Stability analysis of fractional relativistic polytropes

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 polytropic spheres shift from stable to unstable states. The results highlight the critical relativistic parameter ( sigma CR ) where the polytrope's mass peaks, signaling the onset of radial instability. For polytropic indices of n = 1, 1.5, 2, and 3 with a fractional parameter alpha=1, we observe stable relativistic polytropes for sigma values below the critical thresholds of sigma CR= 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.