Models of the Heavens: Ptolemy, Copernicus, Tycho and Kepler

How four models of the heavens were built to save the same sky, what each explained, and four ways to judge one model better than another (an interpretation).

What a model must save

Established A model must reproduce the daily turning of the sky and the Sun's yearly path at uneven speed (The Sky as a Sphere). It must also track the Moon and the five bright planets. Each planet drifts eastwards against the stars but at times runs backwards for weeks or months, in retrograde motion. For Mars, Jupiter and Saturn the middle of the backward arc falls at opposition, when the planet stands opposite the Sun in the sky (Linton, From Eudoxus to Einstein, ch. 3). The lab shows the retrograde loops.

Ptolemy: deferent, epicycle and equant

In the Almagest (about AD 150) Ptolemy keeps the Earth at rest and gives each planet a small circle, the epicycle, whose centre rides on a larger circle, the deferent. His order outwards is Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn (Almagest IX.1). On the side nearest the Earth, the epicycle carries the planet backwards faster than the deferent carries it forwards. To fit the planet's changing speed, Ptolemy put the Earth off the deferent's centre. That alone got the retrograde arcs wrong. So he made the epicycle's centre sweep equal angles in equal times about a third point, the equant. That broke the old rule of uniform motion about a circle's centre. On Mars's deferent of radius 60, the Earth sits 6 units off centre: an eccentricity of 0.10. The equant sits 6 units on the other side (Linton, pp. 76 and 78).

An equant can stay within ten minutes of arc (sixtieths of a degree) of modern theory, even for Mars (Linton, p. 76). The tables in use did worse: in 1563 Ptolemaic tables missed a conjunction (lining-up) of Jupiter and Saturn by nearly a month (MacTutor).

Copernicus: overtaking, order and scale

In De revolutionibus (1543) Copernicus sets the Sun at rest and makes the Earth a planet. The planets' stations (apparent stops) and retrogradations are then "not movements of the planets, but a motion of the earth" (I.9, trans. Rosen). The Earth, at a mean 29.8 km/s, overtakes Mars, at 24.1 km/s, as it passes between Mars and the Sun, at opposition. Mars then seems to slide backwards against the stars. With circular orbits in one plane, a calculation for this page gives Mars a backward arc of 16 degrees. It lasts about 73 days and recurs every 780 days. Real Mars loops from 1997 to 2040 last 60 to 81 days (JPL elements).

With the Sun at the centre, the longer a planet's period, the farther it lies from the Sun. Copernicus saw here "a marvelous symmetry of the universe", which explains why the backward arcs shrink from Mars to Jupiter to Saturn (I.10). Ptolemy's numbers hide the distances. Counting in sixtieths, he gave Venus an epicycle of 43;10 (43 and 10/60) on a deferent of 60, a ratio of 0.719. Venus lies 0.723 astronomical units (AU, about the Earth's mean distance) from the Sun. Mars's ratio, 39;30 over 60 or 0.658, matches 1/1.524 = 0.656, the Earth's distance over Mars's. Ptolemy fitted each ratio separately. For Copernicus, Venus's epicycle is Venus's orbit, and the epicycles of Mars, Jupiter and Saturn are the Earth's. So each ratio gives a distance from the Sun.

Established The gain is in explanation, not prediction. Copernicus kept off-centre circles and epicycles but swapped the equant for small epicycles. A theory with an equant seemed to him "neither perfect enough nor sufficiently in accord with reason" (Commentariolus, written by 1514). The historian Owen Gingerich found "relatively little to distinguish between the accuracy of the Alfonsine Tables and the Prutenic Tables", based on Ptolemy and Copernicus respectively (quoted in the Stanford Encyclopedia).

Tycho: a compromise

Tycho Brahe published his system in 1588. In it the Earth is at rest, the Moon and Sun circle it, and the five planets circle the Sun. Each planet then moves relative to the others as in Copernicus's system; only what is at rest differs. Tycho could find no stellar parallax, a yearly shift in the stars' positions (MacTutor). A moving Earth would cause one, unless the stars were enormously far away. His observations had errors mostly between 0.5 and 1 minute of arc.

The phases of Venus

In an anagram of December 1610, solved in January 1611, Galileo reported that Venus imitates the phases of the Moon (letters in Kepler's Dioptrice, trans. Carlos). He had seen it round, then half lit. A full Venus lies beyond the Sun (Newton, Principia III, Phaenomenon III): only there does its whole sunlit half face the Earth. Ptolemy placed Venus below the Sun and kept its epicycle's centre nearly in line with the Sun as seen from the Earth (Linton, pp. 62 and 78). A calculation for this page shows Venus is then at most about half lit and never round. So a round Venus refutes Ptolemy's arrangement, but not Tycho's, in which Venus circles the Sun and can pass beyond it. A strong test is one on which rivals predict different results (Hypotheses, Predictions and Tests). This one was strong against Ptolemy and silent between Copernicus and Tycho. Network troubleshooting has tests like this. A working ping shows the address you pinged is reachable. It is silent between a wrong DNS answer and a bad certificate.

Kepler: ellipses, areas and a harmonic law

From Tycho's observations of Mars at opposition, Kepler built a circle-and-equant model that gave Mars's direction from the Sun to within about 2 minutes of arc. But Mars's distances put the circle's centre midway between the Sun and the equant. Moved there, the model was out by up to 8 minutes, well beyond Tycho's errors. Kepler wrote that "these eight minutes alone will lead us along a path to the reform of the whole of Astronomy" (Astronomia Nova, chapter 19).

Astronomia Nova (1609) gave two laws. Law 1: a planet moves on an ellipse with the Sun at one focus. Law 2: the line from the Sun to a planet sweeps out equal areas in equal times. He later found they held for the other planets too (MacTutor). Law 3, in Harmonices Mundi V (1619): the square of a planet's period T is proportional to the cube of its semi-major axis a, half the ellipse's longest diameter. See laws 1 and 2 in the lab.

Check law 3. In years and AU, T^2 = a^3: for Mars (a = 1.524, T = 1.881) both sides are 3.54. So T = a^1.5. For the six planets Kepler knew, the best-fit exponent is 1.500 with JPL's mean elements. The lab, with NASA fact-sheet values, gives 1.499. Compare the Academy's Prop. XII.

Kepler's own cause, a force from the rotating Sun, did not survive. Demonstrated, given Newton's laws of motion: in his Principia (1687), a force towards a fixed centre gives equal areas in equal times (Book I, Proposition 1). An ellipse about a focus needs an inverse-square force (Proposition 11), and such ellipses obey the third law (Proposition 15). Kepler also predicted a transit of Mercury across the Sun, which Gassendi saw on 7 November 1631. Its success helped win astronomers over to Kepler's ellipses and area law (Athreya and Gingerich).

What makes a model better

Interpretation. Four tests run through this history.

Fit is agreement with the data within the observers' error. Tycho's data demanded better than an equant's ten minutes.
Simplicity is how much is explained rather than fitted. Copernicus read Ptolemy's separately fitted ratios as distances.
Cause is a reason why. Kepler sought one and Newton supplied it.
Prediction is foretelling what has not been seen, as Kepler did for a transit.

Before Newton, no model won on all four. Osiander's unsigned foreword to De revolutionibus said hypotheses "need not be true nor even probable" if they give a calculus consistent with the observations. Kepler's title, Astronomia Nova "or celestial physics", claims more. Every model also rests on measurements and stated assumptions, as in Eratosthenes Measures the Earth.

Try this

Draw the Earth, Sun and Venus in four positions under Ptolemy's arrangement and four under Tycho's, shading Venus's sunlit half. Find which drawings let the Earth see a full disc.
Check the third law for Saturn: period 10,755.699 days, semi-major axis 1,432.041 million km (NASA fact sheet) or 9.537 AU (JPL). Use 365.256 days a year and 149.598 million km an AU. T^2/a^3 comes to about 0.99 and 1.00. Suggest why two published semi-major axes could differ.