoch is the one towards which all the previous
epochs have pointed, and for which the great discoveries of the
past have laid the foundations. The epoch of Cosmogony has
been especially advanced by the researches and discoveries of
American astronomers, beginning with the photographic work



168 BRIEF BIOGRAPHY AND POPULAR ACCOUNT OF THE

of Keeler in 1899; so that this science of the centuries seems likely
to be peculiarly an American science. It has been considerably
advanced also by eminent European investigators, among whom
Darwin, Poincare, Arrhenius and Stromgren seem to have taken
the leading part.

In order to bring out the most significant facts, we shall con-
sider in succession a number of topics. It is not necessary to dwell
on Laplace's old nebular hypothesis, because it is now universally
abandoned by astronomers, but we may recall very briefly the line
of argument developed in Babinet's criterion, by which the detach-
ment theory was finally overthrown, and the new theory of capture,
or of addition from without, was introduced to take its place.

Such a complete transformation of this great subject neces-
sarily involves new causes heretofore quite overlooked; among
which we should mention the resisting medium and the operation
of repulsive forces in nature. The resisting medium has exercised
vast influence in building up central masses and reducing the size
and eccentricity of orbits, producing incidentally the capture of
satellites; while the repulsive forces have operated to disperse
matter in the form of fine dust from the stars, to produce nebulae,
which finally condense into all manner of planetary and stellar
systems. Even the comets as well as variable and temporary stars,
thus find a simple explanation in accordance with the general laws
of the heavens.

IV. BABINET'S CRITERION SHOWS THAT LAPLACE'S COSMOGONY
RESTS ON A FALSE PREMISE.

In the Comptes Rendus of the Paris Academy of Sciences for
March 18, 1861, Babinet introduced an important criterion show-
ing that Laplace's cosmogony was erroneous, by proving from the
mechanical principle of the conservation of areas, that when the
sun is expanded to fill the orbits of the planets, as imagined by
Laplace, the rotation is much too slow to develop a centrifugal
force adequate to detach the planets. The following table gives
the principal data from Babinet's criterion as now applied to the
planets and satellites of the solar system:



UNPARALLELED DISCOVERIES OF T. J. J. SEE



169



I. TABLE SHOWING THE APPLICATION OF BABINET'S CRITERION TO THE PLANETS

AND SATELLITES WHEN THE SUN AND PLANETS ARE EXPANDED TO FILL

THE ORBITS OF THE BODIES REVOLVING ABOUT THEM.

Solar System.



Planet.


#0

The Sun's Observed
Time of Rotation.


Po

Observed Period
of Planet.




Time of Sun's Rotation
Calculated by Babinet's
Criterion.


Mercury


25.3 days
=0.069267 yrs.




"479'yi
1673
3192
7424
24487
86560
290962
1176765
2888533


s.


0.24085 yr
0.61237
1.00000
1.88085
4.60345
11.86
29.46
84.02
164.78


s.


Venus
The Earth
Mars
Ceres
Jupiter
Saturn
Uranus
Neptune

















Sub-systems.



Planet.


Satellite.


R c

Adopted Rotation
of Planet.


PC

Observed Period
of Satellite.


Ro
Time of Planet's
Rotation Calcu-
lated by Babinet's
Criterion.


The Earth
Mars . . .


The moon

Phobos
Deimos

V
I
II
III
IV
VI
VII
VIII

Inner edge of ring
Outer edge of ring
Mimas
Enceladus
Tethys
Dione
Rhea
Titan
Hyperion
lapetus
Phoebe

Ariel

Umbriel
Titania
Oberon

Satellite


Iday
24.62297 hrs.


27.32166 days

7.6542 hours
30.2983 "

11.9563 hours
1.7698605days
3.5540942 "
7.1663872 "
16.7535524 "
250.618
265.0
930.73

0.236 days
0.6456
0.94242 "
1.37022 "
1.887796 "
2.736913 "
4.517500 "
15.945417 "
21.277396 "
79.329375 "
546.5

2.520383 days

4.144181 days
8.705897 "
13.463269 "

5.87690 days


3632.45 days

190.62 hours
1193.53

64.456 hours
14.60 days
35.900 '
93.933
290.63
10768.8
11602.4
61997.2

0.6228days
2.383 '
4.2902
7.0615
10.822
17.751
34.620
186.05
273.06
1580.1
20712.

33.714 days

65.435 days
176.05
314.83

141.8 days


Jupiter .


9.928 hrs.
























10.641 hrs.


Saturn . .


































Uranus.


10.1112 hrs.
(Cf. A.N., 3992)







Neptune


12.84817 hrs.
(Cf. A.N., 3992)



170



BRIEF BIOGRAPHY AND POPULAR ACCOUNT OF THE



II. TABLE OF DATA RELATING TO THE SOLAR SYSTEM.



Planet.


Centrifugal Force, calculated from
data of Babinet's criterion, pres-
ent orbital centrifugal force being
unity.


Density of Central Body, when
expanded to fill orbit, that of at-
mospheric air at sea level being
unity.


Mercury


0.000000253


0.001776


Venus


0.000000134


0.0002723


The Earth


0.000000098


00001029


Mars


0.000000064


0.00002913


Ceres


0.000000035




TuDiter


0.000000019


0000000732


Saturn


0.000000010


0.000000118


Uranus


0.0000000051


0.0000000146


Neptune


0.0000000032


0.0000000038



III. TABLE OF DATA RELATING TO THE SATELLITE SYSTEMS.



Planet.


Satellite.


Centrifugal Force, calculated
from data of Babinet's criterion,
present orbital centrifugal force
being unity.


Density of Central Body,
when expanded to fill or-
bit, that of atmospheric
air at sea level being
unity.


The Earth ....
Mars


The Moon
Phobos


0.00005657
0001612


0.01965
1151.




Deimos


0000644


7305


Tuoiter .


V


0034408


58.93




I


014694


4.66




II


0009277


1.15




III


0.005820


0.285




IV


0003323


0.0523




VI


0005416


0.000232




VII


00005217


0.000208




VIII


0002254


0.0000169


Saturn


Inner Ring


0.1435






Mimas


048254


16.45




Enceladus


037651


7.61




Tethys


030436


4.11




Dione


023772


1.96




Rhea


017017


0.717




Titan


0.0073449


0.0576




Hyperion


0060716


0.0324




lapetus


0.0025205


0.00232




Phoebe


0.0006962


0.000049


Uranus


Ariel


0.0055888


2.40




Umbriel


00040111


0.88




Titania


0024454


0.200




Oberon


0018287


0.082


Neptune


Satellite


0.0017177


0.43



It should be noticed that the centrifugal force varies as the
square of the velocity divided by the radius. Thus in the case of
the earth, actual revolution in the orbit occupies one year, whereas
the hypothetical nebulous sun expanded to fill the earth's orbit
requires 3,192 years for a rotation; and the rotational centrifugal



UNPARALLELED DISCOVERIES OF T. J. J. SEE 171

force therefore is only 1 : (3,192) 2 of what is required to detach the
earth, or less than a ten millionth part. In the case of Neptune,
the calculated time of rotation for the expanded central nebula
by Babinet's criterion, is 2,888,000 years, whereas Neptune actu-
ally revolves in 165 years. The calculated time of rotation is thus
17,500 longer than the observed time of revolution; and the
rotational centrifugal force is therefore only 1: (17,500) 2 or 1:306,-
250,000 of the centrifugal force required to detach Neptune.

In view of these facts we know that the planets never were
detached from the sun by acceleration of rotation as held by
Laplace and long believed by astronomers; but on the contrary
that they were formed independently, at a great distance, and
have since approached the sun as their orbits have been made
smaller and rounder and rounder under the secular action of a
resisting medium.

In regard to the satellites, the case most favorable to the
detachment theory is offered by the inner ring of Saturn, but even
here the rotating planet, when expanded to fill the ring, gives only
one-seventh of the centrifugal force required for detachment. So
that the theory of detachment is wholly given up, not only for the
planets, but also for all the satellites of the solar system. More-
over, the retrograde satellites of Jupiter, Saturn and Neptune are
easily explained by the modern capture theory, while they cannot
be harmonized with the old nebular hypothesis of Laplace, which
is therefore quite abandoned by all recent investigators.*

* These statements are positive, because Babinet's criterion, based on the
conservation of areas, is incontestable, having in the rotation of bodies the same
dynamical rigor that the law of gravitation has for the orbital motions of the
planets. Curiously enough prominent astronomers occasionally overlook the
decisive import of these elementary mechanical principles. Thus in POPULAR
ASTRONOMY, for October, 1911, p. 467, Professor E. B. Frost, Director of the Yerkes
Observatory, is led to the sad conclusion that "no adequate substitute has been
proposed" for the abandoned theory of Laplace. He adds that these views are
shared by friends whose opinions he values, showing that they are quite unaware of
the notable progress recently made, and that obscurity regarding the significance
of this dynamical principle still is widespread, although carefully treated by me
three years ago when I first called attention to Babinet's neglected work of 1861,
and more fully developed in my Researches, Vol. II, a copy of which was presented
to Professor Frost in October, 1910.



172 BRIEF BIOGRAPHY AND POPULAR ACCOUNT OF THE

There are at least three good reasons why the capture of
satellites is inevitable: (1) The planetary rotations are not
rapid enough to throw the bodies off, even if none of them revolved
in the contrary direction, as observed in the case of the outer
satellites of the systems of Jupiter and Saturn. (2) The density
of the expanded central globes would in all cases be too small to
exert any sensible hydrostatic pressure outward, so that unless
the angular rotation gave adequate centrifugal force, the deficiency
could not be supplied by hydrostatic pressure from the center.
(3) It is found that if set in revolution with the velocity assigned
by Babinet's criterion the satellite in every case would fall into
the planet before half a revolution was accomplished, (cf. Proc.
Am. Phil. Soc., Vol. XLIX, No. 197, Nov., 1910, p. 356) . It there-
fore follows incontestably that the satellites can have been set
revolving in their orbits only by capture, or addition from with-
out, which is clearly indicated also by the retrograde motion of
the outer satellites of Jupiter and Saturn.

V. How THE SATELLITES WERE CAPTURED.

In 1836 the celebrated German mathematician Jacobi com-
municated to the Paris Academy of Sciences an integral of the
problem of three bodies in the restricted case where the system is
made up of a sun attended by a planet revolving in a circular
orbit; and the third body a particle of insensible mass. This is
the case which is of special interest in Cosmogony, and here it has
found its widest application. In 1877 the work of Jacobi was
much extended by Dr. G. W. Hill in his researches on the Lunar
Theory, which have been the starting point of the profound re-
searches of Poincare, Darwin and others on periodic orbits, and
related topics in celestial mechanics.

Dr. Hill showed that in the restricted problem of three bodies,
implied in Jacobi' s integral, there is a partition of the whole space
into three parts, one about each of the large bodies, the sun and
planet, and a larger domain enclosing both bodies within which
the power of control over the particle is vested in the two bodies



UNPARALLELED DISCOVERIES OF T. J. J. SEE 173

individually and collectively, respectively. The closed surface
about the earth includes the orbit of the moon, and the orbits of
the other satellites in like manner are within the closed surfaces
about their several planets; and Dr. Hill remarks that this ar-
rangement is necessary to secure stability. (Hill's Collected
Mathematical Works, Vol. I, p. 330). If a satellite is once within
this region, with the surface of zero velocity closed about it, it
cannot escape, but will always remain attached to the planet, and
its radius vector will have a superior limit. How the moon and
other satellites came within these closed regions Dr. Hill did not
inquire; and subsequent investigators appear to have supposed
that as these bodies cannot now escape from their planets, so also
they cannot have come in from a remote distance, but must have
originated where they now are. This is the view put forth by
Moulton in his discussion of Professor W. H. Pickering's sug-
gestion that Phoebe had been captured by Saturn (Astrophysical
Journal, October, 1905, p. 178) ; but such reasoning is easily shown
to be erroneous by the following considerations:

Jacobi's integral, as originally given by him, is based on the
differential equations for unrestricted motion in empty space,
and no account is taken of the additional terms which must be
added to the differential equations of the motion of the sun, planet,
and particle, when the motion is very slightly conditioned by the
introduction of a nebular resisting medium, such as existed in the
early history of our system, and is now observed to be widely dif-
fused throughout Nature. Jacobi's original integral, therefore,
requires the addition of a secular term to represent the actual
movement of a sun, planet, and particle; and the complete ex-
pression for any particle whose coordinates are x-, y { z if becomes:



The secular term a, t-, makes the constant d increase with
the time.



174 BRIEF BIOGRAPHY AND POPULAR ACCOUNT OF THE

Now the surfaces of zero relative velocity, which define the
closed spaces about the planets, have larger values of d the nearer
we approach to the sun or planet. This is easily seen in the ac-
companying plate from Darwin's celebrated memoir on Periodic
Orbits (Acta Mathematica, Vol. XXI). When the particle or
satellite revolves against resistance, therefore, the second member




CURVES OF ZERO VELOCITY (DARWIN)

This diagram illustrates the hour-glass shaped space through which the

particle may move and drop down nearer the sun or planet,

till it becomes captured by one of the larger bodies.

of (1) increases, and there is a secular shrinkage of the surface of
zero relative velocity. Accordingly the particle drops down nearer
and nearer these centers, and the surface finally becomes closed,
leaving it no longer free to move about both bodies in the hour-
glass shaped space, as formerly, but restricted to the sphere of
influence controlled by the sun or planet individually, as the case
may be. The particle which once revolved about both the sun



UNPARALLELED DISCOVERIES OF T. J. J. SEE 175

and planet can no longer do so, but becomes an inferior planet
(satellite of the sun) or a satellite of the planet.

This is how the satellites of the solar system were captured.
At first they moved principally under the attraction of the sun,
and could pass from the sun's to the planet's domain, through the
neck of the hour-glass shaped space connecting the two spheres of
influence. When the neck is narrow, Darwin says that a particle
which passes from the sun's to the planet's control may revolve
about it hundreds of times before quitting the planet's sphere to
return again to the sun's control. And if resistance is meanwhile
encountered, so that the neck of the surface of zero velocity be-
comes closed, it is clear that the particle never will quit the sphere
of the planet's control, but will abide there permanently as a
satellite.

Thus it incontestably follows that the satellites of Jupiter,
Saturn, and other planets formerly moved about the sun, and
since they were captured have had their orbits reduced in size and
rounded up under the secular action of the resisting medium for-
merly pervading our solar system. Satellites may cross over
the line SJ before coming completely under the planet's control,
in which case they will move retrograde. In such cases the neck
connecting the two spaces is extremely narrow. But as the neck
usually is not so narrow as to produce crossing satellites, most of
them naturally move direct, in accordance with observation. This
is the reason also why the planets have direct rotations on their
axes. The planets have in no case been inverted, as some have
recently supposed, in order to account for the retrograde motion
of the satellites of Jupiter and Saturn.

VI. CAPTURE THEORY OF SATELLITES INDEPENDENTLY CON-
FIRMED BY BROWN AND POINCARE.

The above discussion is substantially that given by the writer
in the Publications of the Astronomical Society of the Pacific for
August, 1909. The subject has since been treated more in detail
and from a slightly different point of view by Professor E. W.



176 BRIEF BIOGRAPHY AND POPULAR ACCOUNT OF THE

Brown, in the Monthly Notices of the Royal Astronomical Society
for March, 1911, p. 453. Brown considers the oscillations of an
asteroid about the triangular points, where Lagrange showed that
there are particular solutions of the problem of three bodies, and
a small body may revolve in stability when the asteroid is sub-
jected to small disturbances, as under the action of a resisting
medium. Brown shows that it will revolve in periodic orbits
about the triangular point, but the constant C of the Jacobian
integral will steadily increase and the periodic orbits increase in
size; and finally it will reach a critical stage corresponding to the
equilibrium point in opposition to Jupiter, where Lagrange showed
that another particular solution of the problem of three bodies
exists. Under disturbances all these solutions are unstable, and
for values of C well beyond this critical value, Brown adds that
the orbits "consist (1) of an inner planetary orbit making com-
plete revolutions round the sun in the positive sense; (2) of an
outer planetary orbit making complete revolutions round the sun
in the negative sense relative to the moving axes; (3) of a satellite
revolving round Jupiter in the positive sense." In other words,
an asteroid passing through these equilibrium points with suitable
velocity corresponding to the critical value of C, may pass under
the control of Jupiter and become a satellite of that planet, as I
demonstrated in 1909 (A.N. 4341-42).

Finally it remains to add, as already pointed out, that the
capture of satellites has been treated also by Poincare in his course
at the Sorbonne during the present year, which is being published
and will appear in November. Astronomers and mathematicians
naturally will be interested in the forthcoming work of this great
master of celestial mechanics. In a recent letter he tells me that
he insists on the capture of planets as satellites under the action
of a resisting medium.

It is worthy of remark that as far back as 1906 Professor Elis
Stromgren, now of Copenhagen, while occupied with the motion of
three bodies made a very similar investigation of the problem of
cusps and loops, and obtained some remarkable results (Astron.



UNPARALLELED DISCOVERIES OF T. J. J. SEE 177

Nachr. No. 4155) which are now found to confirm the capture of
satellites.

From this brief outline it will be seen that the capture theory of
satellites is now established and very generally accepted by the lead-
ing authorities in this difficult branch of mathematical astronomy.

VII. THE ORIGIN OF THE RETROGRADE REVOLUTIONS OF THE

SATELLITES AND OF THE DIRECT ROTATIONS OF

THE PLANETS ON THEIR AXES.

In my work of 1909, quoted above, it was shown how the
retrograde motion of the satellite could arise, by the body crossing
over the line S / in passing through the neck of the hour-glass
space before it came under the control of the planet. It may arise
also in other ways. We have seen above that in passing through
the equilibrium point, in opposition to Jupiter, the satellite usually
will move direct, because there is a space of finite extent in this
region corresponding to the critical value of the Jacobian integral,
and unless the disturbance is considerable, the gradual transition
will give a direct revolution; but if the disturbance be larger, at
this critical stage, the satellite may be driven beyond the center
of the equilibrium point and set revolving in a retrograde direction.
This result is very similar to the case (2) above quoted from
Brown's paper, where he concludes that the asteroid may make
complete revolutions around the sun in the negative sense relative
to the moving axes.

We need not dwell at greater length on the motions of satel-
lites. It is obvious that they may move either direct or retro-
grade, as first pointed out by me in 1909. But if most of the satel-
lites have direct motion, those which are retrograde will be likely
to be destroyed, unless they are so situated as to escape serious
collision. It is remarkable that the retrograde satellites of Jupiter
and Saturn are on the outside of their systems, where they could
easily survive; and this no doubt is the secret of their escape from
destruction. Probably they were captured quite late in the history
of the solar system, when the resisting medium was relatively



178 BRIEF BIOGRAPHY AND POPULAR ACCOUNT OF THE

ineffective, as shown by the rather large surviving eccentricities
of their orbits.

And now just as the satellites usually have direct orbital
revolutions, so also do the meteorites and other small masses of
cosmical dust circulating in the vortices about the planets, of
which the satellites alone are large enough and bright enough to
be visible in our telescopes. When swarms of this dust collide
with the planet, therefore, the tendency is to give the globe a
direct rotation on its axis, because the number of particles having
direct revolutions greatly predominates over those having retro-
grade revolution. This is the secret of the direct rotations of the
planets on their axes; and whenever the down-pour of dust is
appreciable the axial rotations are being accelerated. It is not
strange therefore that the sun, Jupiter and Saturn have equatorial
accelerations, which long proved bewildering to astronomers.

VIII. THE OBLIQUITIES OF THE PLANETS.

The problem of the obliquities of the planets long presented
great difficulty to the astronomer, and was not solved till 1908,
when it was shown by the present writer that it is determined by
the capture and absorption of small bodies revolving about the
sun in planes nearly coinciding with the orbits of the planets. The
countless collisions of these small bodies with a planet like the
earth are illustrated by the meteors vaporized in our atmosphere
to the number of over a billion daily. In a century this dust would
make a layer a millimeter thick all over the globe. The same
down-pour of cosmical dust occurs on the other planets, and often
times the masses are larger than those now swept up by the earth,
as we see by the embedded satellites which produced the lunar
craters.

As they revolve about the sun in orbits, which on the average
are not much inclined to the orbit of the earth, the tendency is to
tilt the earth's axis into a position at right angles to the plane of
the ecliptic. The obliquity thus tends to vanish, as illustrated
by Jupiter, the greatest of the planets, which has thus acquired a



UNPARALLELED DISCOVERIES OF T. J. J. SEE 179

very small obliquity of some 3, from the capture and absorption
of comets, asteroids and satellites. The almost zero obliquity of
Jupiter shows the normal development of the process. As a test
of the theory it was shown by actual calculation that Saturn would
have the present obliquity of some 28 nearly destroyed if the mass
of that planet were trebled, to become equal to that of Jupiter,
by the same process of the capture and absorption of small bodies
moving near the plane of the planet's orbit. The increase of the
obliquities from Jupiter to Saturn and Uranus therefore is natural,
and in accordance with established theory. Within Jupiter's
orbit the obliquities decrease from Mars (24.5) to the earth (23.5)
and Venus (12 or 15); and the rotation periods also decrease
from 24 h 37 m 22 s in the case of Mars, to 23 h 56 m 4 s (sidereal day)
for the earth, and 23 h 21 m for Venus. The order thus exist-
ing among the terrestrial planets shows that they have been
formed by a very regular cause which has produced harmonious


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