4.1 Dynamics of a massive binary in a fixed stellar background

Stars passing within a distance $~ 3 a$ of the center of mass of a hard binary undergo a complex interaction with the two black holes, followed almost always by ejection at velocity $V~ ------- ~ m/m12Vbin$ , the “gravitational slingshot” [191

]. Each ejected star carries away energy and angular momentum, causing the semi-major axis, eccentricity, orientation, and center-of-mass velocity of the binary to change and the local density of stars to drop. If the stellar distribution is assumed fixed far from the binary and if the contribution to the potential from the stars is ignored, the rate at which these changes occur can be computed by carrying out scattering experiments of massless stars against a binary whose orbital elements remain fixed during each interaction [89

, 187 , 87

, 88

, 13 , 148

, 177

, 133

, 134

]. Figure 4

shows an example of field star velocity changes in a set of scattering experiments.

Figure 4:

Distribution of field star velocity changes for a set of scattering experiments in which the field star’s velocity at infinity relative to the binary was $vf0 = 0.5Vbin^ex$ . The binary’s mass ratio was 1:1, and the orientation of the binary’s orbital plane with respect to the $x$ -axis was varied randomly between the scattering experiments. Each plot represents $5× 104$ scattering experiments within some range of impact parameters $[p1,p2]$ in units of $a$ . (a) $[6,10]$ (b) $[2,4]$ (c) $[0.6,1]$ (d) $[0.4,0.6]$ . Solid lines in (a) and (b) are the distributions corresponding to scattering off a point-mass perturber. In (c) and (d), the mean of this distribution (which is very narrow) is indicated by the arrows. The gravitational slingshot is apparent in the rightward shift of the $dv$ values when $p$ is small, due to the randomization of ejection angles (from [133

]).

Consider an encounter of a single field star of mass $m*$ with the binary. Long before the encounter, the field star has velocity $v0$ with respect to the center of mass of the field star-binary system and its impact parameter is $p$ . A long time after the encounter, the velocity $v$ of the field star attains a constant value. Conservation of linear momentum implies that the change $dV$ in the velocity of the binary’s center of mass is given by

The velocity change results in a random walk of the binary’s center-of-mass momentum, as discussed in more detail below. The energy of the field star-binary system, expressed in terms of pre-encounter quantities, is

E = 1m v2 + 1-m V 2- Gm1m2--- 0 2 * 0 2 12 0 2a0 1 ( m ) Gm m = -m* 1 + ---* v20 - ----1--2 (12) 2 m12 2a0

with $V0 = - (m*/m12)v0$ the initial velocity of the binary’s center of mass and $a0$ the binary’s initial semi-major axis. After the encounter,

and $E = E 0$ , so that

( ) 2 2 ( ) d 1- = m*(v----v0)- 1 + -m*- a Gm1m2 m12 m*(v2 - v2) ~~ ---------0-. (14) Gm1m2

Averaged over a distribution of field-star velocities and directions, Equation (14

) gives the binary hardening rate $(d/dt)(1/a)$ [87

, 88 , 148

, 177

The angular momentum of the field star-binary system about its center of mass, expressed in terms of pre-encounter quantities, is

where $l0 =_ pv0$ and $lb0 =_ Lb0/m$ with $Lb$ the binary’s orbital angular momentum. Conservation of angular momentum during the encounter gives

( ) dlb = - m*- 1 + -m*- dl m m12 m*- ~~ - m dl. (16)

Changes in $|lb|$ correspond to changes in the binary’s orbital eccentricity $e$ via the relation $e2 = 1 - l2b/Gm12m2a$ [148

, 177

]. Changes in the direction of $lb$ correspond to changes in the orientation of the binary [134

The results of the scattering experiments can be summarized via a set of dimensionless coefficients $H, J,K, L, ...$ which define the mean rates of change of the parameters characterizing the binary and the stellar background. These coefficients are functions of the binary mass ratio, eccentricity and hardness but are typically independent of $a$ in the limit that the binary is very hard. The hardening rate of the binary is given by

with $r$ and $s$ the density and 1D velocity dispersion of stars at infinity. The mass ejection rate is

with $Mej$ the mass in stars that escape the binary. The rate of change of the binary’s orbital eccentricity is

The diffusion coefficient describing changes in the binary’s orientation is

with $m*$ the stellar mass. Brownian motion of the binary’s center of mass is determined by the coefficients $A$ and $C$ which characterize the Chandrasekhar diffusion coefficients at low $V$ :

The mean square velocity of the binary’s center of mass is $<V 2> = C/2A$ .

The binary hardening coefficient $H$ reaches a constant value of $~ 16$ in the limit $a « ah$ , with a weak dependence on $q$ [87 , 148 , 177 ]. In a fixed background, Equation (17 ) therefore implies that a hard binary hardens at a constant rate:

This is sometimes taken as the definition of a “hard binary”. The time to reach zero separation is

s 1 2s3 2q (1 + q)2 s3 t- th = ----- ---= ----2---= ------------2------ HGr ah HG rm H G rm(12 ) 5 2( s )3 r -1 ~~ 5.2× 10 yr q(1 + q) 200km--s-1- 103M---pc-3- ( ) -1 o . -m12--- × 108M . (23) o .

Orbital shrinkage would occur quite rapidly in the environment of a galactic nucleus if the properties of the stellar background remained fixed.

However if the binary manages to shrink to a separation at which $tgr <~ 1010yr$ , the changes it induces in its stellar surroundings will be considerable. The mass ejected by the binary in decaying from $ah$ to $agr$ is given by the integral of Equation (18 ):

Figure 5

shows $M ej$ as a function of the mass ratio $q$ for $s = 200km s-1$ and various values of $t gr$ . The mass ejected in reaching coalescence is of order $m12$ for equal-mass binaries, and several times $m2$ when $m2 « m1$ . A SBH that grew to its current size through a succession of mergers should therefore have displaced a few times its own mass in stars. If this mass came mostly from stars that were originally in the nucleus, the density within $rinfl$ would drop drastically and the hardening would stop. Without some way of replenishing the supply of stars (and in the absence of other mechanisms for extracting angular momentum from the binary, e.g. torques from gas clouds; cf. Section 8), decay would stall at a separation much greater than $agr$ .

Figure 5:

Mass ejected by a decaying binary, in units of $m12 = m1 + m2$ (solid lines) or $m2$ (dashed lines), calculated by an integration of Equation (24

), with the coefficient $J(a)$ taken from [177

]. Curves show mass that must be ejected in order for the binary to reach a separation where the emission of gravitational radiation causes coalescence on a time scale of $1010yr$ (lower), $109yr$ (middle) and $8 10 yr$ (upper).

In the case of extreme binary mass ratios, $m « m 2 1$ , the assumption that all stars passing a distance $~ a$ from the binary will be ejected is likely to be incorrect; many such stars will pass through the binary system without being appreciably perturbed by the smaller black hole. The concept of “ejection” in extreme binary mass ratios may be misleading; since the typical amount of energy transferred in stellar pericenter passages is small, most of the slingshot stars are not ejected from the nucleus and remain bound to the larger black hole. Figure 5

should thus be interpreted with caution in the regime $m2 « m1$ . The extreme-mass-ratio regime is poorly understood but deserves more study in view of the possibility that intermediate-mass black holes may exist having masses of $~ 103Mo.$ [43 ].

Changes in the binary’s orbital eccentricity (Equation 19 ) are potentially important because the gravity wave coalescence time drops rapidly as $e --> 1$ (Equation 8 ). For a hard binary, scattering experiments give $2 K(e) ~~ K0e(1 - e )$ , with $K0 ~~ 0.5$ for an equal-mass binary [148 , 177 ]. The dependence of $K$ on $m2/m1$ is not well understood and is an important topic for further study. The implied changes in $e$ as a binary decays from $a = ah$ to zero are modest, $De <~ 0.2$ , for all initial eccentricities.