]> The Hydrostatic Equations
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## Geopotential and Geometric Altitude

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The standard atmosphere is defined in terms of geopotential altitude. The idea behind this concept is that a small change in geopotential altitude will make the same change in gravitational potential energy as the geometric altitude at sea level. Mathematically, this is expressed as g dZ = G dH where H stands for geopotential altitude and Z stands for geometric altitude, g is the acceleration of gravity and G is the value of g at sea level. The value of g varies with altitude and is shown in elementary physics texts to vary as

 $g G = ( E Z+E )2$ (1)

where E is the radius of the earth. So,

 $dH= g G dZ= ( E Z+E )2 dZ$ (2)

and integrating yields

 $∫0 H dH= ∫0 Z ( E Z+E )2 dZ$ (3)
 $H= EZ E+Z$ (4)
 $Z= EH E-H$ (5)

While Z and H are virtually identical at low altitudes, you can calculate that Z = 86 km corresponds to H=84.852 km. (Use 6356 km for the radius of the earth). At this altitude, g is 0.9735 times the value at sea level. If you don't like the definition of H as a differential, you can regard H=EZ / (E+Z) as the definition of H and then derive dH/dZ=g/G.

## The Perfect Gas Law

The equation of state of a perfect gas is

 $ρ= MP RT$ (6)

where P is the atmospheric pressure, R is the universal gas constant, rho is the density, T is the absolute temperature and M is the mean molecular weight of air. M is assumed constant (=28.9644) up to 86 km where dissociation and diffusive separation become significant. R is 8.31432 joules K-1mol-1.

## The Hydrostatic Equations

The fundamental equation is

 $dP=-ρgdZ= -ρGdH$ (7)

and using the perfect gas law gives

 $dP=- MP RT GdH$ (8)

This equation leads directly to the calculation of pressure in the standard atmosphere. Within an atmospheric layer, the temperature T is a linear function of the geopotential altitude H.

 $T= Tb +L(H- Hb )$ (9)

where $L$ is the constant gradient of temperature and $Tb$ and $Hb$ are the temperature and geopotential altitude at the base of the layer. The hydrostatic equation then becomes

 $dP=- GM R P ( Tb +L(H- Hb )) dH$ (10)

and the pressure at any value of H within this layer is found by integration of this equation

 $∫ Pb P dP P =- ∫ Hb H GM R( Tb +L(H- Hb )) dH$ (11)

The right hand integral takes different forms, depending upon whether L is zero or not. When L=0, the integral is

 $ln( P Pb )=- GM Tb R (H- Hb )$ (12)

and when L is not zero, the integral is

 $ln( P Pb )=- GM RL ln( Tb +L(H- Hb ) Tb )$ (13)

Writing these equations in exponential form, when L=0, then

 $P Pb =exp(- GM(H- Hb ) RTb )$ (14)

and when L is not zero,

 $P Pb = ( Tb +L(H- Hb ) Tb )- GM RL$ (15)

You can see now why geopotential altitude is used for the definition of the standard atmosphere. If Z were used, then g would appear in the equations instead of G and the variation of g with altitude would have to be included in the integration, making a rather complicated equation.

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