AstrometricBinaries

In many cases a binary system is too far away, or the stars are too close, or one star is so much brighter than the other that we cannot distinguish the two stars visually. In that case we may still infer that the system is binary by several indirect methods. One such method is to detect the presence of an unseen companion by its gravitational influence on the primary star. A binary discovered in this way is termed an astrometric binary.

### Motion around the Center of Mass

The orbits in a binary system are illustrated schematically in the following figure.

 Orbits for binary star systems

Consider the diagram shown to the right. We may define a point called the center of mass between two objects through the equations

m1 r1 = m2 r2

r1 + r2 = R

where R is the total separation between the centers of the two objects.

### Modification of Kepler's Third Law

This requires that Kepler's 3rd Law be modified to read,

( m1 + m2 ) P2 = ( r1 + r2 ) 3 = R3

(The equation in this form is only valid if the orbits are approximately circular; for more elliptical orbits a somewhat more complicated equation must be used.) Because the two stars revolve around the common center of mass, even if we cannot see one of them we may still infer its presence by observing the motion of the star that we can see around a center of mass between the two stars.

### Wobbling Motion on the Celestial Sphere

The combination of the motion around the center of mass and the proper motion on the celestial sphere gives rise to a wobbling motion on the celestial sphere, as illustrated in the following diagram for the Sirius system.

 Wobbling motion of Sirius caused by companion star

Notice that even if we could not see Sirius B, the wobbling path of Sirius A would be a clear sign of the presence of the unseen companion. In fact, Sirius B was discovered first as an astrometric binary in this way, and only later were telescopes sufficiently good to pick out the faint companion in the glare of the primary star. Thus Sirius was originally an astrometric binary, but is now a visual binary.

### Determination of Masses

From the three equations quoted above, it is possible to determine the masses in binary star systems if sufficient information can be obtained from observations. From these equations, we conclude that
1. If the period P and the total separation R between the stars can be determined, the sum of the masses of the two stars can be calculated.
2. If in addition the individual separations r1 and r2 can also be measured, the masses of each of the stars can be determined.
We shall discuss this more quantitatively below. Such methods are extremely important, because they are one of the only ways that we can reliably determine the masses of stars.

### Planets around other Stars

There has been considerable recent interest in possible planets detected around other stars. There are two interesting considerations concerning planets that are appropriate to discuss here. The first is that similar astrometric methods as described here for binary stars can also be used to detect the presence of unseen planets around stars. (The measurements require remarkable precision: The perturbation of the unseen planet may correspond to a shift in the velocity of the star of only 10-20 meters per second.) The second is the issue of whether in binary star systems there are planetary orbits that are (1) stable, and (2) conducive to the evolution of life.

Here is a set of links to the general question of extrasolar planets:

Here is a Java applet that lets you explore the orbits for some newly discovered planet candidates in other star systems.

Here is a Java applet that lets you explore the possibility of a stable planetary orbit in a binary star system, and the implications of that orbit for the amount of planetary heating by the two stars.

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