The number of scientific papers on a given subject tends to grow exponentially. Care will therefore be taken in this section to avoid painting a picture that is too simplistic. But it is hoped that what follows is representative of the general directions Cepheid calibrations have taken over the last one or two decades.

The main advances have been based upon new technologies, in particular highly sensitive, multi-wavelength, charge-coupled device (CCD) observations, and increased use of infrared (IR) wavelengths. With the advent of these technologies, cepheid calibration has shifted towards the empirical, rather than the speculative. Rather than attempt to infer, for example, the value of the coefficient for the colour term, or attempt to deduce reddenings, it has recently proved more profitable to simply observe. This has altered the character of the work. Scientific papers are more likely to commence with a description of the technology applied, or with numerical analysis of data, than with a semi-theoretical pulsation equation. This era is also characterised by the pursuit of an accurate value for the zero point, and a return to PL rather than PLC relations. In the light of the new technologies, evidence has gradually been building to suggest that many PLC relations are mainly reddening corrections, and that even a ‘true’ PLC relation would only have marginal benefits given the random and correlated errors already present. However, it must be emphasised that this issue has not been fully resolved, and remains an area of controversy.

Brodie & Madore (1980) applied numerical simulations to the random and correlated errors in PL and PLC relations to investigate the likelihood of the observed scatter being due to reddening, rather than intrinsic colour. They wrote:

Madore & Freedman (1991) provided a hypothetical example to illustrate how observed scatter might easily arise through reddening. What follows is based upon this example, except that the present author has substituted the actual data of Martin, Warren & Feast (MWF) (1979) from the previous section. First the possibility of a true PLC is eliminated; by imagining that the instability strip is very narrow in both color and magnitude at fixed period. MWF observed a sample of Cepheids with a total spread in reddening of about +/- 0.10 magnitudes (two standard deviations). Differentially obscured, these stars would have a period-colour correlation with a full colour width of 0.2 magnitudes. For an assumed ratio of



this would produce a period-luminosity relation with a width of 0.66 in <V>.

Figure 20 shows MWF’s PL relation from the previous section (Fig. 18), to which the present author has added, by eye, a line of best fit and envelope lines 0.33 magnitudes either side. Approximately 85% of the stars lie on or within the boundaries of this reddening envelope. This does appear to suggest that the scatter might be easily explained in terms of differential reddening alone.

Fig. 20: MWF’s PL relation with predicted spread due to reddening
after Brodie & Madore (1980) and Madore & Freedman (1991)
Click on image to enlarge

Madore & Freedman (1991) wrote:

Multiwavelength observations have become increasingly important in dealing with the reddening problem. There are two main sources of interstellar reddening: reddening caused by dust in our own Galactic foreground, and reddening in the stars’ parent galaxy. The first is relatively straightforward to deal with. A given stellar system occupies a small area of the sky and, to a reasonable approximation, all of the light from it passes through the same column of Galactic dust. The differential reddening effects are therefore acceptably small. However, reddening in the parent galaxy has proved much more problematic because it differentially affects stars according to their location and optical depth. Many authors (e.g. MWF) appear to have placed little emphasis on this. Nevertheless, it has been shown (Freedman 1988, Cardelli et al. 1989) that it is possible to determine estimates of total reddening directly from multiwavelength observations, irrespective of the location of the source.

Figure 21 shows the proportion of extinction at l to total visual extinction, plotted against inverse wavelength (The three curves are for different values of . Note that the extinction becomes negligible, and the curves coincide, at IR wavelengths).

Fig. 21: Extinction-wavelength relation (Cardelli et al. 1989)
Click on image to enlarge

Accordingly, a Cepheid (observed at a particular point in its cycle) will have a different apparent distance modulus at each wavelength. With an appropriate extinction law, Madore found that it was possible to determine the total reddening by simultaneously fitting these distance moduli to a standard one. Figure 22 shows the light curve of a Galactic Cepheid at 7 wavelengths from the optical to the infrared (K = 2.2 microns). Note the reduction in amplitude of the light curves with wavelength.

Fig. 22: Light curves at different wavelengths for a Galactic Cepheid (Madore and Freedman 1991)
Click on image to enlarge

Observations of Cepheids in the longer wavelengths are of particular interest. Whatever the degree of scepticism regarding the validity of PLC relations, no one disputes that the instability strip has a finite width. But the observed width of the strip will certainly be smaller at longer wavelengths. It is probably about 1.6 magnitudes in B, but falls to only 0.5 magnitudes in the infrared K band. There are other advantages to IR observations; the amplitude of a variable star's light curve in the IR is so small that a single photometric measurement of magnitude taken at random (a random-phase observation) may have an accuracy comparable with a whole series of time-averaged measurements at optical wavelengths. (Obviously the IR curves, with their low amplitudes and poorly defined peaks make them unsuitable for period determination, but these are easily obtained in the optical bands). Figure 23 shows the startling effect that observations at longer wavelengths have on the observed PL scatter.

Fig. 23: The PL relation at 7 wavelengths (Madore & Freedman 1991)
Click on image to enlarge

As the wavelength increases there is an obvious reduction in scatter. Note that variables from both the SMC and LMC are combined in these relations and there is no apparent difference in slope between the two, due for example to metallicity, as has been suggested. But it is still not possible to say whether the reduction in scatter in the IR is due to the reduced intrinsic colour width, or the removal of reddening (because scattering in the IR will be much reduced)! As such the PLC debate has not been settled. However, given that the reddening ought to have been reduced by a factor of 50 in the K plot, and that some scatter remains, perhaps here we are, for the first time, genuinely observing the width of the instability strip. Given the quality and possibilities of data like these, it is not surprising that the PLC relation has been largely side-stepped in recent years. Earlier in this paper, the post-Baade period - with its discussions of Q, non-linear PL relations, and endless attempts at colour-correction - was described as being mediaeval in character – this is the justification, the modern picture does appear to be simpler.

That differences in composition will affect the PL relation in some way is beyond dispute, but the size of the effect remains uncertain. The metallicities of stars are, after all, to some extent correlated with their surrounding interstellar medium, which in turn affects extinction. Once again, there is a requirement to decouple the effects of reddening in order to investigate this properly.

Freedman & Madore (1990) using recently developed multiwavelength techniques for reddening correction, applied an empirical test to the problem. They examined three fields situated 3, 10 and 20 kpc from the centre of M31. Abundance analyses of supernova remnants and HII regions suggested that the metallicities of these regions ranged over a factor of 5. Their chief aim was to determine whether metallicities affected the distance moduli for Cepheids (and by implication, the zero points for PL relations). Figure 24 shows the apparent moduli for Cepheids in the three fields plotted against inverse wavelength. The three dashed lines represent the predicted apparent moduli as a result of reddening.

Fig. 24: Apparent moduli for Cepheids in three fields of M31 (Freedman & Madore 1990)
Click on image to enlarge

They concluded that the large observed differences in apparent moduli could be wholly attributed to the different extinctions in the three fields, and that the true distance moduli for the three fields agreed to within +/- 0.13 magnitudes.

They wrote:

Recently the wide field planetary camera on the Hubble Space Telescope has been used to carry out a similar study in M101. Kennicutt et al. (1998) found that there was tentative evidence for a small Z-dependence of the PL relation at V and I wavelengths, whose trend was in the expected direction (metal-rich Cepheids appearing brighter and closer than metal-poor ones). However, they concluded that the overall effects of metallicity dependence on the distance scale would be, "of the order of a few percent or less". It is interesting to note that they recommended improvements to the camera’s capabilities at longer wavelengths to reduce the effect further.

Feast & Catchpole (1997) used parallaxes of Galactic Cepheids determined by the Hipparcos satellite to recalibrate the Cepheid zero point. This generated a great deal of interest in the astronomical community, as it represented an independent calibration of a new kind. There was also the obvious appeal of the application of new space-based technology. However, before their findings are examined, they must first be put into context. It was stated earlier that the zero point of the Cepheid distance scale has barely moved in over 30 years. Figure 25 is adapted from a table of influential calibrations as judged by Webb (1999 p179). The only addition has been Baade’s 1952 adjustment of approximately 1.5 magnitudes (which, in hindsight should have been less than one magnitude!). The graph is only intended to be illustrative, and the data selected may have been subject to unintentional bias, but it nevertheless indicates the stability of the zero point since Sandage & Tammann (1968). As such, the Hipparcos recalibration should perhaps be regarded as a comforting verification, rather than a revolution – it was to all intents and purposes identical to that of 1968, and would prima facie only represent a 4% increase in distance to the LMC as previously determined by Madore & Freedman (1991).

The mean standard error of a single Hipparcos parallax measurement was 1.5 milliarcseconds (mas) – a larger parallax error than most of the individual Cepheids’ parallaxes. The largest individual parallax was 7.56 mas for Polaris, our closest cepheid, which unfortunately is almost certainly a first overtone pulsator and required special treatment. Nevertheless, a large sample of 223 stars was available and a mean parallax was obtained after weighting each one according to its estimated standard error.

Fig. 25: Influential calibrations of the PL relation.
Click on image to enlarge

In this way they obtained the PL relation:

However, there were really two parts to Feast & Catchpole’s paper. After obtaining this relation they turned their attention to considerations of reddening and metallicity in the LMC; arguably matters entirely unconnected with their numerical recalibration. Thus, although their zero point would cause only an incremental change in the distance to the LMC, their reddening estimates caused them to quote an increase in distance to the LMC of 10%, and by implication a full 17% for M31.

Madore & Freedman (1998) responded to these distance estimates with a paper of their own based on multiwavelength observations, which effectively moved the LMC back to its original position. And so the extinction versus intrinsic colour debate continues.

There have been other objections to the parallax estimates themselves. One of Feast & Catchpole’s referees raised concerns about a possible Lutz-Kelker bias in their calculations, but the authors believed its effects to be small. However, this debate has continued. Oudmaijer et al. (1998) re-analysed the data and determined a relatively large bias. They concluded that this restored the zero point to its previously estimated value. The likelihood of errors due to the effects of binary orbits has also been raised. Szabados (1997) wrote:

Figure 26 shows Szabados’ plot of the PL relation for nearby Cepheids based upon Hipparcos data. Open circles are binary Cepheids, filled circles solitary ones. The dotted line is the least squares fit based on the latter. The binaries certainly appear to exhibit more scatter, but basing the mean relation on the solitary stars may not be justifiable.

Fig 26: Hipparcos PL relation for solitary and binary Cepheids.

Whatever the final outcome of this debate, Feast & Catchpole’s recalibration was highly statistical in nature, and such treatments will always be subject to criticism on the bases of error and bias.

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