Understanding the dynamical structure of pulsating stars: The BaadeWesselink projection factor of the δ Scuti stars AI Velorum and β Cassiopeiae^{⋆}
^{1} Laboratoire Lagrange, UMR 7293, UNS/CNRS/OCA, BP 4229, 06304 Nice Cedex 4, France
email: guillaume.guiglion@oca.eu
^{2} Institut de Recherche en Astrophysique et Planétologie, UMR 5277, 57 avenue d’Azereix, 65000 Tarbes, France
^{3} INAF – Osservatorio Astronomico di Brera, via E. Bianchi 46, 23807 Merate, Italy
^{4} Institute of Astronomy of the Russian Academy of Sciences, 48 Pjatnitskaya Str., 109017 Moscow, Russia
^{5} Departamento de Astronomía, Universidad de Concepción, Casilla 160C, CL Concepción, Chile
Received: 23 November 2012
Accepted: 2 January 2013
Aims. The BaadeWesselink method of distance determination is based on the oscillations of pulsating stars. The key parameter of this method is the projection factor used to convert the radial velocity into the pulsation velocity. Our analysis was aimed at deriving for the first time the projection factor of δ Scuti stars, using highresolution spectra of the highamplitude pulsator AI Vel and of the fast rotator β Cas.
Methods. The geometric component of the projection factor (i.e. p_{0}) was calculated using a limbdarkening model of the intensity distribution for AI Vel, and a fastrotator model for β Cas. Then, using SOPHIE/OHP data for β Cas and HARPS/ESO data for AI Vel, we compared the radial velocity curves of several spectral lines forming at different levels in the atmosphere and derived the velocity gradient associated to the spectrallineforming regions in the atmosphere of the star. This velocity gradient was used to derive a dynamical projection factor p.
Results. We find a flat velocity gradient for both stars and finally p = p_{0} = 1.44 for AI Vel and p = p_{0} = 1.41 for β Cas. By comparing Cepheids and δ Scuti stars, these results bring valuable insights into the dynamical structure of pulsating star atmospheres. They suggest that the periodprojection factor relation derived for Cepheids is also applicable to δ Scuti stars pulsating in a dominant radial mode.
Key words: stars: oscillations / stars: atmospheres / stars: variables:δScuti / techniques: spectroscopic
© ESO, 2013
1. Introduction
Determining distances in the Universe is not a trivial task. From our Galaxy to the Virgo Cluster, distances can be derived using the periodluminosity relation (PL) of Cepheids (Riess et al. 2009a,b). However, this relation has to be calibrated, using the BaadeWesselink method of distance determination for instance (Storm et al. 2011a,b). The principle of this method is simple: after determining the angular diameter and the linear radius variations of the star, the distance is derived by a simple ratio. Angular diameter variations can be measured using interferometry (Kervella et al. 2004) or the infrared surface brightness relation (Gieren et al. 1998, 2005). The linear radius variation is measured by integrating the pulsation velocity (hereafter V_{puls}) over one pulsating cycle. However, from observations we have only access to the radial velocity (V_{rad}) because of the projection along the lineofsight. The projection factor, used to convert the radial velocity into the pulsation velocity, is defined by p = V_{puls}/V_{rad}. There are in principle three subconcepts involved in the BaadeWesselink projection factor: (1) the geometric projection factor p_{0}, which is directly related to the limbdarkening of the star (see Sect. 3), (2) the correction f_{grad} due to the velocity gradient between the spectrallineforming region and the photosphere of the star; this quantity can be derived directly from observations by comparing different lines forming at different levels in the atmosphere (see Sects. 2 and 4), and (3) the correction f_{o − g} due to the relative motion between the optical and gas layers associated to the photosphere (see Sect. 4). For a detailed analysis of the pfactor decomposition we refer to Nardetto et al. (2007). The projection factor is then defined by p = p_{0}f_{grad}f_{o − g}. In the following, we apply this decomposition of the projection factor (originally developed for Cepheids) to the δ Scuti stars AI Vel and β Cas. The impact of nonradial modes of δ Scuti stars on the projection factor is a very difficult question studied previously (Dziembowski 1977; Balona & Stobie 1979; Stamford & Watson 1981; Hatzes 1996). The particular cases of AI Vel and β Cas are discussed in the conclusion. This paper is part of the international Araucaria Project, whose purpose is to provide an improved local calibration of the extragalactic distance scales out to distances of a few megaparsecs (Gieren et al. 2005). In this context, δ Scuti stars are extremely interesting since it has been shown recently that they follow a PL relation (McNamara et al. 2007; Poretti et al. 2008).
2. Spectroscopic observations of δ Scuti stars
AI Vel (HD 69213, A9 IV/V) is one of the most often observed doublemode, highamplitude δ Scuti stars. This star pulsates in the fundamental and first overtone radial modes with a wellconstrained period ratio P_{1}/P_{0} of 0.77 (Poretti et al. 2005). In addition to P_{0} = 0.111574 d and P_{1} = 0.0862073 d, Walraven et al. (1992) clearly detected two other periods, tentatively identified as the third and fifth radial overtones. We observed AI Vel using the HARPS spectrograph mounted at the ESO 3.6m telescope. We analysed 26 high signaltonoise ratio (S/N ≃ 140) spectra taken in the highefficiency mode (EGGS, R = 80 000) on the night of January 9 − 10, 2011. We identified 53 metallic unblended spectral lines (ranging from 3780 to 6910 Å) relevant for the determination of radial velocities. Figure 1 (left panel) shows the behaviour of the mean spectral line profile along the pulsation phase. The shifts due to the radial modes clearly dominate the line profile variations.
Fig. 1
Mean profiles of the HARPS spectra of AI Vel (left panel, T_{0} = JD 2 443 176.00) and of the SOPHIE spectra of β Cas (right panel, T_{0} = JD 2 438 911.88). The pulsation phase is given on the yaxis. A strong broadening is clearly seen for β Cas due to its high rotation velocity. We computed the mean line profiles of AI Vel and β Cas spectra by means of a deconvolution process using the LSD software (Donati et al. 1997). 

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The target β Cas (HD 432, F2 III/IV) is a lowamplitude δ Scuti star. We observed β Cas with the SOPHIE instrument (R = 75 000) at the OHP 1.93m telescope on the night of September 30, 2011. We collected 241 highresolution spectra with a mean S/N of 100. We could distinguish only height unblended spectral lines relevant for the spectral analysis because of the strong rotational broadening (Fig. 1, right panel). Riboni et al. (1994) showed that the star is a monoperiodic pulsator at the detection limit of groundbased photometric measurements, with a pulsation period of P = 0.101036676 d. The mode identification is unclear (Rodriguez et al. 1992; Riboni et al. 1994). Today, the distance of β Cas is known to be 16.8 pc from the Hipparcos parallax (van Leeuwen 2007). Thus we could obtain the absolute magnitude M_{V} = 1.14 from the apparent magnitude V = 2.27. The PL relations (McNamara et al. 2007; Poretti et al. 2008) supply a fundamental radial period of about 0.15 d at this M_{V} value. Therefore, the observed period is similar to that expected for the second radial overtone. We attempted a mode identification from our spectroscopic data using the FAMIAS^{1} software. Since we dealt with a fast rotator, we used the Fourier parameter fit method (Zima 2006). Imposing the frequency 1/P, the results from spectroscopy point towards an axisymmetric mode, without a clear indication on the ℓvalue. Since β Cas is seen almost poleon (i = 19.9 ± 1.9°, Che et al. 2011), a lowdegree, axisymmetric, nonradial mode mimics the pulsation behaviour of a radial mode very well. On the basis of these considerations, we treated β Cas as a radial monoperiodic pulsator for our purposes. We also used the mean line profiles of β Cas to estimate the vsini values from the position of the first zero of their Fourier transforms (Carroll 1933). This approach is possible only for objects where the rotational broadening is dominant with respect to the other sources of line broadening (e.g., instrumental effect, microturbulence), which is always the case for β Cas, but not for AI Vel, where we were unable to use the Fourier transform method on the narrower lines (vsini < 10 km s^{1}). The radial velocity values of the observed profiles of β Cas range from 5.3 to 11.6 km s^{1} and the vsini values from 74.0 to 77.5 km s^{1} (Fig. 1, right panel). We could also determine a mean vsini value from the average of all the mean profiles and obtained 75.72 ± 0.14 km s^{1}. This value is consistent with literature values (Bernacca & Perinotto 1970; Uesugi & Fukuda 1970; Schröder et al. 2009). When considering i = 19.9°, this means that β Cas is an intrinsic fast rotator, with a velociy of v_{rot} ≃ 220 km s^{1} which is consistent with Che et al. (2011).
Finally, for both stars, the centroid radial velocity RV_{c} (or the firstmoment radial velocity) and the line depth D are derived as a function of the pulsation phase for each selected spectral line. These data are used in Sect. 4.
3. The geometric projection factor p_{0}
Considering a limbdarkened pulsating star in rotation with a onelayer atmosphere, the projection factor is purely geometric. Thus, . The radial velocity is then defined by (1)where D_{R} is the surface of the stellar disc of radius R, and I(x,y,λ) the limbdarkened continuum intensity distribution considered at the wavelength of observation λ defined by , where u_{λ} is the linear limbdarkening coefficient from Claret & Bloemen (2011). Considering T_{eff} = 7400 K and log g = 3.5, we find u_{R} = 0.474 ± 0.025 in the Rband from Claret & Bloemen (2011). Using Eq. (1), we deduce a value of the geometric projection factor for AI Vel of p_{0} = 1.43 ± 0.01. p_{0} is assumed to be constant with the pulsation phase (Nardetto et al. 2004). This value is higher than what we generally obtain for Cepheids (typically 1.37 to 1.41, see Fig. 2 (top panel)).
The geometrical shape of β Cas and its intensity distribution are distorted by its high rotation rate. The geometric projection factor depends on the inclination of the star’s rotation axis compared to the lineofsight. If the star’s rotation axis is along the lineofsight (i = 0°), the star is observed poleon and is seen as a disc. For i > 0° the star has an ellipsoidal shape.
Fig. 2
Top: p_{0} as a function of the limbdarkening parameter u_{λ}. The red box indicates the uncertainty on p_{0} for the δ Scuti AI Vel. The blue box indicates the typical values of u_{λ} and p_{0} for Cepheids. The dots corresponds to the relation provided by Nardetto et al. (2006). Bottom: p_{0} as a function of the inclination of the fast rotating star β Cas for three different wavelengths (λ = 6000 Å (□), λ = 6500 Å ( △ ), and λ = 7000 Å ( × )). The red box indicates the uncertainty on p_{0} for β Cas. The case of an uniform elonged disc is overplotted ( + ), and we find that p_{0} = 1.5 for i = 0°, as expected for a circular uniform disc. 

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Fig. 3
Modelled intensity distributions of β Cas at 6000 Å for i = 5° (left), i = 50° (middle), i = 90° (right). 

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Using the fundamental parameters of the modified von Ziepel model found by Che et al. (2011) and the rotating stars model by Domiciano de Souza et al. (2002, 2012), we derived the intensity distribution in the continuum for different inclinations of the star (from i = 0° to i = 90° with a step of 5°) and for three wavelengths: λ = 6000,6500 and 7000 Å. Using these intensity maps, we can easily calculate the geometric projection factor. Indeed, for an ellipsoid, V_{rad} is then defined by (2)where a and b are the semimajor and semiminor axis of the ellipse. In Fig. 2 (bottom panel), we show the geometric projection factor (p_{0}) as a function of i. Fig. 3 presents the modelled intensity distributions for several inclinations. This relation is extremely interesting because it shows that the inclination of a fastrotating star can have an impact of more than 10% on the projection factor. Of course, it also depends on the rotation velocity of the star: the higher the rotation velocity (for a given inclination), the lower the projection factor. Using the inclination found by Che et al. (2011), i = 19.9 ± 1.9°, we finally find a geometric projection factor for β Cas of p_{0} = 1.41 ± 0.02 (averaged over the three wavelengths considered).
4. Dynamical structure of the atmosphere
By comparing the 2Kamplitude (defined as the amplitude of the RV_{c} curve, hereafter ΔRV_{c}) with the depth (D) of the 53 spectral lines selected in the case of AI Vel, one can directly measure the atmospheric velocity gradient in the part of the atmosphere where the spectral lines are formed (Fig. 4, topleft). To quantify the impact of velocity gradient on the projection (f_{grad}), we do not need to derive the velocity gradient over the whole atmophere, but only at the location of the forming regions of the spectral lines used to derive the distance of the star. We therefore performed a linear regression according to the relation ΔRV_{c} = a_{0} D + b_{0}. We obtained ΔRV_{c} = [−0.40 ± 0.53] D + [32.87 ± 0.23] km s^{1} (Fig. 4, topright). In principle, f_{grad} depends on the spectral line considered (Nardetto et al. 2007): f_{grad} = b_{0}/(a_{0}D + b_{0}). Here, we find that f_{grad} is typically the same for all spectral lines (f_{grad} = 1.01 ± 0.01), which is consistent with no correction of the projection factor due to the velocity gradient. The uncertainty on f_{grad} is derived from the errors on a_{0} and b_{0}.
Fig. 4
Top: RV_{c} as a function of the MJD in the case of the FeII 5234.625 Å spectral line, and amplitude of the RV_{c} curves as a function of the spectral line depth (D) for AI Vel. Typical error bars are indicated in the upper right of each panel. Bottom: the same for β Cas f the FeII 4508.288 Å spectral line. 

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Figure 4 (bottom left) presents the interpoled RV_{c} curve of β Cas for the FeI spectral line (λ = 4508.288 Å). We clearly see an increase of the amplitude of the radial velocity curve (~4.6 ± 0.9% per cycle). Moreover, the radial velocity curves have several minima and maxima and we can easily deduce a period of pulsation. We find P = 0.10046 ± 0.00054 d. Our value agrees well with that of Riboni et al. (1994).
In Fig. 4 (bottom right), ΔRV_{c} is plotted as a function of D for cycle 2 (see the vertical line in the figure). For β Cas, the range of the spectral line depth is eight times lower compared to AI Vel. The velocity gradient is f_{grad} = 0.64 ± 0.82. The large error bar prevents any physical discussion about the exact value of f_{grad}. We note that f_{grad} = 0.64 implies a huge velocity gradient in the star’s atmosphere, which seems unrealistic. Since it is consistent with our determination, we assumed f_{grad} = 1.0. Considering a_{0} = 0, we obtained b_{0} = 4.07 ± 0.25 with a reduced χ^{2} of 1.0, which provides an uncertainty on f_{grad} of 0.14. The f_{o  g} correction, which is the last component of the projection factor decomposition, cannot be measured from observations. To estimate the differential velocity between the optical and gas layers at the photosphere of the star, we need a hydrodynamic model.
However, modelling the pulsating atmosphere of δ Scuti stars is not an easy task because of (1) cycletocycle variations (nonradial modes) and (2) fast rotation in some cases (as for β Cas). However, f_{o  g} have been studied intensively in the Cepheids (Nardetto et al. 2004, 2007, 2011), and it seems that there is a linear relation between f_{o  g} and log P: f_{o  g} = [ − 0.023 ± 0.005] log P + [0.979 ± 0.005] . Moreover, we have a theoretical value of f_{o  g} for the shortperiod βCephei α Lup (P = 0.2598 d), f_{o  g} = 0.99 ± 0.01, which seems to be consistent with the log P–f_{o  g} relation of Cepheids (Nardetto et al. 2012). For our study, we propose to extend this law for the δ Scuti β Cas and AI Vel. Considering P = 0.11157 d for AI Vel and P = 0.10046 ± 0.00054 d for β Cas (from this paper), we find f_{o  g} = 1.00 ± 0.02 (0.02 is a conservative arbitrary uncertainty) for both stars, which basically means no photospheric correction to the projection factors.
5. Discussion
We can now calculate the projection factor p, using the relation p = p_{0}f_{grad}f_{o  g}. We find p = 1.44 ± 0.05 for AI Vel and p = 1.41 ± 0.25 for β Cas. However, the generalisation of this study to any δ Scuti stars is presently limited since we have to study the impact of multimodes, in particular nonradial ones, on the projection factor. This complicated effect has been studied by several authors for the bisector method of the radial velocity determination (Dziembowski 1977; Balona & Stobie 1979; Stamford & Watson 1981; Hatzes 1996). We assume a star pulsating in two modes, one radial and one nonradial. The projection factor can be defined as p = α_{r}p_{r} + α_{nr}p_{nr}, with α_{r} and α_{nr} the relative contributions of the velocity amplitudes of the radial and nonradial modes to the pulsation (α_{r} + α_{nr} = 1). p_{r} is our previous decomposition of the projection factor in the case of a purely radial mode p_{r} = p_{0}f_{grad}f_{o  g}, while p_{nr} is the projection factor in the case of a purely nonradial mode. Using Eq. (4) of Hatzes (1996), we find that p_{nr} = p_{0}e^{ + kℓ2}, where k = 0.15 in the case of a non radial pmode and k = 1.2 in the case of a nonradial gmode. ℓ is the spherical harmonic degree. This relation is derived for the first moment (i.e. the radial velocity) determination only, which is independent of the star’s rotation. This equation can be used when ℓ = m only (where m is the spherical harmonic order). Additional work is necessary to derive it when ℓ ≠ m. We emphasize also that the higher ℓ( = m), the higher is the nonradial projection factor. This is expected since for high values of ℓ (and this is qualitatively also true when ℓ ≠ m) there are more red and blueshifted velocity zones on the star that cancel each other in the integrated line profile, which leads to a lower amplitude of the nonradial velocity curve, and in turn a high value of the nonradial projection factor (see Eqs. (1) and (2) of Hatzes 1996).
We performed our study on AI Vel and β Cas under the assumption of monoperiodic radial pulsation. Our results (without nonradial correction) are consistent (at the 1σ level) with the periodprojectionfactor (Pp) relation p = [ − 0.071 ± 0.020] log P + [1.311 ± 0.019] by Laney & Joner (2009) applied for classical and dwarf Cepheids (it corresponds to p = 1.38 ± 0.02 for AI Vel and β Cas). To derive these values, Laney & Joner simply compared the distance of the stars obtained from the PL relation with the distances from the photometric version of the BaadeWesselink method. This suggests that the eventual nonradial components of β Cas have probably a negligeable impact on the projection factors (which means α_{nr} ≃ 0). Interestingly, if we use the Pp relation obtained for classical Cepheids by Nardetto et al. (2007), p = [ − 0.064 ± 0.020] log P + [1.376 ± 0.023] (derived with the first moment method^{2}) to derive the projection factors of the two δ Scuti stars, we find p = 1.44 ± 0.01, which is consistent with our values. This seems to show that the Pp relation provided by Nardetto et al. (2007) for single lines is also applicable to δ Scuti stars pulsating in a dominant radial mode. In addition, and as already shown, for fastrotating δ Scuti stars, an intrinsic dispersion of the Pp relation due to the random orientation of the rotation axis has to be considered.
Developed in the framework of the FP6 European Coordination Action HELAS (http://www.helaseu.org/)
We remind that using the crosscorrelation method, one has to use the Pp relation from Nardetto et al. (2009).
Acknowledgments
G.G. and N.N. thank J. Monnier for useful discussions. W.G. is grateful for support from the BASAL Centro de Astrofisica y Tecnologias Afines (CATA) PFP06/2007. E.P. and M.R. acknowledge financial support from the Italian PRININAF 2010 Asteroseismology: looking inside the stars with space and groundbased observations.
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All Figures
Fig. 1
Mean profiles of the HARPS spectra of AI Vel (left panel, T_{0} = JD 2 443 176.00) and of the SOPHIE spectra of β Cas (right panel, T_{0} = JD 2 438 911.88). The pulsation phase is given on the yaxis. A strong broadening is clearly seen for β Cas due to its high rotation velocity. We computed the mean line profiles of AI Vel and β Cas spectra by means of a deconvolution process using the LSD software (Donati et al. 1997). 

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In the text 
Fig. 2
Top: p_{0} as a function of the limbdarkening parameter u_{λ}. The red box indicates the uncertainty on p_{0} for the δ Scuti AI Vel. The blue box indicates the typical values of u_{λ} and p_{0} for Cepheids. The dots corresponds to the relation provided by Nardetto et al. (2006). Bottom: p_{0} as a function of the inclination of the fast rotating star β Cas for three different wavelengths (λ = 6000 Å (□), λ = 6500 Å ( △ ), and λ = 7000 Å ( × )). The red box indicates the uncertainty on p_{0} for β Cas. The case of an uniform elonged disc is overplotted ( + ), and we find that p_{0} = 1.5 for i = 0°, as expected for a circular uniform disc. 

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In the text 
Fig. 3
Modelled intensity distributions of β Cas at 6000 Å for i = 5° (left), i = 50° (middle), i = 90° (right). 

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In the text 
Fig. 4
Top: RV_{c} as a function of the MJD in the case of the FeII 5234.625 Å spectral line, and amplitude of the RV_{c} curves as a function of the spectral line depth (D) for AI Vel. Typical error bars are indicated in the upper right of each panel. Bottom: the same for β Cas f the FeII 4508.288 Å spectral line. 

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In the text 