Planetary and Space Science 50 (2002) 789–806 www.elsevier.com/locate/pss

A simple empirical model ofthe equatorial radial ÿeld in Jupiter’s middle magnetosphere, based on spacecraft *y-by and Galileo orbiter data E.J. Bunce ∗, P.G. Hanlon1, S.W.H. Cowley

Department of Physics & Astronomy, University of Leicester, Leicester LE1 7RH, UK Received 31 May 2001; received in revised form 5 December 2001; accepted 28 March 2002

Abstract

In this paper we consider empirical models ofthe radial ÿeld and azimuthal current in Jupiter’s middle magnetosphere region, at distances in the range 20–45 RJ. We ÿrst ofall compare the model derived previously by Bunce and Cowley (Planet. Space Sci. 49 (2001) 261) using Pioneer, Voyager and Ulysses *y-by data, with a combined data set that now also incorporates data from the ÿrst 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 ÿeld (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 ÿtted 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 ÿeld is taken as asymmetric outside of14 :5RJ, and to fall with radial distance with an exponent which is taken to vary sinusoidally with local time, varying between −1:5 at noon and −1:0 at midnight, such that the ÿeld 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-ÿtsto the 2 h MLT binned data, and representing a worthwhile improvement over the original Bunce and Cowley −2 model. The implied divergence ofthe azimuthal current forthe revised model peaks at ∼ 15 kA RJ near the dawn-dusk meridian at a radial distance of ∼ 23 RJ. The implied diBerence in the total azimuthal current *owing in the current sheet between 20 and 50 RJ at midnight compared with noon is 19 MA, in a total (at dawn and dusk) of59 MA. ? 2002 Published by Elsevier Science Ltd.

1. Introduction signiÿcantly distorts the planetary ÿeld lines at distances of ∼ 10 RJ and beyond (Smith et al., 1974, 1975, 1976; Ness Gledhill (1967) was the ÿrst to postulate that Jupiter’s et al., 1979a, b). However, they also showed that the prin- near-planet equatorial magnetic ÿeld lines would be radially cipal plasma source for the current sheet was not Jupiter’s distended, due to centrifugal forces associated with rapid ionosphere, but the moon Io, which orbits at a jovicentric planetary rotation and ionospheric plasma loading. Subse- distance of ∼ 5:9RJ (Krimigis and Roelof, 1983). The next quently, the ÿrst in situ measurements ofJupiter’s mag- spacecraft to *y past Jupiter was Ulysses in 1992 (Balogh netic environment, made during the Pioneer-10 and -11 and et al., 1992), and more recently the Galileo orbiter arrived in Voyager-1 and -2 spacecraft *y-bys in the 1970s, indeed 1995 to commence a long-term study ofthe Jovian system showed the signatures ofthe radial distension ofthe mag- (Kivelson et al., 1992). The magnetic eBects ofthe equa- netic ÿeld. The earliest studies based on the data from these torial current sheet, or magnetodisc, have been found to be *y-bys demonstrated the existence ofa thin equatorial az- present at all local times investigated by these spacecraft. imuthal current sheet *owing in an eastward direction, which The local time coverage ofthe ÿve Jupiter *y-bys men- tioned above, and the ÿrst 20 orbits ofthe Galileo mis- sion (between 1996 and 1999) are shown in Fig. 1a, where ∗ Corresponding author. Tel.: 0044 116 223 1302; fax: 0044 116 252 the spacecraft trajectories are shown in Jupiter Solar Or- 3555. bital (JSO) coordinates, i.e. X (R ) is positive sunwards, and E-mail address: [email protected] (E.J. Bunce). J 1 Now at Blackett Laboratory, Imperial College, London SW7 2BZ, Y (RJ) is orthogonal to X and in the plane ofJupiter’s orbit. UK. The Pioneer and Voyager *y-bys covered the dawn sector of

0032-0633/02/$ - see front matter ? 2002 Published by Elsevier Science Ltd. PII: S 0032-0633(02)00011-9 790 E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806

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Fig. 1. (a) Trajectories ofthe ÿrst 20 orbits ofthe Galileo orbiter along with the ÿve *y-by spacecraftrelative to Jupiter, shown in Jupiter Solar Orb ital 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 forthis paper, 20–45 R J, is highlighted by the grey annulus in the centre ofthe plot. This ÿgure was kindly provided by Joe Maÿ ofthe Planetary Data System, UCLA. (b) Plot ofthe half-houraverages ofthe magnetic components measured outside the current sheet during the ÿrst 20 orbits ofthe Galileo orbiter and the ÿve *y-bys ofPioneer-10 and -11, Voyager-1 and -2, and Ulysses, fromwhich the VIP4 planetary ÿeld model (Connerney et al., 1998) has been subtracted. The averages have been projected onto the magnetic equatorial plane and rotated through 90◦ to indicate the approximate direction and strength ofthe corresponding current. Those ÿelds measured north ofthe current sheet have been rotated 90 ◦ 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 (RJ), and local time is also shown. The individual spacecraft are identiÿable by comparison with Fig. 1a. At the bottom right of the plot is the scale for 50 nT. the magnetosphere from near noon (Pioneer-11 outbound) to from dawn through to midnight, and some way into the post-midnight (Voyager-2 outbound), while Ulysses passed evening sector. The jovigraphic latitudes ofthese trajecto- through the pre-noon sector inbound and made unique ob- ries were near-equatorial in the main, except for the out- servations ofthe dusk meridian magnetosphere outbound. bound passes ofPioneer-11 and Ulysses, which exited near Presently available data from the Galileo mission extend noon at ∼33◦N and near dusk at ∼37◦S, respectively. Also E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806 791 shown in the ÿgure are the positions ofthe magnetopause upon local time. For example, at distances of ∼40–50 RJ and bow shock as modelled from the Voyager-2 data (Ness the current is approximately twice as strong at a given radial et al., 1979b). The shaded region also indicates the domain distance at midnight than at the same distance at noon. This ofinterest forthis study, that is, the middle magnetosphere phenomenon was ÿrst noticed by Goertz (1978) in a com- region between 20 and 50 RJ. On the dayside, the magne- parison ofthe Pioneer-10 inbound and outbound data. The topause extends on average to ∼65 RJ as shown here, but diBering gradients ofradial ÿeld fall-oBwith distance at is highly variable depending upon the upstream solar wind the two local times (∼1000 MLT inbound and ∼0500 MLT conditions. On the nightside the magnetospheric tail extends outbound for Pioneer-10) were discussed in terms of the to ∼3000 RJ and has a diameter of ∼300 RJ (Ness et al., asymmetrical compressive and conÿning eBect the solar 1979c). wind dynamic pressure has on the magnetosphere, com- As indicated above, it is understood that the dynamics of pressing the *ux tubes on the dayside but allowing them to the Jovian middle magnetosphere are governed by the Io stretch out on the nightside. This stretching further distends plasma source, located deep within the equatorial magneto- the magnetic ÿeld lines, hence increasing the azimuthal cur- sphere at ∼5:9RJ. The current in the equatorial magnetodisc rent, on the nightside. Bunce and Cowley (2001a) favour is then carried (a) by the inertia current ofnear-corotating this interpretation, which then indicates that azimuthal cur- cold torus plasma which slowly diBuses outwards, and (b) rent closure is enforced via radial currents *owing wholly by the pressure-gradient current oflow density hot plasma within the current sheet, *owing away from the planet which slowly diBuses inwards (Hill, 1979; Vasyliunas, at dawn and towards the planet at dusk. Khurana (2001) 1983; Caudal, 1986; and references therein). This azimuthal prefers to attribute the divergence of the azimuthal current current sheet deÿnes what has become known as the Jovian to an Earth-like partial ring current closing via “region-2 “middle magnetosphere” region, which extends from ∼5RJ type” ÿeld-aligned currents, *owing towards the planet at (the inner edge ofthe Io plasma torus) to within ∼15 RJ of dawn, closing through the jovian ionosphere and *owing the magnetopause on the dayside. The radial range ofthe away from Jupiter at dusk. Here, however we focus on the current sheet on the dayside ofthe planet is thus controlled central factofthe azimuthal asymmetry ofthe azimuthal by the dynamic pressure ofthe solar wind, which causes current itself, and leave further considerations of closure to the magnetopause to be compressed or to expand. On the future study. nightside the magnetosdisc merges at larger distances with Whilst previous models ofthe middle magnetosphere the cross-tail currents which are associated with the magne- current sheet have been based upon axial symmetry (e.g. totail, and hence with solar wind-magnetosphere coupling Connerney et al., 1981; Khurana, 1992), and are indeed an (Ness et al., 1979c). excellent indicator ofthe jovian ÿeld in the inner region In the middle magnetosphere, the current disc is located ofthe middle magnetosphere, it is now evident that out- close to the magnetic equatorial plane and thus displays side this region, roughly beyond ∼15–20 RJ, the current is a quasi-sinusoidal north–south oscillation as the magnetic signiÿcantly dependent on MLT as outlined above. Bunce dipole, tilted by ∼10◦ from the spin axis, rotates with the and Cowley (2001a) presented a simple empirical model planet. As the relative full thickness of the current sheet ofthe near-equatorial radial component ofthe ÿeld in the (between 2 and ∼8RJ, for example see Smith et al. (1976), region between 20 and 50 RJ, valid for all magnetic local Goertz et al. (1976), Connerney et al. (1981), Behannon times, based on the *y-by data. This model (herein referred et al. (1981), Acu˜na et al. (1983), Staines et al. (1996), to as the BC model) serves as a useful empirical tool for Dougherty et al. (1996)) is much smaller than the charac- modelling the middle magnetosphere, and in particular for teristic size ofthe magnetosphere, the radial ÿeld undergoes quantifying the divergence of the azimuthal current. How sharp reversals across the current sheet from positive values much current is diverted out ofor into the azimuthal current in the north, to negative values to the south. At the inner *ow, combined with similar information on the radial cur- edge ofthe current sheet, the planetary ÿeld dominates rent derived from the azimuthal component of the magnetic that due to the current sheet alone, but since the dipole ÿeld, provides the necessary information from which the component ofthe planetary ÿeld fallsas r−3 whilst that ÿeld-aligned currents (FACs) connecting to the ionosphere of the current sheet is found to fall oB much less rapidly, can be calculated (Hill, 1979; Vasyliunas, 1983; Khurana between ∼r−1 and r−2 (Barish and Smith, 1975; Goertz and Kivelson, 1993; Bunce and Cowley, 2001b). The na- et al., 1976; Jones et al., 1981; Behannon et al., 1981; ture ofthe FACs connects in turn with other important Connerney et al., 1981; Khurana, 1997; Bunce and Cowley magnetospheric phenomena such as the jovian auroras and 2001a), the current sheet ÿeld becomes dominant beyond the decametric radio emission (Cowley and Bunce, 2001). ∼15 RJ. In this paper we compare the BC model ofthe radial ÿeld In recent independent studies, Bunce and Cowley (2001a) B with newly-available ÿeld data from the Galileo orbiter, using magnetometer data from the ÿve *y-by missions men- as a function of both local time and radial distance. We show tioned above, and Khurana (2001) also incorporating data that while the BC model is generally in good agreement from the Galileo orbiter spacecraft, have shown that the az- with the Galileo data, some reÿnements are nevertheless imuthal current in the outer middle magnetosphere depends suggested that bring the model into better accord with the 792 E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806 combined *y-by and orbiter data set. We thus derive such a over 30 min intervals, and take these values to represent model, using techniques similar to those employed by BC. conditions at the similarly averaged locations outside ofthe As seen in Fig. 1a, inclusion ofthis additional data enhances current sheet. The signature ofthe changing latitude ofthe the overall coverage ofthe middle magnetosphere region. In spacecraft outside the current sheet will be discussed further particular, Galileo signiÿcantly increases the quantity ofdata below. in the dawn and pre-midnight sectors ofthe magnetosphere. Collectively, the data are shown in Fig. 1b. In order to However, the evolution ofthe Galileo orbits have not as yet indicate the overall current *ow in the equatorial regions, provided new data from the dayside middle magnetosphere we show the 30 min averages ofthe total ÿeld vectors pro- as perijove lies within the inner magnetosphere at this local jected onto the magnetic equatorial plane. The vectors have time. then been rotated through 90◦ to indicate the approximate direction ofthe corresponding equatorial current. To take account ofthe reversal in the equatorial ÿeld components 2. Data analysis across the current sheet, those ÿelds measured north ofthe current sheet have been rotated 90◦ anticlockwise, while 2.1. Current sheet ÿeld averages those measured in the southern hemisphere have been ro- tated 90◦ clockwise. In keeping with the previous study by We begin our study by presenting magnetic ÿeld vectors Bunce and Cowley (2001a), every eBort has been made to observed during the ÿve jovian *y-bys and the ÿrst 20 or- ensure that averages were taken only when the spacecraft bits ofthe Galileo mission as discussed above. All data were were outside ofthe current carrying region. Since we are supplied by the Planetary Data System at UCLA, at 10 s res- interested in estimating the total azimuthal current, inclu- olution for Pioneer-11 and Voyager-2, 48 s for Voyager-1, sion ofreduced values obtained when the spacecraftin and 1 min for Pioneer-10 and Ulysses. Due to telemetry fact remained in the current-carrying layer would result in constraints the Galileo data are only available at high time under-estimates ofthe total current. Hence we have chosen resolution approximately halfofthe time, and as such there to exclude those Galileo data from the MRO mode, whose are two distinct time resolutions ofmagnetic ÿeld data. The time resolution was too low to distinguish clearly between real time survey (RTS) mode supplies data at 24 s time res- such times. Ifthe current layer is then considered to be a olution, whilst the memory read out (MRO) mode provides quasi-inÿnite sheet with perturbation ÿelds ofequal mag- 32 min averaged data. nitude but opposite direction on either side, a perturbation The VIP 4 planetary ÿeld model (Connerney et al., ÿeld of10 nT corresponds to an azimuthal sheet current of −1 1981) has been subtracted from the data to leave only those intensity 1:1MARJ , integrated through the full sheet. ÿelds which are due to the external currents (principally The contributions ofindividual spacecraftin Fig. 1b are the equatorial current sheet). In the case ofthe spacecraft identiÿable by comparison with Fig. 1a. The inbound passes trajectories lying close to the jovigraphic equatorial plane, ofPioneer-10 and -11, Voyager-1 and -2, and Ulysses are all the current sheet passes completely across the spacecraft in the pre-noon sector, and the outbound passes are all on the twice per 10 h rotation period. Correspondingly, it can be nightside, with the exception ofPioneer-11 outbound which seen in the magnetic ÿeld data that the radial ÿeld cycles is near noon. The Galileo passes (G1-2, C3, E4, E6, G7, between intervals ofrelatively steady positive and negative C9-10, E11-12, E16-19) included in this study mainly lie values, interspersed with periods ofÿeld *uctuation and between 0900 and 0000 MLT. As described above, all passes reversal when the spacecraft crossed through the equatorial are near-equatorial (within ±10◦ ofthe jovigraphic equator), current sheet. However, in the case ofthe non-equatorial with the principal exceptions being Pioneer-11 inbound and Pioneer-11 inbound (14◦S), Pioneer-11 (33◦N) outbound, outbound, Pioneer-10 outbound and Ulysses outbound as Pioneer-10 outbound (11◦N), and Ulysses (37◦S) outbound noted above. We see in Fig. 1b that the sense ofthe azimuthal passes, the measured ÿeld is generally dominated either currents are eastward, associated with the radial distension by positive or negative radial components depending upon ofthe magnetic ÿeld lines away fromthe planet in the middle the latitude of the spacecraft, the former corresponding to magnetosphere. The larger values ofthe azimuthal current a location north ofthe current sheet and the latter to the on the nightside at a particular distance compared with the south. The radial ÿeld then exhibits depressed values and=or dayside values are evident. Outward radial currents are also enhanced *uctuations indicative ofhot plasma currents at apparent on the dawn side ofthe magnetosphere, consistent ∼10 h intervals when the spacecraft approached the mag- with the magnetic ÿeld line “lagging” out ofmeridian planes. netic equatorial plane. At other times, when the spacecraft However, on the dusk side ofthe magnetosphere the outward were at larger distances from the equator, the ÿelds are in- (“lagging”) currents evolve into inward (“leading”) currents stead stronger and smoothly varying, indicating only weak in the outer region at larger distances beyond ∼40 RJ, which local currents and a consequent location outside ofthe we take to be associated with solar wind induced eBects current sheet. Ignoring periods when enhanced magnetic including that due to the asymmetrical conÿning eBect of variations are present, therefore, we have averaged the ÿeld the solar wind on the magnetosphere, as mentioned in the components from both equatorial and non-equatorial passes introduction. E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806 793

2.2. Latitude-correction of non-equatorial radial ÿeld ofvarying magnetic latitude at the spacecraftis particularly data evident in the *y-by data shown in panel (b), where indi- vidual groups ofpoints formpartial “U”-shaped patterns. Since the equatorial current sheet is ofÿnite spatial ex- These groups ofpoints correspond to averages derived from tent, the radial ÿeld outside the sheet at a given distance individual spacecraft excursions outside of the current sheet will fall slowly with height above the sheet on either side. during the planet’s rotation, such that averages obtained near The ÿeld values which give the best indication ofthe to- the start and end ofeach group correspond to values ob- tal azimuthal current are those obtained at the outer edge tained at lower magnetic latitudes relatively close the edge ofthe sheet, while those obtained at higher latitudes will ofthe current sheet, while those in the middle were obtained thus provide an under-estimate. Bunce and Cowley (2001a) at higher magnetic latitudes at larger distances from the cur- made an approximate correction for this eBect using a simple rent sheet. The eBect offallingradial ÿelds with distance theoretical model, and in this work we follow the same pro- from the current sheet is thus very clear in these *y-by data cedure. The beneÿts ofperformingsuch a correction are du- (in the present case reaching ∼−20◦ magnetic latitude near alistic. First, we reduce the latitude-related “scatter” in the the centre ofeach group), and the need to introduce a lati- radial ÿeld proÿles, thus allowing a more accurate represen- tude correction is correspondingly clear. However, with this tation by least-squares ÿtting. Second, we allow inclusion of introduction, the latitude eBect is seen to be present with re- the non-equatorial data. We are required particularly to cor- duced amplitude in the Galileo data as well, in both panels rect those data from the Pioneer-11 outbound and Ulysses (a) and (b). outbound passes, ifthey are to be included in this study, Panels (c) and (d) then show the eBect ofapplying the but we should also note that much ofthe data in the *y-by latitude correction factor derived from the Connerney et al. proÿles beneÿt from (albeit modest) corrections. As previ- (1981) model which, as indicated above, maps these data ously discussed, the Galileo orbiter data were taken close values to the edge ofthe current sheet. It can be seen that to the jovigraphic equator throughout most ofthe orbits and the “scatter” in both data sets is signiÿcantly reduced, with therefore do not require substantial correction, although for two immediate eBects. First, the Galileo and *y-by data are consistency all data has undergone the same procedure. The brought into much closer agreement with each other. Sec- approach is to simply map the ÿeld measurements to the ond, both data sets are brought into better general (ifnot edge ofthe current sheet using mapping factorsobtained perfect) agreement with the empirical BC model, which, as from the approximate forms of the Connerney et al. (1981) indicated above, was derived from and intended to represent model described in a recent paper by Edwards et al. (2001). the latitude-corrected radial ÿeld at the edge ofthe current For precise details ofthis procedure the reader is directed sheet. All ofthe data we will henceforthanalyse and display to Bunce and Cowley (2001a), as the method adopted for in this paper will thus correspond to “latitude-corrected” correction here is identical. Mapping factors depend on ra- radial ÿeld averages mapped to the edge ofthe cur- dial distance, but are typically ∼1:05 for a latitude of ∼5◦, rent sheet, which will be termed “equatorial” radial ÿeld increasing to ∼1:25 for ∼15◦, such that the corrections are averages. not substantial. We ÿnally note at this juncture that data from the In order to demonstrate the eBect oflatitude correction, inbound portion ofthe Voyager-1 *y-by have been ex- we present in Fig. 2 plots ofthe radial ÿeld versus radial dis- cluded from this study. These data values are found to tance in a log–log format. Throughout this paper we employ be signiÿcantly depressed in magnitude compared with cylindrical coordinates referenced to the magnetic dipole corresponding data from other *y-bys (e.g. Pioneer-11 axis. Thus the “radial ÿeld” is the cylindrical component per- outbound and Voyager-2 inbound), suggesting that the pendicular to the dipole axis, and the “radial distance” is the spacecraft may never have emerged from the current sheet perpendicular distance from that axis. In panels (a) and (b) during this pass. This eBect was noted previously by we show 30 min “current sheet” radial ÿeld averages (i.e. Connerney et al. (1981) in the comparison with their em- the radial ÿeld with the planetary ÿeld subtracted), denoted pirical model. Here, therefore, we will not employ these by B, before “latitude-correction” for the 1 h MLT intervals data. 0600–0700 and 0800–0900 MLT, respectively. The same data is shown after correction in panels (c) and (d). Aver- ages derived from Galileo data are indicated by stars, while 3. Comparison of the combined y-by and Galileo data the *y-by data employed previously by BC are shown by di- with the Bunce and Cowley empirical model amonds. For the intervals shown, *y-by data is present only in panels (b) and (d), where it was derived principally from In this section we will compare 30 min-averaged values the Pioneer-11 inbound pass. In each panel ofthe ÿgure the ofthe “equatorial” radial current sheet ÿeld, denoted here BC empirical model proÿle corresponding to the limits of by B0, with the BC model. Data from both Galileo and the the MLT bin are shown by the solid lines, while the extreme *y-bys will be shown in order to facilitate inter-comparison, proÿles ofthe model are indicated by the dashed lines, for where the Galileo data correspond to orbits G-1 to C-20 noon (lower) and midnight (upper), respectively. The eBect inclusive (between 1996 and 1999). We recall that the BC 794 E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806

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Fig. 2. Log–log plots ofthe 30-min averaged radial ÿeld component B outside the current sheet, versus the perpendicular distance from the magnetic axis , with the internal planetary ÿeld subtracted. Data are shown before they have been corrected for latitude-related eBects for (a) 0600–0700 MLT and (b) 0800–0900 MLT and after “correction” for (c) 0600–0700 MLT and (d) 0800–0900 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 where ’ is azimuth measured positive eastward from noon,
 =0:48 and ÿ =1:26. The model is thus described by four  m(’) 0 simple parameters only. B = A ; (1) 0  In Fig. 3 we thus show model and observed values plotted versus MLT in four radial ranges of width 2:5R, which span where A =41:1nT,0 =18:8RJ, and m is given by J the range ofvalidity ofthe model between 20 and 50 R J. m(’)= cos ’ + ÿ; (2) In panels (a) to (d) these radial ranges are 20.0–22.5, E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806 795

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Fig. 3. Representative plots ofthe “latitude-corrected” radial ÿeld B0, as a function of magnetic local time (MLT), are shown for (a) 20–22:5RJ, (b) 30–32:5RJ, (c) 37.5–40 RJ, and (d) 47.5–50 RJ. 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 forthe two extremes ofradial range shown. At the footofeach 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.

30.0–32.5, 37.5–40.0, and 47.5–50.0, respectively. Data lent BC model value, normalised to the model magnitude, obtained when the spacecraft were south of the current is given in each panel. This value gives a RMS fractional sheet, such that B0 was negative, have been reversed in residual ofthe data in each panel. sign, assuming anti-symmetry in B0 about the centre of In panel (a) ofFig. 3 we see that a majority ofthe data the current sheet. As in Fig. 2, the *y-by data previously points lie within or immediately beside the “band” ofmodel employed are shown by diamonds, while the Galileo data values, though a small proportion lie well outside. As noted are shown by stars. The solid lines indicate the BC model previously, this panel corresponds to the radial range 20– for the extremities of each radial range shown. In addition, 22:5RJ, and therefore lies at the innermost edge of valid- the RMS residual value ofthe data points fromthe equiva- ity ofthe BC model. We note, however, that the Galileo 796 E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806 and *y-by data correspond well, and that the RMS residual Table 1 is 11.1%, such that the model represents a good indicator Comparison ofthe RMS residual ofthe BC model and the Revised BC ofthe radial ÿeld strength in this region. Panel (b) shows (RBC) model for the six radial ranges shown the data and model for the radial domain 30–32:5RJ. Here Radial range (RJ) RMS Residual RMS Residual the BC model ÿts both Galileo and *y-by data well, with a (BC model) (%) (RBC model) (%) residual error of11.2%. The local time asymmetry is now 20–25 11.5 10.9 clearly evident in both sets ofdata, with the radial “current 25–30 10.8 9.0 sheet” ÿeld being approximately 10 nT stronger at midnight 30–35 11.7 8.6 than at noon in this particular radial range. It can also be 35–40 13.4 10.0 seen that the Galileo data and *y-by data are closely similar, 40–45 13.9 10.2 45–50 20.3 15.6 with rather little scatter about the mean values, despite the fact that the contributing data span ∼25 years oftime. This Overall RMS (20–45 RJ) 12.7 9.7 indicates that the radial ÿeld at a given location is a rela- residual tively robust parameter over such intervals to within 10%. Moving out further into the middle magnetosphere, panel (c) The radial range 45–50 RJ (shown in italics) has not been included shows the data and model values between 37.5 and 40 RJ. in the calculation in the overall RMS error. The day–night asymmetry in the ÿeld is still marked, and now the similar 10 nT diBerence between noon and midnight denotes a factor of almost two in the radial ÿeld strength. justifying our restricting further attention to the reduced Once more the two data sets are in close agreement, and the range 20–45 RJ. model represents a good estimation ofthe ÿeld with a resid- In addition to comparing the Galileo and *y-by data and ual error of13.3%. In panel (d) we ÿnally show data be- the BC model at ÿxed radial distance ranges as above, it is tween 47.5 and 50 RJ, the outermost limit ofvalidity ofthe also instructive to divide the data into ranges oflocal time BC model. We notice that some ofthe data fromthe Galileo and study them as a function of radial distance, . In Fig. 4 orbiter do not ÿt well to the BC model in this region, and representative plots ofthe equatorial radial ÿeld versus radial the RMS residual is now 20.7%. It can be seen that while distance are shown in a log–log format for four 2-h ranges certain ofthe Galileo data do followthe model values as ofMLT. Panels (a) to (d) correspond to 0800–1000, 0400– they decrease towards magnetic noon, a large percentage of 0600, 1800–2000, and 0000–0200 MLT, respectively, such the data population do not. Instead, they remain at an ap- that they represent observations from the pre-noon sector, proximately constant value as a function of MLT. Further and the near dawn, dusk and midnight meridians, respec- inspection ofthe individual radial bins shows that this “*at- tively. The formatofthe panels ofthis ÿgure is essentially tening” is ÿrst observed in the 45–47:5RJ radial range, sug- the same as for Fig. 2, with some additional features to gesting that the local time asymmetry in the “current sheet” be described below. Panel (a) represents data mostly from radial ÿeld does not always exist in this region ofthe middle the inbound pass ofPioneer-11 and some points fromthe magnetosphere. We suppose that these variations ofthe ÿeld Galileo data set, whilst panel (b) consists ofcontributions strength (presumably from orbit to orbit) may be a signature from both Pionner-10 outbound and from various Galileo ofthe eBects ofcompressions and expansions ofthe magne- orbits. In panel (c) we see the Ulysses outbound data along tosphere due to changes in the solar wind dynamic pressure, with a solitary average derived from Galileo orbit C20. Fi- causing the ÿeld in the outer regions to change whilst those nally, in panel (d) the majority ofthe data are fromvarious stronger ÿelds closer to the planet, remain relatively unaf- Galileo orbits with a few points from Voyager-2 outbound fected. For this reason, our revised ÿeld model derived below at smaller distances. In each panel the RMS residual is given for the combined data set will be restricted to the radial range for the BC model over the radial range 20–45 RJ, which 20–45 RJ. at ∼10–15%, can thus be seen to be a reasonable measure The residual fractional errors occurring in various radial ofthe ÿeld values. These RMS residuals are collected to- ranges are collected together in the ÿrst two columns ofTa- gether in Table 2 for the 2-h local time intervals, and are ble 1, where we show the RMS fractional residuals in 5 RJ in general less than 15% (with the exception ofthe 1200 radial ranges relative to the BC model values. At the foot –1400 MLT sector). As indicated before, the overall RMS ofthe table the overall RMS fractionalresidual is shown, residual is 12.7%. which for the BC model is ∼12:7% over the radial range In addition to the BC model lines (shown by the dashed 20–45 RJ. Clearly the model provides a reasonably good and lighter solid lines), we also show in Fig. 4 the results of estimate ofthe radial ÿeld in this domain. In Table 1 the a straightforward least-squares ÿt to the logged data points, error for the radial range 45–50 RJ is also given, the ital- indicated by the heavy solid lines. These were ÿtted to the ics indicating that values from this range were not included data only in the radial range 20–45 RJ, for reasons previ- in calculating the overall error for the BC model to the ously discussed. However, we have extrapolated these lines Galileo and *y-by data. The residual error for this range to the edge ofthe plot, as shown by the heavy dashed line, to is seen to be almost twice that for the other ranges, thus cover the whole range ofthe data. It can be seen that the ÿt E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806 797

0800-1000 MLT 0400-0600 MLT 40 40

30 30

20 20 / nT / nT 0 0 ρ ' ρ ' B B

10 10

Residual (BC model) = 15.1 % Residual (BC model) = 10.3 % Residual (least-squares) = 10.0 % Residual (least-squares) = 8.3 % m = 1.60 m = 1.13 A = 4959.9 nT A = 1234.0 nT 5 5 20 30 40 50 60 20 30 40 50 60 ρ / R (a) J (b) ρ / RJ

1800-2000 MLT 0000-0200 MLT 40 40

30 30

20 20 / nT / nT 0 0 ρ ρ ' ' B B

10 10

Residual (BC model) = 13.0 % Residual (BC model) = 14.3 % Residual (least-squares) = 3.3 % Residual (least-squares) = 7.9 % m = 0.91 m = 0.87 A = 610.9 nT A = 494.8 nT 5 5 20 30 40 50 60 20 30 40 50 60 (c)ρ / RJ (d) ρ / RJ

Fig. 4. Log–log plots ofthe radial ÿeld B0, as a function of radial distance , for the four MLT intervals: (a) 0800–1000, (b) 0400–0600, (c) 1800 –2000, and (d) 0000–0200. For each panel, the solid lines indicate the BC model for the outer limits of the local time interval shown, whilst the dashed −m lines show the extremes ofthe BC model (i.e. noon and midnight). The heavy solid indicates the least-squares power-law ÿts B0 = A(nT)(RJ) over the radial range 20–45 RJ, where the values ofthe coeQcient 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 leftofeach plot is the RMS residual ofboth the least-squares ÿts and ofthe BC model values foreach point. generally represents a reasonably good approximation to the within the ÿtting range. From panel to panel the error varies data out to at least 60 RJ. The parameters ofthe ÿt, namely by a small amount, and no panel has an error ofmore than the coeQcient A and the exponent m, where 10%, which is on average signiÿcantly less than the overall 12.7% error for the BC model values. Evidently, a simple B = A(nT)(R )−m; (3) 0 J power law ÿt to the joint data set provides a good description are given in each plot. We also indicate the RMS residual ofthe middle magnetosphere “current sheet” ÿeld. We can ofthe ÿt, which refersspeciÿcally only to the points lying also see from these panels that while the intercepts near 798 E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806

Table 2 by the number ofpoints n, exponent m, and the coeQcient Comparison ofthe RMS residual ofthe BC model and the Revised BC log10 A. The standard error on the m and log10 A values are (RBC) model for the given local time ranges calculated over the radial also shown and were calculated according to the method of range 20–45 R . J Topping (1955). The ÿnal column shows the RMS residual MLT range (h) Residual Residual ofeach ÿt. Overall, the RMS residual forthese “best-ÿt” (BC model) (%) (RBC model) (%) lines (eBectively a 20-parameter ÿt to the data) is found to 0000–0200 14.3 9.1 be 7.7% as given at the foot of Table 3. This compares with 0200–0400 9.1 8.4 the overall value of 12.7% for the simple four-parameter 0400–0600 10.3 8.8 BC model. Although the BC model thus gives a reasonable 0600–0800 13.6 8.5 0800–1000 15.1 10.6 description ofthe overall data set, the factthat the residual 1000–1200 9.0 16.9 values are overall ∼65% greater than those ofthe best-ÿt 1200–1400 18.7 24.3 lines provides motivation to undertake a revision. This will 1400–1600 — — now be attempted in the next section. 1600–1800 — — 1800–2000 12.9 11.4 2000–2200 8.2 5.2 2200–0000 12.6 9.0 4. Revision of the Bunce and Cowley empirical model

Overall RMS residual 12.7 9.7 4.1. Determination of ÿeld model parameters via minimisation of overall RMS error The overall RMS residuals are indicated in the ÿnal row for both BC and RBC models. A dash indicates that insuQcient data were available in that local time sector. Thus far we have shown, in Fig. 4, only the best-ÿt lines to the data in four local time ranges. Now, in Fig. 5, we compare the ÿtted lines from all ten of the 2-h local time ∼20 RJ do not vary greatly, each being close to ∼40 nT, the ÿeld gradients are clearly largest on the dayside, smaller at intervals which provided data over a suQcient range that dawn and dusk, and smallest on the nightside, in accordance the slope m and intercept A relevant to the distance range with the BC model and other papers cited in the introduction. 20–45 RJ can be determined with conÿdence. Bunce and In Table 3 we thus provide for detailed reference the best-ÿt Cowley (2001a) noted that the lines ofbest ÿt to the *y-by coeQcients A and m derived from the 2-h local time ranges data seemed to converge at ∼20 RJ, and hence used this which have suQcient number ofradial ÿeld averages over as a starting point for their model. They assumed that the lines do in fact converge at a certain radial distance  , the the radial range 20–45 RJ. The ÿrst column indicates the 0 local time sector and the second column the radial range of distance within which the ÿeld may be taken as cylindri- the data. This information shows that we have only used cally symmetric, and then fall with distance at various rates those ranges which were deemed to be over a suitable radial depending upon the local time, as described by Eqs. (1) range; those ranges which are not included did not meet and (2). Here we use a model ofthe same form,but deter- this requirement. This is followed in columns 3, 4 and 5 mine the parameters using a slightly diBerent 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 ÿt over the radial range 20–45 RJ. This is accompanied by the corresponding normalised RMS residual in each case

MLT Fit over 20–45 RJ

Range (RJ) nm log A (nT) Residual (%) 0000–0200 21–45+ 103 0:87 ± 0:05 2:69 ± 0:08 7.9 0200–0400 20–45+ 82 1:16 ± 0:04 3:15 ± 0:07 7.6 0400–0600 20–45+ 150 1:13 ± 0:03 3:09 ± 0:06 8.3 0600–0800 20–45+ 236 1:30 ± 0:01 3:29 ± 0:05 8.7 0800–1000 20–45+ 96 1:60 ± 0:04 3:70 ± 0:07 10.0 1000–1200 25–45 21 1:79 ± 0:14 3:92 ± 0:22 8.5 1200–1400 20–30 15 1:37 ± 0:31 3:23 ± 0:48 9.1 1800–2000 22–45+ 26 0:91 ± 0:03 2:78 ± 0:05 3.3 2000–2200 20–40 17 1:10 ± 0:06 3:06 ± 0:09 5.3 2200–0000 22–45+ 116 1:15 ± 0:04 3:13 ± 0:06 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 foreach 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 ÿtting procedure. E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806 799

a selection ofradial proÿles forÿxed 2-h MLT ranges, in a format similar to Fig. 4. In this case, however, the solid 40 lines show the values implied by our revised model (which we term the RBC model) corresponding to the upper and 30 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 corre- sponding to the upper and lower limits ofthe MLT range for 20 our original “BC” model. At the foot of each panel we also show the RMS residuals for both models. It can be seen that

/ nT h / MLT 0 the models provide a very good overall description ofthe ρ 1900 B` 0100 data, with RMS residuals oftypically 10–15%. However, 2100 the residuals ofthe RBC model are generally several per- 2300 0300 centage points lower than that ofthe original model. This ev- 0500 idence is borne out in Table 2, where the RMS residuals are 10 0700 compared for the two models for the various 2-h local time sectors. The RBC model provides an improved description 0900 in most MLT sectors, and, taking all the data together, we ÿnd a RMS residual of9.7% forthe RBC model compared 1300 5 with 12.7% for the original BC model. The four-parameter 1100 RBC model thus provides a description which is almost as 20 30 40 50 60 accurate as those ofthe least squares ÿts at intervals of2-h ρ / RJ MLT, for which the RMS residual was 7.7%. Fig. 5. Plot ofthe ÿtted lines as in Fig. 4, fromthe 10 MLT intervals In Fig. 6b we provide an alternative presentation, where which could be used to determine the dependence on distance in the we show the model values versus MLT in ÿxed ranges of radial range 20–45 RJ. These MLT values are shown on the right hand radial distance, similar to Fig. 3. Here the solid lines show margin. The solid part ofeach line depicts the radial range over which the RBC model, and the dotted lines the BC model. The RMS the ÿt was determined, while the dashed part (i.e. at radial distances residuals for both models are also shown at the bottom of greater than 45 R ) show where the line has been extrapolated outside of J each panel (and over wider radial ranges in Table 1). Again the range. An arrow is drawn at the “hinge” point 0, that is the point ofmaximum convergence, which was determined fromthe least value of the models provide a good overall representation ofthe ÿeld the standard deviation ofthe B0 values weighted to the inverse ofthe in the middle magnetosphere, with the RBC model giving standard error ofeach ÿtted line and normalised to the weighted average, smaller RMS residuals than the BC model in all ranges. while the horizontal bar gives an estimate ofthe error in this value. emphasises the minimisation ofthe RMS fractionalerror. 5. Divergence of the azimuthal current We ÿrst choose a value of 0 (to be iterated), and then deter- mine A by calculating the weighted mean ofthe values ob- 5.1. The azimuthal current and its divergence tained from the ten ÿtted lines shown in Fig. 5. The weights chosen were inversely proportional to the standard error of As previously outlined by Bunce and Cowley (2001a), the gradient ofthe ÿtted lines as given in Table 3. Using a local time asymmetry in the radial ÿeld in the middle these values of 0 and A we then iterate and ÿ to ÿnd the jovian magnetosphere implies a divergence in the equato- pair ofvalues that give the minimum RMS fractionalerror rial azimuthal current. Signiÿcantly larger currents occur at between the model and the data set. We then repeat the pro- midnight at a given distance than at noon. In order to quan- cedure for a range of values of 0 until the value which gives tify the divergence of the azimuthal current we ÿrst need to a global minimum RMS fractional error is found. The global consider the equivalence ofthe radial magnetic ÿeld to minimum is found to have a value of 9.7% at 0 =14:5RJ, the azimuthal current intensity. As described by Bunce and although it is a relatively shallow minimum over the range Cowley (2001a), we ÿnd that, via AmpÂere’s law, the inte- −1 of 0 values ofinterest. The corresponding value of A is grated current intensity (A m ) in the equatorial current 59:7 ± 2:9 nT (where the error given is the standard devi- sheet is given by   ation ofthe values), together with =0:20, and ÿ =1:25. 2 @Bz These parameters then deÿne our revised ÿt to the data. i’ = B0 − D ; (4) 0 @ 4.2. Comparison of model and ÿeld data where, as before, the primed ÿelds indicate that the curl-free planetary ÿeld has been subtracted, D is the half-thickness, In this section we ÿnally check how well the overall model and 0 is the permeability offreespace. In deriving this ex- deÿned in Section 4.1 ÿts the data. In Fig. 6a we ÿrst show pression we have assumed anti-symmetry in the radial ÿeld 800 E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806

(a) 0800-1000 MLT (b) 0400-0600 MLT 40 40

30 30

20 20 / nT / nT 0 0 ρ ρ ' ' B B

10 10

Residual (BC model) = 15.1 % Residual (BC model) = 10.3 % Residual (RBC model) = 10.6 % Residual (RBC model) = 8.8% 5 5 20 30 40 50 60 20 30 40 50 60 ρ ρ / RJ / RJ

(c) 1800-2000 MLT (d) 0000-0200 MLT 40 40

30 30

20 20 / nT / nT 0 0 ρ ' ρ ' B B

10 10

Residual (BC model) = 13.0 % Residual (BC model) = 14.3 % Residual (RBC model) = 11.4 % Residual (RBC model) = 9.1 % 5 5 20 30 40 50 60 20 30 40 50 60 ρ / R ρ J / RJ (a)

Fig. 6. (a) In the same format as Fig. 4, and for the same MLT intervals, we show the radial ÿeld B0, 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 ÿeld B0, as a function of MLT are shown for (a) 22.5–25 RJ, (b) 30–32:5RJ, (c) 35–37:5RJ, and (d) 42.5–45 RJ. The same symbols are used for the Galileo and *y-by data as in previous ÿgures. 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 Bz remains ap- smaller that the ÿrst, such that to within ∼10% the azimuthal proximately constant through the thickness ofthe current current intensity is given by sheet. Bunce and Cowley (2001a) show that for jovian cur- 2B0 rent sheet conditions, the second term in Eq. (4) is much i’ ≈ : (5) 0 E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806 801

50 50 (a) 20-22.5RJ (b) 30-32.5RJ

40 40 /nT /nT 30 30 0 0 ρ ρ B' B'

20 20

10 10

Residual (BC model) = 11.7 % Residual (BC model) = 11.2 % Residual (RBC model) = 10.8 % 0 0 Residual (RBC model) = 8.0 % 00 06 12 18 00 00 06 12 18 00 MLT / h MLT / h 50 50 (c) 3 5 - 37.5RJ (d) 42.5-45RJ

40 40

/nT 30

/nT 30 0 0 ρ ρ B' B'

20 20

10 10

Residual (BC model) = 13.5 % Residual (RBC model) = 10.7 % Residual (BC model) = 14.4 % Residual (RBC model) = 11.3 % 0 0 00 06 12 18 00 00 06 12 18 00 MLT / h MLT / h (b)

Fig. 6. (Continued)

Consequently, our model for the equatorial radial ÿeld Introducing the revised empirical model given by Eqs. (1) just outside the current carrying layer given by Eqs. (1) and (2), we ÿnd
 and (2) above may be approximately but directly converted 2 0 into an empirical model for the azimuthal current, which divi’ ≈ sin ’ ln B0: (7) 0  will therefore undergo the same local time variations as the In Fig. 7a we show a contour map ofthis functionin radial ÿeld. The divergence ofthe azimuthal current is then the equatorial plane (solid lines), labelled with the diver- simply given by gence values in kA R−2. The divergence is exactly zero at 1 @i 2 @B J ’ 0 0 =14:5RJ, within which the model ÿeld is axi-symmetric, divi’ = ≈ : (6)  @’ 0 @’ and also at all radial distances on the noon–midnight 802 E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806

side ofthis range (which allow comparison with the results ofBunce and Cowley, 2001a) should be interpreted with caution. We see that the divergence is negative at dawn, implying a sink ofazimuthal current in that sector, while re- versing to positive at dusk, thus requiring a source ofcurrent. −2 The magnitude ofthe peak divergence in i’ is 15 kA RJ , occurring at a radial distance of ∼23 RJ near the dawn-dusk meridian. In the original BC model the peak magnitude −2 was ∼18 kA RJ occurring at ∼30 RJ near the dawn-dusk meridian. Necessarily, the current overall is divergence-free, and continuity must therefore be maintained either by ra- dial currents *owing wholly within the equatorial current sheet, or via ÿeld-aligned currents which must *ow to- wards 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 direc- tions. The lower three curves in this ÿgure show the total azimuthal current *owing in the model current sheet in the radial ranges 20–30, 30–40 and 40–50 RJ (slightly beyond the outer limit ofthe model), versus MLT. The upper curve shows the sum ofthese, that is the total current *owing be- tween 20 and 50 RJ. These curves have been computed by direct integration ofEq. (5), combined with Eqs. (1) and (2). Each ofthe curves shows, as expected, that the current is maximum at midnight and minimum at noon. Speciÿ- cally, the amount ofcurrent diverted in each case is 6.0, 6.6, and 6:2 MA, the total being 18:8 MA within the range 20–50 RJ. For the original BC model these currents were 8.2, 12.5, and 13:1 MA, respectively, totalling to 33:7MA. The total diverted current is thus a relatively sensitive function of the model employed, but both models indicate values of ∼20–30 MA in a total (e.g. at dawn or dusk) oB ∼59 MA.

5.2. Current stream-function

As indicated above, the “diverted” azimuthal current must *ow either radially in the current sheet itself, or close via Fig. 7. (a) Contours ofthe divergence ofthe azimuthal current in the ÿeld-aligned currents in the ionosphere. In general, both clo- −2 magnetic equatorial plane, in units ofkA R J , derived from the empirical sure paths may be expected to be present. In this case, Bunce model of B derived here. Midnight is marked at the top ofthe plot, with 0 and Cowley (2001b) suggested on physical grounds that the dusk to the right. The dashed rings indicate radial distances of20, 30, 40, total middle magnetosphere current system might best be and 50 RJ, a somewhat extended range ofvalidity. Jupiter is shown in the centre to scale. (b) The total current in MA *owing in various radial viewed as the sum ofa divergence-freecurrent that *ows ranges in the equatorial current sheet versus magnetic local time, obtained wholly within the equatorial current sheet itself, iCS, which from the Revised BC (RBC) empirical model derived here. The current includes all ofthe azimuthal current, together with additional has been integrated in the ranges 20–30, 30–40, and 40–50 RJ, and over radial currents that close wholly via ÿeld-aligned currents in the entire range 20–50 R , as indicated on the right-hand side ofthe plot. J the ionosphere. The divergence-free equatorial current can then be described by a current stream-function ICS having units ofamps, which is such that ICS(; ’)=constant deÿnes meridian, as this is the axis ofsymmetry ofthe model. The a current streamline in the current sheet, while the amount dashed lines in the ÿgure indicate radial distance in the equa- ofcurrent *owing between ICS and ICS +dICS is just dICS. torial plane, starting with 10 RJ and increasing in increments This stream-function is related to the current intensity iCS by of10–50 R J. The range ofdetailed validity ofthe RBC iCS =ˆz ×∇ICS, wherez ˆ is a unit vector perpendicular to the model continues only to 45 RJ, and as such the contours out- current sheet directed northwards. For the model currents E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806 803

12 B0, as a function of radial distance  and magnetic local 100 time MLT. We ÿrst compared the properties ofthe empirical model suggested previously by Bunce and Cowley (2001a) (the “BC” model) derived from the *y-by data from the 80 Pioneer-10 and -11, Voyager-1 and -2, and Ulysses space- craft, with the combined data set obtained from the *y-bys 60 and from the ÿrst twenty orbits of the Galileo orbiter. We 40 ÿnd that the Galileo data exhibit the same local time asym- 20 metry which was described and modelled empirically by Bunce and Cowley (2001a) using the *y-by data alone, and as reported independently by Khurana (2001). This indi- 18 06 cates that the radial ÿelds, and hence azimuthal currents, are weaker on the dayside than those at the same distance on the nightside. Comparison ofthe entire Galileo orbiter and spacecraft *y-by data set with the BC model shows that the model is generally a good representation ofthe radial ÿeld in the middle magnetosphere over the range 20–45 RJ, and overall has a RMS residual of12.7%. Scope forrevis- ing the model is, however, evident from the fact that the overall RMS residual from the “best-ÿt” lines, ÿtted at 2-h 00 MLT intervals is considerably smaller than this at 7.7% (this representing, in eBect the RMS residual ofa 20-parameter Fig. 8. Streamlines ofthe divergence-freecomponent ofthe equatorial ÿt). We thus follow a similar procedure to that described current, ics, determined from the RBC empirical model derived here. by Bunce and Cowley (2001a) and revise the model for the The streamlines are indicated by solid lines and are marked with values larger Galileo and *y-by data set, considering the range 20 showing the total amount ofcurrent carried in the current sheet between –40 R . the streamline concerned and that at radius  =14:5RJ (the innermost J solid line), the inner edge ofthe RBC model. The lines are shown at Overall, we ÿnd that the data are well described by the equal intervals of10 MA, so that the distance between them indicates the function current intensity. The distance from the centre of the planet is shown by
  m(’) the dashed lines, in steps of10 R J, from 10 to 50 RJ, the outer edge of B (; ’)=A 0 ; the region ofinterest. Local times are also indicated and Jupiter is shown 0  to scale in the centre ofthe plot. where A =59:7nT,0 =14:5RJ, and m =0:20 cos ’ +1:25. The ÿeld thus falls as −1:45 at noon, and as −1:05. The over- implied by Eqs. (1) and (2), all RMS residual for this model is 9.7%, now close to the   
m(’)−1 value obtained from the individual least squares ÿts to data 2A0 1 0 ICS(; ’)= 1 − : (8) binned by 2-h intervals ofMLT. This represents a worth- 0 (m(’) − 1)  while improvement over the ÿt provided by the original BC In these equations, the arbitrary zero of I has been set at model. CS The above asymmetry in the ÿeld implies a divergence of radius 0 =14:5RJ. In Fig. 8, the solid lines show contours of I in the magnetic equatorial plane (i.e. current stream- the azimuthal current, with stronger currents implied on the CS nightside at a given radial distance than on the dayside. We lines), where the noon meridian is at the top and dusk to the −2 left. The dashed lines indicate jovicentric distance, and are ÿnd that this divergence has a peak value of15 kA R J near the dawn-dusk meridian at distances close to ∼23 RJ. Over shown at intervals of10 R J out to 50 RJ. The stream con- the range 20–50 RJ, the total diBerence in the azimuthal tours are labelled by the value of ICS in MA, and thus indi- cate the total amount ofcurrent *owing in the current sheet current *owing at midnight compared with noon computed from the RBC model is ∼18:8 MA, compared with a total between that location and 14:5RJ. This diagram shows ex- plicitly how the current streamlines expand outwards on the current (*owing e.g. at dawn and dusk) of ∼59 MA. dayside compared with the nightside, associated with out- An important ÿnal consideration at this juncture concerns ward radial currents at dawn and inward radial currents at the eBect ofthe results presented here on the study ofthe dusk. total equatorial current divergence described by Bunce and Cowley (2001b). The purpose ofthis study was to estimate the magnitude ofthe FACs *owing into or out ofthe cur- 6. Summary and discussion rent sheet from the divergence of the current sheet current intensity, i.e. 1 1 In this paper we have studied the equatorial radial ÿeld j = − divi = − [div(i ˆ) + div(i ’ˆ)]; (9) just outside the jovian middle magnetosphere current sheet, z 2 2  ’ 804 E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806

40 40

30 30

20 20 -2 -2 J J 10 div iρ 10 kA R kA R / / div iρ 0 0 Current Divergence Current Divergence div i div iφ -10 φ -10

-20 -20 10 10

5 5 -2 -2 J 0 J 0 / kA R / kA R

z -5 z -5 j j

-10 -10

-15 -15 P11 In P10 Out -20 -20 20 30 40 50 60 20 30 40 50 60 ρ / R ρ (a) J (b) / RJ

40 40

30 30 div i ρ div iρ 20 20 -2 J -2 J 10

10 kA R kA R / / 0

0 Current Divergence div iφ Current Divergence div iφ -10 -10

-20 -20 10 10

5 5 -2 J -2 J 0 0 / kA R / kA R z -5 j

z -5 j

-10 -10

-15 -15 V1 Out V2 Out -20 -20 20 60 20 30 40 50 60 30 40 50 ρ ρ / RJ (c) / RJ (d)

−2 Fig. 9. The upper plots in each panel ofthis ÿgure show the divergence ofthe radial and azimuthal equatorial currents (kA R J ) as a function of the perpendicular distance from the magnetic axis of the planet  for the four spacecraft passes described in Bunce and Cowley (2001b), i.e. Pioneer-11 inbound, Pioneer-10 outbound, Voyager-1 outbound, and Voyager-2 outbound. The uncertainty estimates are indicated by the dashed lines. The divergence of the radial current, and its uncertainty limits, have been obtained from the ÿtted lines for the four spacecraft passes of Fig. 4, in Bunce and Cowley (2001b). The corresponding quantities for the azimuthal current have been obtained from the Revised BC empirical model. The lower plots show the current density jz normal to the current sheet at its northern surface required for current continuity, as a function of . An equal but opposite current is assumed to *ow out ofthe southern surface.The uncertainties shown by the dashed lines are the square root ofthe sum ofthe squared errors shown in the upper plots ofthis ÿgure.

where jz is the northward current density *owing out ofthe Thus in Fig. 9 we show the re-computed divergence ofthe northward surface of the current sheet (and we assume an equatorial current for the four spacecraft *y-bys employed equal and opposite current *ow from the southern surface). previously, i.e. (a) Pioneer-11 inbound, (b) Pioneer-10 out- The ÿrst term on the RHS was estimated from the radial bound, (c) Voyager-1 outbound, and (d) Voyager-2 out- proÿle of B’ along various *y-by trajectories using a pro- bound. This shows the divergence ofthe azimuthal current cedure analagous to that employed here for the radial ÿeld calculated from the RBC model, marked “divi’”, in the and azimuthal current. The second term was then estimated upper parts ofeach panel. Also shown is the divergence using the BC model, and it is ofimportance to determine ofthe radial current, marked “div i”, calculated from the how much the results are eBected ifthe RBC model is used ÿtted lines to the i data, determined from the B’ data instead. as described in detail in Bunce and Cowley (2001b). The E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806 805

20.0 presented in Fig. 6 ofBunce and Cowley (2001b), with the identical colour coding for the individual spacecraft. Com- P11 Inbound paring this with Fig. 6 in Bunce and Cowley (2001b), we P10 Outbound can immediately conclude that the eBect ofrevising the BC V1 Outbound 15.0 V2 Outbound model to incorporate the ÿrst twenty orbits ofthe Galileo orbiter data, has no signiÿcant eBect overall on the value of j=B for any of the spacecraft passes, although clearly

-1 in detail each have slightly diBerent values. Note here that

nT 10.0 the sign corresponds to the northern hemisphere, such that -2 positive values indicate current *owing from the ionosphere A m to the current sheet. As such, we conclude here that the -13 previous estimation ofthe ÿeld-aligned current density of −2 −12 −2 −1 5.0 ∼0:4 Am (for (j=B)=0:5 × 10 Am nT )at / B) x 10 //

(j the ionospheric heights (particularly from the Voyager data) remains entirely valid.

0.0 Acknowledgements

We thank Prof. M. G. Kivelson for providing the Galileo magnetometer data to the PDS in a timely fashion and Joe -5.0 Maÿ ofthe Planetary Data System, UCLA, forsupplying 20 25 30 35 40 45 50 us with the Galileo magnetometer data and for Fig. la. ρ / RJ EJB was supported during this study by a PPARC Quota Studentship and by PPARC grant PPA=G=0=1999=00181. Fig. 10. Plot ofthe variation of( j=B) versus distance  for the same four SWHC was supported by PPARC Senior Fellowship spacecraft passes, for the Revised BC model. The sign shown corresponds PPA=N=5=2000=00197. to the northern hemisphere, such that positive values indicate current *owing from the northern ionosphere to the current sheet, and vice versa for negative values. The coloured bands indicate the limits of uncertainty, References which follow from the previous ÿgure. The colours also serve as spacecraft identiÿers. Acu˜na, M.H., Behannon, K.W., Connerney, J.E.P., 1983. Jupiter’s magnetic ÿeld and magnetosphere. In: Dessler, A.J. (Ed.), Physics of the Jovian Magnetosphere. Cambridge University Press, Cambridge, value of j , given by Eq. (9) above, is shown in the lower UK, pp. 1. z Balogh, A., Dougherty, M.K., Forsyth, R.J., Southwood, D.J., Smith, parts ofeach panel. We note that, as expected, the diver- E.J., Tsurutani, B.T., Murphy, N., Burton, M.E., 1992. Magnetic ÿeld gence ofthe azimuthal current is somewhat less forthe observations during the Ulysses *y-by ofJupiter. Science 257, 1515. RBC model than for the BC model for each of the space- Barish, F.D., Smith, R.A., 1975. An analytic model ofthe Jovian magnetosphere. Geophys. Res. Lett. 2, 269. craft passes. For Pioneer-11 inbound jz is now seen to be slightly negative (rather than slightly positive when com- Behannon, K.W., Burlaga, L.F., Ness, N.F., 1981. The Jovian magnetotail and current sheet. J. Geophys. Res. 86, 8385. pared with Fig. 5 ofBunce and Cowley, 2001b) but ap- Bunce, E.J., Cowley, S.W.H., 2001a. Local time asymmetry ofthe proximately zero within the estimated errors as previously. equatorial current sheet in Jupiter’s magnetosphere. Planet. Space Sci. Similarly, the Pioneer-10 pass shows that jz is less positive 49, 261. −2 Bunce, E.J., Cowley, S.W.H., 2001b. Divergence ofthe equatorial current (∼2kARJ ) than the value calculated from the BC model. The two Voyager spacecraft have values which are slightly in the dawn sector ofJupiter’s magnetosphere: analysis ofPioneer and Voyager magnetic ÿeld data. Planet. Space Sci. 49, 1089. larger than their previous values, but are not signiÿcantly Caudal, G., 1986. A self-consistent model of Jupiter’s magnetodisc diBerent. including the eBects of centrifugal force and pressure. J. Geophys. We then combine these values of jz with the appropriate Res. 91, 4201. Connerney, J.E.P., Acu˜na, M.H., Ness, N.F., 1981. Modelling the Jovian Bz models as described in Bunce and Cowley (2001b), and hence derive the parameter (j =B)=(j =B ) versus distance current sheet and inner magnetosphere. J. Geophys. Res. 86, 8370. z z Connerney, J.E.P., Acu˜na, M.H., Ness, N.F., Satoh, T., 1998.  for the four passes. This parameter can be used to ob- Cowley, S.W.H., Bunce, E.J., 2001. Origin ofthe main auroral oval tain a direct estimate ofthe ÿeld-aligned current density at in Jupiter’s coupled magnetosphere–ionosphere system. Planet. Space ionospheric heights, since it is expected to be approximately Sci. 49, 1067. conserved along ÿeld lines between the equatorial plane and Dougherty, M.K., Balogh, A., Southwood, D.J., Smith, E.J., 1996. Ulysses the ionosphere. We are therefore interested in quantifying assessment ofthe Jovian planetary ÿeld. J. Geophys. Res. 101, 24 929. Edwards, T.M., Bunce, E.J., Cowley, S.W.H., 2001. A note on the vector any changes that the RBC model implies for this value. In potential ofConnerney et al.’s model ofthe equatorial current sheet Fig. 10 we show this parameter in the same format as it was in Jupiter’s magnetosphere. Planet. Space Sci. 49, 1115. 806 E.J. Bunce et al. / Planetary and Space Science 50 (2002) 789–806

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