REMINIS:ENCE OF A STELLAR PULSATION THEORIST

The author started working in stellar pulsation theory in 1965. Since this time, we have studied the effects of improved radiative transfer and dynamic zoning on models of Cepheids, RR Lyrae, and W Virginis stars. In this paper we discuss the relevant equations and some results in comparisons to observations of W Virginis and longbzriod Cepheids, Some suggestions for the next generation of stellar nulsation codes is given.


Introduction
The first nonlinear stellar pulsation calculations were done in the early 1960s by Christy, Cox, and Hillendahl, These codes were preceded by the development of the method of pseudo-viscosity. The addition of m~lti-group radiative transfer to the Lagrangian hydrodynamic codes was made in the late 1960s, By the 1970s, a dynamically zoned hydro code was available and it seemed obvious that a more complete code could be deveioped including dynamic zoning and radiative transfer, The methods to calculate line transfer in a moving media, Ala Mihalas, were developed in the 70s but now with the new computers (~, e, YMPs and connection machines) there is the possibility of solving these equations coupled with the dynamically-zoned hydrodynamic equations,

Early Hydro
The early hydro codes were of the Richtmyer-Von Newnan pseudo-viscosity variety, These codes used standard explicit differing srhemes Eqs. (l-5) with the possible addition of expansions for time centering. It is interesting to note that the author in his pulsation code (SPECEP) had to add a small amount of linear viscosity to the center zone of the mesh, as did Christy, in order to stabilize the mesh, The standard quadratic pseudo-viscosity term was used in these early codes, with a cutoff added later by Stellingwerf (see Eq, [6]), to reduce unwanted dissipation, Nonlinear

Radiative Transfer
The quthor joined Los Alamos in 1965 to work with A, Cox to qdd 9 radiative transfer scheme to the nonlinear pulsation models, At about the same time J. Castor became a graduate student under Christy at Cal, The calculated light curve for W Virginis using a multi-group radiative.trcnsfer code (SPECEP), V Virginis first proposed by Christy in 1966, The calculated light curve ii.8hown in Fig, 2 with an est~mate of the shock location vs. phase in Fig, 3, The shoulder, as observed in W Virginis (Fig, 4), did ',\ot appear in Christy's diffugion model,

P&E (thro)
The photospheric and shock radii locations from the W Virginis

Dynamic Zones
In association with John Castor, we developed a dyn~mically-zoned stellar pulsation code (DYN) in 1977. The idea of resolving the ionization driving region by adapting zones, in a non-Lagrangian mesh had been studied by Castor in his thesis at Cal Tech (see Fig. 5). The code is implicit because of restricted Courant time steps that occur in these thin zones. An idsa of the improvement obtained with dynamic zoning is shown in Fig. 6. Castor suggests that further improvements can be obtained by using the more robust adaptive mesh-algorithms of l~inkler. The code DYN has been used extensively by Takeuti's group in Tokyo . Recen:ly we have applied DYN to questions of model content (Buchler and Kovacs) [1], resonance in Cepheids (Simon) [2] and longperiod variables with Barnes and Moffet [3]. I will only show re~ults from the later problems to emphasize the importance of dynamic zoning. A particular model of X Cygni was studied because of the carefui observations of it's light curve. Using an evolutionary mass with an appropriate luminosity and effective temperature, we produced the model which is compared to the observations in Fig. 7. The "dip" observed before light maximum appears real and was observed in other stars. The dip in the calculation is due to the transit of the shock through the atmosphere and it is therefore effected by the pseudo-viscosity.  Fig. 7. A uodel of a long-period cepheid (X Cygni) using dynamic zoning (DYN) as compared to the obssrvstions,

5, Conclusions
From these reminiscence, it is obvious that a lot more could be done in modeling stellar pulsations, The addition of radiative transfer to a dynamically-zoned code (DYN for instance) would enhance our ability to study high luminosity to mass stars. A treatment of line transfer in the moving atmosphere (Ala Mihalas) would allow us to make a mere direct comparison to obsemations.
I have not mentioned improved opacities nor equations of state that are being developed; these should be included in future codes. There is a lot to be done but only a few new codes have appeared in recent years (we should not ignore the contributions of Von Sengbush and Stellingwerf using relaxation methods). 6.