SatBase

Thermal Equilibrium

Steady-state temperature of a body from solar flux, absorptivity, emissivity, and radiating area, with planet heating as an option.

Inputs

Heat sources

W/m²
W

Surface

m²
m²

Results

Temperature

Equilibrium temperature 308K
Equilibrium temperature (Celsius) 34.9°C

Heat loads

Absorbed solar 408.3W
Internal dissipation 0W
Total heat load 408.3W
Absorptivity to emissivity ratio 0.375

Method

A single-node radiative balance: the temperature at which a body radiates exactly the heat it takes in. Steady state, one temperature for the whole body.

T = [ (Qsun + Qalb + QIR + Qi) / (ε · σ · Ar) ]1/4 σ = 5.670 × 10⁻⁸ W/m²K⁴, the Stefan-Boltzmann constant. Qalb and QIR are zero with the Sun as the only heat source.

Derivation

1
Absorb the direct sunlight The surface intercepts sunlight on the area it presents to the Sun and keeps the fraction its absorptivity allows. The rest is reflected and plays no part. Qsun = α · S · As
2
Balance heat in against heat radiated A surface radiates in proportion to the fourth power of its temperature. Setting that equal to everything absorbed and dissipated gives the temperature the body settles at. ε · σ · Ar · T⁴ = Qsun + Qalb + QIR + Qi
3
Find how much of the sky the planet fills With a planet in the balance, its angular radius from the body follows from the radius and the altitude, and the view factor is the share of a sphere’s surface that faces it. At the surface that is one half. sin ρ = R / (R + h),   F = (1 − cos ρ) / 2
4
Add the sunlight the planet reflects Albedo is sunlight bounced off the planet, so the surface absorbs it at its solar absorptivity. The correction accounts for the planet being a sphere, lit most strongly at the point under the Sun, with ρ in radians. Qalb = α · S · a · Ka · F · Ar,   Ka = 0.664 + 0.521ρ − 0.203ρ²
5
Add the infrared the planet emits The planet also radiates its own heat. That arrives in the infrared, where a surface absorbs as well as it emits, so the emissivity is the right coefficient here and not the absorptivity. QIR = ε · q · F · Ar
6
Repeat the balance in eclipse In the planet’s shadow the direct sunlight and the albedo are gone. The planetary infrared and the internal dissipation are what is left, and they set the cold case. Tecl = [ (QIR + Qi) / (ε · σ · Ar) ]1/4

Symbols

Input Description Unit
S Solar flux at the body W/m²
Qi Internal dissipation W
α Solar absorptivity of the surface —
ε Infrared emissivity of the surface —
As Sunlit area: projected toward the Sun m²
Ar Radiating area: all surface open to space m²
h Altitude above the planet (planet only) km
a Planet albedo (planet only) —
q Planetary infrared flux (planet only) W/m²
σ Stefan-Boltzmann constant: 5.670 × 10⁻⁸, fixed W/m²K⁴
Derived Description Unit
Qsun Absorbed direct sunlight W
ρ Angular radius of the planet as seen from the body deg
F View factor from the body to the planet —
Ka Albedo correction for a spherical planet —
Qalb Absorbed sunlight reflected by the planet W
QIR Absorbed planetary infrared W
T Equilibrium temperature in sunlight K
Tecl Equilibrium temperature in eclipse K

Reference values

Figures for sanity-checking inputs.

Surface finish Absorptivity Emissivity
Z-93 white paint 0.17 0.92
S-13G white paint 0.20 0.90
Chemglaze Z306 black paint 0.96 0.91
Second-surface mirror, silvered 0.07 0.79
Silvered Teflon, 5 mil 0.08 0.81
Aluminized Kapton, 1 mil 0.38 0.67
Vapor-deposited aluminum 0.08 0.02
Vapor-deposited gold 0.19 0.02
Buffed aluminum 0.16 0.03
Solar cells 0.75–0.91 0.79–0.86
Body Solar flux W/m² Albedo Infrared W/m²
Mercury 9,083 0.068 2,118
Venus 2,601 0.77 150
Earth 1,361 0.294 236
Moon 1,361 0.11 303
Mars 586 0.250 110
Jupiter 50.3 0.343 8.3
Saturn 14.8 0.342 2.4
Uranus 3.69 0.300 0.65
Neptune 1.51 0.290 0.27

Finish values are beginning-of-life averages from NASA RP-1121, good to about 0.02; absorptivity rises over a mission as ultraviolet and contamination darken a surface. Body values are from the NASA planetary fact sheets, with the infrared flux taken from each body's black-body temperature as a whole-surface mean: the day side of Mercury or the Moon emits far more than that and the night side far less, and Jupiter, Saturn and Neptune also radiate internal heat the figure leaves out. The model itself is one node at one temperature in steady state. It has no temperature gradients, no thermal mass, and no orbit averaging: the sunlit figure assumes the Sun, the albedo and the infrared all at once, and the planet terms use a sphere's view of the planet. Treat these results as a first estimate, not a thermal analysis.