They vary their light output on very short timescales.
Active galaxies change brightness on the timescale of hours. This again places an upper limit to the size of the energy source:They are luminous.
Quasars have luminosities 100-1000 times that of the Milky Way. Yet it all comes from a region smaller than the Solar System!They must be massive objects.
Remember that light itself (photons) can impart a force onto a particle (for example, an electron). If a source has luminosity L, then a particle with interaction cross section sigma at a distance R will feel a outward force given by:Let's see. Very massive. Very luminous. Very small. What does this sound like?
This radiative force pushes material out of the nucleus. What holds it in? Gravity:Let's balance these forces And solve for the gravitational mass: What does this mean? If the mass is at least this much, gravity will hold the system together. If the mass is less than this, radiation pressure wins and the system will be pushed driven apart. So unless quasars are wildly out of equilibrium, their luminosity places a lower limit on their mass. This is called the Eddington limit. It actually holds for a wide variety of astrophysical applications.
Using numbers typical for quasars, we get a mass of ~ a few x 108 Msun. Very massive!
But why are there so many types of active galaxies?
