BackIn January 2008 I noticed, purely by chance, that Mercury had returned to almost exactly the same place in the sky as I had seen it just under one year previously. This led me to investigate the "back to the same place" phenomenon more generally.
One must first of all define what one means by "the same place", of course. It could be relative to the stars, relative to the Sun or relative to the sky: these three cases set quite different problems. Also, one has to be aware that the situation for the "inner" planets (Mercury & Venus) and the "outer" planets (Mars to Neptune) is different because of their very different apparent motions as seen from Earth.
For the outer planets there isn't really much to say because (as I show in my theory article 'The behaviour of the outer planets near opposition') their path through the sky is a series of loops or zigzags unrelated to the motion of the Sun. Successive loops will often return a planet to the same place as it occupied some while ago (relative to the stars), but this will not be the same "same place" each time [if you see what I mean!]. About the only meaningful statement one can make is that an outer planet will be in the same place relative to the Sun once every "synodic period". To see how this works imagine the Sun, the Earth and the planet (in that order) aligned in an exact straight line: this is the position called "opposition" [because, as seen from Earth, the planet will be opposite the Sun in the sky]. One year later, the Sun and Earth will be in the same relative position again but the planet will have moved round its orbit a bit. To regain the 3-in-a-line orientation, the Earth must therefore keep going round its orbit slightly longer. The total time taken to regain the alignment is called the synodic period. For the distant planets, the synodic period is only a little more than a year (as they move comparatively slowly) but for fast-moving Mars it is as long as 780days. This periodicity also goes some way towards fulfilling the "same place in the sky" criterion because a planet at opposition is due south at local midnight [ignoring daylight-saving adjustments]. It won't be at the same altitude each time though, because the path the Sun and planets follow across the sky (the ecliptic) moves up and down depending on the date and this will change (considerably for Mars, much less for the other outer planets) from year to year.
The inner planets offer more scope however. Their path through the sky is also looped, interestingly, but due to their proper motion as they orbit the Sun rather than the effect of parallax as in the case of the outer planets. Mercury and Venus swing to and fro from one side of the Sun to the other and also follow the Sun across the sky during the course of a year - this makes it much more likely they will return to the same place in the sky on a periodic basis. They have a synodic period of course, but defined as the time between successive conjunctions (the alignment Sun-planet-Earth) rather than oppositions (Sun-Earth-planet). The period is 116days for Mercury (as it orbits much faster than the Earth) but as much as 584days for Venus because, like Mars, its orbital period is much closer to that of the Earth.
These numbers have a trick up their sleeve though, as they are 0.317 and 1.599yrs respectively. Taking the case of Venus first, it is clear that 5 such periods are incredibly close to 8yrs. This means that after 8yrs not only will Venus be in the same position relative to the Sun [by the definition of synodic period] but the Sun will also be in the same position relative to the sky [and indeed the stars] as a whole number of years have elapsed. Even better, every 8yr period (except those spanning a century) always contains two full leap year cycles and so, over time, the re-alignments will keep in step with real calendar years as well as "averaged" astronomical years. Thus Venus will be in almost exactly the same position in the sky every 8 calendar years: this is the basis of the finder chart given on the Venus page in the Astrophotographs section. Similar relationships exist for Mercury, where 41 synodic periods are very close to 13yrs and 52 synodic periods are as close to 16.5yrs as are 5 Venus periods to 8yrs. Note that a period of 16.5yrs will contain four full leap year cycles 7 times out of 8 (and on this timescale century years affect the calculation anyway) so, as with Venus, series of Mercury alignments also keep in step with calendar years.
Digressing slightly for a moment, these periodicities have a relevance when considering transits of the inner planets across the face of the Sun, which occur at intervals of 8yrs for Venus and series of 13 then 3.5yrs for Mercury. A transit does not happen at every possible interval however, because the orbits of the planets are slightly tilted with respect to that of the Earth and so they usually pass "under" or "over" the Sun instead of directly across it. The most recent transits visible in the UK happened on 7th May 2003, 9th May 2016 & 11th November 2019 for Mercury (illustrating the 13+3.5yr periodicity) and 8th June 2004 for Venus: that of Venus on 6th June 2012 (illustrating the 8yr periodicity) was not visible in the UK. The next (partially) UK-visible transit of Mercury is on 13th November 2032. Unfortunately the next UK-visible transit of Venus is not until 8th December 2125!
Getting back to the more general "same place" problem, note that three Mercury synodic periods are roughly equal to a year (just under 18days short). Given the rapid movement of that planet across the sky it would thus seem possible that at sometime during that fortnight or so it would return quite close to where it was a year previously. Investigation with my astronomy programs showed that for the period early February 2007 to late January 2008 this was indeed the case: there were several pairs of dates separated by about 359days on which Mercury was quite or very close to being at the same place in the sky. The dates on which I observed it were separated by just 352days but if corrections are made for the different viewpoints and the different times of observation [an extra 4mins is equivalent to observing at the same time 1day later] the effective interval rises to 360days, in good agreement with my investigations and explaining why the planet had returned to nearly the same place in the sky. This is not a true periodicity though, as the 359day figure is only applicable to this particular time-period. Another "nearly 1yr" period would have a different return-to-place interval as the value is determined by the precise path across the sky taken by Mercury around the two dates in question, which varies considerably depending on the season and on which part of its rather eccentric orbit we are seeing. Good to have a theoretical explanation for my observations though!
Just to complete the story, the Moon also has a same-place periodicity, best appreciated with reference to Full Moons. These occur at intervals of 29.53days so 37 full Moons take just a fraction over 3days short of 3yrs. Indeed, depending on what value you take for the exact duration of a year (which is a story in its own right!) the interval can be expressed as 3yrs minus 3days 3hrs - nice and easy to remember. These 3days cannot be ignored though (as the Moon moves much further each day compared to the planets) and indeed the 3hrs quickly add up to spoil the situation, so this periodicity is in practice much nearer to the "pseudo-periodicity" of nearly-1yr Mercury alignments than the almost exact periodicity of Venus alignments. Werewolves are advised to be aware of the "3-(3+3)" calculation though!