  That the RR Lyrae star period -- K-band luminosity relation is a promising
tool as a distance indicator in the Milky Way and Local Group of Galaxies is
apparent on observational and theoretical grounds in the literature. Less clear
is the sensitivity of the relation, and consequently the physics of horizontal
branch stars, to differences in stellar environment. In this paper, the first
measurement of the (fundamental) period -- K-band luminosity relation for the
central region of a globular cluster is presented. It is based on a sample of
seven RR Lyrae stars imaged with adaptive optics. In addition, the relation for
the outer region has been reanalyzed, and is found to be in good agreement with
both the previous estimate by Longmore et al. (1990), and with the inner region
relation, especially when irregular and double-mode RR Lyrae stars are
excluded. Importantly, there is no difference between the slope of the inner
and outer region relation within measurement uncertainties, suggesting no
difference in evolutionary state (luminosity). Taking the M3 distance modulus
as 15.0 +/- 0.07mag, the period -- absolute K-band magnitude relation derived
by linear least squares fitting is: M_K = -0.96 (+/- 0.10) - 2.42 (+/- 0.16)Log
Po for the inner region. Excluding irregular variable stars, the outer region
relation is: M_K = -1.07 (+/- 0.10) - 2.38 (+/- 0.15)Log Po. This good
agreement provides further strong support for the near-IR period-luminosity
relation as a distance indicator.
