Astronomy & Celestial Mechanics
the sky, computed
The observed sky

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

Summer is not when the Earth is closest to the Sun — that happens in January. It is when the Sun climbs higher and stays up longer, and both come from a 23.4° tilt.
12 min read 7 figures The sky as a clockSmall perturbations, large consequences

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.

The Sun's altitude through the day at latitude 52°Solar altitude against the hour of the day, at one latitude, for the solstices and the equinox. Where a curve crosses zero is sunrise or sunset, and the width between the crossings is the length of the day.05101520-40-200204060hour (local solar time)June solstice — 61° at noon, 16.5 h of dayequinox — 38° at noon, 12.0 h of dayDecember solstice — 15° at noon, 7.5 h of daybelow the horizon
Fig. 1 The Sun’s altitude through the day at latitude 52°, at the two solstices and the equinox. Each curve is computed from the declination and the latitude; where it crosses zero is sunrise or sunset, and the width between crossings is the length of the day.

One formula, two effects

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

sin⁡(alt)=sin⁡φsin⁡δ+cos⁡φcos⁡δcos⁡H.\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°+23.4° at the June solstice through zero at the equinoxes to −23.4°-23.4° in December, and it enters twice.

At local noon, cos⁡H=1\cos H = 1 and the expression collapses to alt=90°−φ+δ\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⁡φtan⁡δ\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 Sun's altitude through the day at latitude 0°Solar altitude against the hour of the day, at one latitude, for the solstices and the equinox. Where a curve crosses zero is sunrise or sunset, and the width between the crossings is the length of the day.05101520-40-20020406080100hour (local solar time)June solstice — 67° at noon, 12.0 h of dayequinox — 90° at noon, 12.0 h of dayDecember solstice — 67° at noon, 12.0 h of daybelow the horizon
Fig. 2 The equator. The Sun passes nearly overhead at every season, the day is twelve hours long all year, and the two solstice curves are mirror images. There is no thermal season here at all — the wet and dry seasons that exist have a different cause.
The Sun's altitude through the day at latitude 69°Solar altitude against the hour of the day, at one latitude, for the solstices and the equinox. Where a curve crosses zero is sunrise or sunset, and the width between the crossings is the length of the day.05101520-40-2002040hour (local solar time)June solstice — 44° at noon, 24.0 h of dayequinox — 21° at noon, 12.0 h of dayDecember solstice — -2° at noon, 0.0 h of daybelow the horizon
Fig. 3 Inside the Arctic Circle. In June the curve never touches zero — the Sun does not set — and in December it never rises above it. The polar day and polar night are the same formula with tan⁡φtan⁡δ\tan\varphi\tan\delta pushed past 1, where the equation has no solution.

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⁡φtan⁡δ=1\tan\varphi\tan\delta = 1 at the solstice, which is 90°−23.4°=66.6°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.

An orbit at eccentricity 0.017An orbit of eccentricity 0.017. The primary sits at a focus, offset from the centre by 0.017 of the semi-major axis, and the closest and furthest points differ by a factor of 1.03.empty focusrperiapsisapoapsis
Fig. 4 The Earth’s orbit at its true eccentricity. The variation in distance over a year is real and produces a 6.8% swing in received sunlight — an effect that applies to both hemispheres at once and therefore cannot make seasons.

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.

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 sky from latitude 52°The celestial sphere seen from latitude 52 degrees. The pole stands 52 degrees above the horizon, the celestial equator meets the horizon due east and west, and a star at declination 23.44 degrees traces the drawn circle once a day. Everything below the horizon is drawn faint.celestial poleobserverzenithnorthsouthcelestial equatorthe daily circle of a star at δ = 23.44°the pole sits 52° up, because the observer is at latitude 52°faint arcs are below the horizon
Fig. 5 The June Sun’s daily circle at latitude 52°. Most of the circle is above the horizon, which is the long day, and it reaches high, which is the strong Sun. Both come from the circle sitting well north of the celestial equator.
The sky from latitude 52°The celestial sphere seen from latitude 52 degrees. The pole stands 52 degrees above the horizon, the celestial equator meets the horizon due east and west, and a star at declination -23.44 degrees traces the drawn circle once a day. Everything below the horizon is drawn faint.celestial poleobserverzenithnorthsouthcelestial equatorthe daily circle of a star at δ = -23.44°the pole sits 52° up, because the observer is at latitude 52°faint arcs are below the horizon
Fig. 6 The December Sun at the same place. The same construction with the declination negated: a circle mostly below the horizon and a low maximum. Nothing has changed but one angle.

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.

Phases are a viewing angle, not a shadowA satellite at eight points of its orbit. Exactly half of it is lit at every one of them; what changes is how much of the lit half faces the centre. Nothing is in shadow except at an eclipse.sunlightnewwaxing crescentfirst quarterwaxing gibbousfullwaning gibbouslast quarterwaning crescentas seen from the centrehalf lit, always
Fig. 7 The lunar month. The Moon’s orbit is inclined only 5° to the ecliptic, so it follows almost the Sun’s own annual path — which is why the full moon in winter rides where the summer Sun does.

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.

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 ladder from here

Later rungs: the equation of time and the analemma. The obliquity’s slow variation. Milankovitch cycles and the ice-age record. Insolation integrated over a day, and the surprising fact that the polar summer receives more daily energy than the tropics. 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. And obliquity elsewhere: Uranus at 98°, where the seasons are unrecognisable, and Mars at 25° with a far more eccentric orbit, where the two effects are comparable.

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.

Where this goes next

celestial poleobserverzenithnorthsouthcelestial equatorthe daily circle of a star at δ = 20°the pole sits 52° up, because the observer is at latitude 52°faint arcs are below the horizon The sphere that is not there, and why it is still the right model The stars are at wildly different distances and the celestial sphere is a fiction. It is also the most useful fiction in observational astronomy, because for pointing at things, distance is exactly the information to throw away. empty focusabae = 0.6arperiapsisapoapsis The orbit is an ellipse, and the Sun is not in the middle of it Kepler's first law is usually drawn wrong. The interesting content is not the ellipse — it is the focus, and the fact that one of the two is empty. sunlightnewwaxing crescentfirst quarterwaxing gibbousfullwaning gibbouslast quarterwaning crescentas seen from the centrehalf lit, always Phases are not shadows, and eclipses are Half the Moon is lit at every instant of every month. The phase is which part of the lit half faces the Earth — and confusing that with a shadow is the commonest error in astronomy.