The Worst-Case Scenario. The Final Parsec Problem. and. Milos Milosavljevic. California Institute of Technology. Collaborator: David Merritt

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1 The Final Parsec Problem and The Worst-Case Scenario Milos Milosavljevic California Institute of Technology Collaborator: David Merritt NSF AST NASA NAG5-6037, NAG Sherman Fairchild Foundation

2 MBH Binaries Form in Galaxy Mergers Borne et al 2000

3 GALAXY MERGER a hard hard binary G ( M = 8 + σ 1 2 M 2 ) black hole mass (solar mass) binary s semi-major axis (parsec)

4 ) 10 G Gyr ( M 1 + a hard = 8 σ 2 M 2 = 5 64 t gr 3 G M 1 M c 5 a 4 ( M M ) 2 F ( e ) COALESCENCE black hole mass (solar mass) binary s semi-major axis (parsec)

5 The Final Parsec Problem GALAXY MERGER Can the binaries cover this? binary s semi-major axis (parsec) COALESCENCE black hole mass (solar mass)

6 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

7 Why is This Problem Difficult? N-body simulations are required Discreteness produces wrong trends for N 10 Numerical algorithms partially developed and implemented: Aarseth, Hemsendorf, Makino, Merritt, Mikkola, MM, Spurzem, and others. 6 Parameter space: MBH masses Density profiles Flattening/triaxiality Orbit: eccentricity? More than 2 MBHs Factors of two count!

8 Gravitational Slingshot Interaction Velocity of a star can increase or decrease at each encounter. Distribution of velocities following ejection. N ( v eject ) v binary ~ G ( M 1 a + M 2 ) v binary For stars interacting with the binary, the binary is a thermostat with an internal degree of freedom positively coupled to the heat flow.

9 Mass Ejection and Hardening ln a a initial final = M ejected JM bh When binary is hard, J is independent of the separation between the black holes N-body simulations yield: J 0.5 ejected M bh M

10 Hard binary separation is a function of the orbital mass initially inside the loss cone ρ ~ r 2 power-law luminosity density core Gebhardt et al 1996 luminosity density radius orbital mass ~ 10 binary masses Simulations show that initially, the binary shrinks by x10 or more from the equipartition value. (MM & Merritt 2001)

11 power-law core super-hard binary black hole mass (solar mass) Stars inside the loss-cone close to MBHB ejected once binary s semi-major axis (parsec)

12 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 disruptions of stars (Yu 2002) energy However: stars ejected by a MBH binary survive the ejection and can return to the nucleus angular momentum

13 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.

14 Re-Ejection in S. Isothermal Sphere Radial orbit return time at energy E ~ P ( E ) ~ e E / 2 σ 2 Due to the re-ejection, the semi-major axis of a massive black hole binary can shrink by the factor of 2-5 in a Hubble time. inv. semi-major axis time σ m N = + ln 1 + a ( t ) a (0) G ( M + M ) 2 µσ E P ( t E 0 )

15 power-law core re-ejection re-ejection black hole mass (solar mass) 10 Gyr binary s semi-major axis (parsec)

16 Diffusion into the Loss Cone angular momentum energy Equilibrium diffusion: Lightman & Shapiro 1977 Cohn & Kulsrud 1978, etc. Magorrian & Tremaine 1999 Yu 2002 GC WARNING: The above authors assume equilibrium w.r.t. collisional relaxation. Galaxies It can take more than a Hubble time to reach the state of equilibrium, particularly in intermediate and massive galaxies.

17 The Loss Cone: An Initial Value Problem Energy 2 R J / J c 2 ( E ) N t = µ 2 R N Heat equation in cylindrical coordinates The loss cone boundary Angular Momentum

18 Loss cone Out of Equilibrium 1 Myr 10 Myr 100 Myr 1 Gyr 10 Gyr number of stars angular momentum time (Myr) consumption / Mbh

19 evolution of the semi-major axis equilibrium loss cone time dependent loss cone

20 Speculation: Episodic Refilling? E.g. Zhao, Haehnelt, & Rees 2002 loss cone refilled loss cone refilled separation N(L) log(l) Satellite/star cluster infall Star formation episode N(L) log(l) time

21 power-law core equilibrium diffusion black hole mass (solar mass) warning: diffusion and re-ejection are simultaneous 10 Gyr binary s semi-major axis (parsec)

22 power-law core re-ejection GALAXY MERGER hard binary super-hard binary re-ejection COALESCENCE equilibrium diffusion black hole mass (solar mass) warning: diffusion and re-ejection are simultaneous non-equilibrium enhancement binary s semi-major axis (parsec)

23 q N-Body Simulations Fail to Recover the Correct Long-Term Evolution q = orbital period / time to diffuse across the loss cone M32 simulations q pc 0.1 pc loss cone full 1 pc energy energy

24 Spherical Galaxy: A Summary Pinhole-dominated 1 a t Pinhole/diffusion Diffusion-dominated a a 1 N α t α 1 1 N t, 1 3 (Makino 1997) Large-N limit Re-ejection dominated a 1 constant 1 a ln ( 1 + β t ) + γ (MM & Merritt 2002)

25 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.

26 The Mass Deficit Definition: Mass that had to be removed to produce the observed profile from the fiducial pure power-law. MM, Merritt, Rest & van den Bosch 2001 γ min = 2.0 γ min =1.75 γ min =1.5

27 Repeated/Multiple Mergers minor major simultaneous M def M i M > M def i M def ~ 10 M i increasing damage

28 Core in a Minor Merger Mass ratio 100:1, no diffusion

29 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.