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

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 = constant. This result cannot be obtained in Newtonian dynamics which cannot explain the origin of the BTF and BFJ relations. An alternative, modified Newtonian dynamics (MOND) assumes that G = G ₀ is constant in space and derives naturally a characteristic constant acceleration a = a ₀, as well as the BTF and BFJ relations. This is overkill and it comes with a penalty: MOND cannot explain the origin of a ₀. A solid physical resolution of this issue is that G ∝ a ^-1, which implies that in lower-acceleration environments the gravitational 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 be linked to the weak-field limit of the fourth-order theory of conformal Weyl gravity.