Stability Analysis Of Fractional Relativistic Polytropes

TL;DR

This paper investigates the stability of relativistic, fractional polytropic spheres in astrophysics, identifying critical parameters where these celestial structures transition from stability to instability using fractional calculus and energetic principles.

Contribution

It introduces a fractional calculus approach to analyze the stability of relativistic polytropes, extending classical methods to include fractional parameters and relativistic effects.

Findings

Critical relativistic parameter  where instability begins for different polytropic indices.

Stable polytropes exist below specific  thresholds for each index.

Critical  increases as fractional parameter  decreases.

Abstract

In astrophysics, the gravitational stability of a self-gravitating polytropic fluid sphere is an intriguing subject, especially when trying to comprehend the genesis and development of celestial bodies like planets and stars. This stability is the sphere's capacity to stay in balance in the face of disruptions. We utilize fractional calculus to explore self-gravitating, hydrostatic spheres governed by a polytropic equation of state \P=K\rho^{1+1/n}. We focus on structures with polytropic indices ranging from 1 to 3 and consider relativistic and fractional parameters, denoted by \sigma and \alpha, respectively. The stability of these relativistic polytropes is evaluated using the critical point method, which is associated with the energetic principles developed in 1964 by Tooper. This approach enables us to pinpoint the critical mass and radius at which where polytropic spheres shift from stable to unstable states. The results highlight the critical relativistic parameter where the polytrope's mass peaks, signaling the onset of radial instability. For polytropic indices of 1, 1.5, 2, and 3 with a fractional parameter \alpha, we observe stable relativistic polytropes for \sigma values below the critical thresholds of \sigma= 0.42, 0.20, 0.10, and 0.0, respectively. Conversely, instability emerges as \sigma surpasses these critical values. Our comprehensive calculations reveal that the critical relativistic value (\sigma_{CR}) for the onset of instability tends to increase as the fractional parameter {\alpha} decreases.