ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-37-101-2019Magnetodisc modelling in Jupiter's magnetosphere using Juno magnetic field data and the paraboloid magnetic field modelAnalysis of Juno magnetic field data using paraboloid magnetospheric modelPensionerovIvan A.pensionerov@gmail.comBelenkayaElena S.https://orcid.org/0000-0001-6183-4103CowleyStanley W. H.AlexeevIgor I.https://orcid.org/0000-0003-2057-5365KalegaevVladimir V.ParunakianDavid A.https://orcid.org/0000-0002-5468-4060Federal State Budget Educational Institution of Higher Education M.V. Lomonosov Moscow State University, Skobeltsyn Institute of Nuclear Physics (SINP MSU), 1(2), Leninskie gory, GSP-1, Moscow 119991, Russian FederationDepartment of Physics & Astronomy, University of Leicester, Leicester LE1 7RH, UKIvan A. Pensionerov (pensionerov@gmail.com)5February20193711011096July201812July201818January201927January2019This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://angeo.copernicus.org/articles/37/101/2019/angeo-37-101-2019.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/37/101/2019/angeo-37-101-2019.pdf
One of the main features of Jupiter's magnetosphere is its equatorial
magnetodisc, which significantly increases the field strength and size of the
magnetosphere. Analysis of Juno measurements of the magnetic field during the
first 10 orbits covering the dawn to pre-dawn sector of the magnetosphere
(∼03:30–06:00 local time) has allowed us to determine optimal parameters of the
magnetodisc using the paraboloid magnetospheric magnetic field model, which
employs analytic expressions for the magnetospheric current systems.
Specifically, within the model we determine the size of the Jovian
magnetodisc and the magnetic field strength at its outer edge.
Introduction
In this paper we consider magnetic field measurements made by the Juno
spacecraft in Jupiter's magnetosphere, paying particular attention to the
middle magnetosphere measurements where Jupiter's magnetodisc field plays a
major role. The structure and properties of the Jovian magnetodisc have been
described in many papers, starting from the first spacecraft flybys of
Jupiter, discussed for example by and references therein. In
particular, the empirical magnetodisc model presented by
, derived from Voyager-1 and -2 and Pioneer-10
observations, has been employed as a basis in numerous subsequent studies,
including predictions for the Juno mission by .
Detailed physical models have also been constructed by ,
who derived a steady-state MHD magnetodisc model in which both centrifugal
and plasma pressure (assumed isotropic) forces were included, and by
, who incorporated a self-consistent plasma angular
velocity model. have also included the effects of plasma
pressure anisotropy, as observed in Voyager and Galileo particle
measurements, which redistributes the azimuthal currents in the magnetodisc,
changing its thickness.
Here we model the magnetic field observations during Juno's first 10 orbits
for which both inbound and outbound passes are presently available,
corresponding to perijoves (PJs) 0 to 9, using the semi-empirical global
paraboloid Jovian magnetospheric magnetic field model derived by
. We focus on the middle magnetosphere, observed on these
orbits in the dawn to pre-dawn sector of the magnetosphere
(∼03:30–06:00 local time, LT), for which the magnetodisc provides the
main contribution to the magnetospheric magnetic field. In the model, in
which the field contributions are calculated using parameterised analytic
equations, the magnetodisc is described by a simple thin plane disc lying in
the planetary magnetic equatorial plane. We thus search the paraboloid model
magnetodisc input parameters to determine the best fit to the Juno
measurements. We note that the magnetodisc may be regarded as the most
important source of magnetic field in Jupiter's magnetosphere, with a
magnetic moment in the model derived by using Ulysses
inbound data, for example, which is 2.6 times the planetary dipole moment.
Consequently, the magnetodisc plays a major role in determining the size of
the system in its interaction with the solar wind and is thus an appropriate
focus of a study using Juno magnetic field data.
In (a) we show a schematic of Jupiter's magnetosphere in
the magnetic equatorial plane, showing various parameters of the paraboloid
model. In (b) we show the definition of the planetary magnetic
dipole angle Ψ in the JSM system, where XJSM points towards
the Sun and the planetary dipole is contained in the
XJSM–ZJSM plane.
The Jupiter paraboloid model
The paraboloid magnetospheric magnetic field model was developed for Jupiter
by , based on the terrestrial paraboloid model of
and . It contains the internal
planetary field, Bi, calculated from the full order-4 VIP4
model of ; the magnetodisc field, BMD;
the field of the magnetopause shielding currents, Bsi and
BsMD, which screen the planetary and magnetodisc fields,
respectively; the field of the magnetotail current system,
BTS; and the penetrating part of the interplanetary
magnetic field (IMF), kBIMF, where k is the IMF
penetration coefficient. The magnetopause is described by a paraboloid of
revolution in Jovian solar magnetospheric (JSM) coordinates with the origin
at Jupiter's centre:
xRss=1-y2+z22Rss2,
where x is directed towards the Sun, the x–z plane contains the
planet's magnetic moment, and y completes the right-hand orthogonal set
pointing towards dusk. Rss is the distance to the subsolar
magnetopause, where y=0 and z=0. The magnetospheric magnetic field,
Bm, is then the sum of the fields created by all these
current systems:
Bm=Bi(Ψ)+BTS(Ψ,Rss,R2,Bt)+BMD(Ψ,BDC,RDC1,RDC2)+Bsi(Ψ,Rss)+BsMD(Ψ,Rss,BDC,RDC1,RDC2)+kBIMF,
where Ψ is Jupiter's dipole tilt angle relative to the z axis. The
magnetodisc is approximated as a thin disc with outer and inner radii
RDC1 and RDC2, respectively. BDC is the
magnetodisc field at the outer boundary, while the azimuthal currents in the
disc are assumed to decrease as r-2. R2 is the distance to the inner
edge of the tail current sheet, and Bt is the tail current
magnetic field there. The magnetospheric current systems are thus described
by nine input parameters, determining the physical size of the current
systems, and their magnetic field (current) strength (Ψ,
Rss, R2, RDC1, RDC2, Bt,
BDC, k, BIMF). In Fig. we
show sketches illustrating the parameters of the model. On the left we show a
view in the magnetospheric equatorial plane, where we note that in the
physical system, the overlapping model magnetodisc and tail current sheets
merge together on the nightside. On the right we show the planetary magnetic
dipole axis at angle Ψ in the JSM system. As shown by
, the magnetic moment of the model current disc is given
by
MMD=BDC2RDC131-RDC2RDC1.
and determined model parameters
which approximated the magnetic field along the Ulysses inbound trajectory
rather well. These parameters are Rss=100RJ, R2=65RJ, Bt=-2.5nT,
RDC1=92RJ, RDC2=18.4RJ, and BDC=2.5nT. This set of
parameters is used in the present paper as a starting point for fitting
parameters to the Juno data. The dipole tilt angle Ψ changes during
the observations and is calculated as a function of time in the paraboloid
model.
Magnetic field calculations for the first 10 Juno orbits
As indicated above, field calculations have been made using the paraboloid
model for comparison with the data from the first 10 Juno orbits for which
data are presently available for study. The orbits were closely polar, with
large eccentricity, and with apoapsis initially located south of the equator
in the dawn magnetosphere e.g.. In
Fig. we show the perijove 1 trajectory versus time (in
day of year (DOY) 2016) in JSM Cartesian coordinates, specifically showing
the cylindrical and spherical radial distances ρJSM=x2+y2 and r, ZJSM, and the LT. The vertical dashed line shows
the time of periapsis. On later orbits apoapsis moved towards the nightside,
reaching 03:30 LT by perijove 9, and also rotated further into the
Southern Hemisphere.
Juno perijove 1 trajectory in JSM Cartesian coordinates plotted
versus time in DOY 2016, where the vertical dashed line shows the time of
periapsis.
In this paper we confine our attention to the middle magnetosphere, where, as
we now show, the magnetic field is dominated by the magnetodisc and the
planetary field. In the outer magnetosphere the field becomes strongly
influenced by external conditions in the solar wind, and although in some
circumstances these can be reasonably well predicted by MHD models
initialised using data obtained near Earth's orbit e.g., they will typically vary strongly on the timescale of the Juno
orbit (Fig. ), and with them too the outer magnetospheric
field. In Figs. and , for example, we
show the magnitudes of the modelled field from different sources along the
inbound (a) and outbound (b) passes of perijoves 1 and 9, respectively,
plotted versus radial distance. The red lines in these figures show the
internal JRM09 (“Juno reference model through perijove 9”) planetary field
derived by , which employs the well-determined degree
and order 10 coefficients from an overall degree 20 spherical harmonic
fit to the data (plus disc model field) from the first nine Juno orbits. The
black lines show the field of the various magnetospheric current systems in
the paraboloid model as marked, where the model parameters employed are those
derived from Ulysses inbound data by , as outlined in
Sect. . It can be seen from Figs. 3 and 4 that for r<60RJ the contributions to the
magnetospheric field from the magnetopause and tail current systems (which
are oppositely directed near the dawn–dusk meridian) are negligible compared
with the magnetodisc field, being less than 10 % for perijove 1 and less
than 16 % for perijove 9, and may thus be treated approximately inside
this distance. For related reasons we also neglect the penetrating IMF term
in Eq. (), which is unknown when Juno is inside the
magnetosphere, highly variable in direction with time, and typically of
magnitude ∼0.1–1 nT. This
field too, with penetration coefficient k<1, is therefore similarly
negligible in the r<60RJ middle magnetosphere studied here.
Magnitude of the model magnetic fields for the Juno perijove 1
inbound (a) and outbound (b) passes, due to the internal
planetary field (JRM09, in red), and the various model magnetospheric currents
as marked (magnetopause, tail, and magnetodisc, in black).
As for Fig. , but for perijove 9.
As a consequence of these considerations, here we employ the JRM09 model of
the internal field and fit only the magnetodisc parameters to the middle
magnetosphere data. For the small fields contributed by the magnetopause and
tail current systems in this regime, we simply use the Ulysses parameters
from and as sufficient
approximations, i.e. Rss=100RJ, R2=65RJ, and Bt=-2.5nT. However, use of the
Ulysses magnetodisc parameters is found to lead, for example, to a systematic
underestimation of the field along the perijove 1 trajectory, and thus needs
to be modified. Thus only three parameters, RDC1,
RDC2, and BDC, need to be fitted.
Magnetodisc parameters derived for the Ulysses inbound pass and the
first 10 Juno orbits, together with the estimated errors and the minimum
inbound and outbound radial distances available in the Juno passes.
To optimise the model we choose the approach of minimising function S given
by
S(BDC,RDC1,RDC2)=1N∑n=1NBmod(n)-Bobs(n)2Bobs(n)2,
where Bmod(n) is the modelled field vector due to the
current systems, Bobs(n) is the observed residual
field following subtraction of the JRM09 internal field model, n is the
index number of the data point along the trajectory, and the total number of
points is N. S represents a root-mean-square relative deviation of the
modelled magnetic field from the observed field vectors. We used a relative
deviation instead of an absolute value to equalise the influence of all the
data points, noting that the magnetic field varies in magnitude significantly
along the part of the trajectory examined here (see Figs.
and ). Use of the absolute deviation gives good results in
the region closer to the planet where the field magnitude is greater, but a
poorer fit in other parts of the trajectory.
Observed (black) and modelled (red) residual fields in JSM
cylindrical components, together with the residual field magnitude, for Juno
perijove 1. The residual field is the observed field with the JRM09 internal
field subtracted. The fields are plotted versus spherical radial distance
with inbound data shown on the left and outbound data on the right. The same
model field is used for both.
As for Fig. , but for perijove 6.
With regard to the choice of interval employed to minimise S, we note that
use of data from the innermost region is not optimal. The JRM09 internal
planetary field model differs from observations at periapsis (1.06RJ) by 0.3×105nT, which
is reasonable accuracy for describing an observed field of magnitude ∼8×105nT, but does not allow us to distinguish the magnetodisc
field of order 100 nT on this background. We thus restricted the
inner border of the interval to consider r>5RJ only.
However, on most passes examined here, the inner radial limit is set instead
at somewhat larger radii by the data that are presently available for study.
A further limitation on the region of calculation of S in the outer
magnetosphere arises from the fact that the paraboloid model does not display
regions of low field strength during intersections with the magnetodisc, as
is observed in the field at larger distances, due to the use of the
infinitely thin disc approximation (see Sect. ). It is thus
necessary to avoid these regions by excluding parts of the trajectory where
the spacecraft is closer than 4 RJ from the magnetic equator.
We thus minimise S in the inbound and outbound radial ranges between
Rmin and Rmax on each pass to determine the best-fit
magnetodisc parameters. The minimisation was undertaken using the trust
region reflective procedure . The best-fit values are
given, together with the estimated error values and the radial ranges employed, in
Table , where we also compare with the values derived by
from Ulysses inbound data. We estimated parameter errors
by choosing several different starting points for the algorithm in parameter
space and running it with a more generous termination condition in comparison
with the normal runs. Specifically, we stopped the calculation when dS<0.1S, where dS is the change of function S in the algorithm step. We
then estimated the error as (Pmax–Pmin)/2, where
Pmax and Pmin are the maximum and minimum parameter
values obtained in these runs. For all the Juno fits we found that the
best-fit outer disc radius RDC1 was the maximum value of
95 RJ allowed in the fitting process, set by requiring that the
disc radius should be less than the subsolar magnetopause radius
(100 RJ) by a few RJ. This indicates that the
current density in the model disc, varying as r-2, decreases somewhat
too quickly with distance. The values of the inner disc radius
RDC2 lie between 12.5 and 18.7 RJ, usually smaller
than the value of 18.4 RJ, derived from the Ulysses data, while
the field strength parameter BDC varies between 2.6 and
3.1 nT, larger than the Ulysses value of 2.5 nT.
In Figs. and we provide comparisons of the
observed (black) and modelled (red) residual fields for Juno perijoves 1 and
6, respectively, from which the JRM09 planetary field has been subtracted.
Specifically we show the JSM cylindrical field components together with the
residual field magnitude plotted versus radial distance, where the same model
applies to both inbound (left side) and outbound (right side) data. As can be
seen, the fitted models are generally in good accordance with the
observations for the Bρ and Bz components, while the Bϕ
component is not adequately described, because the model does not include
radial currents in the magnetodisc and their closure current via the
ionosphere. It is also seen in Fig. that the field magnitude is
underestimated inside of ∼10RJ, again probably related to
the too-steep radial dependence of the azimuthal current. As the distance
from Jupiter decreases, a sharp increase in the residual field is observed in
the inner region to >100nT, while the model field plateaus at
several tens of nanoteslas (nT). At the closest distances from the planet the increase is probably due
to inaccuracy of the JRM09 model of the internal field, noting that the model
represents only the degree and order 10 terms from an overall degree 20 fit
.
Comparison of the observed residual field (black) and best-fit
magnetodisc model field (blue) in a similar format to
Fig. . We also show the best-fit paraboloid model (red) as in
Fig. .
Approaches for future improvement of the Jupiter paraboloid model
We first compare the fits derived here with those obtained using the
magnetodisc model derived by from Voyager-1 and -2 and
Pioneer-10 field data, but now fitted to Juno perijove 1 data. In this model
the current flows in a planet-centred annular disc of full thickness
5 RJ, with inner (R0) and outer (R1) radii at 5 and
∼50RJ, respectively. The azimuthal current in the disc is
taken to vary as I0/ρ, where ρ is the perpendicular distance from
the planetary dipole magnetic axis. We optimised this model for Juno perijove
1 using the same method as outlined above, to find best-fit parameters
I0=21×106ARJ-1 (μ0I0/2≈185 nT),
R0=6RJ, and R1=67RJ.
Figure shows a comparison of the observed residual fields
(black) with the best-fit Connerney et al. model (blue) in a similar format
to Figs. and , where we also show the best-fit
paraboloid model (red) from Fig. . One important difference
between the model results is the fact that the model
reflects well the observed periodic sharp drops of magnetic field strength
during spacecraft intersections with the disc. The magnetodisc radial
magnetic field component reverses sign above and below the disc, and at its
centre becomes equal to zero. As indicated in Sect. , the
paraboloid model with an infinitely thin disc certainly cannot reproduce this
feature and should thus be improved by use of a disc current of finite
thickness. The Connerney et al. model demonstrates reasonable coincidence
with observations near Jupiter, but at greater distances overestimates the
magnetic field strength, which indicates that at these distances the current
density variation as ρ-1 is too slow.
As indicated above, neither of the magnetodisc models considered here
describe the azimuthal field well at medium and large distances, which shows
short-term modulations of the field between positive and negative values
related to crossings of the current sheet near the planetary rotation period
(see for example the inbound data in Fig. ). This points to the
well-known existence of radial currents in the magnetodisc associated with
sweepback of the field into a “lagging” configuration
e.g.. Neither of models considered here, the
model and the paraboloid model of ,
include these currents, but only the azimuthal current in the
magnetodisc. Such radial currents have been included in the models by
and , and could be a useful
addition to the paraboloid model, together with their field-aligned and
ionospheric closure currents.
Discussion and conclusions
As shown in Figs. and , in the middle
part of the Jovian magnetosphere selected for study here, the main
contribution to the field due to the magnetospheric current systems is the
equatorial magnetodisc. Here we have refined the magnetodisc parameters
within the Jovian paraboloid model to best fit the Juno data from the first
10 orbits in this region, for which both inbound and outbound data are
presently available. Analysis of the field at very close radial distances
requires better knowledge of the internal planetary field, while the field at
large distances is strongly influenced by the solar wind, whose simultaneous
parameters remain unknown and are generally varying rapidly with time on the
scale of the Juno passes.
As the simplest approximation we took magnetopause and tail current parameters
derived using the Ulysses mission data and
changed only the radial and field strength parameters of the magnetodisc. We
found that the best-fit model consistently had a large outer radius
comparable with the subsolar magnetopause distance (taken to be
100 RJ from the Ulysses model), an inner radius usually between
∼12 and 14 RJ smaller than the Ulysses model
(∼18RJ), and a comparable field strength parameter (at the
outer edge of the disc) of ∼2.5 nT.
To further refine the Jovian paraboloid magnetospheric model, it will be
necessary to take into account the finite thickness of the magnetodisc
current, and also to accurately determine its dependence on the radial
distance from the planet. The existence of radial currents in the disc, as
well as their closure via field-aligned currents in the planetary ionosphere,
should also be incorporated.
Those who would like to work with
the paraboloid model may contact Igor I. Alexeev at
alexeev@dec1.sinp.msu.ru.
IAP – Formal analysis, Visualization, Writing – original draft;
ESB – Supervision, Writing – original draft, Writing – review and editing;
SWHC – Supervision, Writing – review and editing; IIA – Methodology,
Software; VVK – Methodology, Software; DAP – Software, Data curation.
The authors declare that they have no conflict of
interest.
Acknowledgements
Work at the Federal State Budget Educational Institution of Higher Education
M.V. Lomonosov Moscow State University, Skobeltsyn Institute of Nuclear
Physics (SINP MSU), was partially supported by the Ministry of Education and
Science of the Russian Federation (grant RFMEFI61617X0084). Work at the
University of Leicester was supported by STFC grant ST/N000749/1. The Juno
magnetometer data were obtained from the Planetary Data System (PDS). We are
grateful to the Juno team for making the magnetic field data available (FGM
instrument scientist John E. P. Connerney; principal investigator of Juno
mission Scott J. Bolton). Edited by: Elias
Roussos Reviewed by: two anonymous referees
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