For all the sophistication astrometry boasts, measuring the mass of a black hole is still tricky business. This isn't to mention as-yet inexplicable phenomena like feedback due to the M-sigma relation and dark matter.

Black holes are abominable anomalies. Any theory that encounters them dare not go head on, but only around it, mathematically speaking. Like the swirling clouds of gases that comprise the accretion disk around a black hole which are digested by the central singularity, the theory too skirts it at an appropriate distance and moves on.

Even to tell if a black hole’s sitting at a particular point in space is a tricky thing. The classic example of a black hole’s power, that it sucks even light in, could be misleading. When observing what could be a black hole from a long way away, light from other sources could get in the way of ruling out a characteristic “dark spot” in space.

Obviously, other, more sophisticated techniques have to be used.

Localised sources of X-rays

One is that when a black hole pulls an object in toward itself, it doesn’t just suck it in like a cuttlefish. This is because the object most likely was moving through space, or had started to move as it neared the black hole. As the pull becomes stronger, the object starts to go around the black hole faster and spirals in toward the centre, undergoing extreme compression and heating. When it’s hot enough, the object – in whatever malformed state – will start emitting X-rays.

“See what that weird star over there’s up to.”

Another technique depends on fortuity; specifically, if there’s a companion star going around the black hole, but not being sucked in… yet. Knowing how long the star takes to go around the black hole once, the shape of the orbit, and the closest the star gets to the black hole itself can be used to adjudge if there’s a black hole in the area. Of course, this involves keeping an eye out for stars that seem to be going around a seemingly empty region of space.

Whip out the math.

In fact, given the radius of the companion star’s orbit, the black hole’s mass can be calculated using Kepler’s laws. Then, assuming that the central object’s got to be much smaller in order to prevent it from colliding with the star, an estimation of the black hole’s maximum radius can be arrived at. Using that, the volume of the black hole can be calculated. So, we’ll have mass, we’ll have volume, and we’ll have the opportunity to judge if it’s a black hole by looking at how much mass is being contained in how much volume.

Stellar kinematics

So much for spotting a singularity; no wonder calculating the mass of a black hole can be a real problem! It’s not like we can assemble a weighing balance of cosmological scales around a black hole and use a counterweight on the other side to compare its mass with. The pretty much sounds like weird science-fiction anyway.

Anyway, for starters, like in our last example, a star’s orbit around the black hole can be used to determine its mass, but this calculation doesn’t yield an accurate measurement but under exceptional circumstances. Why? Because a lot of precise values for hard-to-measure parameters are involved.

First, the radial velocity of the black hole has to be measured precisely. This is the speed with which the black hole is moving toward or away from an observer on Earth. Now, you’d think a black hole would be stationary. This isn’t the case because just like the black hole pulls everything in its vicinity toward itself, the mass of all the objects around it exert a weak gravitational pull toward themselves.

This means lighter black holes would move quite a bit with respect to objects around them, whereas heavier ones would move slightly. Nevertheless, there is some radial velocity, and it is measured using the Doppler Effect. Astronomers on Earth use powerful optical (i.e., light-tracking) telescopes to make this measurement by observing how much light from the black hole’s direction changes in colour, i.e. red-shifts, based on how much the black hole is pushing or pulling it from Earth.

Next, since the Earth itself is located in the Milky Way galaxy, Earth’s rotation speed with respect to the entire galaxy’s rotation speed will have to be accounted for, again, precisely. Third, the same strength of measurement will have to be applied to the companion star whose position and path we’re tracking. I think you see the problem. For approximate values of the black hole’s mass, this method is OK, but for ultrafine measurements, other options are available which involve fewer, but persistent, complexities.

In fact, using this method, the Max Planck Institute for Extraterrestrial Physics and the UCLA Galactic Center group have estimated the mass of the supermassive black hole (SMBH) at the Milky Way’s centre to be about 8.2x1036 kg. That’s about 4.1 million times the mass of our beloved Sun. For this, they used the Keck Telescope, Hawaii, and the European Southern Observatory, northern Chile.

Astrophysical masers and Mysterium

A more sophisticated option is to use a funky concept called astrophysical masers. In 1965, scientists thought individual molecules couldn’t exist in space, so by extension microwave radiation corresponding to such molecules wouldn’t be found in space. Then, Prof. H. Weaver from California tracked some mysterious microwave radiation at about 1665-1670 MHz in space. Because the source couldn’t be pinned down, it was attributed to a hypothetical entity called Mysterium.

However, the wonderment wasn’t to last long. Research revealed that the microwave radiation was coming from some compressed molecules in gas clouds that were being naturally excited to a specific higher energy by their environs, and then stabilising by emitting microwave radiation. By 1974, water, methanol, silicon-oxide, and formaldehyde were classified as such stimulated emitters of microwave radiation (i.e., masers) in space.

And as a gas cloud accretes around a black hole, measuring how these molecules in the cloud mase in different ways can throw light on the gas cloud’s velocity and temperature, the black hole’s mass (which “engines” the cloud around itself), and even the speed at which the black hole is spinning (reflected in temperature variations).

In fact, after repeated measurements from different environments around the same black hole, statisticians can be introduced in the picture, statistical analysis kicked off to narrow down the results, eliminate as many errors as possible, measure the most significant emission frequencies, etc., all done together with velocity analysis. Technically, this process is called reverberation mapping.

The idea of using molecular masers is popular among astronomers and astrophysicists, and is in fact deployed in conjunction with other techniques, such as stellar and ionised-gas kinematics, as well.

The M-σ relation

In 1999, an interesting consequence of reverberation mapping was revealed at a conference at the Institut d’astrophysique de Paris, France, under the name of the Faber-Jackson law, popularly called the M-σ relation (formula from Wikipedia shown below). This law was important in asserting that SMBHs are native components of almost all galaxies because it was observed that the size of the black hole had something to do with the galaxy’s bulge around the centre.

In other words, the more massive the central supermassive black hole was, the more proportionately larger was the galaxy’s bulge’s size. The reason for this “feedback” is still unknown.

There are other problems, too. Apart from the fact that these are pretty much all the techniques we’ve available, the closest black hole to Earth is Sagittarius A*, at the eye of the Milky Way, about 26,000 light-years, or 246 quadrillion km, away. What’s more, cosmological unknowns also exist, such as dark matter which, I’m sorry to use the phrase, we’re hopelessly in the dark about.

Advanced interferometry and the future

Fortunately, we’re living today in an era of rapid advancements, as the cliché goes, in millimetre and sub-millimetre interferometry. While it’s got a fancy name, what interferometers do is simple: They superimpose different waves to see how they combine, in the process revealing their properties. This is like pairing up one stranger with another and seeing how they behave in each other’s presence.

Better interferometers form the basis of better spectroscopic analysis of radiation in space. As the abstract of this paper in Nature notes: "With the next-generation millimetre-wavelength interferometers ... observations [of the dynamics of gas clouds] could be reproduced in galaxies out to 75  megaparsecs in less than 5 hours of observing time."