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## Air Pressure, Density, and Temperature vs. Altitude in Standard Atmosphere Calculator

### International Standard Atmosphere (ISA) and US Standard Atmosphere 1976 Calculator

This air pressure, density, and temperature vs. altitude calculator determines the atmospheric pressure, air density, temperature and the speed of sound for a given altitude and a temperature offset using the International Standard Atmosphere (ISA) and the US Standard Atmosphere 1976 (USSA) models. These models are essentially the same in the interval of 0–86 km. The temperature offset is the temperature deviation from the standard atmosphere 15°C value. For example, if the actual air temperature near the Earth’s surface is 25°C then the offset will be 10°C. The calculator allows the selection of various values of the Earth’s radius used in calculations.

Example: Calculate the atmospheric pressure, air density, temperature and the speed of sound at the traditional cruise flight altitude of 35,000 feet (10,600 meters); the temperature offset is 10°С.

**Input**

Altitude (geometric)

*h*

Temperature offset

*t*_{o}

Radius of Earth, **R**_{⊕}

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**Output**

Pressure

** p** Pa

** p** psi at

Air density

** ρ** kg/m³ (g/L)

Temperature

** t** K °C

Speed of sound

** c** m/s km/h

Earth gravity at this altitude

** g_{e}** m/s²

Geopotential height

** h** km

To calculate, enter the values into the corresponding boxes, select Imperial or metric units and click or tap the **Calculate** button.

International Standard Atmosphere (ISA)

U.S. Standard Atmosphere

Definitions, Constants, and Formulas Used in Calculations

Altitude and Flight Level (FL)

Radius of Earth Selector R⊕

Specific Gas Constant for Dry Air Rsp

Standard Acceleration Due to Gravity

Geopotential Height (Altitude)

Speed of Sound

Gravity vs. Altitude

Temperature vs. Altitude

Pressure vs. Altitude

Air Density vs. Altitude

The Earth’s atmosphere is constantly changing, therefore, hypothetical models were developed as an approximation of what may be expected assuming the air does not contain dust or moisture and there is no winds or turbulence. These models are known as “standard atmospheres”. Their use is necessary for aircraft development, for studying their performance, for comparing the performance of different aircraft and for many other aeronautic and aviation science applications.

A concept of the standard atmosphere was developed to standardize calibration of pressure in altimeters, for studying the performance of aircraft engines where knowledge of air density, pressure and temperature at mean sea level (MSL) and their distribution as a function of altitude is essential. The International Standard Atmosphere is one such model. It is published by the International Organization for Standardization (ISO) as an international standard ISO 2533:1975. Standards organizations in various countries publish their own atmospheric models based on ISA. Another widely used model is the 1976 U.S. Standard Atmosphere (USSA), which uses the same atmospheric model. The differences between the two are only at altitudes higher than 86 km, which are outside the scope of this calculator.

The Earth’s atmosphere is constantly changing

## International Standard Atmosphere (ISA)

The International Standard Atmosphere “is intended for use in calculations and design of flying vehicles, to present the test results of flying vehicles and their components under identical conditions, and to allow unification in the field of development and calibration of instruments.” The use of this atmospheric model is also recommended in the processing of data from geophysical and meteorological observations. It is used as a standard against which one can compare the actual atmosphere and based on the values at mean sea level shown below. All values decrease with increase in altitude:

- Pressure of 101.325 kPa at mean sea level (MSL).
- Temperature of +15°C at MSL
- Density of 1.225 kg/m³ at MSL.

## U.S. Standard Atmosphere

“The U.S. Standard Atmosphere,1976 is an idealized, steady-state representation of the earth’s atmosphere from the surface to 1000 km”. The model is based on existing international standards and is largely consistent in methodology with the International Standard Atmosphere (ISA). The equations used are adopted by the United States Committee on Extension to the Standard Atmosphere (COESA), which represented 29 U.S. scientific, governmental, military and engineering organizations. The U.S. Standard Atmosphere model represents the Earth’s atmosphere pressure, temperature, density and viscosity distributions at various altitudes. The model divides the atmosphere into seven layers to a maximum of 86 km. The main difference between the US Standard Atmosphere (USSA) and ISA is mainly in the assumed temperature distribution at higher altitudes, which are not covered in this converter.

## Definitions, Constants, and Formulas Used in Calculations

### Altitude and Flight Level (FL)

A modern drum-type installed in the Fokker 100 co*ckpit. The altimeter with two small Kollsman windows showing pressure in hectopascals and inches of mercury

Though both the altitude and the flight level are measured in the units of length and distance (meters, kilometers, feet, and miles), they are actually different physical values:

- The
**altitude**is the vertical distance of an object from mean sea level**measured by means of a distance measuring instrument**(for example a laser distance finder or a radar altimeter). - The
**flight level**is a vertical standard “pressure altitude” also called nominal altitude expressed in hecto-feet (hundreds of feet). For example, a pressure altitude at 34,000 feet is referred to as FL340. The flight level is**measured by means of a pressure measuring instrument**(for example, pressure altimeter, which is essentially a calibrated barometer calibrated in the units of altitude). During preparation to take off, the altimeter is set to zero altitude. When a plane climbs high enough (higher than the transition altitude), pilots set the altimeter to the standard pressure (29.921 inches of mercury or 1013.25 hectopascals). During preparation to landing, at low altitudes (typically less than 3000 to 18000 feet above sea level in different jurisdictions) the altimeter is set to the correct local pressure at the airport to show the actual altitude above sea level.

A mechanical altimeter with the barometric adjust knob measures the atmospheric pressure on the static port outside the aircraft. It is calibrated to show the pressure as an altitude above sea level. Before takeoff and landing, the pilot obtains a local barometric reading from the airport and sets it in the small window using the knob.

### Radius of Earth Selector *R*_{⊕}

Four constants are used in the Radius of Earth selector:

Mean Earth’s radius defined in the 1984 World Geodetic System: *R*₁ = 6371.0088 km.

Mean Earth’s radius defined in the 1976 U.S. Standard Atmosphere: *R*₀ = 6356.766 km.

The Earth’s equatorial (semi-major) radius defined in World Geodetic System WGS 84: *a* = 6378.1370 km.

The Earth’s polar (semi-minor) radius defined in World Geodetic System WGS 8:4 *b* = 6356.7523142 km.

A — equatorial, B — polar, and C — mean Earth radii; C = (2A + B)/3

### Specific Gas Constant for Dry Air *R*_{sp}

The Specific Gas Constant *R*_{sp} for dry air is defined as the molar (universal) gas constant divided by the mean molar mass for dry air. The molar gas constant is defined in 1976 U.S. Standard Atmosphere as *R** = 8314.32 N m kmol⁻¹ K⁻¹. The mean molar mass for dry air M = 28.9645 g/mol = 28.9645 kg/kmol. Therefore, the specific gas constant for dry air in J K⁻¹ kg⁻¹ is calculated as

### Standard Acceleration Due to Gravity

The standard acceleration due to gravity is defined by an ISO 80000-3 Quantities and Units Part 3 Space and Time standard as *g*₀ = 9.80665 m/s² or 32.17405 ft/s². Although the actual acceleration due to gravity on Earth is different in various locations, this standard value is always used for metrological purposes.

### Geopotential Height (Altitude)

Earth’s gravity depends on the altitude and latitude. The transportation of geometric height into geopotential height eliminates the variable portion of the acceleration of gravity. The geopotential height is a vertical coordinate referenced to Earth’s mean sea level, an adjustment to geometric altitude, which is what would be measured by a measuring tape. The geopotential height takes into account the variation of gravity with latitude and elevation. In other words, geopotential height is a gravity-adjusted height. Variation in latitude is too small and usually is not taken in to account. The geopotential height is really a measure of the specific potential energy at a given geometric height relative to the Earth’s surface. It is used by meteorologists and in aviation. The relationship between the geopotential height *H* and the geometric height *Z* is given by the following formula (equation 18 in 1976 USSA), which is used in our calculations:

For example, if *Z* = 86 km, which is the maximum geometric height in this calculator, the corresponding geopotential height will be *H* = 84.852 km. In this calculator, the geopotential height is calculated before pressure and temperature calculations.

### Speed of Sound

The speed of sound in air is about 343 m/s or 1.235 km/h or 767 mph. That means the sound can travel through the air one kilometer in about 3 seconds or a mile in about 5 seconds. The speed of sound in air depends mainly on its temperature; dependence on the sound frequency and air pressure is negligible.

Water condensation at transonic speed

The speed of sound in dry air assuming it is an ideal gas at a relatively low pressure and density, which is correct for standard sea-level conditions, and also assuming that its temperature is lower or equal the room temperature is determined by the following formula, which is used in this calculator:

where *γ* is the specific heat ratio discussed below, *R* = 287.052 J·kg⁻¹·K⁻¹ is the specific gas constant and *T* is the air absolute temperature in kelvins.

The specific heat ratio or heat capacity ratio of a gas is denoted by the Greek letter γ (gamma) and it is the ratio of the specific heat at constant pressure *C*_{p} to the specific heat at constant volume *C*_{v}

For dry air at 20°C, γ=1.40

### Gravity vs. Altitude

The gravitational acceleration *G*_{h} dependence on altitude *h* is approximately defined by the following formula, which is used in our calculations:

where

*g*_{0} is the standard gravitational acceleration. For example, the gravitational acceleration *G*_{h} at the geometric altitude is 86 km, which is the maximum geometric height in this calculator, is 0.974 times the standard gravitational acceleration g_{0}, that is, the difference is very small.

### Temperature vs. Altitude

Within the troposphere, the air temperature in Earth’s atmosphere decreases with an increase in altitude. According to the international standard atmosphere (ISA) and 1976 U.S. Standard Atmosphere (USSA), the rate of decrease of temperature (lapse rate) is 6.5 K/km from sea level (0 km) to 11 km or 36,089 feet. In the region from 11 to 20 km or 65,617 feet the temperature is constant and is equal to –56.5°C (–69.7°F or 216.7 K). In the ionosphere, from 20 to32 km or 104,987 feet, the rate of decrease of temperature (lapse rate) is 1.0 K/km, and so on. These values are tabulated below to the altitude of 86 km (geopotential altitude 84.85 km) in Table 4 of the 1976 USSA document.

**Table 1**

Atmospheric Level | Geopotential Altitude Range (km) | Interval (Layer) Number, b | Base Geopotential Altitude above mean sea level (MSL), H_{b} (km) | Base Static Pressure, P_{b} (Pa) | Base Temperature, T_{b} (K) | Base Temperature Lapse Rate per Kilometer of Geopotential Altitude, L_{b} (K/km) |
---|---|---|---|---|---|---|

Troposphere | 0–11 | 0 | 0 | 101325 | 288.15 | -6.5 |

Tropopause (StratosphereI) | 11–20 | 1 | 11 | 22632.06 | 216.65 | 0 |

StratosphereII | 20–32 | 2 | 20 | 5474.889 | 216.65 | +1.0 |

StratosphereIII | 32–47 | 3 | 32 | 868.0187 | 228.65 | +2.8 |

Stratopause (MesosphereI) | 47–51 | 4 | 47 | 110.9063 | 270.65 | 0 |

MesosphereII | 51–71 | 5 | 51 | 66.93887 | 270.65 | -2.8 |

MesosphereIII | 71–84.9 | 6 | 71 | 3.95642 | 214.65 | -2.0 |

7 | 84.852 | 0.3734 | 186.87 | — |

“Base” in the table means a value at the start (or bottom) of the altitude range. The negative temperature lapse means the temperature is decreasing with height and the positive lapse means the temperature is increasing with height. I higher number of lapse indicates greater cooling (or warming) with height.

To calculate the temperature:

- find the geopotential altitude from a geometric height;
- find the interval number,
*b*; - the temperature
*T*_{M}at the geopotential altitude*H*from the surface to 84.85 km is found using seven successive linear equations in different altitude intervals by means of inserting the values from table 1:

Where

*H*_{b} is the base geopotential altitude (see Table 1),

*T*_{b} is the base temperature,

*L*_{b} is the base temperature lapse rate

The temperature *T*_{M} is called the molecular temperature, which is defined as

where *T* is the kinetic temperature, which is the air temperature we usually measure using a thermometer. It is a function of the velocity of the molecules of gases of the Earth’s atmosphere. *M*_{0} is the molecular mass of air at sea level and *M*_{H} is the molecular mass of air at the altitude *H*. At the altitudes below 100 km, molecular mass of air remains constant, therefore the molecular temperature is equal to the kinetic temperature.

**The temperature offset**. Of course, the real atmosphere is never the same as considered in this standard. Variations in temperature affect the air density and consequently the pressure and weight. In cold air, pressure decreases more rapidly than in hot air. On a hot day, the whole atmosphere and the temperature graph will be displaced (see the chart below) — the temperature offset will be added to the temperature curve and the pilots who use barometric instruments to measure their altitudes must be aware that on a hot day the geometric altitude of their airplane will be higher than the pressure altitude displayed on the altimeter. On a cold day, the true altitude will be lower than the altitude displayed on the altimeter.

Air temperature, density, and pressure vs. geopotential altitude. Blue — pressure, violet — density, +20°C offset, orange — density, 0°C offset, green — temperature, +20°C offset, red — temperature, 0°C offset

If an airplane flies into an area where the temperature is much lower than the ISA temperature (+15 °C at sea level), the altimeter will display higher altitude, which is dangerous. To consider these deviations from ISA conditions, this calculator includes the **Temperature Offset** field, which can be used, for example, to analyze or predict aircraft performance on a hot day. Remember that the temperature offset is a temperature interval and when converting between degrees Celsius or kelvins and degrees Fahrenheit or Rankine, only the scaling factor should be used: 1 kelvin = 1°C = 9/5°F = 1.8°F = 1.8°R. You can also use our Temperature Interval Converter for this conversion.

### Pressure vs. Altitude

The barometric formula is used in both ISA and USSA to model how the pressure and density of air changes with altitude. Two different equations are used with the data from Table 1 above for determining the pressure at various height intervals from 0 to 86 km.

If the base temperature lapse rate *L*_{b} is zero, the following equation is used:

If the base temperature lapse rate *L*_{b} is not equal to zero, the following equation is used:

or

In these equations all values with index *b* are used from Table 1:

*P*_{b} is the base static pressure of the layer *b* in Pa

*T*_{b} is the base temperature of the layer *b* in K

*L*_{b} is the base temperature lapse rate of the layer *b* in K/m

*H*_{b} is the base geopotential height of the layer *b* in m

*H* is the geopotential height above sea level in m

*R** = 8.31432·10³ N m kmol⁻¹ K⁻¹ is the universal gas constant

*g*_{0} = 9.80665 m/s² is the gravitational acceleration

*M* = 0.0289644 kg/mol is the molar mass of Earth’s air.

*T*_{M} is the molecular scale temperature at the altitude *H* defined above:

### Air Density vs. Altitude

The air density is the mass of air per its unit volume. It is denoted by the Greek letter *ρ* (rho) and measured in kg/m³ in SI or lb/ft³. ISA (International Standard Atmosphere) and 1976 U.S. Standard Atmosphere define air density at the standard pressure 1013.25 hPa and temperature 15°C as 1.225 kg/m³ or 0.0765 lb/ft³. Air density is affected not only by the temperature and pressure but also by the amount of water in the air. The more water vapor is contained in the air, the less dense it is.

Air density depends on temperature and pressure. At standard temperature and pressure (STP), the value for the air density depends on the standard you use. According to the International Standard Atmosphere (ISA), the density of dry air is 1.225 kg/m³ or 0.0765 lb/ft³. The International Union of Pure and Applied Chemistry (IUPAC) defines the air density differently. According to IUPAC, the density of dry air is 1.2754 kg/m³ or 0.0796 lb/ft³ at 1000 hPa and 0°C.

In this calculator, we consider only dry air. The density of dry air *ρ* is calculated using the ideal gas law using the calculated pressure at a given altitude by the following formula:

where:

*p* is the absolute pressure in Pa,

*T* is the absolute temperature of air in K, and

*R*_{sp} = 287.052 J·kg⁻¹·K⁻¹ is the specific gas constant.

Note that because we consider air as the ideal gas, and only dry air, the result of calculations is only an approximation. The most accurate results can be obtained at low temperature and pressure values (at high altitudes!).

This article was written by Anatoly Zolotkov

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