The dynamics of temperature for the surface of a spherical body without an atmosphere and subject to solar radiation are given by the equation

C(dT/dt) = ¼A(1-a)ψ − AσT⁴

where

• T is the absolute temperature of the surface
• t is time
• C is the heat capacity of the surface
• A is the area of the surface
• a is the albedo
• ψ is the solar intensity
• σ is the Stefan-Boltzmann constant

When there is an atmosphere containing greenhouse gases a certain proportion of the outgoing thermal radiation is absorbed and a portion c is radiated back to the surface. The amount of radiation depends upon the absolute temperature of the atmosphere T_a.

In this case there are two equations for the dynamics of temperature

C(dT/dt) = ¼A(1-a)ψ − AσT⁴ + bcAσT_a⁴
and
C_a(dT_a/dt) = bAσT⁴ − AσT_a⁴

The equilibrium temperatures are the values such that dT/dt=0 and dT_adt=0. The latter condition implies that

bT⁴ = T_a⁴
and thus
T_a = b^1/4T

The substitution of this expression for T_a results in the following equation for the equilibrium subface temperature

¼(1-a)ψ = σT⁴ − b²cσT⁴
which reduces to
¼(1-a)ψ = σ(1−b²c)T⁴
and hence
T⁴ = (¼(1-a)ψ/σ)/(1−b²c)
and
T = (¼(1-a)ψ/σ)^1/4/(1−b²c)^1/4

Among other things the equilibrium surface temperature depends upon the intensity of solar radiation per unit area.

The Variations in the Intensity of Solar Radiation

The parameter denoted here as ψ is sometimes called the solar constant. It is not however constant. It varies inversely with the square of the distance of the Earth's surface from the Sun. At its closest point, perihelion, the Earth is about 91.4 million miles from the Sun. This occurs in early January of each year. At its fartherest point, aphelion, in early July the Earth is about 94.5 million miles from the Sun. The square of the distance thus varies from 8353.96 to 8930.25, a 6.9 percent variation. Thus the Earth gets 6.9 percent more radiant energy in January at perihelion than it does in July at aphelion. The equilibrium temperature varies with the fourth root of ψ so it would be 1.6816 percent higher in January than in July. With an average temperature of 287 K° this would mean a slightly over 4.8 C° difference in the Earth's temperature in January compared with July. No such actual temperature difference prevails because the equilibrium temperature is ever achieved.

The accepted estimates of monthly average surface temperatures (land and sea) for the 20th century are:

The graph of these data is shown below.

The variation in absolute temperature between the high of July and low in January is 1.33 percent. The temperature difference is 3.8 C°. This is smaller by 20 percent than the amount which the monthly variation in solar intensity would account for but a reasonably good approximation. The difference in the magnitudes could be accounted for by the monthly variation in ice and snow cover. The Northern Hemisphere with its greater land area in the high latitudes would have much greater reflectance in its winter than the Southern Hemisphere would have in its winter. However the ice and snow cover are only one component of the factors which affect the albedo of the Earth. The cloud cover is far larger than the polar snow and ice fields.

The fact that the maximum comes in July and the minimum in January instead of vice versa is more of a problem. This could be just a bias in the estimating procedure toward Northern Hemispheric data.

(To be continued.)