Isothermal Atmosphere
We now examine the static equilibrium of a bulk of gas under
gravity, taking into account its compressibility. Equation (2)
14 AERODYNAMICS [CH.
applies, but specification is needed of the relationship between p
and p. The simple assumption made in the present article is that
appropriate to Boyle's law, viz. constant temperature TO, so that />/p
remains constant. From (2) :
*.-*. 9g From (9) : 1 _ J5r 9g P ' Hence : ,. BiQ = - dh. P
Integrating between levels Ax and h2 , where p = pt and p2 respectively,
BTO log (pjpj = h, - h, . . . (10)
The logarithm in this expression is to base e. Throughout this
book Napierian logarithms will be intended, unless it is stated
otherwise. The result (10) states that the pressure and therefore
the density of a bulk of gas which is everywhere at the same temperature
vary exponentially with altitude.
The result, although accurately true only for a single gas, applies
with negligible error to a mass of air under isothermal conditions,
provided great altitude changes are excluded. The stratosphere is
in conductive equilibrium, the uniform temperature being about
- 55 C. The constitution of the air at its lowest levels is as given
in Article 1. As altitude increases, the constitution is subject to
Dalton's law : a mixture of gases in isothermal equilibrium may be
regarded as the aggregate of a number of atmospheres, one for each
constituent gas, the law of density variation in each atmosphere
being the same as if it constituted the whole. Hence argon and
other heavy gases and subsequently oxygen, nitrogen, and neon will
become rarer at higher levels. The value of B for the atmosphere
will consequently increase with altitude, although we have assumed
it constant in order to obtain (10). The variation of B for several
miles into the stratosphere will, however, be small. At greater
altitudes still the temperature increases again.
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