Mathematics – Logic
Scientific paper
Oct 1990
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1990mnras.246..415p&link_type=abstract
Monthly Notices of the Royal Astronomical Society, Vol. 246, NO. 3/OCT1, P. 415, 1990
Mathematics
Logic
8
Scientific paper
We present numerical simulations of the evolution of unstable, radially anisotropic spherical systems, and search for neighbouring equilibria. We show that in all cases considered there are at most three distinguishable non-spherical equilibria. Two of these are axisymmetric (oblate and prolate), leaving at most one triaxial configuration. Not all these equilibria are stable. For very anisotropic initial systems, only the triaxial configuration is stable. As the degree of initial anisotropy is reduced, the triaxial equilibrium disappears and the prolate equilibrium becomes the stable end-point. Decreasing the anisotropy still further we find that the oblate and spherical equilibria merge, causing a second exchange of stability, after which the spherical system becomes metastable. At this stage there are two non-spherical equilibria, both axisymmetric and prolate. One of these is unstable and forms a barrier between the spherical and the other prolate equilibrium. We demonstrate this by pushing the spherical system with increasing amplitude until it is pushed `over the hill' into the prolate solution.
We present an analytic model to describe this neighbourhood of state space. We derive a potential whose surface describes the state space, in which the various equilibria found in the simulations are represented by turning points. This model gives a physical picture of why these equilibria exist, and hence suggests that the state space is topologically the same for any reasonable, well-behaved initial distribution function peaked towards low angular momentum orbits.
We argue, therefore, that these equilibria are the only available equilibria for the end-states of dissipationless, non-rotating collapse calculations starting from reasonably cold initial conditions. Although the final distribution function depends upon initial conditions, and the evolution during collapse, the final configuration will qualitatively resemble the stable equilibrium end-states found in these simulations.
Allen Anthony J.
Palmer P. L.
Papaloizou John
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