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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, A07227, doi:10.1029/2004JA010757, 2005 Global structure of Jupiter’s magnetospheric current sheet Krishan K. Khurana and Hannes K. Schwarzl Institute of Geophysics and Planetary Physics, University of California at Los Angeles, Los Angeles, California, USA Received 23 August 2004; revised 23 November 2004; accepted 25 February 2005; published 23 July 2005. [1] Jupiter’s magnetosphere contains a gigantic sheet-like structure located near its dipole magnetic equator that contains most of the plasma and energetic particles swirling in Jupiter’s magnetosphere. Called the ‘‘current sheet,’’ it behaves like a rigid structure inside a radial distance of 50 RJ where the periodic reversals of the Br component are highly predictable. Beyond a radial distance of 25 RJ, the tilted current sheet lags behind the dipole magnetic equator in proportion to the radial distance of the observer. On the nightside, at radial distances >50 RJ, the current sheet is seen to become parallel to the solar wind flow direction. In this work, we analyze magnetic field observations from all six spacecraft that have explored Jupiter’s magnetosphere (Pioneers 10 and 11, Voyagers 1 and 2, Ulysses, and Galileo) to determine the global structure of Jupiter’s current sheet. We have assembled a database of 6328 current sheet crossings by using an automated procedure which utilizes reversals in the radial component of the magnetic field to identify current sheet crossings. The assembled database of current sheet crossings spans all local times in Jupiter’s magnetosphere under differing solar wind conditions. The new model is based on a further generalization of the hingedmagnetodisc models of Behannon et al. (1981) and Khurana (1992). Four new features of the improved model are that (1) close to Jupiter, the prime meridian of the current sheet (the azimuthal direction in which it attains its highest inclination) is found to be shifted by 2.2 from the VIP4 model current sheet (Connerney et al., 1998). (2) In addition to the delay caused by the wave travel time, the location of the current sheet is further delayed because of the sweep-back of the field lines. (3) The signal delay associated with wave propagation is seen to vary both with radial distance and local time, and (4) the current sheet is allowed to become parallel to the solar wind direction at large distances in the magnetotail in agreement with the observations. The new model is much superior at predicting current sheet crossing times than previously published models (especially in the midnight and dusk sectors). The RMS error of fit between the modeled and observed current sheet crossing longitudes is 19.3. A comparison of the new model with previous models is presented. Citation: Khurana, K. K., and H. K. Schwarzl (2005), Global structure of Jupiter’s magnetospheric current sheet, J. Geophys. Res., 110, A07227, doi:10.1029/2004JA010757. 1. Introduction [2] Magnetic field and energetic particle observations from Pioneer 10 and Pioneer 11 [Smith et al., 1974, 1976; Van Allen et al., 1974, Van Allen, 1976; McDonald and Trainor, 1976] showed the presence of a thin (half thickness 2.5 RJ) equatorial current sheet in Jupiter’s dawn magnetosphere. Further observations from Voyager 1 and Voyager 2 [Ness et al., 1979a, 1979b, 1979c; Bridge et al., 1979a, 1979b] established that the current sheet also contains lowenergy plasma and showed that it merges into the magnetotail current system on the nightside. Analyses using pressure balance argument showed that the average energy density of ions is 30 keV in the current sheet [Walker et al., 1978; Lanzerotti et al., 1980]. Copyright 2005 by the American Geophysical Union. 0148-0227/05/2004JA010757$09.00 [3] The structural studies of the current sheet from Pioneer and Voyager data revealed that at radial distances >25 RJ, the current sheet crossings do not coincide with the expected location of the dipole magnetic equator but are delayed in time [McKibben and Simpson, 1974; Carbary, 1980]. Following a lead provided by the theoretical works of Northrop et al. [1974] and Eviatar and Ershkovich [1976], the delay was modeled by researchers as an information propagation lag time that increased linearly with radial distance [Kivelson et al., 1978; Carbary, 1980; Goertz, 1981; Behannon et al., 1981]. [4] Seven spacecraft (Pioneer 10, 11; Voyager 1, 2; Ulysses, Galileo, and Cassini) have now visited the magnetosphere of Jupiter. Figure 1 shows an equatorial view of the trajectories of these spacecraft. Cassini skimmed the boundaries of Jupiter’s magnetosphere in January 2001 but did not come close enough to provide useful information on Jupiter’s current sheet. The other six spacecraft have A07227 1 of 12 A07227 KHURANA AND SCHWARZL: JUPITER’S CURRENT SHEET A07227 coordinate system (R, q, l) called system III (see Dessler [1983] for a definition), where R, q, and l are the radial distance, colatitude, and west longitude of the observation. System III (sIII) is a left-handed coordinate system. In the pre-Galileo era, the space physics community adopted sIII for use with spacecraft and planetary trajectories. However, in accordance with the SI convention, the magnetic field data were presented in a right-handed spherical coordinate system. In this work, we have rationalized the use of the coordinate systems and use a right-handed coordinate system (R, q, j) for both data and trajectories. The sIII righthanded coordinate system used by us is identical to the sIII system except that its azimuthal coordinate f(= 360 l) increases toward east. In order to minimize confusion, we have also reexpressed the equations from previous publications in the sIII right-handed coordinates. [6] The second coordinate system used is called JupiterSun-orbit (JSO) coordinate system (x, y, z) which has its x axis pointing to the Sun and its z axis is perpendicular to the orbital plane of Jupiter. The y axis is normal to z and x axes and completes the triad. This coordinate system is useful for studying structures and phenomena that are influenced by the solar wind. 3. Data Figure 1. The trajectories of the seven spacecraft that have visited Jupiter in the JSE coordinate system. In this coordinate system, the Z axis is along Jupiter’s rotation axis and the X-Y plane lies in the Jovian equatorial plane. The X-Z plane is defined to contain the instantaneous Sun position vector. The legend describes the line style used for each of the spacecraft trajectories. Cassini did not sample the current sheet of Jupiter. Between them, the other six spacecraft have sampled Jupiter’s current sheet over all local times and a wide range of radial distances. The magnetopause and the bow shock boundaries inferred from Galileo data by Joy et al. [2002] have been superimposed. between them sampled the Jovian current sheet over all longitudes and local times over a broad range of radial distances and solar wind conditions. We use magnetic field observations from these six spacecraft to locate all of the current sheet crossings and develop a global model which is in better agreement with the physics of the problem and is valid over all local times. 2. Coordinate Systems [5] In this work, we use two different coordinate systems, a right-handed Jupiter centered spherical coordinate system rotating with Jupiter and a nonspinning Cartesian coordinate system that uses Sun direction as a reference. Traditionally, for observations from Jupiter, astronomers and planetologists have used a left-handed Jupiter-centered spherical [7] A dominant feature of the magnetic field observations from a low-latitude spacecraft in Jupiter’s magnetosphere (see Figure 2) is the periodic reversal of the radial component at the rotation rate of Jupiter. This periodicity caused by the up and down motion of the tilted magnetic dipole equator and the Jovian current sheet over the spacecraft can be used to determine the location of Jupiter’s current sheet. When the spacecraft is above (below) the current sheet, the radial component of the magnetic field is positive (negative). Thus a change of sign of Br from positive to negative marks a north to south (N ! S) crossing where the spacecraft moves from north of the current sheet to its south. Similarly, a change of Br from a negative value to a positive value marks a south to north (S ! N) crossing. In an average sense, the Bq component is always positive in the equatorial plane and because the Jovian current sheet is thin, the relation r B = 0 implies that Bq does not change appreciably across the current sheet. Modeling of data from the dawn/midnight sector shows [Khurana, 1992] that the half thickness of the current sheet is 2.5 RJ in that sector. In much of the magnetosphere, the azimuthal component of the magnetic field is seen to be out of phase with the radial component (see Figure 2) giving the magnetic field lines a swept-back configuration. Hill [1979] and Vasyliunas [1983] showed that the sweepback of the field lines results from a radial current flowing in the equatorial plane which reinforces corotation on the outflowing plasma. Khurana and Kivelson [1993] analyzed the radial currents in the Jovian equatorial plane and concluded that the plasma corotation in the postmidnight quadrant of Jupiter’s magnetosphere could be maintained up to a radial distance of 50 RJ if the outflow rate in that quadrant does not exceed 2.5 1029 amu/s. Recent works have explored the relationship between the Jovian aurora and the equatorial radial currents [Hill, 2001; Cowley and Bunce, 2001; Khurana, 2001]. 2 of 12 A07227 KHURANA AND SCHWARZL: JUPITER’S CURRENT SHEET A07227 Figure 2. An example of magnetic field data collected by Galileo in the dawn sector. Also marked are the N!S crossings (solid lines) and the S!N crossings (dashed lines) identified by the software used in this work. Please note that the y axis scale for the Bj panel is different from the other three panels. [8] In this work we have used all of the magnetic field data available from Pioneers 10 and 11, Voyagers 1 and 2, Ulysses, and Galileo to understand the global structure of the Jovian current sheet. For Pioneers and Ulysses, the available data set resolution was 1 min. For Voyagers we used the 48 s averaged data. For Galileo, the data from all three telemetry modes (LPW, Dt = 0.333 s; RTS, Dt = 24 s; and MRO, Dt = 32 min) were used. To determine the current sheet crossings, the data from all of the spacecraft were first averaged over running windows of 32 min duration. We identified magnetopause crossings visually and excised the data collected from the magnetosheath and the solar wind regions. Next, a computer program identified 6328 current sheet crossings from the reversals of the Br component. [9] The current sheet behaves like a rigid structure inside a radial distance of 50 RJ where the periodic reversals of the Br component are highly predictable. Figure 3 shows magnetic field data from the dawn and dusk sectors of the Jovian magnetotail over a radial distance range of 40– 85 RJ. Inside of 50 RJ, the current sheet crossings are regular, with only a few crossings qualifying as multiple crossings where the spacecraft encounters Br = 0 more than once during its traversal from one lobe of the magnetosphere to the other. However, as the observations from Figure 3 show, beyond 50– 60 RJ the current sheet becomes ‘‘floppy’’ where multiple crossings of the current sheet are common. The main reason for these oscillations of the current sheet at large distance is that because of the reduced field strength, the equilibrium location of the magnetotail changes strongly in response to changes in the solar wind conditions. In addition to the magnetotail motions, we find that the Br component of the magnetic field becomes extremely irregular in the dusk sector beyond a radial distance of 60 RJ (see Figure 3, right). We do not fully understand the reason for the irregular nature of the Br component on the duskside, but a part of the answer may lie in the fact that the current sheet becomes much thicker (half-thickness > 6 RJ) on the duskside so that the spacecraft spends more of its time in the high b current sheet where natural waves and fluctuations are large. The large thickness of the current sheet on the duskside can be gauged from the facts that the Bq component is stronger on the duskside and the lobe-like 3 of 12 A07227 KHURANA AND SCHWARZL: JUPITER’S CURRENT SHEET A07227 Figure 3. A comparison of magnetic field observations from the postmidnight/dawn sector (radial distance 40– 85 RJ) of Jupiter’s magnetotail with the observations from the dusk/premidnight sector (radial distance 85– 40 RJ). quiet field regions are absent. In this work, we have excluded such chaotic periods from our database of current sheet crossings. Alfven wave velocity. Northrop et al. used Mestel’s [1961, 1968] MHD solution to relate the magnetic field configuration to the plasma flow: u ¼ kB þ mr^ j; 4. Current Sheet Description [10] The height of a rigid tilted magnetodisc located in a dipole equator is given by Zcs ¼ r tan qcs cosðf f0 Þ; ð1Þ where Zcs is the height of the current sheet in sIII (righthand) coordinates, r is the cylindrical radial distance of the observer from Jupiter’s spin axis, qcs is the tilt of the magnetodisc with respect to the planetary equator, and j0 is the azimuthal direction (called the prime meridian) in which the elevation of the current sheet is maximum. From the VIP4 model of Jupiter’s internal field [Connerney et al., 1998], qcs = 9.52 and j0 = 339.4 (east longitude). [11] As discussed above, observations from Pioneer and Voyager spacecraft show that the current sheet crossings at radial distances >25 RJ are delayed from the expected dipole equator crossings in proportion to the radial distance of the observer (see Figure 4, where we show current sheet crossing longitudes from Voyager 1). This has been traditionally understood in terms of a signal delay required for the information about the motion of the dipole to propagate to the outer magnetosphere. Northrop et al. [1974] provide a more accurate description of the situation by using the concept of a wave packet traveling in a subcorotating magnetosphere in the presence of a radial outflow (ur). Northrop et al. showed that the incremental delay dd/dr in the arrival of information about the magnetic equator at a radial distance r is given by dd VA Bf rBðWJ Wm Þ ; ¼þ dr r ur B þ VA Br ð2Þ where WJ and Wm are the angular rotation rates of Jupiter and the magnetosphere, respectively, and VA is the local ð3Þ where k, an arbitrary scalar function, quantifies the relationship between the field-aligned flow and the magnetic field and m is a constant along a field line. In the ionosphere of Jupiter, Bj is close to zero; therefore m can be identified as the angular velocity of the ionospheric plasma (Wi). Following Goertz [1981], equation (2) can then be rearranged with the help of (3) as dd Bf ðWJ Wi Þ : ¼ dr rBr ur þ VA Br =B ð4Þ Equation (4) shows that the delay in the current sheet crossing time arises as a consequence of the nonrigid Figure 4. The longitude of Voyager 1 at the time of N!S (solid boxes) and S!N (solid diamonds) current sheet crossings. Shown in solid lines are the expected locations from a model of Khurana [1992]. 4 of 12 KHURANA AND SCHWARZL: JUPITER’S CURRENT SHEET A07227 A07227 As Jupiter rotates, the current sheet moves over the spacecraft and the spacecraft finds itself south of the current sheet (a N!S crossing). As the figure shows, the hinged current sheet would arrive later than the fully tilted current sheet. The figure also shows that the S!N crossing would be observed sooner for the hinged current sheet than for the fully tilted current sheet. The hinging model also predicts that the situation is reversed for a spacecraft located south of Jupiter’s equatorial plane. For such a spacecraft, the S!N crossings would be delayed more than the N!S crossings. [13] The first model to include both hinging and wave delay was put forward by Behannon et al. [1981], who rewrote equation (1) as r Zcs ¼ a0 tan qcs tanh cosðf f0 Þ; a0 Figure 5. The heights of a fully tilted (solid line) and a hinged (dashed line) current sheet observed at a fixed radial location plotted as a function of time. rotation of Jovian plasma with respect to Jupiter. The incremental delay consists of two effects, the bend-back of the field lines in a partially corotating outflowing plasma (the first term on the right-hand side of equation (4), which is negative), and the time it takes for the information to travel through the subcorotating and outflowing (ur > 0) plasma in the magnetosphere (the second term on the righthand side). Integrating (4) over r, we get f0 ¼ f0r¼1 þ dB þ dwave ; ð5Þ where f0r=1 is the prime meridian of the current sheet near Jupiter and Zr dB ¼ Bf dr rBr where f0 = f0D WJr/U, fD0 is the prime meridian of the dipole, a0 is the hinging distance, WJ is the angular velocity of Jupiter, and U is the wave propagation speed. Near Jupiter, in the prime meridian (where f = f0 and the current sheet achieves its highest elevation), Zcs = r tan qcs, but for r a0, Zcs approaches a constant value, a0 tan qcs. [14] Even though the hinged-plane equation (8) provides good fits for Voyager 1 and Voyager 2 current sheet crossings, Goertz [1976, 1979] and Kivelson et al. [1978] have shown that for Pioneer 10 observations, the current sheet did not appreciably bend away from the dipole equator. Another problem with equation (8) is that different hinging distances are required for Voyager 1 and Voyager 2 flybys to obtain good fits. These inconsistencies were resolved by Khurana [1992], who suggested that the hinging was caused by solar wind forcing and not by the centrifugal force acting on the plasma. He chose to hinge the current sheet at a fixed x (JSO) distance rather than at a fixed radial distance. He generalized equation (8) to ð6Þ Zcs ¼ r tan qcs 1 and ð8Þ x0 x cosðf f0 Þ; tanh x x0 ð9Þ where Zr dwave ¼ 1 ðWJ Wi Þ dr: ur þ VA Br =B ð7Þ f0 ¼ f0D WJ Zr dr vðrÞ ð10Þ 1 On the basis of Pioneer 10 observations, Kivelson et al. [1978] put forward the first computational model for the current sheet structure that included wave delay for current sheet crossings observed beyond a radial distance of 14 RJ. Later, models by Behannon et al. [1981], Goertz [1981], Khurana and Kivelson [1989], and Khurana [1992] generalized the model for use with both Pioneer and Voyager data. However, all of the previous models have ignored the delay, dB, arising from the bend-back of the field lines. [12] In addition to the systematic delay seen in all of the current sheet crossings, it was observed that for spacecraft located north of Jupiter’s equatorial plane, the N!S crossings are delayed more than the south to north (S ! N) crossings. This effect has been understood to be related to the hinging of the current sheet, as illustrated in Figure 5. A spacecraft located at a fixed Jovian latitude and radial distance is initially north of the fully tilted current sheet. and the information propagation velocity varies with cylindrical radial distance r vðrÞ ¼ v0 coth r0 ð11Þ so that f0 ¼ f0D WJ r0 r : ln cosh v0 r0 ð12Þ [15] Our experience with the above model shows that in the dawn sector the model works extremely well (see, for example, Khurana [1997]). However, in the dusk sector, the predicted current sheet crossings are systematically delayed compared with the observations. We have therefore gener- 5 of 12 A07227 KHURANA AND SCHWARZL: JUPITER’S CURRENT SHEET Figure 6. The location of the current sheet for three models in three meridians f = f0, f = f0 + 90 and f = f0 + 180 in which the height of the current sheet is maximum, average, and minimum, respectively. alized the Khurana [1992] model further by including three additional effects. First, in the determination of prime meridian, f0, we now explicitly include its dependence on the field line geometry (see equation (5) above). Second, we let the wave velocity v be a function both of radial distance and local time, Y. Finally, we let the current sheet become parallel to the solar wind direction and not the Jovian equator at large distances. The equation describing the new model is given by 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 x 2 @ Zcs ¼ xH tanh þ y2 A tan qVIP4 cosðf f0 Þ xH x H þ x 1 tanh tan qsun ; x A07227 fashion. Therefore a direct inversion of equation (13) is not possible. We solve the problem by using a two-step procedure. In step one, we determine the prime meridian as a function of radial distance and local time by exploiting the fact that for any two consecutive current sheet crossings, the prime meridian and latitude of the current sheet do not change appreciably. In step two, we use direct substitution to determine the hinging distance of the current sheet. [17] For any two neighboring current sheet crossings, the s/c can be assumed to remain at a fixed location in local time coordinates. Therefore the coordinates x and y of the JSO system can be assumed to be constant in equation (13) for the two crossings. In addition, over the small radial distance and local time covered by the spacecraft over this time, the prime meridian f0 of the current sheet, which is a slowly varying function of radial distance and local time, can also be assumed to remain constant. Equation (13) shows that any two neighboring current sheet crossings occur where the vertical location of the spacecraft equals that of the current sheet, i.e., Zsc1 ¼ Zcs1 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 x 2 ¼@ xH tanh þ y2 A tan qVIP4 cosðf1 f0 Þ xH x H þ x 1 tanh tan qsun ð14Þ x and Zsc2 ¼ Zcs2 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 x 2 ¼@ xH tanh þ y2 A tan qVIP4 cosðf2 f0 Þ xH x H þ x 1 tanh tan qsun : ð15Þ x ð13Þ where xH is the hinging distance and qsun is the angle between the Sun-Jupiter line and the Jovigraphic equator. The functional form of the prime meridian longitude, f0, is described in the next section. Figure 6 plots the location of the current sheet obtained from equation (13) in three pseudo-meridians in which the tilt of the current sheet is maximum, zero, and minimum, respectively, with respect to the Jovian equator (f = f0, f = f0 + 90 and f = f0 + 180). For comparison, we also show the locations of the dipole magnetic equator and the current sheet of Khurana [1992] in the same meridians. For this simulation, we chose qsun positive, i.e., the solar wind flow has a southward component and xH = 47 RJ. As shown in the figure, the hinging variable xH acts in such a way that for x < xH, the current sheet is collocated with the dipole magnetic equator but becomes parallel to the solar wind flow for x xH. 5. Determination of the Prime Meridian Longitude [16] As discussed above, new observations show that both the prime meridian and the elevation angle of the current sheet vary with radial distance and local time in a nonlinear Figure 7. The prime meridian, f0(observed), of the current sheet calculated from all of the viable current sheet crossing pairs. In order to reduce the effect of natural noise on the display, the observations were bin averaged in 10 10 RJ2 bins before plotting. In this and following figure s, the location of the magnetopause corresponds to a solar wind dynamic pressure of 0.21 nP and was obtained from the model of Joy et al. [2002]. 6 of 12 KHURANA AND SCHWARZL: JUPITER’S CURRENT SHEET A07227 A07227 Figure 8. (left) Observed ratio Bf/(rBr) in the Jovian magnetosphere computed from data obtained from all six of the spacecraft that have visited Jupiter. In order to reduce the effect of natural noise on the display, the observed ratios were bin averaged in 10 10 RJ2 bins before plotting. The figure was adapted from Khurana [2001] where we have added additional Galileo data. (right) The modeled ratio Bf/(rBr). As most of current sheet crossing data were obtained from low-latitude s/c whose latitude does not change appreciably over the 5 to 6 hours spanning the two consecutive current sheet crossings, we can assume that Zsc1 = Zsc2. Thus equating (14) and (15), we get cosðf1 f0 Þ ¼ cosðf2 f0 Þ: ð16Þ A general solution of (16) is ðf1 f0 Þ ¼ 2np ðf2 f0 Þ; ð17Þ where n is either 0 or 1. Thus there are eight different solutions depending on the signs used in the left-hand and right-hand sides of equation (17) and the value chosen for n. Four of the solutions are trivial and arise when the same sign is used on both sides of (17) (and correspond to situations when both chosen current sheet crossings are either N!S or S!N crossings). However, when one considers a pair in which a N!S current sheet crossing precedes a S!N current sheet crossing, she gets f1 f0 ¼ ðf2 f0 Þ ) f0 ¼ ðf1 þ f2 Þ=2: ð18Þ Also, for the pair consisting of a S ! N crossing followed by a N ! S crossing, she gets ðf1 f0 Þ ¼ 2p þ ðf2 f0 Þ ) f0 ¼ ðf1 þ f2 Þ=2 þ p; ð19Þ where in (18) and (19), one ensures that the cyclic variable j1 is numerically always greater than the cyclic variable j2 by adding 360 to j1, if needed (because in sIII right-hand system, the longitude of a slow moving spacecraft must decrease with time). Figure 7 shows the observed j0 calculated from equations (18) and (19) used on the complete database of current sheet crossings. Only consecutive nonmultiple crossings were used in this procedure. In case of a data gap, we ensured that the two crossings in the pair were not separated by more than 11 hours. We found 4343 viable pairs which were used in equations (18) and (19). As expected, the observed prime meridian decreases with radial distance for all local times (i.e., the prime meridian is delayed). However, we also see that the radial gradient is not uniform over all local times. At a fixed radial distance, the delays are much more pronounced in the dawn sector compared with their values in the dusk sector. To understand this puzzling behavior, let us turn to equation (5), which states that the delays in current sheet result from a combination of two effects, namely, the field line bend-back (dB) and wave propagation time delay (dwave). Observations [Khurana, 2001] show that the bend-back of the field lines is a strong function of local time. Therefore it is possible that some or all of the asymmetry in the observed f0 is caused by the asymmetry of the field line configuration. Figure 8 (left) shows a global plot of the ratio Bj/(rBr) observed by the six spacecraft in Jupiter’s magnetosphere. The ratio clearly shows that the Table 1. Best Fit Values of the Parameters Obtained From the Observationsa dB Parameters a0, RJ1 r0, RJ r1, RJ a1, RJ1 a2, RJ1 a3, RJ1 a4, RJ1 a1, degrees a2, degrees a3, degrees a4, degrees 7 of 12 a Value dwave Parameters Value 0.0056 33.0 52.8 V0, RJ/Hr r2, RJ 39.4 83.4 0.0056 0.0031 0.0036 0.0061 77.0 23.7 167.2 24.5 b0, degrees b1 b2 b3 b4 b1, degrees b2, degrees b3, degrees b4, degrees r3, RJ 2.2 0.7489 0.6144 0.5414 0.0703 119.9 112.2 108.1 71.9 26.2 Hinging distance, xH = 47 RJ. KHURANA AND SCHWARZL: JUPITER’S CURRENT SHEET A07227 A07227 technique with line search [Kahaner et al., 1989] to optimize the fit for the nonlinear equation (20). The best fit model coefficients are shown in Table 1. The RMS error of fit was 0.002 R1 J . Figure 8 (right) shows a color plot of the best fit model. We next integrated equation (20) with respect to the radial distance to get dB (model) (see equation (6) above): 0 r1 2 r r dB ðmodelÞ ¼ a0 r0 ln cosh þ ðr þ r1 Þe r1 A: þ@ 2r1 r0 !#r 4 X am cos mY am : " m¼1 Figure 9. The modeled delay dB in the current sheet caused by the sweep-back of magnetic field. A plot of the modeled dB is shown in Figure 9. As expected, the delays are close to zero near Jupiter and attain large values in the dawn sector in the outer magnetosphere. The dB delays are quite small (<10) in the dusk sector. [18] Next, we compute dwave (observed) from the equation dwave ðobservedÞ ¼ f0 fVIP4 dB : field is more swept-back in the dawn sector than it is in the dusk sector. Noting that the ratio approaches zero near Jupiter and attains a weak radial and strong local time dependence at large radial distance, we selected the following functional form to fit the ratio: Bj r ¼ a0 tanh rBr r0 0 r1 r@ 1 e r1 A þ r1 4 X !! ; am cos mY am ð20Þ m¼1 where Y is the local time of the observation (expressed in radians and measured from midnight) and a0, r0, r1, am, and am are model parameters. We used a quasi-Newton ð21Þ 1 ð22Þ Figure 10 (left) shows the ‘‘observed’’ dwave delay. The wave delay is similar in magnitude to the delay caused by the sweep-back of the field lines. Close to Jupiter, the wave delay is seen to be small and independent of local time. At large distances, the wave delay displays a pronounced local time dependence. In order to model dwave, we generalized the wave delay function of Khurana [1992] to include a local time dependence at large distances (see equation (12) above): WJ r2 r dwave ðmodelÞ ¼ b0 ln cosh V0 r2 " " # # 4 X r ; ð23Þ 1þ bj cos jY bj tanh r 3 j¼1 Figure 10. The dwave (left) observed and (right) modeled. In order to reduce the effect of natural noise on the display, the observations (left) were bin averaged in 10 10 RJ2 bins before plotting. 8 of 12 A07227 KHURANA AND SCHWARZL: JUPITER’S CURRENT SHEET Figure 11. The modeled prime meridian, f0(model), of Jupiter’s current sheet. This figure should be compared with Figure 7 where we plot f0(observed). A07227 where ds is a segment of the field line. An inversion of (24) for wave velocity V requires a knowledge of global magnetic field (to define the relationship between ds and r), which is not yet possible. We will therefore not try to determine the Alfven wave velocity profile in the magnetosphere. [19] Figure 10 (right) shows the modeled dwave. The RMS error of the fit is 11.9 degrees. In order to reduce the effect of outliers on the fit, any data points that differed from the fit by more than 2.5*RMS were excised from the database. This procedure further eliminated another 1252 data points from our database. A physical examination of the excised crossings revealed that most of them occurred during times when the magnetosphere and the current sheet were highly disturbed. The best fit coefficients obtained from the remaining 3091 current sheet crossings are shown in Table 1. The model successfully reproduces both the local time and the radial distance variations of dwave. [20] Finally, we compute the modeled prime meridian, f0, from f0 ðmodelÞ ¼ f0VIP4 þ dB ðmodelÞ þ dwave ðmodelÞ; where Y is the local time of the observation (expressed in radians and measured from midnight), b0, V0, r2, bj, bj, and r3 are model parameters. The parameter b0 is a measure of the difference between the prime meridians of the VIP4 magnetic equator and the new model near Jupiter. For r r3, the term inside the bracket reduces to 1, and the wave delay becomes independent of local time. However at large r, the bracketed term introduces a local time dependence to the delay. The wave velocity V is related to dwave by Zr dwave ðmodelÞ ¼ WJ ds ; V ð24Þ ð25Þ which is plotted in Figure 11. Comparing f0(model) with f0(observed) plotted in Figure 7, we find that the complex behavior of f0(observed) in the magnetosphere of Jupiter is reproduced quite well in the model. 6. Determination of the Current Sheet Tilt [21] At large distance from Jupiter, the current sheet must become parallel to the solar wind flow in the magnetotail. In our model, the changing tilt of the current sheet is described Table 2. RMS Error of Fit Measured in Degrees 1 Model Applied to Kivelson Behannon Behannon New Data From et al. VG1 VG2 KK 1992 Model Current Sheet Crossing Data Derived From All Distances Pioneer 10 21.3 20.2 26.1 19.9 Voyager 1 21.7 27.5 38.3 22.3 Voyager 2 16.4 18.4 16.4 19.6 Galileo 32.7 32.3 25.6 33.0 All spacecraft 32.4 32.0 25.6 32.7 Figure 12. The RMS error in fitting current sheet crossing longitudes as a function of hinging distance. The minimum occurs for x = 47 RJ. 14.8 16.9 12.5 19.4 19.3 Pioneer 10 Voyager 1 Voyager 2 Galileo All spacecraft Data Derived From R < 70 17.1 17.2 19.7 11.0 11.8 17.9 15.5 18.8 14.6 21.7 24.3 19.3 21.5 24.1 19.2 18.2 11.6 12.2 20.5 20.3 13.3 13.8 12.7 14.9 15.0 Pioneer 10 Voyager 1 Voyager 2 Galileo All spacecraft Data Derived From R < 60 18.0 17.0 20.6 10.3 10.1 12.1 14.1 18.7 14.9 18.6 21.9 17.4 18.4 21.8 17.4 18.6 10.3 12.5 17.3 17.2 14.0 13.9 11.3 12.9 13.1 Pioneer 10 Voyager 1 Voyager 2 Galileo All spacecraft Data Derived From R < 50 20.3 18.4 23.2 8.4 9.3 8.0 13.5 18.7 15.1 14.6 18.8 14.8 14.6 18.7 14.9 20.8 8.0 12.5 13.5 13.5 15.7 11.1 10.9 10.4 10.6 9 of 12 KHURANA AND SCHWARZL: JUPITER’S CURRENT SHEET A07227 A07227 Figure 13. The sIII (right-hand) longitude of Galileo at the times of current sheet crossings as a function of radial distance. Shown are current sheet crossings from four different orbits representative of dawn, midnight, dusk, and noon sectors of the magnetosphere. Open circles mark the N!S crossings, whereas boxes mark the S!N crossings. Also shown in solid lines are the best fit curves from the new model. by the parameter xH, called the hinging distance (see equation (13)). As the prime meridian, f0, is now known through equation (25), we can compute the longitude of a current sheet crossing, fcs (model), directly by substituting a range of values of xH in the following equation: 8 > > > > < 9 > > > x > = H tan q Z x 1 tanh sc sun 0 x ! fcs ðmodelÞ ¼ f cos1 ; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 > > > > > > x 2 > xH tanh xH þ y tan qVIP4 > : ; ð26Þ where we have used the fact that at the time of current sheet crossing, Zcs = Zsc. In the above equation the plus (minus) sign is used for the N!S (S!N) crossings. The xH that minimizes the RMS difference between jcs(observed) and jcs(model) is 47 RJ (see Figure 12). The best fit model has an overall RMS error of fit of 19.3 degrees in predicting the current sheet crossing longitudes. 7. Best Fit Model [22] Table 2 compares the RMS error of fit of the new model with those of the previous models. It is seen that the new model performs much better than the existing models in fitting the current sheet database. When compared with the previous models, which were constructed from data derived exclusively from the dawnside, it is seen that the current model performs as well as or better than the existing models in that local time sector. However, the real improvement in the modeling is observed when comparisons are made in the midnight, dusk, and postnoon sectors from new data derived from Galileo. [23] Another informative way of assessing the new model is to plot the observed and modeled sIII right-hand longitude of the spacecraft during current sheet crossings. These 10 of 12 A07227 KHURANA AND SCHWARZL: JUPITER’S CURRENT SHEET are shown in Figure 13 for four representative orbits (G2 outbound, G7 outbound, C22 outbound, and I33 outbound) selected to highlight dawn, midnight, dusk, and noon sectors of the magnetosphere. A good agreement is seen between the observations and the new model in all local time sectors. 8. Summary and Discussion [24] By using a new analysis technique, we have developed a global model of the structure of Jupiter’s current sheet. The new model form shows that the fast Jovian rotation and the supersonic solar wind provide equally important contributions to the equilibrium location of the Jovian current sheet. Four new features make our model robust and add new physics to it. These are (1) near Jupiter, the prime meridian longitude of the current sheet is larger by 2.2 than that in the VIP4 model, (2) the delay in the current sheet location caused by field line bend-back is found to be significant and is now properly modeled, (3) the wave propagation velocity is allowed to be a function of both radial distance and local time, and (4) the current sheet is made parallel to the solar wind direction at large distances in the magnetotail. [25] The results of this study would be of interest for many future studies. The 5 and 10 hour periodicities observed in many spacecraft measurements like local plasma density, electric current density, particle distribution functions, plasma wave intensity, and total charge on dust grains are caused by the relative motion of the spacecraft with respect to the current sheet. The structural models of the current sheet can therefore be used to organize and further understand these observations. The current sheet models are also required in building global models of the magnetospheric field. In addition, works that require estimates of the thickness of the current sheet in the magnetosphere will benefit from this study because magnetic field and particle density data can be fitted to models like the Harris neutral sheet model using equation (13). [26] Finally, we would like to comment on the 2.2 shift required in the prime longitude of the current sheet near Jupiter. The shift implies that VIP4 model does not adequately represent the orientation of the Jovian dipole during the Galileo epoch. The dipole tilt and longitudinal orientation of the VIP4 model are 9.5 and 159.2, respectively. We find that even though the VIP4 dipole tilt agrees with our model, a more realistic value of its longitudinal orientation is 161.4. There is an obvious explanation for this longitudinal shift. It is quite likely that the Jovian rotation period used to calculate the spacecraft longitudes needs a small revision (requiring a change of 2.2 over the 2 1/2 decades that separate Galileo measurements from Voyager and Pioneer observations). Recently, Z. Yu et al. (Rotation period of Jupiter from the observations of its magnetic field, submitted to Icarus, 2005) performed a singular value decomposition analysis of the internal field of Jupiter to determine its secular variations and to better define the rotation period of Jupiter. Even though they used a different data set (vector magnetic field from Galileo inside of 20 RJ rather than the current sheet crossing locations used by us) and also used a completely different technique (least squares fit to the internal magnetic field) to determine the dipole’s A07227 longitudinal orientation, our value of f0 = 341.6 is virtually identical to the value obtained by them (341.9) for the Galileo epoch. This suggests that the rotation period of Jupiter as defined by IAU (= 9 hours 55 min 29.71 s) requires a very minor correction (6 ms) to be compatible with the Galileo magnetic field observations. The new rotation period consistent with our observations is 9 hours 55 min 29.704 s. [27] Acknowledgments. We gratefully acknowledge many discussions with Margaret Kivelson, which helped us to greatly improve the paper. We would like to thank Steven Joy and Joe Mafi of the Magnetospheric Node of the Planetary Data System for preparing the magnetic field data sets used in this work. We also thank Dianne Taylor for her contribution to this work. This work was supported by the National Aeronautics and Space Administration through Jet Propulsion Laboratory under contract 1238965, and by NASA grants NAG5-8945 and NAG59546. UCLA-IGPP publication 6206. [28] Lou-Chuang Lee thanks Aharon Eviatar and another reviewer for their assistance in evaluating this paper. References Behannon, K. W., L. F. 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