A natural approach to extended Newtonian gravity: tests and predictions across astrophysical scales

In the pursuit of a general formulation for a modified gravitational theory at the non-relativistic level and as an alternative to the dark matter hypothesis, we construct a model valid over a wide variety of astrophysical scales. Through the inclusion of Milgrom's acceleration constant into a gravitational theory, we show that very general formulas can be constructed for the acceleration felt by a particle. Dimensional analysis shows that this inclusion naturally leads to the appearance of a mass-length scale in gravity, breaking its scale invariance. A particular form of the modified gravitational force is constructed and tested for consistency with observations over a wide range of astrophysical environments, from solar system to extragalactic scales. We show that over any limited range of physical parameters, which define a specific class of astrophysical objects, the dispersion velocity of a system must be a power law of its mass and size. These powers appear linked together through a natural constraint relation of the theory. This yields a generalised gravitational equilibrium relation valid for all astrophysical systems. A general scheme for treating spherical symmetrical density distributions is presented, which in particular shows that the fundamental plane of elliptical galaxies, the Newtonian virial equilibrium, the Tully-Fisher and the Faber-Jackson relations, as well as the scalings observed in local dwarf spheroidal galaxies, are nothing but particular cases of that relation when applied to the appropriate mass-length scales. We discuss the implications of this approach for a modified theory of gravity and emphasise the advantages of working with the force, instead of altering Newton's second law of motion, in the formulation of a gravitational theory.