Polar Day, Polar Night and Twilight: Why Sunrise Can Be Missing

Does no sunrise mean a full day of darkness?

This guide builds a deliberately simple geometric model so that those differences can be calculated. It is not a forecast for a city or a particular date. The model holds solar declination fixed during one rotation, assumes an unobstructed geometric horizon and labels time by solar hour angle. Its worked cases explain why an event may be absent without treating absence as a clock time.

Separate the event, the interval and the threshold

Sunrise is an event: the Sun crosses a chosen rising threshold. Daylight duration is an interval between relevant boundaries, or a continuous interval when the Sun never sets. Solar altitude is an angle. These three quantities should not be stored or described as if they were interchangeable.

Ordinary sunrise conventions account approximately for the solar disk and atmospheric refraction. NOAA's equation uses a solar-centre altitude of about −0.833°, corresponding to zenith distance 90.833°. A geometric centre crossing at 0° is a different event. The difference becomes especially instructive when the daily path just grazes a threshold.

The National Weather Service describes twilight using the Sun's geometric centre at 6°, 12° and 18° below the horizon. Those thresholds distinguish civil, nautical and astronomical twilight boundaries. They are not extra sunrise times, and a named twilight boundary need not be crossed on every date at every latitude.

The model and its declared assumptions

Let φ be latitude, positive north; δ be solar declination, positive north; H be solar hour angle; and h be the geometric altitude of the Sun's centre. The relation is sin h = sin φ sin δ + cos φ cos δ cos H. It is the altitude form of NOAA's zenith-angle equation.

Write A = sin φ sin δ and B = cos φ cos δ. In our model H = 15° × (t − 12), where t runs from 0 to 24. Thus t = 12 is the model's upper culmination and t = 0 or 24 its lower culmination. This is a uniform solar-angle clock, not local civil time or a promise that a real transit occurs at 12:00 on a watch.

For each case δ stays constant throughout the calculation. Real solar declination changes with date and slightly during a day. We choose ±23.44° as illustrative near-solstitial values and 0° as a separate equatorial-declination case. No calendar date, longitude, time zone, terrain profile or weather observation is supplied.

First calculate the highest and lowest altitude

For the terrestrial latitude and solar declination ranges used here, B is nonnegative. The maximum sine of altitude is A + B and the minimum is A − B. Therefore hmax = arcsin(A + B) and hmin = arcsin(A − B), with a tiny numerical clamp only to handle floating-point rounding at ±1.

At 70° north with δ = +23.44°, the centre ranges from +3.44° to +43.44°. It stays above the geometric horizon for the full model rotation. With δ = −23.44°, the range is −43.44° to −3.44°. It never reaches 0°, but its highest point is above −6°. That is the key to twilight during a period without sunrise.

Checking the extrema first prevents a common calculation mistake: passing an impossible argument into an inverse cosine and then calling the result a failed sunrise calculation. Sometimes there is simply no crossing to calculate. The geometry can explain the missing event before a clock conversion is attempted.

Find zero, one tangency or two threshold crossings

For a chosen altitude h₀ and nonzero B, rearrange the equation as C = (sin h₀ − A) / B. When −1 < C < 1, there are two crossings. Their hour angles are −H₀ and +H₀, where H₀ = arccos C. The model duration at or above the threshold is 2H₀/15 hours.

If the minimum altitude is greater than the threshold, the Sun stays above it: 24 model hours. If the maximum is lower, the Sun stays below it: zero hours. In those cases no rising or setting event exists for that threshold in this model rotation. Do not invent an event by forcing a genuinely out-of-range C back into the interval −1 to 1.

Equality deserves its own label. When the minimum merely touches the threshold, the duration is 24 hours with a single tangency, not an ordinary sunset followed by sunrise. When the maximum just touches it, the time above the threshold is zero except for the instant of contact, which has zero duration. A tangent touch is not a crossing.

Nine cases that test the difficult boundaries

The two 66° north cases share the same solar path. Its minimum is −0.56°. It crosses the geometric 0° horizon but stays above the −0.833° approximation. Thus changing the event definition can change the classification even though the input latitude and declination are unchanged.

The 66.56° north cases deliberately create tangency because 66.56 + 23.44 = 90. They are exact arithmetic constructions, not a claim that an observed polar-circle boundary is fixed at that latitude under every condition. The positive-declination case touches 0° at its minimum; the negative case touches it at its maximum.

The southern example mirrors the northern one by changing both latitude and declination signs. Their product in A is unchanged, and B is also unchanged, so the altitude curve is identical. This is a symmetry of the declared geometry, not evidence that the hemispheres have identical weather or surface conditions.

See a day with no sunset and a day with no sunrise

Read the lower panel at its highest point. The curve is still below the geometric horizon, so no centre-rise event occurs. Yet it enters the altitude band above −6° around model noon. Farther from noon it passes through the deeper twilight thresholds and spends part of the rotation below −18°.

The upper panel is an equally useful contrast. Its lowest point remains positive. The absence of a sunset there means continuous geometric daylight, not a zero-length day. An event table needs a state describing that condition; a blank time alone does not explain it.

Hours above a threshold are cumulative

Each row asks how long the centre is at or above that altitude. The −18° row includes all the time above −12°, which includes the time above −6°, and so on. Adding those rows would count many hours repeatedly. They are cumulative durations, not separate twilight lengths.

In the winter example, the 0° and −0.833° rows are both zero. The −6° row is positive because the highest altitude is −3.44°. This makes the distinction between no sunrise and no twilight numerical rather than merely verbal.

For a conventional sunrise-based daylight estimate, read the appropriate apparent-disk approximation with its assumptions. For the geometric exercise, read 0°. Keep the threshold in the column label so that two different measures do not both get presented simply as “day length.”

Calculate separate altitude bands by subtraction

These five rows partition the winter model rotation into non-overlapping altitude bands. The −12° to −6° duration is the time above −12° minus the time above −6°. Likewise, time below −18° is 24 minus the cumulative time above −18°. The unrounded five durations sum to 24 hours.

The top twilight band here ends at the geometric 0° centre horizon. It is explicitly a geometric band, not a replacement for the usual civil-twilight interval ending at apparent sunrise. In this particular winter example neither 0° nor −0.833° is reached, so that upper-boundary difference contributes no duration. In other cases it can matter.

A duration below −18° describes the Sun's position. It does not promise a perfectly black sky: moonlight, aurora, artificial lighting and atmospheric conditions are different influences. The exercise computes an angle-based classification, not a measurement of sky brightness.

Why the pole requires a separate calculation

At the geographic pole, cos φ is zero, so B is zero. Dividing by B in the crossing formula is invalid. The original altitude equation remains meaningful: the altitude is constant during this frozen-declination rotation. At the north pole it equals δ; at the south pole it equals −δ.

With φ = 90° and δ = 0°, the entire model path is on the geometric horizon. It is not two ordinary events at arbitrary times. The table counts 24 hours at or above the threshold while explicitly labelling the state as on the threshold throughout. Real declination is not frozen for a whole season, and real refraction adds further complications.

This case explains why a robust method begins with geometry and an event state, then attaches a time only when an event exists. A denominator of zero should not be hidden by substituting a nearby latitude without saying so.

A practical reading checklist for a missing event

1. Confirm the location, date, time zone and the event being requested.
2. Identify whether the threshold concerns the centre, upper limb or twilight.
3. Check whether the solar path stays above, below or touches that threshold.
4. Keep “no event” separate from 00:00, zero duration and missing data.
5. Check twilight thresholds independently if you need light conditions.
6. Use a date-dependent calculation and local horizon information for actual planning.

A report that offers previous or next sunrise is answering a different question from today's sunrise. The first returning sunrise date cannot be obtained from this fixed-declination model. It requires following the Sun's changing declination over successive dates and applying the chosen event convention.

The sunrise and sunset tool is a route to location-based information; this article does not verify its behaviour at every polar coordinate. When a requested event is absent, the explanation and assumptions deserve as much attention as the formatted time.

What the calculations establish, and what they leave out

The nine cases and the graph show how a daily solar path can fail to cross one threshold while crossing another. They also expose tangencies and the pole special case. These are mathematical consequences of the stated equation, not measured observations or a dated ephemeris.

Atmospheric refraction, the visible solar disk, observer height and an obstructed horizon affect observed rising and setting. NOAA's calculation notes describe increased sensitivity when the Sun meets the horizon at a shallow angle. The −0.833° value is a conventional approximation, not a guarantee for every atmosphere or landscape.

NOAA now labels its calculator as no longer actively supported or maintained. The references here document equations and conventions; they are not a recommendation to rely on an unsupported service for a critical schedule. For the basic distinction between sunrise and dawn, continue with sunrise, dawn and twilight.

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