CHAPTER 3 | CYCLES OF THE MOON 39
The Moon’s Shadow
To see a solar eclipse, you have to be in the moon’s shadow. Like
Earth’s shadow, the moon’s shadow consists of a central umbra of
total shadow and a penumbra of partial shadow. Th e moon’s
umbral shadow produces a spot of darkness about 270 km
( 170 mi) in diameter on Earth’s surface. (Th e exact size of the
umbral shadow depends on the location of the moon in its ellipti-
cal orbit and the angle at which the shadow strikes Earth.) Because
of the orbital motion of the moon and the rotation of Earth, the
moon’s shadow rushes across Earth at speeds of at least 1700 km/h
(1060 mph) sweeping out a path of totality (■ Figure 3-8). People
lucky enough to be in the path of totality will see a total eclipse of
the sun when the umbral spot sweeps over them. Observers just
outside the path of totality will see a partial solar eclipse as the
penumbral shadow sweeps over their location. Th ose living even
farther from the path of totality will see no eclipse.
Sometimes the moon’s umbral shadow is not long enough to
reach Earth, and, seen from Earth’s surface, the disk of the moon
is not big enough to cover the sun. Th e orbit of the moon is
slightly elliptical, and its distance from Earth varies. When it is at
apogee, its farthest point from Earth, its angular diameter is
about 6 percent smaller than average, and when it is at perigee,
its closest point to Earth, its angular diameter is 6 percent larger
than average. Another factor is Earth’s slightly elliptical orbit
around the sun. When Earth is closest to the sun in January, the
sun looks 1.7 percent larger in angular diameter; and, when Earth
is at its farthest point in July, the sun looks 1.7 percent smaller. If
the moon crosses in front of the sun when the moon’s disk is
smaller in angular diameter than the sun’s, it produces an annular
eclipse, a solar eclipse in which a ring (or annulus) of light is vis-
ible around the disk of the moon (■ Figure 3-9). An annular
eclipse swept across the United States on May 10, 1994.Linear diameter is simply the distance between an object’s
opposite sides. You use linear diameter when you order a 16-inch
pizza—the pizza is 16 inches across. Th e linear diameter of the
moon is 3476 km.
Th e angular diameter of an object is the angle formed by
lines extending toward you from opposite edges of the object and
meeting at your eye (■ Figure 3-7). Clearly, the farther away an
object is, the smaller its angular diameter.
To fi nd the angular diameter of the moon, you need to use
the small-angle formula. It gives you a way to fi gure out the
angular diameter of any object, whether it is a pizza or the moon.
In the small-angle formula, you should always express angular
diameter in arc seconds and always use the same units for dis-
tance and linear diameter:
angular diameter linear diameter
206,265 distance
You can use this formula* to fi nd any one of these three quanti-
ties if you know the other two; in this case, you are interested in
fi nding the angular diameter of the moon.
Th e moon has a linear diameter of 3476 km and a distance
from Earth of about 384,000 km. Because the moon’s linear
diameter and distance are both given in the same units, kilometers,
you can put them directly into the small-angle formula:
angular diameter 3476 km
206,265 384,000 km
To solve for angular diameter, multiply both sides by 206,265.
You will fi nd that the angular diameter of the moon is 1870 arc
seconds. If you divide by 60, you get 31 arc minutes, or, dividing
by 60 again, about 0.5°. Th e moon’s orbit is slightly elliptical, so
the moon can sometimes look a bit larger or smaller, but its angu-
lar diameter is always close to 0.5°. It is a Common
Misconception that the moon is larger when it is on the
horizon. Certainly the rising full moon looks big when you see it
on the horizon, but that is an optical illusion. In reality, the moon
is the same size on the horizon as when it is high overhead.
Now repeat this small-angle calculation to fi nd the angular
diameter of the sun. Th e sun is 1.39 106 km in linear diameter
and 1.50 108 km from Earth. If you put these numbers into
the small-angle formula, you will discover that the sun has an
angular diameter of 1900 arc seconds, which is 32 arc minutes, or
about 0.5°. Earth’s orbit is slightly elliptical, and consequently the
sun can sometimes look slightly larger or smaller in the sky, but
it, like the moon, is always close to 0.5° in angular diameter.
By fantastic good luck, you live on a planet with a moon that
is almost exactly the same angular diameter as your sun. Th anks
to that coincidence, when the moon passes in front of the sun, it
is almost exactly the right size to cover the sun’s brilliant surface
but leave the sun’s atmosphere visible.
Linear
diameterAngular
diameterDistance■ Figure 3-7
The angular diameter of an object is related to both its linear diameter and
its distance.*Th e number 206,265 is the number of arc seconds in a radian. When you
divide by 206,265, you convert the angle from arc seconds to radians.