Astronomy Answers: AstronomyAnswerBook: The Sky and Horizon

Astronomy Answers
AstronomyAnswerBook: The Sky and Horizon


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1. Elongation ... 2. Is the Bible Really an Astronomy Book? ... 3. The Star of Bethlehem ... 4. The Ecliptic ... 5. The Ecliptic and the Milky Way ... 6. Angles between the Ecliptic, the Galactic Equator, and the Celestial Equator ... 7. Distances on a Sphere ... 8. Zenith and Nadir ... 9. Celestial Pole and Celestial Equator ... 10. Culmination ... 11. The Brightness of the Sun ... 12. Motion of the Stars, Planets, and Moon in the Sky ... 13. The Motion and Look of the Moon ... 14. Rotation of the Face of the Moon ... 15. Standstills or "Stices" ... 16. Can a book of lunar phases for one country be used also in another country? ... 17. Determination of Your Location Using Sun and Stars ... 17.1. Latitude ... 17.2. Longitude ... 18. How Far Can You See? ... 19. How Far is the Horizon?

This page answers questions about the sky and the horizon. The questions are:

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1. Elongation

The elongation measures the position of a planet in the sky, compared to the Sun. There are different ways to calculate the elongation, depending on which information you already have. Also, there are different definitions for the elongation, which in practice yield almost the same results for the brightest planets.

The simplest definition for the elongation of a celestial object is that it is the difference between the geocentric ecliptic longitude of that object and that of the Sun. You can view the longitude as mile posts along the path that the planets and the Sun follow along the sky. The elongation says how many mile posts are between the planet and the Sun, and on what side of the Sun the planet is.

The Sky Positions Calculation Page explains how to calculate the geocentric ecliptic longitude, and also explains about the different definitions of elongation (especially in chapter 11).

If you want to calculate the greatest elongation that a planet may have, then you only need to know the greatest distance that that planet can be from the Sun, measured in Astronomical Units. If that distance is greater than 1, then the planet can reach any elongation (i.e., up to 180 degrees). If the distance is less than 1, then the greatest elongation is equal to the arc sine of the distance.

For example: Venus does not get more than 0.728 AU from the Sun, so the greatest elongation that Venus can reach is \(\arcsin(0.728) = 47°\).

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The Sun is (as seen from the northern hemisphere) due south at noon, so a planet can only be due south at midnight if that planet is directly opposite the Sun in ths sky, so it must have an elongation of 180°, so the Earth must then be between the Sun and that planet, so that planet must be further away from the Sun than the Earth is. Venus is always closer to the Sun than the Earth is (Venus is an inferior planet), so Venus cannot be in the sky at midnight.

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2. Is the Bible Really an Astronomy Book?

Unfortunately we cannot ask the people who wrote or edited the Bible during the centuries of its formation exactly what they meant by each word, so we'll never know for sure what the intended meaning is. It is impossible to prove that your favorite interpretation of the Bible is the correct one. Anybody can come up with a different interpretation. Who can say which one is the correct one? However, we can have an opinion about how likely it seems to us that a particular interpretation is correct.

It may be that the Bible is really a description of astronomical events, but then it is a rather cryptic one. Researchers of the Bible say that different parts of the Bible were written by different people at different times; that the Bible is a compilation and adaptation of different separate documents. It seems unlikely to me that all of those separate authors, who could not have known that their writings would one day form a single book, would all have followed a single plan to describe astronomical events with the same metaphors.

Already in Biblical times divine persons were associated with radiant celestial objects, and associating resurrection (a form of new life) with springtime is not very far-fetched.

The Bible was written by people with their own cultural background. They grew up with stories that may have come from different times and places, and may have changed significantly during their transmission. It is very easy to invent a new interpretation of the Bible, especially an interpretation that is based on hidden layers of meaning (such as that the Bible is really a description of astronomical events). It is very difficult to prove that such an interpretation is correct ― or incorrect. That is why there are so many interpretations of the Bible that don't agree with each other, and that is why there are so many different kinds of religious communities that usually think that they are the only ones to know what the Bible really means, and that all of the other ones are wrong.

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3. The Star of Bethlehem

The Bible contains the story of the Three Kings from the East who were guided to baby Jesus in the stable in Bethlehem by an unusual bright star in the sky. This would have been a few years before the epoch of the Julian calendar. Many people have speculated about the nature of the Star of Bethlehem. Just type "Star of Bethlehem" into a search engine on the internet and you'll find many pages about this topic.

Of possible astronomical explanations for the Star of Bethlehem, a conjunction of planets, or a supernova, or a nova, or a comet are the most popular. In his book "Marking Time: The Epic Quest to Invent the Perfect Calendar", Duncan Steel describes that there were three conjunctions of Jupiter and Saturn in 7 BC, a conjunction of Mars, Jupiter, and Saturn in 6 BC, and a conjunction of Venus and Jupiter in 3 BC. Some people think that the Star of Bethlehem was actually such a conjunction of planets.

A conjunction of planets means that the planets are then reasonably close together in the sky, but even then they are still so far apart than they are visible as separate points of light. During the conjunction of 6 BC, there was always at least 6.5 degrees between any two of the three involved planets, and that is 13 times greater than the diameter of the full Moon. So, even then the planets weren't lined up one behind the other.

Johan Kepler thought that the conjunction of 6 BC would have generated a supernova, just like in his own time in 1604 a supernova closely followed a conjunction of the very same planets. The Star of Bethlehem would then have been the supernova that went with the conjunction of planets of that time. However, nowadays we know that the appearance of supernovas (which are very far beyond our Solar System) has nothing to do with conjunctions of planets (which are inside our Solar System).

Some people think that a conjunction of planets is too mundane to have been the Star of Bethlehem, but that a supernova is sufficiently spectacular to have been that star. However, we have found no records of a supernova around that time, even though such a supernova would have been visible for quite some time from at least half of the planet, and for example the Chinese were very good at writing down reports of such unusual things in the sky. In addition, none of the known supernova remnants (which can be discovered nowadays with powerful telescopes even when the supernovas were thousands of years ago) fits a supernova that would have been visible around that time.

A nova is a less spectacular but more frequently occurring phenomenon than a supernova, but the records of that time do not report anything like a nova, either.

That leaves comets. The Chinese report a comet in 5 BC (which would have been visible also from Europe and the Middle East), and other details of the Star of Bethlehem in the Bible story fit a comet better than a conjunction of planets or a supernova or a nova. To me, a comet seems the most reasonable explanation for the Star of Bethlehem.

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4. The Ecliptic

The ecliptic is the path that the Sun appears to follow between the stars. Usually, this is understood to be "as seen from Earth", but the Sun of course also follows some path between the stars as seen from another planet, and we call that path an ecliptic, too, but the ecliptic of that planet.

The inclination of the orbits of the other planets compared to that of the Earth is only a few degrees in general (except for Pluto, whose inclination is 17°), so the ecliptics of the other planets are usually close to that of the Earth, so they pass through just about the same constellations as the ecliptic of Earth does, namely the constellations of the zodiac.

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The ecliptic makes an angle with the celestial equator, because the rotation axis of the Earth does not make a right angle with the ecliptic, which allows the Sun to get up to 23 degrees north and south of the celestial equator as it moves along the ecliptic. There is no particular reason why the rotation axis of a planet should be oriented such that the ecliptic and the celestial equator are the same. The ecliptica of the other planets is different from their celestial equator, too.

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5. The Ecliptic and the Milky Way

The positions of the two solstices between the stars shift about 50 arcseconds per year (compared to the stars) because of the precession of the equinoxes, and happened to cross the adopted center line of the Milky Way around the beginning of 1999. This happens about once every 13,000 years. In star atlases that use the equinox of J2000.0 the distance between the solstices and the center line of the Milky Way is only about 50 arcseconds or 0.014 degrees, and such a small angle can only be discerned in very large star atlases. There is no physical reason for a connection between the plane of the Milky Way and the plane of the orbit of the Earth, and the precession did not notice that the solstices crossed the center line of the Milky Way.

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6. Angles between the Ecliptic, the Galactic Equator, and the Celestial Equator

The angle between the ecliptic plane and the galactic plane is about 60.2 degrees. The angle between the celestial equatorial plane and the galactic plane is about 62.9 degrees. The angle between the celestial equatorial plane and the ecliptic plane is about 23.4 degrees.

There is no easy relationship between these three angles, just like there is no easy relationship between the distances between three points. If you know that there is 23.4 km between cities A and B, then you don't have enough information to say how much distance there is between A and C, or between B and C. If you also know that there is 60.2 km between A and C, then you still don't have enough information to say precisely how much distance there is between B and C: It could be as little as 60.2 - 23.4 = 36.8 km (if B is exactly between A and C) or as much as 60.2 + 23.4 = 83.6 km (if A is exactly between B and C), or any value in between, depending on the angles inside triangle ABC. Something similar goes for the three celestial planes.

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7. Distances on a Sphere

If you have the longitudes and latitudes (or right ascensions and declinations) of two points on a sphere, then you can calculate the distance between those two points as follows:

\begin{equation} d = \arccos(\sin(b_1) \sin(b_2) + \cos(b_1) \cos(b_2) \cos(l_1 - l_2)) \label{eq:afst} \end{equation}

where \(l_1\) and \(b_1\) are the longitude and latitude of the first point, and \(l_2\) and \(b_2\) the longitude and latitude of the second point (with + or − for east or west longitude and north or south latitude). The result \(d\) is then the (shortest) angular distance between the two points across the sphere. If the angular distance is measured in degrees, and if the radius of the sphere is equal to \(R\) (for example, measured in kilometers), then the distance \(D\) across the sphere (in the same units as \(R\)) is equal to

\begin{equation} D = d R \frac{π}{180} ≈ 0.01745329 d R \end{equation}

For example: The distance between the Netherlands (52° north latitude, 5° east longitude) and Mexico City (19° north latitude, 99° west longitude) is equal to \(d = \arccos(\sin(52°)×\sin(19°) + \cos(52°)×\cos(19°)×\cos(5° - (−99°)) = 83.35°\). The radius of Earth is 6378 km, so this distance corresponds to \(D = 0.01745329×83.35×6378 = 9279 \text{ km}\).

Distances that you calculate across the Earth in this way are not entirely correct, because the Earth is not a perfect sphere. The error will be less than half a percent in general.

If the two points have almost the same longitudes and also almost the same latitudes, then they are close together on the sphere. In that case the results of equation \eqref{eq:afst} might be inaccurate because of rounding errors. You can then use the following alternative formula:

\begin{equation} d = \sqrt{(b_1 - b_2)^2 + \left( (l_1 - l_2)×\sin\left( \frac{1}{2}(b_1 + b_2) \right) \right)^2} \end{equation}

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8. Zenith and Nadir

The point in the sky that is straight above you (or the direction that is straight up) is called the zenith. The point that is straight below you (or the direction that is straight down) is the nadir. The astronomical horizon is exactly midway in the sky between those two points. The height of the zenith is +90° and that of the nadir is −90°.

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9. Celestial Pole and Celestial Equator

A pole is a place where a rotation axis sticks through the surface. A pole of the Earth is a place where the rotation axis of the Earth sticks through the surface, just as if the Earth rotates around a skewer that was stuck through the Earth from one pole to the other pole. Something that rotates always rotates around some line or axis. The points on that axis seem to stay in their place, and all other points seem to rotate around the axis. If you are high above a pole, then it seems as if that pole stays in the same place but all other locations (and the people and other things that are there) move in circles around the pole. The Earth has two poles.

The stars in the sky at night also seem to move in circles around a point. That point is a pole of the sky, also called a celestial pole. It is just as if the sky rotates around an axis that sticks through the celestial poles. There are two celestial poles. The northern celestial pole lies close to the Pole Star (whence its name), so all other stars appear to rotate around the Pole Star and the Pole Star always remains in the same location in the sky (as seen from a fixed location).

The celestial equator is the imaginary line along the sky that is equally far from both poles of the sky. The celestial equator divides the starry sky into a northern half (seen best from the northern hemisphere of the Earth) and a southern half (seen best from the southern hemisphere of the Earth).

Fig. 1: Diagram Earth/Celestial Pole/Celestial Equator
Fig. 1: Diagram Earth/Celestial Pole/Celestial Equator

The correspondence between the Earth's pole, Earth's equator, celestial pole, celestial equator, and zenith are displayed in figure 1, which shows a slice of the (assumed spherical) Earth with rotation axis \(A\), center \(C\), equator \(E\), and pole \(P\).

There is one observer at pole \(P\), and another observer at location \(X\) at latitude \(φ\) (north or south). The plane \(H\) of the horizon is tangent to the surface of the Earth at the location of the observer, and is perpendicular to the direction to the center \(C\) of the planet. The zenith \(z\) always points away from the center of the Earth, straight "up", and is perpendicular to the plane \(H\) of the horizon.

The direction to the visible celestial pole \(p\) from any place on Earth is parallel to the rotation axis \(A\) of the Earth. This direction is indicated in the figure for places \(P\) and \(X\). Seen from pole \(P\), the celestial pole is straight up, in the zenith, at an altitude of 90 degrees. Seen from \(X\), the celestial pole is not in the zenith but rather at altitude \(φ\) above the horizon, in the direction (north or south) of the nearest pole of the Earth. The other celestial pole lies in the opposite direction, \(φ\) degrees below the horizon in the direction (north, south) of the equator.

The plane \(e\) of the celestial equator is parallel to the equator \(E\) of the Earth, as seen from any place on Earth. The intersection of the celestial equator and the slice of the Earth in the figure is indicated for places \(P\) and \(X\). Seen from pole \(P\), the celestial equator coincides with the horizon. Seen from \(X\), the celestial equator makes an angle \(90° - φ\) with the zenith. The horizon and the celestial equator make an angle of \(90° - φ\) and intersect in the east point and west point (which are not shown in the figure). The celestial equator is furthest from the horizon in the north and south, namely \(90° - φ\). The celestial equator is highest in the sky in the direction (north or south) of the Earth's equator, and furthest below the horizon in the opposite direction.

At every moment, exactly half of the celestial equator is above the horizon. The part that is above the horizon begins in the east at the horizon, goes up towards the south (if you're in the northern hemisphere of the Earth) or north (if you're in the southern hemisphere), and then goes back down towards the west at the horizon.

For example, if you are at 52° north latitude, then the celestial equator reaches to 90° - 52° = 38° above the southern horizon. If you are at 30° south latitude, then the celestial equator gets up to 90° - 30° = 60° above the northern horizon. Seen from the equator of the Earth, the celestial equator passes straight overhead.

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The rotational axis of the Earth pays no attention to where the Earth is in its orbit around the Sun ― except perhaps for very small effects that aren't obvious to the naked eye. The rotating Earth is a very large gyroscope. It is not so easy to change the orientation of the axis of rotation of a gyroscope ― and that is what makes gyroscopes useful.

If you apply a force equally to all parts of a gyroscope, then you move the gyroscope but do not change the orientation of its axis of rotation. To change the orientation of a gyroscope, you need to apply opposite forces to different sides of the gyroscope. You need to apply torque (moment of force).

Most of the force of gravity that acts between the Sun and the Earth applies nearly equally or symmetrically to all parts of the Earth, and so does not produce a torque, hence no change in the orientation of the axis of rotation. It does change the motion of the Earth, making it follow a curved path around the Sun, rather than a straight line through space.

The main asymmetrical effect on Earth of the force of gravity between the Earth and the Sun are the tides. These are associated with a torque that can change the orientation of the axis of rotation of the Earth, but the torque is relatively very small so the change in the orientation is very slow. The main effect is the precession of the equinoxes. Smaller effects are called nutation.

The periods of variation of the force of gravity between the Earth and the Sun (i.e., one year) and Moon (one month) play little or no role in setting the periods of variation of the orientation of the rotation axis of the Earth. Physical characteristics of the Earth (such as its moment of inertia are much more important in that regard. The main periods of variation of the orientation of the axis of rotation of the Earth are around 26.000 years (for precession) and 18.6 years (for nutation). Precession has a large effect but is very very slow. Nutation has a much smaller effect but is faster.

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10. Culmination

A celestial body culminates (is in culmination) when it is highest in the sky. At that moment, the body goes through the celestial meridian, which runs from the northern celestial pole via the zenith to the southern celestial pole. The culmination is also called the transit.

To be able to calculate at what time a celestial body is highest in the sky, you need to know the right ascension of that body. To calculate how high the body gets in the sky, you need to know the declination of the body. How you can calculate those for the planets is explained on the Sky Positions Calculation Page. For the Sun, you can read the Sun Position Calculation Page.

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11. The Brightness of the Sun

The light from the Sun must travel through the atmosphere of the Earth to get to your eyes. The atmosphere takes away some of the sunlight, depending on how much air it travels through. When the Sun is low in the sky, then the sunlight travels through much more air than when the Sun is high in the sky, so the air takes much more light away from a low Sun than from a high Sun, and that's why it's painful to look into a high Sun but not to look into a setting Sun.

If you have trouble seeing that light from the setting Sun travels through more air than the light from a high Sun, then you can try the following experiment: Put about an inch of water in a tall dinner plate or low pot or something else that has a flat bottom and is wider than it is tall. Imagine that the water is the atmosphere of the Earth. Now take something thin and long, like a straw. Imagine that the straw is a ray of sunlight. Set the straw straight up in the water so it touches the bottom of the plate. Mark how much of the straw is in the water. That's how much "air" the "ray of light" has to travel through when the Sun is straight up. Now lay the straw as flat as you can, so that one end touches the bottom of the plate as before and the straw also touches the edge of the plate or pan. You'll notice that a much longer stretch of the straw is in the water now, so the "ray of light" has to travel through much more "air" to get to the ground.

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12. Motion of the Stars, Planets, and Moon in the Sky

The stars move along the sky as if they are fixed to the inside of a giant sphere that rotates around an axis through the North Pole and South Pole of the Earth. It seems as if every observer including you) is in the middle of that giant celestial sphere. The starry sky rotates once around its axis in 23 hours and 56 minutes, so after that much time the stars have returned to the exact same spot in the sky (as seen from the same location).

Fig. 2: Sky Motion Diagram
Fig. 2: Sky Motion Diagram

That celestial sphere is displayed in Figure 2, for an observer at φ = 35 degrees north latitude. The sphere is divided into two halves by a disk which represents the ground, and the edge of it is the horizon. The part of the celestial sphere that is below the ground cannot be seen in reality and is drawn in dotted lines. The letters N, O, Z, and W along the horizon indicate the north, east, south, and west points. The observer is in the center of the sphere, where the three straight lines N - Z, O - W, and NP - ZP meet. The northern celestial pole (NP) lies at the northern end of the line NP - ZP and is at 35° above the horizon, for an observer at 35° north latitude (see above). The southern celestial pole (ZP) lies at the other end of the line and is always 35° below the horizon towards the south, from this location. The celestial equator (declination 0°) lies midway between the northern celestial pole and the southern celestial pole.

The celestial sphere seems to rotate around the line NP - ZP. The two celestial poles stay in their places, but all other celestial objects in the sky trace parallel circles on the celestial sphere, of which five are drawn, for declinations +60°, +30°, 0°, −30°, and −60°. The celestial sphere rotates from east to west.

A celestial object at declination 0° (i.e., on the celestial equator) rises due east (O), moves up and sideways towards the south, and then down and sideways until it sets due west (W). Below the horizon it continues down and sideways towards the north, and then up and sideways until the next rising in the east. Exactly half of the path is above the horizon, so such a celestial object is above the horizon exactly half of the time.

Celestial bodies that are closer to the visible celestial pole spend more time above the horizon than below it, and are closer to that pole also when the rise and set. Celestial bodies that are less than |φ| degrees from the visible celestial pole are always above the horizon. For this example, this includes all celestial objects with a declination greater than 90° − 35° = 55°. These celestial bodies are circumpolar. Celestial bodies that are further away from the celestial pole spend more time below the horizon than above it. Celestial objects that are less than |φ| degrees from the invisible celestial pole are always below the horizon and are hence never visible from this location. For this example, this includes all celestial objects with a declination that is more negative than −55°.

Fig. 3: Ecliptic on Celestial Sphere
Fig. 3: Ecliptic on Celestial Sphere

The rotation axis of the Earth does not make a right angle with the orbit of the Earth around the Sun, but rather an angle of about 67°. This means that the Sun does not move along the celestial equator, but rather along a great circle that makes an angle of about 90° − 67° = 23° with the celestial equator. This is shown in figure 3, which shows the celestial sphere, just like the previous figure, but now with the plane of the orbit of the Earth instead of the horizon in the middle. The observer is again at the intersection of the straight lines. NP is again the northern celestial pole and ZP the southern celestial pole. The line NP - ZP is again parallel to the rotation axis of the Earth. The circle ABCD is the path of the Sun along the celestial sphere, which the Sun traverses in that direction each year. Point A is the ascending equinox (vernal equinox), B the northern solstice, C the descending equinox, and D the southern solstice. The celestial equator is midway between the two celestial poles. The ecliptic and the celestial equator intersect at points A and C. Because the ecliptic makes an angle of 23° with the celestial equator, the Sun gets at most 23° away from the celestial equator (at points B and D).

The Sun, Moon, and planets move along the sky with the planets, but also move relative to the stars. The motion between the stars (13 degrees per day for the Moon, about 1 degree or less for the Sun and the planets) is much slower than the rotation of the starry sky (360 degrees per day), so the motion relative to the stars doesn't stand out at first sight. So, the stars, planets, Moon, and Sun move through the sky in just about the same way.

The Sun, Moon, and planets cannot appear everywhere in the sky. As a rule of thumb, if you never see the Sun in a particular direction in the sky (for example, straight up or due north, if you are in Europe) then the Moon and planets cannot be in that direction, either.

The paths that the Sun takes through the sky throughout the year are all parallel to one another. On a summer's day, the Sun travels along a higher path (a "summer's path") and on a winter's day, the Sun takes a lower path (a "winter's path"). After the longest day, the Sun takes a lower path each day, until the shortest day, after which it takes a higher path each day, until the longest day comes around again.

The Moon and the planets travel roughly the same paths as the Sun, but not all the same path at the same time. The Sun is always highest in the sky at noon, but for the Moon or the planets this can happen at any time of day or night. It may happen that on a certain day Mars takes the path through the sky that the Sun takes on 6 November, and that Jupiter on that same day takes the path that the Sun takes on 22 April, and that Mars is highest in the sky around midnight and Jupiter about a quarter of an hour before the Sun.

Everything that rises (Sun, Moon, planets, and many stars) rises in the eastern half of the horizon and sets in the western half. The motion in the western half of the sky is the mirror image of the motion in the eastern half of the sky. Something that rises due east and is highest in the sky exactly 6 hours later sets exactly another 6 hours later due west (except for minor differences especially for the Moon because of its motion relative to the stars).

The phase of the Moon (Full Moon, Last Quarter, New Moon, First Quarter) changes so little in one night that you can pretend that the phase is the same all night long. If the Moon rises as a thin crescent, then it will set as a thin crescent, too.

If you are satisfied with a smaller part of the sky, then you can compare the starry sky with an umbrella that has stars stuck to its underside. The handle of the umbrella is then the rotation axis of the sky (which is the extension into the sky of the rotation axis of the Earth), with the Pole Star or North Star at the spot where the handle touches the cloth. Hold the umbrella over a table such that part of it disappears below the edge of the table. The surface of the table is then the ground, and the edge of the table is the horizon. The handle of the umbrella may rotate around its axis but it must not move (because the rotation axis of the Earth practically doesn't move either).

Now rotate the umbrella in just under a day. The stars then seem to trace circles around the handle (Pole Star). The stars close to the edge of the umbrella sometimes disappear beneath the edge of the table (horizon), but if a star is not too far from the Pole Star, then it is always above the horizon. Such a star is then circumpolar (as seen from that location on Earth).

The Pole Star is now about half a degree from the rotation axis of the sky, so the Pole Star also makes a circle in the sky, but that circle is so small that you usually don't notice it.

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As seen from a location in the northern hemisphere of the Earth, the northern celestial pole is always at the same place in the sky, and the southern celestial pole is never visible. Likewise, as seen from the southern hemisphere, the southern celestial pole is always at the same place in the sky and the northern celestial pole is never visible.

The height above the horizon of the visible celestial pole is equal to your latitude (north or south). Stars and constellations that are less far from the visible celestial pole than your latitude are circumpolar and therefore always above the horizon and visible every clear night. That is the case, for example, with the Great Bear from about 40 degrees north latitude, and with the Southern Cross from about 35 degrees south latitude.

Stars and constellations that are not circumpolar are sometimes above the horizon and sometimes below it, so they are not visible every clear night. Constellations like Orion that are partly on each side of the celestial equator are never circumpolar (as a whole). The constellations of the zodiac are not circumpolar either (except for people far beyond the polar circles, and there aren't many of those). For every constellation there is a time of the year when the Sun is closest to them. Constellations near the Sun are then above the horizon when the Sun is, during daytime, so then they are not visible at night. Six months later, the Sun is at the exact opposite side of the starry sky, so then those constellations are visible at night.

The angle between the handle of the umbrella (the Pole Star) and the surface of the table (horizon) should be equal to the (northerly) latitude of your location. At the North Pole, the umbrella should be straight up, and all visible stars are circumpolar. At the equator, the handle should be on the table, and then no star is circumpolar.

To see the effect of your latitude, you should imagine that the "stellar umbrella" sticks through the Earth from one pole to the other. Anywhere on Earth, someone is standing up straight when his legs point to the center of the planet, so someone who is at the pole thinks that the umbrella is also standing straight up, but someone on the equator thinks that the umbrella is horizontal. Someone between the equator and the pole thinks that the umbrella is at an angle.

You can divide the starry sky of a certain place into three parts: stars that are always above the horizon, stars that are sometimes above the horizon, and stars that are never above the horizon.

The stars that are sometimes above the horizon are visible only in certain seasons, which are different for each star. For example, the constellation of Orion is not visible in summer, and the constellation of the Archer is not visible in winter. This is because in that season the Sun is close to that constellation in the sky, so then that constellation and the Sun are above the horizon during the day, and not during the night.

The Sun is not always near the same stars in the sky because the Earth orbits around the Sun in one year, and therefore it looks from Earth as if the Sun travels along the ecliptic in one year. Relative to the stars, the Earth rotates one more time around its axis each year than relative to the Sun. That extra rotation is because of the orbit around the Sun. Because of this, the stars seem to rotate around the Earth 366/365 times as fast as the Sun does, so one rotation of the stars takes 365/366 days, which is 23 hours and 56 minutes.

The starry sky rotates around its axis in a day (24 hours) minus four minutes. Those four minutes mean that each night at the same time the sky has rotated a little further. That's why some constellations can only be seen in certain seasons, and why circumpolar constellations (such as the Great Bear as seen from the Netherlands and Belgium) are low in the sky at a certain time in some seasons, but high in the sky at the same time in other seasons.

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13. The Motion and Look of the Moon

Just like the Sun, the Moon is above the horizon about half of the time, averaged over a whole year. However, because the distance of the Moon from the Sun in the sky (the elongation) keeps changing, the times during which the Moon is above the horizon also changes from day to day.

If it is sunny some day, then take a ball outside and hold it in the sunlight. In which direction (compared to the direction to the Sun) do you have to hold the Sun so that you cannot see the ball's own shadow on itself? That corresponds to Full Moon. And in which direction so that exactly half of the ball is in its own shadow? That corresponds to First Quarter or Last Quarter. And in which direction so that you can see only the ball's shadow on itself? That corresponds to New Moon.

I think you'll find that the ball is only "full" (i.e., so you can't see the ball's shadow at all) if the ball is almost in the opposite direction as the Sun is. Not exactly in the opposite direction, because then the ball would be inside your shadow, and that corresponds to a lunar eclipse.

And the ball is only "new" (i.e., so all of the ball that you can see is in its own shadow) if the ball is almost or completely in front of the Sun. If you block the sunlight with the ball, then your eyes are in the shadow of the ball, and that corresponds to a solar eclipse.

So, it is Full Moon when the Moon is (almost) exactly opposite the Sun in the sky, but that means that the Moon must be below the horizon whenever the Sun is above the horizon, and vice versa. That's why you can never see a Full Moon during the day (but you can see an almost-Full Moon, just before sunset or just after sunrise).

Depending on the phase of the Moon, the Moon is visible at different times of the day or night. The period of visibility of the Moon is on average about 50 minutes later every day. Between New Moon and Full Moon, the Moon is visible just after sunset (if the weather cooperates, of course) but not just before sunrise. Between Full Moon and New Moon the Moon is visible just before sunrise but not at sunset.

It is New Moon when the Moon is (almost) between the Earth and the Sun. The Moon is then closest to the Sun in the sky and follows about the same path along the sky that the Sun does, rises with the Sun, and sets with the Sun. In the course of a month, the Moon lags ever more behind the Sun, at a rate of about 50 minutes per day.

Two or three nights after New Moon you can see the Moon as a thin crescent just after sunset near the westen horizon, at the same place where the Sun was an hour or two earlier. The outside of the crescent always points at the Sun (along the path of the Moon).

In the next couple of days, the crescent fills out and the Moon sets longer and longer after sunset. About seven days after New Moon the western half of the disk of the Moon is lit and then it is First Quarter. The Moon then lags the Sun by about 6 hours, sets around midnight, and rises again around noon. The path that the Moon then follows along the sky is about one season ahead of the Sun, so the Moon in First Quarter in the spring follows a summer's path along the sky (and is then as high in the sky at sunset as the Sun is at noon on a summer's day).

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About another week later the lunar disk is fully lit, and then it is Full Moon. The Moon is then almost directly opposite the Sun in the sky, lags the Sun by about 12 hours (or is ahead by about 12 hours, that's the same thing then), rises at sunset (roughly in the east) and sets at sunrise (roughly in the west). The Moon then follows a path through the sky that lags the Sun by about two seasons (or is two seasons ahead). The Full Moon in winter takes a summer's path (and is then as high in the sky at midnight as the Sun is at noon in summer), and the Full Moon in summer follows a winter's path (and is then as low in the sky as the Sun is in winter).

In the following week, more and more of the lunar disk gets dark from the western side. About three weeks after New Moon, only the eastern half of the disk is illuminated, and then it is Last Quarter. The Moon then lags the Sun by about 18 hours (or is 6 hours ahead, which is the same thing), rises around midnight and sets around noon. The Moon then follows a path that lags the Sun by one season, so the springtime Last Quarter Moon is as high in the sky at sunrise as the Sun is at noon in winter.

In the next few days, the Moon is again just a crescent, but now with the bright side in the east, and only visible just before sunrise. About 29 and a half days after the previous New Moon there is a next New Moon and then the whole cycle starts over again.

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The locations of moonrise and moonset at the horizon show the same variation during a month that the locations of sunrise and sunset do during a year. If the location of sunrise varies between about northeast and about southeast during a year at your location (the exact range depends on your location), then the location of moonrise also varies between about northeast and about southeast at your location, but during a month rather than during a year.

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The Moon looks full for a day or two each month, but the fullness of the Moon changes continuously, so it is practically impossible to define a "period of fullness" that everyone will agree about. Astronomers use definitions of full moon that mean that "true" full moon is limited to exactly one moment each month (namely the moment when the Moon is most closely opposite the Sun in the sky). Most people will call the Moon full also one day before or after that moment (when "only" 99% of the disk of the Moon still appears bright), but not everyone will agree about how long before or after that moment the Moon should still be called "full". The same goes for new moon, first quarter, and last quarter.

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14. Rotation of the Face of the Moon

As seen from the northern parts of the world, the south pole of the Moon is always lower in the sky than the north pole is, and Tycho crater is always in the lower part of the face of the Moon (in the sky). As seen from the southern parts of the world, the north pole of the Moon is always lower in the sky than the south pole is, and Tycho crater is always in the upper part of the face of the Moon.

Relative to the horizon, the Moon follows a curved path along the sky, from the point where it rises (in the east) upward until it passes through its highest point, and then down again to where it sets (in the west). All of that time, the line connecting the north and south poles of the Moon is perpendicular to its path.

Just after it rises, the Moon moves obliquely up and to the west, so the line connecting the north pole and south pole of the Moon is then not exactly upright, but the upper pole is rotated somewhat to the east (the direction where the Moon rose). And just before it sets, the Moon moves obliquely down and toward the west, so the line connecting the north pole and south pole of the Moon is then again not exactly upright, but has the upper pole rotated somewhat to the west (the direction where the Moon sets). Between moonrise and moonset, the upper pole rotates toward the west and the lower pole toward the east, and the craters on the face of the Moon rotate accordingly. Tycho crater is in the same (upper/lower) part of the Moon when it sets as when it rose.

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15. Standstills or "Stices"

Fig. 4: Declination of Sun and Moon (1)
Fig. 4: Declination of Sun and Moon (1)

Fig. 5: Declination of Sun and Moon (2)
Fig. 5: Declination of Sun and Moon (2)

How long a celestial object stays above the horizon (until it sets again) depends on your geographical latitude and on the declination of the celestial object. If the declination of the object is closest to your geographical latitude (with positive declination corresponding to north latitude and negative declination with south latitude), then that object stays above the horizon the longest.

The declination of the Sun is always between −23° and +23°, so for someone north of the tropics (for example, in Europe) it is the longest day (with the Sun longest above the horizon) when the declination of the Sun is most positive (i.e., +23°), and for someone south of the tropics (for example, in South Africa) it is the longest day when the declination of the Sun is the most negative (i.e., −23°).

The path of the Sun is show as the solid curve in figures 4 and 5. Right ascension (α) is shown along the horizontal axis and declination (δ) along the vertical axis. The first figure holds when the Sun is in the ascending equinox, and the second figure when the Sun is at the northern solstice.

The position of the Sun is indicated with a little circle. When it is New Moon, then the Moon is very close to the Sun. At First Quarter, the Moon is 90° east of the Sun, at I in the figures. At Full Moon, the Moon is 180° from the Sun, at II in the figures. At Last Quarter, the Moon is 90° west of the Sun, at III in the figures.

The Sun or Moon has a standstill or "stice" (solstice for the Sun, "lunastice" for the Moon) when it has reached the highest (right ascension = 6 hours, declination = +23°) or lowest (right ascension = 18 hours, declination = −23°) point on the curve. The Sun moves along the ecliptic once a year, so the Sun has two solstices each year, one at midsummer's day and one at midwinter's day. The Moon moves along the ecliptic once each sidereal month, so the Moon has about 27 standstills each year.

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It depends on the season at which phase of the Moon the Moon has a standstill. In the first of the two figures, the Sun is in the ascending equinox (around 21 March in the Gregorian Calendar) and the Moon has a northern standstill when it is First Quarter and a southern standstill when it is Last Quarter. In the second figure, the Sun is in the northern solstice (around 21 June) and the Moon has a northern standstill when it is New Moon, and a southern standstill at Full Moon.

Fig. 6: Declination of Sun and Moon (3)
Fig. 6: Declination of Sun and Moon (3)

In the preceding two figures we assumed that the Moon follows the exact same path along the sky as the Sun, but that is not correct. The orbit of the Moon makes an angle of about 5° with the orbit of the Earth, and hence with the ecliptic, which is the apparent path of the Sun along the sky. The kind of influence this can have is shown in figure 6. The deviation of the path of the Moon compared to the ecliptic is displayed as the long-dashed curve. If you add that deviation to the path of the Sun (the solid curve), then you get the short-dashed curve, which indicates the path of the Moon along the sky. With this path, the Moon gets further from the celestial equator (which is at δ = 0°) than the Sun does. The intersections of the orbit of the Moon (the short-dashed curve) and the ecliptic (the solid curve) are the nodes of the orbit of the Moon. In the example, the ascending node lies at a right ascension of 2 hours.

We do not have a solar eclipse and a lunar eclipse every month, because the Moon does not follow the exact same path along the sky as the Sun. There can only be an eclipse when the Sun is in a node of the lunar orbit just when the Moon is also in a node of its orbit, and that does not happen every month.

Fig. 7: Declination of Sun and Moon (4)
Fig. 7: Declination of Sun and Moon (4)

However, the orbit of the Moon is not fixed in space. The nodes of the orbit move once around the ecliptic in about 18 years. So, about 9 years after the situation displayed in figure 6 you get the situation shown in figure 7, where the ascending node has moved from right ascension 2 hours to right ascension 14 hours. Now the Moon gets less far from the celestial equator than the Sun does.

The Moon will get further or less far from the celestial equator during a standstill, depending on where the ascending node of the lunar orbit is. If the ascending node of the lunar orbit coincides with the ascending equinox (the vernal equinox, right ascension 0 hours), and the descending node with the descending equinox, then the Moon will be as far as possible from the celestial equator at the standstills, at about 28° (positive or negative). Such a lunar standstill is called a major standstill. If the ascending node coincides with the descending equinox (right ascension 12 hours), and the descending node with the ascending equinox, then the Moon will get the least far from the celestial equator at the standstills (compared to other standstills). Such lunar standstills are called minor standstill.

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The following formulas give the approximate Julian Day Numbers \(J_☊\) at which the mean ascending node of the lunar orbit passes through the ascending equinox (major standstill), and \(J_☋\) at which the mean descending node passes through the ascending equinox (minor standstill):

\begin{align} J_☊ & = 2453906.50 + 6798.3842 k + 0.001271 k^2 ± 0.11 \\ J_☋ & = 2450507.25 + 6798.3828 k + 0.001318 k^2 ± 0.08 \end{align}

Here \(k\) must be a whole number. \(k = 0\) yields a date near the year 2000. The specified standard errors (after the ±) are valid for \(k\) between −53 and +53.

If you prefer estimates measured in years, then use

\begin{align} j_☊ & = 2006.46544 + 18.612962 k + 3.480×10^{−6} k^2 ± 3.1×10^{−4} \\ j_☋ & = 1997.15880 + 18.612958 k + 3.610×10^{−6} k^2 ± 2.3×10^{−4} \end{align}

The dates at which the Moon reaches its greatest northern and southern declinations (in a period of a couple of years around \(J_☊\) or \(J_☋\)) are usually not exactly equal to \(J_☊\) or \(J_☋\), (1) because the above formulas are based on the motion of the mean nodes, and the true nodes move around the mean nodes, and (2) because for a great declination we don't just need the lunar node to be in the right place, but also for the Moon itself to be at the right place relative to that node, and those two rarely if ever coincide.

We are now (at the beginning of 2007) just past a major standstill. The next minor standstill will occur around 2015, and the next major standstill around 2025.

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The Moon rises the furthest to the north along the horizon as seen from the northern hemisphere and the furthest to the south as seen from the southern hemisphere around the time of a major standstill.

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16. Can a book of lunar phases for one country be used also in another country?

The clock times of astronomical phenomena depend on the place from where you watch the sky, but with a few modifications a book of lunar phases from Austria can also be used in Belgium.

Austria uses the same time zone as Belgium does, so clock times of the lunar phases (new moon, first quarter, full moon, last quarter) and of lunar eclipses and of when the Moon is closest in the sky to a particular star or planet are practically the same in Austria as they are in Belgium, because those phenomena depend very little on where you are on Earth.

Phenomena that depend strongly on where the Moon is in the sky relative to the horizon (such as moonrise, transit, and moonset) are more difficult. Austria is about 9 degrees east of Belgium in the same time zone, which means that such phenomena occur on average 9×4 = 36 minutes later in Belgium than they do in Austria. Belgium and Austria are in the same time zone, so the clock time at which those phenomena can be seen from Belgium are on average 36 minutes later than the clock times in Austria.

The times of phenomena that occur at the horizon (rise and set of the Moon) also depend on the latitude of the observer, which is about 3 degrees less in Austria than in Belgium. If the Moon is on a high path across the sky (like the Sun follows in summer), then the Moon is above the horizon a bit longer as seen from Belgium than as seen from Austria, so moonrise is relatively early and moonset is relatively late in Belgium as compared to Austria. And if the Moon is on a low path across the sky (like the Sun follows in winter), then the Moon is above the horizon for a shorter time as seen from Belgium than as seen from Austria, so then moonrise is relatively late and moonset is relatively early in Belgium as compared to Austria. The path of the Moon along the sky varies from low to high and back to low in about a month's time. The greatest time difference (compared to the average) due to the difference in latitude between Austria and Belgium is about a quarter of an hour. The average difference was 36 minutes, so the actual difference can be up to about a quarter of an hour less than 36 minutes and about a quarter of an hour more than 36 minutes.

Summarizing:

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17. Determination of Your Location Using Sun and Stars

17.1. Latitude

Your geographical latitude is equal to the height of the celestial pole above the horizon. The Pole Star or North Star is very close to the northern celestial pole, so in the northern hemisphere your latitude is almost the same as the height of the Pole Star above the horizon.

In the southern hemisphere there is not bright star real close to the celestial pole, but using other bright stars that are a little further away from the pole you can still estimate where the pole is, and hence what its height above the horizon is.

17.2. Longitude

You can only determine your geographical longitude by measuring the the difference between the local time of the place where you are and the local time of a place of which you know the longitude (for example, your house). Because the Earth rotates 360 degrees in 24 hours, 1 degree of longitude corresponds to 4 minutes of time.

You can estimate the local time (specifically for your location, without time zones, so that the Sun is highest in the sky around 12:00 noon) reasonably well by looking at the Sun or stars. If you also know what time it is then at home, then you can determine the time difference and hence the longitude difference.

How can you know what time it is at home if you're far from home? There are several methods: (1) bring an accurate clock from home that shows the time at home. (2) wait for a predictable astronomical phenomenon of which you know at what time it happens at home (or have someone measure it at home) and of which you can determine accurately at what time it happens, for example an eclipse of the Sun or Moon or a certain configuration of the moons of Jupiter.

I once used the first method, combined with checking the height of the Pole Star, to determine my location when I traveled from the Netherlands to California. I estimated from the stars (without using any equipment) what my latitude and the local time in California were, and from the difference between the local time and the Dutch time that my watch showed I estimated my location. I was off by a few hundred kilometers.

The second method (with eclipses) was already used by the ancient Greeks (Ptolemy and Strabo) of 2000 years ago to determine longitudes: have two observers note at what local time the eclipse happens in the two places, subtract the two times, and convert the time difference to a longitude difference. Columbus used a predicted solar eclipse to convince native people in the Carribean that he was very powerful and they should do what he wanted, and he probably brought knowledge of that eclipse along specifically to be able to determine his longitude.

Determining your longitude (which boils down to the problem of making a clock that will keep time accurately even on a ship in stormy seas) has been for centuries one of the most important problems for ships at sea, with big rewards offered for who would solve the problem. The history of this is described, for example, in the book Longitude by Dava Sobel.

Nowadays almost everyone has an accurate clock that shows the official time, i.e., the time of the official time zone, which is the time at a standard meridian. For most of western Europe (excluding the British Isles) the standard meridian is that of 15 degrees East in winter, and of 30 degrees East in summer. The difference between the official time and the local time determines the difference between your latitude and the latitude of the standard meridian for which your clock shows the time.

For example: The Sun is due south always around 12:00 noon local time (it's always between 11:44 and 12:14 local time, depending on the date; see the table). If you determined from your observations that the Sun was due south at 13:10 hours on the clock, then it would follow that the clock time is about 70 minutes ahead of local time, and hence that your location is about 70/4 = 17.5 degrees west of the meridian of the clock time.

Another example: if you read in the newspaper or somewhere else that the Sun is due south today at 12:43 hours (as determined for town X), but on your accurate clock it happens already at 12:40, then you are apparently 3 minutes or 3/4 degrees east of town X. You cannot easily use the times of sunrise or sunset for this, because those depend on your latitude as well.

Measuring the lengths of shadows is not a good way to determine when the Sun is due south, because the length of shadows changes the slowest exactly when the Sun is in the south. It is better to look at the direction of the shadow and to determine when that is due south. You can find the south by looking for the Pole Star at night: it indicates where north is, so south is in exactly the opposite direction.

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18. How Far Can You See?

At night, from a dark spot, if the sky is clear, you can see stars out to a few thousand lightyears, and sometimes another galaxy at 2 million lightyears, even without binoculars or a telescope. With a big telescope, one can see quasars at thousands of millions of lightyears. How far one can see depends on how good your eyes are, where you are, what kind of optical aids (such as telescopes) you have, and if the weather cooperates.

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19. How Far is the Horizon?

If you have a clear view of the horizon, and if the land or sea is flat there, then the distance of the horizon from you depends on how much higher your eyes are than the land or sea at the horizon (and also a little on how high that land or sea itself is).

The formula for the distance \(d\) of your eye to the horizon on a perfectly spherical Earth with radius \(R\), with your eye at height \(h\) above the ground, is equal to

\begin{equation} d = \sqrt{2×R×h + h×h} \end{equation}

where \(\sqrt{.}\) is the square root function. Measure \(R\) and \(h\) in meters or feet, then \(d\) comes out in meters or feet, too. \(R\) = 6,378,000 meters = 20,925,200 ft. For example, with \(h = 2\) (i.e., your eye 2 meters above the ground), \(d = 5051\) m, or about 5 kilometers.

If you simplify the formula, insert the radius of the Earth, and transform \(d\) from meters to kilometers, then you find

\begin{equation} d = 3.571 × \sqrt{h} \end{equation}

In this formula, \(h\) is measured in meters and \(d\) in kilometers. For example, with \(h = 2\) you find \(d = 5.05\) km. The results of this simpler formula deviate less than 1 % from the results of the complete formula if the height is less than 250 km.

Here are a couple of corresponding values for \(h\) and \(d\):

Table 1: Distance of the Horizon

\({h}\) (d\)
1 m = 3 ft 3.6 km = 2.2 mi
2 m = 7 ft 5.1 km = 3.2 mi
5 m = 16 ft 8.0 km = 5.0 mi
10 m = 33 ft 11.3 km = 7.0 mi
20 m = 66 ft 16.0 km = 9.9 mi
50 m = 164 ft 25.3 km = 15.7 mi
100 m = 328 ft 35.7 km = 22.2 mi
200 m = 656 ft 50.5 km = 31.4 mi
500 m = 1640 ft 79.9 km = 49.6 mi
1 km = 0.62 mi 112.9 km = 70.2 mi
2 km = 1.24 mi 159.7 km = 99.2 mi
5 km = 3.11 mi 252.5 km = 156.9 mi
10 km = 6.2 mi 357.2 km = 222.0 mi

For example: From an airplane at 10 km height you can see the Earth out to a distance of about 350 km (if there are no clouds in the way).

The communication tower at Lopik is 375 m high and would therefore be visible (on an otherwise flat and empty Earth) out to 69 km distance as seen from just above the ground. The communication tower at Roermond is 197 m high and from there the horizon is 50 km distant. You can see the tip of the one tower from the other tower if they are less than 60 + 50 = 119 km apart (on an otherwise empty Earth).



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