  S stars are late-type giants that are transition objects between M-type stars
and carbon stars on the asymptotic giant branch (AGB). They are classified into
two types: intrinsic or extrinsic, based on the presence or absence of
technetium (Tc). The Tc-rich or intrinsic S stars are thermally-pulsing
(TP-)AGB stars internally producing s-process elements (including Tc) which are
brought to their surface via the third dredge-up (TDU). Tc-poor or extrinsic S
stars gained their s-process overabundances via accretion of s-process-rich
material from an AGB companion which has since turned into a dim white dwarf.
Our goal is to investigate the evolutionary status of Tc-rich S stars by
locating them in a Hertzsprung-Russell (HR) diagram using the results of Gaia
early Data Release 3 (EDR3). We combine the current sample of 13 Tc-rich stars
with our previous studies of 10 Tc-rich stars to determine the observational
onset of the TDU in the metallicity range [-0.7; 0]. We also compare our
abundance determinations with dedicated AGB nucleosynthesis predictions. The
stellar parameters are derived using an iterative tool which combines HERMES
high-resolution spectra, accurate Gaia EDR3 parallaxes, stellar evolution
models and tailored MARCS model atmospheres for S-type stars. Using these
stellar parameters we determine the heavy-element abundances by line synthesis.
In the HR diagram, the intrinsic S stars are located at higher luminosities
than the predicted onset of the TDU. These findings are consistent with Tc-rich
S stars being genuinely TP-AGB stars. The comparison of the derived s-process
abundance profiles of our intrinsic S stars with the nucleosynthesis
predictions provide an overall good agreement. Stars with highest [s/Fe] tend
to have the highest C/O ratios.
