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 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 ([Formula see text]), 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 [Formula see text] and the fractional parameter ([Formula see text]). For n = 1, the volume decreases when [Formula see text] = 0.1 and then increases when [Formula see text] = 0.2 and 0.3. The volume of the sphere reduces as both [Formula see text] and [Formula see text] 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 ([Formula see text], [Formula see text]). The situation is altered by lowering the fractional parameter; the mass limit increases to Mlimit = 1.63348 M⊙ at [Formula see text] and [Formula see text].