Page:The evolution of worlds - Lowell.djvu/288

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,

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

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

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

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

.

As $R-r=a_{1}(1+e)$ 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 $e=2{\sqrt {\frac {MR}{r}}}-{\frac {RM}{r}}$ approx.