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

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?

The orbital energy of two bodies is defined as the energy required to separate them to infinite distance: $$E = E_1 + E_2= \mu \frac{v^2}{2} - \frac{ G m_1 m_2 }{ r } = \mu \left( \frac{v^2}{2} - \frac{ G m_2 }{ r } - \frac{ G m_1 }{ r }\right) = - \frac{ G m_1 m_2 }{ 2 a } = - \frac{ G m_1 m_2 (1+e) }{ 2 d }$$ where: $\mu$ is the reduced mass, v is the relative (heliocentric) velocity, r is the relative separation, a is the semimajor axis, and d is the apoapsis distance. $E_1$ is the amount of energy applied to the 1st body, and $E_2$ is the amount of energy applied to the 2nd body.
So the energy necessary to knock a planet into a parabolic (escape) orbit is: $$\Delta\text{energy} = \frac{ G m_\text{sun} \, m_\text{planet}(1+e) }{ 2 d } = \frac{ G m_\text{sun} \, m_\text{planet} }{ 2 a }$$
Notice that the energy in your first equation is the energy of the planet ($E_2$) with respect to the center of mass of the system. An additional non-negligible amount of energy ($E_1$) is required to knock the Sun into a parabolic orbit. Also note that the r in your equations is defined as the barycentric speed of the planet (not the heliocentric speed).