  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.
