  Planets in the "Habitable Zones" around M-type stars are important targets
for characterization in future observations. Due to tidal-locking in
synchronous spin-orbit rotations, the planets tend to have a hot dayside and a
cold nightside. On the cold nightside, water vapor transferred from the dayside
can be frozen in ("cold trap") or the major atmospheric constituent could also
condense ("atmospheric collapse") if the atmosphere is so thin that the heat
re-distribution is not efficient, in the case of a single M-type star.
Motivated by the abundance of binary star systems, we investigate the effects
of irradiation from a G-type companion star on the climate of a tidally locked
planet around an M-type star using the 2D energy balance model. We find that
the irradiation from the G-type star is more effective at warming up the
nightside of the planet than the dayside. This contributes to the prevention of
the irreversible trapping of water and atmosphere on the cold nightside,
broadening the parameter space where tidally locked planets can maintain
surface liquid water. Tidally locked ocean planets with < ~ 0.3 bar atmospheres
or land planets with < ~ 3 bar atmospheres can realize temperate climate with
surface liquid water when they are also irradiated by a companion star with a
separation of 1 - 4 au. We also demonstrate that planets with given properties
can be in the Earth-like temperate climate regime or in a completely frozen
state under the same total irradiation.
