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There are four different stable climate states for pure water atmospheres, asmight exist on so-called "waterworlds". I map these as a function of solarconstant for planets ranging in size from Mars size to 10 Earth-mass. Thestates are: globally ice covered (Ts< 245K), cold and damp (270 < Ts< 290K),hot and moist (350< Ts< 550K) and very hot and dry (Ts< 900K). No stableclimate exists for 290< Ts < 350K or 550 < Ts < 900K. The union of hot moistand cold damp climates describe the liquid water habitable zone, the width andlocation of which depends on planet mass. At each solar constant, two or threedifferent climate states are stable. This is a consequence of strongnon-linearities in both thermal emission and the net absorption of sunlight.  Across the range of planet sizes, I account for the atmospheres expanding tohigh altitudes as they warm. The emitting and absorbing surfaces (optical depthof unity) move to high altitude, making their area larger than the planetsurface, so more thermal radiation is emitted and more sunlight absorbed (theformer dominates). The atmospheres of small planets expand more due to weakergravity: the effective runaway greenhouse threshold is about 35Wm-2 higher forMars, 10Wm-2 higher for Earth or Venus but only a few Wm-2 higher for a 10Earth-mass planet. There is an underlying (expansion neglected) trend ofincreasing runaway greenhouse threshold with planetary size (40Wm-2 higher fora 10 Earth-mass planet than for Mars). Summing these opposing trends means thatVenus-size (or slightly smaller) planets are most susceptible to a runawaygreenhouse.  The habitable zone for pure water atmospheres is very narrow, with aninsolation range of 0.07 times the solar constant. A wider habitable zonerequires background gas and greenhouse gas; N2 and CO2 on Earth, which arebiologically controlled. Thus, habitability depends on inhabitance.

Habitability of waterworlds: runaway greenhouses, atmospheric expansion and multiple climate states of pure water atmospheres