Determining Masses Using Velocities

By definition we cannot resolve the two components in a spectroscopic binary. Therefore, we cannot determine the orbits visually in order to apply Kepler's third law in the obvious way to determine masses. On the other hand, the velocity curves for the two components give us information that allows us in the right circumstances to measure, or at least place limits, on the total mass of the system and even the mass of the individual components. The box below describes the details of such an analysis.

Technically Speaking: Masses in Spectroscopic Binaries

In the adjacent figure we illustrate the orbits around the center of mass for a pair of stars in a binary. For simplicity in the discussion we assume the orbits to be circular and to not be tilted with respect to the observer. The period for the revolution is P, subscripts "1" and "2" refer to the two different stars, m denotes a mass, v denotes the magnitude of the velocity (which is constant for a circular orbit), r denotes the separation from the center of mass, and G is the gravitational constant.

Steps in the Analysis
The analysis then involves the following steps. The first two equations relate the magnitude of the velocity to the period for the orbit and the radial velocity for each star by requiring that in one period the star goes once around its circular orbit at constant velocity. From the spectrum we can determine the velocities (Doppler shift) and the period. Thus, the first two equations may be solved to give us the separation r from the center of mass for each star. Then the measured velocities and period can be used in the third equation (which is just a particular form of Kepler's third law) to determine the sum of the masses. Finally, the last equation, which is the definition of the center of mass, can be solved simultaneously with the Kepler's law equation and the known separations r to divide the total mass between the two stars.

Complicating Issues
We conclude from the preceding discussion that by measuring the velocities using the Doppler effect and observing the period, it is possible to determine the masses in the binary. In practice, this method is more complicated than our simple discussion would suggest because the orbits may not be circular, and we generally do not know the tilt angle of the orbits with respect to the line of sight. These complications mean that often we cannot determine the masses completely, but can place limits on them. Even this limited information can be very important since we have few other ways to determine masses of stars reliably.