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That the RR Lyrae star period -- K-band luminosity relation is a promisingtool as a distance indicator in the Milky Way and Local Group of Galaxies isapparent on observational and theoretical grounds in the literature. Less clearis the sensitivity of the relation, and consequently the physics of horizontalbranch stars, to differences in stellar environment. In this paper, the firstmeasurement of the (fundamental) period -- K-band luminosity relation for thecentral region of a globular cluster is presented. It is based on a sample ofseven RR Lyrae stars imaged with adaptive optics. In addition, the relation forthe outer region has been reanalyzed, and is found to be in good agreement withboth the previous estimate by Longmore et al. (1990), and with the inner regionrelation, especially when irregular and double-mode RR Lyrae stars areexcluded. Importantly, there is no difference between the slope of the innerand outer region relation within measurement uncertainties, suggesting nodifference in evolutionary state (luminosity). Taking the M3 distance modulusas 15.0 +/- 0.07mag, the period -- absolute K-band magnitude relation derivedby linear least squares fitting is: M_K = -0.96 (+/- 0.10) - 2.42 (+/- 0.16)LogPo for the inner region. Excluding irregular variable stars, the outer regionrelation is: M_K = -1.07 (+/- 0.10) - 2.38 (+/- 0.15)Log Po. This goodagreement provides further strong support for the near-IR period-luminosityrelation as a distance indicator.Comment: Accepted for publication in Astronomy & Astrophysics, 10 pages, 5  figure

The RR Lyrae star period –K-band luminosity relation of the globular cluster M 3