Why the Sun Looks Flat at Sunset: Refraction Explained

Why can the Sun look flattened near sunset?

Atmospheric Optics explains this unequal lifting and shows how unusual atmospheric layers can create more complicated distortions. Here we calculate a simpler case: a circular solar disc of declared angular size under a published refraction approximation. The original tables and graph illustrate vertical compression, not a photograph or a forecast for a particular sunset.

Separate geometric altitude from apparent altitude

Geometric altitude is the direction an object would occupy without atmospheric bending. Apparent altitude is the observed direction after that bending. In the ordinary conditions modelled here, apparent altitude is higher. Write h for geometric altitude and R(h) for the positive refraction correction.

When R is expressed in arcminutes and h in degrees, the apparent altitude is h + R(h)/60. An arcminute is one-sixtieth of a degree. Omitting that conversion turns a modest correction into an error sixty times too large.

The two limbs of the Sun have different geometric altitudes. They therefore require two corrections. Adding one correction to the centre tells you where a point-like centre appears; it does not by itself tell you the height of an extended disc. This distinction is the key to the worked examples.

Unequal lifting changes height; equal lifting does not

Let the unrefracted vertical diameter be D₀ and let Rᵤ and Rₗ be corrections at the upper and lower limbs. Using the same angular unit throughout gives D = D₀ + Rᵤ − Rₗ. If the lower correction is greater, the difference is negative and the apparent height is smaller.

As a purely invented arithmetic example, take D₀ = 32′, Rₗ = 24′ and Rᵤ = 20′. Both edges move up, but the lower one moves four arcminutes more. The apparent vertical diameter is 28′, a reduction of 12.5%. The four-arcminute difference, not the full 24′, is the amount subtracted from the diameter.

If both corrections were instead 20′, the whole disc would shift upward with no vertical compression in this simple construction. Conversely, raising the lower edge more cannot be described as adding its correction to the diameter. Draw the two endpoints before deciding the sign.

A reproducible approximation with declared units

The British Astronomical Association presents the Sæmundsson approximation with pressure and temperature factors. For this illustration we use it as written:

R(h) = 1.02 × cot[h + 10.3/(h + 5.11)] × (P/101) × [283/(273 + T)].

R is in arcminutes, h and the angle inside cot are in degrees, P is pressure in kPa, and T is temperature in °C. Cot is 1/tan. A programming language usually expects radians for tan, so convert the bracketed angle before evaluating it. This is a documented empirical approximation, not a full ray-tracing calculation through a measured atmosphere.

Fix the geometric solar diameter at 32′, so each limb lies 16′ = 0.266666…° from the centre. For a centre altitude h꜀, evaluate the lower limb at h꜀ − 16/60 and the upper at h꜀ + 16/60. Our centre altitudes range from 0.5° to 10°; even the lowest geometric lower limb is above the horizon. We deliberately avoid extending the exercise to disappearing limbs or below-horizon branches.

Work through the 1° example

Choose h꜀ = 1°, P = 101 kPa and T = 10°C. The lower and upper geometric limb altitudes are approximately 0.733333° and 1.266667°. In the chosen approximation, their refraction corrections are 23.3991′ and 20.2616′.

The lower apparent altitude is about 0.733333 + 23.3991/60 = 1.123318°. The upper is about 1.266667 + 20.2616/60 = 1.604360°. Subtracting gives about 0.481041°, or 28.8625′. The same result comes directly from 32 + 20.2616 − 23.3991.

Relative to the declared 32′ geometric diameter, the reduction is about 9.80%. This is a computed model value. It is not a measurement of a city, a claim about every sunset or a statement that the Sun's physical diameter changed by that percentage.

Eight heights in the same model atmosphere

At 0.5° centre altitude the model height is about 28.1507′, whereas at 10° it is about 31.7360′. The diameter approaches its unrefracted value as the difference between the two limb corrections becomes smaller. Both corrections can still be positive when their difference is small.

Cover the last two columns and calculate them from the limb corrections. Use unrounded values if you want to reproduce the last displayed digit exactly. Four decimal places document the arithmetic; they do not imply that local atmospheric conditions or image edges are known to that precision.

This table calculates vertical diameter only. It does not compute horizontal refraction, a complete apparent ellipse, brightness or colour. Dividing height by 32′ gives a ratio to the declared geometric diameter, not a fully modelled height-to-apparent-width ratio.

What changing pressure and temperature does here

Within this formula, the correction scales directly with P/101 and with 283/(273 + T). At a fixed geometric altitude, lowering pressure reduces the model's limb-correction difference; increasing temperature also reduces that difference through the stated factor. The apparent vertical diameter then lies closer to 32′.

Use the stated pressure unit. 101 kPa equals 1,010 hPa. Entering 1,010 into a field expecting kPa makes the pressure factor ten times too large. The temperature factor uses the formula's stated 273 constant; these displayed results reproduce that approximation rather than silently substituting another version.

A pair of surface pressure and temperature values does not describe the full vertical temperature profile. These factors cannot reconstruct inversions, layered mirages or a jagged solar limb. The nine rows are controlled changes to the formula, not nine guaranteed atmospheric states.

An original graph of apparent vertical diameter

Read the horizontal axis as geometric centre altitude, not clock time. The black reference is the unrefracted 32′ diameter. The curves rise toward it as the centre becomes higher. The differences between curves illustrate the scaling factors; they do not describe the changing weather during one actual sunset.

The first point is 0.5°, not the moment of conventional sunset. A graph that extrapolated these curves until an edge vanished would need additional attention to the horizon, the range of the approximation and the atmosphere. Keeping the plotted domain explicit is part of interpreting the figure correctly.

Flattening and sunset timing are related but different

The Hong Kong Observatory explains the conventional near-horizon allowance as roughly 16′ for the solar radius plus 34′ for average refraction. Together they place the geometric solar centre near −50′ when the apparent upper edge meets the horizon. This is a timing convention using average conditions, not a prediction that every observer sees an identical disc.

Do not add a full 50′ refraction correction and then add the 16′ radius again. The radius is already included in that combined allowance. Also, the generic allowance is not the same operation as applying the chosen formula separately to two positive-altitude limbs in our table.

The equinox daylight guide develops the timing consequence, while dawn versus sunrise distinguishes other boundaries. Astrogyana's local Sun times provide daily timing context; a listed sunset time is not a forecast of the disc's visible shape.

What to check in an existing photograph

1. Use an existing image made with appropriate solar-viewing equipment; do not stare at the Sun to reproduce the measurement.
2. Check whether the horizon or a cloud cuts off part of the lower limb.
3. Confirm that the image has not been stretched differently in its horizontal and vertical directions.
4. Keep the original pixel aspect ratio and note any lens correction or resizing.
5. Measure both edges consistently, recording blur, saturation and uncertainty rather than choosing convenient pixels.
6. Compare the observed vertical compression with the model's limited question, not with a claimed exact weather reconstruction.

A cropped lower edge is missing information, not a measured full diameter. Camera saturation, focus and atmospheric turbulence can also make a boundary unreliable. A striking oval image may be useful for illustration while remaining unsuitable for a precise numerical test.

Follow NASA's solar-viewing guidance. Ordinary sunglasses are unsuitable for direct solar viewing. Optical equipment needs appropriate solar filtration at the front; eclipse glasses behind a telescope or binoculars do not make that arrangement safe. Studying a properly acquired existing image avoids the need for this article's reader to improvise an observing setup.

Keep shape, colour and apparent size separate

A reduced vertical angular diameter is a geometrical claim about the image. A redder colour involves the light transmitted through the atmosphere. A subjective impression that the horizon Sun looks larger is a different perceptual question. One observation should not be offered as the complete explanation of the other two.

Unusual temperature layers can produce shapes beyond this smooth model, and a high observer has a different sightline through the atmosphere. Our graph therefore cannot diagnose every photograph. Its dependable lesson is narrower: compare the corrections at both edges, preserve their units and sign, and distinguish the calculated image from the physical Sun.

For a complementary geometric example, Zero Shadow Day examines the Sun near the zenith rather than the horizon. The contrast helps explain why an observing model must state both where the Sun is and which effect it is trying to calculate.

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