Standard Atmosphere

Determine the atmospheric conditions at an altitude based on the 1976 US Standard Atmosphere Model [1].

Disclaimer

This calculator is for educational and curiosity purposes only. No gaurantee on accuracy is inferred by a result.

Select Desired System of Units

Geometric Altitude, h_G (m)

You must enter a value for altitude that is positive and less than 86000 m (282152 ft).

\(h_G \; (m)\)

\(h \; (m)\)

\(g \; (m/s^2)\)

\(T \; (K)\)

\(p \; (N/m^2)\)

\(\rho \; (kg/m^3)\)

\(\mu \; (kg/m \cdot s)\)

\(a \; (m/s)\)

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Nomenclature

\(a\)	Speed of sound
\(g\)	Local gravitational acceleration
\(h\)	Geopotential altitude
\(h_G\)	Geometric altitude
\(p\)	Absolute pressure
\(T\)	Temperature
\(\mu\)	Dynamic viscosity
\(\rho\)	Density

Standard Atmosphere Calculator Notes

Although the 1976 US Standard Atmosphere Model provides properties up to a geopotential altitude of 1,000 km, this calculator only provides property values up to a 84.852 km (h_G = 86 km). This is the altitude at which the model begins to account for the species fraction of air. At this altitude the variation in temperature as a function of altitude ceases to behave linearly.

Units, Constants and Conversion Factors

The model is defined in terms of the International System (SI) of Units. US units are also offered in this calculator. To provide US units, the user-provided geometric altitude is converted to meters, and all computed properties are calculated in SI units. These computed properties are then converted back to US units. This method was chosen as to eliminate the need to convert the constants required in the computations. Table 1, below, displays the conversion factors used. Only the basic dimension conversion factors were used to convert between SI and US units, with the exception of force. Table 2 provides the constants used in the calculations. These constants are consistent with those listed in the model.

Table 1. List of SI and US standard units along with the conversion factors. The unit for force is a derived unit, all others are basic dimensions.

Quantity	SI Unit, Symbol	US Unit, Symbol	Conversion Factor
Mass	Kilogram, \(kg\)	Slug, \(slug\)	\[kg \left( \frac{1}{14.5939} \right)\]
Length	Meter, \(m\)	Foot, \(ft\)	\[m \left( \frac{1}{0.3048} \right) \]
Time	Second, \(s\)	Second, \(s\)	-
Temperature	Kelvin, \(K\)	Degree Rankine, \(^{\circ}R\)	\[K \times 1.8\]
Force	Newton, \(N\)	Pound, \(lb\)	\[N \times 0.224809\]

Table 2. Constants used in property calculations. These are consistent with those used in Ref. 1.

Constant, Symbol	Value	Unit
Molecular weight of air, \(M\)	28.9644	\[\frac{kg}{kmol}\]
Universal gas constant, \(R^*\)	8.31432 × 10³	\[\frac{N·m}{kmol·K}\]
Ratio of specific heats, \(\gamma\)	1.4	-
Radius of earth, \(r_e\)	6356766	\[m\]
Sea-level gravity, \(g_o\)	9.80665	\[\frac{m}{s^2}\]

Calculation Details

The reader is encouraged to review Ref. 1 for the details of the model. But, a brief overview of the process and equations used to determine the properties is provided here.

The primary quantities determined by the model for the properties provided by this tool are pressure and temperature. Both are a function of geopotential altitude, \(h\). Geopotential altitude can be considered a "gravity-adjusted height" and can be determined by Eq. 1. Temperature is determined by the Lapse rate, \(L\) of the atmosphere. In the region of the atmosphere that is capable of being calculated in this tool, the lapse rate defines six different regions of the atmosphere. The equation to determine temperature is provided in Eq. 2.

\[h = h_G \left(\frac{r_e}{r_e + h_G} \right)\]

(1)

\(T = T_b + L(h - h_b)\)

(2)

In the above equation, \(T_b\) is the temperature at the base, or beginning, of the particular region of the atomosphere in which one is calculating properties for. Likewise, \(h_b\) is the geopotential altitude at which the region begins. All base values for each region are provided in Table 3. Pressure is derived from the hydrostatic pressure equation and becomes a function of the geopotential altitude. Two forms exist. The first, shown in Eq. 3, models the behavior of pressure in regions of the atmosphere where the lapse rate is non-zero. Eq. 4 models the behavior of pressure in regions in which the lapse rate is zero. As one may expect, \(p_b\) is the pressure at the base of the atmospheric region.

\[p = p_b \left[ \frac{T_b}{T} \right]^{\left[ \frac{g_o \cdot M}{R^* \cdot L} \right]}\]

(3)

\[p = p_b \cdot \exp \left[ \frac{-g_o \cdot M(h - h_b)}{R^* \cdot T_b} \right]\]

(4)

Density, \(\rho\), is determined by the ideal gas law once pressure and temperature have been calculated. Similarly, the speed of sound, \(a\), is determined by Eq. 6, which assumes the air behaves ideally.

\[ \rho = \frac{p \cdot M}{R^* \cdot T} \]

(5)

\[ a = \sqrt{ \frac{\gamma \cdot R^* \cdot T}{M} } \]

(6)

Table 3. Base values used for the different regions of the atmosphere, as identified by lapse rate.

Region	\(h_b, m\)	\(L, K/m\)	\(T_b, K\)	\(p_b, N/m^2\)
1	0	-0.0065	288.150	101325.0
2	11,000	0.0	216.650	22632.00
3	20,000	0.0010	216.650	5474.800
4	32,000	0.0028	228.650	868.0100
5	47,000	0.0	270.650	110.9000
6	51,000	-0.0028	270.650	66.93800
7	71,000	-0.0020	214.650	3.956400

Dynamic viscosity, \(\mu\), is determined using Sutherland's Law and is based on kinetic theory for an ideal gas [2]. It can be expressed as

\[ \mu = \mu_{ref} \left( \frac{T}{T_{ref}} \right)^{\frac{3}{2}} \left( \frac{T_{ref} + S}{T + S} \right) \]

(7)

Here, \(T_{ref}\) is a reference temperature and \( \mu_{ref} \) is the corresponding value of viscosity at the reference temperature. Also, \(S\) is termed the Sutherland constant and is specific to a gas. The values for air, and thus the values used in this tool, are provided in Table 4.

Table 4. Constants for air used in Sutherland's law to determine viscosity at altitude.

\( \mu_{ref}, kg/m \cdot s \)	\( T_{ref}, K \)	\( S, K \)
1.716e-5	273.150	110.4

References

[1] Atmosphere, US Standard. "NASA-TM-X-74335." National Aeronautics and Space Administration (NASA), US Government Printing Office, Washington, DC (1976). https://ntrs.nasa.gov/citations/19770009539

[2] Sutherland, William. "LII. The viscosity of gases and molecular force." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 36, no. 223 (1893): 507-531. https://doi.org/10.1080/14786449308620508