Astronomy & Astrophysics manuscript no.

guiglion c ESO 2018

November 6, 2018

Understanding the dynamical structure of pulsating stars:

The Baade-Wesselink projection factor of the δ Scuti stars
AI Vel and β Cas ⋆
G. Guiglion1 , N. Nardetto1 , P. Mathias2 , A. Domiciano de Souza1 , E. Poretti3 , M. Rainer3 , A. Fokin4 , D.
Mourard1 , W. Gieren5

1
Laboratoire Lagrange, UMR 7293, UNS/CNRS/OCA BP 4229, 06304 Nice Cedex 4, France
2
Institut de Recherche en Astrophysique et Planétologie, UMR 5277, 57 avenue d’Azereix, 65000 Tarbes, France
arXiv:1301.2475v1 [astro-ph.SR] 11 Jan 2013

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., Moscow 109017, Russia
5
Departamento de Astronomı́a, Universidad de Concepción, Casilla 160-C, CL Concepción, Chile

Received 23 November 2012; accepted 2 January 2013

ABSTRACT

Aims. The Baade-Wesselink 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 high-resolution spectra of
the high-amplitude pulsator AI Vel and of the fast rotator β Cas.
Methods. The geometric component of the projection factor (i.e. p0 ) was calculated using a limb-darkening model of
the intensity distribution for AI Vel, and a fast-rotator 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 spectral-line-forming 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 = p0 = 1.44 for AI Vel and p = p0 = 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 period-projection factor relation derived for Cepheids is also applicable to
δ Scuti stars pulsating in a dominant radial mode.
Key words. Stars: pulsating – Stars: atmospheres – Stars: variables: δ Scuti – Techniques: spectroscopic

1. Introduction projection factor: (1) the geometric projection factor p0 ,

which is directly related to the limb-darkening of the star
Determining distances in the Universe is not a trivial task. (see Sect. 3), (2) the correction fgrad due to the velocity gra-
From our Galaxy to the Virgo Cluster, distances can be de- dient between the spectral-line-forming region and the pho-
rived using the period-luminosity relation (P L) of Cepheids tosphere of the star; this quantity can be derived directly
(Riess et al. 2009a,b). However, this relation has to be cal- from observations by comparing different lines forming at
ibrated, using the Baade-Wesselink method of distance de- different levels in the atmosphere (see Sect. 2 and 4), and
termination for instance (Storm et al. 2011a,b). The prin- (3) the correction fo−g due to the relative motion between
ciple of this method is simple: after determining the angu- the optical and gas layers associated to the photosphere
lar diameter and the linear radius variations of the star, (see Sect. 4). For a detailed analysis of the p-factor decom-
the distance is derived by a simple ratio. Angular di- position we refer to Nardetto et al. (2007). The projection
ameter variations can be measured using interferometry factor is then defined by p = p0 fgrad fo−g . In the follow-
(Kervella et al. 2004) or the infrared surface brightness re- ing, we apply this decomposition of the projection factor
lation (Gieren et al. 1998, 2005). The linear radius variation (originally developed for Cepheids) to the δ Scuti stars AI
is measured by integrating the pulsation velocity (hereafter Vel and β Cas. The impact of non-radial modes of δ Scuti
Vpuls ) over one pulsating cycle. However, from observations stars on the projection factor is a very difficult question
we have only access to the radial velocity (Vrad ) because of studied previously (Dziembowski 1977; Balona & Stobie
the projection along the line-of-sight. The projection fac- 1979; Stamford & Watson 1981; Hatzes 1996). The partic-
tor, used to convert the radial velocity into the pulsation ular cases of AI Vel and β Cas are discussed in the con-
velocity, is defined by p = Vpuls /Vrad . There are in prin- clusion. This paper is part of the international Araucaria
ciple three sub-concepts involved in the Baade-Wesselink Project, whose purpose is to provide an improved local cali-
⋆ bration of the extragalactic distance scales out to distances
This work uses observations made with the HARPS instru-
ment at the 3.6 m telescope (La Silla, Chile) in the framework of a few megaparsecs (Gieren et al. 2005). In this context, δ
of the LP185.D-0056 and with the SOPHIE instrument at OHP Scuti stars are extremely interesting since it has been shown
(France).

1
 G. Guiglion and collaborators: The Baade-Wesselink projection factor of the δ Scuti stars

recently that they follow a P L relation (McNamara et al. the star is a monoperiodic pulsator at the detection limit of
2007; Poretti et al. 2008). ground-based 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,
2. Spectroscopic observations of δ Scuti stars
the distance of β Cas is known to be 16.8 pc from the
AI Vel (HD 69213, A9 IV/V) is one of the most often Hipparcos parallax (van Leeuwen 2007). Thus we could ob-
observed double-mode, high-amplitude δ Scuti stars. This tain the absolute magnitude MV =1.14 from the apparent
star pulsates in the fundamental and first overtone radial magnitude V =2.27. The P L relations (McNamara et al.
modes with a well-constrained period ratio P1 /P0 of 0.77 2007; Poretti et al. 2008) supply a fundamental radial pe-
(Poretti et al. 2005). In addition to P0 = 0.111574 d and riod of about 0.15 d at this MV value. Therefore, the ob-
P1 = 0.0862073 d, Walraven et al. (1992) clearly detected served period is similar to that expected for the second
two other periods, tentatively identified as the third and radial overtone. We attempted a mode identification from
fifth radial overtones. We observed AI Vel using the HARPS our spectroscopic data using the FAMIAS1 software. Since
spectrograph mounted at the ESO 3.6-m telescope. We we dealt with a fast rotator, we used the Fourier parameter
analysed 26 high signal-to-noise ratio (S/N≃140) spectra fit method (Zima 2006). Imposing the frequency 1/P , the
taken in the high-efficiency mode (EGGS, R=80 000) on results from spectroscopy point towards an axisymmetric
the night of January 9-10, 2011. We identified 53 metal- mode, without a clear indication on the ℓ-value. Since β Cas
lic unblended spectral lines (ranging from 3 780 to 6 910 Å) is seen almost pole-on (i = 19.9 ± 1.9◦ , Che et al. (2011)), a
relevant for the determination of radial velocities. Figure 1 low–degree, axisymmetric, nonradial mode mimics the pul-
(left panel ) shows the behaviour of the mean spectral line sation behaviour of a radial mode very well. On the basis of
profile along the pulsation phase. The shifts due to the ra- these considerations, we treated β Cas as a radial monope-
dial modes clearly dominate the line profile variations. riodic pulsator for our purposes. We also used the mean
line profiles of β Cas to estimate the v sin i values from the
position of the first zero of their Fourier transforms (Carroll
0.013
0.099
0.028 1933). This approach is possible only for objects where the
0.121
0.143
0.056
0.084 rotational broadening is dominant with respect to the other
0.165 0.112
0.140 sources of line broadening (e.g., instrumental effect, micro-
0.188
0.210
0.169
0.197
turbulence), which is always the case for β Cas, but not
0.232 0.225
0.253
for AI Vel, where we were unable to use the Fourier trans-
0.254
0.276
0.282
0.316
form method on the narrower lines (v sin i < 10 km s−1 ).
0.298
0.344
0.373
The radial velocity values of the observed profiles of β Cas
0.320
0.342
0.401
0.429
range from 5.3 to 11.6 km s−1 and the v sin i values from
0.364
0.458
0.486
74.0 to 77.5 km s−1 (Fig.1, right panel ). We could also
determine a mean v sin i value from the average of all the
Phase

0.386 0.514
0.408 0.542
0.431
0.571 mean profiles and obtained 75.72 ± 0.14 km s−1 . This value
0.608
0.453 0.636 is consistent with literature values (Bernacca & Perinotto
0.475 0.665
0.497
0.693 1970; Uesugi & Fukuda 1970; Schröder et al. 2009). When
0.721
0.519 0.749 considering i = 19.9◦ , this means that β Cas is an intrinsic
0.778
0.541
0.563
0.806 fast rotator, with a velociy of vrot ≃ 220 km s−1 which is
0.835
0.585 0.863 consistent with Che et al. (2011).
0.607 0.919
0.950 Finally, for both stars, the centroid radial velocity RVc
0.629
0.651
0.981
(or the first-moment 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.
-40 -20 0 20 40 60 80 -100 -60 -20 20 60 100 140
-1 -1
Radial velocity (km s ) Radial velocity (km s )

3. The geometric projection factor p0

Fig. 1. Mean profiles of the HARPS spectra of AI Vel (left
panel, T0 =JD 2443176.00) and of the SOPHIE spectra Considering a limb-darkened pulsating star in rotation with
of β Cas (right panel, T0 =JD 2438911.88). The pulsation a one-layer atmosphere, the projection factor is purely ge-
V
phase is given on the y-axis. A strong broadening is clearly ometric. Thus, p = Vpuls
rad
= p0 . The radial velocity is then
seen for β Cas due to its high rotation velocity. We com- defined by
puted the mean line profiles of AI Vel and β Cas spectra by
means of a deconvolution process using the LSD software r
1 (x2 + y 2 )
Z
(Donati et al. 1997). Vrad = I(x, y, λ) · Vpuls · 1− dxdy,
πR2 x,y ∈DR R2
(1)
The target β Cas (HD 432, F2 III/IV) is a low- where DR is the surface of the stellar disc of radius R,
amplitude δ Scuti star. We observed β Cas with the and I(x, y, λ) the limb-darkened continuum intensity dis-
SOPHIE instrument (R = 75 000) at the OHP 1.93-m tele- tribution considered at the wavelength of
scope on the night of September 30, 2011. We collected 241 p observation λ
defined by I(x, y, λ) = I0 (1 − uλ + uλ 1 − (x2 + y 2 )),
high-resolution spectra with a mean S/N of 100. We could where uλ is the linear limb-darkening coefficient from
distinguish only height unblended spectral lines relevant for
the spectral analysis because of the strong rotational broad- 1
Developed in the framework of the FP6 European
ening (Fig. 1, right panel ). Riboni et al. (1994) showed that Coordination Action HELAS (http://www.helas-eu.org/).

2
 G. Guiglion and collaborators: The Baade-Wesselink projection factor of the δ Scuti stars

Claret & Bloemen (2011). Considering Teff = 7 400 K and

log g = 3.5, we find uR = 0.474 ± 0.025 in the R-band s
x2 y2
 
1
Z
from Claret & Bloemen (2011). Using Eq. 1, we deduce
Vrad = I(x, y, λ)·Vpuls · 1 − + 2 dxdy,
a value of the geometric projection factor for AI Vel of πR2 x,y ∈DR a2 b
p0 = 1.43 ± 0.01. p0 is assumed to be constant with the (2)
pulsation phase (Nardetto et al. 2004). This value is higher where a and b are the semi-major and semi-minor axis of
than what we generally obtain for Cepheids (typically 1.37 the ellipse. In Fig. 2 (bottom panel ), we show the geomet-
to 1.41, see Fig.2 (top panel )). ric projection factor (p0 ) as a function of i. Fig. 3 presents
The geometrical shape of β Cas and its intensity distri- the modelled intensity distributions for several inclinations.
bution are distorted by its high rotation rate. The geomet- This relation is extremely interesting because it shows that
ric projection factor depends on the inclination of the star’s the inclination of a fast-rotating star can have an impact
rotation axis compared to the line-of-sight. If the star’s ro- of more than 10% on the projection factor. Of course, it
tation axis is along the line-of-sight (i = 0◦ ), the star is also depends on the rotation velocity of the star: the higher
observed pole-on and is seen as a disc. For i > 0◦ the star the rotation velocity (for a given inclination), the lower the
has an ellipsoidal shape. projection factor. Using the inclination found by Che et al.
(2011), i = 19.9 ± 1.9◦ , we finally find a geometric projec-
tion factor for β Cas of p0 = 1.41 ± 0.02 (averaged over the
three wavelengths considered).

Fig. 3. Modelled intensity distributions of β Cas at

6 000 Å for i = 5◦ (left ), i = 50◦ (middle), i = 90◦ (right ).

4. Dynamical structure of the atmosphere

By comparing the 2K-amplitude (defined as the ampli-
tude of the RVc curve, hereafter ∆RVc ) with the depth
(D) of the 53 spectral lines selected in the case of AI Vel,
one can directly measure the atmospheric velocity gra-
dient in the part of the atmosphere where the spectral
lines are formed (Figure 4, top-left). To quantify the im-
pact of velocity gradient on the projection (fgrad ), we do
not need to derive the velocity gradient over the whole
atmophere, but only at the location of the forming re-
Fig. 2. Top: p0 as a function of the limb-darkening param- gions of the spectral lines used to derive the distance of
eter uλ . The red box indicates the uncertainty on p0 for the star. We therefore performed a linear regression ac-
the δ Scuti AI Vel. The blue box indicates the typical val- cording to the relation ∆RVc = a0 D + b0 . We obtained
ues of uλ and p0 for Cepheids. The dots corresponds to ∆RVc = [−0.40 ± 0.53]D + [32.87 ± 0.23] km s−1 (Fig. 4,
the relation provided by Nardetto et al. (2006). Bottom: p0 top-right). In principle, fgrad depends on the spectral line
as a function of the inclination of the fast rotating star considered (Nardetto et al. 2007): fgrad = b0 /(a0 D + b0 ).
β Cas for three different wavelengths (λ = 6 000 Å (), Here, we find that fgrad is typically the same for all spec-
λ = 6 500 Å (△), and λ = 7 000 Å (×)). The red box in- tral lines (fgrad = 1.01 ± 0.01), which is consistent with no
dicates the uncertainty on p0 for β Cas. The case of an correction of the projection factor due to the velocity gra-
uniform elonged disc is over-plotted (+), and we find that dient. The uncertainty on fgrad is derived from the errors
p0 = 1.5 for i = 0◦ , as expected for a circular uniform disc. on a0 and b0 .
Figure 4 (bottom left ) presents the interpoled RVc curve
of β Cas for the FeI spectral line (λ = 4 508.288 Å). We
clearly see an increase of the amplitude of the radial ve-
Using the fundamental parameters of the modified von locity curve (∼ 4.6 ± 0.9 % per cycle). Moreover, the ra-
Ziepel model found by Che et al. (2011) and the rotat- dial velocity curves have several minima and maxima and
ing stars model by Domiciano de Souza et al. (2002, 2012), we can easily deduce a period of pulsation. We find P =
we derived the intensity distribution in the continuum 0.10046 ± 0.00054 d. Our value agrees well with that of
for different inclinations of the star (from i = 0◦ to Riboni et al. (1994).
i = 90◦ with a step of 5◦ ) and for three wavelengths: In Fig. 4 (bottom right), ∆RVc is plotted as a function of D
λ = 6 000, 6 500 and 7 000 Å. Using these intensity maps, we for cycle 2 (see the vertical line in the figure). For β Cas, the
can easily calculate the geometric projection factor. Indeed, range of the spectral line depth is eight times lower com-
for an ellipsoid, Vrad is then defined by pared to AI Vel. The velocity gradient is fgrad = 0.64±0.82.

3
 G. Guiglion and collaborators: The Baade-Wesselink projection factor of the δ Scuti stars

Vel and p = 1.41 ± 0.25 for β Cas. However, the generali-

sation of this study to any δ Scuti stars is presently lim-
ited since we have to study the impact of multi-modes,
in particular non-radial ones, on the projection factor.
This complicated effect has been studied by several au-
thors for the bisector method of the radial velocity de-
termination (Dziembowski 1977; Balona & Stobie 1979;
Stamford & Watson 1981; Hatzes 1996). We assume a star
pulsating in two modes, one radial and one non-radial. The
projection factor can be defined as p = αr pr + αnr pnr , with
αr and αnr the relative contributions of the velocity am-
plitudes of the radial and non-radial modes to the pulsa-
tion (αr + αnr = 1). pr is our previous decomposition of
the projection factor in the case of a purely radial mode
pr = p0 fgrad fo-g , while pnr is the projection factor in the
case of a purely non-radial mode. Using Eq. 4 of Hatzes
2
(1996), we find that pnr = p0 e+kℓ , where k = 0.15 in the
case of a non radial p-mode and k = 1.2 in the case of
a non-radial g-mode. ℓ is the spherical harmonic degree.
This relation is derived for the first moment (i.e. the ra-
dial velocity) determination only, which is independent of
the star’s rotation. This equation can be used when ℓ = m
Fig. 4. Top: RVc as a function of the MJD in the case only (where m is the spherical harmonic order). Additional
of the FeII 5 234.625 Å spectral line, and amplitude of the work is necessary to derive it when ℓ 6= m. We emphasize
RVc curves as a function of the spectral line depth (D) also that the higher ℓ(= m), the higher is the non-radial
for AI Vel. Typical error bars are indicated in the upper projection factor. This is expected since for high values of
right of each panel. Bottom: The same for β Cas f the FeII ℓ (and this is qualitatively also true when ℓ 6= m) there are
4 508.288 Å spectral line. more red- and blue-shifted velocity zones on the star that
cancel each other in the integrated line profile, which leads
to a lower amplitude of the non-radial velocity curve, and
The large error bar prevents any physical discussion about in turn a high value of the non-radial projection factor (see
the exact value of fgrad . We note that fgrad = 0.64 implies Eq. 1&2 of Hatzes (1996)).
a huge velocity gradient in the star’s atmosphere, which We performed our study on AI Vel and β Cas under the
seems unrealistic. Since it is consistent with our determi- assumption of monoperiodic radial pulsation. Our results
nation, we assumed fgrad = 1.0. Considering a0 = 0, we (without non-radial correction) are consistent (at the 1σ
obtained b0 = 4.07 ± 0.25 with a reduced χ2 of 1.0, which level) with the period-projection-factor (P p) relation p =
provides an uncertainty on fgrad of 0.14. The fo-g correc- [−0.071 ± 0.020] log P + [1.311 ± 0.019] by Laney & Joner
tion, which is the last component of the projection factor (2009) applied for classical and dwarf Cepheids (it corre-
decomposition, cannot be measured from observations. To sponds to p = 1.38 ± 0.02 for AI Vel and β Cas). To derive
estimate the differential velocity between the optical and these values, Laney & Joner simply compared the distance
gas layers at the photosphere of the star, we need a hydro- of the stars obtained from the PL relation with the dis-
dynamic model. tances from the photometric version of the Baade-Wesselink
However, modelling the pulsating atmosphere of δ Scuti method. This suggests that the eventual non-radial compo-
stars is not an easy task because of (1) cycle-to-cycle vari- nents of β Cas have probably a negligeable impact on the
ations (non-radial modes) and (2) fast rotation in some projection factors (which means αnr ≃ 0). Interestingly,
cases (as for β Cas). However, fo-g have been studied inten- if we use the P p relation obtained for classical Cepheids
sively in the Cepheids (Nardetto et al. 2004, 2007, 2011), by Nardetto et al. (2007), p = [−0.064 ± 0.020] log P +
and it seems that there is a linear relation between fo-g [1.376 ± 0.023] (derived with the first moment method2 )
and log P : fo-g = [−0.023 ± 0.005] log P + [0.979 ± 0.005]. to derive the projection factors of the two δ Scuti stars,
Moreover, we have a theoretical value of fo-g for the short- we find p = 1.44 ± 0.01, which is consistent with our val-
period β-Cephei α Lup (P = 0.2598 d), fo-g = 0.99 ± 0.01, ues. This seems to show that the P p relation provided by
which seems to be consistent with the log P -fo-g rela- Nardetto et al. (2007) for single lines is also applicable to
tion of Cepheids (Nardetto et al. 2012). For our study, δ Scuti stars pulsating in a dominant radial mode. In addi-
we propose to extend this law for the δ Scuti β Cas and tion, and as already shown, for fast-rotating δ Scuti stars,
AI Vel. Considering P = 0.11157 d for AI Vel and P = an intrinsic dispersion of the P p relation due to the random
0.10046 ± 0.00054 d for β Cas (from this paper), we find orientation of the rotation axis has to be considered.
fo-g = 1.00 ± 0.02 (0.02 is a conservative arbitrary un-
certainty) for both stars, which basically means no pho- Acknowledgements. G.G. and N.N. thank J. Monnier for useful dis-
tospheric correction to the projection factors. cussions. W.G. is grateful for support from the BASAL Centro de
Astrofisica y Tecnologias Afines (CATA) PFP-06/2007. E.P. and M.R.
acknowledge financial support from the Italian PRIN-INAF 2010
Asteroseismology: looking inside the stars with space- and ground-
5. Discussion based observations.
We can now calculate the projection factor p, using the 2
We remind that using the cross-correlation method, one has
relation p = p0 fgrad fo-g . We find p = 1.44 ± 0.05 for AI to use the P p relation from Nardetto et al. (2009).

4
 G. Guiglion and collaborators: The Baade-Wesselink projection factor of the δ Scuti stars

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