At what rate does the temperature away from the Sun decrease?

Question

brodireo1 · Accepted Answer

The absolute temperature, T of the iron varies with its distance, r, from the Sun according to

T \propto \frac{1}{\sqrt{r}} .

My reasoning is as follows.
If the radiant power of the sun is L, the solar energy passing through an imaginary spherical surface of radius r, centred on the Sun, per unit time, per unit area will be

I = \frac{L}{4 π r^{2}} .

So we have an inverse square law of intensity. If we place an iron sphere of radius a at distance r from the Sun, the solar power intercepted will be

P = π a^{2} I = \frac{π a^{2} L}{4 π r^{2}} .

The sphere will reach a temperature at which the rate of solar gain is equal to the rate of loss of energy by radiation. We will assume that the sphere absorbs all the intercepted radiation and that it emits as a 'black body' obeying Stefan's law, that is at a power of

P_{e m} = σ A T^{4} = σ 4 π a^{2} T^{4}

in which

σ

is the Stefan constant and A is the surface area. So putting

P = P_{e m}

we get

T^{4} = \frac{L}{16 π σ r^{2}},

from which follows the relationship given at the top.

At what rate does the temperature away from the Sun decrease?

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