# Total disk height from the sum of m=0,...,3

Click on plot to zoom . . .

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.

Click on plot to zoom . . .

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
*r _{s}≈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.