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
and
(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
10d) Cepheids have absolute
magnitudes MV < -4, and long-period (P 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 26 mag, much brighter than the HST
detection limit of ~ 30 mag, for distance scale work using Cepheids.
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.
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).