Movatterモバイル変換

[0]ホーム

Jump to content

Barometric formula

Edit links

From Wikipedia, the free encyclopedia

Formula used to model how air pressure varies with altitude

For broader coverage of this topic, seeVertical pressure variation.

Thebarometric formula is aformula used to model how theair pressure (orair density) changes withaltitude.

Model equations

[edit]

See also:Atmospheric pressure

Pressure as a function of the height above the sea level

TheU.S. Standard Atmosphere gives two equations for computing pressure as a function of height, valid from sea level to 86 km altitude. The first equation is applicable to the atmospheric layers in which the temperature is assumed to vary with altitude at a non null temperature gradient of $L_{M,b}$ : $P=P_{b}\cdot \left[{\frac {T_{M,b}}{T_{M,b}+L_{M,b}\cdot \left(H-H_{b}\right)}}\right]^{\frac {g_{0}'\cdot M_{0}}{R^{*}\cdot L_{M,b}}}$ .^[1]^: 12

The second equation is applicable to the atmospheric layers in which the temperature is assumed not to vary with altitude (zero temperature gradient): $P=P_{b}\cdot \exp \left[{\frac {-g_{0}'\cdot M_{0}\left(H-H_{b}\right)}{R^{*}\cdot T_{M,b}}}\right]$ ,^[1]^: 12 where:

$P_{b}$ = reference pressure
$T_{M,b}$ = reference temperature (K)
$L_{M,b}$ = temperature gradient (K/m), e.g. -6.5 K/km at sea level. This is thelapse rate with the opposite sign convention.
$H {\displaystyle H}$ =geopotential height at which pressure is calculated (m)
$H_{b}$ = geopotential height of reference levelb (meters; e.g.,H_b = 11 000 m)
$R^{*}$ =universal gas constant: taken to be 8.31432×10³ N·m/(kmol·K)^[1]^: 3, although the actual constant's value in those units rounds to 8.31446.
$M_{0}$ = mean molar mass of air at sea level: 28.9644 kg/kmol as of 1976^[update]^[1]^: 9
- NeitherR^* norM₀ use correctSI units - both usekmol rather thanmol. In particular, care must be taken when replacingM₀ with more current values.
$g_{0}'$ = Thegravitational acceleration in units of geopotential height, 9.80665 m/s²^[1]^: 3

Or converted toimperial units:^[2]

$P_{b}$ = reference pressure
$T_{M,b}$ = reference temperature (K)
$L_{M,b}$ = temperature gradient (K/ft)
$H {\displaystyle H}$ = height at which pressure is calculated (ft)
$H_{b}$ = height of reference levelb (feet; e.g.,H_b = 36,089 ft)
$R^{*}$ =universal gas constant; using feet, kelvins, and (SI)moles: taken to be roughly8.9494596×10⁴ lb_m·ft²/(lb_m-mol·K·s²) by correctly converting the (incorrectly) taken constant from metric to imperial.
$g_{0}$ =gravitational acceleration: 32.17405 ft/s²
$M {\displaystyle M}$ = mean molar mass of Earth's air: 28.9644 lb/lb‑mol (both lbs arepound-mass,notpound-force).

The value of subscriptb ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. In these equations,g₀,M andR^* are each single-valued constants, whileP,L,T, andH are multivalued constants in accordance with the table below. The values used forM,g₀, andR^* are in accordance with theU.S. Standard Atmosphere, 1976, and the value forR^* in particular does not agree with standard values for this constant.^[1]^: 3 The reference value forP_b forb = 0 is the defined sea level value,P₀ = 101 325Pa or 29.92126 inHg. Values ofP_b ofb = 1 throughb = 6 are obtained from the application of the appropriate member of the pair equations 1 and 2 for the case whenH =H_b+1.^[1]^: 12

Subscriptb	Geopotential height above MSL(H)^[1]^: 3		Static pressure		Standard temperature (K)	Temperature gradient^[1]^: 3		Exponent g0 M / R L
Subscriptb	(km)	(ft)	(Pa)	(inHg)	Standard temperature (K)	(K/km)	(K/ft)	Exponent g0 M / R L
0	0	0	101 325	29.9213	288.15	-6.5	-0.0019812	-5.25588
1	11	36 089	22 632.1	6.68324	216.65	0.0	0.0	—
2	20	65 617	5 474.89	1.616734	216.65	1.0	0.0003048	34.1626
3	32	104 987	868.019	0.256326	228.65	2.8	0.00085344	12.2009
4	47	154 199	110.9063	0.0327506	270.65	0.0	0.0	—
5	51	167 323	66.9389	0.0197670	270.65	-2.8	-0.00085344	-12.2009
6	71	232 940	3.95642	0.00116833	214.65	-2	-0.0006096	-17.0813

Density can be calculated from pressure and temperature using

$\rho ={\frac {P\cdot M_{0}}{R^{*}\cdot T_{M}}}={\frac {P\cdot M}{R^{*}\cdot T}}$ ,^[1]^: 15 where

$M_{0}$ is the molecular weight at sea level
$M {\displaystyle M}$ is the mean molecular weight at the altitude of interest
$T {\displaystyle T}$ is the temperature at the altitude of interest
$T_{M}=T\cdot {\frac {M_{0}}{M}}$ is the molecular-scale temperature.^[1]^: 9

The atmosphere is assumed to be fully mixed up to about 80 km, so $M=M_{0}$ within the region of validity of the equations presented here.^[1]^: 9

Alternatively, density equations can be derived in the same form as those for pressure, using reference densities instead of reference pressures.^{[citation needed]}

This model, with its simple linearly segmented temperature profile, does not closely agree with the physically observed atmosphere at altitudes below 20 km. From 51 km to 81 km it is closer to observed conditions.^[1]^: 1

Derivation

[edit]

The barometric formula can be derived using theideal gas law: $P={\frac {\rho }{M}}{R^{*}}T$

Assuming that all pressure ishydrostatic: $dP=-\rho g\,dz$ and dividing this equation by $P {\displaystyle P}$ we get: ${\frac {dP}{P}}=-{\frac {Mg\,dz}{R^{*}T}}$

Integrating this expression from the surface to the altitudez we get: $P=P_{0}e^{-\int _{0}^{z}{Mgdz/R^{*}T}}$

Assuming linear temperature change $T=T_{0}-Lz$ and constant molar mass and gravitational acceleration, we get the first barometric formula: $P=P_{0}\cdot \left[{\frac {T}{T_{0}}}\right]^{\textstyle {\frac {Mg}{R^{*}L}}}$

Instead, assuming constant temperature, integrating gives the second barometric formula: $P=P_{0}e^{-Mgz/R^{*}T}$

In this formulation,R^* is thegas constant, and the termR^*T/Mg gives thescale height (approximately equal to 8.4 km for thetroposphere).

The derivation shown above uses a method that relies on classical mechanics. There are several alterantive derivations, the most notable are the ones based on thermodynamic forces and statistical mechanics.^[3]

(For exact results, it should be remembered that atmospheres containing water do not behave as anideal gas. Seereal gas orperfect gas orgas for further understanding.)

Barosphere

[edit]

The barosphere is the region of a planetary atmosphere where thebarometric law applies. It ranges from the ground to thethermopause, also known as the baropause. Above this altitude is theexosphere, where the atmospheric velocity distribution isnon-Maxwellian due to high velocity atoms and molecules being able to escape the atmosphere.^[4]^[5]

References

[edit]

^^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^mU.S. Standard Atmosphere, 1976(PDF) (Report). Washington, D.C.: U.S. Government Printing Office. October 1976. NTRS 19770009539. Retrieved2025-06-29.
^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.
^Lente, G.; Ősz, K. (2020)."Barometric formulas: various derivations and comparisons to environmentally relevant observations".ChemTexts.6 (2): 13.doi:10.1007/s40828-020-0111-6.
^Bauer, Siegfried; Lammer, Helmut (2013).Planetary Aeronomy: Atmosphere Environments in Planetary Systems. Physics of Earth and Space Environments. Springer Science & Business Media. pp. 4–5.ISBN 978-3-662-09362-7.
^Hargreaves, John Keith (1992).The Solar-Terrestrial Environment: An Introduction to Geospace - the Science of the Terrestrial Upper Atmosphere, Ionosphere, and Magnetosphere. Cambridge Atmospheric and Space Science Series. Cambridge University Press. p. 100.ISBN 9780521427371.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Barometric_formula&oldid=1329091954"

Categories:

Hidden categories:

[8]ページ先頭