Hail Mary Science
Light, heat and climate

Radiative cooling in a vacuum

In a vacuum there is no air to carry heat away, so a hot object can only cool by emitting radiation. Choose the starting temperature and the time, and the model works out how far a 2 kg metal cylinder has cooled.

Key formula

P = ε · σ · A · T⁴ · σ = 5.67·10⁻⁸ W/(m²·K⁴)

Try it

How to use the simulator

In a vacuum there's no air to carry heat away, so a hot object can only cool by radiating, following the Stefan-Boltzmann law (P = σ·A·T⁴). Because power scales with the fourth power of temperature, a very hot object cools quickly at first and ever more slowly afterwards. The model uses a 2 kg metal cylinder with 0.1 m² of surface.

Controls

  • Initial temperature: 50 to 500°C.
  • Time in space: 1 to 120 minutes.

Things to try

  1. Fix the time at 40 minutes and try different starting temperatures. See how close the final results end up: initial differences shrink a lot over time.
  2. Compare the first 5 minutes with the last 5 of a 120-minute run: the drop is much faster at the start.
  3. Remember that the result depends on the object's mass, area and material, which the model fixes. A heavier object, or one with less surface, would stay hot much longer.

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Three ways to lose heat

On Earth a hot object cools by conduction (touching something colder), convection (air or water warms up and carries heat away) and radiation (it emits infrared light). In a vacuum the first two vanish and only radiation is left.

That is why space is not an instant freezer, as it sometimes appears in films. A thermos works exactly this way: there is a vacuum between its two walls so heat can only escape by radiation, which is slow at normal temperatures.

The fourth power

The Stefan-Boltzmann law says radiated power grows with the fourth power of absolute temperature. Doubling the temperature in kelvin multiplies the emission by 16. That is why a very hot object loses heat very fast at first and ever more slowly as it cools.

The factor ε (emissivity) runs from 0 to 1 and depends on the surface: polished metal emits little and matte black paint nearly the maximum. Spacecraft use this to control their temperature with paints, insulating blankets and radiators.

Worked example

Example: a metal cylinder at 500 °C

  1. At 500 °C (773 K), with 0.1 m² of surface and ε = 1: P = 5.67·10⁻⁸ · 0.1 · 773⁴ ≈ 2,000 W.
  2. With 2 kg of metal and a specific heat of 500 J/(kg·K), it loses about 2 degrees per second at first.
  3. At 100 °C (373 K) the power drops to about 110 W: almost 20 times less, even though the temperature in kelvin has only halved.
  4. According to the model, after 5 minutes it would be at about 243 °C and after 40 minutes at about 12 °C. The last few degrees take the longest.

In Project Hail Mary

In the novel, the main character has to handle hot parts and materials in a vacuum and needs to work out how long they will take to cool. This model shows why the process is slower than people tend to imagine.

This tool appears in:Chapter 9The Tunnel

Frequently asked questions

Would an astronaut without a suit freeze instantly?

No. They would lose heat only by radiation and by water evaporating from the skin, which is slow by comparison. The immediate dangers are lack of oxygen and low pressure, not cold.

Why do spacecraft struggle to shed heat?

Because radiating is their only option, and at moderate temperatures that takes large surfaces. The Space Station carries huge white radiators to get rid of the heat from its equipment.

What temperature is space?

The cosmic microwave background is at about 2.7 K (−270 °C), but empty vacuum has no temperature as such because there is no matter. An object in sunlight near Earth can exceed 100 °C, and in shadow drop below −100 °C.

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