spiral structure Astrophysical ghost matter.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 times.
- Introduction of rotation.
It was tempting to try to give a rotational motion to the central cluster of positive masses. But then, no analytical 2D solutions were available. F. Lhanseat decided to introduce empirically the following initial rotation curve (which tends toward solid-body rotation at the center and toward zero at the periphery):
Fig. 11 : Initial rotation curve profile
The centrifugal force tends to destroy the system's stability. To balance the centrifugal force, one can reduce the pressure force (thermal velocity in the rotating positive mass subsystem) or increase the confinement effect by increasing m. However, as shown by F. Lhandseat, increasing this parameter produces an artifact due to the relatively small number of points. If one attempts to balance the centrifugal 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 as follows. The two systems—cluster and halo—cannot be considered as continuous gas masses. They are simply 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 mutually repel each other). This can be compared to a sieve acting on mashed potatoes. A sieve has holes.
Fig. 12-a : The sieve, 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 the holes are small, our spherical sieve efficiently confines the central mass of "mashed potatoes." If the holes are too large, the mashed potatoes will pass through the sieve, as suggested by figures 12-a and 12-b.
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Fig. 12-b : When the holes are too large, the sieve can no longer retain the mashed potatoes: they pass through.
If one reduces the number of points involved in the simulation, the maximum value of m becomes smaller, because the "holes" in this negative matter distribution become larger. Here we reach a fundamental limit of this numerical simulation, due to this artifact. With only 2 × 10,000 points, if m exceeds 5, the cluster passes through the halo and dissipates. With a larger number of points, a stronger confinement effect could have been achieved, but the fundamental limitation of our machine did not allow it.
In any case, F. Lhandseat adjusted the conditions empirically and found that the results appeared satisfactory when the characteristic rotation velocity (maximum value) was about ten times smaller than the mean thermal velocity in the cluster (positive mass subsystem), meaning that rotational energy was lower than pressure energy. Physically speaking, the gravitational force was primarily balanced by the pressure force, not by the centrifugal force. Under such conditions, the epicyclic frequency was equal to 1.
