Chapter 24 — Key Takeaways (Thermal Control)

A one-page re-grounding card: the equations, numbers, and decision rules of keeping a spacecraft in its temperature band between a furnace and a freezer.

The one idea

In vacuum there is no convection and no conduction to the surroundings — the only way a spacecraft exchanges heat with its environment is thermal radiation, which scales as $T^4$. Everything else in the chapter is a consequence. Thermal control is the art of arranging absorbed and radiated power, surface by surface, so each component settles in its allowed band.

Key equations (with symbols and units)

Quantity Equation Symbols / units
Radiated power (Stefan–Boltzmann) $P = \varepsilon\,\sigma\,A\,T^4$ $\varepsilon$ emissivity (–); $\sigma = 5.670\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$; $A$ area (m²); $T$ (K)
Solar flux vs. distance $S(d) = S_{1\text{AU}}\,(1\,\text{AU}/d)^2$ $S_{1\text{AU}} = 1361\ \text{W/m}^2$; $d$ (AU)
Equilibrium temperature $T_{\text{eq}} = \left(\dfrac{\alpha S A_{\text{sun}} + Q_{\text{int}}}{\varepsilon\,\sigma\,A_{\text{rad}}}\right)^{1/4}$ $\alpha$ solar absorptivity; $A_{\text{sun}}$ sunlit area; $A_{\text{rad}}$ radiating area; $Q_{\text{int}}$ internal power (W)
Its scaling (no internal power) $T_{\text{eq}} \propto \left(\dfrac{\alpha}{\varepsilon}\right)^{1/4}\left(\dfrac{A_{\text{sun}}}{A_{\text{rad}}}\right)^{1/4}$ temperature moves as the fourth root of both ratios
MLI heat leak $q = \dfrac{\sigma(T_h^4 - T_c^4)}{(N+1)(2/\varepsilon - 1)} \equiv \varepsilon^{*}\sigma(T_h^4 - T_c^4)$ $N$ shields; $\varepsilon^{*}$ effective emittance
Radiator area $A = \dfrac{Q}{\varepsilon\sigma T^4 - \alpha S - q_{\text{env}}}$ subtract absorbed sun/environment from emission
Cryogenic boiloff $\dot m = Q/h_{fg}$ $h_{fg}$ latent heat (J/kg); LH2 $\approx 446\ \text{kJ/kg}$
Thermal mass $Q = m\,c_p\,\Delta T$ $c_p$ specific heat (Al $\approx 900\ \text{J kg}^{-1}\text{K}^{-1}$)

Numbers worth memorizing

Quantity Value
Solar flux at 1 AU $S = 1361\ \text{W/m}^2$
Stefan–Boltzmann constant $\sigma = 5.670\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$
Earth albedo (Bond) $a \approx 0.30$
Earth outgoing IR (mean) $\approx 240\ \text{W/m}^2$
Deep-space sink $2.7\ \text{K}$ (treat as $0\ \text{K}$)
Grey sphere at 1 AU ($\alpha=\varepsilon$) $278\ \text{K} = +5\ ^\circ\text{C}$
Black 1-sided plate at 1 AU ($\alpha=\varepsilon=1$) $394\ \text{K} = +121\ ^\circ\text{C}$
Blackbody at $394\ \text{K}$ radiates $\approx 1361\ \text{W/m}^2$ (memory hook)
Convert $^\circ\text{C} \to \text{K}$ add $273$

Surface coatings — representative $\alpha,\varepsilon$ (Tier 2)

Finish $\alpha$ $\varepsilon$ $\alpha/\varepsilon$ Runs
Gold foil 0.25 0.04 6.3 very hot
Polished aluminum 0.15 0.05 3.0 hot
Black paint 0.95 0.85 1.1 grey/moderate
White paint 0.20 0.90 0.22 cold
Optical solar reflector (OSR) 0.08 0.80 0.10 coldest — radiator of choice

Same sphere, same orbit, temperature from $-116$ to $+168\ ^\circ\text{C}$ by paint alone. $\alpha$ rises with UV/atomic-oxygen age → use end-of-life $\alpha$ for the hot case.

Which tool — when

Need Reach for
Set a surface's steady temperature Coating ($\alpha/\varepsilon$) — passive, first choice
Keep heat in / environment out Multi-layer insulation (MLI), $\varepsilon^{*} \approx 0.01$–$0.05$
Damp eclipse swings Thermal mass (but mass is the enemy — use sparingly)
Warm a too-cold component Heater (costs power → power budget)
Dump internal waste heat Radiator (size $\propto Q/T^4$; keep out of Sun)
Move heat to a distant radiator Heat pipe (passive, phase-change)
Power-free variable emittance Louver (bimetal blinds)
Reach tens of kelvin Sunshield (staged MLI) + passive radiation
Reach a few kelvin Cryocooler (active — $T^4$ starves passive below ~40 K)

Common pitfalls

  • Thinking in "color," not $\alpha/\varepsilon$. White paint runs cold; shiny metal runs hot.
  • Confusing $A_{\text{sun}}$ with $A_{\text{rad}}$. A sphere absorbs on $\pi r^2$, radiates on $4\pi r^2$.
  • Sizing the hot case with fresh coatings. Use degraded end-of-life $\alpha$.
  • Putting a radiator in sunlight. Absorbed $\alpha S$ can exceed emission and defeat it entirely.
  • Trusting ideal MLI $\varepsilon^{*}$. Seams and struts make flown blankets 10–40× leakier.
  • Designing for the average. Always bracket worst-case hot and worst-case cold separately.

Mission Design / astrotools additions

  • Thermal note added to the MDR: solar flux at your distance; hot-case and cold-case equilibrium temperature of a representative surface; the resulting heater power (→ power budget, Ch. 25) and radiator area (→ mass budget, Ch. 23).
  • thermal.py: solar_flux(au), equilibrium_temperature(alpha, eps, a_sun, a_rad, flux, q_internal), radiator_area(q_reject, t_rad, eps, alpha, flux). Benchmarks: grey sphere → $278\ \text{K}$; black plate → $394\ \text{K}$; MDR panel hot/cold → $+60/-50\ ^\circ\text{C}$.