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
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
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
We shall discuss this more quantitatively
Such methods are extremely important, because they are one of the only ways that we
can reliably determine the masses of stars.
- 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.
- If in addition the individual separations r1 and r2
can also be measured, the masses of each of the stars can be determined.
Planets around other Stars
There has been considerable recent interest in possible planets detected around
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
(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
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