ADIABATIC PROPERTIES OF PULSATING DA WHITE-DWARFS .1. THE TREATMENT OF THE BRUNT-VAISALA FREQUENCY AND THE REGION OF PERIOD FORMATION

We present the first results of a series of investigations aimed at exploring the systematics of the gravity-mode period structure of models of pulsating DA white dwarfs in the adiabatic approximation. In this first paper, we consider the basic issue of the region of period formation in a degenerate star. This is motivated by a paper by Pesnell which challenges the usually accepted concept of envelope modes for white dwarfs. It is essential here to treat carefully the Brunt-Vaisala frequency, which is the oscillation frequency of fluid elements when buoyancy is the restoring force. We show that it must be appropriately transformed in order to obtain reliable numerical results in degenerate stellar models. This is because the Brunt-Vaisala frequency is defined in terms of a difference between two numbers which become nearly equal in highly degenerate matter, thus causing serious numerical problems and systematic errors. We derive an alternative expression which is generally valid for multicomponent, nonideal, partially degenerate, and partially ionized plasmas such as those encountered in white dwarf envelopes. We use this expression to compute the period structure of the same white dwarf considered by Pesnell. For one particular model he investigated we find a period and a weight function distribution that are quite different from his results. However, in a numerical experiment based on the usual relation for the Brunt-Vaisala frequency (which leads to unreliable values in degenerate matter), we recover exactly the results of Pesnell. We conclude that the implicit numerical differencing used in the Lagrangian pulsation code of Pesnell leads to very serious difficulties when used with models of degenerate stars, and reaffirm the correctness of the fundamental result of other investigations: gravity-mode pulsations in white dwarfs are truly envelope modes. The implications of our findings on the work of Cox et al., based on the same Lagrangian approach, are also discussed. We find that the basic period structure of all their models suffers from obvious symptoms of theory breakdown; this implies that their nonadiabatic results must be regarded as questionable.