A&A 438, L9-L12 (2005)
DOI: 10.1051/0004-6361:200500139
A. Mérand1 - P. Kervella1 - V. Coudé du Foresto1 - S. T. Ridgway1,2,3 - J. P. Aufdenberg2 - T. A. ten Brummelaar3 - D. H. Berger3 - J. Sturmann3 - L. Sturmann3 - N. H. Turner3 - H. A. McAlister3
1 - LESIA, UMR8109, Observatoire de Paris-Meudon, 5, place Jules Janssen, 92195 Meudon Cedex, France
2 -
National Optical Astronomical Observatory 950 North Cherry Avenue, Tucson, AZ 85719, USA
3 -
Center for High Angular Resolution Astronomy, Georgia State University, PO Box 3965, Atlanta, Georgia 30302-3965, USA
Received 6 May 2005 / Accepted 7 June 2005
Abstract
Cepheids play a key role in astronomy as standard candles
for measuring intergalactic distances. Their distance is usually
inferred from the period-luminosity relationship, calibrated using the
semi-empirical Baade-Wesselink method. Using this method, the distance
is known to a multiplicative factor, called the projection
factor. Presently, this factor is computed using numerical models - it
has hitherto never been measured directly. Based on our new
interferometric measurements obtained with the CHARA Array and the
already published parallax, we present a geometrical measurement of
the projection factor of a Cepheid, Cep. The value
we determined, p=1.27
0.06, confirms the generally adopted value
of p=1.36 within 1.5 sigmas. Our value is in line with recent
theoretical predictions of Nardetto et al. (2004, A&A, 428, 131).
Key words: techniques: interferometric - stars: variables: Cepheids -
stars: individual: Cep - cosmology: distance scale
Cepheid stars are commonly used as cosmological distance indicators, thanks to their well-established period-luminosity law (P-L). This remarkable property has turned these supergiant stars into primary standard candles for extragalactic distance estimations. With intrinsic brightnesses of up to 100 000 times that of the Sun, Cepheids are easily distinguished in distant galaxies (up to about 30 Mpc distant). As such, they are used to calibrate the secondary distance indicators (supernovae, etc...) that are used to estimate even larger cosmological distances. For instance, the Hubble Key Project to measure the Hubble constant H0 (Freedman et al. 2001) is based on the assumption of a distance to the LMC that was established primarily using Cepheids. Located at the very base of the cosmological distance ladder, a bias on the calibration of the Cepheid P-L relation would impact our whole perception of the scale of the Universe.
The P-L relation takes the form
,
where L is the (absolute) luminosity, P the period,
the
slope, and
the zero point. The determination of
is
straightforward: one can consider a large number of Cepheids in the
LMC, located at a common distance from us. Calibrating the zero-point
is a much more challenging task, as it requires an independent
distance measurement to a number of Cepheids. Ideally, one should
measure directly their geometrical parallaxes, in order to obtain
their absolute luminosity. Knowing their variation period,
would then come out easily. However, Cepheids are rare stars: only a
few of them are located in the solar neighborhood, and these nearby
stars are generally too far away for precise parallax measurements,
with the exception of
Cep.
The most commonly used alternative to measure the distance to a
pulsating star is the Baade-Wesselink (BW) method. Developed in the
first part of the 20th century (Wesselink 1946; Baade 1926), it
utilizes the pulsational velocity
of the surface of
the star and its angular size. Integrating the pulsational velocity
curve provides an estimation of the linear radius variation
over the pulsation. Comparing the linear and angular
amplitudes of the Cepheid pulsation gives directly its distance. The
most recent implementation (Kervella et al. 2004) of the BW method makes
use of long-baseline interferometry to measure directly the angular
size of the star.
Unfortunately, spectroscopy measures the apparent radial velocity
,
i.e. the Doppler shift of absorption lines in the stellar atmosphere, projected along the line of sight and integrated over the stellar disk. This is where p, a projection factor, has to
be introduced, which is defined as
.
The general BW method can be
summarized in the relation:
Until now, distance measurements to Cepheids used a p-factor value
estimated from numerical models. Looking closely at Eq. (1), it
is clear that any uncertainty on the value of p will create the same
relative uncertainty on the distance estimation, and
subsequently to the P-L relation calibration. In other words, the
Cepheid distance scale relies implicitly on numerical models of these
stars. But how good are the models? To answer this question, one
should confront their predictions to measurable quantities. Until now,
this comparison was impossible due to the difficulty to constrain the
two variables
and d from observations, i.e. the angular
diameter and the distance.
Among classical Cepheids, Cep (HR 8571,
HD 213306) is remarkable: it is not only the prototype of its
kind, but also the Cepheid with the most precise trigonometric
parallax currently available, obtained recently using the FGS instrument aboard the Hubble Space Telescope (Benedict et al. 2002). This direct measurement of the distance opens
the way to the direct measurement (with the smallest sensitivity to
stellar models) of the p factor of
Cep, provided that
high-precision angular diameters can be measured by interferometry.
To achieve this goal, interferometric observations were undertaken at
the CHARA Array (ten Brummelaar et al. 2003,2005), in the
infrared K' band (
)
with the Fiber Linked Unit for Optical
Recombination (Coudé du Foresto et al. 2003) (FLUOR) using two East-West baselines
of the CHARA Array: E1-W1 and E2-W1, with baselines of 313 and 251 m
respectively. Observations took place during summer 2004 for E2-W1
(seven nights between JD
and JD
)
and Fall 2004
for E1-W1 (six consecutive nights, from JD
to JD
). The pulsation phase was computed using the following
period and reference epoch (Moffett & Barnes 1985):
P=5.366316 d,
(Julian date), the 0-phase being defined at maximum
light in the V band. The resulting phase coverage is very good for the
longest baseline (E1-W1), while data lack at minimum diameter for the
smaller one (E2-W1)
The FLUOR Data reduction software (DRS) (Coudé du Foresto et al. 1997), was used
to extract the squared modulus of the coherence factor between the two independent apertures. All calibrator stars were chosen in a catalogue
computed for this specific purpose (Mérand et al. 2005) (see
Table 1). Calibrators chosen for this work are all K giants, whereas Cep is a G0 supergiant. The spectral type difference is properly taken into account in the reduction, even
though it has no significant influence on the final result. The
interferometric transfer function of the instrument was estimated by
observing calibrators before and after each
Cep data
point. The efficiency of CHARA/FLUOR was consistent between all
calibrators and stable over the night around 85%. Data that share a
calibrator are affected by a common systematic error due to the
uncertainty of the a priori angular diameter of this
calibrator. In order to interpret our data properly, we used a
specific formalism (Perrin 2003) tailored to propagate these
correlations into the model fitting process. Diameters are derived
from the visibility data points using a full model of the FLUOR instrument including the spectral bandwidth effects (Kervella et al. 2003). The stellar center-to-limb darkening is
corrected using a model intensity profile taken from tabulated values
(Claret 2000) with parameters corresponding to
Cep
(
,
and solar metallicity). The
limb darkened (LD) angular diameter comes out 3% larger than its
uniform disk (UD) counterpart.
Table 1: Calibrators with spectral type, uniform disk angular diameter in K band (in milliarcsecond) and baseline (Mérand et al. 2005).
The theoretical correction for LD has only a weak influence on the p-factor determination, since that determination is related to a diameter variation. For example, based on our data set, a general bias of 5% in the diameters (due to a wrongly estimated limb darkening) leads to a bias smaller than 1% in terms of the p-factor. Differential variations of the LD correction during the pulsation may also influence the projection factor: comparison between hydrodynamic and hydrostatic simulations (Marengo et al. 2003) showed negligible variations. An accuracy of 0.2% on the angular diameters for a given baseline is required to be sensitive to dynamical LD effects. This is close to, but still beyond, the best accuracy that we obtained on the angular diameter with a single visibility measurement: 0.35% (median 0.45%).
Among the various sets of measurements of the radial velocity
available for
Cep, we chose measurements from Bersier et al. (1994) and Barnes et al. (2005). These works offer the best phase coverage, especially near the extrema, in order to
accurately estimate the associated photospheric amplitude. In order
not to introduce any bias due to a possible mismatch in the radial
velocity zero-point between the two data sets, we decided to reduce
them separately and then combine the resulting p-factor. An
integration over time is required to obtain the photospheric
displacement (see Eq. (1)). This process is noisy for unequally
spaced data points: the radial velocity profile was smoothly
interpolated using a periodic cubic spline function.
Fitting the inferred photospheric displacement and observed angular
diameter variations, we adjust three parameters: the mean angular
diameter
,
a free phase shift
and the
projection factor p (see Fig. 1). The mean angular diameter is
found to be 1.475
0.004 mas (milliarcsecond) for both radial
velocity data sets. Assuming a distance of 274
11 pc
(Benedict et al. 2002), this leads to a linear radius of 43.3
1.7 solar radii. The fitted phase shift is very small in both cases (of the order of 0.01). We used the same parameters (Moffett & Barnes 1985) to compute the phase from both observation sets and considering that
they were obtained more than ten years apart, this phase shift
corresponds to an uncertainty in the period of approximately five
seconds. We thus consider the phase shift to be reasonably the result
of uncertainty in the ephemeris.
![]() |
Figure 1: Radial velocity smoothed using splines. A. Radial velocity data points, as a function of pulsation phase (0-phase defined as the maximum of light). This set was extracted using a cross-correlation technique (Bersier et al. 1994). The solid line is a 4-knot periodic cubic spline fit. B. Residuals of the fit. |
Open with DEXTER |
![]() |
Figure 2:
p-factor determination. A. Our angular diameter measurements (points). Crosses correspond to the medium baseline (E2-W1), while circles correspond to the largest baseline (E1-W1). The continuous line is the integration of the 4-knots periodic cubic spline fitted to the radial velocities (Fig. 1). Integration parameters:
![]() ![]() |
Open with DEXTER |
The two different radial velocity data sets lead to a consolidated
value of p=1.27
0.06, once again assuming a distance of 274
.
The final reduced
is 1.5. The error bars account for three independent contributions: uncertainties in the radial velocities, the angular diameters and the distance. The first
was estimated using a bootstrap approach, while the others were
estimated analytically (taking into account calibration correlation
for interferometric errors): for p, the detailed error is p=1.273
.
The error is dominated by the distance contribution (see Table 2).
Table 2: Best fit results for p, with the two different radial velocity sets. The third line is a weighted average of the two individual measurements. Fourth and fith lines are the detailed quadratic contribution to the final error bar. Last line gives the final adopted value with the overall error bar. References are: (1) Bersier et al. (1994); and (2) Barnes et al. (2005).
Until now, the p-factor has been determined using models:
hydrostatic models (Burki et al. 1982) produced the generally adopted
value, p=1.36. First attempts were made by Sabbey et al. (1995) to
take into account dynamical effects due to the pulsation. They
concluded that the average value of p should be 5% larger than in
previous works (1.43 instead of 1.36) and that p is not constant
during the pulsation. Because they increased p by 5%, they claimed
that distances and diameters have to be larger in the same
proportion. More recently Nardetto et al. (2004) computed pspecifically for Cep using dynamical models. Different values
of p were found, whether one measures diameters in the continuum or
in the layer where the specific line is formed. In our case, broad
band stellar interferometry (angular diameters are measured in the
continuum) these authors suggest p=1.27
0.01. Concerning the
variation of p during the pulsation, they estimate that the error in
terms of distance is of the order of 0.2%, smaller than what we would
have been able to measure with our interferometric data set. While our
estimate, p=1.27
0.06, is statistically compatible with this
recent work, marginally with the widely used p=1.36, and not
consistent with the former value p=1.43 at a 2
level. We
note that Gieren et al. (2005) have recently derived an expression of
the p-factor as a function of the period that predicts a value of
1.47
0.06 for
Cep. While this value is in agreement
with the modeling by Sabbey et al. (1995), is is slightly larger than
the present measurement (by 2.4
). As a remark, Gieren et al. obtain a distance of 280
4 pc for
Cep, that is slightly larger than Benedict et al.'s (2002) value 274
11 pc assumed in the present work. Assuming this new distance estimation
with our data would result in a p-factor of 1.30
0.06, bringing the agreement to 2
only.
Our geometrical determination of the p-factor, p=1.27
0.06, using
the IBW method is currently limited by the error bar on the
parallax (Benedict et al. 2002). Conversely, assuming a perfectly known
p-factor, the uncertainty of the stellar distance determined using the
same method would have been only 1.5%, two-times better than
the best geometrical parallax currently available. The value we
determined for p is statistically compatible with the value
generally adopted to calibrate the Cepheid P-L relation in most
recent works. It is expected that the distance to approximatively 30 Cepheids will be determined interferometrically in the near future using particularly the CHARA Array and the VLT Interferometer (Glindemann 2005). In order not to limit the final
accuracy on the derived distances, theoretical p-factor studies
using realistic hydrodynamical codes is necessary. With a better
understanding of the detailed dynamics of the Cepheid atmospheres, we
will be in a position to exclude a p-factor bias on the calibration
of the P-L relation, at a few percent level.
Acknowledgements
We thank P. J. Goldfinger for her assistance during the observations. The CHARA Array was constructed with funding from Georgia State University, the National Science Foundation, the W. M. Keck Foundation, and the David and Lucile Packard Foundation. The CHARA Array is operated by Georgia State University with support from the College of Arts and Sciences, from the Research Program Enhancement Fund administered by the Vice President for Research, and from the National Science Foundation under NSF Grant AST 0307562.
Table 3:
Individal measurements. Columns are (1) date of
observation, JD0 =
;
(2) phase; (3, 4) u-v coordinate in
meter; (5) squared visibility and error; (6) corresponding limb
darkened disk diameter in mas; (7, 10) HD number of calibrators,
prior and after the given data point respectivaly, 0 means that
there was no calibrator; (8, 9, 11, 12) quantities for computing the
correlation matrix (Perrin 2003):
are
errors on the estimated visibility of the calibrators.