A model of the orbit of Mars that predicted every observed position to within eight arcminutes would have been the best astronomy ever done, and the reason it counts as a failure is the whole content of this lesson.
Johannes Kepler arrived in Prague in 1600 to work for Tycho Brahe, whose observations, described in the previous lesson, were good to about one arcminute. He was given Mars because Mars was the problem: the most eccentric orbit of the planets then known apart from Mercury, close enough to be observed well, and the body on which every existing model failed worst. He expected to finish in eight days. It took him five years, and the book that came out of it, Astronomia Nova of 1609, is the first work in which a physical cause is offered for a planetary motion and the first in which a theory is discarded because of a residual too small to see.
Kepler wanted a cause
Kepler is a strange figure to have this role, and it is worth being clear about what he actually believed, because it explains what he did.
His first book, the Mysterium Cosmographicum of 1596, argues that there are exactly six planets because there are exactly five regular solids, and that the spacing of their orbits is set by nesting a cube, tetrahedron, dodecahedron, icosahedron and octahedron between six spheres. This is not a metaphor. He calculated it, compared it with Copernicus's distances, found agreement of a few per cent, and regarded it as the discovery of his life. He was still defending it twenty years later.
Alongside that runs the idea that made him useful. In the same book he noticed that the further a planet is from the sun, the more slowly it moves, and not just in the sense that a longer orbit takes longer: the actual speed along the path is lower. Mercury takes 88 days, Saturn takes over 29 years, and the ratio is far greater than the ratio of the orbit lengths. To Kepler this meant that something emanating from the sun drives the planets and weakens with distance. That single thought sets him apart from everyone before him. Ptolemy, Copernicus and Tycho all wanted a geometry that reproduced the positions. Kepler wanted a physical cause, and he gave Astronomia Nova the subtitle Physica Coelestis, a physics of the heavens, which had previously been a contradiction in terms.
The consequence is methodological. If your model is a calculating device, a residual of eight arcminutes is a triumph. If your model is a claim about a real body being pushed by a real cause, a residual of eight arcminutes is a fact about the world that you do not understand.
The vicarious hypothesis
Kepler's first attack used the classical machinery: a circular orbit, with the sun displaced from the centre by some distance, and an equant point on the far side about which the motion is uniform. Ptolemy had always placed the equant exactly as far from the centre as the earth was, splitting the offset evenly. Kepler let the two distances be free parameters and fitted them to four accurately observed oppositions of Mars.
The result was excellent. The fitted model, which he called the vicarious hypothesis because he did not believe it was true, reproduced the longitudes of Mars, its positions round the zodiac, to within about 2 arcminutes across the whole set of oppositions. That is the accuracy of the data themselves. By every standard then in use the problem was solved.
He kept testing. The model that fitted the longitudes gave a splitting of the offset that disagreed with the value obtained from Mars's latitudes, its distances north and south of the ecliptic, and the disagreement was not tiny. Forcing the two into agreement, and then predicting positions near the middle of the orbit rather than at opposition, left errors of up to 8 arcminutes.
Example. How big is 8 arcminutes, and why is it decisive here when it would have been invisible fifty years earlier?
The full moon is about 31 arcminutes across, so 8 arcminutes is roughly a quarter of the moon's width: a gap you could not see between two stars without an instrument. The question is not whether it is small in absolute terms but how it compares with the error of the data. Tycho's positions are good to about 1 to 2 arcminutes, so an 8 arcminute residual is four to eight times the uncertainty. It is not noise. Had Kepler been working with the older observations, good to perhaps 10 arcminutes, the residual would have been comfortably inside the error bars and the vicarious hypothesis would have been declared correct. Tycho's money is what makes the discrepancy exist as a fact rather than as a possibility.
Now you. Kepler wrote that these eight minutes alone pointed the way to a complete reformation of astronomy. What would the alternative response have been, and could anyone have defended it?
Answer
The alternative was to add another small circle. That was the standard, respectable move: a residual is absorbed by an extra epicycle with the amplitude and period needed to cancel it, and there is no principled limit to how often it can be done. It could certainly have been defended, and it would have produced a model fitting the data as well as Kepler's ellipse. What made Kepler refuse it was not evidence but his commitment to a physical cause: an extra circle is a description of the residual, not an explanation of it, and there is no force in nature that would make a planet ride a small circle carried on a big one. Notice that this is not a purely empirical decision. Two people looking at the same 8 arcminutes could reasonably do different things with them, and what separates them is a prior view about what an astronomical model is for.
Fixing the earth first
Before the orbit of Mars could be found, Kepler had to solve a problem nobody had taken seriously: every observation of Mars is made from a moving platform whose own motion was only approximately known. Copernicus had given the earth a uniform circular orbit with an offset sun, and no equant, on the grounds that the earth was special.
Kepler's method for testing this is the most elegant thing in the book. Mars returns to the same point in its orbit every sidereal period, days. So take observations of Mars separated by exactly that interval, and however many times you repeat it, Mars is in the same place in space. The earth is not: it has moved to a different point on its own orbit each time. Each observation then gives the direction of Mars and the direction of the sun as seen from the earth, and with Mars pinned down as a fixed reference the earth's position can be triangulated. Kepler used Mars as a surveying benchmark to map the earth's orbit from outside.
The answer came back that the earth does have an equant, and that its offset splits evenly, exactly like the other planets. The earth's speed varies in its orbit in the same way and for the same reason. That is a small technical result with a large consequence: the earth is not a special case with special rules, but a planet, behaving like the rest. Whatever drives Mars drives us.
The area law, then the ellipse
With the earth's orbit known, Kepler could convert Tycho's angles into actual distances from the sun to Mars at many points, and the shape that emerged was not a circle with an equant.
He got the speed law first, which is the reverse of the order in which the laws are usually listed. Working from his conviction that the sun drives the planets, he tried the rule that a planet's speed is inversely proportional to its distance from the sun. That is not exactly right, but summing the distances round the orbit to get the time was hopelessly laborious, so he replaced the sum with the area swept out by the line from the sun to the planet, which he could compute geometrically. The result is the area law: the line joining a planet to the sun sweeps equal areas in equal times. He regarded it as an approximate computing device standing in for the real distance law. It is exact, and the distance law is the approximation, which is one of the luckier mistakes in the history of science.
The shape took another three years. He tried an oval, could not compute with it, and eventually noticed a number. The width of the orbit he needed was less than the width of the circumscribed circle by a factor of , and the maximum angular offset between the two ways of computing the planet's position was . He then observed that , the same number. He wrote that he awoke as from sleep. The curve that has that property is an ellipse with the sun at one focus.
Example. Mars has an eccentricity of and a semi-major axis of AU. How different is its orbit from a circle, and how far is the sun from the centre?
The semi-minor axis is , so . The orbit is narrower than a circle by per cent, which no measurement of that era could have detected as a shape. The sun sits at a focus, a distance AU from the centre, which is per cent of the semi-major axis. So what Kepler detected was not that the orbit is an oval; it is very nearly circular. What he detected is that the sun is well off centre and that the planet's speed varies accordingly. Every model since Ptolemy had known that and had handled it with an equant. The ellipse is the correct way of handling it, and the difference between correct and nearly correct is those 8 arcminutes.
Now you. Find the perihelion and aphelion distances of Mars, and the ratio of its speeds at those two points.
Answer
Perihelion is AU and aphelion is AU. At those two points the line from the sun is perpendicular to the motion, so the area swept per unit time is in each case; the area law makes these equal, giving . Mars moves about 21 per cent faster at perihelion than at aphelion. That is a large effect, easily within reach of the observations, which is why speed variation had been modelled since antiquity and why getting its exact form right was the hard part.
The third law
The two laws of Astronomia Nova describe a single orbit. Neither says anything about how one orbit relates to another, and Kepler had been hunting for that relation since 1596, because it is the question the polyhedra were supposed to answer.
He found it on 15 May 1618 and published it in Harmonices Mundi the following year, buried in a book largely concerned with the musical intervals sounded by the planets. The third law states that the square of a planet's period is proportional to the cube of its semi-major axis, so that with the period in years and the distance in astronomical units, exactly.
Example. Jupiter's sidereal period is years. Where should it be?
AU. The modern semi-major axis is AU, so the law is right to within per cent for a planet five times as far out as the earth.
Now you. Saturn's period is years. Predict its distance, and check it against the value Copernicus obtained.
Answer
AU, against a modern value of , so the law is right to a hundredth of a per cent. The previous lesson gave Copernicus's figure for Saturn as , which is per cent low and by far his worst planet. Kepler's law, applied to nothing but a timing anybody can make with a calendar, beats a careful geometrical determination by a factor of several hundred. This is the first appearance of something that becomes the signature of the new astronomy: a law that is not a summary of the measurements but a constraint the measurements have to satisfy, which can then be used to correct them.
What it left unexplained, and who believed it
The three laws are exact, they are simple, and they are entirely without foundation. Nothing in them says why the orbit should be an ellipse rather than an oval, why the areas rather than the arcs should be equal, or why the exponent in the third law should be rather than anything else. Kepler supplied a physical cause and it was wrong: he imagined the rotating sun sweeping the planets round with an immaterial emanation that thins with distance like light, plus a magnetic action pulling each planet alternately in and out to make the orbit oval. He knew it was unsatisfactory.
Nor was the reception rapid. Galileo, who corresponded with Kepler, never mentioned the ellipse in the Dialogue of 1632 and went on using circles to the end. The book is dense, argumentative, and written as a narrative of the author's own failures over five years, which is admirable and unreadable. Astronomers adopted the laws slowly and mostly for a practical reason.
That reason was the Rudolphine Tables, published in 1627, the last thing Tycho's data were used for. Where earlier tables were wrong by degrees, these were wrong by arcminutes, an improvement of one to two orders of magnitude. Kepler used them to predict a transit of Mercury across the face of the sun on 7 November 1631, which Pierre Gassendi observed in Paris, finding Mercury within about a quarter of a degree of the predicted place, while predictions on the older tables were out by degrees. Kepler had died the year before. He also predicted a transit of Venus for 1631 that was not visible from Europe, and missed the one in 1639, which the English curate Jeremiah Horrocks caught by correcting Kepler's own elements.
That is the mechanism by which the new astronomy actually spread: not by convincing anyone of the earth's motion, but by making better tables, which navigators, calendar makers and astrologers wanted regardless of what moved. Meanwhile, in Padua, a mathematics professor had pointed a new instrument at the sky and found things nobody had known were there.