2 Preliminaries

We write $m 1$ and $m 2$ for the masses of the two components of a binary SBH, with $m < m 2 1$ , $q =_ m2/m1$ , and $m12 =_ m1 + m2$ . (We also sometimes write $M$ for the mass of the single SBH that forms via coalescence of two SBHs of combined mass $m12$ .) The semi-major axis of the binary’s Keplerian orbit is $a$ and $e$ is the orbital eccentricity. The binary’s binding energy is

with $m = m1m2/m12$ the reduced mass. The orbital period is

The relative velocity of the two SBHs, assuming a circular orbit, is

A binary is “hard” when its binding energy per unit mass, $|E|/m12 = Gm/2a$ , exceeds $~ s2$ , where $s$ is the 1D velocity dispersion of the stars in the nucleus. The precise meaning of “hard” is debatable when talking about a binary whose components are much more massive than the surrounding stars [87

, 177

]. For concreteness, we adopt the following definition for the semi-major axis of a hard binary:

At distances $r » a$ , stars respond to the binary as if it were a single SBH of mass $M$ . The gravitational influence radius of a single SBH is defined as the distance within which the force on a test mass is dominated by the SBH, rather than by the stars. A standard definition for $r infl$ is

Thus $rinfl = 4(M/m) ah$ . For an equal-mass binary, $rinfl ~~ 16 ah$ , and for a more typical mass ratio of $q = 0.1$ , $rinfl ~~ 50 ah$ . An alternative, and often more useful, definition for $rinfl$ is the radius at which the enclosed mass in stars is twice the black hole mass:

This definition is appropriate in nuclei where $s$ is a strong function of radius; it is equivalent to Equation (5

) when the density of stars satisfies $r(r) = s2/2pGr2$ , the “singular isothermal sphere”, and when $s$ is measured well outside of $r infl$ .

If the binary’s semi-major axis is small enough that its subsequent evolution is dominated by emission of gravitational radiation, then $a oc - a-3$ and coalescence takes place in a time $tgr$ , where [165 ]

---5-----c5 -a4--- tgr = 256F (e)G3 mm2 , ( ) ( 12 ) F (e) = 1 - e2 7/2 1 + 73-e2 + 37e4 . (7) 24 96

This can be written

3 5( )4 t = ----5--- Gm--c- a-- gr 164F (e) s8m212 ah 8 3 ( ) ( ) -8( )4 ~~ 3.07-×-10-yr---q---- --m12-- ----s------ ---a--- . (8) F (e) (1 + q)6 108Mo. 200 km s-1 10- 2ah

This relation can be simplified by making use of the tight empirical correlation between SBH mass and $s$ , the “ $M$ - $s$ relation”. Of the two forms of the $M$ - $s$ relation in the literature [51

, 64

], the more relevant one [51 ] is based on the velocity dispersion measured in an aperture centered on the SBH, which is approximately the same quantity $s$ defined above; the alternative form [64 ] defines $s$ as a mean value along a slit that extends over the entire half-light radius of the galaxy. In terms of the central $s$ , the best current estimate of the $M$ - $s$ relation is [50 ]

with $a = 4.86± 0.43$ . Combining Equations (8

) and (9

) and setting $M = m12$ , $F = 1$ gives

q3 ( s ) -3.14( a )4 tgr ~~ 5.0 × 108 yr-------6 ---------1- ----2-- (1 + q) ( 200 km s) ( 10 )ah 8 q3 m12 -0.65 a 4 ~~ 7.1 × 10 yr-------6 --8---- ----2-- . (10) (1 + q) 10 Mo. 10 ah

Coalescence in a Hubble time ( $~ 1010 yr$ ) requires $a <~ 0.05 ah$ for an equal-mass binary and $a <~ 0.15ah$ for a binary with $q = 0.1$ . Inducing a SBH to decay from a separation $0 a ~~ ah ~~ 10 pc$ to a separation such that $10 tgr <~ 10 yr$ is called the “final parsec problem” [151 ]. Much of the theoretical work on massive black hole binary evolution has focused on this problem.