Hi Larry & Helmut,

    I had a working N-body program which computed the motion
to 1,000,000 softened particles confined to a plane.  The
initial conditions consisted of an initial sprinkling of
particles to form a Gaussian disk of mass = 1, and a scale
length of = 1.  The gravitational constant G = 1.

    The standard Plummer softening had a value of 0.25 which
made a cold disk stable to axisymmetrical instabilities.
(I think a softening of 0.198+ already does this).

    Another way of looking at these calculations is to
think of each particle as a small Kuzmin-Toomre disk of
size 0.25/2 which interact according to Newton's law.

    One further wrinkle was to fake a rigid halo by
saying that a fraction (frac) of the disk would be mobile,
and that the rest (1 - frac) remained fixed.  The forces
from the fixed part were precomputed, and the masses
of the mobile disk particles were reduced by the amount frac.
   

    The forces of the mobile are calculated using a grid
scheme.  First each little K-T disk is pasted onto the grid
using a 8 x 8 Lagrange interpolation scheme (yes you heard
me correctly: they were spread over 64 grid points, not the
nearest 4), then this gridded density is converted to a force
grid with the help of Fourier transforms, and finally the
forces on each particle are calculated by a similar 8 x 8
Lagrange interpolation scheme.  The grid is chosen so fine
that the forces are accurate to about 10 decimal places.
The time to calculate the forces is about 1 second using
a single 2.2 GHz AMD Athlon CPU.

    The overall accuracy was good enough to measure the
pattern speed and growth rate of the most prominent bar
mode to better than 8 decimal places.


    It was easy to add an initial rotating quadrupole
force field which came on and off in a sin(beta*time)**2
fashion over interval 0 < time < 20.  

    There is a set of files in

       www.mso.anu.edu.au/~agris/Larry

which show the density distribution produced by the
little K-T disks.  The initial parameters are coded
in the file names:

   The number after Speed is the pattern speed of the
initial quadrupole field and frac is the fraction of
disk particles.

[agris@mizar ~/Mar]# ll ~/public_html/Larry/

  6033838 Mar  2 10:14  Speed-0.08-frac-0.0g.ps
  6035344 Mar  2 10:14  Speed-0.08-frac-0.1g.ps
  6040839 Mar  2 10:14  Speed-0.16-frac-0.0g.ps
  8778756 Mar  2 10:14  Speed-0.16-frac-0.1g.ps
  6037595 Mar  2 10:14  Speed-0.16-frac-0.2g.ps
  6039431 Mar  2 10:14  Speed-0.16-frac-0.3g.ps
    27704 Mar  1 15:53  gauss_freqs.ps

   The timing information is found in gauss_freqs.ps,
It shows that the central rotation rate of the softened
particles was 0.65295... , which implies a period of
9.9228 time units at the center.

   The quadrupole pattern speed of 0.16 is higher than
the precession rate of the all orbits which are viewed
as central ellipses, namely $(\omega - \kappa/2)$.
The lower speed of 0.8 does a better job of exciting
the wrapping spiral. 

   Notice that the $(\omega - \kappa/2)$ has a maximum
at radius 2.5156 and that the sense of the winding of
the spiral changes here.

   Note also that the first picture shows the unperturbed
disk, and the subsequent ones the differences from it.


   It looks like the frac = 0.0, 0.1, and 0.2 are
stable, but frac = 0.3 has a slow growing "bar" mode.
I have calculated the m = 2 pattern speeds and growth
rates for the frac = 0.3, 0.4, and 0.5 cases:

        frac    omega_p   growth      

        0.3    0.147037  0.001700
        0.4    0.164351  0.011925
        0.5    0.182984  0.034197

and an extrapolation suggests that the growth is
zero or very small when frac = 0.2 .  The pictures
seem to say so.


    Cheers,  Agris