STARS, RR LYRAE TYPE JAMES NEMEC RR Lyrae stars are 10-15-Gyr-old low-mass variable stars that pulsate radially with periods between about 0.2 and 0.9 d. They were first discovered in nearby globular clusters about a century ago. Today, -1500 RR Lyrae stars have been identified in globular clusters, and -6000 isolated field stars are known to be RR Lyraes. Because RR Lyrae stars are luminous and are easily recognized from their periods and from the shapes of their light curves, and because they have a small range in their luminosities, they are useful for distance estimation. They have also been used extensively for studying the structure and evolution of the Galaxy, and are currently being used as probes of other galaxies. The most distant of the known RR Lyrae stars are those discovered recently in the Local Group galaxies M31, M33, NGC 185, and NGC 147, which have distances -700 kpc. As electronic detectors on ground-based telescopes are improved, and the Hubble Space Telescope is optically repaired, it is likely that RR Lyrae stars in galaxies beyond the Local Group will be found. There are two basic types of RR Lyrae stars: Bailey ab-type stars, which pulsate in the fundamental mode with periods between -0.4 and 0.9 d; and Bailey c-type stars, which pulsate in the first-overtone mode with periods between -0.2 and 0.5 d. In general, RRab stars have cooler surface temperatures and are much more common than RRc stars. Furthermore, whereas the RRab stars typically have large amplitudes and asymmetric light curves, the RRc stars tend to have smaller amplitudes and more sinusoidal light curves. In globular clusters, the two types are easily distinguishable if the periods and amplitudes are accurately known. In that case, a plot of amplitude versus period, that is, a P-A diagram, shows that the RRab stars have decreasing amplitudes for increasing periods, whereas the P-A relationship for RRc stars is less clear. In 1939 and 1944, P.Th. Oosterhoff discovered that the periods of RR Lyrae stars in globular clusters can be used to sort the clusters into two groups: type I systems (e.g., M3, M5), in which the mean periods of the ab- and c-type RR Lyrae stars are (***) -0.54 d and <**> -0.3 d, respectively; and type II systems (e.g., M15, M92), in which <***>=0.64 d and -0.37 d. Later it was shown that the two groups can be discriminated by metal abundance. Over the past 50 years much work has gone into confirming the apparent near-absence of globular clusters with <***> -0.60 d, and into explaining the Oosterhoff dichotomy. In addition to ab- and c-type RR Lyrae stars, several other types are now recognized. Included among these are the Blazhko variables, in which the pulsations are characterized by -20-200-d periodic modulation of the amplitude and phase of maximum light. The brightest known Blazhko variable is RR Lyrae itself, which has a fundamental-mode period of 0.567 d, and an amplitude-modulation period of 41 d. Although B1azhko variables are quite common, and tend to be found among the shortest- period RRab stars, the mechanism that produces their amplitude modulations is not well understood. Another type of RR Lyrae star, the double-mode (or d-type) RR Lyraes, of which -40 are known, pulsate simultaneously in the fundamental and the first-overtone modes, with periods in the ratio P* /P* -0.746*0.002. These stars are exceedingly valuable sources of information because their masses can be determined using only the Petersen P*/P* versus P* diagram, and theoretical mass-calibration curves. Masses derived in this way can be compared with less-direct mass estimates based on pulsation theory, stellar evolution theory, etc. It has been suggested that RRd stars might be in a transitional phase in which they are switching pulsation modes from the fundamental to the first-overtone mode as they evolve from ab to c type, or vice versa. One triple-mode RR Lyrae star, AC And, has been identified. It pulsates simultaneously in the fundamental, first-overtone, and second-overtone modes. There has been considerable speculation about the possibility of RR Lyrae stars pulsating in the second-overtone mode; however, such stars have yet to be identified with certainty. It was once common to classify Cepheid variable stars with periods between about 1 and 3 d as long-period RR Lyrae stars. However, the majority of these stars are now known to be either Population II Cepheids that are more luminous than RR Lyrae stars, anomalous Cepheids (recent evidence suggests that these may be coalesced binary systems), or Population I Cepheids that are more massive and younger than RR Lyrae stars. At periods shorter than -0.25 d, the period distribution of RR Lyrae stars overlaps that of young, massive Population I dwarf Cepheids (i.e., * Scuti stars) and old Population II dwarf Cepheids (i.e., SX Phoenicis stars). PULSATIONAL INSTABILITY The physics of the pulsations of RR Lyrae stars is very similar to that of Gepheid variables. The observed light and radial velocity curves of both types of stars are mirror images that is, at maximum light the radial velocity is at a minimum, and at minimum light the radial velocity is at a maximum. Originally this was interpreted as being due to line-of-sight motion in a spectroscopic binary system. However, because this hypothesis failed to provide satisfactory orbital solutions, and failed to explain a number of other observations (such as the irregular variations seen in some light curves and in some periods, the observed spectral-type changes over each light cycle, and the complete absence of spectral features that are expected for binary systems), it was soon rejected. The hypothesis that RR Lyrae stars and Cepheids are single stars that are pulsating radially was proposed as an alternative, and continues to be the most plausible explanation. For a radially pulsating star, the observed velocity is the sum of the radial velocity arising from the projection of the space motion of the star onto the line of sight, and the contribution from the radial pulsations. Considering only the contributions arising from the pulsations, the fact that the light curve and radial velocity curve are mirror images suggests the following interpretation. At maximum luminosity, the expansion of the star is most rapid. As the star dims, the expansion continues, the diameter reaching a maximum when the radial velocity is zero. Alter this the star contracts, and the positive radial velocities measured by an observer are due to the apparent recession of the atmosphere. At minimum light the contraction is most rapid, at which time the higher temperatures associated with the compressed gas lead to an increased flux of radiation emerging from the star, and subsequent brightening. At maximum compression the radial velocity is again zero, and the star is once again on its way to reaching maximum light. In an attempt to account for the observed light and radial velocity curves of Cepheids and RR Lyrae stars, Arthur S. Eddington developed a mathematical model for radial pulsations. This model, while unable to explain fully the observed phase relationships between velocity, luminosity, and temperature, did succeed in explaining the pulsation periods, and provided the foundations of modern pulsational theory. The starting point of Eddington's investigations was J. Homer Lane's pioneering 1869 study of the temperature stratification within a star, and August Ritter's 1878-1889 series of articles on the analysis of adiabatic pulsations of gaseous stars in convective equilibrium. Eddington, following Karl Schwarzschild, assumed radiative rather than convective equilibrium throughout the star. The aim of his model was to explain the pulsations, while balancing the energy continually being liberated in the star and the energy needed to excite and maintain the pulsations, as well as the obvious loss of energy from the surface of the star. A significant finding was that in RR Lyrae stars and Cepheids, it is the outer layers that are subject to the pulsational instabilities. Two of Eddington's pulsation mechanisms, the gamma and kappa mechanisms, remain relevant today. Gamma mechanism: From Ritter's thermodynamical studies, it is known that if the ratio of the specific heats of a gaseous star,**********, where *** the specific heat at constant pressure, and *** the specific heat at constant volume), is less than ** (on average), then the total energy of the star is positive, and an equilibrium configuration is not possible. Therefore, in the adiabatic central regions of stars, where the ideal gas law applies and ****, pulsations do not oocur. On the other hand, * can be lowered to values ** or less, in the hydrogen and helium ionization zones near the cooler, nonadiabatic surface of an RR Lyrae star. Upon compression, because of the smaller *, these ionization zones remain cooler than their surroundings and heat is absorbed when the temperature is high. Upon expansion, when the temperature falls, heat is given off. These conditions, which resemble the valve mechanism of a heat engine, can lead to pulsational instability if the density of material in one (or more) of these zones is sufficiently great to provide enough mechanical energy to drive the pulsations. The * mechanism (named for the symbol used to represent the opacity of the gas) also involves ionization zones and the conversion of radiative energy into mechanical energy. In regions where high temperatures give rise to high opacities, upon compression the temperature rises, causing the opacity to rise, thereby decreasing the flow of radiation through the star. The damming of trapped photons eventually causes the envelope to expand, thus reducing the temperature of the zone and lowering the opacity. This leads to an increased outflow of radiation. Eventually, gravity halts the expansion and the envelope collapses on a free-fall (or dynamical) time scale, and the cycle starts again. It was not until 1953 that the second helium ionization zone was identified as the zone most likely to be responsible for exciting and maintaining the pulsations. The observed phase relationship between luminosity and radial velocity was explained in the 1960s, when nonadiabatic and nonlinear numerical models of RR Lyrae stars were computed, and accurate computer modeling of the light and radial velocity curves became possible. Today, fully hydrodynamic models are being used to calculate synthetic light and radial velocity curves for RR Lyrae stars, which have been found to provide good fits to the observations. PHYSICAL CHARACTERISTICS In color-magnitude diagrams, RR Lyrae stars are readily seen to be horizontal-branch stars in the Cepheid instability strip, with mean B-V colors between 0.17 and 0.42 mag (corresponding to effective temperatures in the range 6000 ****** 9000 K). Unlike Cepheids, which are more luminous than RR Lyrae stars and are found over a large luminosity range, RR Lyrae stars have luminosities in the narrow range 40 *******90 (corresponding to absolute visual magnitudes in the range ************); L* is the solar luminosity. The RRab stars tend to be located on the low-temperature side of the strip, and the RRc stars tend to be hotter. The temperature of the red edge of the RR Lyrae instability strip, which separates the RRab stars from the red horizontal branch stars, and the temperature of the blue edge, which separates the RRc stars from blue horizontal branch stars, is determined by the depths of the narrow hydrogen and helium ionization zones, and the energy transport mechanisms in the outer layers of the star. At the red edge, the pulsation is damped by the onset of efficient convective energy transport, and at the blue edge the instability is damped because the location of the second helium ionization zone is too near the surface to be effective. Given the mean T* and the mean L of an RR Lyrae star, its mean radius, R, can be calculated from the Stefan-Boltzmann law, L=*********, where ** is the Stefan-Boltzmann constant. A typical mean radius is *5 R* (solar radii), characteristic of a giant star. During each pulsation cycle, it is common for an RR Lyrae star to change its radius by ******%. Stellar evolution models have shown that RR Lyrae stars are in an advanced stage of their life, deriving their energy from the nucleosynthesis of helium (via the triple * process) in a small central core, and from hydrogen burning (via the CNO cycle) in a narrow shell outside of the core. This energy is transported by convection in the inner core, and by radiation through the outer regions of the star. Numerical models show the central temperature to be -********* K, and the mass density at the center of the star to be -*************. Depending on the particular assumptions that are used in the models, estimates of the core mass range from 0.47 to 0.51 M*, and total masses are in the range 0.50 M/M***0.80. Most RR Lyrae stars exhibit slowly changing periods. At one time it was hoped that the measured period change rates could be used to determine the time required to evolve across the instability strip; however, the period changes have a stochastic nature, which remains unexplained. Nevertheless, the observed pulsation properties of RR Lyrae stars are sufficiently well understood that they are used to test ideas about the advanced evolution of old low-mass stars. DISTANCES AND LUMINOSITIES The use of RR Lyrae stars as distance indicators stems from their relatively small range in luminosity (or, equivalently, absolute bolometric magnitude M**,). By measuring the mean apparent bolometric magnitude of an RR Lyrae star, m**, and the total extinction caused by interstellar material, A**, the distance to the star, d, follows from the distance modulus equation: (m-M)****************. In practice, RR Lyrae stars are usually observed through blue (B) and visual (V) filters, and the visual apparent magnitude ** and the reddening ***** (O.33Av) are measured. Conversion to bolometric quantities is made by applying bolometric corrections, which are small for RR Lyrae stars. Within a globular cluster that is rich in RR Lyrae stars, the apparent magnitude of the RR Lyrae stars usually varies by only *0.2 mag about the mean magnitude. For this reason, a single absolute visual magnitude, Mv, is usually assumed for all the stars. Then, because the diameter of the cluster is small relative to the distance to the cluster, the distance to the cluster can be assumed to be identical to that of the ensemble of cluster RR Lyrae stars. Given the distance to the cluster, and reddening-corrected apparent magnitudes, the luminosity level of its main-sequence turnoff can be computed, and the cluster age determined by fitting theoretical isochrones to the cluster main-sequence turnoff. RR Lyrae stars have also proved to be useful for determining the distance to the galactic center. By identifying large numbers of RR Lyrae stars in low-extinction "windows" (directions in which the interstelar reddening is relatively small, the most famous of which is Baade's window), and make reasonable assumptions about Mv and ****, the peak of the histogram of the derived distances to the RR Lyrae stars gives the modal distance to the stars. The distance to the galactic center then follows by assuming that the RR Lyrae stars are spherically distributed about the center, and that the modal distance of the RR Lyrae stars coincides with the distance to the center. From recent measurements (which include observations made at infrared wavelengths, where the extinction is much reduced) it has been inferred that the distance to the galactic center is ******** kpc. This estimate agrees well with distance determinations made-using non-RR Lyrae star techniques. The greatest concern in using RR Lyrae stars as distance indicators is the uncertainty in their absolute magnitudes. It has long been known (from theoretical models) that RR Lyrae stars evolve away from their initial positions on the zero-age horizontal branch on a time scale *** yr. During this time they become more luminous and eventually ascend the red giant branch for a second time (as asymptotic giant branch stars). The RR Lyrae stars in the most advanced evolutionary states are presumed to be those that started out with the greatest masses (but not so great that they are no longer RR Lyrae stars). Thus, changes in both luminosity and effective temperature are to be expected over the lifetime of an RR Lyrae star because of this dependence on evolutionary phase. Furthermore, because L and T*, depend on the age of the star; the metal abundance [Fe/H]; the helium abundance Y; the abundances of carbon, nitrogen, and oxygen; and other parameters (such as rotation, strengths of magnetic fields, etc.), knowledge of the nature of this dependency is required before reliable estimates of M* can be obtained. Presently, the best estimates of M* range between *0.2 and *1.0, with metal-poor RR Lyrae stars having greater luminosities and masses than metal-rich RR Lyrae stars. HALO AND THICK-DISK RR LYRAE STARS The majority of the RR Lyrae stars in the Galaxy have yet to be identified. Nevertheless, from careful surveys of selected areas of the sky it is clear that the total number of galactic RR Lyrae stars is very large. The same surveys tell us that the number density of RR Lyrae stars is proportional to ***, where R is the distance from the galactic center. At R=0.6 kpc, the density is -4000 stars kpc**, and at R=1.5 kpc the density is only -260 stars kpc**. Although the number density is very low, RR Lyrae stars are still seen out to R=50 kpc. Studies of the chemical compositions and space motions of the RR Lyrae stars in the Galaxy have shown that they do not constitute a homogeneous stellar population, but rather are a mixture of halo and thick-disk (or old-disk) stars. The halo RR Lyrae stars have metal abundances [Fe/H]****0.8, and are found in metal-poor globular clusters and throughout the Galaxy (the term in brackets is defined under "Heavy Elements" in the entry on Star Clusters, Globular, Chemical Composition). They appear to have similar kinematic properties and compositions to halo subdwarfs. The entire system of halo RR Lyrae stars occupies an approximately spherical volume centered on the galactic center, and the system rotates (relative to a galactic rest frame) very slowly, if at all. Assuming that the field halo RR Lyrae stars are as old as their counterparts in globular clusters, then their ages are **** Gyr. Thick-disk RR Lyrae stars are more metal rich than [Fe/H]**0.8, and are found only in the inner regions of the Galaxy. The spatial distribution of the system of thick-disk RR Lyrae stars is flattened, like an oblate spheroid with an axial ratio c/a**0.6 (where c is the semiminor axis measured in the plane of the Galaxy, and * is the semimajor axis measured perpendicular to the plane). The system of thick-disk RR Lyrae stars rotates faster than the system of halo stars, but both systems lag behind the rapidly rotating stars in the thin disk of the Galaxy. It has been estimated that in the solar neighborhood there are **20 thick-disk subdwarf stars for every halo subdwarf. However, only one in four of the nearby RR Lyrae stars is metal rich. The reason for this discrepancy is that metal-rich horizontal branch stars rarely evolve to hot enough temperatures to enter the instability strip, and thus they are much rarer. For the same reason, it is unusual to find RR Lyrae stars in globular clusters that only have red horizontal branches. The question of the origin of these halo and thick-disk RR Lyrae stars is presently unsolved. Did the metal-poor stars form before the more metal-rich stars? Or, are they all approximately the same age, with the metal-rich RR Lyrae stars having formed out of clouds chemically enriched as a result of nearby supernovae explosions? Were all (or some fraction) of the field RR Lyrae stars once in globular clusters? And if so, did they enter the field as a result from the disruption of entire globular clusters, or are they isolated escapees? The answers to these questions are tied directly to the fundamental question of how the Galaxy formed. Because our present knowledge of the halo and disk stellar populations in the Galaxy is still very incomplete, RR Lyrae stars will continue to play a major role in solving the puzzle. And, when more RR Lyrae stars are discovered in other galaxies it will be of considerable interest to see if they too divide into different population types, providing valuable clues to the histories of these other systems. Additional Reading Baker, N. and Kippenhahn, R.(1962). The pulsations of models of * Cephei stars. Z. Ap. 54 114. Christy, R.F.(1966). A study of pulsation in RR Lyrae models. Astrophys. J. 144 108. Cox, A.N., Hodson, S.W., and Clancy, S.P.(1983). Double-mode RR Lyrae variables in M15. Astrophys. J.266 94. Eddington, A.S.(1930). The Internal Constitution of the Stars. Cambridge University Press, Cambridge. Iben, J., Jr.(1971). Globular-cluster stars: Results of theoretical evolution and pulsation studies compared with the observations. Publ. Astron. Soc. Pac. 83 697. Oort, J.H. and Plaut, L.(1975). The distance to the galactic centre derived from RR Lyrae variables, the distribution of these variables in the Calaxy's inner region and halo, and a rediscussion of the galactic rotation constants. Astron. Ap. 41 71. Sandage, A.R.(1990). The Oosterhoff period effect: Luminosities of globular cluster zero-age horizontal branches and field RR Lyrae stars as a function of metallicity. Astrophys. J. 350 631. See also Star Clusters, Globular, Variable Stars; Stars, Horizontal Branch; Stars, Pulsating, Overview; Stars, Pulsating, Theory; Stellar Evolution, Pulsations.