2014MNRAS.445.1186L

We investigate self-similar hydrodynamics of a general polytropic (GP) gas with spherical symmetry under self-gravity and extend the conventional polytropic (CP) relation n = 2 - γ for the self-similar index n and the polytropic index γ to a general relation n = 2(q + γ - 2)/(3q - 2), where q is a real parameter by specific entropy conservation along streamlines. We derive GP Larson-Penston (LP)-type solutions for q > 2/3 and γ > 4/3; Larson-Penston-Hunter (LPH)-type solutions are also constructed in a GP gas by a time-reversal operation on a GP-LP-type solution and by connecting to a GP free-fall-type solution across t = 0. These GP-LPH solutions describe dynamic processes that a GP gas globule, static and dense initially, undergoes a runaway collapse under self-gravity, forms a central mass singularity, and keeps accreting during a free-fall stage. We apply such GP-LPH-type solutions with variable envelope mass infall rates (EMIRs) for the dynamic evolution of globules and dense cores in star-forming molecular clouds. In particular, a GP-LPH-type solution can sustain an EMIR as low as 10^–8 ∼ 10^–6M_☉/yr or even lower - much lower than that of Shu's isothermal model for a cloud core in Class 0 and Class I phases. Such GP-LPH-type solutions with EMIRs as low as 10^–9 ∼ 10^–8M_☉/yr offer a sensible viable mechanism of forming brown dwarfs during the accretion stage in a collapsed GP globules with 1.495 ≤ γ ≤ 1.50 and 0.99 ≤ n ≤ 1.0. The GP-LPH solutions with 0.94 < n < 0.99 and 1.47 < γ < 1.495 can even give extremely low EMIRs of 10^–12 ∼ 10^–9M_☉/yr to form gaseous planet-type objects in mini gas globules.