I'm trying to understand this "toy model" of the greenhouse effect.
The model predicts the surface temperature of the Earth given the sunlight intensity and the emissivity of the atmosphere, $\lambda$. To do this, they use a two-energy model of light. There's shortwave radiation from the sun, and there's thermal radiation from terrestrial sources. They way they define $S$, the radiation intensity from the sun, allows for the atmosphere to reflect some shortwave light back into space, but the model doesn't allow for the atmosphere to absorb any shortwave light, meaning that all of $S$ makes it to the ground. Then the ground is treated as a blackbody. It logically follows that it emits radiation of intensity $G$ upwards, of which $\lambda G$ is absorbed by the atmosphere. Atmosphere emits $1/2 \lambda G$ back down to the surface because it is also a blackbody.
What I don't understand is the complexity they seem to invoke to write equations describing this. They use equations for surface, atmosphere, and planet.
$$S + \lambda A = G$$ $$\lambda G = 2 \lambda A$$ $$S = \lambda A + (1 - \lambda) G$$
These are 3 equations, but aren't there only 2 unknowns?
My question: how do you draw the necessary system boundaries to get closure for this system? Also, is there a requirement on the temperature of the atmosphere? I think the source argued that it need to be cold for it to have insulation properties, but I don't think that's reflected in the math, since we require (radiation in) = (radiation out). That would seem to imply the temperature of the atmosphere is arbitrary.