Spectroscopic Mode Identification in Slowly Pulsating Subdwarf-B Stars
aa r X i v : . [ a s t r o - ph ] A p r **FULL TITLE**ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION****NAMES OF EDITORS** Spectroscopic Mode Identification in Slowly PulsatingSubdwarf-B Stars
Caroline Schoenaers and Tony Lynas-Gray
Department of Physics, University of Oxford, Denys WilkinsonBuilding, Keble Road, Oxford OX1 3RH, United Kingdom
Abstract.
Mode identification is crucial for an asteroseismological study ofany significance. Contrarily to spectroscopic techniques, methods such as period-fitting and multi-colour photometry do not provide a full reconstruction of non-radial pulsations. We present a new method of spectroscopic mode identificationand test it on time-series of synthetic spectra appropriate for pulsating subdwarf-B stars. We then apply it to the newly discovered slowly pulsating subdwarf-Bstar HD 4539.
1. Introduction
Subdwarf B (sdB) stars are low-mass ( ∼ . M ⊙ ) core helium burning objectswith thin and mostly inert hydrogen-rich residual envelopes. They belong tothe Extreme Horizontal Branch (EHB) (Heber 1986) and remain hot (20000 ≤ T eff ≤ ≤ log g ≤
7) throughout their lifetime (Saffer et al.1994), before evolving towards the white dwarf cooling sequence without expe-riencing the Asymptotic Giant Branch and Planetary Nebula phases of stellarevolution. Han et al. (2002, 2003) use binary population synthesis calculationsto demonstrate the formation of sdB stars through several possible channels, butresulting models require further comparison with observation. Kilkenny et al.(1997) and Green et al. (2003) respectively discovered that some sdB stars un-dergo fast and slow non-radial pulsations. This means asteroseismology can beused to study the internal structure of these stars and so constrain evolutionmodels.After having measuring the pulsation periods, the first step of any seismicmodelling is to identify the pulsation modes that are excited , which is far frombeing trivial. In the case of pulsating sdB stars, most mode identifications upto date have been performed using either period fitting (Brassard et al. 2001;Charpinet et al. 2005; Randall et al. 2006a,b, etc. ) or multicolour amplituderatios (Jeffery et al. 2004, 2006, etc. ). However, these methods do not provide afull reconstruction of the pulsations, and should therefore be complemented byspectroscopic mode identification techniques, such as the one we present here. Since non-radial pulsations are modelled by spherical harmonics Y nℓm , mode identificationconsists in assigning values to the spherical wavenumbers n , ℓ and m , respectively the numberof nodes of the radial displacement, the number of nodal lines on the stellar surface and thenumber of such lines passing through the rotation axis of the star. Schoenaers and Lynas-Gray
2. Spectroscopic Mode Identification
As reported earlier (Schoenaers & Lynas-Gray 2006), line-profile variations (lpv),although weak, are to be expected in pulsating subdwarf-B stars. Furthermore,they have been observed, for instance in PG 1325+101 by Telting & Østensen(2004). A reliable spectroscopic mode identification method is therefore neededto take advantage of these and similar observations.
Nowadays, the most commonly used spectroscopic mode identification methodis the “moment method” (hereafter MM). It was first introduced by Balona(1986a,b, 1987) and later reformulated into computer-friendly terms and ap-plied to Main Sequence pulsators by Aerts et al. (1992) and Aerts (1996) formonoperiodic stars, while Briquet & Aerts (2003) extended the formalism tomultiperiodic stars. In this method, one does not try to identify pulsation modesdirectly from the line-profiles, because this is often impossible for multiperiodicstars, but instead replaces each observed line-profile by its first few momentsand compares their time dependence with that of theoretical moments. Thelatter depending on the wavenumbers ℓ and m of the pulsation mode, as wellas on the pulsation amplitude and on the inclination angle i of the star, thismethod should in principle allow a complete reconstruction of any non-radialpulsation. It has been used successfully for the study of β Cephei and SlowlyPulsating B (SPB) stars (Briquet & Aerts 2003; De Cat et al. 2005). However,this method makes use of a crucial (and not always justified) approximation:except for radial velocity variations, the MM neglects all other variations due tonon-radial pulsations.However, because of non-radial pulsations, different points on the stellar sur-face not only have different radial velocities, but also different temperatures, log g and orientations, and their contributions to a line-profile therefore have differentamplitudes. Hence, lpv do not only come from the Doppler shift in wavelength,but also from the complex temperature and log g behavior on the stellar surface.This is especially the case in sdB pulsators: because they are so dense, any signif-icant displacement of the stellar surface due to pulsation is kept at its minimum,and instead the available energy causes temperature perturbations in the stellarphotosphere. To demonstrate that neglecting temperature, gravity and geomet-ric variations can lead to a mistaken mode identification, even in the simplestcases, we used the MM to identify various monoperiodic pulsations present intime-series of synthetic spectra computed following Schoenaers & Lynas-Gray(2006). On the one hand, it could be seen that the MM always provide, withinits five best solutions, the proper mode identification (as well as very good es-timates of i , A p and v eq ) when only radial velocity variations were taken intoaccount. However, on the other hand, when temperature, gravity and geometricvariations were taken into account (as indeed they should be) this test demon-strated that even in the simple case of monoperiodic pulsations, the current MMfalls short of properly identifying all pulsational characteristics. Clearly, in thecase of pulsating sdB stars, where temperature, pressure and geometric effects Following Briquet & Aerts (2003)’s advice. pectroscopic Mode ID in Slowly Pulsating sdB Stars
Schoenaers & Lynas-Gray (2006) introduced a state-of-the-art code that com-putes times-series of synthetic spectra and lpv. The “synthetic moment” method(hereafter SMM) introduced here takes advantage of this ability to compute“synthetic moments” for these extremely accurate synthetic lpv, and to com-pare them to observed moments: for a given target and observed line-profilevariation, a grid of time-series of synthetic spectra spanning a suitable parame-ter space is computed, the first three moments of each of these synthetic lpv areobtained, and the best fit between observed and synthetic moments is found byminimising a merit function of the form∆ ℓ ′ ,m ′ = ( N obs N obs X j =1 h ( < v > syn ,j − < v > obs ,j ) + | < v > syn ,j − < v > obs ,j | (1)+ ( < v > syn ,j − < v > obs ,j ) / i) / where N obs is the number of observations, odd moments have average zero, andthe synthetic second moment < v > syn ,i is adjusted to have the same averageas its observed counterpart < v > obs ,i .The choice of the parameter space to be searched is crucial: if too limitedone might miss the proper mode identification, but if too broad the computingtime of the synthetic lpv can become prohibitive. One could consider buildingan extensive grid of synthetic lpv by varying these parameters more or less “con-tinuously”, but because most parameters are target-dependent, it is simply moreefficient to compute one grid for each pulsating star. Provided spectroscopic ob-servations of the target are obtained at high enough SNR, T eff , log g , v eq sin i and[X / H] can be determined to good accuracy. Before attempting mode identifica-tion, one should also obtain reliable values of the pulsation period(s) P , eitherthrough (ideally simultaneous) photometry or radial velocity measurements. Ifthe latter are obtained (either directly through spectroscopic measurements orindirectly from the first moment), then good estimates of the physical velocityamplitudes of the pulsation mode(s) can be obtained; if not, then pulsation am-plitudes can in principle remain a free parameter, along with the inclination i ′ and the actual mode identification ( ℓ ′ , m ′ ).Ideally, when dealing with a multiperiodic star, all pulsation modes shouldbe identified simultaneously, but this would require a very large grid of syntheticlpv, which can become prohibitive. It is however possible to attempt modeidentification as follows:1. Use the SMM on the dominant pulsation mode with a suitable primary gridof synthetic lpv to obtain ( ℓ ′ , m ′ ) and i ′ for the five best fitting solutions(minimising ∆ ℓ ′ ,m ′ ). Then, if at all possible, further discriminate betweenthese five best solutions (see next subsection for more details). Schoenaers and Lynas-Gray
2. For each of the possible i ′ and ( ℓ ′ , m ′ ) combinations obtained in Step 1,build a secondary grid of synthetic lpv (one for each ( ℓ ′ , m ′ ) combination)and use the SMM to identify the second pulsation mode.3. Step 2 can be repeated at will until all modes have been succesfully iden-tified. As experience has shown in the case of Main Sequence pulsators (see for exam-ple De Cat et al. (2005)) it is in general safer not to accept straight away thebest solution of the MM (and by extension, of the SMM) as the proper modeidentification, but instead to discriminate between a few best solutions usingsome other mode identification method. Similarly, to confirm the identificationof pulsation modes in PG1716 stars, we suggest a visual inspection of the fitbetween observed and synthetic moments, as well as the use of the IPS (forIntensity Period Search) diagnostic (Schrijvers et al. 1997; Telting & Schrijvers1997a,b; Schrijvers & Telting 1999) to compare the amplitude and phase dia-grams of observed lpv with those of synthetic profiles corresponding to the fewbest solutions of the SMM.
Figure 1 shows in grey-scale a biperiodic lpv, representative of the PG1716 starPG 1338+481 ( T eff = 27 000 K, log g = 5 . M = 0 . M ⊙ , R = 0 . R ⊙ , v eq =5 km · s − , [He/H]= − .
3, [Mg/H]= − Figure 1. Grey-scale plots of synthetic lpv ( left : He I λ λ right : He I λ ℓ -values representative ofPG 1338+481 (Randall et al. 2006a): i = 60 ◦ , ( ℓ , m ) = (1 , P = 3828 s, A p , = 2 . · s − and ( ℓ , m ) = (1 , P = 3530 s, A p , = 1 . · s − . pectroscopic Mode ID in Slowly Pulsating sdB Stars R and ψ T were computed using Randall et al. (2005)’sEquations 46 and 47 Applying the SMM to these synthetic lpv of He I λ provides, for thefirst pulsation mode, the five best solutions listed in the first four columns ofTable 1. Visual inspection of the match between “observed” and synthetic mo- Table 1. Five best identifications obtained by the SMM for the biperiodicpulsation modelled above. The first column is the inclination of the rotationaxis with regard to the line-of-sight (in degrees), the second and third columnsgive the ( ℓ ′ , m ′ )-values of the first mode, and the fourth provides the valueof ∆ ℓ ′ ,m ′ , minimum for the best fitting solution. Columns 5 to 7 give the( ℓ ′ , m ′ )-values and the value of ∆ ℓ ′ ,m ′ for the second mode, provided the firstwas properly identified. For each mode, the proper identification is written inbold font. First mode Second mode i ′ ℓ ′ m ′ ∆ ℓ ′ ,m ′ ℓ ′ m ′ ∆ ℓ ′ ,m ′
30 2 1 5.1651505 1 − − − ℓ ′ , m ′ ) = (1 ,
1) and (3 ,
1) identificationsfor the first mode. Further discrimination can be achieved by comparing the“observed” IPS diagnostic (shown in Figure 2) to those of the ( ℓ ′ , m ′ ) = (1 , ℓ , m ) = (1 ,
1) and i = 60 ◦ fixed, and applying the SMM toidentify the second pulsation mode yields the five best identifications listedin Columns 5 to 7 of Table 1. The IPS diagnostic of the best identification( ℓ ′ , m ′ ) = (1 ,
0) plotted in Figure 4 matches the “observed” IPS diagnostic verywell, and this shows the SMM does provide a reliable mode identification, evenfor multiperiodic stars. Note that Equation 47 of Randall et al. (2005) should read h ψ T i = π − .
30 exp( − P ℓ . / Similar results are obtained when applying the SMM to He I λ λ Schoenaers and Lynas-Gray A m p lit ud e ( - ) Model 9 - F6676 6678 6680-1.0 P h a s e ( π ) A m p lit ud e ( - ) Model 9 - 2F6676 6678 6680-10 P h a s e ( π ) Figure 2. “Observed” IPS diagnostic for the lpv shown in Figure 1. Thetwo leftmost panels show the amplitude ( left ) and phase ( right ) of variationswith the main frequency across the profile, while in the two rightmost panelsthe amplitude ( left ) and phase ( right ) plots are for variations with the firstharmonics of the main frequency. A m p lit ud e ( - ) (1,1) - F6676 6678 6680-10 P h a s e ( π ) A m p lit ud e ( - ) (1,1) - 2F6676 6678 668001 P h a s e ( π ) A m p lit ud e ( - ) (3,1) - F6676 6678 668001 P h a s e ( π ) A m p lit ud e ( - ) (3,1) - 2F6676 6678 668012 P h a s e ( π ) Figure 3. Same as Figure 2, but for possible identifications of the firstpulsation mode: ( ℓ ′ , m ′ ) = (1 ,
1) ( top ) and (3 ,
1) ( bottom ). A m p lit ud e ( - ) (1,0) - F6676 6678 6680-10 P h a s e ( π ) A m p lit ud e ( - ) (1,0) - 2F6676 6678 6680-10 P h a s e ( π ) Figure 4. Same as Figure 2, but for possible identifications of the firstpulsation mode: ( ℓ ′ , m ′ ) = (1 ,
1) ( top ) and (3 ,
1) ( bottom ). pectroscopic Mode ID in Slowly Pulsating sdB Stars N o r m a li s e d f l ux H e I O II N II H e I M g II N o r m a li s e d f l ux H e I O II N II H e I M g II Figure 5. Line-profile variations in HD 4539 observed in August2005 ( two leftmost panels ) on MJD=53611.13918 ( solid line ) andMJD=53612.03206 ( dashed line ), and in August 2006 ( two rightmost panels )on MJD=53958.15420 ( solid line ) and MJD=53952.16671 ( dashed line ).
3. Application to a New Slowly Pulsating Subdwarf-B Star: HD 4539
HD 4539 ( B ∼ .
12, 2000.0 coordinates: α = 0h47m29 . , δ = +09 ◦ ′ ′′ ) isone of the brightest known sdB stars. Its surface parameters (see e.g. Bascheck et al.1972; Saffer et al. 1994) are similar to those of PG1716 stars, and Schoenaers & Lynas-Gray(2007) report the discovery of non-radial pulsations in HD 4539, together withtheir frequency analysis, but without mode identification and subsequent anal-ysis. In Figure 5, we plot lpv observed in HD 4539 in August 2005 and 2006.These lpv are much stronger than the average noise present in our spectra, indi-cating they are real. A careful radial velocity analysis and period search yieldedfour frequencies (Schoenaers & Lynas-Gray 2007), of which we identify the dom-inant one ( P = 9310 s) in this preliminary report. A more thorough analysiswill be presented elsewhere.The wavelength range of our spectra includes He I λ λ Table 2. Five best fitting SMM identifications of P in HD 4539, using He I λ left ) and Mg II λ right ). In each case, the first and secondcolumns give the tentative ( ℓ ′ , m ′ ) identification, the third column providesthe corresponding inclination, and the fourth is the merit function ∆ ℓ ′ ,m ′ . He I λ λ ℓ ′ m ′ i ′ ∆ ℓ ′ ,m ′ ℓ ′ m ′ i ′ ∆ ℓ ′ ,m ′ Schoenaers and Lynas-Gray are the same (although found in a different order) whether the identification isperformed using He I λ λ λ λ P = 9310 s)in HD 4539 can be identified as a ℓ = 1 , m = 1 mode, while the inclination ofthe rotation axis with respect to the line-of-sight is (close to) 85 ◦ . Acknowledgments.
The authors wish to thank the editors for allowinga very late submission. C.S. thanks St Cross College, the MPLS division ofthe University of Oxford, and the LOC for their generous contribution to myexpenses during the Third Meeting on Hot Subdwarf Stars.
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