Perihelion precession for modified Newtonian gravity

We calculate the perihelion precession $\delta$ for nearly circular orbits in a central potential $V(r)$. Differently from other approaches to this problem, we do not assume that the potential is close to the Newtonian one. The main idea in the deduction is to apply the underlying symmetries of the system to show that $\delta$ must be a function of $r·V^{\prime \prime }(r)/V^{\prime }(r)$ and to use the transformation behavior of $\delta$ in a rotating system of reference. This is equivalent to say that the effective potential can be written in a one-parameter set of possibilities as the sum of centrifugal potential and potential of the central force. We get the following universal formula valid for $V^{\prime }(r)>0$ reading $\delta (r)=2\pi ·[\frac{1}{\sqrt{3+r·V^{\prime \prime }(r)/V^{\prime }(r)}}-1]$. It has to be read as follows: a circular orbit at this value $r$ exists and is stable if and only if this $\delta$ is well-defined as real; and if this is the case, then the angular difference from one perihelion to the next one for nearly circular orbits at this $r$ is exactly $2\pi +\delta (r)$. Then we apply this result to examples of recent interest like modified Newtonian gravity and linearized fourth-order gravity. In the second part of the paper, we generalize this universal formula to static spherically symmetric space-times $d{s}^2=-e^{2\lambda (r)}d{t}^2+e^{2\mu (r)}d{r}^2+r^{2}d{\Omega }^2$; for orbits near $r$ it reads $\delta =2\pi ·[\frac{e^{\mu (r)}}{\sqrt{3-2r·{\lambda }^{\prime }(r)+r·{\lambda }^{\prime \prime }(r)/{\lambda }^{\prime }(r)}}-1]$ and can be applied to a large class of theories. For the Schwarzschild black hole with mass parameter $m>0$ it leads to $\delta =2\pi ·[\frac{1}{\sqrt{1-\frac{6m}{r}}}-1]$, a surprisingly unknown formula. It represents a strict result and is applicable for all values $r>6m$ and is in good agreement with the fact that stable circular orbits exist for $r>6m$ only. For $r\gg m$, one can develop in powers of $m$ and get the well-known approximation $\delta \approx \frac{6\pi m}{r}$.