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 NASA NAG5-6037, NAG5-9046 Sherman Fairchild Foundation

Collaborator: David Merritt

MBH Binaries Form in Galaxy Mergers

Borne et al 2000

binary’s semi-major axis (parsec)

2 2 1 hard

8 ) ( σ M M G a + =

GALAXY MERGER

“hard binary”

black hole mass (solar mass)

binary’s semi-major axis (parsec)

2 2 1

8 ) ( σ M M G ahard + =

COALESCENCE

( )

) ( 64 5

2 1 2 1 3 4 5

e F M M M M G a c tgr + =

10 Gyr

black hole mass (solar mass)

The Final Parsec Problem

black hole mass (solar mass) binary’s semi-major axis (parsec)

GALAXY MERGER COALESCENCE

Can the binaries cover this?

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

required

• Discreteness produces

wrong trends for

• Numerical algorithms

partially developed and implemented: Aarseth, Hemsendorf, Makino, Merritt, Mikkola, MM, Spurzem, and others.

6

10 ≤ N

• Parameter space:

MBH masses Density profiles Flattening/triaxiality Orbit: eccentricity? More than 2 MBHs Factors of two count!

Gravitational Slingshot Interaction

Velocity of a star can increase or decrease at each encounter.

) ( eject v N

binary

v

a M M G

v

) ( binary

2 1

~

+

Distribution of velocities following ejection.

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

bh ejected final initial

ln JM M a a =        

When binary is hard, J is independent

• f the separation between the black holes

5 . ≈ J

bh ejected M

M ≈

N-body simulations yield:

Hard binary separation is a function of the orbital mass initially inside the loss cone

“power-law” “core”

2

~

−

r ρ

luminosity density luminosity density radius

Gebhardt et al 1996

• rbital mass ~ 10 binary masses

Simulations show that initially, the binary shrinks by x10 or more from the equipartition value.

(MM & Merritt 2001)

black hole mass (solar mass) binary’s semi-major axis (parsec)

super-hard binary power-law core

Stars inside the “loss-cone” close to MBHB ejected once

The Loss Cone

angular momentum

circular orbit Definition: Domain in phase space consisting of orbits strongly perturbed by individual components

• f 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

|energy|

number of stars

Provided that the galactic potential is sufficiently spherical, the stars that are ejected by slingshot return to the nucleus on radial

• rbits 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

2 /

~ ) ( ~

σ E

e E P E

      ∆ + + + =

∗

) ( 2 1 ln ) ( 4 ) ( 1 ) ( 1

2 2 1 2

E P t E N m M M G a t a µσ σ

time

• 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.

black hole mass (solar mass) binary’s semi-major axis (parsec)

re-ejection

r e

• e

j e c t i

• n

power-law core 10 Gyr

Diffusion into the Loss Cone

|energy|

angular momentum

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

The Loss Cone: An Initial Value Problem

Heat equation in cylindrical coordinates

Energy Angular Momentum The loss cone boundary

N t N

R 2

∇ = ∂ ∂ µ ) ( /

2 2

E J J R

c

≡

Loss cone Out of Equilibrium

number of stars 1 Myr 10 Myr 100 Myr 1 Gyr 10 Gyr angular momentum time (Myr) consumption / Mbh

equilibrium loss cone time dependent loss cone

evolution of the semi-major axis

Speculation: Episodic Refilling?

E.g. Zhao, Haehnelt, & Rees 2002 loss cone refilled loss cone refilled

separation

N(L) Satellite/star cluster infall Star formation episode log(L) N(L) log(L)

time

black hole mass (solar mass) binary’s semi-major axis (parsec) equilibrium diffusion

power-law core

warning: diffusion and re-ejection are simultaneous

10 Gyr

black hole mass (solar mass) binary’s semi-major axis (parsec) equilibrium diffusion

re-ejection

r e

• e

j e c t i

• n

COALESCENCE

super-hard binary

GALAXY MERGER

hard binary

warning: diffusion and re-ejection are simultaneous

power-law core

non-equilibrium enhancement

N-Body Simulations Fail to Recover the Correct Long-Term Evolution

q |energy| M32 simulations

q = orbital period / time to diffuse across the loss cone

loss cone full

5

10

6

10

1 pc 0.01 pc 0.1 pc

q |energy|

Spherical Galaxy: A Summary

• Pinhole-dominated
• Pinhole/diffusion
• Diffusion-dominated
• Large-N limit
• Re-ejection dominated

t a ∝

−1 3 1 1

, ≈ ∝

− −

α

αt

N a t N a

1 1 − − ∝

(Makino 1997)

constant

1 ∝ −

a

( ) γ

β + + ∝

−

t a 1 ln

1

(MM & Merritt 2002)

Observed Galaxies

When photometric data are available, the distribution

• f 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

MM, Merritt, Rest & van den Bosch 2001

. 2

min =

γ 75 . 1

min =

γ

Definition: Mass that had to be removed to produce the

• bserved profile from the

5 . 1

min =

γ

fiducial pure power-law.

Repeated/Multiple Mergers

minor simultaneous major

∑

>

i

M M def

∑

×

i

M M 10 ~

def

∑

≈

i

M M def

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.