International Standard Atmosphere

The Layers of the Atmosphere

The first two major layers of the atmosphere are the troposphere and the stratosphere. These two layers, along with the temperature variations across them are depicted below. The temperature drops in the troposphere at a constant rate, by increasing the altitude, and it reaches a constant value at the tropopause and remains constant in the stratosphere.

To learn more about the different layers in the atmosphere, visit this webpage from NASA.

Temperature Variation in the Standard Atmosphere

Temperature decreases with increasing altitude. The rate is \frac{-6.5^\circ C}{1000 m}, or \frac{-1.98^\circ C}{1000 ft}, up to the tropopause. From the tropopause upward, temperature remains at a constant value of -56.5^\circ C. Note that, the standard temperature at mean sea level is 15.15^\circ C = 288.15 K. Thus, the standard temperature at a given altitude is

(1)   \begin{equation*}T_{ISA} = 15.15^\circ C - 1.98^\circ C \left( \frac{h}{1000 ft} \right)\end{equation*}


where h is the altitude in feet.

Pressure Variation in the Standard Atmosphere

Pressure at sea level is P_0 = 1013.25 hPa, where hPa is hectoPascal and 1hPa = 100Pa. Pressure above the sea level and below the tropopause is obtained using the following equation:

(2)   \begin{equation*}P = P_0 \left( 1 - \frac{\alpha}{T_0} h \right)^{\frac{g_0}{\alpha R}}\end{equation*}


where P_0=1013.25hPa and T_0=288.15K are the standard pressure and temperature at mean sea level, respectively, \alpha=0.0065^\circ C/m is the temperature drop ratio, g_0=9.80665 m/s^2 is the standard gravity acceleration, R=287.053 J/kgK is the gas constants, and h is the altitude in meters.

Furthermore, the pressure above the tropopause is obtained via

(3)   \begin{equation*}P = P_1 e^{\frac{-g_0 \left( h - h_1 \right)}{RT_1}}\end{equation*}


where P_1=226.32hPa and T_1=216.5K are the standard pressure and temperature at the tropopause, and h=11000m.

Density Variation in the Standard Atmosphere

Density is always

(4)   \begin{equation*}\rho = \frac{P}{RT}\end{equation*}


where P is in Pascal, T is in Kelvin, and R=287.053 J/kgK.

Temperature, Pressure, and Density Ratios

The temperature, pressure, and density ratios are defind as

(5)   \begin{equation*}\theta = \frac{T}{T_0} ^\star, \end{equation*}


(6)   \begin{equation*}\delta = \frac{P}{P_0}, \end{equation*}


(7)   \begin{equation*}\sigma = \frac{\rho}{\rho_0} = \frac{\delta}{\theta}, \end{equation*}


respectively.
\star Note that to calculate the temperature ratio, temperatures must be in Kelvin or Rankin.

Temperature, Pressure, and Density Ratios Variations

Below the tropopause

(8)   \begin{equation*}\theta_{ISA} = 1 - 6.87535 \times 10^{-6} h,\end{equation*}


(9)   \begin{equation*}\delta = \theta_{ISA}^{5.2559},\end{equation*}


(10)   \begin{equation*}\sigma = \frac{\delta}{\theta},\end{equation*}


Above the tropopause

(11)   \begin{equation*}\theta_{ISA} = 0.7519,\end{equation*}


(12)   \begin{equation*}\delta = 0.22336 e^{-\frac{h-36089}{20806}},\end{equation*}


(13)   \begin{equation*}\sigma = <meta charset="utf-8">\frac{\delta}{\theta},\end{equation*}


where h is in feet.

7-Minutes Lecture