  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.
