Download 1 - Università degli Studi dell`Insubria

A. Sesana1, F. Haardt1, P. Madau2
The Final Parsec:
Orbital Decay of Massive Black Holes
in Galactic Stellar Cusps
1 Universita`
2
dell'Insubria, via Valleggio 11, 22100 Como, Italy
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
Galaxy formation proceeds as a
series of subsequent halo mergers
(Volonteri, Haardt & Madau 2003)
Z=0
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!!
Z=20
SMBHs 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
GW emission takes over at 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!
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
Z
> MBHB M1>M2 on a Keplerian orbit with
semimajor axis a and eccentricity e
Y
> incoming star with m* <<M2 and velocity v
>The initial condition is a point in a nine dimensional parameter space:
X
> 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
We integrate the nine coupled second order, differential equations
using the explicit Runge-Kutta integrator DOPRI5 (Hairer & Wanner 2002)
> 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
> At the end of each run the program records:
> the position and velocity of each star
> the quantities B and C defined as:
M2/M1=1
e=0
M2/M1=1
e=0
C and B-C distributions vs. x, a rescaled impact parameter defined as
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
Starting from the energy exchange during
a single scattering event we can write:
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
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 -
> If  >1, then
Hard binaries hardens at a constant rate
where  =- 1
only in a flat stellar background!
> The hardening rate is:
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
M2/M1=1
and the interacting mass integrating
e=0
where
M2/M1=1
THE SINGULAR ISOTHERMAL SPHERE (SIS)
> We model, as a studing case, the stellar
distribution as a SIS with density profile
> The MBHB mass is chosen to satisfy
the M-  relation (Tremaine et al. 2002)
> we can factorize F(bax,t) F0 (bax) x (t)
r is related to t simply as dr/dt=31/2
> The umperturbed loss cone mass content is Mlc ~ 3/2  M 2
1- MBHB Shrinking
2-Distribution of Scattered Stars
Partial loss cone depletion
~20% of the interacting stars
returns in the new loss cone
of the shrinked binary
The loss of low angular
momentum stars
Ejected mass
Interacting star distribution
tends to flatten and corotate
with the MBHB
Stellar distribution flattening
and corotation with the MBHB
The ejected mass is of the order
Mej ≈0.7M
3-The Role of Returning Stars
The inner density profile
flatten significatively
The shrinking factor  scales as (M2/M)1/2
and is weakly dependent on e
Total shrinking
Total loss cone depletion
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
LISA binaries (104-107 M⊙) may neede=0.6
extra
help to coalesce within an Hubble time!!!
e=0.9
What can help ?
> MBHB random walk
(e.g. Quinlan & Hernquist 1997, Chatterjee et al. 2003)
M <105M⊙
> 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)
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
Results
>H in the hard stage is proportional to a -/2
>K is typically positive, but the eccentricity evoution of
the binary is modest
>MBHB-star interactions flatten the stellar distribution
>Interacting stars typically corotate with the MBHB
>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
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”