Compact stars in Rastall gravity: hydrostatic equilibrium and radial pulsations

Within the context of Rastall gravity, we investigate the hydrostatic equilibrium and dynamical stability against radial pulsations of compact stars, where a free parameter $\beta$ measures the deviations from General Relativity (GR). We derive both the modified Tolman-Oppenheimer-Volkoff (TOV) equations and the Sturm-Liouville differential equation governing the adiabatic radial oscillations. Such equations are solved numerically in order to obtain the compact-star properties for two realistic equations of state (EoSs). For hadronic matter, the fundamental mode frequency $\omega_0$ becomes unstable almost at the critical central energy density corresponding to the maximum gravitational mass. However, for quark matter, where larger values of $\vert\beta\vert$ are required to observe appreciable changes in the mass-radius diagram, there exist stable stars after the maximum-mass configuration for negative values of $\beta$. Using an independent analysis, our results reveal that the emergence of a cusp can be used as a criterion to indicate the onset of instability when the binding energy is plotted as a function of the proper mass. Specifically, we find that the central-density value where the binding energy is a minimum corresponds precisely to $\omega_0^2 =0$.