Radiative properties of an alltroposphere dry atmosphere

Let's consider an atmosphere for which the convection is so deep that it establishes a dry adiabat throughout the depth of the atmosphere. Thus, T(p) = Ts(p/ps)R/Cp all the way to p = 0. We wish to compute the OLR for this atmosphere, which is done by substituting this T(p) into the grey-gas form of Eq. 4.9 and evaluating the integral for 1+ at t = t0, i.e. the top of the atmosphere. Since the temperature is expressed as a function of pressure, it is necessary to substitute for pressure in terms of optical thickness in order to carry out the integral. We'll suppose that k is a constant, so that t0 — t = Kp/g = T0p/ps. Using this to eliminate pressure from T(p), the integral for OLR becomes r T~ t _ t ' ,

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