4.5. ON ANGULAR MOMENTUM EXCHANGE AND THE ROLE OF RESONANCES: THE ANALYTIC APPROACH

Two papers are the pillars of the analytical work on angular momentum redistribution in disk galaxies - namely Lynden-Bell & Kalnajs (1972) and Tremaine & Weinberg (1984) - while further useful information can be found in, e.g., Kalnajs (1971), Dekker (1974), Weinberg (1985, 1994), Athanassoula (2003), Fuchs (2004), Fuchs & Athanassoula (2005).

In order to reach tractable analytic expressions, it is necessary to consider the disk and the spheroid components separately, and use different approximations in the two cases. For the disk we can use the epicyclic approximation (i.e., we will assume that the disk orbits can be reasonably well approximated by epicycles), while for the spheroid we will assume that the distribution function depends only on the energy, as is the case for spherical isotropic systems. The main results obtained in the papers listed above are:

1. Angular momentum is emitted or absorbed mainly at resonances. It is, however, also possible to emit or absorb away from resonances if the potential is not stationary, but grows or decays with time. Nevertheless, the contribution of the non-resonant material to the total emission or absorption should remain small, unless the growth or decay of the potential is important.
2. In the disk component, angular momentum is emitted from the ILR and at other l < 0 resonances and absorbed at the OLR and at other l > 0 resonances. It is also absorbed at CR, but, all else being equal, at lesser quantities than at the Lindblad resonances.
3. The spheroid absorbs angular momentum at all its resonances.
4. The global picture is thus that angular momentum is emitted from the bar region and absorbed by the CR and OLR in the disk, and by all resonances in the spheroid. Thus, angular momentum is transported from the inner parts of the disk, to the part of the disk outside CR and to the spheroid resonant regions.
5. For both the disk and the spheroid components it is possible to show that, for the same perturbing potential and the same amount of resonant material, a given resonance will emit or absorb more angular momentum if the material there is colder (i.e., has a lower velocity dispersion). Therefore, since the disk is always colder than the spheroid, it will absorb more angular momentum per unit resonant mass. Nevertheless, the spheroid is much more massive than the outer disk, so the amount of angular momentum it absorbs may exceed that absorbed by the outer disk.
6. Since the bar is inside corotation, it has negative energy and angular momentum and as it emits angular momentum it gets destabilised, i.e., it grows stronger. It is thus expected that the more angular momentum is emitted, the stronger the bar will become.