  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.
