Page 151 - Earth's Climate Past and Future

CHAPTER 7 • Astronomical Control of Solar Radiation 127

BOX 7-2 LOOKING DEEPER INTO CLIMATE SCIENCE
Earth’s Precession as a Sine Wave

or a right-angle triangle (A), the sine of the angle ω is a triangle lying within the circle such that the sides of this
Fdefined as the length of the opposite side over the triangle can be measured in a rectangular (horizontal and
length of the hypotenuse (the longest side). Consider a vertical) coordinate system (C). In this conversion, the
circle whose radius is a vector r that sweeps around in a hypotenuse of the triangle is also the radius vector r of
360° arc in an angular motion measured by the changing the circle.
angle ω (B). Note that the circular motion described by The sweeping motion of the radius vector r around the
the angle ω is analogous to actual changes in Earth-Sun circle causes the shape of the internal triangle to change.
geometry. The radius vector r always has a value of +1 because its
The angular motion of the radius vector r around the length stays the same and its sign is defined within the
circle can be converted into changes in the dimensions of angular coordinate system as a positive value.
But the length of the opposite side of the triangle (y) is
defined within the rectangular coordinate system, and it
can change both in amplitude and in sign (positive or neg-
ative). As the radius vector r sweeps around the circle, y
y = +1
increases and decreases along the vertical scale, cycling
Ordinate
Hypotenuse Opposite y =0 Radius r ω y back and forth between values of +1 and –1. When r lies in
the top half of the circle, y has values greater than 0. When
y
ω
it lies in the lower half, y is negative.
Opposite
Sinω = The angular motion of r can be converted to a linear
Hypotenuse
mathematical form by plotting changes in sinω as the
A B y = –1
Ordinate y radius vector r sweeps out a full 360° circle, with the angle
Sinω =
Radius r ω increasing from 0° to 90°, 180°, 270°, and back to
360° (= 0°). As before, sinω is defined as the ratio of the
y =1 length of the opposite side y over the hypotenuse r.
y =0 r =1 ω y =0 y =0
r =1 r =1 ω y = –1 ω r =1 The mathematical function sinω cycles smoothly
r =1
from +1 to –1 and then back to +1 for each complete
y / = 0 y / = 1 y / = 0 y / = –1 y / = 0 revolution of the radius vector r. At the starting point (ω
r
r
r
r
r
= 0°), the length of the opposite side y is 0 and the
1
radius is +1, so the value of sinω is 0/1, or 0. As the angle
Sinω
y
(= / ) 0 ω increases, the length of the opposite side of the trian-
r
gle (y) increases in relation to the (constant) radius
–1
0° 90° 180° 270° 360° of the circle. When ω reaches 90°, sinω = +1 because
C Angle ω
the lengths of the opposite side and the radius
Converting angular motion to a sine wave (A) The sine of (hypotenuse) are identical (1/1). At 180°, sinω has
an angle is the length of the opposite side of a triangle over returned to 0 because the length of the opposite side (y)
its hypotenuse. (B) This concept can be applied to a circle is again 0. For angles greater than 180°, the sinω values
where the hypotenuse is the radius (amplitude = 1) and the
become negative because the opposite side of the tri-
length of the opposite side of the triangle varies from +1 to
angle y now falls in negative rectangular coordinates
–1 along a vertical coordinate axis. (C) As the radius vector
(values below 0 on the vertical axis). Sinω values reach a
sweeps out a full circle and ω increases from 0° to 360°, the
minimum value of –1 at ω = 270° (–1/1). After that,
sine of ω changes from +1 to –1 and back to +1, producing a
sine wave representation of circular motion and of Earth’s sinω again begins to increase, returning to a value of 0
precessional motion. at ω = 360° (= 0°).