This paper describes a method for determining the dynamical interaction between extended halo and spheroid components and an environmental disturbance. One finds that resonant interaction between a galaxy and passing interlopers or satellite companions can carry the disturbance inward, deep inside the halo, where it can perturb the disk. Applied to the Milky Way for example, the LMC and SMC appear to be sufficient to cause the observed Galactic warp and possibly seed other asymmetries. This is a multi-scale interaction in which the halo wake has a feature at roughly half the satellite orbital radius due to a 2:1 orbital resonance. The rotating disturbance then excites an m = 1 vertical disk mode which has the classic integral-sign morphology. A polar satellite orbit produces the largest warp and therefore the inferred LMC orbit is nearly optimal for maximum warp production. Both the magnitude and morphology of the response depend on the details of the disk and halo models. Most critically, a change in the halo profile will shift the resonant frequencies and response location and consequently alter the coupling to the bending disk. Increasing the halo support relative to the disk, a sub-maximal disk model, decreases the warp amplitude. Finally, the results and prognosis for N-body simulations are discussed. Discreteness noise in the halo, similar to that due to a population of 106M⊙ black holes, can produce observable warping. Details are discussed in an associated paper.
Weinberg, Martin D., "Dynamics of an Interacting Luminous Disk, Dark Halo, and Satellite Companion" (1997). Astronomy Department Faculty Publication Series. 10.