[Physics] Calculate temperature of the earth through blackbody radiation

Question

I don't understand the solutions to a problem about blackbody radiation and was wondering if anybody could help me out.

Here is the question:

The sun can be considered as a blackbody radiation source at temperature
T = 5778 K. Radiation from the sun which is incident on the earth is reflected by the
atmosphere such that the intensity hitting the earth's surface is reduced by a factor R.
Some of the radiation emitted from the earth's surface is reflected by the atmosphere such
that only a fraction A leaves the atmosphere. If A = R = 0:1, what temperature would
the earth be?

Then in the solutions they state:
This is obtained by first trying to find the power from the sun which passes through unit area at the earth's radial distance. This is given by:

$\frac{4 \pi r_s^2}{4 \pi d_e^2}\sigma T_s^4$

where $r_s$ is the radius of the sun, $d_e$ is the distance between the earth and sun and $T_s^4 = 5778K$ is the temperature of the sun.

I know that $\sigma T_s^4$ is from the Stefan-Boltzmann law, and that $4 \pi r_s^2$ is the surface of the sun. What I don't understand is why the distance to the earth is important.
Thanks in advance!

Answer 1

The standard procedure (or at least how I think about it) for getting the temperature of a planet from that of the star consists of alternating between power and power per unit area.

• You start with $\sigma T^4$, the total power per unit area of the star.
• To get the total power, multiply by the area of the star.
• To get the power per unit area at the distance of the planet, you have to divide by the area of the sphere over which this energy is (by assumption evenly) distributed. This gets you to the quantity you have.
• Now you can find the total power absorbed by the planet, given the power per unit area it gets and an area¹.
• Then you can divide by an area¹ to get the power per unit area given off by the planet, which is a quantity you can set equal to $\sigma T_\text{planet}^4$ to find its temperature.

So there is a lot of multiplying and dividing by areas, and lots of factors of $\pi$ will cancel when you string it all together. Note one can modify these steps to take additional complexities into account (the $R$ and $A$ of your problem, for example).

Further, note that I used the planet's radius - twice in fact. And if you go through the arithmetic, you should find that it cancels itself. Intuitively, you might expect that an object's temperature (an average thermal energy) only depends on the strength of the heat source and its distance, but not on the object's size. Both you and your pet hamster² get to about the same temperature sitting at equal distances from the fireplace.

¹ Be careful about these two areas. You have to think about what area is appropriate where.

² I have no idea how I came up with this example.