This chapter presents the basic properties of quasi-circular orbits in axisymmetric stellar systems. In fact, axisymmetric models are often sufficiently realistic descriptions – beyond the zeroth-order spherical case – of elliptical and disk galaxies. The associated potentials (with a reflection plane, the equatorial plane) admit circular orbits, and in this chapter we focus on the properties of orbits slightly departing from perfect circularity, describing in some detail the second-order epicyclic approximation. We also derive, in a geometrically rigorous way, the expression of the Oort constants, important kinematical quantities related to the rotation curve of disk galaxies, and in particular to the orbit of the Sun around the center of the Milky Way.

The first variant of Milgrom’s theory (T0) consists simply of his three postulates from 1983. These postulates entail a number of novel predictions, predictions that have subsequently been confirmed by observational astrophysicists. The first is the “baryonic Tully–Fisher relation” (BTFR), a unique relation between the total mass of a galaxy and its asymptotic rotation speed. An even more surprising prediction is the “radial acceleration relation” (RAR), which states that the rotation speed anywhere in a disk galaxy is determined precisely by the observed distribution of matter – but not in the way that Newton’s laws would predict. According to Lakatos’s Methodology, these, and some other, successful novel predictions imply that Milgrom’s postulates constitute a progressive departure from the dark matter hypotheses of the standard cosmological model.