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Generalized multi-polytropic Rankine-Hugoniot relations and the entropy condition

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 $γ=5/3$ 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.