The Final Parsec Problem and The Worst-Case Scenario Milos Milosavljevic California Institute of Technology NSF AST 00-71099 Collaborator: David Merritt NASA NAG5-6037, NAG5-9046 Sherman Fairchild Foundation MBH Binaries Form in Galaxy Mergers Borne et al 2000 GALAXY MERGER “hard binary” G(M + M ) a = 1 2 hard 8σ 2 binary’s semi-major axis (parsec) black hole mass (solar mass) 10G Gyr(M + M ) a = 1 2 hard 8σ 2 5 c5a4 tgr = 3 F(e) 64 G M1M 2 ()M1 + M 2 binary’s semi-major axis (parsec) COALESCENCE black hole mass (solar mass) The Final Parsec Problem GALAXY MERGER Can the binaries cover this? binary’s semi-major axis (parsec) COALESCENCE black hole mass (solar mass) The Worst-Case Scenario: Smooth, spherical galaxies. Gould & Rix, ApJL 532, 2000 • Oversimplified the stellar dynamics near MBHB • Assumed that the calescence in less than 1 Gyr is too long Milosavljevic & Merritt, ApJ 563, 2001 • Ignored collisional relaxation (it may be important) • Simulations lacked the resolution to study long-term evolution Yu, MNRAS 331, 2002 • Assumed a collisionally-relaxed state for the post-merger galaxy • Ignored the repeated/multiple interactions of stars with MBHB Why is This Problem Difficult? • N-body simulations are • Parameter space: required • Discreteness produces MBH masses 6 wrong trends for N ≤10 Density profiles • Numerical algorithms Flattening/triaxiality partially developed and implemented: Orbit: eccentricity? Aarseth, Hemsendorf, More than 2 MBHs Makino, Merritt, Mikkola, MM, Spurzem, and others. Factors of two count! Gravitational Slingshot Interaction Velocity of a star can increase or decrease at each encounter. Distribution of velocities following ejection. G(M1 +M 2 ) N(v ) eject vbinary ~ a vbinary For stars interacting with the binary, the binary is a thermostat with an internal degree of freedom positively coupled to the heat flow. Mass Ejection and Hardening ainitial M ejected ln = afinal JM bh When binary is hard, J is independent of the separation between the black holes N-body simulations yield: J ≈ 0.5 Mejected≈Mbh Hard binary separation is a function of the orbital mass initially inside the loss cone “power-law” ρ ~ r −2 luminosity density orbital mass ~ 10 binary masses “core” Simulations show that initially, the binary shrinks by x10 or more from the equipartition value. Gebhardt et al 1996 luminosity density (MM & Merritt 2001) radius power-law core super-hard binary binary’s semi-major axis (parsec) Stars insi the “loss-cone” de close to MB ejected once HB black hole mass (solar mass) The Loss Cone circular orbit Definition: Domain in phase space consisting of orbits strongly perturbed by individual components of a MBH binary Analogy with the loss cone for the tidal momentum disruptions of stars (Yu 2002) angular However: stars ejected by a MBH binary survive the ejection and can return to the nucleus |energy| Content of the Loss Cone number of stars |energy| Provided that the galactic potential is sufficiently spherical, the stars that are ejected by slingshot return to the nucleus on radial orbits and can be re-ejected. Most of the ejected stars remain inside the loss cone at all times. Consequently, the black hole binary continues to harden even after all stars inside the loss cone have been ejected once. Re-Ejection in S. Isothermal Sphere 2 Radial orbit return time at energy E ~ P(E) ~ eE / 2σ xis a Due to the re-ejection, the r ajo semi-major axis of a -m i massive black hole binary can shrink by the factor of v. sem 2-5 in a Hubble time. in time 2 1 1 4σ m∗ N ∆E t = + ln1+ 2 a(t) a(0) G(M1 + M 2 ) 2µσ P(E0 ) power-law core re-ejection re-ejection binary’s semi-major axis (parsec) 10 Gyr black hole mass (solar mass) Diffusion into the Loss Cone Equilibrium diffusion: Lightman & Shapiro 1977 GC Cohn & Kulsrud 1978, etc. Magorrian & Tremaine 1999 Galaxies Yu 2002 WARNING: The above authors assume angular momentum equilibrium w.r.t. collisional relaxation. |energy| It can take more than a Hubble time to reach the state of equilibrium, particularly in intermediate and massive galaxies. The Loss Cone: An Initial Value Problem Energy 2 2 R ≡ J / J c (E) ∂N = µ ∇ 2 N ∂t R Heat equation in cylindrical coordinates Angular Momentum The loss cone boundary Loss cone Out of Equilibrium 1 Myr 10 Myr 100 Myr 1 Gyr 10 Gyr number of stars angular momentum consumption / Mbh time (Myr) equilibrium loss cone evolution of the time dependent loss cone semi-major axis Speculation: Episodic Refilling? E.g. Zhao, Haehnelt, & Rees 2002 N(L) loss cone refilled log(L) Satellite/star cluster infall Star formation episode separation loss cone refilled N(L) log(L) time power-law core binary’s semi-major axis (parsec) equilibrium diffusion warning: diffusion and re-ejection are black hole mass (solar mass) simultaneous 10 Gyr power-law GALAXY MERGER har d bi nary su per core -har binary’s semi-major axisr (parsec) d b e-e inar jectio y n re- ejec equilibrium diffusion tion warn diffusion and COALESCENCE re-ejectioning: are simultaneous black hole mass (solar mass) non-equilibrium enhancement N-Body Simulations Fail to Recover the Correct Long-Term Evolution q = orbital period / time to diffuse across the loss cone M32 simulations 105 q q 106 0.01 pc 0.1 pc loss cone full 1 pc |energy| |energy| Spherical Galaxy: A Summary • Pinhole-dominated a−1 ∝ t • Pinhole/diffusion −1 −α 1 a ∝ N t, α ≈ 3 (Makino 1997) • Diffusion-dominated a−1 ∝ N −1t •Large-N limit a−1 ∝ constant • Re-ejection dominated a−1 ∝ ln()1+ βt + γ (MM & Merritt 2002) Observed Galaxies When photometric data are available, the distribution of stars near the loss cone cannot be inferred without knowing the binary’s age. The present day rate of diffusion into the loss cone cannot be determined better than to within a factor of 2 (10). Inferences about the binary separation based on the present-day luminosity profiles potentially underestimate the past decay rate, when the stellar cusp could have been denser. The Mass Deficit γ min = 2.0 γ min =1.75 Definition: Mass that had to be removed to produce the γ =1.5 observed profile from the min fiducial pure power-law. MM, Merritt, Rest & van den Bosch 2001 Repeated/Multiple Mergers minor major simultaneous M > M M ~ 10× M M def ≈ ∑ M i def ∑ i def ∑ i increasing damage Core in a Minor Merger Mass ratio 100:1, no diffusion Conclusions Idealized dynamical models suggest that long-lived massive black hole binaries are generically produced in the mergers of intermediate and large-mass galaxies. Massive black hole binaries that form in mergers of low-mass galaxies coalesce in a Hubble time due to an efficient loss-cone refilling. Circumstantial evidence suggests that massive black hole are not ubiquitous. All established physical mechanisms, including the results presented here, aid the coalescence of the black holes. Huge progress has been made (BBR, Hills, Valtonen, Quinlan, Makino, Magorrian & Tremaine, Zier, Merritt, Yu, etc.). However our understanding of the non-equilibrium dynamics of the binary black hole nuclei is not yet complete and uncertainties relevant to LISA remain. .