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

Analytical investigation of the decrease in the size of the habitable zone due to limited CO$_2$ outgassing rate