An average that precession cannot move

The first rung of this anchor established that an average over an orbit is not one number, and that which of the four candidates is meant depends entirely on what the average is for. This rung takes the one that matters most outside celestial mechanics — the average of the sunlight a planet receives — and finds that it is invariant under the very thing that is usually blamed for the ice ages.

The cancellation is exact and it is one line. Flux falls as $1/r^2$. Kepler’s second law makes the time spent in an interval of orbital longitude proportional to $r^2$. Integrate the product over a year and every factor of $r$ is gone.

$$\langle Q\rangle \;=\; \frac{1}{2\pi\sqrt{1-e^{2}}}\int_0^{2\pi} G\big(\varphi,\ \delta(\lambda)\big)\,\mathrm{d}\lambda$$

Neither the eccentricity nor the longitude of perihelion survives into the integrand. The eccentricity reappears once, in the normalisation, as $1/\sqrt{1-e^2}$ — which at Earth’s eccentricity is a gain of 0.014 per cent. The longitude of perihelion does not reappear at all.

What the three orbital elements are #

Three quantities vary and it is worth separating them before using them, because the names overlap confusingly with the astronomy the rest of this collection uses.

Eccentricity varies between about 0.005 and 0.058 with dominant periods near 100,000 and 405,000 years, driven by the secular interactions of the planets — no planet has an eccentricity of its own, and the Earth’s is a superposition of modes belonging to the system.

Obliquity varies between about 22.1 and 24.5 degrees with a period near 41,000 years, and is a dynamical variable with its own equilibria rather than a constant.

The longitude of perihelion measured from the equinox — sometimes called climatic precession — moves at the sum of two rates. The Earth’s axis precesses westward in about 26,000 years, which is the motion that gives the pole star a shelf life; the orbit’s own apsidal line advances eastward in about 112,000. The two combine to move the equinox relative to perihelion in about 21,000 years, and that is the period in the record.

Only the third of these is the subject of the cancellation. The other two do change the annual mean, the obliquity substantially.

The cancellation, drawn #

The exactness deserves emphasis because it is unusual. Most statements of this kind in celestial mechanics are first-order results with a small correction hiding behind them. This one has no correction. The cancellation is between the inverse-square law and the second law, both of which are exact statements about a Keplerian orbit, and it holds at any eccentricity whatsoever — for a comet on a parabola as much as for the Earth.

It is worth noticing what the cancellation is not. It is not the statement that the Earth receives the same energy as a planet on a circular orbit of the same semi-major axis — it does not, it receives $1/\sqrt{1-e^2}$ times as much, which is 0.014 per cent more. And it is not a statement about the whole planet: the global annual mean and the annual mean at every individual latitude are both independent of the perihelion longitude, which is a stronger result than the global one alone and is what the figure shows.

What it means physically is that a planet’s orbit is a device for redistributing a fixed annual allowance. Move perihelion and the planet spends less time far away in one season and more in another; the total is conserved because the two effects are the same effect seen from either end.

One more consequence is worth drawing out because it applies far beyond the Earth. The result holds for any planet on any Keplerian orbit, so a planet’s habitability, judged by its annual energy budget, is unaffected by where its perihelion points — and is affected by its eccentricity only through a factor that is 1.005 even at e = 0.1. That is a useful thing to know when the band a planet has to stay inside is being computed for an eccentric orbit: the flux to use is the time-averaged one, it is a hair above the circular value, and the perihelion phase can be ignored entirely. What cannot be ignored is the instantaneous excursion, since a planet inside the zone on average can be outside it at every perihelion, and a runaway is a threshold rather than an average.

Why the ice ages care about a season #

If precession moves no annual energy, it is reasonable to ask why it appears in every spectrum of an ice-age record with a period of twenty-three thousand years and a large amplitude.

The answer is that an ice sheet does not integrate over a year. It grows in winter and melts in summer, and the two are not symmetric: winter snowfall is limited by how much moisture the atmosphere can carry, which is a weak function of anything orbital, while summer melting is a strong function of how much sunlight arrives. The controlling quantity is summer insolation at the latitude where the ice sheets are, which is around 65 degrees north, and that is the quantity in the first figure.

There is a second reason the annual mean is the wrong quantity, and it is thermodynamic rather than glaciological. A climate responds to a forcing through feedbacks whose strength depends on the state it is in, and the ice–albedo feedback in particular is enormously stronger when there is ice to melt than when there is not. A watt per square metre delivered to a summer with a melting ice margin does something a watt delivered to a January polar night cannot. Averaging over the year throws away exactly the information the nonlinearity needs.

That modulation is the sharpest observational test the theory has, and it passes. The eccentricity itself varies on periods near a hundred thousand and four hundred thousand years, and the precession signal in the record waxes and wanes on exactly those periods, without the precession’s own period changing.

The equator, where it does something else #

The equatorial case also exposes something the high-latitude case hides. The insolation curve at 65 degrees is dominated by the day length, which depends on obliquity and latitude and not at all on the orbit; the orbit enters only through the inverse-square factor. So the shape of the curve against perihelion longitude is a pure sinusoid at every latitude, and the amplitude ratio between latitudes is exactly one. Every latitude on the planet sees the same fractional swing, and what differs between them is only how much a given fractional change matters.

Two hemispheres complicate the picture in a way worth stating plainly. Precession moves the northern and southern summers in antiphase: when perihelion falls in northern summer it necessarily falls in southern winter. So the precession term in a global record is a difference between hemispheres rather than a global forcing, and the reason the northern signal dominates is geographical — the northern hemisphere has the land on which an ice sheet can sit.

There is a well-known difficulty with all of this and it should be stated rather than skirted. The dominant period in the last eight hundred thousand years of the record is a hundred thousand years, which is an eccentricity period — and eccentricity’s direct forcing is a tenth of a per cent. Before about a million years ago the dominant period was forty-one thousand, which is obliquity’s, and the transition between the two regimes happened with no change in the orbital forcing at all. Both facts say that the climate system’s own dynamics, rather than the astronomy, decide which of the available periods it responds to. The astronomy supplies a pacemaker and not a driver, and how a pacemaker of the wrong period comes to set the tempo is not settled.

What eccentricity does on its own #

The eccentricity has a direct effect too, and it is small enough to be worth quantifying next to the indirect one.

So the eccentricity’s own contribution to the ice-age forcing, through the annual mean, is a tenth of a per cent over the whole of its variation — an order of magnitude too small to explain anything in the record. Its contribution through the precession, as the amplitude of the seasonal swing, is ten per cent. The hundred-thousand-year period that dominates the last million years of the record is therefore an eccentricity period appearing in the amplitude of a precession signal rather than a forcing in its own right, which is the single strangest fact in the whole subject: the largest signal in the record corresponds to the smallest forcing in the theory.

There is a neat consequence of the antiphase that is sometimes offered as an objection and is not one. If the two hemispheres receive opposite precession forcing, why is there a global precession signal at all rather than a cancellation? Two answers, both partial. The land distribution is asymmetric, so the ice–albedo feedback is available in the north and not in the south. And the record’s proxy is global ice volume, which is dominated by whichever hemisphere is growing ice — an inherently rectifying quantity, which converts an antisymmetric forcing into a signal at the forcing’s own period rather than cancelling it.

What is actually measured #

Neither of these curves is observed. What is observed is a proxy.

The ice-age record comes from oxygen isotope ratios in the shells of deep-sea foraminifera, which respond to the volume of ice on land and to the temperature of the water the animal lived in, mixed together in proportions that have to be modelled. The chronology comes partly from the assumption that the record contains the orbital periods — which makes testing whether it contains them delicate — and partly from independent dating of a few horizons.

There is a second measurement problem which is about the axis rather than the values. Fitting an orbital period into a sediment record requires a depth-to-time conversion, and the conversion is derived by assuming a sedimentation rate — often by tuning the record until its spectrum contains the orbital periods most sharply. That procedure cannot then be used as evidence that the periods are present. The tests that survive are the ones performed on records with independent chronologies, radiometric dates on volcanic horizons or magnetic reversals, and those are sparser and noisier than the tuned ones.

The orbital elements themselves, by contrast, are known extremely well. Integrating the solar system’s secular evolution backwards gives the eccentricity, obliquity and precession phase to good accuracy for the last twenty million years and usefully for fifty, beyond which the chaos in the inner planets’ orbits makes the phases unrecoverable. So the forcing is the well-determined half of this comparison and the response is not, which is the opposite of the usual situation in observational astronomy. Fifty million years is also roughly where a prediction acquires an expiry date: the inner solar system’s Lyapunov time is a few million years, so two integrations differing by a metre in the Earth’s initial position diverge in phase within a hundred million, and the precession angle at two hundred million years ago is not a number anybody has.

There is one more element in the classical treatment and it is a bookkeeping trap rather than a physical effect. The lengths of the astronomical seasons are not equal — Kepler’s second law makes the Earth linger in the half of the orbit near aphelion — and they change as perihelion precesses. Northern spring and summer together currently run about seven days longer than autumn and winter. Any calculation that sums insolation over “summer” therefore has to say whether summer is a fixed number of days, a fixed range of solar longitude, or the half of the year that is warmer, and the three give different curves. Milankovitch chose the third, the caloric half-year, precisely to avoid the ambiguity — the definition is self-referential but it is at least unique.

The generalisation #

The structure worth extracting is that an exact cancellation is worth more than a small quantity, because it says which of two effects can be ignored at any parameter value rather than only at the present one.

The annual mean’s independence of the perihelion longitude is not an approximation good at small eccentricity. It holds at e = 0.9. So any argument about a planet’s climate that depends on the annual energy budget can drop precession entirely, forever, and any argument that finds precession mattering must be an argument about seasons. That is a much stronger statement than “the effect is small”, and it comes from a structural feature — two exponents summing to zero — rather than from a numerical estimate.

The same shape occurs wherever a conservation law makes two dependences cancel: the second law is angular-momentum conservation in disguise, and its appearance here is the reason the cancellation is exact rather than fortuitous.

It is worth adding what the cancellation does not license. Nothing here says the eccentricity is climatically unimportant — it sets the amplitude of the precession term, which is the largest single forcing in the whole calculation. What it says is that the eccentricity’s importance is entirely indirect, arriving through a seasonal redistribution rather than through an energy budget, and that an argument attempting to warm a planet by making its orbit more elliptical is off by two orders of magnitude.

The corollary is a rule for reading a forcing. Ask what the responding system integrates over. An ice sheet integrates over a summer; an atmosphere over a few weeks; an ocean over centuries. The same orbital variation is a large forcing to the first and none at all to the third, and the difference is entirely in the averaging window rather than in the astronomy.

The obliquity term, for contrast #

Setting the third element beside the second sharpens what is peculiar about precession.

Obliquity does move the annual mean, and by a lot at high latitude: the second figure’s spread between 22.1 and 24.5 degrees is the whole of the obliquity cycle, and it changes the polar annual mean by something like two per cent. It moves it in the same direction in both hemispheres simultaneously, because tilting the axis further points both poles more directly at the Sun at their respective summers. And it moves energy from the tropics to the poles, conserving the global total to within the small change in the planet’s own cross-section presented to the Sun.

So obliquity is a genuine redistribution in latitude, precession is a redistribution in season, and eccentricity is a modulation of the second. Three elements, three mechanisms, three signatures in a record — in-phase between hemispheres at 41,000 years, antiphase at 21,000, and an amplitude envelope at 100,000. A record showing the 21,000-year term in phase between the hemispheres would refute the whole framework, which is the sort of thing worth knowing about a theory that is often accused of being unfalsifiable.

Where the ladder goes next #

The next rung takes the caloric half-year, which is Milankovitch’s own construction and is not the same as the solstice value used here: the sum of insolation over the warmer half of the year, defined by which days are warmer rather than by the calendar, which removes an ambiguity about how to compare years of different lengths. It is a slightly different function of the same three elements and it is the quantity the classical curves actually plot.

Further rungs on this anchor: the obliquity term, whose forty-one-thousand-year period is in phase across both hemispheres and which does move the annual mean; the phase relationship between the three terms and what a record with the wrong phase would rule out; the averaging that makes secular perturbation theory work at all, which is the same procedure applied to the perturbing function rather than to the sunlight; and the equivalent calculation for a planet whose obliquity is not stable, where the insolation history has no periods in it at all.