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General Relativity extensions based on Renormalization Group effects aremotivated by a known physical principle and constitute a class of extendedgravity theories that have some unexplored unique aspects. In this work wedevelop in detail the Newtonian and post Newtonian limits of a realisationcalled Renormalization Group extended General Relativity (RGGR). Specialattention is taken to the external potential effect, which constitutes a typeof screening mechanism typical of RGGR. In the Solar System, RGGR depends on asingle dimensionless parameter $\bar \nu_\odot$, and this parameter is suchthat for $\bar \nu_\odot = 0$ one fully recovers GR in the Solar System.Previously this parameter was constrained to be $|\bar \nu_\odot| \lesssim10^{-21}$, without considering the external potential effect. Here we show thatunder a certain approximation RGGR can be cast in a form compatible with theParametrised Post-Newtonian (PPN) formalism, and we use both the PPN formalismand the Laplace-Runge-Lenz technique to put new bounds on $\bar \nu_\odot$,either considering or not the external potential effect. With the externalpotential effect the new bound reads $|\bar \nu_\odot| \lesssim 10^{-16}$. Wediscuss the possible consequences of this bound to the dark matter abundance ingalaxies.

physical review. d/physical review. d.

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