Our Galaxy is rotating at a rather *fast* pace. We are spinning around the center at a rate of 220 kilometers every second. This is to keep us from falling in towards the mass of the center. (This is why the Earth rotates around the Sun at 30 km/s, to keep from falling into the Sun).
We can use this information, the speed at which we are rotating,
to tell us how much mass we are trying to hold up against. The
faster we go, then there must be more mass interior to our orbit
(or we'd be leaving the Galaxy). Since the Sun has been around
for 5 billion years already, and it takes only 200 Million years
to go around the Galaxy, we've already been around 25 times---safe
to assume we're in a stable orbit. We can derive the speed
of stars at all different radii and determine a Rotation Curve
for the Galaxy:
Strangely enough, the speed of rotation stays constant as you go further out in the galaxy. This is called differential rotation. Astronomers have estimated the mass of the Milky Way Galaxy to be:
Thats 1,000 Billion Stars!