The Final Parsec: Orbital Decay of Massive Black Holes

in Galactic Stellar Cusps

A. Sesana1, F. Haardt1, P. Madau2

1 Universita` dell'Insubria, via Valleggio 11, 22100 Como, Italy

2 University of California, 1156 High Street, Santa Cruz, CA 95064

Como, 20 September 2005

OUTLINE

>Merging History of Massive Black Holes

>MBHBs Dynamics: the “Final Parsec Problem”

>Scattering Experiments: Model Description

>Results: Binary Decay in a Time-Evolvig Cuspy Background: the Study Case of the SIS

>Effects on the Stellar Population >Returning Stars

>Tidal Disruption Rates >Implication for SMBH Coalescence

>Summary

MERGING HISTORY OF SMBHs

Z=0

Z=20

(Volonteri, Haardt & Madau 2003)Galaxy formation proceeds as a

series of subsequent halo mergers

MBH assemby follow the galaxy

evolution starting from seed BHs

with mass ~100M⊙ forming

in minihalos at z~20

During mergers,

MBHBs will

inevitably form!!

SMBHs DYNAMICSSMBHs DYNAMICS

1. dynamical friction (Lacey & Cole 1993, Colpi et al. 2000)

● from the interaction between the DM halos to the formation of the BH binary● determined by the global distribution of matter● efficient only for major mergers against mass stripping

2. hardening of the binary (Quinlan 1996, Merritt 1999, Miloslavljevic &

Merritt 2001)● 3 bodies interactions between the binary and the surrounding stars ● the binding energy of the BHs is larger than the thermal energy of the stars● the SMBHs create a stellar density core ejecting the background stars

3. emission of gravitational waves (Peters 1964)

● takes over at subparsec scales ● leads the binary to coalescence

DESCRIPTION OF THE PROBLEM

We want MBHBs to coalesce after a major merger

Dynamical friction is efficient in driving the two

BHs to a separation of the order

The ratio can be written as

we need a physical mechanism able to shrink the binary

separation of about two orders of magnitude!

GW emission takes over at separation of the order

GRAVITATIONAL SLINGSHOT

Extraction of binary binding energy via three body interactions with stars

Scattering experiments (e.g. Mikkola & Valtonen 1992, Quinlan 1996)

N-body simulations

(e.g. Milosavljevic & Merritt 2001)

resolution problem

> More feasibles

> need a large amount of data for significative statistics

(eccentricity problem)

> warning: connection with real galaxies!

> initial conditions

> loss cone depletion

> contribution of returning stars

> presence of bound stellar cusps

SCATTERING EXPERIMENTS

Y

X

Z

> MBHB M1>M2 on a Keplerian orbit with

semimajor axis a and

eccentricity e

> incoming star with m* <<M2 and velocity v

>The initial condition is a point in a nine dimensional parameter space:

> q=M2/M1, e, m* /M2

> v, b, , , ,

Our choices:

> In the limit m*<<M2: results are indipendent on m

*

we set m* =10- 7M (M=M1+M2)

> we sampled six values of q: 1, 1/3, 1/9, 1/27, 1/81, 1/243

and seven values of e: 0.01, 0.15, 0.3, 0.45, 0.6, 0.75, 0.9 for each q

> we sampled 80 values of v in the range 3x10- 3(M2/M)1/2 < v/Vc < 3x102(M2/M)1/2

> we sampled b and the four angles in order to reproduce a

spherical distribution of incoming stars

> Tolerance is settled so that the energy conservation for each orbit is of the order 10- 2 E*

> Integration is stopped when:

> the star leave ri with positive total energy

> the integration needs more than 106 steps

> the physical integration time is >1010 yrs

> the star is tidally disrupted

We integrate the nine coupled second order, differential equations

using the explicit Runge-Kutta integrator DOPRI5 (Hairer & Wanner

2002)

> At the end of each run the program records:

> the position and velocity of each star

> the quantities B and C defined as:

C and B-C distributions vs. x, a rescaled impact parameter defined as

M2/M1=1 M2/M1=1

e=0 e=0

SEMIANALITICAL MODEL

We consider:

> a MBHB with a semimajor axis a and eccentricity e

> a spherically simmetric stellar background

> (r) = 0(r/r0)- is the power law density profile. (0 is the density at the reference distance r0 from the centre)

> f(v,) is the stellar velocity distribution.

is the 1- D velocity dispersion (in the following we will always consider a Maxwellian distribution)

C and B can be used to compute the MBHB evolution

Writing d2N(b,t)/dbdt=2 b(b,t)v/m* and (b,t)=

0 F(ba x,t) we find:

Weighting over a velocity distribution f(v,) we finally get

H is the HARDENING RATE

Similarly we find the equation for the eccentricity evolution

K is the ECCENTRICITY GROWTH RATE

Starting from the energy exchange during

a single scattering event we can write:

F(bax,t) is a function, to be determined, of the rescaled impact parameter x

and of the time t and depends on the density profile of the stellar distribution

Early studies (Mikkola & Valtonen 1992, Quinlan 1996) assumed F(bax,t) =1

i.e. they studied the hardening problem in a

flat core of density 0 constant in time!!

Warning: connection with real galaxies!

1- Almost all galaxies show cuspy density profiles in their inner regions

r - 0< <2.5

(n.b. faint early type galaxies show steeper cusps that giants ellipticals)

2- In real galaxies there is a finite supply of stars to the hardening process

LOSS CONE PROBLEM

1-HARDENING IN A CUSPY PROFILE

We consider a density profile

r -

where =- 1

> If >1, then

> The hardening rate is:

Hard binaries hardens at a constant rate

only in a flat stellar background!

Eccentricity Growth

K is typically small: eccentricity

evolution will be modest

2-MODELLING THE LOSS CONE CONTENT

Definition: the loss cone is the portion of the space E, J constituded by those

stars that are allowed to approach the MBHB as close as x a,

where is a constant (we choose = 5)

Given (r ) we can evaluate the mass in the

unperturbed loss cone as

and the interacting mass integrating

where

M2/M1=1

M2/M1=1

e=0

THE SINGULAR ISOTHERMAL SPHERE (SIS)

> we can factorize F(bax,t) F0 (bax) x (t)

> The umperturbed loss cone mass content is Mlc ~ 3/2 M 2

> We model, as a studing case, the stellar

distribution as a SIS with density profile

r is related to t simply as dr/dt=31/2

> The MBHB mass is chosen to satisfy

the M- relation (Tremaine et al. 2002)

1- MBHB Shrinking

2-Distribution of Scattered Stars

The loss of low angular

momentum stars

Partial loss cone depletion

~20% of the interacting stars

returns in the new loss cone

of the shrinked binary

Stellar distribution flattening

and corotation with the MBHB

Interacting star distribution

tends to flatten and corotate

with the MBHB

Ejected mass

The ejected mass is of the order

Mej ≈0.7M

3-The Role of Returning Stars

Total shrinking

The shrinking factor scales as (M2/M)1/2

and is weakly dependent on e

Total loss cone depletion

The inner density profile

flatten significatively

Final Velocity Distribution

4-Tidal Disruption Rates

A star is tidally disrupted if it approaches

one of the holes as close as the tidal

disruption radius rtd,i~(m* / Mi)1/3r

*

We can then derive the mean TD rate as:

N TD stars / hardening time

> The TD rate is extremely high during

the hardening phase (respect to TD

rates due to a single BH ~10- 4 star/yr)

> The high TD rate phase is

extremely short

Hard to detect a MBHB via TD

stars

5-Binary Coalescence

As the shrinking factor is proportional to (M1/M)1/2, writing af = x

ah, we finally get

e=0

e=0.9

e=0.6LISA binaries (104-107 M⊙) may need extra

help to coalesce within an Hubble time!!!

What can help ?

> MBHB random walk (e.g. Quinlan & Hernquist 1997, Chatterjee et al. 2003)

> Star diffusion in the loss cone via two body relaxation (Milosavljevic & Merritt 2001)

> Loss cone amplification (loss wedge) in axisimmetric and triaxial potentials (Yu 2002, Merritt & Poon 2004)

> Torques exerted on the MBHB by a gaseous disk (Armitage & Natarajan 2002, Escala et al. 2005, Dotti et al. in preparation)

M <105M⊙

Summary >We have studied the interaction MBHB-stars in detail using scattering experiments coupled with a semianalitical model for MBHB and steller background evolution including: >a cuspy time-evolving stellar background >the effect of returning stars

>H in the hard stage is proportional to a -/2

>K is typically positive, but the eccentricity evoution of the binary is modest

>Interacting stars typically corotate with the MBHB

>MBHB-star interactions flatten the stellar distribution

>A mass of the order of 0.7M is ejected from the bulge on nearly radial corotating orbits in the MBHB plane>LISA binaries may need the support of other mechanisms to reach coalescence within an Hubble time

Results

Future Prospects

Investigate the contribution of other mechanisms to the binary hardening

Evaluate the eventual role of bound stellar cusps

Include this treatment of MBHB dynamics in a merger tree model to give realistic estimations for the number counts of “LISA coalescences”