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AstrophysicsandSpaceScience DOI10.1007/s•••••-•••-••••-• Constraining the degree of the dominant mode in QQ Vir Andrzej Baran1 • John Telting2 • Roy Østensen3 • Maciej Winiarski1 • Marek Droz˙dz˙1 • Dorota Koziel 4 • Mike Reed5 • Raquel Oreiro3 • Roberto Silvotti6 • Micha l Siwak4 • Uli Heber7 • 0 Peter Papics8 1 0 2 n a J (cid:13)c Springer-Verlag•••• 8 2 was to compare the value of the ℓ parameter derived for the main mode in QQVir to previously published ] Abstract Wepresentearlyresultsoftheapplicationof R values derived by using different methods. a method which uses multicolor photometry and spec- S troscopy for ℓ discrimination. This method has been h. successfully applied to the pulsating hot subdwarfBal- Keywords hot subdwarfs; pulsating stars p loon090100001. Here we apply the method to QQVir - o (PG1325+101). This star was observed spectroscop- r ically and photometrically in 2008. Details on spec- 1 The method t s troscopy can be found in Telting et al. (this volume) a [ while photometryandpreliminaryresultsonℓdiscrim- The method we used for mode degree ℓ discrimination ination are provided here. The main aim of this work has been already described and the results of its ap- 1 plication to the main sequence stars presented in sev- v eral papers. This method is very challenging, since it 9 AndrzejBaran 2 requires observations performed in at least three fil- CracowPedagogical University,IowaStateUniversity 2 ters. Moreover, time-series spectroscopy could also be 5 JohnTelting employed. For this reason, the best objects to apply . NordicOpticalTelescope,LaPalma,Spain 1 this method to are bright stars with relatively long pe- 0 RoyØstensen riods and relatively high amplitudes, so they are ac- 0 Institute forAstronomy,K.U.Leuven cessible even with small size telescopes. It is possi- 1 MaciejWiniarski : ble then to achieve good sampling of the brightness v CracowPedagogical University changes to derive amplitudes and periods of the de- i X MarekDroz˙dz˙ tected modes. Although hot subdwarfs (sdB) are rel- r CracowPedagogical University atively faint objects, in 2004 independent photometric a DorotaKoziel and spectroscopic observations of the brightest pulsat- JagiellonianUniversity ingsdBstar,Balloon090910001(Bal09),allowedasuc- MikeReed cessful application of the method (Baran et al. 2008). MissouriState University For the same star, but with high precision multicolor RaquelOreiro photometry, Charpinet et al. (2008) also derived good Institute forAstronomy,K.U.Leuven discriminationforafewmodeswithhighestamplitudes. RobertoSilvotti Fromthesetwopaperswecandrawoneimportantcon- INAF–OsservatorioAstronomicodiTorino clusion. The method may give positive discrimination Micha lSiwak of the ℓ parameter either having high precision three– filterphotometry,orafew(atleasttwo)oflowprecision JagiellonianUniversity photometrysupportedbyspectroscopy. Encouragedby UliHeber the successful application of the method for Bal09 we Dr.Remeis–Sternwarte,Institute forAstronomy,UniversityEr- langen–Nu¨rnberg decided to try with another pulsating subdwarfB star, Peter Papics QQVir. KonkolyObservatory 2 The main goal of performing another observation of QQVir was to decide which (or if any) value of the ℓ parameter for the dominant mode, obtained in previous works, is the correct one. Telting & Østensen (2004) analyzed spectroscopic data and stated that 600 the main mode is consistent with ℓ=0. Quite inter- estingly, Charpinet et al. (2006), based on photometry 500 RV data only, derived ℓ=2 from frequency matching. The Charpinet et al. (2006) result is even more surprising 400 amsoidteinwdiicthateℓs=u2numsuoadlley. hIigthisinitnrtinerseicstainmgpltihtuendetooftuhsee mag [mma] 300 U bcoonthstrkaiinndsthoefvdaalutae,opfhtohteommeotdrye adnegdresepefcotrrotshceopdyo,mtio- Delta 200 B nant mode in QQVir and if possible, either to confirm 100 one of the already obtained values or to exclude both 0 V of them. -100 960 970 980 990 1000 1010 1020 1030 1040 1050 1060 Frequency [mHz] 2 Observations Fig. 1 Light curve of QQVir in three filters. Data in different filters are colored according to the bandpass and From the previous observations we knew that the pul- denotedontheleftside. Reddataarefortheradialvelocity sationalperiodofthedominantmodeisrelativelyshort curveand thescale on Y-axisis not relevant for thesedata ( 2min.). Along with a low brightness ( 13.5mag in ∼ ∼ the V filter), data required for the method were not easy to obtain, particularly with small size telescopes. For this reason, we performed only one filter photometry on each telescope with the exposure time short enough to get appropriate sampling but not necessar- ily long enough to have high precision data. Details on the observations are presented in Table1, while all photometrygatheredinthis projectis plottedinFig.1. This figurealsoincludes spectroscopicdata whichhave 25 been used in this method. Details on spectroscopy can be found in the paper by Telting et al. (2010) in this volume. 20 Data in different filters were taken at different telescopes. This is the reason why there are not the same ma] 15 nspuemctbreurmofofntihghetsstaforranadllsfipletcetrrsa.lrWesipthonrseespofectthetoCCthDe Amplitude [m 10 we used, the highest signal has been obtained in the V filter. These data have been used for precise fre- quencydeterminationofthedominantmode. Thatwas 5 also the reason why observations in the V filter have beenperformedforthelongesttimefromthesiteswith 0 6 6.5 7 7.5 8 8.5 9 easy access, like: Suhora, Mercator and Baker. In ad- Frequency [mHz] dition to the V filter, data in the U filter were taken Fig. 2 Amplitude spectrum of QQVir in V filter. The at Loiano and Nordic Optical Telescope (NOT) during dominant mode is clearly seen at ∼7.25mHz. Significant three nights in total. Fortunately, the dominant mode signalisalsopresentathigherfrequencies. Thisspectrumis is clearly resolved with only one night of data. Some- constrainedtotheregionwherethedominantsignalappears time later, it turned out that, quite luckily, almost at thesametimeasourcampaign,absolutelyindependent data in B filter were taken during the course of a different project. That enables us to use three filter data. ModedegreeinQQVir 3 The B filter data were not crucial, but this method of Table 1 ObservationallogforQQVirdata. Alldatawere mode discrimination improves with the number of fil- taken in 2008. ters used. Date Site Exp [s] Length [h] Filter Although the observations in different filters were 16 Mar Loiano 20 2.1 U not conducted simultaneously, we can use published 17 Mar Loiano 20 5.8 U results to see if the amplitude of the dominant fre- 28 Mar NOT 22 2.5 U quency changes with time. Results obtained by 01 Mar Loiano 15 4.4 B Telting & Østensen (2004) and Silvotti et al. (2006) 13 Mar Konkoly 15 4.8 B have radial velocity and B filter amplitudes consis- 16 Feb Suhora 15 4.2 V tent with ours. Those observations were performed 26 Feb Suhora 15 0.4 V independently and at differing times during 2003. We 15 Mar Mercator 20 0.8 V therefore conclude that the amplitudes are not chang- 21 Mar Mercator 20 1.9 V ing significantly and remain stable to within the er- 22 Mar Mercator 20 1.1 V rors. Moreover, results from the multisite campaign 24 Mar Baker 25 8.1 V of Silvotti et al. (2006) indicate there are no nearby 25 Mar Baker 30 7.5 V frequencies to the dominant one, even at very low am- 27 Mar Mercator 20 2.6 V plitudes. Thissuggeststhatourdatadonotsufferfrom 28 Mar Mercator 20 2.6 V amplitude variability or the effects of nearby frequen- 30 Mar Mercator 20 2.2 V cies. 31 Mar Mercator 20 7.0 V 01 Apr Mercator 20 6.8 V 03 Apr Mercator 20 5.9 V 3 Fourier analysis 04 Apr Mercator 20 4.4 V 05 Apr Mercator 20 0.8 V A prewhitening procedure was applied to derive fre- 20 Apr Suhora 15 1.3 V quencies, amplitudes and phases of the pulsation 24 Apr Suhora 15 4.9 V modes. First, we assumed that amplitudes can vary 25 Apr Suhora 15 7.3 V linearly. As it turned out, the change was insignificant 26 Apr Suhora 15 7.3 V (withintheerrors)andwehavedecidedtoassumecon- 27 Apr Suhora 15 4.3 V stant amplitude and redid the prewhitening process. 28 Apr Suhora 15 3.9 V Phases were calculated relative to epoch 2454552.0 30 Apr Suhora 15 1.2 V which was chosen arbitrarily. Amplitudes and phases 02 May Suhora 15 5.1 V for the dominant mode at frequency 7.2554776 ± 11 May Suhora 15 3.8 V 0.0000004mHz (derived from V filter data) derived 12 May Suhora 15 4.9 V fromthedatainUBVfilters,aswellasfromtheradial 14 May Suhora 15 2.8 V velocity curve are presented in Table2. A raw ampli- 16 May Suhora 15 2.3 V tude spectrum calculated from the data in V filter is showninFig.2. Ithasbeenlimited tothe regionwhere thesignalappeared,whichis6–9mHz. Theamplitude for the dominant modes detected both in the radial- spectracalculatedfromotherfilterslookssimilar,how- velocity and brightness variations. Discrimination of ever the noise level and resolution is worse as less data ℓ, does not depend on the observations alone. It also were obtained. Since the main aim of this work is to depends on the model parameters, in particular effec- derive the most likely value of the dominant mode, the tive temperature, surface gravity, flux and limb dark- other modes are not included in Table2. Nevertheless, eningderivativesovereffectivetemperatureandsurface during the prewhitening process they were taken into gravity. For QQVir we assumed the effective tempera- account to check if their removal has any influence on tureandsurfacegravityderivedbyTelting & Østensen the frequency and/or amplitude of the mode in ques- (2004), Teff = 34800 K, logg = 5.81 dex. Then we tion. If so, they were included in the solution. needed to calculate flux distributions in the range between 300 and 700 nm for a grid of models with Teff ranging from 32500 to 37500 K and logg, from 5.5 to 4 Application of the method 6.0 dex. To do this, we used a grid of models prepared fortheinterestedrangeinTeff andlogg. Theresulting Having derived frequencies, amplitudes and phases we fluxdistributionsweremultipliedbytransmissionfunc- could apply the method to discriminate ℓ parameters 4 tions of the filters to get integrated fluxes in the three photometric bands – UBV – that we used. For the same grid of models, we also calculated the specific intensities as a function of µ = cosθ, where θ is theanglebetweenthe line ofsightandnormaltothe stellarsurfaceinagivenpointofthestellardisc. Using these intensities, the limb darkening law was derived fitting the coefficients c and d of a function I(µ) = I(0)[1 c(1 µ) d(1 √µ)] (1) − − − − which was found to reproduce the calculated changes of I(µ) best. This was done for different combinations of input parameters of the model, thus allowing the calculation of the necessary derivatives. 5 Discussion and conclusions A disadvantageof the method we applied for mode de- 9 gree discrimination is that we cannot derive a value of ℓinadirectway. We mustuseaniterativeprocessand 7 employ a merit function to discriminate which mode is the most probable. The minimum of the function (if it exists) indicates the possible value. In this study we χ2 5 applied this method to discriminate between divergent solutions for the main frequency of QQVir. Results of 3 our calculation are shown in Fig.3. The results clearly show that the best fit was obtained for ℓ>0. However, 1 theminimumofthemeritfunctionisnotverydeepand cannotexclude ℓ=1and ℓ=2. Itcan, however,,exclude 0 1 2 3 4 l ℓ >2. This does not help to achieve the main goal of χ2 0.50 0.94 1.72 9.30 89.80 unique mode identification, but is consistent with previous work. In the case of Bal09, Baran et al. (2008) excluded any ℓ value if the χ2 function was at least 3 Fig. 3 Results of application of the method for mode de- greediscrimination. Theplot shows χ2 asafunction ofthe timeshigherthanthe smallestvalueofthisfunction. If ℓ parameter. In the bottom of the figure the precise values we follow this rule we can also exclude the ℓ 2 value. ≥ of χ2 for each ℓ are given On the other hand, we can use the probability Q that the merit function for the correct model will exceed a givenvaluebychance. TheQvaluesfortheℓ=0,1and 2 are 0.7, 0.4, 0.13, respectively. In this case none of the firstthree ℓ’s canbe excluded. Howeverthe chance that ℓ=0 is wrong is more than 5 times smaller than for ℓ=2. Acknowledgements This work was partially sup- portedbyPolishMNiSWgrantNo. 554/MOB/2009/0. RS wishes to thank A.DeBlasi for having contributed to the Loiano observations. R.O. is supported by the Research Council of Leuven University, under grant GOA/2003/04. ModedegreeinQQVir 5 Table 2 Resultsoftheprewhiteningprocess. Onlyresultsforthemainmodeat7.2554776mHzarepresented. Phasesare given for epoch 2454552.0 filter Amplitude [mma] error [mma] phase [rad] error [rad] U 31.89 1.02 3.425 0.048 B 26.15 0.42 3.438 0.049 V 23.73 0.14 3.434 0.007 Radial Velocity [km/s] error [km/s] phase [rad] error [rad] RV 11.74 0.55 5.633 0.052 6 References Baran, A., Pigulski, A., O’Toole., S., 2008, Mon. Not. R. Astron. Soc., 385, 255 Charpinet, S., Silvotti, R., Bonanno, A., 2006, Astron. As- trophys., 459, 565 Charpinet, S., Fontaine, G., Brassard, P., 2008, Hot sub- dwarf stars and related objects ASP Conference Series, 392, 297 Silvotti, R.,Bonanno, A.,Bernabei, S.et al., 2006, Astron. Astrophys.,459, 557 Telting, J.H. & Østensen R.H., 2004, Astron. Astrophys., 419, 685 Telting, J.H. et al., 2010, this volume ThismanuscriptwaspreparedwiththeAASLATEXmacrosv5.2.