  The pure self-gravitating system in this paper refers to a multi-body gaseous
system where the self-gravity plays a dominant role and the intermolecular
interactions can be neglected. Therefore its total mass must be much more than
a limit mass, the minimum mass of the system exhibiting long-range nature. Thee
method to estimate the limit mass is then proposed. The nonequlibrium
stationary state in the system is identical to the Tsallis equilibrium state,
at which the Tsallis entropy approaches to its maximum. On basis of this idea,
we introduce a new concept of the temperature whose expression includes the
gravitational potential and therefore we call it gravitational temperature.
Accordingly, the gravitational thermal capacity is also introduced and it can
be used to verify the thermodynamic stability of the astrophysical systems.
