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The sign of cluster evolution depends on the orbital energy of the captured stars, not just on their small angular momentum. Let the instantaneous capture rate be $\Gamma >0$, so the remaining stellar mass $M_s=Nm$ has ${\dot{M}}_s=-m\Gamma$, while ${\dot{M}}_H\simeq m\Gamma$. A captured star has negative specific orbital energy ${ϵ}_{\mathrm{cap}}$ measured outside the absorbing region. Removing it increases the energy of the surviving stars by $-m{ϵ}_{\mathrm{cap}}$; growth of the central black hole instead deepens their potential.

To turn this balance into a radius estimate, assume the surviving cluster evolves slowly through homologous virial equilibria of scale $R$. Write its stellar self-energy and black-hole interaction energy aswhere positive $a,b$ depend on the density shape. The virial theorem givesAt the instant of capture, the stellar-system energy budget, excluding feedback and energy carried by other escaping stars, isThe second term is the work of the changing central potential; it must not be confused with the change in the number of stars. Differentiating the virial expression and using ${\dot{M}}_H=m\Gamma$ givesThis estimates the radius response of a general homologous cluster once its structure and capture energies are specified.

For a simpler locally Keplerian subsystem, neglect its stellar self-gravity, so $E=-bG{M}_H{M}_s/(2R)$. Put ${ϵ}_{\mathrm{cap}}=\xi E/M_s$, with $\xi >0$ measuring how tightly bound a captured star is relative to the average orbital energy. The preceding balance reduces toThis is Keplerian cluster radius evolution under stellar capture. Removing representative-energy stars, $\xi =1$, leaves unchanged the characteristic orbits if $M_H$ is fixed; with black-hole growth it contracts them at rate $\dot{R}/R\simeq -{\dot{M}}_H/M_H$. Preferential removal of more tightly bound stars drives expansion if $\xi -1>M_s/M_H$. In that heating-dominated case,so inserting the empty-cone flux gives $t_R\sim t_{\mathrm{rel}}\log (1/{\mathcal{R}}_{\mathrm{lc}})/|\xi -1|$ for the one-zone estimate. Growth-dominated contraction instead has timescale $M_H/(m\Gamma )$.

For a Kepler orbit of semimajor axis $a_{\mathrm{cap}}$, ${ϵ}_{\mathrm{cap}}=-G{M}_H/(2a_{\mathrm{cap}})$ and $\xi =R/(b{a}_{\mathrm{cap}})$. Reaching $R_{\mathrm{sch}}$ does not imply that the orbit had binding energy of order $c^2$: a nearly parabolic star can reach that pericentre with almost zero orbital energy. Therefore expansion is expected when strongly bound central stars are preferentially swallowed, but the supplied information does not uniquely fix the sign or rate for the entire remaining cluster. Captured-energy selection, the density profile and any energy feedback must be specified. The balances above give both the conventional expansion estimate and the contraction case without assuming missing data.