• magnetized plasma
• MIP
• mutual impedance experiments
• mutual impedance probe

The beginning of the space age in the late Fifties marked a crucial step in the understanding of the Solar System, as well as for proposing the Heliosphere as a plasma physics laboratory.
Although the existence of the eight planets was established in the nineteenth century, the modelling of the space environment as a system of orbiting bodies governed by Newtonian gravity was not sufficient to explain a number of phenomena.
In particular, the explanation of cometary tails was challenging at the time, and was one of the first suggestions of the existence of a flux of particles continuously streaming from the Sun, named the solar wind.
This prediction was experimentally verified by the first direct measurement performed by spacecraft in the interplanetary medium.
A measurement performed at the location where the event of interest is taking place is referred to as \emph{in situ} measurement.
The existence of the solar wind was not the only case in which \emph{in situ} measurements were crucial in developing the understanding of space plasmas.
The upper region of the Earth's atmosphere is ionized as a result of the Sun's radiation, forming a plasma called the ionosphere, that was first discovered thanks to its ability to reflect radio wave communications, and later investigated both by remote and \emph{in situ} measurement techniques.
The behavior of both the ionospheric and the solar wind plasmas in the vicinity of the Earth are modified by our planet's magnetic field. This region of influence, where the motion of charged particles is modified by the Earth's magnetic field (and vice versa), is named the magnetosphere.
It was examined by the first space missions, one of their first discoveries being the presence of high-energy particles in localized spatial regions around the Earth, called the Van Allen belts.
Nowadays, a variety of satellites are providing continuous measurements in the ionosphere, the magnetosphere and the solar wind.
The interactions between these three plasmas, and their dynamics, is analyzed both using observational techniques and numerical simulations.

Numerical simulations can provide a complete three-dimensional spatial picture of the time changing plasma, as well as the possibility to explore different plasma conditions.
Given the complexity and multiscale characteristic of the space environment, simulations are however limited by the available computational power. This limitation can be circumvented, for example, by considering limited portions of the system of interest, by performing local simulations instead of global simulations.
The conclusions that are drawn from the analysis of numerical simulations provide important physical insights for the analysis of the observations, and in particular of \emph{in situ} measurements.
These measurements are limited by the fact that they represent a one dimensional trajectory of a three-dimensional system, taken at a fixed time.
Still, the synergy of the these two approaches has been the key to significantly enhance our current understanding of space plasmas in the Solar System.

Several instrumental space techniques have been developed to provide \emph{in situ} plasma measurements of the plasma particles and the electromagnetic fields.
One of such space plasma instruments, called mutual impedance experiment, is dedicated to both \emph{in situ} plasma and electric field diagnostics, and is today largely used on a variety of space missions to retrieve plasma observables, among which are the electron density and temperature.
This type of diagnostic probes the plasma properties through the propagation and measurement of an electric signal within the plasma itself. In particular, they are able to provide information about the characteristic resonant frequencies of the plasma.
Practically, the measure is performed by imposing on one electric antenna (the emitter antenna) an oscillating electric current at a given frequency, while a second electric antenna (the receiver antenna) measures the electric potential difference generated in the surrounding plasma.
Through a spectral analysis, the ratio between the measured electric potential difference and the imposed electric current, expressed in function of the electric current frequency, provides the measurement of the so-called mutual impedance.
The working principle of the mutual impedance experiments is based on the fact that the amplitude of the mutual impedance spectrum presents extremas at the frequencies corresponding to the plasma resonant frequencies, meaning for frequencies at which the electric signal is able to propagate inside the plasma from the emitter to the receiver antenna.
The modeling of mutual impedance experiments relies on the calculation of the electrostatic potential produced by the emitter antenna, which in turn depends upon the inverse of the plasma dielectric constant.
The plasma dielectric constant is a function of several plasma parameters, as for instance the electron density and temperature, and depends on the presence of an external magnetic field.
The modelling of mutual impedance experiments, therefore, consists in solving the inverse problem of finding the electron density and temperature that correspond to the measured spectrum.

The mutual impedance measurement technique was adapted from the field of geophysical survey to near Earth space plasmas' exploration in the Seventies. After being successfully used in this context for the analysis of the ionosphere and magnetosphere, it was later applied in planetary exploration. One such example is the study of the coma of Comet 67P/Churuymov-Gerasimenko by the Rosetta mission.
The state-of-the-art modelling of mutual impedance probe experiments was developed for the analysis of the plasma environment of comet 67P, and is based on the assumption of the absence of an external magnetic field.
Following Rosetta, mutual impedance probe experiments are currently embarked, among others, on two upcoming missions of the European Space Agency: BepiColombo, targeting Mercury and launched in 2018, and JUICE, targeting Jupiter and its moon Ganymede, which will be launched in 2022.
These missions will encounter plasmas for which the effect of the planet or satellite magnetic field cannot be neglected, as it influences the plasma dynamics and therefore the mutual impedance measurement. Therefore, the current state-of-the-art approach to the modelling of mutual impedance experiments is not applicable.

In this context, waiting for the future JUICE and BepiColombo space mission results, the objective of this work is to extend the current state-of-the-art mutual impedance modelling to the case of a magnetized plasma, meaning a plasma for which the ration of plasma frequency to electron cyclotron frequency is close, or less then, one.

In order to do so, the computation of the electrostatic potential of the emitter antenna is extended to the case of a magnetized plasma.
Two numerical techniques are investigated: (i) an adaptive integration scheme and (ii) discrete numerical transforms. Both are tested and validated in the well explored physical regime of unmagnetized plasmas.
Both methods are then tested on the case of interest of magnetized plasmas, by comparing them with analytical approximations. The discrete numerical transform, while providing an improvement in computational speed, is found to be not suited for the case of a magnetized plasma.

To prepare for future applications of the model, two types of experimental data are investigated: (i) the Rosetta spacecraft \emph{in situ} observations and (ii) laboratory experiments data.
The Rosetta spacecraft was mostly operating in the solar wind and in cometary plasmas, both considered as unmagnetized plasmas from an instrumental point of view. Still, during its cruise to comet 67P, it also performed three flybys of the Earth, in 2005, 2007 and 2009. During these maneuvers, the spacecraft entered the Earth's magnetosphere and reached distances of less than six Earth radii from our planet, therefore encountering a strongly magnetized plasma environment. The data collected by the mutual impedance instrument are calibrated, thus making them ready for future analysis using the newly developed instrumental model.
The lack of repeatability for space measurements poses limits on the type of tests that can be performed to validate an instrumental model. To overcome this limitation, an experimental setup using a plasma chamber has been prepared at the \emph{LPC2E} Laboratory in Orléans. In the context of this work, a mutual impedance probe experiment has been performed in this facility in both magnetized and unmagnetized conditions. The measurements performed by the mutual impedance experiment are found to be consistent with the measurements obtained by a Langmuir probe present inside the plasma chamber.