Question

The solar flux S arriving at the outer edge of the atmosphere varies by percent as the Earth moves in its orbit (reaching its greatest value in early January). By how many degrees would the effective temperature of the Earth vary as a result?

Answer 1

4.34 K

Step-by-step explanation:

The % is not given by usually by January the solar flux varies from 3.2 to 3.5%. We can take 3.4% as a good estimate for this case.

The solar flux S, arriving at the outer edge of the atmosphere, is represented by the solar constant of
`1370 W/m^2` . We are assuming that the solar flux varies by ±3.4 percent as the earth moves in its orbit. If we do the operation 1370*0.034=46.58, then the margin for the solar flux would be
`Solar flux =1370 \pm 46.58 W/m^2` .

From theory we have an expression for the energy absorbed by Earth and is given by
`S\pi R^2 (1-\alpha)` and we have also a formula for the energy radiated back to space by earth given by
`4\sigma \piR^2 T^4`, according to the Stefan-Boltzmann Law.

For this case the energy absorbed by Earth needs to be equals the energy radiated back to space since we assume a balance of energy, so we can set equal the two quantities of energy like this:

`S\pi R^2 (1-\alpha)=4\sigma \piR^2 T^4`.

And if we solve for the temperature we have this:

`T=[(S(1-\alpha))/(4\sigma)]^(1/4)`

We need to do another ssumptions for example the average albedo for Earth constant,
`\alpha=0.3`. The Stefan-Boltzmann Constant is
`\sigma=5.67x10^(-8)W/(m^2 K^4)`.

Since we have a variation
`S=1370\pm 46.58 W/m^2` we can do operations in order to find the possible change of temperature, like this:

`T=[(1323.42W/m^2(1-0.3))/(4*5.67x10^(-8)W/(m^2 K^4))]^(1/4)=252.806 K`

`T=[(1416.58W/m^2(1-0.3))/(4*5.67x10^(-8)W/(m^2 K^4))]^(1/4)=257.143 K`

So the possible variation on this case is 257.143-252.806 K=4.34 K