Figure: Top row: Fluctuation energy with orthogonal function index in
units of background potential energy. Bottom row: Cumulative
fluctuation energy with orthogonal function indices less than n.
The left and right columns show the l=m=1 and l=m=2 harmonics
respectively. The spatial profile of the potential functions is
shown in Fig. 4 for l=1, 2. The solid
and dashed line follows from the analytic calculation for the
self-gravitating response and Poisson fluctuations alone,
respectively. The open circles are computed from the time-averaged
expansion coefficients in the n-body simulation.
Figure: As in Fig. 1 but
for the l=m=3 (left) and l=m=4 (right) harmonics.
Figure: Cumulative fluctuation energy for all harmonics of given l
(labeled)
with orthogonal function indices less than n. Shown in units of
background potential energy as in Fig. 1.
We will look at both King models and core-free Hernquist models.
First, we apply equation (28) to a King model (
)
for harmonics l=m=1, 2, 3, 4 and compare with a SCF simulation for
100,000 particles and 
. Figures 1 and
2 show the fluctuation energy per expansion function
plotted against the index of expansion functions (cf. Fig.
4) and the cumulative fluctuation energy less than
given index. One energy unit is equal to the total gravitational
potential energy of the unperturbed sphere and 
. There
are slight systematic differences between the n-body results and the
predictions but all-in-all, the n-body simulation follows the analytic
predictions fairly well and confirms the excess power at low order due
to amplification by self gravity. This excess is illustrated in
Figure 3 which shows the total energy for harmonic
orders l=1-4 derived from the simulation. For l=1, the
enhancement is roughly a factor of 6. The large magnitude for l=1
is due to the stochastic excitation of a weakly-damped mode. For
l=2, the enhancement is roughly a factor of 1.5. For l>4 the
power enhancement due to self gravity is negligible. The size of
coherent structures decrease with increasing harmonic order and
therefore higher harmonics have power at smaller and smaller scales
for which self gravity is less important. The index at peak power
increases with l as expected.
The case for the core-free Hernquist model is shown in Figure 5 for harmonics l=m=1, 2. The profiles are more sharply peaked about n=1 because the expansion functions well matched to the model profile (Hernquist & Ostriker 1992). Especially for the l=m=1 harmonic, the agreement between the expansion and the simulation is better in this case. This is probably due to the choice of expansion functions. For the l=m=2, the power appears systematically high at larger radial order. As for the King model, the power at l=1 has the largest enhancement by self-gravity, now by roughly a factor of 15 and this amplitude is verified by the simulation. The enhancement at l=2 is similar, roughly a factor of 1.5.
The root energy indicates the expected magnitude of the density or
potential fluctuation and can be multiplied by 
to
estimate the magnitude for an N particle simulation. For galaxian
disk embedded in a massive halo, an large-scale 
distortion in
the halo can have interesting consequences for disk evolution (cf.
Weinberg 1997). In order to realistically test dynamical hypotheses
for disk-halo interactions, we need to suppress noise below this
level. This requires live halos with 
particles for S/N>10.
In particular, the fluctuating dipole (l=1) force field
differentially accelerates and bends the disk, causing thickening.
The quadrupole (l=2) in a arbitrary orientation warps the disk,
causing the sort of warp discussed by Weinberg (1995,
1997) but now due to noise rather than a
satellite wake. Even without self gravity, discreteness noise is
sufficient to require 
.
Figure: Potential functions used in fluctuation energy calculation in
Fig. 1.
Figure: As in Figure 1 but for the Hernquist
model.