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The Newtonian gravitational constant $G$ obeys the dimensional relation $[G][M] [a] = [v]^4$, where $M$, $a$, and $v$ denote mass, acceleration, and speed,respectively. Since the baryonic Tully-Fisher (BTF) and Faber-Jackson (BFJ)relations are observed facts, this relation implies that $G\, a = {\rmconstant}$. This result cannot be obtained in Newtonian dynamics which cannotexplain the origin of the BTF and BFJ relations. An alternative, modifiedNewtonian dynamics (MOND) assumes that $G=G_0$ is constant in space and derivesnaturally a characteristic constant acceleration $a=a_0$, as well as the BTFand BFJ relations. This is overkill and it comes with a penalty: MOND cannotexplain the origin of $a_0$. A solid physical resolution of this issue is that$G \propto a^{-1}$, which implies that in lower-acceleration environments thegravitational force is boosted relative to its Newtonian value because $G$increases. This eliminates all problems related to MOND's empirical cutoff$a_0$ and yields a quantitative method for mapping the detailed variations of$G(a)$ across each individual galaxy as well as on larger and smaller scales.On the opposite end, the large accelerations produced by $G(a)$ appear to belinked to the weak-field limit of the fourth-order theory of conformal Weylgravity.

Interposing a Varying Gravitational Constant Between Modified Newtonian Dynamics and Weak Weyl Gravity