Radiative Processes in the Climate System - Climate Dynamics - page 69

the fact that half the earth is receiving no solar energy at any given time, and
also for the oblique angle of incidence of the solar rays away from the subsolar
point. Figure 4.2b shows that the earth intercepts energy from the sun over a
cross-sectional area of p R 2 , where R is the average radius of the earth, so the
total energy received by the earth is p R 2 S 0 . This energy is shared over the sur-
face of the earth, which has an area of 4p R 2 . Thus, the average rate at which
solar energy is incident on each unit area of the earth is

=== SR S

2

0 0

2

S 342 W/m . (4.8)

INC

4

2

4 π R

Finally, to calculate S ABS we need to take into account that the earth does
not absorb all the solar energy incident on it. About 31% of the incident solar
radiation is reflected back to space from clouds, molecules, and particulates
in the atmosphere, and the surface. The fraction of incident solar radiation
reflected by the surface and the atmosphere together is the planetary albedo , a.
Taking the planetary albedo into account, we have

S (1 − α )

0

2

S =−= S (1 α ) = 236 W/m . (4.9)

ABS INC

4

Substituting Eq. 4.9 into Eq. 4.6 allows us to calculate the radiative equilib-
rium temperature of the earth as

R

S (1 − α )

V 14 /

S
S

W
W

0

T = = 254 K . (4.10)

E

4 σ

T X

4.3 HOW CONSTANT IS THE
SOLAR CONSTANT?

Although S 0 is called the solar constant, its value changes. Figure 4.2 shows a
composite of daily (gray lines) and annual mean (black line) values of the solar
constant observed over a period of 35 years by various satellites. The 11-year
sunspot cycle is evident, with high values of the solar constant associated with
high sunspot counts (solar maximum). Differences in annual mean values of
S 0 between solar minimum and solar maximum are about 1 W/m 2 . Day- to- day
variations in the solar constant are larger during solar maxima and subdued
during solar minima. During solar maxima, the sun's rotation can result in
variations in S 0 of up to 4 W/m 2 (about 0.3%) within the 30-day rotation
period as the rotation brings magnetically active regions (sunspots) to face the
earth.
In addition to these variations in solar luminosity on daily, monthly, interan-
nual, and decadal time scales associated with solar activity, the solar constant
changes on millennial time scales and longer. Variations over tens of thou-
sands of years are associated with the Milankovitch cycles (section 3.5), and
solar evolution changes the solar constant over billions of years. According to
models of stellar evolution, the sun was significantly cooler billions of years