No theoretical basis for the period-magnitude relation has yet been found. Since the period (as already explained) depends on the density and the absolute magnitude depends mainly on the mass, it would seem that for a star of given mass there is just one stage in the course of its contraction when it becomes un stable and is thrown into pulsation ; at other times it burns steadily. The theoretical conditions for starting and maintaining a pulsation are bound up with the laws of liberation of sub-atomic energy in the interior of the star, as to which little is definitely known (see STELLAR EVOLUTION and COSMOGONY). It seems likely that ordinary stars may be near the border line between stability and pulsation, and that some small change in the physical constants of the material occurring when a particular temperature or density is reached puts the star over the borderline.
The typical Cepheid variable which has given its name to the class is .3 Cephei. Its period is 5.366 days. It is seven magnitudes brighter than the sun, and is believed to have about nine times the sun's mass. It is a giant diffuse star having a radius of 20 million kilometres, which changes by I-1 million kilometres in the course of the pulsation. The light-range is o.6 mag. visual and 1.o mag. photographic, the difference being due to the fact that the light becomes much bluer at maximum brightness. The "cluster variables" are a sub-class of the Cepheids; they have very short periods usually about 12 hours. The shortest period known is 3.22 hours.
The enormous variation of light is rather misleading to us, because there is no such large variation in the output of heat. The surface-temperature is varying through a range which happens to be especially critical for the sensitiveness of the human eye. The following results relate to Mira Ceti which is the most famous star of this type (discovered by Fabricius about 1596). To the eye at minimum it appears as a faint star of magnitude 9, difficult to distinguish from the numerous other ninth magnitude stars in the field; but the more impartial bolometer, which meas ures the total radiant energy or heat received, singles it out as a very conspicuous object of bolometric magnitude 1.5. At maxi mum the surface temperature has risen a little, causing a moderate increase of radiation and bringing it up to bolometric magnitude 0-2 ; at the same time the character of the radiation becomes more suited to the sensitiveness of the eye, so that an enormous increase of visual brightness occurs. At maximum Mira is gener ally between mag. 3 and 4 (visual), and occasionally it reaches mag. 2.
Accordingly the variation of heat output is not much more than one magnitude, and is much the same as in the Cepheid variables. It is now generally believed that the long-period vari ables are pulsating stars not essentially different from the Cepheids. They are very diffuse stars of great bulk, and this extreme physical condition is probably responsible for the differ ences of behaviour between them and the typical Cepheids. Pease has succeeded in measuring the angular diameter of Mira with an interferometer, obtaining the result o".o56. It is probable that the star has a radius of at least 1•5 astronomical units or 300 times the radius of the sun. A number of minor classes of vari ables are recognized, which differ in material respects from the three chief classes here described. The whole subject of variable stars is now one of the most active branches of stellar research.