$$\def\|{&}\DeclareMathOperator{\D}{\bigtriangleup\!} \DeclareMathOperator{\d}{\text{d}\!}$$

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## 1. Sunrise and Sunset at the Poles

Exactly at the poles, the Sun rises one time each year and sets one time each year, at the equinoxes (around 21 March and 23 September). Tables of sunrise and sunset can be found on the Sun Position Tables Page.

That there are such places on Earth where the Sun stays above the horizon for that long and then stays below the horizon for that long is because the Sun is not always above the equator of the Earth, because the rotation axis of the Earth does not make a right angle with the orbit of the Earth around the Sun. This is also the reason for the seasons.

Sunrise and sunset at the poles take a long time. At the poles, the Sun does not go higher and lower in the sky each 24-hour day, but goes very slowly up in the spring or down in the summer. During a 24-hour period, the Sun goes all the way around the sky at almost the same height. Around the beginning of spring, as seen from a pole, the Sun gets about 0.4 degrees higher in the sky each day. The apparent diameter of the Sun in the sky is about 0.5 degrees, so about 30 hours pass between the moment when the top of the Sun is first visible and the moment when the bottom of the Sun is first visible (assuming the horizon is level and clear in all directions), so sunrise takes about 30 hours at the pole. The same holds for sunset. During those 30 hours, the Sun moves all the way around the horizon, and then some, so if you look at the Sun for all of that time, then you'll have looked in the direction of all continents and oceans.

The direction in which you'd catch the first/last glimpse of the Sun probably depends the most on irregularities in the horizon. If the horizon has a small dip in a particular direction, then you'd probably see the Sun first/last in that direction. If the horizon were perfectly level and flat, then the direction of the first/last glimpse would vary from year to year.

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## 2. Seeing the Sun at Night

It is night when you are in the shadow of the Earth. The umbral shadow of the Earth is at most as wide as the Earth itself, and gets narrower the further it gets from the Earth, so if you go high enough above the Earth, then you get back into sunlight, even when it is night on the ground.

However, you'll have to stand on top of an extremely tall building if you want to see the Sun for a lot longer every day in this way. If you stand on a building of 100 m (300 ft) tall, then you would see the Sun set only about 3 minutes later than you would if you were on the ground (assuming that you have a clear view of the horizon when you are on the ground), and if you want to double the extra time that you can see the Sun, then you have to go find a building that is four times as tall. If you'd want to see the Sun when it is only about 10 degrees below the horizon as seen from the ground (about an hour after sunset), then you'd have to be 100 km (60 mi) above the ground!

The formulas that go with this are as follows. If the Sun is $$φ$$ degrees below the horizon as seen from the ground at your location, and if the radius of the Earth is equal to $$R$$, then the night (the shadow of the Earth) reaches to an altitude $$h$$ above your head:

$$h = R \left( \frac{1}{\cos(φ)} - 1 \right)$$

This formula overestimates the altitude when the angle $$φ$$ is very close to 90 degrees, because this formula assumes that the shadow of the Earth is attached to the Earth like an infinitely long tube, while in reality the shadow of the Earth is attached to the Earth like a cone with a very long and sharp point that reaches to about one and a half million kilometers (about one million miles) from Earth.

If you measure altitude and radius in kilometers, and if you fill in the radius of the Earth already ($$R = 6378$$ km), and if you assume that the angle $$φ$$ is not more than about 15 degrees, then you can approximate the above formula with

$$h = 0.97 φ^2$$

For example, if the Sun is 1 degree below the horizon ($$φ = 1$$), then the night reaches to about $$h = 0.97 × 1^2 = 0.97$$ km above your head. Multiply km by 0.6 to get mi, so 0.97 km is about 0.97×0.6 = 0.6 mi. The Sun sets at a speed of roughly about 10 degrees per hour, so the Sun is one degree below the horizon after only about 6 minutes, and after that time the night has risen a kilometer from the ground already, and it keeps going up faster and faster.

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## 3. The Highest Sun

At a place at latitude $$φ$$ degrees (north or south), the Sun gets at most $$113 - φ$$ degrees or 90 degrees above the horizon, whichever number is smaller.

For places outside of the tropics (i.e., at latitudes above 23 degrees), the greatest height is less than 90 degrees and is reached at noon on midsummer's day, which is around 21 June in the northern hemisphere and around 21 December in the southern hemisphere.

At the poles, the Sun cannot get higher above the horizon than 23 degrees. The Sun attains this greatest height at midsummer's day, which at the South Pole happens around 21 December, and at the North Pole around 21 June.

For places between the tropics (i.e., at latitudes below 23 degrees), the greatest height is equal to 90 degrees and is reached twice a year, about equally long before and after midsummer's day, and further from midsummer's day for places closer to the equator. At the equator, the greatest height is reached at the equinoxes (21 March and 23 September).

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## 4. Day and Night

It is day when you are on the side of the Earth that faces the Sun, and it is night when you are on the side of the Earth that faces away from the Sun. The Earth turns around its own axis (which runs from the North Pole through the center of the Earth to the South Pole) in about 24 hours, and as the world turns you are carried into the sunlight and then into the darkness, and the same the next day, and so on.

You can play Earth-and-Sun yourself at night in your room. Take an apple or any other small round thing to your room and turn off all the lights except for one. It works best if the one light that you turn on is a bulb and not one of those long tubes. You can also use a flashlight (US) or torch (UK). Imagine that the apple is the Earth, and the lamp that's on is the Sun. Put a mark somewhere on the apple to show where you are on the Earth, or pick a mark that's already there. Then rotate the apple around so that the mark is sometimes in the light and sometimes in the shadow. When the mark is in the light, then it is daytime there, and when the mark is in the dark, then it is nighttime. Each time you turn the apple all the way around, one more day has passed on it. That's all there is to it. It works just the same for the real Earth and the real Sun.

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## 5. Equal Length for Day and Night

At the beginning of spring and autumn (usually on 21 March and 23 September) the Sun stands straight above the equator and then it is daytime for 12 hours and nighttime for 12 hours everywhere on Earth. Those days are called the equinoxes. Exactly at the North Pole and the South Pole is is daytime for 6 months and nighttime for 6 months. Those are the polar day and polar night. You can only have 12 hours of daytime and 12 hours of nighttime (in a period of 24 hours) everywhere on Earth if that is also the case at the poles, so this can only happen on the two days of the year that the poles switch between day and night, and that happens only at the beginning of spring and autumn, on 21 March and 23 September. Only then is there a right angle between the rotation axis of the Earth and the direction towards the Sun, and only then are the poles on the boundary line between day and night.

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At the equator, every day and every night is 12 hours long, because the Sun rises straight up and sets straight down, so the circle that the Sun traverses above the horizon and below the horizon is split into two equal parts by the horizon, and so each part takes the same amount of time.

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I wrote above that the length of the day and night are 12 hours at the equator and at the equinoxes, but that does not seem to match with the times of sunrise and sunset that I provide in the associated tables. The discrepancy comes from the use of slightly different definitions of night and day. The sunrise and sunset tables are based on the definition that the day starts or ends when the top of the disk of the Sun touches the horizon (if the ground is level all around you up to the horizon and you are on the ground), taking into account the average refraction of light by the atmosphere.

The claim that night and day are symmetric (i.e., last equally long at the equator and at the equinoxes) is based on the definition that the day starts or ends when the middle of the disk of the Sun touches the horizon, ignoring the effects of the atmosphere.

The effects of looking at the top of the Sun and taking the atmosphere into account are both such that the day becomes longer and the night shorter. However, both of those effects depend on which planet you're on, because the size of the disk of the Sun depends on how far away you are from the Sun, and by how much refraction of light lifts up the image of the Sun depends on the properties of the atmosphere.

To get a basic understanding of night and day and how their lengths change with the seasons, it is best to emphasize the symmetry. To get the best fit with what newspapers and other casual sources of times of sunset and sunrise report, you must take into account the two effects that I mentioned.

In most cases the difference is just a few minutes. Only in situations where the Sun goes up or down very slowly in the sky near sunset or sunrise (i.e., in the polar regions in the right season) does the difference get large, but in any case the Sun is always on the horizon during the period between the sunrise defined by symmetry and the sunrise taking into account the size of the solar disk and the atmosphere.

Not making this difference explicit everywhere is a bit sloppy, but, as you can tell, explaining it takes some time and draws the attention away from the point that I'm trying to make about day and night in the text where I'm being sloppy.

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## 6. Average Length of Day and Night

If you neglect small deviations, then the average length of days and nights over a whole year is 12 hours. The small deviations that make the averages differ a bit from 12 hours are:

• that refraction of sunlight by the atmosphere makes the daytime a couple of minutes longer and nighttime a couple of minutes shorter, depending on your location.
• that the Earth's orbit is not a perfect circle, so the seasons aren't all equally long, so the longer summer days and the shorter winter days (and likewise nights) don't cancel out entirely in the average.
• that sunrise is usually defined as the moment when the top of the Sun, not the middle of the Sun, passes through the horizon. This makes the daytime a few minutes longer, depending on your location.
• that the seasons don't all begin in the middle of a day, so that the amount of daytime and nighttime in the summer half and the winter half of the year isn't exactly equal.

I don't know offhand exactly how large these deviations are, but I expect that it won't be more than a few minutes (except maybe near the poles).

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## 7. The Direction of the Sun

In the Netherlands and Belgium we never see the Sun towards the north, because the Sun seems to rotate around the Earth roughly above the equator, and as seen from here the equator lies to the south. (The Sun doesn't really rotate around the Earth, actually the Earth rotates around its own axis, but that doesn't make a difference in this case.) The same holds also for all other places north of the tropics but south of the northern polar circle.

It is just the opposite in Australia and other regions south of the tropics but north of the southern polar circle: there, the equator is to the north and so the Sun is never due south. And in the tropics (near the equator) the Sun always passes almost straight overhead.

In the polar regions you have polar days, when the Sun does not set for at least 24 hours. During such a polar day the Sun goes around the sky once every 24 hours, and is then one time due north and one time due south.

The difference between the directions of sunrise and sunset is about 180 degrees only at the equinoxes (around 21 March and 23 September). In the summer half of the year the Sun moves more than 180 degrees between sunrise and sunset as measured along the horizon, and in the winter half of the year the Sun moves less than 180 degrees. At 50 degrees north latitude on June 8th, the Sun moves about 128.4 + 128.5 = 256.9 degrees along the horizon, which is far greater than 180 degrees. Half a year later at that same latitude, the Sun moves only about 103 degrees along the horizon between sunrise and sunset.

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An object's shadow always points in the opposite direction from where the source of light is. For example, in the late afternoon the Sun tends to be roughly towards the west, so shadows outlined by sunlight then point roughly towards the east. However, the exact location of the Sun in the late afternoon depends a lot on the season, the exact time of day, and on your geographical latitude, so the direction of a shadow outlined by sunlight is likewise variable. You can find the direction of the Sun at sunrise or sunset on the Solar Position Tables Page, or find how to calculate the direction for any time on the Solar Position Calculation Page.

## 8. When Do I Get Sunlight in my Garden?

Fig. 1: The Sun in the Netherlands

Figure 1 shows where in the sky the Sun is during the year, as seen from a location at 51.6° north latitude and 5.15° east longitude (in the Netherlands). The horizontal axis shows the azimuth, with the south at 0°, the west at 90°, and the east at −90°. The vertical axis shows the altitude above the horizon, with the horizon at 0°.

Each curve that begins at the horizon on the left, then goes up and to the right, and then down and to the right back to the horizon shows the path of the Sun for a whole day (between sunrise and sunset) at the beginning of a calendar month. The first letter of the months is indicated in the middle. The blue curves are for January through June, and the red curves for July through December.

The numbers 5 through 20 along the topmost curve indicate whole hours (o'clock) in Central European (Standard) Time. In summer the clock shows 1 hour later than the numbers displayed in the figure. The 8-shaped closed curves for each hour show the path of the Sun in the sky for the whole year if you look at the Sun's position every day at that hour. That 8-shaped curve is called the analemma. In the blue part the Sun moves up, and in the red part the Sun moves down.

Using Figure 1, you can determine at what time the Sun shines directly into your garden or kitchen window or onto your front door. For this you need to know the orientation of your house relative to the south. Determine which side of your house faces south the most. If you're standing with your back against that side and look straight ahead, then how many degrees west or east of south are you looking? Count degrees to the west of south positive, and degrees east of south negative. So, if you are looking 10 degrees west of south, then the azimuth for that wall is A = +10°, and if you look 10 degrees east of south, then the azimuth A = −10°.

You can determine this direction using a map of the surroundings of your house, or using Figure 1. If you want to use Figure 1 for that, then see at what time the Sun stands directly in front of that wall and look for that time (subtract 1 hour if it is Daylight Savings Time) and the right time of year in the figure. Go down to the horizontal axis, where you can read your azimuth.

Fig. 2: 0621T13

Fig. 3: Sunlight in a Dutch Garden

Figure 2 shows a house with a garden. The long axis of the house is rotated 63° clockwise relative to the north-south line. Let's call the side that is visible most face-on the front side. That side looks in the direction of 27° east of south, so at A = −27°. The left side looks at A = −27° + 90° = 63°, the back side at A = 63° + 90° = 153°, and the right side at A = −27° − 90° = −117°. This is equivalent to A = 153° + 90° = 243°, because you can add or subtract multiples of 360° until the result is between −180° and +180°.

Figure 3 shows those directions with dotted lines, except that A = 153° falls outside of the diagram; the Sun is never directly in front of that side.

We can now see in Figure 3 and in the table of images below at what time the Sun is directly in front of those sides. For example: around June 21 the Sun is directly in front of the right side around 05:40 (standard time, i.e., 06:40 daylight savings time), but at no more than about 8° above the horizon (see Figured 4 and 5). Around 11:40 (standard time) on that day, the Sun is straight ahead of the front side (at about 58° above the horizon; see 6 and 7), so then the left and right sides are just getting/have just lost sunlight. On 21 December that happens already around 10:45 (standard time, 11:45 daylight savings time; at about 12° above the horizon, see 10 and 11). Around 15:30 (standard time, so 16:30 daylight savings time) on 21 June the Sun is directly in front of the left side (8 and 9). On 21 December the Sun is still far below the horizon when it is straight ahead of the right side, and is half an hour past sunset when it is directly in front of the left side.

 Fig. 4: 0621T05 Fig. 5: 0621T06 Fig. 6: 0621T11 Fig. 7: 0621T12 Fig. 8: 0621T15 Fig. 9: 0621T16 Fig. 10: 1221T10 Fig. 11: 1221T11

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## 9. Long Days, Short Days

At the equator, day and night always last about the same amount of time (12 hours). The difference between the length of the night and the length of the day gets bigger the further away you get from the equator in the direction of a pole. At the poles, the difference has reached its greatest possible value. There a summer's day lasts 24 hours without any night, and a winter's night lasts 24 hours without any day.

If in the summer of the northern hemisphere you travel from the North Pole to the South Pole, then the day gets shorter and shorter as you go further to the south, and the night gets longer and longer, until you reach the South Pole, where the Sun then doesn't rise at all.

There is usually a difference between the length of the day and the length of the night because the rotation axis of the Earth is tilted, so that sometimes the North Pole and sometimes the South Pole is tilted a bit towards the Sun.

A globe that you buy in a store usually has its rotation axis tilted over the same angle as the Earth's rotation axis. If you imagine that the sunlight comes horizontally to such a globe, then you can recreate the situation with the real Sun and Earth.

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At any place on Earth there are times when the Sun is above the horizon. The longest period of time that the Sun can be below the horizon is 6 months, and that happens only on the North Pole and South Pole. If you want a place where the Sun never shines, then you have to shield that place from the parts of the sky where the Sun can be. There are islands in the stormy seas around Antarctica where it is almost always cloudy, so you hardly ever see the Sun from there.

In principle, it is daytime for half of the time and nighttime for half of the time, but just not everywhere on Earth at the same time. At the poles it is daytime for half a year and then nighttime for half a year. At the equator it is daytime for about 12 hours every day and nighttime for about 12 hours every day. Except at the equator, daytime is longer and nighttime shorter in summer, and nighttime is longer and daytime is shorter in winter.

The next table shows for the Netherlands at approximately which days of the year the daytime lasts a fixed number of hours. The "h" column mentions the number of hours of daylight. The "m" columns show month numbers (1 = January), and the "d" columns show day numbers. For example, there are 9 hours of daylight around January 30th (m = 1, d = 30) and November 11th (m = 11, d = 11).

h m d m d
8 1 8 12 4
8.5 1 21 11 21
9 1 30 11 11
9.5 2 8 11 3
10 2 16 10 25
10.5 2 24 10 18
11 3 2 10 10
11.5 3 10 10 2
12 3 17 9 25
12.5 3 25 9 17
13 4 1 9 10
13.5 4 9 9 2
14 4 16 8 25
14.5 4 24 8 17
15 5 2 8 9
15.5 5 11 7 31
16 5 21 7 21
16.5 6 4 7 7

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## 10. Variation of the Length of the Day and the Night

The length of the daytime period (how long the Sun is above the horizon) is tied to the maximum height that the Sun reaches in the sky that day: the higher the Sun gets, the longer the day lasts. This maximum height changes with the seasons (because of the tilt of the Earth's axis relative to the Earth's orbit around the Sun), but does not change by the same amount every day. It behaves like a pendulum that swings to and fro: As soon as the pendulum swings past its lowest (average) position, it starts to slow down. When it has slowed down to standstill, then it has reached the greatest distance from the average, and then it starts to move ever faster in the opposite direction, until it swings past its average position again. The pendulum moves fastest near its lowest position, and slowest near its turn-around points. For the length of the daytime period, the equinoxes correspond to the lowest position (the day is 12 hours long), and the solstices correspond to the turn-around points (the day is longest or shortest), so the length of the daytime period changes fastest around the equinoxes and slowest around the solstices.

If the change had to be equally large every day, then it would have to suddenly turn around at the solstices as if it bounced against a wall, but such a "bounce" would have to correspond to a similarly sudden change in the rotation of the Earth or the revolution of the Earth around the Sun, and there is no such change.

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## 11. Rotation of the Sun

The Sun and stars appear to rotate around the Earth about once a day, but that is because the Earth rotates around its axis about once a day. Something similar happens when you sit in a rotating merry-go-round: then, too, it seems like the rest of the world rotates relative to the merry-go-round, instead of the other way around.

The Sun also rotates around its own axis, but not at the same rate everywhere. Near the equator of the Sun, it takes about 25 days to go around once, but near the poles of the Sun it takes about 35 days.

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The Sun travels almost exactly 15 degrees in the sky each hour, but that motion is not only from east to west but also up and down, depending on the time of day, the season, and on where you are on Earth (on your geographical latitude).

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## 12. Sunrise, Transit, and Sunset

In most places, the Sun rises once during each period of 24 hours, and sets once during each 24 hours, except in the polar regions, where the Sun does not rise or set for a longer period, up to 6 months at a time exactly at the poles.

You can find formulas to calculate times of sunrise, transit, and sunset on the appropriate calculation page. If you prefer tables, then you can go to the appropriate tables page. If you live in the Netherlands or Belgium and prefer pictures, then read on below. If you want to find precise times without having to do any calculations, then you can check many newspapers and teletext. Times for a whole year are listed, e.g., in many astronomical and non-astronomical almanacs.

At what time the Sun rises, transits, and sets depends on the season, on where you are, and on the condition of the atmosphere.

You can read the times of sunrise, transit (high noon, when the Sun is due south and highest in the sky -- at least outside the tropics), and sunset for Utrecht and Brussels from Figure 12:

Fig. 12: times of the sun

The months of the year are along the horizontal axis (1 = the beginning of January, 12 = the beginning of December), and the hours of the day along the vertical axis, between 5:00 in the morning and just after 22:00 (10 pm) at night, in the official time of the Netherlands and Belgium. The solid lines are for Utrecht, and the dashed lines for Brussels. The breaks in the curves near the end of March and the end of October indicate the transition to and from Daylight Savings Time.

Sunrise in Utrecht (at 52°5' North latitude and 4°54' East longitude) is between about 5:18 daylight savings time (mid-June) and 8:48 standard time (at the end of December). Sunrise in Brussels (at 50°51' North and 4°21' East) is between about 5:28 daylight savings time and 8:45 standard time. Transit of the Sun occurs in Utrecht between 12:23 and 12:54 (pm) standard time, or 13:23 and 13:54 (1:23 pm and 1:54 pm) daylight savings time. In Brussels those are 3 minutes later. Sunset in Utrecht comes between about 16:27 (4:27 pm) standard time (mid-Decmeber) and 22:04 (10:04 pm) daylight savings time (at the end of June). Sunset in Brussels is between 16:37 standard time (4:37 pm) and 22:00 (10:00 pm) daylight savings time.

The times of sunrise and sunset that are printed on calendars and in newspapers are usually calculated for a central location (often an astronomical observatory that used to have responsibility for keeping national time). For the Netherlands this is usually Utrecht, and for Belgium it is Uccle. However, even in fairly small countries such as the Netherlands and Belgium, the times for other places can be many minutes different from the times calculated for the national central location.

All times of phenomena of the Sun or stars get one minute earlier (in the same time zone) for every quarter of a degree of longitude that you travel to the east. This corresponds to one minute per 27.8 cos φ kilometers (φ is the latitude) that you travel to the east. At 50° latitude (about the latitude of the Netherlands and Belgium), this is one minute earlier for each about 18 kilometers to the east.

Times of sunrise and sunset (but not of high noon) also change if you travel to the north or south, depending on the season and on your latitude, with at most one minute for each 70 (cos φ)2 kilometer. At 50° latitude this is at most one minute for each 29 km, or at most 4 minutes per degree of latitude. This maximum is reached at the beginning of summer and winter. At the beginning of spring and autumn the times of sunrise and sunset do not depend on the latitude.

Figure 13 may help you to determine the time difference because of your location in the Benelux.

Fig. 13: Benelux

The vertical dotted lines with the numbers at the bottom indicate how much the times of sunrise, transit, and sunset change because of the geographical longitude. Each number along the bottom shows how many minutes later sunrise, transit, and sunset happen than on the meridian of 15° east, on which the Central European Time (CET) is based that is used in the Netherlands and Belgium in winter. For example, if it is high noon in Berlin (at 15° east longitude) at 12:00 CET, then it is high noon in Groningen (G on the map, near the vertical dotted line labeled "34") at 12:34 CET, and at 12:43 CET in Brussels (B on the map).

Suppose that the newspaper claims that the Sun will be highest in the sky in Utrecht at 13:50 hours. Then how late will the Sun be highest in the sky in Maastricht (M on the map)? This time does not depend on the latitude, so we need only look at the vertical lines in the diagram. Utrecht is near 39.5 minutes, and Maastricht near 37.5 minutes, so in Maastricht transits (high noons) occur about 2 minutes earlier than in Utrecht. So, if high noon is at 13:50 hours in Utrecht, then it is at 13:48 hours in Maastricht. In the same way, you can find that the Sun transits 13 minutes earlier in Groningen than in Brugge (b on the map).

The horizontal dotted lines with numbers along the left side of the diagram indicate north latitudes. The time of sunrise near 21 Decmeber (midwinter's day) is about 4 minutes later here for each degree that you go to the north, and the time of sunset about 4 minutes earlier. Around 21 June (midsummer's day) things are just the other way around. Around 21 March and 23 September (the equinoxes) you need not worry about the latitudes when calculating times of sunrise or sunset. The latitude difference between Groningen and Brugge is about 2 degrees, so the part of the time difference between a sunrise in Brugge and one in Groningen that depends on the latitude is at most about 8 minutes

The effects due to the longitude and those due to the latitude add up. The next table shows the components due to longitude and latitude and their totals for sunrise, transit, and sunset, for two pairs of cities. For example: The Sun rises 19.5 minutes earlier and sets 1.5 minutes later in Groningen than in Brussels.

Table 1: Time Differences in the Benelux

month longitude latitude rise noon set
Groningen - Brussel March +9 0 +9 +9 +9
June +9 +10,5 +19,5 +9 −1,5
September +9 0 +9 +9 +9
December +9 −10,5 −1,5 +9 +19,5
Amsterdam - Luxemburg March −5 0 −5 −5 −5
June −5 +11,5 +6,5 −5 −16,5
September −5 0 −5 −5 −5
December −5 −11,5 −16,5 −5 +6,5

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## 13. Determining the Moment of MeridianTransit of the Sun

At the meridian transit of the Sun,

1. the Sun is due north or due south (depending on your location), so the shadow of a vertical pole points straight south or north. If you mark a line from the bottom of the pole that goes straight north or south, then the meridian transit occurs when the shadow of the pole falls on that line.
2. the Sun is highest in the sky, so a vertical pole casts the shortest shadow. When the shadow is shortest, then the meridian transit is there.

The first method is better than the second, because the length of the shadow changes the slowest around the time of meridian transit, but the direction of the shadow changes the fastest around the time of meridian transit.

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## 14. Length of Sunrise and Sunset

The amount of time that passes between the moment when the bottom of the solar disk touches the horizon and when the top of the solar disk touches the horizon is not constant, but depends on your latitude, on the season, and on how quickly the conditions in the atmosphere change. We'll assume a quiet and unchanging atmosphere.

The fastest sunset (or sunrise) at any given latitude occurs at the equinoxes (near 21 March and 23 September). The sunset then lasts approximately $$128/\cos φ$$ seconds, where $$φ$$ is the latitude. The slowest sunset occurs at the solstices (near 21 June and 21 December). For latitudes up to 60 degrees, the sunset then lasts approximately $$142/\cos(1.14 φ)$$ seconds. For latitudes greater than 60 degrees, the length of the sunset rises steeply with latitude, until you get to the latitudes where there is a polar night or day, when the Sun doesn't rise or set at all for days or months. For arbitrary dates, there is no simple formula to calculate the length of sunset. The full procedure involves several of the formulas from the relevant Calculation Page.

For example, at a latitude of 40 degrees (either North or South), the fastest sunset takes about $$128/\cos(40°) = 167$$ seconds (2 minutes 47 seconds), and the slowest one about $$142/\cos(1.14×40°) = 203$$ seconds (3 minutes 23 seconds). At a latitude of 50 degrees, the sunset lasts approximately between 199 and 261 seoonds (3 minutes 19 seconds and 4 minutes 21 seconds). At the equator, the sunset lasts between about 128 and 142 seconds (2 minutes 8 seconds and 2 minutes 22 seconds).

The duration of sunset and sunrise is independent of the refraction by the atmosphere that slightly lifts up things near the horizon so that they appear to be higher in the sky than they would have done without any refraction, because you compare two instants of time when different parts of the Sun are at the same altitude in the sky. The same altitude means that the refraction is equally strong (except if the conditions of the atmosphere in that direction have changed in the meantime), so both instants are delayed by the same amount and their difference remains the same.

## 15. Sunrise and Sunset Near the Solstices

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### 15.1. The Shortest and Longest Days

The longest and shortest days (when the Sun stays longest or shortest above the horizon) are the days of the solstices, around June 21st (the northern solstice) and December 21st (the southern solstice) of each year in the Gregorian calendar. The northern solstice yields the longest day in the northern hemisphere and the shortest day in the southern hemisphere, and the southern solstice yields the longest day in the southern hemisphere and the shortest day in the northern hemisphere.

The length of the daytime period is roughly symmetrical around the solstices: the daytime period is about equally long $$x$$ days before the solstice as $$x$$ days after the solstice.

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### 15.2. The Earliest and Latest Sunrise and Sunset

The earliest and latest sunrise and sunset generally occur on different dates, and not on the days of the solstices. Moreover, the precise dates depend on the geographical latitude of the observer.

Fig. 14: Sunrise and Sunset
Figure 14 shows how these dates depend on the latitude $$φ$$. The $$d$$ along the vertical axis is the number of days since 0 January. For comparison: the northern solstice has roughly $$d = 171$$ and the southern solstice roughly $$d = 354$$ (or $$d = −11$$). The curve of "max ↑" shows the date at which the latest sunrise occurs, the "min ↑" is for the earliest sunrise, the "max ↓" for the latest sunset, and the "min ↓" for the earliest sunset.

The seasons of the southern hemisphere are shifted over half a year compared to the seasons of the northern hemisphere, and the same holds roughly for the dates of the earliest and latest sunrises and sunsets.

The difference between the date of the earliest sunrise and the latest sunset (around midsummer) increases when you get closer to the equator, and the same holds also for the difference between the date of the latest sunrise and the earliest sunset (around midwinter). Very close to the equator the Sun is above the horizon about 12 hours every day, so then the differences in the lengths of the daytime period are small.

For example: for $$φ = +52°$$ (for example the Netherlands) the earliest sunrise occurs around day 168 (June 17th), the latest sunset around day 176 (June 25th), the earliest sunset around day 347 (December 13th) and the latest sunrise around day 365 (December 31st). For $$φ = +14°$$ (for example Thailand) the corresponding dates are June 2nd, July 9th, November 21st, and January 23rd: further away from midsummer and midwinter than for $$φ = +52°$$.

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### 15.3. Shift of Sunrise and Sunset Around the Solstices

From the beginning of winter until the end of spring, the Sun rises a bit earlier every day and sets a bit later every day, and from the beginning of summer until the end of autumn, the Sun rises a bit later every day and sets a bit earlier every day. The times of sunrise and sunset then shift in opposite directions: one gets earlier and the other one gets later.

However, near the solstices (21 December, 21 June) sunrise and sunset shift in the same direction. That is because then the length of the daytime period hardly changes, so then the times of sunrise, sunset, and transit shift by practically the same amounts. The time of transit of the Sun (high noon) is not the same every day, because the orbit of the Earth is not a perfect circle and because the rotation axis of the Earth is not perpendicular to the orbit of the Earth around the Sun (which also causes the seasons). Around 21 December, the time of solar transit (and thus also the times of sunrise and sunset) get later by about 25 seconds per day, and around 21 June by about 12 seconds per day.

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## 16. In What Direction Does the Sun Set?

The direction in which the Sun sets changes from day to day. If you watch sunsets from the same location for a year, then you'll notice that the Sun sets a little further towards the south each day between 21 June and 21 December, and a little further towards the north each day between 21 December and 21 June. Around 21 June, the sunsets is furthest to the north, and around 21 December it is furthest towards the south. Around 21 March and 23 September, the setting Sun is almost due west.

The size of the difference between the northernmost and southernmost sunset depends on your geographical latitude. For the Netherlands (at 52° north) the difference is about 80°: around 21 June the Sun sets about 40° north of west there (so about in the northwest), and around 21 December the Sun sets about 40° south of west there (so about in the southwest). If you get closer to the pole, then the difference increases, and if you go closer to the equator, then the difference diminishes.

The same holds for sunrise, but then in the east instead of in the west. Around 21 June, in the Netherlands, the Sun rises in about the northeast, goes through the south, and sets in about the northwest. Around 21 March and 23 September, the Sun rises in the east, goes through the south, and sets in the west. Around 21 December, the Sun rises in about the southeast, goes through the south, and sets in about the southwest.

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Some people think that since the Sun rises in about the east as seen from the northern hemisphere (e.g., North-America or Europe), it must rise in about the west as seen from the southern hemisphere (e.g., Australia or South-America), but that is not correct. The directions "east" and "west" do not suddenly swap places as you cross the equator, and the direction in which the Sun rises does not suddenly swing around to the opposite side of the sky, either. The Earth rotates to the east, so the Sun always seems to rise in that direction.

If a ship could sail straight from the North Pole to the South Pole, then south would always be straight ahead, north always due behind, east always on the left (port), and west always on the right (starboard). When the ship is far to the north of the equator, then the Sun rises to the left (east), reaches its highest point straight ahead (south), and sets to the right (west). As the ship travels south, the point where the Sun is highest in the sky for that day travels north in the sky. When the ship is close to the equator, then the Sun rises to the left (east), reaches its highest point almost straight overhead, and sets to the right (west). When the ship is far to the south of the equator, then the Sun rises to the left (east), reaches its highest point due behind (north), and sets to the right (west).

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## 17. Sunrise Never to the West

The turning of the Earth around its axis is like the turning of a merry-go-round. If you stand on a turning merry-go-round then the surroundings seem to rotate around you in always the same direction. In the same way, the Sun always seems to move through the sky in the same direction and to rise in about the same direction (which is near the east). To let the Sun rise to the west, you'd have to make the Earth turn in the opposite direction. If that happens, then we're likely in for a lot of trouble, because the Earth is so enormously massive that only something even more mindbogglingly massive can make the Earth turn in the other direction. We cannot possibly make that happen ourselves.

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## 18. Sunrise not in the East?

I think that the most likely answer to the question "where on Earth does the Sun not rise in the East?" is "above the polar circles". I can't be sure, because the question is not specific enough. Technically speaking, the Sun rises exactly in the east only on or near the two equinoxes (around March 21st and September 23rd), everywhere on Earth except at the North and South poles. At other times of the year the Sun does not rise due east, but some distance to the north or south of due east. The "some distance" gets bigger the further you go from the equator and the closer you get to either pole. Above the polar circles, the Sun can even rise in the north or the south, and doesn't rise or set at all during a period of time that grows from 0 to 6 months as you travel from the polar circle to the pole.

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## 19. Predicting When Sunrise Or Sunset Are In A Certain Direction

You can use the associated tables to predict when sunrise or sunset are in a certain direction as seen from a certain place. Search through the table of directions of sunrise or sunset for the column that corresponds to the geographical latitude of the place, and read the date from the row in that column where you find the desired direction. Only for sunrise due east and sunset due west no tables are necessary: Those occur at the equinoxes (around 21 March and 23 September).

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## 20. The Range of the Sun

Outside of the tropics, days are short in winter and the Sun then traces only a short arc along the sky, but in summer days are long and the Sun traces a long arc along the sky. The Sun traces its longest arc along the sky on the longest day, and that is when the Sun reaches its greatest declination (in the same direction as your geographical latitude $$φ$$), on the summer solstice. The declination of the Sun is then equal in magnitude to the obliquity $$ε$$ of the ecliptic, which is now equal to 23.44°. The length $$l$$ of the arc of the Sun along the sky above the horizon is then equal to

$$l = 2 \arccos(-\tan(|φ|) \tan(ε))$$

The vertical lines around the $$φ$$ mean that you should omit any minus sign from its value. For example, if you are at latitude 50° (north or south), then the longest arc is equal to $$2×\arccos(-\tan(50°)×\tan(23.44°)) = 242.2°$$. That is the range of an equatorial sundial at that latitude: the shadow "hand" of that sundial won't trace out an angle greater than that.

The associated length of the day is equal to $$l/15$$ if you measure $$l$$ in degrees and the time in hours. The length of the longest day at 50° latitude is therefore equal to 242.2/15 = 16.15 hours.

As measured following the Sun but along the horizon, the distance between the point where the Sun rises and the point where the Sun sets on the longest day is equal to

$$l_\text{h} = 2 \arccos\left( -\frac{\sin(ε)}{\cos(|φ|)} \right)$$

For example, at 50° latitude this is equal to $$2×\arccos(-\sin(23.44°)/\cos(50°)) = 256.5°$$.

If you want to know the shortest arcs, then you should remove the minus signs from the formulas. For example, at 50° latitude, the shortest arc through the sky is $$2×\arccos(\tan(50°)×\tan(23.44°)) = 117.8°$$, the shortest day is 117.8/15 = 7.9 hours long, and the shortest arc along the horizon is $$2×\arccos(\sin(23.44°)/\cos(50°)) = 103.5°$$.

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## 21. Local Noon

At local high noon, the Sun passes through the celestial meridian (the line that goes from the one celestial pole via the zenith to the other celestial pole). At that moment, the Sun is also almost highest in the sky for that day. The continuously and slowly changing declination of the Sun means that the Sun is not actually highest in the sky exactly when the Sun passes through the meridian. This is easy to see: If the Sun is exactly on the meridian then it still moves up or down a little because of the continuously changing declination, so it is just a tad higher in the sky a little bit before or after that moment. The difference is exceedingly small, however. I've made a quick estimate and found about 15 seconds in March and September, when the declination of the Sun varies most rapidly. In 15 seconds, the Sun moves about 0.06 degrees away from the meridian. The height difference is then about 0.00001 degrees, and that is almost impossible to measure. This effect is therefore only a mathematical curiosity, and I hadn't seen anyone paying attention to it before.

A much larger effect is that the Sun does not pass through the meridian at exactly the same clock time every day, even if you have an accurate clock. The difference between the true transit and the mean transit can get up to 15 minutes, and is called the Equation of Time.

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## 22. Flying into the Sunset or Sunrise

I don't think an easy formula exists for calculating when the Sun rises or sets as seen from an airplane at altitude on an arbitrary heading. If the airplane flies straight at a constant speed, then you can calculate where the airplane is at every moment (as described on the Great Circle Calculation Page), and you can also calculate where the Sun is in the sky as seen from that location and time (as described on the Solar Position Calculation Page), but if you then demand that the Sun be rising at just that time and place, then you find an equation that cannot be rewritten as a formula that gives the time. There are no reasonable approximations that make the problem solvable. This means that you can find the solution only by trying different solutions until you find the right one.

The simplest method is to calculate the position of the airplane for some moment, then calculate for that time and place where the Sun is in the sky, and then to check whether the Sun is rising or setting. This is not something you do quickly with a calculator, but it is suitable for a computer program.

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## 23. Formula for the Terminator

The terminator is the line that separates day and night on a planet or moon. There is no single simple formula for its location. With the next procedure you can calculate the location of the terminator on Earth to an accuracy of about 1 degree:

First, calculate the declination $$δ$$ of the Sun. (All angles are measured in degrees.)

\begin{align} M \| = −3.6° + 0.9856 d \\ ν \| = M + 1.9° \sin M \\ λ \| = ν + 102.9° \\ δ \| = 22.8° \sin(λ) + 0.6° \sin^3(λ) \end{align}

where $$d$$ is the number of days since (the beginning of) the most recent December 31st (i.e., $$d = 1$$ for midnight at the beginning of January 1st, $$d = 2$$ for January 2nd, and so on).

Then, at $$t$$ hours UTC on that day, the Sun is straight up from a location at a latitude equal to $$b = δ$$ (northern latitudes are positive and southern latitudes are negative), and a longitude equal to $$l = 180° − 15×t$$ degrees. Add or subtract 360° if the result is not between −180° and +180°. Then eastern longitudes are positive, and western longitudes are negative.

If $$ψ$$ measures the distance (in degrees) along the terminator from a particular one of its intersections with the equator, then the east longitude $$L$$ and north latitude $$B$$ of the point at distance $$ψ$$ on the terminator are

\begin{align} B \| = \arcsin(\cos(b)\sin(ψ)) \\ x \| = -\cos(l)\sin(b)\sin(ψ) - \sin(l)\cos(ψ) \\ y \| = -\sin(l)\sin(b)\sin(ψ) + \cos(l)\cos(ψ) \\ L \| = \arctan(y, x) \end{align}

If you don't have an $$\arctan(y, x)$$ function, then you can use $$\arctan(y/x)$$ instead, but then you have to add 180 degrees to the result if $$x \lt 0$$.

Now let $$ψ$$ run from 0 to 360 degrees, and then $$L$$ and $$B$$ trace out the terminator.

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## 24. Travelling to the Sun

If you could travel from Earth to the Sun without stopping at a speed of 1000 kilometers per hour, then it would take you 17 years! The Sun is really far away.

If you travelled to the Moon in the same way, then that would take you 16 days. The Moon is much closer to us than the Sun is, but is still very far away.

It is not a good idea to fly to the Sun. It can get quite hot here on Earth during the summer, and when you go closer to the source of the heat (which is the Sun), then it gets hotter. Very close to the Sun it is way too hot and then your spaceship would melt and burn.

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## 25. Time Between Successive Moonrises

The time between a moonrise/transit/set and the next moonrise/transit/set can vary greatly. The difference between the times of successive moonrises/transits/sets depends mostly on the following things:

• that the Moon is not always equally far from the Earth and does not orbit around the Earth at a fixed speed. This means that the Moon travels further along the sky during some days than during some other days, and then the next moonrise occurs a bit sooner or later than usual. Moreover, the perifocus ― the place in the sky (relative to the stars) where the Moon travels fastest along its orbit ― is not fixed, but itself makes one lap around the sky in about 8.85 years (relative to the stars).
• that the orbit of the Moon makes an angle with the equator of the Earth, so that the Moon can sometimes get much higher in the sky than at other times (just like the Sun can get much higher in the sky during summer than it can during winter). Also, the angle that the orbit of the Moon makes with the equator of the Earth is not fixed, which causes extra variation. And the nodes ― the points in the sky (relative to the stars) where the orbit of the Moon crosses the orbit of the Sun ― are not fixed, but make one lap around the sky in about 18.6 years (relative to the stars).
• how high the Moon is in the sky, relative to the celestial equator. When the Moon is further north of the equator, then it can get higher in the sky (when seen from the northern hemisphere of Earth), is above the horizon for a longer period of time, and the times of moonrise and moonset are further apart.
• how high the Moon got in the sky yesterday, relative to the celestial equator. When the Moon is on its way to higher positions (further to the north of the celestial equator, when seen from the northern hemisphere of Earth), then the times of moonrise and moonset get further apart than yesterday, so then today's moonrise is relatively a bit earlier than yesterday's, and moonset relatively later. When the Moon is on its way to lower positions, then the times of sunrise and sunset get closer together than they were yesterday, and then today's moonrise is relatively later than yesterday's, and moonrise relatively earlier.

The time difference between successive moonrises/transits/sets depends on many things, and there is no simple rule for it. For example, it is not always greatest around New Moon.

If the obliquity of the ecliptic were 0 degrees (so there weren't any seasons on Earth), and the eccentricity of the orbit of the Moon were 0 (so the lunar orbit was a circle), then the time between successive rises of the Moon would always be 24 hours plus 50 minutes, just like the time between successive transits of the Moon (when the Moon is highest in the sky) and the time between successive settings of the Sun.

Fig. 15: Lunar Rise-Transit-Set 52°

Figure 15 shows the influence (for a location at 52 degrees north latitude, for example the Netherlands) of the most important effect that make the time between successive lunar rises variable. In each of the four shown diagrams, the yellow points are for moonrise, the green points are for transit of the Moon, and the blue points are for moonset. The horizontal axis shows the number of those occurrences since the beginning of the relevant month, and the vertical axis shows the number of minutes beyond 24 hours. Each graph shows the data points for each day from a period of 200 years. The data points for each next month begin at the left side of the graph and run to the right side.

Fig. 16: Lunar Rise-Transit-Set 0°

Figure 16 shows the same for a location on the equator.

Diagram A shows the effect of only the obliquity $$ε$$ of the ecliptic (= 23.44°), when the eccentricity of the lunar orbit is 0 (so the orbit of the Moon is a circle) and the inclination of the lunar orbit is 0° (so the orbit of the Moon is in the plane of the ecliptic). The distances (minus 24 hours) $$t_1$$ and $$t_3$$ between two successive moonrises or moonsets observed from 52 degrees latitude then varies between 18 and 75 minutes, and observed from the equator varies between 45.7 and 55.3 minutes. The distance $$t_2$$ (minus 24 hours) between two successive transits varies between 45.7 and 55.3 minutes, regardless of the geographical latitude. The variation during the month is exactly the same each month, so the curves for all months lie exactly on top of each other. For a place sufficiently far from the equator, we see that if the next moonrise is relatively late, then the next moonset is relatively early, and vice versa. There are 26.50708 rises/transits/sets per month.

Diagram B shows the effect of only the eccentricity $$e$$ (= 0.0549), when the obliquity of the ecliptic is 0° (so there are no seasons on Earth) and the inclination of the lunar orbit is 0°. $$t_1$$, $$t_2$$, and $$t_3$$ then vary between 44.5 and 56.9 minutes, regardless of the place on Earth. If the next moonrise is relatively late, then the next transit and moonset are late as well. There are 26.62552 rises/transits/sets per month. This is more than for diagram A, because the perifocus of the lunar orbit slowly moves with respect to the stars.

Diagram C shows the effect of only the inclination $$i$$ of the orbit of the Moon (= 5.145°), when the obliquity of the ecliptic is 0° and the eccentricity of the orbit of the Moon is 0. $$t_1$$ and $$t_3$$ then vary, observed from 52° latitude, between 43.6 and 56.5 minutes, and observed from the equator between 50.0 and 50.5 minutes. $$t_2$$ varies between 50.0 and 50.5 minutes, regardless of the location on Earth. For a place sufficiently far from the equator, we see that if the next moonrise is relatively late, then the next moonset is relatively early, and vice versa. There are 26.40048 rises/transits/sets per month. That is fewer than for diagrams A and B, because the nodes of the lunar orbit slowly move with respect to the stars, at a different speed than the perifocus moves.

Diagram D shows the effect when $$ε$$, $$e$$, and $$i$$ all have their (average) true value. $$t_1$$ and $$t_3$$ then vary, as seen from 52° latitude, between 8.6 and 94.3 minutes, and as seen from the equator between 40.9 and 61.6 minutes. $$t_2$$ varies, as seen from 52° latitude, between 38.5 and 65.5 minutes, and as seen from the equator between 41.0 and 61.7 minutes. The diagrams were drawn for 25.50707 rises/transits/sets per month. Now the curves for all months do not fall on top of each other, because now three separate effects with slightly different month lengths combine.

In practice, the effect of the eccentricity $$e$$ of the orbit of the Moon is gives slighly more variable than shown in the figure, because the eccentricity of the lunar orbit is itself not constant.

We see that the obliquity of the ecliptic ($$ε$$) is the most important reason for the variation in the time between successive moonrises/transits/moonsets.

Similar effects occur also for the rise/transit/set of the Sun, and lead to the equation of time and to the analemma, but the variation is much less for the Sun than it is for the Moon, because the Sun moves much less far along the ecliptic each day than the Moon does.

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