A program to obtain photometric and spectroscopic (high and medium resolution) observations of a number of southern RV Tauri stars has been undertaken over the past two years at the Mount John University Observatory (MJUO). Eleven RV Tauri stars of both RVa (constant mean magnitude) and RVb (varying mean magnitude) photometric type have been chosen as well as normal and weak metal lined RV Tauri stars.

Most program stars display the alternating deep and shallow semi-regular light variations as well as the light curve - colour curve phase lag characteristic of RV Tauri stars. Fourier analyses of the light curves have revealed the dominant periodicities (see figure 1) and allowed phasing of the spectroscopic observations.

High resolution échelle spectra obtained of these stars around the Hα region display the complex emission and absorption structure of the Hα line at various phases. Metallic lines show emission and line doubling or ‘splitting’ - profiles characteristic of the shock wave that propagates through the line-formation regions of these stars during a pulsational cycle. Spectra at specific phases will be used in an abundance analysis of selected RV Tauri stars.

The blue edge of the instability region of radially pulsating helium stars is located on the H-R diagram nearly vertically at luminosities less than 104L⊙ and corresponds to Teff = 7400K and 1.1 · 104K for stars with mass of 1M⊙ and 0.7M⊙, respectively. The pulsation instability of the stars located along the vertical part of the instability edge is characterised by the increasing order of the pulsation mode with decreasing luminosity. For example, for stars with M = 0.7M⊙ and Tteff = 104K the principal pulsation mode is nearly second overtone for Mbol = –5 mag and is nearly fourth overtone for Mbol = –4 mag. For stars brighter than Mbol = –5 mag the blue edge turns blueward so that the models with Teff as high as 3 · 104K are pulsationally unstable. The sequences of the hydrodynamic models characterized by constant luminosity Mbol < – 5 mag) reveal the decrease of the light amplitude with increasing Teff, whereas the amplitude of the radial velocities of the outer layers is almost independent of Teff. For example, for stars with Teff > 2·104K the light amplitude is less than 0.01 mag, whereas the radial velocity amplitude is in the range from 20 to 50 km/s. The pulsation instability of these stars is driven mainly due to the γ-mechanism. Fourier analysis of the hydrodynamic solution shows that the pulsation motions of the hot helium stars can be represented as a superposition of the running waves (in contrast to Classical cepheids where the pulsation motions are described in the terms of superposition of standing waves). The pulsation constant gradually decreases with increasing Teff, down to Q ≈ 0.012 day at Teff ≈ 3 · 104K.

We have continued to improve and update the OPAL opacity code. Addition of intermediate coupling has further increased the opacity over earlier LS coupling results. A ‘corresponding states’ method has been used to extend the tables in both X and Z. This has allowed the calculation and distribution of extensive opacity tables for several different sets of metal abundance.

The paper gives a brief summary of the work of the Opacity Project.

The beat and bump mass discrepancies have been a long standing, unsolved difficulty in the Cepheid modelling (Cox 1980). The new opacities of Rogers & Iglesias (1992) has provided a partial solution to the problem, bringing the beat masses into good agreement with other mass determinations (Moskalik, Buchler & Marom 1992; herafter MBM). The discrepancy for the bump masses has also been greatly reduced, nevertheless it has not been eliminated entirely.

Recently, the new version of the Livermore tables has been released (Iglesias, Rogers & Wilson 1992). The new calculations take into account previously disregarded spin-orbit coupling in the iron atoms. The inclusion of this effect leeds to a further enhancement of the “metal opacity bump” around 2-5xl05K. In the following we repeat the calculations of MBM to assess the consequences of this enhancement for the bump mass calibration.

The OPAL opacities (Iglesias and Rogers 1991; Iglesias, Rogers and Wilson 1992) are recent revisions to the traditional Los Alamos (LA) opacities and have proven very successful in resolving the Cepheid beat mass, and to a lesser extent, the Cepheid bump mass discrepancies (Moskalik, Buchler and Marom 1991, hereafter MBM). MBM showed that for the beat Cepheids the P1/P0 period ratios, calculated with OPAL, yielded solar masses between 4 and 7, in agreement with other mass determinations. MBM also found that the P2/P0 period ratios were reduced, implying higher bump Cepheid masses, though the conflict with standard evolutionary masses was not completely eliminated. We have made linear nonadiabatic pulsation calculations with another set of recently computed opacities, namely those of the Opacity Project (Seaton 1987, 1992, hereafter OP). Since the two teams have employed different techniques in their calculations, it is of interest to compare pulsation results using the two sets of opacities.

The anomalously low theoretical masses for double-mode RR Lyrae variable stars obtained from accurately observed period ratios recently have been increased by using stellar models constructed with the new Livermore OPAL opacities. These new models appear to the lowest order radial modes to be less concentrated in density, and they thus produce longer periods. Since the fundamental mode is increased in period more than the first overtone, this period ratio from the model is decreased. Cox (1991) showed that now these pulsation-based masses agree with evolution masses, and all RR Lyrae variable stars show little mass loss from the red giant tip in earlier evolution. This investigation has been validated by two additional papers that use more directly the OPAL opacities rather than simple adjustments to the Stellingwerf (1975ab) fit. These new papers (Kovacs, Buchler, and Marom, 1991, and Kovacs, Buchler, Marom, Iglesias, and Rogers, 1992) also display composition effects for Population II mixtures.

My new study uses directly the latest OPAL intermediate coupling opacities for the Oosterhoff I and II (Z=0.0003 and 0.0001, respectively) compositions in frozen-in convection (mixing length/pressure scale height ratio=1.25) models to see if there is any difference between the masses of variable stars in these two globular cluster classes. Derived masses are just over 0.65 and 0.75 M⊙, respectively. The Oosterhoff I mass could be as large as the Oosterhoff II mass only if the iron abundance in the Z mixture is increased by more than 30% over that for the solar mixture.

The Sandage effect of higher periods for variables in Oosterhoff II clusters at lower Z than for the Oosterhoff I clusters might be explained with my new masses by having the Oosterhoff II variables born much to the blue of the instability strip and evolving at increasing luminosity to the red. For Oosterhoff I variables, they can be born at their lower mass and higher Z inside the instability strip at lower luminosity, evolving both to the red and blue with little evolutionary luminosity increase before they become red nonvariables.

We present nonlinear radiation hydrodynamical calculations of Cepheid envelopes carried out using the method described in Dorfi & Feuchtinger (1991, A&A 249, 414). The radiative transport is treated within the grey Eddington approximation and all equations are discretised in conservative form on an adaptive mesh which is solved simultaneously with the physical equations. The resulting system of nonlinear algebraic equations is solved implicitly to avoid the restrictive Courant-Friedrichs-Lewy time step condition. In order to treat schock waves we use the artificial tensor viscosity developed by Tscharnuter & Winkler (1979, Comp.Phys.Comm. 18, 171) which is suitable for spherically symmetric problems. For the opacity we employ the latest OPAL tables (Rogers and Iglesias, 1992, ApJ, submitted); the equation of state corresponds to the standard evolution calculations according to Baker & Kippenhahn (1962, Zeitschr. f. Astrophys. 54,114).

We start from an initial hydrostatic model whose parameters correspond to a typical Cepheid (5 M⊙, 3000 L⊙, 5600 K). The subsequent dynamical evolution is initiated by the excitation of oscillations at the ionization zones in the stellar atmosphere where the opacity changes by several orders of magnitude. After an initial relaxation caused by discretisation errors and numerical noise, a periodic pulsation develops. Each pulsation is accompanied by a shock wave which propagates through the stellar atmosphere. A typical pulsation cycle is covered by about 70 time steps and the obtained periods lie in the range of 1 to 40 days.

We discuss in detail the excitation of the oscillations and the structure of the model during a pulsation cycle.

Can pulsation studies yield information on the operation of element diffusion in stars? Can diffusion explain unusual properties of pulsating variables? Element diffusion theory and recent research relevant to these questions will be reviewed, with emphasis on the Sun and δ Scuti stars. High-degree solar p-modes that are sensitive to helium ionization support a reduced convection zone helium abundance consistent with that expected from diffusion. Intermediate-degree p-modes can be used to probe the structure of the convection zone base and constrain possible diffusion-produced composition gradients. δ Scuti variables have shallow convection zones and relatively short diffusion timescales. Helium diffusion may explain unusual period ratios, influence period changes, and affect the amplitudes and light curve shapes of δ Scuti stars.

Convection undoubtedly plays a strong role in defining the RR Lyrae instability strip, but its effects on amplitude, mode of pulsation, and light curve have always been problematical. One reason is simply that convective models are difficult to compute, and their accuracy is difficult to ascertain. We present results of a new convective survey of RR Lyrae stars that constitute the best models available at this time, with better boundary conditions, physics, zoning, and length of computational run than previous results.

Bump Cepheids display a resonance between the fundamental and second overtone modes in the form of a bump on descending light for periods less than 10 days and on the ascending light curve for longer periods. A long-standing problem has been how to explain this resonance for stellar masses consistent with evolution theory, rather than significantly lower ones. New Livermore OPAL intermediate coupling opacities now produce stellar models that are less density concentrated in the outer 1/4 of the radius, and all radial mode periods are increased. The second overtone to fundamental mode period ratio is reduced. This reduction then requires larger masses for the bump resonance, much closer to those from the old Becker, Iben, and Tuggle and the new Stothers and Chin stellar evolution results with no or very little core convection overshooting and the standard (X=0.70 and Z=0.02) composition. This achievement of the OPAL opacities is not adequate, though, because the period ratios have not quite decreased enough. There is another missing ingredient suggested to me by Norman Simon. A second deep “iron line” convection zone now appears between 145,000 and 205,000 K for higher Cepheid luminosities and masses like 7 and 8 M⊙. This convection is necessary to transport the higher luminosities that cannot be carried by radiation alone. An increase in the ratio of the mixing length to the pressure scale height to about 1.5 for 7 M⊙ and 2.0 or more for 8 M⊙ can give an appropriate convection efficiency and the required deep convection zone structure. Thus OPAL opacities, when used with the full physics of convection models all across the instability strip and nonadiabatic pulsation analyses, actually do explain the bump Cepheid puzzle. Further, the unknown convection efficiency for intermediate mass yellow giants can apparently be calibrated as a function of stellar mass.

An extensive grid of non-linear pulsating models of RR Lyrae stars have been computed. To simulate the outer regions of these variables a non-local and time-dependent treatment of convective transport has been adopted. In this poster we briefly describe some new features of the instability strip (IS).

A simple method for computing the full spectrum of the linear (nonadiabatic) radial modes is implemented and tested on RR Lyrae models. The growthrates of the vibrational modes display unexpected, but physically correct undulations as a function of period, with the 8th through 10th overtones almost unstable. Caution must be exercised when only a small number of meshpoints is used, or when the meshpoints are not well distributed, as some of these overtones may become unstable, clearly an artifact of the differencing. The role that such unstable high overtones can play in hydrodynamic calculations is demonstrated.

The shock phenomena and Hα formation in the atmosphere of RR Lyrae are investigated by means of numerical simulations. The full-amplitude hydrodynamical model is generated, adopting M = 0.578M⊙, L = 62L⊙, Teff = 7175K and Population II composition. The Hα profiles are obtained by solution of the non-LTE line transfer problem for the multilevel hydrogen atom. In the course of pulsations an extended (about 16 static scales) low-density atmosphere is produced with nearly exponential density decrease. Two shocks, propagating outwards, are successively generated during one period. The shock amplitude reaches 140 km/s. No mass loss has been found. These results confirm the earlier results of S. Hill (1972). The main shock develops very high, at mass depths near 0.001 g per area, whereas computations for W Vir yield 100. An analysis of the Fe I and Ti II lines shows that this difference is sufficient to explain the lack of metallic line-doubling in RR Lyrae stars. The bump on the light curve is due to a weak shock, propagating below the photosphere, which is generated with the early shock at the stage of expansion at the H-recombination front. The central intensities, Dopplerian shifts, amplitudes and phases of splitting of the computed Hα profiles agree well with the observed values (Gillet & Crowe 1988). The lack of strong emission also fits the observations, resulting from strong scattering processes near the shock. The predicted Lyman lines show strong emission, reaching maximum at the maximum light.

Cox (1991), Kovacs, Buchler, and Marom (1991), and Kovacs et al. (1992) showed that opacity changes indicated by the Livermore OPAL opacities increase the pulsation masses of double-mode RR Lyrae variables. We calculated limiting amplitude solutions for radiative RR Lyrae models of 0.75 M⊙, 51 L⊙, and Z=0.0001 (Oosterhoff II) to investigate the effects of the mass increase and opacity changes suggested by the OPAL opacities. In particular, we modified the Stellingwerf analytical fit (1975) to the Cox-Tabor (1976) tables to decrease the opacity by 20% between 20,000 and 30,000 K. The Stellingwerf periodic relaxation method was used to converge the models to a limit cycle, and the Floquet matrix analysed to search for a tendency of the fundamental mode to grow from the full-amplitude overtone solution, and the overtone to grow from the full-amplitude fundamental mode solution, thereby predicting double-mode behavior.

We review the recent nonlinear hydrodynamical results and the numerical problems to be solved by the next generation of codes. The behavior of the Fourier parameters and double-mode RR Lyrae pulsation will be discussed in detail. We emphasize the importance of the implementation of the various adaptive-mesh schemes for stellar pulsations. A further goal is the substitution of the artificial viscosity with modern shock capturing methods and a better treatment of radiative transfer. Convection however, must ultimately be included in order to supply the necessary physical mechanism for dissipation.

Double-mode RR Lyrae stars (RRd) represent a fundamental testing ground for the theory of stellar pulsation because they play an important role in understanding the interaction between different pulsation modes. In spite of the fact that from an observational point of view the identification of the fundamental (F) and first overtone (IO) periods needs only few periods of light curve coverage (Sandage et al. 1981, Nemec 1985), the theoretical models of double-mode and mixed-mode variables dating back to the original papers of Cox et al. (1983) and Stellingwerf (1975) were not integrated for long enough times to ensure proper mode identification. Recently Kovacs and Buchler (1992) using a standard radiative hydrodynamical code investigated the theoretical properties of RRd stars in more detail. This paper represents a brief description of the results of few cases computed adopting a non-linear, non-local treatment of convective transport, and long integration times to enable positive mode identification.

We have studied the nonlinear behavior of several sequences of BL Herculis-type models. The question arose whether the 2:1 resonance between the fundamental mode and the second overtone would cause the same systematic variation of Fourier parameters of the pulsation cycle with period ratio P2/P0 as was seen in the classical Cepheids. We find that for the BL Her stars, the behaviour of the light-curve Fourier phases is markedly different from the Cepheid case. In particular, ϕ21 exhibits essentially a featureless, monotone increase throughout the range of P2/P0, which is in qualitative agreement with the observed trend (Petersen & Diethelm 1986). In the velocity curves, on the other hand, the 2:1 resonance is a dominant feature and the progression of the Fourier phases and the amplitude ratios is similar to those witnessed in the Cepheids. However, here the sensitivity to the stellar masses and luminosities is significantly stronger. Our results show that radial velocity observations of the BL Her stars would pinpoint the resonance and put important new constraints on the models.

We argue that triple-mode RR Lyrae pulsation with low amplitudes might be quite common. It is shown that until very accurate abundance data become available, triple-mode RR Lyrae stars are the only hope to estimate reliable stellar parameters from the periods alone.

There are two problems with the Lee et al. (1990, LDZ; Lee 1990, 1991) scenario for the Sandage effect (Sandage 1990 and references therein) which may still warrant further detailed analyses: 1) on the basis of standard tracks, RR Lyrae stars in OoII, M15-like globular clusters can only be well-evolved objects from the blue ZAHB for unrealistically low helium abundance (Y) values; 2) in comparison with the models by Sweigart (1987, SW87), the evolutionary tracks of Lee & Demarque (1990, employed by LDZ) tend to overestimate the effect of evolution away from the ZAHB for the RR Lyrae stars, despite the similar assumptions involved in the calculations of the two sets of models. (See Catelan 1992a, C92A.)