- The orbit of each planet is an ellipse with the Sun occupying one focus.
- The line joining the Sun to a planet sweeps out equal areas in equal intervals of time.
- A planet's orbital period is proportional to the mean distance between Sun and planet, raised to the power .

This theory states that the force of gravitational attraction between two point-like bodies of mass m, m with a distance r from each other is equal to

(4.1) |

Influences like atmospheric drag and radiation pressure play a minor role in celestial mechanics. The references for sections 4.1.1 to 4.1.4 are Nussbaumer 1989 chapter 2 and Fortescue and Stark 1995 chapter 4. In the following x = for any vector .

(4.2) |

(4.3) |

(4.4) |

(from Fortescue and Stark 1995)

Body | ( ) |

Sun | 1.327×10 |

Earth | 3.986×10 |

Venus | 3.249×10 |

Mars | 4.281×10 |

Jupiter | 1.267×10 |

Saturn | 3.794×10 |

Moon | 4.902×10 |

Taking the cross product of (4.4) with yields to

(4.5) |

(4.6) |

(4.7) |

(4.8) |

(4.9) |

The general form of an ellipse is shown in figure 4.1. The shape of the ellipse is characterized by the

(4.10) |

(4.11) |

a | = | semi-mayor axis | = | true anomaly | |

b | = | semi-minor axi | E | = | eccentric anomaly |

p | = | semi-latus rectum | r | = | pericenter radius |

r | = | apocenter radius |

For the restricted problem of two bodies the heavier body occupies one focal point. The point closest to the main body is called

and | (4.12) |

(4.13) |

(4.14) |

To find the position versus time relationship the differentiation of (4.10) and (4.8) gives

Using equation (4.6) and r sin = a sinE yields

(4.15) |

For the restricted problem of two bodies the

(4.16) |

(4.17) |

(4.18) |

and | (4.19) |

The evolution of the velocity along an orbit is shown in Figure 4.2. The figure shows the feature of equation (4.17). The velocity rises to a maximum at the apogee while being low for most of the trajectory.

In the case of a parabolic trajectory the perigee height r is

(4.20) |

From equation (4.11) we see that hyperbolic curves have a negative semi-major axis. The hyperbolic orbit is open and applies to swing-by manoeuvres and escape trajectories. The shape of a hyperbola is given in figure 4.3. The time evolution is calculated in a similar way than for the ellipse. A

(4.21) |

(4.22) |

(4.23) |

(4.24) |

The

The orbit of the satellite is generally defined by the six

(4.25) |

There are two fundamentally different techniques to calculate an orbit more precisely. Special perturbation techniques deal with the direct numerical integration of the equation of motion including all relevant perturbing accelerations. These techniques are straightforward and applicable to any number of bodies and perturbing accelerations. However they require considerable computational effort and don't lead to generally usable formulas. General perturbation techniques on the other hand provide an analytical description of the variation with time of the orbit. Although being more complicated, these techniques provide a better physical insight and allow the development of analytical expressions that require a minimum of computational effort. One of them is the

Spacecrafts are under influence of many forces disturbing the potential. These include gravitation effects, atmospheric drag and radiation pressure.

The terrestrial field differs from the potential.

(4.26) |

J | 1082.6×10 | C | 0 | S | 0 | ||

J | -2.53×10 | C | 1.57×10 | S | -0.904×10 | ||

J | -1.62×10 | C | 2.19×10 | S | 0.27×10 |

The J term is larger than the others. It results from the Earth's oblateness and dominates the gravitational perturbation of the Earth. The Earth's oblateness has two main effects on an orbit around Earth. The

(4.27) |

(4.28) |

Atmospheric drag decreases the satellite energy and thus the orbital height. This decrease has to be watched carefully as it may eventually cause atmospheric reentry and consequent loss of the supervised satellite. The force parallel to the velocity vector is best given by

(4.29) |

A satellite is also influenced by the gravitational pull from other bodies. Orbits around Earth are generally equally influenced by the Moon and the Sun. These perturbations are termed

(4.30) |

A spacecraft moving through the solar system is exposed to a force from the impact of solar radiation on it's surface. Electromagnetic radiation carries a momentum that exerts a pressure

(4.31) |

(from Fortescue and Stark 1995)

Figure 4.6 gives the a comparison of the disturbing accelerations in the vicinity of Earth. The logarithm of the forces normalized with 1 g is shown. The dominant force is the primary inverse square law gravity field, followed by the J perturbation of the Earth's oblateness. The higher order terms are only important for low orbit operations, for example rendezvous strategies. For the surface forces a ratio of 0.005 was taken. The atmospheric density depends on the solar activity (see section 3.1.3). Therefore the curve for the drag is very uncertain and can vary up to one order of magnitude.

The equation of motion for the third body is

(4.32) | |||

By setting velocity and acceleration to zero five points are obtained within the frame of reference at which stationary body will be at equilibrium. These

In the restricted tree body problem for the Earth Sun system the L point is 1.5 million km towards and L is 1.5 million km from Earth away from the Sun. The L point is behind the Sun at 150 million km which is the Sun Earth distance.

(4.33) |

To get a notion for the energy required for a plane change let's consider the following: Injection into geostationary Earth orbit from a geostationary transfer orbit with inclination 7° (launch from Kourou) needs a V of 1.47 where the same manoeuvre from a geostationary transfer orbit with inclination 28° (launch from Kennedy Space Center) needs a V of 1.80. This calculation was made with (4.18), (4.19) and (4.27) and the vector composition shown below.

The transfer between two coplanar circular orbits that requires a minimum of energy is the

The idea behind a

Another way to change an orbit is by progressive change by perturbations. But this method was never used as it requires a lot of numerical computation.

During its life a spacecraft needs different types of propulsion. The biggest thrust (V~10) is required for the launch vehicle to lift the spacecraft from the Earth's surface into a low Earth orbit just above the atmosphere. Manoeuvres follow to bring the spacecraft into the desired operational orbit. These manoeuvres which require a V between 2 and 8 are performed by the launch vehicle, by the on board engine or by both. Finally the orbit and the spacecraft attitude needs to be maintained which needs a V from 0.03 to 0.3.

A rocket engine is characterized by its

(4.34) |

(4.35) |

Different types of thrust engines were conceived but up to now only chemical rockets are widely used. Solid propellant engines have a specific impulse from 200 to 260 s. They have a simple design, but once ignited the combustion proceeds at a given thrust until all the propellant is consumed. The liquid propellant engine on the other hand is more complex in design but flexible in burn duration and thrust level. Monopropellant engines have a specific impulse about 200 s whereas bi-propellant engines have a I of 300 to 400 s. Bi-propellant engines are nearly twice as complex as monopropellant engines.

For orbital manoeuvres the minimum energy consumption is achieved by giving the spacecraft the necessary V within an instant at a certain point. As engine burns take a certain time a bigger burn is required that in this theoretical case. This loss of thrust energy is called

[%] | (4.36) |