BINARY BLACK HOLE DYNAMICS · ∴ The binary’s L decreases, i.e., its eccentricity increases. In...
Transcript of BINARY BLACK HOLE DYNAMICS · ∴ The binary’s L decreases, i.e., its eccentricity increases. In...
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BINARY BLACK HOLE DYNAMICS
DAVID MERRITT
“BINARY BLACK HOLES & DUAL AGN: A WORKSHOP IN MEMORY OF DAVID S. DE YOUNG”
TUCSON, ARIZONA NOVEMBER 29-30, 2012
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I. Final-Parsec Problem
II. Binary Evolution in Rotating Nuclei
III. Post-Binary Evolution in Rotating Nuclei
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3Gualandris & DM (2012)
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eccentriccircular
eccentric
circular
Gualandris & DM (2012)
Separation Between SMBHs
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The mass deficit is defined as
Mass Deficits
Mdef ≡ 4π
� rmax
0[ρinit(r)− ρ(r)] r2dr.
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Nuclear Structure of Bright Galaxies
Mgal ≾ 1010 MSun Mgal ≿ 1010 MSun
core
Graham (2011)
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Graham 2004 ApJ, 648, 976
Observed Mass Deficits
Graham (2004)
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Stalling of the Binary in Spherical Galaxies
DM, Mikkola & Szell (2007)
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Spherical galaxy: Stalling
Non-spherical galaxy: No stalling?
Khan et al. (2011)
Preto et al. (2011)
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• Brownian motion of the binary (enables interaction with larger number of stars)
• Non-stationary solution for loss-cone repopulation
• Secondary slingshot (stars interact with binary several times)
• Gas physics
• Perturbations to the stellar distribution arising from transient events (infall of large molecular clouds, additional minor mergers, …)
• Effects of non-sphericity on the orbits of stars in the nucleus
Possible Ways to Enhance the Hardening Rate
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“Pyramid” Orbits
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Centrophilic Orbits in the Merger Simulations?
increasing smoothness of the galaxy model
Antonini, Vasiliev & DM (2013)
∴ Not clear that efficient hardening of the binary is due to the
presence of centrophilic orbits.
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I. Final-Parsec Problem
II. Binary Evolution in Rotating Nuclei
III. Post-Binary Evolution in Rotating Nuclei
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II. Binary Evolution in Rotating Nuclei
Evolution of a binary SMBH can be described in terms of dimensionless rate coefficients:
H ≡ σ
Gρ
d
dt
�1
a
�
K ≡ de
d ln(1/a)=
σ
GρaH
de
dt
(energy)
(eccentricity)
For a binary SMBH in an isotropic nucleus, K is small (K ≾ 0.15) and positive (eccentricity increasing).
Quinlan (1996)
Mikkola & Valtonen (1992)
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In a rotating nucleus, changes in the binary’s eccentricity can be much larger.
Sesana et al. (2011)
counter-rotating nucleus
co-rotating nucleus
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In a rotating nucleus, the number of retrograde and prograde encounters with the binary is different.
•Stars on prograde orbits interact strongly with the binary; stars on retrograde orbits avoid ejection during their initial encounter.•If the binary is even mildy eccentric, the torque it exerts on a star can cause its orbit to “flip,” from retrograde to prograde (DM, Gulandris & Mikkola 2009).
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Initial vs final stellar orbits in scattering experiments
prograde
retrograde
prograde retrograde
Lbin L★θ
Lbin
Rasskazov & DM (2012)
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In a rotating nucleus, the number of retrograde and prograde encounters with the binary is different.
•Stars on prograde orbits interact strongly with the binary; stars on retrograde orbits avoid ejection during their initial encounter.•If the binary is even mildy eccentric, the torque it exerts on a star can cause its orbit to “flip,” from retrograde to prograde (DM, Gulandris & Mikkola 2009).•When such a star is finally ejected, it is likely to be on a prograde orbit; it has experienced a net increase in L that is in the same direction as the binary’s L.
∴ The binary’s L decreases, i.e., its eccentricity increases.
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In a rotating nucleus, changes in the binary’s eccentricity can be much larger.
Sesana et al. (2011)
counter-rotating nucleus
co-rotating nucleus
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Angular momentum is a vector, and change of the binary’s L implies changes also in its orbital plane.
Diffusion coefficients:
R1 ≡ σ
Gρa�∆θ�
R2 ≡ M12
m�
σ
Gρa�∆ϑ2�
(drift)
(diffusion) Lbinary
“Rotational Brownian motion”Debye (1913, 1929)
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Second-order rotational diffusion coefficient:
● M12/m✶ = 164 ● M12/m✶ = 328 ● M12/m✶ = 655
≈ 30-60 for “hard,” equal-mass binaries
R2 ≡ M12
m�
σ
Gρa�∆ϑ2�
R2 ∝ q−1(1− e2)−1
q ≡ M2/M1 ≤ 1
Milosavljevic & DM (2001) DM (2002)
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The first-order rotational diffusion coefficient:
R1 ≡ σ
Gρa�∆θ�
is zero, by symmetry, in a non-rotating nucleus.
But in a rotating nucleus, one expects the binary’s L to align with that of the stars, at a rate that is independent of m★/M12.
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For instance, assume that stars are ejected from the binary in essentially random directions.
The average change in angular momentum of stars that impinge on the binary is then
�∆L�� = �L�,final −L�,initial�
≈ −�L�,initial�
Since the change in the binary’s L is opposite in sign to ⟨ΔL✶⟩, the binary’s axis of rotation tends to align with that of the nucleus.
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Evolution of the angle θ between Lbinary and Lcusp.
Gualandris et al. (2012)
R1 ∝ sin θ
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I. Final-Parsec Problem
II. Binary Evolution in Rotating Nuclei
III. Post-Binary Evolution in Rotating Nuclei
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Coalescence of two SMBHs results in a spinning remnant.
The spin, S, can continue to evolve due to:
•Accretion of gas•Capture of stars
•Torquing by a gas disk• Torquing by stars
III. Post-Binary Evolution in Rotating Nuclei
} changes the magnitude, direction of S
} changes only the direction of S
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Lense-Thirring torques cause orbital angular momenta, Lj, to evolve as
Lj = ωj ×Lj , ωj =2GS
c2a3j (1− e2j )3/2
.
S = ωS × S, ωS =2G
c2
�
j
Lj
a3j�1− e2j
�3/2
III. Post-Binary Evolution in Rotating Nuclei
The same torques act back on the SMBH, causing its spin, S, to evolve as
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J = S +�
j
Lj ≡ S +Ltot
|S| ≡ S
|Lj | ≡ Lj , j = 1, . . . , N
ωS
Conserved quantities: Not conserved:
Lj = ωj ×Lj , ωj =2GS
c2a3j (1− e2j )3/2
.
S = ωS × S, ωS =2G
c2
�
j
Lj
a3j�1− e2j
�3/2
Ltot,S
III. Post-Binary Evolution in Rotating Nuclei
DM & Vasiliev (2012)
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Spin evolution I.Damped precession
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Spin evolution II.Sustained precession
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III. Post-Binary Evolution in Rotating Nuclei
There is a radius around the SMBH beyond which changes in the Lj are dominated by star-star, rather than spin-orbit, torques.
The “radius of rotational influence,” aK, of a SMBH is defined by
Beyond this radius, the stellar Lj evolve stochastically, on the “resonant relaxation” timescale.
rg ≡ GM•c2
.
DM & Vasiliev (2012)
�1− e2
�3�aKrg
�3
<16χ2
N(a)
�M•m�
�2
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Radius (pc)
Radii associated with SMBH spin
DM & Vasiliev (2012)
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Response of χ≡ S/(GM●/c2) to torquing by stars, some of which lie outside the radius of rotational influence.
m✶ = 0.1 M⊙
m✶ = 1.0 M⊙
m✶ = 10. M⊙
DM & Vasiliev (2012)
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Accretion disk Stars
•Dissipative•Gas near SMBH in thin disk•Torque determined by gas at ~warp radius*
•S aligns with Lgas
•Dissipationless•Stars near SMBH can have any distribution•Torque determined by stars at ~rotational influence radius*•S may, or may not, align with Lstars
*after matter near the SMBH has aligned with it
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S-shaped radio source
Condon & Mitchell (1984)