From Wikipedia
The accepted explanation for the pulsation of Cepheids is called the Eddington valve,[11] or κ-mechanism, where the Greek letter κ (kappa) denotes gas opacity.
Helium is the gas thought to be most active in the process. Doubly-ionized helium (helium whose atoms are missing two electrons) is more opaque than singly-ionized helium. The more helium is heated, the more ionized it becomes.
At the dimmest part of a Cepheid's cycle, the ionized gas in the outer layers of the star is opaque, and so is heated by the star's radiation, and due to the increased temperature, begins to expand. As it expands, it cools, and so becomes less ionized and therefore more transparent, allowing the radiation to escape. Then the expansion stops, and reverses due to the star's gravitational attraction. The process then repeats.
The mechanics of the pulsation as a heat-engine was proposed in 1917 by Arthur Stanley Eddington[12] (who wrote at length on the dynamics of Cepheids), but it was not until 1953 that S. A. Zhevakin identified ionized helium[13] as a likely valve for the engine.
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