A Closed-Form Master Solution for Gravitational Decoupling of a Unified Family of Isotropic Stellar Seeds

Abstract

We present a unification scheme for anisotropic compact stars generated via minimal geometric deformation (MGD), built on a two-parameter seed family $ν(r)=\text{Ln}[C(1+ar^{2n})^{m}]$ whose radial metric potential $μ(r)=e^{-λ(r)}$ is fixed in closed form by the isotropy condition. At $n=1$ the family reproduces Tolman IV, Korkina--Orlyanskii, Heintzmann IIa, and Durgapal IV--V as the particular cases $m=1,2,3,4,5$. The key result is a single closure of the MGD $θ$-sector, through one source function $g(r)$, that integrates the decoupler equation in closed form, in terms of the Gauss hypergeometric function ${}_2F_1$, for \emph{arbitrary} $(n,m)$: the anisotropic extension of the $n=1$ family is thus obtained from one master formula. We derive the effective anisotropic matter content, the Darmois--Israel matching to the exterior Schwarzschild vacuum, and the full set of physical acceptability criteria constraining the parameter space, and map the admissible region in compactness and decoupling strength explicitly for the case $n=1$: Tolman IV and Korkina--Orlyanskii admit physically consistent anisotropic deformations at arbitrarily high compactness, while Heintzmann IIa and Durgapal IV--V are bounded by a compactness ceiling beyond which no value of the decoupling parameter restores causality. Using the same closed-form matter content, we compute the quadrupolar tidal Love number and dimensionless tidal deformability across the admissible region, finding that the decoupling systematically enhances both relative to the isotropic seed at fixed compactness, with a sharp rise near the causal ceiling in the Heintzmann and Durgapal branches.