At the meeting of the Society in November last I mentioned that on the day of the next opposition of the planet Mars, the Earth and Moon, as seen from Mars, would cross the Sun’s disk, a phenomenon which has not happened since the year 1800, and I stated the chief circumstances connected with the case.
[…]
The last occasion when the Earth and Moon crossed the Sun’s disk for Mars occurred on November 8, 1800. The two next transits, near the opposite node, will take place at the times of the oppositions in May 1905 and May 1984.

A. Marth, Monthly Notices of the Royal Astronomical Society (1879) 39: 513

Only a minute to go; getting down to business. For the record: year, 1984; month, May; day, 11, coming up to four hours thirty minutes Ephemeris Time … now.
[…]
There it is … there it is! I can hardly believe it! A tiny black dot at the edge of the Sun … growing, growing, growing …
Hello Earth. Look up at me, the brightest star in your sky, straight overhead at midnight.

Arthur C. Clarke “Transit of Earth” 1971

We’re familiar, here on Earth, with the fact that the planets Venus and Mercury occasionally pass between the Earth and Sun, showing up as tiny shadow discs against the solar surface. I’ve previously written about one such transit of Mercury.

Less familiar, but intuitively obvious once you think about it, is the fact that the Earth and Moon will occasionally cross the solar disc if viewed from the planet Mars, our next neighbour outwards in the solar system.

My first quote, above, is the earliest record I can find of someone discussing this phenomenon, back in 1879, in anticipation of the transit of Earth that would occur that year, late on 12 November.

My second quotation is from the short story “Transit Of Earth”, in which Arthur C. Clarke has a dying astronaut, stranded on Mars, talk us through the last of the four transits mentioned in the MNRAS notice—that of 11 May 1984. The story was first published in Playboy, in January 1971, back in the day when claiming that you subscribed to the magazine just for the stories and articles was almost plausible. When the story was subsequently collected in The Wind From The Sun (1972), Clarke mentioned in the preface that his detailed description of the event was based on an article by Jean Meeus, published in the Journal of the British Astronomical Association (1962) 72: 286. My picture at the head of this post accurately depicts the same event, with the disc of Earth, trailed by the Moon, silhouetted against the Sun.

I can’t find a freely accessible copy of Meeus’s original work on-line, but no matter—he revisited the topic more recently, in an article co-authored with Edwin Goffin, and dedicated to Arthur C. Clarke*: Journal of the British Astronomical Association (1983) 93: 120.

The Sun, Earth and Mars roughly align once every 780 days (on average), as the Earth, in its faster orbit, catches up with the slower-moving Mars. (This is called the synodic period of Mars, for reasons that would provide material for an entire other post.) But Mars doesn’t see a transit of Earth every time that happens, because the orbits of the two planets are tilted relative to each other. It’s by less than two degrees, but that slight inclination usually means that the Earth, as seen from Mars, passes either north or south of the Sun, rather than across its disc. The only places that a transit can occur is the line along which the two orbital planes intersect, which is called the line of the nodes. (See my post about Keplerian Orbital Elements for more on that.) Here’s a diagram showing the shape and size of the two orbits, with the line of nodes marked:

So there are two opportunities for Mars to align precisely enough with Earth for a transit to occur. One is called the descending node, where Mars passes through the plane of the Earth’s orbit heading in a southerly direction. Earth passes through that end of the line of nodes in May, and if Mars is at or near its descending node at that time, a hypothetical Martian could glimpse Earth passing across the solar disc. I’ve marked such a May alignment on my diagram. The other opportunity is at the other end of the line of nodes, at (you’ve guessed it) the ascending node, which Earth reaches in November.

But if there’s an Earth transit one May, there certainly won’t be one the following year. The 780-day duration of the synodic period between Earth-Mars alignments means that, after one year, Earth and Mars will be on almost opposite sides of the Sun, and on the next year Mars will be well past its descending node before Earth arrives in the right vicinity. So for a repeat rendezvous, we need to wait out an interval that is both a round number of years and a round number of synodic periods. That happens after 79 Earth years, which is almost (but not quite) equal to 37 synodic periods. So transits of Earth generally occur in pairs separated by 79 years. And we can see that periodicity in Marth’s calculations at the head of this post—a pair of November (ascending node) transits in 1800 and 1879, and a pair of May (descending node) transits in 1905 and 1984.

Why don’t these transits repeat in another 79 years? Because the match between 79 years and 37 synodic periods isn’t exact. If Mars is exactly at its node for one transit, on the next transit alignment, 79 years later, it will pass through the node three days early. This changes its view of Earth. Here are the May 1905 and 1984 (descending node) transits, with the Moon’s orbit also shown:

In 1905, Mars had not yet reached Earth’s orbital plane, and so had a perspective looking slightly “down” on the Earth, so that the Earth and Moon traversed the southern half of the Sun’s disc. In 1984, Mars was already through the Earth’s orbital plane at the time of transit, so Earth and Moon crossed the northern half of the Sun. You can see how, after another 79-year cycle, the Earth would pass north of the Sun, missing it entirely.

The same applies, in reverse, to transits at the ascending node. Here are the November 1800 and 1879 transits:

This time, Mars is slightly below Earth’s orbital plane in 1800, so that the Earth traverses the northern half of the solar disc. By 1879 Mars is sitting above the Earth’s orbital plane at the time of transit, and the Earth transits the southern part of the Sun.

Notice, also, that the width of the Moon’s orbit is so large compared to the apparent diameter of the Sun that the Earth and Moon sometimes have completely separate transits—one of them leaving the solar disc before the other enters, as happened in 1800.

Once one of these 79-year pairs has occurred, it takes a long time for Earth and Mars to fall back into a good enough alignment for a repeat performance. The pairs normally repeat in a cycle of 284 years, which is a pretty good match for 133 synodic periods. So the usual pattern at one node looks like this:

If we add in the other node, the cycle becomes more complicated:

The asymmetry between the two nodes is a little surprising. One might imagine that the 79-year pairs for each node would fall neatly in the middle of the 205-year gap at the other node. But, again, the transitions need to be whole numbers of years (plus a half, for the difference between May and November) which also produce the correct synodic alignment at the opposite node. And this is complicated by the asymmetry of Mars’s orbit. If you look again at my orbit diagram, you can see that the line of the nodes is almost at right angles to the line of the apsides, joining Mars’s closest approach to the Sun (perihelion) with its farthest excursion (aphelion). Mars moves faster when its closer to the Sun, so it completes the half orbit from descending node to ascending node faster than the return journey from ascending node to descending—the difference is about 80 days.

But even the 284-year cycle isn’t precise. So the location of the 79-year pairs drifts slowly with each 284-year repeat, until one of the pair gets very close to the northern or southern rim of the Sun. Here’s what happens at the descending node in a few hundred years’ time:

The 2473 and 2552 transits are part of the normal 284-year cycle, but they’re so far north they’ve opened up space for an early transit to sneak in, in 2394. So the 79-year pair has become a 79-year triplet, disrupting the usual progression:

The early transit in May 2394 intrudes between the normal 79-year November pair, producing a rapid-fire alternating sequence at 25½ and 53½ year intervals, followed by another “normal” 25½-year gap before the regular May pairing in 2473/2552. But the early transit in 2394 establishes a new rhythm of 79-year pairs, with the next pair appearing in 2678 and 2757.

So such early transits have the potential to disrupt the neat alternation between May and November transits at intervals of 79, 25½, 79 and 100½ years. But Meeus and Goffin’s data provide an illustration of how that disruption can sort itself out. Here’s a long run of Earth transits starting in the year 459:

Because this sequence predates the shift to the Gregorian calendar in 1582, the transits are occurring at the end of April and end of October, rather than mid-May and mid-November; but the 79, 25½, 79, 100½ pattern is ticking along nicely, until an early ascending-node transit occurs in October 664, kicking off a triplet like the one we’ve just seen. This briefly causes a shift to a 79, 21½, 79, 104½ pattern as the descending node transits continue in their regular rhythm—until an early descending-node transit in 1337^† resynchronizes April with the time-shifted November transits, and restores the 79, 25½, 79, 100½ rhythm that we’re currently enjoying.

So Meeus and Goffin’s 3000-year dataset seems to suggest that the current pattern is the standard pattern, disrupted only for a few centuries each millennium when first one node and then the other experiences an early transit. But, actually, over the longer term the current stable situation proves to be unusual—because the line of the nodes and the line of the apsides move slowly, and in opposite directions. This means that, in a few millennia, Mars’s perihelion will be near its ascending node, and its aphelion near the descending node. Because Mars will be moving fast through the ascending node, the time window in which an Earth transit can occur will be correspondingly narrowed, while the window at the descending node will be wider. Looking at John Walker’s Quarter Million Year Canon of Solar System Transits we discover that by the sixth millennium, Earth is reaching Mars’s ascending node in December, and its descending node in June—and there are three or four June transits for every December transit!

And rolling back to the fourth millennium BCE, we find the situation reversed, with Mars’s perihelion near the descending node, and aphelion at the ascending node, so that October transits are three or four times more common than April transits.

So even patterns that last for millennia count as fleeting, by the standards of the solar system.

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* Interestingly, all three have asteroids named after them: 1722 Goffin, 2213 Meeus and 4923 Clarke.
^† Notice that the 1337 transit didn’t kick off a triplet—there was no transit in 1495, because the Earth just barely missed the solar disc in that year.