Spacecraft External Temperatures

References:
Elements of Space Technology for Aerospace Engineers by Rudolph X. Meyer
The Geostationary Satellite by Peter Berlin
Spacecraft Systems Engineering by Peter W Fortescue, John Stark and Graham Swinerd
Spacecraft Thermal Modelling and Testing by Isidoro Martinez
Thermal Systems Design PDF – UMD ENAE 483/788D – Principles of Space Systems Design (1.5 MB PDF)

### Equilibrium Temperature of a Spherical Black-Body

T = [ E / (4 • σ) ]1/4

Where:

T: Temperature of Black-Body in Kelvin.
E:
Irradiance in Watts/m2 from a solar source
σ: Stefan-Boltzmann’s constant, which is: 5.67 x 10-8 W/m2 K4
 EXAMPLE: What is the Black-Body Temperature of a sphere at 0.23 AU from the sun, where E is 25,822 W/m2? [25,822 / (4 • 5.67 x 10-8)]1/4 = 580.88 K

### Equilibrium Temperature of a Isothermal Sphere

NOTE: This is a good way to “eyeball” the effects of different coatings on the thermal control of your spacecraft with a relatively simple equation.

T = [ (α • E) / (4 • ε • σ) ]1/4
or as
T = [ (α / ε) • (E / 4 • σ) ]1/4

Where:

T: Equilibrium temperature of sphere in Kelvin.
α:
Absorptivity
ε: Emissivity.
E: Irradiance in Watts/m2 from a solar source
σ: Stefan-Boltzmann’s constant, which is: 5.67 x 10-8 W/m2 K4

### Equilibrium Temperature for Non-Ideal Radiative Heat Transfer in free space with a Sun

T = [ (E • α • AS + PINT) / ( ε• σ • ARAD) ]1/4

where

E: Irradiance in Watts/m2 from a solar source
α: Absorptivity
ε: Emissivity.
PINT: Internal heat/power generation.
AS: Absorption area in square meters. (The area of the side that faces the sun).
σ: Stefan-Boltzmann’s constant, which is: 5.67 x 10-8 W/m2 K4
 EXAMPLE: The AERCam/SPRINT Satellite, a 30 cm sphere with 200 W of internal power, an absorptivity of 0.2 and an emissivity of 0.8 is launched into free space around earth. We do not consider reflected albedo or any effects that raise it's temperature, just internal power and solar irradance. What is its equilibrium temperature? [ (1366 • 0.2 • 0.0707 + 200) / ( 0.8 • 5.67 x 10-8 •0.2827) ]1/4 = 361.6K

### Equilibrium Temperature for Non-Ideal Radiative Heat Transfer in free space with no Sun

T = [ PINT / ( ε• σ • ARAD) ]1/4

where

ε: Emissivity.
PINT: Internal heat/power generation.
σ: Stefan-Boltzmann’s constant, which is: 5.67 x 10-8 W/m2 K4
 EXAMPLE: The AERCam/SPRINT Satellite, a 30 cm sphere with 200 W of internal power, an absorptivity of 0.2 and an emissivity of 0.8 is launched into the darkness between the stars. What is its equilibrium temperature? [ 200 / (0.8• 5.67 x 10-8 • 0.2827) ]1/4 = 353.4K

 Emissivity/Absorptivity Properties of Selected Materials(Generally from materials at 294K (20.85C) Material Condition Solar Absorptivityα Infrared Emissivityε White Enamel Paint Over Aluminum Substrate 0.250 0.853 White Epoxy Paint Over Aluminum Substrate 0.248 0.924 Black Paint Over Aluminum Substrate 0.975 0.874 Aluminum Paint As Received 0.250 0.250 Graphite Epoxy As Received 0.950 0.750 Fiberglass As Received 0.900 0.900 Aluminum (6061-T6) Polished Metal 0.031 0.039 Stainless Steel (AM350) Polished Metal 0.357 0.095 Titanium (6AL-4V) Polished Metal 0.448 0.129 Teflon, Silvered As Received 0.080 0.660 Teflon, Aluminized As Received 0.163 0.800 Optical Solar Reflector(Quartz Over Silver) As Received 0.077 0.790

### Calculating Thermal Capacities of Materials

Q = m • cp • (ΔT/Δt)

Where:

Q: Input/Output of energy in joules.
m: Mass of the material in kilograms.
cp: Specific heat capacity of the material in J • kg-1K-1
ΔT: Temperature increase (or decrease) in Kelvin.
Δt: Duration of input/output energies in seconds.