Measuring a black hole mass - you're doing it right

1.  The orbit of every planet is an ellipse with the Sun at one of the two foci.
2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Kepler's laws are among the most fundamental laws of astronomy.
Basically, they describe the orbits of planets around the Sun (or any "small" object around a much more massive central object), under the assumption that the gravitational field can be considered Newtonian (even if historically quite the opposite happened, with Newton using Kepler's orbits to derive his laws of gravity).
The orbits of all planets in the Solar system are very well described, to first order, by these laws. Only after centuries of observations of Mercury, the closest to the Sun, it was possible to notice post-Keplerian deviations due to relativistic effects.
When Kepler's laws say "proportional", the proportionality factor involves the masses of the objects (this was found by Newton). For example, in this approximation the third law is actually

where a is the semi-major axis of the orbit (the radius if the orbit is circular), G (=6.67384 × 10^-11 m³ kg^-1 s^-2) the gravitational constant, M the mass of the central object, and T the orbital period.

It is therefore possible to use the orbit of the small object to infer the mass of the big one, if we have an idea of the size of the orbit.

One nice example? The measurement of our Galaxy's central supermassive black hole.
Several groups have been able to measure accurately the mass of this black hole by tracking the movement of several stars around it.
In the animated GIF above, a nice illustration of the procedure. Every orbit is an ellipse having in one of its foci, marked by the red cross in the center of the picture, an "invisible" object. This object, in order to describe the orbits of all the tracked orbiting points, must have a mass of about 4 million solar masses.
Here is a quite complete description of the work done by several groups to obtain this result. 
Enjoy!

Everything you always wanted to know about GR tests in Astrophysics but were afraid to ask

While I was looking for a refreshing reading on GR tests in Astrophysics, I stumbled upon this very nice review by Dimitrios Psaltis.
He's one of the most renowned experts on GR tests with compact objects (the one always invited at conferences to give review talks on the topic, if you know what I mean), and this paper gives a very nice overview of what constraints can be put on GR and alternative theories with astrophysical observations. It dates 2008 but nonetheless there are only a handful of things that need to be updated. For example, there will be no IXO (that was dropped by NASA, then became Athena, and was officially dropped by ESA yesterday, shame on them. Ok, I stop complaining about it. No I don't).

Anyways, if you are interested on this topic, you can read the review. And you should.

It starts by explaining in layman's terms (ok, maybe in young PhD-understandable terms) why it is important to test GR over alternative theories, and what are the differences between them.
An important thing that is pointed out is that our most successful tests of GR in Astrophysics have only shown that it works very well in the limit of very weak gravitational fields (low curvature of the spacetime). Even double neutron star observations like the one on this fantastic double pulsar, that follows GR predictions with an uncertainty of 0.05%, probe a spacetime whose curvature is not very different from the one probed in solar system tests. Wikipedia classifies, incorrectly, those tests as "Strong field tests". (But keep in mind that even weak fields can be used to test alternative theories, as a former post in this blog showed).

Is there a way to probe strong gravitational fields? Yes. We make observations of accreting black holes and neutron stars (the most compact objects present in the Universe), for example, and we measure (possibly relativistic) oscillations, broadened spectral lines, orbital decays and more that can be used to measure the physical properties of the compact object, model-dependently, and test the predictions of GR.

The constraints that we can put today on GR and alternatives are weak, but still significant. Some sporadic observations of particularly interesting sources might improve greatly our knowledge even with current instruments. For example, the recent discovery of a very heavy neutron star permitted to put unprecedented constraints on the equation of state of these objects, and hence to have better estimates on the gravitational pull at their surface. The importance for nuclear physicists of the constraints on neutron star equations of state is even greater, but we'll talk about it in the future.

Let's not talk about canceled missions (I cannot even pronounce her name without dropping a tear, poor Greek goddess) that would have had the features to give nice measures where now we struggle with upper limits, it's too painful. But let us hope that smaller but powerful missions like LOFT have better chance (If you are interested, we host a science meeting in Toulouse in September). Stay tuned.