Tidal Radius, Hills Mass, and Direct Capture

A close stellar encounter has three qualitatively different outcomes: the star can lose an envelope, be fully disrupted, or cross the relativistic capture boundary before producing escaping debris. The useful organizing comparison is therefore not “tidal radius versus horizon.” It is stellar response versus the spin-, inclination-, and orbit-dependent capture separatrix. This page derives the scale competition and identifies where the Newtonian estimate stops deciding the outcome. Generic stellar dynamics and exact geodesic derivations are outside scope.

Required background. Geodesics and Jacobi Fields supplies the geodesic-deviation and tidal-tensor language used to compare disruption with capture.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon and capture concepts; Relativistic Matter and Compact Objects supplies stellar structure and self-gravity; and Black-Hole Astrophysics supplies the source environment and transient interpretation.

From the tidal tensor to a disruption scale

Let the black hole have physical mass $M_{\rm BH}$ and the star have mass $M_*$ and radius $R_*$. Across a star whose size is small compared with its orbital radius $r$, the Newtonian potential $\Phi=-GM_{\rm BH}/r$ produces the tidal tensor

$$C_{ij}=-\partial_i\partial_j\Phi =\frac{GM_{\rm BH}}{r^3} \left(-\delta_{ij}+3n_i n_j\right).$$

Its radial stretching eigenvalue is $2GM_{\rm BH}/r^3$; order-unity conventions differ according to whether one balances that maximum eigenvalue, a mean force, or an actual stellar response. Equating the characteristic tidal acceleration $GM_{\rm BH}R_*/r^3$ with surface self-gravity $GM_*/R_*^2$ gives

$$r_t=R_*\left(\frac{M_{\rm BH}}{M_*}\right)^{1/3}.$$

The cancellation of $G$ is a useful check: the estimate compares two accelerations governed by the same gravitational coupling. It is also a mean-density statement, since $r_t^3/M_{\rm BH}=R_*^3/M_*$. Rossi and collaborators derive the estimate from the tidal tensor and then replace it by a curvature tensor projected on a parallel-propagated tetrad when the encounter is relativistic (Rossi et al. 2021, §2, Eqs. 1–9).

Define the penetration factor

$$\beta\equiv\frac{r_t}{r_p},$$

where $r_p$ is the center-of-mass pericenter in the stated orbital model. A value $\beta=1$ is not a universal full-disruption threshold. A centrally concentrated star can retain a core at larger $\beta$ than a diffuse polytrope, while stellar spin, prior evolution, encounter duration, and relativistic compression change the response. Hydrodynamic calculations therefore supply structure-dependent partial- and full-disruption thresholds $\beta_{\rm p}$ and $\beta_{\rm d}$ rather than a single sharp number.

Capture must be tested before observability

For a parabolic Schwarzschild orbit, the marginally bound circular orbit lies at the Boyer–Lindquist radius $r_{\rm mb}=4GM_{\rm BH}$ in the site units. An orbit below the corresponding angular-momentum separatrix plunges; the horizon radius $2GM_{\rm BH}$ is too late a comparison. In Kerr spacetime, both the capture separatrix and the tidal eigenvalues depend on $a_*=J/(GM_{\rm BH}^2)$, orbital inclination, energy, axial angular momentum, and Carter constant.

The distinction can be checked without a stellar model. In Schwarzschild spacetime, a timelike geodesic arriving marginally bound from infinity has specific energy $\mathcal E=1$ and specific angular momentum $\ell$. Its radial equation is

$$\left(\frac{dr}{d\tau}\right)^2 =1-\left(1-\frac{2GM_{\rm BH}}{r}\right) \left(1+\frac{\ell^2}{r^2}\right).$$

A turning point therefore obeys $\ell^2=2GM_{\rm BH}r^2/(r-2GM_{\rm BH})$. The minimum of this expression occurs at $r=4GM_{\rm BH}$, giving $\ell_{\rm cap}=4GM_{\rm BH}$. Below that angular momentum there is no outer turning point: a plunging orbit has no actual pericenter outside the horizon. At equality the ideal geodesic approaches the unstable circular orbit asymptotically. Thus the commonly drawn “capture radius” is shorthand for a boundary in orbital constants, not a material surface that every trajectory crosses at its closest approach.

The invariant procedure is:

1. specify the incoming constants of motion in the black-hole rest frame;
2. determine whether the geodesic crosses the capture separatrix;
3. along a noncaptured orbit, project curvature onto the star’s freely falling tetrad;
4. evolve a stellar model under those tides and classify surviving core, full disruption, and escaping debris.

Kesden treats geodesic capture and the relativistic tidal tensor for generically oriented Kerr orbits, comparing instantaneous tidal stretching with the star’s Newtonian self-gravity. This is a disruption criterion, not a hydrodynamic evolution of the star in step 4. The calculation shows that direct capture becomes a major rate correction above $M_{\rm BH}\sim10^7M_\odot$ (Kesden 2012, §§II–IV, Eqs. 10–11 and Figs. 3–4). A coordinate pericenter can still label a calculation, but it is not itself the physical disruption criterion.

This is also the seam with finite-size orbital theory. The worldline-breakdown treatment asks when a controlled multipole expansion ceases to approximate an extended body. Here the additional question is astrophysical: whether the resulting matter is captured, remains in a core, or returns as observable debris.

The Hills mass is a scale, not a wall

Write a schematic capture radius as

$$r_{\rm cap}=\kappa(a_*,\iota,\mathcal E)\,GM_{\rm BH},$$

where $\kappa=4$ for the marginally bound parabolic Schwarzschild comparison and $\iota$ is orbital inclination. Setting $r_t=r_{\rm cap}$ gives

$$M_{\rm H} =\frac{1}{\kappa^{3/2}G^{3/2}} \left(\frac{R_*^3}{M_*}\right)^{1/2}.$$

Temporarily restoring $c$ multiplies the numerator by $c^3$. For a solar-type star and $\kappa=4$, this estimate is about $4\times10^7M_\odot$; relativistic tidal response and the precise observable-debris criterion move the coefficient by factors of a few. The scaling is more trustworthy:

$$M_{\rm H}\propto \bar\rho_*^{-1/2}\kappa^{-3/2}.$$

A puffier giant can be disrupted by a more massive black hole, while a compact white dwarf requires a much smaller one. Favorable prograde Kerr orbits reduce $\kappa$ and raise the limiting mass; retrograde or inclined orbits generally do less well. Fully relativistic calculations extend solar-type disruptions toward several $10^8M_\odot$ for nearly maximal favorable spin, but the existence of a disruption is not yet the existence of a detectable flare (Rossi et al. 2021, §5, Eq. 33 and the discussion following it).

Outcome map and limitations

The scale comparison can be summarized without pretending that its boundaries are exact.

Ordering or response	Physical outcome	What still must be modeled
$r_p>r_t$	weak tides or tidal excitation	repeated passages, mode damping, and orbital diffusion
$r_{\rm cap}<r_p\lesssim r_t$ with a surviving core	partial stripping	stellar profile, spin, kick, and subsequent orbit
$r_{\rm cap}<r_p\lesssim r_t$ with no core	full disruption	debris energy distribution and hydrodynamics
Incoming orbital constants lie on the plunging side of the capture separatrix	direct plunge or prompt capture; an exterior turning-point $r_p$ need not exist	Kerr separatrix and whether any debris escapes before capture

The Newtonian calculation assumes $R_*/r_p\ll1$, a nearly parabolic center-of-mass orbit, and a point-mass black-hole field. It cannot determine the order-unity disruption threshold, relativistic compression, nuclear burning in a deep encounter, or the mass of an escaping debris tail. In particular, comparing $r_t$ only with the horizon both overcounts observable events and erases the strong spin-orientation dependence near the Hills scale.

Scale-choice diagnostic

A solar-type star approaches a nonspinning $10^8M_\odot$ black hole on a nearly parabolic orbit. Use the Newtonian $r_t$ and compare it first with the horizon $2GM_{\rm BH}$ and then with the marginally bound capture scale $4GM_{\rm BH}$. Which comparison is physically relevant, and what may be concluded?

Solution

For a solar-type star,

$$\frac{r_t}{GM_{\rm BH}} =\frac{R_*}{GM_*} \left(\frac{M_*}{M_{\rm BH}}\right)^{2/3} \approx2.2.$$

Thus $r_t$ lies slightly outside the Schwarzschild horizon but well inside the $4GM_{\rm BH}$ marginally bound capture scale. The horizon-only comparison would incorrectly advertise room for disruption. The capture comparison says that a parabolic Schwarzschild orbit reaching the Newtonian tidal scale plunges first, so it does not yield the standard returning-debris flare. Because the estimate uses a Newtonian tide in a strong field, it establishes the correct regime choice rather than a percent-level boundary; a Kerr tidal-tensor calculation is required for spin or inclined orbits.

References

• Kesden, M. “Tidal-Disruption Rate of Stars by Spinning Supermassive Black Holes.” Physical Review D 85, 024037 (2012). DOI.
• Rossi, E. M., Stone, N. C., Law-Smith, J. A. P., et al. “The Process of Stellar Tidal Disruption by Supermassive Black Holes: The First Pericenter Passage.” Space Science Reviews 217, 40 (2021). DOI.