Basics of Time Reckoning

Human life is very strongly influenced by astronomical phenomena. At first the day must be mentioned, followed by the year manifested by the change of the seasons. Day as well as year are caused by the earth's own movement. The moon's phases do not have such a great impact on everyday life and are resulting from the movement of earth and moon. These basic processes are the subject of this article.

Coordinates

Some remarks shall be made on coordinates before getting into the movements of moon, earth, and sun. These coordinates are used to describe positions and orbits of heavenly bodies.

Since there is no real fix point in the universe it becomes necessary to define certain points as fixed. Such a point can be chosen as the origin of a coordinate system in which distance and speed can be measured. It is possible to define a coordinate system to have its origin in the centre of our galaxy. Tracking the sun one would find that the sun moves on an elliptical orbit (though very close to a circle) around the galaxy centre. The earth's orbit would be a bit complicated to describe. It would have many loops, the earth wobbling around the sun. Yet, for an observer on earth these coordinates are not of great interest.

Another coordinate system can be defined using the sun's centre as origin. It is used when stating that the earth moves around the sun. The orbits of the planets, including earth, have comparatively simple shapes. With the origin in the sun this system is called heliocentric. Heliocentrically the moon's orbit is waving around the earth's. It is greatly disturbed by the extremely close earth and therefore has no exact elliptical form.

Another coordinate system can be used to describe the moon's movement around earth. Its origin is the centre of gravity of earth and moon. Although this point is not in the earth's centre it still lies deep beneath the earth's surface. For considerations with less required accuracy a coordinate system with the earth's center as origin might even be used. Such a coordinate systen is called geocentric.

Compared to cosmic distances all observers on earth are relatively close to the origin of such a geocentric coordinate system. Hence we experience cosmic events geocentrically.

Real Movement of the Earth

The earth is revolving around the sun on an elliptical orbit only slightly different from a circle (Kepler's first law). Due to the ellkiptical form of the orbit, the earth's distance from the sun is not constant, ranging from 147.1 million kilometres in the point nearest to the sun (the perihelion) to a maximum of 152.1 million kilometres when the earth is in the most distant point of its orbit (the aphelion). The perihelion is passed in January, the aphelion in July. The medium distance between earth and sun is thus 149.6 million kilometres. The orbit of the earth around the sun lies in a plane called the ecliptical plane. Perihelion and aphelion are called the apsides, connected by the line of apsides. The small difference between maximum and minimum distance may illustrate that this is not the reason for the seasons on earth.

The form of the earth's orbit is illustrated in Fig. 1. The figure, as all figures on this site, must not be scaled. The sizes of sun, moon, and planets do not correspond with the real sizes compared to the orbit. The real orbit is much more closer to a circle than in the figure.

The position of the apsides is not constant over long periods. Besides the annual revolution of the earth itself the line of apsides rotates in the same direction. The earth thus "loops" around the sun. The loops although are very close because the line of apsides rotates at a very low speed. The following Fig. 2 shows the resulting form of the orbit. However, the speed of the apsides is very much slower than in the figure.

1-sun, 2-aphelion at the start, 2'-aphelion after about one revolution, 2"-aphelion
after about two revolutions

Fig. 2: Earth's Orbit with rotating Line of Apsides

The earth rotates around its axis once every 24 hours (mean solar time), causing the change of day and night for an observer on the surface. Looking from the north the earth rotates clockwise. The axis is inclined by about 23.5° from the direction perpendicular to the ecliptical plane. Thus, the planet can be considered a rotating top moving around the sun. The direction of the axis of rotation is almost constant as is illustrated in Fig. 3. A further slow movement will be described later.

The inclination of the rotation axis is the reason for the seasons on earth. Because of the axis having an almost constant direction the northern and southern hemispheres are inclined towards the sun for certain periods in the year. While the earth is moving between the vernal and the autumnal equinox, the northern hemisphere is leaning towards the sun. Therefore the sun shines for a longer time and rides higher on the sky. It is northern summer. On the southern hemisphere temperatures stay lower because of the sun ascending less high and shining for a shorter time per day. Here we have southern winter. While the earth is moving in the other half of her orbit, the situation is vice versa. Fig. 4 shows the earth as an observer following her on her track around the sun would see her.

N	N	N	N

S	S	S	S
When the earth passes the vernal equinox on about 21 March (pos. 1 in Fig. 3) north and south poles are on the line between day and night. Except on the poles, the day has a length of 12 hours at any location.	On about 21 June (summer solstice, pos. 2 in Fig. 3) the sun is perpendicular bove the Tropic of Cancer. The north pole is inclined towards the sun as far as possible. South of the southern polar circle there is polar night.	On about 23 September (autumnal equinox, pos. 3 in Fig. 3) day and night have the same length of 12 hours again, as they had half a year before.	The winter solstice occurs on about 21 December (pos. 4 in Fig. 3). The sun is perpendicular above the Tropic of Capricorn, and we have polar night in areas north of the northern polar circle.
Fig. 4: Seasons

The shape of the earth differs slightly from that of a perfect sphere. Neglecting further local irregularities it can be approximated as an ellipsoid. The diameter measured along the rotation axis is about 21 km smaller than that measured on the equator. Fig. 5 illustrates the ellipsoid (the actual shape is much more sphere-like than in the figure).

Compared with an ideal sphere, along the earth's equator mass is concentrated. Because of the inclination of the rotation axis this mass concentration is not within the ecliptical plane. Sun, moon, and the planets are moving close to the ecliptical plane. Gravitational forces of these bodies cause the effect called precession which is a slow rotation of the earth's axis around an axis perpendicular to the ecliptical plane. Fig. 6 illustrates the circumstances leading to the precession.

			The axis of a top has a constant direction as long as there are no external forces acting on the top. With a moment of a force acting on the top its axis would not follow this moment directly but move perpendicular to it. The gravitational forces F mainly of moon and sun are pulling on the top near the ecliptical plane E. They try to tear the mass that is concentrated on the equator into the ecliptical plane. Resulting is the moment M of that gravitational forces. The earth's axis is moving perpendicular to that moment and thus rotating slowly as shown in the sketch.
	Fig. 6: Forces causing Precession

Seen from the north the earth's axis is rotating clockwise, i. e. in the opposite direction of the earth's movement around the sun. Fig. 7 illustrates the effects of the precession. The figure shows a perpendicular view from the north on the ecliptical plane. The earth's north pole can thus be seen in the sketch.

At the start the earth is at the point marked 2. The earth's north pole is inclined as far as possible to the left, which is exactly towards the sun. Hence point 2 is the summer solstice. While moving on along its orbit in direction 2' the axis of the earth is simultaneously rotating because of the precession. The direction of this rotation is marked 2" in the figure. Thus the autumnal equinox occurs at point 3 already. It can be seen, that there the north pole is inclined not exactly to the left anymore. The points of the solstices and equinoxes that would occur without the precession are indicated by dotted lines. Compared with these, the autumnal equinox (as well as the other solstices and equinoxes) occurs earlier. The winter solstice is reached in point 4 where the north pole is inclined away from the sun as far as possible. Compared with the autumnal equinox the earth's axis has rotated further.

1-sun, 2-earth's position on the summer solstice, 2'-direction of the earth's movement around the sun, 2"-direction of the precession movement, 3-earth's position on the autumnal equinox, 4-earth's position on the winter solstice, 5-earth's position on the vernal equinox, 6-earth's position on the summer solstice of the following year (The actual speed of the precession is very much lower than shown in the sketch. The sizes of earth, its orbit, and sun are not signifying the actual proportions.)

Fig. 7: Precession

After one (tropical) year the summer solstice occurs at point 6 already, i. e. earlier than point 2 where it occured if there was no precession. Due to the precession the solstices and equinoxes are moving slowly "backwards" along the earth's orbit. The precession is much slower than the 30° per year shown in the sketch. In fact, the axis rotates about 50 angle seconds per year, which are some 0.014°. The time it takes the solstices and equinoxes to rotate the full 360° is about 25700 years, a so-called Platonic year.

Real Movement of the Moon

Described in the same coordinate system as the earth's orbit (i. e. heliocentrically), the moon would appear to be swinging around the earth on an orbit still bent towards the sun in every point. One may say that the moon is moving around the sun on an orbit strongly disturbed by the (very close and comparatively heavy) earth. For people on earth, however, the moon's movement as observed from earth is much more interesting.

Geocentrically, the moon is moving on an elliptical orbit around the earth (1). In the point nearest to earth (the perigee) its distance is 356410 km, while in the most distant point (the apogee) it is 406740 km away from the earth. The mean distance is 384400 km which is about 30 times the earth's diameter. As for the earth's orbit around the sun there is also a line of apsides for the moon's orbit, which is the line between perigee and apogee.

The plane in which the orbit lies is inclined by slightly more than 5° against the ecliptical plane. The line between the two points where the moon crosses the ecliptical plane rotates slowly in the opposite direction of the moon's movement. This rotation has a period of some 18.6 years. The line of apsides is also rotating. The speed of this rotation is much higher than that of the earth's line of apsides. A full rotation takes just 8.85 years.

Moon's Movement and Moon's Phases

The change of the phase of the moon can easily be observed without any telescope. The light the observer sees is the sunlight which is reflected by the moon's surface. When the moon is between the earth and the sun, it is invisible at night: new moon. On full moon, the earth is between the moon and the sun. The inclination of the moon's orbit causes the moon's shadow to miss the earth in most cases. The moon also misses the umbra of the earth's shadow on most full moons. Eclipses occur only if the moon is near one of the nodes of its orbit on full moon (lunar eclipse) or new moon (solar eclipse). During a (synodic) month the relative positions of earth, moon, and sun are changing continuously. Thus, different areas of the part of the moon surface visible from earth are illuminated by sunlight.

Fig. 8 may serve to illustrate how we come to see the moon in different shapes.

The figure shows a view on the moon's orbit from the north. The moon is moving counter clockwise.

Apparent Movement of Sun and Moon

The movements described so far mostly cannot be directly observed directly from earth. Moreover, the daily paths of sun and moon on the sky seem to lead her around the observer. To the observer, the earth seems to stand fixed.

Looking towards the sky the observer perceives a giant vault having its highest point perpendicular above the observer. The height of this vault seems smaller than its radius. This discrepancy is caused by the human eye measuring angles differently in horizontal and vertical direction. Therefore, a model is introduced of a celestial sphere.

According to this model, the earth is in the center of a gigantic theoretical sphere, The radius of this sphere is taken to be very large so that the earth's radius is negligible compared to it. The celestial sphere radius can even be considere infinit. On the celestial sphere, certain points and lines can be defined.

For the observer, only the upper half of the celestial sphere is visible, the lower one being covered by the earth. The plane which includes the location of the observer and is tangential to the earth's surface intersects the celestial sphere in a circle called horizon. The plane itself is sometimes called horizontal plane.

The plane in which the earth's equator lies intersects the celestial sphere in the celestial equator, an important line on the celestial sphere. In the same way the celestial north and south poles are defined as the points in which the earth's axis meets the celestial sphere. The celestial northern pole is perpendicular above the earth's north pole.

A Circle perpendicular to the celestial equator and going through the celestial poles is called circle of declination. Especially one such circle going throug a certain fixed star is the circle of declination of that star. This circle is going through the star an the two celestial poles is used for defining the length of a sidereal year later.

Comparing these definitions with the geographical coordinate on earth the celestial poles resemble the respective poles of the earth. The circles of declination are corresponding to the meridians on earth and the celestial equator to its celestial counterpart.

Fig. 9 shows the celestial sphere. The ecliptical plane is (arbitrarily) chosen to be horizontal. To be able to show the position of the earth's equator and poles, the size of the earth is strongly exaggerated.

1-vernal equinox, 2-earth's equator, 2'-celestial equator, 2"-equatorial plane,
3-celestial sphere, 4-earth's axis, 5-celestial north pole, 6-apparent path of
the sun in one year, 7-ecliptical plane

Fig. 9: Celestial Sphere

Definition of Year and Month

The vernal equinox can be taken as the starting (and end) point for the definition of a tropical year. The length of it is currently 365.242199 days. That is the time it takes the sun, starting on the vernal equinox, to return to the vernal equinox on her apparent path. For calendars this year is very important, because the seasons repeat with this period. Astronomically, the beginning of spring is defined as the moment, in which the sun passes the vernal equinox.

Besides the tropical year, the sidereal year has some calendrical significance, being employed in the Hindu calendar, for instance. It is the time between two successive passes of the sun throug the circle of declination of a certain fixed star (i. e. 360°). It has a length of 365,256354 days. Thus, it is slightly longer than than the tropical year. The difference is caused by the precession movement (see above) which results (on the celestial sphere) in the vernal equinox moving in the opposite direction than the sun. This is illustrated in Fig. 10.

1-direction of the precession movement of the vernal equinox, 2-direction
of the sun's apparent movement, 3-path and direction of the celestial north
pole's movement, 4-earth's axis

Fig. 10: Precession of the Vernal Equinox

Apart from tropical year and sidereal year there can be defined the anomalistic year. It is the time between two successive passes of the earth through the perihelion (see above).

After the sun the moon is the most significant heavanly body for the observer on earth. It is apparently moving faster on the celestial sphere. The time it takes the moon to go once around the earth is called (lunar) month. As with the year, choosing different start and end points leads to different month lengths.

The sidereal month is the time between two successive passes of the moon through the circle of declination of a certain fixed star. It has a length of 27 days 7 hours 43 minutes and 11.5 seconds. More obvious though is the synodic month. It is defined as the time between two successive conjunctions of moon and sun, i. e. the time between two successive new moons (or full moons). With a length of 29 days 12 hours 44 minutes and 2.9 seconds it is more than two days longer than the sidereal month. The reason for this is the earth simultaneously moving along her orbit around the sun.

Fig. 11 illustrates the difference between sidereal and synodic month. At point 3 the moon appears as full moon, the earth being at point 2. At the same time the moon appears in the same direction as a certain fixed star 6. After a full revolution the moon is at point 3' and again appears in the direction of that fixed star (6'). With the earth having moved to point 2' in the meantime, the moon still has to go to point 3" to appear as full moon again. (Even in that time the earth is moving, but for the sketch this is negligible.)

1-sun, 2, 2'-earth, 3, 3', 3"-moon, 4-earth's orbit, 5-moon's orbit (geocentrically), 6, 6'-direction
towards a certain fixed star; for further explanation see text

Fig. 11: Synodic Month and sidereal Month

Bound Rotation

The moon is pointing only one side towards the earth, the "back" side being invisible for an observer on earth. The reason for this phenomenon is the rotation of the moon around its axis. It takes the moon to rotate around its axis as much time as it needs to travel once around the earth, making a day (lasting from noon to noon) on the moon as long as our synodic month. However, the irregularities in the moon's movement make it wobble around a mean position, making about 59% of the moon's surface visible from earth - at different times, of course. The first pictures of the "back" side were taken by the Soviet sond Luna 3 in 1959, which is why we can find formations as Mare Moscoviense, Tsiolkovski, or Korolev.

Last but not least, the moon is the only extraterrestial heavenly body on which man left his footsteps, quite literally. The US astronauts Neil Armstrong and Edwin Aldrin, on 20 July 1969, were the first men to walk on moon's surface.

1
The earth is not exactly rotating around an axis through its center. Earth and moon rotate around an axis going through their common centre of gravity. Because of the moon's relatively high mass, this point is not in the earth's centre, but deep inside the earth's body. Nevertheless, this has no effect on calendars.

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