This is a presentation and analysis of a minimalist model of radiative heating that incorporates the elements of the greenhouse effect. Consider a blackbody plane under a layer of atmosphere. The atmosphere is partially transparent to the incipient shortwave radiation but absorbs a portion of the longwave radiation generated by the heated blackbody plane. The atmospheric layer radiates as a gray body to the surface and to outerspace.

Let T_s and T_a be the absolute temperatures of the surface and the atmosphere, respectively. Let ε be the emissivity and absorbity of the atmosphere. Let α be the proportion of incipient shortwave absorbed by the atmosphere.

If F is the intensity of the incoming shortwave radiation, then the energy per unit area reaching the surface is (1-α)F. If T_s is the surface temperatre the emission from the surface is σT_s⁴. The surface also receives from the atmosphere energy per unit area of εσT⁴

The atmosphere receives energy from the incoming shortwave radiation of αF and absorbs from the radiating surface energy of εσT_s⁴. It radiates to the surface and to outerspace energy equal to, in total, 2εσT_s⁴.

At equilibrium the net energy flow both to the surface and the atmosphere must be zero so the equations to be satisfied are:

Atmosphere: αF - 2εσT_a⁴ + εσT_s⁴ = 0
 
Surface: (1-α)F + εσT_a⁴ − σT_s⁴ = 0
 

This is a system of two equations in two unknowns, T_s⁴ and T_a⁴. Written in matric form this system is:

Radiative Balance Equation

Let the matrix on the left above be represented as M, the column vector of the fourth powers of temperatures as S and the column vector on the right as A. The system is then MS=A(F/σ) Then the solution for the column vector of the fourth powers of temperatures S is equal to M^-1A(F/σ).

The determinant of M is 2ε−ε², which is (2−ε)ε. The inverse of M is then ) ">

Therefore the solution is

The Solution

(To be continued.)

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