10.6 Stellar Masses

Figure 10.15 Binary Stars The periods and separations of binary stars can be observed directly if each star is clearly seen. (Harvard College Observatory)
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. (More Precisely 1-2)

Binary Stars

Most stars are members of multiple-star systems—groups of two or more stars in orbit around one another. The majority 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. (The Sun is not part of a multiple-star system; if it has anything at all uncommon about it, it is this 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 bright enough to be observed and monitored separately (Figure 10.15). The more common spectroscopic binaries are too distant from us 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 one another and their line-of-sight velocities vary periodically. (More Precisely 2-3)

Figure 10.16 Binary Spectra 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 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 laboratory spectra are shown at top and bottom.)
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 (Figure 10.16), one star is too faint for its spectrum to be distinguished, so we see only one set of lines shifting 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.

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, we observe a periodic decrease of starlight intensity as one member of the binary passes in front of the other (Figure 10.17). By studying the variation of the light from an eclipsing binary system—called the binary’s light curve—we can derive detailed information not only about the stars’ orbits and masses but also about their radii.

Figure 10.17 Binary Light Curves If the two stars in a binary happen to eclipse one another, additional information on their radii and masses can be obtained by observing the periodic decrease in starlight intensity as one star passes in front of the other.
Note that these categories are not mutually exclusive. For example, an eclipsing binary may also be (and, in fact, often is) a spectroscopic binary system.

Measuring Stellar Masses

The period of a binary can be measured by observing the orbits of the component stars, the back-and-forth motion of the stellar spectral lines, or the dips in the light curve—whatever information is available. Observed binary 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, then the orbits of each component can be individually tracked, and the masses of the component can be determined.

For spectroscopic binaries, Doppler-shift measurements give us information only on the radial velocities of the component stars, and this limits the 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 system, 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.

If a spectroscopic binary happens also to be an eclipsing system, then the uncertainty in the orbital 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 mass of the brighter component can be obtained by other means (for example, if it is recognized as a main sequence star of a certain spectral class—see Figure 10.18).

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 (= 203/502) times the mass of the Sun. (Sec. 1.4) 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 10.18 Stellar Masses More than any other stellar property, mass determines a star’s position on the main sequence. Low-mass stars are 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. The symbol "" means "solar mass."
Mass and Other Stellar Properties

Figure 10.18 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.

Figure 10.19 illustrates how the radii and luminosities of main-sequence stars depend on mass. The two plots are called the mass–radius (part a) and mass–luminosity (part b) relationships. Along the main sequence, both radius and luminosity increase with mass. As a (very rough) rule of thumb, radius rises 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 eight (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.

Table 10.3 compares some key properties of several well-known main-sequence stars, arranged in order of decreasing mass. Notice that the central temperature (obtained from mathematical models similar to those discussed in Chapter 9) differs relatively little from one star to another, compared to the large spread in stellar luminosities. The final column in the table presents a very rough estimate of each star’s lifetime, obtained simply by dividing the amount of fuel available (that is, the star’s mass) by the rate at which the fuel is being consumed (the star’s luminosity):

,

and noting that the lifetime of the Sun (see Chapter 12) is about 10 billion years.

Because luminosity increases so rapidly with mass, the most massive stars are by far the shortest lived. For example, according to the mass-luminosity relationship, the lifetime of a 20-solar-mass O-type star is roughly 20/203 = 1/400 that of the Sun, or about 25 million years. We can be sure that all the O- and B-type stars we now observe are quite young—less than a few tens of millions of years old. Their nuclear reactions proceed so rapidly that their fuel is quickly depleted despite their large masses. At the opposite end of the main sequence, the low core density and temperature of an 0.1-solar-mass M-type star mean that its proton–proton reactions churn away much more sluggishly than in the Sun’s core, leading to a very low luminosity and a correspondingly long lifetime. (Sec. 9.5) Many of the K- and M-type stars now visible in the sky will shine on for at least another trillion years. The evolution of stars is the subject of the next two chapters.

 
Figure 10.19 Stellar Radii and Luminosities(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 roughly as the cube of the mass.

Concept Check

How do we know the masses of stars that aren’t members of binaries?