Earth Temperature Homework Solution

[Silviu]

Homework Statement

Calculate the temperature of the Earth assuming that the Sun and the Earth are black bodies. Assume that Earth is in a steady state as far as energy balance is concerned

Homework Equations

$\frac{E}{St} = \sigma T^4$ - Stefan-Boltzman law

The Attempt at a Solution

The power radiated by sun is $P=\sigma T_{sun}^4 S_{sun} = \sigma T_{sun}^4 4\pi R_{sun}^2$. The amount of this received by Earth is proportional to: $\frac{\pi R_{earth}^2}{4 \pi R_{sun-earth}^2}$, with $R_{sun-earth}$ being the distance from sun to earth. The power radiated by Earth is $\sigma T_{earth}^4 4\pi R_{earth}$. As the Earth is in a steady state we have in the end: $\sigma T_{sun}^4 4\pi R_{sun}^2 \frac{\pi R_{earth}^2}{4 \pi R_{sun-earth}^2} = \sigma T_{earht}^4 4\pi R_{earth}^2$ and from here we can get the temperature of the earth, as all the other constants are considered to be known. Is this correct? I obtained a numerical value of about 900K. I am aware that here we ignore the shielding of the atmosphere, the albedo and other effects that would influence the temperature, but it still seems to be pretty high.

 

[Bandersnatch]

The power radiated by Earth is $\sigma T_{earth}^4 4\pi R_{earth}$.

You've made an error here. The radius should be squared.
Correcting that will net you a factor of ~1/9, so if you've got 900K out of the equation, there has to be some other issue elsewhere. You should get a bit below 300K. The equation is otherwise sound, so it has to be either a number crunching error, or bad constants. Check those first.

 

[Silviu]

You've made an error here. The radius should be squared.
Correcting that will net you a factor of ~1/9, so if you've got 900K out of the equation, there has to be some other issue elsewhere. You should get a bit below 300K. The equation is otherwise sound, so it has to be either a number crunching error, or bad constants. Check those first.

Hey, sorry I have squared there on paper I just mistyped it here. Ok, I will check the numbers again. Thank you!