Pattern on a paper plate

ToV frequencyTransits of Venus are predictable, but the pattern seems odd. Two transits occur within eight years, then there is a lapse of 105 1/2 years, then another pair within eight years, then wait 121 1/2 years. How do you explain this frequency?

Kathryn Williamson, a graduate student at Montana State University, has adapted and improved with her illustrations an activity I have done that uses a simple paper plate to explain the conundrum.

Imagine daughter Venus and mother Earth are racing around a central object, starting from the bottom of the circle with Earth-Venus-Sun aligned for a transit of Venus. Speedy daughter laps mother after 1.6 earth orbits (2.6 Venus orbits), or 1.6 earth years. Continuing the race, the fifth time (shown in colors) that daughter catches up to mother–eight years later–they are again near the starting line. So why don’t transits happen every 1.6 years?  Or every eight years?

Five alignments Transit of Venus pentagon

Incline Venus’ orbit a few degrees, and Venus is usually either a little above or below the Sun-Earth line. Only at the nodes, or the hinge of the paper plate orbits, are they near an alignment enough for Venus to transit the half-degree sun.

There’s one more problem. After the fifth conjunction, the two planets are actually just a wee bit short of the starting line. Each fifth alignment is shifted clockwise. Eight years after the race begins at the 2004 transit, they’re still close enough to the node for a second alignment in 2012. But by the 2020 conjunction,Venus has moved up the orbital incline so it would be just above the sun, as seen from earth–no transit.

In drawing the pentagon from the five sets of conjunctions, you can see how one point of the star (“1st pass”) needs to regress to the opposite node before the next 8-year pair of transits occurs. That explains the “half” of 105 1/2 and 121 1/2. Hence, the 8-year transit pairs shift from June to December each century-plus.

Try it for yourself on a paper plate using some colored markers, scissors, and Williamson’s instructions (PDF file). Thanks, Kathryn, for the clarity.

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