Editing Barometric formula

{{short description|Formula used to model how air pressure varies with altitude}}
{{broader|Vertical pressure variation}}

The '''barometric formula''' is a [[formula]] used to model how the [[air pressure]] (or [[air density]]) changes with [[altitude]].

== Pressure equations ==
{{see also|Atmospheric pressure}}
[[File:Pressure air.svg|thumb|300px|Pressure as a function of the height above the sea level]]
There are two equations for computing pressure as a function of height. The first equation is applicable to the atmospheric layers in which the temperature is assumed to vary with altitude at a non null [[lapse rate]] of <math>L_b</math>:
{{anchor|Non-zero lapse rate}}
<math display="block">P = P_{b} \left[ 1 - \frac{L_{M,b}}{T_{M,b}} (h - h_{b})\right]^{\frac{g_{0}' M_{0}}{R^{*} L_{M,b}}}</math>
The second equation is applicable to the atmospheric layers in which the temperature is assumed not to vary{{cn|date=August 2023}} with altitude ([[lapse rate]] is null):
{{anchor|Zero lapse rate}}
<math display="block">P = P_b \exp \left[\frac{-g_0 M \left(h-h_b\right)}{R^* {T_{M,b}}}\right]</math>
where:
*<math>P_b</math> = reference pressure 
*<math>T_{M,b}</math> = reference temperature ([[kelvin|K]])
*<math>L_{M,b}</math> = temperature lapse rate (K/m) in [[International Standard Atmosphere|ISA]]
*<math>h</math> = [[geopotential height]] at which pressure is calculated (m) 
*<math>h_b</math> = geopotential height of reference level ''b'' (meters; e.g., ''h<sub>b</sub>'' = 11 000&nbsp;m)
*<math>R^*</math> = [[universal gas constant]]: 8.3144598&nbsp;J/(mol·K)
*<math>g_0</math> = [[standard gravity|gravitational acceleration]]: 9.80665&nbsp;m/s<sup>2</sup>
*<math>M</math> = molar mass of Earth's air: 0.0289644&nbsp;kg/mol

Or converted to [[imperial units]]:<ref name="conversion">Mechtly, E. A., 1973: ''[https://ntrs.nasa.gov/api/citations/19730018242/downloads/19730018242.pdf The International System of Units, Physical Constants and Conversion Factors]''.  NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C.</ref>

*<math>P_b</math> = reference pressure 
*<math>T_{M,b}</math> = reference temperature ([[kelvin|K]])
*<math>L_{M,b}</math> = temperature lapse rate (K/ft) in [[International Standard Atmosphere|ISA]]
*<math>h</math> = height at which pressure is calculated (ft)
*<math>h_b</math> = height of reference level ''b'' (feet; e.g., ''h<sub>b</sub>'' = 36,089&nbsp;ft)
*<math>R^*</math> = [[universal gas constant]]; using feet, kelvins, and (SI) [[mole (unit)|moles]]: {{val|8.9494596e4|u=lb·ft<sup>2</sup>/(lb-mol·K·s<sup>2</sup>)}}
*<math>g_0</math> = [[standard gravity|gravitational acceleration]]: 32.17405&nbsp;ft/s<sup>2</sup>
*<math>M</math> = molar mass of Earth's air: 28.9644&nbsp;lb/lb-mol

The value of subscript ''b'' ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below.  In these equations, ''g''<sub>0</sub>, ''M'' and ''R''<sup>*</sup> are each single-valued constants, while ''P'', ''L,'' ''T,'' and ''h'' are multivalued constants in accordance with the table below.  The values used for ''M'', ''g''<sub>0</sub>, and ''R''<sup>*</sup> are in accordance with the [[U.S. Standard Atmosphere]], 1976, and the value for ''R''<sup>*</sup> in particular does not agree with standard values for this constant.<ref name="USSA1976">[https://www.ngdc.noaa.gov/stp/space-weather/online-publications/miscellaneous/us-standard-atmosphere-1976/us-standard-atmosphere_st76-1562_noaa.pdf U.S. Standard Atmosphere], 1976, U.S. Government Printing Office, Washington, D.C., 1976. (Linked file is 17 Mb)</ref> The reference value for ''P<sub>b</sub>'' for ''b'' = 0 is the defined sea level value, ''P''<sub>0</sub> = 101 325 [[pascal (unit)|Pa]] or 29.92126 inHg.  Values of ''P<sub>b</sub>'' of ''b'' = 1 through ''b'' = 6 are obtained from the application of the appropriate member of the pair equations 1 and 2 for the case when ''h'' = ''h''<sub>''b''+1</sub>.<ref name=USSA1976/>

{| class="wikitable" style="text-align: center"
|-
! rowspan="2"|Subscript ''b''
! colspan="2"|Geopotential
height above MSL
(h)
! colspan="2"|Static pressure
! rowspan="2"|Standard temperature<br> (K)
! colspan="2"|Temperature lapse rate
! rowspan="2"|Exponent <br> g0 M / R L
|-
! (m) !! (ft)!! (Pa) !! (inHg) !! (K/m) !! (K/ft) 
|-
| 0 || 0 || 0 || 101 325.00 || 29.92126 || 288.15 || 0.0065 || 0.0019812 || 5.25588
|-
| 1 || 11 000 || 36,089 || 22 632.10 || 6.683245 || 216.65 || 0.0 || 0.0 || &mdash;
|-
| 2 || 20 000 || 65,617 || 5474.89 || 1.616734 || 216.65 || -0.001 || -0.0003048 || -34.1626
|-
| 3 || 32 000 || 104,987 || 868.02 || 0.2563258 || 228.65 || -0.0028 || -0.00085344 || -12.2009
|-
| 4 || 47 000 || 154,199 || 110.91 || 0.0327506 || 270.65 || 0.0 || 0.0 || &mdash;
|-
| 5 || 51 000 || 167,323 || 66.94 || 0.01976704 || 270.65 || 0.0028 || 0.00085344 || 12.2009
|-
| 6 || 71 000 || 232,940 || 3.96 || 0.00116833 || 214.65 || 0.002 || 0.0006096 || 17.0813
|}

==Density equations==
{{further|Atmospheric density}}

The expressions for calculating density are nearly identical to calculating pressure.  The only difference is the exponent in Equation 1.

There are two equations for computing density as a function of height. The first equation is applicable to the standard model of the [[troposphere]] in which the temperature is assumed to vary with altitude at a [[lapse rate]] of <math>L_b</math>; the second equation is applicable to the standard model of the [[stratosphere]] in which the temperature is assumed not to vary with altitude.

Equation 1:
<math display="block">\rho = \rho_b \left[\frac{T_b - (h-h_b) L_b}{T_b}\right]^{\left(\frac{g_0 M}{R^*  L_b}-1\right)}</math>

which is equivalent to the ratio of the relative pressure and temperature changes

<math display="block">\rho = \rho_b \frac{P}{T} \frac{T_b}{P_b} </math>

Equation 2:
<math display="block">\rho =\rho_b \exp\left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right]</math>

where
*<math>{\rho}</math> = mass density (kg/m<sup>3</sup>)
*<math>T_b</math> = standard temperature (K)
*<math>L</math> = standard temperature lapse rate (see table below) (K/m) in [[International Standard Atmosphere|ISA]]
*<math>h</math> = height above sea level (geopotential meters)
*<math>R^*</math> = [[universal gas constant]] 8.3144598&nbsp;N·m/(mol·K)
*<math>g_0</math> = gravitational acceleration: 9.80665&nbsp;m/s<sup>2</sup>
*<math>M</math> = molar mass of Earth's air: 0.0289644&nbsp;kg/mol

or, converted to U.S. gravitational foot-pound-second units (no longer used in U.K.):<ref name="conversion"/>
*<math>{\rho}</math> = mass density ([[slug (unit)|slug]]/ft<sup>3</sup>)
*<math>{T_b}</math> = standard temperature (K)
*<math>{L}</math> = standard temperature lapse rate (K/ft)
*<math>{h}</math> = height above sea level (geopotential feet)
*<math>{R^*}</math> = universal gas constant: 8.9494596×10<sup>4</sup>&nbsp;ft<sup>2</sup>/(s·K)
*<math>{g_0}</math> = gravitational acceleration: 32.17405&nbsp;ft/s<sup>2</sup>
*<math>{M}</math> = molar mass of Earth's air: 28.9644&nbsp;lb/lb-mol

The value of subscript ''b'' ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. The reference value for ''ρ<sub>b</sub>'' for ''b'' = 0 is the defined sea level value, ''ρ''<sub>0</sub> = 1.2250&nbsp;kg/m<sup>3</sup> or 0.0023768908&nbsp;slug/ft<sup>3</sup>. Values of ''ρ<sub>b</sub>'' of ''b'' = 1 through ''b'' = 6 are obtained from the application of the appropriate member of the pair equations 1 and 2 for the case when ''h'' = ''h''<sub>''b''+1</sub>.<ref name=USSA1976/>

In these equations, ''g''<sub>0</sub>, ''M'' and ''R''<sup>*</sup> are each single-valued constants, while ''ρ'', ''L'', ''T'' and ''h'' are multi-valued constants in accordance with the table below. The values used for ''M'', ''g''<sub>0</sub> and ''R''<sup>*</sup> are in accordance with the [[U.S. Standard Atmosphere]], 1976, and that the value for ''R''<sup>*</sup> in particular does not agree with standard values for this constant.<ref name="USSA1976"/>

{| class="wikitable"
|-
! rowspan="2"|Subscript ''b''
! colspan="2"|Geopotential
height above MSL
(h)
! colspan="2"|Mass Density (<math>\rho</math>)
! rowspan="2"|Standard Temperature (''T''')<br> (K)
! colspan="2"|Temperature Lapse Rate (''L'')
|-
! (m) !! (ft)!! (kg/m<sup>3</sup>) !! (slug/ft<sup>3</sup>) !! (K/m) !! (K/ft)
|-
| align="center" |0
| align="center" |0
| align="center" |0
| align="center" |1.2250
| align="center" |{{val|2.3768908|e=-3}}
| align="center" |288.15
| align="center" |0.0065
| align="center" |0.0019812
|-
| align="center" |1
| align="center" |11 000
| align="center" |36,089.24
| align="center" |0.36391
| align="center" |{{val|7.0611703|e=-4}}
| align="center" |216.65
| align="center" |0.0
| align="center" |0.0
|-
| align="center" |2
| align="center" |20 000
| align="center" |65,616.79
| align="center" |0.08803
| align="center" |{{val|1.7081572|e=-4}}
| align="center" |216.65
| align="center" |-0.001
| align="center" |-0.0003048
|-
| align="center" |3
| align="center" |32 000
| align="center" |104,986.87
| align="center" |0.01322
| align="center" |{{val|2.5660735|e=-5}}
| align="center" |228.65
| align="center" |-0.0028
| align="center" |-0.00085344
|-
| align="center" |4
| align="center" |47 000
| align="center" |154,199.48
| align="center" |0.00143
| align="center" |{{val|2.7698702|e=-6}}
| align="center" |270.65
| align="center" |0.0
| align="center" |0.0
|-
| align="center" |5
| align="center" |51 000
| align="center" |167,322.83
| align="center" |0.00086
| align="center" |{{val|1.6717895|e=-6}}
| align="center" |270.65
| align="center" |0.0028
| align="center" |0.00085344
|-
| align="center" |6
| align="center" |71 000
| align="center" |232,939.63
| align="center" |0.000064
| align="center" |{{val|1.2458989|e=-7}}
| align="center" |214.65
| align="center" |0.002
| align="center" |0.0006096
|}

==Derivation==
The barometric formula can be derived using the [[ideal gas law]]:
<math display="block"> P = \frac{\rho}{M}  {R^*} T</math>

Assuming that all pressure is [[Hydrostatic pressure|hydrostatic]]:
<math display="block"> dP = - \rho g\,dz</math>
and dividing this equation by <math> P </math> we get:
<math display="block"> \frac{dP}{P} = - \frac{M g\,dz}{R^*T}</math>

[[Integral|Integrating]] this expression from the surface to the altitude ''z'' we get:
<math display="block"> P = P_0 e^{-\int_{0}^{z}{M g dz/R^*T}}</math>

Assuming linear temperature change <math>T = T_0 - L z</math> and constant molar mass and gravitational acceleration, we get the first barometric formula:
<math display="block"> P = P_0 \cdot \left[\frac{T}{T_0}\right]^{\textstyle \frac{M g}{R^* L}}</math>

Instead, assuming constant temperature, integrating gives the second barometric formula:
<math display="block"> P = P_0 e^{-M g z/R^*T}</math>

In this formulation, ''R<sup>*</sup>'' is the [[gas constant]], and the term ''R<sup>*</sup>T''/''Mg'' gives the [[scale height]] (approximately equal to 8.4&nbsp;km for the [[troposphere]]).

(For exact results, it should be remembered that atmospheres containing water do not behave as an ''ideal gas''. See [[real gas]] or [[perfect gas]] or [[gas]] for further understanding.)

==See also==
*[[Hypsometric equation]]
*[[NRLMSISE-00]]

== References ==
<references/>

{{DEFAULTSORT:Barometric Formula}}
[[Category:Vertical distributions]]
[[Category:Atmospheric pressure]]