The Final Parsec Problem and The Worst-Case Scenario Milos - PowerPoint PPT Presentation

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 binary’s semi-major axis (parsec) “hard binary” + G ( M M ) = a 1 2 hard σ 2 8 black hole mass (solar mass)

binary’s semi-major axis (parsec) + G ( M M ) 10 Gyr = a hard 1 2 σ 2 8 5 4 5 c a = t gr F ( e ) ( ) 3 + 64 G M M M M 1 2 1 2 COALESCENCE black hole mass (solar mass)

The Final Parsec Problem GALAXY MERGER binary’s semi-major axis (parsec) Can the binaries cover this? 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 wrong trends for ≤ 6 N 10 Density profiles • Numerical algorithms Flattening/triaxiality partially developed and Orbit: eccentricity? implemented: More than 2 MBHs Aarseth, Hemsendorf, 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 ( M M ) v ~ N ( eject v ) 1 2 binary a 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.

Mass Ejection and Hardening   M a   ejected = ln initial   a JM   final bh When binary is hard, J is independent of the separation between the black holes ≈ J 0 . 5 N -body simulations yield: ≈ M ejected M bh

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

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

The Loss Cone circular orbit Definition: Domain in phase space consisting of orbits strongly perturbed by individual components angular momentum of a MBH binary Analogy with the loss cone for the tidal disruptions of stars (Yu 2002) 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 Radial orbit return time at energy 2 σ E / 2 E ~ P ( E ) ~ e inv. semi-major axis 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. time  ∆  σ 2 m N E 1 1 4 t ∗ = + + ln 1   + µσ 2 a ( t ) a ( 0 ) G ( M M ) 2 P ( E )   1 2 0

power-law core n o i t c e j e - e r binary’s semi-major axis (parsec) re-ejection 10 Gyr black hole mass (solar mass)

Diffusion into the Loss Cone angular momentum Equilibrium diffusion: Lightman & Shapiro 1977 GC Cohn & Kulsrud 1978, etc. Magorrian & Tremaine 1999 Galaxies Yu 2002 WARNING: The above authors assume 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 ( E ) c ∂ N = µ ∇ 2 N ∂ R t 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 time dependent loss cone evolution of the semi-major axis

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

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

power-law core n o GALAXY MERGER i t c warning: e j e - e r diffusion and binary’s semi-major axis (parsec) re-ejection are hard binary simultaneous super-hard binary non-equilibrium enhancement re-ejection equilibrium diffusion COALESCENCE black hole mass (solar mass)

N -Body Simulations Fail to Recover the Correct Long-Term Evolution q = orbital period / time to diffuse across the loss cone simulations M32 5 10 q q 6 10 0.01 pc loss cone full 0.1 pc 1 pc |energy| |energy|

Spherical Galaxy: A Summary • Pinhole-dominated − 1 ∝ a t − − α t ∝ α ≈ 1 • Pinhole/diffusion a N , 1 3 (Makino 1997) − ∝ • Diffusion-dominated − 1 1 a N t 1 ∝ − • Large- N limit a constant ( ) γ − ∝ + β + 1 a ln 1 t • Re-ejection dominated (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 min = 1 . 5 observed profile from the fiducial pure power-law. MM, Merritt, Rest & van den Bosch 2001

Repeated/Multiple Mergers minor simultaneous major ∑ ∑ ∑ > × M def M ≈ M ~ 10 M M def M i def i 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.