  We investigate the formation process of self-gravitating protoplanetary disks
in unmagnetized molecular clouds. The angular momentum is redistributed by the
action of gravitational torques in the massive disk during its early formation.
We develop a simplified one-dimensional accretion disk model that takes into
account the infall of gas from the envelope onto the disk and the transfer of
angular momentum in the disk with an effective viscosity. First we evaluate the
gas accretion rate from the cloud core onto the disk by approximately
estimating the effects of gas pressure and gravity acting on the cloud core. We
formulate the effective viscosity as a function of the Toomre Q parameter that
measures the local gravitational stability of the rotating thin disk. We use a
function for viscosity that changes sensitively with Q when the disk is
gravitationally unstable. We find a strong self-regulation mechanism in the
disk evolution. During the formation stage of protoplanetary disks, the
evolution of the surface density does not depend on the other details of the
modeling of effective viscosity, such as the prefactor of the viscosity
coefficient. Next, to verify our model, we compare the time evolution of the
disk calculated with our formulation with that of three-dimensional
hydrodynamical simulations. The structures of the resultant disks from the
one-dimensional accretion disk model agree well with those of the
three-dimensional simulations. Our model is a useful tool for the further
modeling of chemistry, radiative transfer, and planet formation in
protoplanetary disks.
