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0ACCEPTED FOR PUBLICATION IN THEASTROPHYSICALJOURNALPreprint typeset using LATEX style emulateapj v. 11/10/09

EVIDENCE FOR TEMPERATURE CHANGE AND OBLIQUE PULSATION FROMLIGHT CURVE FITS OF THEPULSATING WHITE DWARF GD 358

M. H. MONTGOMERY1,3, J. L. PROVENCAL2,3, A. KANAAN 4, ANJUM S. MUKADAM 5, S. E. THOMPSON2,3, J. DALESSIO2, H. L.SHIPMAN2, D. E. WINGET1, S. O. KEPLER6, AND D. KOESTER7

(Received January 7, 2010; Accepted April 15, 2010)Accepted for publication in the Astrophysical Journal

ABSTRACTConvective driving, the mechanism originally proposed by Brickhill (1991, 1983) for pulsating white dwarf

stars, has gained general acceptance as the generic linear instability mechanism in DAV and DBV white dwarfs.This physical mechanism naturally leads to a nonlinear formulation, reproducing the observed light curves ofmany pulsating white dwarfs. This numerical model can also provide information on the average depth of astar’s convection zone and the inclination angle of its pulsation axis. In this paper, we give two sets of resultsof nonlinear light curve fits to data on the DBV GD 358. Our firstfit is based on data gathered in 2006 by theWhole Earth Telescope (WET); this data set was multiperiodic, containing at least 12 individual modes. Oursecond fit utilizes data obtained in 1996, when GD 358 underwent a dramatic change in excited frequenciesaccompanied by a rapid increase in fractional amplitude; during this event it was essentially monoperiodic. Weargue that GD 358’s convection zone was much thinner in 1996 than in 2006, and we interpret this as a resultof a short-lived increase in its surface temperature. In addition, we find strong evidence of oblique pulsationusing two sets of evenly split triplets in the 2006 data. Thismarks the first time that oblique pulsation has beenidentified in a variable white dwarf star.Subject headings:convection — stars: magnetic field — stars: oscillations — stars: variables: general — stars:

individual (GD 358)

1. ASTROPHYSICAL CONTEXT

White dwarf stars offer several advantages for astrophysicalstudy. First, they are the evolutionary endpoint of about 97%of all stars and are therefore representative of a large fractionof the stellar population. Second, the source of their pres-sure support is electron degeneracy (Chandrasekhar 1939) sotheir bulk mechanical structure is well understood. Third,nuclear reactions, if any, contribute a negligible amount totheir energy, so their evolution is dominated by simple cooling(Mestel 1952). Finally, they are observed to pulsate in specifictemperature ranges. The pulsators are believed to be typical inevery other way, so what we learnasteroseismologicallyaboutthem should apply to all white dwarf stars (for recent reviews,see Winget & Kepler 2008; Fontaine & Brassard 2008).

In addition to learning about the stars themselves, the rel-ative simplicity of white dwarfs makes them ideal laborato-ries for testing and constraining poorly-understood physicalprocesses. One such process, convection, is an importantenergy transfer process in most stars, yet it remains one ofthe largest sources of uncertainty in stellar modeling. Forin-stance, main-sequence stars at least 20% more massive thanthe Sun have convective cores, and the amount of convec-

1 Department of Astronomy and McDonald Observatory, University ofTexas at Austin, Austin, TX, USA; mikemon@astro.as.utexas.edu

2 Department of Physics and Astronomy, University of Delaware,Newark, DE, USA

3 Delaware Asteroseismic Research Center, Mt. Cuba Observatory,Greenville, DE, USA

4 Departamento de Física Universidade Federal de Santa Catarina, C.P.476, 88040-900, Florianópolis, SC, Brazil

5 Department of Astronomy, University of Washington, Seattle, WA,USA

6 Instituto de Fíisica UFRGS, C.P. 10501, 91501-970 Porto Alegre,RS,Brazil

7 Institut für Theoretische Physik und Astrophysik, Universität Kiel,24098 Kiel, Germany

tive overshoot and mixing is the primary factor that deter-mines their main sequence lifetimes (see, e.g., Di Mauro et al.2003). In addition, red giants and AGB stars have large con-vective envelopes, and the details of convection play a roleinthe evolution of their surface abundances and in their overallevolution (Bertelli et al. 2009).

We have developed a method which uses the pulsationsof white dwarf stars to measure fundamental parameters oftheir convection zones. The physical idea is that the pulsa-tions cause local surface temperature (“Teff”) variations thatlead to local variations in the depth of the convection zone.As the convection zone waxes and wanes it both absorbs andreleases energy, modulating the local energy flux (Brickhill1991; Goldreich & Wu 1999). Due to the extreme temper-ature sensitivity of convection, finite amplitude pulsationscan lead to highly nonlinear light curves (Brickhill 1992;Wu 2001; Ising & Koester 2001). In Montgomery (2005) weshowed how a simple numerical model could be used to ob-tain not only good light curve fits but also information on theaverage depth of a star’s convection zone and the inclinationangle of its pulsation axis.

2. NONLINEAR LIGHT CURVE FITS

Montgomery (2005) demonstrated that by considering thenonlinear response of the convection zone (Brickhill 1992;Wu 2001) one could obtain excellent fits to the light curvesof two (nearly) single mode white dwarf pulsators. Wehave since extended this technique to multi-periodic stars(Montgomery 2007) and have taken into account the nonlinearrelationship between the bolometric and observed flux varia-tions (Montgomery 2008). In this section we describe theseeffects and show how they have been added to our nonlinearlight curve fitting technique.

Our approach is well-summarized in Montgomery (2005)and Montgomery (2008). Briefly, we assume that the con-

2 Montgomery et al.

vective turnover time of fluid elements in the convectionzone is short (. 1 s) compared to the periods of pulsa-tion (& 100 s), so that the convection zone responds almostinstantaneously to the pulsations and is always in hydro-static equilibrium. In addition, we assume that the pertur-bation of the flux at the base of the convection zone is si-nusoidal. Next, we assume that the luminosity changes aredue only to temperature changes andnot to geometric effects;Robinson, Kepler, & Nather (1982) found that the fractionalradius change∆R/R∼ 10−5–10−4 for the DAVs, with temper-ature variations (and the associated changes in limb darken-ing) accounting for well over 99% of the luminosity variation,and this result was confirmed by Watson (1988) for both theDAV and DBV stars. Finally, the surface convection zones ofpulsating white dwarfs are quite thin geometrically, of order10−5 of the radius of the star; this allows us to use the planeparallel approximation and neglect horizontal energy trans-port. With these assumptions we are able to derive a rela-tionship between the local flux at a given (θ, φ) entering theconvection zone at its base,Fbase, and that leaving it at thephotosphere,Fphot:

Fphot = Fbase+ τCdFphot

dt, (1)

where the new timescaleτC ≡ τC(Fphot) describes the changingheat capacity of the convection zone as a function of the localphotospheric flux. We parameterize it as

τC = τ0

(

Teff

Teff,0

)−N

, (2)

whereτ0 is the equilibrium value ofτC, Teff is the instanta-neous effective temperature andTeff,0 is its equilibrium value,and N is a parameter describing the sensitivity ofτC tochanges inTeff. From standard mixing length theory of con-vection we expect thatN ∼ 90 for DAVs andN ∼ 23 forDBVs. It is this extreme temperature sensitivity which is re-sponsible for the large nonlinearities seen in white dwarf pul-sations. For reference, this timescale is closely related to thestandard thermal timescale (τth) at the base of the convectionzone: for the DAVsτC ≈ 4τth (Goldreich & Wu 1999) and forthe DBVsτC ≈ 0.6τth. With the further assumption that theangular dependence ofFbaseis given by a spherical harmonicYℓm, we can calculate the bolometric flux changes at the sur-face of the model, and average them appropriately over thevisible disk of the model.

2.1. Improvements in the Modeling

Since 2005 we have made important technical improve-ments to the light curve fitting code. First, we extended thecode to include the more common multiperiodic case, wheremany modes with differentℓ andmvalues are simultaneouslypresent. Thus, the flux at the base of the convection zone isnow given by a sum over the modes:

δFbase

Fbase= Re

M∑

j=1

A jei(ω j t+φ j )Yℓ jmj (θ,φ)

. (3)

In this formula,A j , ω j , φ j , ℓ j , andmj are the amplitude, an-gular frequency, phase,ℓ, andm values of thej-th mode, andthe total number of modes isM.

Second, we adapted the code to simultaneously fit an arbi-trary number of observations (“runs”). This is a necessary

FIG. 1.— An example of the limb-darkening in DB model atmospheres withlogg = 8.0, for the indicated range of temperatures.

step for applying this technique to multiple runs obtainedfrom Whole Earth Telescope (WET) campaigns as well assuccessive nights of single-site observations. Since the codeallocates and deallocates memory as needed it typically usesonly 8 MB of RAM, independent of the number of runs in-cluded in the fit.

Third, we replaced our simple analytical prescription forlimb darkening with tabulated values based on a grid ofour model atmospheres (for a description of the models, seeKoester 2010). This grid ranges inTeff from 20,000 K to30,000 K in steps of 500 K and in logg from 7.5 to 8.5 insteps of 0.25. From this grid we instantaneously calculate thelocal flux as a function ofTeff andµ ≡ cosθ; thus, variationsin the limb darkening withTeff are automatically included.In Figure 1 we show examples of the limb darkening for alogg = 8.0 He atmosphere white dwarf model as a function ofµ for a range ofTeff values.

Finally, we improved the way in which the local bolometricflux variations are mapped into variations in a given wave-length band. Previously we used a flux “correction factor,”αX, to accomplish this. Denoting byFX the flux in the pass-bandX, then, for small fractional changes in the fluxes,αXwas defined by

δFX

FX= αX

δFphot

Fphot, (4)

whereFphot is the local bolometric flux at the photosphere andδF is the variation in the respective fluxes due to the pulsa-tions. Clearly,αX depends on the wavelength coverage ofthe passband as well as the wavelength distribution of theflux from the source. In previous analyses we estimated thatα ∼ 0.42 for DBVs andα ∼ 0.66 for DAVs. The value forthe DAVs is not that different from what one obtains from aproper calculation assuming a passband centered on 5000 Å.For the DBVs, however, the more detailed calculations yielda value ofα∼0.25–0.35 whichis significantly different fromthe earlier estimates.

Temperature Change and Oblique Pulsation in GD 358 3

FIG. 2.— Top panel: the bolometric flux (Fphot) and the flux in the givenpassband (FX ) as a function ofTeff for µ = 1. These calculations are for alogg = 8.0 DB model, and both fluxes have been normalized to one atTeff =25,000 K. Lower panel: the passband flux as a function of the bolometricflux, again normalized atTeff = 25,000 K. This is the functionf given byequation 5.

Of course,α is not strictly a constant but rather is a functionof Teff and thereforeFphot. For larger amplitudes, departuresfrom linearity between the fluxes become more important, andto do the problem properly we needFX as a function ofFphot,i.e.,FX ≡ FX(Fphot).

To calculate this flux conversion we use the same grid of DBmodel atmospheres described earlier. In each model atmo-sphere, the flux is tabulated as a function of wavelengthλ andviewing angleµ, so limb darkening is automatically included.Using a given passband, we integrate it against the model at-mosphere flux to obtainFX as a function ofTeff, µ, and logg.Since logg is essentially constant during the pulsations, weonly have to interpolate once between sets of models for thegiven value of logg; thus, we haveFX ≡ FX(Teff,µ). Finally,sinceFphot∝ T4

eff, we can express this asFX ≡ FX(Fphot,µ).In practice, we are only interested in relative flux changes,

so we calculate everything relative to a reference flux, whichwe take to be the equilibrium flux of the star. If we denote bya subscript “0” the equilibrium values of the fluxes, then thefunction f we want is defined by

FX

FX,0= f

(

Fphot

Fphot,0,µ

)

. (5)

In the top panel of Figure 2 we show the fluxesFX/FX,0andFphot/Fphot,0 as a function ofTeff, both normalized to one

TABLE 1OBSERVINGRUNS USED FORL IGHT CURVE FITS

Run Name Telescope Instrument Length (hrs)

chin20060527 BAO 2.16m PMT 4.5chin20060528 BAO 2.16m PMT 5.2chin20060531 BAO 2.16m PMT 3.9hawa20060518 0.6m Apogee 2.2hawa20060519 0.6m Apogee 1.7hawa20060520 0.6m Apogee 3.5hawa20060521 0.6m Apogee 7.8hawa20060522 0.6m Apogee 6.6hawa20060523 0.6m Apogee 3.7hawa20060524 0.6m Apogee 5.7hawa20060525 0.6m Apogee 8.8hawa20060526 0.6m Apogee 9.1hawa20060527 0.6m Apogee 7.8hawa20060528 0.6m Apogee 9.1hawa20060530 0.6m Apogee 8.7kpno20060518 KPNO 2.1m Apogee 7.0kpno20060519 KPNO 2.1m Apogee 7.3kpno20060520 KPNO 2.1m Apogee 7.6kpno20060521 KPNO 2.1m Apogee 7.3kpno20060522 KPNO 2.1m Apogee 1.0kpno20060523 KPNO 2.1m Apogee 2.9mcdo20060523 2.1m Argos 7.4mcdo20060524 2.1m Argos 7.2mcdo20060525 2.1m Argos 6.4mcdo20060528b 2.1m Argos 7.2mcdo20060529 2.1m Argos 8.2nord20060607 2.7m ALFOSC 7.1nord20060608 2.7m ALFOSC 8.0nord20060609 2.7m ALFOSC 7.9

at Teff = 25,000 K; the filter X is assumed to have an ef-fective wavelength of∼ 5200 Å. The response of this filterwas obtained from convolving the wavelength response of theALFOSC-FASU CCD, the NOT primary and secondary mir-ror reflectivities, an S8612 filter, and atmospheric absorption.In the lower panel we show the functionf defined in equa-tion 5 as derived from these calculations. We see that whilethis is not a perfectly linear relationship, the deviationsfromlinearity are not dramatic. Thus, while we use the fully non-linear relation in our calculations, we expect this effect tomake only a minor contribution to the overall nonlinearitiesassociated with the pulsations.

3. LIGHT CURVE FITS TO THE 2006 WET RUN

Our recent work with light curve fitting has been limitedto nearly single mode pulsators: the DAVs G29-388 andGD154, and the DBV PG1351+489. This is because (1)mono-periodic data can be folded at the pulsation period, pro-ducing a high S/N “light curve,” and (2) the number of possi-ble mode identifications (ℓ andm values) for a single mode issmall enough that all possibilities can be directly explored.

GD 358 violates both of these conditions. First, due to thenonlinear interaction of its large amplitude modes, the pulseshape obtained by folding its light curve at a mode periodis not the same as the pulse shape which would be obtainedin the absence of other modes (Montgomery 2007). Second,GD 358 has a large number of observed modes (see Table 2)and it is impractical to search all possible combinations ofℓandm values which each mode can take. For instance, if weassume GD 358 to have of order∼ 10 modes, all of whichhaveℓ = 1, then all possible permutations ofm values yield

8 More precisely, while G29-38 is normally multiperiodic, the data setused by Montgomery (2005) was taken at a time when its light curve wasdominated by a single large mode.

4 Montgomery et al.

FIG. 3.— A comparison of the simultaneous fit of Table 3 (solid line) to the light curve from run mcdo20060523 (crosses).

a number of (2ℓ + 1)10 ∼ 60,000 different cases! Since eachfit takes of order an hour on a single processor computer, thisis completely impractical using a standard desktop computingapproach.

Fortunately, GD 358 is well studied, so we have a goodidea what theℓ andmvalues for the main pulsation modes are(e.g., Metcalfe, Nather, & Winget 2000; Winget et al. 1994).Even so, our derived values ofτ0 depend only weakly on theassumed mode identifications, so it is not necessary for themode identification of each mode to be exact. Furthermore,the extended time baseline and excellent coverage of the 2006WET run allows us to obtain very accurate frequencies forthese modes (Provencal et al. 2009). With this as a basis, ourapproach is to assume that the frequencies and mode identi-fications are known and then to make nonlinear light curvefits to a subset of runs in the WET campaign which have highS/N. This implicitly assumes that the pulsations are coherentthroughout the time spanned by the runs; in section 6 we showthat this is not strictly true for some of the modes. High S/Ndata are desirable since we are mainly interested in the non-linear part of the light curve, which itself is smaller than thelinear part.

Many high quality runs were taken during the May 2008WET run, and we list in Table 1 those used in our fits. Weincluded the 12 largest amplitude periodicities from the 2006WET run which were deemed to be independent frequenciesand not linear combinations, and these are given in Table 2.

TABLE 2INDEPENDENTMODES FROM

THE 2006 WET RUN

Period k ℓ m(s)

422.56 8 1 1423.90 8 1 -1463.38 9 1 1464.21 9 1 0465.03 9 1 -1571.74 12 1 1574.16 12 1 0575.93 12 1 -1699.68 15 1 0810.29 18 1 0852.50 19 1 0962.38 22 1 0

Theℓ andm identifications are taken from previous analysesof this star (Winget et al. 1994; Kepler et al. 2003; Metcalfe2003; Provencal et al. 2009) and are based on asymptotic the-ory as well as genetic algorithm fits. To calculate the conver-sion from bolometric to observed passband fluxes, we mustassume values forTeff and logg. We chose the values ofBeauchamp et al. (1999),Teff = 24900 K and logg = 7.91, al-though we have also used those of Castanheira et al. (2005)

Temperature Change and Oblique Pulsation in GD 358 5

FIG. 4.— The variation ofθi , N, andτ0 as a function of time during the 2006 WET run. Whileτ0 andN show no obvious signatures,θi shows variations whichmay be sinusoidal in origin.

(Teff = 24100 K, logg = 7.91) to assess the uncertainties thisintroduces.

We began the fitting process with the highest S/N data takenwith the 2.7m Nordic Telescope (NOT). However, it becameapparent that GD 358’s complex pulsation spectrum requiredan extended baseline of data to constrain the phases of closelyspaced frequencies. Fortunately, GD 358’s brightness andlarge amplitude meant that a large number of individual ob-serving runs met the required S/N.

Our simultaneous fit to the runs in Table 1 yields the fol-lowing parameters:τ0 = 572.9± 6.1 s, N = 23.5± 0.1, θi =50.5±0.2. Figure 3 shows the ability of the fit to reproducethe essential features of the light curve (run mcdo20060523plotted). Additional results for this fit, including the ampli-tude and phase for each mode, are given in Table 3. The givenerror bars are formal and should be treated as lower bounds.

To test the sensitivity of the value ofτ0 to the ℓ identifi-cations we re-computed fits changing the assumedℓ of thelargest amplitude mode (P = 810.29 s) fromℓ = 1 to ℓ = 2.With this assumption, the best fit resulted for an identifica-tion with ℓ = 2,m= 1 and had the following parameter values:τ0 = 569.5±7.6 s,N = 14.5±0.1,θi = 61.3±0.1. This indi-cates that the value obtained forτ0 does not crucially dependon the mode identifications of each mode in the fit.

Examination of the fit for each individual run revealed anapparent modulation of amplitudes, i.e., on some nights thevariations in the light curve were smaller than the fit and on

TABLE 3SIMULTANEOUS FIT TO GD 358 DATA SET:τ0 = 572.9±6.1 S, N = 23.5±0.1, θ = 50.5±0.2

Period ℓ m Amplitude Phase (rad)

962.385 1 0 0.1087± 0.0012 2.4641± 0.0069852.502 1 0 0.1198± 0.0015 3.3007± 0.0075810.291 1 0 0.4581± 0.0049 3.1301± 0.0030575.933 1 -1 0.4838± 0.0051 3.4434± 0.0039574.162 1 0 0.2257± 0.0024 5.5258± 0.0062573.485 1 0 0.1082± 0.0016 3.6576± 0.0115571.735 1 1 0.3728± 0.0040 2.4489± 0.0046465.034 1 -1 0.1408± 0.0021 1.8310± 0.0123464.209 1 0 0.1391± 0.0018 2.4672± 0.0097463.376 1 1 0.2540± 0.0030 0.2052± 0.0073423.898 1 -1 0.2406± 0.0030 2.0455± 0.0083422.561 1 1 0.2537± 0.0031 0.6020± 0.0077

others they were larger. In addition, looking at data fromsingle sites suggested that the change from a smaller to alarger amplitude state alternated on a roughly night to nighttimescale.

To test whether a geometric effect or a change in the back-ground state of the star could be causing the night-to-nightmodulation of the amplitudes, we went back and re-fit eachrun in Table 1 individually. For these fits we fixed the pe-riod and amplitude of each mode to be those given in Table 3but we allowed the inclination angleθi and the phases of the

6 Montgomery et al.

FIG. 5.— A section of the light curve of GD 358 during thesforzandoeventin 1996. The crosses are the data points and the curve is the best nonlinearlight curve fit.

modes to vary. We also allowedτ0 andN to vary. Thus, fromeach run we obtained a best fit value of the parameters at thetime of the run. In Figure 4 we present the results of this pro-cedure; we plot the variations inθi , N, andτ0 as a functionof time. While no clear trends are seen inτ0 andN (lowerpanels), the variations inθi are suggestive of a periodic ori-gin (top panel). Assuming a sinusoidal variation, we haveincluded the best fit sine curve in this plot; it has an ampli-tude of 14.9±1.8 and a period of 2.17±0.01 days. On theother hand, if we ignore the possible origin of these variationsand simply consider them to be separate measures ofτ0, N,andθi , then we obtain fairly conservative limits on the errorof the mean values of these quantities:τ0 = 586.0± 11.7 s,N = 27.4±0.7, andθi = 47.5±2.2. We return to the ques-tion of this modulation in section 6.

4. THE SFORZANDO

During the 2006 WET run, GD 358 was stable in that theamplitudes and phases of its modes were fairly constant overthe 3 week length of the run (except for the modulation de-tailed above and in section 6). However, GD 358 is knownto change its pulsation spectrum on a range of timescales. Aspectacular example of this behavior occurred in 1996. Withina period of 36 hrs, all of the power in the highk range (pulsa-tions with periods& 700 s) disappeared within detection lim-its. At the same time, GD 358 more than doubled its apparentpulsation amplitude, with power appearing almost exclusivelyat lower k, with a period of∼ 420 s. Over the next week,its amplitude decreased to “normal” levels, while the highk power did not return for approximately one month. Thisdramatic change is documented in several papers, most re-cently in Kepler et al. (2003) and Provencal et al. (2009). Theepisode itself is termed the “sforzando” after a musical termfor a sudden and short-lived increase in loudness.

In Figure 5 we show a section of GD 358’s light curve dur-ing the sforzando. Ironically, during this dramatic episodethe light curve was much simpler. While the amplitude waslarger, the star was essentially a single-mode pulsator andtheindividual pulses were more sinusoidal (less nonlinear) thanbefore. Still, GD 358 was in a transient state and not all pulseswere identical. The upper and lower panels of Figure 5 showpulses significantly below and above the fit, while the mid-

FIG. 6.— A comparison of the derived convective parametersτ0 with val-ues expected from ML2/α convection. The labeled points are individual ob-jects and the dashed curves are the calculations. The label “GD 358” repre-sents GD 358 during the 2006 WET run while “GD 358∗” stands for GD 358during itssforzandoepisode.

dle panel seems to represent a repeating, “quasi-static” state.Even with these caveats, the relative simplicity of these dataare ideal for nonlinear light curve fitting.

From a time series analysis of this short section of data, wefind that these pulses have a period of approximately 420.7 s.Assumingℓ = 1 and trying all values ofm, we find the bestfit shown in Figure 5:m= 0, τ0 = 41.6±2.3 s,N = 3.6±0.2,andθi = 56.1±1.1. Forℓ = 1, m = 1 a similar quality fit canbe found, but it has a less plausible inclination angle (∼ 85)that is not consistent with the values ofθi previously found inthis paper; even so,τ0 = 28.1 s for this fit. If we assumeℓ = 2then the best fit hasτ0 ∼ 24.2 s andm= 0, although it requiressuch a large intrinsic amplitude that we consider it unphysical.Summing up, while we have good reasons for preferring theℓ = 1, m = 0 identification for our fits, the overall value ofτ0is not strongly dependent on this identification.

Comparing these results to those of section 3, we find thatτ0 was much smaller during thesforzandothan it was duringthe 2006 WET run. This implies that, for whatever reason,GD 358’s convection zone was thinner during thesforzando.As we can see from Figure 6, this would imply aTeff sev-eral thousand degrees hotter than its normal temperature. ForML2/α = 1.0 convection (e.g., Böhm & Cassinelli 1971), thiswould meanTeff ∼ 27,000 K, which is two to three thousanddegrees hotter than its normal equilibrium state.

Such an increase in temperature would lead to about a 40%increase in the bolometric luminosity. From Figure 2 we seethat such an increase in the bolometric luminosity translatesinto a 15–20% increase in intensity as measured in a pass-band centered at 5200 Å. While such a passband is reason-able for a CCD+ BG40/S8612 filter+ atmosphere, the ob-servations in 1996 were made with phototubes, and these aremuch more blue sensitive. Thus, for a phototube passbandcentered around 3800 Å we expect intensity increases of about20–25%.

As reported in Table 5 of Provencal et al. (2009), such anincrease in intensity was observed in GD 358 during thesforzandoevent. In Figure 7 we plot these data, where the tri-angles are data from McDonald Observatory and the squaresare data from Mt. Suhora Observatory in Poland. Both curves

Temperature Change and Oblique Pulsation in GD 358 7

FIG. 7.— The relative intensity of GD 358 as measured relative toitslevels after thesforzandoepisode. The triangles are data from McDonaldObservatory and the squares are data from Mt. Suhora Observatory. Bothdata sets indicate a jump in intensity near BJED∼ 2450311.

are normalized to a value of one near BJED∼ 2450311. Wesee that these data, although sparse, strongly suggest thatGD 358 underwent a sharp intensity increase and decline ofabout 20–30% during this event. This corroborates our ear-lier interpretation that GD 358 had a thinner convection zonewith a smaller value ofτ0, and that this thinness was due to itssurface layers being temporarily hotter.

We note that similar results and conclusions have previ-ously been made by Weidner & Koester (2003). Using theapproach of Ising & Koester (2001) they were the first tomake numerical simulations of GD 358’s light curve duringthe sforzandoevent. As in our present analysis, they foundthat the shape of the light curve could only be reproduced forhigher effective temperatures than normally assumed for thisstar, suggesting a value ofTeff ∼ 27,000 K.

4.1. The Amplitude of the k= 8 Mode

The physical origin of thesforzandoevent is completely un-known. All we know for certain is that the amplitudes of thehigh-k modes rapidly decreased, leaving power in thek = 8mode, which itselfincreasedin amplitude by approximatelya factor of 30. While we have no model to explain the dis-appearance of power in the high-k modes, in this section weconsider what would happen to the apparent amplitude of thek = 8 mode if the convection zone were suddenly removed.

One effect of the thermal response of the convection zoneis to attenuate the amplitude of the flux variations incidentonits base. It acts as a low-pass filter, reducing the photosphericamplitude of a mode according to

∆Fphot

F0=

1√

1+ (ωτ0)2

∆Fbase

F0, (6)

whereF0 is the equilibrium value of the flux,FbaseandFphotare the instantaneous fluxes at the base of the convection zoneand at the photosphere, respectively, andω is the angular fre-quency of the mode (Goldreich & Wu 1999; Wu & Goldreich1999).

If we assume that the amplitude of thek = 8 mode atthe base of the convection zone was constant throughout thesforzando, equation 6 shows that a decrease inτ0 would nat-urally lead to an increase in observed amplitude. Thus, for

a 420 s mode we would see an increase of approximately afactor of 8 in its apparent amplitude asτ0 goes from∼ 600 sto ∼ 40 s. This is a large factor, but it is still much less thanthe factor of∼ 30 increase that was actually observed. Thus,thek = 8 appears to have increased its intrinsic amplitude byof order a factor of 4 during thesforzando. How it was ableto do this on so short a time scale remains a mystery.

5. THE CONVECTIVE TIME SCALEτ0

In Figure 6 we plot the known determinations of the convec-tive response timescale,τ0, versusTeff for the DBVs. Basedon the discussion in the previous section, GD 358 is plottedtwice: once for the 2006 data and once for the 1996 data.In addition, we show the position of the DBV PG1351+489(τ0 ∼ 100 s, Montgomery 2005), where we have assumed thepure He fit for itsTeff (Beauchamp et al. 1999). As expected,the data indicate an increase in the depth/mass of the convec-tion zone with decreasingTeff.

We also plot lines in Figure 6 showing the predictions ofML2/α convection (Böhm & Cassinelli 1971) for various val-ues ofα. We see that while low values ofα are excluded(α. 0.6), values in the range 1.0 to 1.2 provide a reasonabledescription of howτ0 varies withTeff. We note that the loggvalues determined for these stars are nearly identical, so theydo form an actual sequence inTeff. In general, however,τ0 isalso a function of logg, albeit a somewhat weaker one.

Our ultimate goal is to mapτ0 as a function of bothTeff andlogg for both the DBV and DAV instability strips. This willprovide important reference points for new hydrodynamicsimulations of convection which are starting to come online(e.g. Muthsam et al. 2009). For instance, for a given whitedwarf one can use the measuredTeff and logg values andperform a hydrodynamic simulation of its convection zone.Then, using equations 3 and 5 of Wu (2001) one can com-puteτ0. This value ofτ0 can then be compared with the valuederived from the light curve fits outlined above.

6. THE OBLIQUE PULSATOR MODEL

6.1. The Formalism

In section 3 we found evidence of a modulation of the in-clination angleθi (see Figure 4). This modulation has a for-mal significance level of 14.9/1.8∼ 8σ, which cries out for aphysical interpretation. The most obvious is some form of the“oblique pulsator” model, in which the pulsation axis is in-clined with respect to the rotation axis, and this axis precessesas the star rotates (Kurtz & Shibahashi 1986; Kurtz 1982). Amagnetic field that is inclined to the rotation axis is usuallyinvoked, and the pulsations are assumed to be aligned withthe magnetic axis.

While this hypothesis does introduce several unknowns (amagnetic field, the angle between the magnetic and rotationaxes, etc.) it also makes three testable predictions. If we takef and frot to be the mode frequency in the frame of the starand the rotational frequency, respectively, then anℓ = 1 modewith frequencyf aligned with the magnetic axis will appearas three separate peaks in the Fourier transform of the lightcurve, with frequencies off − frot, f , and f + frot. For claritywe will refer to these peaks as “geometric” peaks as they onlyappear because the pulsation axis spins around the rotationaxis, leading to a periodic apparent amplitude modulation ofthe mode. This amplitude modulation manifests itself in theFourier transform as two additional “geometric” peaks on ei-ther side of the original frequency, with the beating of these

8 Montgomery et al.

TABLE 4FREQUENCY SOLUTION FOR OBLIQUE PULSATION MODEL:

frot = 5.362±0.003µHZ

Frequency (µHz) Amplitude (mma) Phase (rad) ∆Φ/2π Independent Freqs.a

Triplet 1

1736.302± 0.004 16.77± 0.13 0.739± 0.008 1736.302± 0.0011741.664± 0.003 10.81± 0.13 2.832± 0.012 0.510± 0.013 1741.665± 0.0011747.027± 0.004 1.75± 0.13 1.724± 0.075 1746.673± 0.007

Triplet 2

1738.362± 0.005 0.95± 0.13 1.268± 0.138 1737.962± 0.0071743.725± 0.003 5.56± 0.13 0.174± 0.023 0.517± 0.023 1743.738± 0.0021749.087± 0.005 12.55± 0.13 2.117± 0.011 1749.083± 0.001

a These are the unconstrained frequency fits of Provencal et al. (2009).

three peaks producing the periodic amplitude changes. As thissplitting is caused by the rotation of the star, these frequenciesmustbe equally split to within measurement errors. This is thefirst and most stringent condition the model must face.

Second, the oblique pulsator model predicts a specific phaserelationship for the peaks in a given geometric triplet (orfor higherℓ, a (2ℓ + 1)-multiplet). From Kurtz & Shibahashi(1986), anℓ = 1 mode generically has luminosity variationsgiven by

∆L/L =A− cos[(ω−Ω)t +φ]+ A0cos[ωt +φ] + A+ cos[(ω+Ω)t +φ], (7)

whereω andΩ are 2π times f and frot, respectively (see Ap-pendix A for the complete expressions). These componentswill only have the same phaseφ for a particular choice of thezero point of time. For other zero points one can show thatthis phase relation translates to

2Φ0 − (Φ+ +Φ−) = 0, (8)

whereΦ−,Φ0,Φ+ are the measured phases of the respectivecomponents. In general, the productA− ×A+ can be negative.For this case, if we define the amplitudes always to be positiveand absorb any minus signs into the phase for each peak, therelation becomes

2Φ0 − (Φ+ +Φ−) = π. (9)

Defining ∆Φ ≡ 2Φ0 − (Φ+ +Φ−), then from the sign of theamplitudes in equations (A2)–(A4) we see that∆Φ/2π = 0for m= 0 modes and∆Φ/2π = 0.5 for m= ±1 modes. Thesephase relations, while less iron-clad than the equal spacing ofthe triplets, should be satisfied within the errors for geometricpeaks split by oblique rotation.

A third condition/prediction is the relative amplitudes ofthegeometric peaks. Such a calculation makes assumptions con-cerning the nature and strength of the magnetic field, so thisprediction of the model is the least reliable of the three. Inequations (A2)–(A4) in Appendix A we give expressions forthe amplitudes of the various components ofℓ = 1 modes per-turbed by a magnetic field and oblique rotation. The relevantparameters are the inclination angle of the rotation axis,θi ,the obliquity of the magnetic axis,β, andx1. The parameterx1 is given by

x1 ≡ω(1)

0 −ω(1)1

Ckℓ Ω, (10)

whereCkℓ is the rotational splitting coefficient due to the Cori-olis force,Ω is the angular frequency of rotation, andω(1)

0

andω(1)1 are the perturbations to the frequencies of them= 0

andm = 1 intrinsic modes due to the magnetic field, respec-tively. Even though this is clearly a more model dependentstatement, we would still hope that the amplitudes could ap-proximately be fit and/or predicted within the oblique pulsatorformalism.

6.2. The Results

First, we consider a fit to thek = 12 region of the FT, from1730–1750µHz. Provencal et al. (2009) found 6 significantpeaks in this region. We interpret these peaks as resultingfrom 2 components of anℓ = 1 mode, each split into a geo-metric triplet by oblique pulsation. These two original peaksare part of an intrinsic triplet produced by standard rotationalsplitting, but the amplitude of the third member is below ourdetection threshold.

We fit two sets of exactly evenly split triplets to the data set,where the central frequencies of each triplet are 1741.664 and1743.725µHz, and the value of the splitting is 5.362µHz.The values of the fit parameters we obtained are given in Ta-ble 4. In Figure 8 we show the result of pre-whitening thek = 12 region by this solution. The reduction in power of theFT is very significant, showing that evenly split triplets are agood representation of the data. This is a necessary conditionfor the oblique pulsator model to be applicable.

Second, we consider the phase relations within each ofthese equally split triplets. From Table 4 we see that the firsttriplet has∆Φ/2π = 0.510±0.013 and that the second triplethas∆Φ/2π = 0.517±0.023. The oblique pulsator model alsopasses this test with flying colors. In addition, the fact that∆Φ/2π ∼ 0.5 rather than 0.0 implies that the modes are notaxi-symmetric, i.e.,|m| = 1 for both modes. For whatever rea-son, them= 0 member of the original triplet is not present atobservable amplitudes.

Finally, we wish to test the ability of the oblique pulsatormodel to adequately reproduce the amplitudes of the geomet-ric peaks. To calculate the amplitudes we use the analyti-cal expressions given in Appendix A (see Unno et al. 1989;Kurtz & Shibahashi 1986) and to perform the fits we haveused a genetic algorithm (Charbonneau 1995). This allowsus to search themvalues of each intrinsic triplet as well as thevalues of the parametersθi , β, andx1. As we demonstrate be-low, such a fit is more constrained than one would think giventhe six data points and five free parameters.

In Figure 9 we show the result of the fit to the amplitudes inTable 4. The fit is quite impressive. It has the added bonus thattriplets 1 and 2 arerequiredto originate from|m| = 1 intrinsic

Temperature Change and Oblique Pulsation in GD 358 9

FIG. 8.— The effect of pre-whitening by two identically split triplets in theoblique pulsator model. The top panel is the original FT of the k = 12 regionand the lower panel is the result of pre-whitening by the two pairs of triplets.The splitting within each triplet is 5.362±0.003µHz.

modes and they must have opposite signs. This corroboratesour earlier result based on the phases that both triplets origi-nated from|m| = 1 intrinsic modes.

Given the near equality of the number of data points andfree parameters we wished to assess how easily our modelcould reproduce any data set. To do this, we randomly gen-erated amplitudes for 500 pairs of triplets and fit them in thesame way we fit the data, normalizing the residuals by themean squared amplitudes of the two triplets. We found that inonly 11 cases out of 500 were the random amplitudes better fitby our model than the measured amplitudes were. Thus, weconclude that our amplitude fits are significant at the∼ 98%level.

As a final check on this procedure, we use these amplitudes

FIG. 9.— A fit of the calculated amplitudes (open squares) to the observedamplitudes (points) for the different components of “k=12”assuming theoblique pulsator model. The errors on the observed amplitudes are smallerthan the size of the points.

FIG. 10.— Pre-whitening of the same region as in Figure 8 but using thefull oblique pulsator model for the amplitudes, frequencies, and phases. Thus,not only are the frequency splittings constrained, but the amplitudes within atriplet and their relative phases are as well.

together with equations A2 and A4 to pre-whiten thek = 12region of the FT, as shown in Figure 10. Although consid-erably more constrained, this pre-whitening is clearly just aseffective as that shown in Figure 8. Thus, the oblique pulsatormodel passes all tests with flying colors when applied to thek = 12 region of the FT.

6.3. Interpretation of the Oblique Pulsator Fits

The relative amplitudes of the peaks within a triplet splitby oblique pulsation/rotation depend onθi , β, and x1 (seesection 6.1 for a definition of these quantities). From theamplitude fits obtained in the previous section, we find thatθi = 46.3, β = 31.8, andx1 = 5.65. This value ofθi is closeto that obtained from our nonlinear light curve fits, 47.5±2.2

(section 3). This provides an important consistency check onboth methods.

Given thatθi is the angle between the rotation axis and ourline-of-sight and thatβ is angle between the rotation and pul-sation/magnetic axes, the effective inclination angle of amodealigned with the magnetic axis should vary betweenθi −β andθi +β as the star rotates, i.e., between 14.5 and 78.1. Fromthe top panel of Figure 4 we see thatθi varies with about halfof this amplitude. The straightforward explanation for this isthat only thek = 12 modes are affected by oblique rotation,while the other modes are aligned with the rotation axis. Asa result, the value ofθi shown in Figure 4 represents the aver-age inclination angle for all modes. Averaging over all modesyields an inclination angle which varies by about 15 ratherthan the full 32 experienced by thek = 12 modes.

Finally, the parameterx1 is a measure of the relativestrength of the Coriolis and magnetic perturbations; the fitvaluex1 = 5.65 indicates that the magnetic field is the dom-inant perturber. This reinforces the idea that the pulsations ofthese modes are tied to the magnetic axis. In addition, for alarge-scale dipole field simple perturbation theory applied to

10 Montgomery et al.

ℓ = 1 and 2 modes yields

ω(1)|m| ∝

25 + 2

5m2, ℓ = 1187 − 2

7m2, ℓ = 2, (11)

whereω(1)|m| is the perturbation to the frequency of the mode

due to the magnetic field (Montgomery 1994; Jones et al.1989). Together with equation (10), the fact thatx1 is posi-tive means thatω(1)

0 > ω(1)1 , which in turn suggests thatℓ≥ 2.

In reality, this indicates that these modes, at least in the sur-face layers, may not be pureℓ= 1 modes but contain a mixtureof other higherℓ components.

The success of the oblique pulsator model in explaining thek = 12 peaks constitutes evidence that oblique pulsation playsan important role in GD 358. However, there is little con-clusive evidence that other modes in this star are undergo-ing oblique pulsation. On the contrary, the stablek = 8 and9 triplets are consistent only with the traditional case of thepulsation axis aligned with the rotation axis. A possible ex-planation is that the outer turning points for thek = 8 and 9modes are farther from the surface than they are for thek = 12modes, so if the dominant magnetic effects are also confinedto the surface layers then the lowerk modes would not be af-fected by the magnetic field and the higherk modes would.

The magnitudesof the multiplet splittings/fine structurealso present a challenge. For instance, the oblique pulsatormodel requires a rotation frequency of 5.36µHz, whereasthe splittings of thek = 8 and 9 modes, believed to resultfrom standard rotational splitting, give a value of 7.5µHz(Provencal et al. 2009; Winget et al. 1994). Since differentmodes sample different regions of the star’s interior, differen-tial rotation in the radial direction could explain these results.

A further concern is the internal consistency of the obliquepulsator fit. The central components of each of the triplets inTable 4 should themselves be members of an intrinsic tripletwhich is split by rotation. This splitting is observed to be2.06 µHz and should equal 2Ckℓ frot cosβ for solid body ro-tation of the star. This is satisfied ifCkℓ ∼ 0.225 instead ofCkℓ ∼ 0.5 as is expected forℓ = 1 modes. The splitting withina geometric triplet is a direct measure of thesurfacerotationrate, while the difference in frequency of the central com-ponents of these triplets (the “intrinsic” modes) is given by2Ckℓ frot cosβ, where frot represents aradial averageof therotation profile. Thus, differential rotation, with the interiorrotating more slowly than the surface layers, could explainthese values. An alternate explanation is that the magneticfield mixes higherℓ components into the spatial structure ofthek = 12 modes. They would therefore have rotational con-stantsCkℓ indicative of higherℓ modes. SinceCkℓ ∼ 1/ℓ(ℓ+1)for moderate to highk modes, these modes would have corre-spondingly smaller values ofCkℓ.

A further possibility is that the intrinsic modes haveℓ = 2instead ofℓ = 1. In this case only 3 of the 5 possible geo-metric peaks would be large enough to be detectable. First,we note that the predicted phase relations are the same as fortheℓ = 1 case, so the measured phase relations support eithercase equally. Also, sinceCkℓ ∼ 1/6 for ℓ = 2, using the rota-tional splitting of thek = 8 and 9 modes yields a splitting of2Ckℓ frot cosβ ∼ 2.2µHz, which is close to the measured valueof 2.06µHz. In this case, though, the value of the inclinationangle,θi = 22.1, does not agree with that from the light curvefits. Also, theℓ = 2 fits are somewhat less constrained than theℓ = 1 fits: fitting 500 random amplitudes in the same way as

the data, 12% of the fits were better than the fit to the data,making theℓ = 2 fit significant at the 88% level.

Even so, given the ability of the oblique pulsator model todescribe (1) the power in thek = 12 region, (2) the phase rela-tions between the geometric triplets, and (3) the relative am-plitudes within the triplets (to some extent), we believe thatthe above-mentioned inconsistencies do not warrant rejectionof the model but rather point us in the direction of future im-provements. For instance, the amplitude calculations includemagnetic effects in a fairly simple way; more sophisticatedtreatments may be necessary to adequately model the obser-vations. Also, differential rotation in the radial direction maybe able to resolve the seeming discrepancy of theCk,ℓ valuesfor theℓ = 1 case.

7. DISCUSSION

In many respects GD 358 is a “simple” star to analyze.For instance, the spacing between the periods of its multi-plets suggests that we are looking at successive radial over-tone numbers ofℓ = 1 modes. This is based both on the factthat (1) given its distance, mass, and luminosity, onlyℓ = 1modes allow a consistent solution (Bradley & Winget 1994)and (2) in many cases these multiplets are well defined triplets(Winget et al. 1994). Furthermore, as a DBV it has no hydro-gen layer, so fewer parameters are needed to model its struc-ture. Indeed, it is the best studied white dwarf variable andsignificant constraints have been placed on its interior struc-ture (Metcalfe, Nather, & Winget 2000; Metcalfe 2003).

On the other hand, GD 358 is in many respects a com-plicated star to model. For instance, while the amplitudesof its modes can be fairly constant during a WET run, overtimescales of a few months the amplitudes can change sig-nificantly. The most extreme case of amplitude change is thepreviously discussedsforzando, in which dramatic amplitudechanges occurred on a timescale of a day or less. We currentlyhave no theory that adequately describes these changes.

What we do know is that two independent lines of evi-dence indicate that GD 358 was hotter during thesforzandoevent. The first is that the average flux of GD 358 relativeto the comparison stars is larger, and the second is that thenonlinear light curve fits indicate a thinner convection zone—presumably the result of an increase in surface temperature.If we assume that GD 358 remained hotter for approximatelya day then such a temperature increase would require a totalenergy input of approximately 6×1036 ergs. While detailednonadiabatic models of GD 358’s pulsational state just priorto thesforzandowould be required, rough estimates indicatethat this amount of energy may typically be present in thehigherk modes. Thus, if these modes were somehow dampedand deposited their pulsation energy in the surface layers ofthe star over the period of a day, this would explain GD 358’stemporary temperature increase. Such a scenario would ex-plain both the temperature increase and the disappearance ofthe highk modes, although more detailed models would benecessary to check quantitatively the energetics.

We note that an increase in the apparent amplitude of thek = 8 mode is a generic feature of a thinner convection zone(e.g., smaller value ofτ0), although the predicted factor (∼ 8)is still less than what was observed (∼ 30). While specula-tive, it is also possible that some mechanism may have trans-ferred power from the highk to the lowk modes, although wecurrently do not understand how this would proceed. Perhapsleast likely is the possibility that the hotter “equilibrium” stateof thesforzandoallowed mode growth and damping to occur

Temperature Change and Oblique Pulsation in GD 358 11

on timescales given by linear theory and that these timescaleswere many orders of magnitude shorter than expected.

Another possibility is that the event was tied to a magneticphenomenon and that this phenomenonsomehow led to a tem-perature increase as well as a change in the amplitudes ofthe various modes. There are several reasons that this is notcompletely far-fetched. First, the oblique pulsator modelusedhere requires a mechanism to align modes with an axis otherthan the rotation axis, and a magnetic field can provide this.Second, thek = 8 and 9 triplets are asymmetric in the sensethat them = 0 mode is closer to either them = +1 or m = −1mode (see Figures 18 and 19 of Provencal et al. 2009). Per-haps not coincidentally, the asymmetry seen in the splittingsof the k = 8 and 9 modes reversed themselves at the time ofthesforzando, and this change has persisted ever since. Thistimescale of years is consistent with what is seen in the mag-netic cycle of the Sun and other stars (Elsworth et al. 1990;Libbrecht & Woodard 1990). Finally, small but definite shiftsin frequency of order 0.5µHz have been seen on timescalesas short as 3 weeks. These could be a sign of magnetic activ-ity: changes in the surface magnetic field could produce slightperturbations in the mode frequencies (Jones et al. 1989), andthe timescale of a few weeks for these to take place againseems plausible.

A final unresolved issue for GD 358 is differential rota-tion. Taken at face value, the difference in triplet splittingbetween highk and lowk modes implies significant differ-ential rotation (Winget et al. 1994). In addition, the obliquepulsator model as applied in this paper implies differential ro-tation: thek = 12 region requires a surface rotation rate of∼ 2.17 days whereas thek = 8 and 9 splittings give a rota-tion rate of∼ 1.5 days. Thek = 8 and 9 splittings reflect abulk average of the rotation rate whereas the frequency differ-ences within the geometric triplets in thek = 12 region giveus the rate at the surface. In addition, the frequency split-ting between the intrinsic modes in thek = 12 region also im-plies differential rotation, albeit with the interior rotating lessrapidly than the surface. Previously, detailed examinations ofthe frequency splittings in GD 358 as a function ofk havebeen inconclusive (Kawaler, Sekii, & Gough 1999), but suchanalyses did not take into account the possibility of obliquepulsation.

An intriguing possibility is that the pulsations themselveslead to differential rotation. Townsend (2009) has recentlyshown that g-modes in massive stars can transport angu-lar momentum relatively rapidly compared to evolutionarytimescales. This transport occurs predominantly in regionsin which the mode is driven and/or damped and leads to dif-ferential rotation. If this effect occurs in white dwarfs thenit could alter the rotation rate from the surface down past thebase of the convection zone into the radiative damping lay-ers. If the rotation profile of these outer layers is continuallychanging then this would also explain the small shifts in fre-quency which have been detected in GD 358. In addition, amore dramatic shift in the rotation profile could be associatedwith thesforzandoevent. Perhaps the outer part of GD 358’srotation profile experienced a shift during this event and ithaspersisted in its new state for the past several years. This wouldexplain the shift in asymmetry of thek = 8 and 9 triplets thatalso occurred at this time (see Provencal et al. 2009).

8. CONCLUSIONS

In this paper we have extended our nonlinear light curve fit-ting technique to the multiperiodic pulsator GD 358. Our fit tothe 2006 WET data provides a good match to the light curvesand we find that the thermal response time of its convectionzone isτ0 = 572.9±6.1 s. This is considerably larger than thatof the star PG1351+489, for whichτ0 ∼ 100 s (Montgomery2005). This difference inτ0 is consistent with the effec-tive temperatures of these stars: the pure He solution forPG1351+489 yields aTeff which is∼ 2,000 K hotter than thatfor GD 358.

We also obtained a fit to the light curve of GD 358 duringthesforzandoevent in 1996. These fits showed that GD 358hadτ0 ∼ 42±2 s, a value much less than that determined fromthe 2006 data. This suggests that its effective temperaturewasapproximately 2,000 K hotter in 1996 than in 2006, and this isconsistent with the estimate of Weidner & Koester (2003) thatthe light curve shape suggestsTeff ∼ 27,000 K. Independentevidence of GD 358’s brightness relative to comparison starsis also consistent with such a temperature increase at the timeof thesforzando(Provencal et al. 2009). The physical originof this temperature increase will be the subject of future work.

As expected, these data indicate an increase in thedepth/mass of the convection zone with decreasingTeff.A similar trend is given by ML2/α = 1.1 convection(Böhm & Cassinelli 1971), although the slope of the theoret-ical relation appears less steep than that of the data. In addi-tion, lower values ofα . 0.6 are excluded. In general,τ0 isalso a function of logg, albeit a somewhat weaker one. Ourultimate goal is to mapτ0 as a function of bothTeff and loggfor both the DBV and DAV instability strips. These data willprovide insight into the physics of convection, still one ofthelargest uncertainties in stellar modeling. They will also serveas important constraints for new hydrodynamic simulationsofconvection which are starting to come online.

Multiple lines of evidence point to some of GD 358’s modesundergoing oblique pulsation, in particular, the peaks in thek = 12 region. First, these peaks can be fit with two sets ofexactly evenly spaced triplets. Second, the relative phases ofeach of the components within the triplet indicate that eachoriginates as a singlem = −1 or +1 mode aligned with themagnetic axis; as the star rotates, the magnetic axis precessesaround the rotation axis, generating a triplet for each intrin-sic mode. Finally, the oblique pulsator model qualitativelyand quantitatively fits the amplitudes of the peaks seen in theFourier transform. Taken together, this marks the first timethat oblique pulsation has been seen in a white dwarf variable.

Having now identified the characteristics of oblique pulsa-tion in GD 358 we now know what to look for in other whitedwarf variables; we have found preliminary indications of itin other stars and in other data sets of GD 358. As discussedin the previous sections, oblique pulsation may prove to be adiagnostic of both the magnetic field and its changes as wellas a diagnostic of differential rotation. This opens an excitingchapter in the seismology of these objects.

This research was supported in part by the Delaware As-teroseismic Research Center, the National Science Founda-tion under grant AST-0909107, and the Norman HackermanAdvanced Research Program under grant 003658-0255-2007.AJS thanks the National Science Foundation for support un-der grant AST-0607840. The Delaware Asteroseismic Re-search Center (DARC) is grateful for the support of the Crys-

12 Montgomery et al.

tal Trust Foundation and Mt. Cuba Observatory. DARC also acknowledges the support of the University of Delaware,through their participation in the SMARTS consortium.

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APPENDIX

EQUATIONS OF OBLIQUE PULSATION FORℓ = 1 AND 2 MODES

The formulae given below are taken from Unno et al. (1989). Following Kurtz & Shibahashi (1986) we useω instead ofσ forthe angular frequencies of the modes. We useω(0) for the unperturbed frequency of the mode in a non-rotating,non-magnetic star,andω(1)

|m| for the perturbation to this frequency due to the magnetic field (but not due to rotation). The pulsation axis is assumedto be aligned with the magnetic axis, which makes an angle ofβ with the rotation axis, andθi is taken to be the angle betweenthe rotation axis and our line-of-sight. As the pulsation and magnetic axes rotate around the star, we find that a mode of givenfrequency andℓ,m values is split into a triplet of peaks. We tabulate below theresulting time dependence of modes having thegiven values ofℓ andm.

In the following,x1 is defined to be

x1 ≡ω(1)

0 −ω(1)1

CΩ, (A1)

whereC is the rotational splitting coefficient due to the Coriolis force forℓ = 1 modes andΩ is the angular velocity of rotation atthe stellar surface. Sincex1 is the ratio of the rotational splitting to the splitting induced by the magnetic field it provides a usefulestimate of the relative importance of the two effects. Also, since it depends on the difference in magnetic splitting betweenm= 0 and|m| = 1 modes, it is sensitive to the geometry of the magnetic field. With these definitions, the luminosity perturbationsassociated withℓ = 1 oblique pulsation are given below:

ℓ = 1, m= −1:

∆L/L =1√2

(

1−1+ cosβ

x1

)

sin2 β

2sinθi cos

[(

ω(0) +ω(1)1 −CΩcosβ −Ω

)

t +φ]

+1√2

(

1−cosβ

x1

)

sinβ cosθi cos[(

ω(0) +ω(1)1 −CΩcosβ

)

t +φ]

Temperature Change and Oblique Pulsation in GD 358 13

−1√2

(

1+1− cosβ

x1

)

cos2β

2sinθi cos

[(

ω(0) +ω(1)1 −CΩcosβ +Ω

)

t +φ]

(A2)

ℓ = 1, m= 0:

∆L/L =12

(

1+1x1

)

sinβ sinθi cos[(

ω(0) +ω(1)0 −Ω

)

t +φ]

+ cosβ cosθi cos[(

ω(0) +ω(1)0

)

t +φ]

+12

(

1−1x1

)

sinβ sinθi cos[(

ω(0) +ω(1)0 +Ω

)

t +φ]

(A3)

ℓ = 1, m= 1:

∆L/L =1√2

(

1−1− cosβ

x1

)

cos2β

2sinθi cos

[(

ω(0) +ω(1)1 +CΩcosβ −Ω

)

t +φ]

−1√2

(

1+cosβ

x1

)

sinβ cosθi cos[(

ω(0) +ω(1)1 +CΩcosβ

)

t +φ]

−1√2

(

1+1+ cosβ

x1

)

sin2 β

2sinθi cos

[(

ω(0) +ω(1)1 +CΩcosβ +Ω

)

t +φ]

(A4)

More generally for arbitraryℓ andm, we definex|m|

x|m| ≡ω(1)

0 −ω(1)|m|

CΩ, (A5)

whereC is the rotational splitting coefficient for the appropriateℓ values. With these definitions, the luminosity perturbationsassociated withℓ = 2 oblique pulsation are given below:

ℓ = 2, m= −2:

∆L/L =

√

38

sin4 β

2sin2θi

(

1+2(1+ cosβ)

x1 − x2

)

cos[(

ω(0) +ω(1)2 − 2CΩcosβ − 2Ω

)

t +φ]

+√

38

sin2 β

2sinβ sin2θi

(

1+1+ 2cosβ

x1 − x2

)

cos[(

ω(0) +ω(1)2 − 2CΩcosβ −Ω

)

t +φ]

+18

√

32

sin2β (1+ 3cos2θi)

(

1+2cosβx1 − x2

)

cos[(

ω(0) +ω(1)2 − 2CΩcosβ

)

t +φ]

−√

38

cos2β

2sinβ sin2θi

(

1+−1+ 2cosβ

x1 − x2

)

cos[(

ω(0) +ω(1)2 − 2CΩcosβ +Ω

)

t +φ]

+√

38

cos4β

2sin2θi

(

1+2(−1+ cosβ)

x1 − x2

)

cos[(

ω(0) +ω(1)2 − 2CΩcosβ + 2Ω

)

t +φ]

(A6)

ℓ = 2, m= −1:

∆L/L =18

√

32

sinβ sin2θi

(

2− 2cosβ −3sin2β

x1−

(1− cosβ)2

x1 − x2

)

cos[(

ω(0) +ω(1)1 −CΩcosβ − 2Ω

)

t +φ]

+14

√

32

sin2θi

[

1+ cosβ − 2cos2β + sin2β

(

−3cosβ

x1+

−1+ cosβx1 − x2

)]

cos[(

ω(0) +ω(1)1 −CΩcosβ −Ω

)

t +φ]

+116

√

32

sinβ (1+ 3cos2θi)

[

4cosβ −1+ 3cos2β

x1−

2sin2β

x1 − x2

]

cos[(

ω(0) +ω(1)1 −CΩcosβ

)

t +φ]

−14

√

32

sin2θi

[

−1+ cosβ + 2cos2β + sin2β

(

3cosβx1

−1+ cosβx1 − x2

)]

cos[(

ω(0) +ω(1)1 −CΩcosβ +Ω

)

t +φ]

−18

√

32

sinβ sin2θi

(

2+ 2cosβ +3sin2β

x1+

(1− cosβ)2

x1 − x2

)

cos[(

ω(0) +ω(1)1 −CΩcosβ + 2Ω

)

t +φ]

(A7)

14 Montgomery et al.

ℓ = 2, m= 0:

∆L/L =38

sin2β sin2θi

(

1+2x1

)

cos[(

ω(0) +ω(1)0 − 2Ω

)

t +φ]

+38

sin2β sin2θi

(

1+1x1

)

cos[(

ω(0) +ω(1)0 −Ω

)

t +φ]

+116

(1+ 3cos2β) (1+ 3cos2θi)cos[(

ω(0) +ω(1)0

)

t +φ]

+38

sin2β sin2θi

(

1−1x1

)

cos[(

ω(0) +ω(1)0 +Ω

)

t +φ]

+38

sin2β sin2θi

(

1−2x1

)

cos[(

ω(0) +ω(1)0 + 2Ω

)

t +φ]

(A8)

ℓ = 2, m= +1:

∆L/L =18

√

32

sinβ sin2θi

(

2+ 2cosβ −3sin2β

x1−

(1+ cosβ)2

x1 − x2

)

cos[(

ω(0) +ω(1)1 −CΩcosβ − 2Ω

)

t +φ]

+14

√

32

sin2θi

[

−1+ cosβ + 2cos2β + sin2β

(

−3cosβ

x1+

1+ cosβx1 − x2

)]

cos[(

ω(0) +ω(1)1 −CΩcosβ −Ω

)

t +φ]

−116

√

32

sinβ (1+ 3cos2θi)

[

4cosβ +1+ 3cos2β

x1+

2sin2β

x1 − x2

]

cos[(

ω(0) +ω(1)1 −CΩcosβ

)

t +φ]

−14

√

32

sin2θi

[

1+ cosβ − 2cos2β + sin2β

(

3cosβx1

+1− cosβx1 − x2

)]

cos[(

ω(0) +ω(1)1 −CΩcosβ +Ω

)

t +φ]

+18

√

32

sinβ sin2θi

(

−2+ 2cosβ −3sin2β

x1−

(1− cosβ)2

x1 − x2

)

cos[(

ω(0) +ω(1)1 −CΩcosβ + 2Ω

)

t +φ]

(A9)

ℓ = 2, m= +2:

∆L/L =

√

38

cos4β

2sin2θi

(

1+2(1− cosβ)

x1 − x2

)

cos[(

ω(0) +ω(1)2 − 2CΩcosβ − 2Ω

)

t +φ]

−√

38

cos2β

2sinβ sin2θi

(

1+1− 2cosβ

x1 − x2

)

cos[(

ω(0) +ω(1)2 − 2CΩcosβ −Ω

)

t +φ]

+18

√

32

sin2β (1+ 3cos2θi)

(

1−2cosβx1 − x2

)

cos[(

ω(0) +ω(1)2 − 2CΩcosβ

)

t +φ]

+√

38

sin2 β

2sinβ sin2θi

(

1−1+ 2cosβ

x1 − x2

)

cos[(

ω(0) +ω(1)2 − 2CΩcosβ +Ω

)

t +φ]

+√

38

sin4 β

2sin2θi

(

1−2(1+ cosβ)

x1 − x2

)

cos[(

ω(0) +ω(1)2 − 2CΩcosβ + 2Ω

)

t +φ]

(A10)