Does the Newton's gravitational constant vary sinusoidally with time? Orbital motions say no

A sinusoidally time-varying pattern of the values of the Newton's constant of gravitation $G$ measured in Earth-based laboratories over the latest decades has been recently reported in the literature. We put to the test the hypothesis that the aforementioned harmonic variation may pertain $G$ itself in a direct and independent way. We numerically integrated the ad-hoc modified equations of motion of the major bodies of the Solar System by finding that the orbits of the planets would be altered by an unacceptably larger amount in view of the present-day high accuracy astrometric measurements. In the case of Saturn, its geocentric right ascension $α$, declination $δ$ and range $ρ$ would be affected up to $10^4-10^5$ milliarcseconds and $10^5$ km, respectively; the present-day residuals of such observables are as little as about $4$ milliarcseconds and $10^{-1}$ km, respectively. \textcolor{black}{We analytically calculated the long-term orbital effects induced by the putative harmonic variation of $G$ at hand finding non-zero rates of change for the semimajor axis $a$, the eccentricity $e$ and the argument of pericenter $ω$ of a test particle. For the LAGEOS satellite, an orbital increase as large as $3.9$ m yr$^{-1}$ is predicted, in contrast with the observed decay of $-0.203\pm 0.035$ m yr$^{-1}$. An anomalous perihelion precession as large as 14 arcseconds per century is implied for Saturn, while latest observations constrain it down to the $10^{-4}$ arcseconds per century level. The rejection level provided by Mercury is of the same order of magnitude.

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