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## Mutual impedance probes (MIP)

The two permittivity probes that are the focus of this manuscript, namely the SESAME-PP/Philae instrument operated on the nucleus of comet 67P/Churyumov–Gerasimenko (67/C-G) and the PWA-MIP/HASI/Huygens instrument used on Titan, are based on the MIP investigation method. Before going into the detail of the theory behind this method (see Chapter 2), and in the interest of comparing it to the methods previously described, we will briefly summarize the main features of the MIP technique.

### Comparing techniques

The techniques described above represent multiple ways of measuring the complex permittivity of the subsurface of planetary objects. These methods are complementary in two ways. First, since these techniques are not operated in the same frequency domain, they provide complementary information on the investigated subsurface (Figure 23). For example, mutual impedance probes operating at very low frequencies are efficiently used for the detection and estimation of the shallow subsurface water ice content, while high-frequency measurement instruments such as radars are better suited for the detection of liquid water, which has a high dielectric constant (80) in the HF domain. The operating frequencies of the instrument determine the electromagnetic laws that describe the interaction of the electromagnetic waves with matter: high frequencies in the propagation domain and at low frequencies in the diffusion domain.

#### History and theory of surface Mutual Impedance Probes (MIP)

Mutual impedance probes have been used on Earth for many decades to measure the subsurface resistivity in a non-destructive way. They were first introduced by Wenner (1916) and consist of four electrodes. In their early version, a DC current was injected between two transmitting electrodes and the potential difference induced by this current was measured between two receiving electrodes in contact with the ground. The ratio of the received voltage potential over the injected current i.e., the mutual impedance of the quadrupole, yields the conductivity of the subjacent ground down to a depth comparable to the separation between the electrodes (see section 4.2). Compared to the self-impedance technique (presented in Chapter 1, section 4.3), the MIP technique is much less sensitive to the presence of heterogeneities in the vicinity of the electrodes and to the quality of the contact between the electrodes and the medium. Alternatively, the electrodes can be buried at various depths below the surface.

Later, Grard (1990) proposed to use the same technique with AC instead of DC signals in order to measure not only the conductivity, but also the dielectric constant i.e., the complex permittivity of the ground. This technique, which had been successfully applied in space plasmas (Storey et al. 1969) in the frame of many ionospheric and magnetospheric experiments around the Earth (Chasseriaux et al. 1972; Décréau et al. 1982; Décréau et al. 1987); it was subsequently validated on Earth (Tabbagh et al. 1993) and used on the surface of a planetary body. The PWA analyzer (Grard et al. 1995) a unit of the HASI package (Fulchignoni et al. 2002) onboard the ESA Huygens probe that landed on the surface of Titan on January 14, 2005 (Fulchignoni et al. 2005; Grard et al. 2006). The Titan surface data was recently revisited with more accurate numerical models (see Hamelin et al. 2016 & Chapter 5). A laboratory MIP called HP3-PP (Stiegler & Kargl 2004) had been designed to be part of the ExoMars Humboldt surface station, which was ultimately cancelled. Lastly, the SESAME-PP/Philae MIP (Seidensticker et al. 2007) acquired a set of measurement on the surface of the nucleus of comet 67P/C-G (Lethuillier et al. 2016 & Chapter 4).

**Mutual impedance for a quadrupole above a surface: derivation of the surface complex permittivity**

Herein we summarize the theory of the quadrupolar array and show how to derive the relative permittivity 𝜖𝑟 of a planetary surface. This approach was first proposed by Grard (1990) and Grard & Tabbagh (1991). It assumes quasi-static approximation, as the wavelength of operation is much larger than the distance between the electrodes, and neglects magnetic induction. In vacuum, the potential 𝑉 at a distance 𝑟 from a point charge 𝑄 is 𝑉= 𝑄4𝜋𝜀0𝑟 (63).

When this charge is at a height ℎ above an interface separating vacuum from a half-space of relative complex permittivity 𝜖𝑟, the potential distribution can be determined with the image charge theory (see Section 3.1) in which we evaluate the effect of the interface by an image charge located at a distance ℎ under the interface. The charge of the image is equal to (Griffiths 1999):

𝑄′=−𝜖𝑟−1𝜖𝑟+1𝑄= −𝛼𝑚𝑄.

**Comparing simple examples to more realistic cases**

In order to illustrate problems I. and thus the need for the Capacitance-Influence Matrix Method (presented in section 2.4), we build a numerical model of a MIP consisting of 4 cylindrical electrodes with a radius 𝑟=0.05 m (a size comparable to the SESAME-PP/Philae electrodes), forming a square of side 𝑑=0.5 𝑚 and located on the interface (height = 0) between a vacuum and a pure dielectric subsurface (i.e., 𝜖𝑟=𝜖𝑟′ & 𝜎𝑒𝑓𝑓=0, Figure 26).

We then run numerical simulations with the COMSOL Multiphysics© software which is a finite element analysis software that solves the Laplace equation in 3D. The model is meshed and boundary conditions are set (see www.comsol.com for more information) to estimate the mutual impedance measured by the MIP. Equation (72) is used to derive the dielectric constant that is plotted in Figure 27 against the nominal dielectric constant of the subsurface. We note a small difference (maximum 1.8 % for a dielectric constant of 5), between the pin point electrodes model and the cylindrical electrodes model. This demonstrates that representing the electrodes by pin point electrodes yields a good approximation of the dielectric constant of the subsurface, for more accurate results, numerical simulations are necessary.

**Table of contents :**

Acronyms

Notations

Index

Introduction

**Chapter 1: Characterizing subsurface electric properties **

1. Interaction of electromagnetic fields with matter

1.1. Maxwell’s equations

1.2. Frequency dependence of the relative permittivity

1.3. Propagation and diffusion domains

2. Electrical properties of natural matter

2.1. Water ice

2.2. Liquid water

2.3. Rocks

2.4. Chondrites

2.5. Lunar regolith

2.6. Martian analogs

2.7. Europa crust analog

3. Mixing laws

4. Methods for the characterization of subsurface electric properties

4.1. Vertical Electrical Sounding (VES)

4.2. Time Domain Electromagnetic Method (TDEM)

4.3. Self-impedance probes

4.4. Mutual impedance probes (MIP)

4.5. Radars

4.5.1. Radars in reflection

4.5.2. Radars in transmission

4.6. Microwave radiometers

4.7. Comparing techniques

5. Concluding remarks

**Chapter 2: Mutual Impedance Probes, numerical modelling and performances **

1. History and theory of surface Mutual Impedance Probes (MIP)

1.1. History of MIP

1.2. Mutual impedance for a quadrupole above a surface: derivation of the surface complex permittivity

2. Numerical modelling and Capacity-Influence Matrix method

2.1. Application to realistic problems

2.2. Derived complex permittivity

2.3. Comparing simple examples to more realistic cases

2.3.1. Finite size electrodes

2.3.2. Presence of conducting elements close to the electrodes

2.3.3. Influence of the electronics circuit

2.4. The Capacitance-Influence Matrix Method (CIMM)

2.5. Derivation of 𝝐𝒓

3. Validation of the use of numerical models

3.1. Method of image charges

3.1. Simplified model of a MIP

3.2. Comparison

4. Exploring the capabilities of the mutual impedance probes

4.1. Sensitivity of the transmitting electrodes

4.2. Sounding depth

4.3. Heterogeneous subsurfaces

4.3.1. Study cases

4.3.2. Derived permittivity

4.4. Maximizing the scientific output

5. Concluding remarks

**Chapter 3: The SESAME-PP/Philae/Rosetta experiment: modelling approaches and performances **

1. The SESAME-PP/Philae/Rosetta experiment

1.1. Comets and Rosetta’s mission objectives

1.1.1. Comets and their scientific interests

1.1.2. Scientific objectives and description of the Rosetta mission

1.2. Rosetta’s and Philae’s payload

1.2.1. Rosetta’s payload

1.2.2. Philae’s payload

1.2.3. Depth sounded

1.3. The SESAME-PP experiment

1.3.1. The SESAME package

1.3.2. The SESAME-PP experiment and operation modes

2. Modeling SESAME-PP

2.1. SESAME-PP numerical model

2.2. SESAME-PP lumped element model

2.3. Application of the Capacity-Influence Matrix Method

Step 1: Derivation of medium capacitance-influence matrix 𝐾𝑚

Step 2: Derivation of the electronic matrix 𝐾𝑒

Step 3: Solving the numerical model

2.4. SESAME-PP laboratory model

2.4.1. Description of the laboratory replica of SESAME-PP

2.4.2. Description of the Lander replica

2.5. Experimental tests in a controlled environment and validation of the numerical model

2.5.1. General considerations

2.5.2. Three-foot configuration measurements in a controlled environment

2.5.3. Five-electrode configuration in a controlled environment

2.6. Tests in a natural environment and comparison with the numerical model

2.6.1. Dachstein field campaign

2.6.2. Description of the area studied

2.6.3. Three-foot configuration measurements over an icy surface

2.7. Sounding depth of SESAME-PP

3. Concluding remarks

**Chapter 4: Electrical properties and porosity of the first meter of 67P/Churyumov-Gerasimenko’s nucleus as constrained by SESAME-PP/Philae/Rosetta **

1. RDV, landing and escort

1.1. The cruise phase and Rosetta “rendez-vous “with 67P/C-G

1.2. Philae separation and landing at Abydos

1.3. Escort phase

2. Main results from the Rosetta mission

2.1. Nucleus

2.2. Coma

2.3. Context of the SESAME-PP measurements

3. SESAME-PP observations during the cruise, descent and landing

3.1. Cruise

3.2. Separation, Descent, Landing (SDL)

3.3. First Science Sequence (FSS) on the surface

4. Analysis of the SESAME-PP surface data

4.1. Approach

4.2. FSS passive measurements

4.3. Transmitted currents

4.4. Received potentials

4.4.1. Reconstruction of Philae attitude and environment at Abydos

4.4.2. Retrieval of the dielectric constant of the near surface of Abydos

5. Implications for the composition and porosity of the first meter of 67P/C-G’s nucleus

6. Concluding remarks

**Chapter 5: The PWA-MIP/HASI/Huygens instrument, revisiting the data collected on the surface of Titan **

1. The Cassini/Huygens mission and Titan

1.1. The Cassini/Huygens mission in brief

1.2. Titan after Cassini-Huygens

2. The PWA-MIP/HASI instrument

2.1. Description

2.1.1. Transmitting circuit

2.1.2. Receiving circuit

2.2. Numerical geometry model

2.3. Electronic model

2.4. Sounding depth

3. Data collected during descent and on the surface

3.1. Descent measurements

3.2. Surface measurements

4. Data calibration and analysis

4.1. Data calibration and electronic matrix

4.2. PWA-MIP/HASI configuration of operation at the Huygens landing site

4.3. Derived permittivity

4.4. Titan’s first meter surface composition

4.5. The 9539 s event

5. Electrical properties of analogues of Titan’s organic materials

5.1. Tholins

5.2. Description of the measurement bench

5.3. Measurement and derivation of the sample complex permittivity

6. Description of the samples

6.1. Frequency and temperature dependence

6.2. Porosity dependence

6.3. Derivation of the complex permittivity of bulk tholins

7. Constrains on Titan’s subsurface composition

6.2 The liquid inclusion model

7.1. Thin conductive surface layer model

8. Concluding remarks

Conclusion

**Bibliography**