How do astronomers measure the size of celestial objects?

Question

What techniques and tools are available to astronomers in order to measure the size of celestial objects, such as stars or perhaps black holes that don't emit light or reflect starlight?

Answer 1

The main tool to measure the diameter of a star is interferometry combined with a parallax-based distance measurement - a brief review by Kervella (2008) might be useful. The principles behind interferometry are described here.

Interferometry involves measuring the light from a star using two (or more) telescopes that are separated by some distance. Together, the signals from these telescopes can be combined to give an angular resolution that can be (in the best circumstances) equivalent to a telescope with a diameter equal to the telescope separation. These measurements give the angular size of the star, which must then be multiplied by their distances to get a physical diameter.

One of the most successful experiments is the Chara array, which has yielded diameters for many nearby stars. Precisions can be as good as a few percent, but more usually 10% and of order 100 (predominantly nearby) stars have had their radii measured in this way.

A second main direct technique is to use eclipsing binary systems. The measured light curve can be used in an almost model-independent way to estimate the radii of the two stars involved. Of course most eclipsing binaries are close pairs with short orbital periods and with orbital inclinations that allow us to see the eclipse. They are therefore highly prized objects. Radii can be measured with precisions of 1%. A reasonably complete catalogue of the $\sim 100$ known eclipsing binaries with precise radii can be found here.

Another technique is lunar occultation. The passage of a star behind the limb of the moon results in a changing diffraction pattern that can be used to estimate the angular size of the star. Again a distance is required to convert this into an actual diameter.

More distant stars are inaccessible - their angular diameters are simply too small. At the moment only indirect estimates of their radii are possible. For example, if we were to assume that a star radiates as a blackbody, then its luminosity ($L$), radius ($R$) and temperature ($T$) are related by Stefan's law. $$ L = 4\pi R^2 \sigma T^4,$$ where $\sigma$ is the Stefan constant. If the star has a measured flux at the Earth and we know how far away it is, then $L$ can be estimated. If we take a spectrum and estimate its temperature, then the equation above can be rearranged to give the radius in terms of the measured luminosity and temperature. Real stars are more complicated than blackbodies, but the principle is the same.

Neither of the above techniques can work for black holes, and the sizes (event horizon or Schwarzschild radius) of black holes have not yet been directly measured. However, the diameters of the "black hole shadow", which is a small multiple of the event horizon diameter and produced by gravitational lensing of light, have been measured for two supermassive black holes by interferometry at microwave wavelengths by the Event Horizon Telescope. The physics of a black hole is relatively simple(!) and so there is a direct relationship between their Schwarzschild radii and their masses (modified somewhat by rotation). Basically it is 3 km multiplied by the mass in solar units. Therefore a measurement of the black hole mass gives its "radius". The masses of black holes are measured by looking at the motions of stars and gas around them and applying our knowledge of how gravity works. The masses can also be estimated from the gravitational wave signatures that pairs of black holes emit as they merge.

Answer 2

If a celestial body is larger than the resolving power of a telescope, its size can be measured directly. This is the case for most galaxies, molecular clouds in the Milky Way and nearby galaxies, and even for a few nearby stars. EDIT: See discussion by Rob Jeffries on how these measurements are carried out for stars using interferometry.

For more distant stars, we can rely on our understanding of stellar evolution, which tells us pretty accurately the radius, once we know its spectrum (EDIT: or just a assume blackbody radiation and use the formula given by Rob). If the star is a member of a binary system whose orbit we observe roughy edge-on, we can measure how the luminosity declines as one star occults the other, and calculate the radius. This can also be used to measure the sizes of exoplanet. And for stars, the same technique is even possible using our own Moon as occulter. See a description here.

Black holes (BH) that don't emit light, cannot be measured (at least not until we are able to detect gravitational waves), but often BHs are surrounded by a disk of accreting gas, which is heated to million of degrees by friction as it spirals down the drain. Measuring the temperature of this gas tells us the mass of the BH, which is directly proportional to its radius ($R_{\mathrm{BH}} \simeq 3 M/M_\odot$ km).

A nice technique for measuring the size of the quite small region of gas clouds around a supermassive BHs, even though they are billions of lightyears away, is called reverberation mapping. Here, some of the light emitted from the BH's accretion disk travels directly in our direction, while some of if travels in other directions, illuminating the clouds around it. When we measure the light from those clouds, the signal looks like the directly observed signal, but with a delay $t$ corresponding to extra length of the path that the light has taken. Since we know the speed of light $c$, we can calculate the extra distance as $d = ct$, i.e. the size of the system.