Making a planet go backwards

Earth goes around the Sun once a year. Mars takes nearly two years, and travels on a bigger circle further out. So Earth is the faster car on the inside lane, and once every twenty-six months or so it catches Mars up and overtakes it.

Think of overtaking a slow train from a fast one. As you pull level and pass, the slow train seems to drift backwards against the hills behind it. Nobody thinks it has reversed. In the sky the hills are the stars and the slow train is Mars, and that backwards drift is what astrologers and astronomers both call a retrograde.

A picture cannot give you dates, so the essay builds the smallest model that can. Two circles around the Sun, one of radius 1 for Earth and one of radius 1.5237 for Mars, each turning at a steady rate. Four numbers, all published by NASA. Draw an arrow from Earth to Mars, measure its direction, and you have the number a chart prints.

Run the model through the year 2020 and the loop appears in the numbers on its own. Nothing was added to produce it.

Then the turning points. A chart records the direction of that arrow, not its length, so the direction stops changing at the moment Mars is moving straight towards us or straight away. That condition becomes one short formula, worked in full in the essay.

The model says Mars turns 36.4 days before it is directly opposite the Sun, and turns back 36.4 days after. For 2020 that gives 7 September and 19 November. Real Mars turned on 9 September and 14 November.

The last section prices that gap. Circles are the wrong shape: Mars's orbit is slightly squashed, so it speeds up when nearer the Sun. Six real retrogrades in a row ran from 62 to 81 days. The model always answers 72.7, because a circle has no near side. It lands near the middle and misses each end by about a week.