The second law introduced here is Kepler's second law, which can be summarized as gravitational
bodies rotate through equal areas in equal time. In this case, a line drawn from Earth
to Sun sweeps the same area of the orbital plane every day, regardless of where the planet
is in its orbit. Consequently, Earth moves faster when it's near the sun than when it
is far because that's a smaller distance it needs to trace a greater arc to cover the
same area. This means that Earth spends less time close to the sun.
Third and last is the inverse square law. It treats the sun as a point source of light
that distributes energy evenly in all directions. Therefore at a given distance from the sun,
its energy is distributed over a surface area of a sphere; Four Pi r squared. At twice the
distance, its energy is distributed over four times the area and so on. The inverse square
law states that the amount of energy striking an area decreases with the square of the distance
from the sun. Or in mathematical terms, I equals I sub zero over four Pi r squared,
where I equals the solar energy per unit area, I sub zero equals the initial solar energy,
Pi equals 3.14159; R equals the distance from the sun. The distance between the sun and
the Earth varies between 147,000,000 kilometers on January third and 152,000,000 kilometers
on July fourth. Putting these values into this equation means that Earth at its furthest
distance from the sun during the summer in the northern hemisphere receives only 93.5
percent of the solar energy that it receives at its closest distance. Nonetheless the more
acute angle of the northern hemisphere to the sun during July more than compensates
for the further distance. For example, at the Tropic of Cancer, the ratio of solar energy
in June over solar energy in December equals I subzero cosine zero degrees over I subzero
cosine 46.8 degrees, or 146%. Together with the effect of distance, this equals 93.5 percent
and 146 percent, which equals in total a 128 percent change.
As discussed above, Earth's diurnal axis of rotation currently has an angle of 23.4 degrees
with respect to its orbital plane around the sun. The planet, however, wobbles like a spinning
top, and this angle oscillates between 22.1 degree and 24.5 degrees every 41,000 years.
This means that the seasonal difference in the amount of solar energy reaching earth
varies with a period of 41,000 years. At the extremes the current Tropic of Cancer, latitude
23.4 degrees, would receive 142.6 or 149.1%, about a 4.5 percent change in solar energy
from the minimum to maximum tilt. The elliptical shape of Earth's orbit around the sun is characterized
by its eccentricity. Eccentricity in mathematical terms is E = a - p, Over a + p where E is
the eccentricity, A is the aphelion, farthest distance from the sun, P is the perihelion,
closest distance to the sun. Currently, Earth's orbit has an eccentricity of 0.0167, but it
oscillates between 0.005 near circular and 0.0617 more elliptical with an average periodicity
of about 100,000 years. As the eccentricity of Earth's orbit increases the amount of solar
energy reaching Earth fluctuates more from summer to winter because of the greater seasonal
differences both in the distance between Earth and the sun, Inverse Square law, and the time
spent at the aphelion and perihelion, Kepler's second law.
When the eccentricity is near circular, seasonal variation derives solely from the differences
in the angle between the axis of Earth's diurnal rotation and its orbital plane around the
sun; cosine law. At present the summer and winter solstices and the aphelion and perihelion,
furthest to closest, nearly coincide. The solstices are June twenty-first and December
twenty-first, while the aphelion and perihelion are respectively July fourth, and January
third. Earth, however, behaves like a wobbling top. And the alignment of the axis of Earth's
diurnal rotation with distance from the sun oscillates with an average period of 21,000
years. That is, 10,500 years ago, Earth was closest to the sun on July fourth when the
northern hemisphere was tilted toward the sun and furthest from the sun on January third
when the northern hemisphere was tilted away.