## Solution

The luminosity is proportional to the square of the radius, so

We use equation (2.2) to find the luminosity ratio for a 10 mag difference:

Combining these two results to find the ratio of the radii yields

The radius of a white dwarf is 1% of the radius of the Sun!

For any cluster for which we plot an HR diagram, we only know the apparent magnitudes, not the absolute magnitudes. If we know the absolute magnitude for one spectral type, then we can find the distance modulus for stars of that spectral type in the cluster. The distance modulus is the same for all the stars in the cluster, so we can calibrate the whole HR diagram in terms of absolute magnitudes. To obtain a reliable calibration, we would like to carry it out for many stars. We have already seen that there is a growing group of nearby stars for which trigonometric parallax can give us a good distance measurement. In Chapter 13, we will see how we can improve on this sample by looking at the motions of clusters.

Once we know the absolute magnitude for a given spectral type, we have a very useful way of determining distances. For any given star, we measure m, the apparent magnitude. We take a spectrum of the star to determine its spectral type. From the spectral type we know the absolute magnitude, M. Since we know m and M, we know the distance modulus, m — M, and therefore the distance. This procedure is called spectro scopic parallax. The word 'spectroscopic' refers to the fact that we use the star's spectrum to determine its absolute magnitude. The word 'parallax' refers to the fact that this is a distance measurement (just as trigonometric parallax was a distance measurement using triangulation).

Example 3.4 Spectroscopic parallax For a B0 star (M = —3), we observe an apparent magnitude m = 10. What is the distance to the star, d?