The Sun sets before it sets

Every position in the sky quoted by an observer on the ground is wrong, and wrong by a known amount that depends only on how high the object is. The air bends light, the bend is upward, and near the horizon it is larger than the Sun.

Why the lift, and why it grows so fast #

Air is denser at the bottom. Light travelling through a medium whose refractive index increases downward is bent downward — toward the denser side — so a ray arriving from a star curves gently as it comes down, and the observer, extending the last piece of it in a straight line, places the star higher than it is.

At the zenith the ray goes straight down the density gradient and there is nothing to bend. At the horizon it runs almost parallel to the layers, crossing a great many of them at a grazing angle, and the accumulated bending is at its largest. Between the two, the refraction goes roughly as the tangent of the zenith angle, which is why the curve is nearly flat over most of the sky and then turns sharply upward in the last few degrees.

The number at the horizon, 34.5 arcminutes, has a consequence that is easy to state and hard to believe.

The flattening is not subtle — the disc is 84 per cent as tall as it is wide, and anyone who has photographed a sunset has recorded it. It is a differential effect: the refraction curve is steep enough at the horizon that the top and bottom of a half-degree object are lifted by measurably different amounts.

What a sunrise time is a statement about #

Because the Sun is a disc rather than a point, and because it is refracted, the convention adopted by every almanac is that sunrise and sunset occur when the upper limb is at apparent altitude zero. In true geometric terms that puts the Sun’s centre 50 arcminutes below the horizon — 34 of refraction plus 16 of semidiameter.

Converting that into a time requires knowing how fast the Sun’s altitude changes, and that depends on latitude, because the Sun rises vertically at the equator and obliquely everywhere else.

So the day is longer than geometry allows, at every latitude and on every date, by between about seven minutes at the equator and considerably more further north. On the equinox, when a naive calculation gives exactly twelve hours of daylight everywhere on Earth, the observed day is about twelve hours and seven minutes at the equator and twelve hours and sixteen minutes at 50° north. The equinox is not the day on which day and night are equal. That day — the equilux — falls several days earlier in spring and later in autumn, and how many days depends on latitude.

The polar case, where the correction is not a correction #

Push the latitude high enough and the shallow crossing angle stops being a modest amplification and becomes the dominant term. The extreme case is a genuine historical observation. In January 1597 Gerrit de Veer, wintering on Novaya Zemlya at 76° north with Willem Barentsz’s expedition, recorded the Sun’s return about two weeks before it was due. The report was disbelieved for three centuries. It is now understood as a duct: a strong temperature inversion over the ice creates a layer whose gradient bends light along the curvature of the Earth itself, and the light can then travel hundreds of kilometres before escaping upward. The Novaya Zemlya effect produces apparent refractions of several degrees rather than half of one.

That is the honest end of this subject and it belongs at the front rather than as a footnote. Refraction at the horizon is not a constant of nature. It is a property of the temperature profile of the lowest few hundred metres of air along the sight line, and it is the least reliable quantity in positional astronomy.

What is actually measured, and what is assumed #

Every refraction table, including the one drawn here, is computed from a standard atmosphere: a specified surface pressure and temperature, and a specified way the temperature falls with height. Bennett’s formula is a fit to such a model, agreeing with a full ray trace to about 0.07 arcminutes across the range.

The measurements it is checked against are real: differences between the observed and computed positions of stars at low altitude, accumulated over centuries of meridian-circle work. What those measurements establish is that the standard-atmosphere value is a good average. They also establish the spread. Observed horizon refractions range from about 28 arcminutes to over 40 in ordinary conditions, a variation of ±20 per cent, and the spread grows with the strength of any inversion. At 5° altitude the scatter is a few tenths of an arcminute; at the zenith it is negligible.

The practical consequence is a rule every observatory works to: positional measurements below about 20° altitude are not to be trusted at the arcsecond level, not because the correction is unknown but because it is not the same from hour to hour. Astrometric catalogues are built from observations near the meridian for exactly this reason, and the refraction correction is applied with a term for the measured air temperature and pressure at the telescope.

The Moon, where two displacements of similar size disagree #

The Sun is far enough away that its direction is the same from anywhere on Earth. The Moon is not, and near the horizon the two effects that move it are comparable and opposite.

Refraction lifts it by 34 arcminutes. Parallax lowers it by 57. An observer sees the Moon from the Earth’s surface rather than from its centre, and at the horizon the whole Earth radius lies across the line of sight, so the Moon appears displaced downward by its own horizontal parallax — which is what the parallax triangle measures, applied at the shortest baseline and the nearest object in the sky. Net, the rising Moon is about 23 arcminutes lower than the geometry says, where the rising Sun is 34 higher.

That sign difference is not a curiosity. It is why moonrise tables and sunrise tables use different conventions, why the predicted times of lunar eclipses near the horizon carry a larger uncertainty than those at altitude, and why the saros — whose repeats land a third of the way round the world and therefore at a different altitude each time — needs both corrections applied afresh at every apparition rather than carried across.

It also settles a very old question about the Moon illusion. The Moon low on the horizon looks enormous, and refraction is the explanation most often reached for. Refraction makes it smaller, by flattening it exactly as it flattens the Sun; parallax makes it smaller again, by up to 1.7 per cent, because the observer at moonrise is one Earth radius further from it than the observer with it overhead. The horizon Moon is measurably the smallest Moon of the night, and it looks the largest. Whatever the illusion is, it is not in the light.

The correction applied to everything else #

Refraction is the first correction in the standard chain that turns a measured direction into a catalogue position, and it is the only one in that chain that depends on the weather.

Where a star is depends on which frame is being asked, and the conversion from what a telescope points at to what a catalogue records runs: refraction, then aberration, then parallax, then precession and nutation. Every one of the others is computed from a model whose inputs are known exactly — the Earth’s velocity, its axis, the date. Refraction alone requires a thermometer and a barometer at the telescope, and gives an answer whose residual scatter no amount of care removes.

That is why the observational infrastructure of positional astronomy looks the way it does. Meridian circles observed stars at culmination, where the altitude is highest and the correction smallest; modern astrometry at the microarcsecond level is done from space, where there is no atmosphere to model; and the ground-based catalogues that preceded Hipparcos carry systematic errors with a signature that maps onto zenith distance.

Cassini published the first useful refraction table in 1662, and the form of the entries has not changed since — a correction in arcminutes against an apparent altitude, computed for a stated temperature and pressure. Three and a half centuries later the almanacs still print one, still with the same caveat attached, and it is still the number that decides what time the Sun comes up.

What the same air does to brightness #

The other consequence of a long path through the atmosphere is that a good deal of the light does not arrive.

At a typical good site the atmosphere removes about 0.2 magnitudes per airmass in the visual band. At the zenith that is a fifth of a magnitude; at the horizon, thirty-eight airmasses take away roughly ten. The wavelength dependence also produces the one atmospheric phenomenon that is genuinely about refraction rather than scattering. Blue light is refracted slightly more than red, so the atmosphere disperses the Sun’s image into a vertical spectrum some 10 arcseconds tall — a red disc at the bottom, a blue one at the top, overlapping almost completely. At the last instant of sunset the red disc has already gone and only the top edge remains, and its colour is what survives the scattering: the green flash. It is a chromatic aberration produced by a lens two hundred kilometres across.

The correction, used as an instrument #

A systematic that depends on the medium is also a measurement of the medium, and refraction has been turned into one of the most productive remote-sensing techniques there is.

The astronomical version is a stellar occultation. When a planet passes in front of a star, the star does not simply vanish: its light is refracted by the planet’s atmosphere on the way past, so it dims gradually over seconds to minutes as the ray path bends. The shape of that dimming is a direct record of how the refractivity — and therefore the density — varies with height at the planet’s limb, and inverting it gives a temperature and pressure profile over several scale heights.

That is how the atmospheres of Titan, Triton and Pluto were first characterised, and how Uranus’s rings were discovered in 1977 — by unexpected brief dips either side of the main occultation, which were not atmosphere at all.

The terrestrial version is the same physics with the roles exchanged. A satellite in low orbit receives a navigation signal from a satellite on the far side of the Earth, whose radio ray passes through the atmosphere’s limb and is bent by it; the bending shows up as a precisely measurable extra delay. Inverting a sequence of those as the geometry sweeps through gives a temperature profile from the surface to the stratosphere.

The technique delivers thousands of soundings a day, globally, over oceans where no instrument stands, and it is self-calibrating in a way a radiometer is not — what is measured is a delay against an atomic clock, so there is no drifting sensitivity to correct. It is now among the more valuable single inputs to numerical weather prediction, and it is the atmospheric correction of this essay run backwards.

When the gradient reverses #

The standard atmosphere gets denser downwards, which is what bends a ray towards the ground and lifts every image upwards. Reverse the gradient locally and the sign of the effect reverses with it.

Over ground much hotter than the air above it — a desert, a road in summer — the lowest layer is less dense than the layer above, so a ray from just above the horizon is bent upwards into the eye and arrives from below the true direction. That is the inferior mirage: an inverted image beneath the object, which the eye interprets as a reflection in water.

Over a cold surface the opposite happens. Air over ice or cold sea is colder and denser than the air above it, and the gradient is steepened rather than reversed, so rays bend downward more sharply than usual and objects beyond the geometric horizon become visible — looming, and in the extreme a superior mirage that stacks distorted images above the object.

The extreme case has a name and a date. A Dutch expedition wintering at Novaya Zemlya in 1597 recorded the Sun returning a full two weeks before it was due, which requires a refraction of several degrees against the half-degree of this essay’s tables. It was disbelieved for centuries and is now understood as a strong temperature inversion over the ice acting as a duct — a channel in which the ray is trapped and guided round the curvature of the Earth.

Nothing about the physics differs from the sunset in the title, which is the point worth taking. The refraction correction is thirty-four arcminutes because the atmosphere usually has a particular vertical structure, and where it does not, the same equations give an answer that sounds impossible.

Why the radio version measures water and the optical one does not. The two occultation techniques above look like the same experiment at different wavelengths, and the physics differs in one term that matters. Optical refractivity comes almost entirely from the polarisability of dry air; water vapour contributes little, because its permanent dipole cannot follow an optical field. At radio frequencies the dipole follows easily, and water vapour contributes a refractivity several times larger per molecule than dry air does. So a radio ray bent through the atmosphere carries a humidity signal that an optical ray does not — which is why a GNSS occultation returns temperature and water vapour together and has to separate them, and why the same measurement made in the visible would return a cleaner thermometer and no hygrometer at all. The separation is done by taking the two unknowns from different altitudes: above about ten kilometres the air is too cold to hold water, so the refractivity there is dry and gives a temperature outright, and the profile is then integrated downward with that as a boundary condition.

What the picture cannot show #

The time it is not correcting. Nothing here touches the other few minutes that separate clock time from solar time; that is the equation of time, which is up to a quarter of an hour and is about the Earth’s orbit rather than its air. Both corrections are applied to every published sunrise, they are of comparable size at some times of year, and they have nothing to do with each other.

A single number for the horizon. The 34.5 arcminutes used throughout is a convention with a ±20 per cent spread behind it, and every figure here inherits that. A sunset timed to the second against these tables would disagree with the sky by tens of seconds on most days.

The path. The drawings show a lift, not a curve. What actually happens is that the ray follows a shallow arc of a few hundred kilometres, and the “apparent altitude” is the tangent to that arc at the observer’s eye. Nothing here draws the ray, because at true scale the arc’s departure from a straight line is invisible.

The sea horizon is not the same horizon twice. Refraction over water depends on the difference between the water temperature and the air temperature immediately above it, which changes through the day, so the apparent distance to the horizon and the exact moment of sunset vary from one evening to the next at the same place under a clear sky. Navigators have known this for as long as they have taken sextant altitudes of the setting Sun, and it is why the correction tables in a nautical almanac carry a temperature argument that the astronomical ones often omit.

And the horizon itself. Everything above assumes the observer is at sea level looking at a true horizon. A viewpoint 100 metres up sees a horizon dipped by about 17 arcminutes, which is half the refraction again, and shifts every time in this essay. The astronomical horizon is a construction, and the visible one is a different thing that depends on where the observer is standing. The extreme latitude makes the size of the correction against the size of the day explicit.

That is the case in which a correction becomes a phenomenon. At the equator the Sun crosses the horizon at about a quarter of a degree per minute and the refraction advances sunrise by two minutes and a half; at the pole it crosses at a few degrees per year, and refraction advances the return of the Sun by roughly two days. The same number of arcminutes, applied to a different rate, produces effects three orders of magnitude apart in duration.

The polar case is also the one where the number is least reliable. Refraction at the horizon depends on the temperature gradient in the lowest few hundred metres of air, and over a polar ice sheet in winter that gradient is a strong inversion — cold dense air at the surface under warmer air above — which increases the refraction well beyond the standard value. Observers have recorded the Sun returning several days early, and one such observation, from Novaya Zemlya in 1597, has a name.

That is the honest end of the subject. A quantity whose standard value is quoted to three figures in every almanac has a real-world scatter at the horizon of tens of per cent, and no observer at high latitude uses the tabulated number for anything that matters. What is used instead is a measurement of the local temperature profile, or a rule of thumb calibrated on that station’s own history, or an acceptance that the times in the almanac are indicative. The standard refraction is a global average of a quantity that is not globally uniform, and its accuracy is best exactly where it matters least.

Where the ladder goes next #

Two rungs. The first is differential refraction across a telescope’s field, which stretches an image toward the zenith and is why an atmospheric dispersion corrector is standard equipment on any instrument working below 60°. The second is the same physics at a much smaller scale: refraction by turbulent cells rather than by smooth layers, which blurs and moves a stellar image many times a second and sets the resolution of every ground-based telescope ever built — and which the same standard atmosphere has nothing to say about.