Discovery and Asteroseismological Analysis of the Pulsating sdB Star PG 0014+067

THE ASTROPHYSICAL JOURNAL, 563 :1013È1030, 2001 December 20( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

DISCOVERY AND ASTEROSEISMOLOGICAL ANALYSIS OF THE PULSATING sdB STAR PG 0014]0671

P. BRASSARD AND G. FONTAINE

de Physique, de CP 6128, Station Centreville, QC H3C 3J7, Canada ;De� partement Universite� Montre� al, Montre� al,brassard=astro.umontreal.ca, fontaine=astro.umontreal.ca

M. BILLERES

European Southern Observatory, Santiago Headquarters, St. Alonso de Cordova 3107, Vitacura, Casilla 19001, Santiago 19, Chile ; mbillere=eso.org

S. CHARPINET

Observatoire 14 Avenue E. Belin, Toulouse F-31400, France ;Midi-Pyre� ne� es, scharpin=ast.obs-mip.fr

JAMES LIEBERT

Steward Observatory, University of Arizona, Tucson, AZ 85721 ; liebert=as.arizona.edu

AND

R. A. SAFFER

Department of Astronomy and Astrophysics, Villanova University, Villanova, PA 19085 ; sa†er=ast.vill.eduReceived 2001 June 7 ; accepted 2001 August 20

ABSTRACTWe report the discovery of low-amplitude, short-period, multiperiodic luminosity variations in the hot

B subdwarf PG 0014]067. This star was selected as a potential target in the course of our ongoingsurvey to search for pulsators of the EC 14026 type. Our model atmosphere analysis of the time-averaged Multiple Mirror Telescope (MMT) optical spectrum of PG 0014]067 indicates that this starhas K and log g \ 5.77^ 0.10, which places it right in the middle of the theoreticalTeff \ 33,550^ 380EC 14026 instability region in the plane. A standard analysis of our Canada-France-Hawaiilog gÈTeffTelescope (CFHT) light curve reveals the presence of at least 13 distinct harmonic oscillations withperiods in the range 80È170 s. Fine structure (closely spaced frequency doublets) is observed in three ofthese oscillations, and Ðve high-frequency peaks due to nonlinear cross frequency superpositions of thebasic oscillations are also possibly seen in the Fourier spectrum. The largest oscillation has an amplitude^0.22% of the mean brightness of the star, making PG 0014]067 the EC 14026 star with the smallestintrinsic amplitudes so far. On the basis of the 13 observed periods, we carry out a detailed aster-oseismological analysis of the data starting with an extensive search in parameter space for a model thatcould account for the observations. To make this search efficient, objective, and reliable, we use a newlydeveloped period matching technique based on an optimization algorithm. This search leads to a modelthat can account remarkably well for the 13 observed periods in the light curve of PG 0014]067. Adetailed comparison of the theoretical period spectrum of this optimal model with the distribution of the13 observed periods leads to the realization that 10 other pulsations, with lower amplitudes than thethreshold value used in our standard analysis, are probably present in the light curve of PG 0014]067.Altogether, we tentatively identify 23 distinct pulsation modes in our target star (counting the frequencydoublets referred to above as single modes). These are all low-order acoustic modes with adjacent valuesof k and with l\ 0, 1, 2, and 3. They deÐne a band of unstable periods, in close agreement with non-adiabatic pulsation theory. Furthermore, the average relative dispersion between the 23 observed periodsand the periods of the corresponding 23 theoretical modes of the optimal model is only ^0.8%, aremarkable achievement by asteroseismological standards. On the basis of our analysis, we infer that theglobal structural parameters of PG 0014]067 are log g \ 5.780^ 0.008, K,Teff \ 34,500K^ 2690

and If we combine theseM*/M

_\ 0.490^ 0.019, log (Menv/M*

) \ [4.31^ 0.22, R/R_

\ 0.149 ^ 0.004.estimates of the surface gravity, total mass, and radius with our value of the spectroscopic temperature(which is more accurately evaluated than its asteroseismological counterpart, in direct contrast to thesurface gravity), we also Ðnd that PG 0014]067 has a luminosity has an absoluteL /L

_\ 25.5^ 2.5,

visual magnitude and is located at a distance d \ 1925 ^ 195 pc (usingMV

\ 4.48^ 0.12,V \ 15.9^ 0.1). If we interpret the Ðne structure (frequency doublets) observed in three of the 23 pulsa-tions in terms of rotational splitting, we further Ðnd that PG 0014]067 rotates with a period of29.2^ 0.9 hr and has a maximum rotational broadening velocity of km s~1.V sin i [ 6.2 ^ 0.4Subject headings : stars : interiors È stars : oscillations È subdwarfs

1 Based on observations gathered at the Canada-France-Hawaii Telescope, operated by the National Research Council of Canada, the Centre Nationalde la Recherche ScientiÐque de France, and the University of Hawaii.

1013

1014 BRASSARD ET AL. Vol. 563

1. INTRODUCTION

A small fraction of the known hot B subdwarf (sdB) starsin the Ðeld show short-period, multiperiodic luminosityvariations caused by pulsational instabilities. These areknown as EC 14026 stars (after the prototype) and form thenewest family of pulsating stars to have been uncovered.Interestingly, their existence was predicted theoretically byCharpinet et al. (1996) at about the same time the serendip-itous discovery of a Ðrst real pulsating sdB star was made(Kilkenny et al. 1997). This was the result of the indepen-dent observational e†orts by a group of astronomers at theSouth African Astronomical Observatory, which quicklyled to the discoveries of three more variable stars withphotometric and spectroscopic properties quite similar tothose of EC 14026-2647 (Koen et al. 1997 ; Stobie et al.1997 ; OÏDonoghue et al. 1997). Further work by our SouthAfrican colleagues, by ourselves, and by others has, in themeantime, led to additional discoveries, bringing thenumber of publicly known EC 14026 stars to 18.

Most of the EC 14026 stars tend to cluster aroundlog g ^ 5.8 and K in the diagram,Teff ^ 33,500 log gÈTeffalthough the presence of outliers such as Feige 48 atlog g ^ 5.4 and K (Koen et al. 1998c), its spec-Teff ^ 29,000troscopic twin HS 2201]2610 at log g ^ 5.4 and Teff ^29,300 K (Ostensen et al. 2001), and PG 1605]072 atlog g ^ 5.2 and K (Koen et al. 1998b)Teff ^ 31,000increases somewhat the dispersion in that diagram. Thenumber of observed periods in a given EC 14026 star rangesfrom a few to more than 50, although this number is highlycorrelated with the sensitivity of the measurements. Theperiods are typically found in the interval 100È200 s but canbe substantially longer in the low-gravity objects such asFeige 48 (250È380 s) and PG 1605]072 (290È600 s). Theamplitudes of the harmonic oscillations detected in the lightcurves of EC 14026 stars range widely, but typical valuesare of several millimagnitudes. Qualitative comparisonswith model calculations show that the observed periods areconsistent with low-order radial and low-order, low-degreenonradial pressure modes (Charpinet et al. 1997 ; Charpinet,Fontaine, & Brassard 2001a ; Stobie et al. 1997). In low-gravity pulsators such as PG 1605]072, for example, bothpressure and gravity modes are relevant to account for theobserved periods through the phenomenon of mixed modes(Charpinet 1998 ; Charpinet et al. 2001b). The pulsationmodes in EC 14026 stars are excited through an opacitymechanism associated with the presence of iron in theenvelopes of these stars (Charpinet et al. 1997). More detailsabout the properties of EC 14026 pulsators may be found inthe reviews presented by Koen et al. (1998a), Fontaine et al.(1998), OÏDonoghue et al. (1999), and Charpinet et al.(2001a).

Initially motivated by the calculations of Charpinet et al.(1996), we started, in 1996 May, an observational search forshort-period luminosity variations in sdB stars. At theoutset, this was a project well suited for high time resolution““ white-light ÏÏ photometry because the expected periods areshort (thus requiring relatively short sampling times, typi-cally 10 s), and the target stars are relatively faint, evenwithout using Ðlters, when sampled at that rate (a typicalÐeld sdB star has V D 14È15). We carried out the surveypart of our program on small telescopes, mostly on the 1.6m telescope at the Steward Observatory Mount BigelowStation. However, EC 14026 stars are not always easy to

detect, let alone study, with a small telescope, so an integralpart of our program was to gather follow-up observationsof newly found or known pulsators on a midsize telescope,in our case the Canada-France-Hawaii Telescope (CFHT).

We report, in this paper, the discovery and detailedanalysis of low-amplitude, short-period, multiperiodic lumi-nosity variations in the light curve of the sdB star PG0014]067. These observations were obtained at the end ofeach of Ðve consecutive nights during one of our CFHTruns dedicated to other known pulsators. Interestingly, wehad previously observed PG 0014]067 in the survey modeon the 1.6 m telescope, but the real-time light curve dis-played at the telescope did not show obvious signs of varia-bility. In retrospect, the faintness of PG 0014]067 and thelow amplitudes of the luminosity variations that weredetected (see below) explain well this situation. Not sur-prisingly, these results suggest a higher yield of EC 14026pulsators when surveyed at higher sensitivity. Furthermore,our CFHT data have revealed not only the variability ofPG 0014]067 but also a period spectrum that is bothsimple and rich enough that we attempt, for the Ðrst timefor an EC 14026 star, a complete asteroseismologicalanalysis, including detailed mode identiÐcation and deter-mination of the structural properties of the target star. Thishas been made possible through the use of a powerfulperiod matching algorithm based on an optimizationmethod developed recently by P. Brassard et al. (2001, inpreparation).

2. OBSERVATIONS

PG 0014]067 is a rather unassuming star in thePalomar-Green catalog of UV excess objects (Green,Schmidt, & Liebert 1986). IdentiÐed there as an sdB star onthe basis of a low-resolution classiÐcation spectrum, theonly estimate of its magnitude is a rough photographicmagnitude To our knowledge, no photoelectricBphD 15.6.or CCD color photometry is available for that star. It wasreobserved by two of us (R. A. S. and J. L.) 5 years ago aspart of a spectroscopic program aimed at providing reliableatmospheric parameters for a large number of sdB stars inthe northern hemisphere (see Sa†er et al. 1994 ; Sa†er &Liebert 1995). This program was based on high signal-to-noise ratio (S/N) optical spectroscopy obtained with the 2.3m telescope at the Steward Observatory Kitt Peak Stationand, for the fainter objects, with the Multiple Mirror Tele-scope (MMT). As a rather faint target, PG 0014]067 wasselected routinely to be observed during an MMT run,when its position in the sky (near the meridian) was optimal.Details of the experimental setup and the reduction pro-cedure have been described in Sa†er (1991) and Sa†er et al.(1994). The MMT spectrum available to us covers the wave-length range jj3650È4500 at D3 resolution and requiredA�an integration time of 600 s. This is longer than the periodsof the observed brightness variations, and, considering aswell the very low amplitudes of these variations (see below),this spectrum is a meaningful time-averaged spectrum ofPG 0014]067.

Figure 1 shows one of our two best (in a least-squaressense) Ðts to the available hydrogen Balmer lines andneutral helium lines. This Ðt was obtained with the samemodel atmosphere and synthetic spectrum tools describedin Sa†er et al. (1994). The derived atmospheric parametersare K and log g ^ 5.79, with formal 1 p errorsTeff ^ 33,310of 250 K and 0.06 dex, respectively. The derived helium

No. 2, 2001 ASTEROSEISMOLOGICAL ANALYSIS OF PG 0014]067 1015

FIG. 1.ÈModel Ðt (dotted curve) to the available hydrogen Balmer linesand neutral helium lines in our time-averaged optical spectrum of PG0014]067.

abundance, characteristically low in an sdB star, is N(He)/N(H)^ 0.022 with a 1 p deviation of 0.003. To provide ameasure of the external errors, we have reÐtted the samespectrum but using a di†erent model atmosphere grid com-puted in 1995 by P. Bergeron at the deUniversite� Montre� al.With this second grid, we obtain a best Ðt of comparablequality and Ðnd K, log g ^ 5.73, and N(He)/Teff ^ 33,900N(H)^ 0.020, with formal 1 p errors of 300 K, 0.08 dex, and0.003, respectively. The agreement is excellent. If we nowtake the weighted average of these two sets of estimates, weÐnd K, log g \ 5.77^ 0.05, and N(He)/Teff \ 33,550^ 190N(H)\ 0.021^ 0.002. We adopt these values as our bestspectroscopic estimates of the atmospheric parameters ofPG 0014]067, but with the understanding that the trueuncertainties are certainly larger than the formal errorsquoted here. A conservative approach is to adopt uncer-tainties on both and log g twice as large as these formalTeff1 p errors. In this context, Wesemael et al. (1997) haverecently discussed the systematics of atmospheric parameter

determinations for sdB stars, and we refer the reader to thatwork for more details.

For the purposes of our survey for pulsating sdB stars, weselected candidates on the basis of their positions in the

diagram, using lists provided in spectroscopiclog gÈTeffstudies such as the Sa†er-Liebert program mentioned justabove. Given that the atmospheric parameters for PG0014]067 were found to be quite representative of the pre-viously known EC 14026 stars (indeed, they nearly corre-spond to the average values) and, furthermore, given thatthese parameters placed the star right in the middle of thetheoretical instability region discussed by Charpinet et al.(1997, 2001a), we picked PG 0014]067 as a high-priorityobject in spite of its relative faintness. We Ðrst observed PG0014]067 in white-light ““ fast ÏÏ photometric mode duringone of our survey runs, a 10 night stand at the MountBigelow 1.6 m telescope in 1997 October, when the star waswell positioned in the sky. As usual in this survey, thephotometric observations were gathered with LAPOUNE,the portable three-channel photometer. UnderMontre� algood observing conditions, as was the case during that run,it is usually only a matter of a few minutes before the varia-bility of a target star can be established (provided some careis taken by selecting a comparison star of approximately thesame brightness as the target object). A quick look at thereal-time light curves at the telescope generally suffices sincethe (constant) comparison star provides an excellent tem-plate of the atmospheric conditions. In the case of PG0014]067, the conspiracy of faintness and low amplitudesprevented us from detecting the luminosity variations inreal time, and we terminated the observation after about 1hr, our usual criterion in this survey. However, in a routinefollow-up procedure, one of us (M. B.) computed theFourier transform of the light curve of PG 0014]067 anddetected two weak but nevertheless quite suggestive period-icities near D140 and D150 s. It was then a matter ofreobserving the star on a larger telescope.

This opportunity arose in 1998 June during a six-nightCFHT run dedicated to the known pulsators Feige 48(Koen et al. 1998c) and KPD 2109]4401 et al.(Billeres1998). We were able to gather, on the average, about 1.9 hrof data at the end of each of Ðve consecutive nights. Giventhe exceptional observing conditions during that run, possi-bly our very best at the CFHT, we gathered excellent white-light data on PG 0014]067. Table 1 provides a journal ofthese observations. As usual in this business, dark time wasrequired in gathering the light curve. With the improvedsensitivity provided by the CFHT, it was, again, only amatter of a few minutes before we could readily see theluminosity variations in real time and thus conÐrm the sug-gestion of low-level photometric activity obtained throughour previous Fourier analysis of the 1.6 m telescope lightcurve.

TABLE 1

CFHT HIGH TIME RESOLUTION PHOTOMETRIC OBSERVATIONS OF PG 0014]067

Date Start Time Sampling Time Total Number of ResolutionRun Number (UT) (UT) (s) Data Points (mHz)

cfh-068 . . . . . . . 1998 Jun 27 13 :12 10 573 0.175cfh-072 . . . . . . . 1998 Jun 28 13 :20 10 530 0.189cfh-075 . . . . . . . 1998 Jun 29 12 :56 10 675 0.148cfh-078 . . . . . . . 1998 Jun 30 12 :55 10 714 0.140cfh-081 . . . . . . . 1998 Jul 01 12 :14 10 962 0.104


FIG. 2.È““White-light ÏÏ light curve of PG 0014]067, observed on 1998June 29 with LAPOUNE attached to the CFHT (run cfh-075). The lightcurve is expressed in terms of residual amplitude relative to the meanbrightness of the star. Each plotted point represents a sampling time of10 s.

We show, in Figure 2, a typical segment of the sky-subtracted, extinction-corrected light curve of PG0014]067. This is from run cfh-075, but that segment isquite representative since the other runs are of comparablequality. The light curve is expressed in terms of the residualamplitude relative to the mean brightness of the star, andeach plotted point corresponds to a sampling time of 10 s. Itclearly reveals the presence of low-amplitude multiperiodicbrightness modulations. The variations show typical peak-to-peak amplitudes of less than D2%, and they appear tobe dominated by a pseudoperiod of D150 s. It is obvious,however, that several other periodicities are involvedbecause we see clear signs of constructive and destructiveinterference between modes in the light curve. Such abehavior is quite typical of EC 14026 pulsators. We analyze,in the next section, the contents of the complete light curveof PG 0014]067. For the record, we point out that the netaverage white-light count rate for the target star at theCFHT was D13,000 counts s~1, corresponding, throughdirect comparisons with similar stars observed during thesame nights (speciÐcally, Feige 48 and KPD 2109]4401),to a visual magnitude of V \ 15.9^ 0.1.

3. ANALYSIS OF THE LIGHT CURVE

The complete light curve of PG 0014]067 consists of Ðvechunks of average length D1.9 hr separated by importantdaily gaps. The sampling time is 10 s throughout. From thebeginning of run cfh-068 to the end of run cfh-081, theobserving window covers a time interval of 351,832 s. Thiscorresponds to a resolution of 2.84 kHz in the Fourierdomain. Since PG 0014]067 was observed during 34,540 sin that observing window, a duty cycle of 9.82% wasachieved.

We have analyzed this time series on PG 0014]067 in astandard manner by combining Fourier analysis, least-squares Ðts to the light curve, and prewhitening techniques.The procedure is described at some length in one of ourprevious papers et al. 2000), and we refer the inter-(Billeresested reader to that work for further details. Contrary to thecase of the sdB pulsator KPD 1930]2752 discussed in thatpaper, however, it was quite straightforward here to identify

and isolate the periodicities present in the light curve of PG0014]067. In the former case, the light curve showed amultitude of closely spaced frequency components, and thiscomplicated signiÐcantly the task of unraveling the trueperiods of oscillations among that forest of peaks in theFourier domain, each accompanied by its own windowpattern (i.e., the complex sidelobe structure associated withthe daily gaps in the light curve). We identiÐed at least 44short-period harmonic oscillations in the light curve ofKPD 1930]2752, and we interpreted this high density offrequency components in terms of overlapping rotationallysplit p-modes in a fast rotator with s etProt D 8218 (Billeresal. 2000). In contrast, PG 0014]067 appears to be a““ simpler ÏÏ pulsator, probably a slow rotator (see below),and we found it relatively easy to identify 13 bands of fre-quency corresponding to true luminosity variations. Inaddition, we found that at least three of those frequencycomponents have internal structure, closely spaced doubletsin each case, which are probably part of multiplet structurespossibly caused by weak rotational splitting.

Table 2 summarizes the characteristics of the 16 harmo-nic oscillations that we found in the light curve of PG0014]067. Note that we have labeled these oscillations f1through in the table, with plus and minus superscriptsf13for the components of the doublets. We list the frequency,period, amplitude, and relative phase (with respect to anarbitrary point in time) for each of these harmonic oscil-lations. In our method of analysis, there are no formalerrors associated with the frequencies (and periods) sincethose were derived directly from the Fourier spectrum. Theprobable accuracy of the frequencies is, however, of theorder of 1/10 of the formal resolution, i.e., about 0.28 kHz(see et al. 2000). By contrast, the amplitudes andBilleresphases of the harmonic oscillations are derived throughleast-squares Ðts to the light curve, a method that providesformal errors as given in the table. We note that the periodsobserved in PG 0014]067 cover a range from D80 toD170 s, a range that is unusually large for a typical EC14026 pulsator (see, e.g., Table 5.2 of Charpinet 1998 orTable 1 of OÏDonoghue et al. 1999). In particular, the com-ponent which appears to be neither a harmonic nor af13,nonlinear cross frequency of the other oscillations, shows, at80.73 s, the shortest period observed so far in an EC 14026star. Nevertheless, the bulk of these periods are quite char-acteristic of an ““ average ÏÏ EC 14026 star, and it seems clearto us that this relatively wide band of periods was revealedsimply because we observed at much higher sensitivity thanhas been the case so far for most pulsators of this kind.Likewise, we note the relatively low amplitudes of the har-monic oscillations detected in the light curve of PG0014]067, being the strongest component with anf 3~amplitude less than D0.22% of the mean brightness of thestar, making it the EC 14026 pulsator with the smallestintrinsic amplitudes so far. This combination of intrinsiclow amplitudes and faintness (V ^ 15.9) explains well, inretrospect, that the variability of PG 0014]067 was missedin real time at the 1.6 m telescope on Mount Bigelow.

We show, in the upper half of Figure 3, the Fourieramplitude spectrum of the light curve of PG 0014]067 inthe 0È15 mHz bandpass. That part of the spectrum extend-ing beyond 15 mHz, and up to the Nyquist limit of 50 mHz,is Ñat, consistent with noise, and is not illustrated here. Theshort segments of solid lines indicate the positions of the 13frequency components, through (on this scale, thef1 f13


TABLE 2

HARMONIC OSCILLATIONS DETECTED IN THE LIGHT CURVE OF PG 0014]067

Frequency Period Amplitude 1 p Error Phase 1 p Error(kHz) (s) (%) (%) (s) (s)

5896.2 ( f 1~) . . . . . . . 169.60 0.0388 0.0088 108.39 6.175923.2 ( f 1`) . . . . . . . 168.83 0.0481 0.0088 2.34 4.956227.7 ( f2) . . . . . . . . 160.57 0.0390 0.0086 51.45 5.666621.1 ( f 3~) . . . . . . . 151.03 0.2183 0.0087 73.57 0.966630.7 ( f 3`) . . . . . . . 150.81 0.0548 0.0088 55.49 3.836837.5 ( f4) . . . . . . . . 146.25 0.1729 0.0087 96.79 1.177079.1 ( f 5~) . . . . . . . 141.26 0.0393 0.0093 82.82 5.377088.7 ( f 5`) . . . . . . . 141.07 0.1950 0.0090 64.55 1.047150.2 ( f6) . . . . . . . . 139.86 0.1435 0.0094 35.54 1.457286.2 ( f7) . . . . . . . . 137.25 0.0318 0.0087 71.57 5.997670.3 ( f8) . . . . . . . . 130.37 0.0281 0.0087 91.22 6.387952.1 ( f9) . . . . . . . . 125.75 0.0466 0.0086 31.70 3.708552.1 ( f10) . . . . . . . 116.93 0.0312 0.0086 75.59 5.139797.6 ( f11) . . . . . . . 102.07 0.0337 0.0086 78.73 4.169970.3 ( f12) . . . . . . . 100.30 0.1061 0.0086 54.77 1.3012386.8 ( f13) . . . . . . 80.73 0.0462 0.0086 6.88 2.39

three doublets are not resolved and are part of the samestructures). We have also indicated the positions of Ðvehigh-frequency peaks (dotted lines) possibly resulting fromnonlinear superpositions of the basic components listed inTable 2. The dotted lines actually give the values of thefrequency resulting from the following combinations (inorder of increasing frequency) : f 1`] f4, f 1~] f 5~, f 3~] f 3`,

and While we are not completely con-f4] f6, f 5~] f8.vinced that all of these Ðve nonlinear features are reallypresent in the Fourier spectrum owing to the small appar-ent amplitudes of some of the components, we Ðnd itstrongly suggestive that their frequencies correspond closelyto Ðve local maxima in the spectrum, as is clearly illustratedin the Ðgure. In a pulsation context, nonlinear peaks of that

FIG. 3.ÈUpper half : Fourier amplitude spectrum of the complete lightcurve of PG 0014]067 (runs cfh-068 through cfh-081) in the 0È15 mHzbandwidth (periods greater than 66.66 s). Some 150,000 frequency pointswere used in the calculation of this Fourier transform. The amplitude axisis expressed in terms of the percentage variations about the mean bright-ness of the star. The positions of the 13 harmonic oscillations (includingthree doublets unresolved at this scale) that we identiÐed in the light curveare indicated by short solid line segments. The dotted line segments indi-cate possible nonlinear (cross frequencies) structures. L ower half : Fourieramplitude spectrum (plotted upside down) of the residual light curve afterhaving subtracted the 16 harmonic components listed in Table 2.

sort are not, of course, associated with independent normalpulsation modes, but their amplitudes relative to those oftheir parent modes contain information about the geome-tries of the modes that can be potentially exploited as hasbeen demonstrated in the case of pulsating white dwarfs(see, e.g., Fontaine & Brassard 1994).

Our analysis of the light curve of PG 0014]067 leaves,after prewhitening the data of the 16 frequency componentsthat were identiÐed previously, a residual light curve

R(t) \ I(t) [ ;i/1

16aicosC2n

Pi(t [ /

i[ /0)

D, (1)

where R(t) and the original light curve I(t) are expressed inpercentage variations about the mean intensity of the starand and are, respectively, the periods, amplitudes,P

i, a

i, /

iand phases of the harmonic oscillations listed in Table 2.The zero-point phase, is arbitrary and corresponds in/0,practice here to the beginning of run cfh-068. The Fouriertransform of R(t) is plotted (upside down) in the lower halfof Figure 3. Although the spectrum is fairly Ñat, it is prob-able that there are additional real oscillations left over inthe light curve. For instance, we strongly suspect that thethree doublet structures that we uncovered [( f 1~, f 1`),

and are simply the largest components( f 3~, f 3`), ( f 5~, f 5`)]of more complex multiplets. We also suspect that thereremains small residual structure around some of the otherpeaks (identiÐed as single components) in the Fourier spec-trum, but we feel that we have reached the practical limit inthe exploitation of our observations, at least with themethods used here. In this context, we Ðnd that the averagevalue of the Fourier amplitude of R(t) in the 5È13 mHzbandpass (where the 16 oscillations are found) is equal to0.0100%. Using this as an estimate of the noise level in thatbandpass, we have retained in our analysis only com-ponents with amplitudes larger than 3 times that value(with the exception of the component with a borderlinef8amplitude of 0.0281%). We have found, by comparing withmore sophisticated statistical methods, that this practicalthree-to-one rule of thumb is generally sufficient to establishthe reality of a peak in the Fourier domain.

To complement the above analysis based, in part, onprewhitening techniques, i.e., subtraction of harmonic com-


FIG. 4.ÈUpper half : Fourier amplitude spectrum of the complete lightcurve of PG 0014]067 in the 5È13 mHz bandwidth obtained on the basisof 80,000 frequency points. L ower half : Fourier amplitude spectrum(plotted upside down) of the noiseless light curve reconstructed on thebasis of the 16 harmonic oscillations listed in Table 2. Further, the point-by-point frequency di†erence between the ““ observed ÏÏ Fourier spectrumand the ““ computed ÏÏ spectrum is shown, but shifted downward by 0.25%.

ponents in the time domain, we also consider subtraction inthe frequency domain. The upper half of Figure 4 shows theFourier amplitude spectrum of the light curve of our targetstar in the 5È13 mHz bandpass. This is just a portion of thespectrum illustrated in the previous Ðgure, shown in thefrequency interval where detectable photometric activityhas been found. Plotted upside down is the Fourier spec-trum of the noiseless reconstructed light curve based on theset of harmonic oscillation parameters given in Table 2. (Inother words, this is the Fourier spectrum of the last term onthe right-hand side of eq. [1] with the values of the periods,amplitudes, and phases taken from the table.) To obtain aquantitative measure of how well the reconstructed lightcurve reproduces the observed one, we have performed apoint-by-point di†erence in the frequency domain betweenthe two Fourier spectra. The residual spectrum is shown inFigure 4, shifted downward by 0.25% for visualization pur-poses. An examination of the Fourier spectrum, the modelspectrum, and their residual indicates that the light curve issatisfactorily reproduced on the basis of the parametersgiven in Table 2. We note that the residual is quite Ñat, aswould be expected in a successful reconstruction. There isperhaps a suggestion of residual structure consistent withour above remarks, especially in the 6.5È7.5 mHz region,but it is difficult, if not impossible, to extract more quanti-tative information from our current data set with themethods used in this section.

To summarize, using standard tools for this type of data,we have uncovered at least 16 periodicities in the light curveof PG 0014]067. Those include three doublets with veryclose frequency components (*f^ 9.6È27 kHz). The periodsrange from 80.73 to 169.90 s, and the amplitudes range from0.0281% to 0.2183%, making PG 0014]067 the EC 14026star with the lowest intrinsic amplitudes so far. The 16 oscil-lations appear all distinct and are best associated with someof the normal modes of vibration of the star on the acousticbranch (p-modes ; see next section). There is also a hint forthe presence of weak nonlinear structure resulting from thesuperposition of some of the basic oscillations.

4. ASTEROSEISMOLOGICAL INTERPRETATION OF THE

RESULTS

4.1. T he Problem at HandCharpinet et al. (1997 ; see also Fontaine et al. 1998) have

demonstrated that their so-called second generation of pul-sating models of sdB stars had the potential to explain theEC 14026 phenomenon. That approach has been exploitedextensively by Charpinet (1998), who showed that, indeed,pulsation theory could account remarkably well for theobserved class properties of the EC 14026 stars. An updateof this, including the latest discoveries, has been recentlypresented by Charpinet et al. (2001a). The driving mecha-nism in the Charpinet et al. models is a classic i mechanismassociated with an opacity bump due to the local enhance-ment of the iron abundance in the envelopes of sdB starscaused by di†usion processes. This mechanism is able todrive low-order, low-degree p-modes (along with some low-order and low-degree g-modes in the lower gravity models)with periods comparable to those observed. However,beyond general qualitative comparisons, encouraging asthey may be, no attempt has been made so far to explain indetail the observed period structure of an EC 14026 pulsa-tor and, hence, exploit the full potential of asteroseismologyfor these stars. Two problems have to be overcome beforesuch an attempt can be made.

The Ðrst difficulty is particular to this category of starsand is related to the newly developing nature of the Ðeld.Indeed, most EC 14026 stars have not yet been observed athigh enough sensitivity to reveal enough periods fordetailed comparisons with models. In short, when suchcomparisons are attempted, only ambiguous results areobtained and many di†erent models, sometimes with vastlydi†erent parameters, can be found to explain the fewperiods available. Interestingly enough, however, ourprogram carried out at the CFHT to follow up on knownpulsators is revealing much richer observed period spectrathan are generally available. This is also the case whenmultisite campaigns are deployed (e.g., Kilkenny et al.1999). For example, Table 1 of OÏDonoghue et al. (1999)reveals that only two to Ðve periods have been uncovered sofar in the light curves, gathered on small telescopes, of themajority of the known EC 14026 pulsators. There are twooutstanding exceptions : PG 1605]072 with more than 50periodicities detected in its light curve (Koen et al. 1998b ;Kilkenny et al. 1999) and KPD 1930]2752 with at least 44observed short-period oscillations et al. 2000).(BilleresUnfortunately, and perhaps ironically, these period spectraappear currently too rich for easy interpretation. The Ðrststar has been shown (Heber, Reid, & Werner 1999), and thesecond one is predicted et al. 2000 ; Maxted, Marsh,(Billeres& North 2000), to be a fast rotator, giving rise to overlap-ping pulsation modes in the Fourier domain (the rotation-ally split 2l ] 1 components of modes of various indices kand l). This combination of overlapping frequency com-ponents with the difficulty of modeling adequately fast rota-tion by taking into account e†ects beyond Ðrst-order solidbody rotation (appropriate only for slow rotation) currentlymakes it quite a challenge to account in detail for the richperiod spectra of PG 1605]072 and KPD 1930]2752. Incontrast to these extreme situations, from too sparsely totoo highly populated, the period spectrum of PG 0014]067that we uncovered from our CFHT observations is bothrich and simple enough to allow a complete and detailed


asteroseismological analysis on the basis of recently devel-oped tools, as we show below.

The second difficulty has plagued the whole Ðeld of aster-oseismology and is related to the problem of matching in anobjective way the observed periods with the periods com-puted from a model. If O observed periods are available andmust be compared to T theoretical periods coming from amodel (with T at least as large as but generally signiÐcantlylarger than O), then the number of possible matching com-binations is T !/(T [ O) ! When the number of observedperiods O becomes interesting in terms of providing enoughobservational constraints on the problem, the number ofmatching combinations becomes extremely large and thequestion of Ðnding the best combination of them all canonly be answered through statistical methods. For example,a typical sdB model with the values of and log g appro-Teffpriate for PG 0014]067 (see ° 2) has D25 acoustic (p-)modes with l\ 0, 1, 2, and 3 in the period window 80È170 s.Treating the three doublets as ““ single ÏÏ degenerate modes(since the Ðne structure is very likely the result of the slightdeformation of the star away from perfect sphericalsymmetry), we have 13 independent periods in our spectrumof PG 0014]067 covering that period window. Thenumber of matching combinations between the 13 observedperiods and the 25 available theoretical periods is 25 !/12 !^ 3.238] 1016, a formidable number if one is to searchone by one the combination that provides the best match.In a signiÐcant theoretical breakthrough, P. Brassard et al.(2001, in preparation) have recently developed a powerful,fast, and accurate algorithm designed to solve such aproblem in combinatorial analysis. It is based on a stochas-tic method using an evolutionary approach to optimizationproblems. We point out that the asteroseismologicalanalysis presented in this paper would simply not have beenpossible without this new tool.

In the light of these developments, we have adopted herethe so-called forward approach in asteroseismology, i.e., wesearch for a model in parameter space whose theoreticalperiod spectrum could account for the observed periods inPG 0014]067. At each grid point in parameter space, theperiod matching algorithm is used to insure that the bestpossible Ðt is obtained between the observed periods andthe available computed periods. Note that such a ““ best ÏÏ Ðtdoes not have to be a ““ good ÏÏ Ðt in the sense of reproducingwell the observed period spectrum. For instance, in regionsof parameter space where the real star PG 0014]067 is notwell represented, the theoretical periods cannot match wellthe observed periods. To describe the quality of the periodmatch in a quantitative way, we deÐne a goodness-of-Ðtmerit function (see below) that is evaluated at each gridpoint. The optimal model is then found according to thebest value of the merit function.

While we refer the interested reader to P. Brassard et al.(2001, in preparation) for more details on our period match-ing code, we note that the optimization algorithm used herefocuses solely on mode identiÐcation itself. That is to say,full grids of models are computed in parameter space andthe code is applied at each grid point to best identify theobserved periods with theoretical modes. This is in contrastto some recent e†orts using the popular genetic approachdeveloped by Charbonneau (1995), which seek to Ðnd theshortest path to the optimal model in parameter space,given some a priori information on mode identiÐcation. Thedisadvantage of our method is that full grids of models are

needed, including a large number of ““ useless ÏÏ models. Thisis largely o†set, however, by the fact that no a prioriassumption is made as to the mode identiÐcation, and thatthe hypersurfaces of the merit function in parameter spacecan be fully mapped and studied. We note further that P.Brassard et al. (2001, in preparation) discuss three types ofoptimization algorithms for the period matching code, allbelonging to the general family of evolutionary schemes.They use and compare ““ full genetic,ÏÏ ““ naive genetic,ÏÏ and““ post genetic ÏÏ approaches to the optimization problemposed by the period matching question. We adopted heretheir most efficient method, the ““ post genetic ÏÏ algorithm.

4.2. Equilibrium Models and T heir Pulsation PropertiesThe Ðrst step in our search procedure is to compute the

pulsation period spectra of a set of models that (hopefully)sandwich PG 0014]067 in parameter space. Three codesare involved in this step. We Ðrst use the model buildingcode originally described in Brassard & Fontaine (1994)and that was adapted to produce the so-called second-generation models of pulsating sdB stars of Charpinet et al.(1997). These are static envelope structures extending asdeep as For the purposes oflog [1 [ M(r)/M

*]\ [0.05.

pulsation calculations, the central missing nucleus contain-ing D10% of the mass is considered as a ““ hard ball.ÏÏ Thisapproach, in comparison with full evolutionary stellarmodels, gives excellent pulsation results, particularly for thep-modes, as described in detail in Charpinet (1998) andCharpinet et al. (2001b). In addition, the envelope modelshave the great advantage over evolutionary models in thatthey incorporate the results of di†usion calculations givingrise to a nonuniform distribution of iron as a function ofdepth. This distribution results from the competing actionsof gravitational settling and radiative levitation and hasbeen shown to be responsible, through the i mechanism, forthe excitation mechanism in sdB stars (Charpinet et al.1997 ; Fontaine et al. 1998). It is worth pointing out that,while the p-mode periods are completely insensitive to thedetails of the structure of the stellar nucleus, they dependquite sensitively on the structure of the outer envelope,including the thermal and mechanical proÐles produced bydi†usion.

There are four free parameters that need to be speciÐed inthe construction of a Charpinet et al. second-generationequilibrium stellar model : the e†ective temperature theTeff,surface gravity (traditionally given in terms of its logarithm)log g, the total mass of the model and the mass fractionM

*,

of the layers above the outer boundary of theM(H)/M*He/H chemical transition zone between the hydrogen-rich

envelope and the helium-rich core.2 Thus, we seek to Ðndthe model in this four-dimensional parameter space thatbest reproduces the observed periods of PG 0014]067. Forthe e†ective temperature and the surface gravity, we use ourspectroscopic determinations above as guides in our search,but we cover a wide range of values about these estimates.SpeciÐcally, for our Ðrst set of models, a coarse grid

2 Note that is intimately related to the more familiar param-M(H)/M*eter which corresponds to the total mass of the hydrogen-richMenv,envelope. The latter quantity includes the mass of the hydrogen contained

in the thin He/H transition zone, while M(H) does not. They are relatedthrough where C is a small positivelog (Menv/M*

)\ log [M(H)/M*]] C,

term, slightly dependent on the model parameters, determined by the massof hydrogen that is present inside the He/H transition zone itself.


designed for a relatively crude exploration of parameterspace, we consider the ranges K in31,000¹Teff ¹ 36,000steps of 1000 K and 5.65 ¹ log g ¹ 5.95 in steps of 0.05. Weemphasize the fact that any eventual determination of theatmospheric parameters of PG 0014]067 through aster-oseismological means has to be consistent with our spectro-scopic estimates. Since the latter are based on standardwell-understood techniques (see, however, the discussion ofWesemael et al. 1997 on possible systematic errors intro-duced by these methods when applied to sdB stars), thevalidity of our pulsating models would have to be ques-tioned if the results show otherwise.

Constraints on the possible values of andM*

M(H)/M*rely on stellar evolution theory, although a few estimates of

the masses of sdB stars (all consistent with theory but withlarge uncertainties) are available from studies of binarystars containing a B subdwarf component (see, e.g., Koen,Orosz, & Wade 1998d ; Orosz & Wade 1999 ; Wood &Sa†er 1999). The evolutionary calculations of Dorman,Rood, & OÏConnell (1993) indicate that sdB stars are corehelium burning stars on the extreme horizontal branch thatevolve as objects. According to these cal-““ AGB-manque� ÏÏculations, their possible masses are found in a narrow range

with a somewhat uncertain0.40È0.43[ M*/M

_[ 0.52,

lower limit related to the minimum mass required to ignitehelium and a more sharply deÐned upper limit above whichthe models evolve to the AGB. On the basis of these results,we consider for our Ðrst grid of models a range of masses

in steps of 0.01. Likewise, we select0.45¹M*/M

_¹ 0.51

the values of the mass of the outer hydrogen-rich envelopeon the basis of the work of Dorman et al. (1993) and ofadditional detailed evolutionary calculations carried out byB. Dorman in the framework of a collaborative e†ort withsome of us to investigate theoretically the pulsation proper-ties of sdB stars (see Charpinet 1998 ; Charpinet et al. 1996,2000, 2001b). We Ðnd that the range 10~5.0[ M(H)/M

*[

covers completely the interval of envelope masses10~2.5found in the models required to fully map the region of the

plane where real sdB stars are found (see, e.g., Fig.log gÈTeff1 of Charpinet et al. 2000). For our Ðrst grid, however, werestrict the search in a narrower range 10~4.3 ¹

in steps of ThisM(H)/M*

¹ 10~3.7 log [M(H)/M*]\ 0.15.

grid then has 1470 grid points [six values of sevenTeff,values of log g, seven values of and Ðve values ofM*,

M(H)/M*].

At each grid point in parameter space, the pulsationproperties of the local model are then computed with thehelp of two very efficient pulsation codes based on Ðniteelement techniques. The Ðrst one is an updated version ofthe adiabatic code described at length in Brassard et al.(1992). It is used as an intermediate (and necessary) step toobtain estimates of the periods that are then used as Ðrstguesses in the solution of the nonadiabatic eigenvalueproblem. The second one is an improved version of thenonadiabatic code that has been described brieÑy in Fon-taine et al. (1994). It provides the necessary quantities tocompare with the observations (essentially the periods andthe stability coefficients). In order to cover adequately therange of observed periods in PG 0014]067, 80È170 s, wecompute all pulsation modes (be it p, f, or g) in the periodwindow 60È300 s and with degree index l\ 0, 1, 2, and 3.We reiterate the usual argument that modes with l º 4would not be observable (and therefore are not of directinterest here) because of geometric cancellation e†ects on

the visible disk of the star (see also below). This three-stepoperation (computations of a model, computations of theadiabatic pulsation properties, and computations of thenonadiabatic properties) involving, on the average, D40pulsation modes takes about 56 s of CPU time on a SunUltra 10 machine. This makes the process a rather efficientone by asteroseismological standards.

A typical output of that operation is shown in Table 3.While it refers speciÐcally to the optimal model (see thediscussion below), we only wish to emphasize the illustra-tive aspect of the table here in this subsection. For eachmode in the given period window, the table gives the degreeindex l, the radial order index k, the period P (\2n/p

R,

where is the real part of the complex eigenfrequency), thepRstability coefficient (the imaginary part of the complexp

Ieigenfrequency), the logarithm of the so-called kineticenergy of the mode E, and the dimensionless Ðrst-orderrigid rotation coefficient Of course, as is standard, ourC

kl.

equilibrium stellar models are perfectly spherical, and eachmode is 2l ] 1 fold degenerate in eigenfrequency.

TABLE 3

PULSATION CHARACTERISTICS OF THE OPTIMAL MODEL IN THE

60È300 s BANDPASS

P pI

log El k (s) (rad s~1) (ergs) C

kl

0 . . . . . . 9 61.696 6.456 ] 10~4 38.079 0.0080 . . . . . . 8 66.423 4.437 ] 10~4 38.109 0.0080 . . . . . . 7 74.310 8.805 ] 10~5 38.832 0.0080 . . . . . . 6 81.715 [2.412 ] 10~5 39.517 0.0150 . . . . . . 5 90.859 [7.697 ] 10~5 39.723 0.0110 . . . . . . 4 105.304 [5.173 ] 10~5 40.210 0.0180 . . . . . . 3 115.320 [2.159 ] 10~5 40.640 0.0240 . . . . . . 2 138.012 [8.201 ] 10~6 40.959 0.0160 . . . . . . 1 158.911 [3.690 ] 10~7 42.040 0.0410 . . . . . . 0 175.951 [1.467 ] 10~7 42.036 0.0201 . . . . . . 10 61.051 6.370 ] 10~4 38.126 0.0031 . . . . . . 9 65.883 5.196 ] 10~4 38.030 0.0041 . . . . . . 8 73.634 9.945 ] 10~5 38.806 0.0051 . . . . . . 7 80.289 [1.525 ] 10~5 39.414 0.0061 . . . . . . 6 89.999 [7.845 ] 10~5 39.677 0.0061 . . . . . . 5 103.269 [4.519 ] 10~5 40.243 0.0111 . . . . . . 4 113.035 [3.227 ] 10~5 40.458 0.0111 . . . . . . 3 136.699 [8.539 ] 10~6 40.949 0.0141 . . . . . . 2 153.579 [6.813 ] 10~7 41.851 0.0251 . . . . . . 1 175.295 [1.697 ] 10~7 41.989 0.0182 . . . . . . 9 64.950 7.796 ] 10~3 37.761 0.0032 . . . . . . 8 72.354 1.158 ] 10~4 38.767 0.0062 . . . . . . 7 78.383 2.479 ] 10~6 39.223 0.0072 . . . . . . 6 88.657 [7.469 ] 10~5 39.632 0.0092 . . . . . . 5 99.373 [3.957 ] 10~5 40.238 0.0192 . . . . . . 4 110.572 [4.699 ] 10~5 40.283 0.0142 . . . . . . 3 132.749 [7.935 ] 10~6 41.004 0.0362 . . . . . . 2 145.195 [2.666 ] 10~6 41.359 0.0462 . . . . . . 1 174.058 [2.096 ] 10~7 41.924 0.0232 . . . . . . 0 213.848 1.353 ] 10~10 45.038 0.4293 . . . . . . 9 64.953 7.792 ] 10~3 37.761 0.0033 . . . . . . 8 70.157 1.512 ] 10~4 38.647 0.0133 . . . . . . 7 76.466 3.595 ] 10~5 39.008 0.0113 . . . . . . 6 86.385 [5.660 ] 10~5 39.617 0.0203 . . . . . . 5 94.545 [5.053 ] 10~5 40.025 0.0313 . . . . . . 4 108.290 [5.627 ] 10~5 40.192 0.0213 . . . . . . 3 124.027 [7.913 ] 10~6 41.051 0.0793 . . . . . . 2 140.043 [6.375 ] 10~6 41.037 0.0353 . . . . . . 1 171.104 [2.749 ] 10~7 41.875 0.0633 . . . . . . 0 184.078 [5.256 ] 10~9 43.009 0.193


The most important variable is obviously the theoreticalperiod. The periods are sensitive to the global structuralparameters of a model, and the primary goal of aster-oseismology is to infer these parameters for a real starthrough a comparison with the observed periods. Becausenonadiabatic e†ects on the periods are small (but areincluded here), asteroseismology could be carried out onlyat the level of the adiabatic approximation for the sdB stars.Nevertheless, the stability coefficient, a purely nonadiabaticquantity, is also a very useful variable as it provides infor-mation on the driving mechanism and the thermal structureof a star. A positive value of indicates that a mode isp

Idamped, while a negative value indicates that the mode isexcited in the model. For its part, the kinetic energy is asecondary quantity in the present context. It gives ameasure of the energy required to excite a mode of a givenamplitude at the surface of a star. Since this is a normalizedquantity, only its relative amplitude from mode to mode isof interest. The kinetic energy is a useful diagnostic tool fordetermining where pulsation modes are formed ; largervalues of E imply that such modes probe deeper into thestar, in higher density regions. The kinetic energy bears thesignatures of trapping/conÐnement phenomena associatedwith the presence of a thin chemical transition zone betweenthe hydrogen-rich envelope and the helium-rich core.Finally, the rotation coefficient is useful for interpreting Ðnestructure in the Fourier domain, such as the doublets wehave uncovered in the spectrum of PG 0014]067. It alsobears the signature of mode trapping/conÐnement, but thatsignature is weaker than in the case of the kinetic energy.

Table 3 shows that modes are excited within a band ofperiods. This is a general characteristic of all of our pulsat-ing models of sdB stars. In the particular model sum-marized in the table, that band of periods covers the rangefrom D80 to upward of D185 s. Note the signiÐcantdecrease in the absolute value of for the 184.078 s modep

I(l\ 3, k \ 0), meaning that the upper limit of the instabilityregion in period space is nearly reached. Note also that the213.848 s mode (l\ 2, k \ 0) is not excited. These twomodes are outstanding in that they show relatively largevalues of E compared to their adjacent modes, and this istrue also for their values of They are, in fact, so-calledC

kl.

f-modes, separating the acoustic from the gravity mode fam-ilies in period space and showing a mixed character betweenpure p-modes and pure g-modes. They are formed relativelydeep in the star, and they are hard to excite. The g-modebranches for l\ 1, 2, and 3 correspond to modes that allhave periods larger than 300 s for this particular model and,consequently, do not belong to Table 3. As discussed atlength by Charpinet et al. (2000), the g-modes in sdB starsare generally deep core modes and cannot be excited by theenvelope iron opacity bump mechanism that is present inour models. For low-gravity models, which are not ofimmediate interest in the case of PG 0014]067, the distinc-tion between low-order p-modes and low-order g-modesbecomes fuzzy, and modes that are formally assigned to theg-branches can also become excited (Fontaine et al. 1998 ;Charpinet 1998 ; Charpinet et al. 2001b).

Before moving on to a discussion of the search for theoptimal model in parameter space, a remark concerning thel index in the model calculations might be in order here. Aquantitative expression of the visibility argument used torestrict the search to low values of l is obtained by evalu-ating numerically the geometric factor given, for example,

by equation (B7) of Brassard, Fontaine, & Wesemael (1995).Using an Eddington limb darkening law in that equation,we Ðnd that the visibility factor takes on the values 1.0000,0.7083, 0.3250, 0.0625, 0.0208, 0.0078, 0.0078, 0.0023, 0.0039,0.0009, 0.0023, 0.0005, and 0.0015, respectively, for l\ 0, 1,2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. While this favors a priorivalues of l \ 0, 1, and 2 in a pulsating star in general, wenote that the mode densities and period distributions seenin at least two well-observed EC 14026 stars (see Fig. 10 of

et al. 2000 for KPD 1930]2752 and Fig. 7 of Char-Billerespinet et al. 2001a for PG 1047]003) force us to considermodes with l \ 3 as well. Otherwise, there would not beenough theoretical modes available in the observed periodwindows to account for the observations. We therefore feltthat, in this Ðrst e†ort to explain quantitatively the observedperiod spectrum of PG 0014]067, all modes from l\ 0through at least l \ 3 should be considered. Of course, wecannot rule out completely the presence of the odd l\ 4mode (or perhaps even a mode with a larger value of l) withan unusually high intrinsic amplitude in the light curve ofPG 0014]067. Our approach explicitly excludes this possi-bility, however, and we seek to interpret the period distribu-tion of our target star solely in terms of l \ 0È3 modes.

4.3. Search for the Optimal Model in Parameter SpaceWe measure the quality of the period match at each grid

point through a dimensionless merit function, which has afamiliar form

s2 \ ;i/1

13 APobsi [ Ptheoi

pi

B2, (2)

where is one of the 13 observed periods and is thePobsi Ptheoitheoretical value assigned to that observed period at thatgrid point through our period matching algorithm. For thepurposes of this comparison, the three doublets identiÐed inTable 2 are averaged together and considered as degeneratemodes with average periods 150.92, andPobs\ 169.22,141.17 s for the and doublets,( f 1~, f 1`), ( f 3~, f 3`), ( f 5~, f 5`)respectively. This leaves 10 other observed modes whoseperiods are given in Table 2. For the weight functions pi, wehave chosen pi\ p, where p is the inverse of the theoreticalmode density (i.e., the ratio of the width of the periodwindow [here 300[ 60 \ 240 s] to the number of theoreti-cal modes in that window) at that grid point. This choice ismade in order to o†set, at least partly, the built-in bias infavor of models with a higher theoretical mode density (inparticular, the models with the lower gravities ; see, e.g., Fig.3 of Fontaine et al. 1998). Indeed, there is less ““ merit ÏÏ atÐnding a good match to a given set of observed periods ifthe number of available theoretical periods in a givenwindow is higher. The optimal model is the one in param-eter space that has the set of theoretical periods that pro-vides the smallest value of s2.

A detailed examination of the behavior of the merit func-tion s2 at the 1470 points in our four-dimensional grid inparameter space reveals a Ðrst interesting result : the s2hypersurfaces (for example, hypercontours of constantvalues of s2) are very well behaved and suggest the existenceof well-deÐned regions in parameter space where s2 goesthrough a minimum. That we could achieve this result wasnot obvious at the outset since there was no guarantee thatour models are sufficiently realistic. We show, in Figure 5, aslice of the s2 hypercontours in the plane at thoselog gÈTeff


points in four-dimensional space corresponding to thelowest values of s2, speciÐcally, at andM

*/M

_\ 0.49

Keeping in mind that our grid is ratherM(H)/M*

\ 10~4.3.coarse (with six e†ective temperatures and seven surfacegravities), the s2 contours are indeed quite smooth. Theyshow the presence of two elongated ““ valleys ÏÏ in e†ectivetemperature, one centered on the lowest gravity of ourtable, log g \ 5.65, and the other centered on log g \ 5.80.We have veriÐed explicitly, with spotted calculations, thatthe true bottom of the lower gravity valley is located alongthe axis log g \ 5.65, and not at some lower gravity o† thescale of Figure 5. The shapes of the contours reÑect thelarger relative sensitivity of the pulsation periods of sdBmodels on the surface gravity as compared to the e†ectivetemperature (see Fontaine et al. 1998 for a more detaileddiscussion of this point). Models in those two disconnectedvalleys provide, in principle, period Ðts of comparablequality. However, the constraints provided above by ouranalysis of the time-averaged optical spectrum of PG

0014]067 come in handy for deciding which of these twovalleys is the ““ correct ÏÏ one. We show, in Figure 5, an errorbox (dotted rectangle) giving the location of our spectro-scopic estimates of the e†ective temperature and surfacegravity of PG 0014]067, namely, KTeff \ 33,550 ^ 380and log g \ 5.77^ 0.10. We Ðnd it tantalizing that ourspectroscopic estimates of and log g overlap quite wellTeffwith a region of the plane where, according to ourlog gÈTeffmerit function s2, theoretical periods best match theobserved periods. This is the second interesting result thatcould not be guaranteed at the outset. On the basis of ourspectroscopic estimates, we then focus our attention on thelog g \ 5.80 valley. We provide below further evidence thatthis is indeed the good choice.

The dashed rectangle in Figure 5 delimits the regionwhere we now zoom in to search for the optimal model. Forthis quest, we computed a new grid of models with a Ðnermesh in the log g direction in particular (nine values from5.76 to 5.84 in steps of 0.01), but also in the directionTeff

FIG. 5.ÈContours of constant s2 in the plane for Ðxed values of and The latter values correspond to thelog gÈTeff M*/M

_\ 0.49 M(H)/M

*\ 10~4.3.

regions of minimal values of s2 in the hyperspace. The contour levels increase in steps of *s2\ 0.5 from the lowest value, s2\ 1.0 (deep blue), to the highestvalues coded in red. The contours deÐne two elongated blue valleys, one centered on log g \ 5.65 and the other centered on log g \ 5.80, where periodmatches of comparable quality are obtained. The higher gravity models (yellowÈred regions) provide very poor period matches and can be ruled out at theoutset. The dotted rectangle is the error box associated with our spectroscopic estimates of the atmospheric parameters of PG 0014]067. The dashedrectangle shows the region of the plane where we zoom in for a search on a Ðner grid of models (see text).log gÈTeff


(Ðve values from 33,000 to 35,000 K in steps of 500 K). Forthe mass of the model, we keep Ðve values at the sameresolution as before, i.e., 0.48, 0.49, 0.50,M

*/M

_\ 0.47,

and 0.51. Finally, we consider the following values of thehydrogen-rich envelope mass : 10~4.5,M(H)/M

*\ 10~5.0,

10~4.0, 10~3.5, 10~3.0, and 10~2.5. In the latter case, wehave extended the range of values beyond 10~4.3 (the sug-gested optimal value from the previous grid) in order tomake sure that the optimal model is contained within agiven volume of parameter space and does not sit on an““ edge ÏÏ (with the implicit suggestion that the ““ true ÏÏoptimal model may be outside that volume). This is donethough at the expense of some reduced resolution along the

axis.M(H)/M*Within the volume of parameter space occupied by this

second grid of 1350 models, there is a well-deÐnedminimum in s2 corresponding to the optimal model (see themore detailed discussion below). This model is located atthe grid point log g \ 5.78, K,Teff \ 34,500 M

*/M

_\

0.49, and The latter value corre-log [M(H)/M*]\ [4.5.

sponds to a hydrogen envelope mass of log (Menv/M*) \

[4.31. We Ðnd that this optimal model can accountremarkably well, and this is our third outstanding result, forthe observed periods in PG 0014]067. The pulsationproperties of the model are summarized in Table 3 and havebeen brieÑy discussed above. Of particular interest in thistable is the existence of a band of excited periods in therange from D80 to s, a band of instability thatZ185appears to apply to all the values of the degree index lconsidered here. In addition, the modes involved are allacoustic modes (including the f-modes).

Table 4 shows the Ðt provided by our period matchingcode for the optimal model. The 13 observed periods aresimultaneously matched in this single model to 13 theoreti-cal periods that all fall quite close to the observed values(see the footnoted values in the table). The average relativedispersion between these observed and theoretical periodsis 1.1%, and the worst di†erence is less than 2%. On anabsolute scale, the average dispersion in period is less than1.5 s. This accurate and simultaneous Ðt of 13 periods is, byitself, a notable result in the Ðeld of asteroseismology. Inaddition, we Ðnd that, in agreement with our theoreticalexpectations, the 13 observed periods all fall within theband of predicted instability as clearly shown in the table.The observed periods are identiÐed with low-order radial(l\ 0) and nonradial (l\ 1, 2, and 3) p-modes. While weÐnd these results most gratifying, we are intrigued by thedistribution of observed modes. The observed modes do notcover the full range of unstable periods predicted by theory,and, more puzzling, there are ““ holes ÏÏ in their distribution.In short, there appear to be ““ missing ÏÏ modes, and wewonder where they hide in the data. We speciÐcally addressthis question in the next subsection.

Before we do, however, we cannot help but remark thatthere is no particular amplitude hierarchy as a function of lfor the 13 modes identiÐed in Table 4. Because of the geo-metric cancellation e†ects alluded to above, this would bethe case if all the modes had the same intrinsic amplitudes.This is clearly not what is observed, in agreement with othertypes of pulsating stars. While the question of amplitudes isbeyond the realm of linear theory as is well known (note, forinstance, the lack of correlation between the growth rates inTable 3 and the actual apparent amplitudes), we mightnevertheless naively expect, at least statistically, some kind

TABLE 4

FIT PROVIDED BY THE PERIOD MATCHING ALGORITHM FOR THE

OPTIMAL MODEL

Ptheo Pobs o*P o o *P o /PobsStability l k (s) (s) (s) (%)

Stable . . . . . . 0 7 74.31 . . . . . . . . .Unstable . . . 0 6 81.72 81.47 0.25 0.31Unstable . . . 0 5 90.86 91.20 0.34 0.37Unstable . . . 0 4 105.30 105.85 0.55 0.52Unstable . . . 0 3 115.32a 116.93a 1.61 1.37Unstable . . . 0 2 138.01a 139.86a 1.85 1.32Unstable . . . 0 1 158.91a 160.57a 1.66 1.03Unstable . . . 0 0 175.95 . . . . . . . . .Stable . . . . . . 1 8 73.63 . . . . . . . . .Unstable . . . 1 7 80.29a 80.73a 0.44 0.54Unstable . . . 1 6 90.00 90.09 0.09 0.10Unstable . . . 1 5 103.27a 102.07a 1.20 1.18Unstable . . . 1 4 113.04 112.31 0.73 0.65Unstable . . . 1 3 136.70a 137.25a 0.55 0.40Unstable . . . 1 2 153.58a 150.92a 2.66 1.76Unstable . . . 1 1 175.30 . . . . . . . . .Stable . . . . . . 2 7 78.38 . . . . . . . . .Unstable . . . 2 6 88.66 88.04 0.61 0.69Unstable . . . 2 5 99.37a 100.30a 0.93 0.93Unstable . . . 2 4 110.57 110.16 0.41 0.37Unstable . . . 2 3 132.75a 130.37a 2.38 1.83Unstable . . . 2 2 145.20a 146.25a 1.06 0.72Unstable . . . 2 1 174.06 . . . . . . . . .Stable . . . . . . 2 0 213.85 . . . . . . . . .Stable . . . . . . 3 7 76.47 . . . . . . . . .Unstable . . . 3 6 86.39 86.69 0.30 0.35Unstable . . . 3 5 94.55 94.33 0.21 0.22Unstable . . . 3 4 108.29 108.10 0.19 0.18Unstable . . . 3 3 124.03a 125.75a 1.72 1.37Unstable . . . 3 2 140.04a 141.17a 1.13 0.80Unstable . . . 3 1 171.10a 169.22a 1.88 1.11Unstable . . . 3 0 184.08 . . . . . . . . .

a The 13 observed periods are simultaneously matched in this singlemodel to 13 theoretical periods that all fall quite close to the observedvalues.

of amplitude attenuation when l is increased. And yet, thisattenuation is not seen here. We have no explanation forthis puzzle and can only speculate on the nature of thephenomenon. We note that the number of modes is small.Perhaps, in addition, the driving mechanism, through non-linear e†ects, systematically favors l \ 2 or 3 modes overother modes in terms of building up larger intrinsic ampli-tudes. This is pure speculation on our part, however, andonly nonlinear theory will be able to address properly thisissue.

4.4. Search for Additional Pulsation Modes in the L ightCurve of PG 0014]067

We have carried out a search for additional pulsationmodes perhaps still buried in our light curve of PG0014]067. The primary clue to make us suspect that theabove ““ missing ÏÏ modes are probably modes with ampli-tudes lower than the threshold value that we used in ° 3 isthat there is no reason, at least within the realm of linearpulsation theory, for the star to excite, for example, modeswith l \ 1 and k \ 2, 3, 5, and 7 but not those with k \ 4 or6 (see Table 4). In brief, there is no known process, at leastto us, that could act as a mode selection mechanism for thep-modes in sdB stars. This is in contrast to the g-modes,


which are subjected to strong trapping/conÐnement e†ectsdue to the chemical stratiÐcation in these objects, particu-larly the e†ects associated with the H/He transition zone atthe bottom of the hydrogen-rich envelope (Charpinet et al.2000). It is generally believed, since the original suggestionof Winget (1981), that similar trapping e†ects in pulsatingwhite dwarfs (g-mode pulsators that are compositionallystratiÐed) allow these stars to selectively excite only a few ofthe modes available to them. In the case of p-modes,however, such trapping/conÐnement e†ects leave only avery weak signature on their propagation properties andcannot act as efficient mode selection mechanisms. Forinstance, we can observe in Table 3 that the kinetic energy Edecreases relatively smoothly with increasing radial order kfor a given value of l. This is a telltale sign that thesep-modes are relatively insensitive to the chemical layeringand, consequently, such layering cannot act as an efficientÐlter. Strong trapping e†ects would, in comparison, show acharacteristic imprint on this distribution, namely, the pres-ence of several important local minima separated in periodby an amount directly related to the mass or the thicknessof the outer hydrogen-rich layer.

We have tried then to uncover the evidence (if any) thatwould allow us to associate the unassigned theoreticalperiods appearing in the fourth column of Table 4 withlow-amplitude pulsation modes that we may have over-looked in our analysis of ° 3 and that would manifest them-selves as maxima in the Fourier spectrum of the residuallight curve of PG 0014]067. On the short-period side ofthe theoretical instability band, the modes with periods74.31 s (l\ 0, k \ 7), 73.63 s (l\ 1, k \ 8), and 76.47 s(l\ 3, k \ 7) correspond to regions of the Fourier spectrumwhere there is no signiÐcant signal above the noise and, inparticular, no local maxima. The mode with period 78.38 s(l\ 2, k \ 7), stable according to our calculations, is anexception as it could be associated with a relatively largemaximum of amplitude 0.0369% appearing in the Fourierspectrum. However, by accident, this structure also corre-sponds to the possible nonlinear superposition of mode f 1`and mode one of the Ðve nonlinear cross frequency com-f4,ponents that we discussed in ° 3 and Ñagged in Figure 3.The frequency of this maximum, 12.763 mHz, correspondsnearly exactly to the sum 5.923 ( f 1`)] 6.838 ( f4)\ 12.761mHz (78.36 s), suggesting very strongly that it is indeed anonlinear feature (and not an independent normal mode).However, we Ðnd it impossible at this stage to rule outeither of the two possibilities. By the same token, we cannot,with any conÐdence, claim that the 78.38 s pulsation is““ seen ÏÏ in the data (it should not be according to theory).

While we are conÐdent that luminosity variations withperiods shorter than D78 s are not present in our lightcurve of PG 0014]067, at least with amplitudes above thenoise and with the exception of the nonlinear features, thesituation is not so clear cut at the long-period end of thespectrum. Indeed, because of the complexity of the residualFourier spectrum in that region, with many overlappingfeatures, we cannot conÐrm or rule out the possible pres-ence of low-amplitude maxima that could be associatedwith the theoretical modes with periods 175.95 s (l \ 0,k \ 0), 175.30 s (l\ 1, k \ 1), 174.06 s (l\ 2, k \ 1), 213.85 s(l\ 2, k \ 0), and 184.08 s (l\ 3, k \ 0). However, and thisis our fourth noteworthy result, we Ðnd that the 10 unas-signed periods that remain in the theoretical spectrum (seeTable 4) can all be associated with low-amplitude maxima

in the Fourier spectrum. This is illustrated in Figure 6where the Fourier amplitude spectrum of the residual lightcurve of PG 0014]067 (after prewhitening the 16 harmonicoscillations of Table 2) is shown in the 8.5È12.5 mHzbandpass. This is just a segment of the residual spectrumalready shown in Figure 3 (plotted upside down) but illus-trated here on an expanded scale for the needs of our com-parison exercise. Note that, on that scale, the daily aliases ofthe various harmonic oscillations are now apparent. Forthe purposes of our comparison, however, the aliases associ-ated with a single harmonic oscillation form a single““ peak.ÏÏ We show, for each of 10 maxima, the range offrequency corresponding to an interval of period of^0.55% about the central period. This interval is half theaverage relative dispersion that was found in the previoussection for the 13 modes that were identiÐed there. Becausethe density of peaks of comparable amplitudes hasincreased in the Fourier spectrum of the residual lightcurve, we use a more severe selection criterion here forassigning a theoretical period to a low-amplitudemaximum. The various ranges are illustrated by the dottedforks in the Ðgure, where we also indicate the values of thecentral periods. The solid vertical thick line segments corre-spond, for their part, to the frequencies of the 10 unassignedtheoretical modes of Table 4. We Ðnd it most remarkablethat eight of these modes have periods that ““ fall ÏÏ within^0.55% of the periods associated with maxima in theFourier spectrum, while the periods of the other two (112.31and 88.04 s) are located less than 0.70% away from othermaxima. This is either an extraordinary coincidence or thedemonstration that the expected pulsation modes areindeed present in the light curve of PG 0014]067, albeitwith relatively low amplitudes.

FIG. 6.ÈComparison of the frequencies of 10 unassigned theoreticalmodes in Table 4 with the frequencies of 10 low-amplitude maxima in theFourier amplitude spectrum of the residual light curve of PG 0014]067after prewhitening the 16 main harmonic oscillations listed in Table 2. Thedotted ““ forks ÏÏ show the ranges in frequency corresponding to periodranges of ^0.55% about the central values of the periods. The periods ofthe maxima are also indicated. The solid vertical thick line segments thatfall within or close to each fork give the frequencies of the theoreticalpulsation modes. The solid horizontal line is a reference level correspond-ing to an amplitude of 0.02%. The solid horizontal thick line segments givethe positions of the ^0.55% bandwidth associated with each maximumwith amplitudes larger than the reference level of 0.02%.


We can put the last statement on a more quantitativebasis by estimating the chance occurrence probability of amatch (within ^0.55%) between the eight theoretical fre-quencies and eight maxima in the Fourier spectrum.Clearly, this probability must be a function of the ampli-tudes of the maxima. The 10 maxima that we have identiÐedall have amplitudes larger than 0.02% (see the horizontalsolid line in Fig. 6). We note that there are, altogether, 21peaks with maximum amplitudes larger than that referencelevel in the bandpass shown in Figure 6. Each of thesefeatures occupies a bandwidth of ^0.55% about its centralfrequency, and this is illustrated by the short solid horizon-tal thick line segments at the 0.035% amplitude level. Thereis some, but not much, overlap between the individualbandwidths, and the total coverage in frequency is 2.16mHz. This gives an average width *f\ 2.16/21 ^ 0.10mHz. From these results, we estimate that the probability

that one theoretical frequency falls by chance withinP10.55% of the frequency of any one of the peaks withamplitudes º0.02% in the Fourier spectrum shown inFigure 6 is given by the ratio of the frequency bandoccupied by these peaks (2.16 mHz) to the total bandpass ofinterest (4.00 mHz). This gives a rela-P1^ 2.16/4.00 \ 0.54,tively large probability consistent with a simple visualinspection of Figure 6. The probability that a secondP2theoretical period falls by chance on any one of the remain-ing peaks is then given by P2\ (2.16 [ *f )/(4.00 [ *f ) ^0.53, after having removed the bandwidth corresponding tothe Ðrst mode. The probability that a third theoreticalP3period falls by chance on any one of the other remainingpeaks is given by P3\ (2.16[ 2*f )/(4.00 [ 2*f ) ^ 0.52,and so on. We Ðnally Ðnd that the probability P that eighttheoretical frequencies fall by chance within ^0.55% of thecentral frequencies of eight maxima with amplitudesº0.02% in the 8.5È12.5 mHz bandpass is given by P\

a dramatic and signiÐcant decrease.%i/18 P

i^ 3.5] 10~3,

If we draw the line a little higher, say for amplitudesº0.025%, then there are only four maxima high enoughout of nine that can be associated with theoretical modes,again within ^0.55% in periods. In that case, the probabil-ity that one theoretical frequency falls by chance on any oneof the peaks with amplitudes º0.025% in the 8.5È12.5 mHzbandpass is It follows that the prob-P1^ 9*f/4 \ 0.225.ability that four theoretical modes fall by chance on fourmaxima with amplitudes º0.025% in the same bandpass is

We provide in Appendix A aP\ %i/14 P

i^ 1.3] 10~3.

more rigorous derivation of this probabilistic argument.We conclude from this exercise that we have probably

uncovered and identiÐed eight pulsation modes that add tothe 13 already discussed above. We also add the two otherpulsations (112.31 and 88.04 s) with periods falling verynear, although not rigorously within, our ^0.55% criterion,the periods of the two remaining unassigned theoreticalmodes. We have reported, in Table 4, the values of the 10additional observed periods next to the periods of theassigned theoretical modes. We note that the average rela-tive dispersion between these Ðgures is 0.45%, while theaverage dispersion on an absolute scale is 0.43 s. We feelthat the presence of these additional pulsation modesstrengthens considerably the case we make for the optimalmodel as a realistic representation of PG 0014]067. In thiscontext, we have carried out tests with models belonging tothe log g \ 5.65 s2 valley of Figure 5. As indicated above,such models provide, in principle, s2 Ðts to the 13 original

periods of comparable quality, but with di†erent modeidentiÐcations, to those obtained for models found rather inthe log g \ 5.80 valley where we have concentrated oursearch. We Ðnd that there are also ““ missing ÏÏ modes in thedistribution of the 13 observed periods as compared to thetheoretically expected period distributions, but, quite sig-niÐcantly we believe, the majority of these missing modescannot be associated, for the log g \ 5.65 models, to low-amplitude maxima in the Fourier spectrum of the lightcurve of PG 0014]067. This allows us to rule out thatother region of local minimum in the s2 hyperspace.

It is interesting to examine more closely the low-amplitude harmonic oscillations in the light curve of PG0014]067 that we just identiÐed as real pulsation modes.As discussed in ° 3, we have adopted there a practical three-to-one rule of thumb to pick up statistically signiÐcant fre-quency peaks in the Fourier spectrum, whereby a peak isconsidered ““ secure ÏÏ if its amplitude is at least 3 times largerthan the amplitude of the noise in the neighborhood. This isa conservative approach, based on comparisons with moresophisticated methods such as the false alarm probabilityformalism, that has led, in ° 3, to the identiÐcation of the 16harmonic components described in Table 2. This approachdoes not mean, of course, that frequency peaks with ampli-tudes less than this threshold value are hopelessly lost in thenoise. Folding techniques can be used to increase the S/N,and, as an example here, we show in Figure 7 the results offolding the light curve on a period of 110.160 s. This is oneof the 10 harmonic components in Figure 6 that we associ-ate with real pulsation modes ; the results are typical of theother periods shown there. Before folding, we removed fromthe light curve of PG 0014]067 the 16 largest harmoniccomponents through prewhitening subtraction using theparameters listed in Table 2. The light curve has 3454 pointsthat have been distributed in Ðve phase bins (D690 pointsbin~1) in order to optimize the S/N and, at the same time,keep a reasonable phase resolution. The solid curve in

FIG. 7.ÈLight curve of PG 0014]067 folded on the period of 110.160 sand distributed in Ðve phase bins. As usual, the curve is plotted twice forbetter visualization. The points give the observational data with error barscorresponding to the errors of the mean in each bin. The light curve hasbeen prewhitened of its 16 largest amplitude oscillations (Table 2). Thesolid curve is a simple template made of a single sinusoid with an ampli-tude and phase obtained through least-squares Ðtting to the light curve.The residuals shown here are the di†erence between the data points andthe template curve. They have been shifted arbitrarily downward by0.10%.


Figure 7 is a template deÐned by the pure sine wave ofperiod 110.160 s and of amplitude and phase obtainedthrough a least-squares Ðt to the light curve. An exami-nation of the pulse shape, the template, and their residualcertainly suggests that the 110.160 s oscillation in the lightcurve of PG 0014]067 is real. What we Ðnd particularlyconvincing in the identiÐcation exercise that we carried outabove is that the light curve shows 10 such oscillations (ofquality comparable to that illustrated in Fig. 7) with periodsthat are not randomly distributed but fall very close to the10 periods expected from theory in that range.

We end this subsection by presenting in Figure 8 agraphic representation of the period match and the modeidentiÐcation that we have secured for PG 0014]067.Using the data of Table 4, we Ðrst plotted the theoreticalperiod spectrum of the optimal model according to thevalue of the degree index l and the radial order k in theperiod range of interest. These are acoustic modes(including f-modes). Modes that are excited according totheory are represented by solid line segments, while modesthat are damped are represented by dotted line segments. Incomparison, the 23 periods that we uncovered in the lightcurve of PG 0014]067 are shown as thick dashed linesegments. What we Ðnd remarkable here is not only that anexcellent (by asteroseismological standards) match betweenthe predicted and observed periods is achieved but also thatthe predictions of nonadiabatic theory as to the stability ofthe modes appear to be in close agreement with the obser-vations. Indeed, a band of excited periods is expected to bepresent for the values of l considered here, and this is what isobserved. Only the four lowest order modes out of a total of27 that are predicted to be unstable are not seen at a detect-able level in the light curve of PG 0014]067. We note that

FIG. 8.ÈComparison of the observed period spectrum of PG0014]067 (thick dashed line segments) with the theoretical pulsationperiod spectrum of the optimal model. For the latter spectrum, solid linesegments indicate excited modes, while dotted line segments correspond todamped modes. All pulsation modes with l\ 0, 1, 2, and 3 in the periodinterval 60È260 s are illustrated. The values of the radial order index k arealso indicated for each mode. These are acoustic modes (including thef-modes). The g-modes have periods that fall outside the range of interestfor PG 0014]067.

these four modes show the lowest growth rates according toTable 3. This excellent agreement with nonadiabatic theoryconstitutes our Ðfth notable result.

4.5. Estimates of the Uncertainties on the DerivedParameters of the Optimal Model

An essential ingredient of our study is a discussion of theuncertainties associated with the parameters of the optimalmodel. We mentioned above that the s2 hypersurface showsa well-deÐned minimum near the grid point deÐning theoptimal model. We present, in Table 5, the properties ofthat hypersurface in the vicinity of the optimal model inparameter space. The characteristics of a true minimum areevident. Parabolic interpolation (see Bevington 1969) can beused to obtain a better estimate of the location of theminimum, but the derived parameters are so close to thoseof the optimal model on the grid point that this reÐnementis not necessary here. In the immediate vicinity of theminimum we can model the s2 hypersurface in terms of aquadratic expansion taking into account the fact that theÐrst-order derivatives are equal to zero at the minimum andthe fact that the cross terms in the second-order derivativesare, in a Ðrst approximation, also equal to zero since thefour parameters are considered to be independent. Inreality, a close examination of the s2 hypersurfaces indicatesthat there is some correlation between the parameters (the““ principle axes ÏÏ of the s2 hypercontours are not strictlyparallel to the parameter axes), but we neglect this second-order e†ect here. This leads to the following relationshipbetween the variation *s2 and the uncertainties, on the*x

i,

parameters (where, for short-hand notation,xi

x1\ log g,andx2\Teff, x3\ M

*/M

_, x4\ log [M(H)/M

*]) :

*s2\ ;i/1

4 12

L2s2Lx

i2Kmin

(*xi)2 , (3)

where the second-order derivatives are evaluated at the gridpoint corresponding to the optimal model. We estimatethese derivatives on the basis of the data given in Table 5.To obtain a conservative estimate of the uncertainty on agiven parameter, we assume that the variation *s2 is dueexclusively to the variation of that one parameter. Thisgives

*xi\C 2*s2(L2s2/Lx

i2) o min

D1@2. (4)

The question is now to estimate *s2. The merit function s2that we deÐned above is not a standard for which wesstd2can use the usual formulae for estimates of normal errors.

TABLE 5

THE s2 HYPERSURFACE IN THE VICINITY OF THE OPTIMAL MODEL

Tefflog g (K) M

*/M

_log [M(H)/M

*] s2

5.78 . . . . . . 34500 0.49 [4.50 0.53745.77 . . . . . . 34500 0.49 [4.50 1.57055.79 . . . . . . 34500 0.49 [4.50 1.48465.78 . . . . . . 34000 0.49 [4.50 0.56475.78 . . . . . . 35000 0.49 [4.50 0.55185.78 . . . . . . 34500 0.48 [4.50 0.70965.78 . . . . . . 34500 0.50 [4.50 0.71195.78 . . . . . . 34500 0.49 [5.00 3.60525.78 . . . . . . 34500 0.49 [4.00 3.6719


We need to renormalize that function. In this, we follow theprescription of Press et al. (1985 ; see also Bevington 1969)and assume that we have a perfect Ðt. In that case, thestandard at the minimum point is equal to the numbersstd2of degrees of freedom l, here equal to 9 (13 periods minusfour free parameters). This leads to a scale factor S betweenthe standard minimum and the value of our merit func-sstd2tion s2 at the optimal grid point given by S \ 9/0.5374\ 16.7473. We next compute the value of that*sstd2must be added to to cover a range of parameter spacesstd2sufficient to reach a certain conÐdence level in the estimatesof the uncertainties on the parameters. We adopt the 1 plimit (68.3% conÐdence level) and compute using the*sstd2GAMMQ routine of Press et al. (1985) for the truncatedgamma function. For the case l\ 9, not covered in thetable on page 536 in Press et al. (1985), we Ðnd that *sstd2 \

Taking into account the above scale factor S, this10.43.corresponds to *s2\ 0.6228, the value that we now use inequation (4). We Ðnally Ðnd that, at the 68.3% conÐdencelevel, the global structural parameters of PG 0014]067 arelog g \ 5.780^ 0.008 (0.14%), KTeff \ 34,500K^ 2690(7.80%), (3.88%), andM

*/M

_\ 0.490^ 0.019

(4.89%). The latter quan-log [M(H)/M*]\ [4.50^ 0.22

tity is related to the envelope mass log (Menv/M*) \[4.31

(5.10%). As a bonus, we Ðnd that the radius of PG^ 0.220014]067 is (2.68%).R/R

_\ 0.149 ^ 0.004

These results are extremely interesting and deserve somecomments. If we look in particular at the relative errors, weÐnd that the surface gravity is determined to a great accu-racy. Indeed, one of the most signiÐcant contributions ofasteroseismology to the study of sdB stars may rest, in thefuture, with such determinations of log g to unprecedentedaccuracy. This is of great interest in the light of the fact thatsuch accuracy is currently not possible with spectroscopictechniques and that such techniques still su†er from system-atic e†ects whose exact nature is not understood (Wesemaelet al. 1997). The high accuracy that can be reached in thedetermination of log g in the asteroseismological approachis directly related to the relatively large sensitivity of thepulsation periods on the surface gravity (see Fontaine et al.1998). In contrast, the lack of sensitivity of the same periodson the e†ective temperature precludes accurate determi-nations of that parameter through asteroseismologicalmeans. We can already do better with spectroscopic andphotometric techniques. Nevertheless, it is still interestingto obtain an independent determination of even atTeffreduced precision. We note, in passing, that our aster-oseismological determinations of log g and are consis-Tefftent with the estimates of these atmospheric parametersobtained through our analysis of our time-averaged opticalspectrum of PG 0014]067. Finally, our determinations of

and at very interesting levels of accuracy areM*

M(H)/M*unique to the asteroseismological approach. Some masses

of sdB stars are known through the study of binary systemscontaining an sdB component, but the accuracy on thederived masses is relatively poor. For its part, the determi-nation of the mass of the outer hydrogen-rich envelope inan sdB star is a pure product of the asteroseismologicalapproach. There is no other known way, beside theory, toactually constrain this quantity. Systematic determinationsof for a large sample of EC 14026 pulsators mayM(H)/M

*provide, in the future, fascinating insights into the internalstructure and the evolution of these old stars on the extremehorizontal branch.

A Ðnal remark in this subsection is that we are fully awarethat our asteroseismological determinations of the globalstructural parameters of PG 0014]067 are, in an absolutesense, only as good as the constitutive physics that wentinto the construction of our equilibrium models. Futureimprovements in our ability at modeling sdB stars at thelevel of the equation of state, the opacity, the radiative levi-tation calculations, initial conditions, and so on, will neces-sarily lead to improved estimates of these parameters. Howdi†erent from our current values these estimates will beremains to be seen. We point out that solid credibility mustalready be acknowledged in favor of the current availableconstitutive physics since it is able to explain the existenceof 23 pulsation modes in an EC 14026 star with an averageaccuracy better than 0.8% on the periods.

4.6. PG 0014]067 as a Slow Rotator?We already pointed out in ° 3 the existence of Ðne struc-

ture in at least three of the 13 pulsations we initiallyuncovered in the light curve of PG 0014]067. These doub-lets lead to frequency spacings kHz,f 1` [ f 1~\ 27.0 f 3`kHz, and kHz. Prompted by[ f 3~\ 9.6 f 5`[ f 5~ \ 9.6our results of ° 4.4, we have searched for the possible pres-ence of additional Ðne-structure components with ampli-tudes lower than the 0.03% threshold limit, particularly inthe neighborhood of the two largest amplitude pulsations

and We found two likely components : one that we( f 3~ f 5`).name at 6611.6 kHz (151.25 s) with an amplitude off 3~~0.0245% that forms a nearly symmetric triplet in frequencywith the 6621.1 and 6630.7 kHz modes, and the( f 3~) ( f 3`)other that we name at 7118.7 kHz (140.47 s) with anf 5``amplitude of 0.0283% that is part of the 7088.7 kHzcomplex and located 30 mHz away from the latter( f 5`)component.

The simplest interpretation for this structure is to assumethat we have detected the strongest components of rotation-ally split 2l ] 1 multiplets created by the slow rotation ofthe star. Treated as a Ðrst-order perturbation, solid bodyrotation is known to lift the degeneracy of a pulsation modespeciÐed by the doublet of indices (k, l) in a spherical modeland produce 2l ] 1 modes now speciÐed by a triplet ofindices (k, l, m). The modes with adjacent values of m( o*m o \ 1) are separated by a frequency spacing given by

*f \ 1 [ Ckl

Prot, (5)

where *f is expressed in Hz, the rotation period isProtexpressed in s, and is the dimensionless Ðrst-order rota-Ckltion coefficient encountered previously in our discussion of

Table 3.In the context of the rotation hypothesis, it is encour-

aging to realize that the theoretical modes that have beenassigned to the multiplets are consistent with such an inter-pretation. In particular, none of the three multiplets areassociated with radial modes (l \ 0), which, of course,would have constituted an inconsistency since these modesare not degenerate and cannot be split by rotation. Accord-ing to Table 4, the doublet is associated with an( f 1`, f 1~)l \ 3, k \ 1 mode ; the nearly symmetric( f 3`, f 3~, f 3~~)triplet is associated with an l \ 1, k \ 2 mode ; and the

asymmetric triplet is associated with an( f 5``, f 5`, f 5~)l \ 3, k \ 2 mode. The frequency splittings that we Ðndbetween the components of these multiplets are 27.0, 9.6,


9.5, 30.0, and 9.6 kHz. Considering that the values of thequantity are small compared to 1 for the theoreticalC

klmodes that have been assigned to the multiplets (see Table3), this suggests an approximately constant commonspacing between adjacent m components. This also assumesthat the spacings of 27.0 and 30 kHz correspond to thefrequency di†erences between modes separated byo*m o \ 3. Considering as well our resolution of 2.8 kHz,we feel justiÐed to use a straight average of the above spac-ings, which leads to a value *fD 9.5^ 0.3 kHz. This, inturn, leads to an estimate of the rotation period of PG0014]067, hr, since IfProt^ (*f )~1\ 29.2^ 0.9 C

kl> 1.

we now combine this value with our estimate above of theradius of the star, we Ðnd an equatorial velocity of V \

km s~1. It is interesting to compare(2nR)/Prot \ 6.2^ 0.4this value of the maximum broadening velocity in PG0014]067 with the results of Heber et al. (1999), who founda value of V sin i^ 39 km s~1 in the fast rotating pulsatorPG 1605]072. In that latter case, rotational splitting issufficiently important that the m components of the variousmultiplets overlap in frequency space, which, as indicatedearlier, complicates seriously the interpretation of the lightcurve of PG 1605]072. In comparison, PG 0014]067 is tobe considered as a well-behaved, slow rotator. High-resolution spectroscopy of the kind used by Heber and col-laborators would provide a nice test of our result for PG0014]067, namely, km s~1. If this interpreta-V sin i[ 6.2tion is correct, we can infer, in agreement with the work ofHeber et al. (1999), that there is dispersion in rotation forstars of similar extended horizontal branch (EHB) param-eters, which implies that landing on the EHB (rather thanthe regular HB) is not a result of, say, rapid rotation assome evolutionary scenarios would have it.

5. SUMMARY AND CONCLUSION

Our observational survey has so far led to the discoveryof p-mode instabilities in the sdB stars PG 1047]003

et al. 1997), KPD 2109]4401 et al. 1998),(Billeres (BilleresKPD 1930]2752 et al. 2000), and PG 0014]067(Billeres(this paper). The latter addition brings the number ofknown EC 14026 stars to 19. It is noteworthy that PG0014]067 was not immediately recognized as a variablestar in real time at the Mount Bigelow 1.6 m telescope ashad been the case previously. It took the resources of amidsize telescope such as the CFHT to uncover the lumi-nosity variations in the light curve of PG 0014]067. Itturns out that the combination of faintness (V ^ 15.9) andlow intrinsic amplitudes made this case a particu-([0.22%)larly difficult one on a small telescope. In contrast, ourCFHT data revealed not only the clear variability of thestar but also the presence of photometric activity in a rangeof periods from D80 to D170 s, unusually wide for a““ typical ÏÏ EC 14026 pulsator. It seems clear to us that thismust be the result of the higher sensitivity of our obser-vations, which suggests caution when interpreting the sta-tistics of the current sample of known EC 14026 pulsators.Low-amplitude variables must certainly have been missedin the current surveys carried out on small telescopes, andlow-amplitude modes in known pulsators must also havebeen missed.

We have attempted, for the Ðrst time in this Ðeld, a fullasteroseismological analysis of an EC 14026 pulsator. Thiswas made possible because, on the one hand, and despitethe relative faintness of PG 0014]067 and the low intrinsic

amplitudes of its luminosity variations, our CFHT datahave revealed a period spectrum that is both rich andsimple enough for interpretation. Using standard tech-niques that combine Fourier analysis, least-squares Ðts tothe light curve, and prewhitening methods, we have isolated13 harmonic oscillations (including three closely spaced fre-quency doublets) in the light curve of PG 0014]067. Thisnumber of independent modes was deemed sufficiently largeto impose enough constraints on an eventual model and tojustify a search for such a model. On the other hand, we alsotook advantage of the recent work of P. Brassard et al.(2001, in preparation), who developed a robust opti-mization algorithm to best match in an objective way a setof observed periods with a set of theoretical periods, aproblem that has impeded much of the whole Ðeld of aster-oseismology to date. The question of Ðnding in parameterspace the one model that provides the very best theoreticalperiod spectrum to account for an observed period spec-trum has found an answer in such an approach.

We have used the so-called second-generation sdBmodels of Charpinet et al. (1997) in our analysis since thosehave been shown to account remarkably well for the classproperties of the EC 14026 pulsators (Charpinet 1998).These models are speciÐed by four free parameters, thesurface gravity log g, the e†ective temperature theTeff,stellar mass and the fractional mass of the outerM

*/M

_,

hydrogen-rich envelope We computed a ÐrstM(H)/M*.

grid of 1470 models, relatively coarse along the log g andaxes, in order to explore Ðrst a relatively large volume ofTeffparameter space. At each grid point, the nonadiabatic

properties of the local model were computed, notably, thepulsation periods and the stability coefficients. The theoreti-cal periods were then matched to the observed periodsthrough the P. Brassard et al. (2001, in preparation) algo-rithm, and the quality of the Ðt was measured through themerit function s2 deÐned by equation (2). A detailed exami-nation of the behavior of s2 hypercontours in parameterspace revealed that these surfaces are smooth, are wellbehaved, and show regions of minimal values for the meritfunction. We found that one of these regions correspondswell to our spectroscopic estimates of log g and for PGTeff0014]067, and we then zoomed in this region with the helpof a Ðner model grid containing, this time, 1350 models. Anoptimal model, deÐned by the smallest value of s2, wasfound in that grid. This model is able to account remark-ably well for the 13 periods observed in the light curve ofPG 0014]067. A detailed comparison of the theoreticalperiod spectrum of the optimal model with the distributionof the 13 observed periods led us to the realization that realpulsation modes, of lower amplitudes than the thresholdvalue used in our standard analysis of the light curve, maybe present in that light curve. Indeed, we found that 10previously unassigned theoretical modes have periods thatfall within ^0.7% of the periods of 10 maxima found in theFourier transform of the residual light curve. Using prob-abilistic arguments, we demonstrated that these maximalikely correspond to real pulsation modes, reinforcing con-siderably the credibility of the optimal model. Altogether,we identiÐed 23 pulsation modes in the light curve of PG0014]067. These are all low-order acoustic modes withl \ 0, 1, 2, and 3, excited in a band of periods from D80 toD170 s. Remarkably, the average relative dispersionbetween the 23 observed periods and the periods of thecorresponding 23 theoretical modes of the optimal model is


TABLE 6

BASIC PROPERTIES OF PG 0014]067 (V \ 15.9^ 0.1)

Quantity Asteroseismology Spectroscopy

log g . . . . . . . . . . . . . . . . . . . . 5.780 ^ 0.008 (0.14%) 5.77 ^ 0.10 (1.73%)Teff (K) . . . . . . . . . . . . . . . . . 34500 ^ 2690 (7.80%) 33550^ 380 (1.13%)M

*/M

_. . . . . . . . . . . . . . . . 0.490 ^ 0.019 (3.88%) . . .

log (Menv/M*) . . . . . . . . . [4.31 ^ 0.22 (5.10%) . . .

R/R_

(M*, g) . . . . . . . . . . . 0.149 ^ 0.004 (2.68%) . . .

L /L_

(Teff, R) . . . . . . . . . . . 28.5 ^ 10.4 (36.5%) 25.5 ^ 2.5 (9.90%)M

V(g, Teff, M

*) . . . . . . . . 4.43 ^ 0.24 (5.42%) 4.48 ^ 0.12 (2.68%)

d(V , MV) (pc) . . . . . . . . . . 1950 ^ 305 (15.6%) 1925^ 195 (10.1%)

Prot (hr) . . . . . . . . . . . . . . . . . 29.2 ^ 0.9 (3.08%) . . .Veq(R, Prot) (km s~1) . . . 6.20 ^ 0.36 (5.81%) . . .

only of order D0.8%. Perhaps even more remarkably, inview of the well-known shortcomings of nonadiabatic pul-sation theory in many types of stars (notably, the pulsatingwhite dwarfs), there is an excellent agreement here betweenthe expectations of nonadiabatic theory, in particular theexpected period band of unstable modes with adjacentvalues of the radial order k, and the distribution of observedperiods. This gives added conÐdence in ourdetermination of the global structural properties ofPG 0014]067, the ultimate goal of asteroseismology.

We have summarized, in Table 6, the basic properties ofPG 0014]067 as inferred from our asteroseismological andspectroscopic approaches. Owing to di†erent period sensiti-vities, the surface gravity is much better constrainedthrough asteroseismology than spectroscopy, while theopposite is true for the e†ective temperature. The table Ðrstlists our inferred values of log g, andTeff, M

*/M

_,

the four basic parameters that deÐne ourlog (Menv/M*),

sdB models. We also calculated secondary quantities suchas the radius the luminosity the abso-R(M

*, g), L (Teff, R),

lute magnitude and the distance d(V ,MV(M

*, g, Teff), M

V)

given the apparent visual magnitude of V \ 15.9^ 0.1.Note that the absolute magnitude is obtained through themodel atmospheres and synthetic spectra and colors calcu-lated by P. Bergeron. Using our best estimate of the e†ectivetemperature (the spectroscopic value), we Ðnd that PG0014]067 has a luminosity has an ab-L /L

_\ 25.5^ 2.5,

solute visual magnitude and is located atMV

\ 4.48 ^ 0.12,a distance of d \ 1925 ^ 195 pc. In addition, if we interpret

the Ðne structure observed in three of the 23 pulsations interms of rotational splitting, we Ðnd that PG 0014]067rotates with a period of ^29.2 hr and has a maximumrotational broadening velocity of km s~1.V sin i [ 6.2

The results of this paper illustrate quite well, we believe,the power of asteroseismology as applied to sdB stars. Thefact that we were able to Ðnd a model (the optimal model)that is able to represent quite accurately the pulsationproperties of PG 0014]067 is a clear conÐrmation that thebasic constitutive physics used in the construction of thismodel is sound. In this respect, we note, in particular, theradiative opacity tables specially computed by the OPALgroup and the radiative levitation calculations carried outby P. Chayer (see Charpinet et al. 1997). To a certain extent,this is reminiscent of the great success obtained by theOPAL team to reconciliate evolutionary and pulsationmasses for Cepheids. Furthermore, the excellent resultsobtained at the nonadiabatic level, results that may havebeen unexpected on the basis of our experience with othertypes of pulsating stars, are a further proof that the ironbump opacity mechanism of Charpinet et al. (1997) isindeed at the origin of the EC 14026 phenomenon.

We wish to acknowledge the essential scientiÐc input ofPierre Chayer, Ben Dorman, Claudio Iglesias, and ForrestRogers, who have helped us develop our modeling capabil-ities for pulsating sdB stars. We are indebted to Pierre Ber-geron for making available to us his most recent grid ofmodel atmospheres, synthetic spectra, and colors for sdBstars. We also wish to thank our South African colleaguesfor their continuing interest in our work and for their funda-mental contributions to this Ðeld. We thank the Directorand sta† of the Canada-France-Hawaii Telescope as well asthe Director and sta† of the Steward Observatories for sup-porting LAPOUNE as a visitor instrument. We are particu-larly indebted to Marcia Rieke, Bob Peterson, and JimChatham in Arizona, and to Ken Barton and David Wood-worth, the latter for their expert help at the CFHT tele-scope. This work was supported in part by the NSREC ofCanada and by the Fund FCAR G. F. also(Que� bec).acknowledges the contribution of the Canada ResearchChair Program.

APPENDIX A

As discussed in ° 4.4, there are, in addition to the 13 modes identiÐed previously, 10 more modes expected from theorywhose pulsation frequencies fall within ^0.70% of the frequencies of some of the largest amplitude peaks that are still presentin the Fourier spectrum of the residual light curve after prewhitening. The likelihood that such a coincidence is due to chanceonly is very small, and we turn in this appendix to a more formal evaluation of the probability of such an event. This should beseen as a complementary approach to the cruder probabilistic arguments that we used in the main text.

We Ðrst consider that the residual Fourier transform in the 8.5È12.5 mHz bandpass (see Fig. 6) is entirely due to noise and,consequently, that the maxima in the Fourier spectrum correspond to random frequencies. We then ask, what is theprobability, that one such random peak ““ falls ÏÏ within * mHz of a given theoretical frequency? Clearly, this is given byP1,where W is the width of the spectral window of interest (4 mHz in the present case). It can next be shown that theP1\ */W ,probability, P(n, m), that at least one frequency among the n random frequencies available in the bandwidth W falls, within *mHz, on each of m theoretical frequencies is given by

P(n, m)\ ;i/0

n~m n !m !i !(n [ i) !

tn~i,mP1n~i(1[ mP1)i , (A1)

1030 BRASSARD ET AL.

where is StirlingÏs number of the second kind. This equation may be evaluated numerically. This assumes that the widthtn~i,m* is the same for all the m theoretical frequencies and that these bands do not overlap in the Fourier domain (otherwise the

above probability becomes an upper limit). This also assumes that the ““ observed ÏÏ frequency peaks in the transform of theresidual light curve are randomly distributed in frequency space.

In our speciÐc application, we consider the n \ 21 peaks with amplitudes larger than 0.02% in the Fourier transform of theresidual light curve shown in Figure 6. Moreover, as a Ðrst estimate of *, we take the average relative dispersion of ^1.1%found above between the observed and theoretical pulsation frequencies of the 13 modes identiÐed in ° 4.3. This correspondsto *^ 0.21 mHz, leading to The probability that maxima with amplitudes larger than 0.02% in the residualP1^ 0.052.Fourier transform fall simultaneously by chance on all the m\ 10 theoretical frequencies within 0.21 mHz is then given byP(21,10)^ 8.4] 10~3. This coincidence becomes even more unlikely if we consider instead the narrower interval of *^ 0.13mHz (corresponding to which is, after the fact, associated with the largest deviation (^0.70%) that we ÐndP1^ 0.033),between the periods of 10 unassigned theoretical modes in our optimal model and the periods of 10 maxima in the Fouriertransform of the residual light curve. In that case, the probability reduces to P(21,10)^ 2.9] 10~4.

We note that these Ðgures are consistent with those derived in the main text from a cruder probabilistic approach. Weconclude from this that we have probably uncovered 10 additional pulsation modes but with relatively low amplitudes.

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Discovery and Asteroseismological Analysis of the Pulsating sdB Star PG 0014+067