Apparent_retrograde_motion

# Apparent retrograde motion **Apparent retrograde motion** is the temporary reversal of a body's drift against the background stars, seen when the observer's own motion carries them past the body or the body past them. Every superior planet does it: [[Mars|Mars]] creeps eastward among the stars for about twenty-two months, then for roughly seventy-three days slides westward through an arc of about sixteen degrees, then resumes its eastward march. Nothing about the planet changes. What changes is the line of sight, because [[Earth|Earth]] on its faster inner [[Orbit|orbit]] overtakes Mars and swings the sightline backwards. In the microsim below the reader sets one number — the period ratio T_outer/T_inner, 1.88 for Mars — and an epicycle radius, and watches two completely different machines draw the identical loop: [[Ptolemy|Ptolemy]]'s deferent-plus-epicycle sum x = R·cos(W·t) + r·cos(w·t), and [[Nicolaus_Copernicus|Copernicus]]'s two circles viewed from a moving Earth. Both repeat on the synodic period given by 1/S = 1/T_E − 1/T_M, which for Mars is 780 days. On the [[Physics|Physics]] flagship this article opens Part I — History, at the section *Ancient astronomy* (row P1), because it is the oldest quantitative problem in [[Classical_physics|physics]] and the cleanest example of an observation that two rival theories explain equally well. The point of the sim is not that Ptolemy was wrong. It is that from the ground the two models are indistinguishable — the sky returns the same curve for both — and that is why the argument between [[Geocentrism|geocentrism]] and [[Heliocentrism|heliocentrism]] ran for fourteen centuries and was settled in the end by other evidence entirely. ## Etymology and history The retrograde loops were the central technical problem of ancient astronomy, and every serious [[Planetary_system|planetary system]] from Babylon to [[Johannes_Kepler|Kepler]] can be read as an answer to them. They are also the reason the word *planet* exists: the Greek *planētēs*, "wanderer", separates the five naked-eye planets from the fixed stars precisely because the planets do not keep a steady course. ### The word *Retrograde* is from the Latin *retrogradus*, "stepping backwards", and in astronomy it means westward relative to the general eastward drift of the [[Sun|Sun]], [[Moon|Moon]] and planets against the stars. The qualifier *apparent* is load-bearing. Mars never reverses its orbital motion; its [[Velocity|velocity]] around the Sun stays in the same sense throughout. Only the direction from which it is seen reverses. The distinction matters because a handful of real bodies — some outer satellites, and comets on retrograde orbits — do circulate the other way, and for those the motion is not apparent but actual. ### Babylonian period relations Babylonian scribes of the Neo-Babylonian and Seleucid periods predicted the stations — the moments when a planet stops and turns — without any geometrical model at all. They worked from period relations: whole numbers of synodic cycles that close almost exactly on a whole number of years, so that the pattern of loops repeats. The Goal-Year texts use 79 years for Mars, 71 for [[Jupiter|Jupiter]], 59 for Saturn, 8 for [[Venus|Venus]] and 46 for Mercury.[^ossendrijver] The arithmetic behind them is the same relation the microsim uses. With the modern periods, Mars's synodic period is 779.9 days, so 37 synodic cycles come to 28,858 days against 28,855 days in 79 Julian years — a drift of under three days per century and a half (derived).[^b077-astro] Jupiter's 398.9-day cycle gives 65 cycles in 25,927 days against 25,933 (derived); Saturn's 378.1-day cycle gives 57 in 21,551 against 21,550 (derived); Mercury's 115.9-day cycle gives 145 in 16,802 against 16,802 (derived). The Babylonian numbers are not approximations chosen for convenience. They are the best small-integer resonances the sky offers, and they let a scribe predict a station centuries ahead using nothing but a table. ### Ptolemy's deferent and epicycle Around AD 150 [[Ptolemy|Claudius Ptolemy]] set out in the *Almagest* the model that carried the subject for the next fourteen hundred years. Each planet rides a small circle, the epicycle, whose centre rides a large circle, the [[Deferent_and_epicycle|deferent]], centred near a motionless Earth. The planet's place is the vector sum of two circular motions, which in coordinates is x = R·cos(W·t) + r·cos(w·t) and y = R·sin(W·t) + r·sin(w·t). When the epicyclic term runs backwards faster than the deferent term runs forwards, the sum reverses, and the planet loops. This closed-form sum is standard kinematics rather than a result taken from any Portal Book, and the microsim labels it so on screen. Ptolemy's numbers carry a message he could not read. He tabulates each epicycle radius in units of a deferent of sixty parts: 39;30 for Mars, 11;30 for Jupiter, 6;30 for Saturn, 43;10 for Venus, 22;30 for Mercury.[^almagest] Divide by sixty and, for the superior planets, the ratio is the inverse of the planet's distance from the Sun in astronomical units. Mars's 39.5/60 = 0.6583 implies 1.519 AU against the modern 1.524, an error of 0.3 per cent; Jupiter's 11.5/60 implies 5.217 AU against 5.204, an error of 0.25 per cent (derived).[^b077-astro] For Venus and Mercury, where the roles of the two circles are exchanged, the ratio is the planet's distance itself: 43;10 gives 0.719 AU against 0.723 (derived). The [[Heliocentrism|heliocentric]] distances were sitting in the Ptolemaic tables all along, disguised as epicycle sizes. ### Copernicus and the overtaking reading In *De revolutionibus orbium coelestium* of 1543, Copernicus removed the epicycles that produced retrogradation by moving the Earth.[^copernicus] If Earth is a planet on an inner, faster orbit, then near opposition it overtakes the outer planet on the inside of the track, and the sightline to that planet swings backwards for as long as the overtaking lasts. The loop is not a motion of the planet but a [[Velocity|relative-velocity]] effect, and the size of the loop follows from the orbit sizes rather than from a free parameter. That was the argument's real content, and it was an argument about economy, not about evidence. Ptolemy's epicycle radius r and Copernicus's Earth-orbit radius a_E play the same role in the same sum: r/R = a_E/a_P. Tycho Brahe's geoheliocentric system of 1588, with the planets circling the Sun and the Sun circling a fixed Earth, reproduces the same appearances a third time.[^tycho] All three predict the same loops, so no loop can decide between them. What eventually decided it came from elsewhere: Galileo's observation in late 1610 that Venus shows a complete cycle of phases, impossible if Venus never passes beyond the Sun,[^drake] and the stellar parallax that Friedrich Bessel finally measured for 61 Cygni in 1838, which had been the geocentrists' strongest objection for two thousand years.[^bessel] ## Apparent motion The mechanism is one vector identity. The line from Earth to a planet is the line from Earth to the Sun plus the line from the Sun to the planet: r_EP(t) = a_P·(cos 2π t/T_P, sin 2π t/T_P) − a_E·(cos 2π t/T_E, sin 2π t/T_E). Read left to right, that is Copernicus. Read as "a big circle plus a small circle", with the small circle's radius fixed at a_E and its arm always parallel to the Sun–Earth line, it is Ptolemy. The two models are the same two terms in a different order, which is why no observation of longitude alone can separate them. ### The two models on one screen The microsim draws both sums at once from the same controls. The reader sets the period ratio T_outer/T_inner between 1.1 and 12 — Mars sits at 1.88, Jupiter at 11.86 — and the epicycle radius r/R between 0 and 1. Two traces appear: the Ptolemaic sum x = R·cos(W·t) + r·cos(w·t) in one colour, the Copernican difference of two orbit vectors in another, with the planet's apparent longitude plotted against time underneath. The traces coincide exactly whenever r/R is set to 1/(period ratio)^(2/3), the value [[Kepler's_laws_of_planetary_motion|Kepler's third law]] assigns, and they separate visibly when it is not — which is the sim's one honest degree of freedom, because Ptolemy fitted r/R to the observed loop size and did not derive it. Both traces repeat on the synodic period, from 1/S = 1/T_E − 1/T_M for a superior planet and 1/S = 1/T_P − 1/T_E for an inferior one. Mars gives 779.9 days, Jupiter 398.9, Saturn 378.1, Venus 583.9 and Mercury 115.9 (derived).[^b077-astro] A loop counter in the corner ticks once per S. One caveat is marked ILLUSTRATIVE on screen: in a strictly coplanar model the planet retraces the same line, a straight back-and-forth rather than a loop. The sideways excursion that makes the familiar loop or zigzag comes from the small tilt of the planet's orbit — 1.85° for Mars — and the sim adds it as a display term, not as part of either historical model. ### From Earth For an observer on Earth the loop happens around opposition, when the planet is opposite the Sun in the sky and closest to Earth. Retrograde motion begins when the angular rate of the sightline changes sign, that is when the cross product r_EP × ṙ_EP passes through zero. At opposition the sightline's transverse speed is a_P·ω_P − a_E·ω_E; for Mars that is 0.01394 − 0.01720 = −0.00327 AU per day, comfortably negative, so the reversal is guaranteed rather than marginal (derived). Solving the cross-product condition for circular coplanar orbits gives the loops that textbooks quote: Mars reverses for 72.7 days through an arc of 15.9°, Jupiter for 120.6 days through 9.9°, Saturn for 137.5 days through 6.7°, Uranus for 151.7 days through 4.0° (all derived).[^b077-grav] The pattern is systematic and is the sim's payoff: the further out the planet, the longer and shallower its loop, because the overtaking takes longer while the planet's parallactic displacement shrinks with distance. The numbers come from the model, not from a table of observations, and they agree with the sky to within the error introduced by treating the orbits as circles — which is largest for Mars, whose [[Orbital_eccentricity|eccentricity]] of 0.093 makes successive loops visibly unequal.[^nasa-facts] The physical reason the whole scheme works at all is [[Newton's_law_of_universal_gravitation|Newton's inverse-square law]], which forces T² ∝ a³ and so ties every period ratio in the sim to a distance ratio.[^b076-gravity][^b074-central] ### From Mercury An observer standing on Mercury would see the same effect with different numbers, and one extra effect with no counterpart on Earth. Taking Mercury's 87.97-day year as T_inner, the synodic periods become 144.6 days for Venus, 115.9 for Earth, 100.9 for Mars and 89.8 for Jupiter (derived): every planet loops several times a Mercurian year, and the loops are wide because Mercury's orbit is a large fraction of theirs. The extra effect involves the [[Sun|Sun]] itself. Mercury is locked in a 3:2 spin–orbit resonance, rotating exactly three times for every two orbits — 58.65 days against 87.97 — which radar measurements established in 1965.[^pettengill][^nasa-facts] Near perihelion the orbital angular rate exceeds the fixed rotation rate, and the Sun's apparent westward march across Mercury's sky stops and reverses. With Kepler's second law the orbital rate at true anomaly ν is proportional to (1 + e·cos ν)²/(1 − e²)^(3/2); setting that equal to the 3/2 rotation ratio at e = 0.2056 gives reversal for |ν| < 25.4°, which is 4.05 days either side of perihelion, so the Sun backs up for about 8.1 Earth days (derived).[^nasa-facts] This retrogradation has nothing to do with the observer overtaking anything. It is a mismatch between two rates on the same body, and it is the cleanest reminder that "retrograde" names an appearance, not a mechanism. ## See also - [[Deferent_and_epicycle]] - [[Geocentrism]] - [[Heliocentrism]] - [[Kepler's_laws_of_planetary_motion]] - [[Orbital_eccentricity]] - [[Planetary_system]] - [[Ptolemy]] ## References [^b077-grav]: Ling, Samuel J.; Sanny, Jeff; Moebs, William (2016). *University Physics Volume 1*. OpenStax. Chapter 13, "Gravitation", pp. 611–664 (Kepler's laws, orbital periods and the inverse-square law; page to pin). Portal Book 077. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 [^b077-astro]: Ling, Sanny and Moebs (2016), *University Physics Volume 1*, Appendix D, "Astronomical Data", pp. 885–886 (planetary semi-major axes and orbital periods; page to pin). Portal Book 077. Synodic periods, Goal-Year residuals and implied distances quoted here are computed from those data for this article. [^b076-gravity]: Gea-Banacloche, Julio (2019). *University Physics I: Classical Mechanics*. Chapter 12, "Gravity", pp. 239–270 (Kepler's laws from the inverse-square force; page to pin). Portal Book 076. https://open.umn.edu/opentextbooks/textbooks/university-physics-i-classical-mechanics [^b074-central]: Likharev, Konstantin K. (2013). *Essential Graduate Physics, Part CM: Classical Mechanics*. Chapter 3 (two-body central forces and planetary motion), pp. 33–54 (page to pin). Portal Book 074. https://open.umn.edu/opentextbooks/textbooks/part-cm-classical-mechanics [^almagest]: Ptolemy, Claudius. *Almagest*, Books IX–XI (the epicycle and deferent radii for the five planets, in units of a deferent of 60 parts). English translation: Toomer, G. J. (1984). *Ptolemy's Almagest*. London: Duckworth (page to pin). [^copernicus]: Copernicus, Nicolaus (1543). *De revolutionibus orbium coelestium*. Nuremberg: Johannes Petreius. Book I, chapter 10, and Book V (the retrogradations of the superior planets referred to the Earth's motion; page to pin). [^tycho]: Brahe, Tycho (1588). *De mundi aetherei recentioribus phaenomenis liber secundus*. Uraniborg. Chapter 8 (the geoheliocentric arrangement; page to pin). [^ossendrijver]: Ossendrijver, Mathieu (2012). *Babylonian Mathematical Astronomy: Procedure Texts*. New York: Springer (Goal-Year period relations and the arithmetical schemes for planetary stations; page to pin). [^drake]: Drake, Stillman (1978). *Galileo at Work: His Scientific Biography*. Chicago: University of Chicago Press (Galileo's observation of the complete phase cycle of Venus in late 1610 and its bearing on the Ptolemaic arrangement; page to pin). [^bessel]: Bessel, F. W. (1838). "Bestimmung der Entfernung des 61sten Sterns des Schwans." *Astronomische Nachrichten* 16: 65–96. [^pettengill]: Pettengill, G. H.; Dyce, R. B. (1965). "A radar determination of the rotation of the planet Mercury." *Nature* 206: 1240. [^nasa-facts]: NASA Space Science Data Coordinated Archive. *Planetary Fact Sheets* (orbital periods, semi-major axes, eccentricities and rotation periods used for the derived numbers on this page). https://nssdc.gsfc.nasa.gov/planetary/factsheet/ ## External links - [Planetary Fact Sheets](https://nssdc.gsfc.nasa.gov/planetary/factsheet/), NASA Space Science Data Coordinated Archive - [*University Physics Volume 1*](https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1), OpenStax — Portal Book 077, chapter 13 "Gravitation" - [*University Physics I: Classical Mechanics*](https://open.umn.edu/opentextbooks/textbooks/university-physics-i-classical-mechanics), Gea-Banacloche — Portal Book 076, chapter 12 "Gravity"  **Microsim — three.js (Wikitube framework), pending deploy:** *Apparent retrograde motion* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Apparent_retrograde_motion.html` is live.   ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Apparent_retrograde_motion) : [Wikitube](https://en.wikitube.io/wiki/Apparent_retrograde_motion) · pinned revision [1345677920](https://en.wikipedia.org/w/index.php?oldid=1345677920) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P1 · sim pending (matter/Apparent_retrograde_motion).*