How black holes get their kicks! Gravitational radiation recoil from binary inspiral and plunge into a rapidly-rotating black hole.

Transcription

1 How black holes get their kicks! Gravitational radiation recoil from binary inspiral and plunge into a rapidly-rotating black hole. Marc Favata (Cornell) Daniel Holz (U. Chicago) Scott Hughes (MIT) The Astrophysics of Gravitational Wave Sources April 24-26, 2003 University of Maryland

2 As the momentum flux scales like the square of the mass ratio, it is small in the point-mass limit. However, we believe that, with accuracy of order 20%, we can scale up our results to the comparable mass limit. As is shown below, this can result in astrophysically significant kick velocities (of order hundreds of km/s). Introduction and Summary It is well known that in a black hole binary, energy and angular momentum are given off in gravitational radiation. What is less well known is that linear momentum is also emitted by the waves. The loss of momentum from the system causes a recoil or kick velocity to be imparted to the center of mass of the coalesced binary. This is sometimes called the radiation rocket effect. If this effect is large, significant velocity kicks could result, possibly ejecting the black hole from its environment. Our goal is to calculate the magnitude of this kick velocity when one of the holes is rapidly rotating. This is done in the extreme-mass-ratio limit using two methods. In the first method, we solve the Teukolsky equation for the case of a point particle inspiraling into a Kerr hole. [Our results here are restricted to circular, equatorial orbits, but our code can easily handle circular, inclined orbits.] From this, we can compute the momentum flux and the associated recoil. In the point particle and adiabatic (dynamical time << radiation reaction time) limits, this treatment is exact and allows us to compute the momentum flux up to the inner-most stable circular orbit (ISCO). During the plunge, we use a more approximate technique that accurately models the orbital dynamics, but ignores strong field effects on the radiation field.

3 Astrophysical Motivation Black hole mergers are expected to occur in our universe. When two galaxies merge, the supermassive black holes at their centers will (through process like dynamical friction and 3-body interactions) form a binary that will eventually decay by gravitational radiation. Black hole binaries are also expected to form in the dense stellar environments of globular clusters. If the merger of these systems results in a substantial kick velocity, there could be several important implications: 1. If the kick is large enough, the resulting velocity could exceed the escape velocity of the system. For galaxies, this corresponds to kicks ~ km/s, and for globular clusters ~ km/s. 2. A kick imparted to a galaxy s central black hole could effect the distribution of stars and dark matter near the galactic center. 3. Intermediate-mass black holes may be formed by multiple mergers in globular clusters. A significant recoil during one of these mergers may disrupt the formation of these systems.

4 Gravitational radiation recoil/rocket effect: The radiation recoil effect can be understood as follows: In the binary pictured here, the smaller mass m 2 moves with a faster velocity about the system s center or mass than does m 1. The smaller mass is thus more effective at forward beaming its gravitational radiation. Instantaneously, there is more linear momentum ejected from the system in the direction of the smaller mass s motion. This results in a recoil on the center of mass in the opposite direction. [ figure from Wiseman, PRD 46, 1517]

5 Gravitational radiation recoil/rocket effect: Mathematically, one can express the linear momentum flux in terms of mass and current multipole moments of the source. To do this one would proceed as follows [see Thorne RMP 52, 1980]: Express the momentum flux as a surface integral over the energy-momentum tensor for gravitational waves. In the transverse-traceless (TT) gauge, the energy-momentum tensor takes the simple form below. Expanding the gravitational wave field h jk in a sum over multipole moments and plugging into the surface integral then gives the momentum flux as sum over multipoles: dp dt = j GW jk r = TGWnkdA = 32π d 4 dt jab 4 3 d dt ab ε t jpq h TT ik d 3 dt t h pa 3 TT ik d n 3 j dt dω qa 3 +L T = GW µν 1 32π TT TT h ik, µ h ik, ν One can see that at lowest order, the momentum flux results from a beating of the mass quadrupole moment against the mass octupole and current quadrupole moments. The velocity of the center of mass can then be found by integrating the momentum flux along the orbit: dx dt j dpgw = dt M dt j CM 1 tot

6 Previous work: Bonnor & Rotenberg (1961), Peres (1962): First pointed out that gravitational waves can cause a recoil. Cooperstock (1967): Caclulated recoil from two separated emitters. Bekenstein (1973): Calculated recoil from a collapsing star. Moncrief (1979): Recoil from a collapsing star using BH perturbation theory. Fitchett (1983): First to address the problem of recoil in a binary system. Fitchett considered a Newtonian binary and computed the lowest order momentum flux and resulting recoil for circular and elliptical orbits. Oohara & Nakamura; Nakamura & Haugan (1983): plunge into Schwarzschild; Kerr radial in-fall along symmetry axis. Fitchett & Detweiler (1984): Examined circular geodesic orbits around a Schwarzschild hole using perturbation theory. Redmount & Rees (1989): Discussed astrophysical implications of the recoil. Wiseman (1992): extended Fitchett s work to post-newtonian theory.

7 Our approach: Kerr circular, equatorial orbits We essentially extend the work of Fitchett and Fitchett & Detweiler to point masses inspiraling into rapidly spinning holes on circular, equatorial orbits. There are 3 phases to the orbital evolution, and each requires a separate method to calculate the energy, momentum, and angular momentum given off in each: Adiabatic inspiral: The radiation reaction timescale is much longer than the orbital period, so the particle s orbit can be evolved by slowly changing the constants of the motion. In this regime, we use BH perturbation theory to compute the momentum flux up to the ISCO. Plunge into the horizon: Past the ISCO, the orbit becomes unstable on a dynamical timescale, and rapidly plunges into the horizon. We use an approximate technique to determine the radiation emitted during this portion of the orbit. (see below) Ring-down: The merged remnant sheds its remaining perturbation, which is calculated using BH perturbation theory. We have not yet addressed this phase, but we expect it to give the smallest contribution to the recoil. Inspiral and plunge into a Schwarschild BH. The blue line represents the ISCO, the black line is the horizon.

8 Adiabatic inspiral phase In the extreme-mass ratio limit, black hole perturbation theory can be used to calculate the radiation given off to infinity and down the horizon. This is done by solving the Teukolsky equation for the Weyl scalar Y 4 (which encodes information about the radiation field). The energy, momentum, and angular momentum fluxes carried by the radiation are readily expressed in terms of quantities that come from solving for Y 4 : Ψ 4 j dp dt = ( r ia cosθ ) GW 1 de j 1 ψ = n dω = 4π dtdω 4π ψ lmk l m k lmk l m k ωmkωm k ) 4 n j dω imφ iωt H imφ iω t r d Rlm r Slm a e e = Z Slm a mk e e lmk mk ( * ω ω ( ) 2 ( θ; ω) 2 ( θ; ω ) lm In order to generate a worldline for the inspiraling particle, a prescription for computing the radiation reaction on the particle is necessary. In the adiabatic limit (where the radiation reaction timescale is much longer than the orbital timescale) and for circular orbits, this can be done by evolving the constants of the motion as follows: 1) For a geodesic orbit with a specific energy E and orbital angular momentum L z, solve the Teukolsky equation for Y 4. 2) Use this to compute the energy and angular momentum fluxes (de/dt and dl z /dt) to infinity and down the horizon. 3) Update the energy and angular momentum of the particle, which effectively moves it to a new geodesic. 4) Repeating this process allows one to generate an inspiral trajectory. Having calculated Y 4 along this trajectory, it is now easy to compute the momentum flux throughout the orbit. r lmk 1 r lmk ψ lmk

9 Results I: Recoil from adiabatic inspiral [Fig. A.] Center of mass velocity vs. Boyer- Lindquist radius for a/m=0, m/m=0.1. Fitchett s quasi-newtonian result is compared with the result from our Teukolsky equation solver. [Fig. B.] Same as fig. A, but for a rapidly spinning hole with a/m=0.99.

10 In the above figures we have plotted the center of mass velocity (in km/s) for a particle that has spiraled in to a radius r (the Boyer-Lindquist radius in units of the large hole s mass M) on a circular, equatorial orbit as calculated from solving the Teukolsky equation (as described above). We compare our calculation (in blue) with the prediction of Fitchett (in red) [Fitchett assumed a Newtonian circular orbit and included only the lowest order multipoles of the radiation field]. Both plots are for a mass ratio of m/m=0.1 and terminate at the ISCO. Figure A, shows the recoil for inspiral into a Schwarzchild black hole. We can see that the our results agree very well with Fitchett s. At the ISCO (r = 6M) we predict a recoil of 4.7 km/s, while Fitchett would predict 5 km/s. Figure B, shows the recoil for inspiral into a rapidly rotating Kerr hole, with spin parameter a/m=0.99. In this case, the ISCO has moved inward very close to the horizon and we start to see significant deviations from Fitchett s quasi-newtonian predictions. Strong field effects, such as the gravitational redshift and scattering of waves off the background curvature, are much more important near the horizon and tend to suppress the emitted radiation, decreasing the momentum flux and the resulting recoil. At the ISCO (r = 1.46M), we predict a recoil of 257 km/s while Fitchett would have predicted over 1000 km/s.

11 Results II: scaling-up to comparable masses [Fig. C.] Scaled-up velocity at the ISCO of a rapidly rotating hole (a/m=0.99, r=1.46m) as a function of mass ratio.

12 The nature of our perturbation calculations restricts us to the small mass ratio limit. However, we expect that larger recoils will result from mergers of more comparable masses. So we would like to be able to extrapolate our results to large mass ratios. Some guidance on this issue is provided by calculations of the energy flux from head-on collisions of two black holes using both numerical relativity and BH perturbation theory. Detweiler and Smarr found that when the perturbation results were scaled-up to larger mass ratios, they agreed with the numerical relativity calculations. This scaling up was accomplished by replacing the small mass with the reduced mass, and the large mass with the total mass. We can do something similar for the momentum flux. In the extreme mass ratio limit, the momentum flux is proportional to the mass ratio squared (m 1 /m 2 ) 2. In the comparable mass case, Fitchett s quasi-newtonian calculation shows that the momentum flux is proportional to a dimensionless function of the mass ratio f(m 1 /m 2 ) that reduces to (m 1 /m 2 ) 2 in the small mass-ratio limit. Fitchett & Detweiler suggest that the correct way to scale-up perturbation theory results for the momentum flux is to simply replace the factor (m 1 /m 2 ) 2 with f(m 1 /m 2 ) in the momentum flux formula. We believe that this scaling-up will be reasonably accurate (~20%), but only a full-blown numerical relativity calculation will be able to say for certain. In Fig. C, we have scaled-up our value for the recoil at the ISCO of the 0.1 mass-ratio system in Fig. B (257 km/s). We also show what Fitchett s calculation predicts for inspiral up to that radius (r = 1.46M). We can see that maximum recoil velocities of 820 km/s are possible (while Fitchett would have predicted over 3000 km/s). Note also that the recoil peaks for mass ratio around 0.38 and is zero for equal masses.

13 Recoil from transition from inspiral to plunge: Near the ISCO, the orbit no longer decays on a radiation reaction timescale, but on a dynamical timescale. The adiabatic assumption is no longer valid, so we can not use our Teukolsky solver to evolve the orbit past the ISCO. Another approach is necessary for the plunge. Fortunately, Ori & Thorne (2000) have derived equations of motion that describe the orbit in the late adiabatic phase and throughout the particle s plunge into the horizon. Their equations of motion are accurate in the extreme mass-ratio limit. To compute the recoil for the plunge, we combine the Ori-Thorne equations of motion with the Kerr geodesic equations to arrive at a reasonably accurate description of the orbit throughout the plunge. We then use the lowest order terms in the multipole expansion of the momentum flux. We also use simplistic, Newtonian definitions to describe the multipole moments in terms of the orbit: dp dt j GW d jab d 16 d pa d ab = + ε 4 3 jpq dt dt 45 dt dt qa +L ab = µ [ x a x b ] STF etc... Although our description of the radiation field is not strictly valid in the fast-motion, strong field region, we suspect that a more accurate treatment will not change our results by huge amounts. However, we are working to make our description of the radiation from the plunge more accurate.

14 Results III: transition from inspiral to plunge [Fig. D.] Plunge orbit into a Schwarzschild hole with mass ratio = 0.1 [Fig. E.] Recoil velocity as a function of Boyer-Lindquist radius for orbit in Fig. D.

15 Results IV: transition from inspiral to plunge [Fig. F.] Retrograde plunge orbit into a Kerr hole with mass ratio = 0.1 and spin parameter a/m = [Fig. G.] Recoil velocity as a function of Boyer-Lindquist radius for the orbit in Fig. F.

16 In the above figures, we show the orbits and the resulting recoil for two plunge orbits. In both cases the mass ratio is 0.1. In Figs. D & F the solid blue line is the ISCO and the solid black line is the horizon. Figure D. shows the late inspiral and plunge into a non-spinning black hole. Note the abrupt change in character of the orbital motion near the ISCO. Figure E shows the recoil velocity for this orbit. We see that the velocity peaks at around 33 km/s near 4M, and then drops down to about 20 km/s before the particle enters the horizon. Also note at 6M, our plunge code predicts a kick of ~8km/s while our perturbation code predicts ~5km/s. We generally tend to overestimate both Fitchett s prediction and our perturbation code prediction because our plunge code includes the relativistic effects on orbital dynamics (which Fitchett neglects), but ignores mechanisms that tend to suppress the radiation field (redshift and scattering) which the perturbation code includes. Figure F shows the orbit for a retrograde inspiral into a rapidly rotating hole. Notice that right before the horizon, the strong inertial frame dragging forces the particle to turn-around and rotate in the opposite direction. Figure G shows the resulting recoil velocity. Since the frame dragging opposes the motion of the particle, this tends to suppress the recoil for most of the plunge as compared to the a/m=0 case.

17 Conclusions: We have computed the recoil velocity due to linear momentum flux emitted during inspiral and plunge on circular, equatorial orbits around rapidly rotating black holes and have found that astrophysically interesting kicks (greater than the escape velocities of some systems) are possible and depend strongly on the spin parameter. Our calculations of the recoil up to the ISCO were performed in the context of black hole perturbation theory using a Teukolsky equation solver developed by Hughes (PRD 61,64). This provides robust results in the limit of small mass ratios. For the plunge, we have developed a more approximate approach to computing the recoil based on the Ori-Thorne equations of motion. For non-spinning holes, we have found kick velocities of 4.7 km/s up to the ISCO, and 20 km/s into the horizon (all numbers are for a mass ratio of 0.1). For prograde inspiral into a rapidly rotating hole (a/m=0.99), we find a kick velocity of 257 km/s. For retrograde inspiral into a rapidly-rotating hole (a/m=-0.99), we find a recoil of 1 km/s up at the ISCO and 30 km/s at the horizon. If we scale-up our results to large mass ratios, velocities t800 km/s are possible. For future work, we plan to extend our results to the case of circular, inclined orbits and to improve our approximations in the plunge part of the calculation. We would also like to estimate the momentum flux from the ringdown phase and explore in some detail the astrophysical implications of this recoil effect.