Simulations of Disk Accretion onto Black Hole Binaries
Brian D. Farris
1,2
Zoltan Haiman
2
Paul Duffell
1
Andrew MacFadyen
1 Center for Cosmology and Particle Physics, New York University
2 Department of Astrophysics, Columbia University
May 19, 2013
Brian D Farris
LISA Symposium 2014
1
Introduction
Physical Picture
All bulge galaxies have SMBH at center with
M ∼ 105 − 109 M
Galaxy mergers → formation of massive BH binary in
merged remnant.
Separation decreases by:
1
2
3
dynamical friction
gravitational slingshot interactions
gravitational radiation
The Final Parsec Problem is Not a Problem!
Circumbinary disk forms
Brian D Farris
LISA Symposium 2014
Motivation
GWs from SMBH binaries detectable by eLISA during inspiral.
May be detectable by Pulsar Timing Arrays for massive (∼ 108 − 109 M ) binaries at z ≈ 1.
Gaseous accretion flow around binary may be a source of detectable EM radiation
Help with source localization
Standard Sirens (distance from GWs, redshift from EM)
Learn about SMBH merger rates.
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LISA Symposium 2014
General Picture
Viscous stresses in the disk transport angular momentum outward and allow gas to migrate
inward
Tidal torques from the binary add angular momentum and drive gas outward
Viscous stresses balance balance tidal torques at inner edge of disk (redge ≈ 2a)
Binary carves out a low-density cavity surrounded by a circumbinary disk.
Quasi-equilibrium state can be maintained provided tvis tgw (pre-decoupling epoch)
When tgw . tvis (i.e. GW dominated regime), binary inspiral must be included in
simulations.
We focus on pre-decoupling epoch.
Questions
How hollow is the cavity? Is the accretion rate suppressed by the presence of a binary?
Does the binary leave an imprint in the accretion rate?
How is the accreted mass divided between the primary and the secondary?
How are continuum spectra modified by presence of a binary?
Brian D Farris
LISA Symposium 2014
Previous Work
2D
Newtonian
MacFadyen & Milosavljević 1980
Cuadra et al. 2009
Roedig & Sesana, 2012
D’Orazio et al. 2012
Farris et al. 2013
1D
Examples:
Goldreich & Tremaine 1980
Artymowicz & Lubow 1994
Milosavljević and Phinney 2005
Haiman et al. 2009
Tanaka & Menou 2010
Liu & Shapiro 2010
Kocsis et al. 2012
Tanaka 2013
Useful for predecoupling, widely separated
binaries.
Can accomodate high res., many orbits.
3D
Use approximate angle-averaged
tidal torque formulae.
Newtonian:
Useful for probing qualitative
features of accretion.
Relativistic:
Fails to capture important,
nonaxisymmetric features such as
accretion streams.
Shi et al. 2012
Roedig et al. 2012
Bode et al 2011
Farris et al. 2012
Noble et al. 2012
Giacomazzo et al. 2012
Computationally expensive.
Often require excised inner regions, thick disks,
short simulations, etc.
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LISA Symposium 2014
DISCO - Duffell & MacFadyen 2012, 2013
Solves (Magneto)Hydrodynamics equations
Uses conservative, shock-capturing
finite-volume methods
Effectively “Lagrangian”, as cells are able
to move with fluid
Minimizes advection errors, allows for
longer timesteps
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LISA Symposium 2014
normalized total accretion rate
For all mass ratios, accretion rate
is enhanced relative to that of
single BH.
1.8
1.7
Consistent with some previous
studies (e.g. Shi et al. 2012)
Ṁ /Ṁ
1.5
1.4
Consensus emerging that accretion
is not significantly reduced by
presence of binary.
0
1.6
1.3
1.2
Simulations needed to verify that
this holds at smaller h/r .
hṀi/hṀ0 i approaches 1 for small
q, as expected.
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1.0
0.0
0.2
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q
0.6
0.8
1.0
Growth of disk eccentricity
Eccentric inner cavity for large mass ratio
binaries seen in many calculations, e.g.
MacFadyen and Milosavljević 2008
D’Orazio et al. 2012
Shi et al. 2012
Noble et al. 2012
Farris et al. 2012
A fraction of gas in each stream does not
accrete onto BH, but rather is flung
outward.
This fraction impacts cavity wall on the
opposite side from which it entered.
If one stream is slightly larger it will push
the opposity wall more, weakening the
opposite stream.
⇒ the imbalance grows.
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LISA Symposium 2014
Comparison with analytic mini-disk size estimates
Artymowicz & Lubow 1994 (cyan circles)
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LISA Symposium 2014
Minidisk timescale
Accretion timescale
tvis,md
=
=
2 ri2
3 νi
α −1 h/r −2 r
3/2 q −1/2
md
42
tbin
0.1
0.1
0.25a
0.1
tvis,md > tbin ⇒ minidisks are persistent.
caveats
We have assumed h/r ∼ 0.1 everywhere, including minidisks. If they are actually much
hotter the accretion timescale is shortened.
We have assumed α = 0.1 everywhere. MHD simulations have indicated it may be larger
near inner disk edge, and possibly inside minidisks as well.
Binary eccentricity may reduce sizes of minidisks, leading to shorter accretion timescale.
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LISA Symposium 2014
q
2.5
=
0.053
=
2.0
1.0
0
Ṁ/Ṁ
0
2.0
1.5
1.0
0.5
0.5
0.0
400
0.0
400
405
410
415
t/tbin
420
425
405
410
415
t/tbin
420
100
80
80
20
0.5
1.0
ω/ωbin
q
1.6
=
1.5
2.0
0.43
1.4
0
Ṁ/Ṁ
Ṁ/Ṁ
0
1.2
1.0
0.8
0.6
0.4
400
405
410
t/tbin
415
420
425
0.5
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
400
1.0
ω/ωbin
q
=
1.5
2.0
0
2.5
2.0
2.5
410
t/tbin
415
420
425
60
40
0
=
1.5
2.0
2.5
420
425
2.0
2.5
1.0
0.6
405
400
Power
Power
Power
1.5
ω/ωbin
q
1.2
20
ω/ωbin
425
1.0
80
1.0
420
0.8
80
0.5
1.0
1.4
80
0
0.5
0.67
100
20
415
60
100
60
t/tbin
40
100
40
410
20
0
2.5
405
0
0
60
40
Ṁ/Ṁ
20
Power
Power
Power
100
80
60
0.25
1.0
0.5
400
425
=
1.5
100
40
q
0.11
2.5
1.5
Ṁ/Ṁ
0
Ṁ/Ṁ
q
3.0
2.0
405
410
t/tbin
415
60
40
20
0.5
1.0
Brian D Farris
ω/ωbin
1.5
2.0
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LISA Symposium 2014
0
0.5
1.0
ω/ωbin
1.5
Mass Ratio Evolution
dq
d
=
dt
dt
M2
M1
=
M2
M1
Ṁ2
Ṁ1
−
M2
M1
!
10
3
Ṁ2 /M2 > Ṁ1 /M1 ⇒ q increasing
2
1
) (
1
2
10
Consistent with previous studies
(Hayasaki et al. 2007, Cuadra et
al. 2009, Roedig et al. 2011,2012,
Hayasaki et al. 2012)
(
2
Ratio > 1 for all cases. Binary
driven toward equal mass.
Ṁ /M / Ṁ /M
1
)
10
10
10
Possible that ratio < 1 for q less
than some q0 , leading to bimodal
mass-ratio distribution.
Brian D Farris
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-1
0.0
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LISA Symposium 2014
0.4
q
0.6
0.8
1.0
Radiative Cooling
Dynamics are sensitive to tvis of minidisks
Isothermal prescription locks h/r to match that of
circumbinary disk
Need to self-consistently balance viscous heating and shock
heating with radiative cooling
Optically thick disk ⇒ qcool =
4σ 4
T
3τ
Assume electron scattering opacity
Neglect radiation pressure ⇒ P = (Σ/mB )kT
Include viscous heating source term in energy evolution
equation
Test that scheme can reproduce Shakura-Sunyaev solution
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show movie
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Density
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Emission - 2D
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Emission 1D
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spectrum
103
minidisks
non-minidisk
∝ν1/3
total
102
Lν
101
100
100
101
102
hν [a −13/16 M −1/8 M−305/4 eV]
100
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103
Summary & Future Work
Summary
First simulations of circumbinary disk accretion using moving-mesh, finite-volume code.
Ṁ onto binary not reduced. Gas efficiently enters cavity along streams.
For each mass-ratio, persistent mini-disks are formed. Accretion timescale of mini-disks
exceed binary orbital timescale.
Mini-disk sizes in rough agreement with analytic predictions (Artymowicz & Lubow 1994).
Significant periodicity in Ṁ for q & 0.1.
Binary torques can excite eccentricity in inner disk and create overdense lump. Orbital
frequency of lump can dominate Ṁ periodograms.
For each mass-ratio considered, accretion rate onto secondary is large enough to cause q to
increase.
Emission can be enhanced in cavity region relative to single BH case
Continuum spectra steepen in X-rays due to hot emission from minidisks and streams
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LISA Symposium 2014
Extra Slides
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LISA Symposium 2014
Setup
Keep aspect ratio of cells ≈ 1, concentrate
resolution near BHs.
α-law viscosity ν = αcs h, with α = 0.1.
Run simulation for longer than a viscous
time at the cavity edge, so that
quasi-equilibrium state is reached.
tsim & tvis (redge ) ∼ 300
r
edge
2a
3/2
10
δ/a
include cavity in computational domain,
treat accretion by adding sink term to
continuity equation:
dΣ
Σ
=−
dt sink,i
tvis,i
10
10
δr
rδφ
0
-1
-2
10
-1
tbin
Vary mass ratio in range 0.026 ≤ q ≤ 1.0.
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LISA Symposium 2014
10
0
r/a
10
1
10
2
Ṁ/Ṁ
0
shorten tacc in sink prescription by factor of 100
q
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
400
405
410
1.0
t/tbin
415
420
425
2.0
2.5
Power
140
120
100
80
60
40
20
0
=
0.5
1.0
ω/ωbin
1.5
Conclusion
The minidisk accretion timescale is
extremely important in determining
the periodicity of Ṁ.
Total time-averaged accretion rate
mostly unchanged.
Brian D Farris
LISA Symposium 2014
Mini-disks ”dampen” accretion variability
q
10
=
1.0
0
8
(
=
)
Ṁ r a /Ṁ
Compare actual accretion rate with
rest mass flux through surface at
r = a.
Flux through r = a much more
variable.
6
4
2
0
−2
400
405
410
t/tbin
415
420
425
2.0
2.5
200
Exaggerates Ω = 2Ωbin
component.
150
Power
Time averaged Ṁ is unchanged
(expected for quasi-steady state).
100
50
0
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LISA Symposium 2014
ω/ωbin
1.5
Viscosity Code Test
“Cartesian” test performed
on cylindrical grid
Test balance of all viscous
forces and hydrodynamic
fluxes and source terms in
all directions.
t =0
1.0
, 1D high-res
, 2D DISCO
t =8
t =8
0.8
initial state
0.6
=
1.0
P
=
0.1
vx
=
0
=
(x − x0 )2
exp −
σ2
vy
v
y
ρ
0.4
0.2
0.0
0
1
2
viscosity
ν = ν0
ΓP 2
x
ρ
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LISA Symposium 2014
3
x
4
5
6
q=0.053
q=1.0
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Surface Density
initial
=0.026
=0.053
=0.11
q =0.25
q =0.43
q =0.67
q =0.82
q =1.0
q
0.6
q
q
0.5
Σ/Σ0
0.4
0.3
cavity size and max eccentricity vs q
3.5
0.2
3.0
0.1
2.5
rc
0.0
4
2
6
8
10
r/a
14
12
2.0
1.5
1.0
0.16
Eccentricity
0.14
ẽmax
0.12
0.20
q =0.026
q =0.053
q =0.11
0.08
0.06
q =0.25
0.15
0.10
q =0.43
0.04
q =0.67
ẽ1
q =0.82
0.02
0.0
q =1.0
0.10
0.2
0.05
0.00
1
2
3
4
5
r/a
6
7
8
9
10
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LISA Symposium 2014
0.4
q
0.6
0.8
1.0