The Standard Atmosphere

Introduction

The "Standard Atmosphere" is a hypothetical vertical distribution of atmospheric properties which, by international agreement, is roughly representative of year-round, mid-latitude conditions. Typical usages include altimeter calibrations and aircraft design and performance calculations. It should be recognized that actual conditions may vary considerably from this standard.

The most recent definition is the "US Standard Atmosphere, 1976" developed jointly by NOAA, NASA, and the USAF. It is an idealized, steady state representation of the earth's atmosphere from the surface to 1000 km, as it is assumed to exist during a period of moderate solar activity. The 1976 model is identical with the earlier 1962 standard up to 51 km, and with the International Civil Aviation Organization (ICAO) standard up to 32 km.

Up to 86 km, the model assumes a constant mean molecular weight, and comprises of a series of six layers, each defined by a linear temperature gradient (lapse rate). (The assumption of linearity conveniently avoids the need for numerical integration in the computation of properties.) The bottom layer, with a negative lapse rate, represents the earth's troposphere, a region where most clouds form, and with generally turbulent conditions. Higher layers form part of the earth's stratosphere, where winds may be high, but turbulence is generally low.

The model is derived by assuming a constant value for g (gravitational acceleration). Strictly speaking, altitudes in this model should therefore be referred to as "geopotential altitudes" rather than "geometric altitudes" (physical height above mean sea level). The relationship between these altitudes is given by:

h_geometric = h_geopotential x R_earth / (R_earth - h_geopotential)

where R_earth is the earth's effective radius. The difference is small, with geometric altitude and geopotential altitude differing from by less than 0.5% at 30 km (~100,000 ft).

The standard is defined in terms of the International System of Units (SI). The air is assumed to be dry and to obey the perfect gas law and hydrostatic equation, which, taken together, relate temperature, pressure and density with geopotential altitude.

It should also be noted that since the standard atmosphere model does not include humidity, and since water has a lower molecular weight than air, its presence produces a lower density. Under extreme circumstances, this can amount to as much as a 3% reduction, but typically is less than 1% and may be neglected.

Symbols

The following symbols are used to define the relationships between variables in the model. Subscript n indicates conditions at the base of the nth layer (or at the top of the (n-1)th layer) or refers to the constant lapse rate in the nth layer The first layer is considered to be layer 0, hence subscript 0 indicates standard, sea level conditions, or lapse rate in the bottom layer.

h = Pressure/Geopotential Altitude
T = Temperature
p = Pressure
ρ = Density
θ = T/T₀ (Temperature Ratio)
δ = p/p₀ (Pressure Ratio)
σ = ρ/ρ₀ (Density Ratio)
μ = Dynamic Viscosity
ν = μ/ρ = Kinematic Viscosity

Constants Used in Model

(Note: = indicates an exact value; ≅ indicates an approximate value correct to 6 significant figures)
Fundamental Constants:

Absolute Zero = -273.15°C (= -459.67°F)

R_earth = 6356.766 km (≈ 3950 statute miles)
g₀ = 9.80665 m/sec² (≅ 32.1740 ft/sec²) (≅ gravitational acceleration at 45° latitude)

R^* = 8.31432 J/mole·K (universal gas constant)
M = 0.0289644 kg/mole (mean molecular mass of air)
R = R^*/M ≅ 287.053 J/kg·K (≅ 1716.56 ft²/sec²·R) (gas constant for air)
γ= 1.40 (ratio of specific heat capacities of air, c_p/c_v)

S = 110.4 K (Sutherland constant)
β = 1.458 x 10^-6 kg/s·m·R^1/2

T₀ = 15.0°C (= 59.0°F)
p₀ = 101,325 N/m² (= 760mm Hg, ≅ 2116.22 lbs/ft²)

ρ₀ ≅ 1.22500 kg/m³ (≅ 0.00237689 slugs/ft³)
μ₀ ≅ 1.78938 x 10^-5 kg/m·sec (≅ 3.73720 x 10^-7 slugs/ft·sec)
ν₀ ≅ 1.46072 x 10^-5 m²/sec (≅ 1.57231 x 10^-4 ft²/sec)

A positive lapse rate (λ > 0) means temperature increases with height. The temperature at the base of layer n is given by:

T_n = T_n-1 + (h_n - h_n-1)λ_n-1

or:

&theta_n/&theta_n-1 = 1 + (h_n - h_n-1)λ_n-1/T_n-1

Physical Laws Used in Model

Ideal gas law:

p = ρRT

and hence: σ = δ / θ
Hydrostatic equilibrium:

dp/dh = -gρ

In this model, viscosity is a function of temperature only, given by the following empirical relationship, valid at all altitudes:

μ = β·T^3/2 / (T+S)

Derivation of Model Equations

Combining the gas law and hydrostatic equations:

dp/p = (-g/R) dh/T

For the case of a constant lapse rate, λ

T = T_n + (h-h_n)λ_n

and

dT/dh = λ_n

Substituting:

∫ dp/p = -g/(λ_nR) ∫ dT/T

Integrating:

log_e(p/p_n) = log_e(T/T_n)^-g/λ_nR

or

p/p_n = (1 + (h-h_n)λ_n/T_n)^-g/λ_nR

For isothermal layers (λ=0), and using the approximation log_e(1+ε)→ε as ε→0, this becomes:

p/p_n = e^{-(h-h_n)g/RT_n}

These solutions lead to the following model equations:

θ = 1 + hλ₀/T₀
δ = θ ^(-g/λ₀R)
σ= (1 + hλ₀/T₀) ^{(-g/λ₀R)-1}
hence:

θ₁ = 1 + h₁λ₀/T₀
δ₁ = θ₁^(-g/λ₀R)

θ = θ₁
δ = δ₁ e^{-(h-h₁)g/RT₁}
σ= (δ₁/θ₁) e^{-(h-h₁)g/RT₁}
hence:

θ₂ = θ₁
δ₂ = δ₁ e^{-(h₂-h₁)g/RT₁}

θ = θ₂ + (h-h₂)λ₂/T₀
δ = δ₂ (θ/θ₂)^(-g/λ₂R)
σ= δ₂ (θ₂ + (h-h₂)λ₂/T₀) ^{(-g/λ₂R)-1} (1/θ₂)^(-g/λ₂R)
hence:

θ₃ = θ₂ + (h₃-h₂)λ₂/T₀
δ₃ = δ₂ (θ₃/θ₂)^(-g/λ₂R)

θ = θ₃ + (h-h₃)λ₃/T₀
δ = δ₃ (θ/θ₃)^(-g/λ₃R)
σ= δ₃ (θ₃ + (h-h₃)λ₃/T₀) ^{(-g/λ₃R)-1} (1/θ₃)^(-g/λ₃R)
hence:

θ₄ = θ₃ + (h₄-h₃)λ₃/T₀
δ₄ = δ₃ (θ₄/θ₃)^(-g/λ₃R)

θ = θ₄
δ = δ₄ e^{-(h-h₄)g/RT₄}
σ= (δ₄/θ₄) e^{-(h-h₄)g/RT₄}
hence:

θ₅ = θ₄
δ₅ = δ₄ e^{-(h₅-h₄)g/RT₄}

θ = θ₅ + (h-h₅)λ₅/T₀
δ = δ₅ (θ/θ₅)^(-g/λ₅R)
σ= δ₅ (θ₅ + (h-h₅)λ₅/T₀) ^{(-g/λ₅R)-1} (1/θ₅)^(-g/λ₅R)
hence:

θ₆ = θ₅ + (h₆-h₅)λ₅/T₀
δ₆ = δ₅ (θ₆/θ₅)^(-g/λ₅R)

θ = θ₆ + (h-h₆)λ₆/T₀
δ = δ₆ (θ/θ₆)^(-g/λ₆R)
σ= δ₆ (θ₆ + (h-h₆)λ₆/T₀) ^{(-g/λ₆R)-1} (1/θ₆)^(-g/λ₆R)
hence:

θ₇ = θ₆ + (h₇-h₆)λ₆/T₀
δ₇ = δ₆ (θ₇/θ₆)^(-g/λ₆R)

Numerical Solutions

(where h is in feet)

h ≤ 36,089

θ = 1 - h / 145,442
δ = (1 - h / 145,442)^5.255876
σ = (1 - h / 145,442)^4.255876

θ = 0.751865
δ = 0.223361 e^{-(h-36,089)/20,806}
σ = 0.297076 e^{-(h-36,089)/20,806}

θ = 0.682457 + h / 945,374
δ = (0.988626 + h / 652,600)^-34.16320
σ = (0.978261 + h / 659,515)^-35.16320

θ = 0.482561 + h / 337,634
δ = (0.898309 + h / 181,373)^-12.20114
σ = (0.857003 + h / 190,115)^-13.20114

θ = 0.939268
δ = 0.00109456 e^{-(h-154,199)/25,992}
σ = 0.00116533 e^{-(h-154,199)/25,992}

θ = 1.434843 - h / 337,634
δ = (0.838263 - h / 577,922)^+12.20114
σ = (0.798990 - h / 606,330)^+11.20114

θ = 1.237723 - h / 472,687
δ = (0.917131 - h / 637,919)^+17.08160
σ = (0.900194 - h / 649,922)^+16.08160

Non-Standard Conditions

When actual conditions vary from those given by the standard atmosphere model, the expressions given above, do not apply. Two definitions need to be introduced at this point:

• Pressure altitude: the altitude in a standard atmosphere corresponding to a given pressure. (Since an altimeter is a barometer calibrated according to the standard atmosphere model, it reads pressure altitude when set to standard sea level conditions of 29.92" Hg). Note: pressure altitude is independent of temperature.
  
• Density altitude: pressure altitude corrected for actual temperature, or the altitude in a standard atmosphere corresponding to a given density. (Since density altitude represents air density, it is more relevant to aircraft performance than pressure altitude).

The pressure altitude is calculated by inverting the appropriate expression for pressure ratio, δ. Density altitude is calculated by inverting the appropriate expression for σ as a function of geopotential, h. Thus, for non-standard temperatures, the density altitude, h_d, is given by:

h_d = (σ^{-1/(g/λ₀R+1)} - 1) T₀/λ₀

h_d = (-RT₁/g) log_e(σθ₁/δ₁) + h₁

h_d = ( (σ/δ₂) θ₂^(-g/λ₂R) )^{-1/(g/λ₂R+1)} - θ₂) T₀/λ₂ + h₂

h_d = ( (σ/δ₃) θ₃^(-g/λ₃R) )^{-1/(g/λ₃R+1)} - θ₃) T₀/λ₃ + h₃

h_d = (-RT₄/g) log_e(σθ₄/δ₄) + h₄

h_d = ( (σ/δ₅) θ₅^(-g/λ₅R) )^{-1/(g/λ₅R+1)} - θ₅) T₀/λ₅ + h₅

h_d = ( (σ/δ₆) θ₆^(-g/λ₆R) )^{-1/(g/λ₆R+1)} - θ₆) T₀/λ₆ + h₆

Page last updated on March 26, 2011

Copyright © 1991-2011, Graham Gyatt. All Rights Reserved.