We can determine the atmospheres of exoplanets and the composition of galaxies billions of light-years away...but if you ask astronomers to weigh a star, they're stumped. Now there's an awesomely circuitous way to do it, but we'll need some exomoons.
Considering all the amazing things astronomers can do, it seems sort of weird that there's no reliable way to measure the mass of a star. The problem is that the volume and mass of a star, while definitely related, aren't really precise predictors of each other, which means knowing how big a star is still doesn't tell you how much it weighs. Before the discovery of exoplanets, the only stars we could even consider calculating the mass of were those in binary systems, because then astronomers could figure out how the mass of each star affected its gravitational pull on the other.
Exoplanets don't offer much help, because we only observe them when they block the light or affect the orbit of their star, meaning we can only measure these planets in terms of their star. That sort of relative data doesn't help make absolute measurements about the size or mass of the star itself. We do have computer models that can give reasonable approximations of these figures, but we still lack a direct method for figuring out something so basic about these stars.
That's where David Kipping of the Harvard-Smithsonian Center for Astrophysics enters the picture. He's come up with a new method that can offer a direct measurement of a star's mass using Kepler's Third Law, as long as we can observe both a planet and its moon transit in front of a star. Measuring how much the transit of the planet and moon dim the light from the star allows astronomers to get a whole bunch of information, including their orbital periods, the relative distances of their orbits, and their relative sizes compared to the stars.
Because Kepler's Third Law links a planet's orbital period to its distance from its star, this allows astronomers to combine the known data points and figure out the relative densities of the planet and the star. We still wouldn't be able to figure out the density of the moon, because it's last on the orbital chain, but by multiplying the new density numbers by the relative volumes, we could come up with the relative masses of the star and planet.
Then we could use the other exoplanet-hunting technique, which involves measuring a star's gravitational "wobble" caused by the planet's gravity. This wobble, also known as the radial velocity of the star, can be combined with the relative mass information to give the absolute mass of the star (though not the planet). It's a lot of work to get to such a seemingly simple figure, but amazingly it's pretty much the easiest direct measurement technique available.
Now all we have to do is find some exomoons, something we've yet to discover. Kipping is hopeful astronomers can put his technique to use in the near future, however, as NASA's exoplanet-hunting mission - appropriately enough, named Kepler - is expected to discover at least a handful of systems with noticeable exomoons. It had better, as otherwise astronomers are headed back to the drawing board:
"If there was no moon, this whole exercise would be impossible. No moon means we can't work out the exact density of the planet, so the whole thing grinds to a halt. When they're found, we'll be ready to weigh them."