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Conformable Fractional Isothermal Gas Spheres
New Astronomy ( IF 1.9 ) Pub Date : 2021-04-01 , DOI: 10.1016/j.newast.2020.101511
Eltayeb A. Yousif , Ahmed M.A. Adam , Abaker A. Hassaballa , Mohamed I. Nouh

Abstract The isothermal gas sphere is well known as a powerful tool to model many problems in astrophysics, physics, chemistry, and engineering. This singular differential equation has not an exact solution and solved only by numerical and approximate methods. In the present paper and within the framework of the Newtonian hydrostatic equilibrium, we have developed general analytical formulations for the fractional isothermal gas sphere. To obtain analytical expressions for mass, radius, and density, besides the fractional isothermal gas sphere, we used the conformable fractional calculus. Using the series expansion method, we obtained a general recurrences relation, which allows us to determine the series coefficients. The comparison of the series solution with the numerical ones for the fractional parameter α = 1 is good for dimensionless parameter up to x ≃ 3.2, beyond this value, the series diverges. We applied a combination of Euler-Abel and Pade techniques to accelerate the series, therefore accelerated series converge to the numerical desired value. We analyzed some physical parameters of a typical model of the neutron stars such as the mass-radius relation, density, and pressure ratio for different models. We found that the current models of the conformable neutron stars had smaller volumes and masses than both stars in the context of modified Rienmann-Liouville derivatives as well as the integer one.

中文翻译:

适形分数等温气体球

摘要 等温气球是众所周知的天体物理学、物理学、化学和工程学中许多问题建模的有力工具。这个奇异微分方程没有精确解,只能用数值方法和近似方法求解。在本论文中,在牛顿流体静力平衡的框架内,我们开发了分数等温气球的一般分析公式。为了获得质量、半径和密度的解析表达式,除了分数等温气球外,我们还使用了适形分数微积分。使用级数展开方法,我们获得了一般的递推关系,这使我们能够确定级数系数。对于分数参数 α = 1 的级数解与数值解的比较对于无量纲参数是好的,直到 x ≃ 3.2,超过这个值,级数发散。我们结合使用 Euler-Abel 和 Pade 技术来加速级数,因此加速级数收敛到数值所需的值。我们分析了中子星典型模型的一些物理参数,如不同模型的质量半径关系、密度和压力比。我们发现,在修正的 Rienmann-Liouville 导数以及整数 1 的背景下,可整合中子星的当前模型具有比两颗恒星更小的体积和质量。因此加速级数收敛到数值所需的值。我们分析了中子星典型模型的一些物理参数,如不同模型的质量半径关系、密度和压力比。我们发现,在修正的 Rienmann-Liouville 导数以及整数 1 的背景下,可整合中子星的当前模型具有比两颗恒星更小的体积和质量。因此加速级数收敛到数值所需的值。我们分析了中子星典型模型的一些物理参数,如不同模型的质量半径关系、密度和压力比。我们发现,在修正的 Rienmann-Liouville 导数以及整数 1 的背景下,可整合中子星的当前模型具有比两颗恒星更小的体积和质量。
更新日期:2021-04-01
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