The Sun's path, and the tilt that makes the seasons

The Earth is closest to the Sun on about the 3rd of January and furthest on about the 4th of July. Northern hemisphere winter happens at the close approach, and northern summer at the far one.

That single pair of facts disposes of the intuitive explanation of the seasons. Distance is not the cause; it is a small effect running the wrong way. The cause is a 23.4° tilt of the rotation axis, and it produces summer by two mechanisms at once — a higher Sun and a longer day — which multiply together.

One formula, two effects #

The Sun’s altitude at any moment follows from three numbers — the observer’s latitude $\varphi$, the Sun’s declination $\delta$, and the hour angle $H$, which is just the time expressed as an angle:

$$\sin(\text{alt}) = \sin\varphi\sin\delta + \cos\varphi\cos\delta\cos H.$$

Everything about the seasons is in that expression. The declination is what changes over the year, running from $+23.4°$ at the June solstice through zero at the equinoxes to $-23.4°$ in December, and it enters twice.

At local noon, $\cos H = 1$ and the expression collapses to $\text{alt} = 90° - \varphi + \delta$. At 52° north that is 61.4° in June and 14.6° in December — a factor of nearly four in the sine, which is the geometric part of the seasonal difference.

Setting the altitude to zero gives the length of the day, through $\cos H = -\tan\varphi\tan\delta$. At 52° that gives about 16 hours 20 minutes in June and 7 hours 40 minutes in December.

Both effects push the same way, and the total energy received compounds them. A higher Sun spreads a given beam over less ground — the cosine of the angle from vertical — and a longer day delivers it for more hours. The daily total at 52° north is about eight times larger in June than in December, from a tilt of 23.4°.

The three latitudes are the same computation, and the trend across them is the entire climate zonation of the planet. At the equator, no seasonal variation in either quantity. At mid-latitude, both vary substantially. Inside the polar circles the day-length equation stops having a solution, and the answer is 24 hours or zero.

The Arctic Circle is defined by exactly that failure: the latitude at which $\tan\varphi\tan\delta = 1$ at the solstice, which is $90° - 23.4° = 66.6°$. The tropics are defined by the other extreme, the latitudes at which the Sun can reach the zenith. All four named circles on a globe are consequences of one angle.

Why distance loses #

Earth’s orbit has an eccentricity of 0.0167, so the distance to the Sun varies by 3.3% over the year, and the received flux — going as the inverse square — by about 6.8%.

That is not nothing. It is, however, small against a factor of eight, and it acts globally rather than seasonally: in January the whole planet receives 6.8% more than in July, north and south alike. The seasons are opposite in the two hemispheres, so any global effect cannot be their cause. What it does do is make the two hemispheres’ seasons slightly unequal. Southern summer coincides with the close approach, so it is mildly more intense and mildly shorter — the second law again, since the Earth moves faster near perihelion. Southern summer is about five days shorter than northern summer. The much larger ocean fraction in the south more than compensates thermally, but the astronomical asymmetry is there and measurable.

On longer timescales it matters a great deal, because the eccentricity is not fixed. The eccentricity varies over about 100,000 years, the tilt oscillates between 22.1° and 24.5° over 41,000 years, and the date of perihelion precesses round the year over about 23,000. Those three cycles modulate the sunlight reaching high northern latitudes in summer, and their combined signature appears in the ice-age record. Milankovitch worked out the theory in the 1920s from nothing but celestial mechanics; the confirmation came in the 1970s from ocean sediment cores, which showed exactly those periods.

The place that receives the most sunlight in a day #

Two effects multiplying together produces one result that almost nobody predicts correctly, and it is worth computing rather than asserting.

Ask where on Earth the most solar energy arrives in a single day. The answer is not the equator and not the tropics. It is the pole — the summer one — and by a clear margin.

The arithmetic is the same formula integrated over 24 hours instead of evaluated at noon. At the North Pole on the June solstice the Sun sits at a constant altitude of 23.4° and never sets, so the ground receives $\sin 23.4° = 0.397$ of the vertical flux for the whole day. At the equator on the same date the Sun rises, passes within 23.4° of the zenith and sets, and the daily average of $\sin(\text{alt})$ over a twelve-hour day, spread across twenty-four, works out to about 0.29. The pole wins by roughly a third.

The reason a long day beats a high Sun is that the two enter differently. Altitude enters through a cosine of the angle from vertical, which is a factor between 0 and 1 and cannot exceed 1. Day length enters as a straight multiplier and at the pole it doubles. A factor of two on the hours beats a factor of 0.4 against 0.9 on the intensity, and the crossover happens around 65° latitude.

The obvious objection is that the Arctic is not warm, and it is the right objection. Insolation is not temperature. Most of that light falls on snow and ice, which reflect 80% of it straight back; the surface spent the previous six months in darkness and has an enormous cold reservoir beneath it; and the whole calculation ignores the atmosphere, which at a solar altitude of 23° is being traversed at two and a half times the vertical path length. The astronomy is unambiguous and it is not what determines the climate. Both statements are worth holding at once, because the figure above genuinely cannot show the second.

Where the sphere puts it #

The Sun’s path is one particular curve on the celestial sphere, and seeing it there makes the seasons a statement about geometry rather than about weather.

The two figures differ in exactly one number, and between them they contain the whole of winter and summer. The Sun’s daily circle is a circle of constant declination — the same object as any star’s — and the seasons are the slow migration of that circle north and south over the year, at a rate the tilt fixes.

Note what is not different between the two: the observer, the latitude, the tilt, the size of the Earth, and the distance to the Sun. The seasons are a single-parameter phenomenon, and the parameter is the declination.

Two bodies on the same tilted circle #

The Sun’s path is one member of a family, and the Moon runs along nearly the same circle at thirteen times the speed. That near-coincidence produces a rule worth knowing: the full moon is opposite the Sun, so its declination is roughly the negative of the Sun’s. The December full moon climbs as high as the June Sun, and the June full moon skims the horizon. Anyone who has noticed that winter moonlight seems unusually bright has noticed a consequence of the same 23.4° tilt that makes the seasons.

The 5° by which the Moon’s orbit is not aligned with the ecliptic is what makes eclipses rare rather than monthly, and it is invisible in every coplanar figure — including this one and the celestial sphere it sits on.

Why the hottest day is not the longest one #

The geometry gives the insolation and not the temperature, and the two are separated by a delay that is worth accounting for because it is larger than most people expect.

The ground and the ocean absorb heat while more arrives than leaves, and their temperature keeps rising until the two balance. The peak insolation is at the solstice; the peak temperature is when the outgoing radiation has caught up, which is several weeks later.

The size of the lag depends entirely on what is being heated. Land has a low heat capacity and heats a thin layer, so a continental interior lags by two or three weeks. The ocean mixes heat through tens of metres and lags by six to eight, which is why a maritime climate’s hottest month is August or even September and its coldest is February.

The same delay appears in the daily cycle at one three-hundred-and-sixty-fifth of the scale: the Sun is highest at noon and the air is warmest in mid-afternoon.

So the seasons are the geometry of this essay convolved with a thermal response, and the response is a property of the surface rather than of the orbit. That is why the same insolation curve produces such different climates at the same latitude on the two sides of a continent, and why the shape of the annual temperature curve carries information about what is underneath it.

It also means that a record of temperature is a poor instrument for recovering the orbit, and a record of the dates of the extremes is a good one, since the lag is nearly constant from year to year while the amplitude is not.

And the southern-hemisphere case, since the whole argument is about the tilt rather than about the distance and the southern seasons are the test of that.

What the picture cannot show #

Three things this figure and every version of it leaves out, all of them substantial.

Thermal lag. The hottest weeks are not the solstice but six to eight weeks after it, because oceans and land take time to warm. Nothing astronomical happens then at all — the geometry peaked weeks earlier. The insolation curve peaks in June and the temperature curve peaks in late July, and the offset is a property of heat capacity, not of astronomy.

Atmosphere. Sunlight passing obliquely traverses more air and is more strongly scattered and absorbed, so the low winter Sun is weakened beyond the geometric cosine factor. Refraction also lifts the Sun by about half a degree near the horizon, lengthening every day by a few minutes at both ends and by considerably more at high latitude.

Geography. The southern hemisphere is mostly ocean and the northern mostly land, and land changes temperature far faster — a fact with no astronomy in it at all, and larger than the orbital asymmetry it partly cancels. The astronomical asymmetry between the hemispheres is small; the thermal asymmetry is large and has the opposite sign.

The equation of time. Solar noon is not at 12:00, and it wanders by up to sixteen minutes over the year — partly because the Earth’s orbital speed varies with distance, partly because the Sun’s motion along the tilted ecliptic projects unevenly onto the equator. The figures here use solar time, in which noon is by definition the Sun’s highest point. A clock does not agree, and the analemma — the figure-of-eight traced by the Sun photographed at the same clock time through the year — is the shape of the disagreement.

What was actually measured #

The tilt is measured by taking the difference between the Sun’s noon altitude at the two solstices and halving it — an angle on the celestial sphere, obtained with no instrument beyond a shadow. That is a measurement anyone can make with a vertical stick and patience, and it was made accurately in the third century BC.

The same stick gives the latitude, from the noon altitude at an equinox. And the two together gave Eratosthenes the size of the Earth: knowing that the Sun was overhead at Syene at the solstice, and measuring that it was 7.2° from vertical at Alexandria on the same day, the distance between the two cities gives the circumference from a single ratio. His answer was within a few percent, from one angle and one caravan route.

There is an important negative measurement too. If the seasons were caused by distance, the Sun’s angular diameter would be largest in summer. It is not — it is largest in January, by 3.4%, and that is measurable with a pinhole and a tape measure. The observation was available for two thousand years before the explanation was.

The tilt is not guaranteed #

Everything above treats 23.4° as a fact about the Earth. It is better described as the Earth’s current value of a quantity that other planets set very differently, and that on this planet is unusually well behaved for a reason that has nothing to do with the Earth.

The range across the solar system is extreme. Mercury is at 0.03°, effectively upright, and has no seasons of any kind. Mars is at 25.2° — close to Earth’s — but with an orbital eccentricity of 0.093, five times Earth’s, so on Mars the distance effect is not negligible against the tilt effect and southern summers are considerably fiercer than northern ones. Uranus is at 97.8°, tipped past the horizontal, so its poles alternately point almost directly at the Sun; each pole gets forty-two years of continuous daylight followed by forty-two of darkness, and the concept of a tropic breaks down completely. Venus is at 177°, which is to say upside down and rotating backwards.

The Earth’s own value oscillates, between about 22.1° and 24.5° with a period of 41,000 years, and that small oscillation is one of the three Milankovitch terms. What is surprising is how small the oscillation is, and why.

The obliquity of a spinning planet is perturbed by the torques of everything else in the system, and those torques produce a precession whose rate depends on the planet’s own oblateness and spin. When the precession rate happens to land near one of the many frequencies present in the planetary orbits, the obliquity can wander chaotically over tens of degrees. Numerical integrations by Jacques Laskar in the 1990s showed that Mars is in exactly that condition: its obliquity has ranged over roughly 0° to 60° in the past ten million years, which is enough to move its polar ice to the equator and back.

The Earth is not, and the reason is the Moon. Its torque on the equatorial bulge dominates the Sun’s and speeds the axial precession up to one turn in 26,000 years, which carries the Earth’s obliquity clear of the resonances that would otherwise capture it. Remove the Moon from the calculation and the Earth’s tilt becomes chaotic too, with excursions plausibly reaching 85°. A stable seasonal cycle turns out to be a consequence of having an unusually large satellite — the same satellite whose orbit produces the eclipses and the phases, doing a job nobody would have thought to assign to it.

One correction to the framing is worth keeping. The tilt does not cause the seasons on its own; it causes them because the Earth’s axis keeps pointing the same way as the planet goes round. An axis that stayed tilted toward the Sun all year would give a permanent summer in one hemisphere and no seasonal cycle at all. What produces the cycle is the constancy of the axis’s direction in space combined with the orbital motion, and it is that constancy which precession slowly undoes.

The same construction at three more latitudes covers the cases the essay’s own argument distinguishes, and each of them makes a different one of the two effects visible.

The ladder from here #

Later rungs: the equation of time and the analemma. Milankovitch cycles and the ice-age record in detail. The tropics and polar circles as consequences of one angle. Sundials, and the gnomon’s alignment. The Antikythera mechanism. Solar declination through the year, and the sinusoid it very nearly is. Insolation integrated over a season rather than a day, which is what the ice ages actually respond to. Precession of the equinoxes, which moves the solstices around the orbit. And the obliquity of tidally locked planets, where the whole framework has to be rebuilt.

The seasons are the most familiar astronomical phenomenon there is, and surveys consistently find that most people, including most graduates, explain them by distance. The correct explanation requires only a tilted stick and the willingness to check which month the Sun looks biggest.