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, $\rho_{A}=(\epsilon+p_{r}+2p_{t})/c^{2}$, 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 $\rho_{c}\simeq 2.8\times 10^{-15} \mbox{g}/\mbox{cm}^{3} (m/M_{\odot})^{2}(r/\mbox{au})^{-4}$, 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.