Determining the Luminosity Class

The primary way to determine the luminosity class of a distant star is through details of the spectrum. The strength and width of certain spectral lines are sensitive to gas density, which generally is lower in the surfaces of larger stars. Also, the presence of key spectral lines such as those for certain metals in the spectrum of the star implies that one is observing a giant. Another criterion is variability. As we shall discuss in the chapter on star death, variable stars of different types typically inhabit relatively small regions of the HR diagram. Thus, identifying a particular type of variable may also tell us with some accuracy how luminous the star is.

Spectroscopic Parallax

Ideally, one measures the distance to a star through its parallax. The apparent cyclic change in a star's position against more distant background stars over the course of a year is a direct measure of its distance from Earth. Unfortunately, we have seen that this annual change of position is so small that it is difficult to measure in all but the closest stars. Therefore, we require other methods for estimating distances to stars that are too far away for reliable parallax measurements.

The Distance from the Spectrum
One technique that is useful for estimating distances to stars at moderate distances (beyond about 50 parsecs) is the method of "spectroscopic parallax" which has nothing to do with true parallax measurements (see the right panel). In this approach, one determines the spectral type and luminosity class of a star. If these are known, one knows roughly where the star belongs on the HR diagram and hence its true luminosity. Knowing the star's true luminosity (and thus absolute magnitude), and given its apparent magnitude, one can infer its distance immediately using the inverse square intensity law. The box below gives a simple example of determining distance by such a method. The central problem is to determine the luminosity class; how can one tell a main sequence G2V star from a giant G2III star, which is about 50 times more luminous? The primary way is through details of the spectrum, as described in the above box.

The Distance to Alpha Centauri

Let us estimate the distance to the nearby star alpha Centauri using its spectral and luminosity classes. It is observed to be a G2V star, like our Sun, with apparent visual magnitude m = -0.01. Let us take our Sun as a representative G2V star. The Sun's absolute visual magnitude is M = 4.82. Then, if we assume that all G2V stars have the same absolute magnitude, we may use our preceding formulas to write

d(pc) = 10(m - M + 5) / 5 =10(-0.01 - 4.82 +5) / 5 = 1.08 pc

Precision Hipparcos measurements give a parallax of 0.74212 with only 0.2 percent uncertainty. The true distance then is d = 1 / 0.74212 = 1.35 pc, and our simple estimate is too low by about 20 percent (not bad, considering the simplicity of the method). Try an example yourself. Vega is an A0 main sequence star with apparent magnitude 0.04. Systematics of HR diagrams suggest that the absolute magnitude of an A0V star is about 0.7. Applying the same analysis to Vega as that shown above for alpha Centauri, you should find a distance to Vega of 7.4 pc, which is only 5 percent lower than the correct distance of 7.76 pc determined by parallax.

Relative Errors for Parallax and Spectroscopic Parallax
Spectroscopic parallax has moderate (20-30 percent) errors associated with it, but they are independent of distance (provided one can determine the spectrum well). True parallax has smaller errors initially, but they grow rapidly with distance. As a rule of thumb, the errors from spectroscopic parallax are smaller than those from true parallax measured from Earth-based telescopes for distances larger than about 50-100 parsecs. High precision parallax measurements from space have recently made true parallax competitive at the thousand-parsec distance scale, however.

Here is a spectroscopic parallax calculator.