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Massive Brans-Dicke (BD) theory is among the simplest general relativityextensions. It is commonly found as the weak-field limit of other gravitationaltheories. Here we do a detailed post-Newtonian analysis of massive BD theories.We start by expanding the massive BD field equations following theWill-Nodtvedt Parameterized-Post-Newtonian (PPN) formalism, withoutpoint-particle approximations. A single potential that is not present in thestandard PPN formalism is found. This new potential hinders immediate PPNconclusions. To proceed, we do a complete first-order post-Newtonian analysisand explicitly derive all the conserved quantities. After demanding that thereexists a Newtonian limit by requiring the BD mass to be sufficiently large, wefind, as expected, that $\gamma = 1$; but there is no effective $\beta$parameter that can have the same physical role of the standard $\beta$ in PPNformalism. All the others standard PPN parameters can be extended to themassive BD case without issues and are shown to have the same values of generalrelativity. At last, we consider numerical relations on the periastron advanceand the BD mass in two different physical contexts, the orbit of Mercury aboutthe Sun and the orbit of the star S2 about the supermassive black hole in theMilky Way.

A detailed first-order post-Newtonian analysis of massive Brans-Dicke theories: numerical constraints and the $\beta$ parameter meaning