Modeling Kicks from the Merger of Nonprecessing Black Hole Binaries

Abstract

Several groups have recently computed the gravitational radiation recoil produced by the merger of two spinning black holes. The results suggest that spin can be the dominant contributor to the kick, with reported recoil speeds of hundreds to even thousands of kilometers per second. The parameter space of spin kicks is large, however, and it is ultimately desirable to have a simple formula that gives the approximate magnitude of the kick given a mass ratio, spin magnitudes, and spin orientations. As a step toward this goal, we perform a systematic study of the recoil speeds from mergers of black holes with mass ratio q ≡ m1/m2 = 2/3 and dimensionless spin parameters of a1/m1 and a2/m2 equal to 0 or 0.2, with directions aligned or antialigned with the orbital angular momentum. We also run an equal-mass a1/m1 = -a2/m2 = 0.2 case, and find good agreement with previous results. We find that, for currently reported kicks from aligned or antialigned spins, a simple kick formula inspired by post-Newtonian analyses can reproduce the numerical results to better than ~10%.

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arXiv:astro-ph/0702390v2 11 Oct 2007
Draft version February 3, 2008
Preprint typeset using LATEX style emulateapj v. 02/07/07
MODELING KICKS FROM THE MERGER OF NON-PRECESSING BLACK-HOLE BINARIES
John G. Baker, William D. Boggs1, Joan Centrella, Bernard J. Kelly, Sean T. McWilliams1, M. Coleman
Miller2, James R. van Meter
Laboratory for Gravitational Astrophysics, NASA Goddard Space Flight Center, Greenbelt, Maryland 20771
Draft version February 3, 2008
ABSTRACT
Several groups have recently computed the gravitational radiation recoil produced by the merger of
two spinning black holes. The results suggest that spin can be the dominant contributor to the kick,
with reported recoil speeds of hundreds to even thousands of kilometers per second. The parameter
space of spin kicks is large, however, and it is ultimately desirable to have a simple formula that gives
the approximate magnitude of the kick given a mass ratio, spin magnitudes, and spin orientations.
As a step toward this goal, we perform a systematic study of the recoil speeds from mergers of black
holes with mass ratio q ≡ m1/m2= 2/3 and dimensionless spin parameters of a1/m1and a2/m2equal
to 0 or 0.2, with directions aligned or anti-aligned with the orbital angular momentum. We also run
an equal-mass a1/m1= −a2/m2= 0.2 case, and find good agreement with previous results. We find
that, for currently reported kicks from aligned or anti-aligned spins, a simple kick formula inspired by
post-Newtonian analyses can reproduce the numerical results to better than ∼10%.
Subject headings: black hole physics – galaxies: nuclei – gravitational waves — relativity
1. INTRODUCTION
In the past two years, numerical relativity has under-
gone a revolution such that now multiple codes are able
to stably evolve the last few cycles of the inspiral, merger,
and ringdown of two black holes (Br¨ ugmann et al. 2004;
Pretorius 2005; Campanelli et al. 2006a,b; Baker et al.
2006a,b).An important astrophysical output of
such simulations is the net recoil due to asymmet-
ric emission of gravitational radiation, because this
has major implications for the growth of super-
massive black holes in hierarchical merger scenarios
(Merritt et al. 2004; Boylan-Kolchin et al. 2004; Haiman
2004; Madau & Quataert 2004; Yoo & Miralda-Escud´ e
2004; Volonteri & Perna 2005; Libeskind et al. 2005;
Micic et al. 2006) as well as the evolution of seed black
holes and current-day intermediate-mass black holes
(Taniguchi et al.2000;Miller & Hamilton
Mouri & Taniguchi2002a,b;
G¨ ultekin et al. 2004, 2006; O’Leary et al. 2006, 2007).
Analyticalcalculations (Peres
1973;Fitchett1983;Fitchett & Detweiler
Redmount & Rees 1989; Wiseman 1992; Favata et al.
2004;Blanchet et al.2005;
2006) have now been augmented with full numerical
results for non-spinning black holes with different
mass ratios (Herrmann et al. 2007c; Baker et al. 2006c;
Gonzalez et al. 2007a), for black holes with equal masses
and spins initially orthogonal to the orbital plane
(Herrmann et al. 2007a; Koppitz et al. 2007), for black
holes with equal masses and spins initially parallel to the
orbital plane(Gonzalez et al. 2007b; Campanelli et al.
2007b), for black holes with equal masses and spins
initially oriented at some angle between the orbital plane
and the orbital angular momentum(Herrmann et al.
2007b; Tichy & Marronetti 2007), and for black holes
2002a,b;
2004;Miller & Colbert
1962;Bekenstein
1984;
Damour & Gopakumar
1University of Maryland, Department of Physics, College Park,
Maryland 20742-4111
2University of Maryland, Department of Astronomy, College
Park, Maryland 20742-2421
with unequal masses and spins initially either parallel
to the orbital plane or oriented at some angle between
the orbital plane and the orbital angular momentum
(Campanelli et al. 2007a).
The parameter space for kicks with spin is large, so
for astrophysical purposes it is important to have a sim-
ple parameterized formula for the kick that can be in-
cluded in simulations of N-body dynamics or cluster and
galaxy mergers. For non-spinning black holes, the classic
Fitchett (1983) formula v ∝ η2(δm/M) does a reasonable
job (although not perfect; see Gonzalez et al. 2007a for
numerical results), where for black hole masses m1and
m2≥ m1we define M = m1+ m2, δm = m2− m1, and
η = m1m2/M2. Initial explorations of equal-mass spin
kicks also show evidence of simplicity, with fits linear in
spin reproducing the results of Herrmann et al. (2007a)
and Koppitz et al. (2007). This encourages us to explore
a more general class of kicks.
Spins that are aligned (prograde) with the orbital
angular momentum may be of particular interest be-
cause it has been argued that interaction with accre-
tion disks will tend to align spins during “wet” mergers
(Bogdanovic et al. 2007). The only previous numerical
studies for a black hole with an aligned spin were car-
ried out by Herrmann et al. (2007a) and Koppitz et al.
(2007), which considered only cases where the spin of the
second black hole was anti-aligned (retrograde) with the
angular momentum and the masses of the black holes
were equal. Here we compute the kick speeds from a
q ≡ m1/m2= 2/3 mass-ratio set of mergers, with spin
parameters of 0 or 0.2, and directions either aligned (pro-
grade) or anti-aligned (retrograde) with the orbital an-
gular momentum. The symmetry of the configuration
therefore guarantees that the kick direction is in the or-
bital plane. We find a formula that matches all of our
kick speeds, and those of Herrmann et al. (2007a) and
Koppitz et al. (2007), to within ∼ 10%. If the in-plane
kicks can be generalized straightforwardly to more gen-
eral orientations, and added to kicks perpendicular to

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2
the orbital plane, there is the prospect of simple astro-
physical modeling of the gravitational rocket effect for
arbitrary black hole mergers. In § 2 we describe our ini-
tial data and methodology. In § 3 we present our results,
and discuss the implications of these simulations.
2. INITIAL DATA AND METHODOLOGY
In the following, we use geometrized units where New-
ton’s gravitational constant G and the speed of light c
are set to unity so that all relevant quantities can be rep-
resented in terms of their mass-scaling. For example, 1
M⊙is equivalent to a distance of 1.4771× 105cm, and a
time of 4.9272×10−6s. Accordingly we express distances
in terms of M, the initial (ADM) mass of the system.
We simulated inspiraling black-hole binaries of various
mass ratios and spins, with the same initial coordinate
separation in each case.
mass ratio approximated either 2/3 or unity.
simulations were performed with our finite differencing
code Hahndol (Imbiriba et al. 2004), which solves a
3+1 formulation of Einstein’s equations.
mesh refinement and most parallelization was handled
by the software package Paramesh (MacNeice et al.
2000). For initial data we used the Brandt & Br¨ ugmann
(1997) Cauchy surface for black hole punctures, as
computedbythesecond-order-accurate,
elliptic solver AMRMG (Brown & Lowe 2005). We evolved
thisdatawiththe standard
Shibata-Nakamura (Nakamura, Oohara & Kojima
1987; Shibata & Nakamura 1995; Baumgarte & Shapiro
1998; Imbiriba et al. 2004) evolution equations, modified
only slightly with dissipation terms as in H¨ ubner (1999)
and constraint-damping terms as in Duez et al. (2004).
Our gauge conditions were chosen according to the
“moving puncture” method,
(2006). Time-integration was performed with a fourth-
order Runge-Kutta algorithm, and spatial differencing
with fourth-order-accurate mesh-adapted differencing
(Baker & van Meter 2005).
refinement regions was fifth-order-accurate.
To explore the parameter space of non-precessing spin
configurationswith some mass-ratiodependence, we have
performed simulations for seven different data sets of
black holes with unequal masses, as well as one equal-
mass case. The initial-data parameters for these eight
data sets are given in Table 1. The spins in these sim-
ulations were always orthogonal to the orbital plane.
Our choice of initial tangential momenta was informed
by a quasi-circular post-Newtonian (PN) approximation
(Damour et al. 2000).
Numerically, we can only directly specify the “punc-
ture” masses m1pand m2pof the two holes; to determine
each hole’s physical (horizon) mass m, we first locate
its apparent horizon (using an adapted version of the
AHFinderDirect code; see Thornburg 2003), and then
apply the Christodoulou (1970) formula:
m2= m2
In these cases our initial
Our
Adaptive
multigrid
Baumgarte-Shapiro-
as in van Meter et al.
Interpolation between
irr+ J2/4m2
irr, (1)
where J is the magnitude of the spin angular momentum
of the hole, mirr=
?AAH/16π, and AAH is the area of
The parameters relevant to our discussion – the mass
ratio q ≡ m1/m2and the dimensionless spin parameters
ˆ a1≡ a1/m1and ˆ a2≡ a2/m2– are listed in the first eight
the apparent horizon.
TABLE 1
Initial data parameters. Runs are labeled “EQ” for equal
mass and “NE” for unequal mass. J1 and J2 are the spin
angular momenta of the two holes, either aligned (positive)
or anti-aligned (negative) with the orbital angular
momentum. mp1 and mp2 are the directly specified puncture
masses of the holes. P is the initial transverse momentum on
each hole, while L is the initial coordinate separation of the
punctures.
RunJ1(M2)J2(M2)mp1(M)mp2(M)P (M)L (M)
NE−−
NE−+
NE0−
NE00
NE0+
NE+−
NE++
EQ+−
-0.032
-0.032
0.000
0.000
0.000
0.032
0.032
0.050
-0.072
0.072
-0.072
0.000
0.072
-0.072
0.072
-0.050
0.374
0.374
0.374
0.382
0.374
0.374
0.374
0.480
0.586
0.586
0.586
0.584
0.586
0.586
0.586
0.480
0.119
0.119
0.119
0.119
0.119
0.119
0.119
0.124
7.05
7.05
7.05
7.05
7.05
7.05
7.05
7.00
rows of Table 2. We have striven to maintain q = 2/3 in
all of our unequal-mass simulations.
The grid spacing in the finest refinement region,
around each black hole, was hf = 3M/160, with the
exception of our nonspinning unequal-mass case, which
was one of a set of runs described previously (Baker et al.
2006c), and for which we used hf = M/40. The ex-
traction radius was at R = 45M in every case except
for the nonspinning case, where it was at R = 50M.
For one of our new physical configurations (NE++) we
also extracted at R = 40M, finding a final kick within
0.8kms−1of that extracted at R = 45M. Assuming a
radially dependent error that falls off as 1/R, as found in
a similar kick computation (Gonzalez et al. 2007b), this
implies that the kick extracted at R = 45M is within
8% of what would be computed at infinite radius. Also
for this physical configuration we ran a higher resolu-
tion, hf = M/64, to verify that the lower resolution of
hf= 3M/160 would be sufficient. We found satisfactory
convergence of the Hamiltonian constraint (Fig.1) and
consistency of the radiated momentum (Fig.2).
The thrust dPi/dt imparted by the radiation was de-
rived by Newman & Tod (1981), and is computed from a
surface integral of the squared time-integral of the radia-
tive Weyl scalar ψ4times the unit radial vector, as given
explicitly by Campanelli & Lousto (1999):
dPi
dt
= lim
R→∞
?
R2
4π
?
dΩxi
R
????
?t
−∞
dtψ4
????
2?
. (2)
To perform the angular integration in (2), we use
the second-order Misner algorithm described in Misner
(2004) and Fiske et al. (2005). This result is then inte-
grated numerically to give the total radiated momentum
Pi; to obtain the final velocity of the merged remnant
black hole, we divide Piby the final black-hole mass, as
computed from the difference of the initial ADM mass
and the total radiated energy.
3. RESULTS AND DISCUSSION
In Fig. 3 we present the aggregated recoil kick from
each of our simulations. The kicks obtained range from

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3
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
0 50 100 150 200
t (M)
250 300 350 400
||H||1
hf=3M/160 (scaled)
hf=M/64
Fig. 1.— L1 norm of the Hamiltonian constraint for the case
of unequal masses with prograde spins (NE++) at two different
resolutions, computed on a domain extending to |xi| = 48M, which
includes our wave-extraction surface. The finest level, containing
nonconvergent points near the puncture, has been excluded. The
lower resolution, hf= 3M/160, has been scaled such that for third-
order convergence it should superpose with the hf= M/64 curve.
0
20
40
60
80
100
120
140
160
180
0 50 100 150 200
t (M)
250 300 350 400
v (km s-1)
hf=3M/160
hf=M/64
Fig. 2.— The total radiated momentum for the case of unequal
masses with prograde spins (NE++) at two different resolutions.
The curves have been shifted slightly in time to line up the peaks,
for comparison of the merger dynamics. The agreement is better
than 5% for most of the evolution and better than 2% through the
merger and final kick (t > 300M).
∼ 60kms−1, in the case where the larger hole’s spin is
aligned, and the smaller anti-aligned, with the orbital an-
gular momentum, to ∼ 190kms−1, when the alignments
are reversed. The kicks possess a common profile, with
the bulk of the momentum radiated over ∼ 50M before
merger. In all but the equal-mass case (EQ+−), we ob-
serve a sharp monotonic rise in kick over 40M, followed
by a substantial “un-kick”. That is, around the time of
merger and ringdown, we often observe a sudden thrust
in momentum that is directed counter to the momen-
tum that had accumulated during inspiral. In the EQ+−
case, this un-kick is absent. We summarize the final kicks
for each of our configurations in the first eight rows of
Table 2.
It has been noted by Campanelli et al. (2006c) that the
presence of spins on black holes in an inspiraling binary
0 100200 300
t/M
0
50
100
150
200
250
v (km s-1)
NE--
NE+-
NE0-
EQ+-
NE-+
NE00
NE++
NE0+
Fig. 3.— Aggregated recoil kicks from all runs listed in Table 1.
The merger time for each binary matches the peak in its kick pro-
file; the relative delay in merger times between data sets differing
in initial spins is consistent with the results of Campanelli et al.
(2006c).All configurations show a marked ”un-kick” after the
peak, with the exception of the equal-mass case, EQ+−.
can significantly extend or reduce the time to merger, de-
pending on whether the spins are aligned or anti-aligned
with the orbital angular momentum. We observe a sim-
ilar trend in merger times, as illustrated in the peak of
the aggregated recoil kicks. This tendency had been also
been expected based on PN calculations which show that
the last stable orbit is pushed to smaller radius, imply-
ing later merger, for aligned spins (Damour 2001). Note
that, although resolution can also affect merger time, for
these short runs we have sufficiently resolved the dynam-
ics that numerical error in merger time appears negligible
compared to the effect of spin, as demonstrated in Fig. 2.
Our simulation results, together with data reported by
other groups, allow us to consider a simple description of
the total kick for arbitrary mass ratio when the two black
holes spins are aligned or anti-aligned with the system’s
orbital angular momentum.
Severalrecent papers
Koppitz et al. 2007; Choi et al. 2007) have suggested
that the kick velocity resulting from comparable-mass
binary black hole mergers may be approximately
described in terms of a simple scaling dependence
consistent with the scaling in the leading-order post-
Newtonian approximation treatment (Kidder 1995).
For kicks generated by non-spinning black holes of
unequal masses, this produces the Fitchett scaling,
which provides a reasonable approximation to recent
numerical simulation results (Gonzalez et al. 2007a).
Recent papers on spinning black hole mergers suggest
that the kicks from nearly equal-mass mergers may
scale linearly, through the quantity ∆ = qˆ a1− ˆ a2, with
spins that are either aligned or anti-aligned with the
orbital angular momentum (Herrmann et al. 2007a;
Koppitz et al. 2007). For head-on collisions, Choi et al.
(2007) have shown that the PN expressions describe the
scaling of the spin-asymmetry kick and its relation to the
kick induced by mass-asymmetry, correctly predicting
the relative directions of the two kick contributions and
supporting the idea that the effects of asymmetries in
spins and masses can be considered independently.
For inspiraling mergers, with the assumption that the
radial velocity is small compared to the tangential veloc-
(Herrmann et al.2007a;

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4
ity, the PN prediction for the cases we consider is that the
instantaneous thrust generated by the spin-asymmetry
will be aligned with that of the mass-asymmetry, though
the relative size of these effects may vary as the merger
proceeds. Unlike the head-on collision case, the direction
of thrust should vary as the system revolves in inspiral-
ing mergers, so that we cannot infer that the net effects
of spin- and mass-asymmetry will produce collinear con-
tributions to the overall kick. Motivated by these ob-
servations we will consider our data with the hypothesis
that the magnitudes of the kicks induced by spin- and
mass-asymmetries each scale independently with the PN-
predicted scaling, but that the directional alignment of
these two contributions to the kick may differ by some
angle θ. The total kick may then be of the form
v=V0[32q2/(1 + q)5]
?
where K = k(qˆ a1− ˆ a2). The parameter V0 gives the
overall scaling of the kick (note that the factor in brack-
ets becomes unity for q = 1), while k gives the relative
scaling of the kick contributions from spin- and mass-
asymmetries.This expression amounts to a general-
ization of the post-Newtonian-inspired kick formula dis-
cussed by Favata et al. (2004) and Koppitz et al. (2007),
which was consistent with collinearity of the two kick
contributions, θ = 0.
We have tested the simple formula (3) with our
own kick speeds, together with published kicks from
Koppitz et al. (2007) and Herrmann et al. (2007a), for
a total of 17 independent data points. Without more
precise knowledge of the uncertainties for each measure-
ment it is not possible to do a true statistical fit. As
a substitute, however, we have obtained values for the
three free parameters V0, θ, and k by minimizing the to-
tal of a χ2-like quantity ∝?
kick speed for the ith combination of parameters. The
proportionality constant is chosen so that the minimum
of “χ2” is 14, equal to the number of degrees of freedom.
With this procedure, our best fit to all simulations cur-
rently reported gives V0= 276km s−1, θ = 0.58 rad, and
k = 0.85. The minimal regions containing 95% of the
probability for each parameter are V0= 267−294km s−1,
θ = 0.45 − 0.65 rad, and k = 0.8 − 0.89. In Table 2 we
compare the predictions of this formula to our results and
those of other groups who have explored prograde or ret-
rograde spins. We see that this simple formula performs
well, with errors less than ∼ 10% in all cases.
We can estimate the uncertainty in θ, and judge how
strongly we can rule out a constant θ = 0, by mak-
ing the conservative assumption that all the numerical
kick results have 10% statistical errors. A chi-squared
analysis then indicates that at one standard deviation
θ = 0.58 ± 0.8 rad. We also find that, formally, θ = 0
is ruled out at > 5σ and θ = π/2 (sum in quadrature)
is ruled out at > 12σ, but this far from our best values
the error contours are clearly non-Gaussian. It is pos-
sible that a more complicated model (e.g., one in which
θ depends on the mass ratio) is a better representation
than θ=constant, but no motivation for this exists in the
current data.
The success of our fit in describing the existing data
suggests that this simple expression may, for many astro-
×
(1 − q)2+ 2(1 − q)K cosθ + K2,(3)
i(vpred,i− vnum,i)2/vnum,i,
where vpred,iis the predicted, and vnum,iis the measured,
TABLE 2
Predicted versus computed kick speed. Runs labeled
“S0.##” are taken from Herrmann et al. (2007a),
while runs labeled “r#” are taken from
Koppitz et al. (2007).
Runqˆ a1
ˆ a2
vnum
vpred
|∆v|
vnum(%)
NE−−
NE−+
NE0−
NE00
NE0+
NE+−
NE++
EQ+−
0.654
0.653
0.645
0.677
0.645
0.655
0.654
1.001
-0.201
-0.201
0.000
0.000
0.000
0.201
0.201
0.198
-0.194
0.193
-0.195
0.000
0.194
-0.194
0.194
-0.198
116.3
58.5
167.7
95.8
76.9
188.6
83.4
89.8
119.5
58.2
153.1
98.6
71.7
181.9
92.4
92.6
2.7
0.5
8.7
2.9
6.8
3.6
10.8
3.2
S0.05
S0.10
S0.15
S0.20
1.000
1.000
1.000
1.000
0.200
0.400
0.600
0.800
-0.200
-0.400
-0.600
-0.800
96.0
190.0
285.0
392.0
93.8
187.6
281.5
375.3
2.3
1.2
1.2
4.3
r0
r1
r2
r3
r4
1.000
0.917
0.872
0.848
0.841
-0.584
-0.438
-0.292
-0.146
0.000
0.584
0.584
0.584
0.584
0.584
260.0
220.0
190.0
140.0
105.0
274.0
220.8
178.1
141.9
110.4
5.4
0.3
6.3
1.4
5.1
physical simulations, adequately describe the dependence
of the kicks on the portion of mass-ratio and spin param-
eter space that we have studied. However, recent simu-
lations have suggested that the dominant component of
the kick may be out of the orbital plane deriving from
spins which lie in the orbital plane (Campanelli et al.
2007a), a configuration outside the parameter space we
have studied. Gonzalez et al. (2007b) have shown that
such a configuration can produce kicks which may ex-
ceed 2500 km s−1directed out of the orbital plane.
These results apparently confirm the early predictions of
Redmount & Rees (1989) that out-of-plane kicks would
be particularly significant.
Since initial submission of this paper, Campanelli et al.
(2007a) have suggested a combined formula, which gen-
eralizes our formula (3) to include out-of-plane kicks
(v||in their notation) of the kind discussed above. If
kick speeds > 1000km s−1are common in comparable-
mass mergers of black holes with substantial spin, this
comes into apparent conflict with the observation that
essentially all galaxies with bulges appear to have super-
massive black holes in their cores, since galactic escape
speeds tend to be < 1000km s−1(see Ferrarese & Ford
2005 for a review of supermassive black holes and their
correlation with galactic properties). It seems unlikely
that most supermassive black holes have low enough
spins to guarantee small kicks, given evidence such
as broad Fe Kα lines from a number of black holes
(Iwasawa et al. 1996; Fabian et al. 2002; Miller et al.
2002; Reynolds & Nowak 2003; Reynolds et al. 2005;
Brenneman & Reynolds 2006) as well as overall argu-
ments from the inferred high average radiation efficiency
of supermassive black holes (So? ltan 1982; Yu & Tremaine
2002). Bogdanovic et al. (2007) suggest that torques
from gas-rich mergers tend to align the spins of the holes,
and thus lead to small kicks, but no known preferred
alignment exists for gas-poor mergers. More exploration
of the parameter space of spin kicks is clearly necessary.

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5
This work was supported in part by NASA grant 06-
BEFS06-19 and NSF grant AST 06-07428. The simu-
lations were carried out using Project Columbia at the
NASA Advanced Supercomputing Division (Ames Re-
search Center) and at the NASA Center for Computa-
tional Sciences (Goddard Space Flight Center). B.J.K.
was supported by the NASA Postdoctoral Program at
the Oak Ridge Associated Universities. S.T.M. was sup-
ported in part by the Leon A. Herreid Graduate Fellow-
ship.
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