  Previous studies suggested that surface ice could be distributed on close-in
terrestrial exoplanets around M-dwarfs if heat redistribution on the planets is
very inefficient. In general, orbital and atmospheric parameters play an
important role in the climate on terrestrial planets, including the cold-trap
region where the permanent surface water reservoir can potentially be
distributed. Here, we develop a simple coupled land-atmosphere model to explore
the potential surface ice distribution on close-in terrestrial planets with
various orbital and atmospheric parameters, assuming that the planets are
airless or have a thin \ce{N2} atmosphere. We find that the most significant
factors in deciding the surface cold trap region are the spin-orbit ratio and
obliquity. The incident stellar flux and the surface pressure play a limited
role in the thin \ce{N2} simulations for incident flux smaller than Mercury's
and surface pressure lower than 10$^4$ Pa. Our result illustrates the possible
distribution of surface ice on arid terrestrial planets and can help to
understand the climate of these exoplanets.
