  There are four different stable climate states for pure water atmospheres, as
might exist on so-called "waterworlds". I map these as a function of solar
constant for planets ranging in size from Mars size to 10 Earth-mass. The
states 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 stable
climate exists for 290< Ts < 350K or 550 < Ts < 900K. The union of hot moist
and cold damp climates describe the liquid water habitable zone, the width and
location of which depends on planet mass. At each solar constant, two or three
different climate states are stable. This is a consequence of strong
non-linearities in both thermal emission and the net absorption of sunlight.
  Across the range of planet sizes, I account for the atmospheres expanding to
high altitudes as they warm. The emitting and absorbing surfaces (optical depth
of unity) move to high altitude, making their area larger than the planet
surface, so more thermal radiation is emitted and more sunlight absorbed (the
former dominates). The atmospheres of small planets expand more due to weaker
gravity: the effective runaway greenhouse threshold is about 35Wm-2 higher for
Mars, 10Wm-2 higher for Earth or Venus but only a few Wm-2 higher for a 10
Earth-mass planet. There is an underlying (expansion neglected) trend of
increasing runaway greenhouse threshold with planetary size (40Wm-2 higher for
a 10 Earth-mass planet than for Mars). Summing these opposing trends means that
Venus-size (or slightly smaller) planets are most susceptible to a runaway
greenhouse.
  The habitable zone for pure water atmospheres is very narrow, with an
insolation range of 0.07 times the solar constant. A wider habitable zone
requires background gas and greenhouse gas; N2 and CO2 on Earth, which are
biologically controlled. Thus, habitability depends on inhabitance.
