Orbital mechanics

The space flight mechanics is a field of aviation and space technology, and deals with the laws of motion of natural and artificial celestial bodies under the influence of gravity of other bodies and possibly their own drive. It extends the field of celestial mechanics, with which it has the historical background as well as the basic physical laws together.

• 7.1 Vis -Viva equation
• 7.2 Kepler's equation
• 7.3 spheres of influence ' Patched Conics '

History

The movements of the planets - including Earth - the sun was already suspected in antiquity and again postulates of Nicolaus Copernicus in the modern era ( 1543). Based on observations in particular his teacher Tycho Brahe was Johannes Kepler 1608/ 09 set up the eponymous laws of planetary motion. The explanation of the mathematical background and the attraction mediating gravitational force only succeeded Isaac Newton 1687th The separation of the conventional celestial mechanics can be set in 1903 with the discovery of the missiles basic equation that shows the basic driving needs of the aerospace, by Konstantin Tsiolkovsky. Other essential basics of space flight mechanics wore Walter Hohmann and Hermann Oberth together.

Kepler's laws

Kepler described in this original version of the legislation for the familiar planet, but they are universal and are not limited to elliptical orbits. Rather, can be calculated from the equations of motion of the idealized two-body system in the potential theory (see below) derive that body move around the central mass on conic paths as follows:

• At low energy orbit is elliptical ( with the circle as a special case );
• A body away with escape velocity on a parabolic orbit and comes to rest at infinity;
• A body away with even higher energy itself. hyperbolic on a train and has a residual velocity at infinity, the hyperbolic excess or surplus speed

It is important to pay attention to the reference system (soil, sun or planet ); changes to the reference system, so speed and kinetic energy must be converted, as the systems move relative to each other.

From the above-mentioned Equations of motion can be calculated which are essential for space travel speeds. The most important are: 7.9 km / s - velocity of a body in a low orbit around the Earth ( "First Stars in Motion " ) 11.2 km / s - velocity of a body to leave the gravitational field of the earth ( " escape velocity " or " Second cosmic velocity " ) 29.8 km / s - speed of Earth in its orbit around the sun ( heliocentric, ie relative to the sun)

Law of gravitation

The recognized by Newton in 1687 as a fundamental physical force and described gravitational effect of two bodies of masses and allows the determination of the mutual attraction:

Stands for the gravitational constant and the distance between the two masses or their priorities. It is assumed that is greater than the extent of the masses themselves gravity is always attractive; Newton showed that the effect of such a force that is inversely to the square of the distance, which causes all the effects described by Kepler. The law of gravity is therefore, although later discovered the basis for Kepler's laws.

The general two-body problem

The general version of the above two laws leads to the general two-body problem, in which two masses m1 and m2 move in an inertial system. We formulated this caused by the attraction of acceleration of each mass

The acceleration of each mass equal to the second derivative of the position vector

Obtained

One can introduce the focus of both masses and their connecting vector r = r2 -r1 and then obtained from the two equations of motion

And from this

From this equation may be directly the conservation laws are derived. From the cross product with r is obtained after integration

This corresponds to the conservation of angular momentum and is equivalent to the second Kepler 's law. Multiplying contrast with scalar, it follows, using the product rule of differential calculus

C has the dimension of a specific energy (energy per unit mass ), and describes the time- invariable energy of the relative motion of the two masses.

Further, the trajectory of motion is determined ( Hamiltonian integral) by forming the cross product of the two bodies equation with the angular momentum vector h. One then obtains the geometric description of the web in the form

The which derives from an integration constant size describes the shape of the orbit. This results in

• For ellipses ( with the limiting case of a circular orbit in the case );
• For a parabola, and consequently no closed path more; this corresponds to the achievement of escape velocity;
• Hyperbolas for increasing energy.

This description is purely geometrical and still does not provide a calculation of the temporal trajectory. For this you need the Kepler 's equation (see below)

Typical problems of space flight mechanics

It is characteristic of the subsequently presented problems to achieve a given objective mission with a minimum of energy. Away from this energy minimum would require an unrealistically large boot device or can only carry a small payload nonsensical. In most cases, it is necessary to extend the travel time or other conditions ( position of other planets in the case of a Swing - bys, etc. ) accept. In any case, requires a working and in a vacuum actuator, in order to achieve the required speeds. On the practical implementation could therefore think only after the development of appropriate missiles.

Reaching an orbit

Reaching orbit requires to accelerate the satellite to the orbital velocity of the earth in the case of a low web 7.9 km / s. Addition is necessary to put the satellite on an appropriate trajectory from the atmosphere to an altitude of at least 200 km. This requires an appropriate control system and succeeded only in 1957 with Sputnik. The Earth's rotation can be a suitable choice of launch site - possible äquatornah - and take advantage of departure to the east and then reduces the drive requirements easily, ideally by about 400 m / s In practice, one must next losses as the penetration of the atmosphere (air friction), consider lifting work against the gravitational field, deflection and energy expenditure for corrective maneuvers and therefore fix a speed demand of about 9 km / s. The energy demand is for higher orbits higher and may exceed the energy required for the escape velocity.

Body to higher orbits run slower than those on lower; Therefore, there is an excellent web height, in which the rotational speed of the satellite corresponds exactly to the speed of rotation of the earth. Satellite in this orbit appear to stand still seen from the surface, they are therefore referred to as geostationary, which is particularly important for communication and weather monitoring of high interest.

The choice of the orbit depends crucially on the purpose of the satellite. For earth observation are i.d.R. Webs with high inclination or polar orbits of interest in order not to limit the observed area of a band around the equator. For the telecommunications and weather observation, the geostationary orbit is. In the case of communication with locations of high latitude is chosen with advantage Molniya orbits. Navigation systems use medium-high orbits as a compromise between low orbit (extremely many satellites for full coverage required) and geostationary orbit ( poor accuracy and local conflicts with other systems ).

Disturbing effects

Due to the flattened and irregular shape of the Earth and other inhomogeneities of the gravitational field ( geoid ), its atmosphere as well as by other bodies ( in particular the sun and moon ) learn satellites in Earth orbit perturbations, the iA must be compensated, but conversely, can be specifically used for special trains, especially sun-synchronous satellites. Analogously, the same is true in general for satellites around other celestial bodies and also for spacecraft.

In the case of the earth are the most important disturbances

• Caused by the flattening of the earth precession of the orbital plane, which depends on web height and angle;
• Caused by the Earth's atmosphere deceleration, which depends strongly on the orbital altitude and the " density " of the satellite. This disorder is not conservative, that is, the satellite loses energy to the earth. Satellites in low orbits, therefore, have a limited life.

Rendezvous

As is known, a rendezvous maneuver to achieve already located on a known path other satellite so as to couple with it, or perform similar operations. In a broader sense can also flights to other celestial bodies put into this category as face the same problems. The tracks must for this maneuver match what usually is only achieved through several corrections; In addition, accurate timing is required, so as not to miss the target body. As a startup window is defined in this context, the period during which a start must be made to such maneuvers.

Transfer orbit

A transfer orbit is used to change from one orbit to another. This is done by changing the speed, either in a pulsed manner ( in the case of conventional chemical drives) or a longer period ( electrical engines. ) Where the following maneuvers of practical use:

• An acceleration or braking at the pericentre ( the next point to the central body of the web ) is increased or decreased the Apozentrum on the opposite side and vice versa. This can be achieved with relatively small velocity changes significant changes in the orbit height; the Hohmann path is an application of this maneuver;
• An acceleration maneuver at an angle to the path in the line of nodes ( the imaginary intersection of old and new orbital plane ) changes the orbital inclination ( Inklination. ) This maneuver is very energy intensive because the entire velocity vector must be changed, and is therefore in practice only for very small Inklinationsänderungen (and then at the lowest possible speed ) performed.

Escape velocity

The escape velocity is obtained from the consideration of energy conservation, by equating potential energy and kinetic energy. For the earth the above value obtained of 11.2 km / s But this does not say anything about other body is made ​​, in particular, this speed is not enough because of the attraction of the sun to exit the solar system.

Trans Lunar Course

For a flight to the moon less than the escape velocity is required, since the moon is relatively close to the earth and exerts a non-negligible own gravitational effect. Therefore, a rate of about 10.8 ... 10.9 km / s ( depending on the position of the moon ) is sufficient, then one obtains a transfer path in the form of an 8, as in the Apollo program carried out. The velocity vector must be aligned very accurately so as not to miss the moving moon; in practice, one works with several correction maneuvers during the nearly three-day transfers.

Flight to inner planets

On a flight to the inner planets Venus and Mercury must drive power to the orbital velocity of the satellite around the sun, which takes this from the earth, will be applied. For these flights, the escape velocity from the earth is therefore directed in such a way that it counteracts the orbital movement. The energy consumption is still considerably and allows only small space probes to be brought into the inner Solar System. The requirements for the control and navigation are again significantly higher than that for a moon flight. An energy minimum is reached only once per synodic period.

Flight to outer planets

If you want to reach the outer planets Mars, Jupiter, Saturn, Uranus or Neptune, the probe is further accelerated after leaving the earth against the gravitational field of the sun. For this second acceleration profiting from the movement of the earth around the sun, provided in turn, the velocity vectors are aligned correctly. As with the air in the inner solar system and here is the speed demand large, which is why the probes are mostly easy, otherwise are swing - bys for further acceleration of the probes required. Due to the large distances to the outer solar system, these flights can last many years.

Flights on tracks with constant thrust

Electric engines allow (for example, the ion drive ) as well as the possible drive a spacecraft with solar sails, maintain, albeit small, thrust over a very long time, and therefore provide for long-duration missions as an alternative to conventional chemical propulsion dar. The position must be the spacecraft continuously monitors so that the thrust in the desired direction goes. Computationally these trajectories, the spirally move in many turns in or out, is no longer easy to detect, since the traction power is not constant.

Flyby maneuver ( Gravity Assist, swing-by )

A flyby maneuver at a in the reference system moving mass leads to energy exchange between the two bodies and therefore allows to accelerate a satellite ( in a passage "behind" the body ) or decelerate. This is associated with a reversal of the trajectory whose size is related to the degree of deceleration or acceleration. For these maneuvers are the giant planets Jupiter and Saturn as well as the fast-moving Venus, as well as the earth itself interesting. The flyby is done on a planetozentrisch hyperbolic path, the energy gain in the heliocentric system by the different inverse transformation of the speed on the road departing. Such maneuvers can be used in the inner as well as outer solar system for flights, may with several flybys of one or more planets; leaving the ecliptic plane, as in the sun Ulysses, is also possible. A disadvantage is the extension of the flight time, the increased requirements for control and navigation as well as the need to pay attention to the position of another body, which usually leads to significant limitations of the start window.

Applications

The actual application is mission planning, which has for its object for a given target - for example, a planet of scientific interest - for a mission, ie in particular to find a suitable flight path.

This required extensive numerical simulations are required. In the early days, these simulations were limited, a substantially predetermined flight path (usually a Hohmann -like transition) to optimize and find appropriate departure data. The energy reserve of the carrier system determined in such cases, the length of the startup window. From this time the level curves similar plots of the required energy comes as a function of the departure and arrival date. Since then, the numerical options were greatly expanded and also allow for the high demands - to determine imaginative tracks required - such as the visit of a comet.

Mathematical Methods

Vis -Viva equation

The following from the energy conservation Vis -Viva equation ( Latin for " living force " ) is a stable two-body track the speed in relationship to the current orbital radius and the characteristically designed size of the web, the semi-major axis of the conic orbit. It reads:

This is for a circle, for an ellipse, for a parabola and a hyperbola. is the standard gravity parameter, which again is the gravitational constant and the mass of the central body:

Kepler's equation

Named after its discoverer, Johannes Kepler equation relates the Mean anomaly, the eccentric anomaly and the eccentricity of an elliptical orbit related as follows:

It therefore allows to determine the position of a railway track disposed on a known function of time. However, this requires iterative or numerical methods, since the equation is transcendental in. For nearly circular ( close to 0 ) and highly elliptical ( close to 1) paths to numerical difficulties can arise when the two right-hand terms of very different orders of magnitude and are almost the same.

Spheres of influence ' Patched Conics '

The concept of the sphere of influence is assumed that the web of a celestial body at a given time is influenced substantially only by one other body; Interference from other bodies are neglected. When approaching another body, there is a point of the virtual equilibrium, at which the two dynamic forces of attraction (ie, the disturbance and the acceleration of the undisturbed two- body panel ) are equal. For the frequent case in the solar system, the mass of the heavier body ( for example the sun, the index S) is substantially greater than that of the other (here, the target planet index P), this point results to

Where is the question approach distance and the distance between the planet and the sun. On this point to change the approach ( eg, from a ground on a sun orbit ) and performs a transformation so that the tracks continuously into one another ( hence the name of joined conics ). This concept is very simple and can even perform in some cases by hand, but takes it one may significant need for correction in purchasing.