Can the Ground Temperature be Expressed in Terms of the Emission Temperature?

[il27]

Homework Statement

:[/B]Use the energy balance equations you wrote down for the ground layer, the atmospheric layer, and the above the atmosphere layer (space) in part (a) to show that the ground temperature Tg can be expressed in terms of the emission temperature TE as follows:
$$ T_g = (2f)^.25T_E $$
and provide an algebraic expression for f .

Homework Equations

Ground equation:

$$ (1 - \alpha - a)\frac{f_0}{4} + e \sigma T_a^4 = \sigma T_g^4 $$

Layer:

$$ \frac{af_0}{4} + e \sigma T_g^4 = 2e \sigma T_a^4 $$Top of atmosphere:

$$ \frac{f_0}{4} = \frac{ \alpha f_0}{4} + (1-e) \sigma T_g^4 + e \sigma T_a^4 $$

The Attempt at a Solution

I tried adding the layer and ground equation, and i ended up getting the top of atmosphere equation, but i am having trouble just manipulating the equations to get:$$ T_g = (2f)^.25T_E $$

 

[haruspex]

Homework Statement

:[/B]Use the energy balance equations you wrote down for the ground layer, the atmospheric layer, and the above the atmosphere layer (space) in part (a) to show that the ground temperature Tg can be expressed in terms of the emission temperature TE as follows:
$$ T_g = (2f)^.25T_E $$
and provide an algebraic expression for f .

Homework Equations

Ground equation:

$$ (1 - \alpha - a)\frac{f_0}{4} + e \sigma T_a^4 = \sigma T_g^4 $$

Layer:

$$ \frac{af_0}{4} + e \sigma T_g^4 = 2e \sigma T_a^4 $$Top of atmosphere:

$$ \frac{f_0}{4} = \frac{ \alpha f_0}{4} + (1-e) \sigma T_g^4 + e \sigma T_a^4 $$

The Attempt at a Solution

I tried adding the layer and ground equation, and i ended up getting the top of atmosphere equation, but i am having trouble just manipulating the equations to get:$$ T_g = (2f)^.25T_E $$

None of the equations you quote mention T_E. You need to involve a definition of that.

 

[il27]

None of the equations you quote mention T_E. You need to involve a definition of that.

$$ T_E = (\frac{f_0}{4}(1 - \alpha))^.25 $$ is the emission temperature.

 

[haruspex]

$$ T_E = (\frac{f_0}{4}(1 - \alpha))^.25 $$ is the emission temperature.

Try using your three given equations to eliminate T_a and e, then see what you have left.

 

[il27]

Try using your three given equations to eliminate T_a and e, then see what you have left.

should i add all the inputs of the 3 equations and then set them equal to all the outputs of the equation?
which equations should i use?

 

[haruspex]

should i add all the inputs of the 3 equations and then set them equal to all the outputs of the equation?
which equations should i use?

Do you know how to manipulate simultaneous equations? If you have two equations involving some variable x that you want to get rid of, get one of them into the form x=(some expression not involving x) and use that to substitute for x in the other equation.

Edit:

which equations should i use?

I see that you only really have two equations. The third can be derived from the other two. That being so, it does not matter which two you use, the result should be the same. It also means you probably cannot eliminate both T_a and e, so just go for eliminating T_a and see what emerges.