ADIABATIC PROPERTIES OF PULSATING DA WHITE-DWARFS .2. MODE TRAPPING IN COMPOSITIONALLY STRATIFIED MODELS

As part of a series of investigations aimed at understanding the adiabatic gravity-mode period structure of models of pulsating DA white dwarfs, we examine qualitatively and quantitatively the phenomenon of mode trapping caused by compositional layering in these stars. This is motivated by the real possibility that the pulsation modes detected in ZZ Ceti stars are actually trapped in the outer hydrogen layer. We first discuss the various manifestations of compositional layering on the pulsational properties of a representative ZZ Ceti star model. The most important effect, from the point of view of the observations, is the formation of a nonuniform period distribution which bears the signature of the strength and location of the composition discontinuity. The trapped modes are conspicuous in that they show the largest deviations from the mean behavior in the period distribution, and are the most sensitive to the characteristics of the H/He composition transition zone. We thus derive a semianalytic formula relating the periods of trapped modes to model properties, which can readily be used to estimate the hydrogen layer mass in a ZZ Ceti star. We then use a more formal approach by investigating, in the asymptotic limit of high radial overtones, the effects of a true discontinuity (an idealization of the H/He transition zone in a DA white dwarf) in the Brunt-Vaisala frequency profile on the g-mode period spectrum of a stellar model. We find an analytic expression which gives the full period spectrum of the model in terms of only four parameters. As a by-product, we also derive another expression which gives, in terms of the same parameters, the period spectrum of trapped modes only. We show that these four parameters are related to the structural properties of a star. Comparisons with exact numerical calculations indicate that our first analytic expression accounts well for the essential features of mode trapping on the full period spectrum of a stellar model. Similar comparisons show that our second analytic expression reproduces quite accurately the period spectrum of trapped modes, raising the interesting prospect of being able to use a very simple model as a guide for interpreting observed period distributions in ZZ Ceti stars and for inferring some of their fundamental properties. On the basis of this simple asymptotic model, we finally propose a procedure for analyzing future observations of ZZ Ceti stars (the current data are not suitable for this type of analysis) in terms of mode trapping.