Single lapse rate

The simplest example of a nonisothermal atmosphere is one with a single lapse rate. 

This is also physically realistic for planetary atmospheres over moderate ranges of altitude. Substituting into Eq. (4.2.3), replacing z with r as the independent variable, and integrating, yields 

To calculate the outgoing intensity, we apply the multilayer model of Subsection 2.4.a [see Eq. (2.4.4)]. The relation between optical and geometric thicknesses of the individual layers is given by [cf. Eq. (4.1.1)] 

Each layer is assumed to be individually isothermal hence, from the kinetic theory 

There are some differences, however. In particular, lines formed in an atmosphere with a temperature gradient do not exhibit the symmetry in absorption and emission shown in homogeneous isothermal media. According to Eqs. (4.2.5) 

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