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Barometric Formula

The Barometric Formula describes the vertical distribution of the gas particles in the atmosphere of the earth, i.e. the change of air pressure, density and temperature with the altitude. One therefore speaks of a vertical pressure, density and temperature gradient, but due to the weather dynamics within the lower atmosphere can only be described by approximations in a mathematical way.

It can be roughly assumed that the air pressure near sea level decreases by one hectopascal hPa for every eight meters increase in altitude.

The Ideal Gas Law describes how pressure, density and temperature of the atmosphere are coupled. If two of the values are known, the third one can be calculated from them. Due to gravity, the atmosphere gets denser with decreasing altitude. The density is also dependent on the temperature, which can not be easily calculated. Therefore, in the International Standard Atmosphere a mean temperature gradient is determined. In this way, the density and pressure can be calculated uniquely depending on the altitude (or temperature).

Temperature

The International Standard Atmosphere model divides the atmosphere into layers, each with a linear temperature gradient or a constant temperature. The temperature gradient is denoted as the negative lapse rate α. The temperature at any altitude can be calculated as follows:

(1)

$T(h) = T_\mathrm{ref} + \alpha_i\, (h-h_\mathrm{ref})$

where^'

$T(h)$ ^'	=^'	^'Temperature at altitude h in Kelvin
$T_\mathrm{ref}$ ^'	=^'	^'Temperatur at a reference altitude (e.g. sea level)
$\alpha_i$ ^'	=^'	^'Temperature Gradient (negative Lapse Rate) of the i-th layer in K/m (Kelvin per meter)
$h$ ^'	=^'	^'Altitude in m
$h_\mathrm{ref}$ ^'	=^'	^'Refrence altitude (sea level = 0 m)

The temperature gradients and Reference Values are listed in the table at the bottom of the page.

Air Pressure

The formulas for air pressure and air density must distinguish between two cases. In air layers with constant temperature (isothermal), a different formula applies than in layers with a linear temperature gradient. [1] [2]:

Linear Temperature Gradient

From sea level h_ref = 0 m up to an altitude of h = 11 km, the temperature curve is approximately linear. The formula for air pressure for a linear temperature gradient is:

(2)

$p = p_\mathrm{ref}\, \left( {T_\mathrm{ref} + \alpha \cdot (h - h_\mathrm{ref}) \over T_\mathrm{ref}} \right) ^ {-\beta} = p_\mathrm{ref}\, \left( 1 + {\alpha \cdot (h - h_\mathrm{ref}) \over T_\mathrm{ref}} \right) ^ {-\beta}$

with

$\beta = {M \cdot g \over R \cdot \alpha} = { g \over R_\mathrm{S} \cdot \alpha}$

where^'

$p$ ^'	=^'	^'Static pressure at altitude h in N/m²
$p_\mathrm{ref}$ ^'	=^'	^'Static pressure at reference altitude h_ref. For h_ref = 0 m (sea level), p_ref = 101,325 Pa
$T_\mathrm{ref}$ ^'	=^'	^'Temperature at reference altitude h_ref. For h_ref = 0 m (sea level), T_ref = 288.15 K (15 °C)
$\alpha$ ^'	=^'	^'Temperature gradient (negative lapse rate) = −0.0065 K/m
$h$ ^'	=^'	^'Altitude above sea level in meter from 0 m up to 11,000 m
$h_\mathrm{ref}$ ^'	=^'	^'Reference altitude. For sea level h_ref = 0 m
$M$ ^'	=^'	^'Molar mass; for dry air = 28.9644 g/mol
$g$ ^'	=^'	^'Mean gravitational acceleration at sea level = 9.80665 m/s²
$R$ ^'	=^'	^'Universal Gas Constant = 8.31446 J/(mol·K)
$R_\mathrm{S}$ ^'	=^'	^'Spezific Gas Constant; dry air = 287.058 J/(kg·K)

If we set the reference altitude to sea level h_ref = 0 m and assume a mean state for the atmosphere, as described by the International Standard Atmosphere (temperature T_ref = 15°C = 288.15 K, air pressure p_ref = 1013.25 hPa and temperature gradient α = −0.0065 K/m), we obtain the barometric formula for air pressure in the troposphere (valid until h = 11 km altitude):

This formula allows you to calculate the air pressure at a given altitude without knowing temperature and temperature gradient. However, the accuracy in is limited since the calculation is based on a mean atmosphere instead of the current atmospheric state.

The Reference Values are listed in the table at the bottom of the page.

Isothermal

From a reference altitude of h_ref = 11 km to an altitude of h = 20 km the air temperature is constant T_ref. The formula for air pressure, when the temperature is constant, is:

(3)

$p = p_\mathrm{ref} \cdot e^{-(h - h_\mathrm{ref}) / h_\mathrm{s}}$

with

$h_\mathrm{s} = {R \cdot T_\mathrm{ref} \over M \cdot g} = { R_\mathrm{S} \cdot T_\mathrm{ref} \over g} = {\rm const.}$

where^'

$p$ ^'	=^'	^'Static pressure at altitude h in N/m²
$p_\mathrm{ref}$ ^'	=^'	^'Static pressure at reference altitude h_ref in N/m²
$h$ ^'	=^'	^'Height above sea level (elevation)
$h_\mathrm{ref}$ ^'	=^'	^'Reference altitude, above which the tempertature is constant
$R$ ^'	=^'	^'Univeral gas constant = 8.31446 J/(mol·K)
$R_\mathrm{S}$ ^'	=^'	^'Spezific gas constant; dry air = 287.058 J/(kg·K)
$T_\mathrm{ref}$ ^'	=^'	^'Temperature above reference altitude h_ref in Kelvin
$M$ ^'	=^'	^'Molar mass; dry air = 28.9644 g/mol
$g$ ^'	=^'	^'Mean gravitational acceleration at sea level = 9.80665 m/s²

The Reference Values are listed in the table at the bottom of the page.

Air Density

According to the ideal gas law, the air density can be calculated according to the following formula from pressure and temperature. Since the pressure p(h) depends on the temperature gradient (isotherm (3) or linear temperature gradient (2)), this also applies to the density:

(4)

$\rho(h) = {p(h) \cdot M \over R \cdot T(h)} = {p(h) \over R_\mathrm{S} \cdot T(h)}$

where^'

$\rho(h)$ ^'	=^'	^'Air density at sea level h in kg/m³
$p(h)$ ^'	=^'	^'Static pressure at altitude h in N/m²
$M$ ^'	=^'	^'Molar mass; dry air = 28.9644 g/mol
$R$ ^'	=^'	^'Universal gas constant = 8.31446 J/(mol·K)
$T(h)$ ^'	=^'	^'Temperature at altitude h in Kelvin
$R_\mathrm{S}$ ^'	=^'	^'Spezific gas constant; dry air = 287.058 J/(kg·K)

The molar mass is generally dependent on the air composition, which varies at different levels. For heights up to 100 km, however, the composition can be assumed to be constant.

If we substitude p(h) and T(h) with the corresponding formulas above we get the formulas listed in the Layer Model below.

Layer Model

Since in the standard model of the atmosphere a distinction is made between isothermal layers and layers with a linear temperature gradient, there are also two groups of corresponding formulas:

Linear Temperature Gradient

Isotherm (T = const.)

(5)

$p(h) = p_\mathrm{ref} \left( 1 + {\alpha_i\, (h-h_\mathrm{ref}) \over T_\mathrm{ref}} \right) ^ {-\beta}$

$p(h) = p_\mathrm{ref}\cdot e^{- (h-h_\mathrm{ref}) / h_\mathrm{s} }$

(6)

$\rho(h) = \rho_\mathrm{ref} \left( 1 + {\alpha_i\, (h-h_\mathrm{ref}) \over T_\mathrm{ref}} \right) ^ {-\beta - 1}$

$\rho(h) = \rho_\mathrm{ref} \cdot e ^ {-(h-h_\mathrm{ref}) / h_\mathrm{s}}$

width

$\beta = {g \over R_\mathrm{S} \cdot \alpha_i}$

$h_\mathrm{s} = {R_\mathrm{S} \cdot T_\mathrm{ref} \over g}$

(7)

$T(h) = T_\mathrm{ref} + \alpha_i\, (h - h_\mathrm{ref})$

$T(h) = T_\mathrm{ref} = {\rm const.}$

where^'

$p(h)$ ^'	=^'	^'Static air pressure at altitude h in N/m²
$p_\mathrm{ref}$ ^'	=^'	^'Static air pressure at reference altitude h_ref in N/m²
$\rho(h)$ ^'	=^'	^'Air density at altitude h in kg/m³
$\rho_\mathrm{ref}$ ^'	=^'	^'Air density at reference altitude h_ref in kg/m³
$h$ ^'	=^'	^'Altiude above sea level (elevation) in meter
$h_\mathrm{ref}$ ^'	=^'	^'Reference altitude in meter
$T_\mathrm{ref}$ ^'	=^'	^'Reference temperature in Kelvin
$\alpha_i$ ^'	=^'	^'Temperature gradient of the i-th level in K/m
$g$ ^'	=^'	^'Mean gravitational acceleration at sea level = 9.80665 m/s²
$R_\mathrm{S}$ ^'	=^'	^'Spezific gas constant; dry air = 287.058 J/(kg·K)

Usage

In order to determine the static air pressure, the air density or the temperature of the air at a certain altitude, according to the model of the International Standard Atmosphere, one proceeds as follows:

Locate in the table below the layer i for your altitude
Only use values from the matching line of the table!
Look at the column α_i whether this altitude has an isothermal temperature gradient or not
Use only the formulas of the corresponding column above (Isothermal or Linear Temperature Gradient)
In the above formulas, substitute the reference values of the appropriate row in the table below to calculate pressure, density and temperature

For this calculations you can use the Calculator on this page.

Reference Values

For the air layers of the standard atmospheric model, the following reference values are determined:

i	Layer [m]	h_ref [m]	α_i [K/m]	T_ref [K]	ρ_ref [g/m³]	p_ref [Pa]
0	0 - 11,000	0	−0.0065	288.15	1225.00	101,325
1	11,000 - 20,000	11,000	0.0	216.65	363.918	22,632.1
2	20,000 - 32,000	20,000	0.001	216.65	88.0348	5474.89
3	32,000 - 47,000	32,000	0.0028	228.65	13.2250	868.019
4	47,000 - 51,000	47,000	0.0	270.65	1.42753	110.906
5	51,000 - 71,000	51,000	−0.0028	270.65	0.861605	66.9389
6	71,000 - 84,852	71,000	−0.0020	214.65	0.0642110	3.95642

Calculator

The following calculator works up to an altitude of h ≤ 84,852 m with an accuracy of 5 digits. The values are calculated using the formulas on this page, i.e. it is calculated with the Layer Model.

The JavaScript of the calculator can be shown at JavaScript Barometric Formula.

Sources

[1]

Formelsammlung Hydrostatik

- Wikibooks (de)
https://de.wikibooks.org/wiki/Formelsammlung_Physik/_Hydrostatik

[2]

Barometric formula

- Wikipedia
https://en.wikipedia.org/wiki/Barometric_formula

About	Walter Bislin (wabis)
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Created	Tuesday, November 12, 2019

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Changed	Wednesday, August 30, 2023

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Barometric Formula

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