Slightly more than half of the stars in the sky belong to binary systems. For example, Sirius, the brightest star in the sky, has a faint blue companion, and a -Cen, the nearest star, has a fairly bright red companion. In these binary systems, the stars orbit each other, just as planets orbit the Sun and other stars.
Observations of binary stars are very important because they give us a chance to measure the stars' masses -- something we cannot do with single stars. The orbital motions of the stars are determined by a balance between centrifugal force and gravity. If we know the speed of the stars and the distance separating the stars, we can calculate the centrifugal force. But that must be equal to the attractive gravitational force, which is proportional to the product of the two masses divided by the separation distance. This principle of balance of forces yields Kepler's Third Law, which may be expressed in the following form:
m1 + m2 = Ka3/P2
where m1 denotes the mass of the first star, m2, the second star, a is the orbital separation (or, if the orbit is elliptical, the semimajor axis), and P is the orbital period (the time to make one full orbit). K is a constant which has the value K = 1 if all quantities are expressed in solar system units: m1 and m2 in units of the Sun's mass, a in Astronomical Units (1 AU = the distance from Earth to Sun), and P in years.
(Actually, Kepler didn't really understand his Third Law -- he simply discovered that this kind of equation could account for the motion of the planets around the Sun as observed by Tycho Brahe. In 1687, 68 years after Kepler discovered his Third Law, Isaac Newton explained how it was a logical consequence of the balance of gravity and centrifugal force.)
Kepler's Third Law applies to any orbiting system: the Moon's orbit around the Earth; the Earth's (or any other planet's) orbit around the Sun; the orbit of planets around other stars; the orbits of two stars around each other; and even the orbit of the Sun around the center of the Milky Way. If we can observe P, the time it takes for one full orbit, and a, the orbital separation, we can calculate m1 + m2, the sum of the two masses. Try it! For example, take the Earth-Moon system, which has P = 0.075 years (slightly less than a month) and a = .0025 AU. If you plug these values into the above formula, you should find the sum of the mass of the Earth and Moon in units of the Sun's mass.
Note that Kepler's Third Law only tells the sum of the masses of the two orbiting objects. If one object is much heavier than the other, this sum will be almost equal to the mass of the heavier object. Thus, Kepler's Third Law applied to the Earth's orbit tells us the Sun's mass (plus a negligible contribution due to the Earth's mass), and applied to the Moon's orbit tells us the Earth's mass (plus a 1% contribution due to the Moon's mass).
For binary star systems, astronomers can directly measure the orbital separation a if the stars are far enough apart so that their images can be resolved separately. Such systems are called visual binaries. Typically, the stars in such systems are so widely separated that the orbital periods are decades or longer.
But in most binary star systems, the two stars are so close together (often only a few stellar diameters apart) that they cannot be resolved with ground-based telescopes. In those cases, astronomers cannot measure their orbital separation directly, but they can infer the separation from the orbital velocities and orbital period, which can be measured from the Doppler shifts of the absorption lines in the spectra of the stars. Such systems are called spectroscopic binaries.
If the two stars in a spectroscopic binary have comparable luminosities, astronomers can see absorption lines from both stars in the spectrum. Such systems are called double-line spectroscopic binaries. The Doppler shifts of spectral lines of the two stars oscillate with opposite phase, and the heavier stars does not move as fast as the lighter star, as illustrated in this Binary star simulation. For such systems, astronomers can infer the mass ratio, m1/m2, from the ratio of Doppler shifts of the spectral lines from the two stars, as well as the sum of the two masses from Kepler's Third Law. Given both the ratio and the sum, they can infer the individual mass of each star.
If, however, one star is much more luminous than the other, astronomers may only be able to see absorption lines from the more luminous star in the spectrum. We call such systems single-line spectroscopic binaries; with these systems we may only be able to estimate the total mass of the system, not the masses of the individual stars.
However, we see some spectroscopic binaries so nearly edge-on that they actually eclipse each other. We call such systems eclipsing binaries. With eclipsing binaries, we know that sin i @ 1. This fact greatly reduces the uncertainty in inferring the masses.
The most famous eclipsing binary is the star Algol, which the ancients called the "Demon star" because its brightness dips periodically as the cool red star in the binary system eclipses the hotter blue star. There's a very good scientific story about Algol that we'll return to later in this course.
By observing eclipsing binaries, we not only can diminish the uncertainty of the stellar masses resulting from the uncertainty in sin i; we have a new way to infer the radius of each star from the way the light varies during the eclipse. This is illustrated here with this Eclipsing Binary Simulation from Cornell University. By comparing the radius inferred this way from the radius as inferred from the Stefan-Boltzmann Law, we can check the accuracy of these methods and test the validity of the Principle of the Universality of Physical Laws.
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Last modified April 19, 2002
Copyright by Richard McCray