Day-night border

Earth and moon both approximately in dichotomy (half phase at phase angle 90 °; image taken by the space probe Galileo [NASA])

The sun seen from the day-night border (sunrise over the Hoggar in southern Algeria)

The day-night boundary is the boundary between the light side illuminated by a star such as the sun and the unlit shadow side of a celestial body , in particular a planet or moon . In astronomy and meteorology , the sight of this light-shadow boundary is also called a terminator ( Latin for 'delimiter') or separator (Latin for 'separator').

The position of the terminator in relation to a viewer results from the phase angle and determines the phase of the celestial body, its light shape for the viewer. The phases of the moon at the phase angles of 0 °, 90 ° and 180 ° are called full, half and new moon.

Basics

Cause and appearance

Study of the crescent moon by Leonardo da Vinci, red chalk, before 1510

If a light source illuminates a body, all areas of its surface that cannot be directly reached by the light are in shadow . If one assumes, for the sake of simplicity, that parallel rays of light hit a sphere, the sphere is divided into a hemisphere in the light and a hemisphere in the shadow . The dividing line forms a great circle . Due to its spatial position and the perspective distortion, this generally appears as half an ellipse , the other half of which lies on the back of the body. At special viewing angles, the dividing line appears as a straight line or a circle.

The best known example of such a borderline between the light and shadow side of a celestial body is the inside of the crescent moon , which changes its shape with the phases of the moon. Leonardo da Vinci described the nature of moonlight as sunlight reflected by the moon ball - and thus also this line - with the words:

“The moon has no light by itself,
and as much as the sun sees of it, it illuminates so much;
and we see as much of this lighting as much of it sees us "

- Leonardo da Vinci

There is also such a terminator on earth, which separates the sunlit side from the dark night side . However, there are additional periods of time when the sun is just below the horizon, but when the sky is still bright due to the scattering of sunlight in the earth's atmosphere , the twilight . As a result of this, the earth's day-night boundary, viewed from outside, is not a sharp line, but rather a clear blurring. At night it has passed into a zone with a width of up to a few hundred kilometers, the twilight zone .

There is no atmosphere on the moon and therefore no twilight. When the sun has set there, it suddenly becomes dark. The terminator of the moon thus shows sharp light-shadow borders on the surface relief. However, there is also a certain blurring of the terminator due to the roughness of the terrain, as well as the fact that the light source (the sun) is not point-shaped and thus its light intensity gradually increases and decreases during sunrise and sunset. The latter effect is strongest with bodies close to the Sun such as Mercury , because of these bodies the apparent diameter of the Sun is the largest.

Movement and meaning

Because the celestial body orbits its central star on a certain orbit and thereby executes a certain rotation (with a certain inclination of the axis to the plane of the orbit), the day-night boundary moves in a certain way over the surface of the celestial body. It shows where there are lights day or night on the surface of the celestial body . The passage of the day-night boundary through a given location defines the beginning and end of the clear day for this location and thus also the length of the day and the duration of the sunshine .

On earth, the day-night boundary moves west near the equator at around 1,670 km / h and reaches the same position again in the same location after about 24 hours. The time span for this one cycle of day and night is the same for all places on earth if they are not in the polar regions. Due to the inclination of the earth's axis towards the ecliptic , it can happen in the geographical latitudes beyond the polar circles that there is no change between day and night for 24 hours - because the day-night boundary does not pass the location. It is then polar day or polar night .

Image examples

The terminator on Mercury
Terminator of the Moon ( Earth's Moon ) -
details of relief with craters
Double terminator on Mimas (Saturn moon I) - above solar terminator, right through reflection (indirect sunlight) of Saturn
Terminator of Titan (Saturn's moon VI) - a blurred zone, as well as the appearance of its atmosphere.
Cassini , April 2007

Phase angle Φ and phase k

Depending on the angle that the light source and the observer form in relation to the illuminated celestial body, a part of its illuminated half of different sizes becomes visible to the observer as a light form of a certain phase. This relationship can be described mathematically via the angle in space, the phase angle, and also via the visible portion of the illuminated pane surface, the phase.

If the observer were standing at the location of the light source or on the line of the projection axis, the phase angle would be 0 °, he would then see exactly the illuminated half, the pane would appear completely illuminated (illuminated part i.e. 1 of 1 or 1/1), thus the phase would be equal to 1 and would be called “full phase”, for example full moon for the moon.

New moon, on the other hand, or "new phase" is phase equal to 0 (proportion 0/1 or 0 of 1), so no part of the disc is illuminated, the observer sees exactly the unlit half and stands in relation to the light source so that the celestial body is exactly on one Line lies in between, the phase angle is now 180 °.

The phase angle is measured between the light source, the sun (S) and the observer (B), in relation to the celestial body as the object (O) at the vertex - seen from the object, it is the angle at which S and B appear:

{\ displaystyle \ Phi = \ sphericalangle {\ overline {OS}} \, {\ overline {OB}}}

If the distance values are known when viewing a distant object, the phase angle can be calculated from them:

{\ displaystyle \ Phi = \ arccos \; {\ frac {r ^ {2} + \ Delta ^ {2} -R ^ {2}} {2 \ cdot r \ cdot \ Delta}}}

with (illuminated) (not illuminated)

{\ displaystyle 0 ^ {\ circ} <\ Phi <180 ^ {\ circ}}

Let:

{\ displaystyle R}

... distance from the barycenter / light source to the observer

{\ displaystyle r}

... distance from the barycenter / light source to the object

{\ displaystyle \ Delta}

... distance from the observer to the object

Both terminator and phase can be specified via the phase angle.

The phase may, in addition, as already mentioned as the illuminated portion of the apparent total reproduced, or as a lighting defect be construed:

{\ displaystyle k = \ arccos \; {\ frac {(r ^ {2} + \ Delta) ^ {2} -R ^ {2}} {4 \ cdot r \ cdot \ Delta}}}

with (illuminated) (not illuminated)

{\ displaystyle 1> k> 0}

The phase angle is only exactly 0 ° or exactly 180 ° if the three objects are exactly on one line. However, since this does not occur in practice, the phase angle only reaches minimum values close to 0 ° or maximum values of almost 180 °.

If the inclination and eccentricity of the path are neglected, the geometric phase angle discussed here (supplemented by a sign for a value range from −180 ° to + 180 °) corresponds to the phase angle for describing the periodic process. ${\ displaystyle \ Phi}$ ${\ displaystyle \ omega t}$

Phase angle and elongation

The phase angle lies on the line of sight from the celestial body to the observer and corresponds to the elongation under which the observer would be seen when viewed from the object. The elongation seen by the observer, on the other hand - i.e. the angle at which the object and light source appear to him - does not determine the phase, but is used to calculate the apparent distance from the shining star to the celestial body and its visibility . In particular, at phase angles close to 180 ° - and thus very little elongation for the observer - a passage or an occultation can occur.

Main phases and phase cycle

Shining light of the earth in the course of the year or earth phases for a heliocentric stationary observer in space (not true to size )

The phases depend on the position of the observer in relation to the celestial body and the sun as its constellation and can regularly follow one another in a cycle; the duration of an entire phase cycle related to the observation site is called the synodic period .

Depending on the phase angle, different phases appear, which are named after their light shape; the following are also referred to as main phases and represent special constellations (like the syszygies as positions on a common level or on the same line).

Full phase, if the sun and observer are in the same direction in the same plane as seen from the celestial body, the observer only sees the day side of the celestial body and the terminator falls on the outer edge of the disc seen. This phase is called full . For an observer on earth, for example, inner planets are then in upper conjunction , outer planets either in conjunction or in opposition , the moon in opposition. ${\ displaystyle \ Phi _ {\ mathrm {min}} = 0 ^ {\ circ}:}$

Half phase, at a phase angle of 90 ° the observer sees the terminator in such a way that it appears as a straight line. The disc is apparently divided in half, a semicircular day side and a night side. This phase is also called a half-phase or dichotomy ( dĭchŏt ‚ mos 'cut in two '); it does not occur with the outer planets. This position is not to be confused with the quadrature . ${\ displaystyle \ Phi = 90 ^ {\ circ}:}$

New phase, when the phase angle is almost at its maximum, the celestial body and the sun are in the same direction as seen from the observer. Inner planets are then in lower conjunction, the other planets and the moon in conjunction, between sun and observer; they turn their night side to him. The (ecliptical) elongation in the observer is then 0, so this phase usually forms the zero point for calculating synodic periods in technical usage. This phase is called new only in relation to the moon and remains otherwise unnamed; it is only possible when the celestial body steps between the observer and the light source, like the earth's moon or inner planets. The prefix 'new-' is otherwise only used for the term new light , for the narrow crescent moon about a day and a half after the new moon. ${\ displaystyle \ Phi _ {\ mathrm {max}} = 180 ^ {\ circ}:}$

Between these main phases, the terminator has the shape of an elliptical arc and gives the light shape the well-known sickle shape or an outline wrongly called "egg-shaped", both before the full phase (increasing) and after the full phase (decreasing) within a phase cycle.

The phase is usually specified geocentrically - ecliptically : The times for true observation on the earth's surface deviate slightly from it, even for the exact date of the moon, new moon. ${\ displaystyle \ Phi _ {\ mathrm {max}}}$

Geometric figure of the terminator

On a spherical body, the projection of light forms a round surface, the edge of which is a circle. In the case of a point source of light, its diameter increases with increasing distance from the body and in the limit of infinite distance - with parallel incident rays without deflection - would be greatest, that of a so-called great circle . The exposed half of the spherical surface would then be set off from an equally large unexposed hemisphere. The same division would also come about with a flat light source that emits at right angles and as a disk with the same radius as the body is at right angles to it - regardless of the distance. The actual projection conditions can in each case be understood as an approximation to these ideal cases and the incidence of light rays then simplified as approximately parallel.

In celestial bodies without an atmosphere, light is not deflected and upon incidence still after reflection so illuminated on a spherical body having a surface with a circular boundary of the reflected light then an image designs than the sight of the body, in astronomy also disc ( English disk ) called .

Depending on the distance of the observer and the angle at which the illuminated hemisphere is seen, the boundary line - the terminator - appears on the image as a circle, an elliptical arc, a straight line or not at all. The edge of the pane as the outer circumference of the pane and the terminator as the inner division of the pane surface together determine the light shape of an illuminated celestial body, its sight or aspect. Depending on the perspective or phase, the disc appears fully, half or not illuminated when the terminator turns from an elliptical arc to a circle or a straight line or invisible. The main axis of the ellipse , as the arc of which the terminator appears, is equal to the apparent diameter of the disk, the minor semi-axis is calculated approximately to . ${\ displaystyle d}$ ${\ displaystyle d \ cdot k}$

That a fine crescent acts sometimes hörnchenfömig, for Luna cornuta is as if the tips bent inward, is due to glare effects ( irradiation ) and the elevation of contrasts by our visual perception; so the bright narrow moon shape is oversubscribed lifted from their different dark environment, on the one hand the night black background, on the other hand that of earthshine dimly lit portion of the moon. In addition, the moonlight is still scattered on the way through the earth's atmosphere. The squirrel shape of the earth crescent on some images from space is mainly due to refraction and scattering effects - the snow-covered polar zones appear brightened from space and appear as elongated peaks. The squirrel illusion becomes even stronger with Venus, which also has an atmosphere, but whose dark side always remains invisible because it does not receive any reflections from a nearby celestial body.

No elliptical border, no phase, but: lunar eclipse March 3rd 2007, entry into the umbra of the earth

The ideal shadow limit can be calculated as follows (program code: C ):

xscale = cos(2 * PI * ph)
for (i = 0; i < RADIUS; i++) {
	cp = RADIUS * cos(asin((double) i / RADIUS));
	if (ph < 0.5) {
		rx = (int) (CENTER + cp);
		lx = (int) (CENTER + xscale * cp);
	} else {
		lx = (int) (CENTER - cp);
		rx = (int) (CENTER - xscale * cp);
	}
}

World map with day-night boundary, shortly after the (northern) spring equinox: Polar day already rules at the North Pole.

[rx, i]… coordinates of the dark disc
edge [lx, i]… coordinates of the terminator
as Cartesian coordinates (here: whole numbers, pixel by pixel)
with:

ph ……… phase

RADIUS ... radius of the disc

CENTER ... x coordinate of the center point of the pane

Source: J. Walker's Moontool , after Meeus / Duffett-Smith

The terminator or separator then takes on more complex forms as it is projected onto a map. Projected onto a flat map , the light-shadow borderline shows a curved course in the form of a distorted sine curve and can break up into two parallel lines on the day of the equinox .

Incidentally, the actual conditions must be taken into account, under which the day-night boundary only approximately forms a great circle or the terminator an ellipse with a major semi-axis of only almost its radius: because the rays emanating from a point source do not quite illuminate one Half of a sphere, refractions in an atmosphere shift the boundary line, scattering makes it blurred, and two-dimensional expansions of the light source - such as the sun - lead to the incidence of light from different angles or can illuminate more than exactly a hemisphere. For general astronomical considerations, however, rays of light from the sun may be assumed to be incident in parallel because of the great distance compared to the diameter of the illuminated celestial bodies.

Celestial mechanics

On the illuminated celestial body, a star highlights one side in the light and differentiates it from the one in the shadow. The delimitation of the light and the shadow side is viewed from two fundamentally different angles depending on the location of the observer:

In the external perspective, observing the reflected light of an illuminated celestial body, the light-shadow boundary becomes the terminator that determines the appearance of the light figure. Within the circumference of the disc, it sets off light from dark and thus marks the phase. This view depends on the solid angle formed by the star and celestial bodies in relation to the observation location, for example the sun and moon to earth. This angle hardly changes when they rotate around their own axis, but it does change when the moon and earth move around the sun. The phase changed as a result is shown on the observed lunar disc as a displacement of the terminator. A continuous series of such phase changes up to repetition forms a phase cycle, which in this case is also called a month.

In the internal perspective, i.e. directly exposed on the surface of the illuminated celestial body of the light source, the boundary between light and shadow becomes the day-night boundary, which determines whether a location on the surface is exposed and whether direct light can reach the ground. There may be forms whose activity is influenced by light energy - be it that they evade the damaging effect or that they use energetic effects for themselves. Whether it is day or night depends on the angle made by the star and surface of the celestial body in relation to the position of the observer. On earth, for example, it would be the position of the sun above sea level on a beach. This angle changes during the orbital motion of the earth in orbit, and also when it rotates around its own axis. This shifts the day-night boundary on the earth's surface; the continued shift with two alternations between light and dark up to the repetition of a similar exposure phase at the same place then forms a whole day-night cycle, which is also referred to as full day.

The day-night boundary can shift when the body is in motion. If it ran around the light source without its own rotation, the light-shadow boundary would move twice over its surface during one full cycle. For a stationary observer on this body a day-night cycle would have expired, which lasted as long as a year on this orbit. Assuming that the body would now rotate exactly once around itself during one orbital period - and this around an axis that is at right angles to the plane of the orbit - there are two possibilities: If it rotates in the same direction as it orbits, then turns He constantly faces the same side of the rotating light source. With such a bound rotation, its day lasts, if not forever, until the star as a source of light goes out - if the body then still exists. However, if it rotates in the opposite direction to the direction of rotation, then its year consists of two days (and two nights).
Depending on whether or not the direction of rotation of a body coincides with the direction of rotation in the orbit , either one day can be omitted or one day can be added for an orbital period in its year. In other words: The number of complete turns of its own is never equal to the number of days in relation to the central star, but either plus or minus 1 within a year. In the case of the earth, this means that a solar day does not correspond to a whole period of rotation, but about 1/365 of it is missing.

If the axis of rotation of the celestial body were exactly perpendicular to its orbital plane, i.e. with an inclination equal to 0, the day-night border would run exactly through both poles . In this case there would be no difference in the time span of day or night with regard to the geographical latitude on this celestial body, because within a solar day the day-night limit would be exceeded twice for all locations.
However, if the axis of rotation is not exactly perpendicular to the plane of the orbit of the celestial body, the periods of light day or night change from one day-night cycle to the next. The further a place is from the equator , the greater the differences that occur during a walk around the sun. Around the poles there are even areas of the surface within which the day-night boundary is not exceeded at all during some rotation periods. The boundaries of these areas define the polar circles .

Earth terminator

Blurring of the earth terminator
from the ISS , altitude about 390 km (211 NM )

On the way through the Earth's atmosphere light is refracted and scattered, so due to refraction (there refraction ) by about half a degree reached another area of the earth's surface and because of its diversification next twilight zones occur. The light-shadow border on the globe therefore covers a larger area during the day than at night and is also blurred at night.

Part of the light is thrown back from the exposed surface of the earth ( reflection ), as diffuse reflection - depending on the nature of the surface and the angle of incidence, a different proportion. This albedo of the earth is currently around a third (approx. 31%) of the incident sunlight, with considerable differences in the reflectivity, for example, of new snow cover (approx. 85%) and of water levels at angles above 45 ° (approx. 5%). The reflected light then passes through the atmosphere a second time with refraction and scattering, before an image of the day-night boundary is created as a terminator on the earth's disk - seen from outside with a perspective from space.

The Earth's axis of rotation is not perpendicular to the plane of the orbit, but at an angle of about 66.5 °, the skew of the ecliptic is about 23.5 °. While the terminator traverses a point on the equator almost exactly every 12 hours, so that bright day and night last about the same time there, the effects of the ecliptic inclination produce significant differences in the lengths of day and night even in the moderate latitudes.

At the solstices or solstices , the bright days in the northern hemisphere are longest in summer ( summer solstice , around June 21) - like the nights in the southern hemisphere - and the days in winter are shortest ( winter solstice , around June 21). December) - while it is the longest day in the southern hemisphere. About a quarter of a year after these dates, bright day and night are of equal length (equinox or equinox , around March 20 and September 23, respectively). The lengths of the day fluctuate in moderate geographical latitudes, for example from 45 °, approximately between 8 hours and 16 hours, corresponding to the night lengths. From about 66.5 ° north or south latitude, the polar circles , polar days with midnight sun or the polar nights can occur.

Towards the poles, the duration of twilight also increases - at 45 ° latitude it is up to an hour and a half, as astronomical twilight around the winter solstice; Transferred to the earth's surface, this corresponds to a zone of over a hundred kilometers and the so-called white nights therefore also occur in latitudes below the polar circles.

The diverse phenomena at setting and rising of the sun arise as astronomical phenomena through scattering and refraction in those layers of the atmosphere that are intersected by the geometric terminator plane .

Lunar Orbiter 1 took first pictures of the moon and the earth's sickle on August 23, 1966
Seasonal fluctuation of the day-night boundary in relation to the ecliptical plane
Day-night boundary of sunset in the course of the year, but now shown geostationary for Central Europe
Scattered light illuminates the earth's atmosphere as seen from the ISS at the level of the terminator

Terminator of the moon

Details on the terminator of the moon

When the sun has set, it suddenly becomes dark on the moon - but the sunset on the moon takes several hours due to its slow rotation . But the terminator forms a sharp line. From the earth the migration of the day-night boundary of the moon can be followed very well: It lasts for a month , a span of the moon phases from full moon - waning crescent - new moon - waxing crescent - to full moon. During this synodic lunar month , the terminator advances a few degrees every night (an average of around 12.6 ° in 24 hours).

The shadow line of the moon, which is approximately at right angles to the ecliptic, is usually seen from the middle geographical latitudes of the earth in such a way that the moon seems to increase or decrease to the left or right. A crescent moon lying exactly horizontally to the horizon, also called a moon boat , can be seen from regions closer to the equator below almost 29 ° latitude, if the lunar orbit is perpendicular to the horizon. Even at a higher degree of latitude, the orbit plane of the moon, which is inclined by a good 5 ° against the ecliptic, may still reach relatively steep positions to the horizon - fluctuating by around 10 ° over the course of almost 19 years and at most at the great lunar turn - but no more vertical ones and so appears a lying crescent moon is mostly tilted a bit. Only very fine sickles can be seen lying horizontally under special conditions with a high ecliptic latitude of the moon.

Compared to locations in the northern hemisphere, the moon figure now shows itself in the same phase from the southern hemisphere in the reverse horizontal orientation - not because of the terminator, which has almost the same position, but because the observer is upside down compared to observers in the northern hemisphere or vice versa and both have the same shape of the moon light depending on their topocentric reference system with different zenith situate. The waning moon seems to point one to the left and the other to the right.

A common deception common to the moon observer is that, for example, the crescent moon in culmination does not seem to point exactly to the sun, which is then just below the horizon, but supposedly points over it: the straight line connecting the sun and the moon seems to be curved. That is an error of interpretation. In fact, it is the effect of perspective.

If the phase angle is not exactly 0 °, the celestial body does not appear completely illuminated. The moon, for example, is not fully illuminated even at full moon , typical values are 99.96%; only with the smallest ecliptical latitude - as with the lunar eclipses - would it come close to its maximum possible illumination, but then it is eclipsed by the earth's shadow and is therefore not fully illuminated.

Barely visible crescent moon of the rising, waning moon a good day before new moon shortly before sunrise

The theoretically visible very narrow crescent of the phase shortly before or after the new moon is actually difficult to perceive, because the moon is then outshone very close to the sun and thus standing with it in the daytime sky - with the exception of a very short period at dusk. The fine crescent of the waning moon just before sunrise and that of the waxing moon just after sunset can be observed. The sighting of the crescent moon for the last time before or for the first time after the new moon forms the basis of many time and calendar calculations, such as the old light of the ancient Egyptian calendar and the new light of the modern day Islamic calendar. The sun is remote part of the moon can be viewed by us as reflections of sunlight on the Earth and its atmosphere, the Earthshine , shine on the moon indirectly and not entirely leave so the erdzugewandte dark side of the moon in the dark. The fact that the crescent moon can be perceived as croissant-shaped is called Luna cornuta .

The terminator of the moon is of great importance for selenography (lunar mapping): the razor-sharp line breaks up at its edge into numerous edges of the craters and other elevations, which at the day-night border in the shadow area are just illuminated as bright arcs from the take off the unlit moon surface or lie in the dark and appear as darkenings in the day side. Known light-shadow effects are for example the Golden Henkel or Lunar X .

Up until the 2000s, accurate drawings of the shadows cast and their mapping on maps were the only way to determine the three-dimensional topography of the moon. Since the Lunar Orbiter missions and the high-resolution photographic mapping of the moon, only circles of committed amateur astronomers have been working with this method to clarify detailed questions about individual surface shapes. Only with probes such as the Chinese Chang'e-1 2007 and the Lunar Reconnaissance Orbiter 2009, which were equipped with stereo photography and laser scanning for altimetry , the detailed lunar topography is no longer dependent on visual terminator observation.

Detail photo of the moon (mosaic of 5 photos) February 10, 2008, France
The same, overexposed: Clearly recognizable some deviations from the ideal elliptical shape

See also: ComputerHotline - Lune-2008-05-14 - Passage of the moon through a fixed camera (ogg-video, 1:30)

Web links

Wiktionary: Terminator - explanations of meanings, word origins, synonyms, translations

Commons : day-night boundary - collection of pictures, videos and audio files

World Sunlight Map. Shows the current day-night boundary with current cloud images. Also in Mercator , Mollweide , Peters and square projection .
John Walker's Earth and Moon Viewer shows the day and night halves on a map, adjustable to any time and coordinates (online version of Home Planet ).
Desktop Earth. Real-time desktop backgrounds. Automatically updated images of the day-night boundary as desktop wallpaper.

Individual evidence

^ JP Richter: The Notebooks of Leonardo da Vinci . 1886, section 896. Of the Moon , p. 157 (English, Gutenberg eText , sacred-texts.com). , della luna (Italian / English). Retrieved February 21, 2011 . German quoted from
Schlichting: Moon phases in the apple tree . In: Spectrum of Science . No. 9/10 , p. 32 .

↑ ^a ^b ^c ^d Wolfgang Vollmann: Physical changeable star data . In: Hermann Mucke (Hrsg.): Modern astronomical phenomenology . 20th Sternfreunde Seminar, 1992/93. Zeiss Planetarium of the City of Vienna and Austrian Astronomical Association , Vienna 1992, p. 185-196 .

↑ Compare also the quoted Leonardo quote, which in this sense can be understood as a motto.

↑ ^a ^b Norbert Pachner: The main positions of the changing stars . Boards for estimating the visibility conditions. In: Modern Astronomical Phenomenology . 20th Sternfreunde Seminar, 1992/93. S. 153–178 (outdated, still based on Francou Bretagnon: VSOP 87 ).

↑ ^a ^b See: H. Joachim Schlichting: Moon phases in the apple tree . In: Spectrum of Science . No. 9/10 , September 2010, pp. 32 f . ( Spektrum.de ). Ders .: Why the sun (not) burns a hole in the world . In: Spectrum of Science . No.
9/09 , p. 38 f .
↑ John Walker : mooncalc.c . Moontool for Windows - Astronomical Calculation Routines. In: Moontool / Homeplanet . Source code. March 15, 1999, routine UPDATEICON - Update tray icon (English, web link see Wikipedia article on the program).
^ Jean Meeus : Astronomical Algorithms . Willmann-Bell, Richmond 1991, ISBN 0-943396-35-2 (English).
↑ Peter Duffett-Smith: Practical Astronomy With Your Calculator . 3. Edition. Cambridge University Press, Cambridge 1981, ISBN 0-521-28411-2 (English).
↑ See also: File: Sunset from the ISS.JPG , with even greater emphasis on the atmospheric layers.
^ Georg Glaeser and Karlheinz Schott: Geometric Considerations About Seemingly Wrong Tilt of Crescent Moon . In: KoG . No. 13 , 2009 (English, srce.hr [accessed April 10, 2019]). M. Wagenschein : Teaching understanding . Beltz, Weinheim 1992, Darkening Knowledge , p. 63 . Quoted in U. Backhaus: From the observation of astronomical phenomena to own measurements . Koblenz, introduction, p. 2 ( didaktik.physik.uni-due.de (PDF)). and in Ders .: The Movement of the Moon . Lecture manuscript at the MNU conference Bremerhaven. Ed .: University of Duisburg-Essen. November 14, 2005, Figure 2: The sun and moon move across the sky together in the course of a day , p. 3 ( didaktik.physik.uni-due.de (PDF)).
↑ Manfred Holl: History of the moon maps. (Webbook) In: Astronomy-historical topics. Accessed in 2010 .

[leo-1] JP Richter: The Notebooks of Leonardo da Vinci . 1886, section 896. Of the Moon , p. 157 (English, Gutenberg eText , sacred-texts.com). , della luna (Italian / English). Retrieved February 21, 2011 . German quoted from
Schlichting: Moon phases in the apple tree . In: Spectrum of Science . No. 9/10 , p. 32 .

[Vollmann-2] Wolfgang Vollmann: Physical changeable star data . In: Hermann Mucke (Hrsg.): Modern astronomical phenomenology . 20th Sternfreunde Seminar, 1992/93. Zeiss Planetarium of the City of Vienna and Austrian Astronomical Association , Vienna 1992, p. 185-196 .

[Anm._Leo-3] Compare also the quoted Leonardo quote, which in this sense can be understood as a motto.

[Pachner-4] Norbert Pachner: The main positions of the changing stars . Boards for estimating the visibility conditions. In: Modern Astronomical Phenomenology . 20th Sternfreunde Seminar, 1992/93. S. 153–178 (outdated, still based on Francou Bretagnon: VSOP 87 ).