The loop a planet does not make

Mars drifts eastward against the stars at a fraction of a degree a day. Then, every couple of years, it slows, stops, runs backwards for ten weeks, stops again and resumes — tracing a closed loop on a sphere that carries no distances at all.

Nothing in the orbit of Mars does anything of the kind. The orbit is a smooth closed curve traversed in one direction, and the speed along it never reverses. Nor does the Earth’s. The loop is in neither.

It is in the difference between them, seen from one of the two. That is the whole claim, and the rest is the arithmetic of what that difference does when the distances change.

What is in the picture and not in the orbit #

The claim, stated so that it could be wrong: the reversal is a property of a projection, and no property of either orbit changes sign anywhere near it.

Two things are projected away. The first is depth: the sky records a direction and discards the range, which is exactly the operation the celestial sphere performs. The second is the observer’s own motion, because a direction measured from a moving platform contains the platform’s velocity whether or not anyone knows about the platform.

Retrograde motion is what those two projections do to a difference of two vectors: throw away the length, keep the direction, and take it from a point that is itself in orbit. Restore either piece and the loop vanishes. An observer with a range-finder sees Mars recede and approach monotonically about opposition; an observer at rest at the Sun sees Mars circulate at a rate that is never zero. The refutation has been available for four centuries: predict the two dates on which the motion stops, from the two orbits alone, with nothing in the calculation that reverses. The prediction holds, for every planet at once.

The sky is the direction of a difference #

Write the two heliocentric positions as $\vec r_E$ and $\vec r_P$. The observed quantity is the direction of

$$\vec\rho(t) = \vec r_P(t) - \vec r_E(t), \qquad \lambda = \arg \vec\rho,$$

and the rate at which that direction turns is

$$\dot\lambda = \frac{\left(\vec\rho \times \dot{\vec\rho}\right)_z}{|\vec\rho|^2}.$$

A station — an instant at which the apparent motion stops — is therefore a zero of that numerator, which happens when the relative velocity points along the line of sight. That is the whole condition, and it says at once why stations exist: the relative velocity swings through a full turn every synodic period, so it must lie along the sight line twice in each one.

At opposition the geometry finishes by hand. Both bodies lie on one line from the Sun, both velocities are perpendicular to it, and the separation is $a - 1$ in units of the Earth’s radius, so

$$\dot\lambda_{\rm opp} = -\frac{v_E - v_P}{a - 1}.$$

The harmonic law supplies $v_P/v_E = a^{-1/2}$, giving $\dot\lambda_{\rm opp} = -2\pi(1 - a^{-1/2})/(a-1)$ radians per year. For Mars at 1.5237 AU that is 0.358° per day, backwards. And because $a^{-1/2} < 1$ for every planet outside the Earth’s orbit, the sign is negative for all of them: retrograde motion at opposition is not a fact about Mars but a theorem about being overtaken on the inside. The same subtraction, done between two spacecraft rather than two planets, is why closing on a target ahead requires slowing down.

One ratio sets the width, and another sets the time #

Only one number enters that formula: $a$, the ratio of the two orbital radii. Not the eccentricities, not the masses, not the absolute scale of anything. Two orbits in the ratio 1.5237 give the same loop in astronomical units or in kilometres — which makes the loop an instrument pointed at that ratio, reading it two ways at once.

Mars reverses for 72.7 days across 15.94° of longitude. Neptune reverses for 158.5 days across 2.80°. The interval has more than doubled and the arc has shrunk by a factor of 5.7, and the only difference between the two calculations is 1.5237 against 30.07.

The two run in opposite directions for separate reasons. The arc narrows because it is essentially the angle the Earth’s orbit subtends at the planet, which falls as $1/a$. The interval lengthens because the reversal ends only when the planet’s own prograde motion cancels the Earth’s advantage, and a further planet has less motion of its own to contribute: Mars travels at 81 per cent of the Earth’s orbital speed and cancels it soon, Neptune at 18 per cent and barely at all. The same slowing shortens the beat between the orbits, from a synodic period of 779.9 days to 367.5.

The trend has a limit, and the limit is half a year #

Swept across the family, both quantities are monotonic and neither runs away. From 1.05 AU out to 34 AU the arc narrows from 16.5° to 2.5° while the interval lengthens from 57 to 160 days, and the five planets fall along it in order: Mars 15.9° in 73 days, Jupiter 9.9° in 121, Saturn 6.8° in 138, Uranus 4.0° in 152, Neptune 2.8° in 158.

The ceiling on the interval is 182.6 days, half the Earth’s year, and the arc’s floor is zero. Both limits describe the same imaginary planet: one infinitely far away, which does not move, about which the Earth’s own half-orbit carries the sight line one way and the other half carries it back. A body that never moves still appears to reverse, once a year, for exactly half of it.

As a fraction of the synodic period the convergence is plainer still. Mars spends 9.3 per cent of its cycle retrograde, Jupiter 30.2, Saturn 36.5, Uranus 41.1 and Neptune 43.1 — climbing towards the one half the limit demands, because for a body that does not move the synodic period is the year. The same beat between two periods, read as a waiting time instead of an arc, is what decides when a mission may be launched.

The loop is a parallax with the baseline in the wrong place #

At the far end of that sweep the loop has stopped being planetary motion of any kind. A distant body that does not move, observed from the two ends of a 2 AU baseline six months apart, is displaced by an angle inversely proportional to its distance — which is the measurement that reaches the stars, arriving from an entirely unexpected direction.

The arithmetic connects the two cases exactly. Neptune sits at 30.07 AU, about 8,900 times nearer than a star at 1.3 parsecs, so the same construction returns 8,900 times the angle: a half-amplitude of 1.91° rather than 0.77 arcseconds, and a full swing of 3.81°. The figure measures a retrograde arc of 2.80°, and the difference is Neptune’s own prograde motion — 0.95° of longitude in the 158.5 days between its stations, with $3.81° - 0.95° = 2.86°$ recovering the measured arc to within the second-order terms the estimate drops.

Retrograde motion and stellar parallax are therefore one phenomenon at two distances, and what hid the kinship is a factor of ten thousand in the target’s range. The planets’ version is degrees and unmissable; the stars’ version is under an arcsecond and took until 1838.

The epicycle had the geometry right #

The difference $\vec\rho = \vec r_P + (-\vec r_E)$ is a sum of two circular motions, and a sum of two circular motions is a deferent carrying an epicycle. Written in a frame where the Earth is held fixed, the first term is a circle of radius $a$ traversed in the planet’s period and the second a circle of radius 1 AU traversed in the Earth’s.

That is Ptolemy’s construction, in the Almagest of about 150, with the pivot moved. It was never wrong about the geometry — a vector sum is a vector sum — and the fits it produced were good. What it could not explain was its own parameters. The ratio of epicycle to deferent must come out as $1/a$, and in the units where the deferent is 60 the tabulated epicycle radii are 39.5 for Mars, 11.5 for Jupiter and 6.5 for Saturn: 0.658, 0.192 and 0.108 of their deferents, against $1/a$ of 0.656, 0.192 and 0.105. The Mars figure computes that ratio independently and gets 0.656.

The tell is the period, not the radius. Every superior planet’s epicycle takes exactly one year, and the line from epicycle centre to planet stays parallel to the Earth–Sun direction — a rule Ptolemy had to impose three times over, once per planet, and which the difference of two vectors produces for free because that line is the Earth’s orbit. Copernicus’s argument of 1543 was not that the epicycles were absent but that they were all the same one, belonging to the observer.

Venus turns round at the near end #

An inferior planet subtracts the same way and produces a mirror image, because the roles of the two radii swap: the Earth is now the slower body and the planet overtakes on the inside.

Venus therefore reverses at inferior conjunction rather than at opposition, and the numbers are not a mild variation on Mars’s. Its arc is 16.11°, slightly the wider of the two, but it is covered in 42.2 days rather than 72.7, and the loop it closes is 6.96° tall against Mars’s 2.94°, because Venus is only 0.281 AU away when it turns and its 3.39° inclination is projected from close range. Mercury is faster and narrower again: 13.78° in 22.9 days.

That asymmetry is the one that makes an inferior planet show a full set of phases while a superior one never does. Both follow from which side of the Earth’s orbit the planet is on when the two heliocentric longitudes coincide: the far side for Mars, so it is fully lit and the loop is slow; the near side for Venus, so it is dark and the loop is quick.

What is actually measured #

Nothing here is measured directly. What an observer records is a date, an ecliptic longitude and an ecliptic latitude, got by measuring the planet’s angular separation from catalogued stars — an armillary instrument or cross-staff good to some ten arcminutes for Ptolemy, a mural quadrant and sextant good to about one for Tycho Brahe. Two angles per night and never a range, which is exactly the situation in which an orbit must be recovered from directions alone.

The stations are not observed at all; they are interpolated, and badly conditioned by construction. Longitude leaves a station quadratically, so on the drawn Mars geometry the planet has moved 0.40 arcminutes one day after its station, 1.58 two days after and 3.55 three days after. A position good to one arcminute therefore fixes the date to no better than about a day and a half, and one good to five arcminutes to about three and a half — days of uncertainty on the 72.7-day interval before any error in the angle is counted.

The same quadratic that ruins the date protects the value. Because the longitude is barely changing near a station, an error of days in when the planet stopped costs arcminutes in where it stopped, so the 15.94° arc is well determined by observations that cannot pin the interval — which is why the arc, not the duration, is what historical records recover reliably.

What the arc buys is $a$, the only unknown in the geometry. The absolute scale has to arrive from elsewhere entirely, as one distance in kilometres from a transit, a radar echo or a spacecraft, and until it did the solar system was known in proportion and not in size.

What the picture cannot show #

The loop need not close. The track here encloses an area because opposition has been placed at the planet’s greatest ecliptic latitude; put it at a node and the identical computation gives a zigzag enclosing nothing. Real oppositions fall at every phase of that cycle, so Mars’s successive apparitions produce loops, S-shapes and near-straight kinks in turn, and every figure here draws one member of that family rather than the family.

The orbits are circles. Mars’s eccentricity of 0.0934 puts its aphelion 1.21 times its perihelion, so its opposition distance varies from one apparition to the next and so does the size of the loop. The drawing has Mars at 0.525 AU; the real range is wider, and a circle is what an ellipse of small eccentricity looks like until it is measured carefully.

No figure here shows a loop’s true shape. Longitude and latitude are drawn at different scales in every one — 4.6 times for Mars, 7.5 for Neptune, 2.1 for Venus — because otherwise the latitude excursion is invisible. At true scale Neptune’s loop is a line with a thickening in the middle, 2.8° long and 0.11° tall.

Depth is absent by design in the sky plots and by convenience in the plan view. The schematic sky in the vector figure is at no scale, and the sight lines stop there for the drawing’s sake; in reality they run past the planet indefinitely, which is the whole reason the loop exists.

The loop seen from somewhere else #

The construction has one observer in it, and moving the observer is the cleanest way to show that the loop belongs to the projection rather than to the planet.

From Mars, the Earth retrogrades. The subtraction is the same and the roles are exchanged: Mars is now the slower body being overtaken on the inside, so the Earth reverses at its inferior conjunction, close in and quickly, in the way Venus does from here. Nothing about the Earth’s orbit differs between the two accounts.

That is not a hypothetical. Spacecraft at Mars have recorded the Earth as a point of light for years at a time, and its apparent path against the stars does what the geometry requires.

The more instructive case is an observer that is not on any planet. A spacecraft on a heliocentric transfer orbit is subtracting its own position from every planet’s, and its orbit is neither circular nor shared with anything, so the loops it sees are of a shape no planetary observer ever gets: their size and duration change from one apparition to the next as the spacecraft’s own distance changes, and a planet the spacecraft is receding from can retrograde for a substantial fraction of a year.

That has a practical consequence rather than merely a curious one. Optical navigation works by imaging a target body against the background stars and fitting the trajectory to the resulting directions, and the geometry of that fit is exactly the geometry of this essay: what is measured is the direction of a difference of two position vectors, one of which is the thing being solved for.

The stations are as unhelpful there as they are here. A body near a station is a body whose apparent direction is barely changing, so an image taken then constrains the spacecraft’s position weakly in one direction, and navigation campaigns are scheduled to avoid exactly those geometries.

A phenomenon that misled astronomy for fifteen hundred years is a scheduling constraint on a spacecraft’s camera, and both follow from the same statement: what a sky records is a direction, and a direction from a moving observer contains the observer.

Where this ladder goes next #

Later rungs on apparent-motion: elongation and the phase angle as one variable, and greatest elongation as the measurement that gives an inferior planet’s distance without any loop at all. The zodiac as a consequence of small inclinations. Conjunctions between two planets, the same beat taken between two synodic periods rather than one. And the outer planets’ apparitions, recurring at intervals only days longer than a year and so drifting through the calendar for the same reason as the star that rises four minutes earlier each night.

The Moon is the exception that confirms the rule. It moves against the stars at about 13° a day, always eastward, and it never retrogrades — because the Earth’s motion is not subtracted from a lunar position in any interesting way. The Moon orbits the pivot. There is no difference of two large vectors to project, so there is no loop, and the fastest thing in the sky is the one thing that never stops.