The Structure of Our Universe

Cepheid Variable Stars

Cepheid variable stars are stars that periodically pulsate because of an instability in their internal structure. They grow brighter and dimmer on a regular cycle as they pulsate radially, alternately become physically larger and then smaller. Their brightening is partly a consequence of their larger surface area. The interest to astronomers of Cepheid variables comes from two useful properties: these stars are very bright, and their period of pulsation is related to their average luminosity, with the luminosity increasing with pulsation period. Because of these properties, one can determine a distance to a Cepheid variable by measuring its period and its apparent brightness.

Cepheid variables are not main sequence stars. Rather, they are stars that are in the helium burning stage of their lives. They have a relatively narrow mass range, and when plotted by their luminosity versus their color (called a Hertzsprung-Russell diagram), they fall within a narrow band that is called the instability strip.

The sources of instability in a Cepheid variable are the regions where helium and hydrogen become ionized, with the region of helium becoming fully ionized the dominant instability region. The ionization of an atom in a plasma has two effects. First, the heat capacity of the plasma is larger within than outside the temperature range over which ionization occurs. This means that as the star contracts, gravitational potential energy goes into ionizing the plasma rather than in increasing the temperature of the plasma and the radiation field. Second, as the plasma becomes more ionized as a star contracts, the ability of the ions to absorb radiation decreases; this decreases the interaction of the radiation field with the plasma, which lowers the pressure exerted by the radiation field on the plasma. The overall effect is that the pressure in the star increases more slowly than the gravitational force as the star contracts. This means that the star does not have a stable configuration; instead, the star oscillates about the unstable static configuration.

Cepheid variables subdivide into two classes: the classical Cepheid variables, which are population I stars?stars with a high metallicity, and, therefore, of the current generation of stars?and the W Virginis variables, which are population II? stars of a low metallicity stars, and, therefore, early generation stars. The remainder of this articles is concerned only with the classical Cepheids.

The properties of the classical Cepheid variables are well known. These values are taken from Theory of Stellar Pulsation by J.P. Cox.¹

Pulsation Period ()	1 day to 50 days
Luminosity (L)	300 Lsun to 26,000 Lsun
Spectrum	F5 to G5
Radius (R)	14 Rsun to 200 Rsun
Mass (M)	< 3.7 Msun to 14 Msun

The relationship between the absolute magnitude and the period of the classical Cepheid variables has been studied for many many years. The most up-to-date equation is derived from the classical Cepheids observed by the Hipparcos astrometric satellite. From a data set of 220 Cepheid variables (this excludes Polaris) that have well-determined distances from their parallax, researchers have found the following relationship: ^{2, 3}

M0<V>

=

-2.81 log - 1.43.

In this equation, M⁰_<V> is the average absolute magnitude in the V frequency band, and is the period of variability in days. From the definition relating absolute magnitude to luminosity, one finds that the average luminosity of a Cepheid variable increases as ^1.124, or slightly faster than proportionally with the period. The distance of a Cepheid variable is related to its period and apparent magnitude by the following equation:

R

=

5.176100.2 m 0.562 pc.

The distance on the left is in units of 10 pc. The apparent magnitude is given by m. For a galaxy of 1 Mpc distance, which is fairly typical of nearby galaxies, Cepheids with 1,000 day periods will have an apparent magnitude of 18. In comparison, limits of ground telescopes are at the level of magnitude 25 (7 magnitudes fainter). This shows why Cepheids are effective tools, but it also shows that their effectiveness drops rapidly with distance.