One of the most remarkable celestial coincidences visible from the Earth is that the Moon and Sun have almost the same apparent size. Depending on its position in orbit, the Moon can appear either larger or smaller than the Sun, resulting in solar eclipses on Earth occurring in two varieties: total, when the Moon is close enough to appear larger than Sun and completely cover it, and annular, where a more distant Moon fails to completely cover the Sun's photosphere, resulting in a “ring of fire”.

This size coincidence is striking, especially since it hasn't always been the case, nor will it be the case forever. Billions of years ago, the Moon was much closer to the Earth and total eclipses were far more common, yet less spectacular because the Sun's corona and prominences wouldn't have been visible all around the Sun. Eventually, tidal-driven recession of the Moon from the Earth will put an end to total solar eclipses visible from Earth and all subsequent eclipses will be annular. Some have actually argued that the closely comparable apparent sizes of the Sun and Moon have contributed in some way to the evolution of human intelligence, providing an “anthropic” explanation of why we happen to be observing such a marvel at the epoch in geological time when it happens to occur.

In the first week of July, 2004 three completely unrelated celestial phenomena occurred within days of one another: the full Moon, the passage of the Moon through perigee (the moment when it most closely approaches the Earth), and the passage of the Earth through aphelion—its greatest annual distance from the Sun. Thus, the full Moon occurred less than 12 hours after lunar perigee, and only three days later the Earth arrived at aphelion. The coincidence of these events permitted taking photographs of the perigean full Moon and the Sun near aphelion, all with the same camera and optics, to illustrate the difference in the apparent sizes of the Sun and Moon in these circumstances.

Such a close coincidence in the time of full Moon, lunar perigee, and aphelion is a relatively rare event. I checked all years between 2000 and 2100 for full moons within 24 hours of perigee occurring in the first ten days of July (aphelion always falls on July 3 through 6). By this definition, perigean full Moons near aphelion occur 5 times in this century, in 2004, 2031, 2058, 2066, and 2093. You can search for such events in other time periods using our Lunar Perigee and Apogee Calculator. The far more rare four-way coincidence of new Moon, perigee, aphelion, and the Moon's crossing a node of its orbit produces solar eclipses with the longest duration of totality, such as the extraordinary eclipse of July 11th, 1991, whose 6 minute 53 second totality will not be exceeded until the eclipse of June 13th, 2132, with two seconds more totality. Amazingly, there were three solar eclipses in the 20th century with even longer totality: the eclipses of 1937, 1955, and 1973—all comfortably within a single human lifespan—each exceeded seven minutes. To put this into context, the last seven-minute-totality eclipse before 1937 was in the year 1098 (July 1st), and the next won't occur until June 25th, 2150, the opening act for the Big Show on July 16th, 2186, when there's an eclipse with 7 minutes and 29 seconds of totality! This is the solar eclipse with longest totality in the entire 8000 year interval from 3000 B.C. to A.D. 5000. Me, I'm signing up for the expedition right away—this one is sure to sell out!

The tables below give the date and time the pictures of the Sun and Moon were taken, and the distance and angular extent of each body at that time.

Even though the photo of the Sun was taken two days before aphelion, the appearance of the Sun in an image at this scale is indistinguishable from one shot precisely at the moment of aphelion; Earth takes a whole year to traverse its orbit, which has an eccentricity of only 1.67%, so a couple of days don't make much difference. The Sun was rather featureless at the time of this photo—we're heading toward the trough in the solar activity cycle, so there weren't any spectacular naked-eye sunspots, only the modest active region 0639 near the centre of the disc.

Both photos were taken with a Nikon D70 digital SLR camera and 500mm Reflex-Nikkor fixed aperture f/8 catadioptric “mirror lens”. Since the image sensor in this camera is 2/3 the size of a 35mm film frame, the scale of the image in the frame is equivalent to that of a 750mm lens on a film camera. The image of the Sun was taken with an Orion full aperture coated glass solar filter mounted on the front of the Nikon lens. Both pictures were shot in full manual mode with a variety of shutter speeds; I selected the best-exposed images for use here. The camera's self-timer was used for all exposures to minimise vibration. (The infrared remote control would be even better, but I haven't obtained one yet.) The images above are pixel-for-pixel crops from the original camera frames. Cropping, brightness and contrast adjustment, and assembly of the two images into the animations were done with Jasc Paint Shop Pro 7.02 and Animation Shop 3.10.

Catching the Converse Coincidence

Two near-apogean full Moons on either side of the Earth's perihelion passage on January 2nd, 2005 provided opportunities to photograph the converse case: Moon near apogee and Earth near perihelion. I had originally aimed for the full Moon of December 26th, which is only a little more than 24 hours before lunar apogee and a week before the Earth reaches perihelion. Planning a photo like this six months in advance amounts to daring the sky to do its worst, and the last week in December the sky took the dare and delivered a steady menu of clouds, howling wind, and precipitation.

The fall-back opportunity was the full Moon of January 25th, which occurred less than two days after apogee on the 23rd. This isn't as close to perihelion as the December opportunity would have been—about three weeks away from perihelion instead of one, but for practical purposes it's close enough; at the time of lunar apogee on the 23rd, the Earth would be only 156,000 kilometres further from the Sun than at perihelion, which is just 3% of the distance from perihelion to aphelion. The difference in apparent size of the Sun would barely be perceptible.

The weather didn't look very promising in the run-up to the January 23–25 opportunity either, but shortly before sunset on the 23rd, the snow stopped and the Sun popped out between the clouds. I grabbed the camera and lens, already mounted on the tripod in the hope of exploiting such an opportunity, dashed outside, planted the tripod in the snow, and shot a sequence of pictures of the Sun through the Orion solar filter. The two visible sunspot groups (NOAA Active Region 0726 near the centre of the disc and 0725 near the solar limb) helped in focusing. From the extensively bracketed sequence of images, I chose one taken at 1/80 second with CCD sensitivity set to ISO 200. The Sun looks somewhat “lumpy” due to refraction through the turbulent atmosphere so near to sunset.

After shooting the Sun, I turned to the Moon, which was rising in a pellucid sky in the East. While the Moon wouldn't be full until 10:33 UTC on January 25th, it was already 97% full, and only three hours before perigee (18:55 UTC). I whipped off the solar filter, centred the Moon, and shot a sequence of exposures at various shutter speeds. The image I chose from the set was taken at 1/500 second with ISO 200 sensitivity. In both the Sun and Moon photos, I used the D70's self-timer to trip the shutter to minimise vibration. I'd have preferred to use an ML-L3 infrared remote, but that gizmo appears to be fabricated from unobtanium bar stock—it's still on back-order after six months—so the self-timer had to do in its absence.

Final Totality

Total solar eclipse, June 21, 2001 Photo by John Walker

Nothing lasts forever. After some date in the distant future, there will be no more total solar eclipses—only annular eclipses where the Moon fails to completely cover the Sun. Our distant earthbound descendants will be forever deprived of one of nature's greatest spectacles. Why? Because of the tides in the ocean.

Really! As the Moon raises tides in the ocean (and to a lesser extent in the solid Earth), and the tides wash up on the shores of the continents, Earth's rotation gradually slows, transferring its rotational angular momentum to the orbit of the Moon, which in turn causes the Moon to steadily recede from the Earth. The gradual slowing of the Earth's rotation was discovered by Edmond Halley in the 18th century by noting discrepancies between the location of solar eclipses recorded in history and predictions based on the assumption of an unchanging length of day. In 1754, Immanuel Kant correctly identified tidal friction as the cause of the lengthening day, and further predicted lunar recession as a consequence. Lunar recession was subsequently confirmed by precision measurement of the Moon's orbit. The Apollo 11, 14, and 15 lunar landing missions in 1969 and 1971, and the Soviet Lunakhod 2 robotic rover in 1973 placed retro-reflectors at various sites on the Moon which permit Lunar Laser Ranging with a precision of a few centimetres. This has permitted the present-day lunar recession rate to be measured as 3.82±0.07 centimetres per year.[1] If you're fifty years old, the moon is about two metres farther away than when you were born.

Annular solar eclipse, May 10, 1994 Photo by Grant W. McKinney

Paleontologists have been able to estimate length of the day in the distant past through the effect of tides on the formation of beds of sandstone. In the Neoproterozoic era (620 million years ago, well before the Cambrian Explosion of metazoan life), Earth's day was only 21.9±0.4 hours long, and over the entire period from then until the present, the Moon's average rate of recession was 2.17±0.31 cm/year—a little more than half the current rate.[2] It's no surprise that the average recession is slower than today's, since the present configuration of continents and oceans is almost optimal to maximise tidal friction: two large pole-to-pole oceans separated by two landmasses which span most of the latitude range, including the equator where tidal forces are the greatest. When continental drift causes the continents to aggregate into a single landmass, or opens up a world-girdling equatorial ocean, or an ice age reduces sea level and freezes out the tides in northern and southern latitudes, tidal braking and hence lunar recession occur at a reduced rate. All of these circumstances have occurred in the 620 million years since the Neoproterozoic, averaging out to about 2 cm/year.

As the Moon recedes from the Earth, its apparent size diminishes. Already, when the Moon is near apogee, its angular size is always less than the Sun's even at aphelion, so a total solar eclipse cannot occur near lunar apogee. The recession of the Moon increases both perigee and apogee, so eventually even the perigee will be sufficiently distant so the Moon doesn't cover the Sun, and from that point unto eternity, no total solar eclipse will ever been seen from Earth. At some point before that date, when the Moon is near perigee and the Earth near aphelion, final totality will occur—the last total eclipse of the Sun visible from Earth. It will be extremely short, just a fraction of a second, and total only within a tiny spot near the Earth's equator, but any being lucky enough to see it will have glimpsed something never, ever to be seen again from the Earth. How long will it be until that last-ever total solar eclipse?

Final totality will occur when the Moon, at a very close perigee, subtends an angle, as seen from the closest point on Earth, just sufficient to cover the disc of the Sun near apogee. Let's put some numbers on these items, bearing in mind that we're only making a rough estimate. There are so many uncertainties which could affect the Moon's recession rate over the intervals involved that seeking great precision in other quantities is unwarranted. The Sun-Earth distance at aphelion varies slightly from year to year depending on the relative positions of the Sun, Earth, and Moon, and the location of the solar system barycentre (centre of mass), which is affected by the positions of all the planets, principally Jupiter and Saturn. The most distant aphelion in the period 1960–2005 was 1.016759 astronomical units,[3] which we'll round off to 1.017 AU. (An astronomical unit is the mean distance from the Earth to the Sun, equal to 149,597,870.691 kilometres.) Since the Sun is not a solid body and rotates, it isn't a sphere, but rather an ellipsoid flattened at the poles with an equatorial bulge. For a total solar eclipse to occur, every part of the Sun's disc must be covered, so we'll use the equatorial radius of 695,000 km as the size of the Sun. From the radius and distance, a little trigonometry tells us that at an extreme aphelion, the broadest part of the Sun's disc subtends about 0.5236 degrees.

Next, we need to calculate how far from the Earth the Moon must be to subtend that angle in the sky. The Moon's mean radius is 1737.5 km, and once more pounding the problem with the hammer of trigonometry, we find that the Moon subtends the same angle as the Sun at apogee when the Moon is 380,376 km from the observer. (Note that since present-day apogees of the Moon are in excess of 406,000 km, no total solar eclipse can occur at lunar apogee.) Total solar eclipses will cease forever when lunar recession elevates the perigee to 380,376 km and above. Before calculating how long that will take, there's one more detail to attend to. The distance at which the Moon appears the same size as the Sun is the distance from the Moon to the observer, not the centre of the Earth, from which perigee is measured. You can't see the Moon from the centre of the Earth (and, besides, it's awfully hot there), so observers will be on the Earth's surface which, at the point where the vector from the centre of the Earth to the centre of the Moon intersects the Earth's surface, will be one Earth radius closer, which at the equator is 6378 km. Since we're looking for the final total eclipse, we'll take maximum advantage of the Earth's equatorial radius and consider an eclipse which is only total for a moment at the equator, which permits the perigee to be as high as 386,754 km while keeping the observer to Moon distance at 380,376 km.

The Earth-Moon distance at perigee varies depending on the relative positions of the Sun, Earth, and Moon. The closest perigee in the A.D. 1500–2500 period occurs on January 1st, 2257 at 356,371 km, so we'll take that as the present-day minimum.[4] The difference between this figure and the 386,754 km perigee which just permits an equatorial observer to see the Moon as big as the Sun: 30,383 km, is how far the orbit of the Moon will have been raised at the time of final totality. Using the mean recession rate over the last 620 million years of 2.17 cm/year as an estimate of the average rate over comparably long periods in the future, we find that it will take about 1.4 billion years before solar eclipses cease to be visible from Earth. Eclipse tour operators thus have no need to worry about near-term business prospects!

Over deep time like this, many factors can strongly influence the actual outcome. Over the next billion years, the Sun will continue to get brighter and brighter (as it has over all of Earth's history), and may consequently appear somewhat larger than today. In any case, solar brightening will result in extreme changes in the Earth's climate. The Earth may exhaust its inventory of carbon accessible to the biosphere, resulting in photosynthesis shutting down and the collapse of the plant and animal biosphere. A runaway greenhouse effect may result in the loss of the oceans, which would dramatically reduce tidal braking and the rate of lunar recession. And, of course, all of these events may result in the only observers present at the final solar eclipse being single-celled bacteria.[5] Not to worry; descendants of humanity will have moved on to more hospitable climes, but will no doubt arrange a Galaxy Wide Webcast of the event.

Exercises for the Reader

Using similar reasoning, estimate how far in the past the first-ever annular eclipse occurred on Earth. Calculate what fraction of the era in which both total and annular eclipses occur has elapsed so far. Decide whether the creepy-crawlers of the Burgess Shale could have witnessed an annular eclipse.

References

1. Dickey, J.O. et al. “Lunar Laser Ranging: A Continuing Legacy of the Apollo Program,” Science 265, 486 (1994).
2. Williams, G. "Geological constraints on the Precambrian history of Earth's rotation and the Moon's orbit". Reviews of Geophysics 38, 37 (2000).
3. Meeus, Jean. Astronomical Tables of the Sun, Moon, and Planets . 2nd. ed. Richmond: Willmann-Bell, 1995. ISBN 0-943396-45-X.
4. Meeus, Jean. Mathematical Astronomy Morsels . Richmond: Willmann-Bell, 1997. ISBN 0-943396-51-4.
5. Ward, Peter D. and Donald Brownlee. The Life and Death of Planet Earth. New York: Owl Books, 2003. ISBN 0-8050-7512-7.

Astronomy Resources at Fourmilab

• Inconstant Moon: The Moon at Apogee and Perigee
• Solar Eclipses: 1999, 2001, 2008, and 2010
• Lunar Eclipse: November 2003
• Transit of Venus, June 2004
• Transit of Mercury, May 2003
• The Three Inner Planets
• Viewing Venus in Broad Daylight
• Naked Eye Sun-Spotting
• Solar System Live
• Your Sky
• Earth and Moon Viewer
• Comets: Hyakutake and Hale-Bopp

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