Retrograde motion is a derivative, and 0 Aries will lie to you

I maintain a small astronomy calculator site, and one of the pages answers a question that sounds like a database lookup: which planets were retrograde on the day you were born? There's no retrograde column in an ephemeris. Retrograde isn't a stored fact, it's a property of the first derivative of a position. A planet is retrograde when its apparent geocentric ecliptic longitude is decreasing. So you compute longitude at two instants, subtract, check the sign. Easy. The subtraction is where it gets you. Written the obvious way it invents a Saturn reversal in April 1996, and I want to lead with that, because it generalises to anything you compute on a circular quantity. The bug Here's the obvious implementation: const retrograde = lon(t + step) - lon(t) 180 ? d - 360 : d; } angularDelta(0.19549, 359.95541) gives +0.24008 . Right sign, right magnitude. Here's the same 1996 scan with each version of the difference, printing every station the detector finds: Saturn 1996, naive (b - a): { mid: '1996-04-07T00:00', toRetro: true } geocentricEclipticLongitude(body, new Date(ms)); const vel = (ms: number): number => angularDelta(lon(ms + halfStepMs), lon(ms - halfStepMs)); Then bisect. The velocity function used for refinement is the same centred difference with h = step/2 , so it agrees exactly with the grid values at the bracket endpoints and the bracket is guaranteed valid: while (hiMs - loMs > REFINE_MS) { const midMs = (loMs + hiMs) / 2; const vMid = vel(midMs); if (vLo < 0 !== vMid < 0) { hiMs = midMs; } else { loMs = midMs; vLo = vMid; } } REFINE_MS is 5000, so it converges to within five seconds, which is about 15 bisection steps from a two-day bracket. The contract the tests actually enforce is one minute. The extra headroom is free, so I take it. Two details that only showed up once I was running this for real. The scan runs 230 days past each end of the requested range, because Pluto's retrograde is about 185 days long and a period straddling a boundary has to be seen whole rather than clipped. And the Sun and Moon return [] immediately, since their apparent geocentric longitude only ever increases and they can never station. That early return kills a whole class of nonsense output. What the distribution looks like Once station detection is right you can ask a question I couldn't find a published answer to: how many planets is a person typically born with retrograde? I scanned every calendar day from 1900 to 2100, 73,414 dates sampled at 12:00 UTC, counting how many of the eight planets sat between a retrograde station and the next direct station. These get recomputed from scratch on every build of the natal retrograde page on my site, astrocalcs.com, so they're never a table I typed in once and stopped checking. | Retrograde planets | Share of dates | |---|---| | 0 | 6.8% | | 1 | 19.8% | | 2 | 31.0% | | 3 | 25.7% | | 4 | 12.4% | | 5 or more | 4.3% | Mean is 2.3, and 93.2% of dates have at least one. The bit that surprises everyone: a chart with zero retrograde planets (6.8%) is rarer than one with four (12.4%). Per planet, share of dates on which it's retrograde: | Planet | Share | Mean retrograde length | |---|---|---| | Venus | 7.2% | 42 days | | Mars | 9.4% | 74 days | | Mercury | 19.2% | 22 days | | Jupiter | 30.2% | 121 days | | Saturn | 36.4% | 138 days | | Uranus | 41.2% | 152 days | | Neptune | 42.9% | 158 days | | Pluto | 44.1% | 162 days | Why the gradient exists That six-fold spread has a clean physical explanation, and it's the part I find genuinely satisfying. The share is just retrograde arc length over synodic period, the time between successive alignments with Earth. Check it against known synodic periods and the whole table falls out: Venus 42/584 = 7.2%, Mars 74/780 = 9.5%, Jupiter 121/399 = 30.3%, Saturn 138/378 = 36.5%, Pluto 162/367 = 44.2%. Every one lands within a few tenths of the measured column, which is a nice independent check that the detector isn't drifting. Now take the limit. As a planet's orbital period grows, its synodic period with Earth collapses toward one year, because a body that barely moves is one we lap once per orbit of our own. Uranus is at 370 days, Neptune 367, Pluto 367. And consider the endpoint of that limit: a fixed star. The only thing moving is Earth, and Earth still moves the star. Parallax displaces a nearby star by an amount that shrinks with distance, but stellar aberration swings every star through an ellipse roughly 20.5 arcseconds across no matter how far away it is, because that one depends on Earth's velocity rather than its position. Either way the star's apparent ecliptic longitude oscillates about its mean once a year, and a sinusoid decreases exactly half the time. A fixed star is retrograde 50% of the year, by an amount far too small for anyone to care about. That's the ceiling, and it's entirely a description of our own motion. Every real planet falls short of that ceiling by exactly as much as its own prograde motion contributes. Mars moves fast enough to cancel most of the parallactic swing and lands at 9.4%. Pluto crawls, cancels almost nothing, gets to 44.1%. The gradient from Mars outward is a direct readout of how much of what we're seeing is us. Mercury and Venus sit outside that ordering because they're interior to Earth and retrograde around inferior conjunction rather than opposition. Same ratio still governs them. Mercury's synodic period is only 116 days, so it retrogrades three or four times a year, but each pass is short, which gives 19.2%. Venus takes 584 days between passes with a 42-day arc, and that makes Venus retrograde the rarest natal placement of the eight. Error bars Where this is uncertain, and I'd rather say it than have someone find it: Frame convention. Everything here is apparent position referred to the true equinox of date. Software using astrometric or mean-equinox positions will put stations minutes away from mine. Neither is wrong, they're answering slightly different questions, and any comparison across tools has to fix the convention first. Precision isn't accuracy. The bisection converges to five seconds of the model's station. astronomy-engine agrees with JPL Horizons to within 0.05° for planets at the anchor instants I test against. Near a station the planet is barely moving, so a small position error turns into a much bigger timing error than it would anywhere else in the orbit. Don't read those five seconds as five seconds of physical truth. There's more on the frame and the validation set on my methodology page. Sampling resolution. The distribution samples one instant per calendar day. It's a day-resolution proxy for "birth dates", not a time-integrated measure of how much of the century each configuration occupies. The unit suite pins the per-planet shares against an independently computed, time-integrated run over 1950 to 2009 and requires agreement within 0.4 percentage points. A date isn't an instant. Allow for every time of day and every zone from UTC+14 to UTC−12 and a birth date covers roughly a 50-hour window. If a station falls inside it, the answer genuinely depends on birth time, and the honest output is "can't be determined from the date alone" rather than a confident guess. None of this touches astrology, which I have no opinion about here. Whether a planet's apparent direction at your birth means anything is a question about a tradition. Whether it was moving backward is a question about a number going down, and that one has an exact answer, as long as you subtract your angles correctly. Top comments (0)