Pulsations of rapidly rotating stars with compositional discontinuities
aa r X i v : . [ a s t r o - ph . S R ] O c t Precision Asteroseismology. Celebration of the Scientific Opus ofWojtek DziembowskiProceedings IAU Symposium No. 301, 2013W. Chaplin, J. Guzik, G. Handler & A. Pigulski, eds. c (cid:13) Pulsations of rapidly rotating stars withcompositional discontinuities
Daniel R. Reese and Francisco Espinosa Lara , and MichelRieutord , Institut d’Astrophysique et G´eophysique de l’Universit´e de Li`ege ,All´ee du 6 Aoˆut 17, 4000 Li`ege, Belgiumemail: [email protected];be Universt´e de Toulouse, UPS-OMP, IRAP, Toulouse, France CNRS, IRAP, 14 avenue Edouard Belin, 31400 Toulouse, France
Abstract.
Recent observations of rapidly rotating stars have revealed the presence of regularpatterns in their pulsation spectra. This has raised the question as to their physical origin, andin particular, whether they can be explained by an asymptotic frequency formula for low-degreeacoustic modes, as recently discovered through numerical calculations and theoretical consider-ations. In this context, a key question is whether compositional/density gradients can adverselyaffect such patterns to the point of hindering their identification. To answer this question, wecalculate frequency spectra using two-dimensional ESTER stellar models. These models use amulti-domain spectral approach, allowing us to easily insert a compositional discontinuity whileretaining a high numerical accuracy. We analyse the effects of such discontinuities on both thefrequencies and eigenfunctions of pulsation modes in the asymptotic regime. We find that al-though there is more scatter around the asymptotic frequency formula, the semi-large frequencyseparation can still be clearly identified in a spectrum of low-degree acoustic modes.
Keywords. stars: oscillations, stars: rotation, stars: interiors
1. Introduction
Recent observations of pulsation spectra in rapidly rotating stars have revealed thepresence of frequency patterns. For instance, Garc´ıa Hern´andez et al. (2009, 2013) foundrecurrent frequency spacings in two δ Scutis observed by CoRoT, thereby allowing theconstruction of an echelle diagram in the latter case. Similarly, Breger et al. (2012, 2013)found multiple sequences of very uniformly spaced frequencies in a δ Scuti observed byKepler. These observations show that although the pulsation spectra of δ Scuti stars lackthe simple frequency patterns present in solar-type pulsators, regular patterns do existin such stars and need to be explained.Among the various possible explanations, one particularly interesting option is theasymptotic frequency pattern for low-degree acoustic modes ( i.e. island modes) in rapidlyrotating stars, recently discovered through numerical (Ligni`eres et al. 2006; Reese et al.2008, 2009) and theoretical considerations (Ligni`eres & Georgeot 2008, 2009; Pasek et al.2011, 2012). Identifying such a pattern in rapidly rotating stars could yield useful in-formation such as the mean density (Reese et al. 2008; Garc´ıa Hern´andez et al. 2013).However, an open question is up to what extent it is affected by strong gradients orglitches (such as µ gradients, ionisation zones, or boundaries of convective regions), andwhether this can hinder its identification. In order to answer this question, we investigatethe pulsation spectra of rapidly rotating models with sharp discontinuities.1 D. R. Reese, F. Espinosa Lara & M. Rieutord
2. Numerical calculations
We worked with various 3 M ⊙ models, where the surface is rotating at 70 % of theKeplerian break-up rotation rate ( v eq = 340 −
350 km/s). These models were produced bythe 2D multi-domain spectral code ESTER, which self-consistently calculates the rotationprofile, Ω (Rieutord & Espinosa Lara 2009, 2013; Espinosa Lara & Rieutord 2013). Itsmulti-domain approach is well-suited to introducing discontinuities without sacrificingnumerical accuracy, since these can be made to coincide with domain boundaries. Inwhat follows, we worked with five different models: M which is smooth, M6 50 , M6 10 , M7 50 , and
M7 10 . Models
M6 xx have a discontinuity deeper within the star (see Fig. 1,left panel). In all cases, the discontinuities follow isobars. The surface hydrogen contentis decreased by 50% and 90% in models
Md 50 and
Md 10 , and corresponds to a 17% and39% jump in the speed of sound, respectively (see Fig. 1, right panel).
Figure 1.
Left: meridional cross-sections of discontinuities in models
M6 xx (dashed line) and
M7 xx (dotted line), and stellar surface (solid line).
Right: density and sound velocity profiles forthree of the models.
Adiabatic calculations of acoustic pulsation modes were carried out thanks to the TOPcode which fully takes into account the effects of rotation (Reese et al. 2009). Regularityconditions were applied in the centre, the simple mechanical condition δp = 0 was en-forced at the stellar surface, and the perturbation to the gravity potential was made tovanish at infinity. Various matching conditions were needed to ensure that the pertur-bation of the pressure, the gravity potential, and its gradient, remain continuous acrossthe perturbed discontinuity. Furthermore, the fluid domain had to be kept continuousby making sure that the deformation caused by the fluid displacement is the same be-low and above the discontinuity. Similar calculations had previously been carried outin Reese et al. (2011). However, these calculations did not take into account the factthat the matching conditions apply across the perturbed discontinuity, and the resultswere less conclusive because the discontinuity was located deeper within the star, whereacoustic island modes are less sensitive.
3. Results
We first turn our attention to the effects of discontinuities on the eigenfunctions. Fig-ure 2 shows the meridional cross-section of an island mode as well as the sound velocityand mode profile along a heuristically determined path. As can be seen in the rightpanel, the discontinuity modifies the wavelength as well as the amplitude of the oscilla-tions. Further tests confirm that the wavelength scales with the sound velocity. Anothereffect which has already been pointed out in Reese et al. (2011) is a slight deviation ofthe mode at the discontinuity. ulsations of rapidly rotating stars with compositional discontinuities Figure 2.
Left: meridional cross-section of an island mode.
Right: sound velocity and modeprofile along the path shown in the left panel (only part of the profile is shown for legibility).
At low rotation rates, discontinuities affect the frequencies by superimposing an oscil-latory pattern over the usual frequency spectrum ( e.g.
Monteiro et al. 1994). A similareffect takes place here, as illustrated by the semi-large frequency separations shown inFig. 3, although the oscillatory pattern is less regular. One can also calculate the scatterbetween the numerical frequencies and a simplified version of the asymptotic formula (seeEq. (27) of Reese et al. 2009). The scatter, (cid:10) ( ν asymp . − ν ) (cid:11) / / ∆ ˜ n , ranges from 0.0143for model M to 0.0436 for model M7 10 . Even in the best case, the scatter is more thanan order of magnitude larger than the scatter obtained around the main sequence ofequidistant frequencies found in Breger et al. (2012), thereby supporting the conclusionthat this sequence is not caused by an asymptotic behaviour.
Figure 3.
Semi-large frequency separation, ∆ ˜ n = ν ˜ n +1 − ν ˜ n , for axisymmetric ( m = 0) modes,as a function of ˜ n , the spheroidal radial order (see Fig. 2, left panel). One can then investigate whether it is possible to recover the semi-large frequencyseparation, ∆ ˜ n . Figure 4 shows histograms of frequencies differences for three models.The lightly shaded areas show all frequency differences, whereas the dark areas showthe frequency differences from modes with adjacent ˜ n values and the same (˜ ℓ, m ) values.The upper row is based on the original numerical frequencies. In all cases, the semi-largefrequency, ∆ ˜ n separation shows up clearly. However, it turns out that the rotation rateis close to ∆ ˜ n thereby amplifying the signal, due to island mode multiplets (Pasek et al.2012). This can be seen by comparing the light and dark regions in the histograms. Inthe lower row, the frequencies were shifted by 0 . m , thereby mimicking a lower rotationrate. Even in this situation, a peak remains at ∆ ˜ n for all three models.In conclusion, although discontinuities lead to more scatter around the asymptotic D. R. Reese, F. Espinosa Lara & M. Rieutordbehaviour of island modes and may complicate mode identification, they are unable tomask features such as the semi-large frequency separation. Hence, the asymptotic formularemains a viable explanation for stars such as those observed by Garc´ıa Hern´andez et al.(2009, 2013). Figure 4.
Histograms of frequency differences for three models (see text for details).
Acknowledgements
DRR is financially supported through a postdoctoral fellowship from the “Subsidef´ed´eral pour la recherche 2012”, University of Li`ege. FEL and MR acknowledge thesupport of the French Agence Nationale de la Recherche (ANR), under grant ESTER(ANR-09-BLAN-0140).