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| {{unreliable sources|date=January 2011|reason=Units and Constants are flawed. Units used should be reviewed and tables recomputed.}}
| | Nice to satisfy you, I am Marvella Shryock. Bookkeeping is what I do. California is our beginning place. Doing ceramics is what my family and I appreciate.<br><br>My blog post [http://www.rachat-points.com/node/5780543 at home std testing] |
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| The '''barometric formula''', sometimes called the ''[[exponential function|exponential]] [[Earth's atmosphere|atmosphere]]'' or ''[[isothermal]] [[atmosphere]]'', is a [[formula]] used to model how the [[pressure]] (or [[density]]) of the air changes with [[altitude]].
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| ==Pressure equations==
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| {{see also|Atmospheric pressure}}
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| [[File:Pressure_air.svg|thumb|300px|Pressure as a function of the height above the sea level]]
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| There are two different equations for computing pressure at various height regimes below 86 km (or 278,400 feet). The first equation is used when the value of Standard Temperature [[lapse rate|Lapse Rate]] is not equal to zero; the second equation is used when standard temperature lapse rate equals zero.
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| Equation 1:
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| :<math>{P}=P_b \cdot \left[\frac{T_b}{T_b + L_b\cdot(h-h_b)}\right]^{\textstyle \frac{g_0 \cdot M}{R^* \cdot L_b}}</math>
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| Equation 2:
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| :<math>P=P_b \cdot \exp \left[\frac{-g_0 \cdot M \cdot (h-h_b)}{R^* \cdot T_b}\right]</math>
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| where
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| :<math>P_b</math> = Static pressure (pascals)
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| :<math>T_b</math> = Standard temperature ([[kelvin|K]])
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| :<math>L_b</math> = Standard temperature lapse rate -0.0065 (K/m) in [[International Standard Atmosphere|ISA]]
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| :<math>h</math> = Height above sea level (meters)
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| :<math>h_b</math> = Height at bottom of layer b (meters; e.g., <math>h_1</math> = 11,000 meters)
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| :<math>R^*</math> = [[Universal gas constant]] for air: 8.31432 N·m /(mol·K)
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| :<math>g_0</math> = [[Standard_gravity|Gravitational acceleration]] (9.80665 m/s<sup>2</sup>)
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| :<math>M</math> = Molar mass of Earth's air (0.0289644 kg/mol)
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| Or converted to Imperial units:<ref name=conversion>Mechtly, E. A., 1973: ''The International System of Units, Physical Constants and Conversion Factors''. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C.</ref>
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| where
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| :<math>P_b</math> = Static pressure (inches of mercury, inHg)
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| :<math>T_b</math> = Standard temperature (K)
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| :<math>L_b</math> = Standard temperature lapse rate (K/ft)
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| :<math>h</math> = Height above sea level (ft)
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| :<math>h_b</math> = Height at bottom of layer b (feet; e.g., <math>h_1</math> = 36,089 ft)
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| :<math>R^*</math> = [[Universal gas constant]]; using feet, kelvins, and (SI) [[mole (unit)|moles]]: 8.9494596×10<sup>4</sup> lb·ft<sup>2</sup>/(lbmol·K·s<sup>2</sup>)
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| :<math>g_0</math> = Gravitational acceleration (32.17405 ft/s<sup>2</sup>)
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| :<math>M</math> = Molar mass of Earth's air (28.9644 lb/lbmol)
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| 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 <math>R^*</math> are in accordance with the [[U.S. Standard Atmosphere]], 1976, and the value for <math>R^*</math> in particular does not agree with standard values for this constant.<ref name=USSA1976>[http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19770009539_1977009539.pdf U.S. Standard Atmosphere], 1976, U.S. Government Printing Office, Washington, D.C., 1976. (Linked file is very large.)</ref> The reference value for ''P<sub>b</sub>'' for ''b'' = 0 is the defined sea level value, ''P<sub>0</sub>'' = 101325 [[pascal (unit)|pascals]] 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 <math>h = h_{b+1}</math>.:<ref name=USSA1976/>
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| {| class="wikitable"
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| |-
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| ! rowspan="2"|Subscript ''b''
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| ! colspan="2"|Height above sea level
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| ! colspan="2"|Static pressure
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| ! rowspan="2"|Standard temperature<br> (K)
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| ! colspan="2"|Temperature lapse rate
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| |-
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| ! (m) !! (ft)!! (pascals) !! (inHg) !! (K/m) !! (K/ft)
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| |-
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| | align="center" |0
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| | align="center" |0
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| | align="center" |0
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| | align="center" |101325.00
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| | align="center" |29.92126
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| | align="center" |288.15
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| | align="center" |-0.0065
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| | align="center" |-0.0019812
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| |-
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| | align="center" |1
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| | align="center" |11,000
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| | align="center" |36,089
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| | align="center" |22632.10
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| | align="center" |6.683245
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| | align="center" |216.65
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| | align="center" |0.0
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| | align="center" |0.0
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| |-
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| | align="center" |2
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| | align="center" |20,000
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| | align="center" |65,617
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| | align="center" |5474.89
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| | align="center" |1.616734
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| | align="center" |216.65
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| | align="center" |0.001
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| | align="center" |0.0003048
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| |-
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| | align="center" |3
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| | align="center" |32,000
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| | align="center" |104,987
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| | align="center" |868.02
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| | align="center" |0.2563258
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| | align="center" |228.65
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| | align="center" |0.0028
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| | align="center" |0.00085344
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| |-
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| | align="center" |4
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| | align="center" |47,000
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| | align="center" |154,199
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| | align="center" |110.91
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| | align="center" |0.0327506
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| | align="center" |270.65
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| | align="center" |0.0
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| | align="center" |0.0
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| |-
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| | align="center" |5
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| | align="center" |51,000
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| | align="center" |167,323
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| | align="center" |66.94
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| | align="center" |0.01976704
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| | align="center" |270.65
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| | align="center" |-0.0028
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| | align="center" |-0.00085344
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| |-
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| | align="center" |6
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| | align="center" |71,000
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| | align="center" |232,940
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| | align="center" |3.96
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| | align="center" |0.00116833
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| | align="center" |214.65
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| | align="center" |-0.002
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| | align="center" |-0.0006096
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| |}
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| ==Density equations==
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| The expressions for calculating density are nearly identical to calculating pressure. The only difference is the exponent in Equation 1.
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| There are two different equations for computing density at various height regimes below 86 geometric km (84,852 geopotential meters or 278,385.8 geopotential feet). The first equation is used when the value of Standard Temperature Lapse rate is not equal to zero; the second equation is used when Standard Temperature Lapse rate equals zero.
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| Equation 1:
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| :<math>{\rho}=\rho_b \cdot \left[\frac{T_b + L_b\cdot(h-h_b)}{T_b}\right]^{\left(-\frac{g_0 \cdot M}{R^* \cdot L_b}\right)-1}</math>
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| Equation 2:
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| :<math>{\rho}=\rho_b \cdot \exp\left[\frac{-g_0 \cdot M \cdot (h-h_b)}{R^* \cdot T_b}\right]</math>
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| where
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| :<math>{\rho}</math> = Mass density (kg/m<sup>3</sup>)
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| :<math>T</math> = Standard temperature (K)
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| :<math>L</math> = Standard temperature lapse rate (see table below) (K/m) in [[International Standard Atmosphere|ISA]]
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| :<math>h</math> = Height above sea level (geopotential meters)
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| :<math>R^*</math> = [[Universal gas constant]] for air: 8.31432 N·m/(mol·K)
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| :<math>g_0</math> = Gravitational acceleration (9.80665 m/s<sup>2</sup>)
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| :<math>M</math> = Molar mass of Earth's air (0.0289644 kg/mol)
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|
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| Or converted to English gravitational foot-pound-second units:<ref name="conversion"/>
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| Where
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| :<math>{\rho}</math> = Mass density (slug/ft<sup>3</sup>)
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| :<math>{T}</math> = Standard temperature (kelvins)
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| :<math>{L}</math> = Standard temperature lapse rate (degrees Celsius per foot)
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| :<math>{h}</math> = Height above sea level (geopotential feet)
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| :<math>{R^*}</math> = Universal gas constant (8.9494596×10<sup>4</sup> ft<sup>2</sup>/(s·°C))
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| :<math>{g_0}</math> = Gravitational acceleration (32.17405 ft/s<sup>2</sup>)
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| :<math>{M}</math> = Molar mass of Earth's air (0.0289644 kg/mol)
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| 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 <math>\rho_b</math> for ''b'' = 0 is the defined sea level value, <math>\rho_o</math> = 1.2250 kg/m<sup>3</sup> or 0.0023768908 slug/ft<sup>3</sup>. Values of <math>\rho_b</math> 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 <math>h = h_{b+1}</math> <ref name=USSA1976>U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976.</ref>
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| In these equations, ''g''<sub>0</sub>, ''M'' and ''R''<sup>*</sup> are each single-valued constants, while <math>\rho</math>, ''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"/>
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| {| class="wikitable"
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| |-
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| ! rowspan="2"|Subscript ''b''
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| ! colspan="2"|Height Above Sea Level (''h'')
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| ! colspan="2"|Mass Density (<math>\rho</math>)
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| ! rowspan="2"|Standard Temperature (''T''')<br> (K)
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| ! colspan="2"|Temperature Lapse Rate (''L'')
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| |-
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| ! (m) !! (ft)!! (kg/m<sup>3</sup>) !! (slugs/ft<sup>3</sup>) !! (K/m) !! (K/ft)
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| |-
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| | align="center" |0
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| | align="center" |0
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| | align="center" |0
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| | align="center" |1.2250
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| | align="center" |2.3768908 x 10<sup>−3</sup>
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| | align="center" |288.15
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| | align="center" |-0.0065
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| | align="center" |-0.0019812
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| |-
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| | align="center" |1
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| | align="center" |11,000
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| | align="center" |36,089.24
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| | align="center" |0.36391
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| | align="center" |7.0611703 x 10<sup>−4</sup>
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| | align="center" |216.65
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| | align="center" |0.0
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| | align="center" |0.0
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| |-
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| | align="center" |2
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| | align="center" |20,000
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| | align="center" |65,616.79
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| | align="center" |0.08803
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| | align="center" |1.7081572 x 10<sup>−4</sup>
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| | align="center" |216.65
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| | align="center" |0.001
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| | align="center" |0.0003048
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| |-
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| | align="center" |3
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| | align="center" |32,000
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| | align="center" |104,986.87
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| | align="center" |0.01322
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| | align="center" |2.5660735 x 10<sup>−5</sup>
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| | align="center" |228.65
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| | align="center" |0.0028
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| | align="center" |0.00085344
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| |-
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| | align="center" |4
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| | align="center" |47,000
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| | align="center" |154,199.48
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| | align="center" |0.00143
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| | align="center" |2.7698702 x 10<sup>−6</sup>
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| | align="center" |270.65
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| | align="center" |0.0
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| | align="center" |0.0
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| |-
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| | align="center" |5
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| | align="center" |51,000
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| | align="center" |167,322.83
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| | align="center" |0.00086
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| | align="center" |1.6717895 x 10<sup>−6</sup>
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| | align="center" |270.65
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| | align="center" |-0.0028
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| | align="center" |-0.00085344
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| |-
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| | align="center" |6
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| | align="center" |71,000
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| | align="center" |232,939.63
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| | align="center" |0.000064
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| | align="center" |1.2458989 x 10<sup>−7</sup>
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| | align="center" |214.65
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| | align="center" |-0.002
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| | align="center" |-0.0006096
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| |}
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| ==Derivation==
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| The barometric formula can be derived fairly easily using the [[ideal gas law]]:
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| :<math> \rho = \frac{M \cdot P}{R^* \cdot T}</math>
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| When density is known:
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| :<math> P = \frac{\rho \cdot {R^*} \cdot T}{M}</math>
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| And assuming that all pressure is [[Hydrostatic pressure|hydrostatic]]:
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| :<math> dP = - \rho g\,dz\,</math>
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| Dividing the <math> dP </math> by the <math> P </math> expression we get:
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| :<math> \frac{dP}{P} = - \frac{M g\,dz}{R^*T}</math>
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| [[Integral|Integrating]] this expression from the surface to the altitude ''z'' we get:
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| :<math> P = P_0 e^{-\int_{0}^{z}{M g dz/R^*T}}\,</math>
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| Assuming constant temperature, molar mass, and gravitational acceleration, we get the barometric formula:
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| :<math> P = P_0 e^{-M g z/R^*T}\,</math>
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| In this formulation, <math>R^*</math> is the [[gas constant]], and the term <math>R^*T/M g</math> gives the [[scale height]] (approximately equal to 8.4 km for the [[troposphere]]).
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| (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)
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| == Estimating the temperature==
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| Assuming that the only energy source is from the sun, and the [[albedo]] is constant throughout the planet, we can get an estimate of a constant temperature.
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| The solar energy flux at a distance <math>R_\mathrm{AU}</math> can be estimated as:
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| :<math> f_\odot= \frac{L_\odot}{4\pi R^2_\mathrm{AU}},</math>
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| where <math>L_\odot</math> is the [[solar luminosity]]. The actual incoming energy can be estimated as
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| :<math>E_{in}=f_\odot (1-\alpha)\pi R^2_\oplus</math>
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| where <math>\alpha</math> is the [[albedo]] of the planet. <math>R_\oplus</math> is the radius of the planet and <math>R_\mathrm{AU}</math> is the distance to the [[Sun]] in [[astronomical unit]]s.
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| The outgoing energy can be estimated using [[Stefan–Boltzmann law|the Stefan-Boltzmann's law]]
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| :<math>E_{out}=4\pi R^2_\oplus \sigma T^4_{eq}</math>
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| where <math>\sigma</math> is the [[Stefan–Boltzmann constant]] and <math>T_{eq}</math> is the [[temperature]] at [[wikt:equilibrium|equilibrium]].
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| Solving the equation:
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| :<math>E_{in}=E_{out} \,</math>
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| leads to the following estimate of a planet's temperature
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| :<math>T_{eq}=\sqrt[4]{\frac{L_\odot (1-\alpha)}{16 \pi \sigma R^2_\mathrm{AU}}}</math>
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| which for Earth is about 255 K or −18 °C
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| ==See also==
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| *[[NRLMSISE-00]]
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| == References ==
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| <references/>
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| {{DEFAULTSORT:Barometric Formula}}
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| [[Category:Atmosphere]]
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