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National Advisory Committee for Aeronautics, Report - Some Approximate Equations for the Standard Atmosphere

A number of factors were considered before the
adoption of an aeronautical standard atmosphere for
use in the United States. A large majority of aero-
nautical requirements are met in any standard atmos-
phere which simply specifies the pressures, tempera-
tures, and densities at various altitudes. For this
reason it was decided that the standard atmosphere
should be based on some simple but definite relation
which gave pressures, temperatures, and densities in
close agreement with the average observed values.
The Weather Bureau, after comparing various for-
mulas with observed data, concluded that Toussaint’s
formula for linear decrease in temperature with alti—
tude up to the lower level of the isothermal atmosphere
at a temperature of —55° 0. not only gave the most
satisfactory results, but it also had the great advantage
of extended official use abroad.1 For these reasons it
was adopted for official use in this country.

In extended use the standard atmosphere has proved
quite satisfactory, but there are certain special cases
in which simple mathematical relations are essential
in order that integrations may be performed. This
condition has led to some criticism of the present
standard atmosphere, chiefly by those who prefer a
simple exponential expression for the density ratio.
It is possible to write approximate equations of various
forms which will give fair to excellent agreement with
the specified values in the standard atmosphere. Such
approximate equations are sufficiently accurate to meet
all ordinary requirements. The need for a systematic
series of these equations is apparent.

In this report no attempt has been made to assemble
more than a representative series of simple approxi-
mate equations of the forms most frequently required.
It should be understood that many other types may
be used and that greater accuracy can be obtained by
the introduction of additional terms or constants. The
complex forms, however, have such limited application
that their inclusion here is hardly justified.