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Question

The total radiation energy E emitted by a heated surface per unit area varies as the fourth power of its absolute temperature T. The temperature is 6,000 K at the surface of the sun and 600 K at the surface of a given planet.

(a)

How many times more radiation energy per unit area is produced by the sun than by the planet?

(b)

Suppose the radius of the planet is 3,480 mi, and the radius of the sun is 435,000 mi. How many times more total radiation does the sun emit than the planet?

Answer 1

a) The first part asks for a comparison of the radiation output of the 6000K per unit area of the sun to the lower 600K per unit area of the planet and provides that the energy output is the fourth power of the absolute temperature, measured in K (Kelvin). Setting it up as a fraction it can be calculated as: 6000⁴K / 600⁴K = 10,000. That is, the suns radiation energy output per unit area is 10,000 greater than the planets radiation energy output per unit area.

b) The second part provides the radius of the sun and planet and asks for the total radiation comparison of the two. We can find the answer by either calculating the total radiation energy output of each surface and comparing them as a fraction or, we can simply compare the surface areas of the sun to the planet in a fraction and multiply the fractions outcome by the 10,000 times more energy we just calculated the sun has. The surface area of a sphere can be calculated using the formula: 4πr², where r is the radius. Let's answer it both ways and compare. Units have left out the units as we already know the answer will be a unitless comparison.

a) (4π435000² * 6000⁴) / (4π3480² * 600⁴) = 156,250,000 = 1.56 x 10⁸

b) ((4π435000²) / (4*3.14*3480²)) * 10000 = 156,250,000 = 1.56 x 10⁸