Why does the ground stay warm?

One gray radiative layer over a blackbody surface, in instant radiative equilibrium. The layer lets sunlight through, absorbs a fraction ε of the surface’s upward infrared, and emits equal infrared up and down. The knob sets ε from 0 to 1: an effective infrared absorption and emission setting, not a CO₂ concentration. Using one ε for both absorbing and emitting is a gray-body simplification; Kirchhoff’s law actually holds wavelength by wavelength and direction by direction.

Fixed values: solar irradiance S₀ = 1361 W/m² and albedo 0.30, so the absorbed sunlight, averaged over the sphere, is F = (1 − 0.30) × 1361 ÷ 4 = 238.175 W/m². σ = 5.670374419 × 10⁻⁸ W m⁻² K⁻⁴. The balances are: surface, F + D = σT_s⁴; layer, εσT_s⁴ = 2D; top, (1 − ε)σT_s⁴ + D = F. So σT_s⁴ = F ÷ (1 − ε/2), and the layer emits D = εF ÷ (2 − ε) in each direction. At ε = 0 the surface settles at 254.6 K and the layer has no temperature, because it neither absorbs nor emits. At ε = 0.5 the surface is 273.6 K and the layer 230.0 K: the surface emits 317.6 W/m², 158.8 passes through and the layer sends 79.4 up, which adds back to 238.2 at the top. At ε = 1 the surface is 302.7 K.

Every setting shows the settled state. The model has no heat capacity, so it says nothing about how long warming takes; in the film, the rising temperature is a sequence of balanced states as the knob turns. Ribbon widths are to scale (0.34 px per W/m² on the stage), lane positions are schematic, and the chevrons only mark direction. Fluxes are shown to 0.1 W/m² and rounded so each balance adds up on screen; every value shown is within 0.1 W/m² of the model. Not modeled: absorption that varies with wavelength, more than one layer, sunlight absorbed in the air, convection and evaporation, clouds and aerosols, geography, oceans, heat storage and feedbacks. These temperatures belong to this stark teaching model. They are not Earth’s measured temperature, a climate-sensitivity estimate or a forecast.

The 1361 W/m² value follows NASA Goddard’s SORCE-based reference; the same page gives the current TSIS-1 estimate of about 1361.6 W/m². The 0.30 albedo is a round value close to the roughly 29% of sunlight that NASA’s energy budget shows reflected. Both are fixed inputs of this model, not a current measurement of Earth’s energy budget.

Sources: NASA Earth Observatory, Climate and Earth’s Energy Budget; NASA Goddard, Solar Irradiance Science; UCAR Center for Science Education, The Greenhouse Effect and The Energy Budget; OpenStax, University Physics Vol. 3, Blackbody Radiation; NIST Technical Note 910-8, Self-study manual on optical radiation measurements, Chapter 12. Related interactive: PhET’s The Greenhouse Effect, a different photon-and-temperature simulation. No PhET code, interface or artwork is used here.