Jupiter's "analemma" over $83$ years

I previously made a post on Jupiter's "analemma" over $12$ years.

Jupiter's orbital period being $11.86$ years, it does not align nicely with the $12$ year duration chosen in that post. That makes the "analemma" only very approximate. For a more perfect "analemma", a duration of $83$ years is needed (see the table in this post). $83$ years equals $7$ Jovian orbits with an error of only $13.027483$ days or $0.301\%$ of the Jovian year.

Tracing Jupiter's "analemma" over $83$ years yields the following path which is definitely more Lissajous and has a better symmetry.

Jupiter's "analemma" over $83$ years

Planetary orbital period multiples that align to Earth years

Assuming that one Earth orbit takes $365.242199$ days, the following table notes the durations at which the other planetary orbital periods come close to a multiple of Earth's. The associated error, as a percentage of the planet's orbital period, is noted as well.

Planet	Orbital Period	Multiple	Number of days	Earth years	Error (day)	Error (%)
Mercury	87.9691	328	28,853.8648	79	0.268921	0.306
Mercury	87.9691	191	16,802.0981	46	0.956946	1.088
Venus	224.701	13	2,921.113	8	0.824592	0.367
Mars	686.98	42	28,853.16	79	0.973721	0.142
Mars	686.98	25	17,174.50	47	8.116647	1.181
Mars	686.98	17	11,678.66	32	9.090368	1.323
Mars	686.98	8	5,495.84	15	17.207015	2.505
Jupiter	4,332.59	29	125,645.11	344	1.793544	0.041
Jupiter	4,332.59	22	95,316.98	261	11.233939	0.259
Jupiter	4,332.59	7	30,328.13	83	13.027483	0.301
Saturn	10,755.70	29	311,915.3	854	1.537946	0.014
Saturn	10,755.70	9	96,801.3	265	12.117265	0.113
Saturn	10,755.70	2	21,511.4	59	37.889741	0.352

This is closely related to how long a planetary "analemma" takes to form an approximately closed curve. The "Error (%)" column indicates how close it comes to making a true closed curve. This assumes that the orbital elements of the planet are constant over the period in question which we all know is not true.

The above numbers were what I relied on when tracing the analemma curves in previous blog posts. For instance, I chose the $59$ year interval for Saturn (here) as other options ($265$ or $854$ years) were too long. My choice of $12$ years for Jupiter (here) was an early approximation, and I need to regenerate the trace again over an $83$ year interval instead.

Saturn's "analemma" over $59$ years

After the posts on Jupiter's "analemma" over $12$ years and Venus' "analemma" over $8$ years, I did the same for Saturn which has an orbital period of $10755.7$ days ($\approx 29.45$ years). Thus, $265$ Earth years equals $9$ Saturnian years with an error of $12.12$ Earth days. This is too long a duration over which to trace an analemma. The next best match is $59$ Earth years which equals $2$ Saturnian years with an error of $37.89$ Earth days. I picked this option as it would have been impractical to trace Saturn over $265$ Earth years. Over that interval, the analemma trace would likely become impossible to follow.

Please manually select the High-Defintion (1080p) quality if YouTube serves up a lower quality version.

With the $(x,y)$ coordinates in the screenshots known already from Stellarium, I used the same script I had earlier used for Jupiter and Venus to trace out the "analemma" curve for Saturn and overlay it on the first image.

Saturn's "analemma" over $59$ years

Epilogue

This was added to the shared post on Reddit on 26-Oct-2025.