Pairs of stars that are physically distant from each other can
appear close on the sky because of projection.
However, it's estimated that between 20% and 80% of stars
are in *true binaries*, i.e. they are bound to each other
by their gravitational attraction.

In *visual binaries* we see (with a telescope) both stars separately,
and their paths in the sky show that they are orbiting around each other.

In *astrometric binaries* we see one star's position wobble
periodically in response to the gravity of its unseen companion.

In *spectroscopic binaries* we detect the motion of one or both
stars from Doppler shifts of their spectral lines.

In *eclipsing binaries* we see the total light from the pair of
stars change periodically as one star eclipses the other.

Two gravitationally bound stars will follow elliptical orbits around their center of mass; the center of mass may have a proper motion as well, but this can be removed.

By comparing the displacements of the two stars, we can learn the ratio
of their masses,

M_{1}/M_{2}=a_{2}/a_{1}.

Newton's law of gravity (F = G M_{1}M_{2}/R^{2})
and Newton's second law of motion (F=MA)
can be combined to obtain a generalized
form of Kepler's third law (Seeds, p. 87, p. 189):

P^{2} =
4`pi'^{2}a^{3} / [G(M_{1}+M_{2})].

P = period

G = Newton's gravitational constant

a = radius (semi-major axis) of orbit

M_{1},M_{2} = masses of stars

The period P can be determined from the orbit. *If* we know
the distance to the binary, we can measure a and apply this
equation to determine the sum of the masses M_{1}+M_{2}.
Since we also know M_{1}/M_{2},
we can determine the masses individually.

Bottom line: With a visual binary, we can determine the masses if (and only if) we can measure the orbits accurately and we know the distance.

In most binaries, the two stars are too close together for us to see them separately. If we can measure Doppler shifts in the spectrum, we can still measure orbital motions of the stars.

A binary that is detected by Doppler shifts is called a
*spectroscopic binary*.

We would like to apply the law of gravity and law of motion to determine the masses of the stars.

Problem: Doppler shifts only tell us the motion relative to the
earth, and we don't know the *inclination* of the orbit,
i.e. if it is close to edge-on or close to face-on.

In an *eclipsing binary*, the brightness declines periodically
as one star passes in front of the other. (See figure 10-17).

For an eclipse to occur, the orbit of the binary must be nearly edge-on.
This solves the "problem" mentioned before.
With good observations, we can determine the masses of both
stars in an eclipsing binary, *without* knowing the distance.

From the shape and duration of the eclipses, we can find the radii of the stars.

After observing many visual binaries and eclipsing binaries, we can determine masses and radii for a large sample of stars (roughly 100). Then we can plot:

- luminosity vs. mass -- the mass-luminosity relation
- radius vs. mass -- the mass-radius relation

For *main sequence* stars, the luminosity and the radius
both correlate well with the mass.

The luminosity is very sensitive to the mass, roughly
L proportional to M^{3.5}.

The radius is less sensitive to the mass. Roughly
R proportional to M for
low-mass (M<2 M_{sun}) stars and
R proportional to M^{1/2} for
high-mass (M>2 M_{sun}) stars.

(Note: M_{sun} means the mass of the sun.)

Sensible results for the mass-luminosity and mass-radius relation for binary stars support our underlying assumption: that the laws of gravity, atomic physics, and light travel that apply in the solar system also apply to other stars.

The sun fits naturally onto the M-L and M-R relation derived for other stars. This continuity verifies the assumption that the sun is a "typical" star.

Between 20% and 80% of stars are in true, gravitationally bound binaries. Binaries can be detected by positional measurements or by spectroscopic observations of Doppler shifts. Binaries can have many wonderful and weird properties, most of which we will ignore.

Binaries play an essential role in the study of stars because, in special cases, we can apply Newton's laws of gravity and motion to learn the masses of the stars.

We can determine the masses in a visual binary if we know its distance.

In an eclipsing, spectroscopic binary, we can determine the stellar masses and stellar radii without knowing the distance.

The existence of a mass-luminosity relation and a mass-radius relation show that the luminosity and radius of a main sequence star depend primarily on its mass. Approximately:

- L proportional to M
^{3.5} - R proportional to M (low-mass stars)
- R proportional to M
^{1/2}(high-mass stars)

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Updated: 1997 January 11 [dhw]