Consider a planet of mass m moving in a nearly circular orbit of radius R around a star of mass M. There is, in addition to gravitation, a repulsive force on the planet proportional to the distance r from the star, F = Ar. Compute the angular velocity of precession of the periastron (point of closest approach to the star).

This is a complex problem in celestial mechanics. Here's a step-by-step guide to solve it:

1. First, we need to understand the forces acting on the planet. There are two forces: the gravitational force and the repulsive force. The gravitational force is given by F = GMm/r^2, where G is the gravitational constant. The repulsive force is given by F = Ar, where A is a constant.

2. The net force acting on the planet is the sum of these two forces. This gives us F_net = GMm/r^2 - Ar.

3. The force acting on the planet also provides the centripetal force required for the planet to move in a circular orbit. This is given by F = mrω^2, where ω is the angular velocity of the planet.

4. Setting the net force equal to the centripetal force gives us GMm/r^2 - Ar = mrω^2.

5. We can solve this equation for ω to find the angular velocity of the planet. This gives us ω = sqrt((GM - Ar^3)/r^3).

6. The precession of the periastron is caused by the change in the direction of the major axis of the planet's orbit. This change in direction is caused by the repulsive force, which is not directed towards the center of the star.

7. The rate of precession is given by the change in the angle θ per unit time. This is given by dθ/dt = ω.

8. Substituting our expression for ω into this equation gives us dθ/dt = sqrt((GM - Ar^3)/r^3).

This is the angular velocity of precession of the periastron. Note that this result is approximate, as we have assumed that the orbit of the planet is nearly circular. For a more accurate result, we would need to take into account the ellipticity of the orbit.