Astronomy & Celestial Mechanics
the sky, computed
The observed sky

The solar system measured from inside one orbit

Venus never appears more than 47 degrees from the Sun. That single angle gives its orbital radius as a fraction of the Earth's, with no distance measured anywhere — and every other planet gives one up as easily.

Assumes Apparent motion and Harmonic law.

16 min read 8 figures The sky as a clockEverything is a conicMeasuring the unreachable

Nothing in the solar system had ever been measured, in any unit, when Copernicus laid out its dimensions. There was no parallax of anything, no distance to anything, and no way to obtain one. What there was, for anyone who cared to look, was a set of angles — the angle between a planet and the Sun, night after night, year after year — and a set of clocks, being the intervals at which the same configuration returned.

Angles and clocks are enough. They give the radius of every visible planet’s orbit to three significant figures, in units of the Earth’s own, and there is no distance anywhere in the derivation.

The solar system to three figures, and not one distance in it. Left, why an inferior planet's wandering is a measurement. At greatest elongation the sight line from the Earth is tangent to the planet's orbit, so the angle at the planet is a right angle and a/a⊕ = sin ε — no distance anywhere in the argument, only the angle between two directions. Venus reaches 45.4°–47.1°, giving 0.7224 AU against the modern 0.72333. Right, every planet Copernicus could see, derived this way and by the synodic route for the outer ones — 1/P = 1/E − 1/S for the year, then the harmonic law for the distance — plotted against the catalogue. Mercury is the interesting failure: its elongation runs from 17.9° to 27.8° rather than sitting still, so the method returns a range, 0.307 to 0.466 AU, and the true 0.3871 lies inside it. That spread is not an error in the method; it is Mercury's eccentricity being measured by a technique that assumed a circle.
Fig. 1 Left, the construction that does it for a planet closer to the Sun than the Earth. At greatest elongation the sight line from the Earth is tangent to the planet’s orbit, so the triangle Sun–planet–Earth is right-angled at the planet, and a/a⊕=sin⁡εa/a_\oplus = \sin\varepsilon — no distance appears anywhere in the argument, only the angle between two directions. Venus reaches 45.4° to 47.1°, giving 0.7224 AU against the modern 0.72333. Right, all five naked-eye planets derived this way and by the synodic route for the outer ones, plotted against the catalogue. Mercury is the informative failure: its elongation runs from 17.9° to 27.8°, so the method returns a range, and the true value lies inside it.

The inner planets: one angle

The construction in the left panel is the whole method for Mercury and Venus, and its cleanness is the point. As an inferior planet moves round the Sun, its apparent angular distance from the Sun — its elongation — grows to a maximum and falls back. At the maximum, the Earth–planet line is tangent to the planet’s orbit, which is a statement of elementary geometry: the sight line touching a circle is perpendicular to the radius at the point of contact.

So sin⁡εmax⁡=a/a⊕\sin\varepsilon_{\max} = a/a_\oplus, and measuring εmax⁡\varepsilon_{\max} is a matter of watching Venus for a few months with a quadrant and noting when it stops receding from the Sun. That is what Copernicus had; it is what any observer with a clear western horizon has.

Mars through 8.3 months of sky. The geocentric ecliptic longitude and latitude of Mars over 252 days — 8.3 months — centred on opposition, computed as the direction of P − E with both orbits taken as circles: the Earth's of radius 1 AU, the planet's of 1.5237 AU inclined 1.850°. The motion reverses for 72.7 days and backs up 15.94° of longitude, and that interval is centred on opposition to within 0.0000° of longitude. The track closes on itself: over 149 days the planet visits the same point of the sky twice, and the loop it encloses is 15.9° long and 2.94° tall. Longitude and latitude are at different scales: a degree of latitude is drawn 4.6 times a degree of longitude, because the sweep of 51° in longitude and 4.5° in latitude will not share a scale on one page. Circular orbits mean one loop for every apparition, and that is the real cost of this figure: Mars is 0.525 AU away at the opposition drawn here, its true distance at opposition varies with its eccentricity, and the real loops differ in size from one apparition to the next because of it. The marks are 20 days apart, and they crowd where the motion stops.
Fig. 2 The motion the measurement is extracted from. A superior planet’s apparent track against the stars, with its retrograde loop — the loop the planet does not make, since nothing reverses and the Earth merely overtakes. The rung below this one establishes that the loop is a projection artefact. The rung here notices that a projection artefact is a measurement: its timing, its width and its cadence all depend on the ratio of the two orbital radii, so the very feature that made the geocentric system complicated is the one that fixes the heliocentric system’s dimensions.

The outer planets: one clock

A superior planet never has a greatest elongation — it can appear anywhere from the Sun, including exactly opposite it — so the tangent construction is unavailable. The clock does the work instead.

The synodic period SS is the interval between successive oppositions, and it is what an observer measures directly. The sidereal period PP is the planet’s own year, which nobody can watch from the outside. They are related by

1S=1E−1P,\frac{1}{S} = \frac{1}{E} - \frac{1}{P},

with EE the Earth’s year — the ordinary statement that two hands of a clock come back into line at the difference of their rates. Mars’s 779.94 days between oppositions gives a sidereal period of 687.0 days; Jupiter’s 398.88 gives 4,333 days; Saturn’s 378.09 gives 10,754.

And then the harmonic law converts a period into a distance.

Mercury’s failure is a measurement

The scatter in Mercury’s elongations looks at first like observational error and is not. Mercury’s orbit has an eccentricity of 0.2056, the largest of any planet, so its distance from the Sun varies from 0.307 AU to 0.467 AU — and the greatest elongation reached on any given apparition depends on where in that range the planet happens to be.

The tangent construction assumes a circle. Applied to Mercury it returns 0.307 at the low end and 0.466 at the high, which are, to three figures, the perihelion and aphelion distances. The method has not failed; it has measured something it was not asked about. An observer who recorded elongations for a few years and found a range rather than a value would have had the eccentricity of Mercury’s orbit in hand, decades before anybody thought orbits could be anything but circles.

Venus through 4.8 months of sky. The geocentric ecliptic longitude and latitude of Venus over 146 days — 4.8 months — centred on inferior conjunction, computed as the direction of P − E with both orbits taken as circles: the Earth's of radius 1 AU, the planet's of 0.7233 AU inclined 3.395°. The motion reverses for 42.2 days and backs up 16.11° of longitude, and that interval is centred on inferior conjunction to within 0.0000° of longitude. The track closes on itself: over 87 days the planet visits the same point of the sky twice, and the loop it encloses is 16.1° long and 6.96° tall. Longitude and latitude are at different scales: a degree of latitude is drawn 2.1 times a degree of longitude, because the sweep of 52° in longitude and 10.3° in latitude will not share a scale on one page. Circular orbits mean one loop for every apparition, and that is the real cost of this figure: Venus is 0.281 AU away at the inferior conjunction drawn here, its true distance at inferior conjunction varies with its eccentricity, and the real loops differ in size from one apparition to the next because of it. The marks are 20 days apart, and they crowd where the motion stops.
Fig. 3 An inferior planet’s loop, which reverses at the near side rather than the far one. Venus backs up when it passes between the Earth and the Sun, so the reversal happens at inferior conjunction, the planet is a thin crescent throughout, and the arc — 16.1° in 42 days — is wider and much faster than Mars’s. Same subtraction, opposite geometry, and it is the difference that told Copernicus which planets were inside the Earth’s orbit before anything had been measured in miles.

What was actually measured, and what was not

The distinction this essay exists for is between a ratio and a unit, and it is worth stating baldly: Copernicus knew the shape of the solar system and its size in no units at all.

Every quantity above is a ratio to the Earth’s orbital radius. Nothing in angles and clocks can produce a length, because a scale model looks identical from the inside at any scale. Double every distance and every size in the solar system, leave the periods alone by adjusting the Sun’s mass to suit, and not one measurement in this essay changes: the elongations are the same, the synodic periods are the same, the retrograde loops are the same width in degrees. The model is exactly as underdetermined as a photograph of a room with no object of known size in it. To get from the model to a distance requires one measurement of a different kind — a baseline, and a parallax against it.

That measurement was two centuries in coming and was much harder than everything above it. The transits of Venus of 1761 and 1769 were mounted for it: observers dispatched to Tahiti, Hudson Bay and the Cape, timing the moments of contact from stations thousands of kilometres apart, so that the small difference in the transit chord would give the parallax of Venus and thereby the astronomical unit. The result was about 153 million kilometres, roughly 2% high, and it was the best number available for a century.

The Copernican argument, restated

It is worth being clear about what this construction settles and what it does not, because the usual account overstates it.

It does not prove that the Earth moves. Ptolemy’s system can reproduce every elongation and every synodic period, because its epicycles are the Earth’s orbit under another name; the deferent-and-epicycle machinery for an inferior planet has the epicycle’s radius in the same ratio to the deferent’s that the heliocentric model has for the orbits. The observations are identical.

What it does is make the ratios fall out rather than be inserted. In the geocentric system each planet’s epicycle radius is an independent free parameter, adjusted to fit; nothing connects Venus’s to Mars’s, and nothing forbids any value. In the heliocentric system the same numbers are the orbital radii, they are fixed by the synodic periods, and they come out ordered — Mercury, Venus, Earth, Mars, Jupiter, Saturn — with no freedom left. Six free parameters become six predictions, and that, rather than any single observation, is the argument.

There is a second, quieter consequence of the same move, and Copernicus drew attention to it himself. In the heliocentric arrangement the period of a planet increases with its distance, monotonically and without exception — 88 days at 0.39 AU, 12 years at 5.2, 29 years at 9.5. In the geocentric arrangement there is no such rule to break or keep, because the epicycle periods and the deferent periods are separate quantities and the ordering of the spheres was decided by argument rather than by observation. The regularity is the harmonic law waiting to be noticed, and it was noticed as a qualitative fact seventy years before Kepler made it exact. A model that produces an unforced regularity is doing something a model that accommodates every regularity is not.

Phases are a viewing angle, not a shadow. A satellite at eight points of its orbit. Exactly half of it is lit at every one of them; what changes is how much of the lit half faces the centre. Nothing is in shadow except at an eclipse.
Fig. 4 The observation that did settle it, and it needed a telescope. Phases are not shadows — a body’s phase is a matter of how much of its lit hemisphere faces the observer. Venus shows a full set, from a thin crescent when near and large to a small full disc when far and on the other side of the Sun. In the Ptolemaic arrangement Venus stays between the Earth and the Sun and can never be fully lit. Galileo saw the full set in 1610, and that single sequence ruled out the arrangement that every one of the measurements above is compatible with.

The oldest attempt, and the angle it was too sensitive to

The same idea was tried eighteen centuries earlier on the two brightest objects in the sky, and it failed in a way worth studying.

Aristarchus of Samos reasoned that at exactly half phase, the Sun–Moon–Earth angle is a right angle at the Moon: the terminator is seen edge-on precisely when the observer’s line of sight is perpendicular to the sunlight. Measure the Moon’s elongation from the Sun at that instant and the triangle is solved — the ratio of the Sun’s distance to the Moon’s is 1/cos⁡ε1/\cos\varepsilon.

It is the same construction as the opening figure with the roles moved around, and it is exact. Aristarchus measured 87° and concluded the Sun is about nineteen times further than the Moon. The true angle is 89.853°, and the true ratio is 390.

Saturn through 11.8 months of sky. The geocentric ecliptic longitude and latitude of Saturn over 358 days — 11.8 months — centred on opposition, computed as the direction of P − E with both orbits taken as circles: the Earth's of radius 1 AU, the planet's of 9.5367 AU inclined 2.486°. The motion reverses for 137.6 days and backs up 6.80° of longitude, and that interval is centred on opposition to within 0.0000° of longitude. The track closes on itself: over 268 days the planet visits the same point of the sky twice, and the loop it encloses is 6.8° long and 0.46° tall. Longitude and latitude are at different scales: a degree of latitude is drawn 6.7 times a degree of longitude, because the sweep of 10° in longitude and 0.5° in latitude will not share a scale on one page. Circular orbits mean one loop for every apparition, and that is the real cost of this figure: Saturn is 8.538 AU away at the opposition drawn here, its true distance at opposition varies with its eccentricity, and the real loops differ in size from one apparition to the next because of it. The marks are 20 days apart, and they crowd where the motion stops.
Fig. 5 And the far end of the same sequence. Saturn reverses for 138 days and backs up 6.8° — under half of Mars’s arc, over nearly twice the interval — because a more distant planet moves more slowly and the Earth overtakes it more nearly at the Earth’s own speed. The width shrinks and the duration grows along the whole sequence, and both are set by one ratio of orbital radii. Nothing in the picture is measured in kilometres and the ordering of the planets falls out anyway.

That failure is the counterpart of Mercury’s, and the pair make the general point. A construction turns an angle into a distance through some function, and the function’s slope decides how good the angle has to be. For Venus the function is sin⁡ε\sin\varepsilon near 46°, whose derivative is modest, and a degree of error costs 1.5% in the answer. For Aristarchus the function is 1/cos⁡ε1/\cos\varepsilon near 90°, and a degree of error costs a factor of two.

How good the numbers actually were

It is one thing to say the method works and another to compare its output with the answer, and the comparison is worth making because the errors are not where a modern reader expects.

Copernicus’s semi-major axes, in units of the Earth’s, against the modern values: Mercury 0.3763 against 0.3871, Venus 0.7193 against 0.7233, Mars 1.5198 against 1.5237, Jupiter 5.2192 against 5.2029, Saturn 9.1743 against 9.5388.

Three of the five are good to better than half a per cent, from naked-eye angles and a clock. Venus is right to five parts in a thousand, Mars to two and a half, Jupiter to three.

The two failures are both explicable and neither is a failure of the method. Mercury is 2.8 per cent low for the reason the hero figure gives: its orbit is eccentric enough that the elongation method returns a range rather than a number, and the value adopted sits inside that range at the wrong place. Saturn is 3.8 per cent low for a different reason — the derivation runs through the synodic period, and Saturn’s synodic period differs from a year by only 13 days, so the subtraction 1/E−1/S1/E - 1/S is a difference of two nearly equal numbers and any error in either is enormously amplified. It is a badly conditioned calculation, in exactly the sense the Aristarchus section describes.

The method’s accuracy is therefore set by the conditioning of each planet’s own arithmetic rather than by the quality of the observations, which were much the same for all five. That is a modern-sounding statement to make about a sixteenth-century result, and it is what a careful look at the residuals says.

The same instrument, turned on the Earth’s orbit

Every construction above measures another planet’s orbit against the Earth’s, and treats the Earth’s as a given. The same materials — angles and clocks, nothing else — measure the Earth’s own orbit too, and the observation required is almost absurdly simple.

Time the equinoxes and the solstices. Those are the four instants when the Sun crosses the celestial equator and reaches its extremes of declination, and they divide the year into four seasons. If the Earth’s orbit were circular and the Sun at its centre, the four would be of equal length, because the Sun would move along the ecliptic at a constant rate.

They are not equal. Spring runs about 92.8 days, summer 93.6, autumn 89.8 and winter 89.0 — a spread of nearly five days between the longest and the shortest. Hipparchus measured the first two in the second century BC and got 94.5 and 92.5, close enough to establish the effect.

The inequality is the Earth moving faster when it is nearer the Sun, and its size gives the eccentricity. To first order the ratio of the shortest to the longest season fixes ee at about 0.017, and which season is shortest fixes the direction of perihelion — currently in early January, which is why northern winter is the short one.

Two things about this are worth carrying. It is a measurement of the Earth’s orbit made from inside it, using no other body — a construction the essay’s title claims and this is the purest instance of. And it was available, in principle, to anybody with a gnomon and a calendar: the seasons’ unequal lengths were known for two thousand years before anyone had a mechanism to explain them, and the explanation, when it came, was the same eccentricity that spoils Mercury’s elongation measurement two paragraphs above.

The size that changes with the phase

Galileo’s phases settled which arrangement was right, and they carry a quantitative test as well that is rarely mentioned alongside them.

The model says Venus is at 0.277 of the Earth’s distance from the Sun when it passes between the Earth and the Sun, and at 1.723 when it is on the far side — the sum and difference of the two orbital radii. That is a ratio of 6.2 in distance, and therefore a ratio of 6.2 in apparent diameter, since an object’s angular size is its diameter divided by its distance.

The measurement matches: Venus’s disc runs from about 9.9 arcseconds when full and distant to about 62 arcseconds when a thin crescent and near. The planet appears six times larger when it is a crescent than when it is full, which is a startling thing to see and is exactly what the geometry demands.

That combination is what makes the observation decisive rather than merely suggestive. A defender of the old arrangement could accommodate phases by moving the epicycle, and could accommodate a varying angular size by the same move — but not both together in the right proportion, because in the Ptolemaic arrangement the planet is nearest when it is least illuminated and the brightest configuration would then be one the model does not produce.

There is a nice consequence that anybody can check without an instrument. Venus’s apparent brightness is the product of its illuminated fraction and the inverse square of its distance, and those two run in opposite directions as it moves — so the brightness peaks neither at full phase nor at closest approach but in between, at a crescent about a quarter illuminated. The greatest brilliancy comes about five weeks either side of inferior conjunction, and it is a prediction of the same three numbers this essay has been assembling.

None of that requires a telescope to predict — only to see. The brightness peak was known to observers long before anybody could resolve the disc, and it sat in the records as an unexplained regularity of the same kind as the unequal seasons.

The general point is the one this essay keeps arriving at from different directions: a scale model constrains the relations between everything in it, so any two of its quantities can be checked against each other, and the checks are as strong as if the model had a unit.

It also explains why the pre-telescopic astronomers were so exercised by the question of order and so untroubled by the question of size: order is what their instrument could decide, and size is what it could not.

Where the model stops

The construction assumes circles, coplanar orbits, and a Sun at the centre of each — and each assumption costs something measurable.

Circles. Mercury’s case is the extreme; Mars, at e=0.0934e = 0.0934, gives a synodic period that varies by about 12 days between oppositions, so the derived sidereal period wobbles unless several oppositions are averaged — the same eccentricity that shows up as the equation of time on the Earth’s own orbit.

Coplanarity. The planets’ orbits are inclined by up to 7° (Mercury) to the ecliptic, so an elongation measured in the sky is not in general the angle in the orbital plane. For Venus, at 3.4°, the correction is below the measurement error of a pre-telescopic quadrant; for Mercury it is not.

The Sun at the centre. It is at a focus, not a centre, which is Kepler’s correction to exactly this picture and the reason the ratios above are semi-major axes rather than radii.

The further the planet, the longer and the narrower its loop. Retrograde arc width and retrograde duration against semi-major axis, for circular coplanar orbits, with the arc read on the left axis in degrees and the duration on the right in days. Both are measured on the drawn geometry: the stations are the zeros of the geocentric longitude rate, and the arc is the longitude between them. Across the sweep from 1.05 to 34 AU the arc narrows from 16.5° to 2.5° while the interval lengthens from 57 to 160 days, monotonically in both. Mars 15.9° in 73 d, Jupiter 9.9° in 121 d, Saturn 6.8° in 138 d, Uranus 4.0° in 152 d, Neptune 2.8° in 158 d. The limit as the planet recedes is the Earth's own half-orbit, 182.6 days, with an arc of nothing at all: at that end the loop has become pure parallax, and its width is the ratio of the Earth's orbit to the planet's distance.
Fig. 6 And a check on all of it that uses none of the machinery above. The angular width of a superior planet’s retrograde loop, against its orbital radius: wide for Mars, narrow for Saturn, because the loop’s size is set by how far the Earth’s orbit subtends at the planet. That is a fourth independent route to the same radii, using neither the tangent construction nor the synodic relation nor the harmonic law, and it agrees. A quantity that can be obtained four ways and comes out the same each time has stopped being a model parameter and become a measurement.

The retrograde loop is the measurement and its width is the quantity, so both are worth drawing for a body further out.

Jupiter through 10.3 months of sky. The geocentric ecliptic longitude and latitude of Jupiter over 314 days — 10.3 months — centred on opposition, computed as the direction of P − E with both orbits taken as circles: the Earth's of radius 1 AU, the planet's of 5.2029 AU inclined 1.304°. The motion reverses for 120.6 days and backs up 9.95° of longitude, and that interval is centred on opposition to within 0.0000° of longitude. The track closes on itself: over 238 days the planet visits the same point of the sky twice, and the loop it encloses is 10.0° long and 0.42° tall. Longitude and latitude are at different scales: a degree of latitude is drawn 9.7 times a degree of longitude, because the sweep of 14° in longitude and 0.5° in latitude will not share a scale on one page. Circular orbits mean one loop for every apparition, and that is the real cost of this figure: Jupiter is 4.203 AU away at the opposition drawn here, its true distance at opposition varies with its eccentricity, and the real loops differ in size from one apparition to the next because of it. The marks are 20 days apart, and they crowd where the motion stops.
Fig. 7 Jupiter through one retrograde loop. The loop is much narrower than Mars’s and lasts much longer, because it is almost entirely the Earth’s own motion projected on the sky — the further the planet, the more nearly the loop is a picture of the Earth’s orbit.
The further the planet, the longer and the narrower its loop. Retrograde arc width and retrograde duration against semi-major axis, for circular coplanar orbits, with the arc read on the left axis in degrees and the duration on the right in days. Both are measured on the drawn geometry: the stations are the zeros of the geocentric longitude rate, and the arc is the longitude between them. Across the sweep from 1.2 to 10 AU the arc narrows from 16.4° to 6.6° while the interval lengthens from 62 to 139 days, monotonically in both. Mars 15.9° in 73 d, Jupiter 9.9° in 121 d, Saturn 6.8° in 138 d. The limit as the planet recedes is the Earth's own half-orbit, 182.6 days, with an arc of nothing at all: at that end the loop has become pure parallax, and its width is the ratio of the Earth's orbit to the planet's distance.
Fig. 8 The loop’s angular width against the planet’s distance. It falls steadily and approaches a limit set by the Earth’s orbital radius alone, so measuring the width of a loop is measuring a distance — which is the whole of what the Copernican construction delivers.

Where this ladder goes next

This rung turns the apparent motion of the planets from a phenomenon to be explained into an instrument. The rungs above it ask what else the instrument reaches.

The nearest is the same trick applied to a body that is not a planet — the elongation and phase of the Moon, which give the ratio of the Sun’s distance to the Moon’s, and which Aristarchus attempted in the third century BC. His answer, that the Sun is nineteen times further than the Moon, is wrong by a factor of twenty because the angle he needed was 89.85° and he measured 87°; the method is exactly this essay’s, and the failure is a lesson about which angles a construction is sensitive to.

Beyond that, the natural continuation is the unit rather than the ratios: how a baseline on the Earth becomes a parallax, why the triangle that reaches the stars runs out so quickly, and what it takes to tie a scale model to a metre.

Where this goes next

Mars through 8.3 months of sky. The geocentric ecliptic longitude and latitude of Mars over 252 days — 8.3 months — centred on opposition, computed as the direction of P − E with both orbits taken as circles: the Earth's of radius 1 AU, the planet's of 1.5237 AU inclined 1.850°. The motion reverses for 72.7 days and backs up 15.94° of longitude, and that interval is centred on opposition to within 0.0000° of longitude. The track closes on itself: over 149 days the planet visits the same point of the sky twice, and the loop it encloses is 15.9° long and 2.94° tall. Longitude and latitude are at different scales: a degree of latitude is drawn 4.6 times a degree of longitude, because the sweep of 51° in longitude and 4.5° in latitude will not share a scale on one page. Circular orbits mean one loop for every apparition, and that is the real cost of this figure: Mars is 0.525 AU away at the opposition drawn here, its true distance at opposition varies with its eccentricity, and the real loops differ in size from one apparition to the next because of it. The marks are 20 days apart, and they crowd where the motion stops. The loop a planet does not make Mars stops in the sky, backs up for ten weeks, and goes on. Neither orbit reverses anywhere — the loop belongs to the difference of two position vectors seen from one of them, and its width and duration are fixed by the ratio of the two radii and by nothing else. Period against size for the planets, around the Sun. Orbital period against semi-major axis on logarithmic axes. The line has slope exactly three-halves — the harmonic law — and the measured bodies sit on it. The law that links period to size, and weighs everything Kepler found that the square of the period goes as the cube of the orbit. Newton found the constant of proportionality, and that constant is a mass — which is how every mass in astronomy has been obtained since. The fractional distance error, rung by rung. Cumulative fractional uncertainty in a measured distance against the distance itself, both on logarithmic axes. Each rung of the ladder is calibrated against the one below it, so its scatter adds in quadrature to everything already inherited, and the total can only rise: radar to the planets 10⁻⁵%, parallax 1.0%, main-sequence fitting 5.1%, Cepheids 6.5%, Type Ia supernovae 8.2%. Every distance is measured with the last one No single method reaches from a planet to a distant galaxy. The ladder is built rung by rung, each calibrated on the one below, and the errors multiply all the way up.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Angular measurementAstronomical unit (AU)Copernican systemGreatest elongationInferior conjunctionOppositionOrbital radiusRelative scaleRetrograde motionSynodic period