The Locations and Shapes of Jupiter s Bow Shock and Magnetopause

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1 The Locations and Shapes of Jupiter s Bow Shock and Magnetopause Raymond J. Walker 1,2, Steven P. Joy 1,2, Margaret G. Kivelson 1,2, Krishan Khurana 1, Tatsuki Ogino 3, Keiichiro Fukazawa 3 1 Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA Department of Earth and Space Science, University of California, Los Angeles, CA Solar Terrestrial Environment Laboratory, Nagoya University, Toyokawa, Aichi, Japan Abstract. The shape and location of the Jovian bow shock and magnetopause have been studied by using magnetic field observations and global magnetohydrodynamic (MHD) simulations. MHD simulations in which the interplanetary magnetic field (IMF) was set to zero were used to define the boundary shapes and positions and how they depend on solar wind dynamic pressure. Polynomial fits to the simulated boundaries along with spacecraft observations were used to determine the probability of a given position being outside of the bow shock or inside of the magnetopause. The magnetopause and possibly the bow shock have two preferred locations, one representing a compressed magnetosphere and the other an expanded magnetosphere. Variations in the solar wind parameters near Jupiter also show a bimodal distribution but the changes in the solar wind dynamic pressure are not sufficient to account for the observed bimodal distribution of observed magnetopause positions. Internal pressure changes at Jupiter are required. The interplanetary magnetic field also influences the location and shape of the boundaries. In particular, when the IMF is in the B y direction or northward magnetopause reconnection acts to reduce polar flattening. Higher internal pressure at dusk leads to a dawn-dusk asymmetry in the magnetopause position with the boundary being farther from Jupiter on the dusk side. For all the simulations the ratio of the bow shock stand-off distance to that of the magnetopause was less than that at the Earth. INTRODUCTION Solar wind plasma is heated and diverted around planetary obstacles by bow shocks that form upstream of the planets. Studies of the Earth s bow shock indicate that the solar wind magnetosonic Mach number, the interplanetary magnetic field (IMF) and plasma beta influence both the strength and location of the shock [1]. Many years ago Spreiter and colleagues [2, 3] used gas dynamic calculations to demonstrate that the shape of the Earth s bow shock depends on the shape of the obstacle. At the Earth the magnetospheric 95

2 obstacle is determined by pressure balance at the magnetopause between the solar wind and the Earth s magnetic field. At Jupiter on the other hand the magnetospheric obstacle is dominated by an azimuthal equatorial current sheet containing a hot plasma sheet with flows that are atmospherically driven toward corotation. As a result at Jupiter the thermal and dynamic plasma pressures also are important at the magnetopause [4]. The presence of the equatorial current sheet stretches dayside magnetic field lines [5] leading to a more sharply pointed magnetopause at Jupiter than at the Earth. This is thought to result in Jupiter s bow shock being relatively much closer to the magnetopause than is the case at the Earth [6, 3]. The Jovian magnetopause and bow shock locations are highly variable [7, 6]. Jupiter provides us with a relatively handy example of a rapidly rotating magnetospheric obstacle that is very different than that at Earth. Seven spacecraft (Pioneer 1 and 11, Voyager 1 and 2, Ulysses, Galileo and Cassini) have provided observations of Jupiter s magnetopause and bow shock. Together they provide us with a substantial database with which to study the Jovian boundaries. Joy et al. [8] used all of the observations available up to that time to develop probabilistic models of the bow shock and magnetopause. The models were based on a combination of this large data base with results from magnetohydrodynamic (MHD) simulations of the effects of different solar wind dynamic pressures on the bow shock and magnetopause shapes and locations. Surprisingly both the bow shock and the magnetopause had bimodal distributions of location with two most probable positions. In this paper we will review both the simulation and data studies used to determine the probabilistic models of the bow shock and magnetopause shapes and locations. We also will use results from additional simulations to evaluate the possible effects of IMF parameters not included in the previous simulations. In section 2 we briefly review the MHD simulation model and in section 3 we examine simulated boundaries as a function of solar wind dynamic pressure. We review the probabilistic model in section 4. Section 5 contains the simulations of the effects of the IMF on the boundaries. Finally in section 6 we summarize our understanding of the overall configuration of the Jovian magnetopause and bow shock. SIMULATION MODEL Our simulation model of Jupiter's magnetosphere has been described in [9]. In this section we briefly review the simulation model and discuss the runs used to support this study. Starting from a model of the plasma and field configuration near Jupiter, at time t = we placed an image dipole upstream of Jupiter to hasten the formation of the magnetopause and help assure B = throughout the simulation box [1]. We 2 launched an unmagnetized solar wind with a dynamic pressure of ρv sw =.75 npa ( v sw =3 km s -1 ) and a temperature of K from the upstream boundary of the simulation box and solved the resistive MHD equations as an initial value problem by using the Modified-Leap Frog Method described by [11]. The Jovian magnetosphere was modeled on either a point, point or point Cartesian grid with grid spacing of 1.5R J (1R J = 71,492km). In the simulation the magnetic field (B), velocity (v), mass density (ρ) and thermal pressure (p) are maintained at solar wind values at the upstream boundary (x 96

3 = 3R J or 225R J ) while free boundary conditions through which waves and plasmas can freely leave the system are used at the downstream, side, and top boundaries. Symmetry boundary conditions are used at the equator (z = ) for the simulations with zero, northward and southward IMF. For the case with an IMF y-component a full three dimensional box was used and free boundary conditions were used at the bottom boundary. The dipole tilt is set to zero in all of the calculations. At the inner magnetosphere boundary all of the simulation parameters (B, v, ρ, p) are fixed for r < 15R J. The simulation quantities are connected with the inner boundary through a smooth max transition region (15<r<21R J ). The numerical stability criterion is Δt / Δx < 1 where max v g is the maximum group velocity in the calculation domain and v g Δ t is the time step. Since the Alfvén velocity becomes very large near Jupiter we placed the inner boundary of the simulation at 15R J in order to keep the time step from getting too small. The simulation parameters are fixed at the inner boundary. In particular the azimuthal velocity is set to corotate and the pressure and density are set to values determined from the Voyager 1 flyby of Jupiter [12]. This reservoir of plasma provides a source for the Jovian magnetosphere. Typically about AMU s -1 pass through a surface at 22.5 R J and enter the Jovian magnetosphere. The source values for each of the simulations are given in Table 2 of [13]. We have not included mass loading terms in the MHD equations for these simulations. The magnetic field is fixed to values from Jupiter's internal dipole. THE EFFECTS OF SOLAR WIND DYNAMIC PRESSURE ON THE LOCATION AND SHAPE OF THE BOW SHOCK AND MAGNETOPAUSE In Figure 1 we have plotted the thermal pressure in the noon-midnight meridian plane (left) and dawn-dusk meridian plane (right) for four different solar wind dynamic pressures. The IMF was set to zero for these simulations. The pressures range from.45npa to.36npa and were selected by scaling the mean dynamic pressure at Jupiter s orbit,.92npa [6, 4, 8], by factors of two. Both the bow shock and magnetopause move toward Jupiter with increasing pressure. We have used the sharp pressure gradients at the boundaries to tabulate their positions in Table 1. In both the observations and the simulations the distance to the magnetopause and bow shock vary with pressure as a power law between P Dyn -1/4 and P Dyn -1/5 [6, 4, 9, 8]. At noon the ratio of the bow shock distance to the magnetopause distance is between 1.31 and It decreases with increasing pressure. The magnetopause has a marked dawn-dusk asymmetry in which the boundary is closer to Jupiter at dawn than at dusk. The asymmetry increases (Y dawn /Y dusk decreases) with increasing pressure. Within the magnetosphere the equatorial plasma sheet is much thinner at dawn than dusk. This is reflected in an irregularly shaped magnetopause in the = plane (Figure 1, right). However the dawn-dusk asymmetry and irregular shape of the magnetopause are much less evident in the bow shock. The dawndusk asymmetry may actually reverse for small dynamic pressure (Table1). 97

4 Table 1. Distances to the bow shock and magnetopause for the case B 7 =. (npa) Y ave Magnetopause Z Z/Y avs * dawn' Ydusk Y aw Bow Shock Z Z/Y avs * dawn' Ydusk Standoff Ratio x h /x m At Jupiter in addition to the magnetic field, hot internal current sheet plasma and magneto spheric flows contribute to the pressure balance at the equatorial magnetopause. Thus we would expect the magnetopause to be relatively further from Jupiter at the equator than at the poles. This is frequently called polar flattening. We have listed the ratio of the boundary locations along the z-axis to the average value along the y-axis. For low dynamic pressure the simulated magnetopause exhibits polar flattening (Z/Y a vg<l). However this effect goes away for higher pressure (Z/Y avg ~1). Again the bow shock shape is not sensitive to the structure found in the magnetopause. Figure 1. Pressure contours in the noon-midnight merdian plane (left) and dawn-dusk meridian plane (right) for solar wind dynamic pressures of.45npa,.9npa, O.lSnPa, and.36npa. The IMF was set to zero for these simulations. 98

5 EMPIRICAL MODELS OF THE BOUNDARY SHAPES AND POSITIONS There have been several attempts to model Jupiter s bow shock and magnetopause empirically. One approach is to fit actual boundary crossings by assuming that the boundaries are conic sections of revolution [14, 6, 4]. The conic section models are symmetric about the x-axis. In [8] we used the simulation results to argue that observable dawn-dusk asymmetry and polar flattening were probable. We argued even with Galileo orbiter data there are too few boundary crossings to make reliable fits to the data including asymmetries. However, spacecraft collect more information about the boundary locations than just the locations of the actual boundary crossings. For instance if a spacecraft is in the magnetosheath, the bow shock must lie further from the planet and the magnetopause must lie closer to the planet. Therefore we developed probabilistic models of the boundaries that include all of the observations. In [8] we used the global MHD simulations to organize the boundary model. First we identified the boundary locations in the simulations for different solar wind dynamic pressures. We then fit the boundary shapes to a functional form ( z = A + Bx + Cx + Dy + Ey + Fxy ) that does not assume dawn-dusk or x-axis symmetry. We assumed that the parameters A through F were functions of dynamic -1/4 pressure (Pdyn). A linear function of P dyn gives an excellent fit to A, B and C while D, E and F are well fit with a linear function of P dyn. We fit this functional form to boundary positions determined from the simulation results plotted in Figure 1. A number of parameters (the magnetic field, the current density, velocity, pressure etc.) can be used to determine the bow shock and magnetopause positions from the simulations. All of these worked well for the bow shock but the gradient of the speed worked slightly better for the magnetopause along the flanks of the magnetosphere. The boundary fits were optimized for the dayside and within 5R J of the equator. The fits should not be used for x<-25r J [8]. Figure 2 shows the results of the fits to the simulation. The plot contains views in the Y, Z and YZ planes of both boundaries for three dynamic pressures (.2nPa,.98nPa,.227nPa). These represent the 1 th, 5 th and 9 th percentiles of the observed solar wind dynamic pressure at Jupiter s orbit. The plots show the extreme variability in the boundary locations at Jupiter. For instance the magnetopause the standoff distance at the subsolar point varies from over R J to ~5R J. Figure 3 contains the trajectories of 35 Galileo orbits and the Pioneer 1 and 11, Voyager 1 and 2, Ulysses and Cassini flybys. We used magnetometer and plasma wave observations to determine the times at which the Galileo satellite crossed the boundaries while for the other spacecraft we used published crossing times [15, 16, 17, 18]. Then we shaded the trajectories according to whether the spacecraft was in the solar wind (blue), the magnetosheath (green) or the outer magnetosphere (red). Trajectories from which data are not available are black. Even with orbiter data there are relatively few boundary crossings. Despite the relatively few boundary crossings, the Galileo orbiter observations significantly improve the probabilistic determination of the boundary locations. However, spacecraft collect more information about the boundaries than just the location of the actual crossings. For instance if a spacecraft is in the magnetosheath, the shock must lie further from the planet and the magnetopause must lie closer. All of the data can be used 99

6 Equatorial Noon-Midnight Dawn-Dusk 2 a Bow Shock 2 Magnetopause d Y Y b e 3 3 Z 2 Z c f 3 3 Z 2 Z Y -2-2 Y figure 4 Figure 2. Fits to the bow shock and magnetopause shapes in Figure 1 evaluated at the 1th (outer), 5th(middle) and 9th percentiles (inner) of the observed solar wind dynamic pressure in the three axis planes (rows). The distances are in Jovian radii. [8]. to establish the probability of finding the bow shock or magnetopause at different locations. Ten minute samples of the data from all of the spacecraft were collected into bins whose shapes were determined by the fits to the simulation results in Figure 2 and whose subsolar standoff distances varied by 4R J. The fraction of data inside or outside of a boundary was determined for each bin. In Figure 4 the fraction of observations outside the bow shock is plotted in the left column while the fraction inside the magnetopause is plotted on the right. The error bars mark the actual data points. The error bars that are closest together give the probable error of the mean. The outer error bars were determined by randomly selecting 1 subsets of the data each with 1% of the data and repeating the analysis. The error bars give the spread in the results from these

7 calculations. Surprisingly both the bow shock and magnetopause positions have bimodal distributions with two preferred boundary positions. Fits to a bimodal distribution (sum of two Gaussian distributions) give peaks in the bow shock position at 73R J and 18R J with a standard deviation of 1R J in both cases. The magnetopause positions are 63R J and 92R J with standard deviations of 4R J and 6R J respectively. Single Gaussian fits were used to create the solid lines. An F-test was used to compare the variances of the deviations from the Gaussian distribution to those of the bimodal distribution. For the magnetopause the improvement in the fit of the bimodal distribution over the Gaussian distribution was at the 99.9% confidence level while for the bow shock it was at the 89.8% confidence level. We have plotted the equatorial intercepts of the model boundaries in Figure 3. The dark shading shows the region between the 25 and 75 percentile contours of being in the solar wind. The lightly shaded regions denote one sigma (standard deviation) bands about the two peaks in the magnetopause distribution. Finally we analyzed all of the solar wind observations (interplanetary magnetic field, dynamic pressure and Alfvén Mach number) near Jupiter [8]. We found a bimodal distribution of solar wind parameters however the magnitude of the bimodal solar wind pressure distribution was too small to account for the bimodal distributions in the boundary positions. Internal pressure changes also are required ULY O CAS (Y 2 +Z 2 ) 1/2 33 P11 O VG1 i - 1 ULY i P1 i VG2 i P11 i -2-3 VG2 O VG1 O P1 O Solar Wind Magnetosheath Magnetosphere Joy 25/75 BS Joy ±1-σ MP -2 - JA Figure 1 Figure 3. Trajectories of spacecraft near Jupiter shaded to show the region through which the spacecraft were traveling. Trajectory segments in the solar wind are blue, those in the magnetosheath are green and those in the magnetosphere are red. Trajectories for which data were not available at the time of the Joy et al. [22] study are black. The dark gray shading shows the region between the 25 and 75 percentile probabilistic models for the bow shock. The light gray regions on the right show plus or minus one standard deviation about the two preferred locations of the magnetopause. (After [8]) 11

8 Figure 4.The fraction of observations outside of the bow shock boundary surfaces (left column) and the fraction of observations inside of the magnetopause boundary surfaces (right column). Solid lines are single distribution fits and the dashed lines are bimodal distribution fits. The error bars are discussed in the text. (After[8]) THE EFFECT OF THE IMF ON THE SHAPES AND POSITIONS OF THE BOUNDARIES By the time the solar wind reaches Jupiter the spiral of the IMF has wound up so tightly that the magnetic field is mainly in the Jovicentric Solar Equatorial (JSEq) y- direction. However, to simplify our investigation of the influence of the IMF on the Jovian boundaries we will start by assuming that the IMF is oriented in the north-south direction. In Figure 5 the thermal pressures for weak southward (BBz=-.15nT) and northward (B zb =.15nT) IMF have been plotted in the Y= (top two panels) and = (bottom two panels) planes. The locations of the magnetopause and bow shock are tabulated in Table 2. Recall in the following that Jupiter s intrinsic magnetic field is opposite to that of the Earth. For northward IMF reconnection at the dayside magnetopause moves both it and the bow shock closer to Jupiter than when BBIMF=. Conversely for southward IMF the boundaries move away from Jupiter. However the bow shock is relatively closer to the magnetopause for both southward and northward IMF than for the zero IMF case. For northward IMF the dawn magnetopause is closer to Jupiter than the dusk magnetopause while the opposite is true for southward IMF. In both cases Y avg >Z but it is less so for the northward IMF case. In Figure 5 the high latitude magnetosheath for B ZB > has a hat like region of increased thermal pressure. A close examination of the magnetic field lines in the hat shows that this region is on field lines that have been opened by dayside reconnection. Since some of the most dramatic changes occurred when the IMF was northward we have carried out a pair of numerical experiments in order to quantify better the effects of the IMF. In the first experiment we set the dynamic pressure to the mean at Jupiter (.9nPa) and modeled the magnetosphere by assuming that the mean IMF (.8nT), [8] was entirely northward. In Table 3 we have listed the distances to the boundaries from this experiment and from one with the same dynamic pressure but half the magnetic field. 12

9 For the larger IMF the subsolar magnetopause is further eroded while the distance to the northern magnetopause increases dramatically. For the simulation with B Z =.84nT the bow shock exits the top of the simulation box just sunward of = so the Z value is a lower limit. The most dramatic change with larger IMF is that Z/Y avg. >1 for both the magnetopause and the bow shock. This is a direct result of increased dayside reconnection and the addition of open flux to the magnetosheath. Table 2. Boundary positions for southward and northward IMF Standoff BBZ(nT) Magnetopause Bow Shock Ratio Y avg Z Z/Y avg Y dawn /Y dusk Y avg Z Z/Y avg Y dawn /Y dusk b / m Table 3. Boundary locations for constant dynamic pressure and decreasing northward IMF. BBZ(nT) Magnetopause Bow Shock Standoff Ratio Y avg Z Z/Y avg Y dawn / b / m Y dusk avg avg dawn Y dusk >225 > Next we held the northward IMF constant at.42nt and decreased the dynamic pressure by a factor of ~4 from.9npa to.2npa. The boundary locations can be found in Table 4. The largest effect of lowering the pressure for constant magnetic field is to increase Z/Y avg. at both the magnetopause and bow shock. Table 4. Boundary locations for constant IMF and decreasing dynamic pressure. P Dyn (npa) Magnetopause Bow Shock Standoff Ratio Y avg Z Z/Y avg Y dawn /Y dusk Y avg Z Z/Y avg Y dawn /Y dusk b / m In Figure 6 we have plotted the pressure in the YZ plane from a simulation for which the IMF was in the Y-direction (BBy=.42nT) pointing toward dusk. The solar wind dynamic pressure was.9npa. The corresponding fits to the IMF B = bow shock and magnetopause positions are shown with solid and dashed lines respectively [8]. The entire magnetosphere rotates about the sun-jupiter line for B Y. At high latitudes the boundaries are farther from Jupiter than for BBIMF= while nearer the equator they are closer to Jupiter. Reconnection can occur near the equator on the flanks of the magnetopause when the IMF points in the y-direction. This can change the shape of the obstacle. The addition of IMF B y does not change the standoff distance at the bow shock or the magnetopause. The standoff ratio remains

10 Figure 5. Pressure contours in the noon-midnight meridian plane (top two plots) and the dawn-dusk meridian plane (bottom two plots) from simulations with the northward and southward IMFs of.15nt and dynamic pressure of.11npa. SUMMARY AND DISCUSSION We have used a combination of global magnetohydrodynamic simulations and observations to form models of Jupiter s magnetopause and bow shock. Rather than fitting observed boundary crossings, we used all of the spacecraft observations at Jupiter to determine the boundary positions in terms of the probability of being outside of the bow shock or inside of the magnetopause. We used the global MHD models to define 14

11 boundary models, to define the boundary shapes and locations and to determine how they vary with solar wind dynamic pressure. Figure 6. Pressure contours in the dawn-dusk meridian plane for a simulation with a.42nt IMF in the y-direction and solar wind dynamic pressure of.9npa. The solid line gives the bow shock position and the dashed line gives the magnetopause position from the fit to the MHD simulations for.9npa and zero IMF. 2 G29 Solar Wind Magnetosheath 2 G29 Solar Wind Magnetosheath Magnetosphere Magnetosphere (Y 2 +Z 2 ) 1/2 15 C3 I31 Cassini I32 (Y 2 +Z 2 ) 1/2 15 Cassini C3 I32 5 I33 5 I31 I33 J35 A34 J35 A Figure Figure 17 Figure 7. Trajectories of spacecraft near Jupiter not included in the Joy et al. [8] study. The trajectories have been labeled to indicate the region in which the spacecraft was flying. Trajectory segments in the solar wind are blue, those in the magnetosheath are green and those in the magnetosphere are red. There were no data along the black trajectory segments. The gray shading on the left shows the region between the 25 and 75 percentile probabilistic models for the bow shock. The gray regions on the right show plus or minus one standard deviation about the two preferred locations of the magnetopause. Distances are in Jovian radii. 15

12 The magnetopause at Jupiter and possibly the bow shock have two preferred locations, one representing a compressed magnetosphere and the other an expanded magnetosphere. The solar wind dynamic pressure in the neighborhood of Jupiter during the time interval under study also has a bimodal distribution. While this contributes to the bimodal distribution observed in the boundary positions the dynamic pressure changes are too small to account for the large variation in the magnetopause position [8]. Internal pressure changes also are required. The bimodal distribution is less clear for the bow shock. The speed with which the bow shock can adjust to changes in either the solar wind or the obstacle will smear out the observed distribution. In the left column of Figure 7 we have shaded the region between the 25 and 75 percentiles of being in the solar wind from the probabilistic model along with Galileo observations from orbits not included in the original study and observations from the Cassini flyby of Jupiter. Most of the magnetosheath observations are from the region between the 25% and 75% curves. However, some of the observations especially those from the Galileo G29 orbit suggest that the bow shock may extend farther from Jupiter than in the MHD simulations. In the right panel we have plotted two bands of magnetopause positions. The shaded areas are centered on the two preferred locations of the magnetopause. The shading extends plus or minus one sigma (standard deviation) about the two preferred locations. Here the models seem to be in reasonable agreement with the new observations. The boundary shapes on which the probabilistic models are based did not include the IMF. We have examined simulations with both a purely north-south IMF and with an IMF in the y-direction. Although the dynamic pressure has the largest effect the inclusion of a non-zero IMF can make smaller changes in the location of the boundaries. For instance for northward IMF dayside reconnection erodes the position of the magnetopause and as a result the bow shock moves toward Jupiter as well. For southward IMF the reconnection site moves to high latitudes and the boundaries move away from Jupiter. However for all of the simulations the ratio of the standoff distances remains significantly less than the typical values at the Earth. Both the solar wind Mach number and the shape of the obstacle can influence the standoff distance. We compared simulations of the Earth and Jupiter at the same Mach number and found that the ratio at Jupiter ( ) was smaller than at Earth ( ). This indicates that the obstacle shape is responsible for the differences in the standoff ratio and that Jupiter s magnetopause is less blunt than the Earth s. We would expect Jupiter s boundaries to have strong polar flattening because of the equatorial current sheet. However, the simulations suggest that this is not always the case. Clear polar flattening is evident in zero IMF simulations for below average dynamic pressure and when the IMF is southward. For above average dynamic pressure the flattening decreases. For northward IMF dayside reconnection adds flux to the lobes thereby moving the boundaries in the z-direction away from Jupiter reducing or eliminating the polar flattening. At Jupiter s orbit the IMF is primarily in the y-direction. For an IMF in the y-direction we also find that the polar flattening is reduced. In the simulations the magnetopause is generally found closer to Jupiter at dawn than at dusk. This effect becomes smaller for smaller dynamic pressure when the magnetopause is farthest from Jupiter. The dawn-dusk asymmetry seems to be related to higher thermal pressure on the dusk side of the magnetosphere (Figures 1, 5 and 6). 16

13 The bow shock shape and position also are influenced by the IMF. The largest effect seems to be related to changes in the obstacle shape caused by reconnection. This can be seen most dramatically in Figures 5 and 6 and Table 3. The addition of open magnetic flux to the tail lobes and magnetosheath caused the hat in Figure 5 and the large increase in the z position of the shock in Table 3. Similarly the inclusion of an IMF B y caused the magnetopause to twist (Figure 6) and that resulted in a twisted bow shock. For given solar wind dynamic pressure and IMF conditions we have analyzed the bow shock and magnetopause locations and shapes by assuming steady-state conditions. In particular we ran the simulations until quasi-steady configurations resulted. However, the Jovian magnetosphere in the simulations can be very dynamic [19]. Changes in the dynamic pressure and IMF lead to large amplitude waves which distort the boundary shapes as the system responds to the changes. Acknowledgements: We would like to thank Mr. Joseph Mafi for help with the data processing and display. Helpful comments by Dr. Lee Bargatze are gratefully acknowledged. The work at UCLA was supported by grant NAG The work at Nagoya University was supported by grants in aid from the Ministry of Education, Science and Culture. Computing support was provided by the Computer Center of Nagoya University. REFERENCES 1. Farris, M. H., and C. T. Russell, Determining the standoff distance of the bow shock: Mach number dependence and use of models, J. Geophys. Res., 99, 17,681, Spreiter, J. R., A. L. Summers, and A. Y. Alksne, Hydromagnetic flow around the magnetosphere, Planet. Space Sci. 14, 223, Stahara, S. S., R. R. Rachiele, J. R. Spreiter, and J. A. Slavin, A three dimensional gas dynamic model for the solar wind past non-axisymmetric magnetospheres: Application to Jupiter and Saturn, J. Geophys. Res., 94(A1), 13,353, Huddleston, D. E., C. T. Russell, M. G. Kivelson, K. K. Khurana, and L. Bennett, Location and shape of the Jovian magnetopause and bow shock, J. Geophys. Res., 13(E9), 2,75, Engle, I. M., and D. B. Beard, Idealized Jovian magnetopause shape and field, J. Geophys. Res, 85(A2), 579, Slavin, J. A., E. J. Smith, J. R. Spreiter, and S. S. Stahara, Solar wind flow about the outer planets: Gas dynamic modeling of the Jupiter and Saturn bow shocks, J. Geophys. Res., 9(A7), 6275, Smith, E. J., R. W. Fillius, and J. H. Wolfe, Compression of Jupiter s magnetosphere by the solar wind, J. Geophys. Res., 83, 4733, Joy, S. P., M. G. Kivelson, R. J. Walker, K. K. Khurana, C. T. Russell and T. Ogino, Probabilistic models of the Jovian magnetopause and bow shock locations, J. Geophys. Res., 17(A1), 139, doi 1.129/21JA9146, Ogino, T., R. J. Walker, and M. G. Kivelson, A global magnetohydrodynamic simulation of the Jovian magnetosphere, J. Geophys. Res., 13, 225,

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