Universit degli Studi dell Insubria

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					                       A. Sesana1, F. Haardt1, P. Madau2

          The Final Parsec:
Orbital Decay of Massive Black Holes
      in Galactic Stellar Cusps

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

>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
                  MERGING HISTORY OF SMBHs

                                    (Volonteri, Haardt & Madau 2003)
 Galaxy formation proceeds as a
series of subsequent halo mergers                                      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!!
                               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
    ●   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
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!
 Extraction of binary binding energy via three body interactions with stars

Scattering experiments                          N-body simulations
                                                (e.g. Milosavljevic & Merritt 2001)
(e.g. Mikkola & Valtonen 1992, Quinlan 1996)

                                                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
                                                     > MBHB M1>M2 on a Keplerian orbit with
                                                     semimajor axis a and eccentricity e
                                                     > incoming star with m* <<M2 and velocity v

  X                >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
          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              M2/M1=1

                           e=0              e=0

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

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

   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

 We consider a density profile
              r -

> If  >1, then
              Hard binaries hardens at a constant rate
                   only in a flat stellar background!
   where  =- 1

> The hardening rate is:
    Eccentricity Growth

K is typically small: eccentricity
    evolution will be modest
    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



   > 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
                                  The ejected mass is of the order
and corotation with the MBHB                 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

LISA 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)       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)
>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
>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
 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”

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