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 $\alpha$, declination $\delta$ and range $\rho$
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
$\omega$ 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.
