Thermal radiation and the temperature profile

The Sun and energy conversion in the interior provide the sources of heating for the atmosphere, and the distribution of opacity sources determines the local heating rates. Thermal equilibrium is achieved when the temperature is adjusted so that the heating and cooling rates are equal in magnitude at every level. 

The equation of transfer governing the infrared radiation field is found from Eq. (2.5.10) to be 

Multiplying both sides by d/x and integrating over the interval (—1, -1-1) yields 

Because the atmosphere is in a state of radiative equilibrium, the rate of cooling must equal the rate of heating at every level in the atmosphere. This requirement is 

This condition can be combined with Eq. (9.1.6) to write the inhomogeneous term in Eq. (9.1.19) as a function of the independent variable r , and the complete equation, with the help of Eq. (9.1.15), becomes 

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