Radial perturbation to a satellite moving in a circular orbit

Question

A satellite is in a circular orbit of radius $R$ around the earth. I am expected to calculate the period of small radial oscillations of the satellite if it is slightly disturbed from its circular orbit in the radial direction. Then, using Kepler's third law for the period of revolution, I have to determine whether the new disturbed orbit is closed or open.

However, I cannot deduce if the satellite, after the radial perturbation, will follow a kind of simple harmonic motion. I think that if a particle moving in a circular orbit has a radial perturbation, it simply would change its orbit to an ellipse, implying that I should find the new orbit's period?

So, the angular momentum is conserved, $\ell = mv_0R$, and the new expression for the velocity is $\vec{v} = v_r \vec{u_r} + v_0\vec{u_{\phi}}$. Then the new Energy expression is $E = m/2(v_r^2+v_0^2)-\gamma/R$ and I've thought about introducing that expression into the one that relates energy with eccentricity, $E = \gamma^2m(\epsilon^2-1)/(2\ell^2)$ to obtain the eccentricity of the new orbit and then introduce it in the following equation $r(\phi)= c/(1+\epsilon\cos{\phi})$, to then apply Kepler's 3rd Law to obtain the new period. It is correct?

Thanks in advance!

This isn't an answer, but you might find it useful. You are working with an inverse square law. Betrand's Theorem shows what kinds of force laws produce stable closed orbits. See An intuitive proof of Bertrand's theorem — mmesser314, Dec 08 '20 at 00:54
"I cannot deduce if the satellite, after the radial perturbation, will follow a kind of simple harmonic motion." The moral of intermediate undergrad classical mechanics courses is that nearly everything is a harmonic oscillator if the perturbation is small enough. — Michael Seifert, Dec 08 '20 at 01:47

Radial perturbation to a satellite moving in a circular orbit

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