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),
rho(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"
dp
-- = -rho g (1)
dz
Also, treating air as an ideal gas, we have
p = rho R T (2)
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/ft^2
rho - slugs/ft^3
T - degR (degrees Rankine)
R - 1716 ft lbf / (slug degR)
Combinning equations (1) and (2) gives
1 g
--- dp = - ----- dz (3)
p R T
Integrating from z = 0 (sea level, where p = p_0) to z = h
(where p = p_h), we have
h
/ p_h \ 1 / g(z)
ln( ----- ) = - --- | ---- dz (4)
\ p_0 / R / T(z)
0
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/sec^2. 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
h
/ p_h \ g / 1
ln( ----- ) = - --- | -------- dz (5)
\ p_0 / R / T_0 - kz
0
where
k = (59-(-70))/(36089-0) = 3.5745E-3 degR/ft
T_0 = 518.67 degR ( 59 degF )
Evaluating the integral in (5) and solving the resulting expression
for h gives
_ _
T_0 | / p_h \kR/g |
h = --- | 1 - ( ----- ) |
k |_ \ p_0 / _|
With p_0 = 14.696, this expression for h is DEFINED as the "pressure
altitude" corresponding to the ambient pressure p_h, for all values
of p_h greater than or equal to 3.2824, which represents the tropo-
pause. Inserting the values of the constants, this gives the formula
for pressure altitude as a function of ambient pressure
_ _
| / p_h \0.190645 |
h = 145102 | 1 - ( ------ ) | (6)
|_ \14.696/ _|
For pressures less than 3.2824, equation (5) gives
h
/ p_h \ g /
ln( ----- ) = - --- | dz (7)
\ p* / RT* /
h*
where asterisks indicate conditions at the tropopause (on a standard
day)
h* = 36089 ft
p* = 3.2824 psia
T* = 389.67 degR (-70 degF)
Solving (7) for h gives the formula for pressure altitude as a
function of ambient pressure for conditions above the tropopause
RT* / p_h \
h = 36089 - ---- ln( ----- )
g \ p* /
Inserting the values of the constants gives
/ p_h \
h = 36089 - 20783 ln( -------- ) (8)
\ 3.2824 /
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 thr 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
r_e+Z
/
PE = | F(z) dz (9)
/
z=r_e
where r_e 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 r_e denote the radius of the Earth (sea level,
neglecting non-spherical effects), we have r = r_e + z, and so
G M
g(z) = -----------
(r_e + z)^2
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
_ _
| 1 1 |
PE = G M m | --- - ------- |
|_ r_e r_e + Z _|
By definition, the geopotential altitude is
PE
H = --------
m g(0)
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
Z
H = ---------
1 + Z/r_e
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