Page:The evolution of worlds

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NOTES

for the case of no action, the other terms being insensible for the satellites in the table, since in all $\scriptstyle {r<{\frac {R}{400}}}$ .

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 $\scriptstyle {R-r=R}$ approx. as unity, is

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

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

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

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

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

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