Orbital effects of spatial variations of fundamental coupling constants

We deal with the effects induced on the orbit of a test particle revolving around a central body by putative spatial variations of fundamental coupling constants $ζ$. In particular, we assume a dipole gradient for $ζ(\bds r)/\barζ$ along a generic direction $\bds{\hat{k}}$ in space. We analytically work out the long-term variations of all the six standard Keplerian orbital elements parameterizing the orbit of a test particle in a gravitationally bound two-body system. It turns out that, apart from the semi-major axis $a$, the eccentricity $e$, the inclination $I$, the longitude of the ascending node $Ω$, the longitude of pericenter $π$ and the mean anomaly $\mathcal{M}$ undergo non-zero long-term changes. By using the usual decomposition along the radial ($R$), transverse ($T$) and normal ($N$) directions, we also analytically work out the long-term changes $ΔR,ΔT,ΔN$ and $Δv_R,Δv_T,Δv_N$ experienced by the position and the velocity vectors $\bds r$ and $\bds v$ of the test particle. It turns out that, apart from $ΔN$, all the other five shifts do not vanish over one full orbital revolution. In the calculation we do not use \textit{a-priori} simplifying assumptions concerning $e$ and $I$. Thus, our results are valid for a generic orbital geometry; moreover, they hold for any gradient direction (abridged).

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