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The Structure of Self-Gravitating Polytropic Systems with n around 5
Astrophysics · Submitted 30 Oct 2000
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Application of this method to the spherically symmetric stellar polytropes with $n\simeq5$ yields the solutions which describe spatially bound systems if n<5 and the formation of a second core if n>5.
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We investigate the structure of self-gravitating polytropic stellar systems. We present a method which allows to obtain approximate analytical solutions, $\psi_{n+\epsilon}({\bf x})$, of the nonlinear Poisson equation with the polytropic index $n+\epsilon$, given the solution $\psi_n({\bf x})$ with the polytropic index n, for any positive or negative $\epsilon$ such that $|\epsilon|\ll1$. Application of this method to the spherically symmetric stellar polytropes with $n\simeq5$ yields the solutions which describe spatially bound systems if n<5 and the formation of a second core if n>5. A heuristic approximate expressions for the radial profiles are also presented. Due to the duality between stellar and gas polytropes, our results are valid for gaseous, self-gravitating polytropic systems (e.g., molecular clouds) with the index $\gamma\simeq 6/5$. Stability of such systems and observational consequences for both stellar and gaseous systems are discussed.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate the structure of self-gravitating polytropic stellar systems. | contribution 0.52 | 0.86 | 0.03 | asserted 0.79 |
| s1 | We present a method which allows to obtain approximate analytical solutions, $\psi_{n+\epsilon}({\bf x})$, of the nonlinear Poisson equation with the polytropic index $n+\epsilon$, given the solution $\psi_n({\bf x})$ with the polytropic index n, for any positive or negative $\epsilon$ such that $|\epsilon|\ll1$. | method 0.86 | 0.95 | 0.09 | asserted 0.97 |
| s2 | Application of this method to the spherically symmetric stellar polytropes with $n\simeq5$ yields the solutions which describe spatially bound systems if n<5 and the formation of a second core if n>5. | result 1.00 | 0.94 | 0.71 | asserted 1.00 |
| s3 | A heuristic approximate expressions for the radial profiles are also presented. | contribution 0.84 | 0.93 | 0.04 | asserted 0.98 |
| s4 | Due to the duality between stellar and gas polytropes, our results are valid for gaseous, self-gravitating polytropic systems (e.g., molecular clouds) with the index $\gamma\simeq 6/5$. | interpretation 0.93 | 0.63 | 0.09 | asserted 1.00 |
| s5 | Stability of such systems and observational consequences for both stellar and gaseous systems are discussed. | interpretation 0.50 | 0.69 | 0.07 | asserted 0.73 |
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Original abstract
We investigate the structure of self-gravitating polytropic stellar systems. We present a method which allows to obtain approximate analytical solutions, $\psi_{n+\epsilon}({\bf x})$, of the nonlinear Poisson equation with the polytropic index $n+\epsilon$, given the solution $\psi_n({\bf x})$ with the polytropic index n, for any positive or negative $\epsilon$ such that $|\epsilon|\ll1$. Application of this method to the spherically symmetric stellar polytropes with $n\simeq5$ yields the solutions which describe spatially bound systems if n<5 and the formation of a second core if n>5. A heuristic approximate expressions for the radial profiles are also presented. Due to the duality between stellar and gas polytropes, our results are valid for gaseous, self-gravitating polytropic systems (e.g., molecular clouds) with the index $\gamma\simeq 6/5$. Stability of such systems and observational consequences for both stellar and gaseous systems are discussed.
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