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Cepheids variable stars have been fundamental to unlocking the cosmological distance scale since Henrietta Leavitt used them in 1912 to estimate the distances to the Magellanic Clouds. Of the various DIs discussed in this Chapter, the Cepheid method is the only one involved in the Hubble constant but not the peculiar velocity problem. Indeed, it is probably safe to say that the raison d'être for Cepheid observations is the ultimate determination of H0. They will do so, however, in conjunction with, not independently of, the secondary distance indicators discussed in later sections.

Cepheids are post-main sequence stars that occupy the instability strip in the H-R diagram. They pulsate according to a characteristic ``sawtooth'' pattern, with periods that can range from a few days to a good fraction of a year. Cepheids exhibit an excellent correlation between mean luminosity (averaged over a pulsation cycle) and pulsation period. This correlation is shown in Figure 1 for Cepheids recently measured by the Hubble Space Telescope (HST) in the nearby galaxy M101 (solid points), and also for Cepheids in the Large Magellanic Cloud (LMC) as they would appear if the LMC lay at the distance of M101. It is apparent that the correlation is extremely similar for the two galaxies. Modern calibrations of the Cepheid Period-Luminosity (P-L) relation in the V and I bandpasses are

Equation 1 (1)


Equation 2 (2)

(Ferrarese et al. 1996). The absolute zero points of these P-L relations have been obtained by observing Cepheids in the the Large and Small Magellanic Clouds, whose distances are known from main sequence fitting (Kennicut et al. 1995). Equations (1) and (2) show that Cepheid variables are intrinsically bright stars. Even short-period (P appeq 10d) Cepheids have absolute magnitudes MV < -4, and long-period (P appeq 50-100d) Cepheids are 2-3 magnitudes brighter still. It follows that individual Cepheid stars can be observed at relatively large distances. Indeed, with the HST Cepheids can be observed out to the distance of the Virgo cluster and possibly beyond. To be useful as distance indicators, however, Cepheids cannot be merely detected. Because they are found in crowded fields, they must be well above the limit of detectability at all phases in order to be accurately photometered. These stringent requirements place a limit of mV appeq 26 mag, much brighter than the HST detection limit of ~ 30 mag, for distance scale work using Cepheids.

Figure 1a
Figure 1b
Figure 1. Cepheid variable Period-Luminosity (PL) relations for the V and I bandpasses. Data for M101 and the Large Magellanic Cloud are shown. Adapted from Ferrarese et al. (1996).

Cepheids yield distances to their host galaxies by comparison of their absolute magnitudes, inferred from the P-L relation, with their observed apparent magnitudes. Specifically, the distance to the host galaxy is obtained by fitting equations (1) and (2), plus a distance modulus offset µ = 5 log(d/10) (where d is in parsecs), to the observed mV and mI versus log(P) diagram. (The same exercise may of course be carried out in other bandpasses as well.) An important advance has been made in recent years by Freedman, Madore, and coworkers, who have developed a method for correcting for extinction in the host galaxies (Freedman & Madore 1990; Freedman et al. 1991). In brief, the photometry is done in several bandpasses, and the magnitudes corrected for an assumed value of the extinction within the host galaxy. The distance modulus is determined for each bandpass, as described above. The value of extinction which brings the distance moduli in the various bands into agreement is assumed to be the correct one. This technique works best when data for a wide range of wavelengths, including if possible the near infrared, are available.

The great utility of Cepheids has been recognized in the designation of an HST Key Project to measure Cepheid distances for 20 nearby galaxies. This program, led by Wendy Freedman, Robert Kennicut, and Jeremy Mould, produced its first results in late 1994. As of this writing (July 1996), Cepheid distances from the Key Project are available for only a handful of galaxies. Distances for the remaining galaxies are expected to become available over the next few years. The results that have received the greatest attention to date involve the Virgo cluster galaxy M100, in which over 50 Cepheid variables have now been accurately measured (Freedman et al. 1994; Mould et al. 1995; Ferrarese et al. 1996). Fitting the universal P-L relations above to the M100 data yields a distance of 16.1 ± 1.3 Mpc. When combined with a suite of assumptions concerning the morphology and peculiar velocity of the Virgo cluster, this distance suggests a Hubble constant of about 85 km s-1 Mpc-1 (Freedman et al. 1994).

Unfortunately, the Hubble constant estimate obtained from M100 has received undue attention. This is understandable, given that determination of H0 is the long-term aim of the Key Project. And, of course, values of H0 in excess of ~ 75 km s-1 Mpc-1 are difficult to square with most estimates of the age of the universe based on its oldest constituents. But as the Key Project group has emphasized (Kennicutt et al. 1995), a single galaxy in the Virgo cluster with a good Cepheid distance does not allow one to estimate the Hubble constant with any accuracy. In fact, the Virgo cluster is a poor laboratory in which to estimate H0 no matter how many galaxies one has Cepheid distances for. The reasons are simple: Virgo's depth is a good fraction (~ 30%) of its distance, and its peculiar velocity is likely to be a good fraction (~ 20-30%) of its Hubble velocity. The velocity/distance ratio of any single Virgo object, or even group of objects, may therefore be a poor approximation of H0, and it is difficult to gauge the systematic errors that affect it.

Thus, Cepheid variables will not themselves be used to measure H0. Instead, they will be used to obtain accurate distances for several tens of galaxies within about 20h-1 Mpc. These galaxies will in turn serve as calibrators for the secondary distance indicators, such as Type 1a Supernovae and the Tully-Fisher relation, that are applicable in the far field of the Hubble flow (and occupy the remainder of this Chapter). Initial steps in this direction have already been taken by Sandage, Tammann, and coworkers (Sandage et al. 1996), who used HST Cepheid distances (their own, not those of the Key Project) to calibrate historical and contemporary Type Ia Supernovae. When they apply this calibration to distant Type Ia SNe (Tammann & Sandage 1995), they derive H0 = 56-58 km s-1 Mpc-1 (the lower value applies to B-band, and the higher value to V-band, measurements; Sandage et al. 1996). There is considerable controversy, however, surrounding the calibration of the historical photometry used in the SNe Ia calibration. Furthermore, the Sandage group has neglected the correlation between the peak luminosity of SNe Ias and the width of their light curves, an effect which now appears important (Section 6). Until these issues are resolved, and agreement between the Sandage and HST Key Project groups on local Cepheid distances achieved, estimates of H0 based on this approach should be considered preliminary.

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