The maths behind the Sunset Visualiser

All the physics lives in js/physics/ as small pure functions with no dependencies. Nothing in js/render/ or js/main.js computes a physical quantity; those files only draw numbers they are handed. The verification suite checks the physics against published values, closed-form identities, and independent derivations of the same quantity that would disagree if either were wrong.

1Conventions

• Distances are in km, times in hours, and angles in degrees unless stated.
• Time is apparent solar time: the Sun is due south or north at 12:00. The hour angle is H = 15° × (t − 12).
• Earth is a sphere of radius equal to the equatorial radius. In the default scenario the subsolar point runs along the equator, where that radius is exact. Oblateness is covered in §10.
• Default scenario: an observer on the equator (0°, 0°), eye height 1.7 m, on 23 September 2026, the day of the September equinox. The window runs from three hours before geometric sunset (15:00) to sunset (18:00), and the timeline carries on far enough for the disc to go: 5 minutes at the equator, longer nearer the poles where the Sun slides down at a shallower angle.
• Place and date are both adjustable, up to 80° of latitude. Combinations with no sunset at all are refused rather than half-drawn. Every figure below is for the default equatorial equinox case unless it says otherwise; latitude and date change the numbers but not the conclusion (§6.8).

2The Sun’s angular diameter

A sphere of radius r seen from distance d to its centre subtends

Note that this is arcsin, not arctan. The sight lines graze the limb, so they are tangent to the sphere and do not pass through its centre plane. The test suite pins this down with a sphere at d = 2r, which must subtend exactly 60°.

At 1 AU: θ = 0.53291° = 31.97′.

3Where the Sun is

This uses the U.S. Naval Observatory / Astronomical Almanac low-precision formulae, accurate to about 0.01° in ecliptic longitude between 1950 and 2050. With n = days since J2000.0 (2000 January 1, 12:00 TT):

The equinox lands 12 minutes off, which is within the formula’s stated accuracy.

4Where the Sun appears from the ground

4.1 Altitude

For an observer on the equator (φ = 0) at an equinox (δ = 0), this reduces to a = 90° − H. The Sun is at exactly 45° at 15:00 and 0° at 18:00, descending at exactly 15° per hour. That is why the default scenario uses this case: the arithmetic is checkable by hand.

altitudeTopocentricDeg then subtracts diurnal parallax, at most 8.79″ for the Sun. That is negligible, but it is included.

4.2 The central angle

The subsolar point is where the Sun is directly overhead. The angle at Earth’s centre between the observer and that point is

The great-circle ground distance to it is Rψ (with ψ in radians). At 15:00 in the default scenario that is 5,009.4 km. This is a measurable distance on the real Earth, and it matters in §6.

4.3 Sunset

With h₀ = 0 (geometric sunset, the Sun’s centre on the horizon) this gives H₀ = 90°, so sunset is at 18:00. The function returns null when |cos H₀| > 1, which is polar day or night; the app then refuses to build a scenario rather than inventing one.

4.4 The sea horizon

For an eye at height h above a spherical sea:

At 1.7 m the dip is 2.51′ and the horizon is 4.66 km away.

5Globe model: why the Sun doesn’t shrink

5.1 Observer–Sun distance

The observer stands at distance R from Earth’s centre and the Sun is at distance D. The angle between those directions, at Earth’s centre, is ψ. By the law of cosines:

since cos ψ = sin a. The test suite checks two exact cases: with the Sun overhead, d = D − R; with the Sun on the horizon, d = √(D² + R²).

5.2 Two effects change the distance, and both are included

The observer–Sun distances the code produces are 150,107,466 km at 15:00 and 150,106,732 km at 18:00. The net change is 734 km out of 150 million, or 0.0005%.

5.3 The resulting angular size

The unaided eye resolves about 60″, which is more than 6,000 times this change.

5.4 Rate of change

Differentiating θ = 2 arcsin(r/d):

Differentiating d² = D² + R² − 2DR sin a splits the distance rate into exactly the two effects above:

dD/dt and dδ/dt come from differentiating the ephemeris series in §3, and da/dt from differentiating §4.1 with dH/dt = 15° per hour. The ephemeris series run in hours of UTC while the app’s clock is apparent solar time; the two drift apart by about 2 parts in 10,000 through the equation of time, and the analytic form corrects for it. That correction matters here only because the two terms nearly cancel.

6Flat model: why it would

6.1 The model

The ground is a plane. The Sun is a small sphere at constant height h above it, travelling horizontally away from the observer in a straight line. This is the most charitable version of the flat model, and its speed is the measured speed of the subsolar point over the real ground:

At the equator this agrees exactly with the arc length in §4.2: 5,009.4 km = v × 3 h. The suite checks the two routes agree to one part in 10¹².

6.2 Where the flat Sun starts

A flat model has to put the Sun somewhere at 15:00, and there are two defensible ways to do it. The app offers both.

Match the real elevation (the default). Start the Sun at the elevation the real Sun actually has:

Both camera views then begin identical, so every later difference comes from the model alone. The cost is an inconsistency, which the app reports on screen rather than hiding: at h = 1,000 mi the Sun must start 1,609 km away horizontally, but the real subsolar point is 5,009 km away.

Above the real subsolar point. Set x(t) = Rψ(t), the measured ground distance. This has no free parameters, but the Sun then starts at the wrong elevation: 17.81° at 1,000 mi instead of 45°.

The two agree at exactly one height, h ≈ 5,009 km, which is offered as a preset.

6.3 The Sun’s size is derived, not assumed

The flat Sun’s physical radius is set so that it has the real angular diameter at the start of the window:

At 1,000 mi that gives a Sun 21.10 km across. The model is handed the correct starting appearance for free and only has to get the next three hours right.

6.4 What happens next

The Sun shrinks by 21.22′, which is 136,140× the globe model’s change. And since a = arctan(h/x) is positive for any finite x, the flat Sun can never set. It only approaches the horizon asymptotically.

6.5 Rate of change

With the elevation anchor x is linear in t, so dd/dt = xv/d and

Where the shrink is fastest. Since θ ≈ 2r/d, the rate goes as x/(x²+h²)^3/2, which peaks at x = h/√2, where the elevation is arctan√2 = 54.7°. The window starts at 45°, already past that peak, so the flat Sun shrinks fastest at the start and slows down: −16.53″ per minute at 15:00, −2.54″ per minute at 18:00. That is the shape of the curve in the rate chart.

6.6 How high would the flat Sun need to be?

Bisection on h for the lowest height at which the shrink across the window stays under 1′, the naked-eye limit:

At those heights the flat model hides the shrink only by failing to set at all.

6.7 The other flat model: the Gleason map

The local-plane version above is a steelman: it is handed the measured speed of the subsolar point and asked only to move in a straight line. The maps flat-Earth arguments actually use are azimuthal-equidistant discs — the Gleason map and its relatives — with the north pole at the centre and distance from the pole preserved:

Choose that map and nothing is left to fit. The Sun circles above the latitude of its own declination, and the observer sits on his own circle, so the distance between them is the cosine rule about the pole, with the hour angle as the included angle:

Three consequences follow, none of them chosen:

The equator comes out longer by exactly π/2, which is 57%, and everything else follows from that. It is not a subtle discrepancy: it is checkable by flying between two points on the equator and timing it, or by watching where the Sun actually sets with a compass. A 45° error in bearing is the width of half the sky.

The plan-view inset in the flat side panel shows the map itself: the pole, the observer's circle, the Sun's circle, and the line between them, because the elevation view alone cannot show a Sun that swings round rather than receding in a straight line.

Which way round it goes. The Sun's longitude east of the observer is −H: it is overhead at solar noon and falls away westward all afternoon. Viewed from above the north pole, eastward runs counter-clockwise, so westward travel is clockwise, and that is how the inset draws it, with an arrow on the leading edge. The sign matters more than it looks: get it wrong and the picture is a view of the disc from underneath, which would disagree with the bearing the model reports for the same instant. The test suite pins the longitude falling through the whole window and the Sun staying in the western half of the sky.

6.8 What latitude changes

The equator is where the flat model fares worst: the Sun starts highest, and so nearest, and then recedes furthest in relative terms. Moving away from it softens the contrast without coming close to rescuing the model.

Even at 80°, where the Sun only creeps down a shallow slope, the flat model still loses more than a quarter of the Sun’s width while the real one changes by six hundredths of an arcsecond. The globe figures grow with latitude because the Sun spends the window nearer the horizon, where the observer’s rotation carries them away from it most directly. They stay thousands of times too small to see.

7Refraction

Standard air by default, and applied equally to both models. Real sunsets happen in an atmosphere, and the comparison does not depend on refraction either way, because it acts on the vertical axis only (§7.2). “Off (pure geometry)” is one of the presets if you want the bare geometry.

Sæmundsson (1986), from true altitude h — the direction the app uses, since geometry is computed first:

Bennett (1982), from apparent altitude h_a, used to cross-check that the two are mutual inverses:

7.1 How much refraction: the settings

Both formulae are written for 1010 hPa and 10 °C, and scale for other conditions as

Denser air bends light more, so refraction rises with pressure and falls with temperature. Each preset’s label is generated at runtime from the functions themselves, so a label can never advertise a figure the code does not produce.

That is the whole range, and it is narrow: across all ordinary weather the scaling spans only about ±10%.

7.2 What refraction does, and what it cannot do

It lifts the Sun, so at geometric sunset the Sun appears about half a degree above the horizon and real sunset comes a few minutes later. And because it lifts the lower limb more than the upper limb, it squashes the disc vertically: to 85.5% of its width in standard air, and to 42% under the ×4 preset.

Horizon dip. With refraction on, the sea-horizon dip uses the navigational value 1.76′√h[m], which is 2.29′ at 1.7 m against a geometric 2.51′ (Bowditch). On the flat model the horizon is the vanishing line of the plane, at exactly 0°.

7.3 Where the formulae stop working

Both formulae contain an inner term of the form h + k/(h + c), which has a minimum at h = √k − c. Below that altitude the term turns and refraction starts falling with depth, which is not what real air does. Applied to a disc it is worse than wrong: the upper limb would get more refraction than the lower, stretching the Sun vertically instead of squashing it.

Nothing you can see depends on the clamp. Under every preset the Sun’s upper limb is already below the horizon before its altitude reaches the limit — between 0.76° and 0.98° below it, depending on the preset, which the test suite checks. Past that the Sun is hidden by the sea, and the readouts say “not modelled this low” instead of quoting a number.

Below those altitudes the code holds refraction at its maximum modelled value rather than following the formula down. The test suite walks both formulae from 60° down to −6° and requires that refraction never decreases, and checks the disc is never widened at any altitude down to 4° below the horizon under every preset.

This matters in practice because the timeline runs to 4° below the horizon, so that the disc can be seen to set even under the ×4 preset, which lifts it by nearly 3°.

8The camera

8.1 Lens

A pinhole (rectilinear) lens on a full-frame 36 × 24 mm sensor. Focal length refers to the 24 mm height, because vertical framing is what this scene is about; the horizontal field extends to fill the pane.

At 24 mm that is 53.13°.

8.2 Projection

A direction d̂ in local east–north–up coordinates, seen by a camera with basis (forward f̂, right r̂, up û), lands at

The Sun’s outline is built on the sphere and then projected, so the lens distorts it exactly as a real lens would rather than being drawn as a circle and hoped for.

8.3 Framing

To put the horizon a fraction k of the frame height above the bottom edge:

The default lens is the longest common focal length that fits everything from the horizon to the top of the Sun at 15:00 (45° + θ/2 = 45.27°) with 2% margins:

The longest common lens that fits is therefore 24 mm. The horizon then sits 8.31% of the frame height above the bottom, and that fraction is held fixed as the lens is changed.

Away from the equator the Sun sets at a slant, so it tracks sideways as it goes: about 32° of bearing across the window at Melbourne, 38° at London. The frame has to hold that too, so the lens is the shorter of the two limits, measured against a 3:2 frame (real panes are wider, so this is the conservative test), and the camera faces the Sun’s mean bearing rather than its bearing at sunset.

That leaves vertical slack, because a low Sun needs little height. Centring the content would then float the horizon up the middle of the frame and fill the bottom half with sea, so the horizon is capped at 12% of the frame height and the slack goes to the sky instead. The suite checks that the Sun and the horizon are both still inside the frame across the range of latitudes the picker allows.

8.4 Why the loupe is a separate camera

A rectilinear lens stretches objects off-axis. At 24 mm the Sun at 45° is 22.6° off-axis, where a disc is drawn 16.9% taller than on-axis.

That is correct lens behaviour, and the main view reproduces it. But magnifying that frame would make the Sun appear to change size as it crossed it — exactly the effect under test. So the loupe is a second camera aimed straight at the Sun, where the disc is undistorted. The suite verifies that a disc viewed head-on spans exactly 2f tan r in both axes, at any altitude.

It defaults to 1600 mm, a 0.86° field, in which the Sun fills about 62% of the height. Where the pane is wide enough, the loupe is a full-height panel beside the frame rather than an inset on top of it.

8.5 Glow brightness

Two things change how much light arrives. One is the Sun’s angular area, since surface brightness does not change with distance, so flux follows the solid angle. The other is the air.

8.6 Extinction, and why people think the Sun shrinks

Light from a low Sun crosses far more atmosphere than light from a high one. The standard fit for how much more is Kasten & Young (1989):

and the loss is the usual magnitude law, transmittance = 10^−0.4kX. With k = 0.15 magnitudes per airmass the total dimming from overhead to the horizon comes to about 5.5 magnitudes, a factor of roughly 165 — consistent with the drop from ~100,000 lux at midday to ~500 lux at a clear sunset. A single coefficient is an approximation: real extinction is Rayleigh scattering plus aerosol plus ozone, and the aerosol part varies by a factor of several with haze. It is the right trend and the right order of magnitude, not photometry.

Put the two together and the globe model predicts this:

Where the physics stops and the drawing starts. The flux is computed. Turning it into a radius is not. Veiling glare in an eye or a lens falls off roughly as the inverse square of angle — the Stiles–Holladay form — so the radius at which it drops below a fixed visibility threshold goes as the square root of the flux, and that is the rule the renderer uses. Opacity is held constant, because both eye and camera re-expose as the scene darkens; anything else would make the glare fade rather than contract, and an altitude term would make it appear to grow towards sunset as the sky went dark behind it. It is a plausible rule, not a scattering model, and both panes use the identical one.

The progression is not linear, and should not be: from 45° down to about 18° the glare barely changes, because the air in the way has barely changed. Almost all of the collapse happens in the last few degrees, which is exactly when people say the Sun looks like it is shrinking.

9What is decorative

These carry no part of the argument, and both models use the same functions for them:

• Sky and sea colours. They vary with solar altitude to look like a sunset, but they are not a radiative-transfer model.
• Glare shape. The flux behind it is computed (§8.5, §8.6); how that flux becomes a radius and an opacity is a rendering rule.
• Globe side view distances. Earth and Sun sizes and spacing are schematic, labelled as such, with the true-scale sizes printed on the panel. The angles are exact.
• Flat side view Sun size. Drawn at a 4.5 px minimum when its true size would be sub-pixel, with the enlargement factor shown. Its position and line of sight are never adjusted.

10Known simplifications

11Headline results

Default scenario: equator, 23 September 2026, 15:00 → 18:00 apparent solar time, flat Sun at 1,000 mi starting at the real elevation.

The flat model’s change is 136,140× the globe model’s.

12Sources

Every constant and formula in js/physics/ traces to one of these.

Constants

Solar position

• Low-precision formulae for the Sun’s coordinates. U.S. Naval Observatory, The Astronomical Almanac, Section C (“Sun”). Stated accuracy 0.01° over 1950–2050.
• Meeus, J. (1998), Astronomical Algorithms, 2nd ed., Willmann-Bell — ch. 13 (coordinate transformations), ch. 15 (rising and setting), ch. 25 (solar coordinates).

Extinction

• Kasten, F. & Young, A. T. (1989), Revised optical air mass tables and approximation formula, Applied Optics 28, 4735–4738.
• Extinction coefficient and the midday-to-sunset illuminance drop: standard observational figures, used here only to fix the scale of the effect.

Refraction and horizon dip

• Sæmundsson, Þ. (1986), Astronomical refraction, Sky & Telescope 72, 70.
• Bennett, G. G. (1982), The calculation of astronomical refraction in marine navigation, Journal of Navigation 35, 255–259.
• Bowditch, N., The American Practical Navigator (NGA Pub. 9), altitude corrections: dip of the sea horizon, 1.76′√h (h in metres).

Values used to check the code

Rules of thumb, flagged as such where used

• Naked-eye angular resolution of about 1′. This is the definition of standard 6/6 (20/20) Snellen visual acuity: resolving detail that subtends one arcminute. It is used only to ask “could anyone see this change?”, and the conclusion holds with any threshold within a factor of 1,000.