  The habitable zone concept is important because it focuses the scientific
search for extraterrestrial life and aids the planning of future telescopes.
Recent work has shown that planets near the outer edge of the habitable zone
might not actually be able to stay warm and habitable if CO$_2$ outgassing
rates are not large enough to maintain high CO$_2$ partial pressures against
removal by silicate weathering. In this paper I use simple equations for the
climate and CO$_2$ budget of a planet in the habitable zone that can capture
the qualitative behavior of the system. With these equations I derive an
analytical formula for an effective outer edge of the habitable zone, including
limitations imposed by the CO$_2$ outgassing rate. I then show that climate
cycles between a Snowball state and a warm climate are only possible beyond
this limit if the weathering rate in the Snowball climate is smaller than the
CO$_2$ outgassing rate (otherwise stable Snowball states result). I derive an
analytical solution for the climate cycles including a formula for their period
in this limit. This work allows us to explore the qualitative effects of
weathering processes on the effective outer edge of the habitable zone, which
is important because weathering parameterizations are uncertain.
