Document Sample

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 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 (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, Z=20 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 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 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 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 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 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 only in a flat stellar background! where =- 1 > 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 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 e=0 LISA binaries (104-107 M⊙) may need extra e=0.6 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”

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 5 |

posted: | 11/1/2012 |

language: | Unknown |

pages: | 28 |

OTHER DOCS BY liaoqinmei

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.