Anisotropic compact stars in quadratic f(R) gravity and influence of quadratic parameter on properties of compact stars

In this article, we explore an anisotropic model to describe the Buchdahl compact star with a spherically symmetric matter distribution within the framework of modified $f\left(R\right)$ gravity. To achieve this, we consider the Starobinsky model $f\left(R\right)=R+\alpha R^2$, where α is a model parameter. To solve the Einstein Field equations, we adopt the embedded class-I geometry that satisfies the Karmarkar condition. Our analysis includes the generalized Darmois-Israel junction conditions, which are crucial for seamlessly matching the star's interior to the Schwarzschild exterior metric at the boundary in $f\left(R\right)$ gravity. These conditions have been employed to calculate the constant parameters by analyzing observational data from multiple compact stars, including Cen X-3, PSR J1903+327, Vela X-1, and PSR J1614-2230. This novel approach enables us to conduct a detailed analysis of Buchdahl compact star models and assess their physical viability. We also perform various physical tests, including equilibrium conditions and stability to evaluate the models' validity. The analysis of mass and radius reveals that the $M-R$ relationship is based on the Buchdahl limit and is sensitive to variations in the parameter α. With $\alpha =0$ representing General Relativity, the maximum mass and radius for different α values were calculated and compared with observational data from the five aforementioned neutron stars. The range of maximum mass and radii are: $[2.51-2.54M_{\odot }]$ and $8.71-9.33\text{km}$. The obtained values of radii align well with observed neutron star radii in X-ray binaries from spectroscopic studies, confirming the validity of the model.