The Boltzmann-Poisson equation with a central body: analytical solutions in one and two dimensions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916270651211776 |
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| author | Chavanis, Pierre-Henri |
| author_facet | Chavanis, Pierre-Henri |
| contents | We consider an isothermal self-gravitating system surrounding a central body. This model can represent a galaxy or a globular cluster harboring a central black hole. It can also represent a gaseous atmosphere surrounding a protoplanet. In three dimensions, the Boltzmann-Poisson equation must be solved numerically in order to obtain the density profile of the gas [Chavanis {\it et al.}, Phys. Rev. E {\bf 109}, 014118 (2024)]. In one and two dimensions, we show that the Boltzmann-Poisson equation can be solved analytically. We obtain explicit analytical expressions of the density profile around a central body which generalize the analytical solutions found by Camm (1950) and Ostriker (1964) in the absence of a central body. Our results also have applications for self-gravitating Brownian particles (Smoluchowski-Poisson system), for the chemotaxis of bacterial populations (Keller-Segel model), and for two-dimensional point vortices (Onsager's model). In the case of bacterial populations, the central body could represent a supply of ``food'' that attracts the bacteria (chemoattractant). In the case of two-dimensional vortices, the central body could be a central vortex |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20438 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Boltzmann-Poisson equation with a central body: analytical solutions in one and two dimensions Chavanis, Pierre-Henri Statistical Mechanics Astrophysics of Galaxies We consider an isothermal self-gravitating system surrounding a central body. This model can represent a galaxy or a globular cluster harboring a central black hole. It can also represent a gaseous atmosphere surrounding a protoplanet. In three dimensions, the Boltzmann-Poisson equation must be solved numerically in order to obtain the density profile of the gas [Chavanis {\it et al.}, Phys. Rev. E {\bf 109}, 014118 (2024)]. In one and two dimensions, we show that the Boltzmann-Poisson equation can be solved analytically. We obtain explicit analytical expressions of the density profile around a central body which generalize the analytical solutions found by Camm (1950) and Ostriker (1964) in the absence of a central body. Our results also have applications for self-gravitating Brownian particles (Smoluchowski-Poisson system), for the chemotaxis of bacterial populations (Keller-Segel model), and for two-dimensional point vortices (Onsager's model). In the case of bacterial populations, the central body could represent a supply of ``food'' that attracts the bacteria (chemoattractant). In the case of two-dimensional vortices, the central body could be a central vortex |
| title | The Boltzmann-Poisson equation with a central body: analytical solutions in one and two dimensions |
| topic | Statistical Mechanics Astrophysics of Galaxies |
| url | https://arxiv.org/abs/2405.20438 |