The sunniest place is the summer pole

The Earth’s five zones are read off the Sun’s noon altitude against latitude. Noon altitude is the wrong quantity for almost every purpose it is used for.

What a surface answers to is energy, and energy arrives all day. The instantaneous flux on a horizontal surface is the solar constant times the sine of the Sun’s altitude, so the day’s total is that integrated over however long the Sun is above the horizon — and the length of the day varies with latitude in the opposite direction to the altitude.

The two run against each other, and at the solstice the second wins outright.

The integral, and the term that survives at the pole #

The expression has two terms and they do different jobs.

$H_0\sin\varphi\sin\delta$ is the part that grows with day length. At high latitude in summer $H_0$ approaches $\pi$ — the Sun is up for the whole rotation — and $\sin\varphi$ is near one, so the term is large.

$\cos\varphi\cos\delta\sin H_0$ is the part that grows with the Sun’s height. At the equator $\cos\varphi = 1$ and the Sun climbs high, so this term dominates; at the pole $\cos\varphi = 0$ and it vanishes entirely.

Inside the polar day $H_0 = \pi$, $\sin H_0 = 0$, and the whole expression collapses to

$$Q_{\text{pole}} = S_0\left(\frac{\bar a}{r}\right)^2\sin\varphi\sin\delta .$$

which at the pole on the solstice is $S_0(\bar a/r)^2\sin\varepsilon$ — the solar constant times the sine of the obliquity, corrected for distance, and nothing else. The Sun sits at a constant altitude of 23.44° for twenty-four hours and delivers $\sin 23.44° = 0.398$ of the perpendicular flux, continuously.

The equator on the same day gets a higher Sun for twelve hours and nothing for the other twelve. Integrating $\sin$ of altitude over a twelve-hour day with the Sun reaching 66.6° at noon gives $S_0\cos\delta/\pi$, and $1/\pi = 0.318$ is smaller than 0.398.

The whole result is the comparison of $\sin\varepsilon$ with $\cos\varepsilon/\pi$, and it is not close: 0.398 against 0.292 before the distance correction. The factor of $\pi$ from averaging a half-sinusoid over a whole day is what the pole avoids by never setting.

The equinox case, which is a cosine and nearly nothing else #

It is worth checking the expression against a case with an answer known in advance, because the polar result is surprising enough to want a control.

At an equinox the declination is zero, so $\sin\delta = 0$ and the first term disappears entirely. The half-day angle is $\pi/2$ everywhere — twelve hours of day at every latitude — so $\sin H_0 = 1$, and what remains is

$$Q_{\text{equinox}} = \frac{S_0}{\pi}\left(\frac{\bar a}{r}\right)^2\cos\varphi .$$

A pure cosine of latitude, falling to zero at both poles. The figure returns 0.503 of the equatorial value at 60° against a cosine’s 0.500, and the three parts in a thousand are the eccentricity: the two equinoxes are not at the same distance from the Sun, so the March curve and the September one differ slightly and the drawn one is an average of neither.

That agreement is the check. A quadrature that is wrong in its treatment of the day length would not reproduce a cosine at the one declination where the day length is uniform, and a sign error in either term would show as a curve that is not symmetric about the equator.

Why Antarctica is not the warmest place on Earth #

The south pole in December receives 559 W/m² against the north pole’s 524 in June, and it is the coldest place on the planet.

Three things separate the insolation from the temperature and all three are large.

Albedo. Snow and ice reflect 80 to 90 per cent of the sunlight that reaches them. A surface absorbing 15 per cent of 559 W/m² is absorbing less than a tropical ocean absorbing 94 per cent of 385.

Altitude. The South Pole sits on nearly three kilometres of ice. The air above it is thin, radiates to space efficiently, and holds little water vapour, so the greenhouse effect that keeps the rest of the planet forty degrees warmer than its equilibrium temperature is largely absent.

And the ocean. The Arctic is a sea covered in ice and the Antarctic is a continent surrounded by one. Heat transported poleward by the ocean reaches the Arctic and is blocked from the Antarctic by the circumpolar current, which is why the two poles differ by twenty degrees despite the southern one receiving more sunlight.

So the insolation figure is a boundary condition and not a climate, and reading a temperature off it would be reading off the wrong quantity — which is worth saying explicitly, because the plot invites exactly that. The same separation between a forcing and a response runs through every argument about where a planet can hold liquid water, and it is the reason a habitable zone is drawn in terms of flux rather than temperature.

The cancellation Kepler’s law arranges #

That near-equality is exact and it is worth seeing why, because it is one of the few places where a conservation law produces a climate statement.

Equal areas in equal times means the Earth moves fastest at perihelion, so the season containing perihelion is the shortest. The northern winter — December to March — runs 89 days; the northern summer runs 93.6. That inequality is the same one that makes the Sun a bad clock, read as a length of season rather than as a daily offset. The difference is 4.6 days on each half, so the southern hemisphere’s summer is about nine days shorter than the northern one.

Meanwhile the intensity goes the other way: the southern summer receives 6.9 per cent more flux at solstice than the northern one does.

The two effects are not independent. The rate of sweeping area is $\tfrac12 r^2\dot\theta =$ constant, and the flux received is proportional to $1/r^2$, so the energy received per unit of orbital angle is

$$\frac{dE}{d\theta} \propto \frac{1}{r^2}\cdot\frac{dt}{d\theta} \propto \frac{1}{r^2}\cdot r^2 = \text{constant}.$$

The two $r^2$ cancel exactly. Each hemisphere receives precisely the same energy over its summer half-year as the other does, whatever the eccentricity, because the extra intensity and the shorter duration are the same factor.

That is Milankovitch’s first result and it is why the eccentricity alone cannot drive an ice age. What it can do is redistribute energy within a season — a short intense summer melts ice differently from a long mild one — and it is the seasonality rather than the annual total that the ice record responds to.

The polar day is not one day #

There is a detail in the pole’s case that the daily-mean number hides and which matters for anything with a memory shorter than a day.

At the pole during polar summer the Sun does not rise and set; it circles the horizon at a constant altitude, completing a full azimuthal circuit in twenty-four hours. The insolation on a horizontal surface is therefore constant — not averaged to 524 W/m², but actually 524 W/m² at every instant.

Away from the pole but inside the polar circle the Sun circles at a varying altitude, dipping towards the horizon at local midnight and rising at local noon, so the flux oscillates without reaching zero. The amplitude of that oscillation grows towards the polar circle, where the midnight altitude reaches zero for the first time.

So the forcing changes character across the polar circle and not merely in magnitude, from a sinusoid with a long off period to a sinusoid with no off period to a constant. The three regimes behave completely differently for anything storing heat, and the daily mean is the same kind of quantity across all three.

That constancy is also why the pole is the only place on Earth where a solar panel pointed at the horizon and rotated once a day would collect the full 1361 W/m² for twenty-four hours running, which is a larger daily total than anywhere on the equator has ever managed.

The quantity the ice sheets actually answer to #

The daily mean is still not the last word. What an ice sheet responds to is the energy delivered over the melt season, which is the daily insolation integrated over the months when it is above the threshold for melting.

Milankovitch’s own choice was the “caloric half-year”: the 182.6 days of the year with the highest insolation at a given latitude, integrated. The modern choice is usually the June insolation at 65°N, which is where the northern ice sheets grew and retreated, and which is chosen because it is the latitude where a summer’s worth of melting decides whether the previous winter’s snow survives.

The three orbital terms enter that quantity with very different periods and amplitudes. Obliquity varies on 41,000 years and moves high-latitude summer insolation by several per cent; precession varies on about 23,000 and moves it by rather more; eccentricity varies on 100,000 and 400,000 and moves it hardly at all on its own.

Both are attempts to answer the same question: what combination of the orbital parameters does the ice hear? And the answer is that they hear a combination — precession, which decides when in the orbit each hemisphere’s summer falls, multiplied by eccentricity, which decides how much that matters, plus obliquity, which decides how strong the summer is at high latitude.

The eccentricity enters only as a multiplier on the precession term, because of the cancellation above. That is why the 100,000-year eccentricity cycle appears in the ice record with an amplitude far larger than its direct effect on insolation can explain, and why that discrepancy has been an open problem for fifty years.

The same integral, for a different planet #

The expression contains three things that vary from planet to planet — the obliquity, the eccentricity, and the solar constant at that distance — and running it on Mars shows how much of the Earth’s arrangement is contingent.

Mars has an obliquity of 25.2°, almost the Earth’s, and an eccentricity of 0.093, which is five and a half times the Earth’s. The distance ratio between perihelion and aphelion is therefore 1.45 in flux rather than 1.07, and southern summer on Mars occurs near perihelion.

The consequence is a hemispheric asymmetry not of seven per cent but of forty-five. Martian southern summers are short and violent, northern ones long and mild, and the global dust storms that occasionally obscure the whole planet start in the southern summer for precisely that reason. The Martian equation of time is dominated by the orbit rather than by the tilt for the same underlying cause, which is that one number there is five times what it is here.

The cancellation still holds. Each Martian hemisphere receives the same energy over its own summer half-year as the other, exactly as on Earth, because Kepler’s second law does not care about the eccentricity’s size. What differs is the seasonality, and on Mars it differs enough to change the weather rather than the climate record.

A solved orbit, and the atmosphere left out of it #

The solar constant is measured, from space, at 1361 W/m², and it varies by about 0.1 per cent over the eleven-year cycle.

The orbit is measured to a precision far beyond anything this calculation needs: the eccentricity to nine figures, the obliquity to milliarcseconds, the date of perihelion to minutes.

The figure solves Kepler’s equation for the distance rather than approximating the orbit as a circle, and that is not a refinement — the inverse-square factor between perihelion and aphelion is 6.9 per cent and it is the whole of the hemispheric asymmetry. A circular-orbit version of this plot has two identical solstice curves and nothing to say about the ice ages.

What is idealised is the atmosphere, which is absent — the same simplification the twilight calculation has to abandon, and for the same reason: once the Sun is low, the path through the air is the dominant term. The insolation drawn is what arrives at the top of the atmosphere, and about 30 per cent of it is reflected before reaching the surface, with the fraction depending on the path length through the air — which is longest exactly where the Sun is lowest, so the polar advantage at the surface is smaller than the figure shows.

No curve here is a temperature #

No curve here is a temperature. Insolation is the forcing; temperature is what a surface with a heat capacity and a radiative loss does with it, and the two differ in timing as well as in magnitude.

The atmospheric path length is not included. At a solar altitude of 23°, sunlight passes through 2.5 times as much air as at the zenith, so the polar surface receives appreciably less than the top-of-atmosphere figure. The comparison between pole and equator is therefore less favourable to the pole at the ground than it is at the top.

And the daily mean hides the diurnal structure entirely. The equator receives its 385 W/m² as twelve hours near 1000 and twelve hours at zero; the pole receives its 524 as twenty-four hours near 540. Those are very different forcings for anything with a heat capacity, which is what the lag below is about.

The dip at sixty degrees #

There is a feature of the solstice curve that is easy to miss and is the clearest evidence that two terms are competing rather than one dominating.

Between the equator and the pole the June curve does not rise monotonically. It falls from the subtropical maximum, reaches a shallow minimum near 60°N, and then rises again to the polar peak. The minimum is where the day-length term has not yet overtaken the altitude term, and the altitude term is already well past its best.

That is a genuinely unusual shape for a physical quantity plotted against latitude: two maxima and a minimum between them, from an expression with no free parameters. It exists only within a band of obliquities, roughly 15° to 35°, because at low tilt the polar term never wins and at high tilt the subtropical maximum disappears.

The Earth sits inside that band and the minimum lands at about 60°N, which is the latitude of southern Scandinavia and central Canada. It has no climatic consequence worth naming, because the surface’s own thermal response and atmospheric transport smooth over a feature this shallow — but it is a place where a two-term expression visibly has two terms.

A result that has been rediscovered repeatedly #

The polar insolation maximum is not new and is not obscure. It follows from an integral any student can do, it appears in every textbook on climate, and it is reliably surprising.

The reason it surprises is that “the Sun is higher” is a good proxy for “it is warmer” everywhere within ordinary experience, because within ordinary experience the day length varies by a few hours rather than by twenty-four. The proxy fails exactly where day length stops being roughly twelve hours, which is exactly where nobody lives.

It is also worth noting how few quantities in this subject have this property. Most surprising results come from a number being larger or smaller than expected; this one comes from an integral being taken over a different variable than the intuition uses, and the numbers involved are all of order one.

And it fails in the direction that makes the polar regions less strange rather than more. The high latitudes are not places starved of sunlight; they are places that receive a great deal of it over a short season and none at all for the rest, which is a different problem and has different consequences.

Still open: nothing here, and one thing next door #

The insolation calculation itself is settled. Every term is geometry or a measured orbital element, and the answer has not changed in a century.

What is not settled is the step from it to a climate, and the unsolved part of that step is the one named above: why the 100,000-year eccentricity cycle dominates the ice record when its direct effect on insolation is the smallest of the three orbital terms and is cancelled at leading order by Kepler’s second law.

From here, once neither cycle is removed #

The daily mean has removed the rotation and kept the season. Removing neither, and asking what a surface with a heat capacity does with a forcing that varies on both timescales, gives the phenomenon everybody has noticed and nobody computes: the hottest month is not the sunniest one, and the hottest hour is not noon. Both are the same equation, one cycle apart.