The radial velocity of a <Kepler orbit> follows from the <vis-viva equation> after subtracting the tangential component:
$$
v_r^2=\frac{\mu_\star}{a}\frac{(r-q)(Q-r)}{r^2},\qquad
T_c=\frac{2\pi a^{3/2}}{\sqrt{\mu_\star}},\qquad a=\frac{Q+q}{2}.
$$
Each radial interval is crossed twice per <orbital period>, so its fraction of time is
$$
\boxed{\frac{dt_{\rm both}}{T_c}=p_r(r)\,dr=\frac{r\,dr}{\pi a\sqrt{(r-q)(Q-r)}}.}
$$
This is the <phase-mixed radial probability of a Kepler orbit>, normalized to one between $q$ and $Q$. Dividing by the annular area $2\pi r\,dr$ gives the <phase-mixed comet surface density> per <comet>:
$$
\boxed{\Sigma_c(r)=\frac{1}{\pi^2(Q+q)\sqrt{(r-q)(Q-r)}}.}
$$
For $Q\gg a_p>q$,
$$
\boxed{\Sigma_c(a_p)\simeq\frac{1}{\pi^2Q^{3/2}a_p^{1/2}}\left(1-\frac q{a_p}\right)^{-1/2}.}
$$
\b[The printed density has the powers of $Q$ and $a_p$ interchanged.] Its expression is too large by $Q/a_p$ in this limit. The normalized residence-time derivation fixes the corrected expression, which is also required for the later $Q^{1/2}$ ejection-time scaling. For a population of $N_c$ independent identical <comets>, multiply $\Sigma_c$ by $N_c$ to obtain a number <surface density of a disk>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 316 / 1 / iii / Solution

{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>.

Phase-mixed comet surface density

{title2=$p_r(r)$}

For a <Kepler orbit> with <pericentre distance> $q$, <apocentre distance> $Q$, and <semi-major axis> $a=(q+Q)/2$, a uniform <mean anomaly> gives
$$
p_r(r)=\frac{r}{\pi a\sqrt{(r-q)(Q-r)}},\qquad q<r<Q.
$$
The two radial passages per <orbital period> are included, and $\int_q^Qp_r(r)\,dr=1$.

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