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I've been watching a video about dark mater and a lot of the mass is missing in our universe. Astronomers got to this by measuring the speed that stars orbit the center of the galaxy and when they did this it didn't match the amount of gravity/mass in the galaxy.

However I thought that the mass of the black hole in the Center of the galaxy was measured by the stars orbiting it? To me it sounds like a catch 22? But I don't understand it very well. So could someone explain it further?

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For the most part the mass of the super massive black hole (SMBH) at the galaxy center is estimated in much the same way the solar mass is estimated by Newtonian mechanics. In fact general relativistic elements are not used much, where GR contributes heavily for orbits with a few number of Schwarzschild radii, or semi-major axis length. The motion of stars in the galactic center have been observed over a number of years and the velocities and accelerations known. It is then straight forwards to use Newtonian mechanics or $v~=~\sqrt{2GM/r}$.

A good source for this is the UCLA Galactic Center web site:

enter image description here

The data animation of stellar trajectories:

is interesting, just click on the animation hyperlink below. A couple of stars are on highly eccentric orbits which take then somewhat close to the SMBH.

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