Pulsating Variables

The wavelengths of the spectral lines of the pulsating variables change along with the brightness variations (Table 13.1). These changes are due to the Doppler effect, showing that the outer layers of the star are indeed

Table 13.1. The main properties of pulsating variables (N, number of stars of the given type in Kukarkin's catalogue, P, pulsation period in days, Am, pulsation amplitude in magnitudes)






Classical cepheids




< 2

(5 Cep, W Vir)

RR Lyrae


< 1


< 2

Dwarf cepheids




< 1

(5 Scuti)

fi Cephei




> 0.3

Mira variables




> 2.5

RV Tauri




< 4





< 4.5





< 2

pulsating. The observed gas velocities are in the range of 40-200 km/s.

The period of pulsation corresponds to a proper frequency of the star. Just like a tuning fork vibrates with a characteristic frequency when hit, a star has a fundamental frequency of vibration. In addition to the fundamental frequency other frequencies, "overtones", are possible. The observed brightness variation can be understood as a superposition of all these modes of vibration. Around 1920, the English astrophysicist Sir Arthur Eddington showed that the period of pulsation P is inversely proportional to the square root of the mean density,

The diameter of the star may double during the pulsation, but usually the changes in size are minor. The main cause of the light variation is the periodic variation of the surface temperature. We have seen in Sect. 5.6 that the luminosity of a star depends sensitively on its effective temperature, L a Te4. Thus a small change in effective temperature leads to a large brightness variation.

Normally a star is in stable hydrostatic equilibrium. If its outer layers expand, the density and temperature decrease. The pressure then becomes smaller and the force of gravity compresses the gas again. However, unless energy can be transferred to the gas motions, these oscillations will be damped.

The flux of radiative energy from the stellar interior could provide a source of energy for the stellar oscil-

lations, if it were preferentially absorbed in regions of higher gas density. Usually this is not the case but in the ionization zones, where hydrogen and helium are partially ionized, the opacity in fact becomes larger when the gas is compressed. If the ionization zones are at a suitable depth in the atmosphere, the energy absorbed during compression and released during expansion of an ionization zone can drive an oscillation. Stars with surface temperatures of 6000-9000 K are liable to this instability. The corresponding section of the HR diagram is called the cepheid instability strip.

Cepheids. Among the most important pulsating variables are the cepheids, named after S Cephei (Fig. 13.3). They are population I supergiants (stellar populations are discussed in Sect. 17.2) of spectral class F-K. Their periods are 1-50 days and their amplitudes, 0.1-2.5 magnitudes. The shape of the light curve is regular, showing a fairly rapid brightening, followed by a slower fall off. There is a relationship between the period of a cepheid and its absolute magnitude (i. e. luminosity), discovered in 1912 by Henrietta Leavitt from cepheids in the Small Magellanic Cloud. This period-luminosity relation (Fig. 13.4) can be used to measure distances of stars and nearby galaxies.

We have already noted that the pulsation period is related to the mean density. On the other hand the size

Fig. 13.4. The period-luminosity relation for cepheids. The black points and squares are theoretically calculated values, the crosses and the straight line represent the observed relation. (Drawing from Novotny, E. (1973): Introduction to Stellar Atmospheres and Interiors (Oxford University Press, New York) p. 359)

Fig. 13.4. The period-luminosity relation for cepheids. The black points and squares are theoretically calculated values, the crosses and the straight line represent the observed relation. (Drawing from Novotny, E. (1973): Introduction to Stellar Atmospheres and Interiors (Oxford University Press, New York) p. 359)

of a star, and hence its mean density, is related to its total luminosity. Thus one can understand why there should be a relation between the period and the luminosity of a pulsating star.

The magnitudes M and periods P of classical cepheids are shown in Fig. 13.4. The relation between M and log P is linear. However, to some extent, the cepheid luminosities also depend on colour: bluer stars are brighter. For accurate distance determinations, this effect needs to be taken into consideration.

W Virginis Stars. In 1952 Walter Baade noted that there are in fact two types of cepheids: the classical cepheids and the W Virginis stars. Both types obey a period-luminosity relation, but the W Vir stars of a given period are 1.5 magnitudes fainter than the corresponding classical cepheids. This difference is due to the fact that the classical cepheids are young population I objects, whereas the W Vir stars are old stars of population II. Otherwise, the two classes of variables are similar.

Earlier, the WVir period-luminosity relation had been used for both types of cepheids. Consequently the calculated distances to classical cepheids were too small. For example, the distance to the Andromeda Galaxy had been based on classical cepheids, since only these were bright enough to be visible at that distance. When the correct period-luminosity relation was used, all extragalactic distances had to be doubled. Distances within the Milky Way did not have to be changed, since their measurements were based on other methods.

RR Lyrae Stars. The third important class of pulsating variables are the RR Lyrae stars. Their brightness variations are smaller than those of the cepheids, usually less than a magnitude. Their periods are also shorter, less than a day. Like the W Vir stars, the RR Lyrae stars are old population II stars. They are very common in the globular star clusters and were therefore previously called cluster variables.

The absolute magnitudes of the RR Lyrae stars are about MV = 0.6 ± 0.3. They are all of roughly the same age and mass, and thus represent the same evolutionary phase, where helium is just beginning to burn in the core. Since the absolute magnitudes of the RR Lyrae variables are known, they can be used to determine distances to the globular clusters.

Fig. 13.5. The lightcurve of a long period Mira variable

Mira Variables (Fig. 13.5). The Mira variables (named after Mira Ceti) are supergiants of spectral classes M,

5 or C, usually with emission lines in their spectrum. They are losing gas in a steady stellar wind. Their periods are normally 100-500 days, and for this reason, they are also sometimes called long period variables. The amplitude of the light variations is typically about

6 magnitudes in the visual region. The period of Mira itself is about 330 days and its diameter is about 2 AU. At its brightest, Mira has the magnitude 2-4, but at light minimum, it may be down to 12. The effective temperature of the Mira variables is only about 2000 K. Thus 95% of their radiation is in the infrared, which means that a very small change in temperature can cause a very large change in visual brightness.

Other Pulsating Variables. One additional large group of pulsating stars are the semiregular and irregular variables. They are supergiants, often very massive young stars with unsteady pulsations in their extended outer layers. If there is some periodicity in the pulsations, these variables are called semiregular; otherwise they are irregular. An example of a semiregular variable is Betelgeuse (a Orionis). The pulsation mechanism of these stars is not well understood, since their outer layers are convective, and the theory of stellar convection is still poorly developed.

In addition to the main types of pulsating variables, there are some smaller separate classes, shown in Fig. 13.2.

The dwarf cepheid and the 8 Scuti stars, which are sometimes counted as a separate type, are located below the RR Lyrae stars in the cepheid instability strip in the HR diagram. The dwarf cepheids are fainter and more rapidly varying than the classical cepheids. Their light

curves often show a beating due to interference between the fundamental frequency and the first overtone.

The j Cephei stars are located in a different part of the HR diagram than the other variables. They are hot massive stars, radiating mainly in the ultraviolet. The variations are rapid and of small amplitude. The pulsation mechanism of the j Cephei stars is unknown.

The RV Tauri stars lie between the cepheids and the Mira variables in the HR diagram. Their period depends slightly on the luminosity. There are some unexplained features in the light curves of the RV Tauri stars, e.g. the minima are alternately deep and shallow.

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