Keplerian cluster radius evolution under stellar capture ($\frac{\dot{R}}{R}\simeq (\xi -1)\frac{\Gamma }{N}-\frac{m\Gamma }{M_H}$)

For a homologous black hole-dominated cluster, write $E=-bG{M}_{H}Nm/(2R)$, and let a captured star's specific orbital energy be ${ϵ}_{\mathrm{cap}}=\xi E/(Nm)$. Removing negative orbital energy raises the remaining cluster energy, while growth ${\dot{M}}_H=m\Gamma$ deepens its potential. The energy budget gives the displayed radius response. Preferentially removing strongly bound stars can drive expansion; representative-energy removal instead gives the adiabatic contraction due to black-hole growth. The sign is not fixed by the loss cone angle alone.