The Chord an Orbit Cannot Reach
(from the workbook of an almanac-maker — the person whose whole trade is to say, in a printed book, a year ahead of time, exactly where a planet will be on a given night)
Here is the thing nobody tells you when you take up the trade: the sky keeps uneven time.
A planet does not walk its ellipse at a steady pace. It hurries when it is near the Sun and it dawdles when it is far — it sweeps equal areas in equal times, which is a beautiful law and a nuisance to a man with a table to fill. If you want the honest angle, the one a navigator can point a glass at, you cannot simply carve the year into even nights. You have to solve Kepler’s little equation,
the even clock-angle equals the true wheel-angle minus the eccentricity times the sine of the wheel-angle,
and it will not solve. It is transcendental. It sits there and refuses, the way a knot refuses, the way the truth refuses to be short.
So the almanac-makers learned two ways to make it give.
The first way is to turn the crank. Guess the angle, measure how wrong you are, correct, guess again — Newton’s wheel, round and round, and after five turns you have the place of Mars to more decimals than any telescope on Earth can check. It is a workman’s method. It never fails and it never sings. You could do it for a comet, for a grain of dust, for anything that falls.
The second way is the one I love, and it is the one that will break your heart if you let it.
You can write the planet’s place as a chord — a fundamental note plus its overtones, the first harmonic and the second and the third, each a little sine wave, each weighed by a number that Friedrich Bessel worked out while he was staring at exactly this problem. (That is where his functions come from. Not from a drum, not from a wire. From the question of where a planet is.) You sum the overtones and the sum is the orbit. A planet’s place in time is a chord. I have never gotten used to it.
And there is a tidier version still — the one every apprentice reaches for first — where you expand everything in powers of the eccentricity, the flatness of the ellipse. One term for a circle, a correction for a slight oval, a smaller correction for a rounder one. It is so clean it feels like it must be the truth. For the planets it is the truth: even Mercury, the most lopsided of them, has an eccentricity of only about a fifth, and the tidy series sings Mercury without a waver.
But I will tell you what an old computer told me, the first winter I sat at his table.
Hand that pretty series a comet, he said, and it will lie to your face.
He was right, and it took the better part of two centuries to say exactly how right. The tidy expansion has a range. Below a certain flatness — an eccentricity of 0·66274 and a little more — the chord rings true, and every overtone you add pulls you closer to the sky. Above it, the notes stop converging. Add another harmonic and you are further off, not nearer; the series bottoms out at its best and then walks away from the answer, whistling. The place is real, the planet is right there in the dark — but that particular song has no note for it. The instrument only reaches so far.
The strange grace of the number is that it does not care what night you ask about. It is not that some dates are hard. Past that flatness the tidy series fails for nearly every night at once — a wall in the eccentricity, not in the calendar. Pierre-Simon Laplace found the wall in 1827 and, in a detail I find unreasonably comforting, got his own number a little wrong — he said 0·66195; Cauchy came along two years later and fixed the last digits. Even the men who find the walls misjudge them by a hair. The wall did not mind.
So where is the note that the tidy song cannot reach? If you follow the music out past the real world, out where the flatness is allowed to be an imaginary quantity — the way you must sometimes leave the room to understand the room — you find the missing note sitting at a perfectly definite place, and the height of it is exactly the flatness at which the wall stands. The chord runs out precisely where its own shadow first touches it. That is the whole secret, and it is the kind of secret that is no comfort and no use and I would not trade it for a working telescope.
But here is the part the apprentice never expects, and the part I would leave you with if I could leave you only one.
The chord I love — Bessel’s, the ugly one with the special functions and the overtones — the one nobody reaches for first because the coefficients are a nuisance to table — that one has no wall. It sings every ellipse there is, every bound orbit from the roundest planet to the most desperate long-period comet skimming back out toward the dark. The tidy series, the one that looked like the truth, is the one with the range. The unshowy one, the one you have to earn, reaches all the way to the edge.
I have stopped being surprised that this is how it goes. The clean account of a thing is almost always the one with the hidden edge, and the account that costs you something to hold is almost always the one that reaches. It is true of orbits. I have found it true of most of what I have had to compute, and one or two things I never could.
I keep an almanac too, of a smaller sky. Some nights I can give the place exactly. Some nights the tidy song runs out and I have to turn the crank, and stay, and get there the long way. The place is always real. It was only ever the song that had the range.
