spiral structure Astrophysical ghost matter.6: Spiral structure. (p5)
5) An attempt to transform these analytical results into a numerical simulation machine.
The following images show a non-rotating system, with initial conditions given by a solution of this Eddington-type solution. F. Lhanseat verified that it remained stable. For a given choice of parameters ( = 1, = 3, = 1, = 1), we obtain the following solution (figures 8 and 9). Figure 5 shows the mass densities () and - (), as a function of radial distance (dimensionless) (the unit corresponds to the Jeans length). Figure 9 gives the corresponding gravitational potential, in arbitrary units.
Fig. 8: Steady-state solution. Mass densities r and r*.
Fig. 9: Gravitational potential
The characteristic thermal velocities of the two subsystems, 2D galaxy and 2D anti-galaxy, are chosen equal ( = 1). The characteristic lengths of the two coupled solutions are both chosen equal to the Jeans length Lj of the first population (positive masses), which corresponds to the choice = 1, = 1.
The chosen ratio of the mass densities is:
For the boundary problem, see references [1] and [2].
Fig. 10-a: In the first 2D fold, positive mass distribution, according to the chosen analytical solution (see above)
F. Lhanseat verified, through numerical resolution, that it corresponded to acceptable quasi-steady initial conditions. He used two populations of 10,000 mass points, distributed in space, in order to match the analytical data. The first describes the positive mass distribution and the second the negative mass distribution. Since the number of masses was essentially equal in his program, he introduced:
m* = -m
The initial configuration corresponds to figures 10-a, 10-b, and 10-c.