Download - SUPERMASSIVE BLACK HOLE BINARIES AS GALACTIC BLENDERS …nicadd.niu.edu/~bterzic/PHYS652/Article_04.pdf · SUPERMASSIVE BLACK HOLE BINARIES AS GALACTIC BLENDERS HenryE.Kandrup DepartmentofAstronomy,DepartmentofPhysics,andInstituteforFundamentalTheory,Published

SUPERMASSIVE BLACK HOLE BINARIES AS GALACTIC BLENDERS

Henry E. Kandrup

Department of Astronomy, Department of Physics, and Institute for Fundamental Theory,University of Florida, Gainesville, FL 32611

Ioannis V. Sideris

Department of Physics, Northern Illinois University, DeKalb, IL 60115

Balsa Terzic

Department of Astronomy, University of Florida, Gainesville, FL 32611

and

Courtlandt L. Bohn

Department of Physics, Northern Illinois University, DeKalb, IL 60115; andFermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510

Received 2003March 9; accepted 2003 July 8

ABSTRACT

This paper focuses on the dynamical implications of close supermassive black hole binaries both as anexample of resonant phase mixing and as a potential explanation of inversions and other anomalous featuresobserved in the luminosity profiles of some elliptical galaxies. The presence of a binary comprised of blackholes executing nearly periodic orbits leads to the possibility of a broad resonant coupling between theblack holes and various stars in the galaxy. This can result in efficient chaotic phase mixing and, in manycases, systematic increases in the energies of stars and their consequent transport toward larger radii. Allow-ing for a supermassive black hole binary with plausible parameter values near the center of a spherical, ornearly spherical, galaxy characterized initially by a Nuker density profile enables one to reproduce inconsiderable detail the central surface brightness distributions of such galaxies as NGC 3706.

Subject headings: galaxies: evolution — galaxies: kinematics and dynamics — galaxies: nuclei —galaxies: structure — stellar dynamics

1. INTRODUCTION AND MOTIVATION

Understanding the dynamical implications of a super-massive black hole binary near the center of a galaxy isimportant both because of the insights the problem can shedon physical processes associated with a time-dependentpotential and because, even if it itself is not resolvable obser-vationally, the binary can have directly observable effects.

As is well known to nonlinear dynamicists, a time-dependent potential can induce significant amounts oftime-dependent ‘‘ transient chaos,’’ an interval during whichorbits exhibit an exponentially sensitive dependence on ini-tial conditions, and resonant couplings between the naturalfrequencies of the time-dependent potential and the fre-quencies of the chaotic orbits can trigger efficient ‘‘ resonantphase mixing ’’ (Kandrup, Vass, & Sideris 2003). Like‘‘ ordinary ’’ chaotic phase mixing (e.g., Kandrup &Mahon1994; Merritt & Valluri 1996), this resonant mixing canfacilitate a rapid shuffling of orbits on different constant-energy hypersurfaces. Even more importantly, however,because the potential is time-dependent the energies of indi-vidual orbits are not conserved, so that resonant mixing canalso facilitate a shuffling of energies between differentconstant-energy hypersurfaces.

For this reason, resonant phase mixing has importantimplications for collective relaxation in nearly collisionlesssystems (Kandrup 2003), for example, holding forth the pros-pect of explaining from first principles the striking efficacy ofviolent relaxation (Lynden-Bell 1967) found in simulationsand inferred from observations (see, e.g., Bertin 2000). That

large-scale collective oscillations could trigger very efficientviolent relaxation has been shown in the context of onesimple model, namely, orbits of stars in a Plummer spheresubjected to a systematic time dependence that eventuallydamps (Kandrup et al. 2003). The binary black hole problemprovides a complementary example of how smaller scale timedependences can also have a surprisingly large effect.

The binary black hole problem is also interesting becausethe binary can have directly observable consequences. Thefact that energy is not conserved implies the possibility ofreadjustments in the density profile of stars near the centerof a galaxy. In many cases this energy nonconservationmeans that on average, stars near the center gain energy,which implies a systematic transport of luminous matternear the black holes out to larger radii. To the extent, how-ever, that mass traces light, such changes in the density dis-tribution translate into predicted changes in the observedsurface brightness distribution because of the presence ofsuch a binary.

In particular, for reasonable choices of black hole massesand orbital parameters, the binary can actually cause an‘‘ inversion ’’ in the surface brightness profile, so that surfacebrightness is no longer a monotonically decreasing functionof distance from the center. Indeed, the simplest models thatone might envision are adequate to reproduce distinctivefeatures observed in the brightness distributions of suchgalaxies as NGC 3706, as reported in Lauer et al. (2002).

The first half of this paper, x 2, considers the binary blackhole problem as an example of how a time-dependent poten-tial can facilitate efficient phase mixing in a galaxy.We focus

The Astrophysical Journal, 597:111–130, 2003 November 1

# 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.

111

on two sets of models, namely, the pedagogical example of aconstant-density ellipsoid, corresponding to an anisotropicoscillator potential, and more realistic cuspy density profilesconsistent with what has been inferred from high-resolutionphotometry (e.g., Lauer et al. 1995).

One important issue here involves determining as a func-tion of amplitude (i.e., black hole masses) and frequency(i.e., orbital period) when the time-dependent perturbationcan have a significant effect. A second involves determiningthe degree to which the efficacy of energy and mass trans-port reflects the degree of chaos exhibited by the orbits, inboth the presence and the absence of the perturbation. Towhat extent, for example, does efficient energy transportrequire that a large fraction of the orbits in the time-dependent potential be chaotic? Does resonant phase mix-ing rely crucially on the presence of transient chaos?

Another issue involves determining the extent to whichthe bulk manifestations of a black hole binary vary forspherical, axisymmetric, and nonaxisymmetric (e.g., tri-axial) galaxies. Is it true, for example, that spherical andnearly spherical systems are impacted less by the presence ofa supermassive binary since, in the absence of the binary, allor almost all of the orbits are regular? In a similar vein, onewould like to understand the extent to which the effects ofthe binary depend on the steepness of the cusp; perhapsmost importantly, it would seem crucial to determine howthe size of the ‘‘ sphere of influence ’’ of the binary dependson the size of the black hole orbits and their masses. Perhapsthe most important conclusion here is that this sphere canbe much larger than the size of the black hole orbits. Forplausible choices of parameter values, black holes movingalong orbits with size �rh can significantly impact thedensity distribution at radii as large as�10rh 20rh or more.

All these issues have important implications for determin-ing when a supermassive black hole binary might beexpected to have observable consequences. The second halfof the paper, xx 3 and 4, considers these consequences.Section 3 considers the generality of the simple models con-sidered in x 2, which assume circular orbits and equal-massblack holes, and then focuses on direction-dependent effectsthat must be understood to determine how potentiallyobservable quantities depend on the relative orientation ofthe observer and the binary.

Section 4 focuses in detail on one specific observable pre-diction, namely, that supermassive black hole binaries canalter the density distribution near the center of a galaxy.This involves (1) generating N-body realizations of densitydistributions consistent with a Nuker law (Lauer et al.1995), (2) evolving these N-body systems in the fixed time-dependent potential corresponding to the galaxy plus orbit-ing black holes, (3) determining how the initial densitydistribution changes over the course of time, and (4)presuming that mass traces light, integrating the resultingdensity distribution along the line of sight to obtain a sur-face brightness profile. These are not self-consistent compu-tations, but they can at least provide strong indications as towhat the expected effects of the binary would be. The crucialpoint, then, is that such an exercise results generically inbrightness distributions that resemble qualitatively theforms reported in Lauer et al. (2002) and that by fine-tuningparameters within a reasonable range, one can reproducemany of the details of what is actually observed. Section 5summarizes the principal conclusions and discussespotential implications.

2. DYNAMICAL EFFECTS OF SUPERMASSIVEBLACK HOLE BINARIES

2.1. Description of the Experiments

The computations described here involve orbits evolvedin potentials of the form

Vðx; y; zÞ ¼ V0ðx; y; zÞ �M

jr� r1ðtÞj� M

jr� r2ðtÞj; ð1Þ

where V0 is time-independent and r1 and r2 correspond tocircular orbits in the x-y plane, i.e.,

x1ðtÞ ¼ rh sin!t ; y1ðtÞ ¼ rh cos!t ; z1ðtÞ ¼ 0 ; ð2Þ

and r2 ¼ �r1. Some of the computations focus on aharmonic oscillator potential

V0ðx; y; zÞ ¼ 12m

2 ; ð3Þ

with

m2 ¼�x

a

�2

þ�y

b

�2

þ�z

c

�2

: ð4Þ

Others focus on more realistic potentials of the form

V0ðx; y; zÞ ¼ � 1

ð2� �Þ 1� m2��

ð1þmÞ2��

" #; ð5Þ

with � a cusp index assumed to satisfy 0 � � � 2. The axisratios a, b, and c are selected of order unity.

The assumptions that the black holes are in circular orbitsand that they have equal masses might appear an extremeidealization. However, as discussed in x 3, it appears thatrelaxing these assumptions does not change the principalconclusions. This model appears structurally stable towardmodest changes in the orbital parameters of the binary.

For a ¼ b ¼ c ¼ 1, equation (5) reduces to the sphericalDehnen (1993) potential with unit mass, and quite gener-ally, for large r, V ! �1=m. Thus, for axis ratios of orderunity, one can interpret equation (5) as the potential for agalaxy with mass Mg � 1:0. For nonspherical systems,equation (5) yields density distributions different fromMerritt & Fridman’s (1996) triaxial Dehnen models, in thatit is V, rather than �, that is constrained to manifestellipsoidal symmetry.

This potential is unrealistic in that, for large radii, V doesnot become spherically symmetric, and one can also arguethat it is unrealistic in the sense that, assuming mass traceslight, the isophotes become peanutty for axis ratios far fromspherical. Given, however, that one is interested primarilyin physical processes in the central portions of the galaxy,the r ! 1 asymptotic behavior is largely unimportant, andit should be recalled that the isophotes in ‘‘ real ’’ galaxiestend to manifest systematic deviations from ellipticity (e.g.,Kormendy & Bender 1996). This potential has the hugeadvantage that, unlike Merritt & Fridman’s nonsphericalDehnen potential, it can be expressed analytically, thusreducing by 2 orders of magnitude or more the time requiredfor orbital integrations. Moreover, as discussed in x 4, forthe case of spherical symmetry the behavior of orbits in thispotential is very similar to orbits evolved in the potentialassociated with a Nuker law, at least for those choices ofNuker parameters for which the potential can be expressedanalytically.

112 KANDRUP ET AL. Vol. 597

For spherical systems with a ¼ b ¼ c ¼ 1,

MðrÞ ¼ r

1þ r

� �3��

ð6Þ

is the mass within r of the galactic center. For axis ratiosof order unity, equation (7) also provides a reasonableestimate for moderately nonspherical systems.

The computations here involved black hole masses inthe range 0:005 � M � 0:05 and radii satisfying 0:005 �rh � 0:5. Following Merritt & Fridman’s (1996) normaliza-tion for their triaxial Dehnen model, one can translate thedimensionless model into physical units by defining thecorrespondence

t ¼ 1 $ 1:46� 106M�1=211 a

3=2kpc yr ; ð7Þ

where

M11 ¼M

1011 M�; akpc ¼

a

1 kpc:

One can identify an energy-dependent dynamical time tDfollowing either Merritt & Fridman, who related it to theperiod of a specific type of regular orbit, or Kandrup &Siopis (2003), who proposed a prescription based on thetimes between turning points in representative orbits. Thosetwo prescriptions yield results in agreement at the 5% levelor better. More generally, for axis ratios of order unity, atleast for small radii the angle-averaged density and energydistributions are relatively similar to those associated with‘‘ true ’’ maximally triaxial Dehnen models, so that adynamical time tDðEÞ can be estimated to within 20% or sofrom Table 1 in Merritt & Fridman (1996). This implies, forexample, that for � ¼ 1:0, a time t ¼ 512 corresponds toroughly 100tD for stars in the 20% mass shell or, equiva-lently,�8� 108M

�1=211 a

3=2kpc yr.

A realistic value for the frequency can be estimated easily.Suppose, for example, that a ¼ b ¼ c ¼ 1:0. If M5MðrhÞ,whereMðrhÞ is the galactic mass contained within radius rh,the black holes can be viewed as test particles moving in thegalactic potential. This implies that

!2 ¼ r��h ð1þ rhÞ��3 ; ð8Þ

so that, for example, ! ¼ r�1=2h ð1þ rhÞ�1 for � ¼ 1:0. If,

alternatively, M4MðrhÞ, the potential associated with thegalaxy can be neglected, and one is reduced de facto tothe circular, equal-mass two-body problem, for which! ¼ ðM=4r3hÞ

1=2. For � ¼ 1:0, M ¼ 0:01, and rh ¼ 0:05,fiducial values considered in many of the computations, thegalactic potential can be neglected in a first approximation,so that ! � ð20Þ1=2 � 4:47.

However, for much of this section, ! is viewed as a freeparameter so that, for fixed amplitude and geometry, onecan explore the response as a function of driving frequency.This enables one to determine the extent to which theresponse manifests a sensitive dependence on frequency,which can provide important insights into the resonantcouplings generating the response.

We focus primarily on the statistical properties of repre-sentative orbit ensembles, integrated from sets of �1600initial conditions. These were generated by uniformlysampling a specified constant-energy hypersurface asdefined in the limit M ! 0 using an algorithm described in

Kandrup & Siopis (2003). Allowing for the black holeschanges the initial energies, so that one is de facto samplinga slightly thickened constant-energy hypersurface. The ini-tial conditions were integrated forward for a time t ¼ 512.The integrations also tracked the evolution of a small initialperturbation, periodically renormalized in the usual fashion(e.g., Lichtenberg & Lieberman 1992), so as to extractestimates of the largest finite-time Lyapunov exponent.

For each simulation, specified by a, b, c,M, rh, and !, thefollowing quantities were extracted:

1. The fraction f of ‘‘ strongly chaotic ’’ orbits, estimatedas in Kandrup & Siopis (2003). As discussed in Kandrupet al. (2003), because the potential is time-dependent it isoften difficult to make an absolute distinction betweenregular and chaotic orbits, although it is relatively easy toidentify orbits that are strongly chaotic.2. The mean value h�i of the finite-time Lyapunov

exponents for the strongly chaotic orbits.3. The mean value h�Ei of the energy shift

�E � EðtÞ � Eð0Þ for all the orbits at various times t > 0.4. The dispersion ��E associated with these shifts.

The data were also analyzed to determine the functionalforms of h�EðtÞi and ��EðtÞ and to search for correlationsbetween changes in energy and values of finite-timeLyapunov exponents for individual orbits within a singleensemble.

Other integrations tracked phase mixing in initially local-ized ensembles, so as to determine the extent to which suchmixing resembles ordinary chaotic phase mixing in a time-independent potential (e.g., Merritt & Valluri 1996;Kandrup 1998) or resonant phase mixing in a galaxy sub-jected to large-scale bulk oscillations (Kandrup et al. 2003).

2.2. Statistical Properties of Orbit Ensembles

Overall, as probed by the shuffling of orbital energies,there is a broad and comparatively efficient resonantresponse. For fixed values of a, b, c, M, and rh, the range ofinteresting frequencies ! can be 2 orders of magnitude ormore in breadth. Fine-tuning of ! is not needed to trigger anefficient shuffling of energies. However, the resonance canexhibit substantial structure, especially for the case of aspherically symmetric oscillator potential. Superposed on asmooth overall trend, quantities like h�Ei can exhibit acomplex, rapidly varying dependence on !.

Consider, for example, Figure 1, which plots h�Ei and��E as functions of ! at time t ¼ 512 for two oscillatormodels, one spherical and the other triaxial. Both haveM ¼ 0:05 and rh ¼ 0:3. The curves for the two modelshave envelopes with a comparatively simple shape, but forthe spherical model, an enormous amount of substructureis superposed. This substructure reflects the fact that allunperturbed orbits oscillate with the same frequency.Indeed, close examination reveals that the resonances areassociated with integer and (to a lesser degree) half-integervalues of !, harmonics of the unperturbed natural fre-quency ! ¼ 1:0. In an axisymmetric system, there are twonatural frequencies, which can yield a yet more complexresponse pattern. If, however, M and rh are chosen largeenough to elicit a significant response, the resonancestypically broaden to the extent that much, if not all,that structure is lost. Allowing for a fully triaxial systemleads to three unequal frequencies, which yields such a

No. 1, 2003 SUPERMASSIVE BLACK HOLE BINARIES 113

plethora of harmonics that, even for comparatively weakresponses, the short-scale structure is largely lost.

It is evident from Figure 1 that although h�Ei is substan-tially larger for the triaxial than for the spherical model, ��E

is comparable. What this means is that even though thespherical model leads to a smaller systematic shifting inenergies, the energies of orbits in these two models areshuffled to a comparable degree. The observed differences inh�Ei do not reflect the fact that the nonspherical model istriaxial. Rather, they appear again to reflect the fact that,for a spherical system, there is only one characteristic fre-quency for the unperturbed orbits. Modest deviations fromspherical symmetry, be these axisymmetric or not, sufficetypically to yield amplitudes more closely resembling Figure1c than Figure 1d.

As indicated in Figure 2, there is a threshold value ofM below which no substantial response is observed, andsimilarly, the response ‘‘ turns off ’’ for higher energyorbits that spend most of their time far from the binary.However, for interesting choices of M, as probed, forexample, by h�Eð!Þi or ��Eð!Þ, the resonance has a char-acteristic shape. As the frequency increases from ! ¼ 0,h�Ei and ��E exhibit a rapid initial increase, peak at amaximum value, and then begin a much slower decrease.For fixed parameter values, the value of the frequencytriggering the largest response is roughly independent ofmass, but it is true that, for larger M, the relative decreasein h�Eð!Þi and ��Eð!Þ with increasing ! is slower than forsmaller M. This is consistent with the notion that forlarger amplitude perturbations, higher order harmonicsbecome progressively more important. Figure 2 alsoshows that the peak frequency is a decreasing function ofrh. In particular, when the black holes are closer togetherone requires larger ! to elicit a significant response.

Efficient shuffling of energies seems tied unambiguouslyto the presence of large amounts of chaos, as probed by thefraction f of strongly chaotic orbits and, especially, the sizeof a typical finite Lyapunov exponent h�i. Large f and h�ido not guarantee large changes in energies, but they are anessential prerequisite. In some cases, notably nearly spheri-cal systems, f and h�i are very small in the limit ! ! 0. As !increases, however, f and especially h�i also increase, andfor values of ! sufficiently large to trigger an efficientresponse, the ensemble will be very chaotic overall. Forvalues of ! in the resonant region, f and h�i tend to exhibitonly a comparatively weak dependence on !.

Orbit ensembles evolved in spherical, axisymmetric, andtriaxial Dehnenesque potentials exhibit resonance patternsquite similar both to one another and to the patternsobserved in nonspherical oscillator models, although somerelatively minor differences do exist. Figure 3 exhibits datafor two such models, one spherical and the other triaxial,each with � ¼ 1:0, rh ¼ 0:05, and M ¼ 0:01. The modelswere both generated for ensembles of initial conditions withE ¼ �0:70 and hrini � 0:33. The black hole radius rh ¼ 0:05corresponds roughly to the 0.2%mass shell.

The triaxial oscillator model in Figure 1 and theDehnenesque models in Figure 3 are representative in thatmodest changes in the parameters of the binary do not leadto significant qualitative changes in the response. Equallyimportantly, however, they are also robust toward changes inaxis ratio. Axisymmetric and slightly triaxial Dehnenesquemodels (e.g., as nonspherical as a2 ¼ 1:05, b2 ¼ 1:00, andc2 ¼ 0:95) yield results very similar to the spherical case.Other ‘‘ strongly ’’ triaxial models yield results similar to theparticular triaxial model exhibited here.

Even for spherical and nearly spherical systems, the rela-tive measure f of strongly chaotic orbits tends to be large

Fig. 1.—(a) Mean shift in energy h�Ei for all the orbits in a 1600 orbit ensemble with E ¼ 0:87 and hrini � 0:86, evolved in a spherical oscillator potentialwith M ¼ 0:05, rh ¼ 0:3, and a2 ¼ b2 ¼ c2 ¼ 1:0, plotted as a function of frequency !. (b) Dispersion ��E for all the orbits. (c, d ) Same as (a, b) for orbitsintegrated in a potential with a2 ¼ 1:33, b2 ¼ 1:0, and c2 ¼ 0:80 and an ensemble withE ¼ 0:87 and hrini � 0:89.


even for ! ¼ 0, which corresponds to stationary but sepa-rated black holes. One does not require a strong timedependence to generate a large measure of chaotic orbits.However, it is true that, for axisymmetric and other nearlyspherical systems, the size of a typical Lyapunov exponenttends to be considerably smaller than for strongly triaxialsystems.

In significantly triaxial models, the degree of chaos, asprobed by f or h�i, is a comparatively flat function of !.By contrast, in nearly spherical and axisymmetric sys-tems, the degree of chaos, especially as probed by h�i,increases rapidly with increasing ! until it becomescomparable to the degree of chaos exhibited by stronglytriaxial models. The obvious inference is that when thesystem is nearly spherical or axisymmetric, the timedependence associated with the orbiting binary isrequired to give the chaotic orbits particularly largeLyapunov exponents.

An analogous result holds for the mean energy shift h�Ei.Quite generally, h�Ei ! 0 for ! ! 0 and increases withincreasing !. However, the initial rate of increase is typicallymuch larger for significantly triaxial models than for nearly

spherical and axisymmetric systems. This has an importantpractical implication: because larger frequencies arerequired to trigger the resonance in galaxies that are nearlyaxisymmetric, in nearly spherical or axisymmetric galaxiesblack holes of given mass must be in a tighter orbit beforethey can trigger a significant response.

2.3. Shuffling of Energies as a Diffusion Process

Overall, the shuffling of energies induced by the blackhole binary is diffusive, although the basic picture dependson the amplitude of the response. When changes in energyexperienced by individual orbits are relatively small, the dis-persion tends to grow diffusively, ��E / t1=2. The mean shiftin energy typically grows more quickly, being reasonablywell fitted by a linear growth law h�Ei / t. Alternatively,when the response is stronger, it is the mean shift that growsdiffusively, h�Ei / t1=2; whereas the dispersion is well fittedby a growth law ��E / t1=4. Examples of both sorts ofbehavior can be seen in Figure 4.

One might have supposed that since the shuffling ofenergies is associated with the presence of chaos, changes in

Fig. 2.—(a) Mean energy shift h�Ei for the same ensemble used to generate Figs. 1c and 1d, integrated with rh ¼ 0:3, a2 ¼ 1:33, b2 ¼ 1:0, c2 ¼ 0:80, andM ¼ 0:05, 0.0281, 0.0158, and 0.005 (top to bottom). (b) Mean energy shift h�Ei for the same ensembles, M ¼ 0:005 (solid line), 0.0158 (dashed line), 0.0281(dot-dashed line), and 0.05 (triple-dot–dashed line), now expressed in units of the maximum shift h�Emaxi. (c) Mean energy shift h�Ei for integrations witha2 ¼ 1:33, b2 ¼ 1:0, c2 ¼ 0:80,M ¼ 0:05, and rh ¼ 0:4, 0.3, 0.2, and 0.1 (curves peaking from left to right). (d ) Mean energy shift h�Ei expressed in units of themaximum shift h�Emaxi; rh ¼ 0:1 (solid line), 0.2 (dashed line), 0.3 (dot-dashed line), and 0.4 (triple-dot–dashed line).


energy should grow exponentially. This, however, does notappear to be the case, at least macroscopically. The initialresponse of the orbits (t < 5tD or so) may be exponential,but it is evident that overall, the response is diffusive. Time-dependent chaos does not trigger exponentially fast mixingin energies. However, it can still be extremely important inthat it allows comparatively efficient shufflings of energiesthat would be completely impossible in a time-independentHamiltonian system.

One final point should be stressed. That changes in energyare diffusive, reflecting a slow accumulation of energy shifts,corroborates a fact also evident from an examination of

individual orbits: changes in energy experienced byindividual orbits do not result from single close encounterswith the black holes. Instead, they really do reflect reso-nance effects associated with the time-dependent potential.

2.4. Correlations among Orbital Properties for DifferentOrbits within an Ensemble

Orbits with smaller finite-time Lyapunov exponents �tend to exhibit energy shifts that are smaller in magnitudej�Ej. Orbits with large � can experience both large and smallnet changes in energy. As has been observed in other time-dependent potentials (Terzic & Kandrup 2003), the fact that

Fig. 3.—(a) Fraction f of strongly chaotic orbits in a 1600 orbit ensemble with initial energy E ¼ �0:70 and mean initial radius hrini � 0:33, evolved in aspherically symmetric Dehnen potential with � ¼ 1:0, M ¼ 0:01, r ¼ 0:05, and a2 ¼ b2 ¼ c2 ¼ 1:00. (b) Mean value h�i of the largest finite-time Lyapunovexponent for the strongly chaotic orbits. (c) Mean shift in energy h�Ei for all the orbits. (d ) Dispersion ��E for all the orbits. (e–h) Same as (a–d ) for orbitsintegrated in a potential with a2 ¼ 1:25, b2 ¼ 1:00, and c2 ¼ 0:75, again for an ensemble withE ¼ �0:70 and hrini � 0:33.


an orbit is chaotic does not necessarily imply that it willexhibit large, systematic drifts in energy over a finite timeinterval. However, the energy shifts in orbits with small �are invariably small.

When the response is weak, so that the dispersion of theensemble evolves diffusively, changes in energy exhibited byindividual orbits are comparably likely to be positive or neg-ative. However, when the response is stronger, so that themean shift evolves diffusively, the energies of individualorbits tend instead to increase systematically.

When the response is relatively weak and changes inenergy are equally likely to be either positive or negative,the distribution of energy shifts nð�EÞ is typically well fittedby a Gaussian with mean roughly equal to zero. However,

when the response becomes stronger, nð�EÞ becomes dis-tinctly asymmetric and cannot be well fitted by a Gaussian,even allowing for a nonzero mean.

Correlations between the degree of chaos and the degreeof energy shuffling experienced by individual orbits areperhaps best illustrated by extracting energy shifts �E at dif-ferent times ti for individual orbits, computing the meanand dispersion h�Ei and ��E associated with the resultingtime series f�EðtiÞg, and demonstrating how these momentscorrelate with the value of the finite-time Lyapunov expo-nent �. Examples of such an analysis are exhibited in Figure5. The obvious point is that the moments are invariablysmall when � is small, whereas larger � typically implieslarger values of jh�Eij and ��E .

Fig. 4.—(a) Quantity �2�E , where ��EðtÞ is the time-dependent spread in energy shifts associated with an ensemble of orbits evolved in an oscillator potentialwithM ¼ 0:05, rh ¼ 0:3, a2 ¼ 1:33, b2 ¼ 1:0, c2 ¼ 0:80, and ! ¼ 0:5. (b) Quantity h�EðtÞi2 for the same ensemble. (c–d, e–f, g–h, i–j) Same as (a–b) for ! ¼ 1:0,2.0, 4.0, and 8.0, respectively.


2.5. Chaotic and Resonant PhaseMixing

The time-dependent potential associated with the blackhole binary can alter ordinary chaotic phase mixing in atleast two important ways.

The time-dependent potential tends to increase boththe fraction of chaotic orbits and the size of a typicalLyapunov exponent. If a galaxy is in a (nearly) time-

independent (near-)equilibrium state, the relative measureof (at least strongly) chaotic orbits should be relativelysmall, since presumably one requires large measures ofregular (or nearly regular) orbits to provide the‘‘ skeleton ’’ of the interesting structures associated withthose chaotic orbits that are present (Binney 1978). Intro-ducing a time-dependent perturbation leads oftentimes toa significant increase in the relative measure of chaotic

Fig. 5.—(a) Scatter plots relating ��E and �, where ��E represents the dispersion associated with the time-dependent �EðtÞ for an individual orbit over theinterval 0 < t < 512. The orbits are the same that were used to generate Fig. 3, integrated with ! ¼ 0:5. (b) Scatter plots relating h�Ei and �, where h�Eirepresents the mean value of �EðtÞ, computed for the same orbits as in (a). (c–d, e–f, g–h, i–j) Same as (a–b) for ! ¼ 1:0, 2.0, 4.0, and 8.0, respectively.


orbits. Moreover, even when the time-dependence doesnot significantly increase the measure of chaotic orbits, itcan make already chaotic orbits more unstable, thusallowing them to mix more efficiently.

Because energy is no longer conserved, the time-depend-ent potential also allows mixing between different constant-energy hypersurfaces, which is completely impossible in theabsence of a time dependence. Overall, this mixing of ener-gies is not as efficient a process as mixing in configuration orvelocity space. However, the resonant mixing of energiesassociated with chaotic orbits still plays an important role.

An example of such resonant phase mixing is provided inthe left-hand panels of Figure 6, which track an initiallylocalized ensemble with E ¼ �0:70 in a spherical Dehnenpotential with � ¼ 1 and a ¼ b ¼ c ¼ 1:0, allowing forblack hole parameters M ¼ 0:005, rh ¼ 0:05, and! ¼ ð10Þ1=2. The right-hand panels track the same ensemble,evolved identically except that ! ¼ 0. Two points are imme-diate. One is that for the realistic case when ! 6¼ 0, a timet ¼ 64, corresponding to �108M

�1=211 a

3=2kpc yr, is sufficient to

achieve a comparatively well mixed configuration. Achiev-ing a comparable degree of mixing for the ! ¼ 0 systemrequires a time t > 512. The other point is that orbits in theensemble evolved with ! 6¼ 0 have diffused to radii r > 0:3,which is impossible for orbits in the ! ¼ 0 ensemble, forwhich energy is conserved.

3. OBSERVATIONAL CONSEQUENCESOF THE DYNAMICS

3.1. Generality of the IdealizedModel

Attention hitherto has focused on the dynamical conse-quences of a supermassive black hole binary, viewed as theprototype of a time-dependent perturbation acting in agalaxy idealized otherwise as a collisionless equilibrium.The object of this and the following section is to considerinstead potentially observable consequences, the mostobvious of which is a changing surface brightness distribu-tion induced by a readjustment in the mass density as starsare transported to larger radii.

In so doing, one can proceed by viewing the host galaxyas a superposition of orbit ensembles with different energiesE and, for various choices of binary parameters, determin-ing when, for any given value of E, the binary can have anappreciable effect, for example, by generating a large energyshift h�Ei. As described already, the response will be largeonly when the size rh of the binary orbit is sufficiently smallthat the total black hole mass M1 þM2 � MðrhÞ. This,however, implies that in a first approximation, the fre-quency of the binary can be estimated neglecting the bulkpotential of the galaxy. Thus, relaxing the assumptions ofequal masses and strictly circular orbits,

! ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM1 þM2

A3

r; ð9Þ

withA the value of the semimajor axis.Perhaps the most obvious question here is simply, for

fixed E and A, how do quantities like h�Ei depend on thetotal mass Mtot ¼ M1 þM2? The answer is that, at leastfor realistic binary black hole masses, i.e., M1 andM2 < 0:01Mgal, h�Ei is a comparatively smooth, monotoni-cally increasing function of Mtot. For very small masses,there is essentially no response, but beyond a critical mass,the precise value of which depends on properties of the hostgalaxy, the dependence is roughly power law in form, i.e.,h�Ei / Mp

tot, with the power p typically in the range1 < p < 2. Examples of this behavior are exhibited in theleft-hand panels of Figure 7, which show the effects ofincreasing the total mass for five different models, onespherical, one prolate axisymmetric, one oblate axisym-metric, and two genuinely triaxial. This particular set ofexamples again incorporates circular orbits and equal blackhole masses, but as discussed below, these assumptions arenot crucial.

Fig. 6.—(a) The x and y coordinates at t ¼ 0 for an initially localizedensemble of orbits with E ¼ �0:70 and hrini � 0:33, evolved in a sphericalDehnen potential with � ¼ 1:0, rh ¼ 0:05, M ¼ 0:005, and ! ¼ ð10Þ1=2.(b–e) Same as (a), but for t ¼ 8, 16, 32, and 64. ( f–j) Same as (a–e), but forstationary black holes, i.e., ! ¼ 0:0.


A second obvious question is, how small must thebinary orbit be in order to elicit a significant response?Physically, one might suppose that the binary was initial-ized in a comparatively large orbit as the result of amerger of two colliding galaxies, but that the orbit slowlydecayed via dynamical friction, allowing the black holesto sink toward the center of the galaxy. However, withinthe context of such a scenario the crucial issues to deter-mine are (1) when the binary can begin to have a largeeffect, i.e., how small the orbit must be, and (2) when theeffects of the binary turn off again. These issues are

addressed in the right-hand panels of Figure 7, whichexhibit h�Ei as a function of rh for the same five galacticmodels used to generate the left-hand panels.

Two points are evident: (1) The binary has its largesteffect when rh is substantially smaller than the typical radiusof the orbits with the specified energy. The ensembles con-sidered each comprised orbits with initial energy E ¼ �0:70and mean radius hri � 0:33, but the maximum response wasobserved for rh � 0:04, i.e., a size roughly 10 times smaller!This reflects the fact that mass and energy transport hasbeen triggered by a resonance rather than by direct binary

Fig. 7.—(a) Mean shift in energy h�Ei computed for an ensemble of orbits with E ¼ �0:70 and hri � 0:33, evolved in a � ¼ 1:0 Dehnen model witha2 ¼ b2 ¼ c2 ¼ 1 in the presence of a supermassive binary comprising two black holes executing strictly circular orbits with rh ¼ 0:05 and different values ofM1 ¼ M2 � M. (b) Mean shift in energy h�Ei for the same ensemble evolved in the same Dehnen model, again allowing for a binary executing circular orbits,but now with M1 ¼ M2 ¼ 0:01 and variable rh. (c, d ) Same as (a, b), but for a model with a2 ¼ b2 ¼ 0:90 and c2 ¼ 1:21. (e, f ) Same, but for a model witha2 ¼ b2 ¼ 1:21 and c2 ¼ 0:64. (g, h) Same, but for a model with a2 ¼ 1:10, b2 ¼ 1:0, and c2 ¼ 0:90. (i, j) Same, but for a model with a2 ¼ 1:25, b2 ¼ 1:0, andc2 ¼ 0:75. In each case, the frequency ! ¼ ½ðM1 þM2Þ=a31=2, withA the semimajor axis.


scatterings of individual stars with the black holes. Oneneeds a very tight binary orbit to get frequencies sufficientlylarge to trigger a significant response. (2) As noted already,for the triaxial models the effects of the binary turn on atsubstantially larger values of rh than for the spherical andaxisymmetric systems. This would suggest that a black holebinary could have an especially large effect in a strongly tri-axial galaxy: since the range of black hole sizes that can havean appreciable effect is substantially larger, the time duringwhich the resonance will act should presumably be longer.

But how generic are the idealized computations describedin x 2? It might not seem unreasonable to assume that theblack holes follow nearly circular orbits, since dynamicalfriction will tend to circularize initially eccentric orbits, butthe assumption of equal-mass black holes is clearly suspect.

Computations show that varying the eccentricity e withinreasonable bounds has only a comparatively minimal effect.Increasing e from values near zero to a value as large ase ¼ 0:5 will not change quantities like h�Ei by more than25%, and in general, the effect is much smaller even than

Fig. 8.—(a) Mean shift in energy h�Ei computed for an ensemble of orbits with E ¼ �0:70 and hri � 0:33, evolved in a � ¼ 1:0 Dehnen model witha2 ¼ b2 ¼ c2 ¼ 1 in the presence of a supermassive binary comprised of two black holes with mass M1 ¼ M2 ¼ 0:01 executing orbits with semimajor axisA ¼ 0:10 and variable eccentricity e. (b) Mean shift in energy h�Ei for the same ensemble evolved in the sameDehnen model, again assumingM1 þM2 ¼ 0:02and a ¼ 0:10 but now allowing for different ratiosM2=ðM1 þM2Þ. (c, d ) Same as (a, b), but for a model with a2 ¼ b2 ¼ 0:90 and c2 ¼ 1:21. (e, f ) Same, but fora model with a2 ¼ b2 ¼ 1:21 and c2 ¼ 0:64. (g, h) Same, but for a model with a2 ¼ 1:10, b2 ¼ 1:0, and c2 ¼ 0:90. (i, j) Same, but for a model with a2 ¼ 1:25,b2 ¼ 1:0, and c2 ¼ 0:75.


this. This is evident, for example, from the left-hand panelsof Figure 8, which were generated for the same five modelsconsidered in Figure 7.

As is evident from the right-hand panels of Figure 8, thereis a substantially stronger, systematic dependence on themass ratio M2=M1. For fixed Mtot ¼ M1 þM2, the largesteffects arise for M1 � M2, but even here the dependence onthe mass ratio is not all that sensitive. In particular, for allbut the triaxial models, the response is a relatively flat func-tion of M2=Mtot for M2=Mtote0:25, so that for fixedM1 þM2, mass ratios 1

3dM2=M1d1 yield comparableresults. It is true that for fixed semimajor axis A and totalmassMtot, the effect of the binary is significantly reduced forM25M1, but the reason for this is obvious: whenM25M1, the more massive black hole is located very nearthe center of the galaxy. This implies, however, that even ifthe binary has a very high frequency, the more massiveblack hole remains too close to the center to have an appre-ciable effect at large radii. The smaller black hole is typicallyfound at much larger values of r, but its mass is too small tohave a significant effect.

3.2. Systematic Changes in Density

Changes in energy induced by transient chaos lead generi-cally to a readjustment in bulk properties such as density,and to the extent that there is an average increase in energy,this readjustment implies a systematic displacement of starsto larger radii. To see how this effect can proceed, one cansample a constant-energy hypersurface to generate a set ofinitial conditions, integrate those initial conditions into thefuture, and then compare angle-averaged radial densitydistributions �ðrÞ generated at various times t � 0.

The left-hand panels of Figure 9 summarize results for amodel with a2 ¼ 1:25, b2 ¼ 1:0, and c2 ¼ 0:75, assumingcircular orbits with M1 ¼ M2 ¼ 0:01, rh ¼ 0:05, and! ¼ ð20Þ1=2. The ensemble was so constructed thatE ¼ �0:70 and hrini � 0:33. The five panels exhibit thedensity distributions at t ¼ 0, 16, 32, 64, and 128, the lastcorresponding physically to �2� 108M

�1=211 a

3=2kpc yr. The

right-hand panels exhibit analogous data for the sameensemble and potential, but with the black holes held fixedin space, i.e., ! � 0.

The density distribution remains essentially unchangedfor the time-independent ! ¼ 0 potential, but the realisticcase with ! ¼ 20ð Þ1=2 leads to a significant density readjust-ment. (Minor changes in the ! ¼ 0 model reflect a modestreadjustment to the insertion of the fixed binary in an equili-brium generated without a binary.) Already by t ¼ 16(11 binary periods), corresponding to an interval�2:5� 107M

�1=211 a

3=2kpc yr, there is a pronounced decrease in

density in the range 0:3 � r � 0:5 and an increase in densityat larger radii. Initially, the trajectories are restrictedenergetically to r � 0:6. By t ¼ 128, more than 13% of thetrajectories are located at r > 1:0.

3.3. The Size of the Sphere of Influence

Figure 9 demonstrates that a black hole binary can signif-icantly impact orbits that spend most of their times at radii4rh. However, the obvious question is, how much larger?To answer this question one can evolve ensembles with avariety of different initial radii and determine their responseas a function of r. The results of two such investigations aresummarized in Figure 10. In each case, the configuration

corresponded to a spherical Dehnen model witha ¼ b ¼ c ¼ 1:0 and a binary with M1 ¼ M2 ¼ 0:005,rh ¼ 0:25, and ! ¼ 0:2828. The left-hand panels are for amodel with � ¼ 0:0, the right-hand panels for � ¼ 1:0.

The sphere of influence is in fact quite large, extendingout to r � 4, even though rh ¼ 0:25. Moreover, it is evidentthat the ensembles that experience the most shuffling inenergies, as probed by hj�Eji and ��E , are precisely thoseensembles with the largest Lyapunov exponent h�i. Indeed,for the � ¼ 0:0 and � ¼ 1:0 models, the rank correlationsbetween the mean shift hj�Eji and the mean exponent h�ifor different ensembles are, respectively, Rðh�Ei; h�iÞ ¼0:615 and 0.613.

It is also clear that the value of the cusp index � has a sig-nificant effect on the details of the response. The value of �does not have a large effect on the size of the binary sphereof influence, but it does impact the amplitude of theresponse and how that response correlates with radius. Inboth cases, there is a significant response for 0:15 � r � 6:0,but the response in this range, as probed by the degree ofshuffling in energies, is somewhat larger for the cuspymodel. Even more strikingly, however, the cusp appears toreduce both the size of the Lyapunov exponents and thedegree of shuffling at very small radii. For the cuspy modelwith � ¼ 1:0, comparatively little shuffling of energies andcomparatively small amounts of chaos are observed at radii5 rh. The very lowest energy stars tend to be more regularand to be less susceptible to resonant mixing.

3.4. Anisotropy

To what extent does the mass transport induced by asupermassive black hole binary depend on direction? Evenif, for example, the host galaxy is modeled as exactly spheri-cal, the binarybreaks the symmetry and, as such, could intro-duce anisotropies into a completely isotropic ensemble ofstars. This is important, since such anisotropies would implythat changes in visual appearance induced by the binarycould depend appreciably on the observer’s viewing angle.

As a simple example, one can consider the direction-dependent density distributions associated with a uniformsampling of a constant-energy hypersurface, which, assum-ing a spherical potential, implies a spherically symmetricdensity distribution and an isotropic distribution of veloc-ities. One example thereof is exhibited in Figure 11, whichwas generated for a � ¼ 1:0 Dehnen model witha ¼ b ¼ c ¼ 1 containing a binary executing a circular orbitin the x-y plane with the ‘‘ correct ’’ Kepler frequency. Fig-ure 11a exhibits spatial distributions at times t ¼ 0 andt ¼ 512; Figure 11b shows the corresponding velocity distri-butions. At t ¼ 0, the spatial and velocity distributions areequal modulo statistical uncertainties; at late times they dif-fer systematically, but it remains true that nðjxjÞ ¼ nðjyjÞand nðjvxjÞ ¼ nðjvyjÞ. There is clearly a systematic outwardtransport of stars in all three directions, but it is also evidentthat there is a larger net effect on the spatial components inthe plane of the orbit. Similarly, there is a modest shift invelocities that, again, is more pronounced in the x and ycomponents. Figures 11c and 11d contain plots of, respec-tively, the x and y and the x and z coordinates at t ¼ 512.These panels confirm that the final distribution is moreextended in the plane of the binary than in the orthogonaldirection. In particular, it could easily be misinterpreted as adisk or a torus.


But what if the host galaxy is already nonspherical? If, forexample, the galaxy is genuinely triaxial, one might supposethat the binary will have settled into a symmetry plane, butassuming that this is the case, there are at least two obviousquestions. (1) How does the overall response depend onwhich symmetry plane? (2) For a binary oriented in a givenplane, to what extent do observable properties depend onviewing angles?

Both these questions were addressed as before by evolv-ing uniform samplings of constant-energy hypersurfacesthat yield a triaxial number density but are still character-ized by an isotropic distribution of velocities. The result ofone such computation is summarized in Figure 12. Figure12a exhibits the initial density distributions; Figures 12b–

12d exhibit the corresponding distributions at t ¼ 512 forthree different integrations, with the binary oriented in thex-y, y-z, and z-x planes, respectively.

Overall the angle-averaged properties of the different sim-ulations are very similar: the mean short-time Lyapunovexponents h�i for the three different runs agree to within10%, and even smaller variations were observed for quanti-ties like h�Ei. Indeed, the shape of the galaxy seems moreimportant than the orientation of the binary. For all threebinary orientations, one observes that the largest effect is inthe x-direction, which corresponds to the long axis, and thesmallest in the short-axis z-direction. The details of theresponse observed here depend to a considerable extent onboth the shape of the potential and the energy of the initial

Fig. 9.—(a) Initial angle-averaged radial density distribution associated with a 4800 orbit sampling of the E ¼ �0:70 constant-energy hypersurface,subsequently integrated in a pseudo-Dehnen potential with � ¼ 1:0, M ¼ 0:01, rh ¼ 0:05, a2 ¼ 1:25, b2 ¼ 1:00, c2 ¼ 0:75, and ! ¼ ð20Þ1=2. (b–e) Density att ¼ 16, 32, 64, and 128, respectively. The dotted line reproduces the initial distribution. ( f–j) Same as (a–e), but for stationary black holes, i.e., ! ¼ 0:0.


ensemble. In particular, for some choices the response islargest in the short-axis rather than long-axis direction.However, it seems true quite generally that the orientationof the binary is comparatively unimportant. There remainsa dependence on viewing angle, but if anything, this effect issomewhat weaker than for the case of spherical systems.

For axisymmetric systems with the binary oriented in thex-y symmetry plane, one finds generically that distributionsin the x and y directions agree to within statistical uncertain-ties, but that the z-direction distributions differ systemati-cally. In some cases (depending on both shape and binaryparameters), there is more mass transport in the z-direction;in others the effect is more pronounced in the x and ydirections. These differences likely reflect the fact that thismass transport is triggered by a resonance. The unperturbed

orbits have different characteristic frequencies in differentdirections, but this would suggest that the resonant couplingcould well be stronger (or weaker) in one direction than inanother.

4. MODELING LUMINOSITY PROFILESIN REAL GALAXIES

4.1. Basic Strategy

The objective here is to show that the physical effects dis-cussed above, seemingly the inevitable consequence of asupermassive black hole binary in the center of a galaxy,could provide a natural explanation of the fact that, in anumber of galaxies that have been observed using WFPC2(e.g., Lauer et al. 2002), the projected surface brightness

Fig. 10.—(a) Mean energy shift h�Ei, computed for ensembles with different radii, for orbits in a spherical Dehnen model with � ¼ 0:0, a ¼ b ¼ c ¼ 1:0,and black hole parameters M ¼ 0:005, rh ¼ 0:25, and ! ¼ 0:2828. Note that the radius r is plotted on a logarithmic scale. (b) Dispersion ��E for the sameensembles. (c) Mean value h�i for each ensemble. (d ) Dispersion ��. (e–h) Same as (a–d ), but for a model with � ¼ 1:0.


distribution in a given direction is not a monotonicallydecreasing function of distance from the center of thegalaxy.

The computations described here are not completely real-istic. As in x 2, they assume black holes of exactly equal massexecuting exactly circular orbits, and the computed orbits oftest stars are not fully self-consistent since one is neglectingboth changes in the form of the bulk potential as stars aredisplaced from their original trajectories and the slow decayof the binary orbit. They do, however, demonstrate thatallowing for a binary of relatively small size, �10 pc, com-prising black holes with mass d1% the mass of the galaxy,leads generically to luminosity dips of the form that havebeen observed. Moreover, fine-tuning parameters within areasonable range of values allows for the possibility of acomparatively detailed (albeit in general nonunique) fit toobservations of specific galaxies.

The basic program is as follows:

1. Generate N-body realizations of a spherical galaxycharacterized by an isotropic distribution of velocities and aNuker (Lauer et al. 1995) density profile �ðrÞ with specifiedparameter values.2. Insert a black hole binary with specified masses

M1 ¼ M2 ¼ M and radius rh. For realistic values of M and

rh, MðrhÞ is typically small compared with the black holemass, so that one can assume, at least approximately, thatthe black holes are executing a Keplerian orbit withfrequency ! ¼ ðM=4r3hÞ

1=2.3. Next, evolve the initial conditions in the fixed time-

dependent potential comprising the Nuker potential plusthe potential of the orbiting black hole binary and track theradial density distribution �ðr; tÞ.4. Finally, assuming that mass traces light, compute line-

of-sight integrals along the density distribution to obtainintegrated surface densities and hence surface brightnessdistributions as functions of time.

Although this approach does not pretend to be com-pletely realistic, it would not seem totally unreasonable toinsert the binary by hand without allowing for the dynamicswhereby it has evolved into a tightly bound orbit near thegalactic center. When the binary orbit is very large, it willhave a comparatively minimal effect. Energy and masstransport becomes important only at comparatively smallradii, where M � MðrhÞ, and becomes unimportant againwhen the radius becomes too small. Most of the actionhappens for a relatively limited range of radii.

Note, moreover, that the assumption MeMðrhÞ tendsto mitigate the fact that the computations are not fully

Fig. 11.—(a) Direction-dependent spatial distributions nðjxjÞ (solid line) and nðjzjÞ (dashed line) at t ¼ 512 generated for a 4800 orbit sampling of theE ¼ �0:70 hypersurface with � ¼ 1:0,M ¼ 0:01, rh ¼ 0:005, a ¼ b ¼ c ¼ 1, and ! ¼ ð20Þ1=2, along with the distribution nðjxjÞ (dot-dashed line) at time t ¼ 0.(b) Corresponding direction-dependent velocity distributions. (c) The x and y coordinates for the ensemble at t ¼ 512. (d ) The x and z coordinates for theensemble at t ¼ 512.


self-consistent: although the bulk forces associated with thegalaxy cannot be neglected at all radii where the binary hasan appreciable effect, they can presumably be neglected, atleast approximately, at the comparatively small radii nearthe binary where the effect of the black holes is strongest.

4.2. The Initial Form of the Density and Potential

Initial attempts at modeling using a spherical Dehnenpotential yielded results in qualitative agreement withobservations. However, comparatively large systematicdeviations were observed, which appeared to reflect the factthat the transition between the inner and outer power-lawprofiles predicted by a Dehnen potential is too gradual torepresent real galaxies. For this reason, models wereconstructed instead using an initial density distributionsatisfying the more general Nuker law (Lauer et al. 1995)

�0ðrÞ ¼ �cr��ð1þ r�Þð��Þ=� : ð10Þ

Dehnen models were recovered for � ¼ 1 and � ¼ 4. Thecentral density �c was chosen so that the total galactic massMg ¼ 1:0. The associated potential VðrÞ satisfies (in unitswithG ¼ 1)

VðrÞ ¼ �4�1

r

Z r

0

�ð~rrÞ~rr2d~rrþZ 1

r

�ð~rrÞ~rr d~rr� �

: ð11Þ

Unfortunately, this potential can be expressed in terms ofelementary functions only for certain choices of � and �,

which means, generically, that orbits must be computedusing an expensive interpolation scheme. This motivated aneffort to seek fits assuming values of � and � for which Vcan be expressed analytically. For the small number of pro-files considered hitherto, reasonable fits were achieved for� ¼ 2 and � ¼ 4, which, for � ¼ 0, yields a potential

VðrÞ ¼ � 2

�

tan�1 r

r; ð12Þ

MðrÞ ¼ 2

�tan�1 r� r

1þ r2

� �: ð13Þ

Most models were constructed assuming MðrhÞ5M1 þM2, so that the approximation of a Keplerianfrequency is typically very good. However, in an effortto allow for the influence of the galactic potential, themodels allowed for a slightly modified frequency ! ¼½M=ð2rhÞ31=2, whereM ¼ M1 þM2 þ 4MðrhÞ.

4.3. Generating a Surface Brightness Distribution

Configuration space was divided into N ¼ 100 equallyspaced concentric shells i. Each shell corresponded to arange of energies Ei�1 < E < Ei, i ¼ 1; . . . ; N, sampledalong the principal axes in the plane of the binary but per-pendicular to the line connecting them. This was done toensure that energy was a monotonic function of radius, sothat shuffling of energies could be related directly to a redis-tribution of orbits in configuration space. Each shell was

Fig. 12.—Direction-dependent spatial distributions nðjxjÞ (solid line), nðjyjÞ (dashed line), and nðjzjÞ (dot-dashed line) generated for a 4800 orbit samplingof the E ¼ �0:62 hypersurface with � ¼ 1:0, M ¼ 0:01, rh ¼ 0:005, a2 ¼ 1:25, b2 ¼ 1:0, c2 ¼ 0:75, and ! ¼ ð20Þ1=2. (a) Distributions at time t ¼ 0.(b–d ) Distributions at t ¼ 512, allowing for a binary orbiting in the x-y, y-z, and z-x planes, respectively.


sampled to select M ¼ 300 initial conditions, which wereintegrated for a time t ¼ 512. Orbital data were recordedperiodically and the new energies used to reassign orbits to(in general) new shells. IfMiðtÞ denotes the number of orbitsin shell i at time t, then

DiðtÞ ¼MiðtÞM

; ð14Þ

the relative fluctuation in number, can be interpreted as adiscretized version of a radial density fluctuation �ðtÞsatisfying

�ðr; tÞ ¼ 1þ �ðr; tÞ½ �0ðrÞ ; ð15Þ

where �0 is the initial density. The fractional change �ðtÞwas interpolated from DiðtÞ using a smooth-curve fittingroutine.

The resulting density �ðr; tÞ was then integrated along theline of sight to generate a surface brightness

lðr; tÞ ¼ 2

�

Z 1

r

�ð~rr; tÞ~rrffiffiffiffiffiffiffiffiffiffiffiffiffiffi~rr2 � r2

p d~rr : ð16Þ

Here � denotes the mass-to-light ratio, which was assumedconstant for the modeling described here.

4.4. Results

Figure 13 exhibits data for a typical model, correspond-ing to a Nuker law with � ¼ 2, � ¼ 4, and � ¼ 0. The binaryparameters are M ¼ 0:005, rh ¼ 0:15, and ! ¼ 0:6086. Thehalf-mass radius is r ¼ 2:264; 75% of the mass is containedwith r ¼ 5.

It is obvious that the binary induces a distinctive signa-ture, characterized by an inversion in both the mass densityand the surface brightness profile. In some cases, especiallywhen � 6¼ 0, the contents of the innermost shells can remainessentially intact. Aside, however, from those innermostshells, one can identify a well-defined sphere of influencewhere the binary has observable effects. For r < r1, there is asystematic underpopulation of stars and hence a dip in lumi-nosity; for r1 < r < r2, there is a systematic overpopulationor bulge resulting from stars transported outward fromradii r < r2. For r > r2, the density and surface brightnessdistribution remain essentially unchanged. Significantly, thesignature, once established, remains largely unchanged inbulk properties. In particular, the dip is comparablyprominent visually at t ¼ 128 and t ¼ 512.

Figure 14 exhibits a more systematic attempt to modelthe surface brightness of NGC 3706, again starting from aNuker law with � ¼ 2, � ¼ 4, and � ¼ 0. Physical distancewas translated into angular separation assuming a scalingsuch that r ¼ 1 corresponds to 0>15 or, given the distanceestimate given by Lauer et al., r � 24 pc. Once againM ¼ 0:005, but now rh ¼ 0:085, which corresponds to aphysical rh � 2:0 pc and an angular separation�0>014. Theorbital frequency ! ¼ 1:567, which implies an orbital period � 4:00.

In this case, the inner dip extends out to �0>10; the bulgeextends to �0>30. On scales e0>30, i.e., re75 pc, thebinary has only a comparatively minimal effect, so that thesurface brightness remains essentially unchanged.

The perturbed Nuker model is quite successful in model-ing the dip and the outer region, where errors in surfacebrightness correspond typically to �l � 0:005 mag or less.In particular, the dip is much better fitted by the perturbed

model than by an unperturbed Nuker model. Both qualita-tively, in that an unperturbed Nuker model requires amonotonically decreasing surface brightness, and quantita-tively, in terms of the actual error �l, the perturbed modeldoes a much better job. However, both the perturbed andunperturbed models are somewhat less successful inaccounting for the detailed shape of the bulge (although themodel with a binary does a bit better).

There are at least two possible explanations for this lack ofsuccess. Most obvious is the fact that demanding � ¼ 2 and� ¼ 4, so that the potential can be written in terms of elemen-tary functions, limits one’s flexibility in modeling the transi-tion region between the inner and outer (unperturbed)power-law profiles. Allowing for fractional parameter values(which requires that the potential be computed numerically)will likely yield better fits. However, it is also possible thatthis lack of success reflects in part the oversimplistic characterof this kinematic model. In a real galaxy, the binary orbitdecays as the binary transfers energy to the stars, and the factthat rh is not really constant might be expected to have someobservable effects. Attempts to remedy these deficiencies ofthemodel are currently underway.

It should, however, be stressed that the general effects ofthe binary are relatively insensitive to rh, provided only thatM > MðrhÞ. This is evident, for example, from Figure 15,which exhibits surface brightness distributions at t ¼ 256 forboth the model considered in Figure 14 and another modelidentical except that rh ¼ 0:025, a radius only 0.29 times aslarge. There are some differences in detail, but neither fit isclearly superior visually. It is also evident from Figure 14that the basic observable structure develops very quickly.Although the details of the surface brightness profile canvary considerably for times as long as t � 128 or more, theexistence of the dip region is obvious already by t � 32,about 8 binary orbital periods for the rh ¼ 0:085model.

Although the model described here is kinematic, onecan try to describe how it might be manifested in a self-consistent description: when the binary orbit is too large, itsdecay will be dominated by more conventional processes,discussed, for example, in Tremaine &Weinberg (1984) andNelson & Tremaine (1999). However, once the radius is suf-ficiently small that MðrhÞ � 2M, the resonant phase mixingdescribed here, which can be viewed as a variant of Trem-aine’s resonant relaxation, will be triggered. As additionalenergy is transferred to the stars, the binary will continue todecay, and when the size of the orbit becomes too small, theeffect will again turn off.

For the models in Figures 14 and 15, the process shouldturn on at rh;1 � 0:2 and turn off at rh;2 � 0:005. However,during this interval, the binary will have lost an energy�M2=rh;2 � 0:02, several percent of the energy of the galaxyat the time that the process begins. The obvious questions,therefore, are, how long does it take for a binary with radiirh satisfying rh;2drhdrh;1 to pump this much energy intothe stars? and, is this long enough to establish the signatureobserved in Figures 14 and 15? The time required dependsto a certain extent on the precise value of rh. However, ananalysis of the models in Figures 14 and 15, as well as mod-els with somewhat larger and smaller values of rh, indicatesthat the total energy required to establish the observed sig-nature is relatively small. For example, the model withrh ¼ 0:085 entailed an increase in galactic energy of order1% at t ¼ 32 and 3% at t ¼ 512. The model with rh ¼ 0:025yielded 1:5% at t ¼ 32 and 6% at t ¼ 512.


Alternatively, a binary decay rate can be estimated asfollows. Given that the pumping of energies into the stars isdiffusive, the decay time should satisfy =T �ðM2=rhh�EiÞ2; where T is the time over which h�Ei is com-puted. Supposing, however, that rh � 0:01, MðrhÞ � 0:01,h�Ei � 0:01, and T � 10, one infers that � T � 10. Forthe orbit to shrink from rh ¼ 0:2 to rh ¼ 0:005, a factor of 40,would require a few , say t � 50, an interval long enough toestablish a distinctive luminosity dip.

One final point should be noted. Attributing such a lumi-nosity dip to a supermassive binary does not necessarilyimply that the binary should still be present. If, neglectingthe binary, the galaxy can be idealized as a collisionless equi-

librium, one might expect that a dip in surface brightness,once generated, could persist even after the binary hascoalesced, at least for times short compared with the time-scale on which stars at larger radii could be scattered inwardvia collisional relaxation. To the extent that the bulk poten-tial is time-independent, in the absence of collisions energyis conserved, so that an underpopulated region in energyspace cannot be repopulated.

5. DISCUSSION

The computations described here yield several significantconclusions about phase mixing in a time-dependent

Fig. 13.—Computed quantities for a Nuker model with � ¼ 2, � ¼ 4, � ¼ 0, MBH ¼ 0:005, rh ¼ 0:15, and ! ¼ 0:6086. The left column exhibits D, therelative fluctuation in density for different shells; the middle exhibits the interpolated smooth density �; the right exhibits the surface brightness, assuming thatmass traces light. From top to bottom, the rows represent integration times t ¼ 128, 256, 384, and 512. In each case, the dotted lines represent the originalunperturbed values.


potential. Most obvious is the fact that a supermassive blackhole binary can serve as an important source of transientchaos that facilitates efficient resonant phase mixing, shuf-fling the energies of stars (or any other objects) as well asphase-space coordinates on a constant-energy hypersurface.In particular, the effects observed here from a comparativelysmall scale perturbation are quite similar to the effectsobserved when galaxies are subjected to larger scale system-atic oscillations (Kandrup et al. 2003). It is especially strik-ing that even though the perturbation is relatively lowamplitude and concentrated very near the center of the gal-axy, it can have significant effects at comparatively large

radii. All this reinforces the expectation that resonant phasemixing could be a generic physical effect in galaxiessubjected to an oscillatory time dependence.

Contrary, perhaps, to naive expectation, it appears thatthe shuffling of energies is diffusive rather than exponential,so that energy phase mixing is less dramatic than phasemixing of coordinates and velocities. Even though thetime-dependent perturbation can increase both the relativeabundance of chaotic orbits and the degree of exponentialsensitivity exhibited by chaotic orbits, it need not makeorbits exponentially unstable in the phase-space directionorthogonal to the constant-energy hypersurfaces.

Fig. 14.—Modeling NGC 3706 with a Nuker model with � ¼ 2, � ¼ 4, � ¼ 0, rh ¼ 0:085, and ! ¼ 1:567. The left column exhibits the observed surfacebrightness profile ( filled circles), the surface density predicted by an unperturbed Nuker law (dotted lines), and the time-dependent surface density generatedby the binary (solid lines) at times t ¼ 32, 64, 128, and 256 (top to bottom). The right column exhibits the relative error of the fit for a Nuker model without(dashed lines) and with the binary (solid lines).


However, such energy shuffling could still play an impor-tant role in violent relaxation. Indeed, the fact that thisenergy shuffling is not exponential is consistent with self-consistent simulations of violent relaxation (e.g., Quinn &Zurek 1988) that indicate that even though particles arealmost completely randomized in terms of most phase-spacecoordinates, they exhibit a partial remembrance of initialconditions. In particular, particles that start with low orhigh binding energies tend systematically to end with low orhigh binding energies, respectively. If, for example, stars insimulations involving hard, head-on collisions of galaxiesare ordered in terms of their initial and final binding ener-gies, the rank correlation R between the initial and finalordered lists typically satisfies (Kandrup, Mahon, & Smith1993)Re0:6.

That a supermassive binary will cause a systematic read-justment in the density distribution of the host galaxy seemslargely independent of the form of the galactic potential orthe orbital parameters of the binary, although the preciseform of the readjustment does depend on these details. Inparticular, one sees qualitatively similar effects for Dehnenpotentials with different cusp indexes � and for Nuker lawswith different transitional radii properties. Similarly, theeccentricity and the orientation of the supermassive binaryare not crucial, and allowing for unequal but still compara-ble masses does not result in qualitative changes. Irrespec-tive of all these details, when the total binary massM1 þM25MðrhÞ, with rh the size of the binary orbit, starscannot resonate with the binary, and comparatively littlemass transport occurs. However, when M1 þM2 � MðrhÞ,one starts seeing substantial effects that can extend to radii4rh.

One might therefore expect that when its orbit is large,the binary will have only a minimal effect on the bulk prop-erties of the galaxy, but that when as a result of dynamicalfriction (e.g., Merritt 2001) the orbit has decayed to a suffi-ciently small size, it will begin to have an appreciable andobservable effect.

H. E. K. acknowledges useful discussions with ChristosSiopis, who tried to convince him of the importance ofexplaining luminosity dips months before he was ready tolisten. H. E. K., I. V. S., and B. T. were supported in part byNSF grant AST 00-70809. I. V. S. and C. L. B. were sup-ported in part by Department of Education grantG1A62056. We would like to thank the Florida StateUniversity School of Computational Science and Informa-tion Technology for granting access to their supercomputerfacilities.

REFERENCES

Bertin, G. 2000, Dynamics of Galaxies (Cambridge: Cambridge Univ.Press)

Binney, J. 1978, Comments Astrophys., 8, 27Dehnen,W. 1993,MNRAS, 265, 250Kandrup, H. E. 1998,MNRAS, 301, 960———. 2003, in Galaxies and Chaos, ed. G. Contopoulos & N. Voglis(Berlin: Springer)

Kandrup, H. E., &Mahon,M. E. 1994, Phys. Rev. E, 49, 3735Kandrup, H. E., Mahon,M. E., & Smith, H. 1993, A&A, 271, 440Kandrup, H. E., & Siopis, C. 2003,MNRAS, in pressKandrup, H. E., Vass, I. M., & Sideris, I. V. 2003,MNRAS, 341, 927Kormendy, J., & Bender, R. 1996, ApJ, 464, L119

Lauer, T., et al. 1995, AJ, 110, 2622———. 2002, AJ, 124, 1975Lichtenberg, A. J., & Lieberman, M. A. 1992, Regular and ChaoticDynamics (NewYork: Springer)

Lynden-Bell, D. 1967,MNRAS, 136, 101Merritt, D. 2001, ApJ, 556, 245Merritt, D., & Fridman, T. 1996, ApJ, 460, 136Merritt, D., & Valluri,M. 1996, ApJ, 471, 82Nelson, R.W., & Tremaine, S. 1999,MNRAS, 306, 1Quinn, P. J., & Zurek,W. H. 1988, ApJ, 331, 1Terzic, B., &Kandrup, H. E. 2003,MNRAS, submittedTremaine, S., &Weinberg,M.D. 1984,MNRAS, 209, 729

Fig. 15.—(a) Best-fit model with � ¼ 2, � ¼ 4, � ¼ 0, rh ¼ 0:025, and! ¼ 8:968 at time t ¼ 256. (b) Same, except assuming rh ¼ 0:085 and! ¼ 1:567.

130 KANDRUP ET AL.

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