Calibrating the Cepheid period-luminosity relation from the infrared surface brightness technique I. The p-factor, the Milky Way relations, and a universal K-band relation

Aims. We determine period-luminosity relations for Milky Way Cepheids in the optical and near-IR bands. These relations can be used directly as reference for extra-galactic distance determination to Cepheid populations with solar metallicity, and they form the basis for a direct comparison with relations obtained in exactly the same manner for stars in the Magellanic Clouds, presented in an accompanying paper. In that paper we show that the metallicity effect is very small and consistent with a null effect, particularly in the near-IR bands, and we combine here all 111 Cepheids from the Milky Way, the LMC and SMC to form a best relation. Methods. We employ the near-IR surface brightness (IRSB) method to determine direct distances to the individual Cepheids after we have recalibrated the projection factor using the recent parallax measurements to ten Galactic Cepheids and the constraint that Cepheid distances to the LMC should be independent of pulsation period. Results. We confirm our earlier finding that the projection factor for converting radial velocity to pulsational velocity depends quite steeply on pulsation period, p = 1.550-0.186 log(P) in disagrement with recent theoretical predictions. We find PL relations based on 70 MilkyWay fundamental mode Cepheids of M-K = -3.33(+/- 0.09)(log(P)-1.0)-5.66(+/- 0.03), W-VI = -3.26(+/- 0.11)(log(P)-1.0)-5.96(+/- 0.04). Combining the 70 Cepheids presented here with the results for 41 Magellanic Cloud Cepheids which are presented in an accompanying paper, we find M-K = -3.30(+/- 0.06)(log(P) - 1.0) - 5.65(+/- 0.02), W-VI = -3.32(+/- 0.08)(log(P) - 1.0) - 5.92(+/- 0.03). Conclusions. We delineate the Cepheid PL relation using 111 Cepheids with direct distances from the IRSB analysis. The relations are by construction in agreement with the recent HST parallax distances to Cepheids and slopes are in excellent agreement with the slopes of apparent magnitudes versus period observed in the LMC.