The Araucaria Project. The Distance to the Sculptor dwarf spheroidal galaxy from infrared photometry of RR Lyrae stars

We have obtained single-phase near-infrared magnitudes in the J and K bands for a sample of 78 RR Lyrae stars in the Sculptor dSph galaxy. Applying different theoretical and empirical calibrations of the period-luminosity-metallicity relation for RR Lyrae stars in the infrared, we find consistent results and obtain a true, reddening-corrected distance modulus of 19.67 $\pm$ 0.02 (statistical) $\pm$ 0.12 (systematic) mag for Sculptor from our data. This distance value is consistent with the value of 19.68 $\pm$ 0.08 mag which we obtain from earlier V-band data of RR Lyrae stars in Sculptor, and the V magnitude-metallicity calibration of Sandage (1993). It is also in a very good agreement with the results obtain by Rizzi (2002) based on tip of the red giant branch (TRGB, 19.64 $\pm$ 0.08 mag) and horizontal branch (HB, 19.66 $\pm$ 0.15 mag).