A Review of PL Relations for Cepheids
Nial R. Tanvir
Institute of Astronomy, University of Cambridge,
Madingley Road, Cambridge,
United Kingdom.
(nrt@ast.cam.ac.uk)
Cepheids have long played a pivotal role in the extragalactic
distance scale. By providing accurate distances to nearby galaxies,
they enable the determination of zero-points for
various secondary indicators, some of which can in turn
be used to measure the far-field Hubble flow.
The Hubble Space Telescope has increased the range of Cepheid studies
by at least a factor five in distance, meaning that far more galaxies
can now be used to calibrate the secondary indicators (see eg.
Freedman, in this proceedings ).
Slowly but surely, the growing number of galaxies with Cepheid
distances is narrowing the range in estimates of the Hubble constant!
The accuracy of Cepheid distance measurements depends, firstly,
on the quality of the PL relation calibration, and, secondly, on how
well the Cepheids are observed in the target galaxy itself.
Below we outline some of the recent work on Cepheids
with particular emphasis on that pertaining to HST distance
estimates.
For a fuller discussion of many of these issues, see
Tanvir 1997.
Introduction to Cepheids
The Cepheid period-luminosity-colour relation (PLC) is
easily understood in terms of fundamental physics.
The period-luminosity relation (PL) is a projection
of the PLC which is constrained by the boundaries of
the instability strip (eg. Madore and Freedman, 1991).
While PLC relations have very low dispersion,
the PL relations, especially in the optical, have a higher
dispersion, although this is effectively reduced by the application
of a reddening correction (see next section).
HST Cepheid Studies
All HST studies have followed an essentially
similar observing procedure, of monitoring the Cepheids
in the V-band and obtaining I-band observations
on a few occasions.
The I-band observations allow an estimate to be made
of the extinction to the Cepheids themselves,
and in most cases this is applied explicitly as a correction
term to get the true distance modulus.
In effect, a "reddening-free" Wesenheit magnitude,
W(V,I)=V-2.45(V-I), is calculated for each Cepheid
and a PL relation fitted to these.
Fortuitously the effect of dust extinction mimics quite
well the intrinsic luminosity-colour correlation in Cepheid properties,
to wit redder Cepheids at a given period will be
fainter. Thus the method used by most HST studies, of
correcting for extinction, actually produces a distance indicator
with a dispersion of 6% or less in distance for individual Cepheids
(see below).
Since those galaxies studied generally provide fairly large
samples of Cepheids, the uncertainty in distance quickly becomes
dominated by systematic effects.
Calibrating the PL Relations
In recent years most extragalactic Cepheid studies have used
the PL relations obtained from Cepheids in the Large Magellanic Cloud.
The uncertainty of the calibration then comes down
to how well the distance to the LMC itself is known,
and how well the LMC Cepheids have been observed.
Any error introduced in either of these becomes
a systematic error in the determination of the Hubble
constant.
Distance to the LMC
The true distance modulus of the LMC has been estimated by many
methods, with a value of 18.5 commonly adopted for the purposes
of calibrating the PL relations. Results from
the Hipparcos satellite have produced several new estimates of
the LMC distance (summarized
by
Turon and Perryman, in this proceedings) which also bracket
this value, with a slight trend to higher values.
Cepheids in the LMC
There have been many studies of the Cepheids in the
LMC in both the optical and infrared.
Recently Tanvir (1997) produced a new calibration of the PL
relations based on a compilation of the work from
many authors (see figure 1 below). These relations are based on a sample
of 53 Cepheids covering a broad period range
and assume a true distance modulus of the LMC of 18.5.
The formal error on the zero-point of the reddening corrected
W(V,I) relation is 0.02 mag.
Figure 1 shows period luminosity relations for LMC Cepheids
in the V-band, I-band and for reddening corrected Wesenheit
magnitudes, where W(V,I)=V-2.45(V-I). The lines are fitted to the
points with log(P) less than 1.8.
Applying Cepheid PL Relations
Clearly, Cepheids in the LMC have very low internal
dispersion around the reddening corrected PL relation.
So the error in estimating the distance to another
galaxy will be dominated by (a) observational uncertainties in
the photometry of faint Cepheids (b) calibration of the CCD
camera (c) systematic differences in the population of Cepheids
resulting, for example, from variations in metallicity.
Dealing with each of these in turn:
Photometry with WFPC2 Images
Crowded field photometry with WFPC2 images is particularly
difficult due to the very poor sampling of the psf.
An indication that crowding is becoming important is
when the shorter period Cepheids,
which are individually likely to be more
effected by crowding since they are fainter,
scatter above the mean line in the period-luminosity
relation. An example, and thorough discussion, of this can be seen in
Saha et al. (1996) for the galaxy NGC4536.
In these circumstances the problem can be minimized by
restricting the fit to the longer period variables.
Fortunately, in most HST target galaxies, crowding is manifestly
less serious.
Calibration of WFPC2
The basic calibration steps (ie. debiassing, flat-fielding
etc.) of HST data are performed by a pipeline at STScI.
Holtzman et al. (1995) made a careful study of WFPC2 photometry,
and this work is widely used to transform WFPC2 magnitudes
onto the standard photometric systems.
Unfortunately the WFPC2 CCDs themselves exhibit several
undesirable effects associated with charge transfer
efficiency. These are still not fully understood,
and readers are referred to the
WFPC2 instrument page
at STScI for latest details.
Systematic Differences between Cepheid Samples
Perhaps the most important unresolved issue with regard
to using Cepheids as distance indicators is that of
possible systematic differences between populations.
In the past this has been thought to be confined to
differences in metallicity, although one may now worry
about other parameters in the light of the Hipparcos
results showing that even the main-sequences of nearby
clusters shift in the HR diagram in a way which is not
understood (eg.
Turon and Perryman, in this proceedings ).
As far as a simple metallicity dependence is concerned,
constraints are still rather poor.
In practical terms, what is important is not the effect
of metallicity
on PL relations in individual bands, but the effect on
the reddening corrected distance estimate.
Furthermore, metallicities should be treated
differentially with respect to the LMC sample.
In figure 2, we compare the results of Freedman and
Madore (1990) for three fields in M31 (of different
metallicity) with (a) the theoretical predictions
of Chiosi et al. (1993); and (b) the empirical work on the large EROS sample
of Cepheids in the magellanic clouds by Beaulieu et al. (1997).
In this test the comparatively strong effect seen in the EROS
data gives a better fit to the M31 points.
As far as the distance to M31 is concerned using the former
curve to find a correction factor
implies a distance modulus of 24.42, while the
the latter gives 24.51.
Figure 2 shows the distance estimated to three fields
(of differing metallicity) in M31, based on V- and I-band
data of Freedman and Madore (1990) corrected for
reddening as described above. These are compared to
the predicted metallicity
dependence of Chiosi et al.
(1993) shown in red and Beaulieu et al. (1997) in green.
The metallicity of the LMC is taken to be Z=0.008.
The data points are as given in the FM90 paper; no
attempt has been made to disentangle the correlated and
uncorrelated errors for the M31 data.
If the Beaulieu et al. (1997) result is correct (and we note that it has a
large formal error itself) then
an error of 0.2 mag could be made in targeting galaxies
of either particularly high or low metallicity.
As Beaulieu et al. point out, the correction works to make
high metallicity galaxies further and low metallicity galaxies
nearer, thus reducing significantly the remaining disagreements
between different groups on the value of H_0.
Finally we note that Sekiguchi and Fukugita (1997) have recently
presented evidence for a surprisingly large metallicity dependence
for Milky-Way Cepheids between their PL distances and the
distances implied by main-sequence fitting to their host clusters.
What is particularly striking about
this work is the close to 1 to 1 correlation of residuals
from the PL relations in the V, J, H and K bands and
the general trend with metallicity.
Whether these effects are due to metallicity dependence of
the Cepheids or in the main-sequence fitting (or both!)
is not yet clear.
Conclusions
In the HST era
Cepheids are more important than ever as the foundation
of the extragalactic distance scale.
LMC Cepheids exhibit a small dispersion around the
reddening-corrected PL relation suggesting that individual
Cepheids with good photometry should give distances accurate
to about 6% internally.
The remaining concerns for HST studies are refining the
distance to the LMC, calibration uncertainties
with WFPC2 and possible systematic
differences between Cepheid samples.
Metallicity variations compared to the calibrating LMC
sample, for example,
may still be important at the 10% level for distance
determination.
References
Beaulieu et al., 1997, A.&A., 318, L47.
Chiosi et al., 1993, Ap.J.Supp., 86, 541.
Freedman and Madore, 1990, Ap.J., 365, 186.
Holtzman et al., 1995, P.A.S.P., 107, 1065.
Madore and Freedman, 1991, P.A.S.P., 103, 933.
Tanvir, 1997,
in The Extragalactic Distance Scale, eds Livio etal, CUP.
Saha et al., 1996, Ap.J., 466, 55.
Sekiguchi and Fukugita, astro-ph/9707229 .