PROGRESS ON THE FRONT OF PULSATING SUBDWARF B STARS

GILLES FONTAINE1, ELIZABETH M. GREEN2, PIERRE BRASSARD1,STEPHANE CHARPINET3, PIERRE CHAYER4, MALVINA BILLERES5,

SUZANNA K. RANDALL1 and BEN DORMAN6

1Departement de Physique, Universite de Montreal, Canada; E-mail: fontaine@astro.umontreal.ca2Steward Observatory, University of Arizona, Tucson, USA

3Laboratoire d’astrophysique (UMR 5572), Observatoire Midi-Pyrenees, Toulouse, France4Bloomberg Center for Physics and Astronomy, Johns Hopkins University, Baltimore, USA

5European Southern Observatory, Santiego Headquarters, Santiago, Chili6NASA, Goddard Space Flight Center, Greenbelt, USA

Abstract. We briefly review the recent advances that have been made on the front of pulsatingsubdwarf B (sdB) stars. The first family of sdB pulsators, the EC 14026 stars, was discovered a fewyears ago and consists of short-period (∼100−200 s) p-mode variables. The second type of pulsatingsdB’s consists of the PG 1716+426 stars, a group of variables showing long-period (∼1 h) g-modepulsations. The existence of the latter was first reported less than a year ago. While the two types of sdBpulsators differ markedly in their observational characteristics, we recently found a unifying propertyin the sense that the observed modes in these objects are excited through the same driving process, aclassic kappa mechanism associated with the radiative levitation of iron in the stellar envelope.

Keywords: hot B subdwarfs, stellar oscillations

1. The Two Families of sdB Pulsators

The study of subdwarf B (sdB) stars received a formidable boost a few years agowhen it was discovered that many of them show luminosity variations caused bystellar pulsations. Among other things, this has opened the door to the applicationof asteroseismological methods to these stars. Furthermore, these discoveries haveattracted attention to the hot B subdwarfs, a class of objects that seems to have beenlargely ignored by a surprisingly large number of astronomers.

There are two distinct classes of pulsating sdBs. Those of the first type arecommonly named EC 14026 stars, after the prototype. Their existence was firstrevealed in 1997 by a group of astronomers at the South African Astronomical Ob-servatory (Kilkenny et al., 1997) and, totally independently, predicted by Charpinetet al. (1996). The number of known EC 14026 stars has now grown to 31 (seeCharpinet 2001; Kilkenny 2002; Solheim et al., 2004). The distribution of 26 ofthose in the log g-Teff plane is shown by the open circles in Figure. 1. As can beseen, a typical EC 14026 star has a log g value of ∼5.8 and an effective temperatureTeff of about 33 000 K. Such a typical object shows pulsation periods in the rangefrom 100 to 200 s.

Astrophysics and Space Science 291: 379–386, 2004.C© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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Figure 1. Distribution of the pulsating sdB stars in the log g-Teff diagram. The open circles give thelocations of 26 EC 14026 stars, while the filled circles indicate those of 21 PG 1716+426 stars. Theyare superposed on some of Ben Dorman’s EHB evolutionary tracks (thin dotted lines) for modelswith a range of H envelope masses. The thick dotted line represents the zero-age He-burning mainsequence.

Using observations gathered at Steward Observatory, Green et al. (2003) re-cently reported the discovery of a second type of pulsating sdB stars, long-period variables with typical luminosity variation timescales of about 1 h. Theyare now referred to as PG 1716+426 stars, after the prototype. Quite interest-ingly, they appear to be quite common and, at last count, there were 24 knownPG 1716+426 stars (Green et al., 2004). The distribution of 21 PG 1716+426stars in the log g-Teff diagram is shown by the filled circles in Figure 1, fromwhich one can see that typical parameters are log g ∼5.5 and Teff ∼27 000 K.Reports on the preliminary outcome of the first major multisite campaign on aPG 1716+426 star are reported elsewhere in these Proceedings (Randall et al.,2004).

The two classes of pulsating sdB stars clearly occupy different domains inthe log g-Teff plane as shown in Figure 1, although the domains may touch. Theshort periods observed in EC 14026 stars are attributed to p-mode pulsations,while the much longer periods seen in PG 1716+426 stars automatically implythat they are g-mode pulsators. It is worthwhile to point out that the existenceof these two distinct families of pulsators within the more general sdB class is areal blessing because p-modes and g-modes probe different regions of a star, andone can therefore hope to learn different things from their respective study. The

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Figure 2. Distributions of variable and nonvariable sdB stars in the log g-Teff diagram. The leftpanel shows the locations of 70 constant stars (open circles) and of 26 pulsators (filled circles), andclearly demonstrates the coexistence of these two types of stars in the EC 14026 domain. (The solidcontour corresponds to the region where maximum p-mode instability is expected from a theoreticalviewpoint.) In contrast, the right panel, which shows the locations of 26 constant stars (open circles)and of 21 PG 1716+426 stars (filled circles), underlines the concentration of the PG 1716+426 starson the cool side of the sdB range.

p-modes are essentially envelope modes in sdB stars, while the g-modes tend tobe core modes. It may very well turn out that, among the 30-odd types of oscil-lating stars that are currently known, pulsating sdB’s prove to be the best labora-tories for asteroseismology. A particularly encouraging example of the potentialof this method is the analysis of the EC 14026 pulsator PG 0014+067 presentedby Brassard et al. (2001). More recent results are discussed by Charpinet et al.(2004).

Other distinguishing characteristics between PG 1716+426 stars and EC 14026pulsators are the facts that (1) the former have lower peak-to-peak amplitudesthan those observed in the latter (see Green et al. 2003 and Fontaine et al. 2003a)and (2) the PG 1716+426 stars appear to be much more common, as most (ifnot all) of the cooler sdB’s show some long-period photometric activity, whilethe EC 14026 variables form a minority with respect to their nonvariable coun-terparts in their common region of the log g-Teff diagram. This is clearly shownin Figure 2 where the open circles locate constant stars defined as those withamplitude limits of less than ∼1 millimag (1) in the period range 20−1 000 sfor the EC 14026 candidates (Billeres et al., 2002) and (2) in the period interval400−15 000 s for the PG 1716+426 star candidates (E.M.G., unpublished). A cur-rent challenge is to account for the distribution of pulsators/nonpulsators shown inFigure 2.

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2. A Common Driving Mechanism

Why do some sdB stars pulsate? In the case of the EC 14026 stars, Charpinet et al.(1996) first discovered a region of local driving in models of sdB stars associatedwith an opacity bump due to the partial ionization of iron. While this local drivingis insufficient to overcome damping in models with a uniform solar metallicity,Charpinet et al. (1996) reasoned that, since diffusion processes are well knownto produce abundance anomalies in sdB stars, perhaps a local overabundance ofiron could be created in the driving region through such processes. In their first at-tempt, they crudely increased the metallicity while keeping it uniform, and they thusfound unstable p-modes in typical sdB models. To make further progress, however,it was clear that much more sophisticated models were required, models takinginto account diffusion processes on iron. Hence, Charpinet et al. (1997) consid-ered “second-generation models” in which the assumption of uniform metallicityis relaxed. Instead, the local abundance of iron in such models is computed assum-ing an equilibrium between gravitational settling and radiative levitation using thetechnique developed by Chayer et al. (1995).

Some properties of such second-generation models are shown in Figure 3. Itcan easily be seen that large overabundances of iron can be supported by radiativelevitation in certain regions of the envelope of sdB stars, especially for the hottermodels. Moreover, time-dependent calculations (Chayer et al., 2004) indicate thatsuch equilibrium abundances can be reached over timescales much smaller thanthe typical lifetime of sdB stars, a few 108 years. Associated with the nonuniformprofile of iron (solid curve) is the run of the Rosseland opacity (dotted curve). Ofprime interest here, we point out that both the location and shape of the iron opacitypeak are critical in the driving process, a classic kappa mechanism in the jargon ofpulsation theory.

It turns out that, at low effective temperatures, there is not enough iron supportedby radiative levitation; the iron opacity peak is rather broad and located relativelydeep, and no overall driving is possible. By increasing the temperature, the Fe opac-ity peak becomes stronger, narrower, and moves toward the stellar surface. Basically,the peak moves through the critical region for efficient driving, the region wherethe local thermal response timescale is comparable to the pulsation periods driven.At too high effective temperatures, the peak has moved too far up in the envelope,past the region of efficient driving. Hence, the second-generation models naturallylead to the creation of a broad instability strip, as observed for EC 14026 pulsators.

There is no question that the approach of Charpinet et al. (1997) has beentremendously successful at explaining, both qualitatively and quantitatively, theEC 14026 phenomenon (Charpinet et al., 2001). For instance, when the region ofmaximum instability for EC 14026 stars was computed (see, e.g., the contour in theleft panel of Figure 2), only four EC 14026 stars were known. Over the years, withnew discoveries being reported, they have kept falling into the expected region ofthe log g-Teff diagram, proving that the radiative levitation theory must be correct.

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Figure 3. Plot of the iron abundance profile (log N (Fe)/N (H); solid curves) and the correspondingmean Rosseland opacity profiles (log κ; dotted curves) for a series of sdB models with varyingeffective temperature in the range 4.34 ≤ log Teff ≤ 4.62 in steps of 0.04 dex. The surface gravity iskept constant at log g = 5.8. Horizontal dashed lines indicate the solar iron abundance value. Theabcissa shows the logarithm of the fractional mass depth. On this scale, the center of the star wouldhave a value of log q = 0.0.

However, what about the PG 1716+426 stars? At the time of their discovery, it wasnot clear what could drive long-period pulsation modes in the cooler sdB stars, andboth Green et al. (2003) and Fontaine et al. (2003a) pointed out that the drivingmechanism remained unknown for the PG 1716+426 stars. This was based onprevious calculations a la Charpinet et al. (1997, 2001) which had suggested thatthe iron opacity instability, which is so successful at explaining the EC 14026 stars,is apparently unable to drive g-modes in typical models of sdB stars. Implicit to thatstatement, however, was the fact that the modes of interest were those with the lowestpossible values of the degree index for g-modes, l = 1 and 2, corresponding to the

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Figure 4. Theoretical period spectra of sdB models in the 60−5000 s range as functions of theeffective temperature. Each panel refers to a different value of the degree index l. In each of them,a small black dot gives the value of the logarithm of the period of a pulsation mode. Excited modesare identified by small red filled circles whose sizes give a (logarithmic) measure of the modulus ofthe imaginary part of the complex eigenfrequency. The bigger the circle, the more excited the mode.The blue curve connects the periods of the fundamental mode (k = 0) and separates the p-branchof the period spectrum below from the g-branch above. Two islands of instability are found here,corresponding to the EC 14026 (below) and PG 1716+426 stars (above) domains respectively. Formore details see Fontaine et al. (2003b).

more likely visible modes as suggested by canonical wisdom based on geometriccancellation effects on the visible disk.

Inspired by encouraging remarks from the authors of the Dziembowski et al.(1993) paper, we recently reconsidered the problem by extending the Charpinetet al. (1997, 2001) calculations to a much wider range of periods than previouslystudied, from 60 s up to 10 000 s, and for modes with values of the degree indexfrom l = 0 up to and including l = 8 (see Fontaine et al. 2003b for details). Thisencompasses both the p- and g-branch of the period spectrum. Some of our resultsare shown in Figure 4. One can easily distinguish a first island of instability centeredaround 31 000 K and corresponding to short-period, low-order p-modes, with a few,more weakly driven, low-order g-modes at intermediate temperatures. Of course,this is nothing more than the instability region first uncovered by Charpinet et al.(1997), the realm of the EC 14026 stars.

Quite interestingly, in the context of the recent discovery of the PG 1716+426stars, Figure 4 also shows that high-order g-modes are excited in the cooler modelsprovided that the degree index l is equal to or larger than 3. Note that the blue edgeof this second island of instability moves to higher effective temperatures as l isincreased. Note further that we also found excited g-modes with l = 2, but only

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in our coolest model at Teff = 22 000 K and for periods larger than the maximum5,000 s shown here. Modes with l = 1 are not excited in our current models witheffective temperatures larger than or equal to 22 000 K.

Given that the range of periods excited in our second island of instability over-laps remarkably well with the range of observed periods in PG 1716+426 stars,and given that the driving mechanism is the same as that responsible for the EC14026 phenomenon, we feel that we have identified correctly the source of thePG 1716+426 stars’ instabilities. We note that radiative levitation is needed toboost the iron abundance in the driving region for both types of pulsating sdB stars.Pulsation modes cannot be excited in current models of sdB stars if the metallicityis assumed to be uniform and solar (as is customarily the case in standard evo-lutionary models). If, as we propose, the PG 1716+426 stars mostly have excitedmodes with l = 3, 4, and not the canonical values l = 1, 2, then the main definingcharacteristics of these pulsators can be explained with our current models: theirlong periods, their low amplitudes, and their relatively low values of their effectivetemperature. However, details remain to be worked out. For instance, at the quan-titative level, we note that our analysis suggests values of Teff for PG 1716+426stars that are somewhat lower than current spectroscopic estimates, although thisproblem is considerably lessened if Uli Heber (private communication) is correctin pointing out that a typical PG 1716+426 should have Teff ∼24 000 K instead of∼27 000 K as suggested in our Figure 1 above. Finally, there is the real hope thatmulticolor photometry or time-resolved spectroscopy could provide a test of ourtheory through the determination of the l index for some of the observed pulsationmodes in PG 1716+426 stars.

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