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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 problems are discussed, including causes of the limitation of the pulsation amplitude to the observed values, relations between the light and velocity variations, and the interaction between pulsations and convection. A new physical interpretation, due to J. Castor, of the well-known "phase lag discrepancy" between the luminosity and radius variations is described. A number of previously unpublished results are presented, particularly with respect to recent nonlinear stellar pulsation calculations.

I. Introduction

The RR Lyrae variables, classical cepheids, and W Virginis variables constitute a very important class of stellar objects. The well-

365

known period-luminosity relation of classical cepheids makes possible 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 primarily 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 computed 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 observational 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

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 discussions 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 considerably 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

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 pulsations 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 explanation 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, respectively, 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

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 somewhat 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 instantaneous 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 represent δ Cephei, the prototype for variable stars of the classical cepheid variety. Using the assumption that the equilibrium model could be represented by the polytrope η =: 3 ( Eddington s standard model), he calculated a decay time of approximately 8000 years. Since this is a relatively short time compared to the lifetime of a star the probability of observing such a star pulsating would be vanishingly small. A number of assumptions which he made were not valid in the light of later information, the most serious of these being the proposed mass distribution. Current knowledge about the structure of giant and supergiant stars, and the theory of stellar evolution, indicate that there is a much greater mass concentration toward the center of the star (see e.g., Epstein (1950); Hofmeister, Kippenhahn, and Weigert (1964c)) than would result

from Eddington's standard model. Later calculations, using more realistic models (see next paragraph), indicated that the decay times were even shorter. This presented the serious problem of explaining the nature of the mechanism which must be maintaining the observed pulsations at a nearly constant amplitude (in time). Eddington first suggested that one might consider the star as a thermodynamic heat engine in which excess energy would be added when the temperature was high and released when it was low. He visualized a "valve" mechanism which served to satisfy the physical requirements for such a heat engine. Since that time a great deal of theoretical research has been concerned with trying to determine the location of this "valve" mechanism in the star. The first suggestion that Eddington made was that the "valve" was located at the center of the star and was related to the rate of the production of nuclear energy. This required that the temperature dependence of the energy production be sufficiently strong and of the proper phase so that additional energy is added to the system at maximum compression and released upon expansion. This condition would seem to be satisfied by known mechanisms of nuclear energy generation, but a problem arises when one considers the dependence of the pulsation amplitude on radial distance from the center of the star. Calculations which yield this dependence for centrally concentrated stellar models were carried out by Epstein (1950). The result was that the radial variation of the surface layers was of the order of a million times that of the region near the center of the star where the nuclear energy generation was occurring, provided that the adiabatic exponent Γι was greater than and not close to 4/3. This very pronounced increase in amplitude as one proceeds toward the surface of the star normally tends to increase its over-all stability. Subseqeunt calculations of the pulsational stability of a giant model were carried out by Cox ( 1955 ) with the result that the increase in the amplitude of a small radial perturbation due to energy generation effects was found to be very small and completely swamped by the "radiative damping" ( i.e., damping due to the fact that most elements of volume throughout the star tend to lose heat by radiation most rapidly when the star is at maximum compression). Similar conclusions were also reached by Ledoux, Simon, and Bierlaire ( 1955 ) and by Rabino- witz (1957). Cox also estimated that the adiabatic approximations

(losses or gains of heat are small compared to the heat content of a given mass element) which he was using were good out to 0.85 of the stellar radius. These developments led investigators to look to the outer nonadiabatic regions ( where losses and gains of heat by an element of mass are not necessarily small compared to the heat content of that element) of the star in an attempt to locate the source of instability. Eddington ( 1926) had presented another possible interpretation for this "valve" mechanism. He suggested that one could obtain the same effect if, instead of varying the supply of heat, one were to vary the leakage of heat so that at maximum compression the leakage was small while during expansion it increased. At the time of Eddington's suggestion it was not known that hydrogen and helium were the most abundant elements found in stars and so Eddington was suggesting that this "valve" might be located in the region where a "predominant" element such as iron was losing L-electrons to go from a neon-like to a helium-like ion. He was proposing that this should occur rather deep in the star. These findings, i.e., the indication that the "valve" mechanism should be sought in the outer 15 percent of the stellar radius and the realization that hydrogen and helium were overwhelmingly the major constituents of stars, paved the way for the next important developments in the search for the excitation mechanism. It was suggested, first by Zhevakin ( 1953 ) and independently by Cox and Whitney ( 1958 ), that the region of second helium ionization might serve as a suitable site for this "valve" mechanism. Eddington (1941, 1942) had suggested earlier that the region of hydrogen ionization in the very outer layers of the star might be important in maintaining the pulsations; however, it seems clear that he still visualized the variation in the rate of nuclear reactions near the center of the star as providing the energy for the pulsations, while the so-called "valve" served only to decrease the dissipation in the outer regions to such an extent that the energy input from the nuclear reactions predominated over the dissipation and the star became pulsationally unstable. Although stellar models were not adequate at that time to permit a detailed test of this hypothesis, this suggestion nevertheless had a strong influence on much of the research which followed.

III. Effectiveness of Envelope lonization Zones as an Excitation Mechanism for Stellar Pulsations

Interior models of stars in the cepheid instability region of the Hertzsprung-Russell (H-R) diagram indicate that for a typical Population I cepheid the region of second helium ionization (i.e., the layer in the stellar envelope where helium that is already once ionized is 50 percent twice ionized) lies at a depth where the temperature is approximately 42,000oK. Because of the central concentration of mass in these stars only the outer 30 to 50 percent (in radius) of the star is undergoing significant radial pulsations.* This outer region of the star in general contains less than 10-4 to ΙΟ-2 of the total mass of the star. The fact that such a small per- centage of the mass of the star is involved is fortunate in the sense that it allows one to carry out the stellar pulsation computations considering only this outer envelope of the star and a precise knowledge of the interior structure is not necessary. It was therefore possible to calculate accurate stellar envelope models for cepheids even before the recent evolutionary models computed by Iben (1966), Hofmeister, Kippenhahn, and Weigert (1964a,b,c), and others were available. The excitation of the pulsations by second helium ionization depends rather sensitively on the properties of the gas. One such property of the gas is its opacity to radiation. If one considers the Rosseland mean opacity, which is the appropriate mean to use in the region of interest, and assumes a dependence of this opacity on temperature and density of the form

η 8 κ = κ{)ρ Τ- , ( 5 )

where η ^ 1 and s ^ 3.5, then it is evident that normally the opacity decreases with increasing temperature. In terms of a star which is undergoing radial pulsations this means that if the above temperature dependence holds, a given layer in the star would normally be expected to be less opaque and hence to radiate at an

#In stars possessing an extensive outer convection zone which contains an appreciable fraction of the total stellar mass, the mass concentration is considerably less than in the case of a purely radiative envelope. In such convective envelope stars the relative pulsation amplitude decreases inward much more slowly than in radiative envelope stars, and the above statement is then not necessarily correct. Unless we specifically state otherwise, we henceforth consider only stars with predominantly radiative envelopes.

increased rate when it is compressed as compared to when it is expanded. Under some conditions, however, it is possible for the material actually to be most opaque near maximum compression. These conditions may prevail within and near regions where an abundant element is undergoing ionization, and the increase in opacity upon compression may be caused either by the fact that the temperature tends to remain low during compression in such regions (see next paragraph) or by the fact that s may be negative in such regions. Detailed opacity calculations have been carried out for a number of chemical compositions by Cox, Stewart, and Eilers (1965). Opacities based on these calculations have been used in all of the most recent studies of the pulsational characteristics of cepheids and RR Lyrae variable stars. Figure 1, which is a plot of the Rosseland mean opacity as a function of time for a pulsating star model, indicates the behavior of several mass zones, both in regions of ionization and outside of such regions. The net effect of the material being most opaque near maximum compression, which is the case in zones 30, 40, and 50, is that some of the radiation is dammed up at this phase producing an excess pressure during the subsequent expansion phase. Eddingtons "valve" mechanism works then in the sense that the leakage of heat is reduced near maximum compression. The above effect of an ionizing gas has been termed the /c-mechanism by Baker and Kippenhahn ( 1962) who first carried out detailed linearized calculations to illustrate its effectiveness as a driving mechanism for cepheid pulsations. In a region of a star in which an abundant element is partially ionized the adiabatic exponents (one of these is defined in equation ( 3 ) ) are all reduced as compared with a region where the ionization is essentially either zero or complete. Hence, such a region will tend to remain cool relative to its surroundings upon compression. This circumstance arises from the fact that in such a region some of the energy which would otherwise go into increasing the temperature of the gas is used to ionize the material. A layer in which helium is partially doubly ionized (He+ ionization region) will therefore tend to absorb heat during compression, leading to a pressure maximum which comes after minimum volume, and hence provide a driving for pulsations. This effect has been called the γ-mechanism by Cox, Cox, Olsen, King, and Eilers (1966).

t/TT

Fig. 1 — Lag κ versus t/Τί for zones 18, 23, 30, 40, and 50 for one of the shallow classical cepheid envelope models investigated by Cox, et al. ( 1966 ) at limiting amplitude. The vertical line (at t/ll = 0.5) marks the instant of minimum surface radius of the envelope. Zone 30 contains the region of 50 percent He+ ionization in the equilibrium model. Zones 40 and 50 lie above, whereas zones 18 and 23 He below, zone 30. Note that the opacity is smallest at maximum compression in zones 18 and 23, and largest at this instant in zones 30, 40, and 50. The individual curves have been arbitrarily displaced vertically for clarity.

A third effect, which can be important in some stars, but is generally small in cepheids, has been termed the radius effect by Baker ( 1967 ). When the star is compressed the curvature of the outer region of the star is increased and this convergence of the envelope layers tends to trap , radiation which heats up the material creating a pressure maximum which comes after minimum volume. This effect of a pressure maximum which occurs after minimum volume ( the net result produced by all three of the above mechanisms) and the way it leads to driving of pulsations, is clearly illustrated by P-V ( pressure-volume ) plots for selected mass shells. Such plots are shown in Figure 2, taken from the calculations of Cox, et al. ( 1966 ). It is seen from the senses in which the loops are traversed that zone 10 ( in the He+ ionization region ) is driving {$PdV> 0, clockwise sense), while zone 5 (interior to the He+ ionization region) is damping.

a? ae as ω ι.ι 1.2 1.3 1.9 2.0 2.1 2.2 2.3 2.4 2.5 V (cm3 ) χ 10 36

Fig. 2 — F-V plots for zones 10 and 5 for one of the shallow classical cepheid envelope models investigated by Cox, et al. ( 1966 ) at limiting amplitude. The directions of the arrows show that driving is present in zone 10 (coinciding approximately with the region of 50 percent He+ ionization), while damping is present in zone 5 (below the He+ ionization region). In the figure V is the total volume of the zone.

Linearized calculations, which took into account the effects of the nonadiabatic regions near the surface of such stars, were carried out by Baker and Kippenhahn (1962, 1965) and by Cox (1963). These calculations confirmed the effectiveness of second helium ionization as a destabilizing mechanism. Cox computed a variety of models with parameters ranging over the entire instability strip. These models used simple power law opacity relations and did not include the ionization regions of hydrogen and first helium. This study isolated the driving, at small amplitudes, due to second helium ionization alone. It was possible, therefore, to obtain a clear picture of how this mechanism works and how it can lead to an instability strip which has the essential features of the observed strip (see Section IV). Baker and Kippenhahn included the region of hydrogen and first helium ionization in their treatment and found that for a 7 ä)?©, Population I star, based on evolutionary models of Hofmeister, Kippenhahn, and Weigert (1964a,b,c), they could define a region of instability which was, however, open on the low temperature side; i.e., although they found a region of maximum instability there was no complete return to stability for the cooler models. It is well known from other considerations that convection becomes an important mechanism for heat transfer in the outer layers of such stars in just the region of the H-R diagram which approximately coincides with the cepheid instability strip. Indeed, the equilibrium models of Baker and Kippenhahn indicate that convection becomes increasingly important in their cooler models. At present a definitive-time-dependent theory of convection, which is needed to determine the interaction of the convection and the pulsations, is not available and therefore the effects of convection on the driving due to a given region of the star cannot be accurately evaluated ( however, see Section V, d ). These authors have suggested that the influence of convection may determine the effective temperature at which their models should return to stability near the low temperature side of the strip. The question of convection is particularly important when one considers the role of the first helium and hydrogen ionization regions. Baker and Kippenhahn found that the driving of the pulsations in their 7 90îo star was due in large part to these latter regions. They also found that the first harmonic was usually pulsationally unstable as well as the fundamental mode, and that the instability in the first harmonic was due primarily to

the driving in the hydrogen ionization region. It should be noted that in their earlier work Baker and Kippenhahn (1962) did not include convection and they found that the hydrogen ionization region was not as important as the second helium ionization region. The difference is primarily due to the fact that convection leads to a more extensive ionization region in the equilibrium model and hence there is more mass involved in the driving. The linear-stability analysis has been applied by Kippenhahn and collaborators (Kippenhahn 1965) also to main-sequence stars having effective temperatures of about 75000K, which is where the cepheid instability strip would cross the main sequence if extrapolated down this far in the H-R diagram. The result is that such main-sequence stars are found to be so very slightly unstable that they are probably stable. Moreover, the pulsation periods of these stars would be about one hour if they were to pulsate. On the other hand, the dwarf cepheids intersect the main sequence at about this point, but their periods are on the order of 90 minutes or more. If these stars obey the usual period-mean density relation ( see equation (1)), then their periods imply masses of about one- third of the corresponding main-sequence masses. Application of the linear-stability analysis to stars in this region of the H-R diagram having masses of only about 0.4 30?© instead of the main-sequence masses which one would expect, showed that these stars were about 100 times more unstable than the corresponding, more mas- sive stars, and that the periods were now about 90 minutes. If the dwarf cepheids are indeed represented by such models, then these considerations would imply considerable mass loss for these stars at earlier stages of evolution, since such small-mass stars would not have had time to evolve to their present positions in the H-R diagram if their masses had been this small during their entire evolution. Considerations of mass loss, however, are outside the scope of the present paper and we do not pursue them further (see, however. Section IV). The linear-stability analysis has also been applied by Baker (1965) to models of RR Lyrae variables. These calculations yield fairly good agreement with the observations, as far as the left (short period) side of the instability region is concerned; the right (long period) side is not well defined theoretically, again because of the increasing importance of convection in the envelope for the cooler stars. One interesting result of Baker's calculations is that

for models with a low helium abundance (say Y = 0.1, where Y is the mass fraction of helium) almost all the excitation arises from hydrogen and first helium ionization, with a nearly negligible contribution from second helium ionization. One reason for this result is that the hydrogen ionization zone is extremely effective as a driving region, principally because the K-mechanism is especially effective here. This fact is a result of the large negative values of s —8 to —12) in the opacity law in this region. Instability was generally found in both the fundamental mode and first overtone. Next, we should mention the "one-zone model" of Baker (1967). This model was investigated for the purpose of isolating and clarify- ing some of the essential physical features of nonadiabatic pulsations of stars. The essential simplification introduced in the one- zone model is the neglect, insofar as is possible, of the spatial dependence of both the coefficients and the pulsation variables them- selves in the linearized pulsation equations. The linearized partial differential equations are thereby reduced from a system of the fourth order in space with spatially variable coefficients to a system of the zeroth order in space with spatially constant coefficients. Hence the system reduces to a set of ordinary differential equations in the time only. The differential equations are of the third order in time, both in the original system and in the reduced, one-zone model, system. The assumption of periodic solutions of the form eiù>t (ω= complex pulsation frequency) then reduces the system to a cubic equation whose solutions for the real and imagin- ary parts of ω give the possible types of temporal behavior for the model. Physically, the one-zone model may be considered approximately as a single, spherical mass shell, throughout which the physical conditions are spatially constant, lying somewhere in a star. The one-zone model has proved quite useful for gaining physical insight into some of the complexities of nonadiabatic stellar pulsations, and has been applied in a number of investigations (e.g., Unno and Kamijo (1966), Gough (1967a), Unno (1967), and Okamoto and Unno (1967) ). Nonlinear oscillations of the one-zone model have been considered by Baker, Moore, and Spiegel (1966, 1967); Moore and Spiegel (1966); and by Usher and Whitney (1968). Nonradial oscillations of the one-zone model have been discussed by Ishizuka (1967) and Zahn (1968).

The final question that should be discussed here, about which significant and useful information has been provided by the linearized calculations, concerns how well we now theoretically understand the famous period-luminosity relation of classical cepheids. As mentioned earlier, this relation has been used for many years by astronomers as a basic empirical yardstick for establishing distance scales both within the Galaxy and for the nearer external galaxies. This question has been discussed by Hofmeister (1967a). It is clear that once a theoretical instability strip (more precisely: a locus of maximum instability; the exact width of the theoretical instability strip cannot now be calculated reliably for reasons discussed elsewhere in this paper) in the H-R diagram has been established on the basis of evolutionary models and stability calculations, one automatically also has a theoretical period-luminosity relation. The stability calculations give the period and corresponding luminosity at which a given evolving stellar model attains maximum instability in a given crossing of the instability region ( the maximally unstable model, if unstable, presumably corresponds approximately to the middle of the theoretical instability strip). The fact that for a certain range of masses (3-5) < 90î/3)îo ^ 9* a given star evidently may cross the instability strip up to five times, each time with a slightly different luminosity and internal structure, introduces some intrinsic scatter into the theoretical relation. Some scatter is also introduced because of the possibility that different cepheids may have different chemical compositions and also that some cepheids may be pulsating in the first overtone rather than in the fundamental mode. (If the mode in which a given star is pulsating were known, then this particular source of scatter could be removed or greatly reduced with the help of the linearized calculations; see Section V, b.) These sources of scatter have been discussed by Hofmeister. She concludes that this intrinsic scatter amounts to only a few tenths of a magnitude (for a given period) and that, all things considered, the agreement between the theoretical and empirical period-luminosity relations is quite satisfactory.

^Recent calculations by Stothers and Chin (1968) suggest, however, that stars having > 9 may also cross the cepheid strip more than once.

IV. The Nonlinear Calculations

It is possible to consider the stellar envelope as a dynamical system and to follow any instabilities which might develop by integrating, in time, the nonlinear partial differential equations which govern its behavior; these equations are merely statements of the conservation of mass, momentum, and energy. The solution of these equations is carried out by writing the differential equations in finite difference form so that the stellar envelope consists of spherical shells of material with a given shell having a constant mass and a single characteristic set of physical parameters (i.e., temperature, pressure, density, etc.) at a given time. All of the forces acting on the interfaces of a given shell and the flow of energy will then determine the values of these physical parameters for each shell after taking a small step in time. Such calculations have been carried out independently by Cox et al. ( 1966 ), Christy ( 1966b) and Stobie (1968). The details involved in writing the difference equations in the appropriate form and in solving them can be found in these papers (see also Cox, Brownlee, and Eilers 1966; Christy 1967a) and will not be discussed here. The work of Christy, Cox, et αι., and Stobie has confirmed the effectiveness of the region of second ionization of helium in the excitation of radial pulsations. The latter two groups have confined their investigations to stars which should closely approximate the Population I, classical cepheids, while Christy has studied primarily models of RR Lyrae stars but has also looked at some models of cepheids. The behavior of a given stellar model (specified mass, equilibrium luminosity, effective temperature, chemical composition) is determined by first constructing an envelope model which is in hydrostatic and radiative equilibrium (convection may also be included in an approximate way in the Cox, et al. computations so that the requirement in that case is that the total flux be constant) throughout the model. Satisfactory equilibrium can be considered to have been achieved when the flux is constant in space to within 0.1 percent and the acceleration of each interface is of the order of 1 X 10_G of the local gravitational acceleration. The boundary conditions to be applied are such that the portion of the star interior to the envelope is treated as an inert core. The

* Reference is made here to only one of his most recent papers. Reference to earlier papers can be found therein.

boundary conditions can be stated in the following way: the interface between the inert stellar core and the outer envelope is held fixed with time, or

hase (ί) = 0 (6)

The inert stellar core serves primarily to determine the value of the local gravity and to provide a source of constant radiative energy which is fed into the envelope and therefore

■Lbase (^) := -Lbase (^)- (Ό

At the outer surface of the envelope the total pressure vanishes:

^surface (0 =0 . (8)

At this boundary the surface luminosity is determined by using the approximate gray-body law for the temperature distribution:

4 T = (3/4)7V (τ + 2/3) 3 (9)

where τ denotes normal optical depth and Te denotes effective temperature, with the luminosity across the surface given by:

2 4 Lsillta,ce — 4πΚ σΤ6 , (10)

where σ is the Stefan-Boltzmann constant. In general there are two distinct kinds of models that have been studied: the "shallow" envelope models (Cox, et ol. (1966); King, Cox, and Eilers ( 1966) ) which have a temperature at the base of approximately one hundred thousand degrees Kelvin and contain from 10"3 to 10-4 of the total stellar mass, and the "deep" envelope models ( Christy ( 1966b ) ; Cox, Eilers, and King ( 1967 ) ; Cox, Cox, Eilers, and King ( 1967a,b) ) which go to a depth of several million degrees and contain roughly one-half of the total mass of the star. The shallow envelope models were designed to isolate only the driving due to second helium ionization, being too coarsely zoned near the surface to handle adequately any effects due to hydrogen ionization. The fact that these models were shallow (the radius of the inert core was approximately 0.8 to 0.9 of the radius of the star) meant that the periods were typically too short since the zero velocity of the inner boundary forced the pulsation amplitude to vanish too near the surface of the star. This led to a period which was similar to a first overtone pulsation period.

The deep envelope models treat not only the driving due to second helium ionization but are zoned so as to treat the hydrogen and first helium ionization regions and a few optically thin zones above as well. These models go sufficiently deep into the interior (radius of the inert core generally < 0.1 of the stellar radius) to insure that the pulsation amplitude is sufficiently small so that the period calculated for the fundamental mode is the same, to within a fraction of a percent, as if the entire star were pulsating. It is of interest to look first at the results obtained using the shallow envelopes where the driving of only one region is present. One Qan approach the study of the pulsational stability of the models in two ways. First, if the radial pulsations are self-excited then an infinitesimal perturbation should be adequate to induce the pulsation amplitude to begin increasing. In order to find out whether or not this expectation is correct, one can follow the time behavior of the model from the equilibrium configuration. Any motions of the mass zones in this model will be determined by the "noise level" established by the accuracy of the numerical calculations. A convenient characteristic of the model for observing the growth of the pulsations is the maximum kinetic energy of the envelope during a pulsation period. The computational noise level provides an equilibrium model which is typically eight to ten orders of magnitude in kinetic energy below that at limiting (i.e., maximum) amplitude for a pulsationally unstable model. It is possible to follow the growth of the pulsations from the noise level up to limiting amplitude and this has been done both for shallow envelopes and for deep envelopes. The results of these studies indicate that indeed the pulsations are self-excited. It is not prac- tical to do this in all cases however if one wants to explore the behavior of many different stellar models, the limitation being that too large an amount of computing time is required. A computing time of several minutes is required to follow one pulsation period and several thousand pulsation periods may be involved in the growth of a single model to limiting amplitude. It is possible to impose some predetermined velocity distribution on the various zone interfaces and in this way to jump to some larger amplitude. This is the second approach, and it has been found to be quite a satisfactory technique for studying a large range of amplitudes for a given model. If the imposed velocity distribution yields an amplitude which is above the natural limiting amplitude of the

model, the pulsations are found to decay down to limiting amplitude. Figure 3 shows an H-R diagram on which are located a number of shallow envelope models calculated by King, et al. ( 1966 ). The filled circles represent models which were found to be pulsationally unstable and the open circles stable models. It can be seen that the cepheid instability strip is qualitatively reproduced by these models. Certain limitations on this agreement which are set by the inherent

log Te

Fig. 3 — Location of the computed shallow classical cepheid envelope models in the Hertzsprung-Russell diagram. The filled circles indicate those envelopes which were unstable; the open circles represent stable models. Numbers beside the plotted points denote model numbers, and have no physical significance. The dashed lines indicate the region in which the classical cepheids are located, according to Kraft ( 1963 ). The cross-hatched area enclosing the unstable models at each luminosity is an estimate of the width of the instability strip as determined by these models.

assumptions present in the calculation of these models will be discussed in Section V. The existence of the cepheid instability strip in the region of the H-R diagram where cepheid variables are actually found can be explained qualitatively in terms of a very simple physical picture due to one of us (Cox 1963, 1967). Although Âis picture includes only the second helium ionization region as a source of driving and does not explicitly require the presence of convection in the cooler models, it is nevertheless of value because it provides a clarification of the basic physics which is involved. The critical parameter is the depth of the driving region below the surface of the star, for a given period and equilibrium luminosity. We have previously stated that in the deep interior of the star the pulsations are essentially determined by the adiabatic relation between temperature and density, while near the surface conditions are expected to be highly nonadiabatic. In these outermost layers the heat content of the material is small and hence the material cannot as readily alter the flow of radiation. This has been expressed in the literature in terms of a "freezing in" ( in space ) of the flux variation. Because of this these outermost layers of the stellar envelope should be approximately neutral insofar as driving or damping of the pulsations is concerned. On the other hand the interior region where the pulsations are approximately adiabatic ( "quasiadiabatic" ) acts to damp the pulsations except in the second helium ionization region. If this ionization region lies too deep in the star the damping above and below this region will tend to cancel out the driving and the star will be stable. If the region lies too near the surface, in the nonadiabatic region, the flux variation is essentially frozen in and the driving is not effective. We see then that there is an optimum depth at which this driving region should occur. If we take a star of a given mass and equilibrium luminosity and consider changing its equilibrium radius (and therefore its effective temperature) so as to move it across the instability strip from higher effective temperatures (smaller radius) to lower effective temperatures ( larger radius ) we can see the effect of changing the depth of the driving region. To the left of the instability strip (i.e., on the high temperature side) the driving region lies too near the surface of the star and so is in the nonadiabatic region. To the right of the strip (i.e., on the low temperature side) the driving region lies too deep and hence the driving is cancelled out by the

damping above and below this region. The fact that the left side of the instability strip has a positive slope in the H-R diagram can be explained on the basis of this picture by noting that the more luminous cepheids have a lower effective gravity in this outer region and hence a lower effective temperature is necessary in order for the driving region to be at the optimum depth. For the models discussed here (i.e., only one driving region and little or no convection) the same explanation holds for the low temperature edge of the instability strip. It has already been pointed out, however, that the right hand edge may actually be determined by the effects of convection and its presumed damping effect on the pulsations (see Section V, d). Thus far we have discussed only the results obtained using the shallow envelope models. It is found that radial pulsations can be self-excited in such models provided that they have a reasonable chemical composition and that they lie near the observed cepheid instability region in the H-R diagram. The deep envelope calculations provide a number of interesting results. As pointed out earlier these models include a sufficient amount of the stellar mass so that the period can be well established and hence one very important link with the observations is provided. In most cases attempts have been made in these models also to treat the hydrogen plus first helium ionization region in sufficient detail to answer the questions of whether or not this region provides a significant driving for the pulsations and of what effect this region has on the phase relations between velocity and luminosity (see Section V, c). Because the temperature decreases very rapidly with increasing radial distance as one passes out through this ionization region (see Section V, c), it is necessary, from a computational point of view, to include zones which will, during at least part of the pulsation cycle, be optically thin. One must be careful in interpreting any results which depend sensitively on the behavior of these optically thin mass zones since the diffusion approximation for describing the flow of radiative energy is not strictly valid in such regions. ( However, Castor ( 1966 ) has shown that the diffusion approximation actually gives a good approximation to the correct radiative flux under the conditions of interest. ) The most extensive published work using deep envelopes has been carried out by Christy (1966a) and deals primarily with models of RR Lyrae stars. He has obtained a number of interesting

results and mention will be made in a later section of those which the writers consider most significant. Here we shall discuss only the importance of the combined hydrogen and first helium ionization region in driving the pulsations. Christy's computations have been carried out with various combinations of mass, equilibrium luminosity, and effective temperature, and for several different assumed chemical compositions. The primary concern in this study was to determine which combinations led to the best fit with the observations, particularly the observed periods. The equilibrium

luminosity which.he chose for most of his models was L0 = 1.5 Χ 103Γ> ergs/sec, corresponding to a bolometric magnitude of

Mboi,(, = +0^76. The heavy element fractional mass abundance was chosen to be Ζ = 0.002 and the stellar mass, effective temperature, and helium abundance were then varied to determine the best fit with the observations. Christy concludes that the best fit is obtained for a helium abundance of Y = 0.30 and a stellar mass of ^ = 0.5¾¾. This mass, if correct, is lower than previous estimates but is not at variance with any present-day observations of RR Lyrae variable stars. The fact that this conclusion may imply mass loss in the past history (and even at present) of these stars is also borne out by the fact that in some of Christy's models there is evidence of large enough velocities in the outer regions to expel matter from the surface of the star. Faulkner and Iben (1966) have found, on the basis of evolutionary model calculations, that it is possible (with no mass loss) to explain the turn-off points of globular cluster color-magnitude diagrams best if one assumes for the horizontal branch stars a mass of from 0.65 to 0.75 and a hydrogen fractional mass abundance of X = 0.65. These numbers would certainly indicate that the preferred RR Lyrae models (X = 0.70 and SR = 0.5 Wo) obtained by Christy are reasonable. For the above-mentioned RR Lyrae models Christy finds that in general more than half of the excitation of the pulsations is due to the region of second helium ionization and the remainder to the combined region of first helium and hydrogen ionization. This result is illustrated in Figure 4, which is a plot of the work, Wi, that the ith mass shell does on its surroundings during a complete period, versus the zone number i, for one of Christy's unstable RR Lyrae models. W-i is a measure of the contribution of the ith mass shell to the total excitation, and is positive in driving regions, nega-

Wj 0

15 10 15 20 25 30 35 3Θ ZONE th Fig. 4 — The work Wi (on an arbitrary scale) done by the ¿ mass shell on its surroundings around a complete period (see eq. (11)) versus i for one of Christy's (1966a) unstable RR Lyrae models at limiting amplitude. (Para- meter values for this model are as follows: Y = 0.30, Ζ = 0.002, L = 1.50 χ 11 10'« erg/sec, W = 1.15 X 10^ gm, ñ = 3.42 χ 10 cm, Te = 6500° K. ) Positive values of W· imply driving, negative values damping.

tive in damping regions. It can be shown (see, e.g., Cox 1967) that W, is given by the cyclic integral

Wi = § PidVi , (11)

where Pi and Vi are the total pressure and volume, respectively, of the ith mass shell, and the integral is to be evaluated over a complete pulsation period. Christy also finds that there is a relationship, for a given mass- luminosity combination, between the effective temperature and the excitation of pulsations in either the fundamental mode or the first overtone mode. This will be discussed in the next section but is mentioned here since the particular mode which is excited appears to depend on the relative importance of these two driving regions. An investigation of the pulsation properties of classical cepheid models, using deep envelopes, is at present under way by Cox, et al, (1967a, b). This investigation is based on the evolutionary

models of a number of investigators, particularly those of masses = 5, 7, and 9 already studied in the linear theory by Baker and Kippenhahn (1965) and by Hofmeister (1967a) with the composition X = 0.602 and Ζ = 0.044, and includes studies of both very small (essentially linear) and large (nearly limiting) pulsation amplitudes. Some preliminary results of this investigation will be presented and discussed briefly in Section V, b. We note in passing, however, that (see Fig. 6) the left (short period) edge of the instability strip computed in this investigation lies close to the corresponding edge of the observed strip (but somewhat to the right of the corresponding edge obtained by Baker and Kippenhahn ( 1965 ) ; this last effect is mostly a result of^ the fact that the more recent calculations include the contributions of lines to the opacities, whereas Baker and Kippenhahns calculations did not). The right (long period) edge of the instability strip has not been studied in detail because of the unknown effects of convection in the envelope (see Section V, d). Somewhat similar investigations are also being undertaken by Christy (1967b) and by Stobie (1968).

V. Special Problems a) Limiting Amplitude. Cepheids and RR Lyrae variables are normally observed to be pulsating with relatively constant amplitudes (in time). We have found that a mechanism is available for exciting the pulsations at very low amplitudes but have not discussed the growth of the pulsation amplitude and its eventual limit at some constant value. We have indicated earlier that computations carried out in the early 1950^ led to the result that the decay of pulsations due to radiative damping, expressed in terms of the time required for the velocity amplitude to decay by a factor of e, were of the order of a few years or less. It turns out that with the excitation mechanisms that have been described, the growth, again in terms of velocity e-folding time, is of the order of one hundred pulsation periods. This is very different from the time of several thousand years obtained by Eddington, and is again a result of the much greater mass concentration now thought to be present in supergiant stars than was previously believed to be the case. Nonlinear pulsational models have been followed from the onset of pulsations up to limiting amplitude, both in the case of shallow

45 to 44 10 GROWTH OF PULSATIONS IN BK MODEL 7 .43

33 10 0 100 200 300 400 500

t/n0 Fig. 5 — Maximum kinetic energy of the entire envelope of BK Model 7 (parameter values given in the footnote near the beginning of Section V, a) versus time in units of the fundamental period. The switch-over from the first overtone at smaller amplitudes to the fundamental mode at nearly limiting amplitude is shown by the vertical line and the kink in the growth curve. In this particular case the time required for the switch-over to occur was artificially reduced to only a few periods by the introduction of a perturbation into the model when symptoms showed that a switch-over was about to occur.

envelopes (Cox, et al. 1966) and of the more realistic deep envelopes (Cox, et al. 1967; Cox, et al. 1967a, b). Figure 5 indicates such a growth for one of the deep envelopes in terms of the maximum kinetic energy of the stellar envelope. In this model# the limiting amplitude was attained in the fundamental mode, although during most of the growth the model pulsated in the first overtone ( see Section V, b ). Both Christy ( 1966a ) and King ( 1967 ) have investigated the approach to limiting amplitude in terms of the relevant physical parameters. It is found that the

0This model is Model 7 of Baker and Kippenhahn (1965), hereafter occasionally referred to as "BK Model 7." It is characterized by the following parameter values: X = 0.602, Ζ = 0.044, m/mG = 7, logm (L/L0) = 3.663, \ogwTe = 3.726, fí/fí© = 80.5, Π() (fundamental period) = 11¾ 1^ (first overtone period ) = 8^26 ( these are the periods found in the nonlinear calculations of Cox, et al. 1967; Cox, et al. 1967a).

amplitude becomes limited in part as a result of a saturation of the driving mechanisms. This saturation is most easily understood in terms of the /c-mechanism. As was noted earlier (see Fig. 1) we find that in a driving zone the opacity maximum comes, in time, near the instant of maximum compression. It is found that as the pulsation amplitude increases the oscillation becomes so large that the maximum in the opacity no longer coincides with minimum radius in certain of the driving mass zones, but comes slightly later in the cycle. This behavior is a result of certain peculiarities of the temperature, density dependence of the opacity in the second helium ionization region. This lag in the opacity maximum destroys the ability of the zone to drive as efficiently as during the growth of the pulsations and the amplitude becomes limited. There is also the question of a nonlinear increase in the rate of dissipation as the amplitude grows and its effect on the final amplitude. It appears that in the case of the RR Lyrae variables, where shocks may be important at the larger amplitudes, such an increase may play an important role in determining the precise value of the limiting amplitude. The models thus far computed do not provide much evidence for such an effect in the classical cepheids and it appears that the saturation of the driving mechanism may be the most significant factor in limiting the pulsation amplitude. It should be noted that the value of the semi-radius

amplitude (i.e., (¾) 8R/R(h where 8R is the total radial excursion) is quantitatively similar to a mean observed value for classical cepheids of approximately 0.05-0.07 and a slightly higher value for RR Lyrae variables. Very few models of W Virginis variables have been calculated at the present time so that it is not possible to make direct comparisons in this case. It should be pointed out that the presence of a corona in such stars, with progressive waves propagating into it, would lead to some dissipation and would therefore presumably cause the amplitude to become limited at a smaller value than the current models indicate. Unno ( 1965 ) has studied the effect of such waves on pulsational stability using a linearized theory. He finds that with reasonable assumptions for the structure of such a corona (i.e., not too dissimilar from the solar corona ) such dissipation would be relatively unimportant for fundamental and first overtone pulsations. He also comments, how-

ever, that the dissipative effects of a corona would probably be more important if the fully nonlinear pulsations could be analyzed. The presence of convection in one or both of the driving regions could also be a factor in limiting the pulsation amplitude since any flux carried by convection will not be directly affected by the opacity variations. The indication then is that the calculated values of the limiting amplitude of these variable stars are expected to be somewhat larger than the observed values. This appears to be the case, at least in the calculations of Cox, et al. (1967a).

b) Fundamental Versus First Overtone Pulsations. The shallow envelope models of classical cepheids which include only the excitation due to second helium ionization give no indication of first overtone pulsations. On the other hand the calculations of Christy ( 1966a) and of Cox, et al. ( 1967a, b), for both RR Lyrae and cepheid models, show that the excitation of the first overtone can be quite important for certain models. Christy found that for his RR Lyrae models there was a relationship between the effective temperature of the model ( for a given combination of mass, equilibrium luminosity, and chemical composition) and the tenden- cy for either the fundamental or first overtone pulsations to be excited. As the effective temperature of the stellar model is lowered from a high value (i.e., starting near the left edge of the instability region), the models tend to change over from the first overtone mode to the fundamental mode of pulsation. There is a narrow range of equilibrium effective temperature for which both modes appear to be equally excited. Christy has referred to this as the transition effective temperature and has identified the high temperature, first overtone, pulsators with the short period Bailey "type c" RR Lyrae variables. The calculation of models with different equilibrium luminosities has led Christy to propose a theoretical period-luminosity relation for RR Lyrae variables. This relation can be written in terms of the previously-mentioned transition effective temperature, with the

period, nir, corresponding to this effective temperature. The relationship is 0 nir-0.057 L ·" (days) , (12)

where L, the equilibrium (or mean) luminosity, is in units of the solar luminosity. This relationship, if correct, could provide an

important means for determining the absolute magnitudes of RR Lyrae variables. It should be pointed out that Christy has not in general carried his computations, for a given model, from low amplitudes up to limiting amplitude, but has imposed either a fundamental or first overtone velocity distribution at some amplitude and followed the subsequent growth or decay of that particular oscillatory mode. This technique presupposes that if the imposed pulsational mode is physically incorrect for the model in question it will decay in a time sufficiently short to be detected after a few pulsation periods. Christy (1966c) has also suggested that the same period-luminosity relation might hold for cepheids as well as for RR Lyrae variables. The calculations presently being carried out by Cox, et al. (1967a, b), based on evolutionary stellar models (see Section IV), yield some interesting results which bear on the present discussion. Some preliminary results of the calculations are shown in Figure 6, which is a theoretical H-R diagram showing the location of the observed cepheid strip (dashed parallel lines, based on the data given by Kraft ( 1963 ) ) and portions of the evolutionary tracks (solid lines) for the models computed by Baker and Kippenhahn (1965) and by Hofmeister (1967a). The dash-dot line on the right (labeled "envelope convection (He+)") is the line to the right of which convection carries more than 50 percent of the flux in the He+ ionization region. The small amplitude investigations were carried out to test for self-excitation. In these calculations the mode in which a model is self-excited is easily determined by following a model which was initially very nearly in hydrostatic and thermal equilibrium on the computer in time (as described in Section IV). Some of the results of the small amplitude investigation are indicated in Figure 6 by the large symbols ( square or circle ) : Self-excitation in the fundamental or first overtone is indicated, respectively, by a large square or large circle, whereas stability is indicated by a cross. It is seen that in most cases investigated the model was self-excited in the first overtone rather than in the fundamental. In fact, it is interesting to note that the mode (either fundamental or first overtone) which was self-excited in these calculations was the same mode which Baker and Kippenhahn ( 1965 ) found to be the more unstable in the linear theory. This agreement provides a very nice link between the linear and nonlinear calculations.

Π ^ 1 ^ \ 1 -7 DEEP ENVELOPE MODELS ( Menv = 0.5 M, Rcore <0.1 R ; X = 0.602, Ζ « 0.044 ) / I OBSERVED I CLASSICAL I CEPHEIDS -6 / I HOFMEISTER I ! (M«9Mq) /177-178 /

-5

M bol / * KIPPENHAHN ΓΝΓ^ΤΜ-) -4 // / / ENVELOPE . / ^ CONVECTION ( He )

/ / HOFMEISTER -3 /122 / (M«5M^) / / / SELF-EXCITED / FUNDAMENTAL □ -2 / I st. HARMONIC O LIMITING AMR FUNDAMENTAL ° I St. HARMONIC ° STABLE X -I J ^^ ι— 3.9 3.8 3.7 3.6 log Te

Fig.. 6 — Theoretical Hertzspmng-Russell diagram, showing the location of the observed cepheid strip (dashed parallel lines, from Kraft (1963)), portions of the evolutionary tracks computed by Baker and Kippenhahn (1965) and by Hofmeister (1967a), and the line (dash-dot, labeled "envelope convection ( He+ ) " ) to the right of which convection carries more than 50 percent of the flux in the He+ ionization zone. Large symbols (square or circle) indicate some of the results of the small amplitude investigation (Cox, et al. 1967a, b) (see text for further details), small symbols (square or circle) indicate some of the results of the limiting amplitude investigation, and a cross indicates stability. The unlabeled numbers alongside the plotted points are the model numbers as given by Baker and Kippenhahn (1965) and by Hofmeister (1967a).

The growth of pulsations has been followed from very small amplitudes up to an amplitude where there is a changeover from the first overtone to the fundamental mode (see Fig. 5). The amplitude continues to increase until these fundamental mode pulsations become limited. It is found that if one imposes the physically incorrect pulsational mode, there will eventually be a changeover to the correct mode. This transition from one mode to another may, however, take from 50 to 100 pulsation periods, so that with these techniques extensive computational time is required. These times are still so short, however, that the probability of observing a star at an amplitude other than limiting amplitude is negligibly small. Some of the limiting amplitude results of this investigation are also shown in Figure 6 by the small symbols (square or circle): Pulsations which at limiting amplitude are in the fundamental or first overtone are indicated by, respectively, small squares or circles. It is seen that, in every case investigated so far (a total of four), the pulsations at limiting amplitude are in the fundamental mode, even though the model may have been self-excited in the first overtone. On the basis of the models thus far computed, then, there is reason to believe that the observed classical cepheids, corresponding to these models, are pulsating in the fundamental mode. This conclusion also agrees with the analysis of Hofmeister ( 1967b ) who finds, through a study of evolutionary models, that the observed period-luminosity law is best fit by her models if one assumes that the classical cepheids are all fundamental pulsators. It should be cautioned, however, that this conclusion may not actually be correct for cepheids of all periods. For example, cepheid models having fundamental periods less than about 3-5 days have not yet been adequately investigated in nonlinear calculations; it is possible that such short-period cepheids might pulsate in, say, the first overtone rather than the fundamental at limiting amplitude or, as has been suggested by Stobie ( 1968 ), perhaps even in the second overtone. Also, Femie (1968) has argued that the scatter in the empirical period-radius relation for classical cepheids can be considerably reduced if it is assumed that certain cepheids are pulsating in the first overtone. These particular cepheids are not confined to any specific period range. He suggests on these grounds that one cepheid, U Carinae, may even be pulsating in

almost independent of the location of a star within the instability strip. The linearized calculations of Baker and Kippenhahn (1962, 1965) and of Cox ( 1963) yield a phase lag due to the second helium ionization region. In the calculations of Cox, where only this ionization is considered, the phase lag essentially disappears in most models by the time the surface of the star is reached. On the other hand the combined hydrogen and first helium ionization region, which is included in the calculations of Baker and Kippenhahn, leads to a phase lag of nearly 180° (i.e., maximum luminosity occurs at approximately the time when the star is near maximum radius) in these calculations. The computed phase lag in the linear theory appears to be somewhat sensitive to the structure of the outermost layers of the star. In particular, the presence of a considerable amount of convection in the hydrogen ionization zone in Baker and Kippeñhahns static models may have caused their computed lags to be somewhat too large. This conclusion is based on recent unpublished calculations by Baker ( 1968 ) and Castor ( 1968a ) in which the linear, nonadiabatic theory was applied to a model similar to Model 7 of Baker and Kippenhahn (1965), except that convection was entirely suppressed in the more recent calculations; in these calculations the phase lag turned out to be near 90°. The only calculations which have consistently reproduced the correct phase lag are the recent nonlinear calculations of Christy (1966a), Castor (1966), Cox, et al. (1967) and Cox, et al. (1967a,b). The results of the calculations of Cox, et al. (1967a) are indicated in Figure 7, where the radial velocity (astronomical sign convention ) of the outer region of the star ( where the absorption lines are formed) and the bolometric magnitude of the star are both plotted as functions of time. These plots should correspond qualitatively to the observed relation between light and velocity variations, and are seen to have very nearly the correct (i.e., observed) phase relations. It will also be noted that the light curve exhibits structure in the form of a secondary maximum. This structure is well correlated with changes in the velocity curve (see Fig. 7). Some possible implications of this structure will be mentioned later in this subsection. A simplified physical picture of the cause of the phase lag in cepheid and RR Lyrae type pulsating stars has recently been ad-

almost independent of the location of a star within the instability strip. The linearized calculations of Baker and Kippenhahn ( 1962, 1965) and of Cox ( 1963) yield a phase lag due to the second helium ionization region. In the calculations of Cox, where only this ionization is considered, the phase lag essentially disappears in most models by the time the surface of the star is reached. On the other hand the combined hydrogen and first helium ionization region, which is included in the calculations of Baker and Kippenhahn, leads to a phase lag of nearly 180° (i.e., maximum luminosity occurs at approximately the time when the star is near maximum radius) in these calculations. The computed phase lag in the linear theory appears to be somewhat sensitive to the structure of the outermost layers of the star. In particular, the presence of a considerable amount of convection in the hydrogen ionization zone in Baker and Kippenhahn s static models may have caused their computed lags to be somewhat too large. This conclusion is based on recent unpublished calculations by Baker (1968) and Castor ( 1968a ) in which the linear, nonadiabatic theory was applied to a model similar to Model 7 of Baker and Kippenhahn ( 1965 ), except that convection was entirely suppressed in the more recent calculations; in these calculations the phase lag turned out to be near 90°. The only calculations which have consistently reproduced the correct phase lag are the recent nonlinear calculations of Christy (1966a), Castor (1966), Cox, et al. (1967) and Cox, et al. (1967a,b). The results of the calculations of Cox, et al. (1967a) are indicated in Figure 7, where the radial velocity (astronomical sign convention) of the outer region of the star (where the absorption lines are formed) and the bolometric magnitude of the star are both plotted as functions of time. These plots should correspond qualitatively to the observed relation between light and velocity variations, and are seen to have very nearly the correct (i.e., observed) phase relations. It will also be noted that the light curve exhibits structure in the form of a secondary maximum. This structure is well correlated with changes in the velocity curve (see Fig. 7). Some possible implications of this structure will be mentioned later in this subsection. A simplified physical picture of the cause of the phase lag in cepheid and RR Lyrae type pulsating stars has recently been ad-

Fig. 7 — Bolometric magnitude (evaluated at optical depth 2/3) and radial velocity (astronomical sign convention) of the outermost zone versus time near limiting amplitude for BK Model 7 ( parameter values given in the footnote near the beginning of Section V, a). The vertical line indicates the instant of minimum radius.

vanced by Castor (1968b), based on experience with both the linear and the nonlinear calculations. According to this picture the phase lag is a consequence of the "sweeping" of the hydrogen (H )-ionization zone back and forth through mass as the star pulsates. If it is assumed that the material within and above the H-ionization zone always remains in radiative equilibrium (i.e., no convection) at all instants during the motion, then the luminosity, Lo, outside this zone will be purely radiative and will, moreover, be practically constant in space above this zone because of the very low heat capacity of these extreme outermost layers. For stars of the kind to which the picture applies (say with effective temperatures ^ 7000° K) the opacity in these regions is a strongly increasing function of temperature. This behavior of the opacity causes an extremely sharp rise in the temperature as one approaches the H-ionization zone ( characterized by a temperature near 10,000° K) from above. Once levels have been reached where essentially all the H is ionized the temperature gradient becomes much less steep again, but the net result is that the temperature profile is almost a step function, increasing from some 8000° Κ just outside the zone to some 15,000° Κ just inside the zone; this increase in temperature may occur in only about 1/20 of a pressure scale height. Thus, the H-ionization zone is extremely thin under these conditions, and the sharp temperature increase may even be treated approximately as a discontinuity, or "front." Castors original discussion, in fact, is based on such a front description, in which Rankine-Hugoniot type relations are applied across the front. The amount of mass, say Am, instanteously lying above the H-ionization zone can then easily be calculated from the theory of radiative

transfer. The result is that L0 (which is also the emergent luminosity) varies inversely as some power of Am and inversely as

another power of the "effective" gravity ge (the ordinary gravity plus the acceleration of the surface regions ) :

Lo cc 1 (13) {ge)'{Am)ß

where the exact values of a and β are determined by the temperature and density dependences of the opacity in the relevant regions. Typical values are a ^ 1/6, β ^ 1/2.

The first factor on the right side of (13) gives the effect of compression of these outermost regions, through the opacity variations: ge is largest approximately when the star is smallest, the opacity of these outer layers is then largest, and the escape of radiation is then impeded. The result is that this factor tends to make Lo smallest at minimum radius, i.e., to produce a phase lag in Lo of 180° behind minimum radius. The second factor gives the effect of the proximity of the H-ionization zone to the stellar photosphere: the star tends to be most luminous when the amount of mass lying above this zone is smallest. The phasing of Am is

determined by the rate at which the luminosity difference Li—L0 between the luminosities immediately inside ( Li ) and immediately outside (Lo) the H-ionization zone can ionize the neutral H if the zone is moving outward, or deionize the ionized H if the zone is moving inward.

The phase of L0 relative to minimum radius is thus seen to depend to a large extent on the phasing of Am and on the total amount of mass over which the H-ionization zone "sweeps" during a period. The phase and amplitude variations of Am for a given radius amplitude are determined, in turn, by the amount of mass, say Δττίο, lying above the H-ionization zone in the equilibrium state of the star, for a given period and equilibrium luminosity.

If Am0 is "too large," then the H-ionization zone lies so deep in the star that the relative variations of Am during a period are small and can be neglected. According to equation (13), then, in this case Lo is essentially "uncoupled" from Li and is smallest approximately when ge is largest, i.e., when the star is smallest. The phase

lag is then about 180°. When Am0 is this large, however, the star is probably stable and, moreover, convection within and above the H-ionization zone is probably important.

On the other hand, if Am0 is "too small," then the variations of Am are relatively large, but then the amount of heat capacity in the layers within and above the H-ionization zone is so small that this zone is not capable of introducing a significant difference

between L¿ and Lo; L0 is then equal in phase and amplitude to Li (the flux variations are "frozen in"). If we neglect the small phase lag in Li due to the He+ ionization zone and assume that Li is in phase with the compression, then it is clear in this case that Lo

will be largest when the star is smallest, i.e., L0 will have a zero phase lag behind minimum radius. Stars having such small values

of Amo, however, lie considerably to the left (high temperature) side of the instability strip, and, accordingly, are probably stable. ,, Finally, if Δπι0 has an "intermediate value, then the relative variations of Am are appreciable and yet the layers within and above the H-ionization zone possess sufficient heat capacity that a

significant difference between Li and L() can be introduced by this zone. It can be shown that in this "intermediate" case the rate at which the H-ionization zone is moving through mass is very nearly in phase with this zone is moving outward through mass fastest when Li is largest. It follows, then, that Am will be smallest approximately a quarter period after maximum Li, i.e., after mini-

mum radius. If the variations in ge are neglected, then it is clear

from equation (13) that L() will attain maximum approximately a

quarter period after minimum radius; i.e., the phase lag of L() behind minimum radius will be about 90°, as observed. It is a remarkable fact that, as is shown in Castor s analysis, this "intermediate" case happens to apply to stars in the instability strip (i.e., cepheids and RR Lyrae variables), which is approximately the locus of stars which are maximally unstable mostly as a result of the driving by second helium ionization. It is presumably for this reason that unstable stars of this kind possess the 90o-phase lag. Also, the existence of the phase lag implies that a significant fraction of the driving must be due to hydrogen ionization, since if there is enough heat capacity in this zone to introduce a signifi-

cant difference between and L0, then this zone should contribute a significant amount of driving. This conclusion is in agreement with the calculations of Christy ( 1966a ) and of Cox, et al. ( 1967 ) and Cox, et al. (1967a,b). Nonlinear effects must evidently be invoked to explain the observed near constancy of the phase lag, i.e., the fact that it is almost independent of the location of a star within the instability strip. Castor ( 1968b ) has also presented arguments ( which will not be described here) which make it understandable that these nonlinear effects indeed act in such a way as to make the phase lag relatively insensitive to stellar parameters for stars in the instability strip; more precisely, to make the luminosity remain approximately in phase with the outward velocity, as observations and calculations show is the case. This picture of the cause of the phase lag is somewhat over- simplified and is still somewhat tentative. It is, nevertheless, in

qualitative agreement with the results of detailed calculations. Moreover, it predicts that pulsating stars which are too hot to have H-ionization zones should exhibit essentially a zero phase lag. This prediction agrees with observations of the β Cephei stars (Ledoux and Walraven (1958), pp. 398 ff. ), whose effective temperatures are in the range 20,000-25,000° K. We conclude this subsection by mentioning briefly some possible implications of the existence and properties of the secondary maximum which shows up in the calculated light curves of some cepheids (see Fig. 7). It has recently been suggested by Christy (1967c) that the secondary maximum is a result of a shock wave generated in the outer layers traveling down to the core of the star and being reflected back to the surface layers. Christy finds on the basis of this interpretation that in order for the secondary maximum to appear at the proper phase (as compared with the observations) for a variable of a given period, masses for cepheids of only about half the values inferred from evolutionary model considerations are implied. Stobie ( 1968 ) also appears to have arrived at the conclusion that smaller cepheid masses are required to account for the phase of the secondary maximum than would be implied by the evolutionary calculations, although he does not suggest a physical reason for this result. It should be pointed out, however, that the exact details of the structure in the light curve are quite sensitive to the precise details of the calculations in these outer, optically-thin layers ( the velocity curves appear to be less affected ). One of us ( King ) is at present investigating an alternative interpretation of the secondary maximum.

d) Effects of Convection on Pulsational Stability. As stated in an earlier section, the effects of convection on pulsational stability and on other pulsation characteristics of stars are not yet reliably known because of the absence of a definitive theory of time-dependent convective transfer. Some exploratory steps are, however, being made at present in an attempt to answer some of these questions, and the purpose of this subsection is to review briefly some of this work. To the best of our knowledge, there have been three serious recent attmpts to produce theories of time-dependent convection for use in astrophysics. These are due to Cough (1967a, b), Unno (1967), and Castor (1968c). Cough's theory is based on a generali-

zation of the Vitense (1953) and Böhm-Vitense (1958) mixing- length theory to include effects resulting from an arbitrary variation in time of the stellar properties in a convective region. Unno's theory is also based on a generalization of the mixing-length theory, but the basic approach and some of the basic assumptions differ from those of Gough. Castor s theory is based on an application of a Boltzmann transport-type equation to the convective elements, somewhat along the lines indicated by Spiegel's (1963) formula- tion. Castor s theory includes not only time dependence, but also allows for arbitrary spatial variations of physical quantities, such as pressure and temperature, in a convective region on scales which may be small compared with a mixing length. Gough's theory, in linearized form, has been applied to the Baker one-zone model (Gough 1967a) and also in the quasi-adiabatic approximation to distributed envelope models similar to the ones investigated by Baker and Kippenhahn (1965) (Baker and Gough 1967). Both of these investigations suggest that convection has the desired stabilizing effect on cepheid models lying to the right (long-period side) of the instability strip. (On the other hand, the unpublished linearized calculations carried out by Baker (1968) and Castor ( 1968a ), referred to in Section V, c, suggest that too much convection in the model might lead to too large a phase lag; it is not yet clear whether this discrepancy constitutes a real difficulty.) The one-zone model analysis of Gough (1967c) suggests, moreover, that for sufficiently long periods (perhaps in the region of the long period variables) the convection may have a destabilizing effect; but this last conclusion is quite tentative, and quantitative confirmation will have to await application of time- dependent convection theories to detailed models of these stars.

VI. Concluding Remarks

It is evident that quite considerable advancements in our understanding of the causes and nature of stellar pulsation, at least as applied to classical cepheids, RR Lyrae variables, and W Virginis variables, have been made in recent years. It is only a little more than a decade ago that answers were not available to such basic questions as: What is the physical mechanism responsible for stellar pulsations? Why are some stars variable while others are not? What is the underlying physical basis of the famous period- luminosity relation? What is the physical cause of the well-known

"phase lag discrepancy>,? We now appear to have reasonably firm (but not necessarily complete or definitive) answers to all of these questions, as well as to certain others. Some of the outstanding problems remaining in pulsation theory at present concern the interaction between pulsations and convection; the cause of the instability and the nature of the pulsations of the Long Period (Mira) variables and of other types of red variables, the β Cephei variables, and various other classes of pulsating stars such as the RV Tauri variables; the detailed interpretation of the intricacies of the light and radial velocity variations of all the pulsating variables; the interpretation of the magnetic and spectrum variables; and the role played by nonradial oscillations of stars. Whether the interpretation of the eruptive and nova-like variables, such as the dwarf, recurrent, and ordinary novae, falls within the scope of what is ordinarily conceived of as pulsation theory remains to be seen.

Acknowledgments

We are grateful to A. N. Cox, D. D. Eilers, D. O. Gough, N. Baker, R. F. Christy, and J. Castor for useful discussions. We are also indebted to D. O. Gough, N. Baker, J. Castor, and R. S. Stobie for separately communicating to us some of their unpublished results. The work involved in the preparation of this paper was supported in part by NSF Grant number GP-6374 through the University of New Mexico and in part by NSF Grant number GP- 6428 through the University of Colorado. The nonlinear calculations carried out at the Los Alamos Scientific Laboratory and described in this paper were supported by the Atomic Energy Commission.

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