Loss Cones and Capture Rates

A black hole consumes or transforms stars only after dynamics places them on sufficiently low-angular-momentum orbits. At fixed orbital energy, those orbits occupy a small region of phase space called the loss cone. The capture or disruption rate is therefore a boundary-flux problem: one must compare angular-momentum diffusion during an orbit with the angular size of the absorbing region. Slow refilling produces an empty, logarithmically depleted cone; rapid refilling produces an almost isotropic, full cone. Quoting a stellar density without this phase-space calculation does not determine a rate.

Required background. Classical Field Theory supplies the distribution-function continuity equation and absorbing-boundary flux used in the derivation.

Helpful background. Black-Hole Geometry and Mechanics identifies the horizon and relativistic capture boundary; Relativistic Matter and Compact Objects supplies stellar radii and compact-remnant properties; Black-Hole Astrophysics places the loss-cone supply in its transient and population context; and Relativistic Two-Body Dynamics clarifies the conservative and dissipative orbital regimes near capture.

Angular momentum sets the absorbing boundary

Let $\mathcal E$ denote positive binding energy per unit mass, $J$ specific angular momentum, and $J_c(\mathcal E)$ the specific angular momentum of the circular orbit at that energy. It is convenient to use

$$\mathcal R=\frac{J^2}{J_c^2(\mathcal E)}, \qquad 0\le\mathcal R\le1.$$

For a nearly parabolic orbit whose destructive periapse is $r_d$, Newtonian orbital geometry gives

$$J_{\rm lc}^2\simeq2GM_\bullet r_d, \qquad \mathcal R_{\rm lc}(\mathcal E) =\frac{J_{\rm lc}^2}{J_c^2(\mathcal E)}.$$

For a Kepler ellipse with semimajor axis $a\gg r_d$, $J_c^2=GM_\bullet a$ and $\mathcal R_{\rm lc}\simeq2r_d/a$. A main-sequence tidal-disruption boundary has the approximate scale $r_t\simeq R_\star(M_\bullet/m_\star)^{1/3}$. Direct capture instead uses the relativistic separatrix, which depends on black-hole spin and orbital orientation. A luminous tidal disruption is possible only when disruption occurs outside capture; the Newtonian $r_t$ is not a universal absorbing boundary.

Why a cone? At a fixed spatial radius $r$, a velocity making angle $\vartheta$ with the inward radial direction has $J=rv\sin\vartheta$. Capture requires its transverse component to be small enough that $J\lt J_{\rm lc}$, selecting a narrow cone of incoming velocity directions. The corresponding outgoing directions can occur before the next destructive periapse; the star is not removed merely because a diffusion step momentarily crosses the threshold at large radius. In the spherical Kepler limit, isotropy gives a distribution uniform in $\mathcal R$ at fixed energy, so the fraction inside a full cone is $\mathcal R_{\rm lc}$.

The phase-space geometry and its Newtonian boundary are derived in Stone et al. 2020, §3.1, pp. 5–7. Alexander gives the corresponding periapse and angular-momentum construction in Alexander 2017, §3.1, author manuscript pp. 18–20.

Diffusion at fixed orbital energy

Energy changes by order unity on a relaxation time, but a nearly radial orbit needs only a small change in $J$ to cross the boundary. Treat energy as fixed and let $N(\mathcal E,\mathcal R,t)\,d\mathcal E\,d\mathcal R$ be the number of stars in an energy–angular-momentum element. In the spherical, nearly Keplerian diffusion baseline, the equation near the cone can be written

$$\frac{\partial N}{\partial t}=\frac{\partial F}{\partial\mathcal R}, \qquad F=D(\mathcal E,\mathcal R) \frac{\partial N}{\partial\mathcal R}, \qquad D(\mathcal E,\mathcal R)\simeq D(\mathcal E)\mathcal R.$$

Here positive $F$ is defined toward decreasing $\mathcal R$, into the absorbing boundary. This form uses the drift–diffusion relation appropriate to an isotropic equilibrium that is constant in $\mathcal R$; it is not a generic Fokker–Planck equation for any choice of distribution or measure. The dimensionless refilling parameter is

$$q(\mathcal E)= \frac{D(\mathcal E)P(\mathcal E)}{\mathcal R_{\rm lc}(\mathcal E)},$$

where $P$ is the radial orbital period. Thus $q$ compares the angular-momentum diffusion accumulated per orbit with the squared angular size of the loss cone.

For sufficiently rapid refilling, the distribution is nearly isotropic and the geometric full-cone flux is

$$F_{\rm full}(\mathcal E) \simeq\frac{\mathcal N(\mathcal E)\mathcal R_{\rm lc}(\mathcal E)}{P(\mathcal E)},$$

where $\mathcal N(\mathcal E)$ is the total number per unit binding energy. Each radial period removes approximately the fraction $\mathcal R_{\rm lc}$; this explains both the period in the denominator and why increasing diffusion indefinitely cannot increase the rate past this supply limit.

For $q\ll1$, a star that enters is removed before diffusion replaces it. Assume a maintained reservoir at $\mathcal R=1$, an absorbing boundary $N(\mathcal E,\mathcal R_{\rm lc})=0$, and a steady inward flux. Since $F=D(\mathcal E)\mathcal R\,\partial_{\mathcal R}N$ is constant in $\mathcal R$,

$$N(\mathcal E,\mathcal R) =\frac{F(\mathcal E)}{D(\mathcal E)} \ln\frac{\mathcal R}{\mathcal R_{\rm lc}}.$$

Integrating this profile from $\mathcal R_{\rm lc}$ to $1$ and identifying its integral with $\mathcal N(\mathcal E)$ gives

$$F_{\rm empty}(\mathcal E) \simeq \frac{D(\mathcal E)\mathcal N(\mathcal E)} {\ln[1/\mathcal R_{\rm lc}(\mathcal E)] +\mathcal R_{\rm lc}(\mathcal E)-1}.$$

This limit depends linearly on the diffusion rate but only logarithmically on cone size. By contrast, the full flux is linear in cone size and independent of $D$. Their asymptotic ratio is approximately $q/\ln(1/\mathcal R_{\rm lc})$. The distinction is useful: $q\sim1$ marks a diffusion step comparable to the cone width, while flux saturation near the full-cone rate requires $q$ to exceed a logarithmic scale, with practical matching near $q\sim\ln(1/\mathcal R_{\rm lc})$. At intermediate $q$, use a boundary condition accounting for removal at periapse instead of imposing a perfectly absorbing boundary throughout the orbit. The boundary condition, interpolation, and integrated rate are presented in Stone et al. 2020, §3.2, pp. 7–11.

The total event rate is the energy integral

$$\dot N=\int F(\mathcal E)\,d\mathcal E.$$

One may not decide the global regime from $q$ at a single radius. Different energies can lie on opposite sides of the transition, and the dominant contribution often comes from a critical region where refilling and draining compete.

Capture, disruption, plunge, and inspiral

The absorbing condition represents different physics for different objects. A normal star may be tidally disrupted; a compact remnant can plunge; a binary can be separated without either component being destroyed. An extreme-mass-ratio inspiral is not an ordinary prompt loss-cone event. It must remain on a dissipative trajectory for many orbits, winning a race between gradual orbital decay and perturbations that scatter it away or push it into direct plunge.

Relativistic precession also changes the torque coherence that feeds small $J$. It can quench resonant relaxation before the formal Newtonian cone is reached. The capture boundary itself is a spin- and inclination-dependent separatrix rather than $J_{\rm lc}^2=2GM_\bullet r_d$. These effects are organized in Alexander 2017, §§3.1.2–3.2.3, author manuscript pp. 19–24 and in Stone et al. 2020, §3.5, pp. 15–17.

This page derives a steady spherical diffusive baseline. It does not transfer generic galactic dynamics into a capture calculation. A bar or merger matters only after its non-spherical torques and time dependence have been represented in the nuclear distribution and boundary flux.

Limitations and adversarial checks

The first adversarial check replaces spherical diffusion with a flattened or triaxial potential. Centro-philic orbit families can refill low angular momentum collisionlessly, while their finite reservoir can drain. Simply multiplying the spherical rate by a constant hides both the changed phase-space geometry and the time dependence. Section-by-section treatments of anisotropic, time-dependent, and aspherical refilling are given in Stone et al. 2020, §§3.3–3.4, pp. 11–15.

The second check perturbs the diffusion model itself. Massive perturbers, binaries, rare strong encounters, and resonant torques need not produce small uncorrelated steps. Energy diffusion can no longer be frozen if its timescale approaches the supply time, and an imposed steady outer distribution can overfeed a nucleus whose reservoir has not relaxed. Compare a time-dependent two-dimensional calculation in $(\mathcal E,\mathcal R)$ with the one-dimensional steady result before attaching a precise rate.

Useful internal checks are dimensional and limiting. $D$ has units of inverse time, so both flux formulas have units of number per unit specific binding energy per time. Sending $D$ to zero must extinguish the empty-cone flux, while sending $D$ upward cannot make the rate exceed the geometric full-cone supply at fixed $\mathcal N$, $P$, and $\mathcal R_{\rm lc}$. The matched rate should remain finite and continuous across the transition.

Exercises

At one energy, let $\mathcal R_{\rm lc}=10^{-5}$, $P=10^3\,\mathrm{yr}$, and $D=10^{-8}\,\mathrm{yr}^{-1}$. Compute $q$ and use the empty-limit formula as an order-of-magnitude estimate of $F/F_{\rm full}$, identifying its limitation. Does $q=1$ imply a full cone? What changes if $\mathcal R_{\rm lc}$ is reduced by a factor of $100$ at fixed $P$, $D$, and $\mathcal N$?

Solution

Initially $q=DP/\mathcal R_{\rm lc}=1$. Inserting this into the empty-limit formula gives $F/F_{\rm full}\simeq1/[\ln(10^5)-1+10^{-5}]=0.095$; retaining only the leading logarithm gives $0.087$. Both indicate a depleted flux of order one tenth of the full value. Neither is a precision result at $q=1$, because the perfectly absorbing empty-limit boundary requires $q\ll1$.

After shrinking the boundary, $\mathcal R_{\rm lc}=10^{-7}$ and $q=100$, whereas $\ln(10^7)\simeq16.1$. Extending the empty-cone formula would exceed the full rate, signalling that it has left its domain; a matched solution approaches the full-cone supply. That new full-cone rate is itself $100$ times smaller because it is proportional to $\mathcal R_{\rm lc}$. Becoming fuller does not mean that the smaller target consumes more stars per unit time.

References

• Alexander, Tal. 2017. “Stellar Dynamics and Stellar Phenomena Near a Massive Black Hole.” Annual Review of Astronomy and Astrophysics 55: 17–57. DOI.
• Stone, Nicholas C., Eugene Vasiliev, Michael Kesden, Elena M. Rossi, Hagai B. Perets, and Pau Amaro-Seoane. 2020. “Rates of Stellar Tidal Disruption.” Space Science Reviews 216: 35. DOI.