The Arrhenius model of global warming constitutes a very large composite system. It assumes that a layer of gases in the atmosphere (A) absorbs infrared radiation from the Earth's surface and re-emits it, with equal amounts going off into space or back to the Earth's surface; 240 W/m² of the solar energy reaching the Earth's surface is reflected back as infrared radiation (S). The “emissivity value” (λ) of the ground (G) is set equal to that of A. λ characterizes the fraction of radiation that is not absorbed. For example, (1–λ) would be the fraction of IR radiation that is absorbed by the ground, and the surface energy would be S = Aλ + G(1–λ). A similar balance for the atmosphere gives, λG = 2λA. This balance indicates that radiation received from G is balanced by that radiated from A. The factor of 2 on the right-hand side appears because radiation can be toward the ground or toward space.

The equations for radiation are given by the relations: G = σT_G⁴ and A = σT_A⁴ where σ = Stefan-Boltzmann constant 5.6704 x 10^-8 (W/m²-K⁴).

(a) Noting that the average T_G is 300 K, solve for λ.

(b) Solve for T_G when λ = 0. This corresponds to zero global warming.

(c) Solve for T_G when λ = 1. This corresponds to perfect global warming.

(d) List at least three oversimplifications in the assumptions of this model. Discuss whether these lead to underestimating or overestimating T_G.