Black holes in gas clouds

I sometimes have research ideas that I think are cool, but that don’t make sense for me to pursue. I generally just make a note of them and move on. This is the twentieth post in a series describing some of the ideas I’ve accumulated. This idea emerged from a conversation with Yuri Levin.

Black holes in gas clouds

What’s the idea?

Asa black hole moves through a gas cloud it should drag on the gas, depositing kinetic energy. The question is: how much kinetic energy gets deposited, and how fast?

Why is this important?

Black hole mergers impart a kick on the remnant. To the extent that we think there should be black hole binary mergers in AGN disks, the resulting remnant should be moving at a decent fraction of the speed of light through the disk, dragging on its material and depositing kinetic energy.

There shouldn’t be an electromagnetic signature from binary black hole mergers in vacuum, but this sort of effect could produce such a signature by heating or shocking matter in the disk. Making predictions of this effect seems important.

How can I get started?

To my knowledge no one has really studied this, but I could be wrong, so I’d suggest doing a literature review first. After that, getting answers probably requires running new hydrodynamics simulations or post-processing existing ones.

The prototypical event to try to match is the EM counterpart to S190521g. I did some analytic estimates about a year ago to try match this event and couldn’t, but that doesn’t mean I’ve exhausted the possibilities. My estimates are below.

BBH Merger

• The S190521g properties paper describes a binary black hole merger:
  • https://arxiv.org/pdf/2009.01190.pdf
  • $M_1 = 85^{+21}{-14} M\odot$
  • $M_2 = 66^{+17}{-18}M\odot$
  • $M_{\rm r} = 140^{+28}{-16} M\odot$ (remnant)
  • Kick velocity posterior peaks around $300km/s$, but has a half-width from $100km/s$ to $1700km/s$ (by eye from Figure 4)
    • Graham+2020 mention that $v_{\rm kick} \propto M_{\rm r}^{1/3}$, so the way this co-varies with mass slightly increases the available kinetic energy from what a simple point estimate on mass would suggest.
    • This probably co-varies with other properties though. So buyer beware when using this to estimate kinetic energies.

Potential EM Counterpart

• Graham+2020 see an electromagnetic counterpart (AGN flare) to the BBH event S190521g.

  • https://arxiv.org/pdf/2006.14122.pdf
  • Location matches reasonably (within the 90% contour on the sky, near the peak in distance).
  • Flare peaked $50d$ after the merger and lasted for of order $50d$.

• Graham+2020 rule out other explanations by claiming:

  • Not a supernova because the color doesn’t evolve.
    • Suggests a constant-temperature shock.
  • Microlensing and TDE’s have low probability of coincidence ($< 10^{-3}$).

Energy Estimates of Graham+2020 Mechanisms

• The kick posterior gives a total remnant kinetic energy that plausibly ranges from $3\times 10^{49}-8\times 10^{51}\mathrm{erg}$.
• Graham+2020 estimate the flare energy at $10^{51}\mathrm{erg}$ assuming typical quasar colors and isotropic emission.
  • Their shock model then requires both that the remnant kinetic energy is on the high end and that the efficiency is high.
• Graham+2020 give two models (which both get called shock models):
  • Equations (2,3) are for the gas bound to the black hole ramming surrounding material.
    • These give too little total energy by many orders of magnitude ($\sim 10^{45}\mathrm{erg}$), but have the right time-scale
      • $20d$, though this scales as $v_{\rm kick}^{-3}$ and so the time-scale becomes much too short for the larger kicks favoured by the GW data
  • Equations (5,6) are for accretion as the remnant flies through new material that wasn’t originally bound to it.
    • $L \propto \dot{M}{\rm Bondi} c^2 \propto M{\rm r}^2 v_{\rm kick}^{-3} \rho$
    • $t_{\rm dec} \propto v_{\rm kick}^3 \rho^{-1} M_{\rm r}^{-1}$ (deceleration time)
      • Fiducial inputs give $200\mathrm{yr}$ time-scale, which is much longer than observed.
      • This one scales as $v_{\rm kick}^3$, so the larger velocities favoured by the GW data imply even longer times.
    • The net result is an energy output that scales as $L t_{\rm dec} \approx 2\times 10^{55}\mathrm{erg}$, assuming an efficiency of $0.1$ relative to $c^2$, scaling proportional to $M_{\rm r}$ and independent of the kick velocity or surrounding density.
      • This is greater than the total kinetic energy because most of the energy is actually coming from gravitational energy as gas accretes.
    • Under this model, the original black holes (with very low velocities) should have had huge luminosities. Say the sound speed is 10km/s. Then they should have each shone with $L \approx 10^{48}erg/s$. That’s clearly not right…
      • These objects are actually Eddington-limited, not Bondi-limited (i.e. Mdot_Bondi » Mdot_Eddington). For instance using equation (4) and M~100Msun and v~200km/s and rho~1e-10g/cm^3 I get an accretion rate of ~1 M_sun/yr, whereas the corresponding Eddington-limited rate is ~1e-6 M_sun/yr.

Other Mechanisms

• Li, Kocsis, Loeb 2012: https://arxiv.org/pdf/1203.0317.pdf
  • GW’s heat nearby gas.
  • Per unit mass, the deposited energy is $q = 4 G \nu E_{\rm lost} / r^2 c^3 = 4 G \nu \Delta M / r^2 c$.
    • $\Delta M$ is the mass difference between the progenitors and the remnant.
    • $\nu$ is the viscosity in $\mathrm{cm^2,s^{-1}}$
  • Integrating over the surrounding medium a distance $h$ (disk scale height) gives $E_{\rm heat} \approx G \rho h \nu \Delta M / c \approx G \Sigma \nu \Delta M/c$.
    • Using $\Delta M \approx 8 M_\odot$ gives $E_{\rm heat} \approx 10^{17}\mathrm{erg} \nu \Sigma$, where $\nu$ and $\Sigma$ are measured in cgs.
    • Need $\nu \Sigma \approx 10^{34}$ to make this work.
    • Graham+2020 estimate $h \approx 10^{14}\mathrm{cm}$ and $\rho \approx 10^{-10}\mathrm{g,cm^{-3}}$, so $\Sigma \approx 10^4$.
    • Using $h \approx 10^{14}cm$ and a sound speed of $10\mathrm{km/s}$, the viscosity is $\nu \approx 10^{20}\mathrm{cm}$.
    • So $\nu \Sigma \approx 10^{24}$, which is much too small.
    • Also $\nu$ is definitely less than estimated above, because the relevant viscosity is that on time-scales comparable to the merger ($0.1s$), not the low-frequency disk viscosity.
• Could this be due to a supernova in the disk (elides Graham+2020 concerns)?
  • Suggested by Yuri Levin
• Or NS-BH tidal disruption, or NS-NS merger around the same time?