Stability Analysis of Relativistic Polytropes

Abstract

We study the relativistic self-gravitating, hydrostatic spheres with a polytropic equation of state, considering structures with the polytropic indices n=1(0.5)3 and illustrate the results for the relativistic parameters σ=0−0.75. We determine the critical relativistic parameter at which the mass of the polytrope has a maximum value and represents the first mode of radial instability. For n=1(0.5)2.5, stable relativistic polytropes occur for σ less than the critical values 0.42, 0.20, 0.10, and 0.04, respectively, while unstable relativistic polytropes are obtained when σ is greater than the same values. When n=3.0 and σ>0.5, energetically unstable solutions occur. The results of critical values are in full agreement with those evaluated by several authors. Comparisons between analytical and numerical solutions of the given relativistic functions provide a maximum relative error of order 10−3.

Stability analysis of fractional relativistic polytropes

“…meaning the gravitational radius / GM c 2 2 is up to 43.0% of the physical (invariant) radius, while for = n 3.0, it is up to 14.4% of the radius, a notable decrease compared to the limit at = n 1.0. These findings, detailed in tables 5-10 and illustrated in figures 5-8, agree with [2,21] studies. The mass-radius relationship in relativistic polytropes is investigated, revealing a variety of correlations between the gravitational and physical radii of these polytropes for a range of relativistic parameters σ and polytropic indices n. All results are consistent with the findings of several other researchers.…”

“…This research underscores the importance of the critical relativistic parameter, s , CR at which the mass of a polytrope reaches its maximum value, indicating the beginning of radial instability. For a fractional parameter a = 1, we identify critical values s CR = 0.42, 0.20, 0.10, and 0.0 for polytropic indices = n 1, 1.5, 2, and 3, respectively, aligning with [16,21] findings. Decreasing values of a, such as 0.99, 0.98, and 0.97, correspond to the increase in the critical relativistic parameter s .…”

“…Studies by [19,20] conducted a comprehensive analysis of polytropic sphere stability, employing two distinct approaches, namely the energetic and dynamic methods based on the study of radial pulsations. In a recent study by [21], the critical value of the relativistic parameter for relativistic polytropes, spanning polytropic indices ranging from 1.0 to 3.0, was thoroughly examined. The investigation revealed a noteworthy trend: as the polytropic index increases, the critical value of the relativistic parameter diminishes.…”

Ann and Analytical Solutions to Relativistic Isothermal Gas Spheres

“…We performed the calculations for the range of the dimensionless parameter x = 0 − 35; this upper limit of x corresponds to an isothermal sphere on the verge of gravothermal collapse (Antonov 1985). To obtain an accurate result for these analytical physical parameters, we accelerated both the series expansions of the Emden function (u) and the mass function (ν) by the accelerated scheme suggested by Nouh (2004); Saad et al (2021). The radius, density, and mass of the gas sphere are calculated using equations (8).…”

“…Sharma (1990) used the Pade approximation technique to provide straightforward and accessible approximate analytical solutions to the TOV equation of hydrostatic equilibrium, and his results indicate that general relativity isothermal arrangements have a limited extent. Saad (2017) proposed a novel approximate analytical solution to the TOV equation using a combination of the Euler-Abel transformation and the Pade approximation. Besides the numerical and analytical solutions of similar nonlinear differential equations similar to TOV equations, artificial intelligence approaches presented reliable results in many problems arising in astrophysics.…”