3.5 Calibration and Uncertainties
The luminosities of Cepheids are difficult to calibrate with high accuracy because of their low space density, since they are massive stars in a very brief evolutionary phase. The nearest Cepheids are about 200 pc away. Trigonometric parallaxes to these stars at 5% accuracy would require a precision of 0.25 milliarcsec. This is at the frontier of the capabilities of pioneering CCD astrometric programs under ideal conditions according to Monet (1988), and the difficulty in achieving this precision is exacerbated by the brightness of the Cepheids relative to field reference stars (usually a difference of 8 mag or more!). Thus, for even the nearest Cepheids, a precision of 10% from trigonometric parallax programs is probably optimistic.
An alternative to trigonometric parallax is statistical parallax. In this technique, one is still faced with the dramatic brightness differences between the program object and field reference stars for proper motion, but the linear increase in proper motion signal-to-noise with time allows this technique to be competitive. The most complete statistical parallax analysis of Cepheids to date is that of Wilson et al. (1991). Their derived zero-point for the P-L relation is in excellent agreement with that of Caldwell and Coulson (1986), and Feast and Walker (1987), although the uncertainty is roughly three times as large. This agreement between kinematic and photometric calibrations suggest that there is no systematic error at the 0.3 mag level.
In Figure 4, four modern paths to the calibration of Cepheid luminosities are illustrated. The first two involve cluster main-sequence fitting and the last two require angular diameters and radii of field Cepheids. The use of the Magellanic Cloud Cepheids as intermediaries is necessary when the period distribution of the calibrators is significantly different from the period distribution of the extragalactic Cepheids. The final path requires the fewest links and will be based on the most direct measurements of distance, but will not be realized until the completion of the Narrabri 1 km Michelson interferometer (see below).
Figure 4. Four modern paths to extragalactic Cepheid distances. The links indicate observationally determined corrections such as absorption (A) and metallicity ([Fe/H]).
Very recently, the promise of direct distance determination to the LMC has been realized by the precise measurement of the SN1987A circumstellar ring light echo by Panagia et al. (1991). This determination has resulted in a true distance modulus of 18.55 ± 0.13 mag to SN1987A, which corresponds to a distance uncertainty of only 6%. Using an inclination of 27 deg and a position angle for the line of nodes of 170 deg, and assuming that the supernova resides in the disk, we obtain a distance modulus of 18.57 ± 0.13 for the center of mass of the LMC. (The east side is known to be closer from the work of de Vaucouleurs 1980 and Gascoigne and Shobbrook 1978). While this distance is likely to remain the most precise for some time to come, the still open question of a luminosity-metallicity dependence suggests that we retain the (independent) techniques for calibration of Cepheid luminosities, and these are now discussed in more detail. (Note that the interpretation of the LMC Ring data has been called into question by Dwek and Felten 1992.)
A number of galactic Cepheids are found in clusters or associations. A recent list is contained in Table 2 of Feast and Walker (1987). This table reveals a total of 28 potential calibrating Cepheids, of which 17 are believed to be in clusters, 8 are believed to belong to associations, and three are calibrated by geometry or binarity. The cluster Cepheids are predominantly short-period stars (most have periods shorter than 12 days) and the association Cepheids are long-period stars (with periods between 15 and 70 days). Clusters are attractive because they permit a distance and reddening estimate independent of observations of the Cepheid. (Future work may also allow individual metallicity estimates for some clusters.) However, the technique is probably not capable of precisions exceeding ± 0.2 mag in distance modulus (equivalent to 10% in distance) per cluster. There are a number of reasons for this. First, the most obvious and most easily measured stars in the cluster are those near the turnoff. These stars lie on a near-vertical segment of the main sequence where any photometric error will be magnified dramatically into an error in the cluster distance modulus. Second, stars near the turnoff may also lie an unknown distance redward of the actual zero-age main sequence due to evolution. In addition, such stars are frequently fast rotators (of unknown inclination) or binaries. Thus, an irreducible intrinsic scatter is introduced in the cluster main-sequence and different distances may be derived solely from different interpretations of the same data.
Relief from the steep and sparsely populated upper main sequence may be obtained by attempting measurements of lower mass stars but here field contamination overwhelms the CMD. Discrimination may be possible using proper motions or radial velocities since velocity dispersions for galactic clusters are expected to be less than 1 km s-1, but early epoch plates did not go sufficiently deep for effective proper motion measurements, and stars cool enough for modern radial velocity spectrometers are very faint, leading to little progress in this area. Both upper and lower main sequence stars may also suffer from differential reddening across the face of the cluster.
Once the position of the main sequence is deduced from the observations, it is necessary to determine the difference in distance modulus between the cluster in question and a local cluster: the Pleiades or Hyades. Originally, the Hyades was the cluster of choice because the smaller distance (40 pc) resulted in a more accurately determined distance modulus. However, the age of that cluster and its high metallicity made it less than ideal for comparison with young, Cepheid-bearing clusters. In the last decade, the Pleiades has been the cluster of choice, at a true distance modulus of 5.57 ± 0.08 mag (130 pc) according to the field star parallax fit of van Leeuwen (1983). The Pleiades may also be calibrated relative to the Hyades after appropriate determination of the metallicity difference and its effect on the absolute magnitude and color, as discussed by Turner (1979). These two techniques are in good agreement.
If the mean radius of a Cepheid variable can be determined by independent means, then in principle a measurement of the angular size of the Cepheid will determine its distance. Such geometric techniques are referred to as ``Baade-Wesselink'' methods after Baade (1926) and Wesselink (1946) who first described how the light and color curves could be combined with the integrated radial velocity curve to obtain the mean stellar radius. An excellent critique of the application of these methods is given by Gautschy (1987). While the full promise of these techniques is yet to be realised, the problems encountered to date are not fundamental in origin. Feast and Walker (1987) have reviewed the zero-points determined from Baade-Wesselink techniques and compared them to the cluster and association zero-point. They find that Baade-Wesselink luminosities tend to be 0.05-0.15 mag brighter. An unresolved question regarding these techniques is the most appropriate color index for use in the Baade-Wesselink solutions.
The angular radius can be determined from direct measurement or inferred from photometry after correction for reddening. The apparently largest Cepheids have angular diameters in the range 1.0-1.5 milliarcsec. Direct techniques for measuring the angular radius include lunar occultation and Michelson interferometry. Lunar occultations are necessarily restricted to stars near the ecliptic, the most favorable being Gem, X Sgr, and W Sgr. Ridgway et al. (1982) have measured the angular radius of Gem, but with a precision of only 20%. Ground-based Michelson interferometry also shows significant promise with angular resolutions of 4 x 10-5 arcsec being obtainable with the 1 km baseline proposed by Davis (1985). Such observations do not suffer from the severe positional and scheduling restrictions of lunar occultation work and hence long-period Cepheids may be observed.
Indirect techniques for estimating the angular diameter rely on establishing a surface brightness from a color and measuring the flux in some bandpass. The relationship between surface brightness and effective temperature is calibrated by measuring colors of giants whose angular diameters have been accurately measured from lunar occultations. While a correction for absorption is required, the derived angular diameter is very insensitive to the value of the absorption used for the (V - R) relationship. A very clear introduction to this technique is given in Barnes et al. (1977).
The nature of the uncertainties associated with a Cepheid-based distance to a galaxy can best be elucidated by considering individual cases. Here we will consider the existing uncertainties in distances to the LMC (0.051 Mpc), M33 (0.84 Mpc), NGC 2403 (3.2 Mpc), and M101 (7.5 Mpc).
LMC: In this nearby galaxy, high-precision photometry of Cepheids of all periods is possible with 1m-class telescopes. Individual reddening estimates are possible using BVI photometry and the scatter in observed reddenings is found to be of order 0.04 mag. According to Caldwell and Laney (1990), 141 Cepheids in the LMC have published photometry using modern detectors. The error in the distance estimate is therefore dominated by the Galactic calibration errors of order ± 0.13 mag or 7% in distance, comparable with the geometric determination of Panagia et al. (1991).
M33: The recent work of Freedman et al. (1991) is the most comprehensive CCD study of M33 Cepheids to date. Their distance estimate is based on BVRI photometry of 10 Cepheids with good phase coverage. With smaller samples of variables, the finite width of the instability strip in temperature and its manifestation as a non-zero luminosity scatter in P-L relations become increasingly important. The mean magnitudes derived from their data have uncertainties of order 0.1 mag which are too large for meaningful individual reddening corrections. Instead a global reddening correction is used. The final internal error in their distance estimate is ± 0.09 mag or 5% in distance. This uncertainty must be added quadratically to the Galactic calibration error, resulting in a total uncertainty of ± 0.16 mag or 8% in distance.
NGC 2403: Freedman and Madore (1988) have obtained few-phase I-band photometry of eight Cepheids in NGC 2403. Their final estimate of the distance modulus uncertainty is ± 0.24 mag or 12% in distance (including the uncertainty in the LMC distance modulus which they estimate as ± 0.15 mag). Interestingly, the contribution of photometric errors to the distance uncertainty is now similar in magnitude to the effects of the instability strip temperature width.
M101: This galaxy is the most distant object in which Cepheid variables have yet been discovered. Cook et al. (1986) found and characterized two Cepheid variables using CCD photometry. A quantitative error analysis was not attempted in that paper, but following the reasoning of Freedman and Madore (1988) we expect the distance uncertainty to be of order ± 0.3 mag or 15% in distance.
Perhaps the most important lesson to be learned from these studies is that the Cepheid distance uncertainty for a galaxy with single-band photometry of two Cepheids at 7.5 Mpc is only twice that of the very nearby and well-studied LMC. Said another way, if photometry is of sufficient quality to allow period determination, an uncertainty of 15% in distance (or less) is virtually assured.