What ultimately determines a star's position on the main sequence? The answer is its mass and its composition. Mass and composition are fundamental properties of any star. They are set once and for all at the time of a star's birth. Together they uniquely determine both the star's internal structure and its external appearance and even (as we will see in Chapter 20) its future evolution. The ability to measure these two key stellar properties is of the utmost importance if we are to understand how stars work. We have already seen how spectroscopy is used to determine a star's composition. Now let's turn to the problem of finding a star's mass.
As with all other objects, we measure a star's mass by observing its gravitational influence on some nearby body—another star, perhaps, or a planet. If we know the distance between the two bodies, then we can use Newton's laws to calculate their masses. The few extrasolar planetary systems that have recently been detected are still not well enough studied to provide independent stellar mass measurements, and we are a long way from placing our own spacecraft in orbit around other stars. Nevertheless, there are ways of determining stellar masses.
Most stars are members of multiple—star systems—groups of two or more stars in orbit around one another. The majority of stars are found in binary—star systems, which consist of two stars in orbit about their common center of mass, held together by their mutual gravitational attraction. Other stars are members of triple, quadruple, or even more complex systems. Most complex of all are the star clusters, to be discussed in Section 17.10. The Sun is not part of a multiple—star system; if it has anything at all uncommon about it, it may be its lack of stellar companions.
Astronomers classify binary-star systems (or simply binaries) according to their appearance from Earth and the ease with which they can be observed. Visual binaries have widely separated members that are bright enough to be observed and monitored separately, as shown in Figure 17.18. The more common spectroscopic binaries are too distant to be resolved into separate stars, but they can be indirectly perceived by monitoring the back—and—forth Doppler shifts of their spectral lines as the stars orbit each other. Recall that motion toward an observer blueshifts the lines, and motion away from the observer redshifts them. 
(Sec. 3.5)
Figure 17.18 The periods and separations of binary stars can be observed directly if each star is clearly seen. This binary star is known as Kruger 60. At left is a schematic diagram of the orbit, at right actual photographs taken in the years indicated.
In a double—line spectroscopic binary, two distinct sets of spectral lines—one for each component star—shift back and forth as the stars move. Because we see particular lines alternately approaching and receding, we know that the objects emitting the lines are in orbit. In the more common single-line systems, such as that shown in Figure 17.19, one star is too faint for its spectrum to be distinguished, so only one set of lines is observed to shift back and forth. This shifting means that the detected star must be in orbit around another star, even though the companion cannot be observed directly.
Figure 17.19 Binary properties can be determined indirectly by measuring the periodic Doppler shift of one star relative to the other as they move in their orbits. The diagram shows a so-called single—line system, in which only one spectrum (from the brighter component) is visible. (The observer is situated to the left of the diagram. Reference lab spectra are shown at top and bottom.)
In the much rarer eclipsing binaries, the orbital plane of the pair of stars is almost edge—on to our line of sight. In this situation, depicted in Figure 17.20, we observe a periodic decrease of starlight as one component passes in front of the other. By studying the variation of the light from the binary system—the binary's light curve—astronomers can derive detailed information not only about the stars' orbits and masses but also about their radii.
Figure 17.20 If two stars in a binary system happen to eclipse one another, information on their radii and masses can be obtained by observing the periodic decrease in starlight as one passes in front of the other.
These categories of binary—star systems are not mutually exclusive. For example, a single—line spectroscopic binary may also happen to be an eclipsing system. In that case astronomers can use the eclipses to gain extra information about the fainter member of the pair. Occasionally, two unrelated stars just happen to lie close together in the sky, even though they are actually widely separated. These optical doubles are just chance superpositions and carry no useful information about stellar properties.
By observing the actual orbits of the stars, or the back—and—forth motion of the spectral lines, or the dips in the light curve—whatever information is available—astronomers can measure the binary's orbital period. Observed periods span a broad range—from hours to centuries. How much additional information can be extracted depends on the type of binary involved.
If the distance to a visual binary is known, the semi-major axis of its orbit can be determined directly, by simple geometry. Knowledge of the binary period and orbit semi-major axis is all we need to determine the combined mass of the component stars, using the modified form of Kepler's third law. (Sec. 2.7) Since the orbits of both components can be separately tracked, it is also possible to determine the individual component masses. Recall from Section 2.7 that in any system of orbiting objects, each object orbits the common center of mass. Measuring the distance from each star to the center of mass of a visual binary yields the ratio of the stellar masses. Knowing both the sum of the masses and the ratio of the masses, we can then find the mass of each star.
For spectroscopic binaries, it is not possible to determine the semi—major axis directly. Doppler—shift measurements give us information on the orbital velocities of the member stars, but only on their radial components. As a result, we cannot determine the inclination of the orbit to our line of sight, and this imposes a limitation on how much information we can obtain—simply put, we cannot distinguish between a slow—moving binary seen edge—on and a fast—moving binary seen almost face—on (so that only a small component of the orbital motion is along the line of sight).
For a double—line spectroscopic system, individual radial velocities, and hence the ratio of the component masses, can be determined, but the uncertainty in the orbital inclination means that only lower limits on the individual masses can be obtained. For single—line systems, even less information is available, and only a fairly complicated relation between the component masses (known as the mass function) can be derived. However, if, as is often the case, the mass of the brighter component can be determined by other means (for example, if it is recognized as a main sequence star of a certain spectral class—see Figure 17.21), a lower limit can then be placed on the mass of the fainter, unseen star.
Figure 17.21 Mass, more than any other stellar property, determines a star's position on the main sequence. Stars that form with low mass will be cool and faint; they lie at the bottom of the main sequence. Very massive stars are hot and bright; they lie at the top of the main sequence.
Finally, if a spectroscopic binary happens also to be an eclipsing system, then the uncertainty in the inclination is removed, as the binary is known to be edge-on, or very nearly so. In that case, both masses can be determined for a double—line binary. For a single—line system, the mass function is simplified to the point where the mass of the unseen component is known if the brighter component can be identified.
Despite all these qualifications, individual component masses have been obtained for many nearby binary systems. Virtually all we know about the masses of stars is based on such observations. As a simple example, consider the nearby visual binary system made up of the bright star Sirius A and its faint companion Sirius B. Their orbital period is 50 years and their orbital semi—major axis is 20 A.U.—7;.5" at a distance of 2.7 pc—implying that the sum of their masses is 3.2 times the mass of the Sun. Further study of the orbit shows that Sirius A has roughly twice the mass of its companion. It follows that the masses of Sirius A and Sirius B are 2.1 and 1.1 solar masses, respectively.
Figure 17.21 is a schematic H—R diagram showing how stellar mass varies along the main sequence. There is a clear progression from low-mass red dwarfs to high—mass blue giants. With few exceptions, main-sequence stars range in mass from about 0.1 to 20 times the mass of the Sun. The hot O— and B—type stars are generally about 10 to 20 times more massive than our Sun. The coolest K— and M—type stars contain only a few tenths of a solar mass. Because all other stellar properties are set once a star's mass is known, we can say that the mass of a star at the time of formation determines its location on the main sequence.
Figure 17.22 illustrates how a main-sequence star's radius and luminosity depend on its mass. The two plots, called the mass—radius and mass—luminosity relations, are based on observations of binary—star systems. Along the main sequence, both radius and luminosity increase with mass. As a (very rough) rule of thumb, radius increases in direct proportion to mass, whereas luminosity increases much faster—more like the cube of the mass. For example, a 2—solar mass main—sequence star has a radius roughly twice that of the Sun and a luminosity of 8 (23) solar luminosities; a 0.2—solar mass main—sequence star has a radius of about 0.2 solar radii and a luminosity of 0.008 (0.23) solar luminosities.
Figure 17.22 (a) Dependence of stellar radius on mass for main—sequence stars. The radius increases roughly in proportion to the mass over much of the range. (b) Dependence of luminosity on mass. The luminosity increases much faster than the mass.
The rapid rate of nuclear burning deep inside a star releases vast amounts of energy per unit time. How long can the fire continue to burn? We can estimate a star's lifetime simply by dividing the amount of fuel available (the mass of the star) by the rate at which the fuel is being consumed (the star's luminosity):
Because the mass—luminosity relation tells us that a star's luminosity is roughly proportional to the cube of its mass, we can rewrite this expression to obtain, approximately,
For example, O and B stars have masses 10 to 20 times that of the Sun and luminosities thousands of times higher than the solar luminosity. Accordingly, these massive stars can survive only for short times. Their nuclear reactions proceed so rapidly that their fuel is quickly depleted despite their large masses. From the given proportionalities, we see that the lifetime of a 20—solar mass O star is roughly 20/203 = 1/400 of the (10—billion—year) lifetime of the Sun, or about 25 million years. We can be sure that all the O and B stars we now observe are quite young—less than a few tens of millions of years old. Massive stars older than that have already exhausted their fuel and no longer emit large amounts of energy. They have, in effect, died.
At the opposite end of the main sequence, the cooler K— and M—type stars have less mass than our Sun. With their low core densities and temperatures, their proton—proton reactions churn away rather sluggishly, much more slowly than those in the Sun's core. The small energy release per unit time leads to low luminosities for these stars, so they have very long lifetimes. Many of the K— and M—type stars now seen in the sky will shine on for at least another trillion years.
Table 17.5 compares some key properties of several well—known main—sequence stars, arranged in order of decreasing mass. Notice how little the central temperature differs from one star to another, and how large is the spread in stellar luminosities and lifetimes.
| TABLE 17.5 Key Properties of Some Well—Known Stars | ||||||||||||||||||||||||||||||||||||
|
