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Genetic-Algorithm-Based Asteroseismological Analysis of the Dbv White Dwarf Gd 358 T.S

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Genetic-Algorithm-Based Asteroseismological Analysis of the Dbv White Dwarf Gd 358 T.S Accepted for publication in the Astrophysical Journal View metadata,Preprint citation typeset and using similar LAT EpapersX style at emulateapj core.ac.uk v. 20/04/00 brought to you by CORE provided by CERN Document Server GENETIC-ALGORITHM-BASED ASTEROSEISMOLOGICAL ANALYSIS OF THE DBV WHITE DWARF GD 358 T.S. Metcalfe, R.E. Nather, and D.E. Winget Department of Astronomy, University of Texas at Austin Mail Code C1400, Austin, TX 78712 Accepted for publication in the Astrophysical Journal ABSTRACT White dwarf asteroseismology offers the opportunity to probe the structure and composition of stellar objects governed by relatively simple physical principles. The observational requirements of asteroseis- mology have been addressed by the development of the Whole Earth Telescope (WET), but the analytical procedures still need to be refined before this technique can yield the complete physical insight that the data can provide. Toward this end, we have utilized a genetic-algorithm-based optimization method to fit our models to the observed pulsation frequencies of the DBV white dwarf GD 358 obtained with the WET in 1990. This new approach has finally exploited the sensitivity of our models to the core composition, and will soon yield some interesting constraints on nuclear reaction cross-sections. Subject headings: methods: numerical—stars: fundamental parameters— stars: individual (GD 358)—stars: interiors—stars: white dwarfs 1. INTRODUCTION have found the only solution or the best one if more than one is possible. We study white dwarf stars because they are the end- points of stellar evolution for the majority of all stars, and their composition and structure can tell us about their 2. GENETIC ALGORITHMS prior history. We can determine the internal structure An optimization scheme based on a genetic algorithm of pulsating white dwarfs by observing their variations in (GA) can avoid the problems inherent in the traditional brightness over time, using the techniques of high speed approach. Restrictions on the range of the parameter- photometry to define their light curves, and then matching space are imposed only by observational constraints and by these observations with a numerical model which behaves the physics of the model. Although the parameter-space the same way. The parameters of the model are chosen so defined is often quite large, the GA provides a rela- to correspond one-to-one with the physical processes that tively efficient means of searching globally for the best-fit give rise to the variations, so a good fit to the data leads model. While it is difficult for GAs to find precise val- us to believe that our model reflects the actual physics of ues for the best-fit set of parameters, they are very good the stars themselves. at finding the region of parameter-space that contains the Although this procedure is simple in outline, its real- global minimum. In this sense, the GA is an objective ization in practice requires specialized instrumentation to means of finding a good first guess for a more traditional overcome the practical difficulties we encounter in the pro- method which can then narrow in on the precise values cess. The Whole Earth Telescope (WET) observing net- and uncertainties of the best-fit set of parameters. work (Nather et al. 1990) was developed to provide the The underlying ideas for genetic algorithms were in- 200 or more hours of essentially gap-free data we require spired by Charles Darwin’s (1859) notion of biological evo- for the analysis. This instrument is now mature, and has lution through natural selection. The basic idea is to solve provided a wealth of seismological data on the different an optimization problem by evolving the best solution from varieties of pulsating white dwarf stars, so the observa- an initial set of completely random guesses. The theoreti- tional part of the procedure is well in hand. Now we are cal model provides the framework within which the evolu- working to improve our analytical procedures to take full tion takes place, and the individual parameters controlling advantage of the possibilities afforded by asteroseismology. it serve as the genetic building blocks. Observations pro- The adjustable parameters in our computer models of vide the selection pressure. For a detailed description of white dwarfs presently include the total mass, the tem- genetic algorithms, see Metcalfe (1999) and Charbonneau perature, hydrogen and helium layer masses, core com- (1995). position, convective efficiency, and composition transition Initially, the parameter-space is filled uniformly with tri- profiles. Finding a proper set of these to provide a close als consisting of randomly chosen values for each parame- fit to the observed data is difficult. The traditional pro- ter, within a range based on the physics that the parameter cedure is a cut-and-try process guided by intuition and is supposed to describe. The model is evaluated for each experience, and is far more subjective than we would like. trial, and the result is compared to the observed data and More objective procedures are essential if asteroseismology assigned a fitness inversely proportional to the variance. A is to become a widely-accepted astronomical technique. new generation of trials is then created by selecting from We must be able to demonstrate that, within the range this population at random, weighted by the fitness. of different values the model parameters can assume, we Each model is encoded into a long string of numbers analogous to a chromosome with each parameter serving 1 2 METCALFE, NATHER, & WINGET as a gene. The encoded trials are paired up and modi- fied in order to explore new regions of parameter-space. The two basic genetic operators are crossover which em- ulates reproduction, and mutation which emulates hap- penstance. After these operators have been applied, the strings are decoded back into sets of numerical values for the parameters. The new generation replaces the old one, and the process begins again. The evolution continues for a specified number of generations, chosen to maximize the efficiency of the method. 3. THE DBV WHITE DWARF GD 358 During a survey of eighty-six suspected white dwarf stars in the Lowell GD lists, Greenstein (1969) classi- fied GD 358 as a helium atmosphere (DB) white dwarf based on its spectrum. Photometric UBV and ubvy col- ors were later determined by Bern & Wramdemark (1973) and Wegner (1979) respectively. Time-series photometry by Winget Robinson Nather & Fontaine (1982) revealed the star to be a pulsating variable—the first confirmation of a new class of variable (DBV) white dwarfs predicted by Winget (1981). In May 1990, GD 358 was the target of a coordinated observing run with the WET. The results of these obser- Fig. 1.| The numerical thin limit of the fractional helium layer mass for various values of the total mass (connected points) and the vations were reported by Winget et al. (1994), and the linear cut implemented in the WD-40 code (dashed line). theoretical interpretation was given in a companion pa- per by Bradley & Winget (1994b). They found a series of nearly equally-spaced periods in the power spectrum are known to be more massive than the upper limit of which they interpreted as non-radial g-mode pulsations of our search, these represent a very small fraction of the consecutive radial overtone. They attempted to match the total population, and for reasonable assumptions about observed periods and the period spacing for these modes the mass-radius relation all known DBVs appear to have using earlier versions of the same theoretical models we masses within the range of our search (Beauchamp et al. have used in this analysis (see 4). Their optimization 1999). method involved computing a grid§ of models near a first The span of effective temperatures within which DB guess determined from general scaling arguments and an- white dwarfs are pulsationally unstable is known as the alytical relations developed by Kawaler (1990), Kawaler DB instability strip. The precise location of this strip & Weiss (1990), Brassard Fontaine Wesemael & Hansen is the subject of some debate, primarily because of dif- (1992), and Bradley Winget & Wood (1993). ficulties in matching the temperature scales from ultravi- olet and optical spectroscopy and the possibility of hiding 4. DBV WHITE DWARF MODELS trace amounts of hydrogen in the envelope (Beauchamp 4.1. Defining the Parameter-Space et al. 1999). The most recent temperature determinations for the 8 known DBV stars were done by Beauchamp et The most important parameters affecting the pulsation al. (1999). These measurements, depending on various as- properties of DBV white dwarf models are the total stellar sumptions, place the red edge as low as 21,800 K, and mass (M ), the effective temperature (Teff ), and the mass the blue edge as high as 27,800 K. Our search includes all ∗ of the atmospheric helium layer (MHe). We want to be temperatures between 20,000 K and 30,000 K. careful to avoid introducing any subjective bias into the The mass of the atmospheric helium layer must not be 2 best-fit determination simply by defining the range of the greater than about 10− M or the pressure of the over- search too narrowly. For this reason, we have specified the lying material would theoretically∗ initiate helium burning range for each parameter based only on the physics of the at the base of the envelope. At the other extreme, none of model, and on observational constraints. our models pulsate for helium layer masses less than about 8 The distribution of masses for isolated white dwarf stars, 10− M over the entire temperature range we are consid- generally inferred from measurements of log g, is strongly ering (Bradley∗ & Winget 1994a). The practical limit is peaked near 0.6 M with a FWHM of about 0.1 M (Napi- actually slightly larger than this theoretical limit, and is a wotzki Green & Saffer 1999).
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