arXiv:2407.06238v1 Announce Type: new

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=Krho^{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.

## A Roadmap for Astrophysics Research on Gravitational Stability

Astrophysics researchers have long been interested in understanding the gravitational stability of self-gravitating polytropic fluid spheres, as it provides insights into the formation and evolution of celestial bodies like planets and stars. In this study, we employ fractional calculus to investigate the stability of hydrostatic spheres governed by a polytropic equation of state P=Krho^{1+1/n}.

### The Methodology

To evaluate the stability of these relativistic polytropes, we employ the critical point method, originally developed by Tooper in 1964. This method relies on energetic principles and allows us to identify the critical mass and radius at which polytropic spheres transition from stable to unstable states.

### The Findings

The results of our calculations demonstrate that there is a critical relativistic parameter, sigma_{CR}, which determines the onset of radial instability in relativistic polytropes. We find that the mass of the polytrope peaks at this critical value, indicating the shift from stable to unstable states.

Here are the critical values (sigma) for different polytropic indices (n) and fractional parameters (alpha):

- For a polytropic index of 1 and a fractional parameter alpha, a stable relativistic polytrope exists for sigma values below the critical threshold of sigma = 0.42.
- For a polytropic index of 1.5 and a fractional parameter alpha, a stable polytrope exists for sigma values below sigma = 0.20.
- For a polytropic index of 2 and a fractional parameter alpha, a stable polytrope exists for sigma values below sigma = 0.10.
- For a polytropic index of 3 and a fractional parameter alpha, a stable polytrope exists for sigma values below sigma = 0.0.

It is important to note that any value of sigma beyond the critical threshold leads to instability in the polytropic sphere.

### The Future Roadmap

Based on these findings, there are several potential challenges and opportunities for future research in astrophysics:

**Further Exploration of Relativistic Polytropes:**Investigate the stability of self-gravitating polytropic fluid spheres with higher polytropic indices and fractional parameters. Determine the critical relativistic values for instability in these scenarios.**Impact of Other Factors:**Consider the effect of additional factors, such as magnetic fields or rotation, on the stability of polytropic spheres. Explore how these factors interact with the fractional and relativistic parameters.**Comparisons with Observations:**Compare the predictions from theoretical calculations with observed stellar and planetary structures to validate the findings and improve our understanding of celestial bodies.**Applications in Cosmology:**Investigate the implications of these stability findings for the broader field of cosmology. Explore how the stability of polytropic spheres influences the formation and evolution of galaxies and galaxy clusters.

As astrophysics continues to advance, a deeper understanding of the gravitational stability of polytropic fluid spheres will shed light on the fundamental processes that shape our universe.