246 
one, the pressure and density of each gas would be 
distributed according to the well-known exponential 
law of Dalton. Thus if, for a gas, m is the molecular 
mass, P the partial pressure at a height h, and Po that 
at the ground, then 
Such an atmosphere is said to be in isothermal 
equilibrium. 
The terrestrial atmosphere is, however, subject to 
turbulence due to heating and other causes and is 
characterized by convective motions. There is therefore 
a tendency for adiabatic equilibrium to be set up, since 
conduction in gases is very slow. In the ideal case of 
adiabatic equilibrium an element of gas transferred 
from one place to another does not lose or gain any 
heat by conduction and takes up the requisite tem- 
perature and pressure in its new position. The density 
(o) distribution with height in this case is given by 
anf 
where A is equal to Po/po’ and y is the ratio of the 
specific heats of air at constant pressure and at constant 
volume. 
If the troposphere were in ideal adiabatic equilibrium, 
then the lapse rate—the fall of temperature with height 
—would have been 9.8C km. Also, theoretically, an 
atmosphere in ideal adiabatic equilibrium has a natural 
limit. From the expression for p it is easily seen that p 
is equal to zero at h = yPo/[pog(y—1)]. For the ter- 
restrial atmosphere in ideal adiabatic equilibrium, this 
limit would be 27.5 km. However, owing to various 
factors, the troposphere is not in perfect adiabatic 
equilibrium. The actual lapse rate is only 5C km. 
Also, the height of the troposphere varies between 
8 km over polar regions to about 18 km over equatorial 
regions, instead of averaging 27.5 km. The limit of the 
adiabatic atmosphere, under actual conditions, cannot 
be a surface separating a region of perfect vacuum above 
from a region containing gas molecules below. Because 
of thermal agitation, molecules from the atmosphere 
below will constantly be evaporating, as 1t were, across 
the separating surface in much the same manner as 
molecules evaporate from the body of a liquid to the 
space above it. The region above the natural limit will 
therefore contain molecules. The atmosphere in adi- 
abatic equilibrium may thus be considered to be capped 
by a region which we call the outer atmosphere. The 
outer atmosphere is necessarily in isothermal equi- 
librium. 
According to the simple consideration given above, the 
outer atmosphere need be only a few metres thick. 
However, account has to be taken of the fact that 
radiation from the relatively hot gases below is con- 
tinually reaching and heating this isothermal layer 
and that in order for the atmospheric gases to be in 
equilibrium the radiation and the absorption of heat 
by each element must balance. These considerations 
THE UPPER ATMOSPHERE 
lead to the conclusion that the isothermal layer above 
the adiabatic layer, instead of being a few metres thicls, 
extends to great heights. 
Assuming that the atmosphere consists of two layers, 
the lower in adiabatic and the upper in isothermal 
equilibrium, and working out the condition for radiative 
equilibrium, it has been shown that for an atmosphere 
of uniform constitution, the adiabatic state cannot 
extend to a height greater than that given by P = 
P,/2, where Po is the pressure at the surface of the 
earth. For an atmosphere of nonuniform constitution, 
the adiabatic layer may extend to greater heights. 
Under certain assumptions concerning the amounts of 
radiation absorbed by water vapour at different heights, 
it has been shown that the height of the adiabatic layer 
cannot exceed that given by P = P)/4 which, for the 
case of the terrestrial atmosphere, is 10.5 km. This, as 
indicated above, is roughly the height of the tropo- 
sphere. 
Limit of the Outer Atmosphere. The outer atmos- 
phere, being more or less in isothermal equilibrium, has 
no natural limit, though the point of minimum density 
resulting from the balance of the centrifugal and the 
gravitational forces is sometimes spoken of as the 
limit of the outer atmosphere. But long before this 
height is attained (distance 6.6 earth-radii in the equa- 
torial plane) a limit of the outer atmosphere is reached 
due to the rarity of collisions and the escape of molecules. 
We are thus led to enquire into the limit of the outer 
atmosphere or, more popularly, to ask, Where does 
the atmosphere end? The question is of considerable 
importance, since it is closely related to the question of 
the escape of the atmospheric gases from the gravita- 
tional field of the earth. 
A limit of the outer atmosphere may be understood 
from the following considerations. In any region above 
the surface of the earth, the atmosphere in the ordinary 
sense of the term can exist only if the molecules in the 
region are prevented from escaping by collisions with 
the molecules above. Now, as one goes up, the atmos- 
phere becomes increasingly rarefied and the frequency 
of collisions between the gas particles becomes smaller 
and smaller. Ultimately a height is reached where the 
collisions are so few and far between that a molecule 
from the denser atmosphere below has little chance of 
returning to earth by collision with molecules above. 
The height at which this state of affairs prevails may be 
said to be the limit of the atmosphere. There is, of 
course, a transition region of considerable thickness 
from which a particle, projected upwards, may escape 
without collisions. The mean height of this transition 
region is variously estimated to lie between 500 and 
1000 km [98]. 
Escape of Atmospheric Gases—The Problem of 
Helium. This leads to the question of atmospheric 
gases overcoming the gravitational field and escaping 
from the earth. The process of escape may best be 
understood as follows (after Milne [66] and Jones [51]). 
Let us suppose that an observer ascends upwards from 
a layer where molecular density is small but appreciable. 
If the molecules were opaque, the hemispherical sky 
