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8. Obtaining accurate cepheid distances

Once new extragalactic Cepheids are found, at least four issues need to be adequately addressed, all of which are tightly coupled to common sets of observations: (1) Periods have to be determined, (2) Complete light curves have to be delineated, (3) Mean magnitudes must be derived, and finally, (4) Accurate colors are required for reddening determinations. Needless to say, it would be hard to derive (1) the periods (or prove that a star is in fact a Cepheid) without (2) the light curves; and vice versa. Similarly, (3) and (4), accurate mean magnitudes and colors generally depend on correct period-phasing and proper lightcurve fitting. But the requirements for accurate periods are in fact quite different from the requirements for accurate magnitudes. The number of data points required to yield a time-averaged magnitude (of specified precision) increases as the square of the lightcurve amplitude. This makes colors and mean magnitudes based on short wavelength observations more costly in observing time than their longer-wavelength counterparts. On the other hand, for fixed photometric uncertainties, periods increase in accuracy almost linearly with the time interval over which the observations are spaced. Furthermore, periods good to a few percent can be obtained using only moderately accurate photometry after only a dozen or so cycles, thereby making the time constraint a minimal one.

Finally, one must contend with the intrinsic width of the instability strip as projected into the PL relation. Increasing N, the numbers of Cepheids is the most obvious solution here. For the B-band the equivalent dispersion in magnitude in the Cepheid PL relation is ± 0.35 mag. The error in the mean apparent distance modulus decreases like sqrtN. In the absence of reddening then it would appear that for apparent distance moduli alone, a dozen Cepheids will give the requisite accuracy in the mean. But of course the real problem, once again, comes when trying to deal with reddening. And an example using two band-passes only illustrates this graphically. In such a case, the ensemble-averaged extinction essentially comes from differencing the mean apparent moduli found at two different wavelengths. Multiplying this difference by the ratio of total-to-selective extinction appropriate to those two wavelengths and subtracting the product from the mean apparent modulus gives the final true modulus. If 10% in distance is the goal (0.2 mag in true modulus), then for the filter combination VI, simple arithmetic shows that two to three dozen Cepheids are required to establish the mean moduli to such a degree that the reddening corrected modulus has a final error of less than 0.2 mag. Of course, either increasing the number of wavelengths and/or increasing the wavelength baseline will each reduce the final error on the mean without demanding an increase in sample size.

In closing this section, we present our adopted fiducial multiwavelength PL relations. We emphasize that these relations differ slightly from those published by other workers up to this point, because they are derived from sets of data which are now totally self-consistent. Specifically, all of the PL relations are based on the same stars in order to eliminate sample-dependent variations in the solutions. Furthermore, Cepheids with log P > 1.8 are excluded from the least-squares fits due to uncertainties in their reddenings and their evolutionary status. The LMC data set (scaled and dereddened as outlined in the next section) has been chosen as fiducial because of its large sample size, large wavelength coverage and because the LMC is very close to being face-on, thereby minimizing the effects of back-to-front geometry on the solutions. The relations are centered on log P = 1.0, the mid-point of the range of periods considered here. Errors on the quoted coefficients are given after each of the values. Following each of the PL relations, the quantity in square brackets is the rms dispersion about the mean for that relation.

MB = -2.43 (± 0.14) (log P - 1.00) - 3.50 (± 0.06) [± 0.36]
MV = -2.76 (± 0.11) (log P - 1.00) - 4.16 (± 0.05) [± 0.27]
MR = -2.94 (± 0.09) (log P - 1.00) - 4.52 (± 0.04) [± 0.22]
MI = -3.06 (± 0.07) (log P - 1.00) - 4.87 (± 0.03) [± 0.18]

[Note that the RI magnitudes are on the Cousins system, while our JHK magnitudes are on the CIT/CTIO system. There are 32 LMC Cepheids for which BVI photoelectric photometry is available in the range 0.2 < log P < 1.8. R photometry is not available for many of the stars used above; however, R magnitudes were derived using the methodology set out in Freedman (1988b).]

Finally, we give below consistent PL solutions, based on a smaller set consisting of only 25 LMC stars, each of which has BVRIJHK photometry available (given the same conditions outlined above):

MB = -2.53 (± 0.28) (log P - 1.00) - 3.46 (± 0.12) [± 0.40]
MV = -2.88 (± 0.20) (log P - 1.00) - 4.12 (± 0.09) [± 0.29]
MR = -3.04 (± 0.17) (log P - 1.00) - 4.48 (± 0.08) [± 0.25]
MI = -3.14 (± 0.17) (log P - 1.00) - 4.84 (± 0.06) [± 0.21]
MJ = -3.31 (± 0.11) (log P - 1.00) - 5.29 (± 0.05) [± 0.16]
MH = -3.37 (± 0.10) (log P - 1.00) - 5.65 (± 0.04) [± 0.14]
MK = -3.42 (± 0.09) (log P - 1.00) - 5.70 (± 0.04) [± 0.13]

The effective wavelengths for each of the seven bandpasses were chosen to be appropriate for a G-star spectrum (see for example, Bessell 1979) where, for future reference, we have adopted: B(0.444 µm), V(0.550 µm), R(0.653 µm), I(0.789 µm), J(1.25 µm), H(1.60 µm), K(2.17 µm).

Finally, it should be noted that there are external checks on the Cepheid calibration and distance scale derived from it that have been applied to galaxies within the Local Group. An extensive review of these methods (including the use of RR Lyrae stars, red giant luminosity functions, novae and long-period variables, to name just a few) confirms (conservatively at the ± 0.2 mag level) the basic solidity of the Cepheid calibration (van den Bergh 1989, de Vaucouleurs 1991). For details, the interested reader is referred to those reviews and the many references cited therein. Because many of these independent methods that provide checks on the Cepheid distance scale use intrinsically fainter stars, it is unlikely that galaxies significantly beyond the Local Group will be of much use in further refining the agreement (or disagreement) between the various estimators. However, more extensive and more precise observations of those same (faint) distance indicators within these and other Local Group galaxies will be crucial for fine tuning the calibration, and may be especially helpful in establishing the level at which metallicity corrections are needed in Population I and Population II distance indicators alike.

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