Stellar Dynamics around a Massive Black Hole – Ben Bar-Or and Tal Alexander – Weizmann Institute of Science
Stellar Dynamics around a Massive Black Hole – Ben Bar-Or and Tal Alexander – Weizmann Institute of Science
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WIS2015
On Github benbaror / WIS2015
Stellar Dynamics around a Massive Black Hole
Ben Bar-Or and Tal Alexander
Weizmann Institute of Science
The Galactic center is a rich environment
- Extremely dense stellar system
- We can observe individual stars
- Observations indicate a Massive Black Hole M=4×106Msun
- A “lab” for testing general relativity
- Most of the stellar objects are unobservable
- Opportunity for studying statistical physics of a stellar system
Schödel et al. (2007)
The S-stars cluster
Tidal disruption
Extreme mass ratio inspiral
Steve Drasco & Curt Cutler
N-body simulation (N=104t=105yr) Bar-Or et al. (2013)
Statistical mechanics of stellar systems is challenging
Commonly used Approximations:
- Local interactions
- Instantaneous interactions: Markovian (uncorrelated) process
- Weak encounters: central limit
Description by a Fokker-Planck (diffusion) equation:
- Random walk in velocities (energy and angular momentum)
- Slow relaxation trelax∼1010yr
2-body encounter
∂∂tf(E,t)=12∂2∂E2[D2(E)f(E,t)]−∂∂x[D1(E)f(E,t)]
Long-time correlations are important
Resonant Relaxation (Rauch & Tremaine 1996):
- Stochastic residual torques ˙J∝√N
- Short timescales (P<t<Tcoh): angular momentum changes coherently
- Longer timescales (t>Tcoh): random walk in angular momentum
- Relaxation can be fast: TRR≪Trelax
N-body simulation (N=104t=105yr) Bar-Or et al. (2013)
N-body simulation (N=104t=105yr) Bar-Or et al. (2013)
Key question: How to describe resonant relaxation?
Challenges:
- Long range interactions
- Long time correlations
- Multiple timescales: correlated process
Description by a diffusion equation?
The Stochastic approach
- Stochastic Equations of motion: ˙J=−τN(J)ˆeψ(ϕ,θ,ψ)×η(t)
- Markovian approximation - uncorrelated noise: ⟨ηi(t)ηj(t′)⟩=δijδ((t−t′)/Tcoh)
N-ring simulation
Resonant relaxation can be extremely efficient
- Random walk in phase space
- Much faster than 2-body relaxation
- All stars will plunge into the massive black hole
- General relativity is not included
Monte Carlo simulations: 2-body only Bar-Or and Alexander (2015)
Monte Carlo simulations: with resonant relaxation Bar-Or and Alexander (2015)
Relativistic stars precess fast
- Precession frequency diverges with eccentricity νGR(j)=3J2cJ2rgaνr(a)
S2
Ghez et al. (2008), Gillessen et al. (2009) TGR≈2×103P
General relativistic effects restrict the relaxation
- Restricted random walk in phase space
- Protection against direct plunges
- Emission of gravitational waves
Post-Newtonian N-body simulation (N=50) Kupi and Alexander (2012)
Key question: How to describe resonant relaxation?
Challenges:
- Long range interactions ✔
- Long time correlations
- Multiple timescales: correlated process
- Non-relativistic orbits: 2π/νGR≫Tcoh
- Relativistic orbits: 2π/νGR≪Tcoh
Description by a diffusion equation?
- Yes! resonant relaxation can be describe by an effective Fokker-Planck equation for a general correlated noise
Stochastic equations of motion with correlated noise
- Stochastic Equations of motion: ˙J=−τN(J)ˆeψ(ϕ,θ,ψ)×η(t)
- Correlated noise: ⟨ηi(t)ηj(t′)⟩=δijC((t−t′)/Tcoh)
Noise models
Power spectrum Bar-Or and Alexander (2014)
Resonant relaxation can be describe by an effective diffusion equation
- Effective Fokker-Planck (diffusion) equation
- Noise dependent diffusion coefficient: D2(j)∝Sη(νGR(j)) Proportional to the spectral power of the noise at the precession frequency.
- GR Precession frequency diverges with eccentricity νGR(j)∝1/J2
Power spectrum Bar-Or and Alexander (2014)
- Adiabatic invariance νGR=2π/Tcoh
The time evolution depends on the noise properties
- Good match between the stochastic equation of motion (circles) and the effective Fokker-plank (lines)
- Smooth (Gaussian) noise results in adiabatic invariance
- Non-smooth (Exponential) noise allows angular momentum to evolve to J→0 (plunge into the MBH)
Bar-Or and Alexander (2014)
Bar-Or and Alexander (2014)
Bar-Or and Alexander (2014)
Bar-Or and Alexander (2014)
Bar-Or and Alexander (2014)
Bar-Or and Alexander (2014)
Bar-Or and Alexander (2014)
Resonant relaxation is efficiently quenched by GR precession
Analytic model Bar-Or and Alexander (2015)
- Inspiral rate: 1.5×10−6yr−1
Monte-Carlo simulation Bar-Or and Alexander (2015)
Summary
- Statistical mechanics framework for resonant relaxation
- Representation of the background as correlated noise
- Derivation of an effective diffusion equation for a general correlated noise
- Due to general relativity, stellar black holes can inspiral into the massive black hole while emitting gravitational waves