Opacity and Stellar Pulsation

As we have seen in earlier discussion, the rate of energy flow out of a star is governed by the opacity of the internal layers. If something alters that opacity, it can alter the energy flow. This can, in turn, upset the hydrostatic equilibrium.

In normal stars the equilibrium is quickly restored if it is upset, but in pulsating variables one gets a sustained oscillation because of outer layers that can alternatively store and release energy as the star pulses.

Pulsation Timescales

Why do stars like Cepheid or RR Lyrae variables pulsate, and why is there a definite relationship between their period for pulsation and their luminosity? These are stars late in their lives (they are giants or supergiants, so they have left the main sequence) and they pulsate because they go through periods in which they have difficulty in maintaining hydrostatic equilibrium. That is, pulsating variables are not a particular kind of star, but rather a stage in the life of many stars.

Structure and Luminosity
Qualitatively, we may expect that pulsation of a star has to do with its internal structure, and the amount of light emitted by the star is also associated with its structure, so there could be a relation between the period for pulsation and the luminosity. We can put that qualitative idea on firmer ground by considering characteristic timescales for expansion and contraction that we discussed in conjunction with the birth of stars in Chapter 20 (hydrodynamical timescales). Those timescales characterize the response of a star to instabilities in the hydrostatic equilibrium.

Hydrodynamical Response Times
Earlier, we found the very important result that the characteristic time for a star to respond to instabilities is inversely proportional to the square root of the density:

Response Time ~ 1 / (GD)1/2

where G is the gravitational constant and D is the average density of the star. But there is also a direct connection between average densities and the size of the star. For example, supergiant stars have average densities that can be 10 million times lower than a main sequence star, and white dwarfs can be a million times more dense than a main sequence star. But this means that there is also a connection between average densities and the luminosity, because low density stars have large surface area and thus are bright.

Period-Luminosity Relations and Density
By these arguments, we see the basic reason for period-luminosity relations. The time for pulsation is related to the hydrodynamical timescale, which is related to the density. The size of the star and thus its luminosity are related to the density also, since larger stars are less dense. Thus, since the pulsation time and the luminosity are both related to the density, they are related to each other. To understand the details requires more physics, but the basic reason for period-luminosity relations in pulsating variables is that they are really period-density relationships.