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The mass of a star is most precisely measured for binary star systems using Kepler's third law.
Mass is a very important property of stars to measure. Why? It turns out that it is related to the lifetime of a star. The greater a star's mass, the faster it burns its "fuel."
But more importantly perhaps, mass is the single most important factor that will determine the fate of the universe. Will the universe continue to expand or will it one day collapse in "The Big Crunch"? The answer depends on the mass of the universe, and knowing that depends on our ability to measure the mass of stars.
For a two-body system where both bodies are bound by their mutual gravitational forces, each body orbits the center of mass of the system. The center of mass is at the point such that
where d1 is the distance between Star 1 and the center of mass of the system and d2 is the distance between Star 2 and the center of mass of the system.
The simulation below shows two equal mass stars. Click on the red star and drag it back and forth. Notice that the center of mass (represented by the black dot) is always equidistant from the two stars.
Now, click on the buttons to increase the mass of the red star to one that is twice as massive, four times, 10 times, or 100 times as massive. What do you notice about the location of the center of mass?
When one star is much more massive than the other, such as 10 or 100 times as massive, then the center of mass of the system is very close to the center of the more massive star. You can see that in the above simulation by looking at the situations where M is 10 or 100 times more massive than m.
In this case, the orbit of the more massive star would be much smaller than the orbit of the less massive star. It's so small, in fact, that we say that it wobbles as a result of the gravitational pull of the less massive companion star.
To see how the ratio of masses of binary stars affect their orbits, view the simulation shown below. Click the different buttons to view stars of different mass ratios.
According to Kepler's third law, the sum of the masses of the stars (in units of solar masses) is given by
where a is the semi-major axis of a star's orbit and P is the period of the orbit (which is the same for each star). Thus, by measuring two things, (1) the sum of the semimajor axes of the binary stars' orbits and (2) the period of the orbit, we can calculate the sum of the masses of the two stars using Kepler's third law.
But this doesn't tell us the mass of each star. To know that we need one more equation, the center of mass equation. At any instant of time
where v1 is the speed of Star 1 and v2 is the speed of Star 2.
The product of an object's mass and speed is called momentum. Thus, I'll refer to the above equation as "the momentum equation." Now, equipped with Kepler's third law and the momentum equation relating the speeds of the stars, we can calculate the mass of each star in the binary system.
However, there is one difficulty. How do we measure the speed of a star? The answer is by using the Doppler effect for light, and that's the subject of the next topic.
The important thing to know from this lesson is that to measure the masses of stars, we look at stars in binary systems and use Kepler's third law and the momentum equation to calculate the mass of each star.
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