A free parametrized TOV: Modified Gravity from Newtonian to Relativistic Stars

We test a free {\it ad hoc} parametrization of the Tolman-Oppenheimer-Volkoff (TOV) equation. We do not have in mind any specific extended theory of gravity (ETG) but each new parameter introduced has a physical interpretation. Our aim is fully pedagogical rather than a proposal for a new ETG. Given a realistic neutron star equation of state we map the contributions of each new parameter into a shift in trajectories of the mass-radius diagram. This exercise allows us to make the correspondence between each TOV sector with possible modifications of gravity and clarifies how neutron star observations are helpful for distinguishing theories.