هيكل لولبي

spiral structure Matter ghost matter astrophysics.6: Spiral structure.(p5)
5) An attempt to convert these analytical results into numerical simulation machinery.

The following pictures shows a non-rotating system, with initial conditions as given by a solution of this Eddington-like solution. F.Lhanseat checked that it remained steady. For a given choice of parameters ( = 1 , = 3, = 1, = 1 ) we get the following solution (figures 8 and 9). The figure 5 shows the mass densities () and - (), versus radial (adimensional) distance (the unity corresponds to the Jeans length). The 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 in the two sub-systems 2d galaxy and 2d anti-galaxy, are chosen equal ( = 1 ). The characteristic lengths of the two coupled solutions, are both chosen equal to the Jean's length Lj of the first population (positive masses), which corresponds to the choice = 1, = 1
The chosen ratio of the mass densities is :

About 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 checked, through numerical solution, that it corresponded to acceptable initial quasi steady-state conditions. He used two populations of 10,000 mass-points, sprayed over space, in order to fit analytical data. The first describes the positive mass distribution and the second the negative mass distribution. As the number of masses were basicly equal in his program, he introduced :

m* = - m

The intial situation corresponds to the figures 10-a, 10-b and 10-c.