An orbit can look exactly like a circle and still not be one

Draw the Earth’s orbit accurately, at any size that fits on a page, and it is a circle. Not approximately a circle — a circle, to well inside the width of the line. The semi-minor axis is 99.986% of the semi-major axis, so at a hundred millimetres across the two differ by fourteen micrometres, which is a fifth of the diameter of a human hair.

And yet the orbit is not a circle, and the difference matters. The Earth is five million kilometres closer to the Sun in January than in July, receives 6.8% more sunlight at one end of the year than at the other, and moves fast enough at the near end that the two halves of the year differ in length by more than seven days.

Both statements are true because the eccentricity does two entirely different things, and it does them at different orders.

One number, two effects, two orders #

An ellipse of semi-major axis $a$ and eccentricity $e$ has a semi-minor axis

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

and its foci sit at $\pm ae$ from the centre. Those are the two quantities the eccentricity controls, and expanding both for small $e$ shows immediately why they behave so differently:

$$\frac{c}{a} = e, \qquad 1 - \frac{b}{a} = 1 - \sqrt{1-e^2} \approx \frac{e^2}{2}.$$

The offset is linear in $e$. The flattening is quadratic. For any small eccentricity the second is negligible compared with the first, and how negligible grows as the eccentricity shrinks: at $e = 0.1$ the ratio of the two effects is 20, at $e = 0.0167$ it is 120, at $e = 0.001$ it is 2,000.

This is the reason the standard picture of an orbit is drawn wrong in a way that is worse than a simple exaggeration. Textbook diagrams stretch the ellipse to make the shape visible, and in doing so they make the shape look like the content. It is not. The content is where the Sun is, and that survives at any eccentricity whatever.

What the offset does to the distance #

Because the primary sits at a focus, the distance to it runs between

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

so the near and far distances differ by a fraction $2e$ of the semi-major axis — again first order. For the Earth that is 3.34%: perihelion at 147.10 million kilometres in early January, aphelion at 152.10 million in early July, a difference of 5.0 million kilometres.

Five million kilometres is a large number by any human measure and it is invisible in the shape of the orbit, because it is an offset of the centre rather than a distortion of the outline. A circle drawn about a point 1.67% off-centre has exactly this property: it is still a circle, and the distances from that point still swing by $\pm 1.67\%$.

The received sunlight goes as the inverse square of the distance, so it swings by twice as much again:

$$\frac{S_{\max}}{S_{\min}} = \left(\frac{1+e}{1-e}\right)^{2} = 1.069.$$

The Earth intercepts 6.9% more solar power in January than in July. That is not a small number — it is comparable in size to the entire radiative forcing that separates a glacial period from an interglacial one — and it is worth being clear that it does not cause the seasons.

Why this is not what makes the seasons #

The seasons are caused by the tilt of the rotation axis, and the eccentricity argument is disposed of by a single observation: the northern and southern hemispheres have opposite seasons at the same time. A distance effect would warm and cool the whole planet together.

The tilt beats the distance by a wide margin, and the comparison is worth doing properly. At latitude 52°, the daily insolation at the summer solstice is about 2.7 times that at the winter solstice — a 170% swing — against the 6.9% the eccentricity supplies. The obliquity effect is roughly twenty-five times larger, and it has the right sign in each hemisphere separately. What the eccentricity does contribute is an asymmetry between the hemispheres. Perihelion currently falls in early January, so the southern summer coincides with the closest approach and the northern summer with the furthest. Southern summers therefore receive about 7% more sunlight than northern ones, and southern winters correspondingly less — the southern hemisphere has the more extreme year, and would have a noticeably more extreme climate if it were not mostly ocean.

That coincidence is not permanent. The perihelion direction precesses and the axis precesses the other way, so the alignment cycles with a period of about 21,000 years. Eleven thousand years ago the arrangement was reversed. This is one of the three Milankovitch cycles, and the eccentricity itself is another: it varies between about 0.005 and 0.058 on periods near 100,000 and 405,000 years, so the 6.9% swing has been as small as 2% and as large as 23%.

What the offset does to the speed #

The other first-order consequence is in the motion. Equal areas in equal times means the angular rate varies inversely with the square of the distance, so between perihelion and aphelion the angular speed changes by the factor

$$\left(\frac{1+e}{1-e}\right)^{2} = 1.069$$

— the same 6.9%, now in the rate at which the Sun appears to move along the ecliptic. The orbital speed itself, from the vis-viva relation, varies by rather less: 30.29 km/s at perihelion against 29.29 at aphelion, a swing of $\pm 1.7\%$ about the mean, which is first order in $e$ exactly as the offset is. The visible consequence is that the two halves of the year are not equal. The Earth passes from the March equinox to the September equinox — the northern summer half — in 186.4 days, and completes the other half in 178.8. Seven and a half days of difference, and the reason is that the summer half contains aphelion, where the Earth is moving slowest. Anyone with a calendar and a table of equinoxes can measure the eccentricity from that asymmetry alone, and this is exactly how it was first done.

The offset shows up in the clock #

There is one more first-order consequence, and it is the one a reader can check against a sundial.

Because the Earth moves faster near perihelion, the Sun appears to run along the ecliptic faster in January than in July, and a clock keeping uniform time drifts against a sundial. The eccentricity contributes a sinusoidal term to the equation of time with an amplitude of

$$\Delta t \approx \frac{2e}{2\pi} \times 1440\ \text{minutes} \approx 7.7\ \text{minutes},$$

a period of one year, and zeros at perihelion and aphelion. That is a straightforward first-order effect: double the eccentricity and the amplitude doubles.

It is not, on its own, the whole equation of time. The obliquity contributes a second term of larger amplitude — about 9.9 minutes — and twice the frequency, and the observed curve is their sum. But the eccentricity term is the one that breaks the symmetry between the two halves of the year, and it is why the analemma’s two lobes are unequal: the lower lobe, traced through the northern winter when the Earth is near perihelion, is the larger one. The size of this effect is worth putting beside the shape effect one last time. The eccentricity moves solar noon by up to about eight minutes, which anyone can see on a sundial in an afternoon. The departure of the orbit from a circle is fourteen parts in a hundred thousand, which nobody has ever seen at all.

What was actually measured #

Hipparchus measured this eccentricity in the second century BC, and the method is the one just described, run backwards.

He had the lengths of the astronomical seasons — the intervals between the equinoxes and solstices — from his own observations and from Meton and Euctemon before him. Spring, from the vernal equinox to the summer solstice, ran 94.5 days; summer, to the autumnal equinox, 92.5; and the remaining two seasons together 178.25. Four unequal intervals dividing a circle that ought, on a uniform circular motion about a central Earth, to have divided it into four equal ones.

His model was the eccentric circle: the Sun moves uniformly on a circle whose centre is displaced from the Earth. That model is not the ellipse and is not correct, but at $e = 0.0167$ the difference between it and the truth is second order — the flattening again — and so it is far below what he could detect. What his four season-lengths determine are two numbers: the size of the displacement and the direction of it. He obtained an eccentricity of 1/24, or 0.0417, and an apogee longitude of 65.5°.

The eccentricity is two and a half times too large. The reason is instructive, and it is not sloppiness. Hipparchus was measuring the equation of centre — the angular effect — which for the true ellipse is $2e$ in first order and which his model attributes to a displacement of $e_{\text{H}}$ producing an effect of $e_{\text{H}}$. His displacement therefore comes out at about twice the true eccentricity, and the remaining discrepancy is observational: solstice timing is intrinsically hard, because the Sun’s declination is stationary there and a day’s error in the date costs almost nothing in altitude.

Ptolemy repeated the measurement three centuries later, obtained the same numbers, and reported them as confirmation. They were not confirmation; the solar apogee really does move, by about 1.7° per century, so three centuries should have shifted it by five degrees. The agreement was a sign that the second measurement had been made to match the first.

The modern value comes from a different kind of measurement entirely. Radar ranging to Venus and Mercury from the 1960s, and then spacecraft tracking, fix the Earth’s orbit directly in kilometres rather than in angles: the current osculating eccentricity is 0.016708, decreasing by about $4 \times 10^{-9}$ per year. It is quoted to five figures because the orbit is not a fixed ellipse — the other planets perturb it, and the number is a snapshot with a date attached.

The generalisation: what a small parameter is allowed to hide #

The pattern here is not confined to orbits, and it is worth stating in general because it is one of the most common ways a picture misleads.

When a system is described by a small parameter, the quantities that depend on it linearly and the quantities that depend on it quadratically separate by a factor of $1/e$, and that factor can be enormous. The linear quantities are the observable ones; the quadratic ones are invisible. Drawing the system with the small parameter exaggerated — which is what anyone does who wants the reader to see it at all — inflates the quadratic effects far faster than the linear ones, and so systematically directs attention to the part that does not matter.

An orbit at $e = 0.6$, the standard textbook drawing, has a flattening of 20% and an offset of 60%: a factor of three between them. The reader sees an oval and a displaced focus and reasonably concludes that both are important. At $e = 0.0167$ the factor is 120, and only one of them exists.

The same trap sits behind the Earth’s oblateness, where a flattening of 1/298 is routinely drawn as a visible egg; behind small-angle approximations, where the error is third order and so a parallax diagram drawn at a visible angle is wrong in a way the real measurement is not; and behind almost every drawing of a spacecraft trajectory, where the departure hyperbola is drawn at a scale that makes it look like a substantial part of the journey.

The orbits that really are circles #

Nothing in the solar system has an eccentricity of zero, and the reason is worth stating because it is not a coincidence and not a matter of measurement precision.

An eccentricity of exactly zero is a single point in a continuous space of possible orbits, and a system of mutually perturbing bodies is continually being moved about in that space. Venus has the smallest eccentricity of the planets at 0.0068, and it is not stationary — it wanders between about 0 and 0.07 over hundreds of thousands of years, passing through very small values on its way rather than resting at them.

Even the orbits that a dissipative process has actively circularised do not reach zero. Tides drain the eccentricity of a close binary or a close-in planet exponentially, so after enough time the residual should be arbitrarily small; what stops it is that something is usually pumping it back — a companion, a resonance with another body, or the star’s own oblateness — and the equilibrium is a balance rather than an endpoint.

There is also a measurement problem that appears exactly in this limit, and it is the reason orbits are not fitted in the elements a textbook uses.

As the eccentricity goes to zero, the direction of periapsis becomes undefined: there is no long axis, so there is nothing for the argument of periapsis to measure. The two parameters therefore become degenerate, and a least-squares fit in them is ill-conditioned — the solver is trying to determine an angle that the data contain no information about, and the eccentricity is driven to spurious nonzero values because a positive quantity fitted near its boundary has nowhere to go but up.

The remedy is to fit the combinations $e\cos\omega$ and $e\sin\omega$ instead. Those two are well behaved through zero, they carry exactly the same information, and their uncertainties are roughly equal and uncorrelated. Almost every published eccentricity for a nearly circular orbit — planetary, binary or spacecraft — was obtained that way, and the quoted value is derived from the pair rather than fitted directly.

A coordinate singularity in the description is not a property of the orbit, and choosing parameters that do not have one is the same move as choosing an aim point over a periapsis distance: the quantity to work in is the one whose behaviour is uniform where the measurement is being made.

Where the model stops #

The eccentricity is not constant. It is an osculating element — the eccentricity of the ellipse the Earth would follow if every other body vanished right now. Jupiter and Venus between them cycle it between roughly 0.0034 and 0.058 over hundreds of thousands of years, so “Earth’s eccentricity is 0.0167” is a statement about the present epoch and needs one.

The second-order effect is not zero, only small. For a body at $e = 0.9$ the flattening is 56% and dominates the picture completely; the argument here is about small eccentricities and says nothing about comets.

Insolation is not distance alone. The 6.9% figure is the flux at the top of the atmosphere on a surface facing the Sun. What reaches a given patch of ground depends on the solar altitude and the day length as well, and those are set by the tilt.

The figures cannot show the thing they argue about. This is the honest limitation of the whole essay, and it deserves saying plainly: no drawing of Earth’s orbit at true eccentricity can show that it is not a circle, because the difference is below the resolution of any display. The hero figure works only because it plots the quantities rather than drawing the orbit. Every picture of the orbit itself on this page is a picture of a circle, and the argument has to be made in numbers.

The ladder from here #

Later rungs on this anchor: the equation of centre, and the series in $e$ that everything above is the first term of. The Milankovitch cycles worked out in full, with the 405,000-year eccentricity metronome that dates the geological record. Perihelion precession as an observable. Osculating elements against mean elements, and what “the orbit” means when it is changing. Laplace’s proof that the eccentricities of the planets stay bounded, and what Poincaré later showed about the proof.

There is a nice historical coda. The reason Kepler found the ellipse at all is that he was working on Mars, whose eccentricity is 0.0934 — five and a half times the Earth’s, so its second-order flattening is thirty times larger. With Tycho’s data, accurate to about two arcminutes, Mars’s departure from a circular orbit was detectable and the Earth’s was not. Had he been handed the Earth’s orbit instead, the circle would have fitted.