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    Henry E. KandrupDepartment of Astronomy, Department of Physics, and Institute for Fundamental Theory,

    University of Florida, Gainesville, FL 32611

    Ioannis V. SiderisDepartment of Physics, Northern Illinois University, DeKalb, IL 60115

    Balsa TerzicDepartment of Astronomy, University of Florida, Gainesville, FL 32611


    Courtlandt L. BohnDepartment 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


    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


    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, 597:111130, 2003 November 1

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


  • 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 as10rh 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.


    2.1. Description of the Experiments

    The computations described here involve orbits evolvedin potentials of the form

    Vx; y; z V0x; y; z M

    jr r1tj Mjr r2tj

    ; 1

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

    x1t rh sin!t ; y1t rh cos!t ; z1t 0 ; 2

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

    V0x; y; z 12m2 ; 3


    m2 x






    2: 4

    Others focus on more realistic potentials of the form

    V0x; y; z 1

    2 1m2


    " #; 5

    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