# 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 ($$h_{11} = 11000 [m]$$):

$\dfrac{p}{p_{11}} = \dfrac{\rho}{\rho_{11}} = e^{-\tfrac{g}{RT_{11}} (h - h_{11})}$ Figure 2.17: ISA atmosphere. © Cmglee / Wikimedia Commons / CC-BY-SA-3.0.

Introducing now the numerical values, it yields:

$T[k] = 216.65;$

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

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

2.3.3: ISA equations is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by Manuel Soler Arnedo via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.