Tropical timings – Kepler’s ‘trigon’ motion of Jupiter and Saturn

Kepler’s diagram of 40 conjunctions of Jupiter and Saturn

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At the time of publishing previous ‘tropical timings’ posts I hadn’t seen this classic effort from planetary theory pioneer Johannes Kepler, published over 400 years ago, so here it is. Let’s have a closer look.

He described a 40 conjunction model of the orbits of Jupiter and Saturn with the Sun in the diagram shown above. The position of each conjunction is numbered and linked to the next one (1 is mid-right, 2 lower left, 3 upper left, 4 next to 1 and so on), indicating a near triangular movement of the axis for every 3 conjunctions, which he called a trigon.

Our post linking those orbit periods to Earth’s rotating reference frame (link here) described how such a system works, with screenshot examples from Arnholm’s solar simulator (link here).

As noted before (see here), there are 15 solar barycentric orbits (SBO) every 179 years, or per 9 Jupiter-Saturn conjunctions (ref. J. Blizard in that link).

This is the basis of the movement of both these planets and the Sun itself, sometimes described as a tri-lobed epitrochoid. A simple example might look like the animation here [credit: mathcurve.com].

The difference between that and the solar barycentric motion is that the SBO has all sorts of variations in length, time and relative position to the actual barycentre. These variations are mainly due to the combined effects of Jupiter and Saturn, but those of Uranus and Neptune also play a lesser part.

The Sun also returns to, or close to, the barycentre occasionally, typically at intervals of 9 J-S, or twice per 9 J-S (e.g. 7 and 2 J-S). Examples occur in 1632, 1811 and 1990 (179~ year intervals).

The ratio of the mean SBO to mean solar cycle length is very close to 13:14. However both have significant variations in individual period length.

In conclusion, Kepler looks about right as far as the diagram goes. It shows the need for three occurrences of its total period (i.e. 120 J-S conjunctions) to reach a whole number of solar barycentric orbits, since 1 trigon = 5 barycentric orbits (again, using mean values) and 40 J-S isn’t a whole number of trigons of 3 J-S each. Hence number 40 is next to number 3, for example. We can also note in 120 J-S that J = 201 orbits and S = 81 orbits (201-81=120), so this is the period when both planets exceed the imaginary 5:2 ratio (times 40 = 200:80) by exactly one orbit.