Binary Star Systems

In a binary star system, the gravitational force which the first star exerts on
the second is

(329) |

(330) |

where

(332) |

Equation (331) is identical to Equation (212), which we have already
solved. Hence, we can immediately write down the solution:

(333) |

(334) |

(335) |

(336) |

In the *inertial* frame of reference whose origin always coincides with the center of mass--the so-called *center of mass frame*--the position vectors of the two stars are

where is specified above. Figure 20 shows an example binary star orbit, in the center of mass frame, calculated with and . Here, the triangles and squares denote the positions of the first and second star, respectively (which are always diagrammatically opposite one another, as indicated by the arrows). It can be seen that both stars execute elliptical orbits about their common center of mass.

Binary star systems have been very useful to astronomers, since it is
possible to determine the masses of both stars in such a system
by careful observation.
The *sum* of the masses of the two stars, , can be found
from Equation (337) after a measurement of the major radius, (which
is the mean of the greatest and smallest distance apart of the two
stars during their orbit), and the orbital period, . The *ratio* of the
masses of the two stars, , can be determined from Equations (338) and (339) by
observing the fixed ratio of the relative distances of the two stars from the common
center of mass about which they both appear to rotate. Obviously, given the sum
of the masses, and the ratio of the masses, the individual masses themselves can
then be calculated.