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This text is published under the Creative Commons license CC-BY-ND.

I must precise that I am not a professional astronomer. The text below may contain errors, be aware of this. I will not be held responsible for any consequences of your reading of this text. The "NO WARRANTY" paragraphs from the GPL and the Artistic License apply not only to Perl code, but also to English (and French) texts.

The text is often (but irregularly) updated on Github. There are a French version and an English version. Since I am more at ease discussing astronomical subjects in French, the English version will lag behind the French one.

This text is an integral part of the module's distribution package. So you can read it on web pages generated from CPAN (for example https://metacpan.org). But it is not used during the module installation process. So, I guess it will not appear in .deb or .rpm packages.

Although this text is stored in the Astro::Sunrise repository, it also documents the DateTime::Event::Sunrise module, which has a very similar core (astronomical computations) and a different API. Since both modules move at different speeds, it may happen that the text you are reading is not synchronised with the Astro::Sunrise module.

Why This Text? For Whom?

The main purpose of this text is to explain how the sunrises and sunsets are computed. These explanations are much too long to be included into the module's POD section.

For Whom? For My Teddy Bear

Have you read brian's Guide to Solving Any Perl Problem? While most advices deal with debugging Perl code, a few advices have a broader scope and apply to any intellectual problem. One of these advices consists in "talking to the teddy-bear". And do not pretend to talk while just forming sentences in your mind. You must really talk in a clear voice in front of your teddy-bear, with syntactically correct sentences. Actually, in the present case, the topic is so big that I skipped to the next level and I prefer writing (on GitHub) for my teddy-bear.

I write this text to tell my teddy-bear which problems I have encountered while maintaining this module and how I fixed them. But mainly, I write this to give him a detailed description of the precise iterative algorithm, because Paul Schlyter's explanations are not detailed enough for my taste and there is no compilable source available to check this algorithm (unlike the simple version without iteration).

For Whom? For The Next Module Maintainer

Actually, my teddy-bear understands French, so the present English version is not for him. The second person for whom I write is the next module maintainer. I have read Neil Bowers' message about the module authors pledge. I agree with him and I declare that should I stop maintaining my modules for whatever reason, I accept that any volunteer can take charge of them.

What Neil did not explain, is that the new maintainer must obey a few criteria and must have three available resources to take over a module maintenance: be competent in Perl programming, have enough available time to work on the module and be enthusiastic enough to get around to it.

In the case of astronomical module, the competence in Perl programming is not enough, you must also be competent in astronomy. So, if you think you might maintain this module, first read the present text. If you understand why I bother about such and such question, if you can follow my train of thought without being lost, then you are competent enough. If you think I am playing Captain Obvious and if you have instant answers to my questions, then you are the ideal person that could maintain this module. If you do not understand what all this is about, and if sines and cosines put you off, do not consider working on this module's innards.

For Whom? For Bug Reporters

This text is also for those who think they have found a bug in the module or who want to offer an idea to improve the module. Maybe the bug is already known and is waiting for a fix. Maybe the bug was found and the fix is not successful. Maybe the proposed improvement contradicts some other functionality of the module.

For Whom? For Curious Users

Lastly, this text is aimed at any person curious enough to learn a few astronomical facts. I tried to steer away from overly complicated computations. Their place is in the Perl source, not in this text. Yet, you will find here simple computations and mathematical reasoning.

Remarks About The Style

Some chunks of this text appear as a series of questions and answers. This is not a FAQ. Rather, this is a elegant way to give a progressive explanation of some subject. This method has already been used by many other writers, especially Plato, Galileo and Douglas Hofstadter.

Other Remarks

In my explanations, I usually take the point of view of a person living in the Northern hemisphere, between the Tropic of Cancer and the Arctic Circle. For example, I will write that at noon, the sun is located exactly southward, although any schoolboy from Australia, New Zealand, South Africa, Argentina and similar countries perfectly knows that at noon, the sun is northward.

In a similar way, 21st of March is called the vernal equinox or the spring equinox, even if it pinpoints the beginning of autumn in the Southern hemisphere.

But using politically correct sentences would yield convoluted phrases, which hinders the pedagogical purpose of the text and the understanding of the described phenomena.

About minutes and seconds: the problem is that minutes and seconds are both angular units (for longitudes and latitudes) and time units (for instants and for durations). I have adopted three different formats to help distinguish between these cases: 12:28:35 for time instants, 2 h 28 mn 35 s for time durations and 59° 28' 35" for angles (latitudes, longitudes and others). So, even if the hour-part or the degree-part is missing, you will be able to distinguish between a 28' 35" angle and a 28 mn 35 s duration.

Sources

I will give here only the sources that provide lists of numerical values. Books and articles with only a literary description of the subject are too many to be listed here.

Unused Sources

Some sources provide a list of sunsets and sunrises, but I did not use them because they do not explain which algorithm they use or because I cannot control the parameters.

The Almanach Du Facteur

In France, it is (or rather it was) customary to buy almanachs from the postman each year. In each almanach, you find a page giving the sunrise and sunset times for all the days of the year. Unfortunately, the times are given in HH:MM syntax, not including the seconds. In addition, even if you buy a provincial edition, the sunrise and sunset times are given for Paris. Lastly, the algorithm is not specified.

The Institut de Mécanique Céleste et de Calcul des Éphémérides (IMCCE, Institute of Celestial Mechanics and Ephemerides Computation)

This website (also available in english) used to give an HTML form to generate a table giving the sunrise and sunset times for a location and a time span of your choosing. Unfortunately, this webpage disappeared.

There is an available webservice to give the same functionality, but I did not try it.

Used Sources

Paul Schlyter's Website

This site provides a C program ready to compile and use, giving the sunrise and sunset times. This is the basis of the simple algorithm used in Astro::Sunrise. Its precision, as stated by the author, is one or two minutes, but it can be much less precise depending on the location and date, especially when we are close to the beginning or the end of the period when the midnight sun is visible.

Paul Schlyter's website includes also many informations about computing the position of various celestial bodies. This website is very interesting, but I preferred writing my own version, describing the computation of only the sun and not bothering with other celestial bodies.

The U.S Naval Observatory

The US Naval Observatory gives a HTML form to compute the sunrise and sunset times. These times are given in HH:MM format. I would have preferred HH:MM:SS, but I will have to deal with just HH:MM.

This website gives also very interesting informations about celestial computations, but without restricting itself to the sun, like I am doing here.

The NOAA solar calculator

The NOAA's Earth System Research Laboratories provide a solar calculator similar to the USNO.

Stellarium

Stellarium is a PC app to simulate a night sky. If you do not bother with the main view giving a real time sky simulation, you can use it to obtain the coordinates of a given celestial body at a given time when seen from a given Earth location. Here is how you can determine the time of sunrise, sunset or true solar noon (version 0.18.0).

Choose an approximate value for the requested time: 12:00 for the true solar noon or use Astro::Sunrise (basic mode) for sunrise or sunset.
Choose the search criterion: azimuth equal to 180° for true solar noon or real altitude equal to 0 degrees and xx minutes below the horizon. xx must be compatible with the altitude parameter of Astro::Sunrise's functions.

Note: we do not use the apparent altitude given by Stellarium. For the deviation angle of sun rays near the horizon, we use Astro::Sunrise's value, which is 35' or 0.583 degree and we aim at this value in Stellarium. And if we want to take into account the radius of the sun's disk, we will use the average value 15' instead of recomputing the precise value depending on the Earth-Sun distance. In this case, we will aim at a 50' altitude below the horizon, that is, 0.833 degree.
In Stellarium, freeze the time with 7 unless already frozen.
Look for the Sun. The search window is activated by <F3>.
You may prefer a display without the ground and without the atmosphere. Use the flip-flops g and a. And e or z to display or to hide the equatorial and azimutal grids, acording to your current taste.

By the way, when the atmosphere is deactivated, Stellarium no longer displays the apparent coordinates of the Sun. So the choice above, using the real Sun altitude and not its apparent altitude is justified.
Press <F6> to specify the observation location. With a location on the Greenwich meridian, the UTC time will coincide with the mean solar time. Do not forget to tick the personalised time zone check box and to select UTC.

In version 0.18.0, use "Royal Observatory (Greenwich)". In the previous version, I entered a location at 51° 28' 40" N, 5" E and a height of 27 m. The longitude and latitude are the values given by Wikipedia for the Greenwich Observatory and the height is nothing more than the previously entered height. This has no relation with the observatory's real height, or this is a lucky coincidence.
Press <F5> to specify date and time.
While keeping the date-time windows opened, try several time values until you get the best approximation of the target azimut or of the target altitude.

Note: on the previous version I used before 0.18.0, the timezone was not a property of the observation location, it was a global application configuration parameter. I had to hit <F2> to open the configuration menu. In this menu, I selected the Plugins tab, then Time zone. With this, I selected a UTC display instead of a display using the computer's local timezone.

Heliocentrism Or Geocentrism?

Here are two assertions. Are they true of false?

A: The Sun goes around the Earth.
B: The Earth goes around the Sun.

Assertion A is false, everyone agrees. But assertion B is false too.

Oh yes indeed, will you answer, it should read actually:

C: The Earth runs along an elliptic orbit with the Sun located on one focus of the ellipse.

This assertion is false too. Each one of the following assertions is nearer to the truth than assertions B and C (and A).

D

The center of mass of the Earth-Moon binary system runs along an elliptic orbit with the center of mass of the Solar System located on a focus of the ellipse.

And I will point that the center of mass of the Sun is not the same as the center of mass of the Solar System. There are even times when the center of mass of the Solar System is outside the surface of the Sun. The webpage about an HP-41 program states that on 15th March 1983, the distance between both centers of mass was nearly 2.1 Sun radii.

E

The Earth runs along an orbit around the Sun, with noticeable perturbations caused by the Moon, Jupiter, Saturn, etc.

Which is a formulation equivalent to assertion D.

F

The movement of the Earth with the Solar System is a n-body problem, with n ≥ 3. Therefore, there is no analytical solution.

G

The Solar System is a chaotic system. Even if we can predict with a reasonable accuracy what the various orbits will look like within the next hundred million years, this prediction is no longer possible for an interval of one milliard years (one billion years for US).

H

The Earth corkscrews in the general direction of the Hercules constellation with a approximate speed of 220 km/s.

I

The Earth runs along an orbit around the center of the Milky Way, with noticeable perturbations caused by the Sun, the Moon, Jupiter, Saturn, etc.

Assertions B and C are what Terry Pratchett, Jack Cohen and Ian Stewart call lies to children (Science of Discworld, chapter 4, pages 38 and 39). These are false assertions, but simple enough to be understood by a child and which, even if false, leads children to a better understanding of the described phenomena and brings them closer to truth. You cannot tell assertion C to a child and expect him to understand it without telling him first assertion B. And it is worse with assertions D and next.

Moreover, these are what I would call lies to adults. In the beginning, people would consider that the aim of Physics was to build a mathematical representation of the real world, getting closer and closer to the ultimate truth. Then, there was quantum physics including especially de Broglie's work with the duality of wave and particle and the Copenhagen interpretation. Is the ultimate nature of the electron (for example) a wave? No. Is it a particle? No. So what? We do not care about the ultimate nature of the electron. The aim of Physics is to no longer to provide a mathematical representation of the real world, but to build several mathematical models of the real world. We know that intrinsically all models are false, but each one has it usefulness to lead to make computations about the real world.

Please note that I was talking about scientific methods. I was not dealing with electoral campaigns and advertisements. Every sane adult knows for sure that these are ridden with lies.

Other lies to adults, also known as "simplifying hypotheses", you will find in the following:

the light propagates instantly from one place to another,
the celestial bodies outside the Solar System are motionless,
they are located on a sphere call the Celestial Sphere,
UTC time is equal to GMT time
the Earth's surface is a perfect sphere, without any polar flattening and without any equatorial bulge,
the Earth's surface is a perfect sphere, without any mountains, valleys or molehills,
there is even a place in this text where I imply that the duration of an astronomical year is an integer number of days (365, of course),
and, as I have already stated, all interesting locations on Earth are between the Tropic of Cancer and the Arctic Circle.

In some paragraphs, I will temporarily set aside some of these lies. But in most paragraphs most of these lies will be in effect.

Conclusion

All this to explain that in the following text, I will not refrain from using the geocentric model where the Sun turns around the Earth in 24 hours or the geocentric model where the Sun turns around the Earth in 365.25 days.

"It is not necessary that the following hypothesis be true or even resemble the truth. One thing is for sure that they provide calculations in accordance with the actual observations"

Excerpt from Osiander's preface to Copernic's book. This excerpt was reused by Jean-Pierre Petit as a foreword to Cosmic Story. In Copernic's time, Osiander wanted to have heliocentrism accepted by people who were certain that geocentrism was the one and only truth. It is ironical that I use the same quotation to have geocentrism accepted by people who believe that heliocentrism is the one and only truth.

Earth / Sun Movements

Basic Movements

In an heliocentric system pointing at fixed stars, Earth orbits around the Sun in one year. In other words, in a geocentric system, the Sun orbits around the Earth in one year, with an average speed of 0.9856 degrees per day or 0.04107 degrees per hour.

Also, the Earth spins around itself, making one turn in 23h 56mn 4s, with a speed of 4.178e-3 degrees per second, that is, 360.9856 degrees per day or 15.04107 degrees per hour.

Q: I thought that the Earth was spinning in 24h!

A: While the Earth spins, the Sun orbits around it. And what we see is the combination of both movements, which gives a combined speed of 360 degrees per day. What the commoner is interested in is to find the Sun at the same place in the sky at regular times day after day. Only after this is achieved, the commoner becomes a learned person and is interested in knowing and understanding the positions of the Moon, the stars and the planets.

Q: And why did you say "average" two or three times?

A: Because the angular speed of the Sun is not constant. We will get back to this question later.

Coordinates

The ecliptic is the plane where the Earth's orbit around the Sun is located (when using an heliocentric model) or where the Sun's orbit around the Earch is located (when using a geocentric model). We define also the equatorial plane, the plane which contains the Earth's equator. These two planes intersect with a 23° 26' angle. The intersection is a line, named line of nodes. In some cases, it is more convenient to use a half-line than a line. In this case, the line of nodes is a half line starting at the Earth center and aiming at the Pisces constellation. The point where the line of nodes meets the celestial sphere is called vernal point (which is politically incorrect, this is the beginning of autumn in the southern hemisphere).

For a point on Earth, we generally use longitude and latitude. We start from an origin in the Gulf of Guinea, where the Greenwich meridian meets the equator. Then we move along the equator in a first arc and along a meridian in a second arc to reach the point. The angle of the first arc is the longitude, the angle of the second arc is the latitude for a point on the ground. But when we consider a celestial body, the first angle is counted from the line of nodes instead of the Gulf of Guinea and is called right ascension; the second angle is called declination. Because of tradition, an old charter or something, the right ascension is usually expressed as hours, minutes and seconds instead of degrees. The declination still uses degrees. And Paul Schlyter's program uses decimal degrees for the right ascension, like any other angle.

Ecliptic coordinates follow the same principle, but the first arc is drawn along the ecliptic instead of the equator. Likewise, the second arc is perpendicular to the ecliptic. These angles are named ecliptic longitude and ecliptic latitude respectively. The ecliptic longitude is counted from the same origin as the right ascension: the line of nodes. That simplifies a little bit the conversions between the two systems of coordinates. On the other hand, the use of hours, minutes and seconds for the right ascension and of degrees for all other angles is an unnecessary complication in the conversions.

Because of the definition of the ecliptic plane, the ecliptic latitude of the sun is always zero.

Lastly, there is the local coordinate system. For a given celestial body, we project its location to the ground, or rather to the plane that is tangent to the ground. The angle between the North and this location on the tangent plane is called azimuth and the angle between the tangent plane and the line to the celestial body is called altitude.

Sidereal Time

The sidereal time is an ambivalent notion. People are divided into two categories, those who describe it as an angle, and those who describe it as a time. Strangely, there is no flame war about these diverging attitudes, each category of people seeming to ignore "the other side".

In the first category, people consider that despite its name, the sidereal time is an angle. See these French webpages: https://fr.wikipedia.org/wiki/Temps_sid%C3%A9ral https://www.futura-sciences.com/sciences/definitions/univers-temps-sideral-960/ http://users.skynet.be/zmn/docs/temps/TempsSideral2.html https://astrochinon.fr/index.php/documents/nos-dossiers/95-le-temps-sideral-de-greenwich. Even if this notion is measured with hours, minutes and seconds, this is an angle. After all, the right ascension also is measured with hours, minutes and seconds and it is still an angle.

Paul Schlyter is definetely in this category. He even defines the sidereal time as the right ascension of an earth-bound location and in his programs, he measures sidereal times in decimal degrees instead of hms values, just like he does for right ascensions.

The second category considers that the sidereal time is a measure of time. See https://www.localsiderealtime.com/whatissiderealtime.html and https://en.wikipedia.org/wiki/Sidereal_time, as well as https://sites.google.com/site/astronomievouteceleste/5---temps-sideral http://www.astrosurf.com/toussaint/dossiers/heuredetoiles/heuredetoiles.htm http://dictionnaire.sensagent.leparisien.fr/Temps%20sid%C3%A9ral/fr-fr/ in French.

But these descriptions seem to never mention issues linked with the measure of time: How is sidereal time influenced by timezones and daylight-saving time? What about leap seconds? With sidereal time as an angle, we can see that if you take two points on the surface of Earth within a few kilometers from each other in the E → W direction, these points have a different sidereal time. With sidereal time as a time, that would translate as saying that sidereal time and the principle of timezones are incompatible. Am I right or am I wrong? I found nothing about that in the "sidereal time as a time" explanations.

So, between a simple and comprehensive definition on one side and a convoluted and incomplete definition on the other side, I will stick with "sidereal time as an angle" definition.

Other Movements

Before I explain the other movements involved with the Sun and the Earth, let me tell you a little digressive note.

Weather And Climate

I hate these people who, each time snow falls, cry "Where is this global warming scientists talk about again and again?" These people seem to ignore that climate and weather are two different things. When the temperature from a meteorologic station varies by 10 degrees C from one day to the following, this is a mundane meteorological event. When the average temperature for a decade varies by 2 degrees C from one century to the next, this is a catastrophic climate event.

The movements I explain below are more "climatic" and less "meteorogical" than Earth's spin and orbital rotation. Their values over a short timespan are so low that the algorithms computing astronomical positions over a short timespan do not care about them.

Note: weather (but not climate) will come back in a few chapters as a real phenomenon, not as a metaphor.

Equinox Precession

The best known movement with a long timescale is the equinox precession. Presently, the vernal point lies within the constellation of Pisces. But actually, it moves all along the ecliptic, making a whole turn in about 26,000 years.

Nutation

The angle between the equatorial plane and the ecliptic plane varies slightly. In Paul Schlyter's C program, the angle decreases by 356 nanodegrees per day (3.56e-7 °/d, 1.3e-4 °/yr).

Perihelion Precession

There is also the perihelion precession. This movement is best known for Mercury, because it is the most apparent, but all other planets have a perihelion precession, including the Earth.

Other Drifts And Fluctuations

The formulas computing the positions of celestial bodies use some constants. But these values are constant only on a short timespan (astronomically speaking; or, with the metaphor above, on a "meteorological" timespan). But they are variable on a longer timespan (or a "climatic" timespan) For example, everybody knows that the day lasts 24 hours (the mean solar day, not the sidereal day). Yet, as I have read it somewhere, in paleontological times, it used to last 22 hours or so.

The variation of the duration of the day is a tiny variation, but with our modern measure instruments, we can measure it. Since the time when scientists abolished the astronomical standard of time for an atomic standard, it has been necessary to add 27 leap seconds over 47 years to synchronise the atomic timescale with the Earth's spin.

For the moment, all adjustments have consisted in adding a leap second. But it can happen that we would have to synchronise in the other direction by removing a second. So this phenomenon produces fluctuations rather than a slow drift in a single direction.

The Equation Of Time

There are other fluctuations, easier to measure and with a more "meteorological" and less "climatic" timescale. The true solar noon does not occur on the same precise time as the mean solar noon. There are two reasons.

Obliquity of the Earth

First, there is an angle between the ecliptical plane and the equatorial plane, therefore, a constant-speed rotation on the ecliptical plane does not translate to a constant-speed rotation when measured by right ascension on the equatorial plane. The rate of variation of the right ascension is a variable rate.

If we use the same units for the ecliptic longitude and the right ascension (either degrees or hours), then both values are nearly equal, but still different. So, when the ecliptic longitude is 46°20'31", the right ascension is 43°52'36", that is, a 2°27'54" gap. The same happens at longitude 226°20'31". And at longitude 313°32'52", the right ascension is 316°47", that is a gap of 2°27'54", but in the other direction. And the same happens at 133°32'52". These are the maximum values for the gap when using an obliquity of 23°26'. And if you prefer hours, here are the values:

.   longitude   right ascension      gap     longitude   right ascension   gap
.   3h05mn22s      2h55mn30s       -9mn51s   46°20'31"     43°52'36"     -2°27'54"
.   8h54mn11s      9h04mn03s        9mn51s  133°32'52"    136°00'47"      2°27'54"
.  15h05mn22s     14h55mn30s       -9mn51s  226°20'31"    223°52'36"     -2°27'54"
.  20h54mn11s     21h04mn03s        9mn51s  313°32'52"    316°00'47"      2°27'54"

Kepler's Second Law

Second, the rotational speed of Sun itself on the ecliptical plane is not a constant. It obeys Kepler's second law, with a rotational speed more or less inversely proportional to the Earth-Sun distance.

Q: You cannot apply Kepler's second law to a geocentric model!

A: No. Kepler's second law applies to a barycentric model as D above. It applies approximately to an heliocentric model as C. But once we have computed Earth's angular speed on its orbit around the Sun in model C, the computation of the Sun's coordinates and speed in the geocentric model is very simple. Especially, the Sun's angular speed in a geocentric model is equal to the Earth's speed in an heliocentric model.

Here are the Sun's positions for 2017, as given by Stellarium. The software gives the equatorial coordinates and I translate them into ecliptic coordinates.

date       right ascension         declination  ecliptic longitude
4 January  18h59mn1s 284°45'15"    -22°44'43"   -76°24'58" or -76,4162°
5 January  19h3mn24s 285°51'       -22°38'18"   -75°23'58" or -75,3996°
3 July      6h48mn   102°           22°58'35"   101°2'7"   or 101,0355°
4 July      6h52mn8s 103°02'        22°53'39"   101°59'26" or 101,9907°

This translates as a speed of 1.0166 degree per day in January at perigee (when in a geocentric model, that is perihelion in an heliocentric model) and a speed of 0.9552 degree per day in July at apogee (or aphelion). Values are respectively 0.0423°/h and 0.0398°/h.

Equation of Time

The Earth's spin velocity is constant, that is 360.9856 degrees per day but the Sun's orbital speed around the Earth is not. The combination of both speeds is variable and it is not 360 degrees per day. The crossing of the meridian by the Sun is not exactly every 86400 seconds. There is a difference between the Solar Mean Time, where noon occurs every 86400 seconds, no more, no less, and the Solar Real Time, in which noon is defined by the time when the Sun crosses the meridian. The difference between the Solar Mean Time and the Solar Real Time is called equation of time.

Here are the extreme values for the equation of time in 2017, computed by a script using DateTime::Event::Sunrise and refined with Stellarium.

Date          DT::E::S    Stellarium
2017-11-02    11:43:33    11:43:37   -16mn23s  earliest noon value,
2017-02-10    12:14:12    12:14:14   +14mn14s  latest noon value
2017-09-11    11:56:33    11:56:34    -3mn26s
2017-09-12    11:56:11    11:56:13    -3mn47s  biggest decrease: 21 or 22 seconds
2017-12-17    11:56:11    11:56:14    -3mn46s
2017-12-18    11:56:41    11:56:44    -3mn16s  biggest increase: 30 seconds

And here is the curve for the equation of time.

Curve of the equation of time during one year

The Analemma

The irregularity of the Sun's trajectory can be visualised by using the Local Mean Time as a reference and pinpointing the positions of the Sun at noon in LMT. The various Sun positions day after day build an 8-shaped curve, called analemma.

Mean Sun, Virtual Homocinetic Sun

In the following, it is useful to imagine a virtual Sun which would use an constant angular speed (either in equatorial coordinates or ecliptic coordinates, depending on which is more convenient).

The concept of Mean Sun is a virtual Sun like this, calibrated so it crosses the meridian at 12:00 (Local Mean Time) each day, and which minimizes the difference between the real local noon and the mean local noon.

I will also consider several "virtual homocinetic suns" or VHS (no relation with magnetic tapes). These virtal suns are synchronised with the real Sun at some convenient point and then move with a constant angular speed.

Computing Sunrise and Sunset

Computing sunrise and sunset consists in taking in account both the variation of day's length and the equation of time to pinpoint when the Sun reaches the altitude that corresponds to sunrise or sunset.

In the schema below, the variation of day's length results in a bobbing up and down of the sinusoidal curve (and less obviously, a vertical stretch or compression of this curve). The equation of time results in a leftward or rightward shift of the curve.

Evolution of the Sun's trajectory during a year

Q: Wahoo! Impressive!

A: You should not be impressed. I took some liberties with the reality. First, I figured the Sun's trajectory as a sinusoidal curve, because it is easy to compute, but I did not check whether it was the real curve. And I would bet that it is only approximately close to the real curve. Second, the equation of time is very much increased. Instead of a true solar noon varying from 11:43 to 12:15 (in mean solar time), here the variation is multiplied by 4 and the solar noon varies from 11:00, or even less, to 13:00. But without this stretching, you would not have seen anything.

Q: And this figure eight, is this the analemma?

A: No. The analemma gives the position of the Sun as azimuth and height at mean solar noon. In the curve above, the abscisse is the mean time of true solar noon and the ordinate is the height of the Sun at this instant. In other words, the analemma is based on a regular temporal event, the mean solar noon, and plots the correlation between two variable spatial phenomena, the azimuth and the height of the Sun. On the other hand, the curve above is based on a precise spatial event, the azimuth 180°, and plots the correlation between a variable spatial phenomenon, the height of the Sun and a variable temporal event, the true solar noon.

I admit that the ordinates of both curves are very similar notions, and it would be comparing golden apples with Granny Smiths. On the other hand, the abscisses are a spatial angle in one case and a time of the day in the other case, so it would be comparing apples with oranges.

Q: And the similarity of the shapes is juste a coincidence?

A: No, this is no coincidence. Let us start with the ordinates. The curve above, which I will call "pseudo-analemma", gives the height of the Sun at true solar noon, so the Sun is at its highest for the current day. Therefore, the height of the Sun in the analemma is obviously lower than on the pseudo-analemma (except during the 4 days when the curve crosses the Y-axis). But since we are near a point with an horizontal tangent, the variation is very small. For example, we consider an observer at Greenwich observatory on 2nd November 2017. At true solar noon (11h 43mn 37s), the Sun is at 23°37'39" while a quarter of an hour later, at mean solar noon it is at 23°31'40" (values given by Stellarium).

For the abscisses, it is a bit more complicated. Let us use the same example as above. At mean solar noon, the Sun's azimuth is 184°19'08", so on the analemma the dot for 2nd November is on the right of the Y-axis. On the pseudo-analemma, true solar noon occurs at 11h 43mn 37s, so the dot for 2nd November is on the left of the Y-axis. Not only the units of measure are not the same, but there is a change of sign. So the pseudo-analemma and the analemma are, approximately, the symetrical image of the other curve.

See below for a politically correct discussion of the analemma and pseudo-analemma.

Principle of the Iterative Computation

There are two models for the variability of the true solar noon from one day to the next. Let us take the example of an observer in Greenwich in September 2017. On 11th September, the true solar noon occurs at 11:56:34, with an altitude of 42°53'40" and the next day it occurs at 11:56:13 with an altitude of 42°30'47".

With the first model, we consider that the 11:56:34 value applies on the 11th from 00:00:01 until 23:59:59, at which time it instantly jumps to 11:56:13 for the 12th. In other words, the pseudo-analemma is a cloud of 365 discrete points.

Or else, we can consider that the true solar noon is a continuous function and that the pseudo-analemma is a continuous curve. When using the orbital parameters for the 11th at 11:56:34, the computed sunset occurs at 18:23:59. Since the true solar noon varies by 21 seconds over a timespan of 86379 seconds (one day minus 21 seconds), using linear interpolation, we can find that after 23225 seconds (i.e. at 18:23:59), the true solar noon has varied by 5.6 seconds. Likewise, the altitude varies by 22'53" in 86379 seconds and by 6'9" in 23225 seconds. So at 18:23:59, we have a virtual true solar noon of 11:56:28 and a virtual noon altitude of 42°47'31".

So we move a little bit the course of the Sun so it will be at its highest at his virtual point and we recompute the intersection between the new course and the horizontal line corresponding to sunset. The result will not be 18:23:59, but very near to this value and even nearer to the value from Stellarium: 18:23:24.

Implementation of Basic Algorithm

Before describing the precise algorithm, let us talk about the basic algorithm. We will use the example of the sunset at Greenwich, on 4th January, 2018.

This paragraph and the following are based on the Perl code below:

for(0, 1) { 
  say join( " | ",$_, sunrise({ year =>  2018, month =>  1, day => 4,
                                lon  =>    0,  lat   => +51.5, tz  =>  0, isdst => 0,
                                alt  => -.833, upper_limb => 0, precise => $_, polar => 'retval',
                                trace => *STDOUT } ));
}

The basic algorithm begins by computing the day's true solar noon. On 4th Jan, the true solar noon at Greenwich happens at 12:04:56.

Then we apply both Earth's spin (360.9856 degrees per day) and the movement of a VHS ("virtual homocinetic sun"), that is, 0.9856 degrees per day. The result is a combined rotational speed of 360 degrees per day, that is, 15 degrees per hour. And sunset happens when the VHS reaches the target altitude.

So on 4th Jan, the angle between noon and sunset is 59.9746° (59° 58' 28"). We need 3.9983 hours (3 h 59 mn 53 s) to run this angle and the sunset for the VHS occurs at 16:04:50.

Implementation of Precise Algorithm

With the precise algorithm, we keep separate Earth's spin (360.9856 degrees per day) and the Sun's rotational speed around Earth. In addition, this rotational speed is the real speed, spanning from 0.9552 to 1.0166 degree per day.

First iteration. We start from the true solar noon at 12:04:56 and we apply Earth's spin (15.04107 degrees per hour). The first result, a very approximate one, is the instant when Earth's spin brings the Sun to the target altitude. On 4th Jan, this first value is 16:04:11.

For iteration 2, we determine the virtual solar noon that corresponds to the Sun's position at 16:04:11. This virtual solar noon occurs at 12:05:01. With this reference, we apply the Earth's rotation and we get a second value for sunset, 16:04:23 (16.0731615074431 in decimal hours).

For iteration 3, we determine the virtual solar noon that corresponds to the Sun's position at 16:04:23. This new virtual solar noon occurs at 12:05:01. And one more time we apply the Earth's rotation and we obtain a third value for sunset, 16:04:23, differing from the previous value by less than a second: 16.0731642391519 instead of 16.0731615074431. The difference is 2,73e-6 hours, that is 9 ms, so we leave the computation loop.

This is one way to describe the algorithm: the Sun reaches by anticipation its position in the evening and stays there, waiting for the spinning Earth to spin until the Sun disappears below the horizon. Another way to describe the algorithm is as follows:

During iteration 2, between the real solar noon 12:04:56 and the time given by iteration 1, 16:04:11, the Sun orbitates with its real speed of 1.0166 degree per day while the Earth spins at 360.9856 degrees per day. Then, at 16:04:11, the Sun freezes in its track and after that, we adjust the position with only the Earth's spin to reach the required altitude. And the sunset occurs at 16:04:23.

During iteration 3, between the real solar noon 12:04:56 and the time given by iteration 2, 16:04:23, the Sun orbitates with its real speed 1.0166 degree per day. Then at 16:04:23, it freezes, letting the Earth continue its spin. And sunset happens 9 milliseconds later, at 16:04:23. So, there are 3 h 59 mn 27 s when we use the Sun's real orbital speed and 9 milliseconds when we use an obviously wrong orbital speed. In the end, it is better than the basic algorithm, which uses an approximate but still wrong orbital speed, but for the whole span of 3 h, 59 mn and 57 s.

What Happened in Spring 2020?

Let us look a bit farther into the past. In January 2019, I published version 0.98 of Astro::Sunrise, with the precise algorithm implemented as explained in the preceding paragraph. For test data, I did not know how to cross-check with authoritative sources, so I just checked they looked plausible and that the iterative computation stopped after a few iterations. At this time, I did not synchronise DateTime::Event::Sunrise with Astro::Sunrise, because it was not the proper time yet.

Then in April 2020, a user created an RT ticket, explaining that he had compared the results of DateTime::Event::Sunrise with authoritative websites and that the results were not precise enough. Of course the results were not precise, I had not yet synchronised DateTime::Event::Sunrise with Astro::Sunrise. Yet this ticket gave me two things: first, a website which would provide precise day-after-day computations of sunrise, sunset and real solar noon, and second a few round tuits to upgrade DateTime::Event::Sunrise to the same level as Astro::Sunrise, with real tests values.

There was a glitch. The precise algorithm copied from Astro::Sunrise and the test data from the NOAA website were not matching. After tweaking the algorithm and asking for advice on the DateTime mailing-list (https://www.nntp.perl.org/group/perl.datetime/2020/06/msg8241.html), I had to admit that using a 15-degree-per-hour spin speed was giving better results than the normal 15.04107-degree-per-hour value. So I coded the 15 value in DateTime::Event::Sunrise and I published the module.

What Went Wrong?

I still think my explanations of the precise algorithm are correct. I think that the error lies in the implementation of this algorithm. Using as an example the values above, during iteration 2 we should use the Sun at its 16:04:11 position and not moving. Without being certain, I think that actually, the program computes the virtual solar noon for 16:04:11, which is 12:05:01 and that after that it uses the Sun at its 12:05:01 position.

How can I check this hypothesis? I need to "create" a new fixed star at the 16:04:11 position of the Sun. And I do not know how to do that in Stellarium. A few months later, I read the documentation for Stellarium 0.20.1-1. In chapter 13, I found that the user can add unofficial novas to the novas coming from the official star catalogues. This would answer my need. The problem is that I have Stellarium 0.18.0 and that I did not succeed in creating "my" nova in this version. So for the moment, it is impossible to check the implementation of the precise algorithm.

More About The Parameters

Below, I give some detailed explanations about the parameter used when calling the module's functions. These explanations would have been too long if they had been included in the module's POD and a casual doc reader would have been drowned in a deluge of informations.

Choosing The Algorithm, `precise` Parameter

Q: When should I choose the precise algorithm?

A: The short answer is "Never". The long answer is the following:

If you want some twilight, use the basic algorithm.
If you live between the polar circles, use the basic algorithm.
If the date is far from a transition between day+night and either polar night or midnight sun, use the basic algorithm.
If the date is near a transition between day+night and polar night, use the basic algorithm.
If you live in a polar location AND the date is near a transition between day+night and midnight Sun AND you are interested in the visibility of the Sun's disk above the horizon, then you may use the precise algorithm.

Note that if you live a bit southward of the arctic circle (say, Reykjavik), you should use the precise algorithm around the 21st of June, even if midnight sun does not happen there. Same thing aoround the 21st of December if you live a bit northward of the antarctic circle.

Q: And can we know why the use of the precise algorithm is so narrow?

A: Let us go back to the animated picture of the solar course curve that moves along the pseudo-analemma. But instead of using a location at Greenwich, we use a polar location still at longitude zero, but at 76 degrees and 59 minutes from the equator.

Evolution of the Sun's trajectory during a year (arctic variant)

As you can see, around 21st April and 21st August, the solar course is tangent or nearly so with the line of horizon. With these conditions, a variation of 6' of the solar altitude can produce a much bigger variation of the points where the solar course crosses the horizon. For example, on 20th April 2017, at sunset time, we need 8 mn 18 s to achieve this variation of 6'.

Q: Where does this 6' value come from?

A; This is the value I calculated in the chapter "Principle of the Iterative Computation".

On the other hand, if you live in a temperate location far from the arctic circles, the slope of the solar course when crossing the horizon is always a bit steep. For example, at Greenwich, the shallowest slope occurs at each solstice and is about 6 or 7 degrees of altitude per hour. So a variation of 6' shifts the sunrise and sunset times by only 50 seconds or less.

The diagram below shows the effect of a 6' vertical translation on the solar course in two cases: 20th April at 76° 59' N and 21st December at Greenwich. Warning: it is not to scale.

Sun course, comparison between Greenwich on 21/12 and latitude 76 on 20/04

Q: And what happens to people living in polar locations when the date is far from any transition?

A: If the period is day+night, the explanations above about the steep enough slope of the curve still apply. If the period is the polar night or the midnight Sun, then the course of the Sun never crosses the horizon and any variation of altitude, within limits, cannot create an intersection with the horizon.

Q: For the transition with the polar night, the solar course is tangent to the equator, like it is at the transition with the midnight Sun period. So, why do you still advise to use the basic algorithm in this case?

A: The basic algorithm and the precise algorithm both try to estimate the ecliptic longitude and the altitude of the virtual noon sun at the time of sunset. But while the precise algorithm uses the real orbital speed which varies from 0.9552°/d to 1.0166°/d, the basic algorithm uses a constant speed of 0.9856°/d, with an error of ±0.0310°/d. For the transition between day+night and midnight Sun, this error runs for more than 11 hours, which might result in an error of 0.015° on the Sun's ecliptic longitude. But for the transition between day+night and polar night, the error runs for one hour or less, yielding an error on the ecliptic longitude of 0.0013° only. So even if a small error on the sun altitude gives a big error on the sunset time, at the transition with the polar night, you will have a tiny error, not a small one.

Q: And what about the computation of twilights? We can encounter a situation where the solar course is tangent to the horizon, if I may use this word for a line situated 24 degrees below the horizontal plane. And we have a ≈12 hour gap as for the midnight Sun transition, not a 1-hour gap as for the polar night transition.

A: Why do you compute twilight times? Because you want a low enough light level and good enough conditions to observe celestial bodies. Do you think there is a big difference between a night when the Sun is at its lowest at 17°57' below the horizon and a night when it is at its lowest at 18°3' below? In some circumstances, you'd better begin your observations when the Sun is at, say, -15° than to wait for the -18° twilight if you know that the Moon will rise at the same time or if a weather report gives a warning about an incoming overcast layer.

`alt` Parameter (Altitude) and `upper_limb` Parameter

Parameter alt, in decimal degrees, allows you to specify the altitude of the sun corresponding to the event you are looking for: sun rise, sun set or a twilight of some degree.

First, the various twilight values: they are arbitrary values. Nothing to add.

For sunrise and sunset, the usual value is -0.833°, that is, -50'. It comes from two values:

-0.583° or -35' for the refraction,
-0.25° or -15' for the radius of the solar disk.

Actually, in both cases, these are approximate values for variable phenomena. The radius of the sun disk varies between 0.271° (or 16'16") on 3rd January, when the sun is closest from the Earth, and 0.262° (or 15'44") on 3rd July, when it is farthest from the Earth.

And the deviation caused by refraction is highly dependent on the temperature profile in the atmosphere, therefore dependent on the current weather.

You can refine the computation by using value -0.583 for the alt parameter and giving a true value to the upper_limb) parameter (usually 1). In this case, the radius of the sun disk is computed anew for each day. On the other hand, there is no way within the module to refine the computation of the refraction. The answer can only come from outside the module computing sunrises and sunsets.

Q: So using a parameter alt with -0.583 and a parameter upper_limb with 0 is stupid?

A: No, it is just unusual. As stated by Paul Schlyter in his website, the Swedish national almanacs define sunrise and sunset as the instants when the centre of the sun disk reaches the optical horizon, not the instants when the upper limb of the sun disk reaches the optical horizon.

Q: On the other hand, using a parameter alt with -0.833 and a parameter upper_limb with 1 is stupid?

A: I would say neither "stupid" nor "unusual", but "suspicious". Actually, the usual -0.833 value comes from three elements:

0 for the position of the observer relative to the surrounding landscape,
-0.583° or -35' for the refraction,
-0.25° or -15' for the radius of the solar disk, or 0 if you give a true value to upper_limb.

But if you are at the top of a 60-meter tower, the horizon line no longer coincides with the horizontal plane, but with a cone making a 0.25° angle with the horizontal direction. In this case, we have:

-0.25° for the position of the observer relative to the surrounding landscape,
-0.583° for the refraction,
0° for the radius of the solar disk if you give a true value to upper_limb, or -0.25° for a false value.

So this parameter combination is valid, but it is suspicious because it does not correspond to a usual and mundane situation.

Q: After your explanation about the precise parameter, is there a significant difference between alt => -0.833, upper_limb => 0 and alt => -0.583, upper_limb => 1?

A: You guessed, it, there is nearly no difference. The example I will take is sunset at Fairbanks on 3rd January 2020. I take 3rd January because it is the time of the year when the sun is at its largest. According to Stellarium, the diameter is 32'32", so the radius is 16'16". And I take Fairbanks, because near a polar circle, the course of the sun at sunset is much shallower than near the equator. So, for alt => -0.833, upper_limb => 0 the sunset occurs when the center of the sun dist is at -50' and for alt => -0.583, upper_limb => 1 it occurs when the center of the sun disk is at -51'16". Stellarium gives 15:59:12 in the first case and 15:59:37 in the second case. A meagre 25-second difference.

On the other hand, let us move a few hundred kilometers to the North, to 68°01'46" N, 147°42'59" W, beyond the polar circle. The computation with alt => -0.833, upper_limb => 0 shows that the sun stays below the horizon and neither rises nor sets, while the computation with alt => -0.583, upper_limb => 1 gives a sunrise at 12:45:11 and a sunset at 13:05:53.

`year` Parameter

Q: Why does Astro::Sunrise need the year to compute sunrise and sunset? I have seen an algorithm which only needs the month and the day.

A: Let us compute the sunset at the Greenwich observatory at the end of February and at the beginning of March. The times are respectively:

.     26/02     27/02     28/02     29/02     01/03     02/03     03/03
2015 17:46:17  17:35:43  17:37:29            17:39:15  17:41:01  17:42:47
2016 17:47:36  17:35:17  17:37:03  17:38:49  17:40:35  17:42:21  17:44:06
2017 17:47:11  17:36:37  17:38:24            17:40:10  17:41:55  17:43:41
2018 17:46:45  17:36:12  17:37:58            17:39:44  17:41:30  17:43:15
2019 17:46:20  17:35:46  17:37:32            17:39:18  17:41:04  17:42:49
2020 17:47:39  17:35:20  17:37:06  17:38:52  17:40:38  17:42:24  17:44:09
2021 17:47:13  17:36:40  17:38:26            17:40:12  17:41:58  17:43:44
2022 17:46:48  17:36:14  17:38:01            17:39:47  17:41:32  17:43:18
2023 17:46:23  17:35:48  17:37:35            17:39:21  17:41:07  17:42:52

As you can see in this table, when we go forward by 365 days, the sunset time decreases by about 25 seconds. When we go forward by 366 days, the sunset time increases by about 1 mn 20 s. And if we go forward by one civil year, the sunset time seesaws. So, for the 28th of February, which result should your yearless algorithm give? 17:37:03? Or 17:38:26?

Q: That means that my algorithm is bad.

A: No. If you want to know the precise instant when the Sun disappear from our field of view, your algorithm is wrong indeed. On the other hand; if you are only interested in the level of light, your algorithm is OK. I know a person who uses a yearless algorithm to activate automated lights in his living room. For him, turning on the lights at 17:37:03 or 17:38:26 has no importance. Under our latitudes, the light variation in two minutes is negligible. Actually, the weather may have a much important effect. If you have a clear sky or a heavy layer of black thunder clouds, you will have to light later or sooner than the computed time.

Q: By the way, you seem to say that a yearless algorithm would be sufficient to compute twilight times? So why use an algorithm requiring the year?

A: Firstly, because the basic algorithm was already coded and a slightly worse algorithm would be redundant. Then because I do not know which licenses apply to these yearless algorithms, while at the same time, Paul Schlyter's algorithm is in the public domain.

Q: Another thing, how did you get the seconds in the table? Astro::Sunrise does not provide the seconds.

A: Because I have used DateTime::Event::Sunrise instead of Astro::Sunrise, which produces DateTime objects, complete with seconds.

Q: Could we modify Astro::Sunrise to give "hh:mm:ss" results instead of "hh:mm"?

A: We could. But would this precision be meaningful? According to Paul Schlyter, the algorithm precision is about 1 or 2 minutes, except at the beginning and the end of the Polar Day period when the precision is much worse. So it is not worth adding the seconds to the results produced by Astro::Sunrise.

Q: And in the discussion above, why did you keep the seconds, if they are not significant?

A: Because I think that if there is an error, it will be the same error for similar dates, that is, end-February and beg-March within a decade. For instance, we may have a +45 s bias on 2015-02-28 and a -50 s bias on 2015-10-28 and on 2050-02-28, but for all the dates similar to 2015-02-28 in both a YYYY fashion and a MM-DD fashion, the bias will be approximately the same as 2015-02-28. Maybe +43 s or +46 s instead of +45 s, but surely not -50 s. So I can make comparisons with a granularity of 1 second. By the way, the bias values I gave above are complete guesses, they are not the result of a precise computation.

TO BE COMPLETED

Annex: Politically Correct Explanations

Policitally Correct Analemma

First, let us deal with observers located north of the Arctic Polar Circle. They just have to know that the analemma and the pseudo-analemma cross the horizon and are partly hidden by the ground. The hidden part, more or less important depending on the observer's latitude, corresponds to the year period when the polar night is in effect. You can find an example of the arctic pseudo-analemma in the paragraph about the precise parameter.

For observers between the Tropic of Capricorn and the Antarctic Polar Circle, this is more strange. True solar noon corresponds to a right ascension of 0°, when the Sun is exactly northward. As the observer must face north instead of south, he sees the sun crossing the sky in the direction E → N → W, that is, in the direction of decreasing right ascension values. So, when the true solar noon is ahead of the mean solar noon, the point on the analemma will be to the left of the Y-axis, and when the true solar noon is later than the mean solar noon, the point on the analemma will be to the right of the Y-axis.

On the same time, for the pseudo-analemma, there is no reason to change the way the time of day is represented on the abscisses, that is, left to right. Therefore, the analemma and the pseudo-analemma will be more or less superposable, without an intervening symmetry.

For observer to the south of the Antarctic Polar Circle, the situation is the same, with the additional provision that the analemma and the pseudo-analemma will be partly hidden by the ground.

And what about observers located between both tropics? An observer facing south cannot see the whole analemma, he would miss the part around 21st of June, which is located behind his back. And if he faces north, he will miss the part around 21st of December. What to do then? Just lie on the ground. If the observer lies with the head to the north and the feet to the south, the observed analemma will be similar to the curve seen by an observer north of the Tropic of Cancer. If the observer lies with the head to the south and the feet to the north, the situation will be similar to an observer on the south of the Tropic of Capricorn and facing north.