Paper: The spin and magnetic flux of M87*

My paper estimating the spin and magnetic flux of the central supermassive black hole of the galaxy M87—the so-called M87* object—was just accepted for publication in the Astrophysical Journal Letters (ApJL). The black hole in M87* is the first in history to have its event horizon imaged with the very large baseline interferometric apparatus of the Event Horizon Telescope (EHT).

The fact that some image of an event horizon would be announced on April 10th was not a surprise to our group. What caught us by surprise—since we are not members of the Event Horizon Telescope Collaboration (EHTC)—was the fact that the black hole imaged was in M87 and not in our Galactic Center (for more information, check out the six papers outlining this incredible, game-changing discovery).

Anyhow, a couple of days after the results were announced, while I was reading the first paper of the series, I was struck by a particular paragraph at the end of Section 6:

I immediately realized that I could produce a more precise estimate of the black hole spin of M87* based on the power of the relativistic jet. Kerr black holes can be completely described by only two numbers: the mass and the spin. There have been several measurements of the mass of M87* with a better than 10% uncertainty on the mass, for example using stellar or gas dynamics or the size of black hole shadow. Getting the spin however is a completely different story and much more difficult. Measuring the spin from the shadow is currently out of question because the images are not sharp enough to the degree that would allow us to get confident estimates. And many other methods in the literature suffer from issues such as large uncertainties in the data or model parameters.

I thought that perhaps I could contribute an interesting estimate of the spin of M87*. Wouldn’t that be a nice—and hopefully quick—paper? Over the next couple of days, I devoted myself entirely to getting this estimate right and assessing whether it would be worth of publishing. I was nervous because if I got that the spin a* is very low or consistent with zero then the result would not be very interesting and not worth writing a paper about it. What I found surprised me. And led me to write my quickest paper up to date: it took me two weeks from the beginning of the analysis up to having a manuscript submitted to ApJL.

Observations

Before digging into the results, what were the observables? The observables I chose were the total power carried by the jet coming from M87* (the jet power) and amount of mass being fed to the black hole—the mass accretion rate which I will also refer to as Mdot. If I have reliable measurements of these two numbers, then I could use current ideas about how black holes produce jets to tie the observations to an estimate of the black hole spin.

Where do these observables come from? The jet power was estimated by Russell et al. (2013) using Chandra X-ray observations of the hot gas around M87*. From the temperature, density and an idea of the volume of such gas, Russell et al. was able to quantify the average power dumped by the jet over a period of about one million years: about 1E43 erg/s. For reference, the black hole in the center of M87 is putting out ten billion times more energy in its environment than the Sun radiates per second.

I should mention that there are several different ways of estimating the jet power in M87* (for reviews, cf. EHTC paper V). I prefer the estimate from X-ray bubbles because it is more robust against time variability.

The mass accretion rate was estimated by Kuo et al. (2014) based on a clever idea originally proposed by Dan Marrone in the context of Sagittarius A* (Sgr A*). Marrone et al. (2006) figured out that if they have a good measurement of the amount of polarization that radiation suffers when it leaves the surroundings of the black hole, then using simple assumptions one can estimate the gas density near the black hole and Mdot. Concretely, one needs to measure the Faraday rotation measure (RM) and assume that one is observing synchrotron radio emission coming from the inner parts of the accretion flow and that his radiation is polarized by the accretion flow itself as it travels outwards and eventually reaches the observer. Kuo et al. (2014) measured the M87* RM and found that Mdot < 9E-4 Msun/year. In other words, Kuo et al. measured an upper limit to Mdot. This means that in one year, M87* eats up one Jupiter worth of mass (actually, less than that).

Importantly, if the observed polarization in M87* is not due to the accretion flow as a “Faraday screen”, then this will affect the estimates of Mdot. I will return to this point further below.

OK, so we have some amount of energy flowing out of the black hole—the jet power—and some upper limit on the amount of energy flowing into it—the Mdot. How can we put this together and estimate the rotation frequency of spacetime?

Theoretical model

It turns out that black holes (BH) are in many ways similar to a car engine. If you wanted to reverse-engineer the energetics of an engine, you would just need to observe its fuel consumption, how much is lost in the exhaust and how much speed it delivers. Then you would have a good idea of how efficient the engine is and start working out how it could achieve such levels of fuel consumption. For a black hole it works exactly in the same way. If you know how much power it produces by accelerating particles in a jet, and if you know how much gas is being fed to the BH, you can work out how “green” it is. Why is this related to the title of this blog post? Because the level to which a BH is economical is related to how fast it rotates. The BH spin is the turbo in the engine: the larger the value of a* is, the larger is the amount of jet power produced by the BH for a given fixed Mdot.

To begin with, I used a model that specifies the efficiency of jet production η as a function of a*. This model is called Blandford-Znajek mechanism named after the researchers that solved Maxwell equations to first order in the curved spacetime of a BH almost forty years ago, and figured out how BHs can power jets (Blandford & Znajek 1977). The Blandford-Znajek model has a couple of free parameters and I needed to anchor these values otherwise I would not learn much about M87* from applying it to my data. I fixed the fudge factors in the model by using a series of advanced numerical simulations of how magnetized plasmas near event horizons behave as time progresses, which have the technical name of general relativistic magnetohydrodynamic (GRMHD) simulations of accretion onto Kerr BHs.

I based my models on the numerical results of my collaborator Sasha Tchekhovskoy, who is an assistant professor ar Northwestern University. The figure below summarizes how efficient BH engines are at producing jets according to the Blandford-Znajek model and GRMHD simulations.

There are two numbers that control the jet efficiency. The first one is the spin, of course. There is a second number as well: the magnetic flux on the event horizon of the BH. Because jets are powered by a helical twisting of magnetic field lines anchored in the event horizon, the power also depends on the magnetic field. Therefore, by modeling M87*’s data we can learn something not only about the spin but also the magnetic field near the BH.

Main results

The two main results of the paper are the following:

1. I get a robust lower limit on the black hole spin in M87* from the observations: a* > 0.5. This means that the black hole must be rotating at least at half of the maximal possible rotation frequency allowed by general relativity.
2. I find lower limits on the amount of magnetic flux threading the event horizon, 𝜙 > 5 (𝜙 in dimensionless units typical of GRMHD works). This means that the magnetic fields surrounding the BH are quite strong. This disfavors a whole category of accretion flow models known as “SANE” for M87*.

If these bounds were to be violated, then the BH would not be able to pump enough energy into the jet to be consistent with the observed power. Combining these results with the constraints from EHT observations—something that I have not done—should reduce even further the parameter space allowed for M87*.

A few more details about result #1. I actually considered both the cases in which the BH could rotating in the same (prograde) or opposite (retrograde) direction as the accretion flow (however, the angular momentum vectors must be parallel or antiparallel). If M87* is prograde, then the lower limit on the spin is |a∗| ≥ 0.4, otherwise it is |a∗| ≥ 0.5. I was not able to distinguish between the prograde or retrograde scenarios based only on the data available. Hopefully, the upcoming EHT polarimetric observations will shed more light on these issues.

Some comments

Spin

What is the meaning of the spin parameter that I talked about above? The maximal possible rotation frequency allowed by general relativity corresponds to max(a*) = 1. At the maximal spin, the equator of the black hole would be rotating at the speed of light. Above that limit, one interpretation is that the black hole would break-up due to centrifugal forces and a naked singularity would be revealed. Nobody has figured out how to do that—even in theory.

Accretion rate and Faraday rotation measure

I should thank the referee because he/she really helped to improve the quality of the manuscript thanks to the thoughtful comments. One of the interesting points made by the referee was the following:

The upper limit on density relies on the model used by Kuo et al. to relate rotation measure to accretion rate. There are large uncertainties in this estimate! The RM depends not only on density, but also magnetic field strength and geometry, […] along a highly inclined line of sight. […]

This made me think about the underlying assumptions behind the Mdot estimate by Kuo et al. 2014. The idea goes back to Marrone et al. (2006) and requires a model for the density and magnetic field in the accretion flow, relying on the following assumptions:

1. the Faraday rotation is caused by the hot accretion flow in front of a source of synchrotron emission
2. the accretion flow is roughly spherical and characterized by a power-law radial density profile
3. the magnetic field is well ordered, radial, and of equipartition strength

Of course, real accretion flows are messy and turbulent. GRMHD simulations indicate that their magnetic fields are predominantly toroidal rather than radial (e.g. Hirose et al. 2004 and many other works). Marrone et al. argues that the assumption of a radial magnetic field should give only a small error. The outer radius used in the estimate of Mdot should depend on the coherence of the magnetic field. Kuo et al. assume rout=rBondi. If rout<rBondi, then Mdot will be even less than estimated, thereby increasing the lower limits on a* and phi.

Quantifying the impact of the line of sight on Mdot and hence on our estimates of a* and phi is also difficult. Given that the jet in M87 has a low inclination angle to the observer, one possibility that cannot be completely ruled out is that the RM originates from the jet sheath, with the line of sight of the observer not passing through the RIAF. This scenario was explored by Moscibrodzka et al. 2017 using GRMHD simulations. If that is the case, Moscibrodzka et al. concluded that the RM would be consistent with a higher Mdot than we considered (and potentially much higher). This would lower the spin and magnetic flux, as discussed in the letter.

Interestingly, in the models by the EHTC in paper V did not obey the Mdot constraints of Kuo et al. Mdot in their models is a free parameter that is tuned to reproduce the observed compact mm flux. The exact value of Mdot in the general relativistic ray-tracing simulations employed in paper V depends on the electron thermodynamics and spans a wide range.

One feature of current GRMHD simulations tackling the amount of Faraday rotation in RIAFs such as Moscibrodzka et al. (2017) is that they have a very small torus extending up to about 60M. Furthermore, they do not have a high enough spatial resolution to resolve MRI in the outer regions of the disk—which is a crucial ingredient for reliable RM estimates—and do not have long enough durations to establish inflow equilibrium in the outer parts of the disk. While a jet-originated RM is possible, I am afraid that some GRMHD simulations might be underestimating the polarization effects of the RIAF at larger scales and this might impact their conclusions on the amount of Faraday rotation. In conclusion, both models for M87*’s RM—the simple analytical RIAF model a la Marrone and the current round of GRMHD simulations—are incomplete. There is definitely a lot of space for improvements in these calculations.

This work was supported by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) under grant 2017/01461-2.