4.3 Collisional loss-cone replenishment

A binary black hole depletes its loss cone very quickly since stars within the loss cone need only a few close encounters with the binary to be ejected. Whether the binary can continue to exchange energy with stars depends on the efficiency with which stars are re-supplied to the loss cone.

The most commonly invoked mechanism for loss cone re-filling is two-body scattering of stars. A small angular momentum perturbation, for instance from a passing star, can deflect a star with $L >~~ Llc$ into the loss cone. This process has been studied in detail in the context of scattering of stars into the tidal disruption sphere of a single black hole [56 , 119 , 27 ]. The basic equations are similar in the case of scattering into a binary SBH, except that the critical angular momentum increases by a factor $V~ ----- ~ a/rt$ , where $rt$ is the tidal disruption radius. (Other differences are discussed below.) If the binary parameters are assumed fixed, a steady-state flow of stars into the loss cone will be achieved on roughly a two-body relaxation time scale $TR$ , and the distribution function near $Llc$ will have the form

where $R$ is a scaled angular momentum variable, $R =_ L2/L2c(E)$ , $Lc(E)$ is the angular momentum of a circular orbit of energy $E$ , and $-- f$ is the distribution function far from the loss cone, assumed to be isotropic. The mass flow into the central object is $integral m* F(E)dE$ , where

The quantity in brackets is the orbit-averaged diffusion coefficient in $R$ .

A crude estimate of the collisional re-supply rate is given by

[56 ], where $M*(r)$ is the mass in stars within radius $r$ and $rcrit$ is the critical radius at which stars scatter into the loss cone in a single orbital period; beyond $rcrit$ , the diffusion rate drops rapidly with radius. Estimates based on simple galaxy models give $r ~~ 10- 100 a crit$ . To get an idea of the scattering rate, we consider the nucleus of the Milky Way. Equation (4

) gives $-1 ah ~~ 0.32(1 + q) pc$ . Assuming $a = ah$ , $q = 0.1$ and $rcrit ~~ 30 ah$ gives $rcrit ~~ 1.1pc$ . The mass within this radius is $~ 3× 106Mo .$ [66

] and the relaxation time at this radius, assuming stars of a solar mass, is $~ 2× 109yr$ . The scattered mass over $1010 yr$ is then $~ 107Mo .$ . This is comparable to the mass of the Milky Way SMBH, $M ~~ 3 × 106M o .$ [195 ], but the scattering rate would drop as the binary shrinks, suggesting that scattered stars would contribute only modestly to refilling of the loss cone. In more massive galaxies, the nuclear density is lower, relaxation times are longer, and collisional refilling would be even less important. A more detailed calculation of the collisional refilling rates in real galaxies [228

] concludes that few if any binary SBHs could reach coalescence via this mechanism. The author took the presently observed luminosity profiles of galaxies as initial conditions. The stellar density profiles must have been steeper before the binary SBH formed [150

], leading to substantially more rapid decay early in the life of the binary.

Another criticism of standard loss cone theory is its assumption of a quasi-steady-state distribution of stars in phase space near $Llc$ [152 ]. This assumption is appropriate at the center of a globular cluster, where relaxation times are much shorter than the age of the universe, but is less appropriate for a galactic nucleus, where relaxation times almost always greatly exceed a Hubble time [44 ]. (The exceptions are the nuclei of small dense systems like the bulge of the Milky Way.) The distribution function $f (E, L)$ immediately following the formation of a hard binary is approximately a step function,

much steeper than the $~ ln L$ dependence in a collisonally relaxed nucleus (Equation 31

). Since the transport rate in phase space is proportional to the gradient of $f$ with respect to $L$ , steep gradients imply an enhanced flux into the loss cone. Figure 6

shows the evolution of $N (E, R)$ at a single $E$ assuming that the loss cone is empty initially within some $R =_ L /L (E) lc lc c$ and that $N (E, R, t = 0)$ is a constant function of $R$ outside of $Rlc$ . (The loss cone boundary is assumed static; in reality it would shrink with the binary.) Also shown is the collisionally-relaxed solution of Equation (31

). The phase-space gradients decay rapidly at first and then more gradually as they approach the steady-state solution. The total mass consumed by the binary, shown in the lower panel of Figure 6

, is substantially greater than would be computed from the steady-state theory, implying greater cusp destruction and more rapid decay of the binary. This time-dependent loss cone refilling might be particularly effective in a nucleus that continues to experience mergers or accretion events, in such a way that the loss cone repeatedly returns to an unrelaxed state with its associated steep gradients.

Figure 6:

(a) Slices of the density $N (E, R,t)$ at one, arbitrary $E$ , recorded, from left to right, at $100$ , $101$ , $102$ , $103$ , and $104Myr$ (solid curve). Initially, $N (E,R, t) = 0$ for $R < Rlc$ and $N (E, R,t) = const$ for $R > Rlc$ . We also show the equilibrium solution of Equation (31

) (dot-dashed curve). (b) The total number of stars consumed by the loss cone as a function of time (solid curve). The scale has been set to galaxy M32 with initial separation between the MBHs of $0.1pc$ . (From [152

])

There are other differences between loss cones around single and binary SBHs. A star that interacts with a massive binary generally remains inside the galaxy and is available for further interactions. In principle, a single star can interact many times with the binary before being ejected from the galaxy or falling outside the loss cone; each interaction takes additional energy from the binary and hastens its decay. Consider a simple model in which a group of $N$ stars in a spherical galaxy interact with the binary and receive a mean energy increment of $<DE >$ . Let the original energy of the stars be $E0$ . Averaged over a single orbital period $P (E)$ , the binary hardens at a rate

In subsequent interactions, the number of stars that remain inside the loss cone scales as $L2lc oc a$ while the ejection energy scales as $~ a-1$ . Hence $N <DE > oc a1a-1 oc a0$ . Assuming the singular isothermal sphere potential for the galaxy, one finds

[152

]. Hence the binary’s binding energy increases as the logarithm of the time, even after all the stars in the loss cone have interacted at least once with the binary. Re-ejection would occur differently in nonspherical galaxies where angular momentum is not conserved and ejected stars could miss the binary on their second passage. However there will generally exist a subset of orbits defined by a maximum pericenter distance $<~ a$ and stars scattered onto such orbits can continue to interact with the binary.

As these arguments suggest, the long-term evolution of a binary SBH due to interactions with stars may be very different in different environments. (We stress that the presence of gas may substantially alter this picture; cf. Section 8.) There are three characteristic regimes [152 ].

Collisional. The relaxation time $TR$ is shorter than the lifetime of the system and the phase-space gradients at the edge of the loss cone are given by steady-state solutions to the Fokker-Planck equation. The densest galactic nuclei may be in this regime. Resupply of the loss cone takes place on the time scale associated with scattering of stars onto eccentric orbits. The decay time of a binary SBH scales as $|a/a |~ m-* 1~ N$ with $N$ the number of stars. In the densest galactic nuclei, collisional loss cone refilling may just be able to drive a binary SBH to coalescence in a Hubble time. For sufficiently small $T R$ , scattering refills the loss cone in less than an orbital period (“full loss cone”) and the decay follows $-1 a ~ t$ . $N$ -body studies are typically in this regime, as discussed below.
Collisionless. The relaxation time is longer than the system lifetime and gravitational encounters between stars can be ignored. The low-density nuclei of bright elliptical galaxies are in this regime. The binary SBH quickly interacts with stars whose pericenters lie within its sphere of influence; in a low-density (spherical or axisymmetric) nucleus, the associated mass is less than that of the binary and the decay tends to stall at a separation too large for gravitational wave emission to be effective. However evolution can continue due to re-ejection of stars that lie within the binary’s loss cone but have not yet escaped from the system. In the spherical geometry, re-ejection implies $|a/a| ~ (1 + t/t0)/a$ , leading to a logarithmic dependence of binary hardness on time. Re-ejection in galactic nuclei may contribute a factor of $~$ a few to the change in $a$ over a Hubble time.
Intermediate. The relaxation time is of order the age of the system or somewhat longer. While gravitational encounters contribute to the re-population of the loss cone, not enough time has elapsed for the phase space distribution to have reached a collisional steady state. Most galactic nuclei are probably in this regime. The flux of stars into the loss cone can be substantially higher than predicted by the steady-state theory, due to strong gradients in the phase space density near the loss cone boundary produced when the binary SBH initially formed. This transitory enhancement would be most important in a nucleus that continues to experience mergers or infall, in such a way that the loss cone repeatedly returns to an unrelaxed state with its associated steep gradients.

Table 2 summarizes the different regimes. The evolution of a real binary SBH may reflect a combination of these and other mechanisms, such as interaction with gas. There is a close parallel between the final parsec problem and the problem of quasar fueling: Both require that of order $108Mo.$ be supplied to the inner parsec of a galaxy in a time shorter than the age of the universe. Nature clearly accomplishes this in the case of quasars, probably through gas flows driven by torques from stellar bars. The same inflow of gas could contribute to the decay of a binary SBH in a number of ways: by leading to the renewed formation of stars which subsequently interact with the binary; by inducing torques which extract angular momentum from the binary; through accretion, increasing the masses of one or both of the SBHs and reducing their separation; etc.