How Does Doubling Earth's Orbital Radius Affect Its Surface Temperature?

[chris_0101]

Homework Statement

Let us assume that the earth, whose mean surface temperature may be taken to be
280 K, is in thermal equilibrium and radiates energy into space at the same rate that it
receives energy from the sun. If the Earth were moved to an orbit of twice its present
radius, what would be its expected mean surface temperature?

T = 280K
T' = ?

Homework Equations

T' = T (?)

The ? indicates that I have to multiply T by something, but I do not know where to start so I was hoping to have some help on where to start​

The Attempt at a Solution

Don't know how to start this question, if someone could help, that would be great. I do know the answer however and it is 198K.

Thanks

 

[Delzac]

Hmmm, have you heard of stefan's law?

 

[chris_0101]

Yes, I have heard of Stefan's law, which is:

I = (5.670*10^-8)(T^4)​

So if I were to plug the value of temperature into the equation, I is equal to:

I = 348.5099Wm^-2

Even though I have intensity, I still don't know what to do.

 

[Delzac]

Firstly, what is the relationship between power, intensity and distance?

Stefan's law gives you power by the way.

Also, there can only be thermal equilibrium when power emitted equal to power absorbed. If you know the power emitted to that point can you then find the temperature?

 

[chris_0101]

The relationship between power, intensity and distance is:
I = P/(r^2)

yet, I still do not know what to do.

 

[Delzac]

Understand that as you move further away, the power transmitted by the sun to that point decrease.

This thus will affect your thermal equilibrium, since Earth is radiating more power than the sun is transmitting.

So it follows that the Earth will start cooling to the power equal to what the sun is transmitting at the new distance.

Now, all you have to do now is to translate all i have said into equation form. You do not need to know the area, emissivity or stefan Boltzmann's constant to do this by the way.

delzac