Phase-mixed comet surface density
= Phase-mixed comet surface density
{title2=$\Sigma_c(r)$}
With uniformly distributed apsidal orientations, the <phase-mixed radial probability of a Kepler orbit> gives the surface probability per <comet>
$$
\Sigma_c(r)=\frac{1}{\pi^2(Q+q)\sqrt{(r-q)(Q-r)}}.
$$
For $Q\gg r>q$, this is $\pi^{-2}Q^{-3/2}r^{-1/2}(1-q/r)^{-1/2}$. Multiply by the population size for a number <surface density of a disk>.