Energy required to kick a planet orbiting the Sun from an elliptical to a parabolic path

Question

I am trying to solve the following problem from Goldstein's Classical Mechanics:

A planet of mass $M$ is in orbit of eccentricity $e=1-\alpha$ where $\alpha<<1$, about the Sun. Assume that the motion of the Sun can be neglected and only gravitational forces act. When the planet is at its greatest distance from the Sun, it is struck by a comet of mass $m$, where $m<<M$, traveling in a tangential direction. Assuming the collision is completely inelastic, find the minimum kinetic energy the comet must have to change the new orbit to a parabola.

My reasoning:

Using the conservation of energy equation, we can find the energy of the planet in its elliptical orbit. Since the motion is purely tangential at this point, we have $$E=\frac{1}{2}m(r\dot\theta)^2-\frac{k}{r}$$ Because it is an ellipse, we have $E<0$. Therefore, the energy the comet should have is $-E$ (since for a parabola, $E=0$), $$E_{comet}=\frac{k}{r}-\frac{1}{2}m(r\dot\theta)^2$$

All that now remains is to simplify the expression for $E_{comet}$ in accordance with the given data. Am I correct in my general reasoning?

Answer 1

The expression for the energy of a parabolic orbit can be greatly simplified. A parabolic orbit is the lowest energy path that will allow the body to escape. So the sum of kinetic plus potential energy is exactly zero. It should be easiest to treat the elliptical orbit as a perturbative oscillation on a circular orbit. (The frequency of oscillation is exactly the period of the orbit.)