Download Global structure of Jupiter`s magnetospheric current sheet

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
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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].
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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
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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].
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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-
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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
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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].
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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
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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.
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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.
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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).
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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
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KHURANA AND SCHWARZL: JUPITER’S CURRENT SHEET
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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
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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
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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.
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