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Black hole feeding rates
in post-merger galaxies
Eugene Vasiliev
Oxford University
Kirill Lezhnin
Princeton University
Stellar aggregates, Bad Honnef, 9 December 2016
Tidal disruptions: observational status
I Occur in quiescent galactic nuclei;
I Observed as X-ray, UV and optical transients;
I Have distinct lightcurves and spectra;
I A few dozen of events registered so far
[see reviews by Komossa (1505.01093), Kochanek (1601.06787)].
-50 0 50 100 150 200 250 300 350
Time since discovery (days)
1042
1043
Lum
inosi
ty (
erg
/s)
PS1-10jh
p=-1.7 t0=-46.6
p=-5/3 t0=-44.3
exponential
NUV (193 nm)
-0.5 0.0 0.5 1.0 1.5 2.0
u - g
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
g -
r
AGN
SNe
TDFs
[from van Velzen+ 2016]
Tidal disruptions and the loss-cone theory
Tidal disruption radius: rt '(M•M?
)1/3R?.
Direct capture occurs when rt ≤ 4rschw = 8GM•/c2.
Critical angular momentum: Lt ≡√
2GM•rt .
The classical loss-cone theory
Distribution function of stars: N (E , L, t).
Two-body relaxation leads to the diffusion of stars in the phase space;
The Fokker–Planck equation for the diffusion in angular momentum:
∂N (E , L, t)
∂t=
∂
∂L
(D1(L) N (E , L, t) +D2(L)
∂N (E , L, t)
∂L
).
Steady-state profile: N (E , L) ∝ A(E ) + B(E ) ln(L/Lt)and the corresponding flux into the black hole F(E ).
Analytic time-dependent solutionin terms of Bessel series[Milosavljevic & Merritt 2003, Lezhnin & Vasiliev 2015].
0.0 0.2 0.4 0.6 0.8 1.0L/Lcirc
0.0
0.2
0.4
0.6
0.8
1.0
N(L
)
Lt
Steady-state tidal disruption rates
I Take a surface brightness profile of a galaxy nucleus Σ(R);
I Assume a black hole mass M• and mass-to-light ratio Υ;
I Deproject Σ to obtain the density profile ρ(r);
I Compute the isotropic distribution function in energy N (E );
I Compute the diffusion coefficients D(E );
I Integrate the steady-state flux to obtain NTDE =∫F(E ) dE .
105 106 107 108 109 1010
M •/M¯
10-7
10-6
10-5
10-4
10-3
10-2
captu
re r
ate
, M
¯/y
r
Wang & Merritt 2005
core
power-law
105 106 107 108 109 1010
M •/M¯
10-7
10-6
10-5
10-4
10-3
10-2
captu
re r
ate
, M
¯/y
r
Stone & Metzger 2016
core
power-law
Theoretical predictions are higher than the observed rates!
Galaxy mergers and binary supermassive black holes
[image credit: Paolo Bonfini]
I Binary SBH naturally created
in galaxy mergers;
I The two SBHs spiral in and eventually
coalesce due to gravitation-wave emission;
I On their way to coalescence, they eject
stars from the galactic nucleus
(the slingshot effect);
I As a result, a gap in the angular
momentum distribution is formed;
I The flux of stars into a residual
single SBH is suppressed until
this gap is refilled.
The gap takes a while to refill
106 107 108 109 1010
time, yr
10-10
10-9
10-8
10-7
10-6
10-5
10-4
captu
re r
ate
, M
¯/y
r
M • =106 M¯
M • =107 M¯
M • =108 M¯
10-4 10-3 10-2 10-1 100
(L/Lcirc)2
0.0
0.2
0.4
0.6
0.8
1.0
N(L
)
t=0
t=10−3 Trefill
t=10−2 Trefill
t=10−1 Trefill
t=Trefill
106 107 108 109 1010
M •/M¯
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
FT
D/F
SSS
Ratio between time-dependent(suppressed) and stationary capturerates for a sample of nearby galaxieswith estimated black hole masses.
The capture rate is still suppressedafter 10 Gyr for galactic nuclei withM• & 107.5 M�.
Non-spherical galactic nuclei
M!0.13!0.08
M!0.02!0.07
Nuclear cluster
Nuclear Disk
Sgr B2 region
Quintuplet cluster
G0.253+0.016
[Milky Way NSC, Schodel+ 2014]
[NGC 4244 NSC, Seth+ 2008][NSC catalog of Georgiev & Boker 2014]
Loss cone in non-spherical stellar systems
Angular momentum L of any star is not conserved, but experiencesoscillations due to torques from non-spherical distribution of stars.
Many more stars can attain low L and enter the loss cone.
The gap is much less prominent in non-spherical systems
This is the reason that the ”Final-parsec problem” does not exist[Vasiliev+ 2015, Gualandris+ 2016].
The Monte Carlo method for non-spherical systems
I Galactic nuclei have N? � 106 – not amenable to directN-body simulations.
I Difficult or impossible to properly scale the simulationswith affordable N in the presence of both collisional(N-dependent) and collisionless (N-independent) processes.
I Conventional Fokker–Planck or Monte Carlo methodsrestricted to spherical symmetry.
I Solution:
use the Monte Carlo approach,but without orbit-averaging.
The Monte Carlo method for non-spherical systems
I Gravitational potential:
spherical-harmonic expansion of an arbitrary density profile
(similar to the self-consistent field method of Hernquist & Ostriker 1992).
I Orbit integration:
adaptive-timestep, all particles move independently in the global potential.
I Two-body relaxation:
local diffusion coefficients in velocity 〈∆v2‖ 〉, 〈∆v2
⊥〉(r , v) computed from
the smooth distribution function f (E ) (spherical isotropic background),with adjustable amplitude (assigned independently of N);
perturbations to velocity applied after each timestep.
I Massive black hole(s):
capture of stars with r < rt, three-body scattering by a massive BH binary.
I Temporal smoothing:
potential and diffusion coefficients updated after an interval of time
� Tdyn, but � Trel =⇒ reduced parasitic noise (+trajectory oversampling).
Implementations of the Monte Carlo methodName Reference relaxation treatment timestep 1:11 BH2 remarks
Princeton Spitzer&Hart(1971),Spitzer&Thuan(1972)
local dif.coefs. in velocity,Maxwellian background f (r, v)
∝ Tdyn − −
Cornell Marchant&Shapiro(1980)
dif.coef. in E , L, self-consistentbackground f (E)
indiv., Tdyn − + particle cloning
− Hopman (2009) same − + stellar binaries
Henon Henon(1971) local pairwise interaction, self-consistent bkgr. f (r, vr , vt )
∝ Trel − −
− Stodo lkiewicz(1982) Henon’s block, Trel (r) − − mass spectrum, disc shocksStodo lkiewicz(1986) binaries, stellar evolution
Giersz(1998) same same + − 3-body scattering (analyt.)Mocca Hypki&Giersz(2013) same same + − single/binary stellar evol.,
few-body scattering (num.)
Joshi+(2000) same ∝ Trel (r = 0) + − partially parallelizedCmc Umbreit+(2012),
Pattabiraman+(2013)(shared) + + fewbody interaction, single/
binary stellar evol., GPU
Me(ssy)2 Freitag&Benz(2002) same indiv.∝ Trel − + cloning, SPH physical collis.
− Sollima&Mastrobuono-Battisti(2014)
same − − realistic tidal field
Raga Vasiliev(2015) local dif.coef. in velocity, self-consistent background f (E)
indiv.∝ Tdyn − + arbitrary geometry
1One-to-one correspondence between particles and stars in the system
2Massive black hole in the center, loss-cone effects
Features of the Monte Carlo code
– Single-mass systems;
– No binaries;
– No stellar evolution;
– No few-body interactions;
+ Central black hole(s);
+ Non-spherical systems;
vs.
Results: capture rates in post-merger galaxies
106 107 108
M •, M¯
10-5
10-4
10-3
TD
E r
ate
, M¯/y
rSph, isotropicSph, gapAxi, isotropicAxi, gapTri, isotropicTri, gapSph, steady-state
Summary
I Tidal disruptions probe the demographics of massive black holes.
I Galaxy mergers lead to non-spherical remnant shapesand a gap in the angular-momentum distribution of stars.
I The increase in tidal disruption rates due to non-spherical shapeis more important than the suppression due to the gap.
I The rates are higher than simple spherical steady-state estimatesby a factor 2÷ 10.
I Discrepancy with observationally inferred rates still exists!
References:
E.Vasiliev, CQG, 31, 244002, 2014.K.Lezhnin & E.Vasiliev, ApJL, 808, 5, 2015.K.Lezhnin & E.Vasiliev, ApJ, 831, 84, 2016.
The Raga code is available at http://td.lpi.ru/~eugvas/raga