]> 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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