Table Of ContentAstronomy & Astrophysics manuscript no. guiglion (cid:13)c ESO 2013
January 14, 2013
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 deRechercheen Astrophysiqueet Plan´etologie, UMR 5277, 57 avenued’Azereix, 65000 Tarbes, France
3 3 INAF-OsservatorioAstronomico di Brera, ViaE. Bianchi 46, 23807 Merate, Italy
1 4 Instituteof Astronomy of theRussian Academy of Sciences, 48 Pjatnitskaya Str., Moscow 109017, Russia
0 5 Departamento deAstronom´ıa, Universidad deConcepci´on, Casilla 160-C, CL Concepci´on, Chile
2
n Received 23 November2012; accepted 2 January 2013
a
J ABSTRACT
1
Aims. The Baade-Wesselink method of distance determination is based on the oscillations of pulsating stars. The key
1
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 δScuti stars, using high-resolution spectra of
]
R thehigh-amplitude pulsator AI Veland of thefast rotator βCas.
S Methods. The geometric component of the projection factor (i.e. p0) was calculated using a limb-darkening model of
theintensitydistributionforAIVel,andafast-rotatormodelforβCas.Then,usingSOPHIE/OHPdataforβCasand
.
h HARPS/ESOdataforAIVel,wecomparedtheradialvelocitycurvesofseveralspectrallinesformingatdifferentlevels
p 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 derivea dynamical projection factor p.
o Results.Wefindaflatvelocitygradientforbothstarsandfinallyp=p0 =1.44forAIVelandp=p0 =1.41forβCas.
r BycomparingCepheidsandδScutistars,theseresultsbringvaluableinsightsintothedynamicalstructureofpulsating
t
s star atmospheres. They suggest that the period-projection factor relation derived for Cepheids is also applicable to
a δScutistars pulsating in a dominant radial mode.
[
Key words.Stars: pulsating – Stars: atmospheres – Stars: variables: δScuti– Techniques:spectroscopic
1
v
5
7 1. Introduction projection factor: (1) the geometric projection factor p0,
4 which is directly related to the limb-darkening of the star
Determining distances in the Universe is not a trivial task.
2 (seeSect.3),(2)thecorrectionf duetothevelocitygra-
grad
. FromourGalaxytotheVirgoCluster,distancescanbede- dientbetweenthespectral-line-formingregionandthepho-
1
rivedusingtheperiod-luminosityrelation(PL)ofCepheids
0 tosphere of the star; this quantity can be derived directly
(Riess et al. 2009a,b). However,this relationhas to be cal-
3 from observations by comparing different lines forming at
ibrated, using the Baade-Wesselinkmethod of distance de-
1 different levels in the atmosphere (see Sect. 2 and 4), and
termination for instance (Storm et al. 2011a,b). The prin-
v: ciple of this method is simple: after determining the angu- (3) the correctionfo−g due to the relative motion between
the optical and gas layers associated to the photosphere
Xi lar diameter and the linear radius variations of the star, (see Sect. 4).For a detailed analysisofthe p-factordecom-
the distance is derived by a simple ratio. Angular di-
r position we refer to Nardetto et al. (2007). The projection
a ameter variations can be measured using interferometry factor is then defined by p = p0fgradfo−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).Thelinearradiusvariation
(originally developed for Cepheids) to the δ Scuti stars AI
ismeasuredbyintegratingthepulsationvelocity(hereafter
Vel and βCas. The impact of non-radial modes of δ Scuti
V )overonepulsatingcycle.However,fromobservations
puls stars on the projection factor is a very difficult question
we have only access to the radialvelocity (V ) because of
rad 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 βCas are discussed in the con-
velocity, is defined by p = V /V . There are in prin-
puls rad clusion. This paper is part of the international Araucaria
ciple three sub-concepts involved in the Baade-Wesselink
Project,whosepurposeistoprovideanimprovedlocalcali-
brationofthe extragalacticdistancescalesouttodistances
⋆
This work uses observations made with the HARPS instru-
ofafewmegaparsecs(Gieren et al.2005).Inthiscontext,δ
ment at the 3.6 m telescope (La Silla, Chile) in the framework
Scutistarsareextremelyinterestingsinceithasbeenshown
oftheLP185.D-0056andwiththeSOPHIEinstrumentatOHP
(France).
1
G. Guiglion and collaborators: The Baade-Wesselink projection factor of theδScutistars
recently that they follow a PL relation (McNamara et al. thestarisamonoperiodicpulsatoratthedetectionlimitof
2007; Poretti et al. 2008). ground-basedphotometric measurements,with a pulsation
period of P = 0.101036676d. 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 (HD69213, A9IV/V) is one of the most often Hipparcosparallax(van Leeuwen2007).Thuswecouldob-
observed double-mode, high-amplitude δScuti 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 PL relations (McNamara et al.
modes with a well-constrained period ratio P /P of 0.77 2007; Poretti et al. 2008) supply a fundamental radial pe-
1 0
(Poretti et al. 2005). In addition to P0 = 0.111574d and riod of about 0.15 d at this MV value. Therefore, the ob-
P = 0.0862073d, Walraven et al. (1992) clearly detected served period is similar to that expected for the second
1
two other periods, tentatively identified as the third and radial overtone. We attempted a mode identification from
fifthradialovertones.WeobservedAIVelusingtheHARPS our spectroscopic data using the FAMIAS1 software.Since
spectrograph mounted at the ESO 3.6-m telescope. We wedealtwithafastrotator,weusedtheFourierparameter
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=80000) on results from spectroscopy point towards an axisymmetric
the night of January 9-10, 2011. We identified 53 metal- mode,withoutaclearindicationontheℓ-value.Sinceβ Cas
lic unblended spectrallines (ranging from3780to 6910˚A) isseenalmostpole-on(i=19.9±1.9◦,Che et al.(2011)),a
relevant for the determination of radialvelocities. Figure 1 low–degree,axisymmetric,nonradialmode mimicsthe pul-
(left panel) shows the behaviour of the mean spectral line sationbehaviourofaradialmodeverywell.Onthebasisof
profile along the pulsation phase. The shifts due to the ra- theseconsiderations,wetreatedβ Casasaradialmonope-
dial modes clearly dominate the line profile variations. riodic pulsator for our purposes. We also used the mean
line profilesofβ Casto estimatethe vsinivaluesfromthe
positionofthefirstzerooftheirFouriertransforms(Carroll
0.099 0.013 1933). This approachis possible only for objects where the
0.028
0.121 0.056 rotationalbroadeningisdominantwithrespecttotheother
0.143 0.084
0.165 0.112 sourcesof line broadening (e.g., instrumentaleffect, micro-
0.140
0.188 0.169 turbulence), which is always the case for β Cas, but not
0.210 0.197
0.232 0.225 for AI Vel, where we were unable to use the Fourier trans-
0.254 00..228523 form method on the narrower lines (vsini < 10 km s−1).
0.276 0.316
0.298 0.344 The radial velocity values of the observed profiles of βCas
0.320 00..430713 range from 5.3 to 11.6 km s−1 and the vsini values from
Phase000000......444333530864318642 00000000........665554443074185268124689 im7ds4eet.ca0eonrnmtspoiirsno7teefi7n.la5etsmwkameintadhnso−lvibt1tseairn(ianFiteuivgdra.e17lu,5ve.ar7lif2ugrhe±ostm0(p.B1tah4neereknlma)av.cescWr−aa1eg&.eTcPoohufeirlsadinlvlaoatltlhustooee
0.475 0.665 1970; Uesugi & Fukuda 1970; Schr¨oder et al. 2009). When
0.693
0.497 0.721 considering i=19.9◦, this means that βCas is an intrinsic
0.519 0.749
0.541 00..870768 fast rotator, with a velociy of vrot ≃220 km s−1 which is
0.563 0.835 consistent with Che et al. (2011).
0.585 0.863
0.607 00..995109 Finally, for both stars, the centroid radial velocity RVc
0.629 0.981 (or the first-moment radial velocity) and the line depth D
0.651
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
Radial velocity (km s-1) Radial velocity (km s-1)
3. The geometric projection factor p
0
Fig.1. Mean profiles of the HARPS spectra of AI Vel (left
Consideringalimb-darkenedpulsatingstarinrotationwith
panel, T =JD2443176.00) and of the SOPHIE spectra
0 a one-layer atmosphere, the projection factor is purely ge-
of βCas (right panel, T =JD 2438911.88). The pulsation
phase is givenonthe y-a0xis.A strongbroadeningis clearly ometric. Thus, p = VVpruadls = p0. The radial velocity is then
seen for βCas due to its high rotation velocity. We com- defined by
putedthemeanlineprofilesofAIVelandβCasspectraby
means of a deconvolution process using the LSD software
1 (x2+y2)
(Donati et al. 1997). V = I(x,y,λ)·V · 1− dxdy,
rad πR2 Zx,y∈DR puls r R2
(1)
The target β Cas (HD432, F2III/IV) is a low-
where D is the surface of the stellar disc of radius R,
R
amplitude δScuti star. We observed βCas with the and I(x,y,λ) the limb-darkened continuum intensity dis-
SOPHIE instrument (R=75000)at the OHP 1.93-mtele-
tribution considered at the wavelength of observation λ
scopeonthenightofSeptember30,2011.Wecollected241
defined by I(x,y,λ) = I (1 − u + u 1−(x2+y2)),
high-resolution spectra with a mean S/N of 100. We could 0 λ λ
where u is the linear limb-darkening coefficient from
distinguishonlyheightunblendedspectrallinesrelevantfor λ p
thespectralanalysisbecauseofthestrongrotationalbroad- 1 Developed in the framework of the FP6 European
ening(Fig.1,right panel).Riboni et al.(1994)showedthat Coordination Action HELAS (http://www.helas-eu.org/).
2
G. Guiglion and collaborators: The Baade-Wesselink projection factor of theδScutistars
Claret & Bloemen (2011). Considering T = 7400K and
eff
logg = 3.5, we find u = 0.474± 0.025 in the R-band
R
from Claret & Bloemen (2011). Using Eq. 1, we deduce 1 x2 y2
V = I(x,y,λ)·V · 1− + dxdy,
ap v=alu1e.4o3f±th0e.0g1e.opmeitsricaspsuromjeedctitoonbfeacctoonrstfoarntAwIitVhelthoef rad πR2 Zx,y∈DR puls s (cid:18)a2 b2(cid:19)
0 0 (2)
pulsationphase(Nardetto et al.2004).Thisvalueishigher
where a and b are the semi-major and semi-minor axis of
than what we generallyobtain 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 (p ) as a function of i. Fig. 3 presents
0
The geometricalshapeofβ Casanditsintensitydistri- themodelledintensitydistributionsforseveralinclinations.
bution are distorted by its high rotationrate.The geomet- Thisrelationis extremelyinterestingbecauseitshowsthat
ricprojectionfactordependsontheinclinationofthestar’s the inclination of a fast-rotating star can have an impact
rotationaxis comparedto 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 alsodependsontherotationvelocityofthestar:thehigher
observed pole-on and is seen as a disc. For i > 0◦ the star therotationvelocity(foragiveninclination),the lowerthe
has an ellipsoidal shape. projectionfactor.Usingthe inclinationfoundbyChe et al.
(2011), i=19.9±1.9◦, we finally find a geometric projec-
tion factor for βCas of p =1.41±0.02 (averagedoverthe
0
three wavelengths considered).
Fig.3. Modelled intensity distributions of βCas at
6000˚A 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 RV curve, hereafter ∆RV ) with the depth
c c
(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 (f ), we do
grad
not need to derive the velocity gradient over the whole
atmophere, but only at the location of the forming re-
Fig.2. Top: p as a function of the limb-darkeningparam- gions of the spectral lines used to derive the distance of
0
the star. We therefore performed a linear regression ac-
eter u . The red box indicates the uncertainty on p for
λ 0
the δ Scuti AI Vel. The blue box indicates the typical val- cording to the relation ∆RVc =a0D+b0. We obtained
ues of uλ and p0 for Cepheids. The dots corresponds to ∆RVc = [−0.40±0.53]D+[32.87±0.23]kms−1 (Fig. 4,
therelationprovidedbyNardetto et al.(2006).Bottom:p top-right). In principle, fgrad depends on the spectral line
0
as a function of the inclination of the fast rotating star considered (Nardetto et al. 2007): fgrad = b0/(a0D +b0).
βCas for three different wavelengths (λ = 6000 ˚A ((cid:3)), Here, we find that fgrad is typically the same for all spec-
λ = 6500 ˚A (△), and λ = 7000 ˚A (×)). The red box in- tral lines (fgrad =1.01±0.01), which is consistent with no
correction of the projection factor due to the velocity gra-
dicates the uncertainty on p for β Cas. The case of an
0
dient. The uncertainty on f is derived from the errors
uniform elonged disc is over-plotted (+), and we find that grad
p =1.5 for i=0◦, as expected for a circular uniformdisc. on a0 and b0.
0
Figure4(bottom left)presentstheinterpoledRV curve
c
of βCas for the FeI spectral line (λ = 4508.288˚A). 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
ingstarsmodelbyDomiciano 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 ± 0.00054d. 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: InFig.4(bottomright),∆RV isplottedasafunctionofD
c
λ=6000,6500and7000˚A.Usingtheseintensitymaps,we forcycle2(seetheverticallineinthefigure).ForβCas,the
caneasilycalculatethegeometricprojectionfactor.Indeed, range of the spectral line depth is eight times lower com-
for an ellipsoid, V is then defined by paredtoAIVel.Thevelocitygradientisf =0.64±0.82.
rad grad
3
G. Guiglion and collaborators: The Baade-Wesselink projection factor of theδScutistars
Vel and p = 1.41±0.25 for β Cas. However, the generali-
sation of this study to any δScuti 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
pulsatingintwomodes,oneradialandonenon-radial.The
projection factor can be defined as p=α p +α p , with
r r nr nr
α and α the relative contributions of the velocity am-
r nr
plitudes of the radial and non-radial modes to the pulsa-
tion (α +α = 1). p is our previous decomposition of
r nr r
the projection factor in the case of a purely radial mode
p = p f f , while p is the projection factor in the
r 0 grad o-g nr
case of a purely non-radial mode. Using Eq. 4 of Hatzes
(1996), we find that p = p e+kℓ2, where k = 0.15 in the
nr 0
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 misthe sphericalharmonicorder).Additional
of the FeII 5234.625˚A 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 βCas f the FeII ℓ (and this is qualitatively also true whenℓ6=m) there are
4508.288˚Aspectral line. more red- and blue-shifted velocity zones on the star that
canceleachother in the integratedline profile,which leads
to a lower amplitude of the non-radial velocity curve, and
The large error bar prevents any physicaldiscussion about
inturnahighvalue ofthe non-radialprojectionfactor(see
the exact value of f . We note that f =0.64 implies
grad grad Eq. 1&2 of Hatzes (1996)).
a huge velocity gradient in the star’s atmosphere, which
WeperformedourstudyonAIVelandβCasunderthe
seems unrealistic. Since it is consistent with our determi-
assumption of monoperiodic radial pulsation. Our results
nation, we assumed f = 1.0. Considering a = 0, we
grad 0 (without non-radial correction) are consistent (at the 1σ
obtained b = 4.07±0.25 with a reduced χ2 of 1.0, which
0 level) with the period-projection-factor (Pp) relation p =
provides an uncertainty on f of 0.14. The f correc-
grad o-g [−0.071±0.020] logP +[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 βCas). To derive
estimate the differential velocity between the optical and
these values, Laney & Joner simply comparedthe distance
gas layersat the photosphere of the star,we need a hydro-
of the stars obtained from the PL relation with the dis-
dynamic model.
tancesfromthephotometricversionoftheBaade-Wesselink
However, modelling the pulsating atmosphere of δ Scuti
method.Thissuggeststhattheeventualnon-radialcompo-
stars is not an easy task because of (1) cycle-to-cycle vari-
nents of βCas have probably a negligeable impact on the
ations (non-radial modes) and (2) fast rotation in some
projection factors (which means α ≃ 0). Interestingly,
nr
cases(asforβCas).However,f havebeenstudiedinten-
o-g if we use the Pp 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] logP +
and it seems that there is a linear relation between fo-g [1.376± 0.023] (derived with the first moment method2)
and logP: f = [−0.023±0.005]logP +[0.979±0.005].
o-g to derive the projection factors of the two δ Scuti stars,
Moreover,we have a theoretical value of f for the short-
o-g we find p = 1.44±0.01, which is consistent with our val-
period β-Cephei αLup (P = 0.2598d), f = 0.99±0.01,
o-g ues. This seems to show that the Pp relation provided by
which seems to be consistent with the logP-f rela-
o-g Nardetto et al. (2007) for single lines is also applicable to
tion of Cepheids (Nardetto et al. 2012). For our study,
δScuti stars pulsating in a dominant radial mode. In addi-
we propose to extend this law for the δScuti βCas and
tion, and as already shown, for fast-rotating δ Scuti stars,
AI Vel. Considering P = 0.11157d for AI Vel and P =
anintrinsicdispersionofthePprelationduetotherandom
0.10046± 0.00054d for βCas (from this paper), we find
orientation of the rotation axis has to be considered.
f = 1.00 ± 0.02 (0.02 is a conservative arbitrary un-
o-g
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
AstrofisicayTecnologiasAfines(CATA)PFP-06/2007.E.P.andM.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 Weremindthatusingthecross-correlation method,onehas
relation p = p0fgradfo-g. We find p = 1.44±0.05 for AI to use thePp relation from Nardetto et al. (2009).
4
G. Guiglion and collaborators: The Baade-Wesselink projection factor of theδScutistars
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