The orbits of Venus and the Earth looking from above. Drawing Nick Lomb
Congratulations on your Venus book. Excellent.
I notice that there is a 243 year cycle for Transits of Venus
243 x 365.242 = 224.7 x 395
So far so good. The axial rotation period for Venus is 243.1 days.
Is this a coincidence or is there some underlying geometrical fact that I cannot see ?
Answer An interesting and complex question that I address below.
Patterns in the transits of Venus
Let us first look at the patterns in the transits of Venus. We need to note that Venus and the Earth line up with the Sun every 583.92 days or 1.59872 years. This is called the synodic period.
If there was a transit, say the one in June 2004, for another transit to occur, the two planets must not only line up with each other and the Sun, but do so after an integer number of years so that they are back in the right places on each of their orbits.
Venus and Earth fulfil these requirements after five synodic periods = 7.9936 years as this is almost, though not quite, equal to the integer eight. Thus transits of Venus generally occur in pairs eight years apart. However, because of the slight inequality there is no third transit after another eight years.
A more accurate relationship occurs after 152 synodic periods = 243.00544 years or ~395 Venus years. The pattern of Venus transits thus repeats at 243 year intervals (This is the cycle quoted by Lionel in his question above). For example, the first pair of June transits after 8 June 2004 begins on 11 June 2247. Of course, in the meantime there is also a pair of December transits beginning in 2117.
The rotation of Venus
Scientists using radar observations from the 1960s onwards discovered that Venus spins backwards, that is in the opposite direction to its motion around the Sun, at the slow rate of 243.02 days.
They soon realised that means that Venus, almost but not quite, shows the same face towards the Earth each time the planets are lined up with each other and the Sun. Somehow there is a resonance between the motion of the Earth around the Sun and Venus’ spin around its axis. Scientists are unsure why this is the case, but one suggestion is that Venus is more massive on the face turned towards the Earth at those times and consequently it was gravitationally captured by the Earth.
How is it worked out that Venus shows the same face towards the Earth each time they line up? The quoted value of 243.02 days is with respect to distant stars. With a little arithmetic (taking inverses) we can easily convert that value to the rotation period with respect to the Sun or, in other words, to the day on Venus. It is 116.75 (Earth) days. Five of those periods equal 583.75 days, which is almost the same as the 583.92 day synodic period. So each time the planets line up Venus shows almost the same face to the Sun and hence the same face to the Earth, which is always on those occasions on the opposite side of Venus.
Coincidence or not
As Lionel points out it is interesting that transits of Venus repeat in a cycle of 243 years while the rotation period of Venus with respect to the stars is 243 days, The above detailed discussion indicates that there is no obvious connection that gives rise to the same number in each case. However, the calculations all depend on many of the same factors such as the orbital periods of Venus and the Earth so maybe there was a chance that the same number should recur.
Note the values quoted above are from the NASA Venus Fact Sheet.
This blog post is simultaneously published on the Transit of Venus website