What is the mass of a star? This question can only be answered by measuring the influence of the star on another object. The orbits of the planets give us the mass of the Sun to very high accuracy. Similarly, the orbits of stars around one-another give us the masses of those stars. But stellar systems with large numbers of stars are too complex to disentangle all of the gravitational influences, and extrasolar planetary systems currently give us no information about a star's mass, because extrasolar planets are invisible to us (an extrasolar planet is found through its gravitational influence on a star, which gives us a measure of the planet's mass rather than the mass of the star). This leaves the binary and triplet star systems as the vehicles for directly measuring the masses of stars.
A whole industry revolves around measuring the masses of stars in binary systems. One goal of these studies is to empirically link a star's luminosity and spectral type to the stellar mass. This bit of knowledge is critical for testing computer models of stellar evolution and for measuring the mass density of the local Galactic disk from the light emitted by the nearby stars.
One complication presented by binary systems for those interested in understanding stellar evolution is that many binary systems are so compact that their stars heat and tidally-distort each-other. If the stars in a binary system are so close that each star fills its Roche lobe, so that their photospheres touch, the system is a contact binary system; if the stars are so close that one fills its Roche lobe, but the other does not, then the binary system is semi-detached; if neither star fills its Roche lobe, the system is detached. The detached binary star systems can give us insight into the mass, luminosity, and spectral type of field stars if the binary stars are widely separated, so that the evolution, luminosity, and spectrum of each is uninfluenced by its companion.
Observationally, binary stars fall into two categories: spectroscopic binaries and angularly-resolved binaries. An angularly-resolved binary system appears as two separate stars in an image obtained with a telescope. A spectroscopic binary system appears as a single unresolved star in a telescope; its binary nature is betrayed by its spectrum, which shows two sets of spectral lines, with one set Doppler shifted relative to the other. Improvements to telescopes, such as the adoption of adaptive optics (changing a telescope's optics in real time to counter the distortions cause by air turbulence) and optical interferometry, have converted many spectroscopic binary systems into angularly-resolved binary systems.
The orbit of a detached binary system is very simple. As with the Keplerian orbit of a planet, the orbits of the stars in a binary system are ellipses that are confined to a plane. The orbit of one star relative to another is also an ellipse with a semimajor axis related to the orbital period and total mass of the binary system by
a3 = G (M1 + M2) P2/4π2,
where a is the semimajor axis of the ellipse, G is the gravitational constant, M1 and M2 are the masses of the two stars, and P is the period of rotation. The equation that describes the orbit of a planet around the Sun is derived from this equation by setting one of the masses to 0. With a binary star, if we can measure the semimajor axis of the relative orbit, which is the average of the separation between the stars at apoastron and periastron, and if we measure the period of the orbit, we immediately have the mass of the binary system.
If we can find the center of mass for the system, we can calculate the mass of each star from the mass of the binary system. The ratio of the masses of the two stars is simply the inverse-ratio of the distance of each star from the system's center of mass. If one of the stars is much larger than the other, then its distance from the system's center of gravity would be much smaller than the distance to its companion star, and it would trace out a smaller ellipse than the smaller star. At the most extreme, if the mass of the smaller star is negligible compared to the mass of the larger star, the system's center of mass would lie inside the larger star, which would be nearly motionless. At the other extreme, if the two stars in a binary system are of the same mass, then the system's center of mass would lie half-way between the two stars, and each star would trace an ellipse of the same shape and size.
To derive the stellar masses of a resolved binary star, one must map the orbits of the stars on the sky with sufficient accuracy to derive the orbital parameters (this assumes one knows the distance to the system, perhaps from the system's annual parallax, to convert the angles on the sky into physical lengths). The stars must have enough separation that their orbits can be precisely mapped over time, but the separations cannot be so large that the stars take centuries to complete an orbit. For a binary system with one-solar-mass stars to have a period of less than a century require a separation of about 26 AU or less; at 5 parsecs distance, such a system would be separated on the sky by 5 arc seconds or less. For a system with one-solar-mass stars to complete an orbit in 10 years requires a separation of 6 AU, which implies a separation of 1 arc second at 5 parsecs distance. These values improve for more-massive stars, and they worsen for less-massive stars, but the changes are small, since the semimajor axis is proportional to the total mass to the one-third power. Current instruments can resolve the positions of stars to about 1 milli-arc-second, which translates to an accuracy of 0.005 AU, or about a solar radius, for a binary system 5 parsecs away. This implies that the semimajor axis of a binary system and the masses of the stars, or at least the mass of the larger star, can be measured to rather high precision.
The random orientation relative to Earth of a binary star system presents a challenge in deriving stellar masses. If we happen to be looking along the system's orbital axis, so that the motion of the stars on the sky fully represents their motion in space, then we would derive easily the masses of the stars by measuring the positions of the stars on the sky and by deriving a distance to the binary system from the annual parallax of the stars, but generally we are not so lucky. The binary systems are randomly oriented relative to us, so the stars move along our line of sight as well as on the plane of the sky. The elliptical orbits projected on the sky still appear elliptical, but the shape of the projected orbit is generally much different than the shape of the orbit on the orbital plane. An orbit that is circular on the orbital plane could appear as a highly-elongated ellipse on the sky, and an orbit that is a highly-elongated ellipse on the orbital plane could appear as a circular orbit on the sky. The relative positions of the two orbits on the sky is the only clue to whether the binary system is tilted relative to us. For instance, if both stars have circular orbits, but the system is tilted relative to the Earth, the orbits would appear as a pair of centered ellipses. This would differ from a binary with elliptical orbits that is viewed from above, because one ellipse would be offset from the other along the long axis of the ellipses. The problem is in measuring this offset accurately.
Fortunately, we do not need to rely solely on positional data to understand the characteristics of a binary star's orbit; the Doppler shifted lines of each star's spectrum give a complementary measure of a binary's properties. The Doppler shift gives us the velocity of each star along the line of sight. If we saw no variation over time of the Doppler shift of the spectral lines from each star, then we would know we are looking down the orbital axis of the system. More generally, the variation of the velocities of the stars over time gives us information about the size and shape of each star's orbit. Because the Doppler shift can be measured very accurately, it is a vital measurement for accurately deriving the properties of a binary system.
With the spectroscopic binary systems, one has no direct information about the semimajor axis, so one is reduced to inferring the total mass of the system from the Doppler-shifted spectral lines of the stars and the orbit period of the system. The masses of the stars are given by the relative changes in velocity of each star; the velocity of the less-massive star changes more than does the velocity of the more-massive star. What one derives from this information is the total mass of the system times the sin of the inclination angle i of the system relative to the line of sight (MT sin i), as well as the mass of each star times the sine of the inclination angle. This effectively means that we can only derive lower limits on the masses of stars in spectroscopic binary systems.
As an extreme example of how these measurements give one a mass, let us consider a very low-mass star orbited in a circle a very high-mass star. If we were in the system's orbital plane, we would be able to derive the circumference of the smaller star's orbit by simply multiplying the orbital period by the maximum velocity derived from the smaller star's Doppler shifted lines. From this we would derive the semimajor axis, which would give us the mass of the binary system, which in this case is effectively the mass of the larger star.
Under some circumstances, one finds spectroscopic binary systems with stars that eclipse each-other. We are close to the orbital plane of these system, so that the inclination angle i is close to 90°. By modeling how the light from the system changes during the eclipses, one can derive an accurate value for i. In fact, eclipsing spectroscopic binary systems provide some of the most-accurate values for stellar masses. On the other hand, while many of these systems are detached stars, the stars are close together, often only several stellar radii from each other; this is inevitable, as the chances of us being positioned to see an eclipsing binary system drops dramatically as the separation between the stars increases.
There are of order 50 detached eclipsing spectroscopic binary star systems for which stellar masses have been derived. The errors are typically 1%. In contrast the masses in stars of the much-more-common spatially-resolved binary systems were determined until recently to only 5% to 20% accuracy. The recent advances in instrumentation have dropped these errors to below 5%, and in some cases to as low as 0.2%.
Technically one measures GM, the gravitational constant times the mass. In the case of the Sun, the value of GM, which has an error of 4×10−10, is known to much higher precision than the gravitational constant, which has an error of 10−4, so the error in the Sun's mass is the error in the value of G.