Measuring the mass of galaxies.

• Find a spiral galaxy that we view neither face-on nor edge-on, but somewhere in between.
  
• Collect the light from the stars in a band across the galaxy at each distance R from the center of the galaxy.
  
• For each R, measure the doppler shift of the starlight.
• This gives the velocity of the stars at each location, either toward us or away from us.
• Correct for the velocity of the center of the galaxy toward us or away from us and for the tilt of the galaxy to get the velocity of the stars in their orbits about the galactic center.
• Make a graph.
• Here is what we expect, based on the idea that most of the galactic mass is at the center of the galaxy.
  
• Here is what we see.
  

Our conclusion

• Most of the light comes from the middle of the galaxy.
• But the mass is much more spread out.
  • There is more mass inside a big radius R than inside a little radius R.
  • This holds even out to the outer regions of the galaxy, where there are few stars.
• If there is more mass than there is in visible stars, we may presume that the mass is in the galactic halo, which is approximately spherically symmetric -- so for approximate purposes, the assumption of spherical symmetry shouldn't be too far off.

How much mass it there?

• Based on measurement of the amount of light in our galaxy, the mass in stars is somewhat less than 10¹¹ M_sun.
• Based on the Sun's speed around the galactic center, the mass of the matter inside the Sun's orbit is 1 x 10¹¹ M_sun.
• Based on the motion of gas clouds and of globular clusters as far from the galactic center as one can find things to look at, the mass of the Galaxy inside 120 kly is
  6 x 10¹¹ M_sun
• Thus the mass that we can account for (in stars and gas) is only 10 to 20% of the total mass of the Galaxy.