As the space probe is very far from the solar system we may neglect the solar and cosmic background radiation. Without any protecting shields, the heat production of the nuclear energy source is radiated away by the surface of the space probe according to the Stefan-Boltzmann law:

I = σAT 4 ,

where σ is the Stefan-Boltzmann constant, A is the surface area of the space probe and T is its surface temperature. When a thin protecting shield encloses the space probe, the same radiation process occurs at the outer surface of the shield, and so the temperature of the shield must be T. However, the shield also emits inwards, and consequently the surface of the probe absorbs and amount of radiation equivalent to that radiated into space (see figure).

This means that the surface of the probe must re-radiate a total received intensity of 2I at a new temperature T 1 , where

It follows that T 1 = T.

For N protecting shields, the net radiation through them will still be I. Repeated application of our previous argument shows that the space probe radiates (N + 1)I and implies that the temperature of the surface of the probe is T N = T.