We saw last lecture that there were 2 ways to measure temperatuer from a spectrum - the first was to measure the wavelength at which the blackbody emission peaked, the second was to measure the pattern of absorption lines. The blackbody emission comes from very dense material, where the light and electrons have interacted many times so the light has come into equilibrium with the electron motions, and so reflects the temperature of the material. Multiple interactions is characteristic of dense material, and a blackbody spectrum does NOT depend on composition - a block of lead heated to 3000K has the same blackbody spectrum as a block of carbon (eg coal!). But if there has not been many interactions, then the light can instead reflect the individual emission and absorption processes - quantum effects. When we look at a star, the bulk of it is very dense, so should emitt as a blackbody, but the very outer layer is not very dense, so we can see individual emission/absorption events, which carry the imprints of these 'special' allowed energy levels within the atoms. when we look at a cluster of stars, we can measure the apparent brightness of each one. then we can take spectra and use the absorption lines to get spectral type. Plot one against each other on a This is called a Hertzsprung-Russell (HR) diagram. Most stars lie on the 'Main Sequence', where temperature and luminosity are tightly linked: high temperature implies high luminosity and vica versa. We can find a star with a spectrum like the sun, and then assume that it has the same luminosity, and so get its distance. This then tells us the distance to the whole cluster so we can convert the apparent brightness of other types of stars in the cluster into luminosities. Then we can calibrate the whole HR diagram and get it in terms of temperature and luminosity. Then for any main sequence star, all we have to do is use its spectrum to determine its spectral type (so we know what its luminosity should be), and measure its apparent brightness, and then we can use the inverse square law to find its distance (assuming there is no fuzz in the way). This method of finding distance is called spectroscopic parallax
The majority of stars sit along a well defined track of luminosity versus temperature in an HR diagram. Why is this ? The obvious quantity that could cause this is mass - maybe the main sequence is telling us about the mass of stars. And this would make sense from what we know already - if a star has more mass then gravity is stronger pullling in so the pressure from the hot gas pushing out has to be higher to balance the star (hydrostatic equilibrium) i.e. high temperatures. And the nuclear fusion reactions are VERY VERY sensitive to temperature so the luminosity would be much much higher. It makes a lot of sense. But is it right ? How can we test it ?How can we possibly know anything about the masses of stars - they are SO far away! But we do know something about gravity. Ever since Newton, we have been able to calculate the force due to gravity at any point once we know the mass of an object and the distance we are from it. The bigger the mass the bigger the gravity, and the closer we are to it then the bigger the gravity.
The pull of gravity on a planet the depends on the mass of the star and the planets distance from the star. But the planet is in orbit around the star. So its velocity will be determined by the gravitation force. More gravity and the star has to orbit faster because the inward pull is stronger. So if we know the distance of a planet from the star its orbiting around, then we can use its orbital velocity (or orbital period) as a way to measure the mass of the star. This is how we know the mass of the Sun.
But there is a problem with the above approach. Firstly to observe something at the distance of a star it needs to be pretty bright i.e. be a star itself rather than a planet! But then the companion star's gravity is important too - the discussion above assumed that the orbiting object was negligible in mass compared to the star. So now we need to know an additional bit of information which is the ratio of the star masses, which is given by the distance of each star from the center of mass of the system - more on gravity and orbits. So we need to find binary stars - this IS NOT hard: about 50 per cent of stars are in multiple systems. But watching stars go around each other to map out the orbit is slow work - if the two stars are far enough apart to be able to see them both ( visual binaries ) then the orbit is very very long. So it'd be much better to get a closer orbit which have shorter periods. But then how can we resolve the two stars ?
Well, we don't need to. We can watch the spectral lines move via the doppler shift. This gives us the speed at which the star is moving towards us or away from us (it can't do anything for motion across the sky). So we can watch the lines shift forward and back, to give us the orbital period, and we can get the maximum velocity of each star (and we can convert this to distance from the center of mass by understanding gravity) and the ratio of the velocities of the two stars so we know the ratio of their masses. So we then have everything we need to find the mass of each star. See this links to how to measure the masses of stars, and play with the java animation on doppler shift for binary stars. Put in M1=M2=1 with a circular orbit (e=0) and play. Then put M1=10 and see the differences. The massive star moves much more slowly, so its velocity and doppler shifts are small. And another site on spectroscopic binaries.
When we do this for many different types of stars we can convert the HR diagram from luminosity-temperature to luminosity-mass. And sure enough, the main sequence is a sequence in mass as we guessed at the start of the lecture - see also an explanation of the Mass-Luminosity relation
Want to know more about that song ?
pages 112-117 in Kuhn