Kepler's Laws and Binary Stars
About half of all stars in our galaxy are part of binary or multiple-star systems. Kepler's laws apply to these systems just as they do to planets orbiting the Sun, with some important extensions.
How Binary Stars Orbit
In a binary system, two stars orbit their common center of mass (barycenter). Each star traces an ellipse around this point. The more massive star has a smaller orbit; the less massive star has a larger one. The ratio of their distances from the barycenter equals the inverse ratio of their masses.
Kepler's Third Law for Binary Stars
The Third Law in its full Newtonian form applies:
Here, a is the semi-major axis of the relative orbit (the orbit of one star as seen from the other), and M1 + M2 is the total mass. This is essentially the same as the planetary version, except that both masses matter, since stars can have comparable masses.
Measuring Stellar Masses
Binary stars provide the most direct method of measuring stellar masses. If you can observe the orbital period (T) and the semi-major axis (a), you can solve for the total mass. If you can also measure each star's orbit around the barycenter separately, you can determine the individual masses. This is fundamentally the same technique described in How We Weigh Planets, applied to stars.
Types of Binary Stars
- Visual binaries: Both stars can be resolved separately through a telescope. Their orbits are tracked directly.
- Spectroscopic binaries: The stars are too close to resolve, but their orbital motion is detected via Doppler shifts in their spectra (similar to the radial velocity method for exoplanets).
- Eclipsing binaries: One star passes in front of the other, causing brightness dips (similar to the transit method).