Geophysical Altitudes

 

Three distinct kinds of "altitude" are commonly used when discussing the vertical heights of objects in the atmosphere above the Earth's surface. The first is simple geometric altitude, which is what would be measured by an ordinary tape measure. However, for many purposes we are more interested in the pressure altitude, which is actually an indication of the ambient pressure, expressed in terms of the altitude at which that pressure would exist on a "standard day". Finally, there is the so-called geopotential altitude, which is really a measure of the specific potential energy at the given height (relative to the Earth's surface), converted into a distance using the somewhat peculiar assumption that the acceleration of gravity is constant, equal to the value it has at the Earth's surface.

 

The geometric altitude is fairly self-explanatory, but is often difficult to measure accurately in real situations (such as from an airplane), because of irregularities in the terrain. Moreover, for the purposes of operating air-breathing equipment (such as human lungs or jet engines), what really matters is ambient pressure, which of course corresponds to the density of the ambient air. If we let z denote the geometric altitude above sea level, and if we let p(z), r(z), and g(z) denote the atmospheric pressure and density and the acceleration of gravity at the height z, then the rate of change of pressure with respect to geometric altitude is given by the "hydrostatic equation"

 

 

Also, treating air as an ideal gas, we have

 

 

where T(z) is the temperature and R is gas constant for air. One possible set of units for these quantities is

 

z    -   ft

p    -   lbf/ft2

r  -   slugs/ft3

T    -   deg R (degrees Rankine)

R    -   1716  ft lbf / (slug deg R)

   

Combining equations (1) and (2) gives

 

 

Integrating from z = 0 (sea level, where p = p0) to z = h (where p = ph), we have

 

 

Now, for aviation purposes we are typically confined to the region below about 45000 ft geometric altitude, and the acceleration of gravity at that height is not very different than at sea level, so to a good approximation we can treat g(z) as a constant, equal to 32.174 ft/sec2. Also, the "standard day" temperature profile is defined to be 59 F at sea level, and dropping linearly to -70 F at the tropopause, which is at 36089 feet above sea level. Above this altitude the standard day temperature is constant, up to well past 50000 ft. Therefore, over the range from sea level to 36089 we can write equation (4) as

 

 

where

 

Evaluating the integral in (5) and solving the resulting expression for h gives

 

 

With p0 = 14.696, this expression for h is defined as the "pressure altitude" corresponding to the ambient pressure ph, for all values of ph greater than or equal to 3.2824, which represents the tropopause. Inserting the values of the constants, this gives the formula for pressure altitude as a function of ambient pressure

 

 

For pressures less than 3.2824, equation (5) gives

 

 

where asterisks indicate conditions at the tropopause (on a standard day)

 

h* = 36089 ft

p* = 3.2824 psia

T* = 389.67 deg R (-70 deg F)

 

Solving (7) for h gives the formula for pressure altitude as a function of ambient pressure for conditions above the tropopause

 

 

Inserting the values of the constants gives

 

 

A plot of ambient pressure versus pressure altitude is shown below.

 

 

Needless to say, this derivations depends on the particular temperature profile that we have assumed. We used the "standard day" profile based on the 1962 Geophysical Survey. There are also conventional definitions of "Hot Day" and "Cold Day" temperature profiles given in the military and commercial literature, and for any such profile we can integrate equation (4), usually with the assumption that g is constant, to give the pressure variation from sea level (where the barometric pressure is typically 14.696 psia, although the computations can be adjusted for particular barometric pressures if necessary.) This leads to a different definition of pressure altitude for each temperature profile.

 

So far we have assumed that the acceleration of gravity was constant over the range of interest, but in fact there is a slight reduction in g as we go up in altitude. This can be significant when dealing with the energy of an object, especially if we need to know very precisely it's position as a function of actual energy. For this purpose, people sometimes use "geopotential altitude". The potential energy required to move a mass m from sea level to the geometric altitude Z is

 

 

where re is the radius of the Earth. Now, the acceleration of gravity, g, is a function of the distance r from the Earth's center, so if we let re denote the radius of the Earth (sea level, neglecting non-spherical effects), we have r = re + z, and so

 

 

where M is the mass of the Earth and G is Newton's gravitational constant. The force required to raise the object is F(z) = g(z)m, so we can substitute this into (9) and integrate to give

 

 

By definition, the geopotential altitude is

 

 

Substituting for the potential energy and the acceleration of gravity at the Earth's surface, and simplifying, gives the geopotential altitude as a function of the geometric altitude

 

 

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