A uniform distribution of dust in the solar system adds to the gravitational attraction of the Sun on a planet an additional force
F=−mCr,
where m is the mass of the planet, C is a constant proportional to the gravitational constant and the density of the dust, and r is the radius vector from the Sun to the planet (both considered as points). This additional force is very small compared to the direct Sun--planet gravitational force.
(a) Calculate the period for a circular orbit of radius r0 of the planet in this combined field.
(b) Calculate the period of radial oscillations for slight disturbances from this circular orbit.
(c) Show that nearly circular orbits can be approximated by a precessing ellipse and find the precession frequency. Is the precession in the same or opposite direction to the orbital angular velocity?