We have performed the first detailed simulation of the horizontal branch of the Sculptor dwarf spheroidal galaxy by means of synthetic modelling techniques,taking consistently into account the star formation history and metallicity evolution as determined from the main sequence and red giant branch spectroscopic observations. The only free parameter in the whole analysis is the integrated mass loss of red giant branch stars. This is the first time that synthetic horizontal branch models, consistent with the complex star formation history of a galaxy, are calculated and matched to the observations. We find that the metallicity range covered by the star formation history, as constrained by observations, plus a simple mass loss law, enable us to cover both the full magnitude and colour range of HB stars. In addition the number count distribution along the observed horizontal branch, can be also reproduced, provided that the red giant branch mass loss is mildly metallicity dependent, with a very small dispersion at fixed metallicity. The magnitude, metallicity and period distribution of the RR Lyrae stars are also well reproduced. There is no excess of bright objects that require enhanced-He models. The lack of signatures of enhanced-He stars along the horizontal branch is consistent with the lack of the O-Na anticorrelation observed in Sculptor and other dwarf galaxies, and confirms the intrinsic difference between Local Group dwarf galaxies and globular cluster populations. We also compare the brightness of the observed red giant branch bump with the synthetic counterpart, and find a discrepancy -- the theoretical bump being brighter -- similar to what is observed in Galactic globular clusters.
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