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I'm getting hung up on this term. In studying SMBHs, I see that velocity dispersion strongly correlates with mass. Just what is the velocity dispersion? How can the velocity dispersion of the galaxy be expressed in one figure (sigma) if it has to be measured all over the galaxy?

I can imagine the velocity dispersion changes with radius, so "which" point is used? Why exactly is it that higher velocity dispersion is correlated with higher Mass? And is all this referring to the bulge only or to the entire galaxy? So to get velocity dispersion, is "one slit" at the center enough or do you scan across?

I hope someone can answer this without relying heavily on the maths. I am really just trying to understand the concept, because I think that velocity dispersion is real, not just a mathematical construct.

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The answer is below, but to address one of your other inquiries: The correlation only holds for stars in the bulge of the galaxy, not the disk if there is one, but this wasn't entirely clear until the last few years. And it's not the black hole's gravity that's setting the correlation, but something else that connects the growth of the black hole to the growth of stars in the rest of the galaxy. See Chris White's comment. –  kleingordon Jul 30 '13 at 4:48

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The concept behind velocity dispersion comes from statistical mechanics, in which you're generally describing gases or fluids. In gases, you characterize the energy of the system by the temperature, which is a measure of the average thermal (random) motion of particles.

Astronomers do the same thing for galaxies. The "velocity dispersion" is one (or many) ways of measuring the effective average 'thermal' velocity of the system, as an indicator of the 'thermal' energy. I put 'thermal' in quotes, because its only analogous to the classical concept of temperature.

In any case, because of something called 'the virial theorem', the average kinetic energy (i.e. the average thermal motion) is directly related to the potential energy of the system. Knowing the velocity dispersion therefore tells you the potential energy of the system, which tells you roughly how massive the galaxy is. That's the primary insight given by the velocity dispersion.

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Assuming the system is in equilibrium, velocity dispersions really do tell you the depth of the potential well. However, except for the SMBH in our own galaxy, we cannot resolve the stars right next to the black hole. The light we see is dominated by the potential of the stars themselves, and empirically the collective mass of the bulge stars is related to the mass of the black hole itself. –  Chris White Dec 17 '12 at 22:27

The fact that sigma is used in statistics to refer to standard deviation is an important clue. There are lots of velocities in a typical galaxy. They have a mean and a standard deviation, which is called the velocity dispersion in this case. Directly, it reveals how fast things are going relative to each other or relative to their mean. Via Newton's laws, the velocity correlates with the mass. The faster they are going, the more massive the potential well that they are in.

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