# Space flight mechanics

The orbital mechanics is a field of aerospace engineering 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 shares the historical background as well as the fundamental physical laws.

## history

The movements of the planets - including the earth - around the sun were already suspected in antiquity and postulated again by Nicolaus Copernicus in modern times (1543). Based on observations in particular by his teacher Tycho Brahe , Johannes Kepler was able to establish the laws of planetary motions named after him in 1608/09. The explanation of the mathematical background and the gravitational force mediating the attraction was only achieved by Isaac Newton in 1687. The separation from conventional celestial mechanics can be determined by Konstantin Ziolkowski in 1903 with the discovery of the basic rocket equation , which shows the basic need for propulsion in space travel . Walter Hohmann and Hermann Oberth brought together other essential fundamentals of space flight mechanics .

## Kepler's laws

1. Kepler's law
The planets move on elliptical orbits with the sun in one focal point.
2. Kepler's law
A “driving beam” drawn from the sun to the planet sweeps over areas of the same size at the same time.
3. Kepler's law
The squares of the orbital times of two planets behave like the third powers (cubes) of the major orbital half-axes.

In this original version, Kepler described the laws for the planets known to him, but they apply universally and are not restricted to elliptical orbits. Rather, it can be derived from the equations of motion of the idealized two-body system in potential theory (see below) that bodies move around the central mass on conic paths as follows:

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

It is important to pay attention to the reference system (earth, sun or planet); If you change the reference system, speed and kinetic energy have to be converted, since the systems move relative to each other.

The velocities essential for space travel can be calculated from the above equations of motion. The most important are:

7.9 km / s: speed of a body in a low orbit around the earth ( first cosmic speed )
11.2 km / s: speed of a body to leave the earth's gravitational field (escape speed or second cosmic speed )
29.8 km / s: speed of the earth on its orbit around the sun (heliocentric, i.e. related to the sun)

## Law of gravitation

The gravitational effect of two bodies of masses recognized and described by Newton in 1687 as a fundamental physical force and allows the determination of the mutual attraction: ${\ displaystyle m_ {1}}$${\ displaystyle m_ {2}}$

${\ displaystyle F (r) = G {\ frac {m_ {1} m_ {2}} {r ^ {2}}}}$.

${\ displaystyle G}$stands here for the gravitational constant and for the distance between the two masses or their centers of gravity. It is assumed that it is greater than the expansion of the masses themselves. Gravitation always has an attractive effect; Newton was able to show that the effect of such a force, which is inverse to the square of the distance, produces all the effects described by Kepler. The law of gravitation, although discovered later, forms the basis for Kepler's laws. ${\ displaystyle r}$${\ displaystyle r}$

## The general two-body problem

The general version of the above two laws leads to the general two-body problem , in which two masses m 1 and m 2 move in an inertial system. The acceleration of each mass caused by the force of attraction is formulated

${\ displaystyle m_ {1} a_ {1} = G {\ frac {m_ {1} m_ {2}} {r ^ {3}}} r \ quad {\ text {and}} \ quad m_ {2} a_ {2} = - G {\ frac {m_ {1} m_ {2}} {r ^ {3}}} r}$

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

${\ displaystyle a_ {i} = {\ frac {d ^ {2} r_ {i}} {dt ^ {2}}}; i = 1 \ dots 2}$

You get

${\ displaystyle m_ {1} {\ ddot {r}} _ {1} + m_ {2} {\ ddot {r}} _ {2} = 0}$

You can introduce the center of gravity of both masses and their connection vector r = r 2 -r 1 and then get from the two equations of motion

${\ displaystyle {\ ddot {r}} _ {2} - {\ ddot {r}} _ {1} = - G (m_ {1} + m_ {2}) {\ frac {r} {r ^ { 3}}} = {\ ddot {r}}}$

and it

${\ displaystyle {\ frac {d ^ {2} r} {dt ^ {2}}} + {\ frac {\ mu} {r ^ {3}}} r = 0 \ quad {\ text {with}} \ quad \ mu = G (m_ {1} + m_ {2})}$

The conservation laws can be derived directly from this equation. The cross product with r is obtained after integration

${\ displaystyle r \ times {\ dot {r}} = {\ text {const.}} = h}$

This corresponds to the conservation of the angular momentum and is equivalent to Kepler's second law. If, on the other hand, one multiplies with scalar , the result is differential calculation using the product rule ${\ displaystyle {\ dot {r}}}$

${\ displaystyle {\ frac {v ^ {2}} {2}} - {\ frac {\ mu} {r}} = {\ text {const.}} = C}$

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

The trajectory of the motion can also be determined ( Hamilton's integral ) by forming the cross product of the two-body equation with the angular momentum vector h. The geometric description of the path in the form is then obtained

${\ displaystyle r (\ phi) = {\ frac {p} {1+ \ epsilon \ cos \ phi}}}$

The variable derived from an integration constant describes the shape of the path. It surrender ${\ displaystyle \ epsilon \ geq 0}$

• for ellipses (with the limit of a circular path in the case );${\ displaystyle \ epsilon <1}$${\ displaystyle \ epsilon = 0}$
• for a parabola, therefore no longer a closed path; this corresponds to reaching the escape speed;${\ displaystyle \ epsilon = 1}$
• for hyperbolas of increasing energy.${\ displaystyle \ epsilon> 1}$

This description is purely geometrical and does not yet provide a calculation of the path over time. For this one needs the Kepler equation (see below)

## Typical problems in space flight mechanics

It is characteristic of the following problems to achieve a given mission goal with a minimum of energy. Apart from this energy minimum, you would need an unrealistically large launch device or you would only be able to carry an insane small payload. Usually you have to accept an extension of the travel time or other requirements (position of other planets in the event of a swing-by, etc.). In any case, you need a drive that also works in a vacuum in order to achieve the required speeds. The practical implementation could therefore only be thought of after the development of the appropriate missiles .

### Reaching an orbit

Reaching orbit requires the satellite to accelerate to orbital speed, which in the case of Earth on a low orbit is 7.9 km / s. In addition, it is necessary to move the satellite on a suitable trajectory from the atmosphere to an altitude of at least about 200 km. This requires a corresponding control system and was only achieved in 1957 with Sputnik . The earth's rotation can be exploited with a suitable choice of take-off location - as close to the equator as possible - and take-off to the east and then slightly reduces the drive requirement, ideally by around 400 m / s. In practice, losses such as penetration of the atmosphere (air friction), lifting work against the gravitational field, deflection and energy expenditure for corrective maneuvers must also be taken into account and therefore a speed requirement of around 9 km / s must be assumed. The energy requirement is correspondingly higher for higher orbits and can exceed the energy expenditure for the escape speed.

Bodies on higher orbits revolve more slowly than those on lower ones; therefore there is an excellent orbit at which the orbital speed of the satellite is exactly the same as the speed of rotation of the earth. Satellites in this orbit appear to be standing still when viewed from the surface of the earth, which is why they are referred to as geostationary , which is of particular interest for communication and weather observation.

The choice of orbit depends critically on the purpose of the satellite. For earth observation i. d. R. orbits with a high inclination or polar orbits of interest, so as not to limit the observable areas to a band around the equator. The geostationary orbit is suitable for telecommunications and weather monitoring. In the case of communication with places of high geographical latitude, it is advantageous to choose (highly elliptical) Molnija orbits . Navigation systems use medium-height orbits as a compromise between low orbits (extremely many satellites are required for full coverage) and geostationary orbits (poor accuracy and local conflicts with other systems).

### Disturbing effects

Due to the flattened and irregular shape of the earth and further inhomogeneities of the gravitational field ( geoid ), its atmosphere and other bodies (in particular the sun and moon), satellites experience orbital disruptions in the earth's orbit, which i. A. have to be compensated, but conversely can also be used specifically for special orbits, especially sun-synchronous satellites . The same applies to satellites around other celestial bodies and to spacecraft in general.

In the case of the earth, the most important are disturbances

• the precession of the plane of the orbit caused by the flattening of the earth , which depends on the height and inclination of the orbit;
• the deceleration caused by the earth's atmosphere, which depends heavily on the orbit altitude and the "density" of the satellite. This disorder is not conservative ; H. the satellite loses energy to the earth. Satellites in low orbits therefore have a limited lifespan.

### date

A rendezvous is a maneuver to reach another satellite that is already in a known orbit in order to couple with it or to carry out similar operations. In a broader sense, flights to other celestial bodies can also be included in this category, since the same problems arise. The trajectories must match for this maneuver, which i. d. Usually only achieved through several corrections; in addition, precise timing is required so as not to miss the target body. In this context, the start window is the period in which such a maneuver must be started.

### Transfer orbits

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 propulsion) or over a longer period (with electric propulsion). The following maneuvers are of practical use:

• an acceleration or braking maneuver at the pericenter (the point on the path closest to the central body) increases or decreases the apocenter on the opposite side and vice versa. Considerable changes in the web height can be achieved with relatively small changes in speed; the Hohmannbahn is an application of this maneuver;
• an acceleration maneuver at an angle to the path in the junction line (the imaginary line of intersection of the old and new path planes) changes the path inclination (inclination) small inclination changes (and then at the lowest possible speed) performed.

### Escape speed

The escape speed results from the conservation of energy by equating potential energy and kinetic energy. For the earth one gets the mentioned value of 11.2 km / s. However, this does not make a statement about other bodies. In particular, due to the gravitational pull of the sun itself, this speed is not sufficient to leave the solar system.

### Translunar course

For a flight to the moon, less than the escape speed is required, since the moon is relatively close to the earth and exerts its own gravitational effect, which cannot be neglected. Therefore a speed of about 10.8 ... 10.9 km / s (depending on the position of the moon) is sufficient, you then get a transfer orbit in the form of an 8, as carried out in the Apollo program . The velocity vector must be very precisely aligned so as not to miss the moving moon; In practice, several corrective maneuvers are used during the almost three-day transfer.

### Flight to inner planets

During a flight to the inner planets Venus and Mercury , drive power must be applied against the orbital speed of the satellite around the sun, which it takes over from Earth. For such flights, the escape speed from the earth is directed in such a way that it counteracts the circular movement. The energy expenditure is nevertheless considerable and only allows small space probes to be brought into the interior of the solar system. The requirements for control and navigation are again considerably higher than those for a moon flight. An energy minimum is only reached once per synodic period .

### Flight to outer planets

If one wants to reach the outer planets Mars , Jupiter , Saturn , Uranus or Neptune , the probe is accelerated further against the gravitational field of the sun after leaving the earth. For this second acceleration one benefits from the movement of the earth around the sun, provided again that the velocity vectors are correctly aligned. As with the flight into the inner solar system, the speed requirement is great, which is why the probes are mostly light, otherwise swing-bys are required to accelerate the probes further. Because of the great distances into the outer solar system, these flights can take many years.

### Flights on orbits with constant thrust

Electric thrusters (e.g. ion propulsion ) as well as the possible propulsion of a spacecraft by solar sails make it possible to maintain a thrust, albeit a small one, for a very long time, and therefore represent an alternative to conventional chemical propulsion systems for long-term missions of the spacecraft can be permanently controlled so that the thrust goes in the desired direction. Mathematically, these trajectories, which move spirally in many turns inwards or outwards, are no longer easy to record because the orbital energy does not remain constant.

### Flyby maneuvers (Gravity Assist, Swing-by)

A flyby maneuver on a mass moving in the reference system leads to an energy exchange between the two bodies and therefore allows a satellite to be accelerated (when passing "behind" the body) or decelerated. This is associated with a deflection of the flight path, the size of which is related to the degree of deceleration or acceleration. For these maneuvers the large planets Jupiter and Saturn as well as the fast moving Venus as well as the earth itself are interesting. The flyby takes place on a planetocentric hyperbolic orbit, the energy gain in the heliocentric system occurs through the different reverse transformation of the speed on the departing branch. Such maneuvers can be used for flights into the inner and outer solar system, u. U. also with several fly-bys on one or more planets; Leaving the ecliptic plane, as with the Ulysses solar probe , is also possible. The disadvantage is the extension of the flight time, the increased demands on control and navigation as well as the need to pay attention to the position of another body, which i. d. Usually leads to significant restrictions of the start window .

## Applications

The actual application is mission planning , the task of which is to find a suitable mission for a given goal - for example a planet of scientific interest. H. especially to find a suitable trajectory.

Extensive numerical simulations are required for this. In the beginning, these simulations were limited to optimizing an essentially predetermined flight path (usually a Hohmann-like transition) and finding suitable departure dates. In such cases, the energy reserve of the carrier system determines the length of the start window . The height curves of similar plots of the energy required as a function of departure and arrival date come from this time. Since then, the numerical possibilities have been expanded considerably and also make it possible to determine the imaginative orbits required for high demands - such as visiting a comet.

## Mathematical Methods

### Vis-Viva equation

The Vis-Viva equation (Latin for "living force"), which follows from energy conservation , relates the speed to the current orbit radius and the characteristic size of the orbit, the major semi-axis of the conic orbit , for a stable two-body path . It is: ${\ displaystyle v}$${\ displaystyle r}$ ${\ displaystyle a}$

${\ displaystyle v ^ {2} = \ mu \ left ({{2 \ over {r}} - {1 \ over {a}}} \ right)}$.

Here is for a circle, for an ellipse, for a parabola and for a hyperbola. is the standard gravitational parameter , where again the gravitational constant and the mass of the central body are: ${\ displaystyle a = r}$${\ displaystyle a> r}$${\ displaystyle a = \ infty}$${\ displaystyle a <0}$${\ displaystyle \ mu}$${\ displaystyle G}$${\ displaystyle M}$

${\ displaystyle \ mu = GM \}$

### Kepler's equation

The equation named after its discoverer Johannes Kepler relates the mean anomaly , the eccentric anomaly and the eccentricity of an elliptical orbit as follows: ${\ displaystyle M}$ ${\ displaystyle E}$${\ displaystyle \ epsilon}$

${\ displaystyle M = E- \ epsilon \ cdot \ sin E}$

It thus allows the position of a body located on a known path to be determined as a function of time. However, this requires iterative or numerical procedures, since the equation is transcendent in . For almost circular ( close to 0) as well as highly elliptical ( close to 1) orbits numerical difficulties can arise if the two right terms are of very different order of magnitude or are almost the same. ${\ displaystyle E}$${\ displaystyle \ epsilon}$${\ displaystyle \ epsilon}$

### Spheres of Influence 'Patched Conics'

The concept of spheres of influence assumes that the orbit of a celestial body at a given time essentially only one is affected other body; Interferences from other bodies are neglected. If one approaches another body, there is a point of virtual equilibrium at which the two dynamic forces of attraction (i.e. the disturbance and the acceleration of the undisturbed two-body path) are equal. For the common case in the solar system that the mass of the heavier body (e.g. the sun, index S) is considerably greater than that of the other (here the target planet, index P), this point results in

${\ displaystyle r_ {T} = r \ left ({\ frac {m_ {P}} {m_ {S}}} \ right) ^ {2/5}}$

where stands for the approach distance in question and the distance between the planet and the sun. At this point one changes the way of looking at things (e.g. from an earth orbit to a solar orbit) and a transformation is carried out so that the orbits continuously merge into one another (hence the designation of joined conic sections ). This concept is very simple and in some cases can even be carried out by hand, but it takes a u. U. considerable need for correction in purchase. ${\ displaystyle r_ {T}}$${\ displaystyle r}$