  The study aims at a derivation of generalized \RH relations, especially that
for the entropy, for the case of different upstream/downstream polytropic
indices and their implications. We discuss the solar/stellar wind interaction
with the interstellar medium for different polytropic indices and concentrate
on the case when the polytropic index changes across hydrodynamical shocks. We
use first a numerical mono-fluid approach with constant polytropic index in the
entire integration region to show the influence of the polytropic index on the
thickness of the helio-/astrosheath and on the compression ratio. Second, the
Rankine-Hugonoit relations for a polytropic index changing across a shock are
derived analytically, particularly including a new form of the entropy
condition. In application to the/an helio-/astrosphere, we find that the size
of the helio-/astrosheath as function of the polytropic index decreases in a
mono-fluid model for indices less than $\gamma=5/3$ and increases for higher
ones and vice versa for the compression ratio. Furthermore, we demonstrate that
changing polytropic indices across a shock are physically allowed only for
sufficiently high Mach numbers and that in the hypersonic limit the compression
ratio depends only on the downstream polytropic index, while the ratios of the
temperature and pressure as well as the entropy difference depend on both, the
upstream and downstream polytropic indices.
