# The Evolution of Worlds/Notes 5

5

PLANETS AND THEIR SATELLITE SYSTEMS

If we compute the speeds of satellites about their primaries in the solar system and compare them with the velocities in their orbits of the planets themselves, a striking parallelism stands displayed between the several systems.

This is shown in the following table of them:

 Mean Speed, Miles A Second Parabolic Speed at Orbit Ratio Speed Sat. About Primary to Planet's Speed in Orbit of Primary in Orbit of Satellite about primary Miles a second. V v Jupiter 8.1 11.5 Sat. 1 10.7 1.32 2 8.5 1.05 3 6. 7 0.83 4 5.1 0.63 Saturn 6.0 8.5 1 9.0 1.50 2 7.9 1.31 3 8.2 1.36 4 6.3 1.05 5 5.3 0.89 6 3.5 0.59 8 2.0 0.34 Uranus 4.2 5.9 1 3.5 0.82 2 2.9 0.70 3 2.3 0.54 4 2.0 0.47 Neptune 3.4 4.8 1 2.7 0.81

The relations here disclosed are too systematic to be the result of chance. The orbits of all these satellites have no perceptible eccentricity independent of perturbation except Iapetus, of which the eccentricity is about .03.

In view of the various cosmogonies which have been advanced for the genesis of the solar system it is interesting to note what these speeds imply as to the effect upon the satellites of the impact of particles circulating in the interplanetary spaces at the time the system evolved. To simplify the question we shall suppose—which is sufficiently near the truth—that the planets move in circles, the interplanetary particles in orbits of any eccentricity.

Taking the Sun's mass as unity, the distance R of any given planet from the Sun also as unity, let the planet's mass be represented by M and the radius of its satellite's orbit, supposed circular, as r. We have for the space velocity of the satellite on the sunward side of the planet, calling that of the planet in its orbit V and that of the satellite in its orbit round the planet v,

${\displaystyle V-v={\sqrt {\frac {1}{R}}}-{\sqrt {\frac {M}{r}}}}$.

For a particle, the semi-major axis of whose orbit is ${\displaystyle a_{1}}$ and which shall encounter the satellite,
the velocity is ${\displaystyle v_{1}={\left({\frac {2}{R-r}}-{\frac {1}{a_{1}}}\right)}^{\frac {1}{2}}}$.

That no effect shall be produced by the impact of these two bodies, their velocities must be equal, or

${\displaystyle {\sqrt {\frac {1}{R}}}-{\sqrt {\frac {M}{r}}}={\sqrt {{\frac {2}{R-r}}-{\frac {1}{a_{1}}}}}}$

.

As ${\displaystyle R-r=a_{1}(1+e)}$ for the point of impact if the particle be wholly within the orbit of the planet and e the eccentricity of its orbit, we find ${\displaystyle e=2{\sqrt {\frac {MR}{r}}}-{\frac {RM}{r}}}$ approx. for the case of no action, the other terms being insensible for the satellites in the table, since in all ${\displaystyle \scriptstyle {r<{\frac {R}{400}}}}$.

Supposing, now, the particles within the orbit of the planet to be equally distributed according to their major axes, then as the velocity of any one of them, taking ${\displaystyle \scriptstyle {R-r=R}}$ approx. as unity, is

${\displaystyle \scriptstyle {v_{1}={\left({\frac {2}{1}}-{\frac {1}{a_{1}}}\right)}^{\frac {1}{2}}}}$,

the mean velocity of all of those which may encounter the satellite is, at the point of collision,

${\displaystyle {\begin{array}{l}\scriptstyle {\qquad \qquad {\frac {\textstyle {\int _{\frac {1}{2}}^{1}}{\frac {(2a_{1}-1)^{\frac {1}{2}}}{a_{1}^{\frac {1}{2}}}}da_{1}}{\textstyle {\int _{\frac {1}{2}}^{1}}da_{1}}}}\\\scriptstyle {=2{\underset {\frac {1}{2}}{\overset {1}{\Bigl [}}}(2a_{1}^{2}-a_{1})^{\frac {1}{2}}-{\frac {1}{\sqrt {2}}}\log\{(2a_{1}-1)^{\frac {1}{2}}+{\sqrt {2a_{1}}}\}{\Bigr ]}}\\\scriptstyle {=0.754;}\end{array}}}$

that is, just over three-quarters of the planet's speed in its orbit.

If we suppose the particles to be equally distributed in space, we shall have more with a given major axis in proportion to that axis, and our integral will become

${\displaystyle {\begin{array}{l}\scriptstyle {\qquad \qquad {\frac {\textstyle {\int _{\frac {1}{2}}^{1}}(2a_{1}-1)^{\frac {1}{2}}a_{1}^{\frac {1}{2}}da_{1}}{\textstyle {\int _{\frac {1}{2}}^{1}}a_{1}da_{1}}}}\\\scriptstyle {={\frac {8}{3}}{\underset {\frac {1}{2}}{\overset {1}{\biggl [}}}{\frac {4a_{1}-1}{8}}(2a_{1}^{2}-a_{1})^{\frac {1}{2}}-{\frac {1}{16{\sqrt {2}}}}\log {\Bigl [}(2a_{1}^{2}-a_{1})^{\frac {1}{2}}}\\\scriptstyle {\qquad \qquad \qquad \qquad +{\sqrt {2}}\cdot a_{1}-{\frac {1}{2{\sqrt {2}}}}{\Bigr ]}{\biggr ]}}\\\scriptstyle {=0.792{\text{ of the planet's orbital speed.}}}\end{array}}}$

The speed v, then, at which a satellite must be moving round the planet to have the same velocity as the average particle within the planet's orbit, is

${\displaystyle V-v_{1}=v.}$

This velocity is, for the several planets:—

 Distribution ofParticles as theirMajor Axes Distribution ofParticles Equalin Space Miles a second Miles a second Jupiter 2.0 1.6 Saturn 1.5 1.2 Uranus 1.0 0.9 Neptune 0.8 0.7

If the satellite be moving in its orbit less fast than this, its space-speed will exceed that of the average particle; it will strike the particle at its own rear and be accelerated by the collision. If faster, the particle will strike it in front and retard it in its motion round its primary.

From the table it appears that all the large satellites of all the planets have an orbital speed round their primaries exceeding those in either column. In consequence, all of them must have been retarded during their formation by the impact of interplanetary particles and forced nearer their primaries than would otherwise have been the case; and this whether the particles were distributed more densely toward the Sun, as ${\displaystyle {\frac {1}{a_{1}}}}$, or were equally strewn throughout.

For interplanetary particles whose orbits lie without the particular planet's path the mean speed is the parabolic at the planet's distance, given in the third column of the table. This is the case on either supposition of distribution. The orbital speed of the satellite which shall not be affected by collisions with them is, for the several planets:—

 Miles a Second Jupiter 3.4 Saturn 2.5 Uranus 1.7 Neptune 1.4

All the satellites but Iapetus have orbital speeds exceeding this, and consequently are retarded also by these particles.

For particles crossing the orbit (2) the mean velocity would be practically parabolic, 1.4, even if the distribution were as

${\displaystyle {\frac {1}{r'}},r'}$ being the distance from the Sun. The effect would depend upon the angle of approach and in the mean give a greater velocity for the particle than for the satellite within the orbit, a less one without; retarding the satellite in both cases. Thus the total effect of all the particles encountering the large satellites is to retard them and to tend to make them hug their primary.

For retrograde satellites the velocities of impact with inside and outside particles moving direct are respectively:

 Inside Outside Jupiter 2.0 + ${\displaystyle v}$ ${\displaystyle v}$ + 3.4 Saturn 1.5 + ${\displaystyle v}$ ${\displaystyle v}$ + 2.5 Uranus 1.0 + ${\displaystyle v}$ ${\displaystyle v}$ + 1.7 Neptune 0.8 + ${\displaystyle v}$ ${\displaystyle v}$ + 1.4

In both cases the impact tends to check the satellite.

Comparing with these the velocities of impact for direct satellites in a direct plenum:—

 Inside Outside Jupiter 2.0 – ${\displaystyle v}$ 3.4 – ${\displaystyle v}$ Saturn 1.5 – ${\displaystyle v}$ 2.5 – ${\displaystyle v}$ Uranus 1.0 – ${\displaystyle v}$ 1.7 – ${\displaystyle v}$ Neptune 0.8 – ${\displaystyle v}$ 1.4 – ${\displaystyle v}$

the signs being taken positive when the motion is direct, we see that retrograde satellites would be more arrested than direct ones with the same orbital speed round the primary.

In a plenum of direct moving particles, then, the force tending to stop the satellite and bring it down upon the planet is greater for retrograde satellites than for direct ones.

If, therefore, the positions of the satellites have been controlled by the impact of interplanetary particles, the retrograde satellites should be found nearer their planets than the direct ones.