Orbit of the Moon

As the moon's orbit approximately elliptical orbit of the Moon around the Earth is called. Since the Moon is exposed not only the attraction of the earth, but at the same time that of the sun and the other planets, its orbit deviates significantly from a pure Kepler ellipse. Your exact calculation is a complicated task whose solution is the subject of lunar theory as part of celestial mechanics. They gave impetus to numerous important physical and mathematical developments.

• 4.1 Fundamental arguments
• 4.2 variation of the semimajor axis
• 4.3 variation of the eccentricity
• 4.4 Rotation of the line of nodes
• 4.5 variation of the orbital inclination
• 4.6 Rotation of the line of apsides
• 4.7 disturbances in ecliptic longitude 4.7.1 Large inequality
• 4.7.2 evection
• 4.7.3 variation
• 4.7.4 Annual equation
• 4.7.5 Reduction to the Ecliptic
• 4.7.6 Equatorial equation
• 4.7.7 Secular Acceleration

Path geometry

The moon's orbit can be approximated regarded as a Kepler ellipse. Due to the disturbances which are caused mainly by the attraction of the sun, this ellipse, however, changed in a complex way both their form and their position in space.

Form

On average, the Moon's orbit is an ellipse with a semimajor axis of about 383,398 km and an eccentricity of about 0.0555 dar. This would correspond to Perigäumsabstand of 362,102 km and a Apogäumsabstand of 404,694 km from Earth. Due to the aforementioned disorders, however, vary both the semi-major axis and the eccentricity ( → perturbations ), so that larger and smaller Extremalabstände are possible. For more information, see the section Erdabstand.

The moon moves across the web prograde, ie seen from the north pole of the ecliptic from the counterclockwise direction, and for a terrestrial observer from west to east (not to be confused with those caused by the Earth's rotation apparent daily movement, which over him from east to west the sky wearing ). Its average path velocity 1.023 km / s, it varies - as required by the second law of Kepler for the elliptical orbit - between 0.964 km / s and 1.076 km / s

The moon moves in its orbit is not exactly around the center of the earth, but to the barycentre, the common center of gravity of the Earth- Moon system. This focus is - as required by the first law of Kepler - in one of the two foci of the elliptical orbit. Since the lunar mass about 1 / 81.3 is the mass of the Earth, the barycenter is the mass ratio corresponding to an average of 384.4 thousand km / 82.3 ≈ 4670 km from the center of the earth away - so only about 1700 km deep in the Earth's mantle. The earth thus runs once a month at an average distance of 4670 km around the barycenter. Earth's center, center of mass and center of the Moon always lie on a common line and in a common plane of the lunar orbit plane. Since this is inclined by about 5 ° to the ecliptic plane, the center of the Earth runs at times a little above, and at times a little below the plane defined by the course of the barycenter ecliptic plane around the Sun. The ecliptic latitude of the earth (that is, seen from the sun deviation from the ecliptic plane ) can be up to 0.7 ".

Location

The plane of the Moon's orbit is inclined to the orbital plane of the Earth ( the ecliptic plane ) on average by about 5.2 °. However, the inclination varies with a period of 173 days ( half eclipse year ) of about ± 0.15 ° around this average value (→ perturbations ).

The intersection of earth and lunar orbit plane ( the line of nodes ) is not associated with a fixed orientation in space, as it would in the absence of disturbances of the case, but performs due to the interference once in 18.61 years, a full downward rotation of 360 ° along the ecliptic (→ perturbations ). Because of this precess the lunar orbit over the years passes alternately over and under a given Ekliptikabschnitt so that the orbit of the moon through one of these precession cycles averaged with the ecliptic plane coincides. This distinguishes the Earth's moon from most other moons that either circulate in the middle in the equatorial plane of their planet or having captured as moons very strong orbital inclinations (see Laplace plane ).

The position of the line of apses, which describes the orientation of the ellipse in the orbital plane does not remain constant, since the disturbances a Apsidendrehung with a period of 8.85 years cause (→ perturbations ). During this period, the perigee runs once prograde around the web.

Erdabstand

Mean distance

The temporal arithmetic mean of the variable distance between the centers of the Earth and Moon is 385,001 kilometers. This can be seen for instance from the given of Chapront and Chapront Touzé - series expansion of the distance:

If one forms the arithmetic mean over this expression, as fall continues the cosine terms, and remain as rounded average 385,001 km. (For the significance of GM and D → see fundamental arguments. )

However, 384.4 thousand kilometers are given traditionally. This value comes from a different mathematical formulation. The lunar theory of EW Brown gave the distance in kilometers not directly, but as a horizontal parallax of the moon at:

Forming the arithmetic mean here, so also is only the constant term and the average lunar parallax is 0.950 724 5 °. If one calculates from the distance of the moon

We obtain 384,399 km, which corresponds to the traditional rounded usual value.

The two numbers are not identical, because even on the distance itself, and is averaged over time the reciprocal value ( in the form of parallax ), see harmonic mean. From a mathematical point of view both averaging are equally legitimate.

Extreme distances

If the Moon's orbit undisturbed ellipse, so the moon would always go through the same perigee and Apogäumsabstände. However, since the eccentricity of the orbit is subject to periodic changes, different extreme distances, depending on how exactly coincident a Apsidendurchlauf of the moon with a particularly large or small eccentricity. The eccentricity takes all 206 days at a maximum when the semi-major axis of the lunar orbit is in the direction of the sun. Then Perigäumsabstand particularly low and Apogäumsabstand is particularly large. Is the semi-major axis perpendicular to the direction of the sun, then decreases to a minimum, the eccentricity and the Apsidenabstände are less extreme. In addition, these changes in the web is not always the same size and in addition long-term drifts are subject. There is therefore a complicated distribution of perigee and Apogäumsabständen without a clear largest or smallest value could be specified. The more extreme a distance, the more rarely it occurs, but there is almost always an opportunity to find a more extreme value at reasonable search. Several authors therefore call it differently rounded extreme values ​​.

A simplified overview of the distribution of distances occurring, the following table:

As can be seen, the Perigäumsabstände vary significantly stronger than the Apogäumsabstände.

Individual values ​​can also occur outside of the specified rounded borders. As record levels in the period from 1500 BC to 8000 AD can be found:

• Greatest Apogäumsabstand: 406 719.97 km on January 7, 2266
• Smallest Perigäumsabstand: 356 352.93 km on November 13, 1054 BC

A particularly low Perigäumsabstand is reached when the moon passes through the perigee as a full moon, the Earth is at aphelion and the moon at the same time has its maximum distance from the ecliptic (ie largest northern or southern latitude). A particularly large Apogäumsabstand is achieved when the moon passes through the apogee than new moon, the Earth is at perihelion, and the moon has its biggest Ekliptikabstand.

Orbital periods

The moon takes on average about 27.3 days to orbit the Earth in relation to the fixed stars once. After this sidereal (that is, the star -related ) month, he draws from the earth to be seen again on the same star over.

During such month, the earth moves in turn further on their orbit around the sun. This is also the direction in which the sun appears seen from Earth changes. Has the moon reaches its original position relative to the fixed stars again after one sidereal month, he must also put aside about 29 °, to regain the same position relative to the sun and thus again the same moon phase. He needs it in the middle just over two days; the synodic month, which corresponds to a complete cycle of all the phases of the moon ( a lunation ) has, therefore, a mean length of 29.5 days. This is the average time interval at which a moon phase repeatedly (eg from full moon to full moon).

Because of Apsidendrehung ( so ) migrate perigee and apogee along the railway, and in the same direction as the moon itself, so that he still has to travel a bit to the completion of a sidereal month to arrive back at the same vestibule. The anomalistic month of 27.6 days is the length of time between two passages of the moon through the perigee or apogee of its orbit, and thus the actual orbital period ( anomalistic period) of the elliptical orbit.

Because of the precession of the lunar orbital plane (see above) wander the path nodes along the ecliptic, ie the motion of the moon opposite, so that he previously to the same node is returned as the same star. Has the drakonitische month as the time interval between two passages of the Moon through the same node, therefore, on average, a length of only 27.2 days. Since lunar and solar eclipses can occur only in the vicinity of a node passage between two eclipses always an integer number of drakonitischen months.

The mean lengths of the various months are as follows:

The lengths of the months listed are averages. Since the movements of both the moon and the earth are uneven in their elliptical orbits, may vary significantly individual months more or less. The duration of a given synodic month, for example, up to about 7 hours or 6 hours longer be shorter than the mean synodic month. In addition, the monthly mean lengths are due to long-term changes in the Earth and Moon's orbit a slow drift. The exact length of the mean synodic month, for example, is calculated according to

Where T is the time elapsed since the standard epoch J2000.0 number of Julian centuries.

In a period of 223 synodic months go 242 drakonitische months to almost exactly an integer. After this period the moon so returns both at the same moon phase as well as to the same node. In order to repeat the conditions for a solar or lunar eclipse, and 223 synodic months after a given darkness is therefore to be expected again with a darkness. This period of 18 years and 10 1/3 days (or 11 1/3 days depending on the number of leap years included ) is the known as the eclipse period Saros period. Since in a Saros period also almost exactly 239 anomalistic months are included, the moon the same Erdabstand and the same dependent of the anomaly Large inequality has again (see below), so that the second darkness very similar runs as the first.

Cassini's laws

The significant correlations between the self-rotation of the Moon and its orbital motion has been recognized by JD Cassini and published in 1693:

• The moon rotates uniformly about its polar axis; its rotation period is identical to the mean sidereal period of its orbit around the Earth.
• The inclination of the lunar axis to the ecliptic remains constant.
• The descending node of the lunar equator on the ecliptic coincides with the ascending node of the lunar orbit on the ecliptic and precesses together with him.

These laws describe several observational facts. An earthly observer always sees the same side of the moon. This must therefore obviously in the same period of time even rotate around itself, in which he runs once around the Earth; this is the first Cassini law. Since the self-rotation of inertia reasons must be done uniformly always, the circulation along the elliptical path but runs unevenly, advised the two during a rotation at times slightly out of step and compensate not always exact. Therefore, the Moon performs from the perspective of the observer during each orbit, a periodic lateral rotation by up to nearly ± 8 ° libration in longitude. The observer can see from the eastern edge of the moon once something more from the western edge and even a little more. This librational motion was discovered by Hevelius.

On Galileo the discovery goes back to the moon also carries a pitching movement beyond, the libration in latitude: he could at times more from the North Rim and at times to see more from the southern edge of the moon. If the axis of the moon are perpendicular to its orbital plane, it would the observer (which yes is in the orbital plane is also practically ) always appear unchanged; So the observation proved an inclination of the lunar axis relative to the orbit. Would be the axis, eg perpendicular to the ecliptic plane, it would have been good 5 ° inclined to the train and the observer could see a nod to good ± 5 °; but are observed approximately ± 7 °, which are reached always at maximum ecliptic latitude of the moon. It follows that the moon axis has yet to be additionally tilted in the form that the northern hemisphere of the moon on reaching its maximum north latitude is somewhat more inclined away from the earth, and a little further executed inclines when reaching the maximum south latitude of the earth. This configuration is described by the Third Cassini Law. It can also be expressed as follows: the normal to the lunar orbit, the perpendicular to the ecliptic plane and the rotation axis of the Moon are together in one plane, the Ekliptiksenkrechte between the other two lies. Since the moon's orbit precesses running (→ rotation of the line of nodes ), the configuration described but - like watching shows - is preserved, so the axis of rotation of the moon as the lunar orbital plane must perform a precession motion in the same direction and the same speed.

The lunar orbit plane is 5.2 ° inclined to the ecliptic plane and the equatorial plane of the Moon is in turn inclined by 6.7 ° from the lunar orbit plane, but in the opposite direction (Third Cassinisches Act). The inclinations therefore raise almost on, and the moon equator is inclined by only 1.5 ° to the ecliptic. Therefore, the solar radiation is subject to the moon almost no seasonal variation, and the sun is at the lunar poles always near the horizon.

The moon equator is inclined at 6.7 ° to the lunar orbit plane and about 1.5 ° to the ecliptic, the angle between lunar orbit and the ecliptic plane but varies by ± 0.15 ° (→ variation of orbital inclination ). After the Second Cassini's law, the inclination of the moon relative to the ecliptic is constant, therefore the inclination of the moon varies in its own orbit around ± 0.15 °.

How could later show the theoretical celestial mechanics, a describe the dynamic stability of the Cassini laws. The longest axis of the moon as described triaxial ellipsoid (on average) is always oriented towards the ground. This orientation corresponds to a minimum of the gravitational energy and therefore remains stable. An ellipsoid may also (without wobbling ) rotate only stable when it rotates about the axis with the largest or the smallest moment of inertia. In the case of the moon is the longest axis (which has the smallest moment of inertia ) as just described retained by the earth, the rotation of the Moon, therefore, takes place about the shortest ellipsoid axis. This rotation axis can not be exactly perpendicular to the lunar orbital plane beyond, otherwise the earth would be in the equatorial plane of the moon and could not hold a torque which causes the precession of the lunar axis described in the Third Cassini Law. The tendency arises ( for energetic reasons ) so that the precession of the lunar axis with the same rate of movement is the same as the precession of the lunar orbit.

Due to the significant perturbations of the Earth-Moon system does not follow strictly the empirically Cassini's laws, but also leads to fluctuations in the ideal configuration. The Cassini laws are therefore considered only the average.

Perturbations

The orbit of the moon can be approximately described as a Kepler ellipse. However, while an undisturbed Kepler ellipse of two point masses in space would be maintained both their shape and their location, subject to the lunar orbit numerous additional gravitational influences and therefore changes shape and position significantly. The main disturbances are the gravitational effect of the sun and the planets (especially Venus and Jupiter ) and the deviation from the spherical shape of the earth.

Fundamental arguments

For the interpretation and calculation of explained in the following disturbances are needed for each this time to be determined "fundamental arguments " that describe the average position of the moon and the sun with respect to different reference points:

Where t is the number of elapsed days since J2000.0 Standardäquinoktium: t = JD - 2451545.0.

Variation of the semimajor axis

The mean 383 397.8 km semi-major axis overlap numerous periodic fluctuations. The most important are a variation of ± 3400.4 km with a period of 14.76 days and a by ± 635.6 km with a period of 31.81 days.

As the series expansion shows

It takes the leading fluctuation term to the largest positive value when the elongation of D is 0 ° or 180 ° accepts, ie at New Moon and Full Moon. The largest negative values ​​of this term result in the first and last quarter (D = 90 ° or 270 °).

Variation of the eccentricity

The mean value of the eccentricity 0.055 546 also superimpose numerous periodic fluctuations. The most important are a variation of ± 0.014 217 with a period of 31.81 days and a variation of ± 0.008 551 with a period of 205.9 days.

The series expansion

Shows that the eccentricity becomes a maximum when the semi-major axis of the lunar orbit towards the sun shows ( then 2D - 2GM = 0 ° or 360 °, ie D = GM or GM 180 °). This happens on average every 205.9 days (a little over half a year since the apsides is indeed the same prograde to 0.11140 degrees per day moves ). It assumes a minimum when the semi-major axis is perpendicular to the sun (2D - 2GM = ± 180 °).

This fluctuation superimposed on strong fluctuations of shorter period ( with maxima at 2D - GM = 0 ° or 360 °, every 31.8 days ) as well as a number of minor variations.

The eccentricity varies in total between the extreme values ​​0.026 and 0.077.

Rotation of the line of nodes

The node line is not fixed in space but moves in decline along the ecliptic. Therefore, the nodes come to meet the moon, which is why a drakonitischer month (the return to the same node ) is shorter than a sidereal month (return to the same fixed star ).

The average velocity of this movement is relative to the ( self-propelled ) the spring point 19.34 ° per year. For a complete revolution of the line of nodes takes 18.6 years (specifically 6798.38 days, a circulation with respect to the fixed stars takes 6793.48 days ).

This movement also superimpose periodic fluctuations. The largest fluctuation term has an amplitude of 1.4979 ° and a period of 173.31 days. This variation was discovered by Tycho Brahe. It means that the line of nodes for a short time almost stands still when she shows towards the sun.

Add or subtract the inclinations of the lunar orbit on the ecliptic and the inclination of the ecliptic to the equator, depending on the mutual position of the line of nodes of the lunar orbit and the line of nodes of the equator, so that the Moon with a period of 18.6 years, alternately a Deklinationsbereich of ± 28.6 ° ± 18.4 °, or only sweeps.

This precession of the lunar orbit has the same cause as the precession of Earth's axis: the attraction of the sun is trying to draw the moon's orbit inclined to the ecliptic plane. The circular on this web Moon reacts like a top, which responds to the external torque with a pivoting axis of the lunar orbit.

The moon approaches, for example, the descending path node, thus causing the additional force acting in the direction of the ecliptic plane disruptive force that the moon of the ecliptic approaching faster than it would have been the case without disturbance force. The moon therefore pierces the Ecliptic earlier and at a steeper angle. The disturbance force has therefore the line of nodes pushed against the moon and increases the orbital inclination. Since the nodal line is shifted each time in the same direction, resulting in a more stable circulation of the nodes along the ecliptic. The short time enlarged orbital inclination, however, is reduced after passing through the node again. The disturbance force now reduces the rate at which the moon moves away from the ecliptic, so that its path becomes flatter again. Therefore, the orbital inclination learns just a regular fluctuation and does not change as the nodal line resistant to the same effect.

Variation of the orbital inclination

The orbital inclination varies about ± 0.15 ° around its mean. The dominant variation has an amplitude of 0.14 ° and a period of 173.3 days.

The series expansion

Can be inferred that the slope becomes a maximum when the line of nodes of the lunar orbit towards the sun shows ( then 2D - 2F = 0 ° or 360 °, ie D = F or F 180 °). This happens on average every 173.3 days ( a little less than half a year, since the line of nodes at the same time yes moves retrograde at 0.05295 degrees per day). This variation are also smaller fluctuations superimposed, which marked occur in the minima of the slope.

The orbital inclination varies, because the attraction of the sun is trying to reduce the inclination angle by pulling the moon in the ecliptic plane. The effect of the sun is maximum when the angle between the ecliptic and lunar orbit pointing towards the sun, so the line of nodes is transverse to the direction of the sun. It is zero when the line of nodes points toward the sun; then takes the slope again to larger values. The period of 173.3 days is therefore just a half eclipse year and the orbital inclination is always at a maximum when the sun is in the vicinity of a node, ie, in particular at eclipses.

The variation of the orbital inclination was discovered by Tycho Brahe, who - according to earlier suggestions were made - in a letter of 1599 for the first time definitely described.

Rotation of the apsides

The apsides is not fixed in space; it moves prograde along the moon's orbit. The apses therefore move in the same direction as the moon, which is why an anomalistic month (return to the same apsis ) is longer than a sidereal month (return to the same fixed star ).

The average speed of Apsidendrehung is with respect to the vernal equinox 40.7 degrees per year. For a complete revolution of the apses need 8.8 years ( 3231.50 days more precisely, a circulation with respect to the fixed stars takes 3232.61 days )

The major terms of the superimposed fluctuations have an amplitude of 15.448 ° with a period of 31.81 days or 9,462 ° with a period of 205.9 days .. In the course of these fluctuations, the Pergäum can take up to 30 ° from its mean position remove.

The periodic changes in distance and speed of the moon while passing through its elliptical orbit caused by the varying tangential and Zentripetalkomponenten acting on the moon gravity. The disturbances alter these force components. In particular, strengthen and weaken them alternately centripetal forces, the weakening prevails, however, as a closer examination shows. The acceleration of the moon toward Earth is thereby reduced, and after a emanating from perihelion web path to Earth's Moon has not approached again as far as it would have been the case without interference. The moon takes a little longer to get back to reach a perihelion; the perihelion and thus the entire line of apsides has therefore shifted towards the moon's motion along the track.

Disturbances in ecliptic longitude

If the position of the Moon along the ecliptic ( its ecliptic longitude λ ) are calculated taking into account the interference, the modern perturbation theory provides a rich set of correction terms, which - added to uniformly increasing average length LM - gives the correct position. The leading terms of such a statement are:

Similar series expansions are also available for the ecliptic latitude and the orbital radius of the moon.

Since the Moon has attracted the attention of measuring and computing Astronomy at a very early stage, some of the largest error terms have been known for a long time and even have their own names.

" The discovery and clear distinction of all those disorders of the lunar orbit, which are within the limits of accuracy of observation with the naked eye, under the most remarkable achievements of earlier science must be counted. Thus the foundation was prepared, on which Newton's dynamics was able to build to reveal a unifying principle of explanation for a variety of seemingly unrelated effects. "

Great inequality

This term is not a fault as such, but only to the consideration of the non-uniform due to the Bahnelliptizität speed. The moon is slower than in the middle close to perigee faster and in Apogäumsnähe. At a mean anomaly GM of approximately 90 ° or 270 °, it is therefore in each case reaches its maximum deviation from the mean position.

In addition to the above leading term of the perturbation series can be found in the remaining terms of the midpoint equation

Which for a path of the eccentricity e of the difference between the true anomaly and ν describes GM medium anomaly (and hence a solution to the problem is Kepler ). 4 of the above term interference number, the next following term is the midpoint equation. The historically common terms " inequality " or " equation " are to be understood in the sense of " correction ".

Overall, the moon may differ due to the midpoint equation to ± 6.2922 ° from the position of a fictitious uniform current moon. This significant deviation was known to ancient astronomers. The Babylonians described by arithmetic series, the Greek astronomer by an epicycle suitably chosen.

Evection

The periodic disturbances of the eccentricity and the position of the perigee deform the path such that the moon - the deformed sheet following - alternately leads or lags the middle position.

Standing Sun, Earth and Moon in a line ( S-E -M or S -M -E, full moon or new moon) so the sun the earth draws in the first case to more than the moon, and in the second case the moon stronger than the earth. The speed of the moon reduced - in both cases by the distance between the Sun and Earth is enlarged and - after the third Kepler 's law. Standing Earth and Moon so that their connecting line is perpendicular to the direction of the sun (First or Last Quarter ), so both are indeed equally attracted by the sun, but the directions in which they are drawn, are not exactly parallel, they converge to the Sun down. It therefore results in an attraction component which moon and earth approaches to each other, thereby increasing the speed of the moon - turn after the third Kepler 's law - increases. The greatest distance to the undisturbed position of the web is always achieved when the speed of variation just described changes sign and begins to operate in the opposite sense. The procedure just described is still superimposed by a velocity fluctuation caused by the eccentricity of the lunar orbit, so that overall a more complicated evolution of the perturbation results.

In the syzygies (2D = 0 ° or 360 °) of the error term is reduced to -1.274 ° · sin ( GM). So the evection is negative, when the moon is at perigee and apogee between this time (0 < GM <180 ° ) and positive when the moon passes between the apogee and perigee is (180 ° < GM <360 °). In the quadratures (2D = ± 180 ° ) dominate the inverse ratio.

In the intermediate positions of the moon, the course of evection is more complicated, but it is always zero when the sun is in the middle between the moon and the perigee is (D = ½ GM), or from that point 90 ° or 180 ° is removed. Their maximum values ​​of ± 1.274 ° reaches the evection with a period of 31.8 days.

Discovered the evection of Ptolemy, having apparently found Hipparchus already signs of deviations from simple Epizykelmodell. It succeeded Ptolemy, to recognize a pattern in the measured deviations and take by introducing a crank mechanism in its epicycle.

Variation

The variation depends only on the elongation D of the moon, so its angular distance from the Sun and thus indirectly by the moon phases. It disappears when the elongation is 0 °, 90 °, 180 ° or 270 °, ie, at new moon, full moon and the two half moons. Their maximum values ​​of ± 0.658 ° it reaches between these path points, ie in the so-called octants (45 °, 135 °, 225 °, 315 °). It therefore varies with a period of half a synodic month.

The cause of the variation is that, unlike in the evection an integer multiple of 90 ° in the octant of the angle that the line connecting the Earth - Moon the line of action of the sun on the earth and the moon occupies but a slant 'Component includes, which causes a forward and reverse sliding of the moon, with respect to its undisturbed position instead of getting closer or farther.

The amount of variation would have the ancient astronomers certainly permitted to discover them; However, the Greeks used mainly for railway eclipses provisions of the moon, where the variation is zero and is not to notice. It was discovered by Tycho Brahe and was first mentioned in 1595 in a letter to Hagecius.

Annual equation

The Annual equation means that the moon is a bit slower move when the Earth-Moon system is located close to the sun ( the perihelion side half of the Earth's orbit, so there are currently in winter) and slightly faster in the aphelion side half ( so during the summer ). It is subject to a period of one anomalistic year and reached maximum values ​​of ± 0.1864 °.

The Annual equation is caused by the eccentricity of Earth's orbit. The Earth-Moon system is in aphelion, the attraction of the sun is slightly smaller in proportion to the gravity of the earth, and the moon is far less dragged away by the sun from the earth. It is in this situation, the soil that is somewhat closer to, and therefore is faster. At perihelion, however, the attraction of the sun is stronger, the moon is further dragged away from the earth and moves slower. In autumn, the moon so its middle position runs ahead slightly, in spring, it remains something. This is also a variation of the orbital periods of ± 10 minutes is connected.

The Annual equation was independently discovered by Kepler and Brahe.

Reduction to the ecliptic

The reduction to the ecliptic is again no disturbance in the true sense. It is used to take account of the fact that the plane in which the moon moves is inclined to the ecliptic plane along which the ecliptic longitude is counted. The therefore necessary conversion of the path coordinate in the Ekliptikkoordinate can by a coordinate transformation or - done by a series expansion - as here.

The magnitude of the reduction depends on the mutual distance of the two mutually coordinate planes tilted, and thus of the distance along the path of the counted FM of the moon from the ascending path nodes. The reduction to the ecliptic is zero in the path node and in the middle between the nodes (with FM = 90 ° and 270 °). When FM = 45 °, 135 °, 225 ° and 315 ° it becomes maximum. So it varies with a period of half a drakonitischen month.

Ptolemy knew this term, but neglected it because of its smallness.

Equatorial equation

The Equatorial equation assumes maximum amounts of ± 0.0356 ° and has a period of a synodic month.

It comes into being as the Annual similar equation. The new moon is closer to the Sun than the full Moon. It is therefore strongly dragged away by the sun from the earth and is slower than the full moon because of its greater distance. The reason slowly accumulating deviation from the undisturbed position is in the crescent phase is the greatest.

The name of this disorder comes from the fact that they are earth - moon to the Earth-Sun distance depends on the ratio of the distance and therefore makes it possible to determine, from a close examination of the moon's motion, the distance and thus the parallax of the sun. Namely, while the other disorders are primarily dependent on the gravitational force of the sun, that is, on the one hand by their removal, on the other hand but also on their mass can be produced from them without independent determination of the mass of the Sun is not close to the distance. The equatorial equation, however, depends only on the distances and not by the mass of the Sun.

Secular acceleration

Besides the above- periodic disturbances of the moon also non-periodic ( " secular " ) is subject to disturbances which lead over the millennia to a ( positive or negative ) acceleration of the lunar race.

The " gravitational acceleration " is effected, that the eccentricity of Earth's orbit decreases at present. Thus, the gravitational influence of the sun is low on the moon in the middle, which - as with the Annual and the parallactic equation - leads to a slightly faster movement of the moon. This acceleration is 6 " / Jhdt2, so that after an amount of hundreds t 6" is to be added to the length of the moon · t2.

In opposite direction, the " tidal acceleration " effect. The piled-up from the moon on the tides Erdozeanen waves are laterally offset from the Earth's rotation, so that they do not lie exactly in the line connecting the Earth - Moon and in turn exert a torque on the moon. This torque causes the moon to angular momentum and energy, so that it is raised to a higher, more energetic track, but which after the third Kepler 's law corresponds to a lower rotational speed. This deceleration is approximately -26 " / Jhdt2, so that after an amount of hundreds t ½ · 26 · t2 deducted from the length of the moon. The fact that, in contrast to the gravitational acceleration appears a factor ½ is only attributable to relevant conventions. As a result of tidal raising its orbit to the Moon removed per year by 3.8 cm from the ground.

Heliocentric lunar orbit

Looking at the moon as it orbits the sun along with the earth, so he moves on a wavy line along the Earth's orbit. However, the moon's path runs through loops. From the sun as seen from the velocity of the moon is alternately the sum and the difference of the speed of the earth to the sun orbit (30 km / s ) and the speed of the moon on the earth's orbit (1 km / s). The difference can never be negative, so the moon can be seen from the Sun never move retrograde and therefore undergo no loop movement.

The Heliocentric looked lunar orbit contains no curved away from the Sun web sections. As the sun the moon is about twice as strong as the earth attracts the moon's orbit is always concave to the sun, that is, the acceleration, which exerts the sun to the moon is at all times greater than the acceleration, which is caused by the earth and the moon's orbit bends always toward the sun. However, this curvature varies greatly, depending on whether the sun and earth in the same directions ( at full moon ) or pull ( new moon ) in opposite directions. Although the attraction of the sun twice as much effect as the attraction of the earth, the moon, the earth will not snatched. Earth and Moon fall together in the gravitational field of the sun; this would be the same everywhere strong, so they could circle each other unmolested in accordance with their mutual attraction. The fact that the closer each of the two sun slightly more than the more remote attracts leads to the above-described complex variations of the undisturbed case.

Topocentric lunar orbit

For the topocentric, so the present on the surface of the rotating Earth observer is subject to the moon and all other celestial bodies of the daily movement. This apparent movement is caused by the rotation of the Earth and celestial bodies can rise above the eastern horizon and go down behind the western. The actual motion of the moon in its orbit around the earth will be made in the opposite direction and is easy to determine for the attentive observer: the Moon, for example, at a given time in the vicinity of a star, so he has an hour later with respect to the star about moved one moon diameter in an easterly direction, while moon and Star hiked together in the course of daily movement by 15 ° (30 moon diameters ) to the west.

The moon moves relative to the fixed stars in a sidereal month of 27.3 days through 360 ° on the sky, on a day that is on average about 13.2 °, which is slightly more than the width of an outstretched fist. To this good 13 °, and the sky must also turn to bring the moon, for example, from a culmination to the next. To this end, a good 50 minutes to complete on average. To this period of time the moon each day goes (on average) to a later date and as the day before. The new moon comes up in the morning together with the sun. The sunrises are late then every day a little further: Standing in the first quarter of the moon is about to lunch on a full moon night, and in the last quarter to midnight. The same applies to the downfall. From the knowledge of the phase of the moon so can be estimated to rise and set time.

Since the moon always moves in the vicinity of the inclined against the ecliptic equator, he did not sweeps when passing through its orbit a similar north-south range as the sun, but how this once a year, but once a month. The full moon as the most visible phase of the moon is the sun in the sky over always, that is located in the southern Ekliptikabschnitt when the sun is in the northern (summer) and vice versa ( in winter). Therefore, full moons are low in the summer and in the winter sky high. The moon is in the first quarter, it is high in spring and autumn low, etc. From a knowledge of the season and moon phase can be so Kulminationshöhe and estimated to rise and set direction.

Since the orbit of the moon is inclined at 5 ° to the ecliptic, it sweeps though not exactly the same but almost north-south range as the sun. If the line of nodes of its orbit so that adding the orbital inclination relative to the ecliptic and the inclination of the ecliptic relative to the equator, then the moon reaches maximum declinations up to ± 28.6 °; according to paint over his rising and setting transition points a particularly wide range on the horizon ( "Big moon turn", most recently in 2006 ). The winter full moons are particularly high and the summer full moons particularly low. 9.3 years later, the nodal line has rotated 180 °, the inclination of the lunar orbit and the ecliptic are opposite and the moon only declinations of ± 18.4 °. His rise and transition area on the horizon now has the smallest dimension ( "Little Moon turn" ).

This means that the inclination of the moon's orbit varies between 18.4 ° to Erdäqator and 26.6 °.

Because of the inclination of the lunar orbit to the ecliptic, the moon can cover not only stars, which are located on the ecliptic, but a total of stars which lie at a distance of up to ± 6.60 ° on either side of the ecliptic ( the caused by the orbital inclination 5 ° are still the parallax of the moon and its disk radius to add ). In a given month, however, the moon covers only those stars which lie in the immediate vicinity of its current path. As a result of Knotenpräzession the web moves in each cycle a bit, and after a maximum of 18.6 years is the web over every reachable star pulled away.