Isothermal Atmosphere

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.