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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
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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
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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!!
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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
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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
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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
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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
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> 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:
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C and B-C distributions vs. x, a rescaled impact parameter defined as
M2/M1=1 M2/M1=1
e=0 e=0
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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)
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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:
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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
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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!
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Eccentricity Growth
K is typically small: eccentricity
evolution will be modest
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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
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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)
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1- MBHB Shrinking
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2-Distribution of Scattered Stars
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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
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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
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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
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Final Velocity Distribution
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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
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5-Binary Coalescence
As the shrinking factor is proportional to (M1/M)1/2, writing af = x
ah, we finally get
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e=0
e=0.9
e=0.6LISA binaries (104-107 M⊙) may need extra
help to coalesce within an Hubble time!!!
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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⊙
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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
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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”