Hemisphere coupling in Enceladus asymmetric plasma interaction

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1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 11,, doi:10.109/007ja01479, 007 Hemisphere coupling in Enceladus asymmetric plasma interaction Joachim Saur, 1 Fritz M. Neubauer, 1 and Nico Schilling 1 Received 17 April 007; revised 17 July 007; accepted 3 August 007; published 16 November 007. [1] The recently discovered neutral gas plume around the south pole of Enceladus introduces a pronounced north-south asymmetry in the plasma interaction of Saturn s magnetosphere with Enceladus. In this work we present an analytic model for such an asymmetric interaction for both an Alfvénic and a unipolar inductor far field model including the Hall effect. We find that even though both hemispheres of Enceladus are not directly connected through magnetic field lines, both hemispheres are electromagnetically coupled, that is, the perturbations of the velocity and magnetic fields generated around the south polar plume are mapped in a modified way into the northern hemisphere. This hemisphere coupling creates surface currents on the magnetic flux tube determined by field lines tangent to the body of Enceladus. The surface currents are accompanied by a magnetic field discontinuity across the flux tube on both hemispheres, that is, a discontinuity across the northern and the southern wings. The Cassini spacecraft has not yet crossed this boundary, which we here predict. The diameter and other properties of the discontinuity are diagnostic of plume properties and might be even diagnostic of electrodynamic induction in a potential subsurface ocean on Enceladus. Our model is also applicable to other satellite plasma interactions where similar asymmetries can occur such as Io or extra solar planets. Citation: Saur, J., F. M. Neubauer, and N. Schilling (007), Hemisphere coupling in Enceladus asymmetric plasma interaction, J. Geophys. Res., 11,, doi:10.109/007ja Introduction [] A unique feature about Enceladus plasma interaction with Saturn s magnetosphere is the north-south asymmetry imposed by recently discovered gas plume(s) at its south pole [Porco et al., 006; Waite et al., 006; Dougherty et al., 006; Tokar et al., 006; Spencer et al., 006; Hansen et al., 006]. Earlier studies of the sub-alfvénic interaction at Enceladus assumed a symmetric source in the north-south direction, that is, along the direction of the background magnetic field [Saur and Strobel, 005; Pontius and Hill, 006]. Symmetric sub-alfvénic plasma interactions have been examined extensively in the Jovian system. In particular, studies of the interaction at Io traditionally played the leading role to advance our understanding of satellite plasma interaction in general [Piddington and Drake, 1968; Goldreich and Lynden-Bell, 1969; Neubauer, 1980; Goertz, 1980; Hill and Pontius, 1998]. With a symmetric source, both the local interaction (within a few satellite radii of the source) and the far field interaction (the interaction starting several satellite radii away from the satellite) are also symmetric for a symmetric magnetosphere and/or parent planet s ionosphere. In this case, the interaction 1 Institut für Geophysik und Meteorologie, Universität zu Köln, Cologne, Germany. Copyright 007 by the American Geophysical Union /07/007JA01479$09.00 forms two symmetric Alfvén wings or two symmetric currents loops with current closure in the parent planet s ionosphere. [3] At Enceladus with a pronounced nonsymmetric neutral gas environment, we are faced with a new situation. If the center of the neutral gas cloud is located near the south pole, electric currents cannot simply be continued into the northern hemisphere of Saturn s magnetosphere because of the intervening body Enceladus. The interaction of ions and electrons with the neutral gas produces electric current through elastic collisions, charge exchange, and pickup. However, if no neutral gas is located on the northern hemisphere of Enceladus, there will be no current generated in the northern hemisphere. Depending on the size of the southern gas cloud, the northern Alfvén wing is either entirely absorbed for a small southern cloud or, for a large cloud, the inner part of the northern Alfvén wing, which intersects with Enceladus, is void of currents along the Alfvén characteristic. [4] The Alfvén travel time t A, that is, the time which an Alfvén wave takes to travel from Enceladus to Saturn s ionosphere and back to Enceladus, compared to the convection time t conv, that is, the time which a plasma element takes to be convected through Enceladus gas cloud, determines if the interaction is best described with the Alfvén wing model or with the unipolar inductor model (see discussion in the work of Goldreich and Lynden-Bell [1969], Crary and Bagenal [1997], or Neubauer [1998]). With the values in the work of Richardson [1998] and Saur and Strobel [005], 1of11

2 we estimate t A to be on the order of s. The strength of the perturbation by Enceladus grows with the electric conductivity perpendicular to the magnetic field. This conductivity is due to the thin atmosphere/ionosphere of Enceladus, whereas the contribution by Enceladus interior is negligible for the stationary interaction. Note that the ice of the crust of Enceladus acts as an efficient insulator regardless of silicate and alkali salt content and the higher temperatures at the southern polar cap. Thus there are no significant conductive paths through the interior that could potentially affect the interaction for the stationary problem [see Neubauer, 1998]. This is essentially true no matter how high the conductivity in the interior is. Only if substantial temporal variations have to be considered, the expected higher conductivities in the interior, for example, in a potential electrolytically conducting water shell [Schubert et al., 007], have to be taken into account. Assuming then an unperturbed plasma flow at Enceladus and an extension of the south polar gas plume of one Enceladus radius, we find t conv 10 s. If the plasma is not dramatically slowed near Enceladus and the extension of the plasma density away from the magnetospheric equator is not very different than calculated by Richardson [1998], the Alfvén wave launched by Enceladus and reflected at Saturn s ionosphere cannot reach Enceladus and the Alfvén wing model is the appropriate model to describe the far field interaction due to the local gas plume around Enceladus south pole. [5] Pontius and Hill [006] present an analytic model for Enceladus interaction where they focus on charge exchange and pickup in an extended gas cloud around Enceladus. On the basis of plasma measurements by Tokar et al. [006], they model the extended interaction region with a dimension of about 30 Enceladus radii and derive a total mass loading rate of more than 100 kg/s. Pontius and Hill [006], however, neglect the north-south asymmetry of the interaction due to the local plume around Enceladus south pole. This local plume produces a distinct plasma interaction of several Enceladus radii as evident in the magnetic field perturbations [Dougherty et al., 006]. [6] In this paper we present a model that focuses on the physics of an interaction with a north-south asymmetry, which has not been studied previously. We particularly aim at two objectives: First, we develop a general model for the plasma interaction of a satellite with an arbitrarily shaped and extended non symmetric gas cloud with its surrounding plasma. Second, we derive the most simple analytic solution of our model, which still contains the essential properties of Enceladus south polar plume on its plasma interaction. Therefore we neglect in the simple solution the effects of the extended gas cloud. With this model we show how both hemispheres of Enceladus are electromagnetically coupled although they are not directly connected through magnetic field lines. [7] We like to stress that it is not the aim of our paper to model the magnetic field and plasma observations of the three Cassini flybys which have occurred up to now. This would be an entirely different paper. Our model, however, provides the framework for future numerical modelling of the interaction which need to include a detailed description of both, the south polar plume and the extended gas cloud around Enceladus. Our model additionally includes qualitative predictions for future Cassini Enceladus flybys. Cassini has not yet obtained magnetic field measurements on magnetic field lines crucial for Enceladus hemisphere coupling, that is, on field lines tangent to the surface of Enceladus. All three previous flybys were on trajectories with field lines passing by Enceladus. [8] In section of this paper, we first derive a general model for a nonsymmetric sub-alfvénic plasma interaction for an arbitrarily distributed gas cloud around a planetary body. A central equation in our description is the equation for the local electric field, or equivalently the electric potential near the satellite. A detailed derivation for the symmetric case can be found for example in the work of Neubauer [1998], and a model for a combined unipolar inductor and Alfvén wing model of Saur [004]. In section 3 we then analytically solve this general model for simplified conditions, which represent the conditions at Enceladus. In section 4 we discuss properties and implications of our analytic solution, and in the final section 5 of this paper we summarize our results and draw conclusions.. General Model for Nonsymmetric Interaction [9] For satellites with a dilute atmosphere where the electron-neutral collision frequency is below the electron cyclotron frequency, the ionospheric conductivity parallel to the magnetic field lines is very large, in particular compared to the conductivity perpendicular to the magnetic field [Neubauer, 1998]. Therefore the electric field parallel to the magnetic field is shorted out and magnetic field lines are isolines of the electric potential F. Magnetic field lines that intersect with a satellite with radius R, however, can have different values of the electric potentials F N and F S north and south of the satellite because of the usually small conductivity of the crust of the satellite. [10] In the following we derive a model for the electric potential near a planetary satellite for an arbitrarily distributed gas cloud around the satellite. We assume a constant magnetic field B 0 near the satellite, which is a good assumption for interactions which produce small magnetic field perturbations jdb/b 0 j1. Such small magnetic field perturbations usually arise in low Alfvén Mach number interactions such as at Enceladus with jdb/b 0 j 0.03 [Dougherty et al., 006]. We use a coordinate system with the x-axis in direction of the unperturbed plasma flow, the z-axis antiparallel to the unperturbed background magnetic field B 0, the y-axis completes a right handed coordinate system, and points in the case of Enceladus with good precision toward Saturn. This coordinate system is basically identical to the Enceladus Interaction Coordinate System (ENIS) of the Planetary Data System since the orientation of B is nearly perpendicular to Enceladus orbital plane. [11] In the satellite s conductive gas cloud or ionosphere, the electric current perpendicular to the magnetic field is given by j? ¼ s rf with the conductivity tensor s, which contains the Pedersen and Hall conductivity. The conductivities can arise due to elastic collisions, charge exchange, and pickup processes. The divergence of these perpendicular currents leads to ð1þ of11

3 field-aligned electric currents. There are two standard models how the field-aligned electric currents are continued in the far field, that is, far away from the satellite. One is the unipolar inductor model where the field-aligned currents close in the planet s ionosphere (e.g., of Jupiter or Saturn) [Goldreich and Lynden-Bell, 1969; Hill and Pontius, 1998; Pontius and Hill, 006]. The other one is the Alfvén wing model [Neubauer, 1980; Goertz, 1980; Southwood et al., 1980], where Alfvén wave perturbations are radiated away from the satellite to be reflected back and forth by the northern and southern parent planet s ionosphere and where current closure is distributed along the wave path in the planetary magnetosphere. However, when the Alfvén waves do not return to the satellite, the reflected waves cannot react back on the satellite environment with their role being simply described then as a local Alfvénic antenna conductance. The parallel electric currents in the far field can be written after Saur [004] for both models as j N=S z ¼S N=S F F g þ F ¼r ~S N=S F rf with the northern and southern far field conductance S N/S F, which in the case of the unipolar inductor model is equal to the Pedersen conductance of the parent planet s ionosphere S P and equal to the Alfvén conductance S A in the case of the Alfvén wing model. The minus and plus signs correspond to the northern and southern hemisphere, respectively. The associated far field conductances S N/S F can be different for each hemisphere in the unipolar inductor case, for example, due to seasonal variations in the parent planet s ionosphere. The geometric factorp gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F is equal to one in the Alfvén wing model and g F = 4 3=L for the unipolar inductor model [Saur, 004]. In the latter case, g F 1.8 for Enceladus at a radial distance L = 3.9 and under the assumption of dipole magnetic field lines connecting the satellite and the planet s ionosphere. We introduce the tensor ~S F in () to write this equation more compactly. [1] Starting with charge conservations for steady state rj ¼ 0 and integration along magnetic field lines from a lower boundary G l to an upper boundary G u, we find j z ðx; y; G u Z Gu Þ j z ðx; y; G l Þ ¼ ¼ G l Z Gu zðx; y; G l ðþ ð3þ dzr? j? ðx; y; zþ ð4þ As depicted in Figure 1, the interaction with a nonsymmetric ionosphere needs to be divided into three subdomains: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi The exterior domain with potential F(x, y) and r = x þ y > R, that is, on field lines not intersecting with the satellite with radius R (outside the yellow arrows), and two subdomains with r < R, that is, on field lines intersecting with the satellite. The latter two subdomains are the northern tube, that is, north of the satellite with potential F N, and the southern tube, that is, south of the satellite with potential F S. [13] For the exterior domain the electric currents at the upper and lower boundaries G u and G l in (4) are the northern and southern far field electric currents of () (see also Figure 1). For the northern tube the currents at the upper boundary G u are the northern far field currents (in the northern Alfvén wing or the parent planet s northern ionosphere) and the currents are zero at the lower boundary G l, i.e., the surface of the satellite s northern hemisphere because the moon s upper crustal layers are assumed to be nonconducting. Note that in the stationary case the internal conductivity has no influence then [Neubauer, 1998]. For the southern tube the upper boundary G u is the nonconducting surface of the satellite s southern hemisphere and the lower boundary G l is the southern far field current (the southern Alfvén wing or the parent planet s southern ionosphere). [14] The integrals on the right-hand side of (4) need to be executed within the satellites ionosphere. Integration along the magnetic field lines leads to the height integrated electric current J? ¼ S rf with the conductance tensor S of the satellite s ionosphere, which includes the Pedersen conductance S 1 and the Hall conductance S. For the exterior domain r > R, this integral sums up all conductivity s whether north or south of the equatorial plane and is denominated S I. In the northern tube for r < R, the integral is performed from the satellite s surface out to a distance where the neutral density vanishes. This leads to the northern ionospheric conductance S N I.In the southern tube, the integral is performed from the satellite s surface again out to a distance where the neutral density vanishes. This leads to the southern ionospheric conductance S S I. Note, S I, S N I, and S S I are for an arbitrarily distributed atmosphere and thus in general functions of x and y. [15] With the height-integrated conductances and the far field conductance, (4) leads to three differential equations for the electric potential within each of the three subdomains. Outside of the flux tube, we find for the electric potential F r S I þ ~S S þ ~S N rf ¼ 0 for r > R: ð6þ F F Inside the southern flux tube, we find with (4) for the southern potential F S ð5þ r S S þ ~S S rf S ¼ 0 for r < R: ð7þ I F and inside the northern flux tube, we find for the northern potential F N r S N I þ ~S N rf N ¼ 0 for r < R: ð8þ F Now we need to pay attention to the boundary across r = R, that is, within and outside of the flux tube. The first condition is that the electric potential is continuous everywhere and thus also on the boundary (r = R) ofthe 3of11

4 Figure 1. Sketch of the electric current system for a nonsymmetric plasma interaction due to a south polar plume. The asymmetric gas cloud drives a surplus of electric current in the southern hemisphere of Enceladus. A fraction of this surplus is closed in the northern far field through surface currents (in yellow) on magnetic field lines tangent to Enceladus. All other currents driven in the interaction are volume currents (in red). flux tube. Second, the electric field can jump across the flux tube boundary. Integrating (4) radially across the R boundary surface of the tube, that is, performing lim!0 R R+ dr on (4) leads to the jump condition lim ~S N þ ~S S þ ~S!0 F F I rf Rþ r ¼ lim ~S N þ ~S N rf N þ ~S S þ ~S S rf S!0 F I F I r R ð9þ with the subscript r denoting the radial component of each expression. [16] The potential equation is also subject to the boundary condition that the perturbation fades away at infinity F 0 ¼ E 0 y for r!1 ð10þ where E 0 = v 0 B 0 is the unperturbed motional electric field seen in the rest frame of the satellite. 4of11

5 [19] We apply the following approach for the solution of equations (6) to (10) in cylindrical coordinates ^F ¼ X1 ^r k ða k sin k8 þ b k cos k8þþ^r k ðg k sin k8 þ h k cos k8þ k¼0 ð11þ where we use ^r = r/r and ^F = F/F 0 with F 0 = E 0 R. For our particular problem we need to divide the solution domain in four subdomains (Figure ): [0] 1. The external potential with superscript e: ^F e ¼ ^r 1 g1 e sin 8 þ he 1 cos 8 þ ^r sin 8 for r > R: ð1þ [1]. The northern potential within the Enceladus tube with superscript n: ^F n ¼ ^r a n 1 sin 8 þ bn 1 cos 8 for r < R: ð13þ [] 3. The southern potential within the Enceladus tube in a region still without neutral gas with superscript i ^F i ¼ ^r a i 1 sin 8 þ bi 1 cos 8 þ ^r 1 g1 i sin 8 þ hi 1 cos 8 for r p < r < R: ð14þ Figure. Geometry and electric current system of the analytic solution for a south polar plume with radial extension r p. Although the plume atmosphere lies south of Enceladus, both ionospheres of Saturn contribute to current closure. Note that the electric current system in the far field is displayed simplified (for a more detailed description, see Saur [004, Figure ]). [17] In summary, to determine the electric potential near Enceladus one needs to solve simultaneously (6), (7), (8), (9) with the boundary condition (10). Note, this set of equations holds for arbitrary neutral gas distribution and needs to be solved numerically in the general case. 3. Analytic Solution for Enceladus [18] Now we provide a simple analytic solution of the model equations (6) to (10), which is tailored to describe the interaction of Saturn s magnetosphere with a neutral gas plume at the south pole of Enceladus. We assume the plume creates an electric conductance S S I due to pickup, charge exchange, and elastic collisions. For simplicity we assume S S I to be constant within a plume radius r p (R) and zero outside (see Figure ). We assume the neutral gas density to be zero elsewhere and thus the conductance on the northern hemisphere S N I as well as the conductances S I for r > R are zero everywhere. We also assume the interaction to be Alfvénic in the far field as discussed in the introduction. Note that in the case of constant S S I when r p > R, both, the northern and the southern far field can feed electric current with equal proportions in the conductive gas cloud and the asymmetry of the electrodynamic interaction as discussed in this paper is lost. [3] 4. The southern potential within the Enceladus tube in the region with neutral gas with superscript s ^F s ¼ ^r a s 1 sin 8 þ bs 1 cos 8 for r < r p : ð15þ [4] Since the boundary condition (10) needs to be applied and the set of differential equations (6) to (8) are linear, all coefficients in (11) with k 6¼ 1 must be zero. Thus equations (1) to (15) are much simplified compared to (11). [5] Additionally, we need to consider the following boundary conditions between the subdomains: Conditions 1 and, the electric potential needs to be continuous, that is, F e (r = R) =F n (r = R) =F i (r = R) as well as condition 3, F i (r = r p )=F s (r = r p ). Condition 4, at r = R we need to consider (9). Condition 5, at the boundary to the gas plume, that is, r = r p, it holds lim r ð S 1 þ S r pþ r p ¼ 0 ð16þ [see Saur et al., 1999]. Condition 6 at infinity the electric perturbation caused by Enceladus needs to fade away, that is, we need to consider (10) already built into (1). [6] With the condition 1 to 6 of the previous paragraph, we can determine the free coefficients in equations (1) to (15). They read with the abbreviations r = ^r p and as follows D ¼ r S 1 þ S þ 8r S A S 1 þ 16S A ð17þ g1 e ¼ 1=g 1 i ¼ r p r S 1 þ S þ 4S1 S A =D ð18þ 5of11

6 Figure 3. Electric potential for (a) the southern hemisphere and for (b) the northern hemisphere generated by a gas plume near the south pole characterized by r p =0.8R, S A =3.4S,S 1 = 10 S, and S = 10 S. The solid circle represents Enceladus and the dashed circle the extension of the gas plume. Note that the electric potential in the north (Figure 3b) is perturbed without an atmosphere in the north. The dotted vertical lines display the trajectories along which we display the magnetic fields in Figures 4. a n 1 ¼ 3rp þ p r4 S 1 þ S þ 8 6r p S1 S A þ 8S A =D ð19þ h a i 1 ¼ r S 1 þ i S þ 4 r p S1 S A þ 8S A =D ð0þ a s 1 ¼ 4S Aðr S 1 þ 4S A Þ=D ð1þ b s 1 ¼ 4r S S A =D h e 1 ¼ bn 1 ¼ bi 1 ¼ 1=hi 1 ¼ 4^r p S S A =D ðþ ð3þ [7] In Figures 3a and 3b we display the solution for the southern and northern hemisphere for S A =3.4S,S 1 =10S, and S = 10 S, and r p =0.8R, respectively. S A is taken from the model of Saur and Strobel [005], and the values for S 1 and S are order of magnitude estimates based on updated runs of the same model, and r p is chosen quite arbitrarily. With a value for r p =0.8R, 64% of the southern disk of Enceladus and thus roughly 40% of the surface of the southern hemisphere are covered with the plume atmosphere. In the following section, we discuss properties of this solution and their implications for Enceladus plasma and magnetic field environment. 4. Implications and Properties of Nonsymmetric Interaction [8] If the atmosphere of Enceladus is asymmetric with respect to the north and south direction, then field lines which intersect Enceladus lie on different potentials north and south of Enceladus. On these field lines the electric field as well as the flow velocities are different in each hemisphere. Although both hemispheres are not directly connected to each other, they still influence each others electromagnetic field properties through their mutual action on the field lines not intersecting Enceladus. With the solution of section 3, we can estimate the effect of the electromagnetic hemisphere coupling from the south to the north Electric Current [9] The electric current system that arises in Enceladus nonsymmetric interaction is sketched in Figure. The motional electric field E in the rest frame of Enceladus drives an electric current through the south polar plume. Outside the plume the electric current is continued along the Alfvén characteristics in the Alfvén wing model or along the magnetic field into Saturn s ionosphere in the unipolar inductor model. The absence of a similar plume in the northern hemisphere of Enceladus creates an interesting, nonsymmetric current system in the far field. [30] In this paragraph, we continue the discussion on qualitative properties of the current system in the unipolar inductor model for the clearness of the current system in the far field in this picture. The current system in the Alfvén wing model looks similar but closes formally at infinity and is bent back within the Alfvén wing by the angle tan Q A = M A with the Alfvén Mach number M A. The asymmetric plasma interaction creates a surplus of electric current driven around the south polar region compared to the north polar region of Enceladus. Saturn s south polar ionosphere is conductive also outside the flux tube and therefore the electric current system can be continued in two different ways. A fraction of the surplus of the electric current closes directly in Saturn s south polar ionosphere, but an other fraction of the electric current can additionally close through the northern hemisphere of Saturn through hemisphere 6of11

7 coupling currents, that is, within a current sheet on the surface of the Enceladus flux tube that connects both ionospheres of Saturn. The field-aligned electric current system, which connects both ionospheres of Saturn, can arise because the electric field in the northern and southern tubes is different and additionally jumps across the flux tube boundaries and thus requires field aligned electric current. Both ionospheres of Saturn are essentially connected in parallel to carry the current driven in Enceladus south polar plume. [31] Without Hall effect, the density of the surface currents which couples both hemispheres for the model scenario of section 3 with an Alfvénic far field is ^r p J z;hemisphere ¼ E 0 S A sin 8 S 1 at r ¼ R: ð4þ 4S A þ r S 1 The electric current that directly flows into the southern ionosphere of Enceladus is S 1 J z;iono ¼ 4E 0 S A sin 8 at r ¼ r p : ð5þ 4S A þ r S 1 [3] The currents on the flux tube that touches Enceladus in (4) are true surface or delta currents compared to the other currents which feed into Enceladus conductive plume. These delta currents are generated because the electric fields on field lines north and south of Enceladus are different and thus the electric fields across the flux tube enveloping Enceladus jump. This enables a hemisphere coupling current system. Outside of this flux tube (r > R) no hemisphere coupling currents exist even with a neutral gas cloud extending onto field lines not threading through Enceladus. For r > R, magnetic field lines are isopotential lines and thus outside this flux tube the electric fields and currents are symmetric in the northern and southern far field for identical Alfvén or ionospheric conductances in both magnetospheric hemispheres. Thus they have identical divergence and drive identical field-aligned electric currents with the opposite signs in each far field (see equation () and Figure 1). Therefore on each nonintersecting field line 50% of the current is closed in the northern far field and 50% in the southern far field for arbitrarily distributed gas along a field line as long as both far fields have identical Alfvén or ionospheric conductances. This implies that outside of the flux tube tangent to Enceladus no hemisphere coupling current can flow and the hemisphere coupling currents are confined to surface currents along magnetic field lines just touching Enceladus. We point out that the asymmetry of the current loop in the northern and southern far field will create slightly asymmetric magnetic fields in the southern and northern ionosphere of Saturn. [33] In contrast, the currents in (5) which feed into the conductive plume are only surface currents for this simplified mathematical model. In reality, the conductivity of the plume is continuously distributed and therefore the currents which directly feed into the plume are continuously distributed as well. [34] The ratio of the hemisphere-coupling currents and the ionospheric currents (4) and (5) is simply J z;hemisphere ¼ 1 r p ð6þ J z;iono R Which gives for r p = R J z;hemisphere J z;iono ¼ 1 : ð7þ This is the expected limit, which can be understood with the help of Figure. In this case, half of the current driven in the plume is connected to the northern Alfvén wing/ ionosphere of Saturn and half of it to the southern Alfvén wing/ionosphere of Saturn as in the symmetric case. 4.. Electric Field and Flow in the Southern Ionosphere and the Coupling to Northern Hemisphere [35] If there is no atmosphere over the north pole, but only over the south pole, the south polar plume still strongly modifies the north polar plasma and field environment. For our analytic solution with constant ionospheric conductance with a plume radius r p in the southern hemisphere, we find an electric field magnitude in the southern ionosphere E s ¼ 4 p SA ffiffiffiffi E 0 : ð8þ D Although the northern hemisphere does not contain an atmosphere/ionosphere in our model, the electric field in the northern tube is still reduced to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r E n 1 ¼ S 1 þ S þ 4S1 S A r 1 þ 4S A pffiffiffiffi E 0 ð9þ D where we used the abbreviation r 1 =1 ^r p. Note r 1 varies between zero and one depending on the width of the gas plume. Without Hall effect we find and the ratio E s ¼ 4S A 4S A þ r S 1 E 0 ; E n ¼ r 1S 1 þ 4S A r S 1 þ 4S A E 0 ; E n E s ¼ 1 þ r 1 S 1 S A ð30þ ð31þ ð3þ which shows that the electric field is less reduced in the north compared to the south (see Figures 3a and 3b). The ratio depends strongly on the width r p of the plume. If r p extends to one Enceladus radius, the electric field in the north and the south are equally much reduced Electron Flow [36] The electrons follow to good order the E B drift. Isolines of the electric potential are streamlines of the electron flow and displayed in Figures 3a and 3b. The flow is slowed within the south polar plume indicated by the dashed circle with radius r p in Figure 3a. On the flanks, directly outside of the plume the flow is accelerated. Across the flux tube tangent to Enceladus at r = R, indicated by the solid circle, the direction of the flow is not continuous due to the hemisphere coupling. However, the electric potential 7of11

8 Figure 4. Magnetic field in the (a) southern Alfvén wing and (b) northern Alfvén wing along trajectories parallel to the y-direction and shifted 0. R downstream from the center of the wing, which is bent back by the angles ±Q A, respectively (for the locations of the trajectory see also dotted lines in Figure 3). The solid line is for a nonsymmetric interaction with south polar plume with r p =0.8R, S A = 3.4 S, S 1 = 10 S, and S = 10 S, i.e., the same parameters as in Figures 3a and 3b. The dotted line is for a symmetric far field interaction, that is, when r p = R and thus 50% of the current feeding into Enceladus conducting gas cloud is coming from the north. The vertical dashed lines mark the boundary given by the field line tangent to Enceladus. is modified and the flow is slowed also around the north pole (see Figure 3b). It is important to note that the flow is slowed above the north pole despite the absence of neutral gas and its associated force on the plasma around the north pole. The magnitude of the electron velocity is proportional to the electric fields calculated in section Hall Effect and Rotation of the Fields [37] Owing to the Hall effect the electric field and consequently the electron flow can be significantly rotated s in the south by the angle Q twist tan Q s twist E x r S ¼ : E y r S 1 þ 4S A ð33þ This rotation is mapped into the northern tube but less pronounced. We find with r 3 = 3^r p + ^r p 4 tan Q n twist ¼ r 3 S 1 þ S ^r p S S A þ 4 3^r p S 1 S A þ 8S A : ð34þ Both rotations are evident in Figures 3a and 3b Magnetic Field [38] Sufficiently far away from Enceladus, that is, several Enceladus radii, the magnetic field perturbations can be calculated self-consistently for our analytic solution. Close to Enceladus the currents in the gas plume add extra complications. We use equations (14) and (15) in the work of Neubauer [1980] to calculate the magnetic field perturbations in Enceladus Alfvén wings. [39] In Figures 4a and 4b we display the magnetic field on a trajectory along the y-direction shifted by 0. R downstream from the centers of the southern and northern wings, respectively. The locations of these trajectories are shown in Figures 3a and 3b, respectively. The jzj values of the trajectories are arbitrary, as long as the trajectory is sufficiently far away, that is, sufficiently below or above Enceladus so that the magnetic field perturbations due to the ionospheric currents have faded away. We choose to display the magnetic field on a trajectory slightly shifted away from the center of the wings because on a trajectory exactly through the center of the wing some interesting properties of the magnetic field are less visible. The magnitude of the shift of 0. R is quite arbitrary. The solid lines are the magnetic field components which correspond to the electric potentials shown in Figures 3a and 3b, that is, with a south polar plume of the extension r p =0.8R. The dotted lines show the magnetic field components for r p = R. In this case, the interaction is symmetric and the same amount of electric current is closed in both hemispheres, that is, in both wings the magnetic field perturbations are symmetric. Otherwise, all the other parameters, such as conductances, are identical for the dotted and the solid lines. The entirely new feature in a nonsymmetric interaction is the discontinuity of the 8of11

9 magnetic field at r = R for the solid line. This discontinuity is a real physical discontinuity due to the hemisphere coupling, that is, due to the fractional closure of electric current in the northern hemisphere of electric current driven in the plume near the south pole. The discontinuity at r = R has formally infinitesimal width within our description. The real width of this structure will however be finite and cannot be described within magnetohydrodynamics (MHD) anymore and is controlled by kinetic plasma effects. [40] The other discontinuities in Figures 4a and 4b are only model discontinuities. In reality, the latter discontinuities disappear since the real gas plume does have a continuous density distributions and thus produces also a continuous conductivity distribution. Therefore the electric currents in Figures 1 and, which flow into the plume are broadly distributed and thus do not create a magnetic field discontinuity. A striking feature of the hemisphere coupling in the southern wing (Figure 4a) is that the separate magnetic field components are significantly enhanced between r = R and r = r p. This region lies between the surface currents which connect both hemispheres and the currents which feed into the southern plume. Both currents have opposite directions and therefore produce enhanced magnetic field components between R and r p. [41] The asymmetry with respect to y = 0 in Figures 4a and 4b is due to the Hall effect. Without Hall effect B x and B z would be symmetric with respect to y = 0 and the B y would be antisymmetric with respect to y = 0. The Hall effect is in addition particularly important for the B y component. The B y components would be exactly zero within r < r p in the southern wing and zero within r < R in the northern wing without the Hall effect. [4] As with the electric field and the velocity, the hemisphere coupling also generates a magnetic field pertubation in the northern wing despite the fact that there is no direct source on the northern hemisphere (see solid lines in Figure 4b). [43] An interesting question is how does the northern Alfvén wing arise if there is no source in the northern hemisphere and the solid body of Enceladus blocks the Alfvénic perturbations produced by the south polar plume to propagate northward. The velocity perturbation at the south pole due to the momentum exchange with the plume creates an Alfvénic perturbation which can freely propagate in the southern direction, but the perturbation also spreads out in the direction perpendicular to B to field lines not intersecting Enceladus anymore, carried by the MHD fast and slow mode. On these field lines Alfvén waves can propagate in the northern direction establishing a northern wing. [44] It is also interesting to investigate the physical relationship between the different regions of the interaction domain. Therefore it is useful to consider the wave propagation problem after a hypothetical switch-on of the neutral atmosphere at a time t = 0. After t = 0, collisions between the plasma and the neutral gas generate Alfvén waves, and fast mode as well as slow mode MHD-waves. The Alfvén wave signal cannot reach all regions around Enceladus because the group velocity is parallel or antiparallel to the background field. In contrast the fast and slow mode have finite components of group velocities transverse to the magnetic field and can therefore reach all regions by direct propagation and by diffraction around the satellite while the electromagnetic fields diffuse through the interior of Enceladus. The interaction of the wave fields with the satellite surface must couple the wave modes among themselves to enforce the boundary conditions of vanishing current density normal to the surface, that is, j n = 0, being most important. The interplay of the three MHD modes might modify aspects of the discontinuity. However, a full description of the interaction in the vicinity of the satellites requires a full numerical solution of the MHD equations with a proper treatment of the interior, which is outside the scope of this paper Effect of an Extended Gas Cloud [45] The simple solution that we present in section 3 does deliberately not incorporate an extended gas cloud around Enceladus to most clearly demonstrate the new features of the hemisphere coupling. An extended gas cloud arises due to the spreading of the nongravitationally bound south polar plume with a minor contribution from surface sputtering due to energetic ions. The effect of this extended gas cloud has been studied by Pontius and Hill [006]. In our analytic solution of section 3, we could introduce, next to the constant conductances describing the south polar plume within r r p, additional constant exterior conductances within a radius r ex which describes the extended gas cloud. The set of equations (6) to (10) still can be solved analytically, but the solution will be increasingly complex and will not provide significant new insight into aspects of the interaction not previously addressed. Nevertheless, it is still worthwhile to briefly describe in the remainder of this subsection how the extended gas cloud qualitatively influences Enceladus interaction. [46] Assuming the extended gas cloud is of significant dimension, for example, on the order of 10 Enceladus radii, charge exchange, elastic collisions, and pickup will slow the plasma flow and reduce the motional electric field seen in the rest frame of Enceladus as modeled by Pontius and Hill [006]. Note, no plasma velocity measurements are available within about 10 Enceladus radii of Enceladus [Tokar et al., 006]. Therefore the strength of the reduction of the plasma velocity in the close vicinity of Enceladus is not directly constrained through observations. A reduced plasma velocity will reduce somewhat the current driven within the south polar plume. The flow, however, cannot be at halt within the extended gas cloud. Otherwise no magnetic field perturbations with a dimension of only a few Enceladus radii as observed by the Cassini spacecraft would be generated. This local magnetic field perturbation is produced through the interaction of the magnetospheric plasma with the local plume around Enceladus south pole [Waite et al., 006; Porco et al., 006] embedded within the extended gas cloud. In the confined plume the plasma is particularly slowed and a southern Alfvén wing is created. Owing to the hemisphere coupling a northern Alfvén wing is created as well. On characteristics that do not intersect with the solid body of Enceladus, the same amount of electric current from the northern and the southern far field region will be fed into the extended gas cloud. Therefore the extended gas cloud does not directly participate in the hemisphere coupling currents. The extended gas cloud only reduces somewhat 9of11

10 the strength of the plasma interaction in the south polar plume because of the (to some extent) reduced plasma velocity in the gas cloud Parallel Electric Fields [47] Another implication of the hemispheric coupling are high electric current densities in the surface current. Using (4), r p =0.8R, and S 1 = 10 S, we can estimate a maximum surface current of Am 1. Assuming a width of the current sheet of 15 km (i.e., the gyroradius of H O + ), we find a current density of Am. With Enceladus located at 3.9 Saturn radii and assuming a dipole field connecting Enceladus to Saturn, the current density close to Saturn s polar ionosphere is A m. According to Cowley and Bunce [003], the critical limit to drive field-aligned acceleration of electrons in Saturn s magnetosphere is Am. Cowley and Bunce [003] derive this limit based on the model of Knight [1973]. We find that the current density in Enceladus surface currents exceeds this limit and thus can potentially drive electron acceleration to produce a faint auroral footprint in both hemispheres of Saturn s atmosphere. If no filamentation of the Alfvén wing occurs as suggested for Io by Chust et al. [005], the shape of the footprint would be a thin crescent in each hemisphere Electromagnetic Induction in a Potential Ocean [48] The discontinuity in the magnetic field across the surface of the Alfvén wing tube has a well-defined width given exactly by the diameter of Enceladus. Neubauer [1999] showed that electromagnetic induction in a potential subsurface ocean reduces the width of the Alfvén wing. At Enceladus, Saturn s magnetosphere does not contain an obvious periodically varying magnetic field as does Jupiter s magnetosphere at the locations of the Galilean satellites. However, any time variation of the magnetospheric field strength at Enceladus, for example, due to dynamic processes in Saturn s magnetosphere, combined with variations in the plume characteristics, will produce induction if there is an electrolytically conducting ocean present and thus produce a wing diameter less than the diameter of Enceladus. The advantage at Enceladus compared to, for example, Europa is that the width of the wing due to the hemisphere coupling is constrained by the well-known size of Enceladus. At Europa, the width of the wing is determined by the much less known structure of Europa s ionosphere. High-resolution magnetic field measurements through Enceladus Alfvén wings could therefore be diagnostic of an ocean on Enceladus. 5. Summary and Conclusions [49] The south polar gas plume on Enceladus generates a plasma interaction with Saturn s magnetosphere with a pronounced north-south asymmetry. Here we present a model of the asymmetric interaction which we analytically solve for simplified conditions. We show that even though the northern and the southern hemisphere are not directly linked, both hemispheres are still coupled. This coupling creates a magnetic field discontinuity across and electric surface currents along the flux tube which touches Enceladus. The Cassini spacecraft has not yet passed through this predicted discontinuity. The upcoming 61En Enceladus flyby in March 008 with closest approach at an altitude of 9 km at 0.5 degree latitude will pass through it. Measurements along this trajectory will be very diagnostic of the hemisphere coupling at Enceladus and might be potentially useful to probe Enceladus interior for electromagnetic induction even though a complete flyby through both of Enceladus Alfvén wings at higher altitudes are better suited for investigating the interior, quantitatively. [50] Our nonsymmetric model can also be extended to other satellite plasma interactions with a nonsymmetric source. At Io volcanoes can contribute to an asymmetric atmosphere, such as the north polar Tvashtar volcanic complex [Milazzo et al., 005]. For extra solar planets close to their parent star [Ip et al., 004; Preusse et al., 006], the interaction of a sub-alfvénic stellar wind with a day-night asymmetric planetary ionosphere will also create asymmetries in the plasma interaction. [51] Acknowledgments. The authors thank S. Jacobsen for preparing Figure 1. JS and NS appreciate support by Deutsche Forschungsgesellschaft. [5] Wolfgang Baumjohann thanks Martin Volwerk and another reviewer for their assistance in evaluating this paper. References Chust, T., et al. (005), Are Io s Alfvén wings filamented? Galileo observations, Planet. Space Sci., 53, Cowley, S. W. H., and E. J. Bunce (003), Corotation-driven magnetosphereionosphere coupling currents in Saturn s magnetosphere and their relation to the auroras, Ann. Geophys., 1, Crary, F. J., and F. Bagenal (1997), Coupling the plasma interaction at Io to Jupiter, Geophys. Res. Lett., 4, Dougherty, M. K., K. K. Khurana, F. M. Neubauer, C. T. Russell, J. Saur, J. S. Leisner, and M. Burton (006), Identification of a dynamic atmosphere at Enceladus with the Cassini Magnetometer, Science, 311, Goertz, C. K. (1980), Io s interaction with the plasma torus, J. Geophys. Res., 85, Goldreich, P., and D. Lynden-Bell (1969), Io, a Jovian unipolar inductor, Astrophys. J., 156, Hansen, C. J., et al. (006), Enceladus water vapor plume, Science, 311, Hill, T. W., and D. H. Pontius (1998), Plasma injection near Io, J. Geophys. Res., 103, 19,879 19,885. Ip, W.-H., A. Kopp, and J. Hu (004), On the star-magnetosphere interaction of close-in exoplanets, Astrophys. J., 60, L53 L56. Knight, S. (1973), Parallel electric fields, Planet. Space Sci., 1, Milazzo, M., et al. (005), Volcanic activitity at Tvashtar Catnea, Io, Icarus, 179, Neubauer, F. M. (1980), Nonlinear standing Alfvén wave current system at Io: Theory, J. Geophys. Res., 85, Neubauer, F. M. (1998), The sub-alfvénic interaction of the Galilean satellites with the Jovian magnetosphere, J. Geophys. Res., 103, 19,843 19,866. Neubauer, F. M. (1999), Alfvén wings and electromagnetic induction in the interiors: Europa and Callisto, J. Geophys. Res., 104, 8,671. Piddington, J. H., and J. F. Drake (1968), Electrodynamic effects of Jupiter s satellite Io, Nature, 17, Pontius, D. H., and T. W. Hill (006), Enceladus: A significant plasma source of Saturn s magnetosphere, J. Geophys. Res., 111, A0914, doi:10.109/006ja Porco, C., et al. (006), Cassini observes the active south pole of Enceladus, Science, 311, Preusse, S., A. Kopp, J. Büchner, and U. Motschmann (006), A magnetic communication scenario for hot Jupiters, Astron. Astrophys., 460, Richardson, J. (1998), Thermal plasma and neutral gas in Saturn s magnetosphere, Rev. Geophys., 36, Saur, J. (004), A model for Io s local electric field for a combined Alfvénic and unipolar inductor far-field coupling, J. Geophys. Res., 109, A0110, doi:10.109/00ja Saur, J., and D. Strobel (005), Atmospheres and plasma interactions at Saturn s largest inner icy satellites, Astrophys. J. Lett., 60, L115 L of 11

11 Saur, J., F. M. Neubauer, D. F. Strobel, and M. E. Summers (1999), Threedimensional plasma simulation of Io s interaction with the Io plasma torus: Asymmetric plasma flow, J. Geophys. Res., 104, 5,105 5,16. Schubert, G., J. Anderson, B. Travis, and J. Palguta (007), Enceladus: Present internal structure and differentiation by early and long-term radiogenic heating, Icarus, 188, Southwood, D. J., M. G. Kivelson, R. J. Walker, and J. A. Slavin (1980), Io and its plasma environment, J. Geophys. Res., 85, Spencer, J., et al. (006), Cassini encounters Enceladus: background and the discovery of a south polar hot spot, Science, 311, Tokar, R., et al. (006), The interaction of the atmosphere of Enceladus with Satur s plasma, Science, 311, Waite, J., et al. (006), Cassini ion and neutral mass spectrometer: Enceladus plume composition and structure, Science, 311, F. M. Neubauer, J. Saur, and N. Schilling, Institut für Geophysik und Meteorologie, Universität zu Köln, Albertus-Magnus Platz, D-5093 Cologne, Germany. (neubauer@geo.uni-koeln.de; saur@geo.uni-koeln.de; schilling@geo.uni-koeln.de) 11 of 11