  Families of steady states of the spherically symmetric Einstein-Vlasov system
are constructed, which are parametrized by the central redshift. It is shown
that as the central redshift tends to zero, the states in such a family are
well approximated by a steady state of the Vlasov-Poisson system, i.e., a
Newtonian limit is established where the speed of light is kept constant as it
should be and the limiting behavior is analyzed in terms of a parameter which
is tied to the physical properties of the individual solutions. This result is
then used to investigate the stability properties of the relativistic steady
states with small redshift parameter in the spirit of recent work by the same
authors, i.e., the second variation of the ADM mass about such a steady state
is shown to be positive definite on a suitable class of states.
