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Equilibrium and Dynamical Evolution of Self-Gravitating System Embedded in a Potential Well

Isothermal and self-gravitating systems bound by non-conducting and conducting walls are known to be unstable if the density contrast between the center and the boundary exceeds critical values. We investigate the equilibrium and dynamical evolution of isothermal and self-gravitating system embedded in potential well, which can be the situation of many astrophysical objects such as the central parts of the galaxies, or clusters of galaxies with potential dominated by dark matter, but is still limited to the case where the potential well is fixed during the evolution. As the ratio between the depth of surrounding potential well and potential of embedded system becomes large, the potential well becomes effectively the same boundary condition as conducting wall, which behaves like a thermal heat bath. We also use the direct N-body simulation code, NBODY6 to simulate the dynamical evolution of stellar system embedded in potential wells and propose the equilibrium models for this system. In deep potential well, which is analogous to the heat bath with high temperature, the embedded self-gravitating system is dynamically hot, and loosely bound or can be unbound since the kinetic energy increases due to the heating by the potential well. On the other hand, the system undergoes core collapse by self-gravity when potential well is shallow. Binary heating can stop the collapse and leads to the expansion, but the evolution is very slow because the potential as a heat bath can absorb the energy generated by the binaries. The system can be regarded as quasi-static. Density and velocity dispersion profiles from the N-body simulations in the final quasi-equilibrium state are similar to our equilibrium models assumed to be in thermal equilibrium with the potential well.