Page:The Solar System - Six Lectures - Lowell.djvu/98

80

The Solar System

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est perturbative action on the rings is Mimas, his effect being more than three times that of Enceladus and more than twice that of Tethys.

The equations of motion are for x—

$\textstyle {\frac {d^{2}x}{dt^{2}}}=-{\frac {x}{r^{3}}}+{\frac {m'(x'-x)}{(r'-r)^{3}}}-{\frac {m'x'}{r'^{3}}},$

of which the first is the direct force of the central body, whose mass is taken as i, upon m; and the other terms are the perturbing force of m on m'.

Assuming the three to be in conjunction, this last becomes $\textstyle m'({\frac {1}{\rho ^{3}}}-{\frac {1}{r'^{3}}})$where $\textstyle \rho =r'-r$. Supposing $m$ to be 74,000 miles from the centre of Saturn, and Mimas, Enceladus, and Tethys at 117,000, 150,000, and 186,000 miles respectively, and taking the masses as proportionate to their volumes, their radii being taken as 400, 400, and 600 miles, we find for their relative perturbative effects :—

Mimas	.	.	.	.	.	.	.	299
Enceladus	.	.	.	.	.	.	.	82
Tethys	.	.	.	.	.	.	.	110

The action of the others is smaller still. Now the major axis of a part of the ring which has a period commensurate with that of Mimas may be found from the formula —

$\textstyle {\frac {T^{2}}{T_{,}^{2}}}={\frac {a^{3}}{a_{,}^{3}}}$,

Kepler's third law. Beginning, then, with the simplest, and therefore the most potent ratio, $\textstyle {\frac {1}{2}}$, we find 73,600 miles for the major axis of a particle