An analytical model for the evolution of the protoplanetary discs
  We obtain a new set of analytical solutions for the evolution of a
self-gravitating accretion disc by holding the Toomre parameter close to its
threshold, and obtaining the stress parameter from the cooling rate. In
agreement with the previous numerical solutions, furthermore, the accretion
rate is assumed to be independent of the disc radius. Extreme situations where
the entire disc is either optically thick or optically thin are studied
independently and the obtained solutions can be used for exploring the early or
the final phases of a protoplanetary disc evolution. Our solutions exhibit
decay of the accretion rate as a power-law function of the age of the system
with the exponent -0.75 and -1.04 for optically thick and thin cases,
respectively. Our calculations permit us to explore evolution of the snow line
analytically. Location of the snow line in the optically thick regime evolves
as a power-law function of time with the exponent -0.16, however, when the disc
is optically thin, location of the snow line as a function of time with the
exponent -0.7 has a stronger dependence on time. It means that in an optically
thin disc inward migration of the snow line is faster than an optically thick
disc.
