Modeling magnetospheric fields in the Jupiter system

TL;DR

This paper presents new numerical models of Jupiter's and Ganymede's magnetospheres, revealing insights into their magnetic field generation, interactions, and the presence of subsurface oceans, with implications for understanding planetary magnetospheres.

Contribution

The paper introduces advanced numerical models that quantitatively analyze the magnetospheric processes of Jupiter and Ganymede, highlighting differences and evidence for subsurface oceans.

Findings

Jupiter's magnetosphere is strongly rotationally driven with large plasma sources.

Ganymede's magnetosphere develops Alfvén wings and influences auroral oscillations.

Evidence suggests Ganymede has a subsurface ocean based on auroral oscillation analysis.

Abstract

The various processes which generate magnetic fields within the Jupiter system are exemplary for a large class of similar processes occurring at other planets in the solar system, but also around extrasolar planets. Jupiter's large internal dynamo magnetic field generates a gigantic magnetosphere, which is strongly rotational driven and possesses large plasma sources located deeply within the magnetosphere. The combination of the latter two effects is the primary reason for Jupiter's main auroral ovals. Jupiter's moon Ganymede is the only known moon with an intrinsic dynamo magnetic field, which generates a mini-magnetosphere located within Jupiter's larger magnetosphere including two auroral ovals. Ganymede's magnetosphere is qualitatively different compared to the one from Jupiter. It possesses no bow shock but develops Alfv\'en wings similar to most of the extrasolar planets which orbit their host stars within 0.1 AU. New numerical models of Jupiter's and Ganymede's magnetospheres presented here provide quantitative insight into the processes that maintain these magnetospheres. Jupiter's magnetospheric field is approximately time-periodic at the locations of Jupiter's moons and induces secondary magnetic fields in electrically conductive layers such as subsurface oceans. In the case of Ganymede, these secondary magnetic fields influence the oscillation of the location of its auroral ovals. Based on dedicated Hubble Space Telescope observations, an analysis of the amplitudes of the auroral oscillations provides evidence that Ganymede harbors a subsurface ocean. Callisto in contrast does not possess a mini-magnetosphere, but still shows a perturbed magnetic field environment. Callisto's ionosphere and atmospheric UV emission is different compared to the other Galilean satellites as it is primarily been generated by solar photons compared to magnetospheric electrons.

Equations

$${\partial }_t{\rho }_m+\nabla \cdot ({\rho }_m\vec{v})$$

$${\rho }_m{\partial }_t\vec{v}+{\rho }_m\vec{v}\cdot \nabla \vec{v}$$

$${\partial }_t\vec{B}$$

$${\partial }_tϵ+\nabla \cdot (ϵ\vec{v})$$

$$t_{lm}^{\ast }(R_0,t)=0,$$

$$R_{0}\frac{\partial p^{*}_{lm}(r,t)}{\partial r}\bigg{|}_{r=R_{0}}-(l+1)p^{*}_{lm}(R_{0},t)=-\frac{2l+1}{l}G^{*}_{lm}(t).$$

Full text

1 Introduction

The plasma interactions in and around Jupiter’s magnetosphere and around Jupiter’s moons are so rich in various phenomena that a huge class of interactions occurring at other planetary bodies in the solar system and at extrasolar planets are represented by the processes in the Jupiter system. In this chapter, we will present results of new models of the interaction of Jupiter’s magnetosphere with the solar wind and interactions at the Galilean moons with particular focus on Ganymede and Callisto.

Jupiter is the largest planet with the largest magnetic moment in the solar system. It’s interaction with the solar wind generates a planetary magnetosphere, which would appear to an observer on Earth larger compared to how the sun appears to us in the sky if Jupiter’s magnetosphere could be seen with the naked eye. The four large Galilean moons, Io, Europa, Ganymede and Callisto are all located within Jupiter’s gigantic magnetosphere, and are subject to the interactions generated by Jupiter’s magnetospheric plasma. The interaction at these moons differ due to the different properties of the moons. For example, Ganymede possesses an internal dynamo magnetic field which leads to its own mini-magnetosphere within Jupiter’s magnetosphere. The other moons in contrast only possess weak internal magnetic fields generated by electromagnetic induction in electrically conductive layers.

In the next section 2 we categorize the plasma interactions in Jupiter’s magnetosphere based on various parameters. With this background we then describe the interaction of the solar wind with Jupiter in order to better understand Jupiter’s magnetosphere (see section 3). In the subsequent section 4 we investigate time-variable effects in Jupiter’s magnetosphere, which can be subdivided in periodic and non-periodic variabilities. The associated time-variable magnetic fields induce secondary magnetic fields in electrically conductive layers within the moons. Measurements of these induced fields are diagnostic of internal layers, such as saline subsurface oceans. The plasma of Jupiter’s magnetosphere interacts with the atmospheres and ionospheres and the interior of the moons. Properties of this interaction, the formation of the ionospheres and the generation of related magnetic fields will be discussed for Ganymede and Callisto in sections 5 and 7. A novel technique to search for induced magnetic fields from a subsurface ocean within Ganymede based on Hubble Space Telescope observations of its auroral ovals is described in section 6.

2 Characterization and description of the interaction

The plasma interactions of flows past planetary bodies can be characterized into different classes, which we discuss in the next subsection. Afterwards we provide an overview of the magnetohydrodynamic (MHD) approach to describe these interactions.

2.1 Overview of the interaction

The interaction of a planetary body with its surrounding plasma is controlled by two factors: (1) the properties of the plasma flowing past the planetary object (discussed in section 2.1) and (2) the properties of the planetary body itself (discussed in section 2.1). Other overviews of the plasma interaction at Jupiter and its moons can be found, e.g., in neub98; kive04 or krup04.

Mach numbers and nature of interaction

A key property which controls the interaction is the ratio of the relative bulk flow velocity $v_{0}$ between the plasma and the planetary object compared to the group velocities of the three magnetohydrodynamic waves. These ratios are called the fast Mach number $M_{f}$, the Alfvén Mach number $M_{A}$, and the slow mode Mach number $M_{s}$ and refer to the ratios of the flow velocity to the the fast magneto-sonic mode, the Alfvén mode and the slow magneto-sonic mode, respectively (e.g., baum96). If the fast mode is larger than 1, a bow shock forms ahead of the object for nearly all object classes. Exceptions are inert moons, i.e., without atmosphere and intrinsic magnetic fields. For $M_{f}<1$, the bow shock disappears which is always the case if $M_{A}<1$, i.e., if the flow is sub-Alfvénic.

The interaction of all planets in the solar system with the solar wind is such that under average conditions the solar wind flow is super-fast, i.e., $M_{f}>1$ and all planets are surrounded by a bow shock. Only exceptionally, i.e., approximately once every two years, the plasma density in the solar wind is so low that the resultant Alfvén velocity $v_{A}=B/\sqrt{\mu_{0}\rho}$ is faster than the solar wind velocity with the mass density $\rho$ and the magnetic permeability of free space $\mu_{0}$. In this case, the Earth looses its bow shock chan12; chan15. This transition has however never been observed for Jupiter. In the case of Jupiter’s moons, which are embedded within Jupiter’s magnetosphere, the relative flow velocity is smaller than the Alfvén velocity and thus no bow shock forms, but so called Alfvén wings develop (neub80; goer80; sout80). Alfvén wings are standing Alfvén waves in the restframe of the moons. The Alfvén waves are generated because the obstacle slows the flow of the magnetized plasma in its vicinity and generates stresses in the magnetic field. The Alfvén wings can be described as ’tubes’ in direction parallel and anti-parallel to the magnetic field but with additional tilts by an angle $\Theta_{A}\approx\tan^{-1}M_{A}$ with respect to the magnetic field neub80.

Extrasolar planets are observed to orbit their host stars at a wide range of radial distances from approx 0.01 AU to 1000 AU. Because the stellar plasma properties are expected to strongly evolve as a function of distance similar to the solar wind properties of the sun (e.g., park58; preu06; lanz08), the plasma conditions around the observed extrasolar planets are expected to vary strongly. In Figure 1,

we show the expected Alfvén Mach number $M_{A}$ near the 850 extrasolar planets known until 2013 (from saur13). The Mach number is calculated based on measured and estimated properties of the host stars and by applying the park58 model for the radial evolution of the stellar winds. Figure 1 shows that extrasolar planets at orbital distances approximately less than 0.1 AU are typically subject to sub-Alfvénic conditions. At these bodies no bows shock but Alfvén wings form. In case such an extrasolar planet possesses a dynamo magnetic fields as expected chri09, then the solar system analogue of these extrasolar planet is Ganymede. In case they do not possess a dynamo field, their interaction is qualitatively similar to Io, Europa, and Callisto. This comparison only relates to the sub-Alfvénic character of the interaction and the principle nature of the obstacle. The atmospheres of the close-in exoplanets however can be significantly different compared to those of the Galilean satellites. The interaction with these atmospheres and the planetary magnetic field is expected to generate plasma and magnetic field perturbations qualitatively similar to the moons of Jupiter. If the exoplanets possess electrically conductive layers, any time-variable external magnetic field will additionally induce secondary magnetic fields similar to the mechanisms at the Galilean satellites. Extrasolar planets at stellar separation larger than approximately 0.1 AU are on average subject to super-Alfvénic conditions ($M_{A}>1)$ and thus are expected to possess bow shocks if additionally $M_{f}>1$ similar to Jupiter. Thus the interactions in the Jupiter system are text book cases for the interactions at extrasolar planets with the benefit that a huge set of in-situ and remote-sensing observations are available. Thus an improvement of our understanding of the plasma and magnetic field environments in the Jupiter system is helpful as this understanding is also relevant for extrasolar planets.

The planetary body: Nature of the obstacle

The nature of the planetary body, i.e., the obstacle to the flow, shapes the plasma and magnetic field environment in particular very close to the planetary body. The body can be a mechanical obstacle, e.g., due to collisions of the neutral particles in the atmosphere with the plasma or if the body possesses a solid surface. The body can also be an electromagnetic obstacle, e.g., if it possesses an internal magnetic field or if it is electrically conductive. The Galilean satellites are a mix of both mechanical and electromagnetic obstacles. They all possess very dilute atmospheres and solid, plasma absorbing surfaces. Among the Galilean satellites, Ganymede is the strongest electromagnetic obstacle because it possesses an internal dynamo magnetic field, which generates a mini-magnetosphere within Jupiter’s large magnetosphere (kive96b). But electrically conductive layers within all of the moons, e.g. saline oceans, metallic cores or a possible magma ocean, generate induced magnetic fields, which influence the external space and plasma environment around all of theses moons (e.g., khur98; neub98; khur11; seuf11).

2.2 MHD model

The most common approach to describe the interaction of a planetary body within its surrounding plasma is the MHD approach. It describes the plasma as an electrically conductive fluid and is applicable to describe the overall properties of the interaction if typical length scales of the interaction are larger than the ion gyro radius and typical time scales are larger than the ion gyro period.

The MHD approach applies to describe the temporal and spatial evolution of the mass density $\rho_{m}$ the continuity equation, for the plasma bulk velocity $\vec{v}$ the velocity equation, for the magnetic field $\vec{B}$ the induction equation, and an equation for the internal energy $\epsilon$, respectively,

[TABLE]

The continuity equation (1) can include sources $P$ due to ionization of neutral particles with mass $m_{i}$ and losses $L$ due to recombination. The velocity equation (2) includes in case of a mechanical obstacle the collisions and the mass loading in an atmosphere of the planetary body with $\nu_{in}$ the collision frequency for momentum transfer between the ions and the neutrals through elastic collisions and charge exchange. Here we assumed that the velocity of the atmosphere is at rest. The evolution of the magnetic field is described by the induction equation (3), which can include a resistive term characterized by the magnetic diffusivity $\eta$. The evolution of the internal energy density $\epsilon$, which is related to the plasma thermal pressure $p$ through $\epsilon=3/2\,p$ is described through (4). The total thermal pressure $p$ includes the effects of the electron temperature $T_{e}$ and the ion temperature $T_{i}$ by $p=nk_{B}(T_{e}+T_{i})$ with the Boltzmann constant $k_{B}$, and the plasma number density $n$. The MHD approach does however not allow to constrain the $T_{e}$ and $T_{i}$ separately without further assumptions. In (4), next to the work done by the pressure, the collisions of the plasma with neutrals as well as plasma production and recombination are included.

In order to describe the interaction as a well posed problem, initial and boundary conditions need to be specified. For the initial conditions, the unperturbed plasma or approximations to the final solutions are generally used. The outer boundary conditions are chosen so that on the upstream side of the obstacle inflowing conditions are set. They characterize the properties of the plasma flow upstream of the obstacle. Downstream of the obstacle outflowing boundary conditions are applied (e.g., chan13; duli14). The inner boundary for the plasma is located at the surface of the planetary body if the plasma reaches all the way to the surface duli14. For a planet with a very dense atmosphere such as Jupiter, the inner boundary is located below the ionosphere chan13. In case of the moons, the plasma is absorbed at the surfaces of the solid bodies, which are additionally assumed to be no source of plasma. These conditions imply that the radial component of the plasma flow $v_{r}$ can only be negative or zero, which also sets the conditions for the plasma, momentum and energy flow in equations (1), (2), and (4) duli14. In case of Jupiter the inner boundary conditions for the plasma is such that $v_{r}=0$ chan13. The boundary condition for the magnetic field is given by the electrically non-conductive nature of the solid surface of the moons and the neutral atmosphere below Jupiter’s ionosphere. This insulating nature implies that the radial component of the electric current at the boundary needs to vanish, i.e., $j_{r}=0$. The latter condition has non-local effects on the magnetic field and can be implemented by decomposing the magnetic field at the surface into poloidal and toroidal fields, which are expended into spherical harmonics. duli14 showed that for the resulting complex poloidal and toroidal coefficients $p^{*}_{lm}$ and $t^{*}_{lm}$ of the spherical harmonics $Y^{*}_{lm}$ of degree $l$ and order $m$, respectively, the following two equations need to be fulfilled

[TABLE]

[TABLE]

These equations need to be fulfilled at the inner boundary at every time step and thus set the magnetic field boundary conditions at the surface located at $R_{0}$. This description of the inner boundary conditions for the magnetic field also allows that the planetary body possesses internal magnetic fields whose origin lie below the surface. They can be internal dynamo fields as is the case for Jupiter and Ganymede, but they can also be time-variable induction magnetic fields generated in saline electrically conductive subsurface oceans. The internal fields are represented in (6) through their complex Gauss coefficients $G^{*}_{lm}(t)$. In case of Jupiter and its very strong magnetic field, the inner boundary conditions can be approximated by setting the azimuthal and the longitudinal component of the magnetic field to the internal Jovian magnetic field chan13.

The MHD approach is overall a very powerful approach to describe the plasma dynamics in the Jupiter system. Naturally, it does not capture all aspects of the plasma interaction, e.g., if electron scale physics is relevant and/or gyro radii and gyro periods need to be resolved. For example, reconnection at the magnetopauses of Jupiter and Ganymede is an non-ideal MHD effect which is best described with models which resolve ion and electron kinetics, e.g., with full particle-in-cell models. Finite gyro radii effects can play a role in situations where non-thermal, high energy ions are involved or where ions with large velocities within a small background magnetic field are picked-up. In case the latter effects are important, appropriate models are hybrid models or particle-in-cell models. Even though MHD models cannot resolve ion and electron kinetics effects, one of their advantages in numerical simulations is that they generally enable better spatial resolutions compared to kinetic models in case of similar computational resources. This advantage is particular helpful if small atmospheric scale heights need to be resolved.

3 Jupiter’s magnetosphere

Jupiter’s magnetosphere is qualitatively different compared to the magnetosphere of the Earth. The magnetosphere of Jupiter is rotationally dominated throughout a large part of the magnetosphere and the moons of Jupiter are huge internal mass sources (e.g., vasy83; bage11). Io provides approximately 103 kg s*-1* mostly in form of SO2 and Europa approximately 50 kg s*-1* in form of O2 to the magnetosphere (e.g., broa79; saur98; saur03a; mauk03). This mass is subsequently being ionized and picked up by the motional electric field of the fast rotating magnetosphere. Jupiter’s sideric rotation period is $\sim$9.9 h. The plasma thus experiences large centrifugal forces responsible for radial transport of the plasma. Due to conservations of angular momentum the magnetospheric plasma while being transported radially outward is not rigidly corotating anymore. This sets up magnetic stresses and electric current systems which couple the magnetosphere to Jupiter’s ionosphere. Through this coupling angular momentum is being transported from Jupiter’s ionosphere into Jupiter’s magnetosphere to bring the magnetosphere closer to full corotation. Jupiter’s magnetosphere and its internal coupling has been described theoretically and numerically by a series of authors (e.g. hill79; vasy83; hill01; cowl01; sout01; walk98; walk03; mori08; ray10).

Because of the immense importance of Jupiter’s magnetosphere-ionosphere (MI) coupling, chan13 developed a new MHD model of Jupiter’s magnetosphere. This model explicitly includes the ionosphere of Jupiter within the model domain. In Jupiter’s ionosphere ion-neutral collisions transfer angular momentum from the neutrals onto the plasma. The resultant flow generates magnetic stresses between the ionosphere and magnetosphere, which accelerate the magnetosphere in the direction of Jupiter’s rotation, but decelerate Jupiter’s ionosphere in return. In the model of chan13 the MI-coupling is explicitly included and the magnetic field boundary conditions can be physically correctly set below the ionosphere (see section 2.2). The downside of the approach is that for numerical reasons, the radial extension of the ionosphere is strongly exaggerated by 4 Jovian radii (RJ) and the surface of Jupiter has been set to be at 4.5 RJ. Despite the latter assumption, the chan13 model however still uses the closest inner boundary of all published MHD models of Jupiter’s magnetosphere. The model includes the mass loading in the Io plasma torus explicitly. It thus explicitly includes two of the most important features of Jupiter’s magnetosphere: The coupling to the ionosphere and the mass loading in a fast rotating magnetosphere.

The overall density and magnetic field structure of the modelled magnetosphere is shown in Figure 2.

The bow shock is located at 73 RJ and the magnetopause at 69 RJ in agreement with in-situ measurements joy02. The magnetosphere is compressed on the day side and strongly elongated on the night side. Figure 2 also shows an X-point on the night side where a plasmoid of plasma is being released as part of the mass loss processes in the tail of the magnetosphere. The figure also shows that the plasma is concentrated in the equatorial regions of the magnetosphere due to the strong centrifugal forces. The radial density profile of the chan13 model agrees very well with measured density profiles taken by the Galileo spacecraft fran02; bage11.

In Figure 3,

the electric current system which represents the angular momentum transfer between Jupiter’s ionosphere and magnetosphere is displayed. The electric current loop connects Jupiter’s ionosphere through field aligned currents directed away from Jupiter and which are feed into the equatorial plasma sheet of Jupiter’s magnetosphere at radial distances of 20 to 30 RJ. In this plasma sheet the currents are directed mostly radially outward and the related $\vec{j}\times\vec{B}$ forces spin up the magnetosphere. The current closure back to the ionosphere occurs at large radial distances (not clearly visible at low latitudes in Figure (3)) and enter the ionosphere in the polar region. The $\vec{j}\times\vec{B}$ forces slow the plasma in the ionosphere and are in balance with the forces exerted by the ion-neutral collisions, which accelerate the ionosphere.

The modeled azimuthal and radial velocities in chan13 are in good agreement with observations by the Galileo and Voyager spacecraft and the theoretical predictions by hill79. In Figure 4 the modeled azimuthal velocities as a function of radial distance are shown for various mass loading rates in comparison to Voyager measurements.

The corotation breakdown (as defined by hill79, 75% of rigid corotation) in the plane of Figure 3 occurs at a radial distance of 32 RJ, but the plasma starts to subcorotate at approximately 20 RJ. These results are consistent with the locations of the field-aligned currents and matches the theoretical predictions obtained for the same ionospheric conductances and total radial mass transport rates by hill79; hill01.

On field lines with large parallel electric currents pointing away from the planet and on locations along these field lines where the charge carrier density is small, electrons need to be accelerated to large energies to maintain the electric current loop knig73; hill01; cowl01. The resultant energetic electrons precipitate into Jupiter’s ionosphere and excite Jupiter’s main auroral oval, which has been extensively observed, e.g., with the Hubble Space Telescope (HST) (e.g., clar05). Therefore the location in Jupiter’s atmosphere, where field lines with large anti-planetward electric current densities map to, can be associated with auroral emission. In Figure 5 we show the electric current density from the chan13 model mapped along dipole field lines into Jupiter’s atmosphere. The images show that the current system matches to colatitudes of approximately 15∘, which is in good agreement with observations (e.g., clar05). Figure 5 also shows that the electric current is azimuthally asymmetric as expected from a magnetosphere with strong local-time asymmetries. In particular, between 8:00 and 11:00 LT the anti-planetward electric current density has a minimum, which is consistent with a discontinuity in the main oval observed by radi08 within HST observations. The discontinuity in the electric current in the model of chan13 is caused by an asymmetry in the pressure distribution due to the interaction between the rotating plasma and the magnetopause.

4 Time-variable magnetosphere

While we discussed in the previous section mostly steady state components of Jupiter’s magnetosphere, we address here magnetospheric time-variability caused by the solar wind, by variations of the internal sources, by the rotation of Jupiter and by dynamical non-linear processes in the magnetosphere.

The internal plasma sources of Jupiter’s magnetosphere, i.e., the mass loading at the moons might be time-variable which is however observationally not established very well. The model of chan13 shows that if the mass loading rate of Io changes from $1\times 10^{3}$ kg s*-1* to $3\times 10^{3}$ kg s*-1*, the azimuthal velocity profile changes and the break down of corotation occurs further inside (see Figure 4). The change in the mass loading also implies changes in the size of the magnetosphere, the structure of the magnetic field, and in the field aligned auroral current systems. For enhanced mass-loading rates, chan13 find that the auroral becomes more symmetric, while the brightness barely changes.

Changing solar wind conditions impact Jupiter’s magnetosphere as well. An increase in solar wind ram pressure, decreases the size of the magnetosphere on the subsolar direction and stretches it towards the night side. An important question is how the auroral brightness responds to an increased ram pressure. Earlier theoretical work by cowl01, sout01 and cowl03b, suggested that the aurora dims in this case because the decreased size of the magnetosphere moves the magnetospheric plasma somewhat radially inward, leading to increased angular velocities (since the angular momentum is conserved). This is expected to reduce the corotational enforcing electric currents and thus to lead to a reduced auroral brightness. The simulations of chan13 predict that the overall response to increased solar wind ram pressure strongly depend on local time, but lead in general to an overal increased auroral brightness chan17. The primary reason is that the enhanced solar wind ram pressure increases the magnetic stresses in the magnetosphere leading to higher field-aligned electric currents. For deriving these results, the three-dimensional nature of the magnetosphere needs to be considerer. chan17 find that only for a short period of time, during the transition phase from weak to strong solar wind ram pressure the aurora locally dims in the noon region.

Another cause of time-variability are dynamical processes in the magnetosphere. Among these processes are intermittent reconnection on the night side and the release of plasmoids on time scales on the order of 10 hours bage07; chan13; vogt14; chan17. The radial transport within the magnetosphere occurs through flux tube interchange (e.g. kive97). The resultant perturbations of the magnetosphere are stochastic in nature. These perturbations in turn interact with each other generating a turbulent cascade of time-dependent magnetic field and velocity fluctuations saur02.

Another class of time-variability is induced in Jupiter’s magnetosphere due to the approximately 10∘ tilt of Jupiter’s magnetic moment with respect to its spin axis. This tilt makes all properties of Jupiter’s magnetosphere time-variable with a period of the sideric rotation period of Jupiter (9.9 hours) when observed in an inertial rest frame. This time-variability is important for the magnetosphere itself, but it also leads to time-variable magnetic fields at the locations of the moons, which can be used to probe their interior structure (e.g., khur98; neub98; kive00; zimm00).

The time-variability of the magnetic field near the Galilean moons has been explored extensively by seuf11 through considerations of a range of possibly frequencies. The magnetic field model of seuf11 includes (a) the dynamo magnetic field of Jupiter represented by an expansion in spherical harmonics, (b) the magnetic fields of the current sheet and (c) fields due to the magnetopause boundary currents. With this magnetic field model, field components at the location of the moons are calculated and subsequently Fourier-transformed to obtain the time-variable magnetic field amplitudes as a function of their period as shown in Figure 6.

The figure shows three sources of time-variability at Ganymede, i.e., three sets of periods: (i) the rotation period of Jupiter and higher harmonics thereof due to non-dipole components of the interior field and due to non-sinusoidal components of the current sheet field. The synodic rotation period of Jupiter seen in the restframe of the satellites generates the strongest amplitudes of all periods with the maximum in the $B_{r}$ component of $\sim 80$ nT seuf11. The higher order harmonics are already significantly smaller on the order of a few nT or less. (ii) The time-periodic contribution due to the orbital period of Ganymede is fairly small on the order of 1-2 nT due the small inclination $i=0.17^{\circ}$ and small eccentricity $e=0.0011$ of Ganymede. (iii) The solar rotation period of the sun is propagated out through the solar wind and can generate time-variable solar wind ram pressure, which generates time-variable magnetopause currents also called Chapman-Ferraro currents. The related currents cause amplitudes in the latitudinal $B_{\Theta}$ components of less than 1 nT. We will see in Section 5 that these various time-dependent fields generate induced magnetic fields in the interior of Ganymede. Observations of these induced magnetic field with the knowledge of the time-variable inducing fields from Figure 6 or previous studies, e.g., by kive02 can be used to probe the interior of Ganymede and the other moons. seuf11 calculate the amplitudes and the phases of the induced magnetic fields for the three inducing frequencies and for various models of the electrical conductivity structure within Ganymede, e.g., with a poorly conductive surface, a conductive saline subsurface water ocean and very highly conductive metallic core.

5 Ganymede’s magnetosphere

Ganymede is the largest moon in the solar system and comparable in size to Mercury. It is also the only known moon with an intrinsic dynamo magnetic field. Thus it possesses a mini-magnetosphere within Jupiter’s gigantic magnetosphere as studied by a number of authors, e.g., kive98; kive02; kopp02; ip02; paty04; paty06; paty08; jia08; jia09; jia10a.

With the new MHD model developed for Ganymede by duli14 and introduced in section 2.2, we model Ganymede’s plasma and magnetic field environment as displayed in Figure 7.

The MHD model includes Ganymede’s internal dynamo magnetic field after kive02 and induction in a subsurface ocean through the non-conducting boundary conditions given in equations (5) and (6). The model uses appropriate outer boundary and initial conditions given by the Galileo spacecraft measurements during each of its flybys at Ganymede. The model also includes a thin atmosphere, in which ionization with a constant ionization frequency generates an ionosphere as a source term in equation (1). Elastic collisions, charge exchange and ionization slow the flow in the ionosphere, which is implemented in the source terms in the velocity equation (2). In the induction equation (3), we include the resistivity of the ionosphere and anomalous resistivity due to reconnection similar to jia10a.

An appropriate description of the magnetic boundary condition at the surface of Ganymede (see section 2.2) is crucial to correctly describe Ganymede’s magnetic field environment. Commonly applied incorrect boundary conditions are to set the magnetic field at the surface of Ganymede to fixed values given by the internal magnetic field. This approach forces the plasma magnetic fields to be zero and allows electric current to enter through the electrically non-conducting surface of Ganymede. In Figure 8,

we show the magnetic field perturbations at the surface of Ganymede, which would be neglected if the magnetic field at surface is set to the values of the dynamo field. The plasma magnetic field assumes values up to $\sim$120 nT. These values are larger than the time-variable components and the induction effects of an ocean and are about 20% of Ganymede’s dynamo magnetic field. Thus applying incorrect boundary conditions significantly distorts the magnetic field environment around Ganymede.

The relative velocity of Jupiter’s magnetospheric plasma with respect to Ganymede is sub-Alfvénic (e.g., neub98) and thus no bow shock forms. Ganymede’s internal magnetic field generates a mini-magnetosphere with a region of closed magnetic field lines as can be seen as the green shaded region in the top panel of Figure 9.

The closed field line region is shifted towards higher latitudes on the upstream side and is shifted towards the equator on the downstream side due to the magnetic stresses acting on Ganymede neub98. This effect is also well visible in Figure 7. The same figure additionally shows that the plasma flow within the closed field line region is reversed, i.e., in the upstream direction compared to the unperturbed magnetospheric flow.

Due to the sub-Alfvénic nature of the incoming flow Alfvén wings form, which are displayed as blue region in the top panel of Figure 9. The Alfvén wings are also visible in Figure 7 as the large structures north and south of Ganymede where the plasma flow is strongly reduced. The width of the Alfvén wings is significantly larger compared to Ganymede and the width of the closed field region. This is due to the orientation of the internal magnetic field and the external field of Jupiter’s magnetosphere. The width of the wings at the presence of an internal magnetic field shown in Fig. 9 is in agreement with quantitative expressions for the width derived in neub98 and saur13.

In the bottom panel of Figure 9 we quantitatively compare the MHD model results of duli14 shown in green with magnetic field measurements by the Galileo spacecraft taken during the G 29 flyby shown in red. This flyby crossed the northern Alfvén wings and is displayed as a yellow arrow in the top panel of Figure 9. The MHD model fits the amplitude and locations of the wing crossing well. It also quantitatively reproduces the magnetic field measurements of all the other Ganymede flyby by the Galileo spacecraft (e.g., duli14).

6 Ganymede’s ocean

The time-periodic magnetic fields in Jupiter’s magnetosphere as discussed in section 4 can be used to explore electrically conductive layers within the moons of Jupiter (e.g., khur98; neub98; zimm00). These fields establish one of the few currently available methods to search for saline and thus electrically conductive subsurface oceans. The method is based on the fact that water in its solid form possesses an electrically conductivity at least 4 orders of magnitude smaller compared to liquid water with salinities discussed in the context of the Galilean satellites (e.g., seuf11). The magnetic field measurements by the Galileo spacecraft near Ganymede have been searched for signs of induction signals from an ocean by kive02. It was found that the magnetic field measurements from multiple flybys are consistent with an ocean, however the measurements can be fitted quantitatively equally well by unknown quadrupole moments of Ganymede’s dynamo magnetic field kive02. Unfortunately, it is impossible to overcome this uncertainty, i.e., to separate spatial and temporal variability with subsequent flybys along different trajectories.

As discussed in section 4 and visible in Figure 6, the time-variable magnetic field component of Jupiter’s magnetosphere with the largest amplitude is the $B_{r}$ component with values of $\sim$80 nT. In Figure 10

we show a sketch of the magnetic field environment around Ganymede for maximum positive $B_{r}$, i.e., pointing away from Jupiter (dashed lines) and for maximum negative $B_{r}$, i.e., pointing towards Jupiter (solid blue lines). The time-variable exterior component also modifies the open-closed field line boundary (OCFL) region of Ganymede’s magnetosphere as displayed in Figure 10) as well.

Ganymede also possesses two auroral ovals hall98; feld00; mcgr13 similar to all known planetary bodies with an intrinsic dynamo magnetic field and an atmosphere. An example of two auroral images taken with the Hubble Space Telescope (HST) is shown in Figure 11.

The two auroral ovals are located near the region where the open-closed field line region intersects with Ganymede’s atmosphere as shown as red and blue lines on the surface of Ganymede in Figure 10. Because the time-variable external magnetic field modifies the open-closed field line boundary, the location of the auroral ovals are time-variable as well and oscillate in the way depicted by the solid and dashed blue lines on the disk of Ganymede shown in Figure 10. The dashed/solid lines represents the location of the auroral ovals when Ganymede is above/below the plasma sheet of Jupiter’s magnetosphere, respectively. When a saline and thus electrically conductive ocean is present, the time-variable external field will induce secondary magnetic fields which will reduce the primary time-variable external magnetic field. The existence of an ocean will thus also reduce the amplitude of the oscillation of the locations of the auroral ovals. The reduced oscillation amplitudes are shown in Figure 10, where the solid and dashed red lines indicate the locations of the auroral ovals when an ocean is present in contrast to the blue lines when no ocean is present.

With dedicated HST observations obtained in November 2010 and October 2011, saur15 measured the locations of the ovals when Ganymede is maximum above and maximum below the current sheet in the search for a subsurface ocean. Figure 11 shows the HST observations from November 2010. Figure 11 also displays averaged locations of the ovals in green obtained from a polynomial fit to the observed ovals. The blue and the red lines in this Figure show the expected locations of the ovals when an ocean and no ocean are present. The expected locations are calculated with the new MHD model of Ganymede by duli14 described in section 5. Only from visual inspections of Figure 11, it is nearly impossible to distinguish if the MHD runs with or without ocean fit better to the observations. We note that for this purpose the absolute locations of the ovals are not the important quantities, but the changes of the locations, i.e., the oscillation amplitudes (see Figure 10) between the locations when Ganymede is above and below the current sheet. Therefore saur15 calculated these difference and associated them to an average oscillation angle, also called rocking angle $\alpha$. The average rocking angle of the northern and southern ovals from the HST observations in 2010 and 2011 combined was found to be $\alpha=2.0^{\circ}$.

The observed locations of Ganymede’s auroral ovals in the HST data are unfortunately patchy (see, e.g. Figure 11). This patchiness is due to intermittent reconnection near Ganymede’s magnetopause and due to the finite signal to noise ratio of the counts on the individual detector pixels saur15. In order to assess the error when comparing model oscillation amplitudes $\alpha$ with and without ocean to the observations, saur15 introduced a Monte-Carlo test. In the Monte-Carlo test synthetic observations with and without ocean are generated based on the MHD model of duli14. Therefore patchiness produced with a random generator based on the physics of the intermittent reconnection and based on the finite signal to noise of the observations were added to the modeled locations of the ovals. The resultant synthetic images appear visually very similar to the actually observed ovals (see Figure 7 in saur15). Subsequently 1024 synthetic HST campaigns where generated with individually different patchy ovals. These synthetic images were then analyzed in an identical ways compared to the real data. In Figure 12,

the resultant distribution of the rocking angles are shown with and without ocean (red and blue distribution, respectively). It can be seen that both distribution functions barely overlap, which implies that the ocean and the non-ocean hypotheses can be well separated with this approach. The expectation value is 2.2${}^{\circ}\pm 1.3^{\circ}$ for the ocean model and 5.8${}^{\circ}\pm 1.3^{\circ}$ for the model without ocean. The uncertainties are calculated based on the one-sigma area around the expectations values in Figure 12. The observed rocking angle $\alpha=2.0^{\circ}$ is thus consistent with the existence of a subsurface ocean and inconsistent with no ocean present below the surface of Ganymede.

With this new approach the non-uniqueness of the interpretation of the Galileo magnetometer measurements by kive02 could be overcome. The key advantage of the new HST technique by saur15 is to search for an ocean with time-resolved two-dimensional observations of the auroral ovals, i.e. to apply time-dependent ”quasi two-dimensional images” of Ganymede’s magnetic field environment.

The resultant internal structure of Ganymede is shown in Figure 13.

The layer of liquid water is embedded within two layers of water in the solid phase. This structure is consistent with theoretical models by sohl02; huss06; ramb11; vanc14. Based on the calculations in saur15, the ocean needs to have a minimum electrical conductivity of 0.09 S/m when assuming it to be located between 150 km and 250 km depth. This conductivity corresponds to a minimum salt concentration of 0.9 gr of MgSO4 per kilogram ocean water. The measurement also require that the top of the ocean cannot be deeper than 330 km when measured from the surface.

7 Callisto

Callisto is in size and average mass density similar to Ganymede, but structurally only a partially differentiated body (e.g., show99). Callisto also does not possess an intrinsic dynamo magnetic field in contrast to Ganymede. However, it encompasses similar to the other Galilean satellites a thin atmosphere and an ionosphere carl99; klio02; cunn15. The time-variable components of Jupiter’s magnetospheric field induce electric currents within electrically conductive layers, such as a subsurface ocean, creating an induced dipole magnetic field based on studies by neub98 and zimm00. Callisto’s atmosphere and ionosphere interact with the plasma of Jupiter’s magnetosphere, which generates additional magnetic field perturbations. In the following two subsections the formation of Callisto’s ionosphere and the plasma interaction will be discussed.

7.1 Callisto’s ionosphere

The first component of Callisto’s atmosphere to be observed was CO2 with a column density of 0.8 $\times$ 1019 m*-2* carl99. The large ionospheric densities up to 4 $\times 10^{10}$ m*-3* inferred by klio02 imply that an additional atmospheric component is present, which was suggested to be O2. Hubble Space Telescope observations with the STIS camera however only led to upper limits for O2 stro02. But subsequent observations presented by cunn15 with the more sensitive HST/COS camera revealed OI 135.6 nm and 130.4 nm emission with a brightness of 1 to 5 Rayleigh. The authors derived from these observations an O2 column density of 4 $\times$ 1019 m*-2*.

A main difference of the ionosphere and the atmospheric UV emission of Callisto compared to those of the other Galilean satellites is that the plasma density of Jupiter’s magnetosphere at Callisto is so dilute that electron impact is not the primary source of ionization and UV excitation anymore. Callisto’s ionosphere and UV emission is in contrast primarily driven by solar photons cunn15. In order to better understand Callisto’s atmosphere and ionosphere, hart17 developed a new model to simultaneously explain the observed ionospheric electron column densities and the atmospheric UV emissions. This model solves for the electron distribution functions at every location in Callisto’s atmosphere for a prescribed atmosphere which includes O2, CO2, and H2O. It takes into account as the primary source of electrons the solar UV fluxes which are highly time-variable as displayed in Figure 14.

The model solves the Boltzmann equation for the supra-thermal electron population and considers a large set of inelastic collisions between the atmospheric species, which modifies the electron energies. As a loss for electrons recombination is included, which is energy dependent and more effective for lower electron energies. The model neglects spatial transport of the electrons, which is a good assumption for altitudes smaller than 180 km for the thermal electrons and smaller than 45 km for supra-thermal electrons hart17. For the low temperature electrons, i.e., energies approximately less than 0.5 eV, the electron distribution function is Maxwellian due to the importance of electron-electron collisions at these energy ranges. Therefore the model of hart17 describes the electrons in this energy range, referred to as thermal electron range, with a fluid description for the electron particle densities and energy densities.

A resultant electron distribution function from the model of hart17 is shown in Figure 15.

It demonstrates the highly non Maxwellian nature of the distribution function for energies larger than 0.5 eV. This distribution function at these energies can only be calculated with a kinetic model. It shows the imprints of the solar input spectrum and the large class of possible collisional processes with the atmospheric neutrals. At lower temperature the distribution function turns Maxwellian due to the electron-electron collisions.

For a joint interpretation of the observed UV emission and the ionospheric electron densities a kinetic description of the electrons is necessary. The high energy tail of this distribution function excites the UV emission emitted from Callisto’s atmosphere. With a plain Maxwellian distribution with the temperature of the thermal population barely any UV radiation would be emitted.

With the combined kinetic and fluid model, hart17 calculate electron densities in Callisto’s ionosphere. Several examples of the resultant electron density structure are displayed in Figure 16.

The figure also shows line of sight paths of the Galileo spacecraft radio science signals through which ionospheric electron densities were derived by klio02. The radio science observations were all taken around the terminator region and thus provide constraints on Callisto’s ionosphere only in this region. Because the radio science technique constrains the integral electron density along a line of sight only, this effect has to be considered when comparing model results with observations.

With the kinetic-fluid model of hart17 the HST observations of cunn15 and the electron density measurement of klio02 can be jointly explained within a certain range of O2 densities in Callisto’s atmosphere. Based on this comparison, Callisto’s atmosphere has a mean O2 column density of $2.1\pm 1.1\times 10^{19}$ m*−2* and the atmosphere possesses a day night asymmetry. The terminator O2 column density has values of $\sim 0.4\times 10^{19}$ m*-2* and associated subsolar O2 column densities are in the range of $2.4-9.8\times 10^{19}$ m*-2* . The calculations by hart17 also show that the electron density is very sensitive to the relative abundance of H2O in Callisto’s atmosphere due to the thermal electron cooling by rotational state excitation of H2O. For the efficiency of Callisto’s atmospheric UV emission it is found that on average one photon is emitted at OI 135.6 nm per every 170 electron ion pairs generated and per every 60 electron ion pairs produced by secondary electron impact ionization.

7.2 Callisto’s plasma interaction

Callisto with its atmosphere and ionosphere is an obstacle to the dilute plasma of Jupiter’s magnetosphere streaming past the moon. This interaction is characterized by a magnetospheric space plasma environment which is more variable compared to the other Galilean satellites. The magnetospheric field $B_{0}=4-42$ nT, the relative velocity $v_{0}=122-272$ km/s, the ion density $n_{0}=0.01-0.5\times 10^{6}$ m*-3*, the Alfvén Mach numbers M${}_{A}=0.02-1.85$ and the ion gyro radius $r_{g}$ = 34-530 km are highly variable due to the varying position of Callisto with respect to Jupiter’s magnetospheric plasma sheet and due to stochastic effects within the magnetosphere kive04; seuf12.

seuf12 constructed an MHD model of Callisto’s interaction with Jupiter’s magnetosphere similar to the model for Ganymede described in section 5. The model includes the formation of an ionosphere through photoionization and electron impact ionization within a pure CO2 atmosphere and alternatively within an atmosphere composed of CO2 and O2. The model also includes as internal magnetic fields the induced fields from a subsurface ocean within Callisto (see section 4). The resultant magnetic fields from the MHD model of seuf12 in comparison with measurements made by the Galileo spacecraft during the C21 flyby are displayed in Figure 17.

The C-21 flyby was a flyby through the wake of Callisto with a closest approach of about 1000 km. In Figure 17, the observed fields are shown in black and the induced magnetic field combined with the background field are shown in red. The magnetic field including the plasma interaction with with a CO2 atmosphere only is shown as green dashed line and the field from the plasma interaction with a combination of CO2 and O2 is shown as solid green line. The results demonstrate that both induction in a conductive ocean and the plasma interaction produce significant magnetic field perturbations. Similar conclusions have been reached by subsequent modeling of liuz15; liuz16. Therefore an appropriate modeling of the plasma interaction and the induction effects is necessary in the interpretation of the plasma and field measurements obtained by the Galileo spacecraft.

8 Summary

In this chapter, we discussed similarities and differences between the largest and one of the smallest magnetosphere in the solar systems, i.e., those of Jupiter and Ganymede. We introduced two new models for the description of their magnetic field and plasma environments. These models cover two classes of the plasma interaction between a magnetized body with its surrounding space plasma, i.e., sub-Alfénic and super-fast interactions, which have counterparts at extrasolar planets. We also showed how HST observations in conjunction with MHD modeling of Ganymede’s auroral ovals can be used to search for a subsurface ocean within Ganymede. We additionally investigated the non-Maxwellian nature of the electron distribution function in Callisto’s ionosphere to constrain its atmosphere based on HST and Galileo spacecraft measurements.