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 \(M=4\times10^{6}M_{\mathrm{sun}}\)
  • A “lab” for testing general relativity
  • Most of the stellar objects are unobservable
  • Opportunity for studying statistical physics of a stellar system

Schoedel07_f1.png
Schödel et al. (2007)

The S-stars cluster

2013orbits_animfull.gif

Tidal disruption

rxj1242_ill_300dpi.jpg

Extreme mass ratio inspiral

ordinaryinv.png Steve Drasco & Curt Cutler

\(N\)-body simulation (\(N=10^4\, t = 10^5 \mathrm{yr}\))
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 \(t_{\mathrm{relax}} \sim 10^{10}\,\mathrm{yr}\)

\(2\)-body encounter

\[\frac{\partial}{\partial t}f\left(E,t\right)=\frac{1}{2}\frac{\partial^{2}}{\partial E^{2}}\left[D_{2}\left(E\right)f\left(E,t\right)\right]-\frac{\partial}{\partial x}\left[D_{1}\left(E\right)f\left(E,t\right)\right] \]

Long-time correlations are important

Resonant Relaxation (Rauch & Tremaine 1996):

  • Stochastic residual torques \(\dot{J} \propto\sqrt{N}\)
  • Short timescales (\(P < t < T_\mathrm{coh}\)):
    angular momentum changes coherently
  • Longer timescales (\(t > T_\mathrm{coh}\)):
    random walk in angular momentum
  • Relaxation can be fast: \(T_{RR} \ll T_{\mathrm{relax}}\)

2body.png

mass_wires.png

\(N\)-body simulation (\(N=10^4\, t = 10^5 \mathrm{yr}\))
Bar-Or et al. (2013)

\(N\)-body simulation (\(N=10^4\, t = 10^5 \mathrm{yr}\))
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?

mass_wires.png

The Stochastic approach

  • Stochastic Equations of motion: \[\dot{\boldsymbol{J}}=-\tau_N(J)\hat{e}_\psi(\phi,\theta,\psi)\times\color{yellow}{\boldsymbol{\eta}(t)}\]
  • Markovian approximation - uncorrelated noise: \[\left\langle \eta_{i}(t)\eta_j(t^\prime)\right\rangle =\delta_{ij}\delta((t-t^\prime)/T_\mathrm{coh})\]

Ylm_Dnlm_blackbg.png

\(N\)-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

Jorbit.png

j_a_noRR.png Monte Carlo simulations: \(2\)-body only
Bar-Or and Alexander (2015)

j_a_NRR.png Monte Carlo simulations: with resonant relaxation
Bar-Or and Alexander (2015)

Relativistic stars precess fast

  • Precession frequency diverges with eccentricity \[\color{yellow}{\nu_{GR}(j)}=3\frac{J_c^2}{J^{2}}\frac{r_{g}}{a}\nu_{r}\left(a\right)\]

prec.png

S2

gill09f3.png
Ghez et al. (2008), Gillessen et al. (2009) \[T_{GR}\approx2\times10^{3}P\]

General relativistic effects restrict the relaxation

  • Restricted random walk in phase space
  • Protection against direct plunges
  • Emission of gravitational waves

j_a_rel.png 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: \(2\pi/\nu_{GR} \gg T_\mathrm{coh}\)
    • Relativistic orbits: \(2\pi/\nu_{GR} \ll T_\mathrm{coh}\)

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: \[\dot{\boldsymbol{J}}=-\tau_N(J)\hat{e}_\psi(\phi,\theta,\psi)\times\color{yellow}{\boldsymbol{\eta}(t)}\]
  • Correlated noise: \[\left\langle \eta_{i}(t)\eta_j(t^\prime)\right\rangle =\delta_{ij}C((t-t^\prime)/T_\mathrm{coh})\]

Noise models

Power spectrum acf_ft_L.png
Bar-Or and Alexander (2014)

Resonant relaxation can be describe by an effective diffusion equation

  • Effective Fokker-Planck (diffusion) equation
  • Noise dependent diffusion coefficient: \[ D_2(j) \propto\color{yellow}{{S_\eta}{\color{yellow}(\nu_{GR}(j))}}\] Proportional to the spectral power of the noise at the precession frequency.
  • GR Precession frequency diverges with eccentricity \[\color{yellow}{\nu_{GR}(j)}\propto 1/J^{2}\]

Power spectrum acf_ft.png
Bar-Or and Alexander (2014)

  • Adiabatic invariance \(\nu_{GR} = 2\pi/T_{\mathrm{coh}}\)

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 \(J\to0\) (plunge into the MBH)

cdf0.png
Bar-Or and Alexander (2014)

cdf1.png
Bar-Or and Alexander (2014)

cdf2.png
Bar-Or and Alexander (2014)

cdf3.png
Bar-Or and Alexander (2014)

cdf4.png
Bar-Or and Alexander (2014)

cdf5.png
Bar-Or and Alexander (2014)

cdf6.png
Bar-Or and Alexander (2014)

Resonant relaxation is efficiently quenched by GR precession

GC_model.png Analytic model
Bar-Or and Alexander (2015)

  • Inspiral rate: \(1.5\times10^{-6}\,\mathrm{yr}^{-1}\)

j_a_RR.png 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