Page:On the Influence of Carbonic Acid in the Air upon the Temperature of the Ground.pdf/20

Stefan's law of radiation, which is now generally accepted, holds good, or in other words that the quantity of heat (${\text{W}}$) that radiates from a body of the albedo $(1-\nu )$ and temperature ${\text{T}}$ (absolute) to another body of the absorption-coefficient $\beta$ and absolute temperature $\theta$ is ${\text{W}}=\nu \beta \gamma ({\text{T}}^{4}-\theta ^{4})$, where $\gamma$ is the so-called radiation constant ($1.21\cdot 10^{-12}$ per sec. and cm.²). Empty space may be regarded as having the absolute temperature $0$^[1].

Provisionally we regard the air as a uniform envelope of the temperature $\theta$ and the absorption-coefficient $\alpha$ for solar heat; so that if ${\text{A}}$ calories arrive from the sun in a column of 1 cm.² cross-section, $\alpha {\text{A}}$ are absorbed by the atmosphere and $(1-\alpha ){\text{A}}$ reach the earth's surface. In the ${\text{A}}$ calories there is, therefore, not included that part of the sun's heat which by means of selective reflexion in the atmosphere is thrown out towards space. Further, let $\beta$ designate the absorption-coefficient of the air for the heat that radiates from the earth's surface; $\beta$ is also the emission-coefficient of the air for radiation of low temperature–strictly 15°; but as the spectral distribution of the heat varies rather slowly with the temperature, $\beta$ may be looked on as the emission-coefficient also at the temperature of the air. Let the albedo of the earth's crust be designated by $(1-\nu )$, and the quantities of heat that are conveyed to the air and to the earth's surface at the point considered be ${\text{M}}$ and ${\text{N}}$ respectively. As unit of time we may take any period: the best choice in the following calculation is perhaps to take three months for this purpose. As unit of surface we may take 1 cm.², and for the heat in the air that contained in a column of 1 cm.² cross-section and the height of the atmosphere. The heat that is reflected from the ground is not appreciably absorbed by the air (see p. 252), for it has previously traversed great quantities of water-vapour and carbonic acid, but a part of it may be returned to the ground by means of diffuse reflexion. Let this part not be included in the albedo $(1-\nu )$. $\gamma$, ${\text{A}}$, $\nu$, ${\text{M}}$, ${\text{N}}$, and $\alpha$ are to be considered as constants, $\beta$ as the independent, and $\theta$ and ${\text{T}}$ as the dependent variables.

Then we find for the column of air $\beta \gamma \theta ^{4}=\beta \gamma \nu ({\text{T}}^{4}-\theta ^{4})+\alpha {\text{A}}+{\text{M}}$. (1)

The first member of this equation represents the heat

1. Langley, 'Prof. Papers,' No. 15, p. 122. "The Temperature of the Moon," p. 206.