Symbols & Variables

Complete reference of symbols, physical constants, and variables used in ISO 2533 (Standard Atmosphere) and ISO 5878 (Reference Atmospheres for Aerospace Use).

ISO 2533

Standard Atmosphere

A deterministic model defining temperature, pressure, density, and derived properties from −5 km to 80 km altitude.

Atmosphere Layers

The Standard Atmosphere divides the atmosphere into layers, each with a defined temperature gradient $β_{s}$ . The profile below shows how temperature varies with altitude.

Layer	Altitude Range	$T$ (K)	$β_{s}$
Below Sea Level	-2 km – 0	301.15	`-6.5 × 10−3`
Lower Troposphere	0 – 11 km	288.15	`-6.5 × 10−3`
Troposphere	11 km – 20 km	216.65	Isothermal
Tropopause	20 km – 32 km	216.65	`+1.0 × 10−3`
Lower Stratosphere	32 km – 47 km	228.65	`+2.8 × 10−3`
Upper Stratosphere	47 km – 51 km	270.65	Isothermal
Stratopause	51 km – 71 km	270.65	`-2.8 × 10−3`
Mesosphere	71 km – 80 km	214.65	`-2.0 × 10−3`

Fundamental Constants

Fixed physical and conventional values that define the Standard Atmosphere. All ISA calculations derive from these constants.

g_{n}

Standard Gravity9.80665 m/s²Standard gravitational acceleration at mean sea level

N_{A}

Avogadro Constant6.0226e+26 kmol⁻¹Number of particles per kilomole of substance

p_{n}

Standard Pressure101,325 PaAtmospheric pressure at mean sea level (1 atm)

T_{n}

Standard Temperature288.15 KTemperature at mean sea level (15 °C)

ρ_{n}

Standard Density1.225 kg/m³Air density at mean sea level

R^{*}

Universal Gas Constant8.31432 J/(kmol·K)Universal molar gas constant

r_{0}

Effective Earth Radius6,356,766 mNominal Earth radius for altitude conversion

κ

Heat Capacity Ratio1.4 —Ratio of specific heats cₚ/cᵥ for dry air

Derived Constants

M

Molar Mass of Air0.029 kg/kmolDerived: M = ρₙR*Tₙ/pₙ

R

Specific Gas Constant287.05 J/(kg·K)Derived: R = R*/M

Computed Properties

All atmospheric properties calculable by the model, organized by category. Each property is available at any altitude within the model range (−5 km to 80 km).

Altitude

H

Geopotential Altitudem, ft

h

Geometric Altitudem, ft

Temperature

T

TemperatureK, °C, °F, °R

θ

Temperature Ratio—T / Tₙ

β_{s}

Temperature GradientK/mLapse rate

Pressure

p

PressurePa, mbar, mmHg

δ

Pressure Ratio—p/pₙ

Density

ρ

Densitykg/m³

σ

Density Ratio—ρ/ρₙ

\sqrt{σ}

√ Density Ratio—

Motion & Viscosity

g

Gravitym/s²

a

Speed of Soundm/s

μ

Dynamic ViscosityPa·s

ν

Kinematic Viscositym²/s

Other Properties

λ

Thermal ConductivityW/(m·K)

H_{p}

Pressure Scale Heightm

γ

Specific WeightN/m³

n

Air Number Densitym⁻³

\bar{v}

Mean Particle Speedm/s

ω

Collision Frequencys⁻¹

l

Mean Free Pathm

V_{m}

Mole Volumem³/mol

T_{M}

Molecular TemperatureK

Key Equations

The fundamental relationships used in the Standard Atmosphere. For complete derivation details, see the ISO 2533 page.

Geopotential ↔ Geometric Altitude

Geopotential altitude $H$ accounts for gravity variation with height, while geometric altitude $h$ is the physical distance from sea level. At sea level they are equal; at high altitudes, geometric exceeds geopotential.

\begin{matrix} H = \frac{r_{0} \cdot h}{r_{0} + h} & geometric → geopotential \\ h = \frac{r_{0} \cdot H}{r_{0} - H} & geopotential → geometric \end{matrix}

Temperature

In gradient layers, temperature varies linearly; in isothermal layers it remains constant.

\begin{matrix} T (H) = T_{b} + β_{s} \cdot (H - H_{b}) & gradient layer \\ T (H) = T_{b} & isothermal layer \end{matrix}

Ideal Gas Law

Density is derived from pressure and temperature via the ideal gas equation:

ρ = \frac{p}{R \cdot T}

ISO 5878

Reference Atmospheres for Aerospace Use

Observed atmospheric conditions by latitude zone and season — temperature profiles, wind distributions (Rice model), and humidity data for real-world aerospace applications.

Wind Distribution

ISO 5878 models wind speed distributions using the circular normal (Rice) distribution. Wind is characterized by its vector components and the statistical spread around the mean.

Wind Vector Components

3Observed

V_{x}

Mean Zonal Windm/sEast–west component (positive eastward)

V_{y}

Mean Meridional Windm/sNorth–south component (positive northward)

σ_{r}

Vector Mean Std Deviationm/sStandard deviation of the vector mean wind

Derived Wind Quantities

4Calculated

V_{r}

Vector Mean Wind Magnitudem/sVr = √(Vx² + Vy²)

σ

Per-Component Std Deviationm/sσ = σr / √2

V_{sc}

Scalar Mean Wind Speedm/sExpected value from Rice distribution

θ

Mean Wind Directionradarctan(Vy / Vx)

Wind Distribution

4Calculated

f (ν)

Probability Density Functions/mRice (circular normal) distribution

ν

Wind Speed Variablem/sRandom variable in the Rice PDF

I_{0}

Modified Bessel Function—Zero-order, first kind

V_{p}

Percentile Wind Speedm/sp% = {1, 10, 20, 80, 90, 99}

Key Equations

Vector mean wind magnitude

V_{r} = \sqrt{V_{x}^{2} + V_{y}^{2}}

Rice probability density function

f (ν) = \frac{2 ν}{σ^{r}} \exp (\frac{- (ν^{2} + V^{r})}{σ^{r}}) \times I_{0} (\frac{2 ν V_{r}}{σ^{r}})

Simplification for zones above 20°N

When $V_{y}$ does not exceed 6% of $V_{x}$ , it is assumed that $V_{y} = 0$ , so:

V_{r} = | V_{x} |

Humidity

ISO 5878 provides humidity profiles for each latitude zone and seasonal model. The humidity mixing ratio is the primary characteristic because it remains constant during vertical or horizontal air movements unless condensation or evaporation occurs.

Humidity Measures

5Observed + Calculated

r

Humidity Mixing Ratiog/kgPrimary characteristic; mv / ma

e^{'}

Vapour PressurehPaPartial pressure of water vapour

e^{'}_{w}

Saturation Vapour PressurehPaEquilibrium vapour pressure

t_{d}

Dew-Point Temperature°CTemperature for saturation at constant pressure

U

Relative Humidity%U = 100 × e′ / e′w

Auxiliary Variables

m_{v}

Mass of Water Vapourkg

m_{a}

Mass of Dry Airkg

a

Saturation CoefficientK7.5 K (t ≥ 0°C) or 9.5 K (t < 0°C)

b

Saturation CoefficientK237.3 K (t ≥ 0°C) or 265.5 K (t < 0°C)

Key Equations

Humidity mixing ratio

r = \frac{m_{v}}{m_{a}}

Vapour pressure from mixing ratio

e^{'} = \frac{r}{621.98 + r} \times p

Saturation vapour pressure

For −20°C < t < 30°C:

e^{'}_{w} = 6.107 \times 10^{(a \cdot t) / (b + t)}

Relative humidity

U = 100 \times \frac{e^{'}}{e^{'}_{w}}

Reference Atmosphere Parameters

Unlike ISO 2533's single global model, ISO 5878 provides separate atmospheric profiles for five latitude zones, each with seasonal (January/July) variations. The following parameters define the surface conditions for each zone.

Reference Atmosphere Parameters

6Calculated + Observed

φ

Latitude°Geographic latitude of the zone

g_{0} (φ)

Sea-Level Gravitym/s²Gravity at sea level, varies by latitude

r_{φ}

Nominal Earth RadiusmEffective Earth radius at latitude φ

T (φ, s)

Temperature ProfileKBy latitude zone φ and season s

p (φ, s)

Pressure ProfilePaBy latitude zone φ and season s

ρ (φ, s)

Density Profilekg/m³By latitude zone φ and season s

Latitude Zones & Seasons

15°

Tropical Zone—Annual model (no seasonal distinction)

30°N

Subtropical Zone—January & July seasonal models

45°N

Mid-Latitude Zone—January & July seasonal models

60°N

Subarctic Zone—January & July seasonal models

80°N

Arctic Zone—January & July seasonal models

Stratospheric Regimes

2Observed

T_{cold}

Cold Regime TemperatureK~223 K (60°N) or ~232 K (80°N) at 45 km

T_{warm}

Warm Regime TemperatureK~267 K at 45 km (60°N and 80°N)

Surface Conditions by Latitude Zone

Calculated $g_{0}$ and $r_{φ}$ from geophysical formulae. Observed Temperature and pressure from meteorological station data.

Zone	Name	$g_{0}$ (m/s²)	$r_{φ}$ (m)	$T$ Dec/Jan (K)	$T$ Jun/Jul (K)
15°	Tropical	9.78381	6,337.84	299.650	299.650
30°N	Subtropical	9.79324	6,345.65	283.150	297.150
45°N	Mid-latitude	9.80665	6,356.77	272.650	291.150
60°N	Subarctic	9.81911	6,367.10	256.150	282.150
80°N	Arctic	9.83051	6,376.56	248.950	276.650

Wind Data Latitude Bands

Wind characteristic data is organized into four latitude bands (distinct from the five zones used for temperature and pressure profiles).

Band	Name	Coverage
0°–20°N	Tropical	Low latitudes
20°–40°N	Subtropical	Mid latitudes
40°–60°N	Temperate	Higher mid latitudes
60°–80°N	Subarctic/Arctic	High latitudes

Next Steps

Use these symbols in the interactive calculator, explore the full standard documentation, or integrate the library into your project.

Open Calculator ISO 2533 ISO 5878 Library Docs