The Rotation Curve of the Milky Way 

Now, what about measuring how fast the disk is rotating? How could you measure that?

Observations show that 

So we can use this to derive...
 

...the orbital period of the Sun...
...and the mass of the Galaxy.

(Review Question: how did we get R0?)

This is similar to (but a bit more than) the observed mass in stars and gas. So everything is fine! Or is it??

This analysis is only for the Sun's galactocentric radius (R0). Can we do this throughout the Galaxy and get v(R)?

The Tangent-Point Method

Look at gas clouds in the Milky Way. Using 21-cm radio emission, we can get their radial velocity via the doppler shift.

Imagine looking at some line of sight through the galaxy and observing the gas clouds:
 

Observing Geometry
Observed Velocities

 

So v(C) = v(Rmin) = v(R0sin(l)).
 
We can do this mapping for all R<R0. At R>R0, the geometry becomes ambiguous, and we need to actually know the true distance to whatever object we are measuring the velocity of. This is harder, although still possible.

From this, we construct the rotation curve of the Milky Way: