Assume an ideal-gas atmosphere of constant mean molecular mass and approximately constant gravity. If temperature decreases linearly with altitude, write

d T / d z = - Γ

\frac{d l o g P}{d z} = - \frac{μ m _{H} g}{k _{B} T}, \frac{d l o g P}{d l o g T} = \frac{μ m _{H} g}{k _{B} Γ} .

The endpoint conditions determine the power law directly:

T (P) = T_{t} (\frac{P}{P _{t}})^{a} = T_{b} (\frac{P}{P _{b}})^{a}, a = \frac{lo g ( T _{b} / T _{t} )}{lo g ( P _{b} / P _{t} )} .

ii

Solution

Differentiate the result of part i and use hydrostatic equilibrium:

\frac{d T}{d z} = a T \frac{d l o g P}{d z} = - \frac{a μ m _{H} g}{k _{B}} .

Γ = - \frac{d T}{d z} = \frac{μ m _{H} g}{k _{B}} \frac{lo g ( T _{b} / T _{t} )}{lo g ( P _{b} / P _{t} )} .

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