Matter ghost matter astrophysics.6:
Spiral structure.(p6)
Fig.
10-b) : Its associated negative mass distribution.
.
Fig. 10-c : Superposition
of the two.
ss distribution.
F.Lhandseat showed that this distribution
of conjugated positive and negative masses was stable over a large number
of Jeans' time.
- Introducing rotation.
It was temptating to try to give
some rotation movement to the central positive mass cluster. But then no analytical
2d results were available. F.Lhanseat decided to introduce empirically the
a priori following initial rotation curve (which tends to solid body rotation
at the center, and to zero at the periphery). :
Fig. 11 : Initial rotation
curve profile
Centrifugal force tends to destroy
the stability of the system. If we want to balance the centrifugal force we
can reduce the pressure force (thermal
velocity in the rotating positive mass sub-system) or increase the confinement
effect by raising m . But, as shown by F.Lhandseat,
the increase of this parameter produces an artefact due to the relatively
low number of points. If one tries to balance centrigugal force with
m >
5 the halo-like structure and the cluster cross each other. Then the halo
transforms into a cluster and vice-versa.
The explanation is the following.
The two sets : cluster and halo, cannot be assimilated to continuous masses
of gas. They are just limited sets of points. Due to its repulsive action,
the (self-attractive) halo tends to compress the cluster (the positive mass
cluster and the negative mass halo repel each other). We can compare it to
a strainer acting on mashed potatoes. A strainer owns holes.
Fig. 12-a : The strain, with small holes, balances
the pressure due to the weight
of the mashed potatoes.
The efficiency of the compression process depends on the diameter of these
holes. If it is small, our spherical strainer confines efficiently the central
mass of "mashed potatoes". If the holes are too large, the mashed potatoes
will pass through the strainer, as suggested on figures 12-a
and 12-b.
.
Fig. 12-b: When the
holes are too large the strain cannot keep the mashed potatoes
: it goes through.
If one reduce the number of points involved in the simulation, the maximum value
of m becomes smaller, for the "holes" in this negative matter distribution become
larger. Here we reach a fundamental limit of this numerical simulation, due
to this artefact. With only 2 x 10,000 points, if 
exceeds 5, the culster goes through the halo, and dissipates. With a larger
number of points, stronger confinement effect could be achieved but the basic
limit of our machine did not afford it.
Anyway, F.Lhandseat adjusted the
conditions empirically and found that the things looked good when the characteristic
rotation velocity (the maximum value) was about ten times smaller than the
mean thermal in the cluster (positive mass sub-system), which meant that the
rotational energy was smaller than the pressure energy. Physically talking,
the gravitational force was mainly balanced by the pressure force, not by
the centrifugal force. In such conditions the epicyclic frequency's value
was 
= 1.
