  We find a new solution to calculate the orbital periastron advance of a test
body subject to a central gravitational force field, for relativistic theories
and models beyond Einstein. This analitycal formula has general validity that
includes all the post-Newtonian (PN) contributions to the dynamics and is
useful for high-precision gravitational tests. The solution is directly
applicable to corrective potentials of various forms, without the need for
numerical integration. Later, we apply it to the Scalar Tensor Fourth Order
Gravity (STFOG) and NonCommutative Geometry, providing corrections to the
Newtonian potential of Yukawa-like form $V(r)=\alpha \frac{e^{-\beta r}}{r}$,
and we conduct the first analysis involving all the PN terms for these
theories. The same work is performed with a Schwarzschild geometry perturbed by
a Quintessence Field, leading to a power-law potential $V(r)=\alpha_q {r}^q$.
Finally, by using astrometric data of the Solar System planetary precessions
and those of S2 star around Sgr A*, we infer new theoretical constraints and
improvements in the bounds for $\beta$. The resulting simulated orbits turn out
to be compatible with General Relativity.
