Earth's Temperature

Let's ask the question, how hot is the earth?

Of course, we know the answer. But answering will give us a hint about why.
We don't need to be very exact; we just need a rough estimate.

First, recall how we measure temperature.

Next, think about how thermal energy is transferred.

The basic idea:

How much is the energy in?

• Amount of energy received from Sun per square meter per second:
  342 watts/meter²
• Of this, 35% is reflected back into space. So 65% of 342 watts/(m² s) is absorbed:
  E_in = 220 watts/meter²

How much is the energy out?

Here is how we would calculate in a physics class. (But for our astronomy class, it is sufficient to understand the general idea. It is, however, important to understand the scientific reasoning because the issue of the temperature of the earth is pretty significant.)

• To be precise, we want the energy radiated per second per square meter from an ``ideal'' hot surface called a black body .
• It depends on the temperature T (measured in Kelvins).
• It involves a constant
  (sigma) = 5.7x10^-8 watts/(m² K⁴).

Just set these equal:

(sigma) T⁴ = 220 watts/meter²

so

T⁴ = 220 watts/meter²/(sigma)

or T⁴ = [220 /(5.7x10^-8)] K⁴ = 3.9 x 10⁹ K⁴

We just need the value of T such that T⁴ is 3.9 x 10⁹:

T = 250 K

That's pretty close, although 300 K would be closer.

Why are we warmer than the blackbody law would suggest?

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