LONG-WAVE RADIATION 41 
the simultaneous change in /ty. Hence 
SSApv => ko. 
Tf this expression is mserted into the square-root for- 
mula (page 35), we obtain 
L = ~/Sms/Ap. 
The theory of line broadening by atomic collisions 
demands a simple proportionality of 6 with p. Conse- 
quently, the absorption L must vary as 1/p. Schnaidt 
[46] emphasizes that the measurements by Falcken- 
berg show a proportionality of the absorption coefficient 
k with 1/p. This means that, when an exponential law 
is used for absorption, x/p appears as a factor of m in 
the exponent. However, if the square-root law is used, 
the logical result is that L <«+/mv/p or L « V/p. 
Thus, there exists a contradiction between theory and 
observation. Nowadays the tendency is to place more 
confidence in theory than in measurements. Neverthe- 
less, a repetition of the measurements would be desir- 
able. 
At the various wave lengths the transition from the 
absorption in the center to that on the flanks occurs 
with entirely differmg thicknesses of vapor strata, be- 
cause of the extraordinarily great differences of the 
absorption coefficients in the water-vapor spectrum. 
This leads to a complicated interspersion of absorption 
amplification and attenuation by the air pressure at 
the same level of the atmosphere. However, Moller 
[85] has shown by a rough calculation that, even on 
the assumption that 6 « p, the factor that must be 
applied to the vapor density of the atmosphere for use 
of the standard absorption equations (9), (13), and 
(15) has a value within the limits of p and 1/p up to 
100 mb (16 km). At still greater altitudes this factor 
increases again, because at the extremely low vapor 
content of the stratosphere only the center of the very 
strongest lines absorb. Moller proposes, instead of the 
factor p = p/po, a more complicated one, namely 
pw’ = 0.985 (p/po)®*? + 0.015 (p/po)1. 
This factor has a minimum at around 100 mb and 
mereases at higher levels. However, in the practical 
calculation of the cooling it is found that at these alti- 
tudes the radiation effect of water vapor becomes 
negheible owing to the greatly diminished vapor con- 
tent, and that the radiation of other absorbers pre- 
dominates. Therefore, the rigorous application of the 
correction factor pu’ is unnecessary, as long as there are 
no better observations available for altitudes above 
100 mb which would require more accurate calculations. 
Nevertheless, calculations to determine the effect of 
air pressure on changes of the shape of the lines have 
not been in vain; for, during the past few years, critics, 
on the basis of the necessary simplifications regarding 
this effect, questioned repeatedly the validity of cal- 
culations by means of the radiation diagrams [39]. 
Outgoing Atmospheric Radiation 
By the use of the correction factor » for the effect of 
the air pressure, the radiation diagrams become suitable 
for investigations of the free atmosphere. For a given 
level z, we can calculate the downward radiation R; 
from the atmospheric layers above it and, correspond- 
ingly, the upward radiation U,; from the ground and 
from the atmospheric layers below this level. In the 
radiation diagram the ground is treated as an isothermal 
gas stratum having the temperature 7’) of the ground 
and an infinitely great content of water vapor and 
CQ:. The difference H,; = U, — fk; is then the net 
radiation which penetrates the reference level in an 
upward direction. The same calculation for another 
level z. furnishes H,. The excess radiation H, — E,, 
emitted by the air column Az = 2, — % with air den- 
sity p, causes a cooling 
af 1m Es 
ot PCp 22 — BI 
== =>. (16) 
Roberts’ first studies [42] of the radiation flow # 
indicated that it increases with altitude. Normally this 
increase is very uniform with altitude in a cloudless 
atmosphere having a contimuous vertical distribution 
of temperature and water vapor. However, the air 
density decreases with altitude so that the cooling rate 
which amounts to about 1C per day near the ground 
increases to two or three times that amount higher up 
in the troposphere. Only close to the ground and near 
the tropopause do special conditions prevail (see be- 
low). 
It seems logical to interpret the cooling of the free 
atmosphere as a radiation into space. That, however, 
is not possible, for the shielding by the layers of water 
vapor above it is too great. It is, rather, a process simi- 
lar to heat conduction. Basically, radiation, just as heat: 
conduction, tends to equalize temperature differences. 
Therefore, we may also try to set up for radiation an 
equation that has a form similar to that for heat con- 
duction, namely 
aT /at = K &T/aw?, (17) 
where KK is a “virtual coefficient of conduction of the 
heat radiation,” w is the mass of water vapor 
/ pudz, 
and p» the density of the water vapor. This possibility 
was long ago developed theoretically by Falckenberg 
and Stoecker [18], and was later used as the basis of 
practical estimates by Brunt [9, 10]; here we shall use it 
only for an interpretation of the cooling. Substitution 
of equation (18) into (17) gives 
oT /oat = yore Opw/ 02, (y =a 
(18) 
w= 
—06T/02) 
which is negative, because dp,,/dz < 0. In other words: 
The vapor masses at an equal geometrical distance 
above and below a given altitude are, to be sure, colder 
or warmer by the same temperature difference; but the 
