We present analytic models for the formation and evolution of tidal tails and related structures following single or multiple impulsive disturbances in galaxy collisions. Since the epicyclic approximation is not valid for large radial excursions, we use orbital equations of the form we call p-ellipses (a class of precessing ellipses). These have been shown to provide accurate representations of orbits in logarithmic and power-law halo potentials.
In the simplest case of an impulsive collision yielding a purely tidal disturbance the resulting tidal tails have simple structure. Scalings for their maximum lengths and other characteristics as non-linear functions of the tidal amplitude and the exponent of the power-law potentials are described. The analytic model shows that azimuthal caustics (orbit crossing zones of high density also seen in numerical models) are produced generically in these tails at a fixed azimuth relative to the point of closest approach. Long tails, with high-order caustics at their base, and ocular waveforms are also produced at larger amplitudes.
The analysis is then extended to non-linear disturbances and multiple encounters, which break the symmetries of purely tidal perturbations. The p-ellipse orbital solutions are similar to those in the linear tidal case. However, as the strength of the non-linear terms is varied the structure of the resulting forms varies from symmetric tails to one-armed plumes. Cases with two or more impulse disturbances are also considered as the simplest analytic models distinguishing between prograde and retrograde encounters. The model shows explicitly how tail growth differs in the two cases. In the prograde case a specific mechanism for the formation of tidal dwarf galaxies at the end of tails is suggested as a consequence of resonance effects in multiple or prolonged encounters. Qualitative comparisons to Arp Atlas systems suggest that the limiting analytic cases are realized in real systems. For example, we identify a few Arp systems which have multiple tidal strands meeting near the base of long tails. These may be swallowtail caustics, where dissipative gas streams are converging and triggering star formation. Ultraviolet and optical images reveal luminous knots of young stars at these ‘hinge clump’ locations.