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If the solar system were immersed in a uniformly dense spherical cloud of weakly-interacting massive particles (WIMPs) then objects in the solar system would experience gravitational forces from both teh Sun and the cloud of WIMPs such that

$F_r = -\frac{k}{r^2} - br$

Assume that the extra force due to the WIMPs is very small (that is *b* « k/r^{3}). Work to first order in *b*.

(a) Find the frequency of radial oscillations for a nearly circular orbit.

(b) Find the average angular velocity.

(c) Find the rate of precession of the perihelion to lowest order in *b* using the results of (a) and (b).

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Answer

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