Page:The evolution of worlds - Lowell.djvu/292

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

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.

6

On the Induced Circularity of Orbits through Collision

Since the moment of momentum is the velocity into the perpendicular upon its direction, in the time $dt$ it is:—

$vpdt=hdt=r^{2}d\theta$

The whole moment of momentum from perihelion to perihelion is therefore:—

${\begin{aligned}\int _{0}^{360^{\circ }}r^{2}d\theta =\,&{\frac {a^{2}\cdot (1-e^{2})^{2}}{1-e^{2}}}\\&{\underset {0}{\overset {360^{\circ }}{{\Biggl [}\;}}}{\frac {-e\sin \theta }{1+e\cos \theta }}+{\frac {2}{(1-e^{2})^{\frac {1}{2}}}}\tan ^{-1}\left({\sqrt {\frac {1-e}{1+e}}}\cdot \tan {\frac {\theta }{2}}\right){\Biggr ]}\end{aligned}}$