To understand why it's hotter inside a car than its surrounding when exposed to sunlight, let's try a simplificate model that is equivalent to what happens with the atomosphere. To start, let's suppose that the Earth's surface is flat and that it behaves like a black body for the sunlight (forget about the atmosphere for a moment). There will be a radiation flow over the surface of the Earth coming from the Sun (1350W/m²) that will produce a heating that will eventually reach an equilibrium temperature given by Stefan-Boltzmann's law. Let's now introduce in our model the atmosphere, under the form of a glass a few meters over the Earth's surface that let pass all the radiation from the Sun but that behaves like a black body for the Earth's radiation. Determine the equilibrium temperature of this situation.
[itex]P/A=\varepsilon \sigma T^4[/itex].
The Attempt at a Solution
I calculated the temperature without considering the atmosphere with the given equation. This gave me about 392.81K which seems somehow big to me but possible after all.
Now I consider the atmosphere. I've made a sketch but I get confused on how to solve the problem. Basically I consider a "ray" of sunlight passing through the atmosphere, reaching the Earth. The Earth will then emit as if it was a body at 392.81 K. The atmosphere will aborb this radiation and re-emit it since it behaves as a black body. I can continue this process ad infinitum and don't reach anything.
If I remember well, the solution contained infinite series. Can someone help me a bit?