Dear Peaker,
What must be understood is that the reason why the heliocentric model was accepted back in the days has everything to do with Kepler’s “mathemagics”. It was Kepler who introduced the fancy notions of “variable orbital speeds” and “elliptical orbits”. In such manner, he was able to ‘stretch and squeeze’ Tycho Brahe’s observational data at will…
As illustrated in my below diagram (from the JS Orrery - a Copernican simulator), Kepler must have felt satisfied when he reached a ‘solution’ to the mysterious “SHORT ESI” of ca. 546 days: as you can see, Copernican simulators show the vectors between Earth & Mars pointing in the same GENERAL DIRECTION on 7 successive occasions (the LONG ESI’s of ca. 707 days) and, the eight time (after only 546 days) - again in the same GENERAL DIRECTION. The problem is, the Copernican model would have Earth & Mars each time, in wholly different positions - perpendicularly to that star (in my example, the star Deneb Algedi). Note that, interestingly enough, the 8th time around (see positions 7 and 8) the Earth will find itself at the exact opposite side of its (supposed) orbit around the Sun! So there you have it: Kepler’s “mathemagics” (somehow) did the trick!
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In stark contrast (and, as you may admit, far more ‘elegantly’), the TYCHOS model has Earth & Mars re-aligning at the exact same line of sight towards that star…on all 8 occasions! And this, “in spite” of the fact that the Tychosium simulator has DONE AWAY with Kepler’s fancy “variable speeds and elliptical orbits”! In other words, the critics of the TYCHOS model would have to chalk all of this up (i.e. that perfectly straight line of sight towards that star AT ALL TIMES) to some sensational / miraculous / astronomical coincidence… 
And yes, as you go to Wikipedia (or any other ‘official’ source) you’ll only find that “687-day” value for Mars’s sidereal period. Yet, in reality, Mars
never reconjuncts with any given star in 687 days - which is just the averaged value of 707 + 707 + 707 + 707 + 707 + 707 + 707 + 546 = 5495 / 8 ≈ 687.
And no, to my knowledge, this issue (of the SHORT ESI of 546 days - as of the Mayan Codex) is not being discussed in any modern astronomy literature… Of course, Kepler must have known about it though! As for Tycho Brahe, I would imagine that the strange motions of Mars tortured him for years - or else he wouldn’t have given Kepler his observational data of Mars. The two didn’t like each other and were always bickering, but perhaps Brahe was hoping that Kepler would help him solve the pesky Mars riddle… Instead, Kepler eventually flipped his master’s excellent and near-perfect Solar System model on its head!