The idea of stellar magnitudes goes back to Hipparchus, who divided stars into six classes depending upon their apparent brightness. More recently it was ascertained that between a 1st magnitude star and a 6th magnitude star there is a 100-fold decrease in apparent luminosity. We are dealing here with a "logarithmic" scale. A star of the 1st magnitude is about 2.5 times as luminous as a star of the 2nd magnitude, a 2nd magnitude star is about 2.5 times as luminous as a 3rd magnitude star, and so on. [(2.5)5 = 100 approximately]
We have been able to infer a great deal about some of the nearby stars. A striking relation appears to hold between most stellar masses and "absolute magnitude." The absolute magnitude of a star is defined as the magnitude it would have if it were moved from its actual position to a point 10 parsecs from the earth. One parsec is about 3.26 light years, not much less than the distance to the closest star. At 10 parsecs the Sun itself would appear as a 5th magnitude star.
From the apparent magnitude of a star, the absolute magnitude can be determined providing we know the distance to the star. Then, if one plots the absolute magnitude versus the mass for those stars whose distance and mass are both known, 90% of the points lie close to a rather well defined curve. Stars which lie close to this curve are designated as "main sequence" stars. Main sequence stars with greater masses have far greater absolute luminosities, for the magnitude scale is a logarithmic one.
The mass-luminosity relation provides a way to infer the masses of stars which are identified as main sequence stars, and whose absolute magnitudes are known. But how can we be sure that a star belongs to the main sequence, and how can we determine a star's absolute luminosity if we do not know its distance?
From the spectral class of a star it is possible to infer its surface temperature. Thus, for example, the Sun's surface temperature is about 6,000oK. If one plots the known absolute magnitudes of the nearby stars versus spectral class (or temperature), the points lie near a well-defined curve in 90% of the cases. The 10% which don't lie on the main-sequence curve are very strange stars indeed, the "white dwarfs."
The immediate neighborhood of the Sun contains a number of white dwarf stars, including the companion of Sirius, which is a binary star system. While these white dwarfs are very hot stars, they are not very luminous. From this it may be inferred that their surface area must be very small, corresponding to radii comparable to that of the earth. Nevertheless, it may be inferred from their orbits that they are roughly as massive as the Sun. It must be concluded that white dwarfs are exceedingly dense, perhaps 200,000 times the density of water. We shall have to discuss later how such strange stars, which constitute 10% of the nearby stars, could have originated. In all probability this fate awaits the Sun some 4 or 5 billion years hence.
If one includes in the H-R diagram stars which are farther from the Sun, the sample would favor brighter stars. It turns out that among these stars are some like Betelgeuse and Antares which have low surface temperature but which are very luminous. These are the "red giants." It is believed that only 1% of stars are in the red giant class, but since they are intrinsically bright, they make their presence known. We shall also have to discuss how such stars can be formed. On its way to becoming a white dwarf, our Sun will pass through a red giant phase.
Suppose we turn now to stars at a considerable distance from the earth, so that their distances cannot be ascertained by parallax. The spectra of such stars can still be classified, and a surface temperature deduced. Furthermore, the fine details of the spectra often permit one to distinguish an M-type red giant from an M-type main sequence star. Thus, one can often assign such a star a place in the H-R diagram. As a result one can infer what the absolute luminosity of the star is. Comparing the absolute luminosity with the apparent luminosity a distance may be deduced.
Distances determined in this way do not always agree with distances inferred from trigonometric parallax. The procedure works well only for stars near the lower end of the main sequence (fainter stars) because of the dispersion near the top end. Hence distances to bright objects very far away still elude us. How do you suppose that such distances can be ascertained?