2.3.3. The Third Rung: Determining the Zeropoints of Standard Candles

The Cepheid Variable Period-Luminosity Relation

The relationship between the period of pulsation and the intrinsic luminosity of Cepheids has been known for 75 years. In general, the calibration of the zero-point of the Cepheid PL relation based on main-sequence fitting to young Galactic clusters is not reliable. This is due to poorly determined photographic magnitudes and, especially, poorly determined foreground reddenings. In the early 80's, a vigorous program of Stromgren and H photometry of 8 open clusters which contain Cepheid variable stars was carried out by E. Schmidt. The choice of photometric system here was designed to give maximum sensitivity to reddening variations across the cluster as well as the mean metallicity of the cluster. Schmidt's (1984) redetermination of the distance moduli of these eight clusters with well-established Cepheid members yielded an astounding result: The zeropoint of the Cepheid PL relationship was too bright by 0.5 mag. Hence, any galaxy whose distance was determined by measuring the Periods of Cepheids located in that galaxy, would have a distance which was too large by 0.5 mag (28%). That better photometry of stars in open clusters would yield a systematic error that large is worrisome testimony to the poor quality of the previously obtained photographic magnitudes and reddening estimates. In recent years, there has been some question about the accuracy of the H photometry obtained by Schmidt but a better recalibration of these open clusters has not yet appeared. Because of these discrepant results, and the generally poor quality of photographic photometry, most practitioners of the extragalactic distance scale ironically do not use the zeropoint of the Cepheid PL relationship as defined by open clusters in our Galaxy but instead, rely on the one obtained by deriving the distance to the LMC, a fundamentally different galaxy than the Milky Way. We will explicit consider how the distance to the LMC is determined a bit later on in this Chapter.

Possible Problems with Cepheids

There are several nearby galaxies within the 4 Mpc Cepheid horizon and in the 1970's and early 80's there were vigorous photographic campaigns aimed at detecting and measuring Cepheid variables in these galaxies. Cepheid magnitudes were generally determined at either Blue or Visual wavelengths, both of which are sensitive to small amounts of reddening in the host galaxy itself. This is a vexing problem for observations of Cepheids in that they are usually located in or near dusty spiral arms. A breakthrough in the credibility of the Cepheid magnitude measurements occurred in the mid 80's with the work of Barry Madore and Wendy Freedman. Thanks to improvement in detector technology, it became possible to make measurements of Cepheid Variables in the near-IR part of the spectrum where the sensitivity to host galaxy reddening was down by an order of magnitude. Furthermore, the amplitude of the variability increases as you go to longer wavelengths and thus Cepheids are more easy to detect, in time series observations of galaxies, at these longer wavelengths. Not surprisingly, when IR measurements of Cepheids were compared to B or V-band measurements different distances to the same galaxy were found. Unfortunately, there is a paucity of data available to calibrate the near-IR Cepheid PL relationship. For instance, the most practical band in which to make the measurements is I-band and, to date, there are only 22 Cepheids in the LMC with measured I-band magnitudes (although the MACHO observations, described later, will add significantly to this database) compared to the hundreds that have been measured (photographically) in the B and V-bands.

The use of the Cepheid PL relationship as a distance indicator is the best example of the Population I distance ladder as, ultimately, the zero point of the relationship is derived through main-sequence fitting of young clusters. However, there has been some concern that the Cepheid PL relationship may be dependent on metallicity. If this is the case, the PL relationship would have a different zeropoint for the LMC than say for M31 as the LMC has a significantly lower metallicity. Theoretically, a small dependence would be expected as the atmospheric opacity of these stars can be dominated by electron scattering. This will also affect the observed color of the Cepheids which suggests that there should be an observable Period-Luminosity-Color relationship. Observationally, it is quite difficult to test for the dependence of the Cepheid PL relationship on metal abundance. The best test done to date is to examine the radial dependence of the PL relationship in the Andromeda Galaxy, where there is a detectable abundance gradient. While observations have failed to find any significant difference in the PL relationship as a function of radius in M31, its overall abundance gradient is not very steep. Furthermore, the observations of Freedman and Madore (1990) have recently been challenged by Gould (1994a) who suggests that indeed the data does reveal a metallicity dependence on Cepheid luminosity. In particular, Gould suggests that metal poor Cepheids are less luminous than metal rich Cepheids at a given period, although his argument is not very convincing as the sample size is small. Still this issue remains somewhat unresolved and could be important at the 10% level.

RR Lyrae Absolute Magnitude Scale

With analogy to the Cepheid PL scale, it is possible, but much more difficult, to use a Population II distance scale ladder which is based on the Population II main-sequence to calibrate the absolute magnitude of RR Lyrae variable stars. These stars are found in globular clusters and in the general Galactic halo field star population. Their absolute magnitudes, however, are around +0.5, which limits their applicability to distances of 0.5 - 1.0 Mpc. But, since these stars are found in galactic globular clusters, then they can be used to determine the distances to these clusters which in turn leads to a determination of the Globular Cluster Luminosity Function (GCLF) in our Galaxy. If we make an assumption that the GCLF is invariant from galaxy-to-galaxy (this assumption will be explored in detail later in this chapter), then the Population II distance scale can be extended out to distances as large as 20 Mpc due to the large intrinsic brightness of globular clusters.

Seeking the galactic calibration of the absolute magnitudes of RR Lyrae stars has been an ongoing endeavor for 30 years. Throughout that time there have been various lines of evidence which suggest that the absolute magnitude has a small dependence on metal abundance, although this evidence has never been entirely clear. Two recent determinations are from Carney (1992) who derive:

(5)

and Simon and Clement (1993) who derive:

(5a)

The other preferred calibration is to assume no metallicity dependence and a zeropoint of +0.60 mag, that is

(6)

It would be desirable to reach resolution between equations 5,5a and 6. Note that equation 6 is recovered in the case where [Fe/H] = -2.6 for equation 5 and [Fe/H] = -1.1 for equation 5a. Most galactic RR Lyraes come from a stellar population with [Fe/H] -1.5. A better calibration of M_v for RR Lyrae stars is potentially achieved by determining distances to the host globular clusters via main sequence fitting. Unfortunately, nearby Population II main-sequence stars are quite rare and, to date, only 1 such star has an accurate parallax measurement. At the moment, it is unknown how many stars in the HIPPARCOS data base are population II main sequence stars but, one expects several dozen good candidates out of their sample of 100,000 nearby stars. Hence, over the next 5 years we should see a more reliable calibration of the Population II main sequence, and hence improved distance determinations to globular clusters from main-sequence fitting.

The Baade-Wesselink Technique

We note finally that for any pulsationally driven variable star, there is a dynamical way for determining the luminosity of the star. This technique, called the Baade-Wesselink technique, works on the principle that the pulsational period is driven by the dynamical timescale of the star which depends on ^-1/2. This means that luminosity variations are directly proportional to variations in stellar radius. An accurate radial velocity curve as a function of phase is required for determination of the radius and this is the observationally difficult part of the technique. Once the radius is known, the luminosity of the star can be determined from the well known relation between luminosity, effective temperature and radius:

(7)

The effective temperature of the star is determined from its color or absorption line spectrum. To date the method has been applied to a small sample of RR Lyrae and Cepheid variable stars with mixed results. In particular, application of the method to RR Lyrae stars by Storm et al. 1994 yields a value +0.2 mag fainter than the zeropoint of equation 6.

Brightest Red Supergiants

Calibration of M-supergiant luminosities in open clusters in our Galaxy suggests that they reach a maximum luminosity of M_v -9. At these luminosities, M-supergiants can be detected as individual stars out the distances of at least 10 Mpc. The overall population of these stars in a galaxy depends strongly on its star formation rate and, for instance, an actively star forming galaxy like M101 (see Figure 2-4) would be expected to be rich in M-supergiants. There are, however, several practical difficulties associated with the use of these stars as distance indicators:

Like Cepheids, their calibration rests generally on poor photographic photometry and main-sequence fitting to open clusters which have reddening variations across them.

It is completely unclear if there is a threshold number of stars that a galaxy must have in order for it to host at least one brightest Red Supergiant. Hence, statistical population effects may come into play; a galaxy like the Milky Way with some 10¹¹ stars may have a few brightest Red supergiants, but dwarf galaxies like IC 1613 and NGC 6822 with only 10⁸ stars, while having supergiants, may not have the brightest red supergiant.

These stars are located in active regions of star formation in external galaxies and hence will be somewhat reddened.

Using only UBVR photometry, it is difficult to unambiguously distinguish between a foreground M-dwarf in our Galaxy and a distant M-supergiant in another galaxy. Indeed, there have been cases where foreground M-dwarfs were mistakenly assumed to be M-supergiants in another galaxy (in this case M101 - see Humphreys et al. 1986). Infrared photometry is needed to clearly distinguish between an M-dwarf and M-supergiant star.

Figure 2-4: CCD Image of M101 taken by the author in blue light. M101 is a typical high surface brightness actively star forming spiral galaxy which is rich in Cepheids located in and near the spiral arms.