The phase space volume $g(E)$ of constant energy surface

Question

Consider a system with two particles described by a Hamiltonian of the form $$H(P,Q,p,r)=\frac{P^2}{2M}+\frac{p^2}{2\mu}-\frac{Gm^2}{r}$$ where $(Q,P)$ are coordinates and momenta of the center of mass, $(r,p)$ are the relative coordinates and momenta, $M = 2m$ is the total mass, $\mu= m/2$ is the reduced mass and $m$ is the mass of the individual particles. Now the volume $g(E)$ of the constant energy surface $H = E$ is typically expressed as $g(E)=\frac{d\Gamma(E)}{dE}$ and $\Gamma(E)=\int d^3Qd^3P d^3p d^3r \Theta(E-H(P,Q,p,r))$.
I am trying to evaluate this as, $$ \begin{align} \Gamma(E)&=\int d^3Qd^3P d^3p d^3r \Theta(E-H(P,Q,p,r)) \\ &=\frac{4}{3}\pi R^3 \int_{a}^{r_{max}} 4\pi r^2 dr d^3P d^3p \Theta \Big(E+\frac{Gm^2}{r}-\frac{P^2}{2M}-\frac{p^2}{2\mu}\Big) \end{align} $$ I am not sure how to evaluate this. Any help is highly appreciated.
P.S. The answer is $$\Gamma(E)=\frac{64}{9}\pi^5 m^3R^3\int_{a}^{r_{max}} r^2dr \bigg(E+\frac{Gm^2}{r}\bigg)^3$$

Answer 1

Bring the spherical coordinates for $P$ and $p$:

$$\begin{align} \int d^3P &= \int P^2dP \int \sin\theta_P d\theta_P \int d\phi_P\\ \int d^3p &= \int p^2dp \int \sin\theta_p d\theta_p \int d\phi_p \end{align}$$

Then your integrand does not depend on the angles, and you are therefore left with the square of the solid angle times the integral on $P$ and $p$,

$$(4\pi)^2\int_{\frac{P^2}{2M}+\frac{p^2}{2\mu}\le E+\frac{Gm^2}{r}} P^2p^2dPdp$$

The domain of integration is an ellipse, so reparametrise as

$$\begin{align} P&=\sqrt{2M\left(E+\frac{Gm^2}{r}\right)} \chi\cos\varphi\\ p&=\sqrt{2\mu\left(E+\frac{Gm^2}{r}\right)} \chi\sin\varphi \end{align}$$

and the change of variable then works out as

$$\int_{\frac{P^2}{2M}+\frac{p^2}{2\mu}\le E+\frac{Gm^2}{r}} dPdp = 2\sqrt{M\mu}\left(E+\frac{Gm^2}{r}\right)\int_{\chi=0}^1 \int_{\varphi=0}^{2\pi} \chi d\chi d\varphi$$

Then insert the integrand $P^2p^2=4M\mu\left(E+\frac{Gm^2}{r}\right)^2\chi^4\cos^2\varphi\sin^2\varphi$

and you are finished.