For a two body system with hamiltonian $H=\frac{P^2}{2M}+\frac{p^2}{2\mu}-\frac{Gm^2}{r}$ and assuming minimal distance between particles r>a, and some large volume V containing the system, I am trying to find the microcanonical density of states using $\Omega(E)=\frac{d\Gamma(E)}{dE}$ with $\Gamma(E)=\int{d^3Pd^3pd^3Qd^3r\Theta(E-H(P,p,Q,r))}$ where $\Theta(x)$ is the Heavyside function.
I have tried calculating this by first using the fact that integration over P would give the volume of a sphere with radius $R=\sqrt{2M(E-\frac{p^2}{2\mu}+\frac{Gm^2}{r})}$, and also that integrating over Q would give V, so that: $$ \Gamma(E)=\frac{4\pi}{3}V\int{d^3pd^3r}((2M(E-\frac{p^2}{2\mu}+\frac{Gm^2}{r}))^\frac{3}{2}\Theta(E-\frac{p^2}{2\mu}+\frac{Gm^2}{r}) $$
Now I tried using spherical coordinates for p with radius $R_2=\sqrt{2\mu(E+\frac{Gm^2}{r})}$ and got the integral: $$ \Gamma(E)=(\frac{4\pi}{3})^2V\int{d^3r}\Theta(E+\frac{Gm^2}{r})\int_0^{R_2}{((2M(E-\frac{p^2}{2\mu}+\frac{Gm^2}{r}))^\frac{3}{2}p^2dp} $$
But here I think something is wrong, because this doesn't appear to give the sort of answer I am looking for. Was my first step correct? If so, how do I continue?
