  Gravitational instability is one of considerable mechanisms to explain the
formation of giant planets. We study the gravitational stability for the
protoplanetary disks around a protostar. The temperature and Toomre's Q-value
are calculated by assuming local equilibrium between viscous heating and
radiative cooling (local thermal equilibrium). We assume constant $\alpha$
viscosity and use a cooling function with realistic opacity. Then, we derive
the critical surface density $\Sigma_{\rm{c}}$ that is necessary for a disk to
become gravitationally unstable as a function of $r$. This critical surface
density $\Sigma_{\rm c}$ is strongly affected by the temperature dependence of
the opacity. At the radius $r_{\rm c}\sim 20$AU, where ices form, the value of
$\Sigma_{\rm c}$ changes discontinuously by one order of magnitude. This
$\Sigma_{\rm c}$ is determined only by local thermal process and criterion of
gravitational instability. By comparing a given surface density profile to
$\Sigma_{\rm c}$, one can discuss the gravitational instability of
protoplanetary disks. As an example, we discuss the gravitational instability
of two semi-analytic models for protoplanetary disks. One is the steady state
accretion disk, which is realized after the viscous evolution. The other is the
disk that has the same angular momentum distribution with its parent cloud
core, which corresponds to the disk that has just formed. As a result, it is
found that the disks tend to become gravitationally unstable for $r\ge r_{\rm
c}$ because ices enable the disks to become low temperature. In the region
closer to the protostar than $r_{\rm c}$, it is difficult for a typical
protoplanetary disk to fragment because of the high temperature and the large
Coriolis force. From this result, we conclude that the fragmentation near the
central star is possible but difficult.
