Page:The evolution of worlds - Lowell.djvu/293

$=2\pi a^{2}\cdot (1-e^{2})^{\frac {1}{2}}$,

which is twice the area of the ellipse.

The energy in the ellipse during an interval $dt$ is

${\frac {1}{2}}mv^{2}dt={\frac {1}{2}}m\mu \left({\frac {2}{r}}-{\frac {1}{a}}\right)dt$

from the well-known equation for the velocity in a focal conic. The integral of this for the whole ellipse is

$\int _{0}^{T}{\frac {1}{2}}mv^{2}dt=\int _{0}^{360^{\circ }}{\frac {1}{2}}{\frac {m\mu }{h}}\left(2r-{\frac {r^{2}}{a}}\right)d\theta =m\mu ^{\frac {1}{2}}\pi a^{\frac {1}{2}}$

Since

$\int rd\theta =\int {\frac {a\cdot 1-e^{2}}{1+e\cos \theta }}d\theta ={\frac {2a\cdot 1-e^{2}}{(1-e^{2})^{\frac {1}{2}}}}\tan ^{-1}\left({\sqrt {\frac {1+e}{1-e}}}\tan {\frac {\theta }{2}}\right)$

and $\int r^{2}d\theta$ is given above.

By collision a part of this energy is lost, being converted into heat. The major axis, a, is, therefore, shortened. But from the expression $\scriptstyle {2\pi a^{2}\cdot (1-e^{2})^{\frac {1}{2}}}$ for the moment of momentum we see that this is greatest when e is least. If, therefore, a is diminished, e must also be diminished, or the moment of momentum would be lessened, which is impossible.

7

Capture of Satellites

See has recently shown (Astr. Nach. No. 4341–42) that a particle moving through a resisting medium under the attraction of two bodies revolving round one another in circles may eventually be captured by one of them though originally under the domination of both. The argument consists in introducing the effect of a resisting medium