Subtitle: Dumbo the Elephant

The following movie (click on the image) shows the time evolution of the disk "flapping". The scaling for each harmonic and the total is the same as in the previous tab. The vertical motion of the satellite continuously excites the m=0 "bobbing" (with a little glitch during the disk plane passage). Notice that the m=1 asymmetry "turns" to face the orbital plane of the satellite; its pattern speed is close to the radial frequency of the satellite. The m=2 pattern speed is faster than the orbit but also close to a resonance (looks like 3:2 to me). Because the m=2 pattern speed is faster, one gets a large change in m=2 amplitude as one gradually changes the satellite orbital frequency (as I described earlier). My point here is that the amplitude m=2 component can be either large or small depending serendipitously on the satellite. Not surprising given the ampltidue of the responses seen previously, the m=1 dominates and m=0 and m=2 modulates.

A technical note: the movies are a bit jerky because of a slight change I made during the rendering; the computer is redoing them now but the basic results will be the same of course.

First frame from m=0 movie
Full response (left) compared with m=0 response (right)
First from from m=1 movie
Full response (left) compared with m=1 response (right)
First frame from m=2 movie
Full response (left) compared with m=2 response (right)
First frame from m=3 movie
Full response (left) compared with m=3 response (right)
First frame from the movie
Full response only

Martin Weinberg <>
Last modified: Sat Nov 5 23:33:36 EST 2005