## Sunlight and the Earth’s Temperature

Now let’s get back to science. Energy flows onto the Earth from the Sun, and energy is essential for life. Yet energy also flows out from the Earth to the cold of outer space. The inflow of energy from the Sun is actually slightly less than the outflow of energy from the Earth, because some of the outflow is heat escaping from the Earth’s interior. If the inflow were always even slightly greater than the outflow, the Earth’s temperature would rise without limit.

We may calculate the Earth’s temperature from the solar surface temperature and the amount of sky the Sun occupies. The Sun’s absolute surface temperature is about 6 000 kelvins.

The Sun’s disk as seen from the Earth has an angular diameter of about half of one degree or 30 minutes of arc. Consider a map of the sky. The equator is 360 degrees around. Therefore 720 suns would fit on the equator with the edges of one touching the edges of the suns on either side. Then it is 90 degrees from the equator to the North Pole, and another 90 degrees to the South Pole. Therefore 360 suns would fit on a longitude line in the sky. The product of 360 and 720 is 259 200. That is more than the number of suns it would take to fill up the sky, because the latitude circles get smaller as one approaches the poles. On the other hand, once the sky was full of suns with their edges touching, one would need to cut up additional suns to fill in the spaces between the circles. It takes calculus to work this out precisely. The sky would be packed full with 210 000 suns. Under such conditions the Earth would have the same temperature as the surface of the Sun, 6 000 kelvins. Even a few extra suns, let alone 210 000 of them, would make the Earth intolerably hot. Happily for us there is only one Sun.

Outer space has an absolute temperature of less than 3 kelvins. This temperature is so low it may be taken as zero for the purposes of our calculation. Radiant energy flow is proportional to the fourth power of the absolute temperature, that is, to the square of the square of the absolute temperature. The total inflow has to be averaged over the whole sky. The main inflow is from the Sun. The remaining inflow from all the stars and planets in the rest of the sky is practically zero. The average inflow is therefore 6 000 kelvins squared and squared again, divided by 210 000. The average outflow is almost equal to the average inflow. Therefore the Earth’s temperature is the square root of the square root of the average inflow. We can simplify the calculation if we use the square root of the square root of 210 000, which is equal to 21.4. Then if we divide 6 000 kelvins by 21.4, the result we get is 280 kelvins. Subtracting 273º C from this, we get the Earth’s average temperature as 7º C or 45º F. That is a bit chilly but not freezing. Since the average Earth temperature is a little higher than this, some of the Earth’s warmth must still be coming from its internal heat. Besides the inflow of heat from the Sun, radioactive materials in the Earth’s core disintegrate and produce additional internal heat, which eventually flows out into space. The Earth’s core is not as hot as the interior of a star, so the core cannot synthesize new radioactive materials. The disintegration is irreversible and contributes continually to the Earth’s increasing entropy.

The Earth absorbs sunlight a little better than it emits heat, because of the greenhouse effect of the atmosphere. That raises the Earth’s temperature slightly.

The foregoing is an equilibrium calculation. If all the incoming energy from the Sun flowed away immediately to the rest of the universe, there would be no change of entropy on the Earth. In perfect balance the net inflow of heat would be zero, so the change in entropy would also be zero. However, some of the Sun’s heat drives winds and makes turbulence in the Earth’s atmosphere. The winds sandblast the mountains and also bring rains that wear the mountains down. No one can ever exactly reverse the random irregularities of turbulent flow. These two processes are therefore irreversible.

We may calculate the Earth’s temperature from the solar surface temperature and the amount of sky the Sun occupies. The Sun’s absolute surface temperature is about 6 000 kelvins.

The Sun’s disk as seen from the Earth has an angular diameter of about half of one degree or 30 minutes of arc. Consider a map of the sky. The equator is 360 degrees around. Therefore 720 suns would fit on the equator with the edges of one touching the edges of the suns on either side. Then it is 90 degrees from the equator to the North Pole, and another 90 degrees to the South Pole. Therefore 360 suns would fit on a longitude line in the sky. The product of 360 and 720 is 259 200. That is more than the number of suns it would take to fill up the sky, because the latitude circles get smaller as one approaches the poles. On the other hand, once the sky was full of suns with their edges touching, one would need to cut up additional suns to fill in the spaces between the circles. It takes calculus to work this out precisely. The sky would be packed full with 210 000 suns. Under such conditions the Earth would have the same temperature as the surface of the Sun, 6 000 kelvins. Even a few extra suns, let alone 210 000 of them, would make the Earth intolerably hot. Happily for us there is only one Sun.

Outer space has an absolute temperature of less than 3 kelvins. This temperature is so low it may be taken as zero for the purposes of our calculation. Radiant energy flow is proportional to the fourth power of the absolute temperature, that is, to the square of the square of the absolute temperature. The total inflow has to be averaged over the whole sky. The main inflow is from the Sun. The remaining inflow from all the stars and planets in the rest of the sky is practically zero. The average inflow is therefore 6 000 kelvins squared and squared again, divided by 210 000. The average outflow is almost equal to the average inflow. Therefore the Earth’s temperature is the square root of the square root of the average inflow. We can simplify the calculation if we use the square root of the square root of 210 000, which is equal to 21.4. Then if we divide 6 000 kelvins by 21.4, the result we get is 280 kelvins. Subtracting 273º C from this, we get the Earth’s average temperature as 7º C or 45º F. That is a bit chilly but not freezing. Since the average Earth temperature is a little higher than this, some of the Earth’s warmth must still be coming from its internal heat. Besides the inflow of heat from the Sun, radioactive materials in the Earth’s core disintegrate and produce additional internal heat, which eventually flows out into space. The Earth’s core is not as hot as the interior of a star, so the core cannot synthesize new radioactive materials. The disintegration is irreversible and contributes continually to the Earth’s increasing entropy.

The Earth absorbs sunlight a little better than it emits heat, because of the greenhouse effect of the atmosphere. That raises the Earth’s temperature slightly.

The foregoing is an equilibrium calculation. If all the incoming energy from the Sun flowed away immediately to the rest of the universe, there would be no change of entropy on the Earth. In perfect balance the net inflow of heat would be zero, so the change in entropy would also be zero. However, some of the Sun’s heat drives winds and makes turbulence in the Earth’s atmosphere. The winds sandblast the mountains and also bring rains that wear the mountains down. No one can ever exactly reverse the random irregularities of turbulent flow. These two processes are therefore irreversible.