The Geometry of Eclipses

Solar and lunar eclipses are some of the most amazing light shows in the sky, based on an intricate geometrical dance of the heavenly bodies and their shadows. To get a glimpse of what makes eclipses possible, you first have to start with the appearance of a shadow behind a large body like the Earth or Moon when illuminated by a even bigger light source like the Sun.

The Structure of a Shadow

When the Earth blocks the light coming from the Sun, the shadow out behind it has a dark and a light part. If you were standing in the dark area (called the "umbra"), the Earth's disk would completely shut off the light from the Sun and you would be in near total darkness. If you stand in the lighter part of the shadow (called the "penumbra"), the Earth's disk is off to the side or else too small to completely block out the Sun. There is a portion of the Sun, appearing as a crescent or a thin ring, that is still delivering some light to your position, producing only a partial or semi-dark shadow.

The illustration at the left shows how these various parts of the shadow are produced. The umbra (labelled "U") is a dark cone behind the Earth that comes to a point. The penumbra (labelled "P") is the lighter cone that expands out behind the Earth. The middle part of the penumbral area is an extension of the umbra, sometimes called the "antiumbra" and indicated by the letter "A". Notice the green surface with the two dotted circles that I drew in, cutting the shadow in a cross sectional view. If you examine the shadow from the vantage point of this cross section, you'll see the dark circular umbra in the center, surrounded by the lighter penumbra.

If you are in the umbra, looking back at the Sun, all you will see is the night time half of the Earth. The Sun is totally obscured. This is called a "total eclipse".

Standing in the penumbral part of the shadow, the Sun is only partially blocked out. A crescent of sunlight is still visible, creating a "partial eclipse".

And if you are in the antiumbra, you see a thin ring (called an "annulus") of sunlight surrounding the Earth's backside. From this distance, the Earth's disk is smaller than the Sun's, making a total eclipse impossible. This kind of event is known as an "annular eclipse".

Actually, to make the first illustration large enough to see on your monitor, I had to shrink the distance between the Sun and Earth quite drastically and distort their relative sizes. If you draw the shadows for the Earth and Moon to scale, they become pencil thin cones as in the next drawing. (Each pixel in this drawing represents 2500 kilometers. The Sun would be about 21 meters or 70 feet to the right of the Earth.) Notice also that the Moon is above the Earth's shadow, due to a 5+ degree tilt of the Moon's orbit from the ecliptic. Unless the Moon is near one of the lunar nodes (the spots on the zodiac where the Moon's orbit crosses the Earth's orbit) at full moon, it won't pass through the Earth's shadow (creating a lunar eclipse). Getting the Moon to pass through the Earth's shadow is like threading a needle. Similarly, the Moon's shadow at new moon (when the Moon is between the Earth and the Sun) will pass above or below the Earth, except when the Sun and Moon are near the lunar nodes (creating a solar eclipse). The role of the nodes is crucial for understanding eclipses and why they happen so infrequently. Note also that the length of the Moon's umbra is nearly the same as the average distance from the Earth to the Moon. This fact is important for solar eclipses, as we'll see.

Eclipse Seasons

As I just mentioned, the plane of the Moon's orbit around the Earth is tilted about 5+ degrees to the plane of the Earth's orbit around the Sun (which is synonymous with the ecliptic circle). These two planes intersect each other at two spots on the ecliptic known as the lunar nodes. As seen from the Earth, the Sun moves around the ecliptic, while the Moon moves in its own orbital plane. When the Sun and Moon are in the same zodiac position (new moon) near the nodes, the Moon can eclipse the Sun. When the Sun and Moon are in opposite zodiac positions (full moon) near the nodes, the Moon passes through the Earth's shadow and a lunar eclipse is formed.

So the times each year when the Sun is near one of the nodes are crucial times for forming eclipses. It takes 346+ days for the Sun to move from the North Node all around the zodiac and back. Half that period, 173+ days, is the time for the Sun to move from one node to the other. Consequently, every 173 days, eclipses are possible.

There is a "fudge factor" of about 18 degrees around the nodes where the shadow geometry is precise enough for eclipses to happen. If the Sun and Moon are quite close to the nodes (10-12 degrees), total eclipses are formed. As you get closer to 18 degrees, they become partial solar or appulse lunar eclipses. Beyond 18 degrees, the shadows just don't line up and no eclipse is possible.

Since the Sun moves about 1 degree a day, that 18 degree fudge factor translates into 18 days either side of the Sun's passage of a lunar node. Consequently, there's a 36 day window centered on the Sun's passage of either the North or South lunar node when eclipses can be formed. Whenever the Sun is in one of these windows, there will be 1 to 3 eclipses formed. These month long windows are known as "eclipse seasons". I suspect that part of the astrological influence attributed to the lunar nodes is due to the power of the eclipses they produce.

Solar Eclipses

A solar eclipse occurs when the new moon passes between the Earth and the Sun, casting a shadow on the Earth's surface. For people on the Earth's surface as the shadow moves on by, it appears as though the Moon blots out the Sun, either totally or partially.

A total solar eclipse is probably one of the most dramatic and awesome celestial spectacles mankind can witness. There are two conditions that must be met for a total eclipse to happen. First, the Sun and Moon at new moon must be near enough to one of the lunar nodes for the central axis of the Moon's shadow to hit the Earth. If the Moon is too far from the node, the lunar umbra will be too far north or south and will miss the Earth (although the penumbra may nick the polar regions). Secondly, the Moon must be in the part of it's orbit where it is closer than average to the Earth (near "perigee"). Since the length of the lunar umbra is somewhat less than the average Earth-Moon distance, it is too short to actually touch the Earth during most of the lunar orbit. This "near perigee" condition is the difference between a total and an annular eclipse. Inside the umbral shadow, an observer would see the Sun's surface totally blackened, leaving only the pearly outer atmosphere (the solar corona). The umbral shadow moves quickly across the Earth, so totality lasts only a few seconds to as much as 7.5 minutes. The "path of totality" traced out by the umbra is quite small, usually much less than 300 kilometers (180 miles) wide.

In an annular eclipse, the Moon is too far from the Earth in its orbit for the umbra to touch the Earth's surface. The lunar disk appears smaller at this distance, too small to completely cover up the Sun's disk. Consequently, a thin annular ring of sunlight is still visible at maximum eclipse.

If the new moon is farther from the node (up to18 degrees), the umbra is simply too far north or south and completely misses the Earth. However, the much wider penumbra can still fall across the polar regions, creating a partial eclipse.

Under rare circumstances, when the Earth-Moon distance is just slightly longer than the lunar umbra, a hybrid (or annular/total) eclipse can happen. At the beginning and ending phases of the eclipse, the umbra still misses the Earth, making an annular eclipse. However, in the middle part of the eclipse, the umbra barely touches the Earth and the eclipse flips over to being total.

Lunar Eclipses

A lunar eclipse occurs when the full moon enters the Earth's shadow. Since the Moon "shines" only by reflecting light from the Sun, the full moon seems to disappear when it is eclipsed. Depending on which parts of the Earth's shadow the full moon passes through (which depends on how close the full moon is to the lunar nodes), a lunar eclipse can be one of three main types.

In a total lunar eclipse, the Moon passes close enough to the center of the Earth's shadow so the entire body of the Moon moves into the umbra at mid-eclipse. While in the umbra, the Moon dims from its usual bright white self to a darkened color (usually dark black, red or dull orange, depending upon weather conditions on Earth). The Moon must be within 10-12 degrees of the node for a total lunar eclipse to happen.

If the Moon grazes the edge of the umbra, halfway in and halfway out, a partial lunar eclipse occurs. At maximum eclipse, the part of the Moon in the umbra is totally dark, while the part in the penumbra is only slightly dimmed, producing a bright crescent effect.

In the situation where the Moon misses the umbra entirely and only enters the penumbra (either grazing it or being completely immersed in it), an appulse (or penumbral) lunar eclipse is formed. The Moon's surface is only slightly dimmed during an appulse. In fact, the casual observer may not even notice this kind of eclipse, since the change in appearance is so subtle. The astrological effect of an appulse is far from subtle, however. The Moon must be within 16-18 degrees of the node for these later kinds of eclipses to happen. Beyond 18 degrees, no lunar eclipse is possible.

Back to the main Eclipse page.