General polytropic dynamic cylinder under self-gravity

We explore self-similar hydrodynamics of general polytropic (GP) and isothermal cylinders of infinite length with axial uniformity and axisymmetry under self-gravity. Specific entropy conservation along streamlines serves as the dynamic equation of state. Together with possible axial flows, we construct classes of analytic and semi-analytic non-linear dynamic solutions for either cylindrical expansion or contraction radially by solving cylindrical Lane–Emden equations. By extensive numerical explorations and fitting trials in reference to asymptotes derived for large index n, we infer several convenient empirical formulae for characteristic solution properties of cylindrical Lane–Emden equations in terms of n values. A new type of asymptotic solutions for small x is also derived in the Appendix. These analyses offer hints for self-similar dynamic evolution of molecular filaments for forming protostars, brown dwarfs and gaseous planets and of large-scale gaseous arms or starburst rings in (barred) spiral galaxies for forming young massive stars. Such dynamic solutions are necessary starting background for further three-dimensional (in)stability analysis of various modes. They may be used to initialize numerical simulations and serve as important benchmarks for testing numerical codes. Such GP formalism can be further generalized to include magnetic field for a GP magnetohydrodynamic analysis.