David Shortt • May 22, 2012

As many readers probably know, on June 5-6, 2012 there will be a rare transit of Venus as the planet crosses the face of the Sun as viewed from Earth. The last transit occurred in June 2004, and the next one won’t be until December 2117. Transits of Venus follow a peculiar pattern -- two transits 8 years apart, then 105.5 years with no transits, then two transits 8 years apart, then 121.5 years with no transits, for a total cycle of 243 years—and thereby hangs a tale.

Since Venus’ orbit is closer to the Sun than Earth’s, Venus has a shorter orbital period and “laps” Earth at regular intervals. Astronomically speaking, such an event is called an inferior conjunction. It is a curious fact that the orbital periods of Venus and Earth are very close to an 8:13 resonance, meaning that for every 8 Earth years, Venus orbits the Sun almost exactly 13 times. There are then 5 inferior conjunctions every 8 years, giving what is called a “synodic period” between inferior conjunctions of 1.6 years. If the orbital planes of the two planets coincided, then every 1.6 years there would be a transit of Venus. Of course, things are not so simple. First, Venus’s orbit is inclined with respect to Earth’s by about 3.4 degrees. Second, the orbital periods are not exactly in resonance. Third, the orbits are not perfect circles. These deviations from perfection give rise to rich and complex behavior.

The figure shows all 152 inferior conjunctions during one cycle of 243 years. (The formal definition of conjunction specifies that two objects have equal right ascension. This is not quite the same as the moment of minimum angular separation which is important for illustrating transits and near-transits, so the figure uses minimum separation. Most of the conjunctions occur within a few hours of minimum separation.) Everything is plotted relative to the Sun, whose disk is the orange circle at center left. The colored bars show the path of Venus for about 10 hours surrounding the moment of minimum separation, and I used a cyclic 5-color palette to show the 5 groups of conjunctions in each 8-year cycle, starting with the conjunction in March 2001 (the first magenta bar in the upper right). Each bar is labeled with the month and year. The font is a bit small (but readable if you zoom the figure to 100%) to avoid too many font collisions in dense parts of the plot. North is up and east is to the left.

The tilt of Venus’ orbit with respect to Earth’s means that transits can only happen if an inferior conjunction occurs at just the right part of Venus’s orbit, namely when Venus happens to be crossing the plane of Earth’s orbit during the conjunction. The crossing can occur at just two places on opposite sides of the Sun, called nodes, along the line where the planes of Venus’ and Earth’s orbits intersect. A transit is not very likely because, due to the tilt and the distances involved, Venus as viewed from Earth can be up to 10 degrees north or south of the Sun in the sky during an inferior conjunction. This is many times the Sun’s apparent diameter of 0.5 degrees, so alignment is unlikely by chance. If the planets do happen to be aligned properly for a transit during a given conjunction, then at the next conjunction 1.6 years later they will not be—Venus will pass well north or south of the Sun.

If the two planets’ orbits were in precise 8:13 resonance, then 8 years after a transit both planets would return to exactly the same positions in their orbits, ready for another transit. Since the planets are not in perfect resonance, this is not quite the case. Earth’s sidereal period (i.e. with respect to the distant stars) is 365.256 days, and Venus’ is 224.701 days. The difference between 8 Earth orbits and 13 Venus orbits is about 0.9 days, so the alignment slips out of phase by this amount every 8 years. This doesn’t sound like much, but it’s enough so that we can have at most two transits 8 years apart, but after 16 years the alignment will have shifted so that Venus will miss the Sun. As the decades pass by and subsequent conjunctions sample different parts of Earth’s and Venus’s orbit, the positions of minimum separation slowly “creep” around a pretty figure-8 pattern. Finally the planets and Sun come into alignment again when Earth is on the opposite side of the Sun from the previous transits (the opposite node), and once more we have two transits 8 years apart (the Dec. 2117 and 2125 transits are shown in the figure). The precise time it takes for this to happen depends on the inexactness of the 8:13 resonance. If the resonance were perfect, there would be only 5 bars in the figure and probably no transits at all. In reality we experience over a century with no transits, followed by pairs of transits 8 years apart. Also, the orbits of both Venus and Earth are not perfect circles, and this introduces some asymmetry in the long transit-free periods, resulting in them being 105.5 years and 121.5 years.

Finally, a second near-resonance, this time 243:395, plays a role in the repeating pattern shown in the figure. This resonance is even closer to perfection than the 8:13 resonance. After 243 years, the whole pattern of 152 conjunctions and 4 transits repeats almost exactly, with a discrepancy of only 0.5 days. Transits in June 2247 and 2255 (not shown in the figure) will overlay the 2004 and 2012 transits almost perfectly, and the basic pattern shown in the figure will be valid for about half a millennium. Over centuries, though, the 0.5-day discrepancy causes a slow shift in the positions of the transits with respect to the disk of the Sun, and eventually the pairs of transits will be replaced by two single transits in each 243-year cycle.

In conclusion, the upcoming June transit is just the latest event in a long, complex dance among Earth, Venus and the Sun. The 8:13 near-resonance coupled with the tilt of Venus’ orbit with respect to Earth’s gives us pairs of transits 8 years apart separated by long gaps without transits. The 243:395 near-resonance makes the entire pattern of 152 inferior conjunctions repeat after 243 years. That’s a beautiful dance!

Note on methods: I created the figure from calculations based on algorithms described in the excellent book *Astronomical Algorithms* by Jean Meeus. I wrote subroutines in high-level code which generated a file of positions for Venus relative to the Sun around the time of closest approach at each inferior conjunction. I then used Mathematica to read this file, make the plot, and produce an encapsulated Postscript file. Finally I rasterized it and touched up the graphic in Photoshop.

Wendy Hagen Bauer: 2012/05/22 12:45 CDTwalfy: 2012/05/24 01:24 CDT