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Steady States of Rotating Stars and Galaxies
Analysis of PDEs · Submitted 12 Mar 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Here it is proven that a curve of solutions exists, parametrized by the rotation speed, with a fixed mass independent of the speed.
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The abstract, sentence by sentence
A rotating continuum of particles attracted to each other by gravity may be modeled by the Euler-Poisson system. The existence of solutions is a very classical problem. Here it is proven that a curve of solutions exists, parametrized by the rotation speed, with a fixed mass independent of the speed. The rotation is allowed to vary with the distance to the axis. A special case is when the equation of state is $p=\rho^\gamma,\ 6/5<\gamma<2,\ \gamma\ne4/3$, in contrast to previous variational methods which have required $4/3 < \gamma$. The continuum of particles may alternatively be modeled microscopically by the Vlasov-Poisson system. The kinetic density is a prescribed function. We prove an analogous theorem asserting the existence of a curve of solutions with constant mass. In this model the whole range $(6/5,2)$ is allowed, including $\gamma=4/3$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A rotating continuum of particles attracted to each other by gravity may be modeled by the Euler-Poisson system. | background 0.97 | 0.07 | 0.02 | hedged possibility 0.98 |
| s1 | The existence of solutions is a very classical problem. | background 1.00 | 0.04 | 0.02 | asserted 0.99 |
| s2 | Here it is proven that a curve of solutions exists, parametrized by the rotation speed, with a fixed mass independent of the speed. | result 0.89 | 0.94 | 0.74 | asserted 1.00 |
| s3 | The rotation is allowed to vary with the distance to the axis. | contribution 0.40 | 0.47 | 0.04 | asserted 0.71 |
| s4 | A special case is when the equation of state is $p=\rho^\gamma,\ 6/5<\gamma<2,\ \gamma\ne4/3$, in contrast to previous variational methods which have required $4/3 < \gamma$. | result 0.47 | 0.72 | 0.10 | asserted 0.97 |
| s5 | The continuum of particles may alternatively be modeled microscopically by the Vlasov-Poisson system. | method 0.62 | 0.13 | 0.03 | hedged possibility 0.94 |
| s6 | The kinetic density is a prescribed function. | method 0.78 | 0.22 | 0.02 | asserted 1.00 |
| s7 | We prove an analogous theorem asserting the existence of a curve of solutions with constant mass. | result 0.88 | 0.95 | 0.29 | asserted 1.00 |
| s8 | In this model the whole range $(6/5,2)$ is allowed, including $\gamma=4/3$. | result 0.91 | 0.85 | 0.15 | asserted 0.99 |
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Original abstract
A rotating continuum of particles attracted to each other by gravity may be modeled by the Euler-Poisson system. The existence of solutions is a very classical problem. Here it is proven that a curve of solutions exists, parametrized by the rotation speed, with a fixed mass independent of the speed. The rotation is allowed to vary with the distance to the axis. A special case is when the equation of state is $p=\rho^\gamma,\ 6/5<\gamma<2,\ \gamma\ne4/3$, in contrast to previous variational methods which have required $4/3 < \gamma$. The continuum of particles may alternatively be modeled microscopically by the Vlasov-Poisson system. The kinetic density is a prescribed function. We prove an analogous theorem asserting the existence of a curve of solutions with constant mass. In this model the whole range $(6/5,2)$ is allowed, including $\gamma=4/3$.
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When this paper was almost complete, we learned of very recent work [ref] by Jang and Makino, who also studied the EP model using an IFT approach in the case of the power law $p=Cρ^γ$ and constant rotation speed.
On slowly rotating axisymmetric solutions of the Euler-Poisson equations · Informal statement of results · 0.82
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