Adiabatic Survey of Subdwarf B Star Oscillations. III. Effects of Extreme Horizontal Branch Stellar Evolution on Pulsation Modes

THE ASTROPHYSICAL JOURNAL SUPPLEMENT SERIES, 131 :223È247, 2000 November2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

ADIABATIC SURVEY OF SUBDWARF B STAR OSCILLATIONS. I. PULSATION PROPERTIES OFA REPRESENTATIVE EVOLUTIONARY MODEL

S. CHARPINET

Canada-France-Hawaii Telescope, P.O. Box 1597, Kamuela, HI 96743 ; charpinet=cfht.hawaii.edu

G. FONTAINE AND P. BRASSARD

de Physique, de Succ. Centre-Ville, Canada, H3C 3J7 ;De� partement Universite� Montre� al, Montre� al, Que� bec, fontaine=astro.umontreal.ca,brassard=astro.umontreal.ca

AND

BEN DORMAN

Raytheon ITSS, Laboratory for High Energy Physics, Code 664, NASA/GSFC, Greenbelt, MD 20771 ; ben.dorman=gsfc.nasa.govReceived 1999 October 11 ; accepted 2000 February 18

ABSTRACTWe present the Ðrst results of a large, systematic adiabatic survey of the pulsation properties of models

of subdwarf B (sdB) stars. This survey is aimed at providing the most basic theoretical data with whichto analyze the asteroseismological properties of the recently discovered class of pulsating sdB stars (theEC 14026 stars). Such a theoretical framework has been lacking up to now. In this paper, the Ðrst of aseries of three, an adiabatic pulsation code is used to compute, in the 80È1500 s period window, theradial (l\ 0) and nonradial (from l\ 1 up to l \ 3) oscillation modes for a representative evolutionarymodel of subdwarf B stars. Quantities such as the periods, kinetic energies, Ðrst-order rotational splittingcoefficients, eigenfunctions, and weight functions are given by the code, providing a complete set of veryuseful diagnostic tools with which to study the mode properties. The main goal is to determine howthese quantities relate to the internal structure of B subdwarfs, a crucial and necessary step if one wantsto eventually apply the tools of asteroseismology to EC 14026 stars. All modes (p, f, and g) were con-sidered in order to build the most complete picture we can have on pulsations in these stars. In thatcontext, we show that g-modes are essentially deep interior modes oscillating mainly in the radiativehelium-rich core (but not in the convective nucleus), while p-modes are shallower envelope modes. Wedemonstrate that g-modes respond to a trapping/conÐnement phenomenon induced mainly by the He/Hchemical transition between the H-rich envelope and the He-rich core of subdwarf B stars. This pheno-menon is very similar in nature to the g-mode trapping and conÐnement mechanisms observed in pul-sating white dwarf models. We emphasize that p-modes may also experience distortions of their perioddistribution due to this He/H transition, although these are not as pronounced as in the g-mode case.These phenomena are of great interest as they can potentially provide powerful tools for probing theinternal structure of these objects, in particular, with respect to constraining the mass of their H-richenvelope. The results given in this Ðrst paper form the minimal background on pulsation mode charac-teristics in sdB stars. Upcoming discussions on additional mode properties in subdwarf B star models(Paper II and Paper III of this series) will strongly rely on these basic results since they provide essentialguidance in understanding mode period behaviors as functions of B subdwarf stellar parameters and/orevolution.Subject headings : stars : interiors È stars : oscillations È subdwarfs

1. INTRODUCTION

We have recently carried out an exploration of theasteroseismological potential of stellar models on theextreme horizontal branch (EHB) and beyond. This wasmade possible thanks to signiÐcant progress in our abilityto compute increasingly sophisticated and realistic modelsfor this relatively neglected phase of stellar evolution (see,e.g., Dorman 1995 for a review). The models of interestrepresents low-mass objects with outer(M [ 0.5 M

_)

H-rich envelopes too thin to reach the AGB after corehelium exhaustion. Such models cannot sustain signiÐcantH-shell burning during core helium-burning evolution. Thecore helium-burning phase is identiÐed with subdwarf B(sdB) stars (Heber et al. 1984 ; Heber 1987). The sdB starshave atmospheric parameters in the ranges 24,000 K [

K and (see Sa†er et al. 1994Teff [ 40,000 5.1[ log g [ 6.2and references therein).

During the postÈcore-He-exhaustion phase, the modelscontract and the H-burning shell Ðnally ignites. With insuf-

Ðcient hydrogen energy to force a red giant star envelope,the stars live in the phase (Greggio &““ AGBÈManque� ÏÏRenzini 1990) for a period similar to the equivalent early(i.e., prethermal pulsing) AGB evolution phase. These post-EHB, He-shell burning models are associated with the Ðeldsubdwarf O (sdO) stars (Dorman, OÏConnell, & Rood 1995,and references therein). The majority of sdO stars clusteraround K and log g D 5.5 (Dreizler 1993).Teff D 45,000Ultimately, the models join the white dwarf cooling tracksnear K and are identiÐed, in the early whiteTeff D 80,000dwarf phase, with the low-gravity DAO white dwarfs(Bergeron et al. 1994).

In the absence of any information as to the possiblevariability of EHB and post-EHB stars, observationally orotherwise, we initially focussed our attention on a stabilityanalysis of relevant models in the hope of uncovering anydestabilization mechanism that could excite pulsationmodes in these evolved stars. We found that both radial andnonradial (p, f, and g) low-order and low-degree modes

223

224 CHARPINET ET AL. Vol. 131

could be excited in some models in the sdB phase of evolu-tion through the action of an efficient driving mechanismdue to the i-e†ect associated with iron ionization(Charpinet et al. 1996, 1997b). On this basis, we made theprediction that a subclass of sdB stars should show lumi-nosity variations resulting from pulsational instabilities.Likewise, we discovered that g-mode instabilities develop inlate post-EHB models as a result of a potent v-mechanismassociated with the presence of an active H-burning shell.The v-process is able to drive low-order and low-degreeg-modes with typical periods in the range 40È125 s in thesemodels identiÐed with low-mass DAO white dwarfs. Hence,we suggested looking for brightness variations in such starsas well (Charpinet et al. 1997c).

While the jury is still out on the possibility that someDAO white dwarfs may be pulsating (see, e.g., Handler1998), the independent discovery of a Ðrst batch of fourpulsating sdB stars by a team of scientists at the SouthAfrican Astronomical Observatory (Kilkenny et al. 1997 ;Koen et al. 1997 ; Stobie et al. 1997 ; OÏDonoghue et al.1997a, 1997b) gave our group added conÐdence in the basicvalidity of the approach of Charpinet et al. (1996, 1997b).This also led us to further reÐnements of the physicaldescription of the iron bump mechanism responsible fordriving pulsation modes in models of sdB stars (Charpinetet al. 1997a). Additional nonadiabatic investigations alongsimilar lines carried out by Charpinet (1998) and Charpinetet al. (1999 ; Charpinet 2000a, in preparation) have estab-lished the existence of a theoretical instability strip in whichall the currently known pulsating sdB stars fall. Moreover,there is an excellent qualitative agreement between theperiods of the expected driven modes and the observedperiods. Low-order radial, and low-order and low-degreenonradial (p, f, and g) modes are involved.

The observational discovery of pulsating sdB stars haspaved the way for a real application of asteroseismologicaltechniques to this newest class of pulsating stars. It providesthe needed impetus for a complete, detailed, and systematicsurvey of the pulsation properties of models of sdB stars.We present, in a short series of three papers, the results ofthe Ðrst survey of the sort. Our principal goal has been toprovide the necessary theoretical framework within whichto interpret the observed pulsation properties of the EC14026 stars, particularly the observed period spectra andthe (eventual) measurements of rates of period changes. Wehave restricted our analysis to the adiabatic approximationbecause the period spectra of the models can be most effi-ciently and quite accurately studied within that approx-imation. In the present study (Paper I), we focus on thestructure of the pulsation spectrum in a typical B subdwarfmodel. The main objective is to describe the major proper-ties of eigenmodes in subdwarf B stars and their relation-ships with the internal structure of these stars. In Paper II(S. Charpinet 2000b, in preparation), we will concentrate onthe e†ects of model parameters (such as log g, theTeff,H-rich envelope mass and the total mass M) on theMenv,period distribution, a step that is crucial and necessary in aneventual application of asteroseismology to observed pulsa-tors. Finally, in Paper III (S. Charpinet 2000c, inpreparation), we will explore, through observable quantitiessuch as the rates of period change the relationship(P0 ),between evolution and pulsations in sdB stars. In particu-lar, interesting constraints may potentially be obtained onthe status of the central helium-burning processes throughsuch an approach.

In the next section (° 2), we provide a description of theevolutionary models used in our adiabatic survey. In partic-ular, we give the appropriate details on the representativemodel that we have chosen. In ° 3, we present and discussthe adiabatic quantities derived from the pulsation calcu-lations carried out with our representative model. The fullperiod spectrum including radial and nonradial acousticand gravity modes is discussed in detail. We Ðnally providea summary in ° 4.

2. STELLAR STRUCTURES FOR HOT B SUBDWARFS

2.1. Model ComputationsOne critical aspect of asteroseismological calculations is

the modeling of the stellar structures upon which thenumerical evaluation of the pulsation quantities, such as theeigenfrequencies, depends quite strongly. It is thereforeessential to rely on models believed to describe as realisti-cally as possible the actual stellar structures. For hot Bsubdwarfs, we have Ðrst relied on a batch of equilibriummodels consisting of complete, evolved structures extractedfrom the extreme horizontal branch (EHB) evolutionarytracks described in the next subsection. In the course of ouradiabatic survey, we have found it useful to supplementthese models by less realistic, nonevolving, static structuresin order to untangle the e†ects of various model parameterson the pulsation properties. This will be addressed in PaperII. We point out here that our two groups of equilibriummodels (referred to as ““ Ðrst generation models ÏÏ in Char-pinet et al. 1997a) are inadequate to study the question ofthe stability of modes as they do not incorporate di†usionprocesses in their calculations. The latter proved critical innonadiabatic investigations as demonstrated by Charpinetet al. (1997a). However, for the adiabatic survey discussed inthis series of papers, our evolutionary models remain themost appropriate and, indeed, the most realistic currentlyavailable, as evolutionary models handling the couplingbetween di†usion and evolution do not yet exist for sdBstars.

Details on how the evolutionary models are built areprovided in Dorman (1992a, 1992b), and Dorman, Rood, &OÏConnell (1993). For the convenience of the reader webrieÑy recall the main steps of their construction as well asthe constitutive physics used for the computations. Initialstructures on the zero age extreme horizontal branch(ZAEHB) are derived, as described in Dorman, Young-Wook, & VandenBerg (1991), from red giant models asclose as possible to the tip of the RGB where ignition of thehelium Ñash occurs (for details on these RGB models, seeVandenBerg 1992). The He-rich remnant of the prior H-burning phase is extracted to form the core (of Ðxed mass

of the horizontal branch stars. Abundance (both ofMc)

helium and heavier elements) appears to be the mainparameter inÑuencing the value of which is otherwiseM

c,

tightly constrained by the underlying physics of the degen-erate helium gas and remains almost independent, at leastto a good approximation, of the mass that the star initiallyhad on the main sequence (Sweigert & Gross 1978). Castel-lani & Castellani (1993) also showed that is una†ectedM

cby changes in the H-rich envelope mass due to mass lossprocesses during the RGB phase of evolution, prior to thehelium Ñash. Sets of models close to the zero age horizontalbranch (ZAHB) and the ZAEHB are then obtained by

No. 1, 2000 ADIABATIC SURVEY OF SDB STAR OSCILLATIONS. I. 225

adjusting the mass of the H-rich envelope surround-(Menv)ing the helium core. This is done by scaling the envelope ofa 0.90 model previously calculated from the tip of theM

_RGB in order to preserve the composition proÐle in theH-burning shell region.1 We deÐne the location of the theo-retical ZAHB (and therefore ZAEHB) to be the location ofthe model in the H-R diagram after 10 small evolutionarytime steps of 105 yr each to ensure that envelope hydrostaticequilibrium is achieved.

The basic physics ingredients used for EHB and post-EHB evolution computations are similar to those men-tioned in Dorman et al. (1993 and references therein), withthe exception of updated radiative and conductive opacities(see below). These include : the Caughlan & Fowler (1988)tables for nuclear reaction rates involving the triple-a chainfor helium and the p-p and CNO chains for the H-burningshell located at the bottom of the H-rich envelope ; the useof the Eggleton-Faulkner-Flannery (EFF) equation of state(Eggleton, Faulkner, & Flannery 1973) which is equivalent,for the physical conditions typically encountered in HB andEHB star models, to a partially degenerate, partially rela-tivistic ideal gas (nonideal e†ects, such as Coulomb inter-actions, are not taken into account as they remain relativelysmall2È!D 0.3 for a typical He-burning coreÈuntil thewhite dwarf cooling phase is reached) ; a treatment for semi-convection described by Dorman & Rood (1993) andDorman (1995) ; a treatment for convection based on themixing length theory with a \ 1.5 (VandenBerg & Poll1989). In addition, and di†ering from the Dorman et al.(1993) model calculations, we used more recent opacitytables described by Rogers & Iglesias (1992) and computedin 1993 December, which adopted the element mix referredto as ““ Grevesse & Noels 1993.ÏÏ During helium Ñashes, itwas also necessary to rely on the new low-temperature opa-cities provided by D. R. Alexander (1995, private communi-cation, described in Alexander & Ferguson 1994) andcalculated for the same mix of elements. These are forced tomatch smoothly with the OPAL opacities within the hydro-gen ionization zone, where the material is strongly convec-tive and the radiative opacities are irrelevant. Anotherdi†erence in the input physics comes from the use of theItoh et al. conductive opacities (Itoh et al. 1983 ; Itoh,Hayashi, & Kohyama 1993 ; Itoh & Kohyama 1993 ; Itoh,Hayashi, & Kohyama 1994 ; Itoh & Kohyama 1994).

Our evolutionary models were computed assuming auniform solar composition in the H-rich envelope(Y \ 0.29911, Z\ 0.017178 and X \ 1 [ Y [ Z\0.683712) appropriate for Ðeld subdwarf B stars. Of course,this may not be suitable for objects located in environmentshaving di†erent metallicities (such as globular clusters). Butwe point out that the amount of metallicity is not of primeimportance in the context of our adiabatic survey. Moreo-ver, as mentioned above, a truly consistent treatment of thechemical composition in the H-rich envelope of sdB starsshould implement di†usion processes. Those would lead tononuniform metallicity proÐles which are largely indepen-dent of the metallicity of the local environment.

1 Note that for the sequences with the smallest envelope masses (i.e.,close to the helium main sequence), the ZAEHB models are constructed byreproducing the layers immediately above the helium core, as these area†ected by hydrogen burning.

2 We stress that, however, nonideal e†ects cannot yet be properlyincorporated in the relativistic, partially degenerate regime.

2.2. Evolutionary Tracks

Seven distinct sequences, spanning the evolution from theZAEHB to the white dwarf phase, were constructed toprovide the stellar structures needed for our pulsation com-putations. These were chosen to cover the whole log g

region where subdwarf B stars are normally found.[ TeffFive of these sequences correspond to the evolution ofobjects with a Ðxed core mass of di†eren-M

c\ 0.4758 M

_tiated from each other by varying initial ZAEHB envelopemasses of, respectively, 0.0002, 0.0012, 0.0022, 0.0032,(Menv)and 0.0042 Two additional sequences with a slightlyM

_.

di†erent core mass and having envelope(Mc\ 0.4690 M

_)

masses of, respectively, 0.0001 and 0.0007 were addedM_later to improve our mapping of the sdB region in the log g

plane. A summary of the main model characteristics[ Tefffor one of these evolutionary tracks (the sequence withis provided inM \M

c] Menv\ 0.4758 ] 0.0002 M

_)

Table 1. The evolution starts at the ZAEHB (model 1) andproceeds toward the white dwarf cooling curve (sampled bythe last available models). The major events occurringduring the evolutionary process are mentioned whereappropriate. For each model of the sequence, the age (inMyr since the ZAEHB), surface parameters such as theluminosity the e†ective temperature the(log L /L

_), (Teff),log of the surface gravity (log g), and core parameters such

as the central helium mass fraction temperature(Yc), (T

c),

density (given as and pressure (given as aslog oc), log P

c)

well as the position of the boundary of the convective corewhere is the mass of the(log q

cc\ log [1[ (M

cc/M)], M

ccconvective core) are provided. Similar data for the six othersequences will be made available and discussed in detail inPaper III, where we focus on the relationship between evol-ution and pulsation properties. The current sequence is,however, fully representative of EHB and post-EHB stellarevolution, which is all we need for our present study.

The coverage of the sdB region in the plane islog g[ Teffillustrated by Figure 1 where the relevant parts of the sevenevolutionary tracks (dotted curves) are presented, in order ofincreasing envelope mass from high to low alongTeff Teff,with a sample of 213 subdwarf B stars for which atmo-spheric parameters are known (open circles ; sample fromR. A. Sa†er 1995, private communication ; see also Sa†er etal. 1994). As shown in that Ðgure, subdwarf B stars aremainly identiÐed with the helium core burning phase ofEHB evolution whose typical secular timescale is estimatedto be D108 yr (see Table 1). We note, however, that some ofthe low surface gravity and high e†ective temperatureobjects present in the sample may correspond to moreadvanced evolutionary epochs identiÐed with the very earlystages of the post-EHB phase characterized by stablehelium shell burning. The timescale for post-EHB evolutionis somewhat shorter (of the order of 3] 107 yr). Hence,once helium is exhausted at the center of the star, evolutionvery quickly drags the models away from the subdwarf Bstar area, crossing regions where subdwarf O (sdO) stars areencountered before entering the white dwarf graveyard.These more advanced post-EHB phases are not shown inFigure 1, as they represent objects that are no longer Bsubdwarfs and are, therefore, beyond the scope of this seriesof papers.

Based on Figure 1, we selected as representative of all thesubdwarf B stars, approximately 150 models (models areindicated by Ðlled squares or triangles along each track)


TABLE 1

MAIN MODEL CHARACTERISTICS FOR THE M \ 0.4758] 0.0002 EVOLUTIONARY SEQUENCEM_

SURFACE

CENTER

AGE T effNUMBER (Myr) log L /L

_(K) log g Y

clog T

clog o

clog P

clog q

cc

Helium Core BurningÈEHB Phase Begins

1 . . . . . . . . 1.79 1.156 31462 5.898 0.928 8.066 4.353 20.257 [0.1132 . . . . . . . . 12.06 1.184 31399 5.866 0.811 8.072 4.348 20.244 [0.1403 . . . . . . . . 18.98 1.200 31351 5.847 0.750 8.075 4.342 20.234 [0.1584 . . . . . . . . 21.73 1.206 31331 5.840 0.727 8.076 4.340 20.230 [0.1655 . . . . . . . . 28.83 1.221 31292 5.823 0.673 8.080 4.333 20.220 [0.1846 . . . . . . . . 46.87 1.259 31253 5.783 0.537 8.090 4.321 20.201 [0.2157a . . . . . . . 61.90 1.294 31310 5.751 0.418 8.100 4.317 20.193 [0.2398 . . . . . . . . 73.01 1.324 31454 5.729 0.317 8.110 4.322 20.196 [0.2529 . . . . . . . . 81.14 1.350 31679 5.715 0.236 8.121 4.333 20.209 [0.19610 . . . . . . 87.19 1.371 31968 5.710 0.175 8.132 4.350 20.230 [0.26911 . . . . . . 91.79 1.387 32300 5.713 0.129 8.142 4.371 20.256 [0.24812 . . . . . . 95.14 1.398 32632 5.719 0.099 8.151 4.391 20.282 [0.27513 . . . . . . 97.94 1.407 33014 5.730 0.073 8.160 4.415 20.312 [0.16314 . . . . . . 100.12 1.415 33416 5.743 0.054 8.169 4.440 20.345 [0.27815 . . . . . . 101.71 1.419 33779 5.757 0.041 8.177 4.462 20.374 [0.16016 . . . . . . 103.17 1.424 34223 5.775 0.030 8.185 4.489 20.409 [0.19017 . . . . . . 104.28 1.427 34616 5.792 0.022 8.193 4.512 20.440 [0.20018 . . . . . . 106.49 1.439 36479 5.871 0.005 8.223 4.622 20.585 [0.13219 . . . . . . 107.23 1.495 39863 5.970 0.000 8.254 4.819 20.835 [0.011

Central Helium ExhaustionÈThe Convective Core Vanishes

20 . . . . . . 107.37 1.524 41473 6.009 0.000 8.247 4.924 20.956 0.00021 . . . . . . 107.60 1.522 43301 6.086 0.000 8.245 5.045 21.103 0.000

Helium Shell BurningÈPost-EHB Evolution Begins

22 . . . . . . 108.59 1.606 43771 6.021 0.000 8.177 5.159 21.215 0.00023 . . . . . . 116.79 1.833 47107 5.921 0.000 8.106 5.336 21.436 0.00024 . . . . . . 124.78 2.036 53333 5.935 0.000 8.100 5.517 21.700 0.00025 . . . . . . 129.01 2.186 61358 6.028 0.000 8.080 5.682 21.946 0.00026 . . . . . . 131.66 2.202 72008 6.290 0.000 8.043 5.839 22.181 0.00027 . . . . . . 133.08 1.816 75458 6.757 0.000 8.015 5.954 22.356 0.000

Cooling TrackÈWhite Dwarf Evolution Begins

28 . . . . . . 133.59 1.160 63616 7.116 0.000 8.006 6.023 22.462 0.00029 . . . . . . 134.52 0.663 53340 7.307 0.000 7.956 6.089 22.561 0.00030 . . . . . . 136.77 0.103 42215 7.461 0.000 7.859 6.151 22.654 0.00031 . . . . . . 141.17 [0.412 33135 7.555 0.000 7.756 6.186 22.705 0.00032 . . . . . . 150.06 [0.902 25908 7.618 0.000 7.647 6.206 22.734 0.00033 . . . . . . 172.62 [1.377 20240 7.664 0.000 7.509 6.218 22.752 0.00034 . . . . . . 230.67 [1.845 15764 7.698 0.000 7.331 6.227 22.764 0.00035 . . . . . . 332.43 [2.308 12255 7.724 0.000 7.145 6.232 22.772 0.000

a Reference model used in this study as a representative structure of all the subdwarf B stars.

from the seven evolutionary sequences (about 20 models persequence), for which pulsation calculations were carriedout. The global results obtained for that set of models willbe presented in detail in Paper III. For the purposes of thecurrent paper, namely to discuss the general properties ofeigenmodes in pulsating sdB stars, we focussed our atten-tion on a single, but representative, model of these stars.

2.3. Internal Structure of a Representative ModelStructure 7 belonging to the sequence with M

c\ 0.4758

and (marked by a footnote in TableM_

Menv\ 0.0002 M_1 and a Ðlled square mixed with a cross in Fig. 1) was

chosen as our reference model. From Table 1, we can seethat it has a surface gravity log g ^ 5.75 and an e†ectivetemperature K which,Teff ^ 31,310 (log Teff ^ 4.50)

according to its relatively central position in the log gplane (see Fig. 1), can be considered as represen-[ log Tefftative of most of the subdwarf B stars. In addition, we note

that, with an age of D62 Myr, this model is approximatelyhalf way through the helium core burning phase of EHBevolution, thus keeping it far from the two extremesbetween which sdB stars are positioned (i.e., between theZAEHB and the point of helium core exhaustion).

Figure 2 illustrates the internal structure of the referencemodel through a mosaic of proÐles representing severalphysical quantities. These are shown as functions of theposition inside the star expressed in terms of the fractionalmass depth (log q \ log [1[ m(r)/M], where m(r) is themass contained inside a sphere of radius r, while M \ m(R)is the total mass of the star, R being its radius). All the given


FIG. 1.ÈEHB evolutionary tracks covering the region where sdB stars are observed (open circles : data from R. A. Sa†er 1995, privatelog g [ Teffcommunication). The seven tracks correspond, respectively, from left (high to right (low to objects with a ZAEHB envelope massTeff) Teff), Menv \ 0.0001,0.0002, 0.0007, 0.0012, 0.0022, 0.0032, and 0.0042 The core mass is for all sequences except the Ðrst and third (from the left) which haveM

_. M

c\ 0.4758 M

_In a given sequence, each Ðlled square (for the sequences with or Ðlled triangle (for the sequences withMc\ 0.4690 M

_. M

c\ 0.4758 M

_) M

c\ 0.4690 M

_)

represents a model available for the pulsation calculations. The reference model used in this paper is indicated by the Ðlled square mixed with a cross.

symbols mentioned have their usual physical meaning (seecaption of Fig. 2) and we will not describe in detail here thebehavior of each of these quantities. However, it is inter-esting and relevant to point out some of the structural fea-tures typical of a sdB star that are particularly important inthe pulsation calculations. One of these is the stratiÐednature of the subdwarf B stars, which is best illustrated inthe panel showing the proÐles of the hydrogen (X), helium(Y ), and carbon (C) mass fractions (upper right corner). Twochemical transition zones are expected in these stars. Onemarks the division between the H-rich envelope and theHe-rich core (hereafter referred to as the H/He transition)and is located, in our reference model, near log q ^ [4.0(associated with the drop in X and the increase of Y in thecorresponding panel of Fig. 2). The inner chemical tran-sition zone is slowly building up as these stars follow theirslow but irreversible EHB evolution, where they steadilyconvert helium to carbon/oxygen in their center. A carbon/oxygen to helium transition zone (hereafter called theC-O/He transition) is therefore built up at the edge of theHe burning convective core located, for the reference model,

near log q ^ [0.24 (decrease of Y while C increases in Fig.2 ; see also Table 1). We note, for this model approximatelypositioned at mid-life on the EHB, that about half (in mass)of the available helium has been converted to carbon. Thesetypes of chemical transition regions a†ect, through trappingand conÐnement phenomena, oscillation mode propertiesin other stratiÐed pulsating stars such as white dwarfs (see,e.g., Brassard et al. 1992a, 1992b). Therefore, one mayexpect a priori to observe similar behaviors for eigenmodesin subdwarf B stars (this is explored below).

Another structural peculiarity of all evolutionary EHBmodels is the presence of a fairly massive and extendedhelium convective nucleus (D0.18 for our referenceM

_model) whose existence is betrayed in Figure 2 by compar-ing the adiabatic temperature gradient with the radi-+adative temperature gradient in the context of the+

r,

Schwarzschild criterion for convective instability at thecenter of the star. The extreme sensitivity of the triple-areactions to temperature, leading to a particularly conÐnedarea where the nuclear energy is released, creates the condi-tions for such a convective core to develop as radiation


FIG. 2.ÈProÐles of various physical quantities plotted as functions of the fractional mass depth (log q \ log [1[ m(r)/M], where M is the total mass ofthe star) taken from our reference model. The center of the star is at log q \ 0. From top to bottom and left to right, we show the proÐles of (all quantities arein cgs units) : the mass log m(r) inside a sphere of radius r ; the radius r ; the density log o ; the pressure log P ; the temperature log T ; the adiabatic (+ad),radiative and real (+) temperature gradients ; the and compressibilities ; the and adiabatic exponents ; the N(H II)/N(H), N(He II)/N(He),(+

r), so sT !1, !2, !3and N(He III)/N(H) ionization fractions ; the Rosseland mean opacity log i ; the and opacity derivatives ; the hydrogen (X), helium (Y ), and carbon (C)i,o i,Tmass fractions ; the total and radiative luminosities expressed as fractions of the surface luminosity the nuclear energy rate for(L tot/L surf) (L rad/L surf) L surf ; vHecentral helium burning ; and the nuclear energy rate for hydrogen shell burning.vH

alone cannot carry all the energy Ñux out of that region.The impact of the convective core on pulsations, if any,could present a major interest for the asteroseismologicalstudy of the deep internal structure of B subdwarfs. Thisprospect will be investigated throughout the present andupcoming papers. We also mention the presence of a thinconvective layer located near log q ^ [12.0, which isassociated with a He II/He III partial ionization zone (see thepanel showing the He III/He and He II/HeN(X

i)/N(X),

curves). It is responsible for a small opacity bump (log i inFig. 2), also located near log q ^ [12.0, which had orig-inally caught our attention on the potential of pulsationalinstabilities in subdwarf B stars (see Charpinet et al. 1996).It was, however, demonstrated in the same paper that thesecondary opacity bump located near log^ [9.2, due toheavy metal partial ionization, is, in fact, responsible for thedestabilizing e†ect.

The main features of the internal structure of subdwarf Bstars are also well reÑected through two fundamental quan-tities of stellar pulsation theory. One of these is the Lambfrequency deÐned, for any given value of l, by the relationL

l

Ll2\ l(l ] 1)C

s2

r2 , (1)

where represents the local adiabaticCs(r) \ [(!1P)/o]1@2

sound speed. The second fundamental quantity is thefrequency N deÐned asBrunt-Va� isa� la�

N2\ gA 1!1

d ln Pdr

[ d ln odrB

, (2)

where g \ Gm(r)/r2 is the local gravitational acceleration.The frequency can be rewritten in the moreBrunt-Va� isa� la�practical form (from the standpoint of numerical calcu-


lations ; see Brassard et al. 1991)

N2\ g2oP

sT

so(+ad [ +] B) . (3)

All the symbols mentioned here have their standardmeaning except B, called the Ledoux term, which containsthe speciÐc contribution due to composition gradientsa†ecting the frequency :Brunt-Va� isa� la�

B4 [ 1sT

;i/0

Nc~1sXi

d ln Xi

d ln P,

with sXi

4A L ln PL ln X

i

Bo,T,XjEi

. (4)

corresponds to the number of chemical elements presentNcin the gas and the sum is made over independentN

c[ 1

species due to the additional constraint £i/1Nc X

i\ 1.

We show, in Figure 3, the runs of the fre-Brunt-Va� isa� la�quency and the Lamb frequencies (for l\ 1, 2, and 3) corre-sponding to our reference model. The insert Ðgure alsodisplays the run of the Ledoux term (B) which appears inthe frequency. That last quantity has a zeroBrunt-Va� isa� la�value everywhere in the star except near the very localizedHe/H and C-O/He transitions characterized by steep gra-dients of chemical abundances. These gradients lead,according to equation (4) and as reÑected in Figure 3, topronounced and very localized peaks. Two of these struc-tures are observed in the reference model : one of them,located at log q ^ [4.0, is identiÐed with the boundarybetween the helium core and the hydrogen-rich envelope,while the other, of smaller size and located nearlog q ^ [0.24, corresponds to the transition between theC-O enriched convective nucleus and the helium radiativecore. It is particularly apparent from Figure 3 that thesestructures, well known to produce the conÐnement and

(log N2) and Lamb for l\ 1, 2, and 3FIG. 3.ÈBrunt-Va� isa� la� (log Ll2 ;

only) frequency proÐles (shown as logarithms of their squared value)obtained for the reference model. The Ledoux term proÐle (B) is alsoprovided in the insert panel.

trapping e†ects observed for g-modes in the chemicallystratiÐed white dwarfs, are the main contributions to the

frequency in those particular regions of sub-Brunt-Va� isa� la�dwarf B star structure. We should therefore expect the samephenomena to strongly a†ect the pulsation modes in theseobjects too. The frequency also reÑectsBrunt-Va� isa� la�several structural features of sdB stars that we already men-tioned earlier. In particular, convection zones, satisfying theSchwarzschild criterion appear as negative+ad[ +¹ 0,divergences in log N2. In Figure 3, the two obvious wells inthe proÐle of log N2 are therefore related to the convectivecore (near the center) and the narrow convection region dueto He II/He III partial ionization (near the surface).

The Lamb frequencies shown in the same Ðgure do notreÑect the major aspects of the structure of sdB stars asobviously as the frequency does. We can stillBrunt-Va� isa� la�recognize the main chemical transition near log q ^ [4.0(He/H) which appears as a steep jump of about 0.2 dex inthe proÐles of This feature is produced by a rapidlog L

l2.

change in the adiabatic sound speed due to a steep(Cs)

variation in the density proÐle across the chemical tran-sition zones. Otherwise, the behavior of directly reÑectsL

l2

equation (1) by showing a divergence at the center of thestar (r ] 0) while at the surface. Moreover, we noteL

l2] 0

that the proÐles of are shifted upward with increas-log Ll2

ing values of l, in agreement with the dependence on theangular index expressed in relation (1).

In the context of stellar pulsations, it is instructive to lookagain at Figure 3 while keeping in mind the general resultsobtained from a ““ local approximation ÏÏ analysis describedin ° 15.2 of Unno et al. (1989). In particular, we note that theg-mode propagation zone (located where withp2\ N2, L

l2 ;

p 4 2n/P standing for the angular mode frequency), islocated relatively deep inside the star, approximatelybetween (this is somewhat dependent[0.2[ log q [[5.0on the particular frequency of each g-mode). This provides aÐrst hint that g-modes may essentially be deep core modesin these stars. The opposite seems to be true for p-modeswhose propagation zone (where spans thep2[N2, L

l2)

superÐcial regions of the star, especially the envelope. Thistherefore constitutes a Ðrst hint that p-modes may mostlybe superÐcial envelope modes. We point out that such asituation is usually typical of nondegenerate stars asopposed to the case of degenerate objects such as whitedwarfs for which p-modes are deep modes while g-modesare envelope modes (see Unno et al. 1989). Finally, we alsonote from Figure 3 that an approximate ““ mean value ÏÏ forthe frequency taken from the regionsBrunt-Va� isa� la�between log q ^ [0.2 and log q ^ [10 is roughlylog N2D [3.2. This corresponds to a period PD 250 s.Because we expect from the conclusions of the ““ localapproximation ÏÏ theory and the respective positions of thepropagation zones to Ðnd p-modes and g-modes eigen-frequencies with values respectively greater and smallerthan this ““ mean value,ÏÏ we can already estimate, withoutany pulsation calculation, that the locus between p-modesand g-modes periods will appear around PD 250 s for ourreference model. This estimate will be compared, in the nextsection (see Table 2), with the results obtained from thedetailed numerical computations of the pulsation periods.

3. GENERAL PROPERTIES OF PULSATION MODES

Pulsation calculations were conducted using a recentversion of the adiabatic pulsation code described exten-

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sively in Brassard et al. (1992c). This code is based on aGalerkin Ðnite element method that solves the complete setof four real, coupled oscillation equations resulting from theadiabatic approximation. Minor adjustments to the code,originally applied to the computations of g-modes in whitedwarf models, were implemented in order to optimize thesearch for periods of not only gravity modes, but also f- andp-modes in subdwarf B star models. Computations weremainly undertaken within a 80È1500 s period window,which is the most suitable period range accessible tomodern fast photometric techniques. It also covers theperiod range in which the EC 14026 stars are now known tooscillate. However, we occasionally pushed the calculationsbeyond those limits when we found it necessary for ourexploration of the mode properties. To ensure a goodspatial resolution of the mesh while keeping computationaltimes reasonably short, we used 800 quadratic elements3 toperform these adiabatic calculations. Tests indicated thatnumerical results do not signiÐcantly depend on that partic-ular choice since almost identical period values (di†eringtypically by only D10~2%È10~3%) were obtained within arange of 800È10,000 elements.

We restricted our survey to radial (l\ 0) and nonradial(with l\ 1, 2, and 3) modes. Due to the fact that the disks ofpulsating stars cannot be resolved from Earth (only theintegrated light from the stellar surface is observed), cancel-lation e†ects are expected to prevent the detection of modeswith higher values of l (typically when using photo-lZ 4)metric techniques (see Balona & Dziembowski 1999 andreferences therein). Such modes, although they may well be

3 We remind the reader that this is not the same as 800 zones.

excited in the EC 14026 stars, are of much less interest astheir apparent amplitudes would be too small for them tobe seen in the photometric light curves.

3.1. Adiabatic QuantitiesNumerical results obtained from the adiabatic calcu-

lations described above and carried out for our referencemodel are given in Table 2, where radial (l \ 0) and non-radial (for l \ 1, 2, and 3) modes with periods within the80È1500 s window are listed. For each value of l, the modesare ordered, from top to bottom, by increasing period (P,given in seconds). The p-modesÈwhich possess the shortestperiodsÈare therefore located in the upper part of Table 2.They are characterized by their decreasing periods when theradial order k increases (from k \ 1 to k \ 7 for this partic-ular model in the given period window). Tabulated belowthe p-modes, we Ðnd the unique mode with k \ 0 associatedto each value of l (except l \ 1, since this mode correspondsto a translation of the entire star and has a frequency whichis zero by deÐnition). For l \ 0, it corresponds to the funda-mental radial mode, which behaves like any other acousticmode. For nonradial oscillations with l º 2, these modesare the so-called Kelvin modes (or f-modes), which usuallyhave a somewhat mixed character lying between those ofacoustic and gravity modes. Finally, g-modes are displayedin the bottom part of Table 2 as they possess the longestperiods, these characteristically increasing with growingvalues of k (from k \ 1 up to k \ 14, for g-modes with l\ 3found in that particular model within the chosen periodwindow). As already noticed in our previous papers (see,e.g., Charpinet et al. 1996)Èand this appears to be typical ofall the subdwarf B star models (see Paper II)ÈTable 2shows that a rich spectrum of pulsation modes (both in

TABLE 2

ADIABATIC QUANTITIES FOR A REPRESENTATIVE SUBDWARF B STAR MODEL

l\ 0 l\ 1 l\ 2 l\ 3

P P P Pk (s) log Ekin (s) log Ekin C

kl(s) log Ekin C

kl(s) log Ekin C

kl

7 . . . . . . . . . . . . . . 82.98 40.92 0.0295 . . . . . . . . . . . . . . . . . .6 . . . . . . . . 88.52 40.93 92.47 40.97 0.0187 89.19 40.92 0.0281 85.69 40.94 0.03925 . . . . . . . . 97.90 41.22 105.99 41.36 0.0277 99.15 41.27 0.0485 94.58 41.06 0.03714 . . . . . . . . 112.39 41.36 118.18 41.59 0.0231 113.11 41.38 0.0338 109.39 41.33 0.04153 . . . . . . . . 129.58 42.02 142.08 42.01 0.0185 132.60 42.02 0.0928 123.66 41.83 0.08252 . . . . . . . . 148.31 42.14 160.89 42.79 0.0244 150.12 42.22 0.0679 145.37 42.05 0.04671 . . . . . . . . 176.24 43.49 208.24 43.10 0.0184 185.82 43.66 0.2804 172.44 43.27 0.1431

0 . . . . . . . . 209.12 43.12 . . . . . . . . . 208.06 43.12 0.0262 206.19 43.09 0.0140

1 . . . . . . . . . . . . . . 425.09 48.14 0.4529 277.24 46.66 0.1343 224.09 44.98 0.05122 . . . . . . . . . . . . . . 671.44 47.84 0.4537 406.90 47.28 0.1238 306.59 46.59 0.04083 . . . . . . . . . . . . . . 958.60 47.28 0.4583 569.27 47.02 0.1262 418.65 46.69 0.04404 . . . . . . . . . . . . . . 1248.31 46.36 0.4680 733.89 46.19 0.1357 532.40 45.96 0.05385 . . . . . . . . . . . . . . 1465.47 45.19 0.5000 853.19 45.09 0.1693 610.41 44.96 0.08936 . . . . . . . . . . . . . . . . . . . . . . . 931.69 45.40 0.1567 668.64 45.39 0.07167 . . . . . . . . . . . . . . . . . . . . . . . 1082.23 46.07 0.1497 775.64 46.06 0.06668 . . . . . . . . . . . . . . . . . . . . . . . 1246.14 46.33 0.1522 891.19 46.33 0.06939 . . . . . . . . . . . . . . . . . . . . . . . 1407.05 46.41 0.1541 1006.65 46.39 0.071610 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1100.36 46.50 0.072311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.58 46.24 0.073712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1273.72 45.70 0.075013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387.66 44.77 0.072414 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1451.01 44.04 0.0592


terms of the density and type of modes) populates our selec-ted period window. We also note that the locus betweenp-modes and g-modes, roughly given by the f-mode periods,is located near 210 s, very close to the expected value of 250s crudely derived in ° 2.3 from our inspection of the Brunt-

proÐle.Va� isa� la�Along with the pulsation periods, Table 2 also provides

two additional useful quantities. The Ðrst one is the totaloscillatory kinetic energy of the mode given as(Ekin ;

in Table 2) deÐned in terms of its displacementlog Ekineigenfunctions by (see, e.g., Brassard 1991)

Ekin412

p2PV

on(r) Æ n*(r)dV

\ 12

p2P0

R[m

r2] l(l] 1)m

h2]or2 dr , (5)

where

n(r)\Cmr(r), m

h(r)

LLh

, mh(r)

1sin h

LL/DY

ml (h, /)eipt, with

mh(r)\ 1

p2rAP@

o] '@

B(6)

represents the Lagrangian displacement eigenvector associ-ated with the mode (P@ and '@ are the Eulerian pertur-bations of, respectively, the pressure and the gravitationalpotential). Because the amplitudes of the displacementeigenfunctions are arbitrary in our computationsÈoscillation amplitudes cannot be provided by the lineartheory of pulsations, hence they are arbitrarily normalizedat the surface of the model (see ° carries no3.2.4)ÈEkinabsolute information and only relative comparisonsbetween eigenmodes should be considered as meaningful.With that in mind, one can still recognizes several instruc-tive features from the values given in Table 2. First, we Ðndthat p-modes have, as a general trend, signiÐcantly lowerkinetic energies than g-modes (by approximately 4 orders ofmagnitude). This means the latter have more ““ inertia,ÏÏ orrequire more energy than the former assuming they havethe same surface amplitudes (which is implicitly the casehere since we impose an arbitrary amplitude normalizationto all modes). In equation (5), integration over the entirestar of all the local contributions to the kinetic energy ofmodes is performed on the relative amplitudes of theLagrangian displacement eigenfunctions weighted by a““ or2 ÏÏ factor. Therefore, because the gas density stronglyincreases from D10~5 g cm~3 near the surface to D103 gcm~3 in the core (see Fig. 2), the denser deeper regions ofthe star tend to have much more inertia than the superÐciallayers, thus contributing to signiÐcant increases of the Ekinvalues if the amplitudes of the modes are sufficiently large inthose inner zones. Hence, the signiÐcant di†erencesobserved between the p-mode and g-mode valueslog Ekinsuggest that g-modes oscillate with larger relative ampli-tudes in the deeper regions of the star compared to p-modes.This behavior is fully consistent with the comments wemade in ° 2.3 concerning the locations of the respectivep-mode and g-mode propagation zones, providing a secondhint that g-modes and p-modes are interior and envelopemodes, respectively. In addition to these general comments,we point out that the kinetic energy is not uniformly distrib-uted among the g-modes, as variations of up to 2 orders of

magnitude can be observed, for example between modesl \ 3, k \ 10, and k \ 14. This behavior looks very similarto the signature of trapping/conÐnement phenomenaobserved for g-modes in the compositionally stratiÐed whitedwarf stars, providing a strong hint that those e†ects occurin subdwarf B stars as well, as one might expect from theirchemical stratiÐcation. In parallel, we note that there is nosuch obvious trend in the p-mode spectrum as the kineticenergies of the modes shown in Table 2 steadily decreasewith the increasing value of k (however, this is explored ingreater detail in ° 3.3).

For nonradial modes, Table 2 also provides the dimen-sionless Ðrst-order rotation coefficients suitable for slow,C

klrigid rotation of angular speed X \ )z (z being the unitvector of the symmetry axis of the nonradial pulsationmodes). Rotation is responsible for frequency (or, equiva-lently, period) splitting between modes having the samequantum numbers k and l, but di†erent indices m (di†erentm modes are degenerate and all have identical periods inperfectly spherically symmetric models). That splitting isgiven, to Ðrst order, by the relation (see, e.g., Brassard 1991)

pklm

\ pkl(0)]m(1[ C

kl)), with

Ckl

\ p22Ekin(k, l)

P0

R[2m

rmh] m

h2]

klor2 dr , (7)

where stands for the degenerate angular eigenfrequencypkl(0)

obtained when considering a system without rotation (thecorresponding periods are tabulated in Table 2). For p-modes, these coefficients tend to be small, with valuesaround 10~1È10~2 (except for the two modes with k \ 1,l \ 2 and 3). Therefore, if we ignore, for a moment, thosetwo apparently peculiar and isolated modes, coefficientsC

klcan be neglected in equation (7) for p-modes, leading toroughly equally spaced (by *p^ )) multiplets in the fre-quency domain. For g-modes, the situation is less uniformas a correlation between the values of and the angularC

klindex l is obvious in Table 2. This calculated trend is consis-tent with the asymptotic theory predictions for g-modesthat (see Brickhill 1975), which leads toC

klasy ^ [l(l ] 1)]~1

0.17, and 0.083 for, respectively, l\ 1, 2, and 3 ;Cklasy ^ 0.5,

close to the values presented in Table 2 for most cases. Ineach g-mode spectrum with a given l, we also note somemode to mode Ñuctuations in the values of which areC

kl,

correlated to the variations observed for kinetic energies.

3.2. T he Gravity Mode SpectrumWe saw that a signiÐcant number of gravity modes are

present in the interesting 80È1500 s period window. Hence,they o†er an interesting potential for asteroseismologicalstudies of subdwarf B stars. To date, however, the modesobserved in all known EC 14026 stars show periods consis-tent with those of radial and nonradial p-modes ; no g-modehas been unambiguously detected so far in these objects.Nevertheless, we claim that it is still worth considering themfor at least the three following reasons :

1. They constitute one major aspect of the whole pulsa-tion mode structure in subdwarf B stars that is useful toexplore in order to develop a more global understanding ofmode behavior in these stars.

2. Some g-modes may be unstable, and thus be observ-able, especially in pulsating B subdwarfs such as PG1605]072 which has a surface gravity signiÐcantly lower


than that of the bulk of the EC 14026 stars (Fontaine et al.1998 ; Charpinet 1998).

3. In certain regions of parameter space, g-modes arefound to interact with p-modes through period collisionsand mode bumping processes (see Paper II), commonlyreferred to as avoided crossings (see Aizenman, Smeyers, &Weigert 1977 ; Roth & Weigert 1979). In such cases, itbecomes clear that p-mode properties (periods, eigen-functions, etc.) will be directly a†ected by g-modes, hencejustifying that we take a closer look at their behavior.

3.2.1. Kinetic Energies and Period Spacings

In the previous subsection, we mentioned that g-modeshave a nonuniform kinetic energy distribution that can berelated to trapping and conÐnement phenomena alreadyencountered in white dwarfs. This kinetic energy behavior ismore strikingly illustrated in the upper panel of Figure 4 forg-modes with di†erent values of l (l\ 1, 2, and 3) havingperiods between 80 and 3000 s, and spanning values of theradial order from k \ 1 to k D 30 (for l\ 3 ; see the Ðgurecaption for details). Similarities with the white dwarfg-mode structure is particularly apparent when Figure 4 iscompared to Figure 1 of Brassard et al. (1992b). In thewhite dwarf context, g-modes located at local minima of thekinetic energy proÐles (for example, modes with k \ 5 andl\ 1, 2, and 3 in Fig. 4) are called trapped modes, whileg-modes close to local maxima (for instance, modes withk \ 10 and l\ 1, 2, and 3) are referred to as conÐnedmodes. Brassard et al. (1992b) show that the He/H chemicaltransition between the H-rich envelope and the He-rich

FIG. 4.ÈIllustration of the g-mode spectra associated with l\ 1 (dottedcurve), 2 (dashed curve), and 3 (solid curve) for a representative sdB modelwith K, log g \ 5.75,Menv \ 0.0002 M

_, M

c\ 0.4758 M

_, Teff \ 31,310

and an age of D62 Myr since the ZAEHB. The upper panel shows therelative kinetic energy of modes (given in the logarithmic form vs.log Ekin)their periods (expressed in seconds). The lower panel plots the periodspacing between two consecutive g-modes given in(*P\ P

k`1,l [ Pk,lseconds) vs. the period Each curve starts with the mode k \ 1,(P\ P

k,l).which has for any given value of l, the shortest period among the g-modesbelonging to that series. The periods of g-modes increase with radial orderk.

layer is mainly responsible for the trapping e†ect in DAwhite dwarfs (the C-O/He chemical transition between thedegenerate core and the He-rich layer has a much smallere†ect). We demonstrate below that this terminologyÈdeveloped for the white dwarf situationÈremains fullyappropriate for subdwarf B stars as well because theseobjects experience g-mode trapping/conÐnement e†ectsthat have similar physical origins. It is interesting to notethat, as for g-modes in white dwarfs, the minima in log Ekinare very narrow, corresponding to a very well identiÐedmode, while the maxima are rather broad, spanning aperiod range corresponding to 3È4 modes with higherkinetic energies. Therefore, mode selection e†ects are moreefficient in trapping situations (as opposed to conÐnementsituations). These properties are, again, very similar to whatis observed from white dwarf g-mode calculations.

Another manifestation of the trapping/conÐnemente†ects becomes apparent when period spacings betweenadjacent modes (i.e., with *k \ 1) are considered. The lowerpanel in Figure 4 provides that information (*P\ P

k`1vs. k is the radial order) for the same g-modes[ Pk

Pk;

(with l \ 1, 2, and 3) discussed previously. The white dwarfcounterpart corresponding to that panel in Brassard et al.(1992b) is given by their Figure 3, showing once more thatstrong similarities exist between the two plots. Trappedmodes and their immediate neighbors are located at theminima in *P, leading, in principle, to an observable signa-ture of that phenomenon. Interestingly, one of the minimaobserved in the lower panel (the second one starting fromthe left and corresponding to modes with k \ 9È10) is notassociated with a minimum in the upper panel as expectedfor trapped modes, but rather corresponds to a localmaximum. We will argue later that these particular modesare indeed subject to a ““ double conÐnement ÏÏ e†ect fromboth the He/H and C-O/He chemical transitions. This con-Ðguration seems also to provoke a concentration of periods(smaller *P) but does not appear frequently in the g-modespectrum (in the reference model, it occurs only once insidethe entire period window covering g-modes from k \ 1 tok \ 30).

3.2.2. Asymptotic Description of g-Mode Trapping

In Brassard et al. (1992a), a semianalytic approach basedon TassoulÏs asymptotic theory (Tassoul 1980) was devel-oped in order to study the e†ectsÈon g-modesÈof a dis-continuity in the frequency generated byBrunt-Va� isa� la�chemical stratiÐcation in white dwarfs. Since subdwarf Bstars are also compositionally stratiÐed, this formalism isrelevant in the present context as well. However, somemodiÐcations to the Brassard et al. (1992a) approach arerequired in order to account for some fundamental structur-al di†erences between white dwarfs and hot B subdwarfs.Thus, it is necessary to proceed through the entire formal-ism in order to derive asymptotic expressions suitable forsdB stars. We recall that the Tassoul (1980) formalism wasderived from the Cowling approximation (the perturbationsof the gravitational Ðeld are neglected in the adiabatic pul-sation equations ; see Unno et al. 1989) and that asymptoticexpansions are, strictly speaking, valid only for sufficientlyhigh radial order modes. This should be kept in mind as wealso consider low-order g-modes (with k D 1) in our study.However, the accuracy of the asymptotic approach remainssufficient for our purpose of gathering additional insight onthe behavior of g-modes in sdB stars.


Asymptotic solutions for g-modes can be derived bydividing a stellar model into as many zones as there areradiative and convective regions present in the star. Formost subdwarf B stars that are still in the central helium-burning phase, the appropriate generic structure is a two-zone model featuring a convective inner region (theconvective nucleus of the star, see ° 2.3) surrounded by aradiative outer region including both the radiative heliumcore and the hydrogen-rich envelope (Tassoul 1980 ; here-after, references to the equations from that paper will begiven with the preÐx ““ T ÏÏ). Note that a thin convective layeris seen in the H-rich envelope due to He II/He III partialionization which, rigorously, should be treated as a thirdzone in the asymptotic model. However, in order to keepour description as simple as possible, and because the e†ectsof that convective region on g-modes indeed appear to bevery small, the third region has not been taken into account.In addition, two structuresÈnow referred to as““ discontinuities ÏÏÈin the frequency shouldBrunt-Va� isa� la�a†ect the behavior of gravity modes through trapping andconÐnement :

1. The C-O/He chemical transition located at the bound-ary of the central convective core, which will be neglecteddue to its small trapping inÑuence (see ° 3.2.3) ;

2. The He/H transition located at the bottom of theH-rich envelope, which actually provides most of the trap-ping and will be treated as proposed in Brassard et al.(1992a ; hereafter, references to equations from that latterpaper will be given with the preÐx ““ B ÏÏ).

Solutions in the convective inner region of the two-zonemodel are given by (T80) near x \ 0 and by (T96) for x ¹x1(where is the position of the convective nucleusx14x

cboundary ; and x indicates the position in the model as afraction of the total radius R of the star, i.e., x \ r/R). Con-tinuity conditions between those two solutions are directlygiven by (T117) and (T118) (where is an arbitrary con-k

istant which depends on amplitude normalization, while k11and are constants associated to the solution neark12x \x1) :k12 \ 0 , (8)

k11 \ kiexp

AJl(l] 1)p

P0

x1 oN o

xdxB

. (9)

In the outer radiative zone, solutions appropriate for x ºare given by (T97). To Ðrst order, for the andx1 S1 S2quantities deÐned by Tassoul (1980), we have :

S1\ Q1(v1b)k11 ] Q3(v1b)k12 (10)

\ k11Av1b

3B1@2C

J~(1@3)Av1b

pB

] J(1@3)Av1b

pBD

4 S1~ (11)

S2\ Q2(v1b)k11 ] Q4(v1b)k12 (12)

\ k11Av1b

3B1@2C

J~(2@3)Av1b

pB

[ J(2@3)Av1b

pBD

4 S2~ (13)

where condition (8) was used to cancel all terms proportion-al to The quantities are the usual Bessel functionsk12. J

n(x)

of the Ðrst kind, while the argument is deÐned asv1b

v1b(x)\ Jl(l] 1)Px1

x oN o

xdx . (14)

The solution near the surface of the star (x \ 1) comes from(T82) leading, to Ðrst order, to

S1\ k0 v0(1@2) Jne`1Av0

pB

4 S1` (15)

S2\ [k0 v0(1@2) Jne

Av0pB

4 S2` (16)

where is an arbitrary constant, is the polytropic indexk0 neof the superÐcial layers (close to the surface), and is givenv0by the relation

v0(x) \ Jl(l ] 1)Px

1 oN o

xdx . (17)

In the asymptotic theory, g-modes with p2> N2(corresponding to high radial order modes) are considered.Therefore, expansions of Bessel functions appropriate forhigh values of the argument can be applied to S1`, S1~, S2`,and leading toS2~,

S1`P k0 sinAv0

p[ n

en

2[ n

4B

(18)

S2`P [k0 cosAv0

p[ n

en

2[n

4B

(19)

S1~P k11 sinAv1b

p] n

4B

(20)

S2~P [k11 cosAv1b

p]n

4B

. (21)

We can now match these two sets of solutions at x \ xH,where the He/H discontinuity in the fre-Brunt-Va� isa� la�quency is located. Following Brassard et al. (1992a), weidealizeÈfor simpliÐcation purposesÈthe chemical tran-sition by a true discontinuity characterized by the values ofN, respectively, above (N`) and below (N~) the discontin-uity along with a parameter a deÐned as a24 N`/N~.Then, the continuity conditions between these solutions canbe written, at x \xH,

S1` \ aS1~, aS2` \ S2~ , (22)

or in the more useful form

S2~S1~

\ a2 S2`S1`

, (S1~)2] (S2~)2\ 1a2 (S1`)2] a2(S2`)2 .

(23)

We can reexpress the continuity conditions (eq. [23]) byusing, along with equations (18)È(21), the relations

p \ 2nP

, v1b(xH) \ 2n2%0,lrad [ 2n2

%H,l, and v0(xH)\ 2n2

%H,l,

(24)

with P corresponding to the period, while quantities %H,land are deÐned as%0,lrad

%0,lrad4%0rad

Jl(l ] 1), with %0rad4 2n2

APrc

R oN o

rdrB~1

,

(25)


%H,l4%H

Jl(l] 1), with %H 4 2n2

APrH

R oN o

rdrB~1

.

(26)

We can then write the continuity conditions (eq. [23]) as

cotCA 2P

%H,l[ 2P

%0,lrad] 12B n

2D

\ a2 cotCA 2P

%H,l[ n

e[ 1

2B n

2D

(27)

and

k112 \ k02a2Csin2

A Pn%H,l

[ nen

2[ n

4B

]a4cos2A Pn%H,l

[ nen

2[ n

4BD

. (28)

These two equations can be compared to relations (B37)and (B38) of Brassard et al. (1992a) to which they are quitesimilar except for the left-hand side term in the transcen-dental equation (27). This di†erence is a direct consequenceof the presence of the convective nucleus in B subdwarfsmodels that is missing in white dwarf stars. Therefore, it isworth pointing out that the convective core directly a†ectsthe global structure of the g-mode period spectrum gener-ated by these equations. Despite that di†erence, the presentset of equations clearly suggests that g-modes in subdwarf Bstars should show behavior very similar to thoseÈfullyexplored in Brassard et al. (1992a)Èof g-modes in whitedwarfs. Without undertaking such a detailed study for sdBobjects, we can still stress a few interesting properties. Wenote that for a model without a discontinuity in the Brunt-

frequency (a \ 1), equation (27) is straightforwardlyVa� isa� la�solved by equalizing each cotangent argument (modulo n).The extracted uniform period spectrum for g-modes is thengiven by the same equation evaluated in Tassoul (1980)(T127) :

Pk,l \

Ak ] n

e2B%0,lrad\

Ak ] n

e2B %0rad

Jl(l] 1). (29)

It is interesting to note that, contrary to the uniform periodspectrum obtained for white dwarf g-modes (see, e.g., eq.[B41]), the reduced periods from equation (29)ÈdeÐned as

independent of the angular index l.Pk,l[l(l] 1)]1@2Èare

Hence, in subdwarf B stars, all g-modes with a given radialorder k should have approximately the same reducedperiod. If we now consider the opposite limit, correspond-ing to perfect trapping (a ] O), the solution is obtained bylooking for zeros of the right cotangent in equation (27). It isthen straightforward to show that the period spectrum isgiven by (see also eq. [B42])

Pi,l\

Ai] n

e2

] 14B%H,l\

Ai] n

e2

] 14B %H

Jl(l] 1), (30)

where i represents the trapped mode order (i\ 1, 2, . . . ,respectively, for the Ðrst, second, . . . , trapped mode) or,equivalently, the number of nodes of the radial displace-ment eigenfunction located in the H-rich envelope of thestar. Again, equation (30) clearly indicates that all trappedmodes having the same order i should possess approx-imately the same reduced period. Still following Brassard et

al. (1992a) by noting that a perfectly trapped mode mustsatisfy simultaneously equations (29) and (30), we obtain therelation between indices i and k :

i \ k%0rad%H

] ne2A%0rad

%H[ 1B

[ 14

. (31)

Of course, the last equation represents an idealized case thatis impossible to reach in realistic situations, since k and i arenecessarily integer numbers, while are indepen-%0rad, %H, n

edent real quantities. Perfect trapping of g-modes thus doesnot exist in our models. Nevertheless, we can still obtaintwo interesting pieces of information directly from equation(31). First, the relation between i and k does not depend on l.Hence, trapped modes with a given index i, but with di†er-ent values of l all have the same radial order k. Second, onecan approximately derive the spacing *k between two con-secutive trapped modes (i.e., for which *i \ 1) with theuseful relation

*k ^%H%0rad

. (32)

Considering now equation (27) for an arbitrary value of thea parameter, one can show that the optimal condition forg-mode trapping (the one that minimizes the kinetic energy)is achieved when all terms in equation (27) are closeÈbutnot strictly equalÈto zero (Brassard et al. 1992a). Havingthese terms be strictly equal to zero at the same time wouldcorrespond to perfect trapping situations which, as we justmentioned, cannot be reached in real models due to con-straints on the indices k and i. An approximate, but generalformulation giving the period spectrum can be obtained byusing Taylor expansions of the relation (27) near trappedmodes (e.g., where all cot^ 0). Keeping only Ðrst-orderterms, we end up with the relation

Pki,l\

1

Jl(l ] 1)

Ck ] i

Aa2[ 1

B] a2n

e2

] a24

] 14D

]%0rad%H

(a2[ 1)%0rad] %H. (33)

As before, k represents the usual radial order deÐning eacheigenmode, while i is the index giving the trapped modeorder. We point out that relation (33) becomes equations(29) and (30) in the limits of a ] 1 and a ] O, respectively,as it should. Again, we notice from equation (33) that thereduced periods are independent of the angular index l,contrary to the white dwarf situation, where the periods aregiven by the corresponding equation (B48). From relation(33), we can also derive the period spacing between consecu-tive trapped modes

Pi`1 [ P

i\ *k ] a2[ 1

Jl(l ] 1)

%0rad%H(a2[ 1)%0rad] %H

, (34)

and assuming that the mean value of *k between thosemodes is given by equation (32), we Ðnally obtain

Pi`1 [ P

i\ %H,l , (35)

as one should have expected from equation (30). This result,identical to what we obtained in the white dwarf context,shows that the spacing between reduced periods of trappedmodes is constant and independent of l. T hus, the quantity


depends both on the envelope structure of the star and on%Hthe position of the He/H transition. It is therefore a directrHfunction of the mass of the H-rich envelope. From relations(35) and (32), the ““ mean ÏÏ period spacing between two con-secutive modes can be estimated as

Pk`1 [ P

k^

Pi`1[ P

i*k

\ %0,lrad . (36)

This result also is very similar to the white dwarf counter-part except for the fact that the constant is evaluated%0radonly for radiative regions of the model and then depends onboth the position of the convective core boundary and(r

c)

on the structure of the surrounding radiative layers. Quiteremarkably, we note that the mean period spacing and,from equation (33), the g-mode periods in general, althoughthey seem to depend on the location of the convective coreboundary through the quantity do not show any con-%0rad,tributions in their associated asymptotic relations from thestructure of the inner convective core itself (i.e., nowhere inthe asymptotic equations there is any integration over thesecentral layers), meaning that details of the core physics arenot critical. The interesting suggestion is that the convectivecore has almost no direct inÑuence on g-modes, except forthe location of its outer boundary. Further investigationson this topic are given in ° 3.2.5. In this context, it is alsoworth mentioning that the di†erences that can existbetween white dwarfs and B subdwarfs concerning theirg-mode period structure are mainly caused by the presenceof this convective core in the B subdwarfs. However, thisconvective core, triggered by central helium burning, doesnot survive to the entire phase of B subdwarf star evolutionand disappears when central burning fades out. Conse-quently, the structure of g-modes becomes ““ white dwarflike ÏÏ at some point in the advanced stages of EHB evolu-tion and the equations derived in Brassard et al. (1992a) canthen be directly applied.

Figure 5 illustrates the g-mode spectrum of the referencemodel already shown in Figure 4 but, this time, in a slightlydi†erent way that allows direct comparisons with theasymptotic properties derived previously. The kinetic ener-gies and the reduced period spacings between adjacentg-modes (same value of l and *k \ 1) are provided in termsof the reduced periods P[l(l] 1)]1@2. As before, only valuesof l\ 1, 2, and 3 (corresponding, respectively, to dotted,dashed, and solid curves) for modes with k \ 1 to k ^ 25are considered (as this range in k is sufficient to allow com-parisons with the predictions from asymptotic theory).

One can readily see from Figure 5 that all the asymptoticbehaviors derived using the simple generic model discussedearlier are observed in the detailed numerical calculations.For instance, modes with the same radial order k but withdi†erent values of l have, as predicted, approximately thesame reduced periods. As expected from the assumptions ofasymptotic theory, this property is more closely obeyed forhigher order modes, although it is remarkable that devi-ations from the theory at low values of k remain rathersmall. In the same vein, trapped modes located at localminima in the plot (modes with k \ 5, 14, and 20log Ekincorresponding to i\ 1, 2, and 3, respectively ; i being thetrapped mode order) also approximately have, as expected,the same reduced period. The next trapped modes corre-sponding to i\ 4, not shown in Figure 5, are modes withk \ 26 and we veriÐed that the trapping/conÐnement

FIG. 5.ÈSame as Fig. 4 except that reduced periods (Pk,l[l(l ] 1)]1@2,

given in seconds) are now considered instead of the usual periods. Theperiod window has also been extended to cover the Ðrst three trappedmodes encountered in the period spectrum. Two relevant asymptoticvalues, s and s (respectively, horizontal long-dashed%0rad^ 367 %H ^ 2020line and solid segment in lower panel), calculated from the unperturbedquantities of the reference model, are also provided.

pattern reproduces itself very regularly every *k \ 6 forhigher order modes. In terms of the reduced periods, thespacing between trapped modes is observed to be constantand roughly s. For modesP

i`1 [ Pi[l(l ] 1)]1@2D 2000

with sufficiently high radial orders, this compares very wellto the asymptotic value s derived from numeri-%H ^ 2020cal integration of equation (26) using appropriate physicalquantities available from the reference model structure (rHwas taken to be the radius where the He/H Ledoux peakreaches its maximum value ; that choice is, however, notcritical for our comparisons purposes). We recall that thevalue of directly depends on the position of the%H rH,transition between the He-rich core and the H-richenvelope, and therefore is an explicit function of Menv.Hence, period spacings between trapped g-modes couldbecome a powerful tool for measuring this quantity, provid-ing one could detect sufficient numbers of g-modes in pul-sating B subdwarfs.

3.2.3. Origin of g-Mode Trapping

The mode trapping and conÐnement calculated in sdBstar models obviously show very similar properties withtheir white dwarf counterparts. Therefore, one might expectthat they should have the same physical origins. Theasymptotic results derived in the previous subsection usingthe two-zone model strongly suggest that the He/H chemi-cal transition zone between the H-rich envelope and thehelium core is deeply involved in those phenomena, as is thecase for white dwarfs. This can be unambiguously demon-strated by conducting a direct numerical experiment. Fol-lowing Brassard et al. (1992c), we recalculated the entireperiod spectrum for two artiÐcially modiÐed versions of thereference model. One of these models is obtained by forcing


the value of the Ledoux term to zero near the C-O/Hechemical transition. This leads to the fre-Brunt-Va� isa� la�quency proÐle shown in Figure 6 (labeled ““ B(C-O/He)\ 0 ÏÏ) which does not exhibit the near central Ledoux peakobserved in the reference model (labeled ““ Ref.ÏÏ in theÐgure).

The second modiÐed model is obtained by setting theLedoux term to zero in all regions. This eliminates both theC-O/He and He/H transition peaks from the Brunt-Va� isa� la�frequency proÐle (see the curve labeled ““B\ 0 ÏÏ in Fig. 6).Since gravity mode properties are known to be closelyrelated to the proÐle of the frequency inside aBrunt-Va� isa� la�star, eliminating the Ledoux components from that quan-tity strongly minimizes impacts of the chemical transitions,and therefore their expected trapping e†ect on g-modes.This is clearly illustrated in Figure 7, where the kineticenergy and period spacing (*P) proÐles for(log Ekin)g-modes with l\ 3 are plotted versus their correspondingperiods (P) for each of the three models shown in Figure 6.

For the reference model, we recognize the typical struc-ture of trapping/conÐnement phenomena shown in Figure 4and Figure 5 and discussed in earlier sections. The same setof g-modes plotted for the artiÐcially modiÐed ““ B(C-O/He)\ 0 ÏÏ model exhibits a shape very similar to the unmodiÐedg-mode spectrum. This clearly demonstrates that theC-O/He chemical transition has little inÑuence on the mainmode trapping structure observed in sdB models. We point

frequency proÐles for the reference modelFIG. 6.ÈBrunt-Va� isa� la�(upper curve, also shown in Fig. 3), and two artiÐcially modiÐed versions ofthat model. The Ðrst version has the Ledoux term B forced to zero in theC-O/He chemical transition zone (model ““ B(C-O/He)\ 0 ÏÏ correspondingto the middle curve), while the second one has B forced to zero everywherein the star (model ““ B\ 0 ÏÏ corresponding to the lower curve). For clarity,the two lowest proÐles have been translated vertically to avoid overlaps.

FIG. 7.ÈThis Ðgure is similar to Fig. 5 except that now only g-modeswith l\ 3 belonging to the reference model (solid curves), the ““ B(C-O/He)\ 0 ÏÏ model (dotted curves), and the ““B\ 0 ÏÏ model (dashed curves) areshown. Values of the asymptotic quantities s and s%0,3rad ^ 106 %

H,3 ^ 583(respectively, the horizontal, long-dashed line and the solid segment inlower panel) are also given.

out, however, that it still has a slight impact on the periodsof g-modes (revealed by the small period shifts observed inFig. 7). We note also that it is responsible for a second-ordertrapping/conÐnement e†ect illustrated here by modeshaving k D 10, and especially visible in the *P versus Pdiagram where a secondary structure in the reference proÐleis missing in the ““ B(C-O/He) \ 0 ÏÏ model. In ° 3.2.1, weinterpreted these modes to be subject to a ““ doubleconÐnement ÏÏ e†ect both by the He/H and C-O/He tran-sitions ; the contribution of the C-O/He transition is wellidentiÐed here from our numerical experiment. As thisfeature is observed only once in the entire period windowshown in Figure 7, double conÐnement clearly occurs muchless often than standard trapping/conÐnement and has asmaller overall impact on g-mode behavior. The proÐlesassociated with the third model (modiÐed to have B\ 0everywhere inside the star) are characterized by the absenceof the trapping structures. Compared to the ““ B(C-O/He)\ 0 ÏÏ model previously discussed, only the inÑuence of theHe/H transition was minimized in the ““B\ 0 ÏÏ model. Thisclearly conÐrms our expectations that most of the trappingand conÐnement e†ects seen in B subdwarf models are con-trolled, as is also the case in white dwarfs, by the chemicaltransition between the helium-rich core and the hydrogen-rich envelope. As predicted by the asymptotic theory,removal of the main chemical discontinuity leads to auniform period spectrum for g-modes which is closelyapproached by our ““ B\ 0 ÏÏ model. This is illustrated bycomparing, for instance, the predicted asymptotic quantity

corresponding to the mean *P value between%0,3rad ^ 106 sadjacent modes to the calculated *P versus P curve for the““B\ 0 ÏÏ model (Fig. 7, lower panel). The match is excellentconsidering that the small wavy structure still observedaround the asymptotic mean value in the g-mode spectrum


of the ““ B\ 0 ÏÏ model comes from the fact that, by takingB\ 0 in the frequency, we minimized, butBrunt-Va� isa� la�did not entirely suppress, the trapping inÑuence of thechemical transitions. Physical ““ discontinuities ÏÏ at thesetransitions, producing minor but nonzero trapping/conÐnement e†ects, also occur in other physical quantitiessuch as the density (see the corresponding proÐle given inFig. 2). These cannot be easily removed from our numericalstellar models and produce the remaining ““ ripples ÏÏ in the*P and Ðgures.log Ekin

3.2.4. Gravity Mode Displacement Eigenfunctions

Looking at g-mode displacement eigenfunctions allowsus to obtain additional useful information on the globalproperties of g-modes, especially in the context of trappingand conÐnement e†ects. Figure 8 shows a plot of both theradial upper panels) and horizontal lower panels)(y1 ; (y2 ;displacements, where and arey14 m

r/r y24 (P@/o] '@)/gr

two of the dimensionless eigenfunctions introduced byDziembowski (1971). In Figures 8a and 8b, three represen-tative g-modes with l \ 3 are plotted : one trapped (k \ 14),one conÐned (k \ 9), and one ““ normal ÏÏ mode (k \ 12). Theamplitudes of the eigenfunctions have been arbitrarilynormalized at the surface of the model by the constraint

The position, near log q ^ [3.94, of they12] y22\ 1.maximum of the Ledoux peak associated with the He/Htransition zone which is responsible for the trapping ofg-modes is also indicated.

It is apparent, from Figure 8, that the trapped modeeigenfunctions have a node located very close to the He/Hinterface. For it is positioned slightly above the tran-y1,sition (log q ^ [3.98), while it lies slightly below for y2(log q ^ [3.76). This observation is consistent with Bras-sard et al. (1992a) discussion showing, in the similar contextof white dwarfs, that it corresponds to the optimal conÐgu-ration minimizing the kinetic energy, and therefore

FIG. 8.ÈDisplacement eigenfunction proÐles and see text for details) for a sample of representative g-modes (having l\ 3). The depth in the model(y1 y2 ;is expressed, as usual, in terms of the internal mass fraction log q. Panels (a) and (b) show, respectively, and for a trapped mode (k \ 14 ; solid curve), ay1 y2““ normal ÏÏ mode (k \ 12 ; dashed curve), and a conÐned mode (k \ 9 ; dotted curve). Panels (c) and (d) show, respectively, and for two g-modes (withy1 y2l\ 3) having approximately the same period (D1450 s). One of them has been calculated using the reference mode (trapped mode with k \ 14 ; solid curve),while the other comes from the modiÐed ““ B\ 0 ÏÏ model (mode with k \ 12 ; dotted curve). The location of the He/H transition is indicated in all panels by avertical, long-dashed curve.


resulting to the trapping of g-modes in the envelope of thestar. The origin of lower kinetic energies for the trappedmodes is also clearly revealed in Figure 8 as a consequenceof their signiÐcantly smaller relative amplitudes in thedeepest regions of the star (namely below the He/H chemi-cal transition, in the helium-rich core). We recall that, dueto their much higher densities, those deeper regions usuallypossess a very signiÐcant weight when contributing tokinetic energies of modes. Therefore, trapped g-modes insdB stars, much like their white dwarf counterparts, aremodes that are almost entirely reÑected back to the surfacefrom the interface between the helium-rich core and thehydrogen-rich envelope in which they are literally““ captured.ÏÏ This is also well illustrated in a slightly di†erentway in Figures 8c and 8d. The eigenfunctions for twog-modes with nearly identical periods and belonging,respectively, to the reference model and the ““B\ 0 ÏÏ modelpresented in ° 3.2.2 are compared. The Ðrst one (solid curves)is the same trapped mode presented in Figures 8a and 8b,while the second one (dotted curves) would have met theconditions of trapping if the He/H transition e†ects had notbeen minimized by removing the Ledoux component fromthe frequency in the model. As expected, theBrunt-Va� isa� la�eigenfunctions of the two modes almost perfectly match inthe H-rich envelope, above the He/H transition. Thepinching e†ect responsible for the decrease of the ampli-tudes of trapped modes in the helium core occurs at thetransition location only for the mode taken from the refer-ence model, while the corresponding mode in the ““ B\ 0 ÏÏmodel remains una†ected.

A reversed situation exists for the conÐned mode shownin Figures 8a and 8b. This time, and each possesses ay1 y2node located, respectively, below (log q ^ [3.65) andabove (log q ^ [4.00) the He/H transition region.However, contrary to trapped modes, it is impossible to getboth nodes simultaneously very close to the transitionregion, which makes the conÐnement process much less effi-cient than the trapping e†ect in selecting modes. Largerthan normal relative amplitudes in the deeper helium coreregions are observed for these conÐned modes, explainingwhy they have larger kinetic energies. g-modes that we con-sidered as ““ normal ÏÏ are neither trapped nor conÐned, andtherefore they have eigenfunction propertiesÈtheir relativeamplitudes in the helium core region, for instanceÈlying inbetween those just discussed for trapped and conÐnedmodes. These modes have (in terms of trapping or conÐne-ment efficiency) the worst possible conÐguration of nodepositions relative to the He/H transition region (herelog q ^ [3.75 and log q ^ [4.25 for, respectively, andy1and they therefore remain almost una†ected by itsy2),reÑective e†ect. Since nodes of g-mode eigenfunctionsmigrate toward the surface of the star as the radial order kincreases (see, e.g., Fig. 13), trapping situations alternatewith conÐnement conÐgurations on the regular basisobserved in Figures 4, 5, and 6.

3.2.5. g-Mode Period Formation Regions

To complete our exploratory tour of gravity modeproperties in subdwarf B stars, we now need to evaluatewhere, inside the star, the regions providing the dominantcontribution to the periods are located. We already showedin our earlier discussions, several indications suggestingthat g-modes are essentially interior modes. These hints areas follows :

1. From Figure 3, the structure of the andBrunt-Va� isa� la�Lamb frequency proÐles indicate a propagation zone forg-modes which is located deep inside the star.

2. In Table 2, g-modes have the largest kinetic energies ofall the calculated modes suggesting that they oscillate withsigniÐcant relative amplitudes deeper in the star than othermodes (i.e., f- and p-modes).

3. Figure 8 conÐrms that most g-modes have rather largerelative amplitudes deep inside the star, well below theH-rich envelope.

However, a more deÐnitive answer can be acquired bylooking at the ““ weight functions ÏÏ associated with a sampleof representative g-modes.

It has been shown (see, e.g., Unno et al. 1989) that p2, thesquared value of the eigenfrequency for a given mode, canbe expressed as an integral form of the corresponding eigen-functions :

p2 PP0

RF(y1, y2, y3, y4 ; r)dr , (37)

where are deÐned in ° 3.2.4, and andy1, y2 y34'@/gr,are the other dimensionless eigenfunctionsy44 g~1d'@/dr

introduced by Dziembowski (1971). In terms of these quan-tities, one can demonstrate that (see, e.g., eq. [105] of Bras-sard 19914 ; see also Kawaler, Winget, & Hansen 1985)

F(mr, P@, '@ ; r) \

Cmr2N2] (P@)2

!1 Po]A P@!1P

] mrN2gBD

or2 .

(38)

Hence, the relative value of the weight function F taken at agiven position r inside the star describes the local contribu-tion of a given region to the period of a mode. The weightfunction therefore constitutes a very powerful diagnostictool for establishing what are the main mode period forma-tion regions.

Figure 9 illustrates the proÐles (each normalized to amaximum value of 1) of the weight functions correspondingto 3 g-modes (with l \ 3) representative of a trapped (k \ 5),conÐned (k \ 3), and ““ normal ÏÏ (k \ 4) modes. We notethat, because each weight function has been normalizedseparately, one can only compare the three plotted curveson a qualitative basis (i.e., by their respective shapes). Thisis, however, fully appropriate in the context of our dis-cussion. The location of the C-O/He and He/H chemicaltransitions are indicated in the Ðgure, as they constituteimportant reference points in our model. For the conÐnedmode (dashed curve), the maximum of F is identiÐed withthe C-O/He transition and the largest amplitude portion ofthe weight function is clearly located in the deepest regionof the helium-rich radiative core (i.e., above the C-O/Hetransition). By comparison, contributions of the outermost,

regions appear to be small, and even almostlog q [[2.0negligible in the H-rich envelope (above the He/H tran-sition zone). Hence, these regions, and especially the H-richenvelope of the star, should have only a very small impacton conÐned g-mode periods. These are therefore deep inte-rior modes that mainly oscillate near the bottom of the

4 We note that the relation given in this reference is erroneous. Eq. (38)is the appropriate expression for the weight function and constitute anerratum to eq. (105) of Brassard (1991).


FIG. 9.ÈWeight function proÐles (see text for details), each normalizedto a maximum value of 1, associated with three representative g-modes(with l\ 3) corresponding to a trapped mode (k \ 5 ; solid curve), a““ normal ÏÏ mode (k \ 4 ; dashed curve), and a conÐned mode (k \ 3 ; dottedcurve). The properties of these three modes, such as their respective periods,are given in Table 2. Two small vertical segments indicate the Ledoux peakpositions of the C-O/He and He/H chemical transitions, respectively.

radiative helium coreÈwhere they appear to be literallytrappedÈsurrounding the central convective region. Themode qualiÐed as ““ normal ÏÏ (dotted curve) has a weightfunction proÐle structure which is very similar in shape tothe conÐned mode weight function. The main di†erencecomes from the fact that the shallower regions, particularlyin the H-rich envelope, contribute somewhat more to the““ normal ÏÏ g-mode period. This slight di†erence shows thatthe distinction between conÐned and ““ normal ÏÏ modes israther fuzzy. The reason for this is clearly the poor modeselection capacity provided by the conÐnement process,which makes the conÐned modes not fundamentally di†er-ent from standard (or ““ normal ÏÏ) g-modes. This alsoexplains the semiuniform g-mode *P proÐles shown inFigure 5. By contrast, due to the signiÐcantly more efficientselection e†ect of the trapping process, the trapped g-modeshown in Figure 9 reveals properties very di†erent from thetwo other classes of gravity modes. Its weight function (solidcurve) exhibits a maximum identiÐed, not surprisingly, tothe He/H transition zone. Moreover, although contribu-tions of the regions deeper than log q ^ [2.0 are still sig-niÐcant, the period forming region is concentrated in thelayers surrounding the base of the H-rich envelope. Hence,trapped g-modes form a class of modes clearly distinct fromthe other types of g-modes as they mainly oscillate near thebase of the H-rich envelope.

In addition to these comments, it is of interest to stresstwo additional properties emerging from Figure 9.

First, because g-modes cannot oscillate in convectiveregions (N2\ 0), no contributions from the convectivenucleus in our reference model) to the(0[ log q Z[0.24period is seen in the weight function proÐles. Consequently,one can envision that the detailed structure of those centralregions, representing as much as D40% of the star (in

mass), has almost no direct impact on the periods ofg-modes (see next paragraph and further investigations dis-cussed in Paper II). Hence, when we are considering thosemodes as interior modes, one should keep in mind that,strictly speaking, this means they mainly oscillate below theH-rich envelope, in the radiative layers of the helium core.

Second, the C-O/He transition zone, although wedemonstrated that it has only a small inÑuence on thetrapping/conÐnement structure, clearly contributes quitesigniÐcantly to the g-mode periods. Indeed, even the weightfunction of the more superÐcial trapped mode shows asmall but non-negligible, peak at that location in the model.This qualitatively explains the small shifts observed for theperiods when we considered the artiÐcially modiÐed ““ B(C-O/He)\ 0 ÏÏ model in ° 3.2.2.

We point out that this contribution of the C-O/He tran-sition region to the g-mode periods provides us with apotentially interesting diagnostic tool for the core evolutiontreatment in EHB model computations. The core evolutionis a difficult-to-study phenomenon which, apart from thegross properties such as lifetimes derived from stellar popu-lation counts, has very few observational constraints. Itsmodeling carries, therefore, some controversies (seeDorman 1995 and references therein). If some of the EC14026 stars can develop observable g-modes, this may allowus to verify if our basic understanding of the core is correct.Moreover, since all horizontal branch (red HB, blue HB,and EHB) cores evolve essentially identically, this wouldhave implications well beyond the study of sdB star pulsa-tions.

Some of the conclusions derived from the discussion ofFigure 9 can easily be reinforced by conducting numericalexperiments. Table 3 provides g-mode periods with l\ 1, 2,and 3 calculated for the reference model (labeled andPref)for two additional models built from that structure. One ofthese models was constructed by removing most of thecentral convective core layers while keeping the entireC-O/He transition zone (the truncation was made nearlog q ^ [0.15 corresponding to a D71% removal of theconvective nucleus). A ““ hard ball ÏÏ central boundaryconditionÈrequiring the displacement eigenfunctions to bezero where the model has been cut o†Èwas used for pulsa-tion calculations (we will discuss impacts of the choice ofcentral boundary conditions in the context of the use ofstatic envelope models presented in Paper II). The periodsobtained with this model are labeled (““ ncc ÏÏ for ““ noPnccconvective core ÏÏ). The second modiÐed model wasobtained by removing most of the H-rich envelope layerswhile keeping the He/H transition zone in its entirety (thetruncation was made near log q ^ [4.25). The periods cal-culated from this model are labeled (““ nenv ÏÏ for ““ noPnenvenvelope ÏÏ).

Comparison of with brings a direct conÐrma-Pncc Preftion of the proposition that the central convective nucleushas, by itself, a very small impact on g-mode periods. Theobserved variations between these quantities are generallywell within 0.5% (and very often less than 0.1%), reachingoccasionally 1% usually for the modes having the lowestradial orders (see, e.g., the mode with k \ 1 and l\ 2). Onenoticeable exception, though, is the mode with k \ 1 andl \ 1 for which a period change as large as D11.5% isobserved. Although the exact origins of this occurrence areunclear (this mode seems to be unusually conÐned near theC-O/He transition region and may respond more strongly


TABLE 3

g-MODE ADIABATIC PERIODS FOR THE REFERENCE AND TWO MODIFIED SdB MODELS

l\ 1 l\ 2 l\ 3

Pref Pncc Pnenv Pref Pncc Pnenv Pref Pncc Pnenvk (s) (s) (s) (s) (s) (s) (s) (s) (s)

1 . . . . . . . 425.09 479.66 424.97 277.24 280.23 277.11 224.09 225.72 223.672 . . . . . . . 671.44 675.61 671.36 406.90 408.08 406.82 306.59 307.23 306.473 . . . . . . . 958.60 959.84 958.27 569.27 569.76 569.01 418.65 418.91 418.374 . . . . . . . 1248.31 1249.02 1244.78 733.89 734.19 731.28 532.40 532.56 529.805 . . . . . . . 1465.47 1465.87 1434.71 853.19 853.40 832.99 610.41 610.55 593.676 . . . . . . . . . . . . . . . . 931.69 932.06 928.70 668.64 668.86 666.667 . . . . . . . . . . . . . . . . 1082.23 1082.62 1084.51 775.64 775.86 777.078 . . . . . . . . . . . . . . . . 1246.14 1246.75 1249.26 891.19 891.49 893.269 . . . . . . . . . . . . . . . . 1407.05 1408.96 1410.10 1006.65 1007.44 1008.8810 . . . . . . . . . . . . . . . . . . . . . . . . 1100.36 1104.37 1101.3111 . . . . . . . . . . . . . . . . . . . . . . . . 1167.58 1170.87 1170.1312 . . . . . . . . . . . . . . . . . . . . . . . . 1273.72 1274.56 1278.9713 . . . . . . . . . . . . . . . . . . . . . . . . 1387.66 1387.98 1400.2514 . . . . . . . . . . . . . . . . . . . . . . . . 1451.01 1451.12 1523.73

to changes in the central boundary position), such a di†er-ence can still be considered relatively small. In the samevein, comparison of with carries the conÐrmationPnenv Prefthat the upper envelope regions located above the He/Htransition marginally inÑuence g-mode periods in general,except for trapped modes. The calculated variations areglobally less than 0.1% for conÐned and ““ normal ÏÏ modes,while they can almost reach 5% for trapped modes (themode with k \ 14 and l\ 3, for instance). Therefore,although the absolute amplitudes of these di†erencesdepend somewhat on the position where the models havebeen truncated (very insensitive, though, as long as parts ofthe C-O/He and He/H transitions are not removed), theirrelative values associated with the three di†erent types ofg-modes reÑect quite well the properties uncovered so farfor the gravity mode oscillations.

3.3. T he Acoustic Mode SpectrumAs indicated previously, acoustic oscillations currently

represent the most interesting and practical case for poten-tial sdB seismology. Indeed, they are the only pulsationmodes clearly identiÐed so far with the periods observed inthe known EC 14026 stars.5 An exploration of their globalproperties, similar to what has been carried out previouslyfor g-modes, is therefore not only a highly desirable step,but also a sine qua non condition toward a better under-standing of the properties of EC 14026 stars.

3.3.1. Kinetic Energies and Frequency Spacings

Contrary to g-modes, p-modes do not exhibit anyobvious hints of trapping and/or conÐnement e†ects (seeTable 2). The values of their kinetic energies have a mono-tonic behavior from mode to mode throughout the entirespectrum. This is illustrated in Figure 10, where, for eachvalue of l taken from 0 to 3, both logarithm oflog EkinÈthethe kinetic energyÈand *lÈthe frequency spacing given inmHz between two consecutive p-modes (*k \ 1)Èareplotted against their respective eigenfrequency l\ 1/P (also

5 However, one has to keep in mind that the behavior of g-modes isworth being studied as those modes may interfere with p-modes undercertain conditions ; see Papers II and III.

given in mHz). For our discussion of p-mode properties, wedecided to follow standard practice and use eigen-frequencies instead of eigenperiods as the former rendergraphical representations and comparisons with the analy-tic asymptotic relations more practical. Accordingly, this isthe frequency of acoustic modes, and not their period, thatincreases with the radial order k. Moreover, the asymptotictheory for chemically uniform stars predicts an uniformacoustic spectrum of the form6 (see eq. [T65] of Tassoul1980)

lkl

^Ak ] l

2] n

e2

] 14BA

2P0

R drC

s

B~1, (39)

with the local adiabatic sound velocity, given by theCs(r),

relation

Cs\A!1P

oB1@2

. (40)

Hence, in the limit of high radial order, one should expect tosee a constant frequency spacing between adjacent p-modesgiven by the relation

*l4 lk`1[ l

k\A2P0

R drC

s

B~1. (41)

Figure 10 shows that the kinetic energy proÐle of p-modes isvery di†erent from its g-mode counterpart (see, e.g., Fig. 4).No obvious important local minima or maxima that can beattributed to trapping and/or conÐnement e†ects areobserved. As mentioned before, instead, we observe a ratherslow, monotonic decrease of the kinetic energy as kincreases in the low radial order (or, equivalently, lowfrequency) section of the energy spectra (associated witheach value of l). The decrease is followed by a relatively wideglobal minimum located near k D 15, which precedes a

6 This expression holds also for radial (l\ 0) acoustic modes but onehas to take into account that each radial mode of a given order k corre-sponds to a nonradial p-mode of order k ] 1. Therefore, in relation (39),when l\ 0, the number k should be the radial order of the mode given inTable 2 plus one.


FIG. 10.ÈFor each value of l between 0 and 3, proÐles of the relative kinetic energy (shown in its logarithmic form and the frequency spacinglog Ekin)between consecutive modes in mHz) vs. p( f)-mode eigenfrequencies in mHz) are provided for both the reference model ( Ðlled(*l\ lk`1[ l

k(l\ l

k\ 1/P

k,

circles, solid curve) and the modiÐed ““B\ 0 ÏÏ model (open circles, dotted curve). Each curve associated with a given value of l begins with the mode with k \ 0(except for l\ 1, where k \ 1) which has the smallest frequency (or longest period) among the p-modes belonging to that series. Then, the frequencies(periods) of p-modes increase (decrease) as their radial order k increases. The asymptotic value *l^ 1.4 mHz calculated from the unperturbed quantities ofthe reference model is also indicated as a horizontal, long-dashed line in each ““*l-l ÏÏ panel.

slow, and apparently linear, rise in the region of highervalues of k (or frequencies). Superimposed on the globalshape, one can notice small Ñuctuations that marginally,but perceptibly, a†ect the kinetic energies of p-modes(variations are D0.1 dex in log Ekin).The associated *l-l diagrams are somewhat more sensi-tive to those perturbations in the p-mode spectrum. Overall,the behavior observed for p-modes is very close to the pre-dictions of the asymptotic theory requiring, for large valuesof k, *l to be constant, at least for chemically uniform stars.From equation (41) evaluated from the unperturbed quan-tities of the reference model, we obtain a value of *l^ 1.4mHz, independent of l, that matches reasonably well thenumerically calculated spectrum. The largest deviationsfrom the asymptotic value, if we do not consider for themoment the omnipresent Ñuctuations, are observed formodes with low values of the radial order (mainly Ofk [ 6).course, this is not surprising, since the domain of validity forthe asymptotic developments applies to high values of k.

Fluctuations around the asymptotic values are approx-imately on the order of 0.15 mHz (about 10% of the mean*l value). To test the e†ects of the Ledoux discontinuitieson p-modes in order to evaluate if they play any role inproducing these perturbations or in inÑuencing the globalstructure of the proÐle, the p-mode spectrumlog Ekinbelonging to the artiÐcially modiÐed ““ B\ 0 ÏÏ model is alsoprovided in Figure 10 (open circles and dotted lines).Although the kinetic energies of modes are slightly higherand the frequencies are sometimes perceptibly a†ected(especially for low-order p-modes) in this model, the experi-ment reveals that the Ledoux discontinuities have aminimal impact on the p-mode spectrum structure. Fluctua-tions in *l are still observed with identical magnitudes andthe overall shape of the proÐle remains unchanged.log EkinAs tempting as it may seem, one cannot, however, con-clude from this experiment alone that the chemical tran-sition regions have no e†ect at all on p-modes. Indeed,contrary to g-modes that are very sensitive to discontin-


uities in the frequency (which actually is theirBrunt-Va� isa� la�natural oscillation frequency), acoustic waves are moreclosely related to the Lamb frequency (their natural oscil-lation frequency) which is directly connected to the localadiabatic sound speed This last quantity, as a function(C

s).

of the density, also has discontinuities due to the chemicaltransitions, although they do not appear so clearly in theproÐle. One can therefore reasonably state that discontin-uities in the Lamb frequencies due to these chemical tran-sitions may play a greater role than the Ledouxdiscontinuities on perturbing the p-mode spectrum. Indeed,we claim that the observed ÑuctuationsÈto which we willnow refer to as the ““ microtrapping e†ect ÏÏÈare actuallycaused by the He/H transition zone, which is also responsiblefor most of the trapping/conÐnement e†ects we uncoveredfor g-modes. Unfortunately, there is no easy numericalexperiment comparable to those undertaken for g-modes(i.e., the construction of an equivalent to the ““ B(C-O/He) ÏÏand ““B\ 0 ÏÏ models) that can be applied to the p-modesituation because there is no way to isolate the contribu-tions of the chemical transitions in the Lamb frequencyexpression given by equation (1) as it was possible for the

frequency (i.e., the Ledoux term). However,Brunt-Va� isa� la�indirect methods to conÐrm that hypothesis, such as explor-ing the e†ects of changing the location of the He/H tran-sition (i.e., changing the mass of the H-rich envelope), areavailable. However, because we have not yet introduced allthe tools needed to do so, further discussions on this topicare postponed to Paper II. Also postponed to Paper II arediscussions on some evidence that the global shape of thekinetic energy spectrum of p-modes observed in Figure 10 isactually a model artifact due to the perfectly reÑectivesurface boundary condition a†ecting the acoustic modeshaving sufficiently high radial orders. This has to be kept inmind when considering high order p-modes in sdB starmodels. We stress, however, that this problem does nota†ect calculations of low-order p-modes which have beenidentiÐed so far to the periods observed in the known EC14026 stars.

3.3.2. Acoustic Mode Displacement Eigenfunctions

Some additional insight on p-mode properties can beobtained by looking at their displacement eigenfunctions.Figure 11 provides proÐles of and for a sample ofy1 y2representative acoustic modes with values of l between 0and 3. Modes with k \ 0 (except for l\ 1), 1, 5, 15, and 25are plotted in order to sample all three regimes uncoveredfrom the kinetic energy spectrum proÐles (the initialdecrease, the Ñat bottom region, and the following linearrise ; see ° 3.3.1 and Fig. 10). As we did for g-modes, theeigenfunctions have been arbitrarily normalized at themodel surface by the constraint y12] y22\ 1.

From Figure 11, one reason why no strong trapping/conÐnement e†ects manifest themselves in the p-mode spec-trum becomes obvious : These modes oscillate withsigniÐcant relative amplitudes only in the H-rich envelopeof the star, well above the He/H transition zone (whoseposition is indicated in each panel of Fig. 11 by the long-dashed vertical line). Therefore, one can expect to observeonly minimal inÑuence from this particular transitionregion on p-mode properties. This also explains the rela-tively low kinetic energy of acoustic modes compared to thegravity modes since the latter show larger relative ampli-tudes in the deeper, denser regions of the star. Relative

amplitudes of p-mode oscillations are obviously very smallin the helium core, demonstrating that they are mainlyenvelope modes in contrast to the g-modes. The low radialorder modes (k \ 0, and 1) are clearly the deepest amongthe p-mode family ; one can still observe some amplitude intheir displacement eigenfunctions below the He/H tran-sition region, but nothing in the He-rich core, at the scale ofFigure 11, is noticeable for the three higher order modes(k \ 5, 15, and 25) presented here. The initial decrease in thekinetic energy spectrum of p-modes Ðnds a natural explana-tion by the overall shift toward the surface of the relativeoscillation amplitude when k increases, as the less denseregions require much less kinetic energy to move. However,near k D 15 the nodes of p-mode eigenfunctions, becausethey migrate upward in the model when the radial orderincreases, accumulate near the perfectly reÑective surface(imposed by the choice of boundary conditions) in the pul-sation calculations. This boundary is located where thecomputations of the stellar model structures have beenstopped (log q ^ [13.0 in our reference model). In thatcontext, we stress that, in stellar structure calculations, thecomputed surface of the model does not, of course, corre-spond to the physical surface of the star (rigorously locatedat log q \ [O). Hence, by compressing the nodes of theireigenfunctions near the surface of the model, this arbitrarycuto† forces sufficiently high order p-modes to developlarger amplitudes deeper and deeper inside the H-richenvelope as k increases. This, then, induces the trend of anoverall rise in kinetic energy. This surface e†ect, as alreadymentioned in ° 3.3.1, is an artifact of the modeling, and onlya†ects high-order p-modes.

3.3.3. p-Mode Period Formation Regions

To complete our discussion of p-mode properties, wehave to clearly establish what regions of the star make thedominant contribution to their periods (or frequencies). Thehints gathered so far strongly suggest that p-modes are““ shallow ÏÏ envelope modes probably only marginally sensi-tive to the deeper regions belongings to the helium-richcore. These hints are as follows :

1. The propagation zone of p-modes (illustrated in Fig. 3)is mainly located close to the surface of the model.

2. The kinetic energies are, from Table 2, signiÐcantlylower than those of the deep interior g-modes.

3. From Figure 11, the relative amplitudes of p-modeeigenfunctions remain very small in the core of the star asmost of the oscillations occur in the H-rich envelope, wellabove the He/H transition region.

As before for the g-mode case, a deÐnite answer to thisquestion is explicitly given by the weight function associ-ated to each p-mode. ProÐles of this quantity normalized tounity at the maximum are provided in Figure 12 for asample of representative acoustic modes with k \ 0, 1, 5and l \ 0, 1, 2, and 3.

Recalling that regions where the weight function of agiven mode shows large relative values contribute signiÐ-cantly to the mode period (frequency), we note, by qualit-ative comparison with Figure 9, that p-mode periods, unlikeg-mode periods, are mainly formed in the outer layers of theH-rich envelope. p-modes can therefore unambiguously becharacterized as shallow envelope modes mainly sensitiveto the outermost regions of the star. However, somewhatunexpectedly considering our previous hints, some non-


FIG. 11.ÈDisplacement eigenfunction proÐles and for a sample of radial (l\ 0) and nonradial (l\ 1, 2, and 3) p( f )-modes with k \ 0, 1, 5, 15,(y1 y2)and 25 (except in the case l\ 1 for which there is no k \ 0 mode) representative of all the regimes identiÐed in Fig. 10 for acoustic modes. The locationof the He/H transition zone is indicated in each panel by vertical, long-dashed line.

negligible contributions of the regions located below theHe/H transition are observed. Hence, although this was notobvious from the eigenfunction proÐles plotted in Figure 11(which showed very small amplitudes below the envelope),the helium-rich core still has some non-negligible inÑuenceon p-mode periods. The p-mode weight functions even showa (very) slight contribution of the convective core region,below the C-O/He transition zone indicated in Figure 12,while we demonstrated earlier that the deep interiorg-modes have almost no sensitivity to that particularregion. This is because the p-modes, unlike the g-modes, canpropagate in convective zones. On a more speciÐc basis, wenote that modes with k \ 0 and k \ 1 are formed deeper inthe star compared to the k \ 5 mode, which are consideredto be representative of the higher radial order acousticmodes in this discussion. In particular, weight functions ofmodes with k \ 1, l\ 0, 2, and 3 exhibit relatively largeamplitudes below the He/H transition zone, in the radiativecore, near the C-O/He transition region, as well as in theconvective core. Hence, these regions should contributeslightly more to the periods of these modes than they do for

the higher order p-modes. This pattern was not found to berecurrent in the acoustic spectrum for larger values of k. It istherefore likely to a†ect only very low-order modes, prob-ably because the p-mode nodes are ““ pushed ÏÏ closer to thesurface by increasing values of l and k (see, e.g., Unno et al.1989) One last interesting comment coming out from theanalysis of Figure 12 is the signiÐcant role played by theHe/H transition in the periods of all p-modes. Most often,the weight function maxima are clearly associated to thatparticular region which should therefore possess some inÑu-ence on p-mode periods. More interestingly, we found thatthis structure does not disappear in the weight functionproÐles of p-modes calculated from the modiÐed ““B\ 0 ÏÏmodel (those are not illustrated here). Indeed, this reinforcesour proposition that the small 10% frequency Ñuctuationsobserved in the p-mode frequency spacing spectrum (see° 3.3.1) are generated by the He/H transition zone throughdiscontinuities present in quantities other than the Brunt-

frequency. These quantities are probably the LambVa� isa� la�frequency and therefore the local adiabatic sound speed.However, we refer the reader to Paper II for details.


FIG. 12.ÈWeight function proÐles, each normalized to a maximum value of 1, associated with three representative p( f)-modes. Each panel corresponds toa given value of the degree index l, between 0 and 3. Properties of these modes, such as their periods, are provided in Table 2. Two small vertical segmentsindicate the positions of, respectively, the C-O/He and He/H transitions.

ConÐrmation of these deductions can be worked out bylooking at the results of a complementary numerical experi-ment provided in Table 4. Periods of p-modes have beenrecalculated for two additional structures constructed fromthe reference model used so far. In order to test the overall

inÑuence of the convective core, one of these new modelswas built by removing most of the central convective corelayers while keeping the whole C-O/He transition zone (thesame model used in ° 2.2.5 in the g-mode context). Theperiods for this model are again labeled in Table 4. ThePncc

TABLE 4

p-MODE ADIABATIC PERIODS FOR THE REFERENCE AND TWO MODIFIED SdB MODELS

l\ 0 l\ 1 l\ 2 l\ 3

Pref Pncc Pnrc Pref Pncc Pnrc Pref Pncc Pnrc Pref Pncc Pnrck (s) (s) (s) (s) (s) (s) (s) (s) (s) (s) (s) (s)

7 . . . . . . . . . . . . . . . 82.98 81.76 61.04 . . . . . . . . . . . . . . . . . .6 . . . . . . 88.52 84.36 61.18 92.47 91.76 68.84 89.19 89.08 68.64 85.69 85.68 68.415 . . . . . . 97.90 93.64 69.01 105.99 104.74 79.08 99.15 98.99 78.83 94.58 94.58 78.554 . . . . . . 112.39 108.26 79.28 118.18 117.22 93.03 113.11 113.05 92.71 109.39 109.39 92.323 . . . . . . 129.58 120.99 93.28 142.08 141.29 112.81 132.60 132.43 112.34 123.66 123.66 111.732 . . . . . . 148.31 144.32 113.15 160.89 159.66 143.28 150.12 150.05 142.42 145.37 145.39 141.251 . . . . . . 176.24 165.42 143.86 208.24 208.22 202.69 185.82 185.60 200.22 172.44 172.51 196.350 . . . . . . 209.12 208.90 204.14 . . . . . . . . . 208.06 208.10 336.81 206.19 206.29 279.99


second model constructed in the course of our experiment isthe reference model with almost all the helium-rich coreremoved (the truncation was made near log q ^ [3.30),leaving only the H-rich envelope and a small chunk of thehelium core so we could keep the entire He/H transitionzone. Periods obtained with this model are labeled Pnrc(““ nrc ÏÏ for ““ no radiative core ÏÏ) in Table 4. As mentioned in° 2.2.5, a ““ hard ball ÏÏ central boundary condition wasapplied in the pulsation calculations when models su†eredamputation of their central regions (which is the case for thetwo modiÐed models presented here).

A global comparison between the periods given in Table4 for each of the three models conÐrms several conclusionsderived from our previous analysis of p-mode weight func-tions. First, the period di†erences found between the fullreference model and the equivalent model having a trun-cated convective core are generally very small (most oftenless than 1% for nonradial acoustic modes). This conÐrmsthat the innermost regions of the star have, as expected, avery small direct inÑuence on p-mode periods. However andinterestingly, the radial (l\ 0) acoustic modes experience asmall systematic shift of their periods (from D2.7% toD6.1%) between those two models, although no obviousdi†erences with their nonradial counterparts can be seenfrom Figure 12. Since the radial mode periods (especiallythe fundamental mode period) essentially sample the soundtravel time from the surface to the center, a plausible expla-nation is that removing the central parts of the modelshorten the travel distance of the radial waves, thusreducing their periods. Nonradial acoustic modes may beless sensitive to these model truncations because theircentral turning point moves out with increasing values of l.Some experiments we made with the central boundary con-ditions revealed that this small shift can be eliminated inthese models by using a ““ soft ball ÏÏ condition (imposing thistime the radial derivative of the displacement eigenfunctionsto be zero at the model central cut o†) instead of the ““ hardball ÏÏ condition applied in these calculations. It is, how-ever, not entirely clear at this point whether these““ improvements ÏÏ depend or not on the particular modelchosen for our present discussion. More on this particulartopic will be provided in Paper II when we will discuss thereliability of pulsation computations using static envelopemodels. Second, comparisons between the unmodiÐed refer-ence structure and the model for which almost all thehelium core has been removed clearly conÐrm, as expectedfrom the weight function proÐles, that signiÐcant contribu-tions to the p-mode periods (up to D30%È40%) are provid-ed by the inner region of the star located below the He/Htransition region, although p-modes are mainly envelopemodes. Exceptions are the k \ 0, l\ 0 and k \ 1, l \ 1modes exhibiting di†erences of only D2.5%, which indeedremarkably agree, from Figure 12, with their respectiveweight function proÐles showing almost no contributionsbelow the He/H transition region.

3.4. Propagation DiagramIn order to completeÈand somewhat summarizeÈthis

extended overlook of the main properties of pulsationmodes in subdwarf B stars, we provide in Figure 13 theso-called propagation diagram (see, e.g., Unno et al. 1989)for the representative model we have used so far as a refer-ence structure in our discussion. The construction of thiskind of diagram is based on the proÐle plots of both the

FIG. 13.ÈPropagation diagram for our representative reference model.p-modes, up to k \ 8, f-, and g-modes up to k \ 5 (all for l \ 2) are shownas horizontal dotted lines positioned in the Ðgure according to their eigen-frequencies (expressed in the logarithmic form log p2, where p \ 2nl\2n/P). Modes are labeled according to their type and radial order ; e.g., themode named corresponds to the gravity mode with k \ 5. For each ofg5these modes, the node positions of are indicated as small Ðlled circles.y1Finally, proÐles of both the frequency (given as log N2) andBrunt-Va� isa� la�the Lamb frequency for l\ 2 (given as are provided to complete thelog L22)propagation diagram representation.

and Lamb frequencies (hence, similar to Fig.Brunt-Va� isa� la�3) on top of which some additional information from thedetailed pulsation calculations are superimposed. These arethe eigenfrequencies and node positions of the radial dis-placement eigenfunctions for a sample of p-, f-, and(y1)g-modes. Figure 13 shows p-modes with radial order k up to8, the f-mode, and g-modes with k up to 5 (all these modeswere chosen to have l \ 2) as horizontal, dotted lines posi-tioned in the diagram according to their respective fre-quency or period (transformed for plotting purposes intothe quantity log p2, with p \ 2nl\ 2n/P). For any givenmode, the location of each node of is displayed as a smally1Ðlled circle drawn on the corresponding dotted line.

This propagation diagram qualitatively illustrates mostof the general properties we uncovered for pulsation modesduring the discussions presented in the previous sections.From the node positions, one can see that p-modes oscillatein the outermost region where (the acousticp2[ L 22, N2mode propagation zone), while g-modes oscillate in theinnermost region where (the gravity mode pro-p2\ L 22, N2pagation zone). This is, of course, consistent with ourgeneral Ðnding that p-modes are relatively shallow envelopemodes in contrast to g-modes, which are relatively deepradiative core modes. The Ðgure clearly shows that for thelatter, new nodes7 in eigenfunctions appear, when ky1increases, close to the wall observed in log N2 associatedwith the outer edge of the central convective core. We stress

7 By convention, nodes are counted from the surface to the center of thestar.


that g-mode nodes are always located outside this region, inaccordance with the fact that they cannot propagate in thecentral convective core (they become strongly evanescent inthose layers). While increasing k, we also observe that theexisting nodes migrate toward the surface, although stayingtrapped inside the g-mode propagation zone. Eventually,they reach the H-rich envelope of the star. In this context,mode appears to be the Ðrst g-mode to have its outer-g5most node located very close to the He/H transition zone(near log q ^ [4.0 ; betrayed by the Ledoux peak in log N2and the discontinuity in This is consistent with thelog L 22).fact that it has been identiÐed with a mode trapped (the Ðrstone in the spectrum) by this chemical transition zone (see°° 3.2.2 and 3.2.3). p-mode nodes are mainly found aroundthe He/H transition and extend outward quite far in theenvelope and inward relatively deep in the helium core ashigher values of the radial order are considered. Inter-estingly, we note that the He/H transition zone is located inthe same area where nodes of acoustic modes are typicallyencountered, thus making the ““ microtrapping ÏÏ idea dis-cussed in ° 3.3.1 even more plausible as perturbations of theperiods are more likely to occur when nodes of eigen-functions are located in the vicinity of the chemical discon-tinuity.

4. CONCLUSION

In this paper, the Ðrst of a series of three dedicated to acomplete and thorough adiabatic survey of the pulsationproperties of subdwarf B stars, we focus our attention onthe oscillation modes encountered in a representative evolu-tionary model. Our primary goal is to lay down the mostbasic theoretical framework needed to understand thebehavior of p-, f-, and g-modes in these objects. This frame-work represents the minimal knowledge required to pursuefurther fundamental investigations such as the dependenceof the pulsation properties of sdB stars on model param-eters and the coupling between these properties and theevolutionary status of these stars. These topics will beaddressed in Papers II and III. We note that our survey isthe Ðrst systematic e†ort aimed at understanding theoreti-cally the pulsation characteristics of sdB stars at the mostbasic level. In the light of growing interest in EC 14026stars, our survey is certainly timely. It represents the Ðrststep toward applying the tools of asteroseismology to theseexciting, excited stars.

In the present e†ort, we speciÐcally investigated the non-radial pulsation mode properties of a representative modelof a sdB star, and we studied how these properties arerelated to the internal structure of the model. Even thoughonly low-order p-modes have been unambiguously identi-Ðed in the observed pulsators so far, we chose to discuss thefull theoretical pulsation spectrum, including g-modes. Webelieve that such an approach has the merit of providing abroader perspective and a better understanding of thepotential asteroseismological properties of sdB stars. Fur-thermore, the possibility that g-modes may be excited andobservable in some sdB pulsators, particularly the lowsurface gravity ones, remains very real.

We found that, in the band of periods of interest forhigh-speed photometric techniques, there exists a rich spec-trum of pulsation modes in a typical sdB star. These includeradial and nonradial low-order acoustic and gravity modes.We established that p-modes (radial and nonradial) havetheir periods largely formed in the outer regions of the star,

especially in the H-rich envelope, making them envelopemodes. However, we also found some signiÐcant contribu-tions to these periods (sometimes up to D40%) comingfrom layers located below the H-rich envelope, mainly inthe He-rich radiative core. Hence, although p-modes shouldbe considered as envelope modes with larger relative ampli-tudes in the shallower regions of these stars, the deeperlayers may still have signiÐcant e†ect on their behavior.However, we found several indications that the centralconvective nucleus itself does not really contribute tothe formation of p-mode periods. Therefore, in theasteroseismological context, one should not expect to beable to probe directly the detailed structure of those inner-most regions with p-modes.

We also determined that, in contrast to p-modes,g-modes globally are deep interior modes ; their periods aremainly formed below the H-rich envelope, making theminsensitive to the structure of the shallower regions.However, outstanding exceptions are regularly seen amongthe g-modes. We indeed demonstrated that g-modes areresponsive to a resonant phenomenon caused by the struc-tural chemical stratiÐcation in subdwarf B stars. A closecomparison with the case of the chemically stratiÐed DApulsating white dwarfs, documented in Brassard et al.(1992b), revealed very strong similarities with the periodpatterns observed in our reference subdwarf B star model.Based on asymptotic theory and on the results obtainedfrom direct numerical experiments, we have shown that theHe/H transition zone between the H-rich envelope and theHe-rich core is indeed responsible for trapping and conÐn-ing g-modes in a way directly analogous to the way gravitymodes are trapped and conÐned in white dwarfs. Wedemonstrated that the trapped modes are signiÐcantly dif-ferent from the other g-modes regarding their globalproperties ; their periods are mainly formed around thechemical transition zone between the H-rich envelope andthe He-rich core, while the ““ normal ÏÏ g-modes are formedmuch deeper. Trapped g-modes are therefore shallowermodes compared to other g-modes, but they are still deeperthan the envelope p-modes. If g-modes are observed in sdBpulsators, this would provide an interesting aster-oseismological diagnostic tool as the trapping and conÐne-ment e†ects a†ecting their periods are directly related to themass of the hydrogen-rich envelope (further investigationson this topic are provided in Paper II). Of potential interesttoo, we further found that the actual extent of the centralconvective nucleus has a weak signature on the g-modetrapping pattern. We also determined that the C-O/Hetransition region between the convective core and the radi-ative core has a signiÐcant impact on the g-mode periods.This may provide, again if g-modes are observed in pulsa-ting sdB stars, an interesting way to constrain the still con-troversial modeling of the helium core evolution inhorizontal branch stars. However, we emphasized that theg-mode periods are not sensitive to the detailed structure ofthe convective central region itself, because g-modes do notpropagate there.

Our investigations have revealed that trapping pheno-mena leaves a much weaker imprint on the p-mode periodspectrum. We found out that the He/H transition zone pro-duces small but signiÐcant perturbations in the acousticmode frequency spectrum, which we refer to as micro-trapping. We showed, in particular, that the He/H tran-sition itself brings a non-negligible contribution to the


p-mode weight functions. We will address this question ofmicrotrapping more fully with the tools developed in PaperII. Overall, we found out that, with respect to the questionof regions of period formation, subdwarf B stars are typicalof nondegenerate stars with some additional features intro-duced by their internal chemical stratiÐcation.

This work was supported in part by the NSERC ofCanada and by the Fund FCAR S. C. also(Que� bec).acknowledges the support of the Canada-France-HawaiiTelescope Corporation, the INSU/CNRS, and the MAE(France).

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Adiabatic Survey of Subdwarf B Star Oscillations. III. Effects of Extreme Horizontal Branch Stellar Evolution on Pulsation Modes

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