The orbit is an ellipse, and the Sun is not in the middle of it

Almost every drawing of a planetary orbit is wrong in the same two ways. It exaggerates the ellipse, and it puts the Sun in the middle.

The second error is the interesting one. An ellipse has a centre and it has two foci, and the entire content of Kepler’s first law is which of the three the Sun occupies. Put it at the centre and the orbit becomes symmetric, the planet moves at a constant speed, and nothing else in celestial mechanics follows. Put it at a focus and everything does.

The one number that decides the shape #

An ellipse is fixed by two lengths: the semi-major axis $a$, half the long way across, and the semi-minor axis $b$, half the short way. Astronomy almost never uses $b$. It uses the eccentricity

$$e = \sqrt{1 - b^2/a^2},$$

which runs from 0 for a circle to just under 1 for a sliver, and which has a far more useful geometric meaning: the two foci sit at $\pm ae$ from the centre, so $e$ is the offset, measured in units of the orbit’s own size.

That definition is what makes the eccentricity the operational quantity. It gives the closest and furthest distances immediately,

$$r_{\min} = a(1 - e), \qquad r_{\max} = a(1 + e),$$

and their ratio, $(1+e)/(1-e)$, is the factor by which the primary’s apparent size and the sunlight received swing over one orbit. For the orbit drawn above, that factor is four.

There is a second definition, older and in some ways better: an ellipse is the set of points whose distances to the two foci add to a constant. That is the string-and-two-pins construction, and it is not a coincidence that the constant is $2a$. It also explains why the empty focus is not a fiction — it is doing geometric work in every point of the curve, even though there is nothing there.

How small the real ones are #

The figure above is drawn at an eccentricity no planet has. This is the standard and slightly dishonest choice, made because an orbit drawn at a real eccentricity is indistinguishable from a circle.

Earth’s eccentricity is 0.0167. The orbit deviates from a perfect circle by about one part in seven thousand — far less than the width of the line drawn for it. Mars, the planet whose orbit Kepler solved, reaches 0.093, and even that is a curve nobody would call visibly elongated.

So the shape is not what was discovered. The offset is. At a distance of 2.5 million kilometres from the centre, the Sun’s apparent diameter changes by 3.4% over the year, the Earth receives 6.8% more sunlight in January than in July, and both are straightforwardly measurable. Kepler had none of those measurements. What he had was Tycho Brahe’s positions of Mars, accurate to about two arcminutes, and a stubborn refusal to accept an eight-arcminute residual from a circular fit. Those eight arcminutes are the whole of the first law.

Comets are where the shape becomes visible. Halley’s comet has $e = 0.967$: it comes inside Venus’s orbit and retreats past Neptune’s, and the ratio of those two distances is about sixty. The same curve, the same law, the same two foci — only the number is different.

What the offset does that the shape does not #

Suppose, for a moment, that Kepler had found an ellipse with the Sun at the centre. Almost nothing in the subject would work.

A body on such a path would be symmetric in every respect: two closest approaches per orbit rather than one, two most distant points, and a speed pattern repeating twice a year. There would be no periapsis in the singular. Above all, there would be no consistent way to relate the distance to the speed, because the same distance would occur at four different points of the orbit with no way to distinguish them.

With the Sun at a focus, the orbit has one closest point and one furthest point, and the distance rises and falls exactly once per revolution. That single-valuedness is what makes the sweeping of equal areas a workable law, what makes the vis-viva relation between speed and distance a function rather than a mess, and what lets an orbit be specified by six numbers instead of a table.

The focus is also where the physics enters. The force points at the primary. A curve whose defining point coincides with the source of the force is a curve about which dynamical statements can be made; a curve centred somewhere the force does not come from is a curve about which they cannot.

Why this curve and not another #

Kepler found the ellipse empirically, from data, and could not say why. The answer took Newton, and it is worth stating in the direction that makes it surprising.

An inverse-square attraction toward a fixed point produces closed orbits — paths that return exactly to where they started, cycle after cycle. That is a stronger condition than it sounds. Almost no force law does it. Under a slightly different exponent the orbit still has a nearest and a furthest distance, but the direction of closest approach rotates a little each time, so the path is a rosette that never closes. Only two force laws in all of mechanics produce closed orbits for every bound trajectory: the inverse square, and a force proportional to distance.

So the ellipse is one member of a family, and its neighbours are the trajectories of comets and of interstellar visitors. The family is exactly the conic sections, known since Apollonius in the third century BC as slices of a cone — an entirely geometric result with no mechanics in it whatsoever, waiting nineteen hundred years for something to be the shape of.

The closure of the ellipse has a consequence that was noticed only much later, and it is a good example of a null result carrying information. If the exponent were not exactly two, the orbits would precess. They do precess, very slightly, and almost all of it is caused by the pull of the other planets. What is left over after that accounting — 43 arcseconds per century for Mercury — is not caused by anything Newtonian, and it is one of the residuals that changed physics.

The timing, which the shape does not give #

Knowing the path is not knowing the motion. A body on this ellipse could in principle traverse it at any speed pattern at all, and the picture would look the same.

The second law supplies the timing, and the figure above is a good demonstration that it is genuinely independent information: the same curve, marked with when. The body races through the close approach and loiters at the far end, and the marks show it.

Getting from a time to a position is harder than it looks, and it is the point at which celestial mechanics stops being geometry. The relation involves solving

$$M = E - e\sin E$$

for $E$, which has no solution in elementary functions — it is transcendental, and Kepler said as much when he wrote it down. Every position in the figure above was obtained by Newton’s method, iterating until the residual vanished, which is exactly what an almanac has done since the seventeenth century.

What was actually measured #

Nothing in this essay was observed directly. The ellipse is an inference, and it is worth being specific about what it was inferred from.

Kepler had angles. Brahe’s instruments gave the direction to Mars against the fixed stars, to a couple of arcminutes, over about twenty years — no distances at all, and no way to obtain one. Turning a table of directions into a shape required a trick of great cunning: Mars returns to the same place in its orbit every 687 days, so observations separated by that interval see the same point from two different positions of the Earth. Two directions to one point from two known places give a distance, by triangulation. Kepler built the orbit of Mars out of a few dozen such triangles, and then, having the orbit of Mars, ran the argument backwards to get the Earth’s.

The result was a table of distances in units of the Earth–Sun distance. Not in kilometres — nobody had a value for that until the transits of Venus in the eighteenth century — and the astronomical unit remained a scale factor on an otherwise complete solar system for two hundred years. The shape came first and the size came much later, which is a pattern that repeats all the way up the distance ladder.

The ellipse nobody draws #

There is a second ellipse in every two-body system, and it is the one the primary traces. Neither body is stationary, so the ellipse drawn throughout this essay is a relative orbit rather than a path through space. For a planet round a star the distinction is a technicality; for a pair of comparable masses it is the whole measurement, since the ratio of the two individual ellipses gives the ratio of the masses and nothing else does.

The primary’s own small ellipse is also what makes planets detectable at all. It is a wobble of a few metres per second, and it is measurable.

How circular is normal #

The essay so far has treated the solar system’s near-circular orbits as the ordinary case and the comets as the exception. There is now enough data to ask whether that is true, and the answer is that it is not.

The solar system’s planets have a mean eccentricity of about 0.06, and if Mercury is set aside, about 0.04. Nothing has to be that circular. An orbit is specified by six numbers and the eccentricity is free to take any value from 0 to just under 1; the observed values are clustered hard against the bottom of that range.

Planets around other stars are not. The catalogue of exoplanets with well-determined orbits has a median eccentricity of roughly 0.2 and a substantial population above 0.4, with individual objects past 0.9 — HD 20782 b goes round its star on an orbit of $e = 0.95$, swinging from beyond Mars’s distance to inside Mercury’s. Those are orbits that no solar-system planet resembles and that would have been called cometary if found here.

Two caveats keep this from being straightforward. The first is a selection effect running the wrong way for once: a highly eccentric planet produces a large, sharply peaked radial-velocity signal that is easy to detect but easy to mis-fit, and eccentricities recovered from sparse data are biased upward — a circular orbit sampled badly can be fitted with a spurious $e$ of 0.2 or so. Careful reanalyses have moved a number of published eccentricities down toward zero. The second is that short-period planets are circularised by tides, so the observed distribution is a mixture of a primordial one and a processed one, split at a period of a few days.

Even after both corrections the conclusion survives. Eccentric planetary orbits are common and the solar system is quiet. The favoured explanation is that eccentricity is what planet–planet scattering leaves behind: a system that forms several giant planets close enough to perturb each other ejects some and leaves the survivors on eccentric orbits, while a system whose giants stayed apart keeps its circles. On that account the near-circularity of the Earth’s orbit — the 0.0167 that took eight arcminutes of residual to detect — is not a default. It is evidence that nothing catastrophic happened here.

The same reasoning applies to the solar system’s own small bodies, where the distribution runs the other way. Comets and trans-Neptunian objects have eccentricities spread across the whole available range, because nothing has circularised them and a great many have been scattered. The planets are the exception within their own system as well as among other systems, and the quiet orbits described in this essay are a property of eight objects rather than of orbits in general.

Why the curve closes at all #

The first law says the orbit is an ellipse, and an ellipse is a closed curve — the body returns to exactly where it started, with exactly the velocity it started with, for ever. That is a much stronger statement than it looks, and it is worth separating from the shape.

A bound orbit under any attractive central force oscillates in radius between a minimum and a maximum, and it sweeps out an angle while doing so. There is no reason for that angle to be exactly $2\pi$. If it is slightly more, the periapsis advances a little each revolution and the path is a rosette that never repeats; if slightly less, it regresses. The generic bound orbit under a generic central force fills an annulus rather than tracing a curve.

Bertrand’s theorem, proved in 1873, says that exactly two force laws avoid this. One is the inverse square. The other is the linear restoring force of a spring, which gives an ellipse centred on the origin rather than focused on it. Every other power law, and every non-power law, precesses.

That the solar system runs on one of the two is what made the first law discoverable. Had gravity gone as $r^{-2.1}$, Brahe’s twenty years of Mars observations would have shown a slowly turning ellipse, and no fit to a fixed conic would have worked at any accuracy. The eight arcminutes of residual that overturned the circle would instead have been a drift, and the shape and the drift would have had to be untangled together.

The closure also carries the same information as a conserved quantity, and it is one an inverse-square force has and nothing else does. Alongside energy and angular momentum, the Kepler problem conserves a vector pointing along the major axis towards periapsis — the Laplace–Runge–Lenz vector — whose direction is fixed and whose length is the eccentricity. A precessing orbit is one whose periapsis direction is not conserved, so the extra conservation law and the closure are the same fact.

Which makes the exceptions informative rather than embarrassing. Mercury’s 43 arcseconds a century is the failure of that conservation, and every measured precession in the solar system is a measurement of how far the real force departs from the exact inverse square — from a planetary perturbation, from an equatorial bulge, or from relativity.

Where the model stops #

The ellipse is the exact solution to a problem nobody has: two point masses, alone in the universe, attracting each other with an inverse-square force. Four assumptions, and every one of them fails somewhere.

Two bodies only. Add a third and the orbit is no longer closed and no longer an ellipse. The solar system’s planets perturb each other continuously, and the resulting slow changes in the orbital elements are the entire subject of perturbation theory. Neptune was found in those perturbations before it was found in a telescope.

Point masses. A body with an equatorial bulge does not pull quite like a point, and the deviation makes satellite orbits precess — usefully, as it happens, since a Sun-synchronous orbit is one tuned to precess at exactly one turn per year.

Newtonian gravity. Mercury’s extra 43 arcseconds per century.

A fixed primary. Neither body is actually still; both orbit their common centre of mass, and the ellipse drawn here is really the relative orbit of one about the other.

There is also a limit of the drawing rather than the model. Every figure on this page shows the orbit face-on, in a plane, which is a viewpoint no observer occupies. Real orbits are tilted, and three of the six orbital elements exist purely to say how — a tilt that is invisible here and that dominates the practical business of predicting where anything will appear.

One more member of the family sits between the two the essay has drawn.

The ladder from here #

Later rungs on this anchor: the focal-property construction, and why the string-and-pins definition is the useful one. The auxiliary circle and the eccentric anomaly. Kepler’s equation, and the iterative methods that solve it. The six orbital elements. The ellipse in three dimensions. Precession, and the ellipse that will not stay still. Mercury’s residual. Ellipses that are not orbits — in whispering galleries, in optical resonators, in lithotripsy. And the close binaries where the two bodies deform each other and the curve stops being a conic at all.

Kepler spent five years on Mars and later wrote that the eight arcminutes he could not explain away had led him to reform the whole of astronomy. They had. He also thought the orbits were spaced by nested Platonic solids for most of his career, which is worth remembering about anybody’s best ideas.