# 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)
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]:

From sea level href = 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)
with
where'
 $p$ ' =' 'Static pressure at altitude h in N/m2 $p_\mathrm{ref}$ ' =' 'Static pressure at reference altitude href. For href = 0 m (sea level), pref = 101,325 Pa $T_\mathrm{ref}$ ' =' 'Temperature at reference altitude href. For href = 0 m (sea level), Tref = 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 href = 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 href = 0 m and assume a mean state for the atmosphere, as described by the International Standard Atmosphere (temperature Tref = 15°C = 288.15 K, air pressure pref = 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 href = 11 km to an altitude of h = 20 km the air temperature is constant Tref. The formula for air pressure, when the temperature is constant, is:

(3)
with
where'
 $p$ ' =' 'Static pressure at altitude h in N/m2 $p_\mathrm{ref}$ ' =' 'Static pressure at reference altitude href 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 href 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)
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:

Isotherm (T = const.)

(5)
(6)
width
(7)
where'
 $p(h)$ ' =' 'Static air pressure at altitude h in N/m2 $p_\mathrm{ref}$ ' =' 'Static air pressure at reference altitude href in N/m2 $\rho(h)$ ' =' 'Air density at altitude h in kg/m3 $\rho_\mathrm{ref}$ ' =' 'Air density at reference altitude href 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:

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

## Reference Values

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

i Layer [m] href [m] αi [K/m] Tref [K] ρref [g/m3] pref [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 h84,852 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

Formelsammlung Hydrostatik - Wikibooks (de)
https://de.wikibooks.org/wiki/Formelsammlung_Physik/_Hydrostatik
Barometric formula - Wikipedia
https://en.wikipedia.org/wiki/Barometric_formula

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