The geometry of the longest day

The apparent motion of the Sun across a year is a deterministic, quasi-periodic signal. Its position is set not by chance but by two fixed geometric facts: the tilt of Earth’s rotation axis relative to the plane of its orbit, and the eccentricity of that orbit. The summer solstice is not an event so much as an extremum of this signal, the instant at which one of its components reaches a turning point. The geometry that locates that extremum also fixes the length of the day at every latitude, explains the origin of the word solstice, and accounts for the discrepancy between the time told by a clock and the time told by a sundial.

Declination: the Sun’s signed height

It helps to picture the sky as the inside of a vast dome, the celestial sphere, with the stars fixed to it and the Sun sliding slowly across it over the year. A point on this dome is located by two angles, much as a point on Earth is located by latitude and longitude. The one that matters here is the declination $\delta$ , the angle of the Sun north (positive) or south (negative) of the celestial equator, which is simply Earth’s equator projected outward onto the sky. Declination is, in effect, the Sun’s latitude. When it is high the Sun climbs higher at noon and lingers longer above the horizon, the everyday signature of summer; when it is low the days are short and the Sun stays low, which is winter. Its rise and fall over the year is the quantity to track.

Earth’s axis does not stand upright in its orbit but leans over by the obliquity of the ecliptic, $\varepsilon \approx 23.44^\circ$ . This single tilt is what produces the seasons. As Earth proceeds along its orbit, the Sun appears to move once around a great circle, the ecliptic (the yearly path the Sun traces against the background stars), inclined by $\varepsilon$ to the celestial equator. Parametrizing the Sun’s progress by its ecliptic longitude $\lambda$ , measured from the vernal equinox, spherical trigonometry gives the declination directly [1]:

\sin\delta = \sin\varepsilon \, \sin\lambda .

At the equinoxes ( $\lambda = 0^\circ$ and $180^\circ$ ) the declination is zero; the Sun lies on the celestial equator. At the June solstice ( $\lambda = 90^\circ$ ) the right-hand side is maximal, so $\delta = +\varepsilon$ , and six months later, at the December solstice, it reaches $\delta = -\varepsilon$ . The declination therefore oscillates between $\pm\varepsilon$ with a one-year period, and the solstices are precisely its extrema.

Schematic of the celestial sphere: the equator and the tilted ecliptic crossing at the equinoxes, with the Sun at the June solstice. — Fig. 1: The celestial sphere viewed obliquely. The Sun travels once a year around the ecliptic (red), tilted by the obliquity ε from the celestial equator (gray); the two cross at the equinoxes, where the declination is zero. The declination δ is the Sun's angular height above the equator, and it reaches its extreme values of +ε at the June solstice and −ε at the December solstice.

Solar declination over the year, a smooth oscillation between plus and minus the obliquity. — Fig. 2: Solar declination δ through the year. It oscillates between +ε and −ε, with the equinoxes at the zero crossings; the June solstice (dashed) is the maximum, where the curve flattens.

Treating $\lambda$ as advancing uniformly in time, an adequate approximation for a nearly circular orbit, makes the declination a smooth, regular rise and fall over the year [2]. The small departures from that smoothness are exactly what the later sections draw on.

The length of the day

Declination is not merely a coordinate; through one more application of spherical geometry it fixes how long the Sun stays above the horizon. For an observer at geographic latitude $\varphi$ , the Sun rises and sets when its altitude is zero, which occurs at the hour angle $H_0$ satisfying [1]

\cos H_0 = -\tan\varphi \, \tan\delta .

The hour angle is just a way of measuring time as an angle: it counts how far the Sun has moved from the overhead meridian as the Earth turns, at the steady rate of $15^\circ$ per hour (a full $360^\circ$ turn in $24$ hours). The Sun stands above the horizon for the arc from $-H_0$ before noon to $+H_0$ after it, so the length of the day in hours is

D = \frac{2 H_0}{15^\circ} = \frac{2}{15^\circ}\arccos\!\big(-\tan\varphi\,\tan\delta\big).

At the equinoxes $\delta = 0$ , giving $\cos H_0 = 0$ , hence $H_0 = 90^\circ$ and $D = 12$ hours at every latitude. Away from the equinoxes the day length depends on the sign of the product $\tan\varphi\,\tan\delta$ : in the northern hemisphere ( $\varphi > 0$ ) a positive declination lengthens the day, and the longest day occurs exactly when $\delta$ is greatest, at the June solstice. For Athens ( $\varphi \approx 38^\circ$ N) the solstice declination $\delta = \varepsilon$ gives $\cos H_0 = -\tan 38^\circ \tan 23.44^\circ \approx -0.339$ , so $H_0 \approx 109.8^\circ$ and $D \approx 14$ h $38$ min.

Length of daylight through the year at the equator, Athens, and 60 degrees north. — Fig. 3: Length of daylight through the year at three latitudes. All curves cross 12 hours at the equinoxes and reach their northern-hemisphere maximum at the June solstice (dashed); the amplitude grows with latitude, and the maximum is visibly flat.

When $|\tan\varphi\,\tan\delta| > 1$ the equation for $H_0$ has no solution: the argument of the arccosine leaves $[-1, 1]$ , and the Sun either never sets or never rises. This first happens at the polar circles, $\varphi = 90^\circ - \varepsilon \approx 66.6^\circ$ , on the solstice, the geometric definition of the midnight Sun.

Why the Sun stands still

The word solstice derives from the Latin sol (sun) and sistere (to stand still). The name records an observation: near the solstice the Sun’s daily extreme height changes imperceptibly from one day to the next, and the length of the day is nearly constant. This is a direct consequence of $\delta$ being at an extremum, and it is worth making quantitative, because an extremum is a point of vanishing first derivative.

Differentiating $\sin\delta = \sin\varepsilon\,\sin\lambda$ with respect to $\lambda$ (in radians, as differentiation of a sine requires) gives

\frac{d\delta}{d\lambda} = \frac{\sin\varepsilon\,\cos\lambda}{\cos\delta}.

At the solstice $\lambda = 90^\circ$ , so $\cos\lambda = 0$ and $d\delta/d\lambda = 0$ : the declination is stationary. To leading order the approach to the extremum is quadratic. Writing $\lambda = 90^\circ + u$ and expanding, $\sin\lambda = \cos u \approx 1 - u^2/2$ , so the declination falls below its peak by an amount of order $u^2$ . A week on either side of the solstice corresponds to $u \approx 6.9^\circ$ , about $0.12$ radians, of ecliptic longitude, which (using the radian value in the expansion) lowers the declination from $23.44^\circ$ to about $23.26^\circ$ , a change of under a fifth of a degree. Propagated through the day-length formula at the latitude of Athens, the longest day and the day a week before or after it differ by roughly one minute. The Sun does not literally stand still, but the rate of change of its declination passes through zero, and the length of the day is, to first order, flat across the surrounding weeks. The flat tops of the curves in Fig. 3 are this stationarity made visible.

The equation of time: when clocks and sundials disagree

The signal analyzed so far has been the declination, the Sun’s north-south motion. Its east-west motion carries a second, subtler structure. Mean solar time, the time kept by a clock, advances as though the Sun crossed the meridian at a perfectly uniform rate. The true Sun does not. The difference between apparent solar time (a sundial) and mean solar time is the equation of time, and over a year it ranges from about $-14$ to $+16$ minutes.

It splits into two contributions of comparable size but different period [3]. The first is the obliquity. Even if Earth’s orbit were a perfect circle, traversed at constant angular speed along the ecliptic, projecting that uniform motion onto the celestial equator (where time is measured) would not be uniform, because the ecliptic is tilted. This projection introduces a component with a period of six months. The second is the eccentricity: Earth’s orbit is an ellipse, and by Kepler’s second law the planet moves faster near perihelion (early January) than near aphelion, so the Sun’s apparent motion speeds up and slows down once per year. This component has a period of one year.

Written out, with $N$ the day of the year, the two add to a compact approximation:

E(t) \approx \underbrace{A_2 \sin 2\beta}_{\text{obliquity}} \; + \; \underbrace{A_1 \sin(\beta + \phi_1)}_{\text{eccentricity}}, \qquad \beta = \frac{2\pi (N - 81)}{365},

where the obliquity term has amplitude $A_2 \approx 9.9$ minutes and the eccentricity term $A_1 \approx 7.7$ minutes. Each on its own is a smooth, symmetric wobble; because the two repeat at different rates they drift in and out of step, reinforcing here and partly cancelling there, and that is what gives the full curve its lopsided shape, dipping and peaking by different amounts.

The equation of time decomposed into a semiannual obliquity component, an annual eccentricity component, and their sum. — Fig. 4: The equation of time (solid) as the sum of a twice-yearly obliquity term (dashed) and a once-yearly eccentricity term (dotted). The two are of comparable size; adding them produces the lopsided annual curve.

The analemma

The two motions can be displayed together. Photographing the Sun from a fixed location at the same clock time on many days through the year, or simply plotting declination against the equation of time, traces a closed figure-eight, the analemma [4]. Its height is the Sun’s north-south swing, from $+\varepsilon$ at the top to $-\varepsilon$ at the bottom; its width is the half-hour spread of the equation of time. The vertical motion repeats just once a year, while the sideways motion repeats partly twice a year, and a slow up-and-down combined with a faster side-to-side is exactly what draws a figure-eight. The two loops come out unequal because the once- and twice-yearly contributions to the sideways motion are differently timed, the same lopsidedness already seen in Fig. 4.

The solstice occupies the top of the figure, the single point where the upper loop turns over. That turning point is the same stationarity established earlier: it is where $d\delta/d\lambda$ vanishes, and it is the longest day.

A signal like any other

The longest day is the visible consequence of a small number of geometric constants: an axial tilt of $23.44^\circ$ , an orbital eccentricity of a few percent, and the spherical trigonometry that connects them to the horizon. Declination, day length, the equation of time, and the analemma are not separate phenomena but different projections of one deterministic signal, and the methods that make sense of them, finding a peak by asking where a rate of change vanishes, or splitting a complicated curve into a few simple repeating pieces, are the same ones used to read any other. A radio antenna, a beating heart, and the turning sky give off signals of utterly different origin, yet the same handful of methods reads them all; the mathematics does not care where the pattern comes from.

References

[1] W. M. Smart, Textbook on Spherical Astronomy, 6th ed. Cambridge University Press, 1977.

[2] P. I. Cooper, “The absorption of radiation in solar stills,” Solar Energy, vol. 12, no. 3, pp. 333–346, 1969.

[3] M. Müller, “Equation of time: problem in astronomy,” Acta Physica Polonica A, vol. 88, supplement, pp. S-49–S-66, 1995.

[4] J. Meeus, Astronomical Algorithms, 2nd ed. Willmann-Bell, 1998.