= Comet energy diffusion
{title2=$x=1/a$}
Model successive uncorrelated planetary kicks as a <random walk> in inverse <semi-major axis> $x=1/a$, proportional to negative <specific orbital energy>. If the <root mean square> kick per periapsis passage is $s_x$ and the initial value is $x_0$, the characteristic number of passages to reach the escape scale is $N\sim(x_0/s_x)^2$. For $s_x=10M_p/(a_pM_\star)$ and passage interval $P(a_0)$,
$$
t_{\rm ej}\sim\frac{P(a_0)}{100}\left(\frac{a_p}{a_0}\right)^2
\left(\frac{M_\star}{M_p}\right)^2.
$$
The quantity $s_x$ has inverse-length units, not the units of a variance-per-time <diffusion coefficient>. With $\langle(\Delta x)^2\rangle=2D_xt$, that coefficient is $D_x=s_x^2/(2P)$. This estimate assumes many independent encounters; changing <orbital periods> and resonant correlations can invalidate the clock. It is not an exact mean <first-passage time>.
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