# Laplace plane

The Laplace plane referred to in the celestial mechanics averaged over long times the orbital plane of a body ( eg a planet, moon, or satellite), which moves in an orbit around a central object (such as the sun or a planet ).

The Laplace plane of most large moons of our solar system, especially the large gas planets practically coincides with the equatorial plane of the respective central planet. An exception is the Earth's moon, whose Laplace plane is described with great accuracy by the ecliptic. The artificial satellite in Earth orbit higher and some (mostly smaller ) moons of other planets possess a Laplace plane that lies between these two levels and therefore needs to be explicitly calculated. Pierre- Simon Laplace in 1805 had reference plane first introduced to describe the web properties of Saturn's moon Iapetus, the largest body in the solar system, in which this level is significantly different both from the equatorial plane and the ecliptic.

## Introduction

A body moves around a central object, he or she may at any time be assigned to an orbital plane, ie the plane in which both the distance vector from the central object to the body, and the velocity vector of the moving body is. The orbital angular momentum vector of the body is just perpendicular to the instantaneous orbital plane. Subject to the movement of the body no torque, so the orbital angular momentum of the body does not change and the orbital plane of the body is correspondingly constant over time. This is for example the case if the central object is exactly spherically symmetric and no external forces act on the system.

In reality, however, the central objects not exactly spherically symmetric ( the planets are more or less flattened ), nor is the movement free from external forces ( the sun, other planets, etc.). Therefore, acting on the orbiting body torques that lead to a change in the orbital plane, which, although small in the course of a few rounds around the central object is in most cases, but lead over time to a precession of the orbital angular momentum. In the case of planetary, lunar and satellite orbits can be used to model this effect over longer periods of time by the action of a constant torque to a fast-moving roundabout. The sheet plane varies then with a certain time period by a time -averaged over a long track plane ( Laplace plane ). The angle of inclination of the instantaneous orbital plane against the Laplace plane remains approximately constant, only the position of the planes to each other changes.

## Example: Iapetus

In contrast to the satellites of the other gas planets, a moon of Saturn was with the discovery of Iapetus in 1671 announced early on that orbits its host planet in quite a large margin ( 3.5 million miles). In the analysis of the path of this moon the French mathematician Laplace found that they over short periods of time (a few years), as the web of all the other moons and planets, extending in a plane. About average periods - in the case of Iapetus several decades - this orbital plane, however, is variable: an effect that is known at the Earth's moon since ancient times, as it is closely linked to the seasonal occurrence of solar and lunar eclipses. Although Iapetus was first observed in 1805 for 134 years, saw Laplace, that the orbital plane of the Moon over long periods of time ( several thousand years ) is making a gyro -like tailspin. For the Earth's moon this rotation, the so-called lunar nodes that time was well known, the period of rotation is 18.6 years. If one considers instead of the orbital plane, the pole of the orbital plane, ie the point on the celestial sphere at which an imaginary line that is perpendicular to the orbital plane ( the normal ) pierces through this sphere, so performs this pole has a circular motion on the celestial sphere. When Earth's moon this circle has a diameter of about 10 ° ( twice the orbital inclination of the Moon) and its center is the pole of the ecliptic (ie the Earth's orbital plane ). In Iapetus the circle has a diameter of about 15 °, but is not at its center the pole of Saturn's orbital plane. Instead, the center point, as you can see in the diagram, such as between the pole of Saturn's orbital plane and the pole of the equatorial plane of Saturn ( ie the piercing point of the rotation axis of the planet through the celestial sphere ). All the other then-known moons ( these were the four Galilean moons, six more moons of Saturn and two moons of Uranus ) have orbits that tend to hardly more than a degree to the equatorial plane of the central planet. For high precision measurements, it is found that, although also perform circular movements around the pole and the equatorial plane of the planet, the railway poles of these moons.

The center of the circle on which moves Bahnpol a moon, the pole of the plane of the path -averaged over a long period, which itself is time- invariant: the Laplace plane. In the near planets, moons massive sun and distant gas planet, this coincides with the equatorial plane. When Earth's moon falls these, as well as in 1888 discovered Saturn 's moon Phoebe outer (planetary distance is about 13 million miles), very closely with the pole of the orbital plane of each planet together. The case of the moon Japetus shows that, however, a zone is the average distance from the planet in which the Laplace plane assumes an intermediate form. Today, in the 21st century, hundreds of planets moons are known and artificial satellites have been launched in planetary orbits, generalize their exact path dynamics the special case of Iapetus. This fact calls immediately after a clarification of the question as to the exact dynamics of the orbital planes looks in the general case.

## Sky Mechanical Statement

The ideal case of a small body moving around a spherical central object, is described by the Kepler two-body problem or the Einzentrenproblem. Because the gravitational field of the spherical central object is radially symmetrical, no torque acts on the rotating body and the angular momentum of the body is constant in this case. This requires on the one hand, the validity of the second law and Kepler other hand, that the movement of the small body in a time-invariant level of the plane of the web takes place.

The orbits of real bodies in orbit around a central object, such as planets orbiting around the sun, moons in orbit around their planet, or artificial satellites, can only approximately be treated as a two-body problem. Deviations from the spherical shape of the central object and the presence of other massive bodies outside the system lead to a disturbed two-body system. Such a faulty system can indeed be further described for short periods by the orbital elements of a Kepler ellipse, but the disturbances lead to a temporal change of the orbital elements. In particular, introduces a torque acting on the orbiting body, the temporal change of the orbital plane, which in the celestial mechanics in general by the orbital inclination (inclination ) i and the position angle (argument ) of the ascending node Ω is expressed in terms of a fixed reference plane.

The nature of the torques occurring T is often such that they are quite small in relation to the orbital angular momentum L, ie, the angular momentum does not change much during one rotation of the body around the central object, in formulas :, where ω is the angular frequency of the body in the circulation is. In this case, the body can be as quick centrifugal view, the rotational axis pointing in the direction of the orbital angular momentum. By the torque axis of rotation changes its direction in such a way that it travels periodically short around a central axis of rotation, which in turn long periodically migrates to a Präzessionspol. The plane perpendicular to the direction of the long time Präzessionspols can be considered as the mean plane of the web and is referred to as the Laplace plane.

Planets, moons and satellites move in ellipses often relatively low eccentricity to their central body, and the torques acting on the body are usually caused by the mechanisms described in the following three paragraphs.

### Deviations of the central body of the spherical shape

As long as both bodies in a two- body system possess the exact spherically symmetric structure, the gravitational field in the resulting equivalent Einzentrenproblem is exactly radially symmetric and it affects no torque between the orbiting bodies. However, deviations from the spherical shape lead to the occurrence of a torque and thus the temporal change of the orbital plane. In celestial mechanics context, the dominant source of this torque is the quadrupole moment of the central body, which results in large part by the flattening along the rotation axis of the body. The torque averaged over one revolution resulting period is perpendicular to the axis of rotation of the central body and perpendicular to the current angular momentum of the rotating body. Characterized the magnitude of the angular momentum does not change, but only the direction of precession about the rotation axis of the central body. The conservation of angular momentum caused it by the way an effect on the intrinsic angular momentum of the central body, which thus, in general, but much more slowly precesses around the Präzessionspol of the orbiting body. This effect of the Moon, for example, one of the main reasons for the Lunisolar precession of Earth's axis.

When assigned to the axis of rotation of the central body a direction vector to which can be viewed neglecting the reaction just described about not too long periods of time as a constant, shows an exact calculation that the angular velocity of the orbital motion [A 1] precesses around the rotation axis. [A 2] In this case, the dot does not change in time, that is, the orbital inclination i on the equatorial plane of the main object, which is thus the Laplace plane, does not change. The argument of the orbit node then moves at an angular velocity of

The radius R of the central body and the semimajor axis a of the orbit.

For planets, you can generally accept as spheroids, the quadrupole moment can be carried out of the moment of inertia I and the flattening f calculated. [A 3] For example applies to the earth f ≈ 1/298 and I ≈ 0.33 M R2 so J2 ≈ 0.0011 in good agreement with the more accurate value J2 = 0.001082.

Plugging this into the formula above and calculate the period of the node rotation for a satellite to erdnahem, nearly equatorial orbit, we obtain a node migration of about 10 ° per day, contrary to the direction of rotation of the satellite, ie, the node is running in about 36 days once around the equator of the earth.

For Jupiter we find J2 ≈ 0.0147 and a radius of R ≈ 71.5 thousand kilometers. The moon Io orbits at a distance of about a ≈ 421,000 km around the planet and the railway junction migrates corresponding to about 47 ° per year and takes about 7.6 years for a complete revolution. , In good agreement with the measured 7.42 years The already low orbital inclination with respect to Jupiter's equatorial plane of 0.05 ° remains constant in good agreement with the model presented here, a fact that does not apply to artificial satellite in Earth orbit or for the Earth's moon.

The inclination with respect to the Earth's equator varies with a period of 18.6 years between 18 ° and 28.5 ° for the Earth's moon. The flattening of the earth, however, would only cause a rotation of the nodes of about 2.1 ° per millennium with a constant inclination, so that the dynamics of the lunar nodes must have a different cause; they will now be presented.

### External disturbances

If a two-body system introduced into an environment interact in the other objects with the two bodies, it can be under certain conditions, the motion of the two bodies treat perturbatively. It is with this approach assumes that the two-body system can be further described over short periods in the form which is known from the unperturbed problem. Over the medium and longer periods is constant orbital parameters will develop dynamically, however, actually. In this section, some special case of disturbance of a Kepler ellipse will be described, wherein the interest is again during the orbital elements and the argument of the inclination i ascending node Ω. We assume for the purpose by the assumptions that the circumferential body can be described as a fast roundabout again and that the disturbing body quickly move in relation to the rate of change of the orbital elements. As an example of such a model of the Earth's moon was chosen, which moves with about 13 times the angular velocity around the earth, with which the Hauptstörkörper, namely the sun moves relative to the Earth-Moon system. The amendment of the relevant path element Ω takes place again with about 18 -fold smaller angular velocity.

Obtained in such cases, an acceptable result when one considers such a system as fast gyro, to which a force acts emanating from a mass distribution that emerges from a temporal averaging of the disturbing masses relative about their paths to the two- body system. If it is a single dominant bluff body moving relative to the system at a constant distance and constant speed, so you can start from a one-dimensional mass ring with appropriate mass M and radius R. The torque which causes the mass ring to the rotating body, producing a change of direction of the angular momentum and hence of the orbital plane. [A 4]

The ascending node of the orbit therefore moves with an angular velocity of

Where M is the mass of the bluff body, and r is its distance to the system. i now denotes the path angle ( ) relative to the orbital plane of the bluff body, which is the Laplace plane here and the normal is perpendicular to it. In the previous section applies here that the amount of the angular velocity, and orbital inclination is constant in time.

If the bluff body, as in the case of the Earth- Moon system, the central body of a larger system (eg solar system ), the mass and the distance of this body can be eliminated by the third Kepler's law in the above formula and obtained

Where ω0 here is the angular velocity of the bluff body ( sun). In the case of the Moon, you can now see directly that (neglecting the orbital inclination ), the lunar nodes move with an angular velocity which is about 4/3 · ω/ω0 ≈ 1.33 × 13.4 ≈ 17.8 times slower than the relative angular velocity of the sun - in other words, the node rotate every 17.8 years through a full 360 °. The observed value of 18.6 years is only achieved by more precise calculation of the lunar orbit. The orbital inclination of the moon is in this migration of the node, apart from short-period fluctuations, relative to the ecliptic ( Laplace plane ) is constant about 5 °. If, however, as the data of Jupiter's moon Io a, we obtain that the nodes would need nearly 40,000 years for a full orbit - an effect that is smaller by almost four orders of magnitude than the induced by the oblateness of the central planet.

### Combination of the two cases

In many celestial mechanics relevant cases are the two effects are of comparable magnitude just described. In such a case, the axis of rotation of the central body and the angular momentum of the outer bluff body in parallel so as to add the two effects, and we get: [ A-5 ]

The precession takes place in turn in a circle around the common direction of the axis of rotation of the central body and the orbital angular momentum of the bluff body. The same applies to the superposition of several external disturbances, all of which occur in the same plane. While this assumption for external disturbances is often justified - Sun and Moon interfere with satellites in Earth orbit, both roughly in the ecliptic, the small outer moons of Jupiter are also disturbed by Sun and Saturn as in the ecliptic, etc. - soft the equatorial planes of the planets often much of the from the ecliptic plane. The Erdäqutor, for example, 23.5 °, 26.8 ° of Saturn equator to the ecliptic is inclined at Uranus and the planes are almost vertical to each other, whereby, for example, both artificial satellite in Earth orbit, as well as the large Saturn's moon Iapetus, for both effects are of comparable size, can not be described by the above equation. Instead, leads the superposition generally involves. a complicated dynamics with a precession and a pole which is located between the axis of rotation of the central body and the orbital angular momentum of the bluff body. [A 6] This leads further to the fact that the orbital inclination is not constant and in terms of the Laplace plane, but varies periodically between a minimum and a maximum value. [A 7] How strong is this variation of the orbital inclination, depends not only on the magnitudes of the two torques considerably also on the angle between the two on. Particularly serious this effect is therefore the planet Uranus, its equatorial plane is nearly perpendicular to its orbital plane. [A 8]

## Example: Jupiter's moons

In the system of the many moons of Jupiter, the various effects described above can be understood by way of example. As you can see in the diagram below which the poles of the Laplace levels of inner moons all placed close to the axis of rotation of the central planet at the bottom left of the diagram. The Laplace plane of Callisto ( IV), which has a distance of nearly 1.9 million km to Jupiter is, it already clearly drawn in the direction of the orbital plane of Jupiter and thus the plane in which the sun acts as a bluff body.

The next outer moon Themisto (XVIII ), which has about 7.5 million km distance from Jupiter, has a Laplace plane, which is hardly affected even primarily from the orbital plane and the equatorial plane ( top right of the diagram). In this area, arrange all the other outer moons of Jupiter. However, it is immediately clear that only the interplay of the equator and the orbital plane of Jupiter is not sufficient to explain the Laplace levels of these moons. These are rather in a large cloud around the orbital plane of Jupiter. This is due to the torques acting in direction of the web of the three poles on outside major planets Saturn, Uranus and Neptune, which are therefore also plotted. This effect is also very nice in the first diagram of this article recognize for the Saturn 's moon Iapetus, whose Laplace plane is clearly drawn from the line connecting the axis of rotation and Bahnpol towards the web poles of the planet further out lying.

A particularly extreme case is the Laplace plane of ananke (XII ), which on the plane of the path of Jupiter addition is inclined almost three degrees and lies beyond the plane of the path of Saturn in the diagram. Among all the known Jupiter moons Ananke is thus a stark outlier represents, which can only be explained by the occurrence of especially here perturbations.