Calibrating the Cepheid period-luminosity relation from the infrared surface brightness technique II. The effect of metallicity and the distance to the LMC

Context. The extragalactic distance scale builds directly on the Cepheid period-luminosity (PL) relation as delineated by the sample of Cepheids in the Large Magellanic Cloud (LMC). However, the LMC is a dwarf irregular galaxy, quite different from the massive spiral galaxies used for calibrating the extragalactic distance scale. Recent investigations suggest that not only the zero-point but also the slope of the Milky Way PL relation differ significantly from that of the LMC, casting doubts on the universality of the Cepheid PL relation. Aims. We want to make a differential comparison of the PL relations in the two galaxies by delineating the PL relations using the same method, the infrared surface brightness method (IRSB), and the same precepts. We furthermore extend the metallicity baseline for investigating the zero-point dependence, by applying the method to five SMC Cepheids as well. Methods. The IRSB method is a Baade-Wesselink type method to determine individual distances to Cepheids. We apply a newly revised calibration of the method as described in an accompanying paper (Paper I) to 36 LMC and five SMC Cepheids and delineate new PL relations in the V, I, J, & K bands as well as in the Wesenheit indices in the optical and near-IR. Results. We present 509 new and accurate radial velocity measurements for a sample of 22 LMC Cepheids, enlarging our earlier sample of 14 stars to include 36 LMC Cepheids. The new calibration of the IRSB method is directly tied to the recent HST parallax measurements to ten MilkyWay Cepheids, and we find a LMC barycenter distance modulus of 18.45 +/- 0.04 (random error only) from the 36 individual LMC Cepheid distances. In the J, K bands we find identical slopes for the LMC and Milky Way PL relations and only a weak letallicity effect on the zero points (consistent with a zero effect), metal poor stars being fainter. In the optical we find the Milky Way slopes are slightly shallower than the LMC slopes (but again consistent with no difference in the slopes) and small effects on the zero points. However, the important Wesenheit index in V, (V - I) shows a metallicity effect on the slope and on the zero point which is likely to be significant. Conclusions. We find a significant metallicity effect on the W-VI index gamma(W-VI) = -0.23 +/- 0.10 mag dex(-1) as well as an effect on the slope. The K-band PL relation on the other hand is found to be an excellent extragalactic standard candle being metallicity insensitive in both slope and zero-point and at the same time being reddening insensitive and showing the least internal dispersion.