Plasma and dust interaction in the magnetosphere ofSaturn

JONAS OLSON

Doctoral ThesisStockholm, Sweden 2012

TRITA-EE 2012:018ISSN 1653-5146ISBN 978-91-7501-343-5

KTH Rymd- och plasmafysikSkolan för elektro- och systemteknik

SE-100 44 Stockholm

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläg-ges till offentlig granskning för avläggande av teknologie doktorsexamen i fysikaliskelektroteknik måndagen den 28 maj 2010 klockan 10.00 i Sal F3, Lindstedtsvä-gen 26, Kungl Tekniska högskolan, Stockholm.

© Jonas Olson, april 2012

Tryck: Universitetsservice US AB

iii

Abstract

The Cassini spacecraft orbits Saturn since 2004, carrying a multitude ofinstruments for studies of the plasma environment around the planet as wellas the constituents of the ring system. Of particular interest to the presentthesis is the large E ring, which consists mainly of water ice grains, smallerthan a few micrometres, referred to as dust. The first part of the work pre-sented here is concerned with the interaction between, on the one hand, theplasma and, on the other hand, the dust, the spacecraft and the Langmuirprobe carried by the spacecraft. In Paper I, dust densities along the trajectoryof Cassini, as it passes through the ring, are inferred from measured electronand ion densities. In Paper II, the situation where a Langmuir probe is lo-cated in the potential well of a spacecraft is considered. The importance ofknowing the potential structure around the spacecraft and probe is empha-sised and its effect on the probe’s current-voltage characteristic is illustratedwith a simple analytical model. In Paper III, particle-in-cell simulations areemployed to study the potential and density profiles around the Cassini asit travels through the plasma at the orbit of the moon Enceladus. The lat-ter part of the work concerns large-scale currents and convection patterns.In Paper IV, the effects of charged E-ring dust moving across the magneticfield is studied, for example in terms of what field-aligned currents it sets up,which compared to corresponding plasma currents. In Paper V, a model forthe convection of the magnetospheric plasma is proposed that recreates theco-rotating density asymmetry of the plasma.

iv

Sammanfattning

Rymdsonden Cassini befinner sig i omloppsbana kring Saturnus sedan2004 och bär med sig en mångfald av instrument för att studera plasmat ochringarna som omger planeten. Av särskilt intresse i denna licentiatuppsats ärden stora E-ringen. Denna utgörs huvudsakligen av mikrometerstora (ellermindre) dammpartiklar, bestående av is. Den första delen av det arbete sompresenteras här behandlar interaktion mellan, å ena sidan, plasmat och, åandra sidan, dammet, rymdsonden och Langmuirprob som denna är utrustadmed. I den bilagda Paper I utvinns dammtätheter längs Cassinis bana genomE-ringen ur mätta elektron- och jontätheter. I Paper II betraktas situationendär en Langmuirprob befinner sig i potentialgropen som omger en rymdsond.Här betonas vikten av att ta hänsyn till potentialstrukturen kring rymdsondoch prob, och en enkel analytisk modell används för att illustrera hur pro-bens ström-spänningskaraktäristik kan påverkas av denna potentialstruktur.I Paper III studeras täthets- och potentialprofilerna runt Cassini numerisktmed particle-in-cellsimuleringar för parametrar som modellerar hur rymdson-den rör sig relativt plasmat vid månen Enceladus bana. Den senare delen avarbetet behandlar storskaliga strömmar och konvektionsmönster. I Paper IVstuderas effekterna av att laddat damm i E-ringen rör sig vinkelrätt mot mag-netfältet, bland annat med avseende på vilka parallellströmmar denna rörelseger upphov till, vilka jämförs med motsvarande plasmaströmmar. I Paper Vframläggs en modell för konvektionen hos magnetosfärens plasma som åter-skapar den co-roterande täthetsasymmetrin hos plasmat.

Acknowledgements

A majority of the work presented herein has been carried out together with NilsBrenning, co-advisor for my doctoral studies. I have learnt a lot about physics andabout how to be a good person. Nils has taught me about ths subject as well assprinkled me with literary quotes and references. We have discussed music andwe have sung together. Nils been a shining example how to be a scientist, havinga scientific mind as well as being honest. Nils and has also shared his ideas onorganizing one’s work and on writing. More than once, we – the two of us – havehad the opportunity to discuss, or, perhaps, explore, the use of punctuation. Deep,non-obvious insights about how to present an argument to a reader, that he hasconfided to me during our work, is, if I remember correctly, that one is supposedto tell the truth (at least as a last resort) and that one is not supposed to write ingerman, or possibly the other way around.

Svetlana Ratynskaia, main advisor and probe specialist among other things,has a most attentive advisor, always eager to see that no obstacles were in my way.Having great concern for the progress of her students, she has always made herselfavailable for discussion, paperwork, pulling strings, turning the world upside-downand making other arrangements whenever necessary. Not only is she herself a greatsource of knowledge about the multiple subjects she is involved with, she has alsobeen able to engage experts in different areas to be our collaborators, which havebeen enormously useful. I am afraid I do not speak Russian yet, but I do knowmore about Russia than I did before, thanks to our discussions sometimes driftingoff topic.

Lars Blomberg, co-advisor, has been the one to turn to when in distress or whenneeding to learn “how things are done” within the KTH. With a calm attitude andcheerful encouragement, he has made things work out time after time. I stronglysuspect his schedule is far more full than you would think from seeing him takinghis time to offer much-needed support and assistance.

Several collaborators have been involved in different aspects of this work andprovided very important knowledge about the state of their respective fields andinsight in available techniques and current practises. I wish to acknowledge the con-tributions from Victora Yaroshenko, Wojciech Miloch, Jan-Erik Wahlund, MichikoMorooka and Herbert Gunell.

I appreciate the help from Anita Kullen, Tomas Karlsson and Michael Raadu,

v

vi ACKNOWLEDGEMENTS

who has all provided insightsful comments on material included in this thesis.I am also very glad for my friends within the department, from other parts of

KTH as well as outside it all. It has been nice to share thoughts about work, toshare an interest unrelated to work or to share a sigh and a tired look with fewwords but much understanding. Thank you for urging me not to work too muchand for cheering me on when working too much is unavoidable anyway.

My dear family, you have provided me invaluable help so many times I cannotremember them all. Always so encouraging, always helping with practical mattersthat are in the way, always so caring, you are the best support I could have.

I thank you.

List of Papers

This thesis is based on the work presented in the following papers.

I. V.V. Yaroshenko, S. Ratynskaia, J. Olson, N. Brenning, J.-E. Wahlund, M.Morooka, W.S. Kurth, D.A. Gurnett, G.E. Morfill“Characteristics of charged dust inferred from the Cassini RPWS measure-ments in the vicinity of Enceladus”Planetary and Space Science 57, 1807–1812 (2009).

II. J. Olson, N. Brenning, J.-E. Wahlund, H. Gunell“On the interpretation of Langmuir probe data inside a spacecraft sheath”Review of Scientific Instruments 81, 105106 (2010).

III. J. Olson, W. J. Miloch, S. Ratynskaia, V. Yaroshenko“Potential structure around the Cassini spacecraft near the orbit of Ence-ladus”Physics of Plasmas 17, 102904 (2010).

IV. J. Olson, N. Brenning“Dust-driven and plasma-driven currents in the inner magnetosphere of Sat-urn”Physics of Plasmas 19, 042903 (2012).

V. J. Olson, N. Brenning“The magnetospheric clock of Saturn: a self organized plasma dynamo”Manuscript submitted to Nature.

The respondent’s contribution to the papers is as follows: Paper I: Derivedanalytical expressions, extracted measurement data from the database and per-formed numerical calculations. Paper II: Performed numerical calculations, ex-tracted measurement data from the database and authored article text (shared

vii

viii LIST OF PAPERS

with co-author). Paper III: Improved existing numerical code, performed the sim-ulations, interpreted the results (shared with co-authors) and authored article text(shared with co-authors). Paper IV: Derived analytical expressions, performed nu-merical calculations and authored part of the article text. Paper V: Constructedand ran the model.

Contents

Acknowledgements v

List of Papers vii

Contents ix

List of Figures xi

1 Introduction 11.1 The E ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Cassini spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The geysers of Enceladus . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Sheaths 72.1 Basic principle of sheaths . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Sheaths in different regimes . . . . . . . . . . . . . . . . . . . . . . . 8

3 Orbital motion limited model 113.1 Floating potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Applicability to the Cassini Langmuir probe . . . . . . . . . . . . . . 15

4 Particle-in-cell simulations 194.1 The PIC technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 PIC simulations of Cassini . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Magnetospheric plasma 275.1 Parallel currents driven by perpendicular currents . . . . . . . . . . . 275.2 Corotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6 Results and discussion 316.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

ix

x CONTENTS

6.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7 Conclusions 377.1 Papers I to III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2 Papers IV and V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Bibliography 39

List of Figures

1.1 Saturn, with a few of its inner rings, as seen by the Hubble Space Tele-scope. (Image credit: NASA/ESA/E. Karkoschka (University of Arizona)) 1

1.2 The ejection of material through the cracks in the surface of Enceladusis seen as a plume in this image captured by Cassini. The radius ofEnceladus is 252 km. (Image credit: NASA/JPL/Space Science Institute) 2

1.3 Cassini during assembly. The large white disc antenna on top is fourmetres in diameter. Several booms and wire antennas, used for measure-ments, were extended from the spacecraft once in space and are thus notvisible here. On the left side of Cassini, the Huygens probe can be seenwith its gold-coloured, cone-shaped heat shield. (Image credit: NASA) . 4

1.4 The “tiger stripes” on the surface of Enceladus. Through these cracks,Enceladus ejects the material that makes up most of the E ring. [Imagecredit: NASA] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Sketch of the sheath structure close to an infinite, conducting wall. (a)In the presheath, both electron and ion densities drop from their bulkvalues, but they do not differ from each other (i.e., quasi-neutralityholds). In the sheath, they continue to decrease – the electron densitymore rapidly than the ion density – leaving the sheath with a positivecharge density. (b) The larger part of the drop in potential (in this figurecalled Φ) between the bulk plasma and the wall occurs in the sheath,which therefore also represents most of the ion acceleration. However,the presheath acceleration alone is enough for the ions to reach the Bohmspeed. (Image credit: Ref. [1]) . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Some sheath regimes classified by the relations between characteristicprobe dimension d, mean free path ` and Debye length λD. . . . . . . . 10

xi

xii List of Figures

3.1 Current to a spherical probe or other object according to the OMLmodel (equation (3.1)) for a negative particle species. (a) The collectedcurrent as a function of the probe potential. In the repulsive region (tothe left of the plasma potential), the current depends exponentially onthe potential, whereas in the attractive region (to the right of the plasmapotential), the dependence is a straight line. (b) The derivative of thecurve in panel (a). Here, transition between the repulsive and attractiveregions are more easily seen, with a “knee” arising where the probe isat the plasma potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 An example of how the currents collected by a spherical object accordingto the OML model depends on the probe potential. For an increasinglynegative potential, more and more electrons are unable to reach thesurface of the object and the collected electron current decays exponen-tially (with the electron temperature as the decay constant). At thesame time, the ion current increases linearly. Because the ion current issmall and largely constant, compared to the dramatic variations in elec-tron current, the floating potential is found where the electron currenthas become small like the ion current, i.e., at a few electron tempera-tures negative. In this example, a photoelectron current has also beenincluded. It is constant for negative potentials and thus has the sameeffect as stronger ion current. . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Current-voltage characteristic of the Cassini Langmuir probe, measurednear the orbit of Enceladus. From the derivative in the lower panel, itis seen that the curve deviates from the ideal OML model of figure 3.1.The glitch at −19 V of the derivative curve appears on many of themeasured sweeps and is thought to be an instrumental defect. . . . . . . 16

4.1 A grid cell of a two-dimensional PIC simulation. The charge (and mass)of the simulation particle is distributed over the four grid points that arethe corners of the grid cell in which the particle resides. More charge andmass goes to the closer corners. Specifically, the portion of the particleascribed to each grid point is proportional to the area of the region withthe same label, in this figure, as the grid point. Thus, in the illustratedexample, most of the charge and mass is given to grid point B and Cgets the least. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Resulting potential structure from a 2D PIC simulation with the plasmaflowing from left to right. The disc representing Cassini is 3.5 m indiameter. The defining parameters of this simulation case are n0 =7 × 107 m−3, kBTe = kBTi = 2.5 eV and vd = 30 km/s. This parametercombination is here used as the reference case, to be compared withthe other simulations, where these three parameters are varied, one at atime. Figures 4.2 to 4.6 all depict the same region, though their scalesdiffer due to their different Debye lengths. . . . . . . . . . . . . . . . . . 23

List of Figures xiii

4.3 Potential from a simulation case with the same density and drift speedas in figure 4.2, but with a lower temperature kBTe = kBTi = 1 eV. . . . 24

4.4 Potential from a simulation case with the same temperature and driftspeed as in figure 4.2, but with a lower density n0 = 3.5× 107 m−3. . . . 24

4.5 Potential from a simulation case with the same density and temperatureas in figure 4.2, but with a lower drift speed vd = 12 km/s. . . . . . . . . 25

4.6 Potential from a simulation case with the same density and temperatureas in figure 4.2, but with a higher drift speed vd = 54 km/s. . . . . . . . 25

5.1 Parallel currents caused by spatial variations in dust density. A neg-atively charged dust slab (shaded) moves with speed vd relative to aplasma (here depicted in the plasma rest frame). Positive and negativecharge densities are created at the trailing and leading edge, respectively,where the density gradient along the direction of motion is non-zero.These drive field-aligned currents which close across the magnetic fieldin some distant load Σload. . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2 Radial current in a corotating magnetosphere. The radial current in theequatorial plane, due to, for example, pickup of new ions, closes via thefield lines and the ionosphere, and transfers momentum from the planetto the magnetosphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6.1 The model of the potential structure used in Paper II, plotted alongthe common axis of the spacecraft and the probe. The minimum UMis proposed to play an important role for the electron collection by theprobe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Chapter 1

Introduction

Of all the planets in the solar system, Saturn puts up the richest display of a ringsystem (figure 1.1), inspiring much awe and admiration and making it somethingof the prototypical illustration of a planet.

Figure 1.1: Saturn, with a few of its inner rings, as seen by the Hubble SpaceTelescope. (Image credit: NASA/ESA/E. Karkoschka (University of Arizona))

1.1 The E ring

The large and diffuse E ring in the Saturnian ring system was not discovered untilthe twentieth century. For comparison, the more clearly visible rings were observedalready in the seventeenth century. The inner edge of the E ring has a radius of 3RS,

1

2 CHAPTER 1. INTRODUCTION

where the Saturn radius RS ≈ 6×107 m, and at the outer edge, one can put 8RS asthe limit of its extent. The constituents of the ring are microscopic ice grains, a fewmicrometers or less across. These have their source on the moon Enceladus, whichorbits Saturn at 4RS in the ring plane. From cracks in the surface at the south poleof Enceladus, the material that populates the E ring shoots out like a geyser. Thishas been photographed by the Cassini spacecraft, orbiting Saturn (figure 1.2).

Figure 1.2: The ejection of material through the cracks in the surface of Enceladusis seen as a plume in this image captured by Cassini. The radius of Enceladus is252 km. (Image credit: NASA/JPL/Space Science Institute)

1.2 The Cassini spacecraft

Cassini is part of the Cassini–Huygens mission, which is a joint effort by the Amer-ican (NASA), European (ESA) and Italian (ASI) space organisations to study the

1.3. THE GEYSERS OF ENCELADUS 3

giant gas planet Saturn, along with its moons, rings and plasma environment.Cassini refers to the orbiter, currently circling Saturn, whereas Huygens is thename of a probe, carried by Cassini, that was released from its carrier to make itsown way down to the surface of the Saturn moon Titan, studying its atmospherealong the way. A picture of Cassini, with the Huygens probe attached, is presentedin figure 1.3. It shows the spacecraft being handled at Kennedy Space Center inpreparation for its launch.

Cassini is equipped with many instruments for studying different aspects ofthe Saturnian environment. Imaging devices take pictures in infra-red, visible,ultraviolet and even microwave wavelengths. A dust detector senses the microscopicdust grains of for example ice, that hits it. Spectrometers register the impactsof electrons, ions and neutrals and gives information on their energy spectra. Amagnetometer allows Cassini to measure the magnetic field, which has its sourceinside Saturn and permeates the ring system and plasma disc, which lies in theequatorial plane of the planet. Yet other instruments have antennas to pick upradio and plasma waves. The spectrum of such waves can provide informationabout the plasma, e.g. by observing resonant frequencies [2]. The upper hybridfrequency, for example, depends on the electron density and magnetic field strength,so that by measuring the magnetic field, the electron density can be determined. Ofparticular interest to this thesis is the Langmuir probe [3]. It consists of a sphere,mounted at the end of a boom which holds it out about 1.5 m away from Cassini.The sphere is biased to different potentials and, at the same time, the currentcollected by the sphere is measured. By sweeping the potential, the current-voltagecharacteristic of the probe is found, from which information about the plasma canthe be extracted.

1.3 The geysers of Enceladus

The moon Enceladus, orbiting Saturn at a radius of 4RS, is the main source ofmaterial for the E ring as well as for the neutral gas torus at a similar distancefrom Saturn. Enceladus contributes to the E ring the estimated 1 kg/s of matterthat is necessary to maintain it [5] through cryovolcanism. A large amount of theejecta is also recaptured by Enceladus as the orbits of the moon and the ejectaeventually cross. This manifests itself as plumes of, for example, water (the mainconstituent of the ring) that erupts from cracks in the surface of the moon. [4] Theplumes are faintly visible in figure 1.2 The cracks from which the plumes emanateare about 130 km long [4] and famously referred to as the “tiger stripes” (figure 1.4)due to their visual appearance.

4 CHAPTER 1. INTRODUCTION

Figure 1.3: Cassini during assembly. The large white disc antenna on top is fourmetres in diameter. Several booms and wire antennas, used for measurements, wereextended from the spacecraft once in space and are thus not visible here. On the leftside of Cassini, the Huygens probe can be seen with its gold-coloured, cone-shapedheat shield. (Image credit: NASA)

1.3. THE GEYSERS OF ENCELADUS 5

Figure 1.4: The “tiger stripes” on the surface of Enceladus. Through these cracks,Enceladus ejects the material that makes up most of the E ring. [Image credit:NASA]

Chapter 2

Sheaths

2.1 Basic principle of sheaths

When a plasma stands in contact with an object, the plasma particles will collidewith its surface and be collected by it. Such an object may be for example a wall,confining the plasma, or a probe, immersed in the plasma. As the particles arecollected by the surface, and thereby removed from the plasma, they contributetheir charge to the object and at the same time deprive the plasma of it. If theobject is made of a conducting material, its charges will redistribute over it so asto maintain a single potential throughout it. If on the other hand the material isan insulator, charges would rather stick close to where they impacted the surface.

Consider the situation where an infinite, conducting plane has just been broughtinto contact with an infinite plasma. If electrons and ions have the same tempera-ture, the electrons, due to being lighter, will have a much higher thermal speed thanthe ions. A situation where the ion temperature is much higher than the electrontemperature, so that the ion thermal speed can compete with the electron thermalspeed, is quite unnatural and is disregarded here. Because of their higher speed,the electrons are the first ones to collide with the wall and many of them will havedone so before the ions have move significantly at all. As electrons are lost fromthe plasma to the wall, they leave behind a net positive charge which causes theplasma to get a positive potential compared to the wall by typically a few electrontemperatures.

There is of course no step-like change in potential, when going from the plasmato the wall. Rather, the potential transitions smoothly in a region near the edgeof the plasma from its higher value in the bulk of the plasma to its lower valueat the wall. This region is called the sheath and extends a few Debye lengths intothe plasma. The potential gradient in the sheath region makes it more difficult forfurther electrons to reach the wall and the electron flux is therefore reduced. Atthe same time, it helps accelerate ions to the wall.

The usual way to model the densities is for electrons to rescale the background

7

8 CHAPTER 2. SHEATHS

density n0 with a Boltzmann factor,

ne = n0eeϕ/(kBTe), (2.1)

and for ions to use conservation of energy and the continuity equation, arriving at

ni = n0√1− 2eϕ

miv20

, (2.2)

where v0 is the flow speed of the ions as they enter the sheath. In the subsequentsolving for the potential from Poisson’s equation

d2ϕ

dx2 = −e(ni − ne)ε0

, (2.3)

with equations (2.1) and (2.2) in place for ne and ni, respectively, it turnsout any physically relevant solution requires v0 to be at least

√ekBTe/mi. The

acceleration of ions to this speed, called the Bohm speed, is accomplished by thepresheath region, located between the bulk plasma and the sheath. In the presheath,which can be much thicker than the actual sheath, there is thus a non-zero electricfield, but the potential drop across the presheath is much less than that across thesheath. A sketch of the behaviours of densities and potential in the bulk plasma,the presheath and the sheath is shown in figure 2.1.

2.2 Sheaths in different regimes

Objects like a dust grain, a probe and a satellite will all have a sheath around themwhen exposed to a plasma, though they will differ in their quantitative description.The precise shape of the sheath depends on the relationship between characteristicparameters such as the object’s characteristic dimension (which can be thought ofas the linear size of the region disturbed by the probe and is typically similar tothe size of the probe itself or a few times larger, depending on the probe shape [8]),the Debye length and the mean free path ` of the plasma particles.

We can categorise the different regimes, somewhat crudely, as in figure 2.2. Inthe present thesis, three types of objects that interact with the plasma – Cassini,its Langmuir probe and the dust particles of the E ring – are considered. In allthree cases, collisions can be neglected (i.e., l � d holds). The microscopic dustfurther fall well into the thick sheath regime, as it is much smaller than the Debyelength, which is of the order of 1 m or larger. This is also the case for the probe(whose radius is 25 mm. The Cassini spacecraft itself, however, is of the order ofa Debye length and the sheath can therefore neither be considered to be thick northin. This intermediate case is more difficult to treat analytically, which is whynumerical simulations are used for this problem in Paper III.

2.2. SHEATHS IN DIFFERENT REGIMES 9

Figure 2.1: Sketch of the sheath structure close to an infinite, conducting wall. (a)In the presheath, both electron and ion densities drop from their bulk values, butthey do not differ from each other (i.e., quasi-neutrality holds). In the sheath, theycontinue to decrease – the electron density more rapidly than the ion density –leaving the sheath with a positive charge density. (b) The larger part of the drop inpotential (in this figure called Φ) between the bulk plasma and the wall occurs inthe sheath, which therefore also represents most of the ion acceleration. However,the presheath acceleration alone is enough for the ions to reach the Bohm speed.(Image credit: Ref. [1])

10 CHAPTER 2. SHEATHS

Current collection regimes

Frequent collisions, ℓ ≪ d(fluid description applicable)

Collisionless, ℓ ≫ d

Thick sheath, λD ≫ d(OML sometimes applicable)

Thin sheath, λD ≪ d Neither thick nor thin sheath(analytical treatment difficult)

Figure 2.2: Some sheath regimes classified by the relations between characteristicprobe dimension d, mean free path ` and Debye length λD.

Chapter 3

Orbital motion limited model

The orbital motion limited (OML) model [7] describes the currents collected by anisolated body in a plasma by making use of the conservation of energy and angularmomentum of each electron and ion that approaches it. By isolated, we here meanthat other bodies are far enough away, or otherwise insignificant enough, that theirinfluence on the currents collected by the studied body is small. If a dust cloudis studied, for example, it should not be too dense if OML theory is to apply. Inits basic formulation, OML considers a spherical body with small a radius a� λDand also assumes that the mean free path of both ions and electrons are largeenough that neither of them undergo collisions on their way from the undisturbedbackground plasma to the body. Furthermore, it disregards the possibility of aneffective potential barrier or, equivalently, an absorption radius larger than theactual radius of the body. The effective potential is a concept that arises whenthe equations describing the three-dimensional motion of plasma particles in thepotential field around the body are reformulated into a one-dimensional version,whose only coordinate is the radial distance from the body [6]. This one-dimensionalmotion takes place in a potential field that is called the effective potential andmight set up a potential barrier outside the body, such that all plasma particlesthat are able to overcome this potential barrier are destined to be collected. Insuch a situation, there are no particles that barely miss the collecting surface, andinstead, the potential barrier acts as the absorption radius [7].

By making these assumptions, and using the sign convention that current leavingthe probe counts as positive, OML arrives at the following current contribution bya Maxwellian particle species with density n, charge q, mass m and temperature T[6]:

I(ϕ) ={I0(1− qϕ/(kBT )) qϕ < 0I0e−qϕ/(kBT ) qϕ > 0

(3.1)

11

12 CHAPTER 3. ORBITAL MOTION LIMITED MODEL

where ϕ is the potential of the body, relative to the plasma,

I0 = q√

8πa2nvT (3.2)

is the random current (collected by an uncharged body),

vT =√kBT

m(3.3)

is the thermal speed, a is the radius of the body and kB is the Boltzmann constant.The convention used here is that a current flowing to the body is considered positive.As seen from equation (3.1), the current has an exponential dependence on thepotential in the repulsive region (qϕ < 0) and a linear dependence (plus a constantterm) in the attractive region (qϕ > 0). This functional shape is illustrated infigure 3.1.

The equivalent current expression can also be constructed for a drifting Maxwelliandistribution with drift speed vT [6]. For attraction (qϕ < 0), this becomes

I(ϕ) = q√πa2nv2

T

vd

(√π

(1 + 2

(ξ2 + qϕ

kBT

))erf(ξ) + 2ξe−ξ

2), (3.4)

where ξ = vd/(√

2vT ), and for repulsion (qϕ > 0),

I(ϕ) =√πa2nv2

T

vd

(√π

(12 − ξ+ξ−

)(erf(ξ+)− erf(ξ−)) + ξ+e−ξ

2− − ξ−e−ξ

2+

),

(3.5)

where ξ± =√qϕ/(kBT )± vd/(

√2vT ).

3.1 Floating potential

If an uncharged body is placed in a plasma consisting of electrons and ions, it willat first collect an electron current that is larger than the ion current. When OMLapplies, this can be understood from equation (3.3), where the thermal speed vTwill be larger for electrons than for ions. As the body collects this negative charge,however, it gets driven negative in potential, which means that fewer electronsmanage to reach its surface and some extra ions are collected. This way, the bodypotential reaches a stable equilibrium, where the electron and ion currents canceleach other, by being equal in absolute value and opposite in sign. This potentialis called the floating potential ϕ0. In the general case with an arbitrary number ofplasma species, the definition of ϕ0 can be written∑

s

Is(ϕ0) = 0, (3.6)

3.1. FLOATING POTENTIAL 13

I (a)

0−kBT/e

probe bias ULP − Upl

dI/d

ULP

(b)

Figure 3.1: Current to a spherical probe or other object according to the OMLmodel (equation (3.1)) for a negative particle species. (a) The collected currentas a function of the probe potential. In the repulsive region (to the left of theplasma potential), the current depends exponentially on the potential, whereas inthe attractive region (to the right of the plasma potential), the dependence is astraight line. (b) The derivative of the curve in panel (a). Here, transition betweenthe repulsive and attractive regions are more easily seen, with a “knee” arisingwhere the probe is at the plasma potential.

where Is(ϕ) is the current contribution from species s. The cases studied in thisthesis involve two species: electrons and positive, singly charged ions. Furthermore,the floating potential is negative and with use of the OML model of equation (3.1),equation (3.6) becomes

e√

8πa2nvTi

(1− eϕ0

kBTi

)− e√

8πa2nvTeeeϕ0/(kBTe) = 0 (3.7)

or more simply

vTi

(1− eϕ0

kBTi

)= vTeeeϕ0/(kBTe). (3.8)

14 CHAPTER 3. ORBITAL MOTION LIMITED MODEL

−11 −10 −9 −8 −7 −6 −5 −4 −3

probe bias ULP − Upl (V)

−1

0

1

2

3

4

5I

(nA)

total current

electron current

ion current

photoelectron current

Figure 3.2: An example of how the currents collected by a spherical object accordingto the OML model depends on the probe potential. For an increasingly negativepotential, more and more electrons are unable to reach the surface of the object andthe collected electron current decays exponentially (with the electron temperatureas the decay constant). At the same time, the ion current increases linearly. Becausethe ion current is small and largely constant, compared to the dramatic variationsin electron current, the floating potential is found where the electron current hasbecome small like the ion current, i.e., at a few electron temperatures negative. Inthis example, a photoelectron current has also been included. It is constant fornegative potentials and thus has the same effect as stronger ion current.

Note that both ion and electron current are proportional to a2, which thereforedisappears from the equation. The floating potential is thus independent of the sizeof the body, which can be a useful property when one wants to estimate the floatingpotential of, say, dust grains based on knowledge about the floating potential of aspacecraft or a probe. Because the electron current depends exponentially on thepotential and the ion current only linearly, the floating potential will settle downat “a few electron temperatures negative”, i.e., ϕ0 ∼ −kBTe/e, for a wide range ofparameters. This rule of thumb can be invalidated if there are other currents alsocontributing to the balance. Such currents arise for example if the body is exposedto sunlight, producing photoelectrons, or is hit by energetic electrons, knocking

3.2. APPLICABILITY TO THE CASSINI LANGMUIR PROBE 15

out secondary electrons. Both of these currents drive a body more positive than itwould otherwise be and its floating potential can even become positive with respectto the ambient plasma.

In figure 3.2, three types of currents collected by a probe are plotted versusthe probe potential. In addition to the ordinary electron and ion currents, bothmodelled with OML, a photoelectron current has been included. Such a current ofphotoelectrons leaving the probe is constant for negative probe potentials as everyelectron that overcomes the work function leaves the probe surface.

A set of floating potentials of an object, experiencing ion collection, electroncollection and photoelectron emission has been tabulated in table 3.1, for differentdensities, temperatures, drift speeds and photoelectron currents. The calculationsapply for a sphere of radius 25 mm and this size, as well as the photoelectroncurrent, has been chosen to imitate the situation of the Cassini Langmuir probe.[3, 11] The first line in the table is the same case as plotted in figure 3.2 and theother lines deviate from this case in one parameter at a time.

n kBT vd Iph φfloat5× 107 m−3 3 eV 40 km/s 500 pA −8.2 V3× 107 m−3 3 eV 40 km/s 500 pA −7.5 V5× 107 m−3 2 eV 40 km/s 500 pA −5.1 V5× 107 m−3 3 eV 60 km/s 500 pA −7.6 V5× 107 m−3 3 eV 40 km/s 1 nA −3.5 V

Table 3.1: Floating potential (φfloat) calculations for a sphere according to OML,with the addition of a photoelectron current Iph. The plasma has density n anddrift speed vd. Ions and electrons share the temperature T . The first parametercombination listed here is also illustrated in figure 3.2, where we can see the de-pendence of the different currents on the considered object. The curve for the totalcurrent crosses zero when the object potential is −8.2 V, compared to the plasmapotential, as indicated in this table.

3.2 Applicability to the Cassini Langmuir probe

The Langmuir probe on Cassini belongs in such a parameter regime that OMLtheory could potentially be used to model its current-voltage characteristic. Forexample, it is much smaller than the Debye length and the plasma around it iscollisionless on the relevant length scale.

There are, however, circumstances that complicate this picture so that actualmeasured sweeps [9] do not follow OML in its unmodified form. Figure 3.3 showsan example of a sweep by the Langmuir probe, captured near a passage of the orbitof Enceladus at 2005-07-14 19:45:18. Though it is difficult to judge from lookingat the current curve, the derivative (constructed by taking the difference between

16 CHAPTER 3. ORBITAL MOTION LIMITED MODEL

consecutive points) reveals that it does not really follow plain OML. Whereas thederivative in OML has one knee, the measured curve could perhaps be said to haveseveral of them.

−40 −30 −20 −10 0 10 20 30 40−1

0

1

2

3

4x 10

−7 2005−07−14 19:45:18

Ubias

(V)

I (A

)

−40 −30 −20 −10 0 10 20 30 40−5

0

5

10

15

20x 10

−9

Ubias

(V)

dI/d

U (

S)

Figure 3.3: Current-voltage characteristic of the Cassini Langmuir probe, measurednear the orbit of Enceladus. From the derivative in the lower panel, it is seen thatthe curve deviates from the ideal OML model of figure 3.1. The glitch at −19 V ofthe derivative curve appears on many of the measured sweeps and is thought to bean instrumental defect.

The interpretation of this kind of curves has involved photoelectrons emittedfrom the probe as well as photoelectrons emitted from the spacecraft and capturedby the probe and has allowed for more than one electron population, shifted inenergy relative to each other. During the present work, however, it was concludedthat the precise shape of the potential structure around spacecraft and probe, wouldalso have a significant influence on the shape of the sweep curve, and should be takeninto account when interpreting the measurements. This is discussed in Paper II of

3.2. APPLICABILITY TO THE CASSINI LANGMUIR PROBE 17

this thesis.

Chapter 4

Particle-in-cell simulations

4.1 The PIC technique

Simulating a plasma in the straightforward way by keeping track of every particleand letting every particle exert a force on every other works in principle, but un-fortunately, the computational resources needed quickly becomes too large. A cubewith a side of a few metres, situated in the plasma disc of Saturn could contain 109

or more particles. In the laboratory, the relevant sizes are much smaller, but onthe other hand, the densities is much higher. Even worse, as every particle inter-acts with every other, the number of forces to calculate scales as the square of thenumber of particles. To reduce the computational effort required, other simulationapproaches are needed.

A fluid description of the plasma, with one fluid for each particle species, is onesuch approach. A fluid viewpoint typically requires collisions to be frequent (i.e.,the mean free path being short), though it can sometimes be substituted by someother condition that plays a similar role, such as plasma particles being tightly tiedto magnetic field lines (i.e., having a small gyroradius).

The simulations presented in this thesis do not involve frequent collisions. Quitecontrary, collisions are completely absent, so a fluid simulation is not applicable. In-stead, a technique called a particle-in-cell (PIC) simulation has been employed. [10]In a PIC simulation, the simulation space (which may be one-, two- or three-dimensional, as the situation requires) is discretised into a grid. In the presentwork, this grid is two-dimensional and regular, i.e., it consists of rectangles of equalsize. One can also use an unstructured grid, where the grid cells differ from eachother in shape and size. An unstructured grid can therefore use small grid cellswhere high resolution is important and large grid cells where the spatial variationof the studied quantities are slow anyway. Each time step, the charge of everyparticle is divided between the grid points that enclose the grid cell the particleresides in. For a regular grid, this means in one dimension, two grid points, in twodimensions, four grid points and in three dimensions, eight grid points. The charge

19

20 CHAPTER 4. PARTICLE-IN-CELL SIMULATIONS

is not distributed equally over those grid points, but more charge is given to gridpoints closer to the position of the particle. If the particle happens to be preciselyat a grid point, for example, all of its charge will be put there. See figure 4.1 for anillustration of the two-dimensional case. With all charge concentrated to the gridpoints, Poisson’s equation is recast into a difference equation, which is then solvedto find the electric potential φ, but again only on the grid points. On a regular gridin two dimensions, this difference equation is

φi−1,j − 2φi,j + φi+1,j

∆x + φi,j−1 − 2φi,j + φi,j+1

∆y = ρi,j , (4.1)

where i and j enumerate the grid points, ∆x and ∆y are the separations betweenadjacent grid points in the x- and y-directions, respectively, and ρ is the chargedensity. Finally, the force on each particle is then found from the potentials of itsneighbouring grid points, and its velocity and position are updated.

grid points

A B

C D

A

D

B

C

simulationparticle

Figure 4.1: A grid cell of a two-dimensional PIC simulation. The charge (andmass) of the simulation particle is distributed over the four grid points that are thecorners of the grid cell in which the particle resides. More charge and mass goesto the closer corners. Specifically, the portion of the particle ascribed to each gridpoint is proportional to the area of the region with the same label, in this figure,as the grid point. Thus, in the illustrated example, most of the charge and mass isgiven to grid point B and C gets the least.

4.2. PIC SIMULATIONS OF CASSINI 21

To reduce the memory usage and the number of computations needed in thesimulation, it is also often necessary to lump several particles (of the same species)together by collecting their masses and charge into a single superparticle and simu-late such particles instead. Though being heavier and more strongly charged, theyretain the all-important charge-to-mass ratio. The advantage of having charge andpotentials only on the grid points, however, remain the strongest point of PIC.

4.2 PIC simulations of Cassini

The work presented in this thesis (in Paper III) employ particle-in-cell simulationsthat has been designed to be relevant for the situation where Cassini crosses plasmadisc of Saturn at the orbit of Enceladus, which is embedded in the E ring. Thesimulation code allows for introducing an object (representing Cassini, in our case)into the simulation box and simulate an electron species and an ion species. [12,13] Each species follows a drifting Maxwellian as their ambient distribution. Theelectrons and ions can be given different temperatures and drift speeds, though inthe present work, they are set to be equal for the two species.

Throughout the duration of the simulation, new particles are injected at theboundary of the simulation box. A particle is followed until (1) it leaves the sim-ulation box by crossing its boundary or (2) it hits the surface of the object and iscollected by it. In both cases, the particle is deleted from the simulation. The po-tential of the object itself is determined by the charge it collects. To self-consistentlyallow the potential of the object to develop and settle at the proper floating po-tential, different approaches can be used depending on whether or not the object isinsulating or conducting. For an insulating object, a simple way is to let the chargesstay at the point where they struck the surface. However, Cassini, like other space-craft, are rather to be seen as conducting and thus the charge it collects should beredistributed over its surface so as to maintain a single potential throughout thebody.

The redistribution of charge over a conducting body is non-trivial and therefore,the code used in this work solves the problem of finding the spacecraft potential ina different way, using the fact that at the floating potential, the current collected byan object in a plasma is zero. The simulation is first run with the object potentialfixed to some value, and the amount of collected charge is kept track of. At the endof this simulation run, the sign of the net charge collected by the object is studied.If the charge is positive, it is concluded that the initial guess for the potential wastoo negative and vice versa. The code then fixates the object at a new potential fora second run and in this way performed a binary search for the floating potentialuntil it has been locked into a small enough interval.

The majority of these simulations are made in two dimensions, rather than three,which greatly reduces the computing time. By these means, several situations,characterised by different combinations of ambient density, temperature and plasmadrift speed are studied. One central parameter combination is also run as a 3D

22 CHAPTER 4. PARTICLE-IN-CELL SIMULATIONS

simulation, which is then compared to its two-dimensional counterpart. We verifythat the 2D setup gives results similar to those of the more trustworthy 3D case,and conclude that the 2D version is relevant for the study. In 2D, the complicatedgeometry of Cassini is represented by a disc-shaped object, and in 3D, by a sphere.

The resulting quantities of interest, calculated by both the 2D and 3D sim-ulations, are electron density, ion density and electric potential, as functions ofthe position within the simulation box. Figures 4.2 to 4.6 exhibits the potentialstructure for five parameter combinations, illustrating how each parameter affectsthe situation. In all cases, the object has a diameter of 3.5 m and is placed at(x, y) = (0, 0). The object is stationary and the plasma flow is in the positivex-direction. The entire simulation box is not shown – only the region where asignificant perturbation takes place.

The reference case, whose potential we find in figure 4.2, is defined as havingambient density n0 = 7× 107 m−3, temperature kBTe = kBTi = 2.5 eV and plasmadrift speed vd = 30 km/s. This is the parameter combination that has also beensimulated in three dimensions, as presented in Paper III. From these parameters wesee that the directed energy of an ion is 84 eV, i.e., much larger than the thermalenergy, but the directed energy of an electron is a negligible 0.0026 eV.

As noted in Paper III, the potential structure resembles an ordinary Debyeshielding, but with its downstream part pulled out into a tail. The length of the tailvaries clearly, as can be expected, with drift speed. For the lower drift of 12 km/s,it almost vanishes. The “head” part of the shielding, i.e., that which is not the tail,changes size with the Debye length as either the density or temperature is varied.Note also that the floating potential is different between the different cases, beingparticularly sensitive to the temperature.

4.2. PIC SIMULATIONS OF CASSINI 23

−5 0 5 10 15 20

−6

−4

−2

0

2

4

6

−7 −6 −5 −4 −3 −2 −1 0

Figure 4.2: Resulting potential structure from a 2D PIC simulation with the plasmaflowing from left to right. The disc representing Cassini is 3.5 m in diameter. Thedefining parameters of this simulation case are n0 = 7 × 107 m−3, kBTe = kBTi =2.5 eV and vd = 30 km/s. This parameter combination is here used as the referencecase, to be compared with the other simulations, where these three parameters arevaried, one at a time. Figures 4.2 to 4.6 all depict the same region, though theirscales differ due to their different Debye lengths.

24 CHAPTER 4. PARTICLE-IN-CELL SIMULATIONS

−10 −5 0 5 10 15 20 25 30

−10

−5

0

5

10

−2 −1 0

Figure 4.3: Potential from a simulation case with the same density and drift speedas in figure 4.2, but with a lower temperature kBTe = kBTi = 1 eV.

−5 0 5 10 15−5

0

5

−8 −7 −6 −5 −4 −3 −2 −1 0

Figure 4.4: Potential from a simulation case with the same temperature and driftspeed as in figure 4.2, but with a lower density n0 = 3.5× 107 m−3.

4.2. PIC SIMULATIONS OF CASSINI 25

−5 0 5 10 15 20

−6

−4

−2

0

2

4

6

−9 −8 −7 −6 −5 −4 −3 −2 −1 0

Figure 4.5: Potential from a simulation case with the same density and temperatureas in figure 4.2, but with a lower drift speed vd = 12 km/s.

−5 0 5 10 15 20

−6

−4

−2

0

2

4

6

−6 −5 −4 −3 −2 −1 0

Figure 4.6: Potential from a simulation case with the same density and temperatureas in figure 4.2, but with a higher drift speed vd = 54 km/s.

Chapter 5

Magnetospheric plasma

5.1 Parallel currents driven by perpendicular currents

A charged partile in motion in a magnetic field will follow a curved trajectory, ratherthan a straight line. In particular, if the field is homogeneous and the velocity isperpendicular to it, the trajectory will be a circle.

The radius rg of this circle is called the gyro radius and is given by

rg = mv⊥|q|B

, (5.1)

where m is the particle mass, v⊥ is its speed perpendicular to the field, q is itscharge and B is the strength of the magnetic field. When the gyro radius of aspecies of plasma particles in a magnetic field is small, compared to the lengthscales of interest, the species is said to be magnetized. As seen from equation (5.1),this happens to a larger extent for particles with a larger charge-to-mass ratio |q|/m.Also a magnetized particle is, however, free to move along the magnetic field. Avelocity component parallel to a homogeneous field thus gives a helical, rather thancircular, trajectory.

Despite being strongly charged, dust has a much lower |q|/m than the ions andelectrons of the plasma. If a dust grain were to have the same charge-to-massratio as a plasma ion, it would have to consist entirely of ions, assuming that thedust consists of the same material as the free ions. The small electron mass givethe electrons an even larger charge-to-mass ratio. Dust is seen as unmagnetizedthroughout the present thesis.

Dust, moving freely across the magnetic field, carries its charge with it and thusdrives cross-field currents in a way the plasma do not. If the dust charge densityis not the same everywhere, this leads to a build-up of space charges. A spatiallyvarying dust-charge density may be due to a varying dust number density or varyingdust-charging conditions (e.g. ne and Te). The plasma, being magnetized and tiedto the magnetic field lines, cannot follow the dust motion to neutralize the space

27

28 CHAPTER 5. MAGNETOSPHERIC PLASMA

charges. However, because they are free to move along the magnetic field, a regionof, say, positive charge will attract elections along the magnetic field and repel ions.

Figure 5.1 illustrates this for the case of a finite dust slab (i.e. non-homogeneousdust density) moving across a magnetic field. At the leading edge of the slab, thenegative dust overlaps with the neutral plasma, creating a negative space charge.At the trailing edge, the negative dust has withdrawn from the previously neutraldust-plasma combination, leaving a positive space charge behind. Currents flowaway from the positive region, along the field lines, through some distant load andto the negative region. A forced current across the magnetic field, here due todust-carried charge, can thus set up parallel currents.

This effect is central in Paper IV, where it is the mechanism that makes the dustring in the equatorial plane drive currents along magnetic field lines and throughthe ionosphere.

++

--

Figure 5.1: Parallel currents caused by spatial variations in dust density. A nega-tively charged dust slab (shaded) moves with speed vd relative to a plasma (heredepicted in the plasma rest frame). Positive and negative charge densities are cre-ated at the trailing and leading edge, respectively, where the density gradient alongthe direction of motion is non-zero. These drive field-aligned currents which closeacross the magnetic field in some distant load Σload.

5.2. COROTATION 29

5.2 Corotation

Magnetospheric corotation [14] is a central concept in the picture of Saturn studiedin Papers IV and V. It refers to a magnetospheric plasma in rigid motion with theplanet it belongs to.

For corotation to take place, it is necessary for the neutrals in the atmosphereof the planet to exchange momentum, through collisions, with the plasma particlesof the ionosphere. That way, the ionosphere is dragged along with the rotation ofthe atmosphere, which is assumed to follow the rotation of the rest of the planet.Furthermore, the field lines of the magnetosphere need to be frozen-in in the plasma.This enables momentum transfer to propagate from the ionosphere out to the entiremagnetosphere (as far as the field is frozen-in).

The rotating motion v of the plasma across the magnetic field B corresponds toan electic field E = −v×B and the motion can be seen as a drift due to this field.Because the plasma moves in a circular orbit, there is also a centrifugal force thatacts upon it. It is stronger for the ions than for the electrons, due to their differencein mass. This force too contributes a (much smaller) drift in the azimuthal directionthat constitutes a net current (a ring current), due to ions and electrons driftingwith different speeds.

When the magnetospheric plasma is in rigid rotation with the planet and thereis no force acting to slow it down, no currents flow between the ionosphere andthe magnetosphere. However, when the magnetospheric plasma experiences a dragfrom, for example, friction against neutrals or pick up of newly created ions, field-aligned currents start to flow and connect the planet with its magnetosphere. Infigure 5.2, a current system of this kind is illustrated.

30 CHAPTER 5. MAGNETOSPHERIC PLASMA

+

+

+

+

+++

+

--

-

--

-

- -

+-

ionizationregion

(a)

(b)

+ -

Figure 5.2: Radial current in a corotating magnetosphere. The radial current in theequatorial plane, due to, for example, pickup of new ions, closes via the field linesand the ionosphere, and transfers momentum from the planet to the magnetosphere.

Chapter 6

Results and discussion

6.1 Paper I

The data obtained by the Cassini Radio and Plasma Wave Science (RPWS) in-strument during the shallow (2005-02-17) and the steep (2005-07-14) crossings ofthe E ring revealed a considerable electron depletion in proximity to Enceladus’sorbit (the difference between the ion and electron densities can reach ∼ 70 cm−3).Assuming that this depletion is a signature of the presence of charged dust particles(i.e., that the missing electrons have been captured by the dust grains), the maincharacteristics of dust down to sub-micron sized particles are derived. Assuming apower law size distribution, with a lower size limit amin, the index is found to beµ ∼ 5.5− 6 for amin = 0.03µm and µ ∼ 7.3− 8 for amin = 0.1 µm. The calculatedaverage integral dust number density is weakly affected by values of µ and amin,though proportional to ni − ne. For a ∼ 0.1 µm, both flybys gave the maximumdust density about 0.1−0.3 cm−3 in the vicinity of Enceladus. These results implythat the dust structure near Enceladus is characterized by a vertical length scaleof about 8000 km.

6.2 Paper II

If a Langmuir probe is located inside the sheath of a negatively charged spacecraft,the potential U1 at the probe is different from the ambient plasma potential Upland the probe characteristic can become strongly modified. We have constructeda simplified model to study this probe-in-sheath problem in the parameter rangeof a small probe (with radius rLP � λD) where the orbit motion limited (OML)probe theory usually applies. We model the spacecraft and the probe as spheresat different potentials and use the resulting potential structure for reasoning aboutthe electron collection by the probe. The potential, according to this model, alongthe common axis of the spacecraft and the probe is illustrated in figure 6.1. Wepropose that the probe characteristics I(ULP) is suitably analysed in terms of three

31

32 CHAPTER 6. RESULTS AND DISCUSSION

position

U1

UM

ULP

USC

Upl

potential

2rSC ℓboom

2rLPwithout probe

with probe

(c)

Figure 6.1: The model of the potential structure used in Paper II, plotted alongthe common axis of the spacecraft and the probe. The minimum UM is proposedto play an important role for the electron collection by the probe.

regions of applied probe potential ULP. In region I, defined by ULP < U1, the curveis close to OML theory using the local density ne1 and the ambient temperature Te.In a transition region II, U1 < ULP < U2, the key factor to determine the shape ofI(ULP) is the depth of a local potential minimum UM, located radially straight outfrom the probe, that acts as a potential barrier and prevents low energy electronsfrom the ambient plasma to reach the probe. In region III, finally, ULP > U2,the depth of this barrier is small (UM � kBTe/e) and I(ULP) again approachesOML theory, now with the ambient ne0 and Te. The main finding in this workis that the probe characteristic in the transition region II departs fundamentallyfrom OML in the following respects: (1) there is an extended region of negatived3I/dU3

LP, a feature which can be used to identify such a region II, (2) althoughthe ambient plasma potential Upl falls into region II, there is no obvious way toidentify it, (3) there is no exponential part of I(ULP) that can be used to obtainTe, instead (3) the curve shape depends strongly on the probe size in a way that,for reliable quantitative evaluation, remains to be understood and separated fromthe dependence on ne0 and Te.

The dependence on probe size is such that in region II the slope dI/dULP de-creases and U2 increases with decreasing rLP. We have tentatively applied oursimplified probe-in-sheath model to real probe data from the Cassini spacecrafttaken in the dense plasma of the Saturnian magnetosphere. We propose that it

6.3. PAPER III 33

can reproduce probe characteristics better than OML, but not good enough fordetailed quantitative results, e.g. such as the identification of bi-Maxwellian elec-tron distributions. For more accurate results, models are needed that include arealistic, self-consistently obtained, potential structure around the spacecraft, howit is modified by a variable probe potential, and a better understanding (proba-bly through particle simulations) of the electron collection to the probe in such apotential structure.

6.3 Paper III

We present the results of numerical simulations of potential structure around an ob-ject in a streaming plasma for parameters relevant for the Cassini spacecraft passingthrough the plasma disc of Saturn (orbit of Enceladus). Two- and three-dimensionalparticle-in-cell codes have been used allowing the potential of the simulated space-craft body to develop self-consistently through the charge it collects. Dependenceof the wake structure on plasma density, electron temperature and ion drift speedis discussed. Implications of the results to the problems of dust charging as well asinterpretation of Cassini Langmuir probe characteristics in the vicinity of Enceladusis pointed out.

6.4 Paper IV

General equations for dust-driven currents and current systems in magnetized plas-mas have been derived and, as a concrete example, applied to the E ring of Saturnat radial distances 3RS < R < 5RS. In this environment dust-driven currents arefound to be maintained by dust with radii of approximately a > 10 nm.

An azimuthal current JD,ϕ is carried by negatively charged dust, orbiting at theKepler speed, in its motion relative to the plasma component. This JD,ϕ acts asa current generator and is coupled to two secondary dust-driven current systemswhich both rotate with the magnetospheric plasma. In the first of these, azimuthalvariations in JD,ϕ drive magnetic-field-aligned currents currents JD,||, down to op-posite sides of the ionosphere of Saturn, in which they close as Pedersen currentsJD,PC across the polar cap. In the other secondary current system, plasma E/Bdrifts in a two-cell potential pattern in the equatorial plane create spatially limitedregions with radial dust currents JD,R. The edges of such regions drive opposedpairs of field-aligned current down to the ionosphere where they close, not acrossthe polar cap in this case, but over limited ranges in latitude.

These three dust-driven current systems are embedded in three systems ofplasma-driven currents Jp. First, a centrifugally driven ring current Jp,ϕ overlapsthe ring current JD,ϕ inside the E ring but also extends further away from Saturn.Second, divergences in Jp,ϕ drive a pair of field-aligned currents closing across thepolar cap as a current Jp,PC. This current system overlaps and coincides with theone that is driven by JD,ϕ. Third, the pickup to corotation speed of ionized matter

34 CHAPTER 6. RESULTS AND DISCUSSION

in the neutral gas torus gives radial plasma currents Jp,R and another field-alignedcurrent system. This has similarities to the second type of secondary dust-drivencurrents, but has a different azimuthal dependence.

The dust-driven, and plasma-driven, current systems described above have beenquantitatively assessed from a data set for the E ring of Saturn. The main uncertainparameter in this data set is the amount, and the size distribution, of small dust(a < 900 nm). This has been treated by a power law extrapolation dnD/da = a−µ

from the known distribution of larger dust, using the same exponent µ = 4.5 butwith a lower size limit amin that is unknown and to be determined. From magneticdata on the ∆BR perturbations during a crossing of the equatorial plane, the sumJD,ϕ+Jp,ϕ has been estimated. This, together with Jp,ϕ estimated separately fromplasma parameters, gives an approximate constraint on the fraction of the electronsthat can be trapped on the dust as ∆ne/ne < 10−4.

Assuming the power law exponent µ = 4.5, the value corresponds to a smallsize cutoff of the dust distribution at a = 300 nm. This limiting case is used as areference case for estimates of the possible strengths of the three dust-driven currentsystems described above. (It should be noted that the small dust (300 nm < a <900 nm) here represents only 40% of the total dust mass density, and consequentlyno large ad hoc assumed extra mass ejection from Enceladus is required to explainits presence.) The dust ring current JD,ϕ is for the reference case found to beof the same order as the plasma-driven Jp,ϕ, although only a fraction ∆ne/ne ≈10−4 of the electrons in the E ring are trapped on the dust. This demonstratesthe important fact that already a small mixing of dust into a plasma can have asignificant impact on the total currents driven.

The azimuthal variations in the plasma density causes overlapping dust-drivenand plasma-driven transpolar current systems. For the reference case the dustcontribution in this current loop is only about 3%. However the trends with plasmadensity are opposite: a decrease in ne would decrease Ip,PC and increase ID,PC. Areduction by a factor of 10 in ne would make them of the same order, ID,PC ∼ Ip,PC.

Regarding the radial currents in the equatorial plane, the dust-driven componentJD,R is an order of magnitude larger than the pickup plasma current, but bothprobably are below the limit where they can be resolved within the 3 nT variationstypical in this region.

6.5 Paper V

The problem of explaining the asymmetrical distribution of magnetospheric plasma [15]around Saturn is approached by proposal of a plasma dynamo related to the neutralgas torus along the orbit of Enceladus. The dynamo involves a convective plasmaflow that passes through the neutral gas torus where ionization is higher than out-side the torus. During its transit through the neutral torus, neutrals are ionised sothat the plasma density increases from n1 to n2, giving n2 = n1ettr/τi , where ttr isthe transit time and τi is the time constant for ionisation. During the remaining

6.5. PAPER V 35

part of the convective flow loop. Part of the plasma is lost to the solar wind andthe density of the plasma element we follow can be seen as having decreased again.

The proposed dynamo model connects the two-cell convective pattern put for-ward by Gurnett et al. [15] with the “plasma tongue” model by Goldreich andFarmer [16], which describes a plasma loss to the solar wind as a narrowing tongue ofstreaming plasma. The primary parameters to the model are the height-integrated,ionospheric conductivity ΣP in the polar caps of Saturn, the fraction Kret of out-flowing plasma that returns as inflow for another loop in the convection pattern(i.e. the fraction that is not lost to the solar wind), and the ionization time con-stant τi. The response of the model to variations in these parameters are tested.Slow, seasonal variations in ΣP are expected due to the changing solar angle to thepolar caps. On a time scale of weeks or months (i.e. faster than the ΣP variations)we expect that the varying injection rate of Enceladus may change the neutral gasdensity and thereby τi. Even faster, on the time scale of a Saturnian rotationalperiod 10.8 h, Kret may fluctuate due to interaction between the plasma tongueand the solar wind.

Chapter 7

Conclusions

7.1 Papers I to III

A object immersed in a plasma is a situation that arises in many places throughoutthe solar system. Objects of different sizes cause different effects in the plasma.A large object in a streaming plasma creates a wake of depleted plasma densityand perturbed potential downstream of itself. By attracting ions, the object canalso focus them into a local region of higher-than-ambient density. Small objectsare not without influence either, though. Even a point-like charge in a streaming,supersonic, plasma effects a Mach cone, for example.

In Paper I, a method to detect the presence of dust through measurements ionand electron densities was explored. In this case, the micron-sized, or smaller, dustacts as the object which is thus much smaller than the Debye length.

In Paper II, a Langmuir probe on a spacecraft was considered and the effectsof the potential structure around spacecraft and probe illustrated with a simpleanalytical model. This model was used to generate the expected current-voltagecharacteristics of such a probe. In this case, the probe falls into the same regimeas the above dust, due to being much smaller than the Debye length. However, itwas concluded that the precise size of the probe is still important for the shape ofits current-voltage characteristic.

In Paper III, numerical simulations of an object representing the Cassini space-craft were presented. The spacecraft diameter is several times the Debye length,and therefore in a regime where analytical calculation of for example its floatingpotential is difficult.

7.2 Papers IV and V

Large-scale structures at Saturn –magnetospheric current systems and a convectionpattern – have been proposed based on observations by Cassini instruments as wellas on theoretical considerations.

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38 CHAPTER 7. CONCLUSIONS

In Paper IV, all three studied types of dust-driven currents are within some-what more than an order of magnitude from the strength of the correspondingplasma-driven currents. Depending on the component considered they are lowerthan, equal to, or larger. Considering that both plasma and dust densities varywith the geyser activity at the south pole of Enceladus, it is concluded that boththe dust-driven and the plasma-driven contributions to the current system associ-ated with the E ring need to be retained for a complete description.

The calculations of dust current in the present article involves uncertain pa-rameters, especially with regard to the dust size distribution. Improved knowledgeabout dust sizes and charging would directly enable more accurate current cal-culations. In addition, refined magnetic field measurements can establish tighterupper bounds for the dust-carried charge density which in turn limits the possibleamount of charged dust that is too small to be detected by the dust detectors onboard Cassini. The different profiles of dust and plasma across the ring plane shouldmake it possible to discern between the contributions to ∆B from the plasma andthe dust-driven currents Jp,ϕ and JD,ϕ.

In Paper V it is demonstrated that a rather inflexible model (i.e. havingfew degrees of freedom), and whose parameters are furthermore locked to observeddata, can reproduce the asymmetric distribution of plasma density. Without fur-ther changing its parameters, it also leads to values for the ionization time constantand mass outflow to the plasma tongue that are consistent with independent treat-ments [17, 16].

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