ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-34-641-2016Optimization of Saturn paraboloid magnetospheric field model parameters
using Cassini equatorial magnetic field dataBelenkayaElena S.elena@dec1.sinp.msu.ruhttps://orcid.org/0000-0001-6183-4103KalegaevVladimir V.CowleyStanley W. H.ProvanGabrielleBlokhinaMarina S.BarinovOleg G.KirillovAlexander A.GrigoryanMaria S.Federal 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, UKElena S. Belenkaya (elena@dec1.sinp.msu.ru)26July201634764165618April20165July20166July2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/34/641/2016/angeo-34-641-2016.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/34/641/2016/angeo-34-641-2016.pdf
The paraboloid model of Saturn's magnetosphere describes the
magnetic field as being due to the sum of contributions from the internal
field of the planet, the ring current, and the tail current, all contained
by surface currents inside a magnetopause boundary which is taken to be a
paraboloid of revolution about the planet-Sun line. The parameters of the
model have previously been determined by comparison with data from a few
passes through Saturn's magnetosphere in compressed and expanded states,
depending on the prevailing dynamic pressure of the solar wind. Here we
significantly expand such comparisons through examination of Cassini
magnetic field data from 18 near-equatorial passes that span wide ranges of
local time, focusing on modelling the co-latitudinal field component that
defines the magnetic flux passing through the equatorial plane. For 12 of
these passes, spanning pre-dawn, via noon, to post-midnight, the spacecraft
crossed the magnetopause during the pass, thus allowing an estimate of the
concurrent subsolar radial distance of the magnetopause R1 to be made,
considered to be the primary parameter defining the scale size of the
system. The best-fit model parameters from these passes are then employed to
determine how the parameters vary with R1, using least-squares linear
fits, thus providing predictive model parameters for any value of R1
within the range. We show that the fits obtained using the linear
approximation parameters are of the same order as those for the individually
selected parameters. We also show that the magnetic flux mapping to the tail
lobes in these models is generally in good accord with observations of the
location of the open-closed field line boundary in Saturn's ionosphere, and
the related position of the auroral oval. We then investigate the field data
on six passes through the nightside magnetosphere, for which the spacecraft
did not cross the magnetopause, such that in this case we compare the
observations with three linear approximation models representative of
compressed, intermediate, and expanded states. Reasonable agreement is found
in these cases for models representing intermediate or expanded states.
The availability of empirically determined models of magnetospheric magnetic
fields is a valuable resource in many areas of related research, providing
knowledge in particular of the mapping of features and phenomena along field
lines between the magnetosphere and the planetary ionosphere. Magnetic
models of Saturn's environment, the subject of the present paper, have
included the global empirical model presented by Khurana et al. (2006) based
on early Cassini magnetic field data, together with a sophisticated
axisymmetric self-consistent model of the magnetodisk field and plasma
populations derived by Achilleos et al. (2010). In this study, however, we
focus on the global “paraboloid model” of Saturn's magnetosphere, in which
the outer magnetopause boundary is taken to form a paraboloid of revolution
about the planet-Sun line (Alexeev et al., 2006). This model has been used,
in particular, to investigate the dependence of the magnetospheric magnetic
structure and the origins of the aurorae on the direction and strength of
the interplanetary magnetic field (IMF), employing data from the Cassini
spacecraft and the Hubble Space Telescope as inputs (Belenkaya et al.,
2006a, 2007, 2008, 2010, 2011, 2013, 2014). In addition to the internal
planetary field, taken to be the three-term axisymmetric model derived by
Burton et al. (2010) in the most recent work, this model contains
representations of the field due to the ring current and the tail current,
all contained within the paraboloid magnetopause by the surface current
flowing on that boundary, to which a penetrating field due to the IMF can be
added. A primary parameter describing the system is clearly the distance to
the subsolar magnetopause determined by pressure balance across the boundary
between the magnetospheric magnetic and plasma pressure on one side and the
magnetosheath magnetic and plasma pressure on the other, the latter pressure
being determined by the dynamic pressure of the upstream solar wind flow. In
turn, the distance to the magnetopause will then modulate the size and
strength of the ring current and tail components of the magnetospheric
system (e.g., Bunce et al., 2007, 2008). However, the model parameters have
so far been determined through comparison with only a small sample of
magnetic field observations. Specifically, Alexeev et al. (2006) derived a
set of model parameters appropriate to the expanded magnetosphere observed
during the Saturn orbit insertion pass of the Cassini spacecraft, while
Belenkaya et al. (2006b) similarly derived a set of parameters appropriate
to the compressed magnetosphere observed during the Pioneer11 flyby.
Belenkaya et al. (2008) also proposed a set of parameters appropriate to
conditions intermediate between these two.
In this paper we significantly expand the comparison of the model with
observations by employing Cassini magnetic field data from 18 passes through
the magnetosphere from apoapsis to periapsis, or vice versa, that cover all
principal local time (LT) regimes. The results are used to empirically
determine best-fit equations of how the model parameters depend on system
size, taken for simplicity to depend linearly on the distance to the
subsolar magnetopause.
Saturn paraboloid model
As indicated in Sect. 1, in addition to the internal planetary field, the
Saturn paraboloid model also contains parameterized field components due to
the ring current and the tail current. The fields of all of these components
are then confined inside a magnetopause outer boundary, on which a surface
current flows such that the normal field component on the boundary is zero.
This boundary is taken to be a paraboloid of revolution about the planet-Sun
line given in Kronian solar magnetospheric (KSM) coordinates by
xR1=1-y2+z22R12,
where x is directed from the planet's centre towards the Sun, the x–z plane
contains the planet's spin (and magnetic) axis, and y completes the right-hand
orthogonal set pointing towards dusk. In this expression R1 is the
distance to the subsolar magnetopause along the x axis where y=z=0, and we
note that the boundary flares to a distance of 2R1 on the
dawn-dusk meridian where x=0. Calculations in the paraboloid model are
performed in KSM coordinates. In addition to the Burton et al. (2010)
internal field coefficients and the distance to the subsolar magnetopause
R1, the other model parameters are (i) the tilt angle ψ between
the planet's spin (and magnetic) axis and the KSM z axis depending on planetary
season, (ii) the radial distances to the outer and inner boundaries of the
ring current Rrc1 and Rrc2, respectively, together with
the radial component of the ring current magnetic field at the outer
boundary Brc1, and (iii) the distance to the inner edge of the tail
current sheet R2 and the characteristic tail field at this distance
Bt. We note that the azimuthal current within the ring current system is
taken to vary inversely as the square of the radial distance from the
planet, and that, following Belenkaya et al. (2006a), the total current
carried is given by
Iϕ=2Brc1Rrc1μ0Rrc1Rrc2-1,
where μ0 is the permeability of the vacuum. We note that the model
does not include a representation of the quasi-steady internal field-aligned
current system associated with subcorotation of the magnetospheric plasma
(Hunt et al., 2014, 2015), which in general will cause a small deviation in
LT of the field mapping between the equatorial plane and the ionosphere.
Inclusion of related effects associated with the rotating “planetary period
oscillation” (PPO) current system is also beyond the scope of the model. In
this study we also ignore the small variable field related to the
penetration of the IMF into the magnetosphere.
The parameter sets discussed by Alexeev et al. (2006) for the expanded
magnetosphere, by Belenkaya et al. (2006b) for the compressed magnetosphere,
and Belenkaya et al. (2008) for the intermediate case are shown for future
reference in Table 1. The subsolar magnetopause distances R1 for these
cases are 17.5, 22, and 28 RS, respectively, associated with solar wind
dynamic pressures psw of ∼ 0.08, ∼ 0.03, and
∼ 0.01 nPa. Here RS is Saturn's 1 bar equatorial radius, equal to
60 268 km. It can be seen that as the solar wind dynamic pressure falls and
the magnetosphere expands, the inner edge of the tail current and the outer
edge of the ring current both move to increasing distances from the planet,
while the inner edge of the ring current remains fixed. Similar conclusions
on the behaviour of the ring current were reached from a study of Cassini
magnetic field data by Bunce et al. (2007, 2008). The fields at the outer
edge of the ring current and the inner edge of the tail both decrease as the
system expands. In Table 1 we also show the total current flowing in the ring
current given by Eq. (2). This increases as the system expands, from
∼ 4 MA for the compressed system to ∼ 14 MA for the expanded
system.
Paraboloid model parameter sets for compressed, intermediate, and
expanded Saturn magnetospheric states, following Alexeev et al. (2006) and
Belenkaya et al. (2006b, 2008), together with the total current flowing in
the ring current given by Eq. (2).
Start and end times in year and decimal DOY, corresponding to
Cassini apoapsis and periapsis, or vice versa, for each of the passes
employed in this study.
Cassini passYearStart timeEnd time(decimal DOY)(decimal DOY)Rev 17 inbound2005294.00302.96Rev 17 outbound2005302.96317.22Rev 18 inbound2005317.22331.47Rev 18 outbound2005331.47345.19Rev 19 inbound2005345.19358.89Rev 19 outbound2005/2006358.895.59Rev 23 outbound2006119.00130.69Rev 24 inbound2006130.69142.38Rev 24 outbound2006142.38161.96Rev 25 inbound2006161.96181.55Rev 25 outbound2006181.55193.22Rev 26 inbound2006193.22204.91Rev 145 inbound201141.3851.57Rev 145 outbound201151.5765.53Rev 146 inbound201165.5379.49Rev 146 outbound201179.4993.45Rev 163 inbound201278.0087.90Rev 163 outbound201287.9096.80Cassini magnetic field data and determination of model parameters
As also indicated in Sect. 1, in this paper we employ magnetic field data
from a set of representative equatorial Cassini orbits that span a broad
range of LTs within Saturn's magnetosphere. Specifically, we employ data
from Revs 17–19 in late 2005 whose inbound and outbound passes explored the
pre-noon and pre-dawn sectors, respectively, together with Revs 145, 146,
and 163 in early 2011 and early 2012, whose inbound and outbound passes
similarly explored the post-dusk and post-noon sectors, respectively. We
note that Cassini Rev (orbit revolution) numbers are defined from apoapsis, via periapsis, to
the next apoapsis. In addition we also employ data from the outbound pass of
Rev 23 to the inbound pass of Rev 26 (six passes in total) in mid- to
late-2006 that explored the fields in the nightside magnetosphere. For
definiteness, the start and end times of these passes, corresponding to apoapsis and periapsis, respectively, or vice versa, are
given as year and decimal day of year (DOY) in Table 2. For reasons
discussed below we have chosen to focus on Revs lying close to the planetary
equatorial plane. These orbits are shown colour-coded in Fig. 1, where we
view the planet's equatorial plane from the north. In this kronocentric
solar magnetic (KSMAG) coordinate system z points along the planet's
spin (and magnetic) axis positive northward, the x–z plane contains the Sun, and (as
in KSM) y points towards dusk (e.g., Arridge et al., 2008). A magnetopause
and bow shock corresponding to a typical solar wind dynamic pressure of
0.03 nPa are also shown, derived from the models of Kanani et al. (2010) and
Masters et al. (2008), respectively.
Plot of Cassini orbits whose data are employed in this study,
colour-coded as shown at the top of the figure, corresponding to the inbound
and outbound passes of Revs 17–19, 145, 146, and 163, together with the
apoapsis passes from the outbound pass of Rev 23 to the inbound pass of Rev
26. These orbits are all near-equatorial and are shown looking down on
Saturn's x–y equatorial plane from the north, where the associated z axis is
aligned with the planet's spin (and magnetic) axis, the x–z plane contains the Sun,
and y points towards dusk (KSMAG coordinates). The dashed lines show the
magnetopause and bow shock for a typical solar wind dynamic pressure of
∼ 0.03 nPa, according to the models of Kanani et al. (2010)
and Masters et al. (2008), respectively.
As indicated above, we consider the model parameters described in Sect. 2 to
depend primarily on the subsolar magnetopause distance R1. Since there
is no upstream spacecraft to monitor solar wind conditions at Saturn, here
we simply use the observed positions of the magnetopause on the inbound and
outbound passes of each Rev. The subsolar distance of the magnetopause is
then estimated from the observed position on each pass using the paraboloid
surface given by Eq. (1), specifically using the last position inbound and
the first position outbound in the case of multiple boundary crossings, and
this value is then taken to apply to the whole of the pass. We thus assume
that the magnetopause remains fixed at the value determined from the
crossing throughout each pass. However, since each pass lasts typically for
a few days, this assumption certainly represents a rough approximation. We
note that although for consistency we employ the paraboloid approximation
(Eq. 1) to determine the subsolar magnetopause distances from the observed
positions, the values do not differ greatly from those obtained from the
more detailed Kanani et al. (2010) model shown in Fig. 1. This is explored
in more detail in the Appendix, where we show that since the Kanani et al.
model flares away from the subsolar position a little more strongly than for
the parabola employed here, the subsolar distances determined from the
latter are a little larger than those determined from the Kanani et al.
model, but only by ∼ 10–15 %. Such differences are not
considered significant given the other assumptions outlined above. With the
subsolar magnetopause distance for a particular pass so determined, we then
iterate the other model parameters, starting from the most appropriate set
shown in Table 1, until an optimal fit is obtained, as discussed further in
Sect. 4 below.
While the procedure described above is appropriate to the Revs which span
the dawn, noon, and dusk sectors, for which the orbits intersect the
magnetopause as shown in Fig. 1, it is clearly not appropriate to the
nightside passes on Revs 23–26 for which this is not the case. Instead, in
these cases we compare these data with a set of models representative of
compressed, intermediate, and expanded conditions, specifically using
updated model parameters determined in Sect. 5 from fitting to the data
derived from Revs 17–19, 145, 146, and 163. These results will be presented
in Sect. 6.
Fits for Revs 17–19 and 145, 146, and 163
In Fig. 2 we show data for two representative Revs employed in this study,
specifically for Rev 18 centred in the dawn sector (Fig. 2a) and 145 centred
in the dusk sector (Fig. 2b). From top to bottom the panels of these figures
show (i) an electron count rate spectrogram obtained by the Cassini Plasma
Spectrometer-Electron Spectrometer (CAPS-ELS) instrument covering the energy
range ∼ 0.6–28 keV employed to help plasma regime
identification, (ii) the three components of the magnetic field in spherical
polar coordinates referenced to the planet's spin (and magnetic) axis from which
the Burton et al. (2010) model of the internal planetary field has been
subtracted (the “residual” field), radial component Br,
co-latitudinal component Bθ, and azimuthal component Bϕ,
and (iii) the spacecraft trajectory in KSMAG coordinates mapped to the
x–y, x–z, and y–z planes together with the Kanani et al. (2010) magnetopause and
Masters et al. (2008) bow shock models for 0.03 nPa as in Fig. 1. Time along
the bottom of the upper panels is given in DOY for the year in question,
2005 for Rev 18 in Fig. 2a and 2011 for Rev 145 in Fig. 2b, and beginning of
day markers are shown by circles on the trajectory plots at the bottom of
the figure. The vertical dashed lines in the upper panels show the last
magnetopause crossing inbound, and the first crossing outbound, from which
the radial distance of the subsolar magnetopause R1 has been estimated
for each pass as discussed in Sect. 3.
(a) Plot showing thermal electron, magnetic field, and trajectory
data for Cassini Rev 18. From top to bottom the panels show (i) a CAPS-ELS
electron count rate spectrogram covering the energy range ∼ 0.6–28 keV
colour coded as shown on the right (the intense fluxes at
few eV energies are principally spacecraft photoelectrons while the fluxes
seen over the whole energy band closest to periapsis are due to penetrating
radiation belt particles), (ii) the spherical polar radial r, co-latitudinal
θ, and azimuthal φ components of the magnetic field
referenced to the planet's northern spin (and magnetic) axis, from which the
three-term Burton et al. (2010) model of the internal planetary field has
been subtracted (except for the φ component since the planetary
field has no measurable azimuthal field), and (iii) the spacecraft
trajectory in KSMAG coordinates mapped to the x–y, x–z, and y–z planes together
with beginning of day markers (red circles) and the Kanani et al. (2010) and
Masters et al. (2008) magnetopause and bow shock models for 0.03 nPa as in
Fig. 1. Time along the bottom of the upper panels is in DOY 2005. The
vertical dashed lines show the last magnetopause crossing observed inbound,
and the first crossing observed outbound. (b) Plot showing thermal electron, magnetic field, and trajectory
data for Cassini Rev 145, in the same format as (a). In this case time
along the bottom of the upper panels is in DOY 2011.
Of the three spherical polar components of the residual field shown in Fig. 2,
here we choose to focus on modelling the co-latitudinal component
Bθ which is orthogonal to the equatorial plane of the spacecraft
orbit, positive southward. This component, taken together with the
co-latitudinal planetary field, specifically describes the distribution of
magnetic flux passing through the planetary equatorial plane. As previously
demonstrated, e.g., by Vogt et al. (2011) in the case of Jupiter's
magnetosphere, a reliable mapping from the equator to the ionosphere, where
the field is dominated by the planetary field, is then assured by
conservation of magnetic flux, irrespective of the exact behaviour of the
field lines in between. The latter behaviour depends, for example, on the
detailed position and width of the equatorial ring current sheet, which
strongly influences the observed equatorial values of the residual radial
field Br, as well as the azimuthal field Bφ associated with
plasma sub-corotation sweepback of the field lines (whose associated
field-aligned current is not included in the present model as indicated in
Sect. 2). However, this will not similarly affect the Bθ component
near-normal to the equatorial current sheet, whose value will be
approximately unvarying through the sheet structure as guaranteed by Gauss's
law for the magnetic field (div B=0). For these reasons, modelling
studies of the near-equatorial residual Br and Bφ field
components optimally require data from inclined spacecraft orbits that cut
north-south through the current sheet (e.g., Kellett et al., 2009), rather
than the equatorial orbits selected here for optimal modelling of the
equatorial Bθ component germane to the mapping problem.
It is evident from the data shown in Fig. 2 that the residual co-latitudinal
field Bθ is most strongly influenced by the field of the ring
current, which results in the large negative (northward) values close to
periapsis. Given the radial distance of the subsolar magnetopause R1
determined from the observed boundary position on a particular pass as
described above, we thus first iterated the ring current parameters to
determine an improved fit to the data, starting from the nearest applicable
parameter set given in Table 1. The tail parameters were then in turn
adjusted, and the process repeated until a satisfactory overall fit to each
pass was achieved. The model parameters obtained are termed the
“selected” parameters, and are given for the inbound and outbound passes
of each Rev in Table 3, together with the radial distance of the subsolar
magnetopause for that pass, the mean dipole tilt, and the total current in
the ring current given by Eq. (2). We note in passing that Table 3 also
shows parameter values for each pass determined from linear fits to the
overall parameter data, the “linear” values in the table, which will be
derived and discussed in Sect. 5. For the “selected” parameters, however,
it can be seen, for example, that for Rev 18 shown in Fig. 2a the
magnetosphere was significantly expanded on the inbound pass, R1≈37RS, but relatively compressed on the outbound pass, R1≈19RS, with an outer ring current radius and field strength that
respond accordingly, while the inner ring current radius is relatively
unchanged. On the other hand, for Rev 145 the subsolar magnetopause was
almost unchanged in position on the two passes, at R1≈27RS
intermediate between the two values for Rev 18. The fitted ring current
parameters were also determined to be similar on the two passes, again
intermediate between those obtained for Rev 18 inbound and outbound.
Plots showing the residual co-latitudinal field Bθ
versus radial distance R (RS) together with paraboloid model results for
(a) Rev 18 inbound, (b) Rev 18 outbound, (c) Rev 145 inbound, and (d) Rev
145 outbound. The data in each panel are shown in black, while coloured lines
show model profiles for the “selected” parameters on each pass as given in
Table 3, namely the field due to the ring current (light blue), the tail
current (magenta), and the magnetopause current (dark blue), together with
the total residual field given by the sum of these components (green). The
red line similarly shows the total residual field corresponding to the
linear approximation parameters appropriate to the estimated subsolar
magnetopause radial distances for each pass (Eq. 4), also given in Table 3.
The black vertical dotted line shows the magnetopause limit of
applicability of the model.
Selected and linear model parameters for the inbound and outbound
passes of Revs 17–19, 145, 146, and 163.
Illustration using data from the inbound and outbound passes of
Revs 18 and 145 of the calculation of the “smooth fitted” field profiles
described in Sect. 4, used to evaluate the errors between the field profiles
and the “selected” parameter and “linear approximation” model field
profiles. The black lines show the observed field values as in Fig. 3, while
the green lines show the smooth fitted profiles.
The fits obtained on these two Revs are shown in Fig. 3, where Fig. 3a and b
correspond to the inbound and outbound passes of Rev 18, and Fig. 3c and d
to the inbound and outbound passes of Rev 145. In each case the black
line shows the observed residual Bθ field plotted versus radial
distance from the planet, with the vertical black dotted line showing the
magnetopause position, representing the outer limit of the fit. The fitted
model is shown by the green line, representing the sum of contributions due
to the ring current shown in pale blue, the tail current shown in magenta,
and the magnetopause current shown in dark blue. The overall fit seems
reasonable in all cases. The red line shows the profile obtained from the
“linear approximation” parameter set that combines together the
“selected” results in Table 3 into a single model parameterised by the
subsolar magnetopause distance R1, as will be discussed in Sect. 5.
To quantitatively assess the uncertainties in the fit, we first define a
“smooth fitted” field profile {Bθ} that eliminates
small-scale fluctuations in the residual field which the model does not aim
to reproduce. The observed field is divided into three intervals depending
on radial distance. In the first interval, within 15 RS of the planet,
the residual field increases rapidly with radial distance from negative
values closer to the planet, and is approximated by parabolic fits to the
field taken 200 data points (1 min resolution) at a time. In the second
interval, between 15 and 25 RS, the field increases less quickly with
distance, and is approximated by linear fits taken 400 data points at a
time. In the third interval, at distances within the magnetopause exceeding
25 RS, the field has little overall trend, and is approximated by
constant terms taken 800 data points at a time. The fits are determined by
the ordinary least squares procedure. Examples are presented in Fig. 4a–d,
where the black lines show the original data for the inbound and outbound
passes of Revs 18 and 145 as in Fig. 3, while the green lines show the
smooth fitted profiles for each pass determined using the above procedure.
The magnitude of the differences between the “selected” (and “linear”)
model fits and the data have then been determined using these smooth fitted
profiles. The absolute error Δ and the relative error δ
between the model BθM and the smooth fitted field {Bθ}
in any interval of data are taken to be given by
Δ=BθM-Bθ,
and
δ=ΔBθ+3σ,
where the average indicated by the angle brackets is taken over the interval concerned. Parameter σ in
Eq. (3b) is the standard deviation of the data from the smooth fitted field
in the interval. This has a typical value of ∼ 0.5 nT, and is
only significant in this equation when the mean smooth fitted value
approaches zero, when it prevents a divergence in the value of δ. We
note that for a normal statistical distribution, deviations of the data from
the mean value exceed 3σ with a probability of only 0.27 %.
Results are presented in Table 4, where we show the mean values of the
modulus of the absolute and relative errors of the “selected” parameter
fits for four radial ranges on the 12 passes employed in this study,
specifically for radial distances < 10, 10–17, 17–25, and
> 25 RS, chosen after consideration of the behaviour of the
modelled field with radial distance from the planet. The percentage errors
in each region are seen to be typically ∼ 30 %. The table
also shows the errors for the modified model parameters (“linear
approximation”) derived in Sect. 5 below by combining the results from the
individual passes.
Linear approximation paraboloid model parameters
We now combine the model parameter values determined from each of the passes
for Revs 17–19, 145, 146, and 163, as given by the “selected” values in
Table 3, to form an overall parameter set that depends linearly on the
radial distance of the subsolar magnetopause R1. Figure 5a shows the fit
values for the radial distance parameters plotted versus R1, where the
red circles and triangles show ring current outer and inner radius
parameters Rrc1 and Rrc2, respectively, while the blue
circles show tail current parameter R2. The upper and lower red lines
then show linear least squares fits to the Rrc1 and
Rrc2 data, respectively, while the blue line shows a linear least
squares fit to R2. The formulae for these lines are
Rrc1=0.85R1+2.57,Rrc2=-0.02R1+8.49,R2=1.29R1-20.27,
where all distances are in units of RS. Similarly in Fig. 5b we show
the fit values for the model field strength parameters, where the red
circles show the ring current field parameter Brc1 and the blue
circles the tail field parameter Bt. The red and blue lines then show
the linear least squares fits given by
Brc1=-0.07R1+3.05,Bt=-0.09R1+9.05,
where R1 is in units of RS and the fields are given in nT. For
purposes of comparison, the open symbols in Fig. 5a and b also show the
parameters corresponding to the initial set given in Table 1, whose values
have not been included in the fits. It is seen that the tail field strengths
are comparable, though the inner edge of the tail current is at somewhat
smaller distances in the present study, while ring current radial distances
are also comparable, though the ring current field is somewhat weaker in the
present study.
Returning to Fig. 3, the red lines in each panel show the Bθ
profiles obtained using the linear fit parameters given by Eq. (4)
appropriate to the inferred values of R1 on each pass, as shown by the
“linear” values in Table 3. It can be seen that these are generally
similar to the individually fitted profiles (green lines in each panel), and
that they similarly fit the data reasonably well. The mean values of the
absolute and relative errors of these profiles are given by the “linear
approximation” values in Table 4, and can be seen to be typically
∼ 30–50 %, of the same order as for the individually
selected fit parameters.
Mean values of the absolute |Δ| and
relative |δ| fit errors for the inbound and outbound
passes of Revs 17–19, 145, 146, and 163 in four radial ranges, for both
selected and linear approximation parameters.
In Fig. 5c the red circles similarly show the total ring current values
Iϕ given by Eq. (2) using the selected fit parameters (given in
Table 3) plotted versus R1, to which a third order polynomial has been
fitted, given by
Iϕ=0.001R13-0.114R12+3.724R1-33.895,
which has a maximum value of ∼ 5.7 MA at ∼ 27.5 RS.
In this expression, R1 is in units of RS and Iϕ
in MA. The open red circles showing the values from Table 1 are of
comparable magnitude for the compressed and intermediate cases as shown,
while the value for the expanded model goes off-scale at ∼ 14 MA and
is omitted from the plot. The green circles similarly show the total
ring current determined from the linear approximation fit parameters (also
given in Table 3), for which the corresponding fit is
Iϕ=-0.0007R13+0.0362R12-0.2363R1+0.4569,
which exhibits only minor changes with respect to Eq. (5a) based on the
selected fit parameters, and similarly has a maximum of ∼ 5.8 MA at
∼ 28.7 RS.
“Selected” model parameters for each pass as given in Table 3
are shown plotted versus the estimated distance to the subsolar magnetopause
R1 (RS) on each pass, together with least-squares fitted lines.
Panel (a) shows the radial distance parameters of the model, where the
filled red circles and triangles show the outer and inner boundaries of the
ring current Rrc1 and Rrc2, respectively, while the filled
blue circles show the radial distance of the inner edge of the tail current
R2. The red and blue lines then show linear least squares fits to these
data, given by Eqs. (4a)–(4c). The corresponding open symbols show the
parameters corresponding to the models proposed by Belenkaya et al. (2008)
given in Table 1, which are not included in the fits. Panel (b) similarly
shows the model field parameters, where the filled red circles show the ring
current field strength Brc1 and the filled blue circles the tail
field parameter Bt. Least squares linear fits are shown by the red and
blue lines, as given by Eqs. (4d) and (4e), while the open symbols again
show the values of the Belenkaya et al. (2008) parameters given in Table 1,
not included in the fits. Panel (c) shows the total ring current values
Iφ for each pass given by Eq. (2) using the selected fit
parameters (red circles) to which a third order polynomial has been fitted
(red line), given by Eq. (5a). The green circles show the total ring current
values determined from the linear approximation fit parameters (also given
in Table 3), for which the corresponding third order fit (green line) is
given by Eq. (5b). The open red circles again show the values corresponding
to the Belenkaya et al. (2008) parameters given in Table 1, though the value
for the expanded model is off-scale at ∼ 14 MA and has been
omitted.
(a) and (b) show results of model field line calculations between the
ionosphere and 200 RS down tail for (a) Rev 18 inbound and (b) Rev 18
outbound, as illustrated in Figs. 2–4. The upper panels show field lines
traced in the noon-midnight meridian plane in KSM coordinates, for the
selected model parameters for these passes on the left and the linear
approximations on the right, as given by the parameters above the field line
plots. The IMF was assumed equal to zero. The middle panels show the
northern ionospheric projections of the field lines that reach
x=-200RS viewed looking down on the north pole again for the
selected parameters on the left (green line) and the linear approximations
on the right (red line). These are compared in the lower panel.
(c) and (d) show results of model field line calculations between the
ionosphere and 200 RS down tail for (c) Rev 145 inbound and (d) Rev 145
outbound, in the same format as (a) and (b).
Comparing the results in Fig. 5 with those of the earlier ring current study
by Bunce et al. (2007), who employed data from the first 2 years of
Cassini data together with Pioneer11 and Voyager data, it can be seen
that the overall trends are similar. Over the same radial range of subsolar
magnetopause distances as investigated by these authors, ∼ 17–27 RS,
we find a similar increase in the radial distance of the
outer edge of the current sheet, a similar unresponsiveness in the radial
distance of inner edge of the current sheet located near ∼ 7 RS,
and a similar increase in the total current, though only between
∼ 3 and ∼ 6 MA in our case, compared with
∼ 9 to ∼ 15 MA as determined by Bunce et al. (2007).
Beyond these radial distances in the present study, however, the
total ring current is then found to fall to smaller values again at subsolar
magnetopause distances up to ∼ 37 RS.
To be sure our fitted models provide a reasonable result not only along the
specific Cassini trajectories employed in the study but also throughout
Saturn's magnetosphere, we have calculated model field lines between the
ionosphere and 200 RS down Saturn's tail. Results are shown for our
representative Revs in Fig. 6, where Fig. 6a and b show the models for Rev
18 inbound and outbound, respectively, while Fig. 6c and d similarly show
the models for Rev 145 inbound and outbound, respectively. In these figures
the upper panels show field lines traced in the x–z plane (noon-midnight
meridian) in KSM coordinates, for the selected parameters on the left and
for the linear approximations on the right (Table 3). The IMF in these
calculations was assumed equal to zero, so that the field lines in the more
distant tail beyond typically ∼ 100 RS become
near-parallel to the x axis with essentially no closed flux crossing the
equator at these distances and beyond, such that the “open” magnetic flux
in the tail lobes becomes near-constant. It is then of interest to calculate
the region in the polar ionosphere to which this tail lobe flux maps, and to
consider its relation to the observed location of the open-closed field
boundary in Saturn's magnetosphere and to the auroral oval. The middle
panels of Fig. 6 thus show the northern ionospheric projections of the field
lines that reach x=-200RS, this distance lying well within the
regime of near-constant tail lobe flux in all cases. Again, results for the
selected parameters are shown on the left (green line) and for the linear
approximations on the right (red line), which are then compared in the lower
panel of Fig. 6. It can be seen that these regions are generally, but not
always, similar to each other.
It can further be seen from the results in Fig. 6 that in most cases the
ionospheric region mapping into the model tail extends to ∼ 10∘
from the northern pole, but with an offset towards the
nightside, so that it is located a degree or two poleward of this
co-latitude on the dayside, and a degree or two equatorward of this
co-latitude on the nightside. We then note that Jinks et al. (2014) found
that the open-closed boundary lies on average at ∼ 13.3∘
in the Northern Hemisphere (as shown in Fig. 6), in a
multi-instrument study of Cassini data in the dusk to midnight sector,
compatible with these results. They also found that the poleward boundary of
the main upward field-aligned current region, presumed associated with the
main auroral region, lay on average ∼ 1.8∘
equatorward of this in the Northern Hemisphere, thus typically at
∼ 15.1∘ in this LT sector, compatible with the
results of Hunt et al. (2015). Given a similar small equatorward
displacement of the auroral oval from the open-closed boundary in the
near-noon sector, we would thus expect its poleward boundary in the Northern
Hemisphere to lie just equatorward ∼ 10∘ on the
basis of most of the results in Fig. 6. This expectation is then compatible
with the study of northern dayside UV auroral emissions observed in Hubble
Space Telescope data by Belenkaya et al. (2014), where the poleward boundary
(in “non-storm” cases) was found typically to be near ∼ 11∘
co-latitude, similar to the poleward half-power points found
in this LT sector by Carbary (2012) from a statistical study of Cassini UV
emission data. Overall, we thus conclude that the tail magnetic flux in our
models generally provides a good representation of the open tail flux in
Saturn's magnetosphere, closely related to the size of Saturn's auroral
ovals.
Application of linear approximation model parameters to nightside
data
In this section we now test the ability of the model to account for magnetic
field observations on the nightside of Saturn. To do this we employ data
from Revs whose apoapsides lay in the tail, and consider full apoapsis
passes from the outbound pass of one Rev to the inbound pass of the next,
recalling that Cassini Revs are defined from apoapsis to apoapsis.
Specifically we employ data from three apoapsis passes, from the outbound
pass of Rev 23 to the inbound pass of Rev 26. As discussed in Sect. 3, since
the spacecraft did not cross the magnetopause during these Revs we have no
means to estimate the radial distance of the subsolar magnetopause
R1. Instead we compare these data with three representative models
corresponding to compressed, intermediate, and expanded conditions with
subsolar magnetopause distances of 17.5, 22, and 28 RS, respectively,
as in Belenkaya et al. (2008), but now using the updated linear
approximation parameters given by Eqs. (4a)–(4e). For definiteness these
parameters are given in Table 5. We note that we have also examined these
data in relation to the original parameter sets for these R1 values
given in Table 1, but find that these generally give less satisfactory
results due mainly to over-estimates of the strength of the ring current
field (see Fig. 5b).
Linear approximation model parameters for compressed, intermediate,
and expanded states, corresponding to subsolar magnetopause distances of
17.5, 22, and 28 RS, respectively.
Plots showing the residual co-latitudinal field Bθ
versus radial distance R (RS) together with paraboloid model results for
(a) Rev 24 outbound, and (b) Rev 25 inbound. Model results are shown by the
red, green, and blue lines for the compressed, intermediate, and expanded
linear approximation models, respectively, whose parameters are given in
Table 5.
Results are exemplified in Fig. 7a and b, where we show the residual field
Bθ plotted versus radial distance for the outbound pass of Rev
24 and the inbound pass of Rev 25, respectively. We note that while apoapsis
occurred at ∼ 70 RS on this orbit (Fig. 1), here we
concentrate on the magnetic data within 50 RS. Model results are shown
in these figures by the red, green, and blue lines for the compressed,
intermediate, and expanded linear approximation models, respectively, given
in Table 5. It can be seen that the models for intermediate and expanded
states are in reasonable accord with the data for both passes shown. This is
quantified in Table 6 where we give the mean values of the moduli of the
absolute Δ and relative δ errors
given by Eqs. (3a) and (3b), respectively, in four radial ranges for each of
the three apoapsis passes investigated, in a similar manner to Table 4.
Overall it can be seen that the errors for the intermediate and expanded
models are generally comparable, and smaller than those for the compressed
model.
Mean values of the absolute |Δ| and
relative |δ| fit errors for the apoapsis passes from
Rev 23 outbound to 26 inbound in four radial ranges, for three assumed
states of the magnetosphere, compressed, intermediate, and expanded, using
the linear approximation model parameters in Table 5.
Empirical models of magnetic fields are of great value in many aspects of
magnetospheric physics, in particular allowing the mapping of field lines
between the planet's ionosphere and the outer regions, which is important,
for example, in studies of the origins of planetary auroras. The paraboloid
model of Saturn's magnetosphere has been employed for such purposes in a
number of recent studies (e.g., Belenkaya et al., 2010, 2011, 2013, 2014),
though the parameters of the model have previously been determined by
comparison with magnetic field data from only a few passes through the
magnetosphere (Alexeev et al., 2006; Belenkaya et al., 2006b, 2008). In
addition to the internal field of the planet, these parameters describe the
spatial size and current carried by the ring current and tail current
systems, confined by a magnetopause surface current which flows on a
paraboloid of revolution about the planet-Sun line. The subsolar radial
distance of the magnetopause, governed physically by pressure balance at the
boundary depending on the dynamic pressure of the upstream solar wind, is
taken to set the basic spatial scale of the system, on which the other
parameters may depend. A small penetrating component of the IMF may also be
added, but for simplicity has not been employed in the present study.
Since there is no monitor of interplanetary conditions upstream of Saturn,
here we have employed data from six Cassini Revs which crossed through the
magnetopause on both inbound and outbound passes. The position of the
subsolar boundary was then estimated and applied to the data from each pass
individually. These 12 passes span LTs from pre-dawn via noon to
post-dusk. We focus on modelling the co-latitudinal field component that
defines the magnetic flux passing through the equatorial plane, and
determine the parameters that best fit the data on each pass. Overall errors
for the residual field, with planetary field subtracted, are typically
∼ 30 %. The best-fit parameters from each pass are then used
to determine linear dependencies of the parameters on the subsolar radial
distance to the magnetopause, thus providing formulae for these parameters
for any value of the radial distance. The variation of the ring current
parameters is found to show similar behaviour to that determined previously
by Bunce et al. (2007). However, the parameters of the tail current system
show significant fluctuations about the fitted lines, and differences from
values determined in previous studies, perhaps indicative of the occurrence
of internal dynamics unconnected with the scale size of the system set by
the solar wind. Overall, the fits to the residual field obtained using these
linear approximation parameters are found to be comparably good relative to
those determined from the individually selected parameters. In addition, the
“open” flux in the tail lobes in these models is found usually to be in
reasonable accord, when mapped to the ionosphere, with the
empirically determined location of the open-closed field boundary in the
ionosphere, as well as with the location of the auroral oval whose poleward
boundary is typically located a degree or two equatorward (Carbary, 2012;
Jinks et al., 2014; Belenkaya et al., 2014; Hunt et al., 2015).
We then investigated the field data on six passes through the magnetospheric
tail, for which the spacecraft did not cross the magnetopause. In these
cases we compared the observations with three linear approximation models
representative of compressed, intermediate, and expanded states. Reasonable
agreement was found in these cases with models representing intermediate or
expanded states.
Data availability
Calibrated data from the Cassini mission are available from the NASA
Planetary Data System (PDS) at the Jet Propulsion Laboratory (https://pds.jpl.nasa.gov/).
Comparison of subsolar magnetopause distances estimated from
observed boundary crossings using the paraboloid and Kanani et al. (2010)
models
In the analysis undertaken in this paper we are required to estimate the radial
distance of the subsolar magnetopause from a local observation of the
boundary at some general point (xmp, ymp, zmp) in KSM
coordinates. The values employed here are straightforwardly determined from
solution of the quadratic in R1 formed by Eq. (1). Here we compare these
values with those estimated from the more detailed empirical model derived
by Kanani et al. (2010). In this model the magnetopause surface is taken to
be given by
R=R021+cosθK,
where R is radial distance from the planet, θ is the co-latitude
angle from the KSM x axis (directed towards the Sun), and R0 is the
subsolar magnetopause distance. The latter parameter together with exponent
K are defined in terms of the solar wind dynamic pressure DP (in nPa) as
R0=a1DP-a2,
and
K=a3+a4DP,
where the empirically determined coefficients are a1=10.3RS,
a2=0.2, a3=0.73, and a4=0.4. We thus note that the ratio
between the magnetopause distance at the subsolar point and, e.g., on the
dawn-dusk meridian is 2K≈1.66 in the Kanani et al. (2010) model
(for moderate pressures), compared with 21/2≈1.41 for the paraboloid
model given by Eq. (1). The Kanani et al. model magnetopause thus flares
away from the subsolar point to a somewhat greater degree than the
paraboloid model. Consequently, given a boundary observation at some general
point (xmp, ymp, zmp), or equivalently at some radial distance
R and co-latitude angle θ with respect to the x axis, the estimate of
the subsolar distance to the boundary using the paraboloid model will be
slightly larger than for the Kanani et al. (2010). Substituting for
DP from Eq. (A2) into Eq. (A3), and hence into Eq. (A1), the estimate
from the Kanani et al. (2010) model is obtained by iterative solution for
R0 of
R=R021+cosθa3+a4a1/R01/a2.
Examination of the differences between the radial distances to the subsolar
magnetopause determined from the paraboloid model, R1 (Eq. 1), and the
Kanani et al. (2010) model, R0 (Eq. A4), for each of the 12 passes
employed in this study (inbound and outbound on Revs 17–19, 145, 146, and
163), shows that these are typically ∼ 2–4 RS, with an
averaged value of ∼ 3.3 RS. However, since the averaged
value of R1 is ∼ 27 RS, this represents a relative
difference of only ∼ 10–15 %.
Competing interests
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 RFBR grant 16-05-00760.
Work at Leicester was supported by STFC consolidated grants ST/K001000/1 and
ST/N000749/1. We thank I. I. Alexeev for access to his paraboloid model
codes. The topical editor, E. Roussos, thanks J. Carbary and one anonymous referee for help in evaluating this paper.
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