Stochastic analysis of compact stars under composite polytropes

The study of dense matter has been greatly advanced by progress in theoretical high-energy simulations and modern observational astronomy. Neutron stars serve as natural laboratories for probing matter under extreme density and strong gravitational fields. While polytropic equations of state are widely employed, conventional models based on a single polytropic index cannot adequately represent the stratified, layered nature of realistic compact-star interiors. To address this limitation, we develop a composite relativistic polytropic model in which the polytropic index varies smoothly with radius. By coupling the Einstein field equations with a generalized composite polytropic equation of state, we derive the composite Tolman–Oppenheimer–Volkoff (CTOV) system. The resulting nonlinear equations are solved using a Monte Carlo-based numerical integration method, which efficiently handles stiffness while enabling probabilistic exploration of the parameter space and natural uncertainty quantification. Our results demonstrate that increasing the relativistic parameter σ significantly reduces both the Emden function and the enclosed mass function, producing more compact stellar configurations. Sharper core–envelope transitions (ε = 0.01) yield systematically higher compactness than smoother transitions (ε = 0.03). The derived mass–radius relations reproduce the observed diversity of neutron stars, successfully matching both low-mass, large-radius systems such as PSR J0030 + 0451 and high-mass compact pulsars such as PSR J1614–2230. Importantly, the maximum-mass analysis shows that stiff composite configurations (nc = 1, ne = 2, xc = 0.7) can support gravitational masses up to Mmax ≈ 3.47 M $$_{ \odot }$$ for ε = 0.01 and Mmax ≈ 2.71 M $$_{ \odot }$$ for ε = 0.03, with corresponding minimum radii in the range Rmin ≈ 10.6–13.2 km, consistent with current observational constraints. These findings confirm that composite polytropes provide a flexible, physically motivated framework for modeling stratified compact stars and for constraining the dense-matter equation of state.