As Mars passed Jupiter early today, a gong sounded, or so I whimsically imagined. But how often does that happen?
In this graph, longitude reads from right to left because the planets travel eastward – leftward as seen from the northern hemisphere. The names, and the thickness of the lines, are proportional to the planets’ sizes. The slopes of the lines result from the planets’ speeds. Mars hurries (relatively) around the Sun, Neptune takes its time. The lines between the zodiacal constellations slope slightly, because precession slowly icrease their longitude.
You can see Mars crossing in front of Jupiter on 2015 Dec. 1, and again in early 2018 and mid 2020. And revealed is a gathering of the three brightest of these planets in 2018, and a tighter gathering of them in 2020, in the direction of eastern Sagittarius.
Here is my calculation of the Mars-Jupiter heliocentric conjunctions during this span.
2455700.586 2011 May 19 2: 4 20.4 0.38
2456522.032 2013 Aug 17 12:45 93.6 1.43 821.446
2457357.571 2015 Dec 1 1:42 160.8 0.59 835.540
2458181.408 2018 Mar 3 21:48 223.3 -0.89 823.837
2458986.142 2020 May 16 15:24 287.4 -1.40 804.734
The figures after the date are the longitudes at which the events happen (today, 161 degrees, for instance, in Leo as seen from the Sun); and the latitude-difference (today, Mars passes only 0.59 degree north of Jupiter as seen from the Sun). The last column is the lapse in days from one event to the next. That was found by subtracting their Julian dates, the long figures at the left – an example of the usefulness of the Julian-date system. The lapses average 821 days, close to 2.25 years.
The lengths of these intervals result from the lengths of the planets’ orbits – their sidereal periods, 1.88 years for Mars and 11.86 for Jupiter. And the closeness of the conjunctions – the latitude-differences – result from where the planets are in their sloping orbits. Mars in its roughly 2-degree-inclined orbit ascends northward through the ecliptic at about longitude 49 degrees. Jupiter’s orbit is inclined only about 1 degree to the ecliptic (Earth’s orbital plane), but essentially it is not inclined at all: it closely coincides with the “invariant plane” of the whole planetary system, which is not surprising, since Jupiter’s gravity dominates the system.
That’s enough for now; I was going to make the planetary gongs sound, but I’ll do that tomorrow.
If the sidereal periods of Mars and Jupiter don’t vary than why do there heliocentric conjunction periods vary? I can see that if Mars has more perihelions for a cycle then the heliocentric conjunction period would be less and vice versa for aphelions. But in the long run shouldn’t the average be 816.38 years?
The intervals between the heliocentric conjunctions vary because the speeds of the planets change in different parts of their orbits – more so for Mars, whose orbit is more eccenrric. The conjunctions fall at various points relative to the planets’ perihelia.
So I’ve run a calculation of the dates of heliocentric conjunction over a longer span, 1950-2050. The intervals vary from 798.920 days to 835.540, and average 816.895 (as against the 821-day average I gave for the shorter span). So your 816.38 was close, though I don’t know how you arrived at it!