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Observational constraints on spatial anisotropy of G from orbital motions

A phenomenological anisotropic variation ΔG/G of the Newtonian gravitational coupling parameter G, if real, would affect the orbital dynamics of a two-body gravitationally bound system in a specific way. We analytically work out the long-term effects that such a putative modification of the usual Newtonian inverse-square law would induce on the trajectory of a test particle orbiting a central mass. Without making any a-priori simplifying assumptions concerning the orbital configuration of the test particle, it turns out that its osculating semi-major axis a, eccentricity e, pericenter \varpi and mean anomaly M undergo long-term temporal variations, while the inclination I and the node Ωare left unaffected. Moreover, the radial and the transverse components of the position and the velocity vectors r and v of the test particle experience non-vanishing changes per orbit, contrary to the out-of-plane ones. Then, we compute our theoretical predictions for some of the major bodies of the solar system by orienting the gradient of G(r) towards the Galactic Center and keeping it fixed over the characteristic timescales involved. By comparing our calculation to the latest observational determinations for the same bodies, we infer ΔG/G <= 10^-17 over about 1 au. Finally, we consider also the Supermassive Black Hole hosted by the Galactic Center in Sgr A^* and the main sequence star S2 orbiting it in about 16 yr, obtaining just ΔG/G <= 10^-2 over 1 kau.