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the two greatest successes of white dwarf asteroseismology have come from the analysis and interpretation of whole earth telescope ( wet ) data on the objects gd  358 and pg  1159 @xcite .
the temporal spectra of both objects show well - defined multiplet structures as well as many consecutive radial orders of the same @xmath2 .
this allows us immediately to estimate the rotational frequency , @xmath3 , and the mean period spacing , @xmath4 , with the latter quantity putting strong constraints on the mass of the best - fitting model .
furthermore , the variations from equidistant period spacing , i.e. , `` mode trapping '' , give us information ( at least in the case of gd  358 ) about the radial structure of the star , such as the thickness of its surface he layer .    in a detailed analysis of the frequency distribution in gd  358
, @xcite found evidence of a chemical transition zone located at a mass depth of @xmath5 , which they interpreted as the c / he boundary .
a he layer thickness of @xmath610@xmath7 is thinner than expected on naive evolutionary grounds , as well as thinner than suggested for the da white dwarfs by the asteroseismological work of @xcite . while the fit of bradley & winget explained the trapping features in the neighborhood of 700 sec reasonably well , it did not do as well for periods near 500 sec .
they found evidence that these periods near 500 sec might be better fit if an additional chemical transition zone at a depth of @xmath610@xmath8 is assumed .
independent of the above considerations , if the theory of diffusion of chemical species in stars is correct , then at least a partial separation of different isotopes of the same element must occur in white dwarfs , as was pointed out to one of us by clayton ( 1988 , private communication ) . at the time
, there were no known observational consequences ; this has since changed with the advent of white dwarf asteroseismology .
while the galactic number ratio of to has been measured to be of order @xmath9 in many astrophysical environments @xcite , this does not necessarily hold for the stellar cores which are presumably the white dwarf progenitors .
in fact , standard evolutionary theory would suggest quite low levels of in these objects @xcite .
our view is that given the theoretical uncertainties in the late stages of stellar evolution , we should not rule out any possibilities , and indeed should seek to make as many independent measurements as possible .
these considerations led to two independent motivations for considering diffusion @xcite .
first , it is a process which may be a generic feature of white dwarf cooling , and , as such , should be considered as part of our asteroseismological analyses .
second , it might allow us to place the mode trapping data for gd  358 in a different context : this star could have a c/ transition zone at @xmath10 and a thinner / transition zone at @xmath11 .
if this is the case , then the dbv s and the dav s would again have the same order of magnitude he layer thicknesses . to determine the plausibility of these hypotheses , we first examine the relevant diffusion timescales , and then the effect which such a layering structure has on our fits of gd  358 .
the process of diffusion is of major importance , since we are aware of no other effect which could lead to the spatial separation of and .
we therefore examine the relevant timescales for such diffusive processes , to see if it is plausible for a significant fraction of the to have separated from in the elapsed evolutionary times for these objects .
fortunately for us , @xcite have already examined the related problem of c diffusion in a background of normal he , i.e. , . in their analysis , they treated c as a trace element .
this is an excellent approximation for our case as well , since we naively expect that the is only about one part in @xmath12 of the .
applying their equations for this case for a 0.612 @xmath13 model , we obtain the result shown in figure  [ diffuse ] .
we see that if we assume a @xmath14 of 25,000  k for a typical dbv , then diffusive equilibrium between and should prevail down to approximately the @xmath15 mass point .
thus , if the layer is as thick or thicker than this , and the abundance is @xmath16 that of the abundance , we would expect a layer of thickness @xmath17 .
one clear prediction of this is that the spectra of db s in this temperature range should show more or less pure , at least for layers which are thicker than @xmath18 .
this is currently in the process of being tested using high - resolution spectroscopic observations ( koester , private communication ) .
we mention as an aside that the case of may be unique in that it is a trace isotope which is _ lighter _ than the dominant species , so that it can diffuse upwards and produce a thin , but not asteroseismologically negligible layer . for trace isotopes which sink
, there should be no such signature .      in this preliminary exploration
, we will only consider / zones with equilibrium profiles , although the profiles could in fact be less sharp than this .
the treatment of the chemical profiles in the transition zones in our present and previous work is based on the work of @xcite .
essentially , we use equation ( a6 ) of @xcite , which assumes that an equilibrium distribution has been reached and that one of the elements may be treated as a trace element .
this is certainly a valid assumption for the initial stages of diffusion , given the expected abundance ratio of to .
to include the effects of a layer we have made a relatively simple modification to the envelope routines in our evolutionary code .
since we are interested in modeling a db , we are free to use the array space normally reserved for the h profile and use it for the profile . to this end , we have replaced the h equation of state ( eos ) and opacities with those appropriate for . in doing this ,
we have taken the eos and opacities to be equal to those of at 3/4 the density ( to correct for the isotopic mass ratio ) .
this approximation should be more than sufficient for our purposes .
a comparison of the region of period formation for the cases of a /h and a / envelope is given in @xcite .
we now wish to examine the effect which a layer could have on the pulsation frequencies .
first , in the top panel of figure  [ fit0.61 ] , we show one of the best - fit models for the star gd  358 from @xcite .
the filled circles connected by solid lines show the observed mode trapping structure , and the open circles connected by dotted lines show the results from the best - fit model .
this model has @xmath19 , @xmath20 24,044 k , and @xmath21 .
the lower 3 panels show the effect a thin layer of has on the mode trapping structure .
if the layer is as thick as @xmath22 , corresponding to @xmath23 , then we see that the mode trapping structure in the vicinity of 700 sec is significantly altered .
fits_emu.tab unfortunately , this is the region in which the fit was already quite good , whereas in the region around 500 sec , where the fit was poor , there is very little change .
however , this analysis clearly demonstrates that the inclusion of a layer can have a measurable effect on the calculated mode trapping structure .
in an effort to test asteroseismologically the viability of the hypothesis of a layer , we performed an extensive set of calculations . using a genetic algorithm @xcite ,
the parameters @xmath24 , @xmath14 , @xmath25 , and @xmath26 were varied to produce pulsational models whose @xmath27 , @xmath28 periods were compared with those inferred from the observations .
the region of parameter space explored is given by : @xmath29 the best - fit models from these runs are listed in table  [ results ] . the column labeled
@xmath30 gives the residuals ( standard deviation ) of the calculated and observed periods , and the column labeled @xmath31 gives the residuals of the calculated and observed period _
spacings_. for the upper four models we have taken the fitness criterion to be @xmath32 , whereas for the lower two models we took it to be @xmath33 $ ] ; thus , for these last two entries , we are fitting not just the periods but the spacings between consecutive periods .
we have chosen this criterion for the models with a more realistic core composition based on our suspicion that the period _ spacings _ will be a more sensitive diagnostic of the layer than the periods themselves will be .    as a reference to previous fits ,
the fit of @xcite for a c core model has @xmath34  sec .
thus , the present best fits , both with and without a layer , having period residuals of @xmath35 and @xmath36  sec , respectively , represent a significant improvement over previous fits .
we see from table  [ results ] that the fits with o / c cores are significantly better than those with pure carbon cores , independent of whether a layer is present or not , a result previously found by @xcite . in figure
[ fit ] , we display the mode trapping diagrams for these two fits .
both fits reproduce the periods quite well , with the model reproducing the period _ spacings _ much better ( @xmath37 sec compared to @xmath38 sec ) .
this indicates that the period spacing may be a better diagnostic for the fine structure produced by the / transition zone than just the periods themselves .
we now seek to understand the relative importance of the o / c and / transition zones . in the asymptotic limit of high radial overtones and large periods ,
the frequency of a given mode is given by a simple radial integral of the brunt - visl frequency .
using a `` period formation '' diagram , we can show the relative weight which a given region has in determining a mode s period .    in figure  [ pform2 ]
, we show such a diagram for the case of our best - fit model .
the three peaks which are labeled correspond to the o / c , c/ , and / transition zones . using the equilibrium diffusion coefficients for the / transition zone
, we see that it is of relatively minor importance in determining the mode frequencies .
in contrast , the o / c and c/ transition zones are both quite pronounced and should significantly affect the periods of the modes calculated in the models ; this is borne out by the major improvement in the standard deviation of the periods , @xmath30 , with the inclusion of an o / c chemical profile .
the period _ spacings _ , on the other hand , are a differential quantity , and therefore sensitive to even small deviations in the background structure .
it is therefore not surprising that the inclusion of a layer results in a major reduction in the resduals of the period spacings .
we now attempt to quantify the statistical significance of the improvement of the fit when the parameter corresponding to the layer thickness is added .
following @xcite , we apply the bayes information criterion ( bic ) for @xmath39 data points .
we find that the addition of a parameter should be accompanied by a decrease in the residuals of at least @xmath610% in order to be considered statistically significant .
we note that this is a necessary but certainly not sufficient condition for the validity of adding a parameter .    examining the best - fit o / c core models
, we see that the standard deviation of the fit to the period spacings improved from 1.92  sec to 1.32  sec with the addition of .
this is a decrease of 30% and should therefore be considered statistically significant .
finally , we again note that the models with o / c cores yield markedly lower residuals , regardless of which criterion is used ( periods or period _ spacings _ ) .
this result has recently been obtained by @xcite , who report the results of such fits in an extensive parameter space of white dwarf models .
from this preliminary analysis , we have shown that diffusion theory applied to white dwarfs predicts that any initially present in the dbv s down to a depth of @xmath610@xmath40 should have diffused upward to produce a surface layer of .
in addition to being detectable spectroscopically ( koester , private communication ) , such a layer can significantly affect the asteroseismological fits , and therefore needs to be included in such analyses .
we find that although the inclusion of a layer results in only a marginal improvement to the fits to the periods , the fit to the period _ spacings _ is significantly improved .
this is because the period spacings are more sensitive to the fine structure which a / transition zone produces than the periods themselves are .    finally , we find that an o / c core ( essentially , the transition in the core from an o / c mixture to pure c ) fits the observed pulsational spectrum of gd  358 _ much _ better than a pure c core , in agreement with @xcite , who first obtained this result .
this gives us the hope of someday being able to constrain the prior nuclear burning history of gd  358 and other pulsating white dwarfs .
we would like to thank jim truran , don clayton , and hugh van horn for useful discussions on this topic .
this work was supported in part by the austrian fonds zur frderung der wissenshaftlichen forschung , project number s7304 , by the national science foundation under grant ast-9876730 , and by the national aeronautics and space administration under grant nag5 - 9321 .
, d.  e. , nather , r.  e. , clemens , j.  c. , provencal , j. , kleinman , s.  j. , bradley , p.  a. , wood , m.  a. , claver , c.  f. , frueh , m.  l. , grauer , a.  d. , hine , b.  p. , hansen , c.  j. , fontaine , g. , achilleos , n. , wickramasinghe , d.  t. , marar , t. m.  k. , seetha , s. , ashoka , b.  n. , odonoghue , d. , warner , b. , kurtz , d.  w. , buckley , d.  a. , brickhill , j. , vauclair , g. , dolez , n. , chevreton , m. , barstow , m.  a. , solheim , j .- e . , kanaan , a. , kepler , s.  o. , henry , g.  w. , & kawaler , s.  d. 1991 , , 378 , 326    , d.  e. , nather , r.  e. , clemens , j.  c. , provencal , j.  l. , kleinman , s.  j. , bradley , p.  a. , claver , c.  f. , dixson , j.  s. , montgomery , m.  h. , hansen , c.  j. , hine , b.  p. , birch , p. , candy , m. , marar , t. m.  k. , seetha , s. , ashoka , b.  n. , leibowitz , e.  m. , odonoghue , d. , warner , b. , buckley , d. a.  h. , tripe , p. , vauclair , g. , dolez , n. , chevreton , m. , serre , t. , garrido , r. , kepler , s.  o. , kanaan , a. , augusteijn , t. , wood , m.  a. , bergeron , p. , & grauer , a.  d. 1994 , , 430 , 839
we consider envelopes of db white dwarfs which are not composed of pure @xmath0he , but rather a mixture of @xmath1he and @xmath0he . given this assumption , the same diffusive processes which produce a relatively pure h layer overlying a he layer in the da s should work to produce a @xmath1he layer overlying a @xmath0he layer in the db s . 
 we examine the relevant timescales for diffusion in these objects , and compare them to the relevant evolutionary timescales in the context of the dbv white dwarfs . 
 we then explore the consequences which separation has on the pulsational spectra of dbv models . 
 since gd  358 is the best - studied member of this class of variables , we examine fits to its observed pulsation spectrum . 
 we find that the inclusion of a layer results in a modest improvement in a direct fit to the periods , while a fit to the period _ spacings _ is significantly improved .