Re: how do you work out the mass of star by its spectrum?

You are on the right track. From the spectrum, astronomers can determine the elements that exist in the outer layers of a star. To get the mass is more complicated since it is difficult to get the percentage of each element and that percentage would only apply to the outer layers of a star. The interior has been changed by nuclear reactions. In general the mass of the companion of a black hole is estimated

1. By determining its spectral class from its spectrum (i.e., seeing what pattern of spectral lines exist in the spectrum). That gives us the surface temperature of a star.

2. After that we estimate its luminosity class, i.e., is the star a supergiant, a giant or a dwarf. Again that is determined by analyzing the exact shape of specific spectral lines that are sensitive to the pressure in the outer layers of a star. Supergiants have very low pressures in the outermost layers and dwarfs (stars on the mainsequence) have higher pressures. In general that means that the spectral lines in supergiants are narrower than in dwarf stars.

3. Once we know the temperature and luminosity (total energy output of a star), we can then estimate the mass of a star from evolutionary tracks of stars of different masses. i.e., from the location of a star in the H-R diagram (the temperature-luminosity diagram) of stars. The estimate of the mass of the star should be correct to 10% to 20% or better.

References: Any introductory astronomy text book that deals with stellar evolution and black holes.