This paper presents results from photometric and statistical-parallax analysis of a sample of 850 field RR Lyrae (RRL) variables. The photometric and spectroscopic data for our sample of RRLs are obtained from (1) our new spectroscopic observations (for 448 RRLs) carried out with the Southern African Large Telescope; (2) our photometric observations using the 1.0-m telescope of the South African Astronomical Observatory, and (3) literature. These are combined with accurate proper motion data from the second data release of the Gaia mission (DR2). This study primarily determines the velocity distribution of solar neighbourhood RRLs, and it also calibrates the zero-points of the RRLs' visual V -band luminosity–metallicity (LZ or MV –[Fe/H]) relation and their period–luminosity–metallicity (PLZ) relations in the Wide-field Infrared Survey ExplorerW 1 and Two-Micron All-Sky Survey Ks bands. We find the bulk velocity of the halo RRLs relative to the Sun to be (U 0, V 0, W 0)Halo =(− 16 ± 7, −219 ± 7, −6 ± 5) km s−1 in the direction of Galactic centre, Galactic rotation, and North Galactic pole, respectively, with velocity-dispersion ellipsoids (σ VR , σ V ϕ, σ V θ)Halo = (153 ± 7, 106 ± 4, 101 ± 4) km s−1. The corresponding parameters for the disc component are found to be (U 0, V 0, W 0)Disc = (− 19 ± 5, −46 ± 5, −14 ± 3) km s−1 and (σ VR , σ V ϕ, σ V θ)Disc =(49 ± 4, 38 ± 4, 25 ± 3) km s−1. The calibrated PLZ in W 1-, Ks -, and V -band LZ relations are |$\langle \, M_{W1}\rangle =$| |$-0.824+0.124[\mathrm{ Fe/H}]-2.381\log \, P_F$| , 〈 MKs 〉 = |$-0.804+0.101[\mathrm{ Fe/H}]-2.33\log \, P_F$| , and |$\langle \, M_V\rangle =1.041+0.232[\mathrm{ Fe/H}]$| , respectively. The calibrated PLZ and LZ relations are used to estimate the Galactic Centre distance and the distance modulus of the Large Magellanic Cloud (LMC), which are found to be 7.99 ± 0.49 kpc and 18.46 ± 0.09 mag, respectively. All our results are in excellent agreement with available literature based on statistical-parallax analysis, but are considerably more accurate and precise. Moreover, the zero-points of our calibrated PLZ and LZ relations are quite consistent with current results found by other techniques and yield an LMC distance modulus that is within 0.04 mag of the current most precise estimate. [ABSTRACT FROM AUTHOR]
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