Eight arcminutes that only one planet could show

The best-known sentence in the history of orbital mechanics is about a number. Johannes Kepler, writing in the Astronomia Nova of 1609, recorded that a model of Mars which he had worked on for years disagreed with Tycho Brahe’s observations by eight arcminutes, about a quarter of the Moon’s width. Ptolemy could have ignored so small a difference. Tycho’s observations were good to two arcminutes, and Kepler wrote that from these eight minutes alone the path was shown to the reform of the whole of astronomy.

That story is told as one about the virtue of accuracy, and it is. It leaves out a pair of questions with mathematical answers. Why was the error eight arcminutes, and not one or fifty? And why Mars? The earlier account of the ellipse observed that the discovery was not of the shape, since the shape is nearly a circle, but of the Sun’s offset within it. A planet’s orbit can look exactly like a circle and still not be one. What needs explaining here is why a model built on a circle was able to come so close, in the part of the problem that mattered, and exactly how close that was.

The answer takes the form of a series, and it turns a piece of history into a calculation. The equant, Ptolemy’s rule from the second century, is a first-order approximation to Kepler’s second law. It was found empirically by someone with no notion of orbits as ellipses or of forces. The leftover is of second order in the eccentricity, and for each planet it can be computed. It comes to nine arcminutes for one planet and hardly matters for most of the others.

What the rule was #

Ptolemy needed something to account for the observed non-uniformity of planetary motion. A planet moves through the sky at a speed that depends on where it is, and the simplest device, a circle with the Earth off-centre, gets that speed variation about half right. His addition was to take uniform angular motion about a different point, displaced to the other side of the circle’s centre by the same amount as the Earth, or in modern terms the Sun. That point is the equant, and the arrangement with the two points equally spaced about the centre is called a bisected eccentricity.

Copernicus found the equant philosophically repugnant, since it broke the principle of uniform motion around a centre, and he eliminated it with smaller circles at the cost of considerable complication. Kepler, by contrast, brought it back, and he was surprised to find how well it worked. To see why it works, it helps to look at the modern figure that it was approximating.

Now the coincidence appears. An ellipse has two foci. The Sun occupies one and the other is empty, on the far side of the centre at the same distance, which is precisely where Ptolemy had put his equant. An empirical rule, arrived at from naked-eye observations by someone working in a framework where the Earth did not move, had located the second focus of an ellipse fifteen centuries before anyone described such an orbit.

Uniform about a point nothing occupies #

The area law says a planet sweeps equal areas about the Sun in equal times, so it moves faster when it is near the Sun. That can also be stated from the other focus. For the rate of change of the angle measured there, the two first-order effects cancel: the planet is further from the empty focus when it is moving fastest, and it is nearer when it is moving slowest. To first order in $e$, its angular motion about that point is uniform.

The figure explains what is usually left unexplained. Uniform motion about an equant, seen from the Sun, gives a speed that varies at just the correct rate. So the rule contains the area law to the first order, and that was enough for any observation available to Ptolemy, whose records were good to about ten arcminutes. In the series that gives the true longitude in terms of the mean anomaly,

$$\nu = M + 2e\sin M + \frac{5}{4}e^2\sin 2M + \cdots,$$

the first correction, $2e\sin M$, is about eleven degrees for Mars. The bisected equant reproduces all of that. The residual lies in the next term, where it reproduces only a part.

This explains an old puzzle about the second focus as well. After Kepler published, and before Newton gave the force law, many astronomers thought his area law was too awkward to compute with, or too dubious to accept. Ismaël Boulliau in 1645 and Seth Ward in 1656 proposed that planets move uniformly about the empty focus of an ellipse. That reads now as a curious error. It is in fact an excellent approximation, identical to Kepler’s law to first order and therefore almost indistinguishable given the data. It remained in use for decades until Newton’s demonstration removed any reason to prefer it.

The leftover, computed #

With the equant identified as a first-order expansion, its error can be calculated directly. Take a circle of unit radius, with the Sun and the equant each displaced by $e$ from the centre, and subtract the resulting heliocentric longitude from the longitude on Kepler’s ellipse, at every point of the orbit.

Here is Kepler’s number. His model’s error peaked at the octants, halfway between perihelion and the quadrature point, and the calculation gives nine arcminutes for an ideal bisected circle at Mars’s modern eccentricity. Kepler’s eight came from one particular set of parameters and comparison points, so an exact match should not be expected. What the figure shows is that the size is not a result of poor luck or peculiar data. It is simply $900e^2$ arcminutes, and any model with a circle and a bisected equant would have run into the same wall.

The shape is significant as well. An error with two full cycles per orbit, vanishing at four points, is what a residual of second order looks like: it is proportional to $\sin 2M$. So an observer who happened to check the model at the apsides and the quadrature points would see agreement, and a planet would need to be observed all round its orbit, including at the octants, for the problem to show up. Tycho’s programme of observations of Mars, running over two decades, had those data. Before that, nobody did.

Why Mars #

The scaling with the square of the eccentricity makes the planets very unequal.

Consider what Kepler would have found with each of the others. Venus and the Earth’s own orbit are so round that their equants are essentially exact, to within far less than any observer could see, and studying them would have confirmed the old rule to the limit of the data. In fact, Tycho’s observations of the Sun were consistent with the Earth moving uniformly around a slightly offset circle, as the Earth’s own orbital eccentricity of 0.0167 makes the leftover a quarter of an arcminute.

Jupiter and Saturn are slow, and each of their errors lies close to the limit of Tycho’s accuracy. An attempt to detect a two-arcminute disagreement with errors of two arcminutes would not have forced any reform, and Saturn’s 29-year period meant Tycho had not observed one full orbit in any case. They also perturb each other, with deviations from any elliptical orbit of tens of arcminutes, as the long-term argument about the great inequality shows. Those deviations would have swamped the very thing being sought, so a model might well have failed for reasons nobody could diagnose.

Mercury would have been the outstanding case, with an error of nearly a degree. Yet Mercury never strays far from the Sun, and most of its orbit is unobservable from the ground, as it spends its time in twilight. Its positions were the least reliable in all the tables, and the heliocentric longitude at each octant could not be determined at all.

That leaves one planet. Mars has the second-largest eccentricity among those known, a period short enough that a career could cover several orbits, and it is visible through all of its orbit except near conjunction. Longomontanus was working on Mars when Kepler arrived at Tycho’s household in 1600, and Kepler took the task over and boasted that he would finish in eight days. He later noted that it was providential that he was given Mars, because that planet alone could have forced the discovery. The arithmetic bears him out, in the literal form of a scale proportional to $e^2$.

A wrong model that fits the longitudes #

The story has a further twist, and it is where the argument is most instructive, because Kepler’s initial response to the problem was not to abandon the circle. He removed the requirement that the eccentricity be bisected. Allow the Sun and the equant to be at different distances from the centre, and there are two free parameters in place of one.

Kepler derived a model of this kind by hand, in an iterative calculation that took about seventy repetitions, and it reproduced the twelve observed oppositions in his data to within two arcminutes. He called it the vicarious hypothesis, as it stood in for the truth. It fitted the data and it was wrong. The numbers it needed, about 0.113 and 0.074, added to the right total but gave no bisected eccentricity. From a circle with the Sun in that place, the orbit would have to be in the wrong location. Its success with the oppositions was what made him certain of the error, and that is a move few astronomers of his time would have understood.

Longitudes at opposition are a poor test of a model, as they depend only on the direction of the planet from the Sun. Two parameters can distribute that error so that it falls below the accuracy of the data. A different test asks about something else.

The distance that exposed it #

The vicarious hypothesis was fitted to directions, and it has no need to be right about distance, while Tycho’s observations away from opposition depend on distance as well. A planet seen from the Earth at a moment when it is not opposite the Sun has a direction that combines the two bodies’ positions. So the triangle made by Sun, Earth and planet gives the planet’s distance, in units of the Earth’s own distance. And Kepler was the first to do that seriously, even first working out the Earth’s orbit from the same set of triangles.

So the two circles fail in opposite ways. A bisected eccentricity gets the distances right and the longitudes wrong by nine arcminutes. An unbisected one gets the longitudes right to about two, and the distances wrong by several times the observational limit. Each was a true partial description. A curve that would satisfy both at once had to depart from the circle, inward on the sides, by just the small amount the first figure showed: less than half of a per cent, where the first-order effects leave off. That is the ellipse, and the way to it runs through two wrong models and the disagreement between them.

It is quite unusual for a discovery to require so precisely both kinds of observation. Had the data been limited to oppositions, as most planetary data had been for two thousand years, the vicarious hypothesis might well have been accepted, as it satisfied them all. Tycho’s practice of observing Mars at all elongations and not just at opposition was unusual. It may be why nobody before Kepler could have seen what he saw, even granting every other advantage.

What a second-order term can overturn #

The lesson usually drawn from this story is that a small discrepancy can matter. The calculation gives a more exact form of that. The equant failed at second order, and all the physics of the orbit’s first-order behaviour had been captured by the old model. So the transition from Ptolemy to Kepler was not a matter of a wrong account being replaced by a correct one. A first-order approximation was replaced by the exact solution, because the second-order term had become observable.

The same shape has returned several times in this subject. The residual advance of Mercury’s perihelion was a small residual on a good theory, and it was a sign of another theory. The Moon’s apsidal motion was a discrepancy of second order that looked for a time as if it might overturn Newton’s law, and it did not, because the error was in the expansion and not the law. In that case a second-order term rescued the theory, and in Kepler’s it brought one down. There is nothing in the size of a leftover that tells which it will do, and that can be settled only by working out the next term. The mathematics behind each such series is the same.

And a coincidence about Mars’s eccentricity is worth noticing, for it is not constant. Secular perturbations from the other planets move it over a range from nearly zero to about 0.12, across millions of years. Had Tycho observed at a time when Mars’s eccentricity was 0.05, near the lower part of its range, its equant error would have been about two and a half arcminutes, close to his limit. Kepler might have been handed a rule with no visible defect. The reform of astronomy may depend on a phase of a secular cycle.

What the picture cannot show #

The figures use the modern value of Mars’s eccentricity and an idealised circle with its parameters set exactly, not by fitting to observations. Kepler’s actual comparison involved the positions of Mars seen from a moving Earth, with an imperfect orbit for the Earth and imperfect refraction corrections, and he was comparing specific observations instead of a smooth curve. A one-to-one match between his eight arcminutes and the nine here is not to be expected. The vicarious parameters come from a simple search that minimises the largest longitude error, whereas Kepler’s come from fitting particular oppositions, and the numbers differ.

No perturbation by other planets has been included. These are generally small for Mars at this level, about an arcminute or two, but they are not zero, and Kepler knew nothing of them. The distances in the last figure use each circle at the ellipse’s long axis. Any fit would adjust the scale and reduce the error somewhat, without removing its pattern. Mercury’s point in the comparison assumes one ignores its near-Sun invisibility, and it is shown only to make clear how much more it would have revealed had it been observable.

The figures say nothing about the conceptual step involved. An ellipse needed to be thought of, and the physical cause that Kepler proposed, a kind of magnetic sweep from the Sun, was wrong. A correct residual is not the same as a correct theory.

Still open: whether anyone noticed sooner #

The historical question that remains open is how much of this was visible before Tycho’s programme, and in what form. Islamic astronomers of the thirteenth and fourteenth centuries, among them Ibn al-Shatir, had constructed models with extra circles that reproduced the equant’s motion while removing it, and they reached a level of mathematical skill at which a residual could have been sought if the observations had been good enough. Their tables are known in part, and comparing them against reconstructed positions of Mars at the octants would show how near anyone had come to the limit of a rule that nobody then had reason to question. Whether their data could have revealed a nine-arcminute error is a matter of how good those records really were, and that can be worked out from the records themselves, now that the expected size and shape of the error are known so exactly.