## 1.1 Electrodynamic equations

A current flowing though a conductive wire in the presence of a magnetic field leads to a Lorentz force. This simple fact is utilized by an electrodynamic tether propulsion. This force can be described quantitatively for a homogeneous magnetic field by the equation

(1.1)

in which represents the magnitude of the electric current and the length vector of the tether, pointing in the direction of the current.

For the simple model explored below it is assumed for Lorentz force calculation that and are always perpendicular to each other. There is no directly closed current loop in this system, as this would nullify the Lorentz force on its way back. Instead, it is closed by other ions and electrons in the planet’s ionosphere, which means those particles, not connected to the spacecraft, experience the resulting reaction force.

The tether can be operated in two modes, either as a generator or as propulsion. This is made possible by using the voltage induced by the orbital velocity of the tether (generator mode), to let a current flow though the wire. The power

can be extracted from the tether in this case. While this is usually larger than the power consumption of the ion engine in a practical tether propulsion system, the ion engine still needs to be supplied to keep the electrons flowing.

In contrast to this, propulsion mode requires to counteract the induced voltage by a power supply to reverse the current direction. This can be accomplished by supplying a voltage equal to , so manifests as a power consumption additional to that of the ion engine.

Lastly, there is also a power need to compensate for the resistance R of the tether. This appears as consumption load both in generator and propulsion mode.

## 1.2 Finite element mechanics equations

Now that the basic electrodynamic principles have been covered, this chapter will describe the finite element model used to calculate how the Lorentz force influences the system in combination with gravity gradient forces. The model used is as follows: There is one end mass, that represents the spacecraft. Then there is a (usually smaller) mass at the other end. The tether itself is divided into n segments, each with their own mass and length pieces, resulting in segmented gravity gradient and Lorentz forces acting at their center points.

These two types of forces balance out each other at an angle α between the line through the center of gravity of the orbiting object and the planet and the tether pushed out of this vertical line by Lorentz forces. This limits the effective maximal Lorentz force that can be applied befor the system becomes unstable through too big an α in function of tether length, mass distribution and size of the gravity gradient.