  The collisionless quasi-equilibrium state realized after the cold collapse of
self-gravitating systems has two remarkable characters. One of them is the
linear temperature-mass (TM) relation, which yields a characteristic
non-Gaussian velocity distribution. Another is the local virial (LV) relation,
the virial relation which holds even locally in collisionless systems through
phase mixing such as cold-collapse. A family of polytropes are examined from a
view point of these two characters. The LV relation imposes a strong constraint
on these models: only polytropes with index $n \sim 5$ with a flat boundary
condition at the center are compatible with the numerical results, except for
the outer region. Using the analytic solutions based on the static and
spherical Jeans equation, we show that this incompatibility in the outer region
implies the important effect of anisotropy of velocity dispersion. Furthermore,
the velocity anisotropy is essential in explaining various numerical results
under the condition of the local virial relation.
