Mass of the Milky Way Galaxy

The mass of an object is one of the most fundamental properties of the object. As such, reliable ways to measure mass are exceedingly important.

• One way is to simply measure the optical emission (luminosity of a galaxy) and to (independently) determine the average mass of a star in the galaxy. This will then lead to an estimate for the mass of the galaxy if all of the luminosity is in fact due to the stars and all material radiates in a similar manner. This method has problems because it is sometimes difficult to know whether all mass produces radiation.

• Another way to determine masses is to use our knowledge of physics (in particular, gravity). Since all mass is expected to interact gravitationally whether or not we can see it, we should be able to measure the mass of objects even if they are invisible to our eyes by their effects on nearby objects. This is the tact we now pursue.
  Rotation Curves ===> Masses

Recall

The gist of the method is to note that all objects are in orbit around the center of our Galaxy. To see how this allows us to infer the mass of our Galaxy, consider the Earth in its orbit in our Solar System.
• The Earth is attracted to the Sun by the force of gravity. If the Earth were stationary, the force of gravity would draw the Earth into the Sun in less than 1 year and we would be gone.

• The Earth overcomes this tendency by moving around the Sun with a speed such that the centrifugal force just balances the inward tug of gravity.

• The mass of the Sun and Earth and the distance between the Earth and Sun determine the strength of the gravitational attraction. The speed of the Earth's motion and the distance of the Earth from the Sun determine the strength of the centrifugal force. For a given size orbit, the mass of the Sun determines how fast the Earth must move to stay in orbit. (This is embodied in Kepler's 3rd law of Planetary Motion.) To turn this around, note that if we can measure how fast the Earth moves in its orbit and the size of its orbit, then we can infer the mass of the Sun! The simple expression for the mass of the Sun is
  M(Sun) = v²D/G

  where G is the gravitational constant, v is the orbital speed, and D is approximately the distance from the Earth to the Sun.

This is the basis of the method used to determine the mass of the Milky Way galaxy (and other disk galaxies). However, before we move on, we have one tweak to consider.


A. In the Solar System, the planets orbit about the Sun. None of the planets orbit within the body of the Sun. The planets closer to the Sun feel stronger gravitational tugs due to the Sun and thus must travel faster to maintain their orbits. For example, the Earth travels with a speed of ~30 km per second. Pluto travels with a speed of ~4.6 km per second. This is typical of a sytem where bodies orbit outside of the attracting mass.

B & C. Suppose we lived in a spherical system where the mass was spread out and we orbited inside of the body. Newton showed that only the mass contained within our orbit contributed to the gravitational force on us. So, if we approached the center of the object, the gravitational force would, in general, vanish. In such an extended object, the orbital speed would not fall-off as quickly as one moved away from the center of the body and, in fact, the orbital speed could stay the same and could even increase in some cases.

For the Milky Way, we find that the orbital speed increases and then remains roughly constant. This implies that we are still within the body of the Milky Way, even at the largest distances from the center of the Milky Way. We have not yet located the edge of the Milky Way; the Milky Way extends much further than the edge of the disk of stars (the visible disk)!

• The mass contained in the Milky Way (in the visible disk) is 2x10¹¹ M(Sun).

• the mass contained in the Milky Way galaxy (out to as far as we can see HI gas) is 6x10¹¹ M(Sun).

Oh my, the mass where we can see stars is only 1/3 of the mass of the entire Galaxy. This suggests that a lot of mass in the Milky Way is in some form which does not radiate large amounts of light (Dark Matter). This interesting result keeps popping up throughout the rest of the course as we discuss mass determinations for the Universe. Today, we believe that ~25 % of the Universe is made up of Dark Matter! Recall that we believe that ~70 % of the Universe is Dark Energy, so that only around 5 % of the Universe is made of stuff that we have identified.