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:

hgeometric = hgeopotential x Rearth / (Rearth - hgeopotential)

where Rearth 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)
Earth:
Rearth = 6356.766 km (≈ 3950 statute miles)
g0 = 9.80665 m/sec2 (≅ 32.1740 ft/sec2) (≅ gravitational acceleration at 45° latitude)
Gas Properties:
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, cp/cv)
Viscosity (empirical constants):

S = 110.4 K (Sutherland constant)
β = 1.458 x 10-6 kg/s·m·R1/2

Sea Level Conditions (by definition):

T0 = 15.0°C (= 59.0°F)
p0 = 101,325 N/m2 (= 760mm Hg, ≅ 2116.22 lbs/ft2)
Hence:
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)
Definition of Layers in Model:
 Layer Base Geopotential Altitude, hn (km) Base Geopotential Altitude, hn (ft) Lapse Rate, λn (K/km) Type 0 0 0 -6.5 1 11 36,089.2 0 Isothermal 2 20 65,616.8 +1.0 Inversion 3 32 104,986.9 +2.8 Inversion 4 47 154,199.5 0 Isothermal 5 51 167,322.8 -2.8 6 71 232,939.6 -2.0 7 84.8520 278,385.8 -

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

Tn = Tn-1 + (hn - hn-1n-1

or:

&thetan/&thetan-1 = 1 + (hn - hn-1n-1/Tn-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·T3/2 / (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 = Tn + (h-hnn

and

dT/dh = λn

Substituting:

∫ dp/p = -g/(λnR) ∫ dT/T

Integrating:

loge(p/pn) = loge(T/Tn)-g/λnR

or

p/pn = (1 + (h-hnn/Tn)-g/λnR

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

p/pn = e-(h-hn)g/RTn

These solutions lead to the following model equations:

h ≤ h1
θ = 1 + hλ0/T0
δ = θ (-g/λ0R)
σ= (1 + hλ0/T0) (-g/λ0R)-1
hence:
θ1 = 1 + h1λ0/T0
δ1 = θ1(-g/λ0R)
h1 < h ≤ h2 (Isothermal)
θ = θ1
δ = δ1 e-(h-h1)g/RT1
σ= (δ11) e-(h-h1)g/RT1
hence:
θ2 = θ1
δ2 = δ1 e-(h2-h1)g/RT1
h2 < h ≤ h3 (Inversion)
θ = θ2 + (h-h22/T0
δ = δ2 (θ/θ2)(-g/λ2R)
σ= δ22 + (h-h22/T0) (-g/λ2R)-1 (1/θ2)(-g/λ2R)
hence:
θ3 = θ2 + (h3-h22/T0
δ3 = δ232)(-g/λ2R)
h3 < h ≤ h4 (Inversion)
θ = θ3 + (h-h33/T0
δ = δ3 (θ/θ3)(-g/λ3R)
σ= δ33 + (h-h33/T0) (-g/λ3R)-1 (1/θ3)(-g/λ3R)
hence:
θ4 = θ3 + (h4-h33/T0
δ4 = δ343)(-g/λ3R)
h4 < h ≤ h5 (Isothermal)
θ = θ4
δ = δ4 e-(h-h4)g/RT4
σ= (δ44) e-(h-h4)g/RT4
hence:
θ5 = θ4
δ5 = δ4 e-(h5-h4)g/RT4
h5 < h ≤ h6
θ = θ5 + (h-h55/T0
δ = δ5 (θ/θ5)(-g/λ5R)
σ= δ55 + (h-h55/T0) (-g/λ5R)-1 (1/θ5)(-g/λ5R)
hence:
θ6 = θ5 + (h6-h55/T0
δ6 = δ565)(-g/λ5R)
h6 < h ≤ h7
θ = θ6 + (h-h66/T0
δ = δ6 (θ/θ6)(-g/λ6R)
σ= δ66 + (h-h66/T0) (-g/λ6R)-1 (1/θ6)(-g/λ6R)
hence:
θ7 = θ6 + (h7-h66/T0
δ7 = δ676)(-g/λ6R)

#### Numerical Solutions

(where h is in feet)

 Layer n θn δn σn = δn / θn 0 1 1 1 1 0.751865 2.23361E-01 2.97076E-01 2 0.751865 5.40330E-02 7.18652E-02 3 0.793510 8.56668E-03 1.07959E-02 4 0.939268 1.09456E-03 1.16533E-03 5 0.939268 6.60635E-04 7.03351E-04 6 0.744925 3.90468E-05 5.24172E-05 7 0.648780 3.68501E-06 5.67991E-06

h ≤ 36,089

θ = 1 - h / 145,442
δ = (1 - h / 145,442)5.255876
σ = (1 - h / 145,442)4.255876
36,089 < h ≤ 65,617 (Isothermal)
θ = 0.751865
δ = 0.223361 e-(h-36,089)/20,806
σ = 0.297076 e-(h-36,089)/20,806
65,617 < h ≤ 104,987 (Inversion)
θ = 0.682457 + h / 945,374
δ = (0.988626 + h / 652,600)-34.16320
σ = (0.978261 + h / 659,515)-35.16320
104,987< h ≤ 154,199 (Inversion)
θ = 0.482561 + h / 337,634
δ = (0.898309 + h / 181,373)-12.20114
σ = (0.857003 + h / 190,115)-13.20114
154,199 < h ≤ 167,323 (Isothermal)
θ = 0.939268
δ = 0.00109456 e-(h-154,199)/25,992
σ = 0.00116533 e-(h-154,199)/25,992
167,323 < h ≤ 232,940   (see ERRATA)
θ = 1.434843 - h / 337,634
δ = (0.838263 - h / 577,922)+12.20114
σ = (0.798990 - h / 606,330)+11.20114
232,940 < h ≤ 278,386   (see ERRATA)
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, hd, is given by:

σ ≥ σ1
hd = (σ-1/(g/λ0R+1) - 1) T00
σ 1 > σ ≥ σ2 (Isothermal)
hd = (-RT1/g) loge(σθ11) + h1
σ 2 > σ ≥ σ3 (Inversion)
hd = ( (σ/δ2) θ2(-g/λ2R) )-1/(g/λ2R+1) - θ2) T02 + h2
σ 3 > σ ≥ σ4 (Inversion)
hd = ( (σ/δ3) θ3(-g/λ3R) )-1/(g/λ3R+1) - θ3) T03 + h3
σ 4 > σ ≥ σ5 (Isothermal)
hd = (-RT4/g) loge(σθ44) + h4
σ 5 > σ ≥ σ6
hd = ( (σ/δ5) θ5(-g/λ5R) )-1/(g/λ5R+1) - θ5) T05 + h5
σ 6 > σ ≥ σ7
hd = ( (σ/δ6) θ6(-g/λ6R) )-1/(g/λ6R+1) - θ6) T06 + h6

ERRATA

1) The sign used in front of the h in the numerical solutions below should have been - not + for the altitude ranges:
167,323 < h ≤ 232,940
and
232,940 < h ≤ 278,386
These were corrected on 4/24/2009.
Many thanks to R. Wayne Wilson of Texas for pointing this out.
Fortunately these were documentation errors only; the Atmosculator code has always used the correct signs.

2) Under Gas Properties, the equation for the Specific Gas Constant in terms of the Universal Gas Constant should read:
R = R*/M and not R = R*·M
This was corrected on 3/26/2011.
Many thanks to Quentin Minster for pointing this out.
This was only a typo in the presentation of the formula, and did not result in an error in any numerical values.

Page last updated on March 26, 2011