# Track disruption

An orbit disruption is a deviation of the actual flight path of a celestial body from the flight path calculated using a model . In addition to the change in the path, the mechanism causing it or the disturbance itself is referred to as path disturbance. In addition to irregular orbit changes, a basic distinction is made between periodic orbit disturbances, which describe fluctuations around a mean value, and secular orbital disturbances , which represent long-term monotonous changes in the orbit elements.

Which influences are to be considered as path disturbances therefore depends on the underlying model. In the simplest case, the orbit is approximated as a two-body problem with the help of Newton's law of gravitation and the orbit is described by the six orbit elements of the Kepler orbit . For a more precise orbit analysis of asteroids, for example, the gravitational forces of the sun, all planets, the moon and other larger celestial bodies are routinely taken into account, as are relativistic effects. The numerical values ​​of the six orbit elements (or satellite orbit elements ) change due to an orbit disruption . In doing so, they generally also become time-dependent. The precession of the node lines , the rotation of the apses and the drift in the orbit axes and orbital times are then typical secular orbital disturbances. In satellite geodesy missions , the model takes into account disturbance variables due to the irregular earth's gravity field , the high atmosphere, direct and indirect solar radiation, and the influence of the moon and sun (directly and indirectly via tides). These path disturbances are periodic.

## Model of the Kepler Railway

Elliptical orbit of m around the central body M, according to Kepler's first law, M is at a focal point of the ellipse.

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

Johannes Kepler published more precise knowledge about the mathematical representation of the orbit of a celestial body in the early 17th century in the form of Kepler's laws, which were later named after him . These laws could later be theoretically derived by Isaac Newton with the help of his law of gravitation and thus justified.

Thus, elliptical orbits replaced the previously assumed circular planetary orbits. For celestial bodies of higher energies there are other Kepler orbits than solutions to the problem of how one celestial body moves around another (which is usually much more massive ): parabolas or hyperbolas . Kepler's model described the orbits much better than the previous theories. But his attempt to make the orbits of the celestial bodies predictable was based on certain simplifications: The actually existing multi-body problem is reduced to a two-body problem. Gravitational or other forces due to the presence of other celestial bodies do not occur. The celestial bodies considered are viewed as point masses with a central field. Further force effects from electrical or magnetic fields , from matter flows, radiation pressure , air resistance, etc. are not considered. Relativistic effects discovered much later are also not taken into account.

## Classification according to the causes

### Gravitational forces

There are different types of disturbances due to gravitational influences, which can already be understood on the basis of Newton's law of gravitation. The cause is the inhomogeneous mass distribution of the earth or the gravitational influence of other celestial bodies.

#### Gravity field variations

Gravity on the earth's surface compared to the ideal earth ellipsoid

The description of the earth as point mass is simplistic. In a better approximation, the earth can be seen as an oblate ellipsoid . Thus, the assumption of the spherically symmetric mass distribution on which the calculation of the Kepler orbit was based does not apply. Up to a height of several earth radii, even open orbits ( fly-bys ) are measurably disturbed by the flattening of the earth. The flattening of the sun, which is much less pronounced compared to the earth, also contributes to a very small part to the perihelion rotations of the planetary orbits.

Near Earth satellites
Orbital elements of a celestial body

Low-flying satellites ( LEO ) sense gravity anomalies through seas, mountains and irregularities in the mass distribution in the Earth's interior. Satellite geodesy takes advantage of this. Even the tides are taken into account.

The irregular gravitational field of the earth exerts a tipping moment on the orbit plane of the orbit of a satellite close to the earth, which the orbit plane evades by a precession movement . This evasive movement means that the ascending node or the nodal line is not fixed, but rotates slowly in the equatorial plane and the right ascension of the ascending node changes continuously. The orbit plane rotates around the z-axis of the astronomical coordinate system. At the same time, the apsidal line rotates around the center of the earth in the plane of the orbit - also due to the influence of gravity. This results in perigee rotations . ${\ displaystyle \ Omega \}$

A geostationary satellite holds its position in only four positions, and only two of them are stable: 105 ° W and 75 ° E. Small path disturbances cause a drift to the stable positions, so such disturbances must be constantly corrected.

The effect of this path disruption can also be used positively. It can be prepared by appropriate choice of the inclination of a sun-synchronous orbit generated or the perigee be maintained over a fixed ground point. Another effect due to the earth's gravity field inhomogeneities is the east-west drift .

#### Influence of other heavenly bodies

The perihelion of the orbit of a planet. The eccentricity of the path and the amount of rotation are schematically exaggerated.

If a celestial body is exposed to a force of attraction which, strictly inversely, decreases quadratically with the distance from the central body, it would move in a classical approximation on a Kepler ellipse, whose shape, position and orientation in space would remain unchanged. The apsidal rotation is a progressive rotation of the entire path in the plane of the path. In the case of the planetary orbits, the influence of the other planets is the main cause of the perihelion rotations: The sum of the gravitational influences of the other planets and the effect of the general theory of relativity (see below) results - in a coordinate system at rest with respect to the fixed star background - a rosette-like movement the planets: the anomalistic period does not exactly correspond to the sidereal period .

With increasing observation accuracy, deviations of the orbit of the planet Uranus from the Kepler orbit could be precisely determined at the beginning of the 19th century . An eighth planet of the solar system, which was still unknown at the time, was considered to be the cause, the position of which could be predicted due to the Uranus orbital disturbances. In 1846 this finally led to the discovery of Neptune . In the same way, the dwarf planet Pluto was discovered in 1930 due to the orbital disturbances it caused.

In the long run, tidal friction not only slows down the earth's rotation, but also increases the lunar orbit . The mean distance between the moon and the earth grows by about 3.8 cm annually.

The influence of the direct force effect by other celestial bodies on an object in Earth orbit depends on the orbit parameters. The main roles are played by the moon and the sun. This influence is greater in the area of geosynchronous orbits than for more strongly bound satellites. The gravitational force can be broken down into a component parallel and one perpendicular to the plane of the orbit. The vertical component causes a secular, the parallel a periodic disturbance. The secular disturbance influences the orbital elements , and analogously to the flattening of the earth . For geo- stationary satellites, that is , only an effect on the inclination remains. Without compensatory corrections , a north-south drift of ${\ displaystyle i}$${\ displaystyle \ Omega}$${\ displaystyle \ omega}$${\ displaystyle e = i = 0}$

${\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} t}} i = 0 {,} 76 ^ {\ circ} / {\ text {year}}}$

### Relativistic Effects

Gravitational field of a homogeneous, non-charged and non-rotating sphere - complete Schwarzschild solution .

In Albert Einstein's general theory of relativity , gravity is interpreted as a force of inertia ; accordingly, it is a geometric property of the curved four-dimensional space - time . Einstein's field equations are used to determine the curvature of spacetime at a point . They were formulated in such a way that in the limit of weak gravity they agree with Newton's law of gravity. Within the solar system, where weak fields or slight curvature of space-time are involved, there are only minor deviations from the predictions of Newton's law of gravitation and thus from the Kepler orbits. In the case of strong curvature, as it is caused by the strong concentration of large masses in a small space, new types of phenomena occur, such as those shown in pulsars in orbit around other stars: The Hulse-Taylor double pulsar also changes its orbit because it contains energy radiates in the form of gravitational waves.

#### The time dilation

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

#### The curvature of space

Because of the general theory of relativity, the law of force for celestial bodies fundamentally deviates from the idealized inverse-quadratic behavior of Newton's law of gravity (even if only to a very small extent), so that a further contribution to the perihelion of the planets arises.

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 transits for a particularly accurate measurement of Mercury's orbit and found, based on the improved data, that Mercury's perihelion was slightly more than expected. According to calculations based on Newton's law of gravitation, it should be around 530 arc seconds (") per century, with around 280" being due to the influence of Venus , around 150 "being due to disturbances from Jupiter and around 100" being due to the other planets. The observed perihelion rotation (modern value: 571.91 "per century) was, however, significantly greater, the discrepancy is 43.11". Only the relativistically calculated proportion of 42.98 "for the perihelion rotation agrees quite well with the observed excess.

#### The Lense Thirring Effect

Another relativistic effect is the lens-thirring effect . To put it clearly, a rotating mass causes space-time to be twisted by carrying along with it. When a celestial body rotates on its orbit, it leads to an additional precession . The satellite mission Gravity Probe B confirmed this effect in 2004/05. Another consequence of the effect would be that the orbital plane of stars orbiting near a supermassive black hole would be made to precession around the axis of rotation of the black hole. This effect has not yet been proven, but proof is expected in the next few years. By comparing the precession of two stars on different orbits, it should in principle be possible to confirm the " no-hair-theorem " of the general theory of relativity.

### Non-gravitational forces

#### Interaction with the atmosphere

Orbit altitude of the Mir space station during its 15-year mission: At the beginning of the 22nd sunspot cycle, the Mir was moved to a higher orbit in order to reduce the effort required to raise the orbit.

With artificial satellites in low orbits, the earth's atmosphere should not be neglected. The interaction is not a flow; the particles hit individually. Since they ricochet and also knock out surface atoms, depending on the inclination of the surface hit, a force occurs across the direction of flight. The largely parallel force component leads to a constant loss of orbital energy, which must be compensated for during longer missions in order to avoid a crash, see the diagram for the Mir space station at an altitude of 300 to 400 km. The Hubble space telescope at an altitude of about 600 km, on the other hand, sank by only about 80 km in 19 years.

The extent of the effect can only be roughly calculated in advance, as the extent of the high atmosphere depends heavily on solar activity . A rough calculation is made with the help of the braking force : ${\ displaystyle F_ {D}}$

${\ displaystyle F_ {D} = {\ frac {1} {2}} c_ {w} A \ rho v ^ {2}}$

in which:

• ${\ displaystyle c_ {w}}$ - is the drag coefficient for free molecular flow and lies between 2.2 and 2.5
• A - Area perpendicular to the direction of flight
• ${\ displaystyle \ rho}$ - Air density at the appropriate level
• v - speed of the object

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

Also, the radiation pressure can lead to perturbations. When a body absorbs , reflects or emits electromagnetic radiation (including light ) , the corresponding impulse transfer affects its path. In the case of absorption and emission, the radiation pressure is equal to the irradiance divided by the speed of light . With complete reflection, the radiation pressure is twice as great.

The (averaged) intensity of solar radiation is in the area of ​​the earth's orbit . With the speed of light , the radiation pressure is calculated as follows: ${\ displaystyle S = 1372 \; {\ text {W}} / {\ text {m}} ^ {2}}$${\ displaystyle c}$${\ displaystyle p}$

${\ displaystyle p = (1 + r) {\ frac {S} {c}} = (1 + r) \ cdot 4 {,} 6 \ cdot 10 ^ {- 6} \, \ mathrm {N} / \ mathrm {m} ^ {2}}$

It depends on the size of the irradiated area and its reflection factor and acts as a force that is continuously directed away from the radiation source. With a typical GPS satellite , the radiation pressure of the sun leads to an acceleration in the order of magnitude of 10 −7 m / s² and thus a position error of a few hundred meters per day. ${\ displaystyle r}$

In addition to direct solar radiation , the sunlight reflected from another celestial body such as the earth may also have an effect .

In the Poynting-Robertson effect , the radiation pressure of the sun affects the orbits of small particles in interplanetary matter in such a way that they move closer and closer to the sun.

In addition to light from the environment, the electromagnetic radiation emitted by satellites for communication can lead to significant orbit disruptions. Particularly in the case of communication satellites with high transmission power and directional antennas (for example parabolic antennas , phased array antennas ), this disturbance becomes noticeable with an acceleration of the order of 10 −9 m / s².

In addition to the direct radiation pressure, the radiation absorption can also be felt indirectly. The resulting heating of the object leads to an additional, mostly uneven emission of thermal radiation. Such disturbances are particularly relevant for smaller bodies with their high surface-to-mass ratio.

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

In the Jarkowski effect , the different degrees of warming of the sides of an asteroid facing or facing away from the sun have such an effect that the different degrees of heat radiation in the different spatial directions cause a different force on the celestial body. The strength of the effect also depends on the rotation, the surface properties (especially the albedo ) and the thermal conductivity of the asteroid. As a result, for example, the orbits of asteroids that rotate in the direction of their trajectory around their own axis (prograd) are pushed outward and those whose natural rotation runs counter to the trajectory (retrograde) are changed inward. A corresponding effect with regard to the state of rotation of a celestial body is the YORP effect .

#### Electromagnetic interaction

The Earth's magnetosphere (artist's representation, not to scale)

Electromagnetic interaction plays a special role for electrically charged or magnetic objects . Even outside the magnetosphere of those celestial bodies that themselves have a planetary magnetic field, there is an interplanetary magnetic field through the sun , which is amplified by the solar wind . Due to the interaction between the magnetic fields or with the electrical charges of the objects that are moved relative to them, further forces act that can cause a path disruption.

## Calculation of disturbed paths

If the disturbing forces are known quantitatively, a disturbed path can be calculated. Conversely, by closely observing a path, disruptive forces can be inferred. In both cases, one of the following calculation methods is used.

### Analytical procedures

If the situation is approximately a two-body problem, i.e. the body in question is essentially tied to a dominant mass point and disturbances can be summarized in a few simple terms, a so-called disturbance calculation is possible. This is based on the Kepler orbit and integrates the interfering influences or the relativistic corrections over one cycle. This results in rates of change of the orbital elements, such as a perihelion. The procedure is also suitable for several bodies that interfere with one another.

Situations without orbital resonances are particularly simple, i.e. when the periods of rotation are not in relation to one another in the ratio of small whole numbers. The mere proximity to a resonance condition is less critical, the lower the level of the disturbance, because then the phase of the disturbance changes before a significant effect has occurred. Without resonances, the interfering bodies can be applied as mass rings smeared over their respective tracks. The result obtained above for only one cycle then applies to many cycles.

### Numerical methods

If the above requirements are not met, the paths are calculated using the small step method, e.g. B. with a Runge-Kutta method .

If a large number of mass points have to be taken into account, for example when two galaxies meet, a finite element method with less computational effort can possibly provide meaningful results.

## literature

• Dino Boccaletti, Giuseppe Pucacco: Theory of Orbits: Volume 2: Perturbative and Geometrical Methods . Springer, 2010, ISBN 978-3-642-08222-1 .
• Andrea Milani , Giovanni Gronchi: Theory of Orbit Determination . Cambridge University Press, 2009, ISBN 978-0-521-87389-5 ( google.de [accessed November 18, 2017]).
• Ernst Messerschmid , Stefanos Fasoulas: Space systems. An introduction with exercises and solutions. 3rd, revised edition. Springer, Berlin a. a. 2009, ISBN 978-3-540-77699-4 .
• Ulrich Walter : Astronautics. (= Physics Textbook ). Wiley-VCH-Verlag, Weinheim 2008, ISBN 978-3-527-40685-2 .
• Wiley J. Larson, James R. Wertz (Eds.): Space Mission Analysis and Design. 3. Edition. 8. Pressure. Microcosm Press et al. a., El Segundo CA et al. a. 2006, ISBN 0-7923-5901-1 ( Space Technology Library ).
• Udo Renner, Joachim Nauck, Nikolaos Balteas: Satellite technology. An introduction. Springer, Berlin a. a. 1988, ISBN 3-540-18030-3 .
• Roger R. Bate, Donald D. Mueller, Jerry E. White: Fundamentals of Astrodynamics . Courier Dover Publications, 1971, ISBN 0-486-60061-0 ( google.com [accessed December 18, 2012]).