the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Magnetodisc modelling in Jupiter's magnetosphere using Juno magnetic field data and the paraboloid magnetic field model

### Ivan A. Pensionerov

### Elena S. Belenkaya

### Stanley W. H. Cowley

### Igor I. Alexeev

### Vladimir V. Kalegaev

### David A. Parunakian

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.

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 Barbosa et al. (1979) and references therein. In particular, the empirical magnetodisc model presented by Connerney et al. (1981), 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 Cowley et al. (2008, 2017). Detailed physical models have also been constructed by Caudal (1986), who derived a steady-state MHD magnetodisc model in which both centrifugal and plasma pressure (assumed isotropic) forces were included, and by Nichols (2011), who incorporated a self-consistent plasma angular velocity model. Nichols et al. (2015) 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 Alexeev and Belenkaya (2005). 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 Alexeev and Belenkaya (2005) 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.

The paraboloid magnetospheric magnetic field model was developed for Jupiter
by Alexeev and Belenkaya (2005), based on the terrestrial paraboloid model of
Alexeev (1986) and Alexeev et al. (1993). It contains the internal
planetary field, *B*_{i}, calculated from the full order-4 VIP4
model of Connerney et al. (1998); the magnetodisc field, *B*_{MD};
the field of the magnetopause shielding currents, *B*_{si} and
*B*_{sMD}, which screen the planetary and magnetodisc fields,
respectively; the field of the magnetotail current system,
*B*_{TS}; and the penetrating part of the interplanetary
magnetic field (IMF), *k**B*_{IMF}, 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:

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. *R*_{ss} is the distance to the subsolar
magnetopause, where *y*=0 and *z*=0. The magnetospheric magnetic field,
*B*_{m}, is then the sum of the fields created by all these
current systems:

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
*R*_{DC1} and *R*_{DC2}, respectively. *B*_{DC} is the
magnetodisc field at the outer boundary, while the azimuthal currents in the
disc are assumed to decrease as *r*^{−2}. *R*_{2} is the distance to the inner
edge of the tail current sheet, and *B*_{t} 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,
*R*_{ss}, *R*_{2}, *R*_{DC1}, *R*_{DC2}, *B*_{t},
*B*_{DC}, *k*, *B*_{IMF}). In Fig. 1 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
Alexeev and Belenkaya (2005), the magnetic moment of the model current disc is given
by

Alexeev and Belenkaya (2005) and Belenkaya (2004) determined model parameters
which approximated the magnetic field along the Ulysses inbound trajectory
rather well. These parameters are *R*_{ss}=100 *R*_{J}, *R*_{2}=65 *R*_{J}, ${B}_{\mathrm{t}}=-\mathrm{2.5}$ nT,
*R*_{DC1}=92 *R*_{J}, *R*_{DC2}=18.4 *R*_{J}, and *B*_{DC}=2.5 nT. 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.

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. Connerney et al., 2017). In
Fig. 2 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 ${\mathit{\rho}}_{\mathrm{JSM}}=\sqrt{{x}^{\mathrm{2}}+{y}^{\mathrm{2}}}$ and *r*, *Z*_{JSM}, 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.

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. Tao et al., 2005; Zieger and Hansen, 2008), they will typically vary strongly on the timescale of the Juno
orbit (Fig. 2), and with them too the outer magnetospheric
field. In Figs. 3 and 4, 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 Connerney et al. (2018), 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 Alexeev and Belenkaya (2005), as outlined in
Sect. 2. It can be seen from Figs. 3 and 4 that for *r*<60 *R*_{J} 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. (2), which is unknown when Juno is inside the
magnetosphere, highly variable in direction with time, and typically of
magnitude ∼0.1–1 nT (Nichols et al., 2006, 2017). This
field too, with penetration coefficient *k*<1, is therefore similarly
negligible in the *r*<60 *R*_{J} middle magnetosphere studied here.

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 Alexeev and Belenkaya (2005) and Belenkaya (2004) as sufficient
approximations, i.e. *R*_{ss}=100 *R*_{J}, *R*_{2}=65 *R*_{J}, and ${B}_{\mathrm{t}}=-\mathrm{2.5}$ nT. 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, *R*_{DC1},
*R*_{DC2}, and *B*_{DC}, need to be fitted.

To optimise the model we choose the approach of minimising function *S* given
by

where ${\mathit{B}}_{\mathrm{mod}}^{\left(n\right)}$ is the modelled field vector due to the
current systems, ${\mathit{B}}_{\mathrm{obs}}^{\left(n\right)}$ 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. 3
and 4). 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.

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 (1.06
*R*_{J}) by 0.3×10^{5} nT (Connerney et al., 2018), which
is reasonable accuracy for describing an observed field of magnitude $\sim \mathrm{8}\times {\mathrm{10}}^{\mathrm{5}}$ nT, 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*>5 *R*_{J} 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. 4). It is thus
necessary to avoid these regions by excluding parts of the trajectory where
the spacecraft is closer than 4 *R*_{J} from the magnetic equator.

We thus minimise *S* in the inbound and outbound radial ranges between
*R*_{min} and *R*_{max} on each pass to determine the best-fit
magnetodisc parameters. The minimisation was undertaken using the trust
region reflective procedure (Branch et al., 1999). The best-fit values are
given, together with the estimated error values and the radial ranges employed, in
Table 1, where we also compare with the values derived by
Alexeev and Belenkaya (2005) 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 d*S*<0.1*S*, where d*S* is the change of function *S* in the algorithm step. We
then estimated the error as (*P*_{max}–*P*_{min})∕2, where
*P*_{max} and *P*_{min} 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 *R*_{DC1} was the maximum value of
95 *R*_{J} allowed in the fitting process, set by requiring that the
disc radius should be less than the subsolar magnetopause radius
(100 *R*_{J}) by a few *R*_{J}. 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
*R*_{DC2} lie between 12.5 and 18.7 *R*_{J}, usually smaller
than the value of 18.4 *R*_{J}, derived from the Ulysses data, while
the field strength parameter *B*_{DC} varies between 2.6 and
3.1 nT, larger than the Ulysses value of 2.5 nT.

In Figs. 5 and 6 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 *B*_{ρ} and *B*_{z} components, while the *B*_{ϕ}
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. 5 that the field magnitude is
underestimated inside of ∼10 *R*_{J}, 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 >100 nT, 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
(Connerney et al., 2018).

We first compare the fits derived here with those obtained using the
magnetodisc model derived by Connerney et al. (1981) 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 *R*_{J}, with inner (*R*_{0}) and outer (*R*_{1}) radii at 5 and
∼50 *R*_{J}, respectively. The azimuthal current in the disc is
taken to vary as *I*_{0}∕*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
${I}_{\mathrm{0}}=\mathrm{21}\times {\mathrm{10}}^{\mathrm{6}}$ $A{R}_{\mathrm{J}}^{-\mathrm{1}}$ (${\mathit{\mu}}_{\mathrm{0}}{I}_{\mathrm{0}}/\mathrm{2}\approx \mathrm{185}$ nT),
*R*_{0}=6 *R*_{J}, and *R*_{1}=67 *R*_{J}.
Figure 7 shows a comparison of the observed residual fields
(black) with the best-fit Connerney et al. model (blue) in a similar format
to Figs. 5 and 6, where we also show the best-fit
paraboloid model (red) from Fig. 5. One important difference
between the model results is the fact that the Connerney et al. (1981) 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. 3, 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. 6). 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. Hill, 1979). Neither of models considered here, the Connerney et al. (1981) model and the paraboloid model of Alexeev and Belenkaya (2005), include these currents, but only the azimuthal current in the magnetodisc. Such radial currents have been included in the models by Khurana (1997) and Cowley et al. (2008, 2017), and could be a useful addition to the paraboloid model, together with their field-aligned and ionospheric closure currents.

As shown in Figs. 3 and 4, 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 (Alexeev and Belenkaya, 2005; Belenkaya, 2004) 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 *R*_{J} from the Ulysses model), an inner radius usually between
∼12 and 14 *R*_{J} smaller than the Ulysses model
(∼18 *R*_{J}), 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.

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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