2.3.3: ISA equations

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Considering Equation (2.3.1.1), Equations (2.3.1.2)-(2.3.1.3), and Equation (2.3.2.2), the variations of \(\rho\) and \(p\) within altitude can be obtained for the different layers of the atmosphere that affect atmospheric flight:

Troposphere (\(0 \le h < 11000 [m]\)): Introducing Equation (2.3.1.1) and Equation (2.3.1.2) in Equation (2.3.2.2), it yields:

\[\dfrac{d p}{dh} = -\dfrac{p}{R (T_0 - \alpha h)} g.\]

Integrating between a generic value of altitude \(h\) and the altitude at sea level (\(h = 0\)), the variation of pressure with altitude yields:

\[\dfrac{p}{p_0} = (1 - \dfrac{\alpha}{T_0} h)^{\tfrac{g}{R\alpha}}.\label{eq2.3.3.2}\]

With the value of pressure given by Equation (\(\ref{eq2.3.3.2}\)), and entering in the equation of perfect gas (2.3.1.1), the variation of density with altitude yields:

\[\dfrac{\rho}{\rho_0} = (1 - \dfrac{\alpha}{T_0} h)^{\tfrac{g}{R\alpha} - 1}.\]

Introducing now the numerical values, it yields:

\[T[k] = 288.15 - 0.0065 h [m];\]

\[\rho [kg/m^3] = 1.225 (1 - 22.558 \times 10^{-6} \times h [m])^{4.2559};\]

\[p [P a] = 101325 (1 - 22.558 \times 10^{-6} \times h[m])^{5.2559}\]

Tropopause and Inferior part of the stratosphere (\(11000 [m] \le h < 20000 [m]\)): Introducing Equation (2.3.1.1) and Equation (2.3.1.3) in Equation (2.3.2.2), and integrating between a generic altitude (\(h > 11000 [m]\)) and the altitude at the tropopause (\(h_{11} = 11000 [m]\)):

\[\dfrac{p}{p_{11}} = \dfrac{\rho}{\rho_{11}} = e^{-\tfrac{g}{RT_{11}} (h - h_{11})}\]

Introducing now the numerical values, it yields:

\[T[k] = 216.65;\]

\[\rho [kg/m^3] = 0.3639e^{-157.69 \cdot 10^{-6} (h[m] - 11000)};\]

\[p [P a] = 22632 e^{-157.69 \cdot 10^{-6} (h[m] - 11000)}.\]