Constraining some $r^{-n}$ extra-potentials in modified gravity models with LAGEOS-type laser-ranged geodetic satellites

We focus on several models of modified gravity which share the characteristic of leading to perturbations of the Newtonian potential $\propto K_2{r}^{-2}$ and $\propto K_3{r}^{-3}$. In particular, by using existing long data records of the LAGEOS satellites, tracked on an almost continuous basis with the Satellite Laser Ranging (SLR) technique, we set preliminary constraints on the free parameters $K_2,K_3$ in a model-independent, phenomenological way. We obtain $\left|K_2\right|\lesssim 2.1\times 10^6~$\text{m}$^4~$\text{s}$^{-2},~ -2.5\times 10^{12}~$\text{m}$^5~$\text{s}$^{-2}\lesssim K_3 \lesssim 4.1\times 10^{12}~$\text{m}$^5~$\text{s}$^{-2}.$ They are several orders of magnitude tighter than corresponding bounds existing in the literature inferred with different techniques and in other astronomical and astrophysical scenarios. Then, we specialize them to the different parameters characterizing the various models considered. The availability of SLR data records of increasing length and accuracy will allow to further refine and strengthen the present results.