## The Earth should emit as much energy as it receives from the Sun

This is something that I wrote quite a while ago. Basically the Earth has to emit as much energy as it receives from the Sun or else it will get warmer. The physics of this is quite well understood. The orgininal question was about using rockets to boost the earth into a higher orbit and therefore counteract the effects of global warming. This was from an 11 year old who was doing an Earth Education programme.

First of all I thought that as you moved away from the Sun then the energy from the Sun would decrease with the square of the distance from the Sun (x).

This is not true since it depends on how much energy the Earth absorbs compared to how much it emits. So we have to look a bit deeper.

# Stefan Boltzman Law

This states that energy flux density (the energy radiated per unit surface energy per unit time) is

where σ is the Stefan–Boltzmann constant (5.67×10^{-8}Js^{-1}m^{-2}K^{4})

The total power emitted by the Sun (Js^{-1}) is therefore

(where r_{Sun} is the radius of the Sun and T_{Sun} is the temperature of the Sun) since the area of a sphere is

similarly for the Earth

.

How much of the Suns energy does the Earth absorb?

The surface area of the sphere that at the distance of the Earth’s orbit from the Sun is where x is the distance from the Sun to the Earth. The Earth takes up of this area. It will therefore only absorb

of the power emitted where γ is a measure of how well the Earth adsorbs the Suns energy.

The power emitted from the Sun to the Earth is therefore

The Earth must emit as much energy as it adsorbs. Therefore

Now we take the square root of both sides

Taking the square root again we have

γ is the opposite of the albedo effect – the amount that the Earth reflects which is about 30%. Therefore γ is .7

The temperature of the Sun is about 5778K

r_{Sun }is 6.98E+08m

x is 1.50E+11m (about 150 million km – 93 million miles)

This gives a temperature of 255K about -18ºC. (lucky we have the greenhouse effect)

More importantly even if the Earth moved 5 million km nearer to the Sun then the temperature would only change by about five degrees. If we used my first approximation then it would be 18 degrees warmer.

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