Analysis of the fractional relativistic polytropic gas sphere
Abstract Many stellar configurations, including white dwarfs, neutron stars, black holes, supermassive stars, and star clusters, rely on relativistic effects. The Tolman–Oppenheimer–Volkoff (TOV) equation of the polytropic gas sphere is ultimately a hydrostatic equilibrium equation developed from th...
| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
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Nature Portfolio
2023-08-01
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| Series: | Scientific Reports |
| Online Access: | https://doi.org/10.1038/s41598-023-41392-y |
| _version_ | 1797276522665476096 |
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| author | Mohamed S. Aboueisha Mohamed I. Nouh Emad A. -B. Abdel-Salam Tarek M. Kamel M. M. Beheary Kamel A. K. Gadallah |
| author_facet | Mohamed S. Aboueisha Mohamed I. Nouh Emad A. -B. Abdel-Salam Tarek M. Kamel M. M. Beheary Kamel A. K. Gadallah |
| author_sort | Mohamed S. Aboueisha |
| collection | DOAJ |
| description | Abstract Many stellar configurations, including white dwarfs, neutron stars, black holes, supermassive stars, and star clusters, rely on relativistic effects. The Tolman–Oppenheimer–Volkoff (TOV) equation of the polytropic gas sphere is ultimately a hydrostatic equilibrium equation developed from the general relativity framework. In the modified Riemann Liouville (mRL) frame, we formulate the fractional TOV (FTOV) equations and introduce an analytical solution. Using power series expansions in solving FTOV equations yields a limited physical range to the convergent power series solution. Therefore, combining the two techniques of Euler–Abel transformation and Padé approximation has been applied to improve the convergence of the obtained series solutions. For all possible values of the relativistic parameters ( $$\sigma$$ σ ), we calculated twenty fractional gas models for the polytropic indexes n = 0, 0.5, 1, 1.5, 2. Investigating the impacts of fractional and relativistic parameters on the models revealed fascinating phenomena; the two effects for n = 0.5 are that the sphere’s volume and mass decrease with increasing $$\sigma$$ σ and the fractional parameter ( $$\alpha$$ α ). For n = 1, the volume decreases when $$\sigma$$ σ = 0.1 and then increases when $$\sigma$$ σ = 0.2 and 0.3. The volume of the sphere reduces as both $$\sigma$$ σ and $$\alpha$$ α increase for n = 1.5 and n = 2. We calculated the maximum mass and the corresponding minimum radius of the white dwarfs modeled with polytropic index n = 3 and several fractional and relativistic parameter values. We obtained a mass limit for the white dwarfs somewhat near the Chandrasekhar limit for the integer models with small relativistic parameters ( $$\alpha = 1$$ α = 1 , $$\sigma = 0.001$$ σ = 0.001 ). The situation is altered by lowering the fractional parameter; the mass limit increases to Mlimit = 1.63348 M⊙ at $$\alpha = 0.95$$ α = 0.95 and $$\sigma = 0.001$$ σ = 0.001 . |
| first_indexed | 2024-03-07T15:29:23Z |
| format | Article |
| id | doaj.art-1836ea902e684bb08453a41bdbf82b96 |
| institution | Directory Open Access Journal |
| issn | 2045-2322 |
| language | English |
| last_indexed | 2024-03-07T15:29:23Z |
| publishDate | 2023-08-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | Scientific Reports |
| spelling | doaj.art-1836ea902e684bb08453a41bdbf82b962024-03-05T16:30:40ZengNature PortfolioScientific Reports2045-23222023-08-0113112610.1038/s41598-023-41392-yAnalysis of the fractional relativistic polytropic gas sphereMohamed S. Aboueisha0Mohamed I. Nouh1Emad A. -B. Abdel-Salam2Tarek M. Kamel3M. M. Beheary4Kamel A. K. Gadallah5Astronomy Department, National Research Institute of Astronomy and GeophysicsAstronomy Department, National Research Institute of Astronomy and GeophysicsDepartment of Mathematics, Faculty of Science, New Valley UniversityAstronomy Department, National Research Institute of Astronomy and GeophysicsDepartment of Astronomy and Meteorology, Faculty of Science, Al-Azhar UniversityDepartment of Astronomy and Meteorology, Faculty of Science, Al-Azhar UniversityAbstract Many stellar configurations, including white dwarfs, neutron stars, black holes, supermassive stars, and star clusters, rely on relativistic effects. The Tolman–Oppenheimer–Volkoff (TOV) equation of the polytropic gas sphere is ultimately a hydrostatic equilibrium equation developed from the general relativity framework. In the modified Riemann Liouville (mRL) frame, we formulate the fractional TOV (FTOV) equations and introduce an analytical solution. Using power series expansions in solving FTOV equations yields a limited physical range to the convergent power series solution. Therefore, combining the two techniques of Euler–Abel transformation and Padé approximation has been applied to improve the convergence of the obtained series solutions. For all possible values of the relativistic parameters ( $$\sigma$$ σ ), we calculated twenty fractional gas models for the polytropic indexes n = 0, 0.5, 1, 1.5, 2. Investigating the impacts of fractional and relativistic parameters on the models revealed fascinating phenomena; the two effects for n = 0.5 are that the sphere’s volume and mass decrease with increasing $$\sigma$$ σ and the fractional parameter ( $$\alpha$$ α ). For n = 1, the volume decreases when $$\sigma$$ σ = 0.1 and then increases when $$\sigma$$ σ = 0.2 and 0.3. The volume of the sphere reduces as both $$\sigma$$ σ and $$\alpha$$ α increase for n = 1.5 and n = 2. We calculated the maximum mass and the corresponding minimum radius of the white dwarfs modeled with polytropic index n = 3 and several fractional and relativistic parameter values. We obtained a mass limit for the white dwarfs somewhat near the Chandrasekhar limit for the integer models with small relativistic parameters ( $$\alpha = 1$$ α = 1 , $$\sigma = 0.001$$ σ = 0.001 ). The situation is altered by lowering the fractional parameter; the mass limit increases to Mlimit = 1.63348 M⊙ at $$\alpha = 0.95$$ α = 0.95 and $$\sigma = 0.001$$ σ = 0.001 .https://doi.org/10.1038/s41598-023-41392-y |
| spellingShingle | Mohamed S. Aboueisha Mohamed I. Nouh Emad A. -B. Abdel-Salam Tarek M. Kamel M. M. Beheary Kamel A. K. Gadallah Analysis of the fractional relativistic polytropic gas sphere Scientific Reports |
| title | Analysis of the fractional relativistic polytropic gas sphere |
| title_full | Analysis of the fractional relativistic polytropic gas sphere |
| title_fullStr | Analysis of the fractional relativistic polytropic gas sphere |
| title_full_unstemmed | Analysis of the fractional relativistic polytropic gas sphere |
| title_short | Analysis of the fractional relativistic polytropic gas sphere |
| title_sort | analysis of the fractional relativistic polytropic gas sphere |
| url | https://doi.org/10.1038/s41598-023-41392-y |
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