Thermal evolution of protoplanetary disks: from $\beta$-cooling to
  decoupled gas and dust temperatures
  Aims: We explore the long-term evolution of young protoplanetary disks with
different approaches to computing the thermal structure determined by various
cooling and heating processes in the disk and its surroundings. Methods:
Numerical hydrodynamics simulations in the thin-disk limit were complemented
with three thermal evolution schemes: a simplified $\beta$-cooling approach
with and without irradiation, in which the rate of disk cooling is proportional
to the local dynamical time, a fiducial model with equal dust and gas
temperatures calculated taking viscous heating, irradiation, and radiative
cooling into account, and also a more sophisticated approach allowing decoupled
dust and gas temperatures. Results: We found that the gas temperature may
significantly exceed that of dust in the outer regions of young disks thanks to
additional compressional heating caused by the infalling envelope material in
the early stages of disk evolution and slow collisional exchange of energy
between gas and dust in low-density disk regions. The outer envelope however
shows an inverse trend with the gas temperatures dropping below that of dust.
The global disk evolution is only weakly sensitive to temperature decoupling.
Nevertheless, separate dust and gas temperatures may affect the chemical
composition, dust evolution, and disk mass estimates. Constant-$\beta$ models
without stellar and background irradiation fail to reproduce the disk evolution
with more sophisticated thermal schemes because of intrinsically variable
nature of the $\beta$-parameter. Constant-$\beta$ models with irradiation can
better match the dynamical and thermal evolution, but the agreement is still
incomplete. Conclusions: Models allowing separate dust and gas temperatures are
needed when emphasis is placed on the chemical or dust evolution in
protoplanetary disks, particularly in sub-solar metallicity environments.
