RR Lyrae variables in Messier 53: Near-infrared Period--Luminosity
  relations and the calibration using Gaia Early Data Release 3
  We present new near-infrared, $JHK_s$, Period--Luminosity relations (PLRs)
for RR Lyrae variables in the Messier 53 (M53 or NGC 5024) globular cluster.
Multi-epoch $JHK_s$ observations, obtained with the WIRCam instrument on the
3.6-m Canada France Hawaii Telescope, are used for the first time to estimate
precise mean-magnitudes for 63 RR Lyrae stars in M53 including 29
fundamental-mode (RRab) and 34 first-overtone modes (RRc) variables. The
$JHK_s$-band PLRs for RR Lyrae stars are best constrained for RRab types with a
minimal scatter of 22, 23, and 19 mmag, respectively. The combined sample of RR
Lyrae is used to derive the $K_s$-band PLR, $K_s = -2.303 (0.063) \log P +
15.212 (0.016)$ exhibiting a $1\sigma$ dispersion of only $0.027$ mag.
Theoretical Period--Luminosity--Metallicity (PLZ) relations are used to predict
parallaxes for 400 Galactic RR Lyrae resulting in a median parallax zero-point
offset of $-7\pm3~\mu$as in {\it Gaia} Early Data Release 3 (EDR3), which
increases to $22\pm2~\mu$as if the parallax corrections are applied. We also
estimate a robust distance modulus, $\mu_\textrm{M53} = 16.403 \pm 0.024$
(statistical) $\pm 0.033$ (systematic) mag, to M53 based on theoretical
calibrations. Homogeneous and precise mean-magnitudes for RR Lyrae in M53
together with similar literature data for M3, M4, M5, and $\omega$ Cen are used
to empirically calibrate a new RR Lyrae PLZ$_{K_s}$ relation, $K_s = -0.848
(0.007) -2.320 (0.006) \log P + 0.166 (0.011) {\rm[Fe/H]}$, anchored with {\it
Gaia} EDR3 distances and theoretically predicted relations, and simultaneously
estimate precise RR Lyrae based distances to these globular clusters.
