Variability of the Earth’s orbit: The astronomical theory - Ice Ages and Interglacials: Measurements, Interpretation, and Models - page 249

Note that the distance of the Earth from the Sun is given by:

A ð 1 e 2 Þ

1 þ e cos L s
where A is the half-length of the major axis of the Earth's orbit. If we integrate
this over dL s from 0 to 2

R ¼

, and divide by 2 , we obtain the average value of r :

r ¼ A ð 1 e 2

1 = 2

Þ

Since total annual solar input to the Earth is proportional to the inverse
square of r, it follows that total annual solar input to the Earth is proportional to

ð 1 e 2 Þ 1 = 2

Thus, total annual solar input to the Earth depends slightly on eccentricity—
not on obliquity or the longitude of precession. As we showed in Figure 9.3 , the
full range of variability of e over hundreds of thousands of years is from about
0.01 to about 0.05. Over this range of eccentricity, the variation of total annual
solar input to the Earth is 0.24%. Since the average value of solar intensity
impinging on a square meter of Earth is 342W, the variability of solar input to
the Earth over long time periods due to variations in eccentricity is about
0.0024 342 ¼ 0.8W/m 2 of forcing. This is not large enough to be the cause of ice
ages.
The longitude of perihelion varies with a period of 22,000 years. It is
actually more complicated than that but this simple model is sucient for our
purposes. The effect of a change in the longitude of perihelion is to change the
season that prevails when the Earth is closest to the Sun. If the longitude of
perihelion (L p ) occurs near 90 , perihelion will occur at the NH summer solstice
(June 21) and, thus, solar irradiance will be a maximum in northern summer
(southern winter). If the longitude of perihelion occurs near L p ¼ 270 , perihelion
will occur at the NH winter solstice (December 21) and, thus, solar irradiance will
be a maximum in northern winter (southern summer). Clearly, when L p is near
90 , solar input to high northern latitudes will be near a maximum in summer if
there is significant eccentricity in the Earth's orbit. Today, with the longitude of
perihelion occurring near L p ¼ 250 , solar input to high northern latitudes is near
a maximum in northern winter and a minimum in northern summer.
The combination of slowly changing eccentricity and rapidly changing
longitude of perihelion has a significant effect on the variability of peak solar
intensity at higher latitudes in summer and determines the positions of peaks and
valleys of solar intensity with time at higher latitudes in summer. Slowly changing
eccentricity acts as an amplitude envelope for the more rapidly varying longitude
of perihelion. Eccentricity affects peak solar intensity in summer when the Earth is
closest to the Sun, and the longitude of perihelion determines the season during
which closest approach occurs.
In order to show the dependence of peak solar intensity in summer at high
latitudes on the combination of high eccentricity and the longitude of perihelion