f3804 Ghost matter astrophysical matter. 5: Results of numerical 2D simulations.
VLS. About a possible scheme for galaxy formation. (p4)
If we study an intermediate initial configuration, we find the result shown in figures 11 and 11bis, that is, a kind of emulsion, stable over long periods. The relative stability of this pattern could come from the fact that any mass concentration of one species forms a potential barrier with respect to the other, and vice versa. Note that this method could be extended to a 3D hypersphere, whose metric is:
(5) ds² = dr² + R² ( dq² + sin²q dj²)
Given two points M₁ (r₁ , q₁ , j₁) and M₂ (r₂ , q₂ , j₂), we can calculate the two geodesic arc lengths d and d' that connect them as well as the gravitational force. However, these spherical or hyperspherical descriptions induce curvature effects. If we want to study a phenomenon whose characteristic scale is L, in a portion of such a 2D or 3D closed universe, assuming that curvature effects can be neglected, we must work with very large 2D or 3D spheres (R >> L), which requires a large number of mass points, far beyond the capabilities of current systems.
Fig. 11 : An emulsion corresponding to Vth = Vth cr.
Fig 11bis : the same with two different shades.
Returning to the simpler classical method, as in [11] and [12], let us introduce a spatial truncation: we limit the interaction calculations to the neighboring mass points located in the dotted square (figure 12), whose side is equal to that of the basic cell.
Fig. 12 : Spatial truncation for spatially periodic system.
The results are similar. If we fill the unit square with a single self-attractive species, with uniform mass density r and uniform thermal velocity field, corresponding to a 2D Maxwell-Boltzmann distribution:
(6)
we find a critical value Vth. See figures 13a and 13b.
Fig. 13: 2D gravitational instability with spatial truncation and a single species. 