Total disk height from the sum of m=0,...,3
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The figure above shows the shape of the warp for both the model and the data. We plot only m = 0,1,2 from the data, ignoring the weak but significant m = 10, 15 terms. The overall agreement is quite good, although there are some differences. The model does not, for example, have a minimum that is as extensive in Galactic azimuth as the data. Lines of maximum descent for m = 1,2 in both the simulations and the HI data analysis are also shown in the figure. They are separated by about 20° in the model, a bit more than, but close to the 12° in Levine (2005). Both sets of lines are oriented in the same sense: close to φ = 90°. Neither set of lines shows evidence of significant variation with galactocentric radius.
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The figure above shows the amplitudes of the first three harmonics, m = 0, 1, 2, in both the simulations and the data. We increased the LMC mass by 33% for a better fit. As in the data, the m=1 in the simulations is the strongest, and increases nearly linearly out to the edge of the disk. At large R, where the inertial is small, the linear theory is expected to overpredict the warp Tsuchiya (2002). The simulations also show a weak response of m = 0,2 out to about 15 kpc, and then increasing nearly linearly, but with asympimately the same amplitude for both, providing a reasonable representation of the data.
The amplitude ordering of these vertical harmonics has a natural physical explanation. The force from the dark-matter density wake and the satellite itself may described by three-dimensional harmonics and these affect the warp height as follows. A spherically symmetric m=0 distortion in the halo does nothing to the warp. An m=1 distortion in the halo will tend to accelerate the disk in the vertical direction in general; the differential response to this body acceleration of the disk and other odd harmonics results in an m=0 "dishing" of the disk. An m=2 distortion in the halo will attract the disk upward and downward in a reflection symmetric way causing the classic integral-sign warp. Higher-order symmetries may be deduced by similar geometric considerations. The power in the halo excitation drops off as an inverse power of the harmonic order and only the lowest order terms have features well inside the satellite orbit. Therefore, the lowest-order terms are dominant. Conversely, the existence of these higher-order harmonics with a power-law drop off is a natural consequence of this tidal theory and is consistent with the data.
We fixed the disk and the halo mass inside of the virial radius while adjusting the satellite orbit and halo concentration and found the following trends. First, the halo wake and its pattern speed are determined by the halo concentration. The disk bending modes have a natural set of frequencies for a given halo. These will be maximally excited when forced by the halo wake at or near harmonics of this natural frequency. The ratio of m=2 amplitude to the m=1 amplitude is maximized for a halo concentration c≈10. For an NFW profile, this puts rs≈30 kpc. This is very close to the ΛCDM estimates for the Milky Way concentration. Secondly, the orientation of the response depends on whether the nearest resonance is larger or smaller than natural frequency. Therefore changing the satellite orbit, which changes the forcing frequency, affects both the amplitude of the amplitude and orientation of the warp response. In short, the warp depends on a "clockwork" of frequency relationships which depends on the satellite orbit, the dark-matter halo and the disk. A pericenter larger than the current 49 kpc estimate shifts the position angle (PA) so that the warp peaks closer to PA=180 degrees. Similarly, smaller pericenter increases the amplitude also shifts the PA. We conclude that our model "prefers" our current fiducial MC model. We are not claiming that our fiducial model is the most probable amongst the distributions of allowed values but that a plausible choice of parameters corresponds to many features of the observed data.