Question

Imagine a small satellite of mass m that describes a circular orbit. If at a point in the orbit it receives a small radial impulse when applying its motors very briefly (ignore the loss of mass), then the orbit changes a small distance δ, such that the satellite begins to oscillate with respect to its orbit r_0, it is say r (t) = r0 + δ (t), (δ (t) ≪r0)

You may recall from your mechanical course that the gravitational force is a central force, implying the conservation of the angular momentum L in orbit (Kepler's second law: law of areas); so we can treat the problem as if it were a one-dimensional problem, only the radial part, where the original potential is replaced by the effective potential,

(a) Find the equilibrium position, that is, the radius r_0 of the circular orbit

(b) Find the period T of the radial oscillation.
(c) Write an equation for small radial disturbances around the circular orbit. Identify the frequency. Show that the equation represents a simple harmonic oscillator of the radial disturbance δ = r (t) −r_0.

Answer 1