Cooling a Hot Plate in Vacuum

What is modelled

In vacuum there is no air to carry heat away, only thermal radiation, which depends on the fourth power of the temperature. A hot plate therefore cools very fast at first and then slowly, and a cool one barely cools at all. Sunlight on one face adds a heat input, so the plate can never cool below an equilibrium set by the ratio of absorptivity to emissivity.

Equations

• Heat balance per area ρ c d dT/dt = −2 ε σ (T⁴ − Ts⁴) + α S with S = 1361 W/m² when sunlit, T_s = 3 K.
• In shade 1/T³ = 1/T0³ + 6 ε σ t / (ρ c d); time to reach T_t: t = (ρ c d / 6 ε σ)(1/Tt³ − 1/T0³).
• Sunlit equilibrium Teq = (α S / 2 ε σ + Ts⁴)1/4.
• Stefan-Boltzmann σ = 2π⁵k⁴ / 15 h³c² = 5.67 × 10−8 W/m²K⁴.

Checks run in the page

• The Planck spectrum integrated over wavelength and divided by T⁴.
• Runge-Kutta cooling against the closed-form time to reach the target.
• The radiated energy integrated over the run against the heat the plate lost.
• A sunlit plate integrated until it settles against the equilibrium formula.

Honest simplifications

The plate is isothermal (thin enough that conduction across it is fast), emissivity and absorptivity are constants rather than depending on temperature and wavelength, the only environment is cold space and the Sun, with no Earth infrared or albedo, and nothing else heats or shades it. The time-lapse speed is shown in the readout.

• Press Polished (low emissivity) and see how slowly a shiny plate sheds heat.
• Press Sunlit plate floor: the plate stops cooling at the equilibrium temperature however long you wait.
• Press Thick copper block to see mass slow the cooling: time scales with the heat stored per area.