SPECTROSCOPIC PARALLAX

In Chapter 2 we introduced the first "rung" on a ladder of distance-measurement techniques that will ultimately carry us to the edge of the observable universe. That rung is radar ranging on the inner planets. (Sec. 2.6) It establishes the scale of the solar system to great accuracy and, in doing so, defines the astronomical unit. Earlier in this chapter we discussed a second rung in the cosmic distance ladde—stellar parallax—which is based on the first. Now, having used the first two rungs to determine the distances and other physical properties of many nearby stars, we can use that knowledge to construct a third rung in the ladder: spectroscopic parallax. (This unfortunate name is very misleading, as the method has nothing in common with stellar (geometric) parallax other than its use as a means of determining stellar distances.) As illustrated schematically in Figure 17.16, this new rung expands our cosmic field of view still deeper into space.

Figure 17.16 Knowledge of a star's luminosity and apparent brightness can yield an estimate of its distance. Astronomers use this third "rung" in our distance ladder, called spectroscopic parallax, to measure distances as far out as individual stars can be clearly discerned—several thousand parsecs.

We have already discussed the connections between absolute brightness (luminosity), apparent brightness (energy flux), and distance. Knowledge of a star's apparent brightness and distance allows us to determine its luminosity using the inverse-square law. But we can also turn the problem around. If we somehow knew a star's luminosity and then measured its apparent brightness, the inverse-square law would give us its distance from Earth.

Consider an analogy. Most of us have a rough idea of the approximate brightness and size of a red traffic signal. Suppose we are driving down an unfamiliar street and see a red traffic light in the distance. Our knowledge of the intrinsic luminosity of the light often enables us immediately to make a mental estimate of its distance. A normal traffic light that is relatively dim must be quite distant (assuming it's not just dirty); a bright one must be relatively close. A measurement of the apparent brightness of a light source, combined with some knowledge of its intrinsic properties, can yield an estimate of the source's distance. For a star, the trick is to find an independent measure of the luminosity without knowing the distance. The H—R diagram can provide just that.

The main sequence represents a fairly close correlation between temperature and luminosity for most stars, with the exception of a few giants and dwarfs. Thus, the main sequence tells us about the average properties of most stars. Let's imagine for a moment that the main sequence is a line, rather than a somewhat fuzzy band, in the H—R diagram and that all stars lie on the main sequence. From a star's spectrum, we can determine its surface temperature or spectral type. If the star lies on the main sequence, then there is only one possible luminosity corresponding to that temperature. We can read the star's luminosity directly off a graph such as Figure 17.14, and then determine its distance by measuring the energy flux at Earth and using the inverse-square law. The existence of the main sequence allows us to make a connection between an easily measured quantity (temperature) and the star's luminosity, which would otherwise be unknown. The term spectroscopic parallax refers to this whole process of using stellar spectra to infer distances.

Spectroscopic parallax can be used to determine stellar distances out to several thousand parsecs. Beyond that, spectra and colors of individual stars are difficult to obtain. The "standard" main sequence is obtained from H—R diagrams of stars whose distances can be measured by (geometric) parallax, so the method of spectroscopic parallax is calibrated using nearby stars. Note that, in using this method, we are applying the "principle of mediocrity" that we discussed in Chapter 2. (Interlude 2-2) Specifically, we are assuming (without proof) that distant stars are basically similar to nearby stars, and that they fall on the same main sequence as nearby stars. Only by making this assumption can we expand the boundaries of our distance-measurement techniques.

Of course, the main sequence is not really a line in the H—R diagram: it has some thickness. For example, the luminosity of a main-sequence G2-type star (such as the Sun) can range from about 0.5 to 1.5 times the luminosity of the Sun. The main reason for this range is the variation in stellar composition and age from place to place in the Galaxy. As a result, there is an uncertainty in the luminosity obtained by this method and so some uncertainty in the distance. Distances obtained by spectroscopic parallax are probably accurate to no better than 25 percent. Although this may not seem very accurate—a cross-country traveler in the United States would hardly be impressed to be told that the best estimate of the distance between Los Angeles and New York is somewhere between 3000 and 5000 km—it illustrates the point that in astronomy even something as simple as the distance to another star can be very difficult to measure. Still, an estimate with an uncertainty of ± 25 percent is far better than no estimate at all.

LUMINOSITY CLASS

If a star happens to be a red giant or a white dwarf, its distance determined by spectroscopic parallax will be incorrect. We could simply argue that since roughly 90 percent of all stars are on the main sequence, the assumption that a star is a main-sequence star is valid 9 out of 10 times, but, in fact, astronomers can do much better. Recall from Chapter 4 that the width of a spectral line can provide information on the density of the gas where the line formed. (Sec. 4.4) The atmosphere of a red giant is much less dense than that of a main-sequence star, and this in turn is much less dense than the atmosphere of a white dwarf. By studying the width of a star's spectral lines, astronomers can usually tell with a high degree of confidence whether or not it is on the main sequence.

Over the years, astronomers have developed a system for classifying stars according to the width of their spectral lines. Because the line width is particularly sensitive to density in the stellar photosphere, and the atmospheric density in turn is well correlated with luminosity, the class in which a star is categorized has come to be known as its luminosity class. This classification provides a means for astronomers to distinguish supergiants from giants, giants from main-sequence stars, and main-sequence stars from white dwarfs by studying a single spectral property—the line broadening—of the radiation received.

The standard stellar luminosity classes are given in Table 17.3. Their locations on the H—R diagram are indicated in Figure 17.17. Now we have a way of specifying a star's location in the diagram in terms of properties that are measurable by purely spectroscopic means; spectral type and luminosity class define a star just as surely as do temperature and luminosity. The full specification of a star's spectral properties includes its luminosity class. For example, the Sun, on the main sequence, is of class G2V, Vega is A0V, the red dwarf Barnard's Star is M5V, the red supergiant Betelgeuse is M2Ia, and so on.

Figure 17.17 Stellar luminosity classes in the H—R diagram. Note that a star's location can be specified by its spectral type and luminosity class instead of by its temperature and luminosity.

Consider, for example, a K7—type star (Table 17.4) with a surface temperature of approximately 4000 K. If the widths of the star's spectral lines tell us that it lies on the main sequence (that is, it is a K7V star), then its luminosity is about 0.1 times the solar value. If its spectral lines are observed to be narrower than lines normally found in main—sequence stars, the star may be recognized as a giant, with a luminosity 20 times that of the Sun (Figure 17.17). If the lines are very narrow, the star may instead be classified as a K7Ib supergiant, brighter by a further factor of 150, at 3000 solar luminosities. In this way the observed width of the star's spectral lines translates directly into a measure of the star's physical state. Knowledge of luminosity classes allows us to use spectroscopic parallax with some confidence that we are not accidentally counting a red giant or a white dwarf as a main—sequence star and making a huge error in our distance estimate as a result.