Comet energy diffusion ($x=1/a$)

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 $s_x=10M_p/(a_p{M}_{\star })$ and passage interval $P(a_0)$,The quantity $s_x$ has inverse-length units, not the units of a variance-per-time diffusion coefficient. With $⟨(\Delta x{)}^2⟩=2D_{x}t$, 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.

At fixed stellar mass $M_{\star }$, planetary mass density ${\rho }_p$, age $t_{\star }$ and initial ratio $\kappa =a_0/a_p$, the order-unity Safronov number comparison and comet energy diffusion give boundariesTheir logarithmic slopes in planetary mass versus radius are $-3/2$ and $3/4$. Above both, strong kicks and a short estimated diffusion time favour ejection; below the escape boundary, repeated weak encounters favour collision statistically. Below the age boundary, the stated diffusion mechanism has not completed on its characteristic clock. Collision lifetimes require additional cross-section and encounter-rate information, so this map alone cannot establish retention or accretion within the age.