Sunset from a height

Eye height, and the geometric dip of the horizon. The dip reuses the horizon distance from the shared geometry module, on the GRS80 mean radius, …. The page then says how much later the Sun’s centre sets than it does at sea level. The rate is 15° of hour angle per hour. Refraction is off in one result and on in another, and the two are labelled apart. The place is the latitude you type, or a click on the map. For that place and date, the page also says how long the sun is up at sea level, and how long the night is.

Eye height (m) Latitude (°, north positive) Date

29 February is counted as the following day, day 60, and 31 December stays day 365 even in a leap year.

100 m is a round figure for a cliff. It is not a surveyed height. Day 81 is the day this approximation gives declination 0°. It is not a claim that the equinox falls on that day. The date becomes that day of the 365-day year.

Click the map to set the place. The latitude field takes the latitude of the click. The marker stays on that place. The Stockholm presets put the marker at 18.0686° E, and the equator preset puts it at longitude 0°. The hour angle does not use that longitude.

The muted line is the horizontal. The blue line is the geometric dip. The brown line is the dip with the shared refraction. The dips in the drawing are enlarged. The cards, not the slope, are the angles.

Horizontal
Geometric dip, refraction off
Dip with the shared refraction

Geometric dip

Refraction off: —

Later sunset

Refraction off: —

Later sunset

Shared refraction on: —

Sun up

At sea level: —

Night

At sea level: —

Hold Ctrl and scroll to zoom the map. The scroll wheel alone moves the page.

What this page shows

From an eye at height h the horizon lies at the arc given by the shared horizon distance. The dip below the horizontal is that arc divided by the radius. At sea level the dip is zero.

At sea level the centre sets when its altitude is 0°. From the height, the centre sets when its altitude is minus the dip, because the horizon is that far below the horizontal. The difference of those two hour angles, divided by 15° per hour, is how much later sunset is.

15° per hour is the rate of change of hour angle in mean solar time. It is the same at every latitude. It is not 15° of altitude per hour. The altitude changes more slowly near the horizon at a high latitude, so the same dip is a longer time there. The latitude enters the hour angle. It does not replace the 15°.

The map marker is the place. A click sets that place. Typing a latitude moves the marker to that latitude and leaves it on the longitude it already had. A preset moves the marker to the preset’s place. The hour angle uses the latitude. It does not use the longitude of the marker.

For the chosen latitude and date, the sun-up card is how long the centre is above the horizon at sea level, and the night card is the rest of the 24 hours. The declination is the same approximation as the midnight-sun page. It is not an almanac. The eye height makes sunset later by the extra time in the later-sunset cards. That extra time is not added again into the sun-up card, and it is not a clock time of sunset. The longitude of the marker is not turned into an hour of the day.

Refraction off uses the geometric radius. Refraction on uses the shared coefficient κ, which enlarges the radius to R / (1 − κ) and reduces the dip. That is the terrestrial ray curvature from the geometry module. It is not the 34′ astronomical refraction on the midnight-sun page. The two results are labelled apart.

Assumptions

The declination is the same approximation as the midnight-sun page, δ = 23.44° × sin(360° × (n + 284) / 365). It is not an almanac. 21 June is day 172 of a 365-day year. Day 81 gives declination 0° in the formula. That is 22 March in that year, and it is not stated as the equinox.
The time is local solar time of the centre. There is no equation of time, no upper limb, and no 34′ allowance in these cards. The sun-up time and the night are durations at sea level. They are not a clock time, and longitude is not used to name an hour.
The sun-up time uses the stated declination approximation, the same one as the midnight-sun page. Night is 24 hours minus the time the centre is up. 29 February is counted as the following day, day 60, and 31 December stays day 365 even in a leap year.
If the centre does not set, the delay is left blank and the reason is written out. At a pole the altitude does not change through the day.
100 m is a round figure for a cliff or a tower. It is not a surveyed height of a named place. Stockholm’s latitude is the city position 59.3293° N already used on this site.
The drawing exaggerates the eye height. The dip in the cards is the calculated angle. The eye height makes sunset later by that dip. The sun-up card stays the sea-level duration.
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 hour angle.

Sources

The horizon distance and the refraction coefficient are the shared geometry module. κ enlarges the radius to R / (1 − κ). The dip is that horizon arc divided by the radius in use.
The hour-angle rate is 15° per hour of mean solar time, which is 360° in 24 hours. Latitude changes the hour angle of sunset, not this rate.
Cooper, P. I. (1969). The absorption of radiation in solar stills. Solar Energy, 12, 333–346. The declination is that approximation with 23.44° in place of 23.45°, as on the midnight-sun page.
Moritz, H. (1980). Geodetic Reference System 1980. The radius is the mean radius from the shared module.
Stockholm, 59.3293° N, 18.0686° E, the city position already used on this site. The longitude is where the marker sits for the Stockholm presets. The equator preset sits at longitude 0°. Neither longitude enters the hour angle.