Time (days)

Radial velocity and light curves for 8 Cep, the prototypical Cepheid. (a) Apparent magnitude as a function of time within the period. (b) Temperature as a function of time. (c) Radius, relative to the minimum radius, as a function of time. (d) Radial velocity of the surface as a function of time. Note that the radial velocity is one quarter cycle (90°) out of phase with the radius.

ceding section. To see how a Cepheid oscillates, lets consider the oscillations of a normal star. These oscillations are radial. They involve inward and outward motions of the outer layers of the star. Suppose we are able to perturb a star by decreasing its radius R. The density then increases, and the pressure increases. The excess pressure will make the outer layers expand back. However, just as a swing overshoots its lowest point as it returns from its maximum height, the star can overshoot its equilibrium radius R0. Now the star is larger than its equilibrium radius, and the pressure decreases, allowing material to fall back. This process then repeats itself.

In the above analysis, we have ignored the effects of opacity. In a normal star, the opacity decreases as the temperature increases. Now, we again start the perturbation by reducing R below R0. This causes P and T to increase. The increase in T decreases the opacity. The reduction of the opacity allows some of the excess pressure to be relieved by allowing heat to flow out of the denser regions as radiation. This reduces the tendency of the star to overshoot. If we had started with a perturbation in which R > R0, P and T would have decreased. The opacity would have increased, and the tendency to fall back too fast would be reduced. The result of the opacity is to quench the oscillation.

For a narrow range of conditions, the opacity increases as the temperature increases. The source of the opacity is the ionization of He+ to form He+ +. If we now start with a perturbation in which R < R0, the pressure and temperature increase. Now the opacity also increases, so the excess pressure is not relieved, except by driving the star back. The tendency to overshoot is enhanced. Similarly, with R > R0, the pressure and temperature decrease. The opacity also decreases, reducing the pressure even further. The material falls back quickly and overshoots. This oscillation can continue indefinitely, rather than being quenched. These are the conditions that produce a Cepheid variable.

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