  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.
