Abstract
Hipparcos parallaxes have recently become available for a sample of Galactic Cepheids, and we have used these new distances to calibrate the Cepheid periodluminosity (PL) relation at six wavelengths (BV IJHK). Comparing these calibrations with previously published multiwavelength PL relations we find agreement to within 0.07 ± 0.14 mag, or 4 ±7% in distance. Unfortunately, the current parallax errors for the fundamental pulsators (ranging in signal-to-noise = π/σ π from 0.3 to 5.3, at best) preclude an unambiguous interpretation of the observed differences, which may arise from a combination of true distance modulus, reddening and/or metallicity effects. We explore these effects and discuss their implications for the distance to the Large Magellanic Cloud (LMC) and the Cepheid-based extragalactic distance scale. These results suggest a range of LMC moduli between 18.44 ±0.35 and 18.57 ±0.11 mag; however, other effects on the Cepheid PL relation (e.g., extinction, metallicity, statistical errors) are still as significant as any such reassessment of its zero point.

1. Introduction Feast and Catchpole (1997 = FC97 hereafter) have recently published the first results on parallaxes to Galactic Cepheids based on measurements from the Hipparcos satellite. They list data for the 26 highest signal-to-noise Cepheid parallaxes; and after an extensive series of reductions (see their Table 2 ) they conclude that the best fit PL relation for the visual bandpass is M V = − 2.81 log(P ) − 1.43, with a standard error on the Hipparcos zero point of ±0.10 mag, adopting the slope from prior work on LMC Cepheids. The authors go on to apply this V-band solution to determining the distance modulus of the LMC corrected for E(B − V ) = 0.074 mag. Adding a metallicity correction of +0.042 mag and adopting < V > o − log(P ) from Caldwell & Laney (1991) gives (m − M V ) LM C o = 18.70 ± 0.10 mag. In this Letter we go beyond the V-band PL relation and explore the implications of the Hipparcos data for the multiwavelength calibrations of the Cepheid PL relation from the blue (B-band) out into the near infrared (2.2µm K-band).
Comparison with V-Band Period-Luminosity Relations
In Figure 1 we compare differentially four calibrations (heavy dotted lines) of the V-band Cepheid PL relation with the FC97 Hipparcos-based relation (solid horizontal lines). The first two comparisons (in the upper two panels) are with the relations given by Madore & Freedman (1991; hereafter MF91) , derived from self-consistent sets of LMC Cepheid data whose stars either had complete BV RI observations (MF91.1 containing 32 Cepheids) or complete BV RIJHK observations (MF91.2 containing 25 stars) 1 These first two solutions indicate the sensitivity of slopes and zero points to sample selection, which are considerable, but within the quoted statistical uncertainties: ±0.11 and ±0.20, respectively for the slopes, and ±0.05 and ±0.09 mag, for the zero points. So as to make the subsequent comparisons consistent, the original Sandage & Tammann (1968) calibration (ST68.1 in the lower left panel) has been placed on the modern Hyades/Pleiades Galactic cluster distance scale by applying a single offset of +0.13 mag derived from the average difference between the absolute magnitudes of the Cepheids used in the 1968 calibration updated to Feast and Walker (1987) , their Table 2 . This distance scale corresponds to a Hyades modulus of 3.27 (see Pel 1985) and uses the Pleiades main sequence, at a modulus of 5.57 (van Leeuwen, 1983) to effectively correct for the over-metallicity of the Hyades with respect to the older Galactic clusters in which the Cepheid calibrators are found. 2 Finally, the FW87 calibration itself is plotted 1 Tanvir (1997) has pointed out that there may be a small correction amounting to 0.04 mag to the published I-band magnitudes of these LMC Cepheids due to the originally sparse sampling and consequent averaging of their light and color curves. For the past five years we have been obtaining new VI CCD observations of the LMC calibrators at Las Campanas and now also at Siding Springs Observatories. These new data are designed to address those concerns.
Comparison with V-Band Period-Luminosity Relations
in the lower left panel. In all panels the dashed horizontal lines represent the fiducial Hipparcos calibration flanked by thin parallel lines at ±0.10 mag.
Comparison with V-Band Period-Luminosity Relations
The error bars of all of the plotted previously published relations overlap with errors quoted for the Hipparcos solution (a formal uncertainty was not given by ST68, so we have arbitrarily assigned them an error of ±0.05 mag). However, the offsets are not randomly distributed, with each of the solutions appearing to be systematically fainter in V with respect to the Hipparcos calibration by about 0.1 mag. We discuss the significance and possible implications of this difference in the following sections.
Multiwavelength Period-Luminosity Relations
In Madore & Freedman (1991) we published fiducial PL relations in seven bandpasses: BV RIJHK. These were all based on selecting self-consistent sets of previously published LMC Cepheid data, scaled to an LMC true distance modulus of 18.50 mag and applying a single line-of-sight reddening correction using E(B − V ) = 0.10 mag. Thirty-two stars were available for a calibration of BV RI PL relations; 25 stars were used for an alternative set of BV RIJHK calibrations. In the following we compare those multiwavelength PL relations with the Hipparcos sample of Galactic Cepheids, individually corrected for foreground reddening and scaled to their geometric parallax distances.
Multiwavelength Period-Luminosity Relations
We have collected from the literature multiwavelength (BV IJHK) mean magnitudes for as many of the Hipparcos-calibrating Cepheids as have been published (notably for the infrared Wisniewski & Johnson 1968 , Welch et al. 1984 , Laney & Stobie 1992 and reference therein). These form rather disjoint subsets. After eliminating the suspected overtone pulsators listed by FC97, the total available sample with parallaxes drops from 26 to 20. Of these only 7 have mean magnitudes published at all six wavelengths, while 10 and 13 Cepheids, respectively have either BV IJK or BV JK magnitudes in common. We have analyzed these four groups of stars independently, but self-consistently, in the following way.
Multiwavelength Period-Luminosity Relations
Using the Hipparcos parallaxes and Galactic reddenings adopted by FC97 from Fernie, Kamper & Seager (1993) scaled to the various wavelengths using the extinction law of Cardelli, Clayton & Mathis (1989) , we derived absolute magnitudes for each of the Cepheids in each of the observed wavelengths. (We note that these corrections for interstellar extinction are not inconsiderable, ranging up to 2 mag in B for several stars). The resulting PL relations are shown Hipparcos trigonometric parallaxes. Following the Venice Meeting in June 1997 the value had changed only slightly to 5.33 ±0.06 mag (C. Turon, private communication) If adopted, this Pleiades modulus would make the Galacticcluster-based calibrations approximately 0.3 mag fainter than the FC97 solution plotted in Figure 1 . At this point in time the Galactic cluster zero point appears to be in a state of flux, and we will not comment on it further, except to note that the Hipparcos calibration will undoubtedly converge on a more accurate zero point than we have access to at this precise moment.
Multiwavelength Period-Luminosity Relations
in the six panels of Figure 2 . Error bars are one-sigma uncertainties from the quoted parallaxes. Note the highly correlated nature of the individual data points about the fiducial lines. And too, we remind the reader that the computation of distances and their related errors from observed parallaxes is non-trivial (Brown et al. 1997) , as distances are not linearly related to parallaxes, and parallax errors can subtly bias samples. A full treatment of this issue is beyond the scope of this paper, but we note that selection biases at least are minimized for stars having the smallest reported errors. As discussed by Brown et al., given the true parallax distribution the expected biases follow naturally; however, corrections to the observed parallaxes require assumptions about the true distribution, and detailed modeling. Fortunately for this application the Cepheid sample is not parallax-selected; the objects being chosen in advance based on their optical variability, periods and apparent magnitudes.
Multiwavelength Period-Luminosity Relations
The differences between these individual (trigonometric) absolute magnitudes and the predicted BV IJHK magnitudes derived from the mean PL relations of MF91 (solid lines in Figure 2 ) are each plotted in Figure 3 against the corresponding B-band residual. The (B − V ) intrinsic color residuals are plotted against the B-band residuals in the upper right panel. The individual residuals at a given wavelength contain random contributions from the parallax uncertainties, reddening errors, and finally the intrinsic (temperature-induced) magnitude residuals which reflect the finite width of the Cepheid instability strip. The observed residuals are however extremely large (nearly 5 mag peak-to-peak) and are almost certainly dominated by the (achromatic) errors in the parallaxes, given the strict unit-slope correlation of the mag-mag residuals, and the total lack of any correlation between the magnitude-color residuals (Figure 3 ).
Multiwavelength Period-Luminosity Relations
Wavelength-dependent offsets between the six mean solutions independently will reflect (1) errors in the adopted true distance to the LMC (which set all of the zero points in the MF91 multiwavelength PL relation calibrations), (2) reddening errors in the adopted extinction to the LMC sample of calibrating Cepheids, and finally (3) intrinsic differences between the LMC and Galactic Cepheids, for example, due to metallicity.
Multiwavelength Period-Luminosity Relations
Our first solution considers the largest data set (in terms of parallaxes) but the one that is most restricted in terms of wavelength coverage: it consists of 19 Cepheids observed in B and V . Weighted by the square of the signal-to-noise ratio in the Hipparcos parallax, the residuals were summed and averaged at each of the two wavelengths giving mean offsets between the LMC calibration and the Galactic Cepheids. The variance in each mean offset was then calculated from the average of the squares of these same residuals again inversely weighted by the variance in the individually quoted parallaxes. The differences are ∆B = +0.23 ± 0.35 mag and ∆V = +0.16 ± 0.28 mag, in the sense that the LMC Cepheid calibration appears to be too faint with respect to the Galactic calibration. (Further restricting the sample to only those 12 stars with π/σ π > 2.0 changes ∆B to +0.22 ± 0.24 mag and ∆V to +0.15 ± 0.17mag.)
Multiwavelength Period-Luminosity Relations
If the (statistically marginal, but apparently systematic) differences in the B and V solutions were to be ascribed to reddening alone, then the Galactic data and the LMC calibration can be reconciled by invoking an increase of ∆E(B − V ) = 0.07 mag in the adopted mean reddening to the LMC Cepheid sample. This is consistent with a similar suggestion regarding the LMC Cepheid calibration made recently by Bohm-Vitense (1997) based on different data. This reddening solution has the consequence that it would also require the distance modulus of the LMC to be revised downwards by -0.06 mag to 18.44 mag; the uncertainty on this offset being at least as large as the uncertainty in the individual moduli (±0.3 mag), depending on the degree of correlation in those cumulative uncertainties. This particular path, of a reddening solution, cannot be considered definitive. Other possibilities are: (1) the LMC true modulus should be increased by (0.23 + 0.16)/2 = +0.20 mag, without any change to the foreground reddening, or (2) that there are differential metallicity corrections amounting to −0.23 and −0.16 mag that need to be applied at the B and V wavelengths, respectively. Of course any suitably contrived linear combination of the above three effects could also be invoked. More constraints on the problem are obviously needed.
Multiwavelength Period-Luminosity Relations
An alternative possibility is that some of the wavelength-dependent effects seen in the comparison of Galactic (high metallicity) data with the LMC (lower metallicity) data could be due to chemical composition differences between the two samples. Taken at face value the dependence of the apparent V modulus on metallicity would be very large, ∆V/∆[F e/H] = 0.16/0.15 = 1.1(±1.9) mag/dex, assuming that the full offset in V noted in the above comparison is due to metallicity, and adopting a metallicity underabundance of 1.4× between the LMC and the Solar neighborhood (see, for example, FW87). However, we note that this effect is basically indistinguishable from reddening in its form (as evidenced by our first set of solutions), and that the offset (whatever its origin) when treated as reddening leads to a true distance modulus for the LMC that is unchanged, from previous assumptions, at 18.50 mag. Given this apparent degeneracy between reddening and metallicity, and the current large uncertainties in the parallaxes, assessing the dependence on metallicity from these data alone will remain problematic.
Multiwavelength Period-Luminosity Relations
To obtain added leverage on the solution, moving to the infrared has numerous well known advantages, as first articulated in McGonegal, McLaren, McAlary & Madore (1982) : reddening effects are known to decrease with wavelength, in a well defined and calibrated manner; and simultaneously, metallicity effects are also expected to decrease in amplitude with increased wavelength.
Multiwavelength Period-Luminosity Relations
Our second solution is based on 13 Cepheids each having BV HK data in common. This four-color solution gives a derived reddening increase for the LMC Cepheid sample of +0.04 ± 0.08 mag, with no formal offset in the derived 18.50 ± 0.13 mag true modulus for the LMC. Our next approximation employs 10 Cepheids each now having BV IJK mean magnitudes. Here the formal solution for the true modulus for the LMC is 18.53 ± 0.14 mag, with a corresponding increase in the mean reddening of +0.06 ± 0.07 mag. Finally, we have analyzed a sample of 7 Galactic Cepheids, each having BV IJHK photometry, to obtain one last solution: ∆E(B − V ) = 0.07 ± 0.07 mag with (m − M ) LM C = 18.57 ± 0.11 mag. The fit to this final set of observations is shown in Figure 4 ; the χ 2 weighted residual fitting surface being shown as an inset. The individual apparent moduli discussed here, and their errors, are summarized in Table 1 .
Multiwavelength Period-Luminosity Relations
Finally, if we now adopt the metallicity correction of ∆V = 0.04 mag advocated by FC97 and assume that the effects at JHK are negligible, (and eliminate B and I from the solution given that metallicity corrections for these filters are not well defined at this time) we find for this 4-color solution ∆E(B − V ) = 0.06 ± 0.11 mag with (m − M ) LM C = 18.57 ± 0.11 mag. This is virtually indistinguishable from the full BV IJHK solution given above.
Discussion and Conclusions
We have used the Hipparcos parallaxes of nearby Galactic Cepheids to explore corrections to the multiwavelength Period-Luminosity relations for LMC Cepheids. The latter are based on an LMC data set scaled to a true distance modulus of 18.50 mag and an adopted foreground reddening of E(B − V ) = 0.10 mag. Although the current uncertainties in the parallaxes are large and still dependent upon the specific subsets of the Cepheids chosen for the comparison, the agreement is good, indicating that to within ±0.14 mag (or, 7% in distance) the previously adopted zero point is substantially correct. Based on different subsamples of data either having BV , BV JK, BV IJK or BV IJHK photometry, LMC moduli, ranging from 18.44 to 18.57 mag are derived. These results, summarized in Table 2 , differ from the value of 18.70 mag of FC97, which are based soley on the reddening-corrected V photometry of Caldwell & Laney (1991) , externally adjusted for metallicity. The Hipparcos data alone do not allow us to discriminate between metallicity effects and the physically distinct possibility of added reddening to the LMC.
Discussion and Conclusions
To alleviate the ambiguity posed by the need to simultaneously solve both for reddening and metallicity effects on the Cepheid distances we are currently deriving OB-star reddenings along the individual lines of sight to several dozen LMC Cepheids. This will allow us to decouple the reddening determinations from metallicity effects, and go beyond the use of a single mean (foreground + internal) reddening for the LMC calibrating Cepheid sample. Preliminary reductions indicate that the variance from field to field is large (ranging from E(B − V ) = 0.00 up to 0.40 mag) while still indicating that an average value of < E(B − V ) > = 0.10 mag is appropriate for the LMC calibrating Cepheids. Details will be presented in Madore, Freedman & Pevunova (1998 in preparation) .
Discussion and Conclusions
We close by noting that at least three other very recent determinations of the true modulus to the LMC fall on either side of the value 18.50 mag adopted by MF91 in setting a zero point for the Cepheid distance scale. Both Reid (1997) and Gratton et al. (1997) derive large LMC moduli (18.65 ± 0.10, and 18.63 ± 0.06 mag, respectively) using Hipparcos-based calibrations of the Galactic globular cluster and RR Lyrae distance scale. On the other hand, Gould & Uza (1997) have re-analyzed the SN 1987A supernova "light echo" and derive an upper limit of µ LM C < 18.37 ±0.04 mag for the LMC true distance modulus; although they note that if the ring is slightly elliptical (b/a ∼ 0.95) this upper limit increases to < 18.44 ± 0.05 mag. A value of 18.56 ± 0.05 mag has been derived by Panagia et al. (1996) from the same data. Until these differences are fully understood and resolved, and given the remaining uncertainties in the Hipparcos Cepheid parallax data we prefer to adopt a true distance modulus of 18.50 mag for the LMC, but now bounded by an uncertainty of ± 0.15 mag, defined to fully encompass the above range of recently published values. This value is consistent with other estimated distances to the LMC based on a wide variety of methods (for a comprehensive modern review see Westerlund 1997) . Viewed in that perspective the Hipparcos data confirm the Cepheid distance scale at better than the ±10% level (95% confidence).
Discussion and Conclusions
BFM was supported in part by the Jet Propulsion Laboratory, California Institute of Technology, under the sponsorship of the Astrophysics Division of NASA's Office of Space Science and Applications and by the NASA/IPAC Extragalactic Database. We acknowledge and appreciate having had illuminating correspondence and discussions with Drs. M. Feast, A. Gould, N. Reid, N. Tanvir, C. Turon and F. van Leeuwen, in the course of preparing this paper. In each panel the solid sloping line is not a fit to the data, but rather it is the published calibration of Madore & Freedman (1991) flanked by thin parallel lines representing the 2-σ limits quoted by them as being the intrinsic width of the instability strip at each wavelength. Fig. 3. -B-band residuals from the multiwavelength Period-Luminosity relations in Figure 3 are sequentially plotted as a function of residuals from each of the other five PL relations and (upper right panel) against the (B-V) color residuals. The total lack of correlation in the latter instance is unexpected except in the limit where the residuals are dominated by distance errors in the derived parallaxes. This latter situation is apparently the case given the strong (unit-slope) correlations of the residuals in each of the other panels, regardless of wavelength. Fig. 4 .-Apparent modulus plots for LMC Cepheids observed at BV IJHK scaled to the Hipparcos zero point and using the published multiwavelength PL solutions of Madore & Freedman (1991) . The solid line is a weighted χ 2 fit of a reddening line to the data; the broken line indicates the one-sigma limits on that solution. Inset (top left) shows the χ 2 surface indicating the minimization solution for the modulus and reddening and the interdependence of their associated errors. 

{'paper_id': '32799298', 'title': 'The Cepheid distance scale', 's2_url': 'https://api.semanticscholar.org/CorpusID:32799298', 'source': 'S2ORC'}