Nanophotonic Radiative Sky Cooling (2D)

Sky & Sun

Solar reflectance R_solar 96%

Atmospheric-window emissivity ε_win 90%

Ambient air temperature 30°C

Convection coefficient h 8 W/m²K

Live Readouts

Surface temp

30.0°C

Sub-ambient ΔT

0.0°C

Net radiative cooling power

0 W/m²

Window-band fraction F(T_s) of Planck curve

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How it works

This 2D version replaces the flat "one fixed emissivity times ΔT⁴" bookkeeping with a real spectral calculation: at every frame it numerically integrates Planck's law over wavelength to find exactly what fraction of the surface's thermal glow actually falls inside the 8-13 µm atmospheric window — the left chart draws that integral directly, shading the escaping (orange) versus reabsorbed (brown) area under the curve.

M(λ,T) = 2πhc² / { λ⁵ [exp(hc/λkT) − 1] }      [Planck spectral exitance]
F(T)   = ∫₈¹³ M(λ,T) dλ  /  σT⁴                [window-band fraction, Simpson's rule]

P_solar = (1 − R_solar) · S                     [absorbed sunlight, S ≈ 900 W/m², day only]
P_window = ε_win · [ F(T_s)·σT_s⁴ − F(T_sky)·σT_sky⁴ ]
P_other  = ε_other · [ (1−F(T_s))·σT_s⁴ − (1−F(T_amb))·σT_amb⁴ ]
P_conv   = h · (T_amb − T_s)

dT_s/dt = ( P_solar + P_conv − P_window − P_other ) / C_eff

Cross-check against the 3D twin: re-deriving this ODE from the stated steady-state balance and testing it numerically (Planck-integral verified to 1.2×10⁻⁵ against standard blackbody-fraction tables; see verification script) turned up a sign error in the flat-emissivity 3D model — it subtracts the absorbed solar term from the rate of change instead of adding it, so in that version a more reflective coating paradoxically ends up warmer, and reflectance sweeps run backwards. This 2D model uses the corrected sign shown above, so higher R_solar and higher ε_win both correctly push the steady state colder, exactly as the passive-cooling literature (and Test 7 in the verification script) predicts.

F(T) — because it comes from the real Planck curve rather than a constant, it shifts with temperature (Wien's law): a cooler surface's peak sits closer to the 8-13 µm band, so F rises slightly as T_s falls — a genuine physical detail the flat-emissivity model cannot show.
R_solar — fraction of sunlight the coating bounces away; every percent left over is absorbed heat (P_solar).
ε_win — how strongly the coating emits specifically inside the window; higher means more of F(T_s)·σT_s⁴ actually escapes to space.
T_sky ≈ T_amb − 30 K (the window's effective view of space); the non-window band exchanges instead with the atmosphere near T_amb (ε_other fixed at 0.15).

Escaping through window → space Reabsorbed by atmosphere Sky / ambient reference curves