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Solar System constraints on Renormalization Group extended General Relativity: The PPN and Laplace-Runge-Lenz analyses with the external potential effect

General Relativity extensions based on Renormalization Group effects are motivated by a known physical principle and constitute a class of extended gravity theories that have some unexplored unique aspects. In this work we develop in detail the Newtonian and post Newtonian limits of a realisation called Renormalization Group extended General Relativity (RGGR). Special attention is taken to the external potential effect, which constitutes a type of screening mechanism typical of RGGR. In the Solar System, RGGR depends on a single dimensionless parameter $\bar ν_\odot$, and this parameter is such that for $\bar ν_\odot = 0$ one fully recovers GR in the Solar System. Previously this parameter was constrained to be $|\bar ν_\odot| \lesssim 10^{-21}$, without considering the external potential effect. Here we show that under a certain approximation RGGR can be cast in a form compatible with the Parametrised Post-Newtonian (PPN) formalism, and we use both the PPN formalism and the Laplace-Runge-Lenz technique to put new bounds on $\bar ν_\odot$, either considering or not the external potential effect. With the external potential effect the new bound reads $|\bar ν_\odot| \lesssim 10^{-16}$. We discuss the possible consequences of this bound to the dark matter abundance in galaxies.