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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.

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 Rieman Liouville (mRL) frame, we formulate the fractional TOV (FTOV) equations and introduce an analytical solution. Using power series expansions to solve the fractional TOV equations yields a limited physical range to the convergent power series solution. Therefore, the two techniques of Euler-Abel transformation and Pade approximation have been combined to improve the convergence of the obtained series solutions. For all possible values of the relativistic parameters (\sigma), 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 \sigma and the fractional parameter (\alpha). For n=1, the volume decreases when \sigma=0.1 and then increases when \sigma=0.2 and 0.3. The volume of the sphere reduces as both \sigma and \alpha 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 (\alpha=1, \sigma=0.001). The situation is altered by lowering the fractional parameter; the mass limit increases to Mlimit=1.63348 M at \alpha=0.95 and \sigma=0.001.