Regulated relative Kepler phase volume and shell law

Use SI units, positive masses $m_1,m_2$ in $\mathrm{k}\mathrm{g}$, reduced mass $\mu =m_1{m}_2\mathrm{/}(m_1+m_2)$, gravitational parameter $A=G{m}_1{m}_2$ in $\mathrm{J}\mathrm{m}$, relative position $q\in {\mathbb{R}}^3$ in $\mathrm{m}$, relative momentum $p\in {\mathbb{R}}^3$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, and a fixed action reference $h_{\ast }>0$ in $\mathrm{J}\mathrm{s}$. Assume $r_0>0$ and $R>16r_0$. The relative phase domain is ${\mathrm{\Gamma }}_{\mathrm{r}\mathrm{e}\mathrm{l}}=\{(q,p):r_0\le \mathrm{\mid }q\mathrm{\mid }\le R,p\in {\mathbb{R}}^3\},d{\nu }_{\mathrm{r}\mathrm{e}\mathrm{l}}=\frac{d^{3}q{d}^{3}p}{h_{\ast }^3},$ with no center-of-mass variables or factor $1\mathrm{/}2!$. The relative Hamiltonian is $h(q,p)=\frac{\mathrm{\mid }p{\mathrm{\mid }}^2}{2\mu }-\frac{A}{\mathrm{\mid }q\mathrm{\mid }}\mathrm{.}$

For $E=-e<0$ with $A\mathrm{/}R<e\le A\mathrm{/}(16r_0)$, put $\delta =e{r}_0\mathrm{/}A\in (0,1\mathrm{/}16]$ and $J(\delta )={\int }_{\delta }^1{s}^{1\mathrm{/}2}(1-s{)}^{3\mathrm{/}2}ds,\mathrm{\Omega }(E)={\nu }_{\mathrm{r}\mathrm{e}\mathrm{l}}\{(q,p)\in {\mathrm{\Gamma }}_{\mathrm{r}\mathrm{e}\mathrm{l}}:h(q,p)\le E\}\mathrm{.}$ The radial integration formula of Radial integration over a bounded Euclidean shell gives the explicit phase volume $\mathrm{\Omega }(-e)=\frac{(4\pi )(4\pi \mathrm{/}3)(2\mu {)}^{3\mathrm{/}2}{A}^3}{h_{\ast }^3}{e}^{-3\mathrm{/}2}J(\delta )\mathrm{.}$ This is finite and positive on the stated branch.

Define $S^2=\{u\in {\mathbb{R}}^3:\mathrm{\mid }u\mathrm{\mid }=1\}$ and $p_{\ast }(r,E)=\sqrt{2\mu (E+A\mathrm{/}r)}$ for $r_0\le r\le A\mathrm{/}e$. Let $d\sigma$ be the angular area measure of total mass $4\pi$ from Radial integration over a bounded Euclidean shell. Push the finite measure $(\mu p_{\ast }\mathrm{/}{h}_{\ast }^3)d^{3}qd\sigma (u)$ on $\{r_0\le \mathrm{\mid }q\mathrm{\mid }\le A\mathrm{/}e\}\times S^2$ forward by $(q,u)\mapsto (q,p_{\ast }(\mathrm{\mid }q\mathrm{\mid },E)u)$; call the resulting measure ${\mathrm{\Sigma }}_E$. Its candidate normalized shell law is $P_E={\mathrm{\Sigma }}_E\mathrm{/}{\mathrm{\Omega }}^{\mathrm{\prime }}(E)$. The separate result Finite regulated Kepler shell measure is normalized ↗ proves ${\mathrm{\Omega }}^{\mathrm{\prime }}(E)>0$, finite total mass, and $P_E(\{h=E\})=1$. Thus the energy-shell preparation is explicitly defined; it is not inferred from phase-volume counting alone.

The entropy convention attached to this model is Gibbs volume, $S_G(E)=k_B\log \mathrm{\Omega }(E)$, with $1\mathrm{/}{T}_G=d{S}_G\mathrm{/}dE$ and $C_G=dE\mathrm{/}d{T}_G$ where defined. Other finite-system entropy conventions are distinct.