Page:On the Influence of Carbonic Acid in the Air upon the Temperature of the Ground.pdf/21

radiated from the air (emission-coefficient $\beta$, temperature $\theta$) to space (temperature $0$). The second one gives the heat radiated from the soil (1 cm.², temperature ${\text{T}}$, albedo $1-\nu$) to the air; the third and fourth give the amount of the sun's radiation absorbed by the air, and the quantity of heat obtained by conduction (air-currents) from other parts of the air or from the ground. In the same manner we find for the earth's surface $\beta \gamma \nu ({\text{T}}^{4}-\theta ^{4})+(1-\beta )\gamma \nu {\text{T}}^{4}=(1-\alpha )\nu {\text{A}}+{\text{N}}$. (2)

The first and second members represent the radiated quantities of heat that go to the air and to space respectively, $(1-\alpha )\nu {\text{A}}$ is the part of the sun's radiation absorbed, and ${\text{N}}$ the heat conducted to the point considered from other parts of the soil or from the air by means of water- or air-currents.

Combining both these equations for the elimination of $\theta$, which has no considerable interest, we find for ${\text{T}}^{4}$ ${\text{T}}^{4}={\frac {\alpha {\text{A}}+{\text{M}}+(1-\alpha ){\text{A}}(1+\nu )+{\text{N}}(1+1/\nu )}{\gamma (1+\nu -\beta \nu )}}={\frac {\text{K}}{1+\nu (1-\beta )}}$. (3)

For the earth's solid crust we may, without sensible error, put $\nu$ equal to $1$, if we except the snowfields, for which we assume $\nu =0.5$. For the water-covered parts of the earth I have calculated the mean value of $\nu$ to be $0.925$ by aid of the figures of Zenker^[1]. We have, also, in the following to make use of the albedo of the clouds. I do not know if this has ever been measured, but it probably does not differ very much from that of fresh fallen snow, which Zöllner has determined to be $0.78$, i. e. $\nu =0.22$. For old snow the albedo is much less or $\nu$ much greater; therefore we have assumed $0.5$ as a mean value.

The last formula shows that the temperature of the earth augments with $\beta$, and the more rapidly the greater $\nu$ is. For an increase of 1° if $\nu =1$ we find the following increases for the values of $\nu =0.925$, $0.5$, and $0.22$ respectively: –

$\beta$.	$\nu =0.925$.	$\nu =0.5$.	$\nu =0.22$.
0.65	0.944	0.575	0.275
0.75	0.940	0.556	0.261
0.85	0.934	0.535	0.245
0.95	0.928	0.512	0.228
1.00	0.925	0.500	0.220

This reasoning holds good if the part of the earth's surface

1. Zenker, Die Vertheilung der Wärme auf der Erdoberfläche, p. 54 (Berlin, 1888).