Measuring theMass 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.

### Kepler's Laws and Masses

The determination of masses in binary systems generally uses Kepler's 3rd Law,

( m1 + m2 ) P2 = ( d1 + d2 ) 3 = R3

where P is the orbital period, m1 and m2 are the respective masses, and R = r1 + r2, 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 = m1 + m2 of the binary system. Then, if we know enough about the orbits to determine the distances d1 and d2 separately, the second equation can be used to determine the individual masses m1 and m2. (The above equations assume that the orbits are circular. If they are more elliptical, the analysis is similar but becomes more complicated.)

### Binary Masses with Limited Information

In practical applications of mass determination we are often faced with insufficient information to apply the preceding method. This is typically because of some combination of two problems:
• We may not be able to map the orbits exactly (obviously true if the binary is astrometric and we see only one star).

• Even if the orbits can be mapped, they correspond to the 2-dimensional projections on the celestial sphere of the true 3-dimensional orbit and further information is required to construct the true orbit.
In these instances, we often can only determine only the sum of the masses rather than the individual masses, or we may only be able to place limits on the masses rather than actually determine them.

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).

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