Highresolution spectroscopy for Cepheids distance determination
V. Impact of the crosscorrelation method on the pfactor and the velocities^{}^{,}^{}
N. Nardetto^{1}  W. Gieren^{1}  P. Kervella^{2}  P. Fouqué^{3}  J. Storm^{4}  G. Pietrzynski^{1,5}  D. Mourard^{6}  D. Queloz^{7}
1  Departamento de Astronomía, Universidad de
Concepción, Casilla 160C, Concepción, Chile
2  Observatoire de
ParisMeudon, LESIA, UMR 8109, 5 Place Jules Janssen, 92195 Meudon
Cedex, France
3  Observatoire MidiPyrénées, Laboratoire
d'Astrophysique, UMR 5572, Université Paul Sabatier, Toulouse 3,
14 avenue Edouart Belin, 31400 Toulouse, France
4 
Astrophysikalisches Institut Postdam, An der Sternwarte 16, 14482
Postdam, Germany
5  Warsaw University Observatory, AL. Ujazdowskie
4, 00478 Warsaw, Poland
6  OCA/CNRS/UNS, Dpt. Fizeau, UMR6525,
Avenue Copernic, 06130 Grasse, France
7  Observatoire de
Genève, Université de Genève, 51 Ch. des Maillettes, 1290
Sauverny, Switzerland
Received 15 April 2009 / Accepted 14 May 2009
Abstract
Context. The cross correlation method (hereafter CC) is widely used to derive the radial velocity curve of Cepheids when the signal to noise ratio of the spectra is low. However, if it is used with an inaccurate projection factor, it might introduce some biases in the BaadeWesselink (BW) methods of determining the distance of Cepheids. In addition, it might affect the average value of the radial velocity curve (or velocity) important for Galactic structure studies.
Aims. We aim to derive a periodprojection factor relation (hereafter Pp) appropriate to be used together with the CC method. Moreover, we investigate whether the CC method can explain the previous estimates of the ``Kterm'' of Cepheids.
Methods. We observed eight galactic Cepheids with the HARPS^{} spectrograph. For each star, we derive an interpolated CC radial velocity curve using the HARPS pipeline. The amplitudes of these curves are used to determine the correction to be applied to the semitheoretical projection factor. Their average value (or velocity) are also compared to the centerofmass velocities derived in previous works.
Results. The correction in amplitudes allows us to derive a new Pp relation:
.
We also find a negligible wavelength dependence (over the optical range) of the Pp relation. We finally show that the velocity derived from the CC method is systematically blueshifted by about
km s^{1} compared to the centerofmass velocity of the star. An additional blueshift of 1.0 km s^{1} is thus needed to totally explain the previous calculation of the ``Kterm'' of Cepheids (around 2 km s^{1}).
Conclusions. The new Pp relation we derived is a reliable tool for distance scale calibration, and especially to derive the distance of LMC Cepheids with the infrared surface brightness technique. Further studies should be devoted to determining the impact of the signal to noise ratio, the spectral resolution, and the metallicity on the Pp relation.
Key words: techniques: spectroscopic  stars: atmospheres  stars: oscillations  stars: variables: Cepheids  stars: distances
1 Introduction
The BaadeWesselink (hereafter BW) method of determining the distance of Cepheids was recently used to calibrate the periodluminosity (PL) of Galactic Cepheids (Fouqué et al. 2007). The basic principle of this method is to compare the linear and angular size variation of a pulsating star in order to derive its distance through a simple division. The angular diameter is either derived by interferometry (for e.g. Kervella et al. 2004; Davis et al. 2008) or using the infrared surface brightness (hereafter IRSB) relation (Gieren et al. 1998, 2005a). However, when determining the linear radius variation of the Cepheid by spectroscopy, one has to use a conversion projection factor from radial to pulsation velocity. This quantity has been studied using hydrodynamic calculations by Sabbey et al. (1996), and more recently Nardetto et al. (2004, 2007).
Following the work of Burki et al. (1982), we showed in Nardetto et al. (2006, hereafter Paper I) that the first moment of the spectral
line is the only method which is independent of the spectral line
width (average value and variation) and the rotation velocity of the
star. The centroid radial velocity (
), or the first
moment of the spectral line profile, is defined as
We thus used this definition of the radial velocity in paper two of this series (Nardetto et al. 2007, hereafter Paper II), to derive a semitheoretical periodprojection factor (hereafter Pp) relation based on spectroscopic measurements with the HARPS high resolution spectrograph. This relation was derived from the specific Fe I 4896.439 Å spectral line which has a relatively low depth for all stars at all pulsation phase (around 8% of the continuum). It was shown that such a low depth value is suitable to reduce the uncertainty on the projection factor due to the velocity gradient between the photosphere (corresponding to angular diameter measurements) and the lineforming region (corresponding to the radius estimation from spectroscopic measurements).
In the crosscorrelation method (hereafter CC method), a mask (composed of hundreds or thousands) of spectral lines is convolved to the observed spectrum. The resulting average profile is then fitted by a Gaussian. In such a method, there is first a mix of different spectral lines forming at different levels (more or less sensitive to a velocity gradient). Second, the resulting velocity can be dependent on the abundances or effective temperature (through the line width), or the rotation of the stars. Third, in Paper III of this series (Nardetto et al. 2008), we derived calibrated centerofmass velocities of the stars of our HARPS sample. By comparing these socalled velocities with the ones found in the literature (generally based on the CC method) and in particular in the Galactic Cepheid Database (Fernie et al. 1995), we obtained an average correction of km s^{1}. This result shows that the ``Kterm'' of Cepheids stems from an intrinsic property of Cepheids. But, it shows also that the crosscorrelation might introduce a bias (up to a few kilometers per second) on the average value of the radial velocity curve.
After a careful definition of the projection factor (Sect. 2), we apply the crosscorrelation method to the Cepheids of our HARPS sample (Sect. 3), in order to derive a periodprojection factor relation appropriate for the CC method (Sect. 4). As the HARPS pipeline also provides crosscorrelated radial velocities for each spectral order, we take the opportunity to study the wavelength dependence of the projection factor law (Sect. 5). Finally, we quantify the impact of the CC method on the velocities (Sect. 6).
2 Definition of the ``CC projection factor''
In this section, we recall some results obtained in Paper II and we
define the projection factor suitable for the crosscorrelation
method. In Paper II, we defined the projection factor as:
where is the amplitude of the pulsation velocity curve associated with the photosphere of the star. is the amplitude of the radial velocity curve obtained from the first moment of the spectral line. Because of the atmospheric velocity gradient, depends on the spectral line considered. Using a selection of 17 spectral lines, we thus derived an interpolated relation between and D, where D is the line depth corresponding to the minimum radius of the star:
This relation was then used to quantify the correction ( ) to be applied to the projection factor due to the velocity gradient (see Eq. (3) of Paper II). The Fe I 4896.439 Å spectral line (which forms close to the photosphere) was found to provide the lowest correction. The amplitude of the radial velocity curve corresponding to the Fe I 4896.439 Å spectral line was finally used (see in Table 5 of Paper II) to derive the semitheoretical Pp relation. It is defined (Eq. (3)) as , where a_{0} and b_{0} are indicated in Table 3 of Paper II. is derived from the interpolation of the line depth curve at the particular phase corresponding to the minimum radius of the star (i.e. when corrected from the velocity 0). and are given in Table 1 of this paper.
Table 1: The Cepheids studied listed with increasing period.
The projection factor suitable to the crosscorrelation method
(hereafter
)
is then simply:
where is the amplitude of the radial velocity curve obtained with the crosscorrelation method, and the correction factor to be applied. Our definition of is independent of the velocities.
Figure 1: Interpolated radial velocity curves based on the crosscorrelation method are presented for each Cepheid in our sample. Uncertainties are too small to be seen (around 0.5 km s^{1}). The horizontal lines near extrema give an indication of . The short horizontal lines are the velocities (see Sect. 6) corresponding to the CC method (solid line), the centerofmass velocity of Paper III (dotted line) and from Fernie et al. (1995, dashed line). 

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3 The CC method applied to HARPS observations
We consider eight Cepheids which have been observed with the HARPS spectrometer (R=120 000): R Tra, S Cru, Y Sgr, Dor, Gem, RZ Vel, Car, RS Pup. Information about observations (number of measurements, pulsation phases) can be found in Paper I.
We apply the HARPS pipeline to our data in order to calculate the crosscorrelated radial velocities (Baranne et al. 1996; Pepe et al. 2002). The basic principle of the CC method is to build a mask, made of zero and nonzero valuezones, where the nonzero zones correspond to the theoretical positions and widths of thousands of metallic spectral lines at zero velocity, carefully selected from a synthetic spectrum of a G2 star. A relative weight is considered for each spectral line according to its depth (derived directly from observations of a G2 type star). An average spectral line profile is finally constructed by shifting the mask as a function of the Doppler velocity. The corresponding radial velocity is derived applying a classical minimization algorithm between the observed line profile and a Gaussian function. The whole profile is considered in the fitting procedure, not only the line core. The average value of the fitted Gaussian corresponds to the crosscorrelated radial velocity (hereafter ). The HARPS instrument has 72 spectral orders. The pipeline provides averaged over the 72 spectral orders, or independently for each order. We first use the averaged values and the corresponding uncertainties.
The curves are then carefully interpolated using a periodic cubic spline function. This function is calculated either directly on the observational points or using arbitrary pivot points. In the latter case, a classical minimization process between observations and the interpolated curve is used to optimize the position of the pivot points (Mérand et al. 2005). For Y Sgr and RS Pup, pivot points are used due to an inadequate phase coverage. When the phase coverage is good (which is the case for all other stars), the two methods are equivalent (Fig. 1). From these curves we are finally able to calculate (Table 2). The statistical uncertainty on is set as the average value of the uncertainty obtained for all measurements over a pulsation cycle of the star.
4 A Pp relation dedicated to the CC method
Table 2: The projection factor ( ) and the velocities ( [CC]) derived from the CC radial velocity curves.
Figure 2: a) The correction factor induced on the projection factor by the crosscorrelation method is shown as a function of the logarithm of the period of the star. b) The periodprojection factor (p) relation from Paper II (crosses and solid line) and the corrected relation suitable for the crosscorrelation method (diamonds and dashed line). 

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Figure 3: a) Wavelength dependency of the amplitude of the crosscorrelated radial velocity curves for each star in our sample. The corresponding linear relation are defined as: . b) The corresponding slopes ( ) as a function of the period. 

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From and obtained for all stars we derive the correction factor using Eq. (2). The result is plotted as a function of the period in Fig. 2a. No particular trend is found. However, the correction factors are clearly statistically dispersed around a mean value of .
Following our definition (
),
the corrected projection factors suitable for the CC method are
given in Table 2. The relation between the period and
remains clear according to the statistical
uncertainties:
The corresponding reduced is 1.2. We refer to this relation in the following using . We recall that the Pp relation we found in Paper II dedicated to the FeI 4896 spectral line was: . These two relations are shown in Fig. 2b. The impact of the crosscorrelation method on the zeropoint of the Pp is thus significant, while the slope increases only slightly (in absolute value) from 0.064 to 0.08.
We have several possible explanations for these results. The crosscorrelation induces two biases:
 1.
 The crosscorrelated radial velocities are
derived using a Gaussian fit, making the result sensitive both to
the spectral line width (i.e. the effective temperature and
abundances) and the rotation velocity projected on the line of
sight. These two quantities, independently, and even more the
combination of both, are not expected to vary linearly with the
logarithm of the period. This might explain why no clear linear
relation is found between
and the period of the
star. However, the mean values of the correction factors (around
)
have a non negligible impact on the zeropoint of
the Pp relation, which decreases from 1.376 to 1.31 (5%).
 2.
 The crosscorrelation method implies a mix of different
spectral lines forming at different levels. In the Pp relation,
the only quantity sensitive to the line depth is
(as defined in Paper II) which compares the amplitude of the
pulsation velocity corresponding to the lineforming region, and the
photosphere. It is thus an estimate of the velocity gradient within
the pulsating atmosphere of the star. The Pp relation was derived
in Paper II for the 4896 spectral line which forms very close
to the photosphere (), while the crosscorrelated radial
velocity is a mix of thousands of spectral lines forming at
different levels, with an average depth of around
.
The crosscorrelation method is thus more sensitive to the velocity
gradient (because the average line depth is large), which may
explain the increase (in absolute value) of the slope from 0.064to 0.08. Moreover, in Paper II we provided a very rough estimate
of the Pp relation associated with the crosscorrelation method,
considering only the impact of the velocity gradient (which
means discarding the bias related to the Gaussian fit). We found
(see Sect. 7 of
Paper II). The slope we find here (0.08) is consistent with this
previous rough estimate of 0.075.
5 Wavelength dependence of the projection factor
With the data at hand, we check for a possible dependence of the
projection factor on the wavelength range used for the
crosscorrelation radial velocity measurement. For each order, we
derive the crosscorrelated interpolated radial velocity curves, and
then the corresponding amplitudes
.
Orders 59, 68 and 72 are not considered due to
instrumental characteristics and/or unrealistic results. For all
stars,
is plotted as a
function of the wavelength, defined as the orders' average values
(Fig. 3a). We find linear relations between these two
quantities:
where and are listed in Table 3. For consistency with the previous section the quantities have been slightly shifted in velocity in such a way that:
where is derived from Table 2 and 502.2 nm is the wavelength averaged over all orders.
We also find a relation between
and the
logarithm of the period of the star:
From these results we can make two comments. First, the amplitude of the crosscorrelated radial velocity curves decreases with wavelength. From hydrodynamical modelling, we know that the spectral lines form over a larger part of the atmosphere in the infrared compared to optical (Sasselov et al. 1990). This effect might help us to understand our result: the more extended the line forming regions are, the lower the amplitude of the radial velocity curves. Second, this effect is greater for longperiod Cepheids than shortperiod Cepheids. A reason might be that the mean radius, the size of the lineforming regions and the velocity gradient increase with the logarithm of the period.
Figure 4: Corrections to apply to the relation (Eq. (5)) in the blue ( nm) and in the red ( nm). 

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Table 3: Coefficients of the linear relations between the amplitude of the radial velocity curve and the wavelength.
In order to quantify the wavelength dependency of the
relation, we define two correction factors
(
and
). We find the
following correcting relation as a function of the logarithm of the
period:
and
The reduced values are respectively 0.3 and 1.3. These relations are shown in Fig. 4. We find that such corrections are currently irrelevant given our statistical uncertainties on the relation (Eq. (5)).
6 The CC velocity and the Kterm of Cepheids
Interestingly, for each Cepheid in our sample, we found in Paper III a linear relation between the velocities (derived using the first moment method) of the various spectral lines and their corresponding asymmetries. Using these linear relations, we provided a physical reference to derive the centerofmass velocity of the stars ( [N08]): it should be zero when the asymmetry is zero. These values are consistent with an axisymmetric rotation model of the Galaxy. Conversely, previous measurements of the velocities found in the literature (for e.g. Fernie et al. 1995: the Galactic Cepheid Database, hereafter [GCD]) were based on the crosscorrelation method, and by using generally only a few measurements over the pulsation cycle. These results led to an apparent ``fall'' of Galactic Cepheids towards the Sun (compared to an axisymmetric rotation model of the Milky Way) with a mean velocity of about 2 km s^{1}. This residual velocity shift has been dubbed the ``Kterm'', and was first estimated by Joy (1939) to be 3.8 km s^{1}.
Figure 5: [CC] as a function of [GCD] (diamond) and [N08] (triangles). The solid and dashed line are the corresponding linear interpolation. 

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We aim to understand why such a 2 km s^{1} error was obtained before. An
hypothesis is that the crosscorrelation method is biased by the
dynamical structure of the atmosphere of Cepheids. To verify this
hypothesis, we have the unique opportunity to compare quantitatively
and in a consistent way
[N08],
[GCD] and the velocities derived from
our HARPS crosscorrelated radial velocity curves (hereafter
[CC]). The comparison is done by plotting
[CC] as a function of
[GCD]
and
[N08] respectively (Fig. 5). The data
are presented in Table 2 and the resulting linear relations
are respectively:
and
(12) 
The reduce values are respectively 3.0 and 3.8.
Several conclusions can be drawn. The slope of these relations are similar and close to one, which means that there is no particular trend of the velocity with the period of the star, or at least, it remains negligible here. As in Paper III, we find a systematic difference of km s^{1} between [N08] and [GCD], which is consistent with the Kterm of Cepheids. However, the velocities we derive in this Paper using the crosscorrelation method are systematically lower by ( ) km s^{1} than the ones found in the literature ( [GCD]), and they are systematically larger by km s^{1} than the calibrated centerofmass velocities ( [N08]). As a consequence, the crosscorrelation method alone cannot explain alone the Kterm. The CC method is sensitive to the dynamical structure of the Cepheids' atmosphere in such a way that it is responsible for 50% of the Kterm. An additional term is required to explain the presence of such offsets in previous determinations of the gammavelocity. It could be related, for instance, to the quality of observations in the past (Joy et al. 1939) or to the different methods used to derive the velocity (Pont et al. 1994).
7 Conclusions
By comparing the amplitude of our crosscorrelated radial velocity curves with previous results based on the first moment method (Paper II), we derived a new Pp relation applicable to radial velocities measured by the crosscorrelation method. This relation is crucial for the distance scale calibration, and in particular to derive the distances of LMC and SMC Cepheids (Gieren et al. 2005a; Gieren et al. 2009, in preparation). We also find a slight dependence of the Pp relation on the wavelength. Considering our current uncertainties this effect is negligible, but it might become significant in the near future. The next steps are to test the impact of the signal to noise ratio, the spectral resolution and the metallicity on the projection factor. The latter point will require a large sample of Cepheids with wellmeasured metallicities. These studies (including this work) are part of the international ``Araucaria Project'' whose purpose is to provide an improved local calibration of the extragalactic distance scale out to distances of a few Megaparsecs (Gieren et al. 2005b). Moreover, the fact that the crosscorrelation method overestimates the amplitude of the radial velocity curve and underestimates the velocity (compared to the calibrated values presented in Paper III) might have some implications for other kinds of pulsating stars, for e.g. in asteroseismology.
Moreover, we show in Paper III that the Kterm of Cepheids vanishes if one considers carefully the dynamical structure of Cepheid atmosphere. From the results presented in this paper, we can state that the crosscorrelation method might not be totally responsible for the Kterm found in the previous studies (only 50% seems to be a consequence of the crosscorrelation method). There seems to be another contribution whose nature should be investigated.
Acknowledgements
Based on observations collected at La Silla observatory, Chile, in the framework of European Southern Observatory's programs 072.D0419 and 073.D0136. This research has made use of the SIMBAD and VIZIER databases at CDS, Strasbourg (France). N.N. and W.G. acknowledge financial support from the FONDAP Center of Astrophysics 15010003, and the BASAL Center of Astrophysics CATA. N.N. acknowledges the Geneva team for support in using the HARPS pipeline.
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Footnotes
 ...velocities^{}
 Based on observations made with ESO telescopes at the Silla Paranal Observatory under programme IDs 072.D0419 and 073.D0136.
 ... ^{}
 Tables 4 and 5 are only available in electronic form at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsweb.ustrasbg.fr/cgibin/qcat?J/A+A/502/951
 ... HARPS^{}
 High Accuracy Radial velocity Planetary Search project developed by the European Southern Observatory.
All Tables
Table 1: The Cepheids studied listed with increasing period.
Table 2: The projection factor ( ) and the velocities ( [CC]) derived from the CC radial velocity curves.
Table 3: Coefficients of the linear relations between the amplitude of the radial velocity curve and the wavelength.
All Figures
Figure 1: Interpolated radial velocity curves based on the crosscorrelation method are presented for each Cepheid in our sample. Uncertainties are too small to be seen (around 0.5 km s^{1}). The horizontal lines near extrema give an indication of . The short horizontal lines are the velocities (see Sect. 6) corresponding to the CC method (solid line), the centerofmass velocity of Paper III (dotted line) and from Fernie et al. (1995, dashed line). 

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In the text 
Figure 2: a) The correction factor induced on the projection factor by the crosscorrelation method is shown as a function of the logarithm of the period of the star. b) The periodprojection factor (p) relation from Paper II (crosses and solid line) and the corrected relation suitable for the crosscorrelation method (diamonds and dashed line). 

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In the text 
Figure 3: a) Wavelength dependency of the amplitude of the crosscorrelated radial velocity curves for each star in our sample. The corresponding linear relation are defined as: . b) The corresponding slopes ( ) as a function of the period. 

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In the text 
Figure 4: Corrections to apply to the relation (Eq. (5)) in the blue ( nm) and in the red ( nm). 

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In the text 
Figure 5: [CC] as a function of [GCD] (diamond) and [N08] (triangles). The solid and dashed line are the corresponding linear interpolation. 

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In the text 
Copyright ESO 2009