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Publications of the Astronomical Society of the Pacific PUBLICATIONS OF THE ASTRONOMICAL SOCIETY OF THE PACIFIC Vol. 80 August 1968 No. 475 PULSATING STARS DAVID S. KING Department of Physics and Astronomy University of New Mexico and JOHN P. COX Department of Physics and Astrophysics and Joint Institute for Laboratory Astrophysics University of Colorado Received March 13, 1968 The current status of our understanding of the causes and nature of stellar pulsation, as applied to classical cepheids, RR Lyrae variables, and W Virginis variables, is reviewed. The historical development of these ideas is briefly surveyed, with emphasis on the search for the excitation mechanism ( s ). Tests of the idea that the instability is due to the ionization of one or more of the elements H, He, and He+ in the stellar envelope, by means of both the linear and the more recent nonlinear calculations, are described and summarized. Some comparisons of theory and observation are made, particularly with respect to the period-luminosity relation, location of variables in the Hertzsprung-Russell diagram, fundamental versus first overtone mode, and phase relations between light and velocity curves. A number of special prob- lems are discussed, including causes of the limitation of the pulsation ampli- tude to the observed values, relations between the light and velocity variations, and the interaction between pulsations and convection. A new physical inter- pretation, due to J. Castor, of the well-known "phase lag discrepancy" be- tween the luminosity and radius variations is described. A number of pre- viously unpublished results are presented, particularly with respect to recent nonlinear stellar pulsation calculations. I. Introduction The RR Lyrae variables, classical cepheids, and W Virginis vari- ables constitute a very important class of stellar objects. The well- 365 © Astronomical Society of the Pacific · Provided by the NASA Astrophysics Data System 366 D. S. KING AND J. P. COX known period-luminosity relation of classical cepheids makes pos- sible the use of these stars as indicators of distance within our own galaxy as well as for nearby extragalactic systems. Recent theoretical work also points to the possibility of a better understanding of the interiors of giant and supergiant stars through a study of their pulsation characteristics. The question of the physical nature of such pulsations is therefore of great interest to a very large seg- ment of astronomy. The discussion which follows will pertain pri- marily to the types of variable stars just mentioned, although much of the theory appears to be equally valid for such stars as the δ Scuti variables, RV Tauri variables, and the dwarf cepheids. No attempt will be made here to give a detailed survey of the extensive observational material which is available for such stars. However, for orientation we list in Table I some of the more basic data for the RR Lyrae variables, the classical cepheids, and the W Virginis variables. The ranges in median absolute magnitude given in this table are approximate and include some allowance for observational uncertainties. In Table II we list some properties of classical cepheids. In this table the mean luminosities (L) and effective temperatures ( Te ) have been computed from the period- luminosity and period-color relations given by Kraft ( 1963 ). The masses (¾)^) are estimates based on the evolutionary tracks com- puted without mass loss by Hofmeister, Kippenhahn, and Weigert (1964a,b,c); Kippenhahn, Thomas, and Weigert (1965); and Hof- meister (1967a). The Q values are computed from the listed values of and R. Further observational facts will be introduced where appropriate during the discussion of various theoretical problems. Excellent and detailed summaries of the basic observa- tional material have been provided by, e.g., Payne-Gaposchkin and Gaposchkin (1963), Payne-Gaposchkin (1954), Ledoux and Wal- TABLE I Pulsating Variables Range of Median Kind of Range of Characteristic Population Median Absolute Star Periods Period Type Spectral Type Magnitude RR Lyrae IbS to 24h 0^5 II A2 to F6 -f 0.5 to -fl Classical Cepheids 1* to 50d 5d to 1(^ I F6 to K2 -0.5 to-6 W Virginis 2d to 45d 12d to 20d II F2 to G6( ?) 0.0 to-3 © Astronomical Society of the Pacific · Provided by the NASA Astrophysics Data System PULSATING STARS 367 TABLE II Properties of Classical Cepheids From To 1« 50d 380L,Ό 31?000Lo F5 G5 6900oK 5400oK 14¾ 200Rq 3.7%Í ivt© imQ 0^037 0-066 raven (1958), and Kukarkin and Parenago (1963); these may be consulted for further details. Two very recent observational dis- cussions of the period-luminosity relation of classical cepheids are by Sandage and Tammann (1968) and by Femie (1967). The theoretical study of pulsating variable stars has been con- siderably advanced in the past ten years. This progress can be attributed in large part to the availability of modem electronic computers which have made it possible to treat the equations appropriate to radial pulsations in their fully nonlinear form. One of the main purposes of this paper is to review some of the more important aspects of this work. Other recent reviews of stellar pulsation theory are provided by Ledoux and Walraven (1958); Ledoux and Whitney ( 1961 ) ; Ledoux ( 1963, 1965 ) ; Zhevakin (1963); Christy (1966b, 1967d); Cox (1967); and Cox and Cox (1967). Earlier reviews have been provided by Eddington (1926), Chapter 8; and Rosseland ( 1949 ). See also the proceedings of the 1965 I.A.U. Bamberg Symposium, The Position of Variable Stars in the Hertzsprung-Russell Diagram. One can characterize the state of our present knowledge by point- ing to the steps which led to it: (a) the establishment of the nature of the light and radial velocity variations and the search for the mechanism ( or mechanisms ) responsible for the maintenance of the pulsations; (b) studies involving the linearized pulsation equations which tended to confirm the effectiveness of "envelope ionization mechanisms" (in particular, the second helium, first helium, and hydrogen ionization zones) in exciting pulsations; (c) solutions of the fully nonlinear pulsation equations; further confirmation of the effectiveness of these envelope ionization mechanisms as sources © Astronomical Society of the Pacific · Provided by the NASA Astrophysics Data System 368 D. S. KING AND J. P. COX of excitation (or "driving") for pulsations in the envelopes of stars; (d) the study of a number of interesting problems, both linear and nonlinear, such as the growth rate of the pulsation amplitude, the approach to limiting amplitude, the phase relations between the light and velocity variations, and the interaction between pul- sations and convection. The plan of this paper will follow the above list of topics. II. Establishment of the Nature of the Variations and the Search for the Excitation Mechanism The concept of radial pulsations of stellar masses as the explana- tion of the observed cepheid phenomena was first given a firm mathematical foundation by Eddington (1926). He assumed that the oscillations were very small and essentially adiabatic, and de- rived and studied solutions of the adiabatic wave equation for such oscillations. Perhaps the most important and far-reaching result of these studies was the theoretical derivation of the well- known "period-mean density" relation which is apparently obeyed by actual pulsating stars: Π(ί5/ρο)4 = Ç , (1) where Π is the period; ρ and /3© are the mean densities of, respec- tively, the star and the sun; and Ç is the "pulsation constant," a slowly varying function of the properties and internal structure (particularly the central mass concentration) of the stellar model. For the fundamental mode of the homogeneous (uniform density) model the value of Q is Ç = 27r/[(3r1 —4) (4/3)^]^ , (2) where G is the constant of gravitation and T1 is one of the adiabatic exponents ( assumed constant throughout the star in equation (2)): ζ dlnP ν ζ 0 ν · (3) where Ρ and ρ are total pressure and density, respectively. The value of Q given by (2) is the largest possible value for a star of given Γι whose density does not increase outward, and is 0^116 for Γι == 5/3. For more realistic distributions of density the theoretical values of Q generally lie in the range (K03-CW6 for Tt = 5/3. The © Astronomical Society of the Pacific · Provided by the NASA Astrophysics Data System PULSATING STARS 369 theoretical values of Q are insensitive to effects of nonadiabaticity and nonlinearity ( Q is increased on the order of a few percent for extremely large pulsation amplitudes where the radial excursion is comparable to the radius itself). Empirical values of Q are some- what uncertain, but generally appear to lie in the range 0^03-0^08, in good agreement with theoretical values. Physically, equation (1) may be interpreted roughly as a statement that the pulsation period is of the order of the time required for a sound wave to travel through the diameter of the star. Hence, small, compact stars have shorter periods than do large, tenuous stars. Eddington also studied the problem of the maintenance of the pulsations and the related problem of the dissipation of pulsation energy. He obtained an expression for the total dissipation of pulsational energy in a star which was subjected to a small radial perturbation. This dissipation is given for a star which in its un- perturbed state is in both hydrostatic and thermal equilibrium by; (δΤ/Τ) dp (4) vol where the volume is that of the entire star and the cyclic integral extends over one pulsation period of the star, δ Τ is the instan- taneous temperature excess of a volume element over its average or equilibrium value and dÇ is the instantaneous gain of heat per unit volume. Eddington used this equation to calculate the time of decay for the pulsations of a model which was supposed to repre- sent δ Cephei, the prototype for variable stars of the classical cepheid variety.
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