Abstract We seek for self-similar solutions describing the time-dependent evolution of self-gravity systems with either spherical symmetry or axisymmetric disk geometry. By assuming self-similar variable x ≡ r/at where a is isothermal sound speed we find self-similar solutions extending from the initial instant t = 0 to the final stage t → ∞ using standard semi-analytical methods. Different types of solutions are constructed, which describe overall expansion or collapse, envelope expansion with core collapse (EECC), the formation of central rotationally supported quasi-equilibrium disk as well as shocks. Though infinitely many, these self-similarity solutions have similar asymptotic behaviors which may impose diagnosis on the velocity and density structures in astrophysical systems.
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