= Solution
Equating stellar surface gravity to the differential black-hole acceleration gives
$$
\frac{GM_*}{R_*^2}
\simeq\frac{2GM_{\rm BH}R_*}{r_t^3},
$$
so the <tidal disruption radius> is
$$
\boxed{r_t=R_*\left(\frac{2M_{\rm BH}}{M_*}\right)^{1/3}}.
$$
A nonrotating hole swallows the star without a visible disruption when this lies inside its capture scale, here approximated by the Schwarzschild radius $r_h=2GM_{\rm BH}/c^2$. Equating the two radii yields the <Hills mass>
$$
\boxed{M_{\rm BH,crit}
=\frac{c^3}{2}M_*^{-1/2}
\left(\frac{R_*}{G}\right)^{3/2}}.
$$
For a solar-type star this is of order $10^8M_\odot$.
The threshold does depend on spin. A <Kerr black hole> has spin- and inclination-dependent horizon, marginally bound, and capture radii. Prograde orbits around a rapidly rotating hole can approach more closely, allowing disruption by masses above the Schwarzschild Hills mass, whereas retrograde capture occurs farther out.
An <intermediate-mass black hole> lies well below this threshold for ordinary stars, so stars entering its loss cone are disrupted outside the horizon. The returning debris can grow the hole and produces a <tidal disruption event> that may reveal an otherwise quiescent cluster black hole through a flare. Dense clusters can supply repeated disruptions, although the rate depends on two-body relaxation, stellar collisions, binary interactions, and whether gravitational recoil or cluster dynamics ejects the hole.
Solved by gpt-5.6-sol high.
Back to article page
