Midnight sun and polar night

What this page shows

The altitude of the Sun’s centre is sin h = sin φ sin δ + cos φ cos δ cos H. φ is the latitude, δ is the declination, and H is the hour angle. H is 0° at local noon and 180° at local midnight. The lowest centre is the midnight altitude. The highest is the noon altitude.

Geometric midnight sun means that lowest altitude is above the horizon. Geometric polar night means the highest altitude is below the horizon. If the centre crosses the horizon, the Sun rises and sets. A value of exactly zero is named as touching the horizon, not folded into the other two.

The declination is δ = 23.44° × sin(360° × (n + 284) / 365), with n the day of the year. Cooper’s 1969 formula used 23.45° in that sine. This page uses 23.44°, the same round figure of the J2000.0 mean obliquity as the Eratosthenes page. The formula’s year has 365 days. It is not an almanac. 21 June is day 172 and 21 December is day 355. 29 February is counted as the following day, day 60, and 31 December stays day 365 even in a leap year.

The sun-up time is how long the centre is above the horizon. Night is 24 hours minus that time. Midnight sun is 24 hours of sun and no night. Polar night is no sun and 24 hours of night. In between, the time comes from the hour angle at the horizon, at 15° per hour. With the upper-limb box off, that horizon is 0°. With the box on, the sun-up time uses the same refraction and semidiameter as the upper-limb card. The geometric day card does not change.

The places on the solstices

Longyearbyen is the IMAGE magnetometer station LYR, 78.20° N, 15.82° E. On 21 June the geometric day there is midnight sun. On 21 December it is polar night.

Nordkapp is the North Cape cliff, 71°10′21″ N, 25°47′04″ E. On 21 June the geometric day is midnight sun. On 21 December it is polar night.

Stockholm is the city position used on the other pages, 59.3293° N, 18.0686° E, not a surveyed pin. On 21 June and on 21 December the Sun rises and sets. It is neither midnight sun nor polar night.

McMurdo Station is the NSF position 77° 50.89′ S, 166° 40.11′ E. On 21 December the geometric day is midnight sun. On 21 June it is polar night.

Union Glacier station is the camp at 79.78130113945241° S, 83.31334156282651° W. It is on this page because the trip there was called The Final Experiment, and several flat-earthers were invited. On 21 December the geometric day is midnight sun. On 21 June it is polar night. The longitude is west, so the marker uses −83.31334156282651°. The altitude does not use it.

Choosing a preset moves the marker to that place. A click sets the place as well. The marker keeps the longitude of the click, and the latitude field takes the latitude. Typing a latitude moves the marker and leaves the longitude where it was. The altitude uses the latitude and the date. It does not use the longitude.

Assumptions

• The geometric cards use the centre of the Sun and a horizon at 0°. They do not include refraction or the semidiameter.
• The upper-limb line is off unless the box is ticked. It adds 34′ of horizon refraction and the semidiameter, and classifies those shifted altitudes on their own. 34′ is the Nautical Almanac’s conventional allowance at the horizon. The semidiameter is atan of the nominal solar radius over 1 AU, which is about 0.27°. Neither number is hidden inside the geometric altitudes.
• That shift is the usual sunrise allowance. It is not a refraction table for other altitudes, and it is not the terrestrial coefficient κ used on the curvature pages.
• At a pole the altitude is the declination, or minus the declination in the south, for the whole day.
• Local solar time is the hour angle. The page does not apply the equation of time, a time zone, or a longitude. The sun-up time and the night are durations. They are not a clock time.
• The sun-up time uses the stated declination. It is not an almanac. Night is 24 hours minus the time the sun is up. Refraction and the semidiameter stay out of that time until the upper-limb box is ticked.
• The drawing is the altitude through the day. It is not a sky chart.
• Hold Ctrl and scroll to zoom the map. The scroll wheel alone moves the page. A click sets the place. The map requests tiles from OpenStreetMap. The coordinates are used only in this browser. The longitude of the marker is not part of the altitude.

Sources

• Cooper, P. I. (1969). The absorption of radiation in solar stills. Solar Energy, 12, 333–346. The sine and the 365-day count are that approximation. The 23.44° in place of 23.45° is the round obliquity used on the Eratosthenes page.
• The Nautical Almanac. Sunrise and sunset are given for the upper limb, with a conventional refraction of 34′. The semidiameter here is computed from the nominal solar radius and the astronomical unit, and it comes out near the Almanac’s 16′.
• IAU 2015 Resolution B3, nominal solar radius 6.957 × 10⁸ m. IAU 2012 Resolution B2, the astronomical unit 149,597,870,700 m.
• IMAGE magnetometer network, station LYR, Longyearbyen, 78.20° N, 15.82° E. space.fmi.fi/image.
• The North Cape cliff, 71°10′21″ N, 25°47′04″ E.
• U.S. National Science Foundation, McMurdo Station, 77° 50.89′ S, 166° 40.11′ E. nsf.gov/od/opp/ail/mcmurdo-station.
• Stockholm, 59.3293° N, 18.0686° E, the city position already used on this site.
• Union Glacier station, the camp, 79.78130113945241° S, 83.31334156282651° W. The longitude is west, and the marker stores it as a negative number where east is positive. These digits are the place used on the page. They are not a new measurement.
• Map tiles © OpenStreetMap contributors.