General formulae for the periapsis shift of a quasi-circular orbit in static spherically symmetric spacetimes and the active gravitational mass density

We study the periapsis shift of a quasi-circular orbit in general static spherically symmetric spacetimes. We derive two formulae in full order with respect to the gravitational field, one in terms of the gravitational mass $m$ and the Einstein tensor and the other in terms of the orbital angular velocity and the Einstein tensor. These formulae reproduce the well-known ones for the forward shift in the Schwarzschild spacetime. In a general case, the shift deviates from that in the vacuum spacetime due to a particular combination of the components of the Einstein tensor at the radius $r$ of the orbit. The formulae give a backward shift due to the extended-mass effect in Newtonian gravity. In general relativity, in the weak-field and diffuse regime, the active gravitational mass density, ${\rho }_A=(\epsilon +p_r+2p_t)\mathrm{/}{c}^2$, plays an important role, where $\epsilon$, $p_r$, and $p_t$ are the energy density, the radial stress, and the tangential stress of the matter field, respectively. We show that the shift is backward if ${\rho }_A$ is beyond a critical value ${\rho }_c\simeq 2.8\times {10}^{-15}g\mathrm{/}c{m}^3(m\mathrm{/}{M}_{\odot }{)}^2(r\mathrm{/}au{)}^{-4}$, while a forward shift greater than that in the vacuum spacetime instead implies ${\rho }_A<0$, i.e., the violation of the strong energy condition, and thereby provides evidence for dark energy. We obtain new observational constraints on ${\rho }_A$ in the Solar System and the Galactic Centre.