  Although the Earth's orbit is never far from circular, terrestrial planets
around other stars might experience substantial changes in eccentricity that
could lead to climate changes, including possible "phase transitions" such as
the snowball transition (or its opposite). There is evidence that Earth has
gone through at least one globally frozen, "snowball" state in the last billion
years, which it is thought to have exited after several million years because
global ice-cover shut off the carbonate-silicate cycle, thereby allowing
greenhouse gases to build up to sufficient concentration to melt the ice. Due
to the positive feedback caused by the high albedo of snow and ice,
susceptibility to falling into snowball states might be a generic feature of
water-rich planets with the capacity to host life. This paper has two main
thrusts. First, we revisit one-dimensional energy balance climate models as
tools for probing possible climates of exoplanets, investigate the dimensional
scaling of such models, and introduce a simple algorithm to treat the melting
of the ice layer on a globally-frozen planet. We show that if a terrestrial
planet undergoes Milankovitch-like oscillations of eccentricity that are of
great enough magnitude, it could melt out of a snowball state. Second, we
examine the kinds of variations of eccentricity that a terrestrial planet might
experience due to the gravitational influence of a giant companion. We show
that a giant planet on a sufficiently eccentric orbit can excite extreme
eccentricity oscillations in the orbit of a habitable terrestrial planet. More
generally, these two results demonstrate that the longterm habitability (and
astronomical observables) of a terrestrial planet can depend on the detailed
architecture of the planetary system in which it resides.
