How can I know the source size using the spectrum variability? When reading some astrophysical papers I saw things like this:
"The sources are variable on very short timescales, implying a compact emission region"
How can we conclude or measure the source size, or emission region size (or its compactess) from the variability of the spectrum?
 A: If the source is very large, then an E.M. Wave from the more distant part of the source lags behind that from the near side.  A very short single pulse can only come from a small source.
A: If a source varies significantly, i.e. changes it's brightness by a large fraction, then that implies there is some sort of coherent change in the source properties. For example it might be that the source region becomes hotter/colder or bigger/smaller.
Coherent changes that can affect the whole of the source region can only propagate at a maximum speed equal to the speed of light. This means that if the source brightness changes significantly on a timescale $\tau$ then the source region cannot be any bigger than $c\tau$.
It is these kinds of argument that were used to support the idea of pulsars being neutron stars with size $\sim 10$ km, because variability is seen on timescales as short as $\sim 0.1$ ms. Similarly, the rapid variations of emission on timescales of a few $r_s/c$ from some quasars also suggests a very compact emission region associated with accretion onto a supermassive black hole.
