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, $JH{K}_s$, Period–Luminosity relations (PLRs) for RR Lyrae variables in the Messier 53 (M53 or NGC 5024) globular cluster. Multi-epoch $JH{K}_s$ 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 $JH{K}_s$-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, $K_s = -2.303 (0.063) \log P + 15.212 (0.016)$exhibitinga$1\sigma$dispersionofonly$0.027$ 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 $-7\pm 3\mu$as in {\it Gaia} Early Data Release 3 (EDR3), which increases to $22\pm 2\mu$as if the parallax corrections are applied. We also estimate a robust distance modulus, ${\mu }_{\text{M53}}=16.403\pm 0.024$ (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.