Celestial mechanics in Newtonian-like gravity with variable $G$

A Newtonian-like theory inspired by the Brans-Dicke gravitational Lagrangian has been recently proposed in Ref. arXiv:2009.04434(v4). This work demonstrates that the modified gravitational force acting on a test particle is analogous to that derived from the Manev potential. Specifically, an additional term $\propto r^{-3}$ emerges alongside the conventional Newtonian component. We analyse the predicted expression for the pericenter advance and the Roche limit and use them to constraint the theory's single free parameter $\omega$ which is analogous to the Brans-Dicke parameter. At the same time this theory is able to solve the advance of Mercury's perihelion, we also show that there is no relevant impact on the Roche limit in comparison to the well known Newtonian results.