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The ACS LCID Project. I. Short-Period Variables in the Isolated Dwarf Spheroidal Galaxies Cetus & Tucana

(abridged) We present the first study of the variable star populations in the isolated dwarf spheroidal galaxies (dSph) Cetus and Tucana. Based on Hubble Space Telescope images obtained with the Advanced Camera for Surveys in the F475W and F814W bands, we identified 180 and 371 variables in Cetus and Tucana, respectively. The vast majority are RR Lyrae stars. In Cetus we also found three anomalous Cepheids, four candidate binaries and one candidate long-period variable (LPV), while six anomalous Cepheids and seven LPV candidates were found in Tucana. Of the RR Lyrae stars, 147 were identified as fundamental mode (RRab) and only eight as first-overtone mode (RRc) in Cetus, with mean periods of 0.614 and 0.363 day, respectively. In Tucana we found 216 RRab and 82 RRc giving mean periods of 0.604 and 0.353 day. These values place both galaxies in the so-called Oosterhoff Gap, as is generally the case for dSph. We calculated the distance modulus to both galaxies using different approaches based on the properties of RRab and RRc, namely the luminosity-metallicity and period-luminosity-metallicity relations, and found values in excellent agreement with previous estimates using independent methods: (m-M)_{0,Cet}=24.46+-0.12 and (m-M)_{0,Tuc}=24.74+-0.12, corresponding to 780+-40 kpc and 890+-50 kpc. We also found numerous RR Lyrae variables pulsating in both modes simultaneously (RRd): 17 in Cetus and 60 in Tucana. Tucana is, after Fornax, the second dSph in which such a large fraction of RRd (~17%) has been observed. We provide the photometry and pulsation parameters for all the variables, and compare the latter with values from the literature for well-studied dSph of the Local Group and Galactic globular clusters.