Astronomy & Celestial Mechanics
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Orbits

A hotter summer that is also a shorter one

When the Earth passes closest to the Sun in northern summer, the northern summer is brighter — and, by Kepler's second law, shorter by exactly the right amount to deliver almost the same energy. Measured by the energy it delivers above a melting threshold rather than by its brightest day, a summer barely feels the precession cycle at all and feels the tilt of the axis several times more strongly.

Assumes Orbital averages, Obliquity and Seasons.

14 min read 6 figures The sky as a clockTwo bodies, and no moreSmall perturbations, large consequences

The Earth’s orbit is an ellipse whose closest point to the Sun slowly moves around the calendar. Today perihelion falls in early January, in the depth of northern winter; eleven thousand years ago it fell in July. The annual mean sunlight at any latitude is exactly independent of that: the second law makes the Earth move fastest where the Sun is brightest, and over a whole year the two cancel identically. But a single day’s sunlight is not independent of it at all. On the June solstice at 65°N, the latitude at which the northern ice sheets grew, the sunlight is about seven per cent greater when perihelion falls in June than when it falls in December.

That single-day number is the one most often plotted in the curves of the astronomical theory of the ice ages since the 1970s: June or July insolation at 65°N, swinging with the twenty-one-thousand-year precession cycle, modulated by the eccentricity, with a smaller contribution from the forty-one-thousand-year cycle of the axial tilt. The premise is that ice sheets survive or melt according to their summers. The question this essay asks is what a summer is — and the answer changes which astronomical cycle matters.

A hotter summer that is also a shorter one. Daily-mean sunlight at 65°N through one year beginning at the March equinox, for an orbit of eccentricity 0.0167 and obliquity 23.44°, drawn twice: with perihelion at the June solstice, when the northern summer is closest to the Sun, and with perihelion at the December solstice, half a precession cycle later. The first summer peaks 6.9% higher, and it is also shorter, because the Earth moves fastest near perihelion: days above 275 W m⁻² number 139 against 144. The energy delivered on those days — the shaded area above the line — is 5.055 against 4.989 GJ m⁻², a difference of 1.3%. The second law makes the intensity and the duration trade against each other almost exactly, so a summer measured by how much energy it delivers barely feels precession at all.
Fig. 1 A year of daily sunlight at 65°N with perihelion in June and with perihelion in December. The June-perihelion summer peaks seven per cent higher and has five fewer days above 275 watts per square metre; the energy it delivers on those days is only 1.3 per cent more.

Brighter and briefer

The two curves in the opening figure are the same latitude, the same tilt, the same orbit, differing only in where on the orbit the northern summer falls. With perihelion in June, the summer Sun is 3.4 per cent closer than average and the sunlight is correspondingly brighter; the peak is 6.9 per cent above the December-perihelion case. That is the effect the classical curves record.

The other thing that happens is visible in the width of the curves. The June-perihelion summer rises and falls faster. The Earth moves through the part of its orbit around perihelion more quickly than through the rest, sweeping equal areas in equal times, so the arc of solar longitude that makes up the northern summer is covered in fewer days. The days above 275 watts per square metre — a reasonable threshold for the sunlight that can melt snow at that latitude — number 139 in one case and 144 in the other. The brighter summer is the briefer one.

The energy delivered on those days is what an ice sheet feels, if it responds to the total heat it receives while it is warm enough to melt rather than to the heat of its sunniest day. That energy differs between the two cases by only 1.3 per cent. The seven per cent in intensity has been almost entirely paid for by the loss of five days.

The second law as a calendar

The trade can be seen directly in the length of the seasons.

The northern summer half-year, lengthened and shortened by precession. The number of days from the March equinox to the September equinox — the northern spring and summer — against the longitude of perihelion, for eccentricities 0.0167 and 0.05. Equal angles of solar longitude take unequal times, because the Earth moves fastest at perihelion, so the half-year containing perihelion is the shorter. Today perihelion falls in early January and the northern warm half-year lasts 186.4 days, about 7.6 more than the southern. With perihelion in June it would be 178.7. At e = 0.05 the swing is ±11.6 days. The swing in duration has the same fractional size as the swing in intensity and the opposite sign, which is the second law written as a calendar.
Fig. 2 The number of days from the March equinox to the September equinox against the longitude of perihelion, for the present eccentricity and for a more eccentric orbit. Today the northern spring and summer last 186.4 days; with perihelion in June they would last 178.7. The swing in duration has the same fractional size as the swing in intensity and the opposite sign.

The two equinoxes divide the orbit into halves of equal solar longitude, 180 degrees each, but not of equal time. The half containing perihelion is covered faster. Today, with perihelion in January, the southern spring and summer contain it, and the northern warm half-year is the longer: 186.4 days against 178.8, a difference of more than a week, which the calendar records as the fact that March 20 to September 22 is longer than September 22 to March 20. When perihelion falls in June, the northern half-year shrinks to 178.7 days.

The fractional change in duration is first order in the eccentricity, like the fractional change in intensity, and it has the opposite sign. That is not a coincidence but the second law. The rate at which the Earth sweeps solar longitude goes as the inverse square of its distance; the intensity of sunlight goes as the inverse square of the same distance; so the sunlight received per degree of solar longitude is independent of the distance altogether. Any quantity that integrates over a fixed range of solar longitude — a season defined by the Sun’s position rather than by the calendar — receives exactly the same energy whatever the orbit’s orientation.

Milankovitch’s own half-year

Milankovitch himself did not plot a single day. He worked with what he called the caloric half-year: the half of the year, 182.6 days long, in which the insolation at a given latitude is higher than on any day of the other half. The sum of the sunlight over that half is his summer, and the curves he published in the 1920s and 1930s are of that sum.

The caloric half-year sits between the two extremes. It is a fixed length of time rather than a fixed arc of solar longitude, so the second law does not cancel within it exactly: when perihelion falls in the northern summer, the half-year of highest insolation covers slightly more than half the orbit’s arc, since the Earth sweeps that part faster, and it gathers slightly more of the year’s light. But it is long — half the year — so most of the cancellation survives, and the caloric summer’s precession swing is a fraction of the solstice’s. The simplification from Milankovitch’s half-year to a single day, made later for convenience, moved the theory towards the extreme in which precession looks most important.

A summer counted three ways

Whether a summer feels precession depends on how it is counted, and the three natural ways give three very different answers.

What precession does to a summer, by how the summer is counted. The change in four measures of northern summer at 65°N as the longitude of perihelion runs through a full precession cycle, relative to their values with perihelion at the March equinox, for eccentricity 0.0167 and obliquity 23.44°. June-solstice insolation, the quantity in the classical curves, swings by ±3.3%. The energy delivered over the whole year does not move at all, ±0.000%, because the second law cancels intensity against time exactly. The energy delivered on days above 275 W m⁻² — a threshold for melting — swings by ±0.7%, and above 400 W m⁻² by ±5.4%. The higher the threshold, the more a summer measure behaves like a single day and the more precession it feels; the lower, the more it behaves like a year and the less. Which of these an ice sheet responds to decides which astronomical cycle its history should show.
Fig. 3 The change in several measures of northern summer as perihelion moves through a full precession cycle. June-solstice insolation swings by ±3.3 per cent. The whole year’s energy does not move at all. The energy on days above 275 watts per square metre swings by less than one per cent; above 400, by about five.

Counted as the brightest day, a summer feels precession fully: ±3.3 per cent at the present eccentricity, the full first-order effect. Counted as the whole year, it feels nothing: the second law cancels it exactly. Counted as the energy delivered on days warm enough to matter — the energy above a threshold — it falls between, and where it falls depends on the threshold.

A low threshold makes the summer long, covering most of the arc between the equinoxes, and the second law’s cancellation is nearly complete: at 275 watts per square metre the swing is below one per cent. A high threshold makes the summer short, a few weeks around the solstice, and it behaves more like a single day: at 400 watts per square metre the swing is about five per cent, larger even than the solstice’s because the number of days above so high a threshold changes steeply with the peak’s height. The measure an ice sheet actually integrates is a question for glaciology, not for orbital mechanics. What the orbital mechanics says is that the answer to “how much does precession matter?” depends almost entirely on that question.

This is a point that was made clearly only in 2006, nearly a century after Milankovitch, by working out the summer energy above a melting threshold rather than the solstice insolation — and it has a direct consequence for the geological record.

The tilt does what the orbit cannot

The obliquity is different in kind. A larger tilt lifts the summer Sun higher at high northern latitudes throughout the season, not only on the longest day, and it does so without any compensating change in the season’s length, because it changes the geometry of the Sun’s path rather than the Earth’s speed along its orbit.

The tilt moves a summer's energy; the orbit's orientation barely does. The energy delivered at 65°N on days above 275 W m⁻², against the obliquity across its range over the last few million years, 22.1° to 24.5°, with perihelion fixed at the June solstice. It rises by 8.5% across that range, steadily, because a larger tilt lifts the summer Sun higher at high latitudes for the whole season and not only on the longest day. The shaded band is the range precession spans at the present obliquity, ±0.7%. Measured this way, the obliquity cycle of forty-one thousand years moves the summer by several times what the twenty-one-thousand-year precession cycle does — the opposite of the ranking the June-solstice curves give.
Fig. 4 The energy delivered at 65°N on days above 275 watts per square metre, as the obliquity runs across its range of the last few million years, 22.1 to 24.5 degrees. It rises by 8.5 per cent across the range. The shaded band is the whole range swept by precession at the present tilt, less than one per cent either way.

Across the obliquity’s range the summer energy changes by 8.5 per cent. Across the whole precession cycle it changes by less than one per cent either way. Measured as summer energy above a melting threshold, the forty-one-thousand-year cycle of the tilt matters several times more than the twenty-one-thousand-year cycle of perihelion — the opposite ranking to the one the solstice curves give.

The mechanism is the geometry that puts the most summer sunlight on Earth at a pole: at high latitude in summer the Sun never sets, and the daily total depends on how high it circles, which the tilt sets directly. A degree more tilt raises the midnight Sun and the noon Sun alike, and it does so on every day of the season. There is nothing for the second law to trade against, because the Earth’s speed along its orbit is unchanged.

The tilt also acts on both hemispheres in the same sense. A larger tilt makes both polar summers warmer and both polar winters colder, while precession makes one hemisphere’s summer warmer and the other’s cooler at the same time. A climate forcing that is in phase in both hemispheres is much easier to reconcile with ice ages that were global than one that is in antiphase, and that has been an awkward point for the precession-dominated picture since the first long records were read.

Obliquity stays ahead until a summer is a few weeks long

The ranking can be followed continuously as the threshold changes.

Obliquity leads at every threshold, and precession closes in. How strongly the summer energy at 65°N responds to each astronomical cycle, against the insolation threshold that defines the summer: the half-range swept by a full precession cycle at the present obliquity, and the half-range swept by the obliquity's variation from 22.1° to 24.5°. At a threshold of zero — the whole year — precession does nothing and obliquity does little. As the threshold rises, the summer shortens towards a single day and precession's influence grows, closing on obliquity's only at the highest thresholds, where a summer is a few weeks long. Between them lies the range of thresholds relevant to melting snow and ice, and in it obliquity dominates — which is one proposed reason the ice ages of one to three million years ago paced themselves at the forty-one-thousand-year obliquity period rather than at the precession period the classical curves emphasise.
Fig. 5 The sensitivity of summer energy at 65°N to a full precession cycle and to the obliquity’s full range, against the threshold that defines the summer. At zero threshold precession does nothing; it grows as the summer shortens, but for thresholds across the range relevant to melting the obliquity’s influence stays larger, and precession closes on it only when the summer has shrunk to a few weeks.

At zero threshold the summer is the whole year and precession has no effect at all. As the threshold rises, the summer shortens towards the solstice, the cancellation weakens, and the precession sensitivity grows steeply. But the obliquity sensitivity grows too, because the energy above a high threshold responds sharply to anything that raises the peak, and across the whole range of thresholds relevant to melting snow and ice the obliquity stays ahead. The two come together only at thresholds so high that the summer has shrunk to a few weeks of the most intense sunlight — the limit in which a summer is a solstice, and the classical curves are recovered.

So the classical curves are not wrong. They are the limit of a very short summer. The question is whether an ice sheet’s mass balance behaves like that limit or like the integral over a longer season, and the evidence from how melting depends on temperature suggests the integral: snow melts on every day warm enough, and the total melt over the season is what matters.

A world that paced itself by the tilt

The geological record has a feature that the classical theory has always struggled to explain, and the summer-energy argument explains it naturally.

For most of the last three million years — from about three million until about a million years ago — the global ice volume recorded in deep-sea sediments oscillated with a period of forty-one thousand years, very regularly, with almost no power at the precession periods near twenty-one thousand years. If the ice sheets responded to solstice insolation, where precession dominates, the record should show precession. It shows the tilt.

Two explanations have been offered. One is that precession’s effect is present in each hemisphere but opposite in the two, so that the northern and southern ice sheets cancel in the global record while the tilt, acting on both in the same sense, adds. The other is the one drawn here: the ice sheets responded to summer energy above a melting threshold, and for any reasonable threshold that quantity is dominated by the obliquity. The two explanations are not exclusive, and the summer-energy one has the attraction of needing nothing but the orbit and a threshold.

Then, about a million years ago, the record changed: the ice ages became longer, larger and more irregular, with a period near a hundred thousand years, and precession’s signature appeared in their terminations. Neither explanation accounts for the change on its own. It has to be something about the ice sheets — their size, the ground they sat on, the carbon dioxide in the atmosphere — that altered how they responded to the same orbital forcing. The orbit did not change character a million years ago. The climate’s response to it did.

The two cycles are easy to confuse because both are called precession in different contexts. The orbit’s perihelion moves around the calendar because the axis itself precesses among the stars and the orbit’s own orientation slowly turns, and the combination brings perihelion back to the same season every twenty-one thousand years or so. The tilt’s cycle is separate: the axis’s angle to the orbit’s plane nods between two limits every forty-one thousand years, held within them by the Moon. The first moves summer sunlight between hemispheres; the second changes it in both at once.

Why the eccentricity still matters

None of this makes the eccentricity irrelevant. The swing in solstice insolation is proportional to the eccentricity, and so are the smaller swings in summer energy at any threshold.

A hotter summer that is also a shorter one. Daily-mean sunlight at 65°N through one year beginning at the March equinox, for an orbit of eccentricity 0.05 and obliquity 23.44°, drawn twice: with perihelion at the June solstice, when the northern summer is closest to the Sun, and with perihelion at the December solstice, half a precession cycle later. The first summer peaks 22.2% higher, and it is also shorter, because the Earth moves fastest near perihelion: days above 275 W m⁻² number 135 against 148. The energy delivered on those days — the shaded area above the line — is 5.124 against 4.927 GJ m⁻², a difference of 4.0%. The second law makes the intensity and the duration trade against each other almost exactly, so a summer measured by how much energy it delivers barely feels precession at all.
Fig. 6 The same comparison at an eccentricity of 0.05, near the largest the Earth’s orbit reaches. The June-perihelion summer now peaks 22 per cent higher and lasts thirteen fewer days above the threshold; its energy is 4 per cent greater. The cancellation still removes most of the difference, but at this eccentricity what survives is no longer small.

At the present eccentricity of 0.0167 the orbit is nearly circular and the cancellation leaves little. When the eccentricity is near 0.05 — as it has been several times in the last million years — the intensity swing is three times larger, the duration swing three times larger, and what survives their near-cancellation is three times larger too: a four per cent difference in summer energy rather than one. Precession therefore matters most in epochs of high eccentricity, which is the modulation the eccentricity’s hundred-thousand- and four-hundred-thousand-year cycles impose. That modulation is real and is seen in the record; it is a modulation of an effect that the second law has already made small.

What the calculation leaves out

The figures treat 65°N as a single latitude with no atmosphere. Sunlight reaching an ice sheet is reduced by clouds and reflected by the ice itself, and a real threshold for melting is a threshold in the energy absorbed at the surface rather than in the sunlight at the top of the atmosphere. Both depend on the season and on the ice’s own state, and a model that tracks them is a model of the ice sheet rather than of the orbit.

And they treat the response as instantaneous. The surface does not reach its highest temperature on the day of greatest sunlight: the hottest month is not the sunniest, because the ground, the ocean and the air store heat and release it weeks later. An ice sheet’s melting season therefore lags the sunlight, and the days that matter for melt are shifted towards late summer, when the precession effect on intensity has a different phase from the one at the solstice. A melt model driven by a lagged temperature rather than by the sunlight itself integrates over a window that is displaced as well as extended, and the displacement changes how precession and tilt weigh against each other. The lag is one more reason the relevant summer is an integral rather than a day: a system with heat capacity cannot respond to a single day at all.

They also treat the threshold as fixed. A warmer climate lowers the effective threshold, since less sunlight is needed to bring the surface to melting, and a colder one raises it. If the threshold drifts with the climate, then so does the relative importance of the tilt and the orbit’s orientation, and a climate that cooled through the last three million years would have moved gradually from a tilt-dominated regime towards one in which precession mattered more — which is at least suggestive, given the change in the record a million years ago, and is not established.

Still open: which summer an ice sheet counts

The orbital side of the problem is now exact: for any definition of a summer, the dependence on each orbital element can be computed to any precision required, and the same arithmetic that makes the annual mean immune to precession sets how immune each shorter season is. The open question is on the other side — which integral of the sunlight the ice sheets actually responded to, and whether it was the same integral through the whole of the ice-age record. The record itself constrains it: an ice sheet responding to solstice sunlight should leave a precession signal in its volume, one responding to summer energy an obliquity signal, and the relative power at the two periods in a well-dated record measures, in effect, the threshold the ice was integrating above. Reading it requires records whose timescales were not themselves tuned to the orbital curves, which the oldest ones were. The ice-sheet models that couple a surface energy balance to the ice dynamics are the tool for answering it, and their answers depend on processes — the albedo of melting snow, the effect of dust on ice, the ice sheets’ own altitude — that are much less certain than the orbit. The mean and the osculating orbit are a second place where the choice of average decides the answer, and the series that separates them makes the difference a matter of kilometres for a satellite.

Where this goes next

June sunlight at 65°N, against where perihelion sits. Daily-mean insolation at latitude 65 degrees north on the June solstice, at an obliquity of 23.44 degrees, against the longitude of perihelion measured from the March equinox — the angle that precesses right round in about twenty-one thousand years. Three eccentricities are drawn. The swing is ±10.0 per cent at e = 0.05 and ±1.0 per cent at e = 0.005, in proportion to the eccentricity, because the Sun–Earth distance on a fixed date carries e cos of the precession angle and that is first order. Over the same range of eccentricity the annual mean at this latitude moves by 0.12 per cent, because the annual mean carries 1/√(1−e²) and that is second order. The two together are the whole of the precession term in Milankovitch's theory: the eccentricity does almost nothing to how much sunlight the Earth receives and a great deal to when it arrives, and the ice sheets of the northern hemisphere respond to the summer they might melt in rather than to the year's total. It also explains why the precession signal disappears when the orbit is nearly circular: multiply a large angular swing by a vanishing eccentricity and there is nothing left, which is what the innermost curve here is. An average that precession cannot move Sunlight arrives as the inverse square of the distance and time passes as its square, so the two cancel exactly in a year's integral. The longitude of perihelion therefore changes the annual mean insolation at every latitude by precisely nothing — and changes June at 65°N by ten per cent. Equal areas in equal times, at eccentricity 0.65. Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach. Equal areas in equal times, which is angular momentum in disguise Kepler's second law is a statement about the area a radius line sweeps. It looks like an odd thing to have noticed, and it turns out to be a conservation law arriving eighty years early. Two equilibria become four, and three worlds sit near the join. The Cassini equilibria of a spin axis, drawn against the ratio of its own precession rate to the rate at which its orbit plane turns, for an orbit inclination of 1.5 degrees. Each column of dots is the full set of obliquities at which the two precessions keep step at that ratio, found by root-finding rather than by tracing a remembered curve. Below α cos ε/|g| = 1.135 there are two such obliquities and above it there are four, and the figure checks both counts on either side of the join. The three marked bodies are placed by their own measured precession constants: the Earth with the Moon at 2.67, safely on the four-state side; the Earth without it at 0.86; and Mars at 1.06. Two of the three sit within a few tenths of the bifurcation, which is the whole reason their obliquities are not constants: near the join the equilibria are close together, the libration around them is wide, and a body pushed between neighbouring resonances wanders. The Moon's contribution to the Earth's precession constant is what moves the first mark away from that region, and the second mark is the same planet with that contribution removed. This is a two-frequency model of a many-frequency system, and the real chaos comes from the overlap of resonances it does not contain. A tilt that is not a constant The Earth's axis leans by 23.4 degrees, and that lean is what makes the seasons. It is also a dynamical variable with its own equilibria, its own resonances and its own chaos — and on Mars the same variable has swung between nearly zero and sixty degrees without anything having to happen.

About the same objects

Not linked from either essay — found by the objects both name.

The objects this essay names

Each one links to every other essay that touches it.

Caloric half-yearEccentricityIce agesInsolationKepler second lawLongitude of perihelionMilankovitch cyclesObliquityPrecessionSummer energy