A planet is stable when the starlight it absorbs equals the thermal radiation it emits to space (outgoing longwave radiation, OLR). For a moist, water-vapor-rich atmosphere this balance is not simple blackbody cooling — as the surface warms, more water evaporates, and that water vapor becomes so opaque in the infrared that the emitting layer rises in the atmosphere and its temperature stops tracking the surface. The result is the Simpson–Nakajima limit: OLR saturates near a fixed ceiling (≈282 W/m² for a CO₂-free atmosphere) no matter how hot the surface gets.
S(d) = L·S₀/d² · (1−A)/4 absorbed flux (global average)
OLR(T) = OLR_max·(1 − exp(−σT⁴ / OLR_max))
OLR_max = 282 − 25·(CO2 loading) [W/m²]
Equilibrium exists only if S < OLR_max:
T_eq = [ −OLR_max/σ · ln(1 − S/OLR_max) ]^(1/4)
Boundary in the (distance, CO2) plane, two equivalent forms:
d_edge = √( S₀·L·(1−A)/4 / OLR_max ) solved for distance
CO2_edge(d) = (282 − S(d)) / 25 solved for CO2 loading
Instead of a 3D orbiting planet, this version plots the whole parameter space at once: the main panel is a live-computed stability field over orbital distance (horizontal) versus greenhouse-gas loading (vertical), colored stable/warning/runaway by the same ratio S/OLR_max the 3D version uses to tint its planet. The dashed white curve is the analytic boundary CO2_edge(d) — solved for CO2 rather than for distance, an independent algebraic path to the same physical edge, so if the curve does not sit exactly on the color transition the model has a bug. The white square is the current parameter point; the inset (top-right) is the OLR(T) curve with the current absorbed flux S and equilibrium temperature marked, same as the source model.
- Orbital distance / luminosity — set the absorbed stellar flux S via the inverse-square law; changing luminosity or albedo redraws the entire field, since both shift S at every distance.
- Albedo — reflected sunlight never enters the balance; higher albedo lowers S and pushes the whole field toward stability.
- Greenhouse-gas loading — extra CO₂ adds opacity beyond what water vapor already saturates, pulling the OLR ceiling down and shrinking the safety margin (moving up the vertical axis moves toward runaway).
- Once S crosses OLR_max there is no stable surface temperature — this is the physical inner edge of the habitable zone (Kasting 1988; Kopparapu et al. 2013).