Skip to content

Atmosphere Theory

International Standard Atmosphere (ISA)

The International Standard Atmosphere (ISA) is a static atmospheric model of how the pressure, temperature, density, and viscosity of the Earth's atmosphere change over a wide range of altitudes or elevations.

Altitude Conversion

Atmospheric properties are calculated based on geopotential altitude \(H\), which accounts for the variation of gravity with height. The relationship between geopotential altitude \(H\) and geometric height \(Z\) is:

\[ H = \frac{R_e \cdot Z}{R_e + Z} \]

where \(R_e \approx 6,356,766\) m is the effective radius of the Earth for atmospheric calculations (ISO 2533:1975).

Implementation Note: Geometric vs. Geopotential Altitude

While the ISA model is formally defined in terms of geopotential altitude (\(H\)), the current lightaero implementation uses geometric altitude (\(Z\)) as the direct input for all atmospheric property calculations.

This simplification avoids the conversion step but introduces a small error that increases with altitude (approximately 0.16% at 10,000 m). For most preliminary research applications, this error is considered negligible.

Layers

The model covers two layers:

  1. Troposphere (0 to 11,000 m): Linear temperature lapse rate.
  2. Tropopause (11,000 to 20,000 m): Constant temperature (isothermal).

Governing Equations

Troposphere (\(h < 11,000\) m)

\[ T = T_0 - L h \]
\[ p = p_0 \left( \frac{T}{T_0} \right)^{\frac{g_0}{R L}} \]
\[ \rho = \frac{p}{R T} \]

Tropopause (\(h \ge 11,000\) m)

\[ T = T_{tp} \]
\[ p = p_{tp} \exp\left( -\frac{g_0 (h - H_{tp})}{R T_{tp}} \right) \]
\[ \rho = \frac{p}{R T_{tp}} \]

Dynamic Viscosity

The dynamic viscosity \(\mu\) is calculated using Sutherland's Law, which accounts for the effect of temperature on the viscosity of a gas:

\[ \mu = \mu_0 \frac{T_0 + S}{T + S} \left( \frac{T}{T_0} \right)^{3/2} \]

For air, the standard constants (ICAO Doc 7488) are:

  • \(\mu_0 = 1.716 \times 10^{-5}\) Pa·s
  • \(T_0 = 273.15\) K
  • \(S = 110.4\) K

Speed of Sound

The speed of sound \(a\) in the atmosphere is a function of the local air temperature:

\[ a = \sqrt{\gamma R T} \]

where \(\gamma = 1.4\) is the ratio of specific heats for air.

Constants

  • \(T_0 = 288.15\) K
  • \(p_0 = 101325.0\) Pa
  • \(R = 287.05287\) J/(kg·K)
  • \(g_0 = 9.80665\) m/s\(^2\)
  • \(L = 0.0065\) K/m

References

  • ISO 2533:1975, "Standard Atmosphere", International Organization for Standardization.
  • ICAO Doc 7488/3, "Manual of the ICAO Standard Atmosphere", 3rd Edition, 1993, International Civil Aviation Organization.