Towards a fundamental calibration of the cosmic distance scale by Cepheid variable stars

MNRAS 430, 546 557 (2013) doi:10.1093/mnras/sts655 Towards a fundamental calibration of the cosmic distance scale by Cepheid variable stars G. P. Di Benedetto INAF Istituto di Astrofisica Spaziale e Fisica Cosmica di Milano, via Bassini 15, I-20133 Milano, Italy Accepted 2012 December 18. Received 2012 November 26; in original form 2012 July 28 ABSTRACT This paper updates the calibration of the cosmic distance scale by Cepheid variable stars making reference to a geometric Baade Wesselink (BW) distance indicator which adopts pulsation linear radii and photometric angular diameters from period radius and surface brightness relations, respectively, calibrated by fundamental data of Galactic Cepheids. The large number of fundamental-like BW distances (pulsation parallaxes) predictable for Cepheids by using the period and the (V, I) intensity mean magnitudes from HIPP photometry are then applied to calibrate the Galactic period luminosity relations and the reddening-free Wesenheit relation in the (V, I) passbands well suited to obtain metallicity-corrected distance moduli μ O = (18.486 ± 0.047) mag or D = (49.80 ± 1.10) kpc from Large Magellanic Cloud Cepheids of the Optical Gravitational Lensing Experiment map and μ O = (29.281 ± 0.051) mag or D = (7.18 ± 0.17) Mpc from Hubble Space Telescope (HST) Cepheids in the inner field of the maser host galaxy NGC 4258, this value resulting in fair good agreement with the highweight geometric water maser distance modulus. The Cepheid distance moduli of galaxies from the HST SN Ia Calibration Programme are found to be affected by significant systematic errors. Removing these errors leads to an upward revision of the Hubble constant at the value H O = (72.4 ± 6.0) km s 1 Mpc 1, in fair good agreement now with the result of H O = (72 ± 8) km s 1 Mpc 1 originally derived in the HST Key Project Programme as well as with the most recent accurate determination of H O = (73.8 ± 2.4) km s 1 Mpc 1 obtained by HST near-infrared observations of Cepheids. Key words: stars: variables: Cepheids distance scale. 1 INTRODUCTION Since the formulation of the Leavitt law (Leavitt 1908), Cepheid variable stars have become a powerful method for measuring distances to nearby galaxies and Cepheid distances lied at the heart of several Hubble Space Telescope (HST) extragalactic distance scale programmes aiming at the determination of the Hubble constant H O. However, in spite of the correlation between galaxy distances and their recession velocities dated many decades ago (Hubble 1929), measurements of reliable distances have remained a challenging observational problem for achieving accurate values of H O and a factor-of-two controversy persisted up to the launch of the HST. The HST Key Project (Freedman et al. 2001) and the SN Ia Calibration Programme (Sandage et al. 2006) have calibrated H O via Cepheids. Their great amount of careful work with final estimates of H O = (72 ± 8) and (62.3 ± 1.3 r ± 5.0 s )kms 1 Mpc 1, respectively, solved decades of extreme uncertainty about the scale and age of the Universe, providing the values of H O with a difference of only E-mail: pdibene2@gmail.com 15 per cent. Currently, there is a general consensus that most of this difference is caused by systematic errors which could amount to as much as 10 per cent with the largest contribution introduced by Cepheids. In this respect, sources of relevant systematics are suspected to be the shape and the zero-point of the Cepheid period luminosity (PL) relation and its possible dependence on metallicity. In recent years, several efforts have been made trying to improve the most critical distance scale calibration by Galactic Cepheids (Sandage, Tammann & Reindl 2004; Saha et al. 2006; Benedict et al. 2007; Fouqué et al. 2007; van Leeuwen et al. 2007; Di Benedetto 2008, hereinafter Paper 1; Storm et al. 2011a), and then to determine whether or not the metallicity has been responsible for the systematic errors still affecting the values of H O determined by the two HST teams. In spite of these attempts, the achieved results have remained controversial, the major discrepancy being related to the value of the log P coefficient of the reddening-free Wesenheit relation in the (V, I) passbands currently adopted for representing the HST Cepheid distances. While Freedman et al. s Key Project carries the (implicit) assumption that this coefficient is universal, that is independent of metallicity, Sandage et al. introduce a mild metallicity dependence on this coefficient, which therefore C 2013 The Author Published by Oxford University Press on behalf of the Royal Astronomical Society

Cosmic distance scale calibration by Cepheids 547 will be somewhat different for Galactic and Large Magellanic Cloud (LMC) Cepheids. One reason of discrepancy is certainly due to the quite small number of Galactic calibrating distances available for sampling the long-period range of Cepheids most relevant in the HST measurement of H O. In a few years, astrometry by Global Astrometric Interferometer for Astrophysics (GAIA) is expected to essentially eliminate this problem, providing as much as thousand of accurate trigonometric parallaxes covering the whole period range of Galactic Cepheids. However, as the fundamental step by GAIA is still in the future, the overall cosmic distance scale calibration by Cepheids is revised here by adopting geometric Baade Wesselink (BW) distances (pulsation parallaxes) to calibrate the Galactic PL relations as well as the reddening-free Wesenheit relation in the (V, I) passbands. It will be shown that a comparison of Cepheid distance moduli at the same reference period P = 10 d, where changes in the log P coefficient induce negligible systematic errors (Paper 1), allows us to reconcile among each other most of the results of Galactic distance scale calibration determined by different methods (Section 2). The same approach is also applied to extragalactic Cepheids in the LMC (Udalski et al. 1999) (Section 3.1), in NGC 4258 (Macri et al. 2006) (Section 3.2), this galaxy being recently included as a fundamental calibrator in the near-infrared (IR) HST observations of Cepheids (Riess et al. 2011), and in all target galaxies observed by the HST (Section 3.3). It then becomes possible to prove that the values of H O determined by the two HST teams can be closely reconciled between each other independently of the log P coefficient of the adopted Wesenheit relation. 2 THE CALIBRATION OF THE GALACTIC DISTANCE SCALE Table 1. Fundamental parameters and predicted photometric diameters of Galactic Cepheids. 2.1 The Baade Wesselink distance indicator of Galactic Cepheids The geometric BW method (Baade 1926; Wesselink 1946) allows us to make Cepheid variable stars into potentially fundamental distance indicators by using first principles. However, this full promise of the method is quite difficult to be realized, since it would require the Cepheid pulsation cycle to be sampled simultaneously by interferometric and spectroscopic techniques. At present, the most direct realization of the method uses these techniques independently between each other, providing the mean interferometric angular size (unit of mas) and the mean spectroscopic linear radius R (unit of R ) by averaging over each Cepheid pulsation cycle (Kervella et al. 2004a). The distance d (unit of pc) is then obtained by applying the simple relation d = 9.3038 (R/ ), (1) which provides the BW distance modulus of individual stars by the distance indicator BW = 5logd 5 = 5logR 5log 0.157. (2) Unfortunately, also the potentiality of this direct approach for fundamental distances can be exploited for only the brightest and angularly largest Galactic Cepheids as reported in Table 1, being a challenging objective for the modern long-baseline interferometry techniques measuring. Therefore, for extended applications, where even individual radii R may not be achievable by the spectroscopic technique, the BW method requires to be implemented by indirect methods using well-calibrated relationships for predicting R and (Paper 1). Among several implementations appeared in the literature, the spectroscopic and interferometric techniques developed at near-ir wavelengths have been proved to provide the most reliable and accurate individual measurements of R and, respectively, allowing, in the mean time, the calibration of useful scales to estimate these parameters. Nonetheless, the results are known to be uncertainties for the following reasons. (i) The spectroscopic linear radii R and the related period radius (PR) relation to predict them depend on the p-factor adopted to convert observed radial velocities to pulsation velocities. The results obtained by using a p-factor recently suggested (Nardetto et al. 2007) may differ by up to 8 per cent from those adopting a conventional p-factor. HIPP Star log P π d μ e O f VI log R g VI (mas) (mas) (mag) (mas) (mag) 47854 l Car 1.551 2.992 ± 0.012 a 2.03 ± 0.16 8.51 ± 0.17 3.012 ± 0.084 2.213 ± 0.036 34088 ζ Gem 1.006 1.747 ± 0.061 b 2.74 ± 0.12 7.83 ± 0.10 1.779 ± 0.050 1.848 ± 0.023 26069 β Dor 0.993 1.891 ± 0.024 b 3.26 ± 0.14 7.45 ± 0.09 1.845 ± 0.052 1.788 ± 0.022 88567 W Sgr 0.881 1.312 ± 0.029 b 2.30 ± 0.19 8.25 ± 0.18 1.207 ± 0.034 1.763 ± 0.038 87072 X Sgr 0.846 1.471 ± 0.033 b 3.17 ± 0.14 7.51 ± 0.10 1.402 ± 0.039 1.681 ± 0.023 89968 Y Sgr 0.761 2.13 ± 0.29 8.51 ± 0.30 0.862 ± 0.024 1.668 ± 0.060 110991 δ Cep 0.730 1.475 ± 0.061 c 3.71 ± 0.12 7.16 ± 0.07 1.516 ± 0.042 1.645 ± 0.019 93124 FF Aql 0.650 2.64 ± 0.16 7.92 ± 0.13 0.872 ± 0.024 1.556 ± 0.029 102949 T Vul 0.647 2.06 ± 0.22 8.52 ± 0.23 0.593 ± 0.017 1.509 ± 0.048 30827 RT Aur 0.572 2.31 ± 0.19 8.24 ± 0.18 0.650 ± 0.018 1.493 ± 0.038 11767 α UMi 0.754 3.123 ± 0.008 c 7.72 ± 0.12 5.56 ± 0.03 3.137 ± 0.087 1.640 ± 0.014 97804 η Aql 0.856 1.839 ± 0.028 b 1.787 ± 0.050 87495 Y Oph 1.234 1.437 ± 0.040 b 1.502 ± 0.042 a From Kervella et al. (2004a). b From Kervella et al. (2004b). c From Mérand et al. (2006). d From (HST+HIPP) parallaxes (van Leeuwen et al. 2007). e From relation: μ O = 5log (1000/π) 5 LK. f From the SB relation: 5 log VI = S V V O with S V = 2.590 + 3.077 (V I) O (Kervella et al. 2004b). g From relation: 5 log R VI = 5log VI + μ O + 0.157.

548 G. P. Di Benedetto (ii) The photometric predictions of are currently based on the surface brightness (SB) relation given by 5log = S V (V A V ) = α + βc V + γ C A V, (3) where S V is the visual SB scale calibrated by angular diameter measurements, V and C the visual intensity mean magnitude and colour of a Cepheid, respectively, β the colour coefficient of the SB scale, A V the visual absorption and γ C = [1 (β E C /A V )] a coefficient measuring the overall sensitivity of the inferred diameter to this absorption with E C the colour excesses of the star. Up to now, only SB scales calibrated by angular sizes of non-variable stars have been adopted to predict, while the application of scales well suited for Cepheid variable stars still remains to be done. Giving these uncertainties, the PR and SB relations available for Cepheids will be briefly revisited below by comparing their predictions with fundamental data, in attempting to remove residual errors still affecting BW distances determined by equation (2). To do this, photometric data of Galactic Cepheids will be referred to the intensity mean magnitudes B, V, I, K and extinctions E(B V) from the Hipparcos (HIPP) photometry (van Leeuwen et al. 2007, N = 249 stars). Furthermore, to be consistent with the adopted calibrations of the SB scales, the values of E(B V) are converted into the visual absorption A V and colour excess E(V I) by applying the following laws (Laney & Stobie 1993; Laney & Caldwell 2007): A V = [3.07 + 0.28 (B V ) O + 0.04 E(B V )] E(B V ) E(V I) = 1.25[1.00 + 0.06(B V ) O + 0.014 E(B V )] E(B V ). (4) 2.1.1 Angular diameters of Galactic Cepheids It was already recognized a long time ago that reliable and accurate predictions of angular diameters of non-variable stars were within the reach of SB techniques by applying fundamental SB scales calibrated as a function of the near-ir colour (Di Benedetto 1993). Recently, fundamental SB scales of pulsating Cepheids have also been calibrated by using interferometric angular diameter measurements along the cycle of most of these variable stars reported in Table 1 (Kervella et al. 2004c). At near-ir wavelengths, this fundamental relation is given by S V = (2.677 ± 0.006) + (1.336 ± 0.008)(V K) O, (5) where (V K) O is the intrinsic near-ir colour in the Johnson Carter magnitude system derived by adopting the selective absorption E(V K) = A V /1.10 (Laney & Stobie 1993). It is relevant that the values of S V from equation (5) converted into the parameter F V = 4.2207 0.1S V closely reproduce the SB scale of non-variable supergiants of F, G, K, M spectral types originally calibrated at near- IR wavelengths for application to Cepheids (Di Benedetto 1994). Indeed, over the years, spectroscopic techniques have taken great advantage from this calibration to derive BW distances to pulsating Cepheids upon the basic assumption that these stars followed the same SB scale as that of non-variable stars (Fouqué & Gieren 1997; Gieren, Fouqué & Gómez 1998; Storm et al. 2004; Barnes et al. 2005; Fouqué et al. 2007). Now, a fundamental SB relation calibrated by Cepheids is also available in the (V, I) passbands more useful for HST extragalactic works on the cosmic distance scale. It is given by (Kervella et al. 2004c) S V = (2.590 ± 0.007) + (3.077 ± 0.024)(V I) O, (6) Figure 1. Comparison of Cepheid angular diameters by SB techniques with fundamental ones by interferometry. Top: optical photometry. Bottom: near-ir photometry. Crosses: omitted Cepheids α UMi, W Sgr. where (V I) O is the intrinsic optical colour in the Johnson Cousins magnitude system dereddened by the colour excess E(V I) from equation (4). For applications below, it becomes important to check in advance the predicting capabilities in angular sizes of relations (5) and (6), if the photometry is referred to the intensity mean magnitudes of a Cepheid, rather than to the individual values along the pulsation cycle as currently done. Fig. 1 compares these photometric predictions φ VI by the optical SB scale (6) (top panel) and φ VK by the near-ir one (5) (lower panel) with the interferometric angular diameters φ reported in Table 1 together with the fundamental parameters π and μ O available for Galactic Cepheids. The residuals in each diagram are plotted with their observational errors (1/ln10)(δφ/φ). Points to note are as follows. (i) The Cepheid W Sgr shows deviations as large as 4σ in both diagrams. The star is known to be a spectroscopic binary (Szabados 2003; Evans, Massa & Proffitt 2009). This is enough to drop it from the analysis, because of a possible influence of a companion. (ii) The high-precision fundamental diameter of l Car is reproduced at a level better than 1.0 per cent by the optical estimate φ VI = 3.013 mas and about 2.0 per cent by the near-ir one φ VK = 2.944 mas. (iii) The high-precision fundamental diameter of the Polaris α UMi (Mérand et al. 2006) is reproduced at a level better than 1.0 per cent by the optical estimate φ VI = 3.137 mas. Unfortunately, its fundamental size cannot be used in the near-ir diagram, due to the heavily saturated K-band observations (van Leeuwen et al. 2007), which makes the SB angular diameter estimate an outlier at 40σ. (iv) The fundamental diameters of ζ Gem, β Dor, δ Cep and η Aql are reproduced at a level better than 3.0 per cent by the optical estimates φ VI. (v) The average shifts in the optical and near-ir diagrams are found to be (0.03 ± 1.10) and (2.95 ± 0.85) per cent, respectively, the near-ir shift resulting significant at 3.5σ. Therefore, giving high confidence on the reliability of angular diameters derived by the optical SB scale (6) using intensity mean magnitudes of Cepheid variable stars, the photometric predictions φ VI through relation (3) will be applied below to well represent fundamental sizes φ of these stars with a conservative

Cosmic distance scale calibration by Cepheids 549 error of 3 per cent obtained from the observational shifts in the top panel of Fig. 1. These optical SB estimates are reported in Table 1. 2.1.2 Linear radii of Galactic Cepheids Linear radii of pulsating Cepheids are currently determined by applying spectroscopic techniques which combine radial velocity measurements with photometric observations in several magnitude colour combinations. These radii can be then applied to calibrate the PR relation of the form 5logR = s(log P 1.0) + z (7) useful for predicting pulsation radii by the Cepheid period. Spectroscopic radii depend on the projection factor, or p-factor, whose value is a key parameter for converting radial velocity to pulsation velocity, and then to obtain reliable radii and coefficients of relation (7). Unfortunately, at present, there is no general consensus about the best representation of the p-factor as a function of the Cepheid period and several linear laws have been proposed with coefficients in considerable disagreement among each other (Ngeow et al. 2012). An alternative approach to avoid the problems related to the calibration of the p-factor is to rely on the fundamental linear radii R recently available by combining the HST parallax measurements of Benedict et al. (2007) with the near-ir interferometric angular diameters of Kervella et al. (2004c). The resulting fundamental PR relation over the period range log P > 0.70 is found to be (Groenewegen 2007) 5logR = 3.430 (log P 1.0) + 9.100 ± 0.180 ± 0.040 (8) with an rms scatter σ (5 log R) = 0.100 mag around the ridgeline corresponding to a fractional error of σ (R) 5.0 per cent. A PR relation was also obtained by Groenewegen adopting the (HST + HIPP) parallaxes of van Leeuwen et al. (2007). The resulting coefficients (s, z) were statistically indistinguishable from those of equation (8), but the individual radii showed a dispersion greater by 20 per cent around the ridgeline. The fundamental equation (8) will be adopted throughout this paper along with equation (6) to estimate fundamental-like BW distances (pulsation parallaxes) to Cepheids by means of equation (2). The comparison between the coefficients of the currently published spectroscopic PR relations and those from the fundamental equation (8) is reported in the last two columns of Table 2. The sets of spectroscopic linear radii were determined using near-ir Table 2. Coefficients of spectroscopic PR relations and comparison with those of the fundamental equation (8). R s z s z (mag) (mag) (mag) (mag) R ML 3.755 a 9.105 a 0.325 ± 0.180 0.005 ± 0.040 R LS 3.750 b 9.125 b 0.320 ± 0.180 0.025 ± 0.040 R BAY 3.760 c 9.102 c 0.330 ± 0.180 0.002 ± 0.040 R BLS 3.850 d 9.050 d 0.420 ± 0.180 0.050 ± 0.040 R IRSB 3.680 e 8.990 e 0.250 ± 0.180 0.110 ± 0.040 a From N = 40 ML radii and p = 1.36. b From N = 28 LS radii and p = 1.39 0.03 log P. c From N = (38 2) BAY radii and p = 1.39 0.03 log P. d From N = (38 2) bisector LS radii and p = 1.39 0.03 log P. e From N = (38 2) BAY radii and p = 1.366 0.075 log P. photometry to minimize the effects of variations in gravity and microturbulence. The different spectroscopic techniques include the maximum likelihood (ML) method (Laney & Stobie 1995, N = 40 radii) which adopts a single value p = 1.36 for all Cepheids; the least-squares (LS) approach (Gieren et al. 1998, N = 28 radii), the LS approach by a bisector fitting procedure applied in the phase region from 0.0 to 0.8 (Storm et al. 2004; Barnes et al. 2005, N = 38 radii) and the Bayesian (BAY) calculations (Barnes et al. 2005, N = 38 radii), all adopting a factor p = (1.39 0.03 log P )asa function of the Cepheid period; the Barnes et al. BAY radii scaled to the infrared surface brightness (IRSB) radii (Fouqué et al. 2007) by adopting the factor p = (1.366 0.075 logp ) based on improvements in modelling investigations (Nardetto et al. 2007). Note that the overtone Cepheid SU Cas has been excluded from the Barnes et al. sample of N = 38 radii as well as EV Sct because of its low amplitude and limited high-precision IR data available. As can be seen, all spectroscopic slopes s are found to be systematically steeper than the fundamental one. Either the underestimate of the PR slope s, giving the large error affecting it, or the p-factor itself adopted by each spectroscopic technique or both can be responsible for the observed systematic effect. For example, by assuming the p-factor to be responsible for steeper spectroscopic slopes, one could attempt to obtain the corrected p-factor for each Cepheid by using the ratios of the spectroscopic radii to the independent pulsation radii derived from the fundamental PR relation (8). If these ratios are plotted on the log P plane for the sample of R BAY, a linear LS fit to the data would yield the factor p = (1.60 ± 0.07) (0.23 ± 0.07)log P in reasonably good agreement with the recent empirical law, where the slope and intercept have been constrained by using LMC and Galactic Cepheid distances, respectively (Storm et al. 2011a). Note, however, that this law is resulting in sharp contradiction with the most accurately determined interferometric p-factor of δ Cep (Mérand et al. 2005; Groenewegen 2007). On the other hand, if the PR slope is assumed to be shallower by about 1σ,i.e. 0.2 mag, the same ratios would provide p = (1.48 ± 0.07) (0.12 ± 0.12)log P in better agreement with the theoretical law of Nardetto et al. (2007), which is best supported by the most recent empirical redetermination of individual p-factors by Joner & Laney (2012). However, more important for the discussion below is the result obtained for the intercepts z. Indeed, despite the large systematic which can affect the slopes s, the corresponding values of z are found to agree with the fundamental one to within the quoted error (except the intercept z from IRSB radii), proving that at least around the period P = 10 d spectroscopic techniques provide results likely consistent with the fundamental one. For applications of the BW realization by equation (2), it is mandatory to check how well the N = 11 linear radii R VI derived by adopting the SB angular diameters φ VI along with the (HST + HIPP) parallaxes of van Leeuwen et al. (2007) reported in Table 1 reproduce the values of R obtained from the fundamental PR relation (8) by sampling the Cepheid period. The last column of Table 1 lists the values of log R VI and the top panel of Fig. 2 shows these data with errors (1/ln10)(δR VI /R VI ) as a function of log R. For log R VI 1.40 mag corresponding to log P 0.40, the mean residual in the lower panel is (0.01 ± 0.01) mag with an rms scatter of 0.030 mag. More important, for log R VI > 1.60 mag or log P 0.70 which notably include the Cepheids α UMi, δ Cep and l Car with the most accurate fundamental radii, the mean residual becomes (0.00 ± 0.01) mag, i.e. a value closely consistent with the zero level and a reduced rms scatter of 0.023 mag. This implies an error of 0.12 mag for BW distance moduli (pulsation parallaxes) predicted by equation (2) sampling the period P and

550 G. P. Di Benedetto Figure 2. Comparison of Cepheid linear radii predicted by optical photometry with those from the fundamental PR relation. The line indicates the loci of equal radii. the optical colour (V I) of a Cepheid. If only the HST parallaxes of Benedict et al. (2007) are adopted with or without the additional more accurate HIPP parallax of α UMi, the mean residual for log P 0.70 would become (0.01 ± 0.01) mag still consistent with the zero level. Therefore, giving the good agreement between predicted and fundamental data according to the results displayed in Figs 1 and 2, the main conclusion is that the pulsation linear radii R from the fundamental PR relation (8) and photometric angular diameters φ VI estimated by relation (3) using the fundamental SB scale (6) along with the intensity mean magnitudes in the (V, I) passbands can be confidently applied below to calibrate the coefficients of the geometric BW distance indicator (2). 2.1.3 The BW distance indicator of Galactic Cepheids With reference to SB and PR relations sampled as a function of the colour and period, respectively, the geometric BW distance indicator (2) can be explicitly rewritten in the form of a period luminosity colour (PLC) relation given by BW = V O + D BW (log P 1.0) β BW C O + Z OBW, (9) where D BW = s, β BW = β and Z OBW = (z α 0.157). Therefore, by adopting the independently calibrated fundamental scales (6) and (8), relation (9) in the (V, I) passbands becomes BW=V O +3.430 (log P 1.0) 3.077 (V I) O +6.353 (10) with errors σ (D BW ) = 0.180 mag, σ (β BW ) = 0.024 mag, σ (Z OBW ) = 0.040 mag and the rms scatter σ (BW) = [σ 2 (5 log R) + σ 2 (5 log φ)] 1/2 = 0.117 mag. The same precepts can also be applied to derive a BW relation in the (V, K) passbands using the near-ir SB scale (5). The values of (D BW,Z OBW ) in equation (9) can also be directly determined by adopting the thermal coefficient β BW = β available via the fundamental SB scale (6) and sets of primary distance calibrators μ O which, at present, can be either the fundamental distance moduli from (HST + HIPP) parallaxes in Table 1 or those from the HST (Benedict et al. 2007) and HIPP (van Leeuwen et al. (2007) alone or from the N = 22 zero-age main sequence (ZAMS) cluster distance moduli Figure 3. Calibration of the optical BW luminosity function by fundamental distance moduli μ O. The line indicates the weighted LS fit to luminosity data. already selected in Paper 1. In these cases, the calibration should be performed in the log P plane by an LS fit to luminosity data of the form M BW = (V O μ O β BW C O ) = 5logφ C + α μ O. For values of μ O from parallaxes, a weighted LS fit to optical data M BW = 5logR VI + 2.590 + 0.157 should be applied by including a photometric uncertainty of 0.015 mag expected to affect the magnitudes V and I of each data point. As an example, data of M BW are plotted in Fig. 3 for the (HST + HIPP) parallaxes of Table 1. Table 3 compares the solutions for the Galactic distance scale calibration using equation (9). It is reassuring that, despite the large errors and the shifts affecting the slopes D BW, the corresponding values of Z OBW at log P = 1.0 are found to be closely consistent among each other to less than 1σ of each quoted error, including the calibration by the ZAMS cluster distances based on the updated Pleiades distance modulus of (5.63 ± 0.02) mag (An et al. 2007). For the first three solutions, the agreement is within about 0.01 mag, confirming the period P = 10 d as the most valuable reference point to compare results obtained from independent calibrations. Interesting enough is also the comparison between the optical BW distances from equation (10) and the near-ir BW results from the updated IRSB spectroscopic technique (Storm et al. 2011a). For a number of N = 64 Galactic Cepheids with optical and near-ir Table 3. Calibration of coefficients of the BW distance indicator BW = V O + D BW (log P 1.0) 3.077 (V I) O + Z OBW. Calibrators D BW Z OBW σ r (mag) (mag) (mag) R, φ a 3.430 ± 0.180 6.353 ± 0.040 0.117 μ b O 3.803 ± 0.111 6.350 ± 0.037 0.173 μ c O 3.621 ± 0.190 6.343 ± 0.036 0.101 μ d O 3.754 ± 0.233 6.382 ± 0.050 0.121 μ e O 3.757 ± 0.225 6.381 ± 0.040 0.095 a From fundamental PR and SB relations. b From N = 22 ZAMS distance moduli (Paper 1). c From N = 11 (HST+HIPP) parallaxes of Table 1. d From N = 10 HST parallaxes (Benedict et al. 2007). e From N = 10 HST parallaxes + α UMi.

Cosmic distance scale calibration by Cepheids 551 intensity mean magnitudes from the Storm et al. list, the LS fit to the distance residuals in the range of periods 0.5 log P 1.7 (omitting six outliers with 2.5σ clipping) gives rise to BW(GAL) = (IRSB G BW G ) = (0.02 ± 0.03) + (0.01 ± 0.10) (log P 1.0) with an rms error of 0.24 mag. It seems to be quite surprising that these optical and near-ir distances, achieved by two fully independent BW methodologies with the near-ir one related to an adopted p-factor law, show such excellent agreement between each other, being indeed the residual at log P = 1.0 of only 0.02 mag and no more than 0.03 mag at any period in the range 0.5 log P 1.7. Moreover, the most relevant (HST + HIPP) parallaxes of l Car and δ Cep are reproduced at a level 3 per cent by each of the BW distances and the most accurate HIPP one of α UMi at the same level of uncertainty by the optical BW distance. All of this provides a likely reciprocal support for either of the different techniques to obtain reliable Galactic Cepheid distances, proving also that the large number of N = 249 HIPP Cepheids can be now available as fundamental-like secondary calibrators (pulsation parallaxes) each affected by a systematic error of 0.12 mag ( 6 per cent of accuracy) by using equation (10). It should be finally noted that the actual BW realization was already adopted in Paper 1, where the linear and angular sizes of Cepheids were estimated by spectroscopic PR relations and the near- IR SB scale of non-variable stars, respectively. Now, both the PR and the optical SB scales given by relations (6) and (8), respectively, rely only on fundamental sizes of Cepheids. 2.1.4 The PL relations of Galactic Cepheids By adopting the N = 249 HIPP Cepheids with fundamental-like BW distances from relation (10) as secondary calibrators, one can confidently update the calibration of the PL relations of the form M X = X O BW = D X (logp 1.0) + Z OX, (11) where M X is the absolute magnitude, X O is the intensity mean magnitude in the band X = B, V, I, H, K corrected for interstellar reddening and (D X, Z OX ) are the PL coefficients determined by an LS fit in the log P plane to values (X O BW). Once the PL relations have been calibrated, they can also be linearly recombined to notably obtain reddening-free Wesenheit luminosity functions M W (Madore 1982) applied in the measurement of the Hubble constant by HST Cepheids (Freedman et al. 2001; Sandage et al. 2006; Riess et al. 2009). For two passbands (V, X), this function is defined by M W = (V O μ O ) β W (V X) O, (12) where μ O is the calibrating distance modulus and β W = A V /E(V X) is the ratio of total to selective absorption chosen to realize the reddening-free condition. In this case, equation (12) can be approximated by the luminosity relation M W = (V μ O ) β W (V X), (13) where observational photometry can be directly adopted. It enables us to derive unknown Cepheid distance moduli by the useful reddening-free PLC relation given by W = V D W (log P 1.0) β W (V X) Z W, (14) where the coefficients (D W, Z W ) follow from an LS fit in the log P plane to data M W = D W (log P 1.0) + Z W with Z W independent of D W at log P = 1.0. Note that for the reddening laws (4), the reddening-free condition is found to be best satisfied by β W = 2.52, being γ C A V 0.01 mag for all HIPP Cepheids within the Table 4. Calibration of PL luminosity functions of the form M = V O BW β (V X) O = D (log P 1.0) + Z O by fundamental-like BW distances of Galactic Cepheids. X O β D Z O σ r (mag) (mag) (mag) B O 1.00 2.217 ± 0.189 3.362 ± 0.043 0.226 V O 1.00 2.599 ± 0.186 4.085 ± 0.042 0.177 I O 1.00 2.869 ± 0.183 4.822 ± 0.041 0.134 H O 1.00 3.167 ± 0.182 5.662 ± 0.041 0.107 K O 1.00 3.199 ± 0.182 5.717 ± 0.041 0.103 I O 2.45 3.261 ± 0.182 5.891 ± 0.041 0.103 I O 2.52 3.279 ± 0.182 5.942 ± 0.041 0.103 I O 2.55 3.288 ± 0.182 5.964 ± 0.041 0.103 K O 1.10 3.259 ± 0.182 5.880 ± 0.041 0.103 period range. The Galactic PL coefficients and those of M W calibrated by adopting μ O = BW are reported in Table 4. As can be seen, the errors affecting the coefficients of equation (10) due to the small number of primary fundamental calibrators remain as a systematic limiting the final accuracy of the PL coefficients themselves. However, it should be stressed that the actual methodology of calibration allows us to closely relate the coefficients of each PL distance indicator, notably those of W in equation (14), to pulsation and thermal properties of Cepheids represented by the meaningful physical components in the geometric BW distance indicator (2). One source of systematic error is expected to come from the adopted photometry and can affect the calibration of the PL and Wesenheit coefficients. For example, instead of the intensity mean magnitudes from the HIPP tabulation, one could prefer deriving properly calibrated values from the photometry recently published for Galactic Cepheids (Storm et al. 2011a). Indeed, for the above selected sample of N = 64 stars, the V- andi-band PL coefficients, notably the values of Z O, are found to be higher by about 0.10 mag. However, those of the Wesenheit function with β W = 2.52, given by D W = 3.259 and Z W = 5.920 and relevant for the discussion below, remain practically unchanged at the values reported in Table 4. In recent times, several works have been devoted to determining the values of D W and Z W for Galactic Cepheids. Table 5 summarizes the results of previous calibrations. The calibrators include Cepheids in clusters with ZAMS distances from main-sequence Table 5. Comparison between coefficients of Galactic Wesenheit luminosity functions and those of the corresponding relations in Table 4. Solution β W D W Z W D W Z W (mag) (mag) (mag) (mag) (mag) M a 1 2.52 3.884 6.044 0.605 ± 0.182 0.102 ± 0.041 M b 2 2.52 3.625 5.886 0.346 ± 0.182 0.056 ± 0.041 M c 3 2.52 3.745 5.957 0.466 ± 0.182 0.015 ± 0.041 M d 4 2.45 3.34 5.86 0.079 ± 0.182 0.031 ± 0.041 M e 5 2.45 3.29 5.866 0.029 ± 0.182 0.025 ± 0.041 M f 6 2.55 3.477 5.891 0.189 ± 0.182 0.073 ± 0.041 M g 7 2.45 3.605 5.881 0.344 ± 0.182 0.010 ± 0.041 a From Sandage et al. (2004) (ZAMS calibrators). b From Sandage et al. (2004) (BW calibrators). c From Sandage et al. (2004) (BW + ZAMS calibrators). d From Benedict et al. (2007) (HST parallaxes). e From van Leeuwen et al. (2007) (HST + HIPP parallaxes). f From Fouqué et al. (2007) (BW calibrators + HST parallaxes). g From Paper 1 (BW calibrators).

552 G. P. Di Benedetto fitting (solution M 1 ), BW distances from spectroscopic techniques (solutions M 2, M 3, M 6, M 7 ) and fundamental HIPP and/or HST parallaxes (solutions M 4, M 5, M 6 ). A comparison with the actual Galactic calibration is reported in the last two columns in the form of corrections ( D W, Z W ) which should be applied to each pair of the oldest coefficients for reproducing those from Table 4. As can be seen, the solutions give rise to somewhat controversial results. Points to note are as follows. (i) The corrections D W for all solutions, except M 4 and M 5 related to the HST and/or HIPP parallaxes, differ from 1σ to 3σ from the zero level, indicating that the corresponding slopes D W are too steep for representing the Galactic coefficient of W in equation (14). For BW calibrators, these slope differences are directly related to the same systematic errors as those affecting the BW distances from spectroscopic techniques. Instead, the slope variation related to M 1 (and then to M 3 ), as significant as 3.3σ, is probably due to the adopted sample of ZAMS calibrators (see also Table 3), where Cepheids in OB associations were not distinguished from the cluster variables, resulting in PL slopes much too steep (Joner & Laney 2012). (ii) Only the correction Z W for the solution M 1 differs as much as 2.5σ from the zero level, indicating that the corresponding Wesenheit function provides distance moduli affected by significant systematic errors. These errors compensate between each other in the overall solution M 3 which happens to agree with the absolute calibration by fundamental-like pulsation parallaxes. 3 THE CALIBRATION OF THE EXTRAGALACTIC DISTANCE SCALE 3.1 The Cepheid distance modulus to the LMC 3.1.1 The PL distance moduli to the LMC Current approaches to the cosmic distance scale by HST Cepheid observations (Freedman et al. 2001; Sandage et al. 2006) have adopted LMC PL relations to determine Cepheid distances to N = 31 external galaxies. In the following, I shall refer to the LMC PL relations originally derived by Udalski et al. (1999) in the B, V, I passbands adopting Cepheids from the Optical Gravitational Lensing Experiment (OGLE) and by Groenewegen (2000) in the H, K passbands combining OGLE with Deep Near Infrared Survey (DENIS) and 2MASS IR data. These calibrated LMC PL relations are reported in Table 6 for a more straightforward comparison with the Galactic PL relations of Table 4. As can be seen, the optical and near-ir slopes D W and D W of the Galactic and LMC Wesenheit luminosity functions, respectively, are found to agree remarkably well among each other, despite the large error quoted for D W probably responsible for the differences between the Galactic and LMC PL components. This close agreement would indicate that optical and near-ir Wesenheit slopes are likely to be independent of metallicity, as also expected from recent theoretical investigations (Bono et al. 2010). The last column of Table 6 lists the LMC distance moduli (Z O Z O)at P = 10 d based on the absolute Galactic calibration of Table 4. It shows that the PL distance moduli change as a consequence of a different amount of bias probably due to the LMC metallicity affecting each distance indicator. The distance moduli from the optical Wesenheit functions are of major interest. They are found to be consistent between each other to within the quite small error of less than 0.01 mag, proving that the results are largely independent of the total-to-selective absorption ratios β W currently adopted in the literature. Therefore, with reference to the optical Wesenheit function with the value β W = 2.52, the unbiased distance modulus to the LMC measured at log P = 1.0 is given by W 1 (LMC) = (18.536 ± 0.042) + ν W, (15) where ν W is introduced to take into account a possible systematic error due to metallicity affecting the determination of the LMC distance calibrated by the Galactic Wesenheit relation. The near-ir Wesenheit LMC distance is found to be slightly shorter than the optical ones, probably due again to an effect of metallicity reduced at IR wavelengths. One could prefer the use of the most recent OGLE-III map to derive the LMC distance from classical Cepheids (Soszyński et al. 2008). In this case, the value of Z = (12.579 ± 0.006) mag at log P = 1.0 derived by the authors for β W = 2.55 would yield a biased Wesenheit distance modulus greater by only 0.006 mag than that from equation (15). In recent years, several samples of LMC Cepheids were observed to determine the distance modulus to the LMC. Table 7 reports these sets of Cepheids adopted here with their published observational photometry in the period range 0.4 log P 2.0. Cepheid periods are also selected with values log P 1.0, this range being most relevant in the HST observations, since it provides data with the highest signal-to-noise ratio. Biased LS distance moduli (W 1 ν W ) of the LMC are determined at P = 10 d by using optical and near-ir Wesenheit functions of Table 4 with β W = 2.52 and 1.10, respectively. The optical LS value from equation (15) is adopted here as a reference to determine the shifts W 1 = [W 1 ν W ) 18.536] mag. As can be seen, all these residuals scaled at P = 10 d, and then independent of the Wesenheit slope D W according to equation (14), are found to be consistent with the zero level to within the random errors quoted for (W 1 ν W ), except the optical one from the HST Key Project sample with log P 1.0, probably due to photometry of poor quality. Average distance moduli W are also obtained from each sample. They are readily achievable by applying the following relation: W = W 1 + (D LS D W )(log P 1.0), (16) where W changes as a function of the differential slope (D LS D W ) derived from each LS fit and of the period distribution of Cepheids through log P. The shifts (W W 1 ) show that for (D LS D W ) 0, i.e. for observational slopes D LS close to the adopted Wesenheit value D W, the average processor and the LS fitting approach give rise to the same results, i.e. W W 1, whereas for D LS much steeper Table 6. PL luminosity functions M = V O β (V O X O ) = D (logp 1.0) + Z O of LMC Cepheids and biased LMC distance moduli at log P = 1.0. X O β D Z O σ r (Z O Z O) c (mag) (mag) (mag) (mag) B a O 1.00 2.439 ± 0.046 14.929 ± 0.031 0.240 18.291 ± 0.053 V a O 1.00 2.779 ± 0.031 14.287 ± 0.021 0.160 18.372 ± 0.047 I a O 1.00 2.979 ± 0.021 13.615 ± 0.014 0.107 18.437 ± 0.043 H b O 1.00 3.236 ± 0.033 12.812 ± 0.023 0.168 18.474 ± 0.047 K b O 1.00 3.246 ± 0.036 12.786 ± 0.025 0.168 18.503 ± 0.048 I O 2.45 3.269 ± 0.014 12.641 ± 0.010 0.076 18.532 ± 0.042 I O 2.52 3.283 ± 0.014 12.594 ± 0.010 0.076 18.536 ± 0.042 I O 2.55 3.289 ± 0.014 12.573 ± 0.010 0.076 18.537 ± 0.042 K O 1.10 3.293 ± 0.036 12.636 ± 0.025 0.168 18.516 ± 0.048 a From Udalski et al. (1999). b From Groenewegen (2000). c From Galactic calibration of Table 4.

Cosmic distance scale calibration by Cepheids 553 Table 7. Biased LMC distance moduli at log P = 1.0 by Wesenheit relations in the (V, I) and (V, K) passbands. N (W 1 ν W ) a W 1 logp (D LS + 3.279) (W W 1 ) σ r (mag) (mag) (mag) (mag) (mag) logp 0.4 657 b 18.533 ± 0.003 0.003 0.656 0.021 ± 0.013 0.004 0.068 34 c 18.546 ± 0.017 0.010 1.207 0.152 ± 0.038 0.034 0.103 98 d 18.540 ± 0.010 0.004 0.919 0.029 ± 0.026 0.001 0.097 52 e 18.555 ± 0.028 0.019 1.283 0.320 ± 0.097 0.094 0.202 29 f 18.522 ± 0.021 0.014 1.083 0.110 ± 0.053 0.009 0.114 48 g 18.506 ± 0.014 0.030 1.178 0.049 ± 0.045 0.012 0.094 logp 1.0 51 b 18.543 ± 0.011 0.007 1.175 0.024 ± 0.085 0.007 0.080 23 c 18.643 ± 0.023 0.113 1.473 0.334 ± 0.090 0.161 0.109 40 d 18.537 ± 0.015 0.001 1.309 0.005 ± 0.077 0.005 0.096 41 e 18.567 ± 0.034 0.031 1.382 0.345 ± 0.141 0.135 0.219 17 f 18.547 ± 0.034 0.011 1.367 0.158 ± 0.155 0.058 0.140 35 g 18.537 ± 0.016 0.001 1.318 0.124 ± 0.085 0.042 0.100 a See the text for details. b From Udalski et al. (1999). c From HST Key Project (Madore & Freedman 1991; Freedman & Madore 2010). d From Sebo et al. (2002). e From Sandage et al. (2004) by omitting Cepheids with log P > 2.0. f From Storm et al. (2011b). g From Persson et al. (2004) and Sebo et al. (2002). than D W significant systematic errors begin to bias the values of W, especially those obtained by averaging Cepheid samples over the long-period range log P 1.0 most relevant in the HST observations. In this case, the solution W 1 adopted for LMC in equation (15) is likely to be preferred, taking also into account the large uncertainty still affecting the slope D W as reported in Table 4. 3.1.2 The BW distance modulus to the LMC According to the BW (equation 1), the determination of the geometric distance to the LMC by Cepheids requires the knowledge of linear and angular sizes of each star. Gieren et al. (2005) have determined the near-ir spectroscopic linear radii R LMC of 13 LMC Cepheids spanning a period range from 3 to 42 d by applying the bisector LS fitting procedure in the phase region from 0.0 to 0.8 (Storm et al. 2004). The unweighted LS fit to this set of radii provides the following spectroscopic PR relation: 5logR LMC = 3.700 (log P 1.0) + 9.055 ± 0.085 ± 0.044 (17) with an rms scatter of σ (5 logr i ) = 0.160 mag around the ridgeline. The comparison with the near-ir spectroscopic PR relation from Galactic radii R BLS (see Table 2) obtained by applying exactly the same technique as that adopted for the LMC radii shows that the value z LMC = (9.055 ± 0.044) mag at P = 10 d agrees remarkably well, indeed exactly, with the independently determined Galactic reference point z = 9.050 mag. The latter point is important, because it proves that at least around the period P = 10 d a Cepheid PR relation is likely to provide the same pulsation radii for LMC as for Galactic Cepheids. This means that I am fully justified in adopting below the value of z LMC = z = (9.100 ± 0.040) from the fundamental Galactic PR relation (8) as the most likely absolute calibration at P = 10 d for also the LMC PR relation useful for deriving reliable linear radii of LMC Cepheids as a function of the period. It should be emphasized that any shift s in the slope of the LMC PR relation (17) with respect to that of the Galactic one (8) becomes unimportant in the current determination of z LMC, since it would imply difference between linear radii of LMC and Galactic Cepheids only for periods log P 1.0. Accurate stellar angular diameters can be predicted by applying SB techniques (Di Benedetto 2005). However, the available SB relation (3) provides unbiased estimates for only Galactic Cepheids, being the fundamental SB colour scale (6) calibrated for these stars alone (Kervella et al. 2004c). Therefore, its application to extragalactic LMC Cepheids is likely to require suitable corrections to take into account the effects of metallicity. With reference to equation (6), relation (3) can be represented in the log P plane by linearly recombining the LMC (PL V, PL I ) relations of Table 6 according to the standard HST procedure, where the coefficient of recombination is now the thermal one β = 3.077 from the fundamental SB scale (6). Hence, the biased photometric angular diameters of LMC Cepheids as a function of the period can be represented by the following relation: 5logφ = 2.590 + 3.077 (V I) O V O = 3.394 (log P 1.0) 9.629 ± 0.017 ± 0.012 (18) and the unbiased ones by 5 log φ LMC = 5logφ + BW,where BW is a function of metallicity that in the more general form can be represented by BW = μ BW (log P 1.0) + ν BW. (19) Therefore, relation (2) with the linear and angular diameters estimated as a function of the period by relations (17) and (18), respectively, the former with z LMC = (9.100 ± 0.040), gives rise to the following unbiased distance modulus to the LMC measured at P = 10 d: BW 1 (LMC) = (18.572 ± 0.042) + ν BW, (20)

554 G. P. Di Benedetto where ν BW represent the unknown bias due to metallicity affecting the determination of the LMC Cepheid angular sizes by equation (18). Comparison with the result from equation (15) shows that the optical geometric BW distance modulus to the LMC based on the use of equation (1) with meaningful physical components can be fully reconciled with the Wesenheit one by adopting ν BW = (ν W 0.036) for the LMC metallicity correction. The LMC BW distance from equation (20) is based on the assumption that the absolute calibration of the PR relation (17) given by the coefficient z is the same for Galactic as for LMC Cepheids at P = 10 d. This, in turn, relies on a small sample of only 13 LMC Cepheids with published linear radii. Recently, a larger sample of N = 36 Cepheids became also available with BW distances to the LMC obtained by the updated spectroscopic IRSB method (Storm et al. 2011b). For a subset of N = 30 Cepheids with optical and near-ir intensity mean magnitudes in common, the BW distance residuals for LMC are given by BW(LMC) = (IRSB L BW L ) = (0.07 ± 0.04) + (0.17 ± 0.10) (log P 1.0). The comparison between BW 1 (LMC) and BW 1 (GAL) (see Section 2.1.3) at log P = 1.0 leads to a rough estimate of ν BW = (BW L 1 BWG 1 ) = (0.09 ± 0.05) mag, and then of ν W = (0.05 ± 0.05) mag, for the metallicity corrections to the optical BW and Wesenheit LMC distances, respectively, provided that the metallicity correction for the IRSB distance to the LMC is assumed to vanish, i.e. (IRSB L 1 IRSBG 1 ) 0. It is perhaps worth noting that, to within the quoted error, the achieved value of ν W happens to be consistent either with the vanishing metallicity correction suggested in a recent investigation (Bono et al. 2010) or with the value from the oldest empirical scale (Sakai et al. 2004). 3.1.3 The metallicity-corrected distance modulus to the LMC Predicting an accurate and reliable value for the magnitude of ν W has proven to be a challenging objective and considerable uncertainty still remains in the corrections obtained by applying the available abundance scales. In this respect, the HST Key Project (Freedman et al. 2001) adopted [O/H] nebular abundances derived from the spectra of H II regions (Zaritsky, Kennicutt & Huchra 1994) and an empirical scale obtained by comparing published extragalactic Cepheid distances with the tips of the red giant branch (TRGB) distances assumed to be unaffected by abundance differences (Sakai et al. 2004). The difference between Cepheid and TRGB distances was found to decrease monotonically with increasing Cepheid abundance, consistent with a metallicity dependence of the Cepheid distance moduli of δ(m M)/δ[O/H] = (0.25 ± 0.09) mag dex 1, where Cepheid distances were derived by using the (PL V, PL I )relations of Table 6. Based on this calibration and on a metallicity difference of δ[o/h] = [O/H] GAL [O/H] LMC = 8.70 8.50 = 0.20dex, the metallicity correction is given by ν W = δ(m M) = (0.05 ± 0.02) mag. This correction applied to equation (15) provides the following true distance modulus to the LMC: W 1 (LMC) = (18.486 ± 0.047) mag. (21) This means an LMC distance D = (49.80 ± 1.10) kpc as accurate as about ±2 per cent with the major contribution of error coming from the absolute calibration of the Galactic distance scale. This accurate distance to the LMC relies on the best empirical support which includes only fundamental data presently available for Cepheids and supersedes the value in Paper 1 uncorrected for metallicity and based on a more uncertain Galactic calibration. 3.1.4 Comparison with other determinations of the distance modulus to the LMC Recent calibrations of the optical Wesenheit relation W LMC by trigonometric parallaxes of Galactic Cepheids (Benedict et al. 2007; van Leeuwen et al. 2007) have obtained a true distance modulus to the LMC shorter by about 0.10 mag than the canonical distance adopted in the HST Key Project (Freedman et al. 2001) and than the value from equation (21). The authors have indeed derived biased distance moduli of (18.50 ± 0.04) and (18.52 ± 0.03) mag in good agreement with the value (15), starting from a common cleaned selection of N = 581 LMC OGLE Cepheids and HST and (HIPP + HST) parallaxes, respectively. However, their final true distance moduli were (18.40 ± 0.05) and (18.39 ± 0.05) mag, respectively, by applying metallicity corrections of ν W = (0.10 ± 0.03) and (0.13 ± 0.04) mag, respectively. These values differ significantly from ν W = (0.05 ± 0.02) mag I have adopted to derive the unbiased distance modulus (21) according to a more statistically significant coefficient of δ(m M)/δ[O/H] = (0.25 ± 0.09) mag dex 1 based on the comparison of the TRGB and Cepheid distance moduli for 20 fields in 18 galaxies (Sakai et al. 2004). This coefficient has been recently used again for a review of the Hubble constant (Freedman & Madore 2010). In the past, because of the uncertainty in the Galactic calibration and the difficulty in establishing an accurate size of the metallicity effect, the distance modulus to the LMC was also obtained by using several methods independent of Cepheids. Sandage et al. (2006) in their HST programme aimed at determining the Hubble constant from the Cepheid-calibrated luminosity of Type Ia supernovae have adopted an LMC distance modulus of μ O (LMC) = (18.54 ± 0.02) mag obtained by averaging dozens of previous determinations, excluding those related to Cepheids (Tammann, Sandage & Reindl 2003). The difference with respect to the value (21) yields [μ O (LMC) W 1 (LMC)] = (0.05 ± 0.05) mag not significant to within the quoted error. Two other recent determinations are noteworthy being achieved by the only direct method available for measuring distances to stars in the LMC. Pietrzyński et al. (2009) have analysed a double-line eclipsing binary system in LMC consisting of two G-type giants with very similar effective temperatures. By applying the fundamental SB (V K) colour relation well established for non-variable stars of spectral types later than AO (Di Benedetto 1998; Groenewegen 2004; Kervella et al. 2004; Di Benedetto 2005), they have derived a distance modulus of μ O (LMC) = (18.50 ± 0.06) mag. Bonanos et al. (2011) have also obtained a distance modulus of μ O (LMC) = (18.52 ± 0.07) mag to the O-type eclipsing binary LMC-SC1-105 in the LMC, by applying atmosphere models to estimate the effective temperatures of the binary components. Both these results based on a geometric method are found to be in good agreement with the unbiased value given by equation (21). The most recent precision measurements of nearby red clump starsinthek band (Laney, Joner & Pietrzyński 2012) have provided an LMC distance with almost negligible systematic errors induced by metallicity, and then well suited for a further check of the present results obtained by Cepheids. The uncorrected K-band red clump LMC distance modulus is found to be (18.505 ± 0.019) mag and the corrected value to be (18.475 ± 0.021) mag. The difference with respect to the uncorrected PL distance modulus in the K band reported in Table 6 and to the corrected optical Wesenheit value (21) are K = 0.002 mag and VI = 0.011 mag, respectively, both shifts being well within the smallest error quoted for each red clump distance.