Measuring the Mass of Stars |
An important applications of binary systems is that under favorable circumstances they provide one of the only ways to determine reliable masses for stars.
( m_{1} + m_{2} ) P^{2} = ( d_{1} + d_{2} ) ^{3} = R^{3}
where P is the orbital period, m_{1} and m_{2} are the respective masses, and R = r_{1} + r_{2}, and the "seesaw equation" for the center of mass:
where R is the total separation between the centers of the two objects.
From the first of these equations, if the period P and the average separation R are known, we can solve for the total mass M = m_{1} + m_{2} of the binary system. Then, if we know enough about the orbits to determine the distances d_{1} and d_{2} separately, the second equation can be used to determine the individual masses m_{1} and m_{2}. (The above equations assume that the orbits are circular. If they are more elliptical, the analysis is similar but becomes more complicated.)
In the case of astrometric binaries, we can often find families of solutions for masses or limits on the masses if certain assumptions are made about the system such as the mass of the primary (which can often be estimated indirectly from systematics of its luminosity and spectral type).