9 Spin Evolution during Mergers

Coalescence of a binary black hole results in a spinning remnant.³ Angular momentum conservation implies

where $S1$ and $S2$ are the spin angular momenta of the two SBHs just before the final plunge, $Lorb$ is the orbital angular momentum of the binary before the plunge, $S$ is the spin of the resulting black hole, and $Jrad$ is the angular momentum carried away by gravitational waves during and after the coalescence [54 ]. The simplest case to treat is extreme mass ratio mergers, $q =_ m /m « 1 2 1$ , for which the binary can be described as a test particle of mass $m =_ m 2$ orbiting a black hole of mass $m1 =_ M » m$ , and both $S2$ and $Jrad$ can be ignored. The change in the larger hole’s spin is computed by adding the smaller hole’s energy and orbital angular momentum at the last stable orbit (LSO). The latter varies from $V~ -- LLSO/m = 12M$ for circular equatorial orbits around a non-spinning hole to $LLSO = M (9M )$ for prograde (retrograde) orbits around a maximally-spinning hole, $S = M 2 1$ . The much larger value of $L LSO$ in the case of retrograde capture implies that a rapidly-rotating hole will typically spin down if capture occurs from random directions [34 , 69 , 226 , 227

, 91

]. The change in spin assuming $q « 1$ is

where $^a =_ |S |/M 2 1$ , $L z$ is the orbital angular momentum parallel to $S$ , and $^L =_ L/mM$ . The first term in Equation (49

) describes conservation of spin angular momentum of the larger hole as its mass grows, $-2 ^a oc M$ , while the second term describes the increase in spin due to torquing by the smaller body. The change in spin after a single coalescence is illustrated in Figure 13

as a function of $q$ and initial spin; the upper (lower) curves represent prograde (retrograde) captures from equatorial orbits, and the dashed lines are for capture over the pole. The bias toward spin-down is evident; retrograde capture from the equatorial plane produces a nearly (but never completely) non-spinning remnant when $q ~~ 2.5 ^a$ , $q < 0.23$ , and rapid final rotation ( $^a >~~ 0.9$ ) requires both a large initial spin and a favorable inclination. On the other hand, if the larger hole is slowly rotating initially, $^a <~ 0.5$ , mass ratios $q >~~ 0.3$ always result in spin-up. The oft-repeated statement that “mergers spin down black holes” reflects a preconception that SBHs are likely to be formed in a state of near-maximal rotation (e.g. [14

, 61 ]).⁴

Figure 13:

Final spin $^a$ of a remnant black hole in terms of its original spin, for mass ratios $q = 0.1$ (red), $0.3$ (green) and $0.5$ (blue). The change in spin was computed using the test-particle approximation for $L LSO$ [16 , 91

, 227 ]. Upper (lower) curves correspond to prograde (retrograde) capture from the equatorial plane; dashed curves are for capture over the pole. Capture of a low-mass secondary is likely to spin down the larger hole unless the latter is slowly rotating initially. Capture of a massive secondary results in spinup unless infall is nearly retrograde or the original spin is large.

Successive mergers from random directions with fixed $q$ (i.e. secondary mass grows proportionately to primary mass) lead to a steady-state spin distribution $N (^a)$ that is uniquely determined by $q$ . For small $q$ , this distribution can be derived from the Fokker-Planck equation [91 ]:

Figure 14

shows $N (^a)$ for various values of $q$ , computed via Monte-Carlo experiments (not from the Fokker-Planck equation) using the test-mass approximation for $LLSO$ . The Gaussian form of Equation (50

) is seen to be accurate only for $q <~ 0.1$ . For $q >~~ 1/8$ , the distribution is skewed toward large spins.

Figure 14:

Steady-state spin distributions produced by successive capture from random directions at fixed mass ratio $q$ , for $q = (1/32, 1/16,1/8, 1/4,1/2)$ . Curves were generated using Monte-Carlo experiments based on the test-particle approximation, $q « 1$ ; hence the curve for $q = 1/2$ should be viewed as illustrative only.

Accurate calculation of spin-up during a merger of comparably massive black holes requires a fully general-relativistic numerical treatment. Adopting various approximations for the radius of the innermost stable circular orbit (ISCO) for comparably massive binaries [28 , 17 , 80 , 31 ], and assuming that mass and angular momenta are conserved during coalescence, gives a remnant spin in equal-mass mergers of $^a ~~ 0.8 -0.9$ . Baker et al. [9 , 10 ] present full numerical calculations of equal-mass mergers with and without initial spins. In the absence of initial spins, 3% of the system’s mass-energy and 12% of its angular momentum are lost to gravitational radiation, and the final spin is $^a ~~ 0.72$ . Coalescence of initially spinning holes from circular orbits in the equatorial plane yields $^a ~~ 0.72 + 0.32 ^s$ with $^s$ the initial spin parameter of the two holes (assumed equal); Baker et al. considered initial spins in the range $- 0.3 < ^s < 0.2$ , where negative/positive values indicate spins aligned/counteraligned with the orbital angular momentum. Extrapolating this result toward $^s = 1$ suggests that prograde mergers of black holes with initial spins $^s > 0.85$ will result in a maximally-spinning remnant.

Confronting these predictions with observation is problematic for a number of reasons: Merger histories of observed SBHs are not known, SBH spins are difficult to determine observationally, and other mechanisms, such as gas accretion, can act efficiently to spin up SBHs [14 ]. However there is circumstantial evidence that mergers played a dominant role in determining the spins of at least some SBH. If SBH spins were the product of gas accretion, the jets in active galaxies should point nearly perpendicularly to the disks of their host galaxies [182 ]. In fact, there is almost no correlation between jet direction and galaxy major axis in Seyfert galaxies [211 , 101 , 194 ]. Among the possible explanations [101 ] for the misalignment, perhaps the most natural is that SBH spins were determined by the same merger events that formed the bulge, long before the formation of the gaseous disk, and that subsequent spin-up by gas accretion from the disk plane has been minimal [134 ].

Coalescence of two black holes during a merger should result in a “spin-flip”, a reorientation of the spin axis of the more massive black hole. In the test-particle limit, the reorientation angle is

with $m$ the cosine of the angle between the orbital angular momentum vector and the spin axis of the larger hole. When $q >~~ 0.2$ , the spin orientation is overwhelmed by the plunging body in a retrograde merger, even if the initial spin of the larger hole was close to maximal. Hence, even “minor mergers” (defined, following galactic dynamicists, as mergers with $q < 0.3$ ) are able to produce a substantial reorientation. In fact there is a class of active galaxies which exhibit radio lobes at two, nearly-orthogonal orientations, and in which the production of plasma along the fainter lobes appears to have ceased [115 , 32 ]. These “X-shaped” or “winged” radio galaxies, of which about a dozen are known, are plausible sites of recent (within the last $~ 108yr$ ) black-hole coalescence [137

, 233 ]. Furthermore the implied coalescence rate is roughly consistent with the expected merger rate for the host galaxies of luminous radio sources [137

]. Alternative models have been proposed for the X-shaped sources, including a warping instability of accretion disks [174 ], backflow of gas along the active lobes [116 ], and binary-disk interactions before coalescence [120 ].⁵ It is likely that all of these mechanisms are active at some level and that the time scale for realignment influences the radio source morphology, with the most rapid realignments producing the classical X-shaped sources, while slower realignment would cause the jet to deposit its energy into a large volume, leading to an S-shaped FRI radio source [137 ]. If the black holes are spinning prior to coalescence, they will experience spin-orbit precession, on a time scale that is intermediate between $tgr$ and the orbital period. To PN2.5 order, the spin angular momentum of either hole evolves as [98 ]

where $^n$ is a unit vector in the direction of the displacement vector between the two black holes. The evolution equation for $S2$ is given by interchanging the indices. The magnitude of each spin vector remains fixed (to this order), and each spin precesses around the total angular momentum vector $J = Lorb + S1 + S2$ . When the two black holes are comparably massive, the orbital angular momentum greatly exceeds the spin angular momentum of either hole until just prior to coalescence. As the binary shrinks, the spins have a tendency to unalign with $Lorb$ . Ignoring spin-spin effects, the precession rate for equal-mass holes in a circular orbit is

The precession rate is lower than the orbital frequency:

where $R = 2m12$ , but higher than the radiation reaction time scale:

unless the binary is close to coalescence. If the dominant source of energy loss during the late stages of infall is gravitational radiation, the spin direction will undergo many cycles of precession before the black holes coalesce. This does not seem to happen in the X-shaped radio sources, based on the apparently sudden change in jet direction; if the coalescence model for the X-sources is correct, the final stages of infall must occur on a shorter time scale than $tgr$ . This might be seen as evidence that shrinkage of the binary is usually driven by gas dynamics, not gravitational radiation losses, prior to the final coalescence.

When the black hole masses are very different, $q« 1$ , the ratio of spin of the larger hole to $Lorb$ is

where $S1 = f1m2 1$ . The two quantities are approximately equal when the separation measured in units of the larger hole’s Schwarzschild radius is equal to $- 2 q$ . When this separation is reached, the binary orbit rapidly changes its plane, and a new regime is reached where the spin of the smaller black hole precesses about the spin of the larger hole. The precession rate of the larger hole is given by

in both regimes, and

The spin direction of a black hole formed via binary coalescence is also affected by torques that reorient the binary prior to coalescence. The role of torques from gaseous accretion disks was discussed above; another source of torques is perturbations from passing stars or gas clouds [134 ]. A single star that passes within a distance $~ 3 a$ of the binary will exchange orbital angular momentum with it, leading both to a change in the binary’s orbital eccentricity as well as a change in the orientation of the binary’s spin axis, as discussed above. Referring to Equations (17 ) and (20 ), the reorientation rate is related to the hardening rate via

where $t- 1 = a(d/dt)(1/a) harden$ . Scattering experiments [134 ] give $L/H ~~ 4$ for a hard, equal-mass binary. The implied change in the binary’s orientation after shrinking from $a ~~ ah$ to $-3 a ~~ 10 ah$ is

The reorientation begins to be significant if $m*/m12 >~~ 10-3$ , which may be the case for intermediate-mass black holes.