Perturbation (astronomy)

A web disorder is a deviation of the actual trajectory of a celestial body from the calculated according to a model theoretical trajectory. What is meant by a path failure, therefore, depends on the underlying model.

To describe the trajectory only the Newtonian law of gravitation is usually based on and in the simplest case of the two-body problem considers the Keplerian orbits. For a more detailed path analysis example of asteroids, however, the gravitational forces of the sun, all the planets, the moon and other celestial bodies larger relativistic effects are routinely taken into account, as well. For missions of satellite geodesy in the model disturbances by the irregular Earth gravity through the upper atmosphere, the direct and indirect solar radiation, and the influence of the moon and sun are taken into account ( directly and indirectly through tides ).

Through a path interference, the numerical values ​​of the six orbital elements (or satellite orbit elements ) change. It gets ( almost) each of the orbital elements of one or more time-dependent terms.

In addition to the change in the web also the causative mechanism or the disturbance itself is known as a web disorder.

• 3.2.1 The time dilation
• 3.2.2 The curvature of space
• The Lense- Thirring 3.2.3 Effect

• 3.3.1 interacts with the atmosphere,
• 3.3.2 radiation pressure
• 3.3.3 heat radiation
• 3.3.4 Electromagnetic interaction

• 4.1 Analytical Methods
• 4.2 Numerical Methods

Model of the Keplerian orbit

The observation of the night sky with the measurement and calculation of the positions of astronomical objects in the collection and item tables is one of the oldest branches of astronomy. Even today, many paths of astronomical objects are periodically published in the ephemeris of renowned observatories. Then as now, perturbations should be integrated as possible. In order to be as directly usable for the observer on the earth effects such as aberration taken into account.

Better insight into the mathematical representation of the path of a celestial body published Johannes Kepler in the early 17th century in the form of later named after him kepler 's laws. These laws could be derived later by Isaac Newton with his law of gravitation and theoretically justified.

This occurred elliptical orbits in place of the previously adopted circular planetary orbits. For celestial bodies of higher energies, there are other Keplerian orbits as solutions of the problem, such as a celestial body to another ( which is much more massive in general ) moves: parabolas or hyperbolas. Kepler's model described the tracks much better than the theories before. But his attempt to make the orbits of the heavenly bodies to predict, based on certain simplifications: The actually existing multi- body problem is reduced to a two-body problem. Gravitational or other forces by the presence of other celestial bodies, so do not occur. The considered celestial bodies are considered as point masses with central field. More dynamic effects by electric or magnetic fields, currents through matter, radiation pressure, air resistance, etc. are not considered. Discovered much later relativistic effects are not taken into account.

Classification according to time dependence

Ground- faults can be classified into:

• Periodic disturbances - these lead to fluctuations around a mean value, so balanced in terms of resources. One again differs in long-and short-period disturbances.
• Secular disorders - these are " progressively " ( within the meaning of linearly growing), can add up over longer periods thus to large contributions.

Completely irregular may be perturbations in the vicinity of irregularly shaped celestial bodies or in clouds of matter.

Classification for the causes

Gravitational forces

There are different types of interference caused by gravitational influences that are already understandable based on the Newtonian law of gravitation. The reason is the inhomogeneous mass distribution of the earth or the gravitational influence of other celestial bodies.

Gravity field variations

The description of the Earth as a point mass is simplistic. Better approximation in the earth can be viewed as an oblate ellipsoid. Thus, the assumption of a spherically symmetric mass distribution does not apply, which was taken as a basis in calculating the Keplerian orbit. Up to a height of several earth radii even open orbits ( flybys ) be measurable disturbed by the Earth flattening. The flattening of the sun, which is much less pronounced in comparison to the earth, contributes to a very small part of the Periheldrehungen of the planetary orbits.

Low flying satellites ( LEO ) felt gravity anomalies by seas, mountains and irregularities in the distribution of mass within the earth. This takes advantage of the satellite geodesy. The tides are taken into account yourself.

On the orbital plane of the orbit of a near-Earth satellites, the irregular gravitational field of the earth exerts a tilting moment, the orbital plane evades by a precession movement. This evasive movement causes the ascending node and the node line is not fixed, but slowly rotates in the equatorial plane and thus the right ascension of the ascending node is constantly changing. The orbital plane rotates about the z- axis of the coordinate system astronomical. At the same time the line of apsides rotates in the orbital plane - also by the forces of gravity - at the center of the earth. This occurs Perigäumsdrehungen.

Only four positions holds a geostationary satellite 's location, and only two of them are stable: 105 ° W and 75 ° E. Small perturbations cause a drift to the stable positions, such interference must therefore be constantly corrected.

The effect of this path interference can also be used favorably. It may by appropriate selection of the inclination sun -synchronous orbit or generates a perigee are held over a fixed ground point. Another effect due to the gravitational field inhomogeneities of the earth is the east-west drift.

Influence of other celestial bodies

When a celestial body is subjected to an attractive force decreases quadratically strictly inversely with distance from the central body, he would move in a classical approximation to a Kepler ellipse whose shape, position and orientation in space would remain unchanged. Under Apsidendrehung refers to a progressive rotation around the rail in the plane of the web. In the case of planetary orbits, the influence of the other planets is the main cause for the Periheldrehungen: The sum of the gravitational influences of other planets and the effects of general relativity (see below ) results - in a with respect to the fixed star background stationary coordinate system - a rosette-like movement the planet: the anomalistic period is not exactly the sidereal.

With increasing accuracy of observation beginning of the 19th century deviations of the orbit of the planet Uranus from the Keplerian orbit could be determined accurately. The cause of a then still unknown eighth planet of the solar system was considered, the position of which could be predicted on the basis of the Uranus perturbations. 1846 ultimately led to the discovery of Neptune. Similarly, the dwarf planet Pluto was discovered in 1930 due to the perturbations caused by him.

In the long run, the tidal friction not only causes a deceleration of the Earth's rotation, but an enlargement of the lunar orbit. The mean distance between the Moon and the Earth is growing by about 3.8 cm per year.

The influence of direct force action by other celestial body to an object in orbit depends on the orbit parameters. The essential role of the moon and sun. This influence is greater than for stronger bound satellites in geosynchronous orbits range. The gravitational force can be decomposed into a component parallel and one perpendicular to the plane of the web. The vertical component causes a secular parallel a periodic disturbance. The secular perturbation influences analogous to the Earth flattening the orbital elements, and. For geostationary satellites, so only remains an effect on the inclination. Without balancing corrections arises with a north-south drift of

An Inclined geosynchronous orbit.

Relativistic effects

In general relativity, Einstein's theory of gravitation is interpreted as an inertial force, it is therefore a geometric property of four-dimensional curved space-time. For the determination of the ruling on a point of curvature of space-time are the Einstein's field equations. They have been formulated so that they coincide in the limit of weak gravity with the Newtonian law of gravitation. Within the solar system, where it is weak fields and small curvature of space-time, there are only small deviations from the predictions of the Newtonian law of gravitation, and thus of the Keplerian orbits. In case of strong curvature, as it is caused by excessive concentration of large mass in a small space, novel phenomena occur, as they show, for example, pulsars in orbit around other stars: the Hulse -Taylor double pulsar changed its course also be because he energy radiates in the form of gravitational waves.

The time dilation

At the time dilation it is not a path failure in the strict sense, but it has implications for the technical operation of satellites, so that this effect is important for the analysis of perturbations. The time dilation depends on two variables, on the one hand from the place in the gravitational field and on the other by the speed of the observed object. The time signal from the GPS satellites must be corrected accordingly: In this altitude ( MEO ), the gravitational relativistic effect ( general relativity ) has a greater impact than the speed-related relativistic effect ( special relativity ), so the clocks of the satellites are relative to watches faster on the earth.

The curvature of space

Because of the general theory of relativity, the law of force for celestial bodies differs fundamentally from the idealized inverse- square behavior of the Newtonian law of gravitation from (even if only to a very limited extent ), so that a further contribution to the Periheldrehungen the planet.

The best-known example of such relativistic effects is the perihelion of Mercury. In the mid-19th century, Urbain Le Verrier used observations of Mercury passes for a particularly accurate measurement of Mercury's orbit and it was clear from improved data that Mercury's perihelion advance something was stronger than expected. According to the calculations based on the Newtonian law of gravitation, they should " be per century, with about 280 about 530 arc seconds ()" was attributable to the influence of Venus, about 150 " to interference from Jupiter and about 100 " to the rest of the planet. The observed perihelion ( modern value: 571.91 " per century ) but was significantly larger, the discrepancy amounts to 43.11 ". Only the relativistically calculated proportion of 42.98 " to perihelion agrees quite well with the observed excess.

The Lense- Thirring effect

Another relativistic effect is the Lense- Thirring effect. To put it clearly causes a rotating mass by entrainment of spacetime twisting of spacetime. When a celestial body rotates in its path, which thus leads to an additional precession. The satellite mission Gravity Probe B has confirmed in 2004 /05 this effect. Another consequence of the effect would be that the orbital plane of the stars that orbit close to a supermassive black hole, would be brought to a precession around the axis of rotation of the black hole. This effect could not be demonstrated, a proof is not expected in the coming years. By comparing the precession of two stars on different orbits, it should in principle be possible, the "no -hair theorem " to confirm the general theory of relativity.

Non- gravitational forces

Interaction with the atmosphere,

For artificial satellites in low orbits the earth's atmosphere is not negligible. During the interaction it is not a flow; the particles take on individually. As they bounce off and knock out surface atoms, depending on the degree of slope occurs also taken a force transverse to the direction of flight. The vast parallel force component leads to a steady loss of orbital energy, which must be compensated for longer missions in order to avoid a crash, see the diagram to the space station Mir in 300 to 400 km altitude. The Hubble Space Telescope at about 600 km altitude, however, decreased in 19 years to just 80 km.

The magnitude of the effect is only predict coarse because the expansion of the upper atmosphere is strongly dependent on the solar activity. A rough calculation is done using the braking force:

Where:

• - Is the drag coefficient for free molecular flow and is 2.2 to 2.5
• A - area perpendicular to the direction of flight
• - Air density in the appropriate amount
• V - velocity of the object

The disturbance can also be used for the benefit of a space mission as aerobraking maneuver.

Radiation pressure

Also, the radiation pressure can lead to perturbations. When a body electromagnetic radiation ( including light ) absorbed or reflected, then the corresponding momentum transfer effect on its orbit. Absorption and emission of radiation pressure is equal to the irradiance divided by the speed of light. With total reflection of the radiation pressure is twice as large.

In the area of ​​the Earth's orbit is the ( average ) intensity of solar radiation. The speed of light to calculate the radiation pressure as follows:

It is dependent on the size of the irradiated surface and its reflection factor and results in a force which is directed away from the radiation source continuously.

In addition to direct solar radiation, where appropriate, acts also from another planet like earth reflected sunlight.

When Poynting - Robertson effect, the solar radiation pressure on the orbits of small particles of interplanetary matter affects so that this more and more closer to the sun.

Heat radiation

In addition to the direct radiation pressure, the radiation absorption also makes indirect noticeable. The resulting heating of the object results in an additional, usually non-uniform emission of thermal radiation. Such disorders are particularly relevant for small bodies, with their high ratio of surface area to mass.

The anisotropic emission of thermal radiation (from the front of the missile ) is also considered a cause of the Pioneer anomaly, the deviation of the path of the early 1970s launched probes Pioneer 10 and Pioneer 11

When Yarkovsky effect, the different degrees of heating of the sonnenzu - or - opposite sides of an asteroid cast so that the different sized heat radiation in the different spatial directions causes a force on the celestial body. The strength of the effect depend also on the rotation, the surface properties (especially the albedo ) and the thermal conductivity of the asteroid. As a result, the orbits of asteroids that rotate in the direction of their flight path around its own axis, for example, ( prograde ), pushed outward and those running their own rotation opposite to the trajectory ( retrograde), changed inside. A corresponding effect in terms of the rotational state of a celestial body is the YORP effect.

Electromagnetic interaction

For electrically charged or magnetic objects, the electromagnetic interaction plays a special role. Even outside the magnetosphere derjeniger celestial bodies that hold a planetary magnetic field itself, there is the sun an interplanetary magnetic field, which is amplified by the solar wind. Due to the interaction between the magnetic fields and the electric charges moving relative to the objects appear more forces that can cause a train fault.

Calculation of disturbed tracks

If the disturbing forces are quantitatively known, can be a perturbed orbit calculated. Conversely, it can be concluded by the precise observation of a railway to disturbing forces. In both cases, one of the following calculation methods is used.

Analytical Methods

If the situation represents approximately a two-body problem, ie the considered body is bound to a dominant point mass essentially and interference effects can be summarized to a few simple terms, then a so-called perturbation theory is possible. This is based on the Kepler track, and integrates the interference or the relativistic corrections via a round. This results in rates of change of the orbital elements, such as a perihelion. For several bodies that interfere with each other, the method is also suitable.

Are particularly simple situations without track resonances, ie when the orbital periods are disproportionate small whole numbers to each other. The mere proximity of a resonant condition is even less critical, the lower the level of interference, as then the phase of the disturbance has changed before a significant effect is occurred. Without interfering resonances, the body can begin as smeared on their respective web mass rings. The result obtained above for only one lap then applies to many rounds.

Numerical Methods

If the above conditions are not met, the paths are calculated by the method of small steps, such as a Runge- Kutta method.

If a large number of mass points to consider, such as the encounter between two galaxies, possibly a finite element method can provide meaningful results with less computational effort.