A simple empirical model ofthe equatorial radial eld in Jupiter s middle magnetosphere, based on spacecraft y-by and Galileo orbiter data

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1 Planetary and Space Science (2) A simple empirical model ofthe equatorial radial eld in Jupiter s middle magnetosphere, based on spacecraft y-by and Galileo orbiter data E.J. Bunce, P.G. Hanlon 1, S.W.H. Cowley Department of Physics & Astronomy, University of Leicester, Leicester LE1 7RH, UK Received 31 May 1; received in revised form December 1; accepted 28 March 2 Abstract In this paper we consider empirical models ofthe radial eld and azimuthal current in Jupiter s middle magnetosphere region, at distances in the range 4 R J. We rst ofall compare the model derived previously by Bunce and Cowley (Planet. Space Sci. 49 (1) 261) using Pioneer, Voyager and Ulysses y-by data, with a combined data set that now also incorporates data from the rst twenty orbits of the Galileo orbiter. The overall RMS fractional residual is found to be 12.7%, such that the model does provide a good description of the combined data set. In particular, it is shown that the Galileo data also exhibit the same local time asymmetry as found in the y-by data, in which the radial eld (and azimuthal current) are stronger at a given radial distance on the nightside compared with the dayside. However, it is also shown that ifthe combined data are separated into 2 h bins oflocal time and then tted to individual power law curves, the overall RMS fractional residual is reduced to 7.7%, thus showing scope for improvement in the empirical model. Based on the combined data set, in our revised model the eld is taken as asymmetric outside of14: R J, and to fall with radial distance with an exponent which is taken to vary sinusoidally with local time, varying between 1: at noon and 1: at midnight, such that the eld becomes increasingly asymmetric with increasing distance. The overall RMS residual for this four-parameter model is found to be 9.7%, only slightly higher than that ofthe free-ts to the 2 h MLT binned data, and representing a worthwhile improvement over the original Bunce and Cowley model. The implied divergence ofthe azimuthal current for the revised model peaks at 1 ka R 2 J near the dawn-dusk meridian at a radial distance of 23 R J. The implied dierence in the total azimuthal current owing in the current sheet between and R J at midnight compared with noon is 19 MA, in a total (at dawn and dusk) of9 MA.? 2 Published by Elsevier Science Ltd. 1. Introduction Gledhill (1967) was the rst to postulate that Jupiter s near-planet equatorial magnetic eld lines would be radially distended, due to centrifugal forces associated with rapid planetary rotation and ionospheric plasma loading. Subsequently, the rst in situ measurements ofjupiter s magnetic environment, made during the Pioneer- and -11 and Voyager-1 and -2 spacecraft y-bys in the 197s, indeed showed the signatures ofthe radial distension ofthe magnetic eld. The earliest studies based on the data from these y-bys demonstrated the existence ofa thin equatorial azimuthal current sheet owing in an eastward direction, which Corresponding author. Tel.: ; fax: address: emma.bunce@ion.le.ac.uk (E.J. Bunce). 1 Now at Blackett Laboratory, Imperial College, London SW7 2BZ, UK. signicantly distorts the planetary eld lines at distances of R J and beyond (Smith et al., 1974, 197, 1976; Ness et al., 1979a, b). However, they also showed that the principal plasma source for the current sheet was not Jupiter s ionosphere, but the moon Io, which orbits at a jovicentric distance of :9 R J (Krimigis and Roelof, 1983). The next spacecraft to y past Jupiter was Ulysses in 1992 (Balogh et al., 1992), and more recently the Galileo orbiter arrived in 199 to commence a long-term study ofthe Jovian system (Kivelson et al., 1992). The magnetic eects ofthe equatorial current sheet, or magnetodisc, have been found to be present at all local times investigated by these spacecraft. The local time coverage ofthe ve Jupiter y-bys mentioned above, and the rst orbits ofthe Galileo mission (between 1996 and 1999) are shown in Fig. 1a, where the spacecraft trajectories are shown in Jupiter Solar Orbital (JSO) coordinates, i.e. X (R J ) is positive sunwards, and Y (R J ) is orthogonal to X and in the plane ofjupiter s orbit. The Pioneer and Voyager y-bys covered the dawn sector of /2/$ - see front matter? 2 Published by Elsevier Science Ltd. PII: S (2)11-9

2 R J 6 79 E.J. Bunce et al. / Planetary and Space Science (2) G(throughC) P P11 U V1 V2 V2 BS V2 MP X (R J ) (a) Y (R J ) nt (b) Fig. 1. (a) Trajectories ofthe rst orbits ofthe Galileo orbiter along with the ve y-by spacecraft relative to Jupiter, shown in Jupiter Solar Orbital coordinates. X points positive sunwards, and Y is orthogonal to X and in the plane ofjupiter s orbit. The solid line indicates the Galileo orbiter and the dashed lines indicate the y-by spacecraft. The individual y-by spacecraft are distinguishable by the varying symbols shown in the key. A heavy dashed line depicts a model bow shock, and a model magnetopause is shown by the heavy solid line. Both model positions are derived from the Voyager-2 data. The region ofinterest for this paper, 4 R J, is highlighted by the grey annulus in the centre ofthe plot. This gure was kindly provided by Joe Ma ofthe Planetary Data System, UCLA. (b) Plot ofthe half-hour averages ofthe magnetic components measured outside the current sheet during the rst orbits ofthe Galileo orbiter and the ve y-bys ofpioneer- and -11, Voyager-1 and -2, and Ulysses, from which the VIP4 planetary eld model (Connerney et al., 1998) has been subtracted. The averages have been projected onto the magnetic equatorial plane and rotated through 9 to indicate the approximate direction and strength ofthe corresponding current. Those elds measured north ofthe current sheet have been rotated 9 anti-clockwise, while those measured to the south have been rotated in a clockwise sense. Dashed lines indicate the distance from the centre of the planet (R J ), and local time is also shown. The individual spacecraft are identiable by comparison with Fig. 1a. At the bottom right of the plot is the scale for nt. the magnetosphere from near noon (Pioneer-11 outbound) to post-midnight (Voyager-2 outbound), while Ulysses passed through the pre-noon sector inbound and made unique observations ofthe dusk meridian magnetosphere outbound. Presently available data from the Galileo mission extend from dawn through to midnight, and some way into the evening sector. The jovigraphic latitudes ofthese trajectories were near-equatorial in the main, except for the outbound passes ofpioneer-11 and Ulysses, which exited near noon at 33 N and near dusk at 37 S, respectively. Also

3 E.J. Bunce et al. / Planetary and Space Science (2) shown in the gure are the positions ofthe magnetopause and bow shock as modelled from the Voyager-2 data (Ness et al., 1979b). The shaded region also indicates the domain ofinterest for this study, that is, the middle magnetosphere region between and R J. On the dayside, the magnetopause extends on average to 6 R J as shown here, but is highly variable depending upon the upstream solar wind conditions. On the nightside the magnetospheric tail extends to R J and has a diameter of R J (Ness et al., 1979c). As indicated above, it is understood that the dynamics of the Jovian middle magnetosphere are governed by the Io plasma source, located deep within the equatorial magnetosphere at :9 R J. The current in the equatorial magnetodisc is then carried (a) by the inertia current ofnear-corotating cold torus plasma which slowly diuses outwards, and (b) by the pressure-gradient current oflow density hot plasma which slowly diuses inwards (Hill, 1979; Vasyliunas, 1983; Caudal, 1986; and references therein). This azimuthal current sheet denes what has become known as the Jovian middle magnetosphere region, which extends from R J (the inner edge ofthe Io plasma torus) to within 1 R J of the magnetopause on the dayside. The radial range ofthe current sheet on the dayside ofthe planet is thus controlled by the dynamic pressure ofthe solar wind, which causes the magnetopause to be compressed or to expand. On the nightside the magnetosdisc merges at larger distances with the cross-tail currents which are associated with the magnetotail, and hence with solar wind-magnetosphere coupling (Ness et al., 1979c). In the middle magnetosphere, the current disc is located close to the magnetic equatorial plane and thus displays a quasi-sinusoidal north south oscillation as the magnetic dipole, tilted by from the spin axis, rotates with the planet. As the relative full thickness of the current sheet (between 2 and 8 R J, for example see Smith et al. (1976), Goertz et al. (1976), Connerney et al. (1981), Behannon et al. (1981), Acuña et al. (1983), Staines et al. (1996), Dougherty et al. (1996)) is much smaller than the characteristic size ofthe magnetosphere, the radial eld undergoes sharp reversals across the current sheet from positive values in the north, to negative values to the south. At the inner edge ofthe current sheet, the planetary eld dominates that due to the current sheet alone, but since the dipole component ofthe planetary eld falls as r 3 whilst that of the current sheet is found to fall o much less rapidly, between r 1 and r 2 (Barish and Smith, 197; Goertz et al., 1976; Jones et al., 1981; Behannon et al., 1981; Connerney et al., 1981; Khurana, 1997; Bunce and Cowley 1a), the current sheet eld becomes dominant beyond 1 R J. In recent independent studies, Bunce and Cowley (1a) using magnetometer data from the ve y-by missions mentioned above, and Khurana (1) also incorporating data from the Galileo orbiter spacecraft, have shown that the azimuthal current in the outer middle magnetosphere depends upon local time. For example, at distances of R J the current is approximately twice as strong at a given radial distance at midnight than at the same distance at noon. This phenomenon was rst noticed by Goertz (1978) in a comparison ofthe Pioneer- inbound and outbound data. The diering gradients ofradial eld fall-o with distance at the two local times ( MLT inbound and MLT outbound for Pioneer-) were discussed in terms of the asymmetrical compressive and conning eect the solar wind dynamic pressure has on the magnetosphere, compressing the ux tubes on the dayside but allowing them to stretch out on the nightside. This stretching further distends the magnetic eld lines, hence increasing the azimuthal current, on the nightside. Bunce and Cowley (1a) favour this interpretation, which then indicates that azimuthal current closure is enforced via radial currents owing wholly within the current sheet, owing away from the planet at dawn and towards the planet at dusk. Khurana (1) prefers to attribute the divergence of the azimuthal current to an Earth-like partial ring current closing via region-2 type eld-aligned currents, owing towards the planet at dawn, closing through the jovian ionosphere and owing away from Jupiter at dusk. Here, however we focus on the central fact ofthe azimuthal asymmetry ofthe azimuthal current itself, and leave further considerations of closure to future study. Whilst previous models ofthe middle magnetosphere current sheet have been based upon axial symmetry (e.g. Connerney et al., 1981; Khurana, 1992), and are indeed an excellent indicator ofthe jovian eld in the inner region ofthe middle magnetosphere, it is now evident that outside this region, roughly beyond 1 R J, the current is signicantly dependent on MLT as outlined above. Bunce and Cowley (1a) presented a simple empirical model ofthe near-equatorial radial component ofthe eld in the region between and R J, valid for all magnetic local times, based on the y-by data. This model (herein referred to as the BC model) serves as a useful empirical tool for modelling the middle magnetosphere, and in particular for quantifying the divergence of the azimuthal current. How much current is diverted out ofor into the azimuthal current ow, combined with similar information on the radial current derived from the azimuthal component of the magnetic eld, provides the necessary information from which the eld-aligned currents (FACs) connecting to the ionosphere can be calculated (Hill, 1979; Vasyliunas, 1983; Khurana and Kivelson, 1993; Bunce and Cowley, 1b). The nature ofthe FACs connects in turn with other important magnetospheric phenomena such as the jovian auroras and the decametric radio emission (Cowley and Bunce, 1). In this paper we compare the BC model ofthe radial eld B with newly-available eld data from the Galileo orbiter, as a function of both local time and radial distance. We show that while the BC model is generally in good agreement with the Galileo data, some renements are nevertheless suggested that bring the model into better accord with the

4 792 E.J. Bunce et al. / Planetary and Space Science (2) combined y-by and orbiter data set. We thus derive such a model, using techniques similar to those employed by BC. As seen in Fig. 1a, inclusion ofthis additional data enhances the overall coverage ofthe middle magnetosphere region. In particular, Galileo signicantly increases the quantity ofdata in the dawn and pre-midnight sectors ofthe magnetosphere. However, the evolution ofthe Galileo orbits have not as yet provided new data from the dayside middle magnetosphere as perijove lies within the inner magnetosphere at this local time. 2. Data analysis 2.1. Current sheet eld averages We begin our study by presenting magnetic eld vectors observed during the ve jovian y-bys and the rst orbits ofthe Galileo mission as discussed above. All data were supplied by the Planetary Data System at UCLA, at s resolution for Pioneer-11 and Voyager-2, 48 s for Voyager-1, and 1 min for Pioneer- and Ulysses. Due to telemetry constraints the Galileo data are only available at high time resolution approximately halfofthe time, and as such there are two distinct time resolutions ofmagnetic eld data. The real time survey (RTS) mode supplies data at 24 s time resolution, whilst the memory read out (MRO) mode provides 32 min averaged data. The VIP 4 planetary eld model (Connerney et al., 1981) has been subtracted from the data to leave only those elds which are due to the external currents (principally the equatorial current sheet). In the case ofthe spacecraft trajectories lying close to the jovigraphic equatorial plane, the current sheet passes completely across the spacecraft twice per h rotation period. Correspondingly, it can be seen in the magnetic eld data that the radial eld cycles between intervals ofrelatively steady positive and negative values, interspersed with periods ofeld uctuation and reversal when the spacecraft crossed through the equatorial current sheet. However, in the case ofthe non-equatorial Pioneer-11 inbound (14 S), Pioneer-11 (33 N) outbound, Pioneer- outbound (11 N), and Ulysses (37 S) outbound passes, the measured eld is generally dominated either by positive or negative radial components depending upon the latitude of the spacecraft, the former corresponding to a location north ofthe current sheet and the latter to the south. The radial eld then exhibits depressed values and=or enhanced uctuations indicative ofhot plasma currents at h intervals when the spacecraft approached the magnetic equatorial plane. At other times, when the spacecraft were at larger distances from the equator, the elds are instead stronger and smoothly varying, indicating only weak local currents and a consequent location outside ofthe current sheet. Ignoring periods when enhanced magnetic variations are present, therefore, we have averaged the eld components from both equatorial and non-equatorial passes over min intervals, and take these values to represent conditions at the similarly averaged locations outside ofthe current sheet. The signature ofthe changing latitude ofthe spacecraft outside the current sheet will be discussed further below. Collectively, the data are shown in Fig. 1b. In order to indicate the overall current ow in the equatorial regions, we show the min averages ofthe total eld vectors projected onto the magnetic equatorial plane. The vectors have then been rotated through 9 to indicate the approximate direction ofthe corresponding equatorial current. To take account ofthe reversal in the equatorial eld components across the current sheet, those elds measured north ofthe current sheet have been rotated 9 anticlockwise, while those measured in the southern hemisphere have been rotated 9 clockwise. In keeping with the previous study by Bunce and Cowley (1a), every eort has been made to ensure that averages were taken only when the spacecraft were outside ofthe current carrying region. Since we are interested in estimating the total azimuthal current, inclusion ofreduced values obtained when the spacecraft in fact remained in the current-carrying layer would result in under-estimates ofthe total current. Hence we have chosen to exclude those Galileo data from the MRO mode, whose time resolution was too low to distinguish clearly between such times. Ifthe current layer is then considered to be a quasi-innite sheet with perturbation elds ofequal magnitude but opposite direction on either side, a perturbation eld of nt corresponds to an azimuthal sheet current of intensity 1:1 MAR 1 J, integrated through the full sheet. The contributions ofindividual spacecraft in Fig. 1b are identiable by comparison with Fig. 1a. The inbound passes ofpioneer- and -11, Voyager-1 and -2, and Ulysses are all in the pre-noon sector, and the outbound passes are all on the nightside, with the exception ofpioneer-11 outbound which is near noon. The Galileo passes (G1-2, C3, E4, E6, G7, C9-, E11-12, E16-19) included in this study mainly lie between 9 and MLT. As described above, all passes are near-equatorial (within ± ofthe jovigraphic equator), with the principal exceptions being Pioneer-11 inbound and outbound, Pioneer- outbound and Ulysses outbound as noted above. We see in Fig. 1b that the sense ofthe azimuthal currents are eastward, associated with the radial distension ofthe magnetic eld lines away from the planet in the middle magnetosphere. The larger values ofthe azimuthal current on the nightside at a particular distance compared with the dayside values are evident. Outward radial currents are also apparent on the dawn side ofthe magnetosphere, consistent with the magnetic eld line lagging out ofmeridian planes. However, on the dusk side ofthe magnetosphere the outward ( lagging ) currents evolve into inward ( leading ) currents in the outer region at larger distances beyond R J, which we take to be associated with solar wind induced eects including that due to the asymmetrical conning eect of the solar wind on the magnetosphere, as mentioned in the introduction.

5 E.J. Bunce et al. / Planetary and Space Science (2) Latitude-correction of non-equatorial radial eld data Since the equatorial current sheet is ofnite spatial extent, the radial eld outside the sheet at a given distance will fall slowly with height above the sheet on either side. The eld values which give the best indication ofthe total azimuthal current are those obtained at the outer edge ofthe sheet, while those obtained at higher latitudes will thus provide an under-estimate. Bunce and Cowley (1a) made an approximate correction for this eect using a simple theoretical model, and in this work we follow the same procedure. The benets ofperforming such a correction are dualistic. First, we reduce the latitude-related scatter in the radial eld proles, thus allowing a more accurate representation by least-squares tting. Second, we allow inclusion of the non-equatorial data. We are required particularly to correct those data from the Pioneer-11 outbound and Ulysses outbound passes, ifthey are to be included in this study, but we should also note that much ofthe data in the y-by proles benet from (albeit modest) corrections. As previously discussed, the Galileo orbiter data were taken close to the jovigraphic equator throughout most ofthe orbits and therefore do not require substantial correction, although for consistency all data has undergone the same procedure. The approach is to simply map the eld measurements to the edge ofthe current sheet using mapping factors obtained from the approximate forms of the Connerney et al. (1981) model described in a recent paper by Edwards et al. (1). For precise details ofthis procedure the reader is directed to Bunce and Cowley (1a), as the method adopted for correction here is identical. Mapping factors depend on radial distance, but are typically 1: for a latitude of, increasing to 1:2 for 1, such that the corrections are not substantial. In order to demonstrate the eect oflatitude correction, we present in Fig. 2 plots ofthe radial eld versus radial distance in a log log format. Throughout this paper we employ cylindrical coordinates referenced to the magnetic dipole axis. Thus the radial eld is the cylindrical component perpendicular to the dipole axis, and the radial distance is the perpendicular distance from that axis. In panels (a) and (b) we show min current sheet radial eld averages (i.e. the radial eld with the planetary eld subtracted), denoted by B, before latitude-correction for the 1 h MLT intervals 6 7 and 8 9 MLT, respectively. The same data is shown after correction in panels (c) and (d). Averages derived from Galileo data are indicated by stars, while the y-by data employed previously by BC are shown by diamonds. For the intervals shown, y-by data is present only in panels (b) and (d), where it was derived principally from the Pioneer-11 inbound pass. In each panel ofthe gure the BC empirical model prole corresponding to the limits of the MLT bin are shown by the solid lines, while the extreme proles ofthe model are indicated by the dashed lines, for noon (lower) and midnight (upper), respectively. The eect ofvarying magnetic latitude at the spacecraft is particularly evident in the y-by data shown in panel (b), where individual groups ofpoints form partial U -shaped patterns. These groups ofpoints correspond to averages derived from individual spacecraft excursions outside of the current sheet during the planet s rotation, such that averages obtained near the start and end ofeach group correspond to values obtained at lower magnetic latitudes relatively close the edge ofthe current sheet, while those in the middle were obtained at higher magnetic latitudes at larger distances from the current sheet. The eect offalling radial elds with distance from the current sheet is thus very clear in these y-by data (in the present case reaching magnetic latitude near the centre ofeach group), and the need to introduce a latitude correction is correspondingly clear. However, with this introduction, the latitude eect is seen to be present with reduced amplitude in the Galileo data as well, in both panels (a) and (b). Panels (c) and (d) then show the eect ofapplying the latitude correction factor derived from the Connerney et al. (1981) model which, as indicated above, maps these data values to the edge ofthe current sheet. It can be seen that the scatter in both data sets is signicantly reduced, with two immediate eects. First, the Galileo and y-by data are brought into much closer agreement with each other. Second, both data sets are brought into better general (ifnot perfect) agreement with the empirical BC model, which, as indicated above, was derived from and intended to represent the latitude-corrected radial eld at the edge ofthe current sheet. All ofthe data we will henceforth analyse and display in this paper will thus correspond to latitude-corrected radial eld averages mapped to the edge ofthe current sheet, which will be termed equatorial radial eld averages. We nally note at this juncture that data from the inbound portion ofthe Voyager-1 y-by have been excluded from this study. These data values are found to be signicantly depressed in magnitude compared with corresponding data from other y-bys (e.g. Pioneer-11 outbound and Voyager-2 inbound), suggesting that the spacecraft may never have emerged from the current sheet during this pass. This eect was noted previously by Connerney et al. (1981) in the comparison with their empirical model. Here, therefore, we will not employ these data. 3. Comparison of the combined y-by and Galileo data with the Bunce and Cowley empirical model In this section we will compare min-averaged values ofthe equatorial radial current sheet eld, denoted here by B, with the BC model. Data from both Galileo and the y-bys will be shown in order to facilitate inter-comparison, where the Galileo data correspond to orbits G-1 to C- inclusive (between 1996 and 1999). We recall that the BC

6 794 E.J. Bunce et al. / Planetary and Space Science (2) MLT 8-9 MLT B ' ρ / nt B ' ρ / nt (a) Before "latitude-correction" 6 (b) Before "latitude-correction" MLT 8-9 MLT B ' ρ / nt B ' ρ / nt (c) After "latitude-correction" 6 (d) After "latitude-correction" 6 Fig. 2. Log log plots ofthe -min averaged radial eld component B outside the current sheet, versus the perpendicular distance from the magnetic axis, with the internal planetary eld subtracted. Data are shown before they have been corrected for latitude-related eects for (a) 6 7 MLT and (b) 8 9 MLT and after correction for (c) 6 7 MLT and (d) 8 9 MLT. The Galileo data are shown by stars and the y-by data are indicated by the diamonds. The solid lines indicate the BC model, whilst the dashed lines indicate the extremes ofthe model, i.e. noon (upper) and midnight (lower). model is given by the simple function ( ) m( ) B = A ; (1) where A =41:1 nt, =18:8R J, and m is given by m( )= cos + ; (2) where is azimuth measured positive eastward from noon, =:48 and =1:26. The model is thus described by four simple parameters only. In Fig. 3 we thus show model and observed values plotted versus MLT in four radial ranges of width 2: R J, which span the range ofvalidity ofthe model between and R J. In panels (a) to (d) these radial ranges are. 22.,

7 E.J. Bunce et al. / Planetary and Space Science (2) R J R J B ' ρ / nt B ' ρ / nt (a) Residual = 11.1 % MLT / h (b) Residual = 11.2 % MLT / h R J R J B ' ρ / nt B ' ρ / nt (c) Residual = 13.3 % MLT / h (d) Residual =.7 % MLT / h Fig. 3. Representative plots ofthe latitude-corrected radial eld B, as a function of magnetic local time (MLT), are shown for (a) 22: R J, (b) 32: R J, (c) 37. R J, and (d) 47. R J. The same symbols are used for the Galileo and y-by points as indicated in Fig. 2. In each case, the two solid lines indicate the BC model for the two extremes ofradial range shown. At the foot ofeach panel the RMS residual ofthe BC model (expressed as a percentage) is indicated. From this point, all averages shown have been corrected for latitude related variations.. 32., 37.., and 47.., respectively. Data obtained when the spacecraft were south of the current sheet, such that B was negative, have been reversed in sign, assuming anti-symmetry in B about the centre of the current sheet. As in Fig. 2, the y-by data previously employed are shown by diamonds, while the Galileo data are shown by stars. The solid lines indicate the BC model for the extremities of each radial range shown. In addition, the RMS residual value ofthe data points from the equivalent BC model value, normalised to the model magnitude, is given in each panel. This value gives a RMS fractional residual ofthe data in each panel. In panel (a) offig. 3 we see that a majority ofthe data points lie within or immediately beside the band ofmodel values, though a small proportion lie well outside. As noted previously, this panel corresponds to the radial range 22: R J, and therefore lies at the innermost edge of validity ofthe BC model. We note, however, that the Galileo

8 796 E.J. Bunce et al. / Planetary and Space Science (2) and y-by data correspond well, and that the RMS residual is 11.1%, such that the model represents a good indicator ofthe radial eld strength in this region. Panel (b) shows the data and model for the radial domain 32:R J. Here the BC model ts both Galileo and y-by data well, with a residual error of11.2%. The local time asymmetry is now clearly evident in both sets ofdata, with the radial current sheet eld being approximately nt stronger at midnight than at noon in this particular radial range. It can also be seen that the Galileo data and y-by data are closely similar, with rather little scatter about the mean values, despite the fact that the contributing data span 2 years oftime. This indicates that the radial eld at a given location is a relatively robust parameter over such intervals to within %. Moving out further into the middle magnetosphere, panel (c) shows the data and model values between 37. and R J. The day night asymmetry in the eld is still marked, and now the similar nt dierence between noon and midnight denotes a factor of almost two in the radial eld strength. Once more the two data sets are in close agreement, and the model represents a good estimation ofthe eld with a residual error of13.3%. In panel (d) we nally show data between 47. and R J, the outermost limit ofvalidity ofthe BC model. We notice that some ofthe data from the Galileo orbiter do not t well to the BC model in this region, and the RMS residual is now.7%. It can be seen that while certain ofthe Galileo data do follow the model values as they decrease towards magnetic noon, a large percentage of the data population do not. Instead, they remain at an approximately constant value as a function of MLT. Further inspection ofthe individual radial bins shows that this attening is rst observed in the 4 47:R J radial range, suggesting that the local time asymmetry in the current sheet radial eld does not always exist in this region ofthe middle magnetosphere. We suppose that these variations ofthe eld strength (presumably from orbit to orbit) may be a signature ofthe eects ofcompressions and expansions ofthe magnetosphere due to changes in the solar wind dynamic pressure, causing the eld in the outer regions to change whilst those stronger elds closer to the planet, remain relatively unaffected. For this reason, our revised eld model derived below for the combined data set will be restricted to the radial range 4 R J. The residual fractional errors occurring in various radial ranges are collected together in the rst two columns oftable 1, where we show the RMS fractional residuals in R J radial ranges relative to the BC model values. At the foot ofthe table the overall RMS fractional residual is shown, which for the BC model is 12:7% over the radial range 4 R J. Clearly the model provides a reasonably good estimate ofthe radial eld in this domain. In Table 1 the error for the radial range 4 R J is also given, the italics indicating that values from this range were not included in calculating the overall error for the BC model to the Galileo and y-by data. The residual error for this range is seen to be almost twice that for the other ranges, thus Table 1 Comparison ofthe RMS residual ofthe BC model and the Revised BC (RBC) model for the six radial ranges shown Radial range (R J ) RMS Residual RMS Residual (BC model) (%) (RBC model) (%) Overall RMS ( 4 R J ) residual The radial range 4 R J (shown in italics) has not been included in the calculation in the overall RMS error. justifying our restricting further attention to the reduced range 4 R J. In addition to comparing the Galileo and y-by data and the BC model at xed radial distance ranges as above, it is also instructive to divide the data into ranges oflocal time and study them as a function of radial distance,. In Fig. 4 representative plots ofthe equatorial radial eld versus radial distance are shown in a log log format for four 2-h ranges ofmlt. Panels (a) to (d) correspond to 8, 6, 18, and MLT, respectively, such that they represent observations from the pre-noon sector, and the near dawn, dusk and midnight meridians, respectively. The format ofthe panels ofthis gure is essentially the same as for Fig. 2, with some additional features to be described below. Panel (a) represents data mostly from the inbound pass ofpioneer-11 and some points from the Galileo data set, whilst panel (b) consists ofcontributions from both Pionner- outbound and from various Galileo orbits. In panel (c) we see the Ulysses outbound data along with a solitary average derived from Galileo orbit C. Finally, in panel (d) the majority ofthe data are from various Galileo orbits with a few points from Voyager-2 outbound at smaller distances. In each panel the RMS residual is given for the BC model over the radial range 4 R J, which at 1%, can thus be seen to be a reasonable measure ofthe eld values. These RMS residuals are collected together in Table 2 for the 2-h local time intervals, and are in general less than 1% (with the exception ofthe 1 1 MLT sector). As indicated before, the overall RMS residual is 12.7%. In addition to the BC model lines (shown by the dashed and lighter solid lines), we also show in Fig. 4 the results of a straightforward least-squares t to the logged data points, indicated by the heavy solid lines. These were tted to the data only in the radial range 4 R J, for reasons previously discussed. However, we have extrapolated these lines to the edge ofthe plot, as shown by the heavy dashed line, to cover the whole range ofthe data. It can be seen that the t

9 E.J. Bunce et al. / Planetary and Space Science (2) MLT -6 MLT B ' ρ / nt B ' ρ / nt (a) Residual (BC model) = 1.1 % Residual (least-squares) =. % m = 1.6 A = nt 6 (b) Residual (BC model) =.3 % Residual (least-squares) = 8.3 % m = 1.13 A = nt MLT - MLT B ' ρ / nt B ' ρ / nt (c) Residual (BC model) = 13. % Residual (least-squares) = 3.3 % m =.91 A = 6.9 nt 6 (d) Residual (BC model) = 14.3 % Residual (least-squares) = 7.9 % m =.87 A = nt 6 Fig. 4. Log log plots ofthe radial eld B, as a function of radial distance, for the four MLT intervals: (a) 8, (b) 6, (c) 18, and (d). For each panel, the solid lines indicate the BC model for the outer limits of the local time interval shown, whilst the dashed lines show the extremes ofthe BC model (i.e. noon and midnight). The heavy solid indicates the least-squares power-law ts B = A(nT)(R J) m over the radial range 4 R J, where the values ofthe coecient A and m are shown in each panel. The line is simply extrapolated over the full data coverage range, indicated by the heavy dashed portion ofthe line. At the bottom left ofeach plot is the RMS residual ofboth the least-squares ts and ofthe BC model values for each point. generally represents a reasonably good approximation to the data out to at least 6 R J. The parameters ofthe t, namely the coecient A and the exponent m, where B = A(nT)(R J ) m ; (3) are given in each plot. We also indicate the RMS residual ofthe t, which refers specically only to the points lying within the tting range. From panel to panel the error varies by a small amount, and no panel has an error ofmore than %, which is on average signicantly less than the overall 12.7% error for the BC model values. Evidently, a simple power law t to the joint data set provides a good description ofthe middle magnetosphere current sheet eld. We can also see from these panels that while the intercepts near

10 798 E.J. Bunce et al. / Planetary and Space Science (2) Table 2 Comparison ofthe RMS residual ofthe BC model and the Revised BC (RBC) model for the given local time ranges calculated over the radial range 4 R J. MLT range (h) Residual Residual (BC model) (%) (RBC model) (%) Overall RMS residual The overall RMS residuals are indicated in the nal row for both BC and RBC models. A dash indicates that insucient data were available in that local time sector. R J do not vary greatly, each being close to nt, the eld gradients are clearly largest on the dayside, smaller at dawn and dusk, and smallest on the nightside, in accordance with the BC model and other papers cited in the introduction. In Table 3 we thus provide for detailed reference the best-t coecients A and m derived from the 2-h local time ranges which have sucient number ofradial eld averages over the radial range 4 R J. The rst column indicates the local time sector and the second column the radial range of the data. This information shows that we have only used those ranges which were deemed to be over a suitable radial range; those ranges which are not included did not meet this requirement. This is followed in columns 3, 4 and by the number ofpoints n, exponent m, and the coecient log A. The standard error on the m and log A values are also shown and were calculated according to the method of Topping (19). The nal column shows the RMS residual ofeach t. Overall, the RMS residual for these best-t lines (eectively a -parameter t to the data) is found to be 7.7% as given at the foot of Table 3. This compares with the overall value of 12.7% for the simple four-parameter BC model. Although the BC model thus gives a reasonable description ofthe overall data set, the fact that the residual values are overall 6% greater than those ofthe best-t lines provides motivation to undertake a revision. This will now be attempted in the next section. 4. Revision of the Bunce and Cowley empirical model 4.1. Determination of eld model parameters via minimisation of overall RMS error Thus far we have shown, in Fig. 4, only the best-t lines to the data in four local time ranges. Now, in Fig., we compare the tted lines from all ten of the 2-h local time intervals which provided data over a sucient range that the slope m and intercept A relevant to the distance range 4 R J can be determined with condence. Bunce and Cowley (1a) noted that the lines ofbest t to the y-by data seemed to converge at R J, and hence used this as a starting point for their model. They assumed that the lines do in fact converge at a certain radial distance, the distance within which the eld may be taken as cylindrically symmetric, and then fall with distance at various rates depending upon the local time, as described by Eqs. (1) and (2). Here we use a model ofthe same form, but determine the parameters using a slightly dierent procedure that Table 3 This table contains the m and A values and their standard errors, for individual 2-h local time ranges, for the least-squares t over the radial range 4 R J. This is accompanied by the corresponding normalised RMS residual in each case MLT Fit over 4 R J Range (R J ) n m log A (nt) Residual (%) :87 ± : 2:69 ± : :16 ± :4 3:1 ± : :13 ± :3 3:9 ± : : ± :1 3:29 ± : :6 ± :4 3:7 ± : :79 ± :14 3:92 ± : :37 ± :31 3:23 ± : :91 ± :3 2:78 ± : : ± :6 3:6 ± : :1 ± :4 3:13 ± :6 7.8 Overall residual 7.7 The overall residual error is shown at the foot of the table. In addition, n indicates number ofpoints used in each local time range and for completeness, the total range ofdata available is given for each local time range. The plus sign indicates that the data actually extends well beyond the value ofthe radial range which has been included in the tting procedure.

11 E.J. Bunce et al. / Planetary and Space Science (2) B`ρ / nt 6 h / MLT Fig.. Plot ofthe tted lines as in Fig. 4, from the MLT intervals which could be used to determine the dependence on distance in the radial range 4 R J. These MLT values are shown on the right hand margin. The solid part ofeach line depicts the radial range over which the t was determined, while the dashed part (i.e. at radial distances greater than 4 R J ) show where the line has been extrapolated outside of the range. An arrow is drawn at the hinge point, that is the point ofmaximum convergence, which was determined from the least value of the standard deviation ofthe B values weighted to the inverse ofthe standard error ofeach tted line and normalised to the weighted average, while the horizontal bar gives an estimate ofthe error in this value. a selection ofradial proles for xed 2-h MLT ranges, in a format similar to Fig. 4. In this case, however, the solid lines show the values implied by our revised model (which we term the RBC model) corresponding to the upper and lower limits ofthe MLT range concerned, while the dashed lines show the noon (upper) and midnight (lower) limits of the model. The dotted lines similarly show the values corresponding to the upper and lower limits ofthe MLT range for our original BC model. At the foot of each panel we also show the RMS residuals for both models. It can be seen that the models provide a very good overall description ofthe data, with RMS residuals oftypically 1%. However, the residuals ofthe RBC model are generally several percentage points lower than that ofthe original model. This evidence is borne out in Table 2, where the RMS residuals are compared for the two models for the various 2-h local time sectors. The RBC model provides an improved description in most MLT sectors, and, taking all the data together, we nd a RMS residual of9.7% for the RBC model compared with 12.7% for the original BC model. The four-parameter RBC model thus provides a description which is almost as accurate as those ofthe least squares ts at intervals of2-h MLT, for which the RMS residual was 7.7%. In Fig. 6b we provide an alternative presentation, where we show the model values versus MLT in xed ranges of radial distance, similar to Fig. 3. Here the solid lines show the RBC model, and the dotted lines the BC model. The RMS residuals for both models are also shown at the bottom of each panel (and over wider radial ranges in Table 1). Again the models provide a good overall representation ofthe eld in the middle magnetosphere, with the RBC model giving smaller RMS residuals than the BC model in all ranges. emphasises the minimisation ofthe RMS fractional error. We rst choose a value of (to be iterated), and then determine A by calculating the weighted mean ofthe values obtained from the ten tted lines shown in Fig.. The weights chosen were inversely proportional to the standard error of the gradient ofthe tted lines as given in Table 3. Using these values of and A we then iterate and to nd the pair ofvalues that give the minimum RMS fractional error between the model and the data set. We then repeat the procedure for a range of values of until the value which gives a global minimum RMS fractional error is found. The global minimum is found to have a value of 9.7% at =14: R J, although it is a relatively shallow minimum over the range of values ofinterest. The corresponding value ofa is 9:7 ± 2:9 nt (where the error given is the standard deviation ofthe values), together with = :, and = 1:2. These parameters then dene our revised t to the data Comparison of model and eld data In this section we nally check how well the overall model dened in Section 4.1 ts the data. In Fig. 6a we rst show. Divergence of the azimuthal current.1. The azimuthal current and its divergence As previously outlined by Bunce and Cowley (1a), a local time asymmetry in the radial eld in the middle jovian magnetosphere implies a divergence in the equatorial azimuthal current. Signicantly larger currents occur at midnight at a given distance than at noon. In order to quantify the divergence of the azimuthal current we rst need to consider the equivalence ofthe radial magnetic eld to the azimuthal current intensity. As described by Bunce and Cowley (1a), we nd that, via Ampere s law, the integrated current intensity (A m 1 ) in the equatorial current sheet is given by i = 2 [ ] B z ; where, as before, the primed elds indicate that the curl-free planetary eld has been subtracted, D is the half-thickness, and is the permeability offree space. In deriving this expression we have assumed anti-symmetry in the radial eld

12 8 E.J. Bunce et al. / Planetary and Space Science (2) (a) 8- MLT (b) -6 MLT B ' ρ / nt B ' ρ / nt Residual (BC model) = 1.1 % Residual (RBC model) =.6 % 6 Residual (BC model) =.3 % Residual (RBC model) = 8.8% 6 (c) 18- MLT (d) - MLT B ' ρ / nt B ' ρ / nt Residual (BC model) = 13. % Residual (RBC model) = 11.4 % 6 (a) Residual (BC model) = 14.3 % Residual (RBC model) = 9.1 % 6 Fig. 6. (a) In the same format as Fig. 4, and for the same MLT intervals, we show the radial eld B, as a function of radial distance for both Galileo orbiter (stars) and the y-by spacecraft (diamonds). In each panel the Revised BC model (RBC) is shown by the solid lines for the upper and lower limits ofthe MLT interval shown. Once again the dashed lines indicate the RBC model limits at noon and midnight. The dot-dashed lines show the original BC model values for the same upper and lower local time values for comparison. At the bottom of each panel, the RMS residuals are given for both the original BC model and for the RBC model, again for comparative purposes, (b) Plots of the radial eld B, as a function of MLT are shown for (a) R J, (b) 32: R J, (c) 3 37: R J, and (d) R J. The same symbols are used for the Galileo and y-by data as in previous gures. The solid lines indicate the RBC model for the outer limits of the radial range shown, whilst the dashed lines indicate the original BC model values for the same distances. Once more, the RMS residual values are shown for both BC and RBC models. on either side ofthe current sheet, and that B z remains approximately constant through the thickness ofthe current sheet. Bunce and Cowley (1a) show that for jovian current sheet conditions, the second term in Eq. (4) is much smaller that the rst, such that to within % the azimuthal current intensity is given by i 2B : ()

13 E.J. Bunce et al. / Planetary and Space Science (2) (a) -22.RJ (b) -32.RJ B' ρ /nt B' ρ /nt Residual (BC model) = 11.7 % Residual (RBC model) =.8 % MLT / h (c) 3-37.RJ Residual (BC model) = 11.2 % Residual (RBC model) = 8. % MLT / h (d) 42.-4RJ B' ρ /nt B' ρ /nt Residual (BC model) = 13. % Residual (RBC model) =.7 % MLT / h (b) Residual (BC model) = 14.4 % Residual (RBC model) = 11.3 % MLT / h Fig. 6. (Continued) Consequently, our model for the equatorial radial eld just outside the current carrying layer given by Eqs. (1) and (2) above may be approximately but directly converted into an empirical model for the azimuthal current, which will therefore undergo the same local time variations as the radial eld. The divergence ofthe azimuthal current is then simply given by divi @ : (6) Introducing the revised empirical model given by Eqs. (1) and (2), we nd divi 2 ( ) sin ln B : (7) In Fig. 7a we show a contour map ofthis function in the equatorial plane (solid lines), labelled with the divergence values in ka R 2 J. The divergence is exactly zero at =14: R J, within which the model eld is axi-symmetric, and also at all radial distances on the noon midnight

14 82 E.J. Bunce et al. / Planetary and Space Science (2) meridian, as this is the axis ofsymmetry ofthe model. The dashed lines in the gure indicate radial distance in the equatorial plane, starting with R J and increasing in increments of R J. The range ofdetailed validity ofthe RBC model continues only to 4 R J, and as such the contours outside ofthis range (which allow comparison with the results ofbunce and Cowley, 1a) should be interpreted with caution. We see that the divergence is negative at dawn, implying a sink ofazimuthal current in that sector, while reversing to positive at dusk, thus requiring a source ofcurrent. The magnitude ofthe peak divergence in i is 1 ka R 2 J, occurring at a radial distance of 23 R J near the dawn-dusk meridian. In the original BC model the peak magnitude was 18 ka R 2 J occurring at R J near the dawn-dusk meridian. Necessarily, the current overall is divergence-free, and continuity must therefore be maintained either by radial currents owing wholly within the equatorial current sheet, or via eld-aligned currents which must ow towards the planet at dawn and away from the planet at dusk. In Fig. 7b we give an indication ofthe overall current which must be diverted into one or other ofthese directions. The lower three curves in this gure show the total azimuthal current owing in the model current sheet in the radial ranges, and R J (slightly beyond the outer limit ofthe model), versus MLT. The upper curve shows the sum ofthese, that is the total current owing between and R J. These curves have been computed by direct integration ofeq. (), combined with Eqs. (1) and (2). Each ofthe curves shows, as expected, that the current is maximum at midnight and minimum at noon. Specically, the amount ofcurrent diverted in each case is 6., 6.6, and 6:2 MA, the total being 18:8 MA within the range R J. For the original BC model these currents were 8.2, 12., and 13:1 MA, respectively, totalling to 33:7 MA. The total diverted current is thus a relatively sensitive function of the model employed, but both models indicate values of MA in a total (e.g. at dawn or dusk) o 9 MA..2. Current stream-function Fig. 7. (a) Contours ofthe divergence ofthe azimuthal current in the magnetic equatorial plane, in units ofka R 2 J, derived from the empirical model of B derived here. Midnight is marked at the top ofthe plot, with dusk to the right. The dashed rings indicate radial distances of,,, and R J, a somewhat extended range ofvalidity. Jupiter is shown in the centre to scale. (b) The total current in MA owing in various radial ranges in the equatorial current sheet versus magnetic local time, obtained from the Revised BC (RBC) empirical model derived here. The current has been integrated in the ranges,, and R J, and over the entire range R J, as indicated on the right-hand side ofthe plot. As indicated above, the diverted azimuthal current must ow either radially in the current sheet itself, or close via eld-aligned currents in the ionosphere. In general, both closure paths may be expected to be present. In this case, Bunce and Cowley (1b) suggested on physical grounds that the total middle magnetosphere current system might best be viewed as the sum ofa divergence-free current that ows wholly within the equatorial current sheet itself, i CS, which includes all ofthe azimuthal current, together with additional radial currents that close wholly via eld-aligned currents in the ionosphere. The divergence-free equatorial current can then be described by a current stream-function I CS having units ofamps, which is such that I CS (; )=constant denes a current streamline in the current sheet, while the amount ofcurrent owing between I CS and I CS +di CS is just di CS. This stream-function is related to the current intensity i CS by i CS =ẑ I CS, where ẑ is a unit vector perpendicular to the current sheet directed northwards. For the model currents