25.3 The mass of the galaxy

Learning objectives

By the end of this section, you will be able to:

• Describe historical attempts to determine the mass of the Galaxy
• Interpret the observed rotation curve of our Galaxy to suggest the presence of dark matter whose distribution extends well beyond the Sun’s orbit

When we described the sections of the Milky Way, we said that the stars are now known to be surrounded by a much larger halo of invisible matter. Let’s see how this surprising discovery was made.

Kepler helps weigh the galaxy

The Sun, like all the other stars in the Galaxy, orbits the center of the Milky Way. Our star’s orbit is nearly circular and lies in the Galaxy’s disk. The speed of the Sun in its orbit is about 200 kilometers per second, which means it takes us approximately 225 million years to go once around the center of the Galaxy. We call the period of the Sun’s revolution the galactic year . It is a long time compared to human time scales; during the entire lifetime of Earth, only about 20 galactic years have passed. This means that we have gone only a tiny fraction of the way around the Galaxy in all the time that humans have gazed into the sky.

We can use the information about the Sun’s orbit to estimate the mass of the Galaxy (just as we could “weigh” the Sun by monitoring the orbit of a planet around it—see Orbits and Gravity ). Let’s assume that the Sun’s orbit is circular and that the Galaxy is roughly spherical, (we know the Galaxy is shaped more like a disk, but to simplify the calculation we will make this assumption, which illustrates the basic approach). Long ago, Newton showed that if you have matter distributed in the shape of a sphere, then it is simple to calculate the pull of gravity on some object just outside that sphere: you can assume that gravity acts as if all the matter were concentrated at a point in the center of the sphere. For our calculation, then, we can assume that all the mass that lies inward of the Sun’s position is concentrated at the center of the Galaxy, and that the Sun orbits that point from a distance of about 26,000 light-years.

This is the sort of situation to which Kepler ’s third law (as modified by Newton) can be directly applied. Plugging numbers into Kepler’s formula, we can calculate the sum of the masses of the Galaxy and the Sun. However, the mass of the Sun is completely trivial compared to the mass of the Galaxy. Thus, for all practical purposes, the result (about 100 billion times the mass of the Sun) is the mass of the Milky Way. More sophisticated calculations based on more sophisticated models give a similar result.

Our estimate tells us how much mass is contained in the volume inside the Sun’s orbit. This is a good estimate for the total mass of the Galaxy only if hardly any mass lies outside the Sun’s orbit. For many years astronomers thought this assumption was reasonable. The number of bright stars and the amount of luminous matter (meaning any material from which we can detect electromagnetic radiation) both drop off dramatically at distances of more than about 30,000 light-years from the galactic center. Little did we suspect how wrong our assumption was.