the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Statistical analysis of magnetopause crossings at lunar distances

### Johannes Z. D. Mieth

### Dennis Frühauff

### Karl-Heinz Glassmeier

Different magnetopause models with a diverse level of complexity are in use.
One thing that they have in common is that they are mainly based on near-earth observations; i.e. they
use measurements at distances of about ±10 Earth radii along the GSM
*x* axis. Only very few observations of magnetopause crossings at larger
distances are used for model fitting. In this study we compare position and
normal direction predictions of the Shue et al. (1997) magnetopause model with
actual observations of magnetopause crossings identified using the ARTEMIS
spacecraft at lunar distance, about 60 Earth radii. We find differences in
the location prediction between model and actual observation but good
agreement in predictions about the magnetopause normal direction.

The magnetopause plays an important role for space weather processes as it is
the primary interaction zone between the solar wind (SW) plasma and the Earth's
magnetosphere. The magnetopause is defined as the boundary between SW
and magnetospheric plasma which can not be penetrated by the SW
(Baumjohann and Treumann, 1996). In the case
of an equilibrium magnetopause this is
the plane where the SW pressure is balanced by the Earth's magnetic
field pressure (e.g. Glassmeier et al., 2008). In 1997 Shue and co-workers
presented a very simple model to predict the magnetopause (MP) position under
different SW conditions (Shue et al., 1997). Additionally to the
location prediction it is also possible to deduce the MP normal direction.
Using data of magnetopause crossings of the ISEE 1 and 2, AMPTE/IRM, and IMP
8 satellites they modelled the magnetopause radial distance *r* with the
functional form $r={r}_{\mathrm{0}}{\left[\mathrm{2}/\left(\mathrm{1}+\mathrm{cos}\mathit{\theta}\right)\right]}^{\mathit{\alpha}}$. Here *r*_{0}, *θ*, and *α* denote the standoff
distance, the angle between the Sun–Earth line and the direction of *r*, and
the magnetopause flaring parameter, respectively (Fig. 3).
Shue et al. (1997) modelled the MP location to be only dependent on the *B*_{z}
component of the interplanetary magnetic field (IMF) and the SW
dynamic pressure *D*_{p}. This functional model is mathematically
axially symmetric around the *x* axis in solar-wind-aberration-corrected
geocentric solar ecliptic (GSE) and geocentric solar magnetospheric (GSM)
coordinates (Hapgood, 1992). Measurements used for the determination of
the fitting parameters are mainly from distances of ±10 Earth radii
(*R*_{E}) on the *x* axis, with only a few data points expanding up to
about 30 *R*_{E} downtail. As precise the Shue model is, it
requires further observations from higher latitudes as well as crossings
further downtail from the Earth to provide a more realistic 3-D magnetopause
model (Shue and Song, 2002). Extensions of the Shue model were thus presented by Lin et al. (2010) and Wang et al. (2013), for example. However, all the proposed models
are still characterized by using only a very limited number of measurements
at greater distances downtail. This is where our study contributes. By using
plasma and magnetic field measurements from the ARTEMIS mission we validate
the Shue model at radial distances of about 60 *R*_{E} downtail.

The Acceleration, Reconnection, Turbulence, and Electrodynamics of
the Moon's Interaction with the Sun (ARTEMIS) Mission
(Angelopoulos, 2010) provides long-term measurements of the plasma
environment in the terrestrial magnetosphere at lunar distances, about
60 *R*_{E}. Since 2011 the two spacecraft THB and THC orbit the Moon
and provide excellent measurements of the plasma environment there. The THB
and THC spacecraft originate from the THEMIS mission
(Angelopoulos, 2008), a NASA Medium-Class Explorers (MIDEX) mission,
launched on 17 February 2007 and designed to investigate the trigger
mechanisms and evolution of magnetospheric substorms. Five identical
spacecraft were put into Earth's orbit to line up along the magnetotail.
After the primary mission phase the two outermost spacecraft were lifted into
a lunar orbit. Since May 2011 both probes are in stable equatorial and
eccentrical orbits.

## 2.1 Observations

Our study covers a time span of 5 years, starting January 2011 and lasting until December 2015. Different types of data products are used to determine magnetopause position and direction. The electrostatic analyzer (ESA) (McFadden et al., 2008) provides ion and electron flux density over a broad energy band from only a few eV up to 30 keV. We use time-resolved ion energy data with a resolution of about 3 s in this study. In order to generate this data set, measurements with higher temporal resolution are integrated over a spin period of the spacecraft. The plasma data are complemented by measurements from the ARTEMIS fluxgate magnetometer (FGM) (Auster et al., 2008), providing vector magnetic field data which we average over the spin period of about 3 s.

## 2.2 Data processing

Since the MP behaviour depends on the instantaneous properties of the SW, measurements of the same are required. Such measurements are provided
by NASA/GSFC's OMNI data set through OMNIWeb, from which we extracted 1 min
SW magnetic field and plasma data for the desired time range. The
magnetic field information as well as the SW velocity is provided in
an aberration-corrected GSE coordinate system. In contrast, the extracted
position information for a MP crossing underlies SW aberration. To calculate
the individual aberration angle for each crossing, the SW velocity
*v*_{SW}(*t*_{0}) within a time range of a few hours before the
crossing is extracted from OMNI. From this the time
*t*_{SW}(*v*_{SW}(*t*_{0})) can be calculated
which the SW needs to propagate from the Bow Shock Nose (BSN) to the
MP position along the *x* axis. The actual SW properties can then be
extracted when the condition

is fulfilled. Afterwards position data and magnetic field data of each crossing can be corrected by the aberration angle and subsequently transformed into GSM coordinates. However, as OMNI data are prepared for the situation before the BSN, the transitional conditions at the BSN have to be taken into account. To do so, Rankine–Hugoniot conditions are applied to the SW velocity by multiplying with a factor of 1∕4. OMNI SW data are also taken as the input parameter for the Shue model.

To determine the MP normal direction, minimum variance analysis (MVA) (e.g. Paschmann and Daly, 1998) is applied to the magnetic field data within 5 min before and after any identified MP crossing, which will be defined below. As the MVA analysis only provides the orientation but not the direction of the normal, we assume the magnetopause normal to be always directed outwards of the MP, into the direction of the magnetosheath.

## 2.3 Identifying MP crossings

Time periods of possible MP crossings are manually selected from the
available ESA and FGM data sets when the spacecraft is located near the MP
position, as predicted by the Shue model. Here, “near” means about ±10 *R*_{E} on the *x**y* plane around the predicted position. The
actual crossings are subsequently identified by visual inspection of ESA and
FGM measurements. The magnetosheath plasma is characterized by a significant
energy flux around 1 keV. This flux almost instantly ceases once
the MP has been crossed (Paschmann et al., 1993); see
Fig. 1. Furthermore, the particle number density,
as derived from the energy spectrum, also exhibits discontinuous changes when
crossing the MP. In this way the precise crossing times and conditions are
determined. Usually multiple crossings of the MP are also detected during the
spacecraft motions into and out of the magnetotail. Like in Shue et al. (1997)
the innermost crossing is selected for further analysis in the current study.
In order to extend the analysis further, the outermost crossing is also
considered separately. As the innermost crossing we denote the last (first)
MP crossing of an inbound (outbound) pass through the boundary region. In the
case of the outermost, it is exactly the opposite.

A total of 227 innermost and outermost
crossing transitions are found this way. For 225 of these SW data are available.
Figure 2 displays the spatial distribution of the
MP positions determined on an *x**d* plane. The *x* axis points in the same direction as in the GSM or GSE coordinate system, whereas $d=\pm \sqrt{{y}^{\mathrm{2}}+{z}^{\mathrm{2}}}$. The sign is equal to the sign of the *y* component so
that in- and outbound passes can be distinguished and either position can be
visualized better. Using *d* as a measure for the distance of the MP crossing
from the *x* axis supports the view of the model as axial symmetric and
removes any projection errors in case of a projection onto any of the GSM
planes. Shown as a red dot are the mean positions of each independent point
clouds.

Figure 3 shows all necessary variables of the Shue model
and our convention to compare actual positions with it. As the Shue model
fits empirical data, fitting parameters *a*_{1} to *a*_{7} for the standoff
distance *r*_{0} and the flaring parameter *α* come with uncertainties
(Shue et al., 1997).

Equations (2) and (3) are Eqs. (10) and (11) in Shue et al. (1997).

We interpret this uncertainty as a measure of the standard deviation of the predicted MP position. If any MP position derived from ARTEMIS observations falls into this standard deviation, we regard this MP position as compatible with the model. This leads to a minimum and maximum modelled MP location, depending on whether the minimal or maximal error is added to the fitting parameters. These are shown as dashed lines in Fig. 3. Using the best fit parameters without any fitting errors lead to the mean MP location (solid line).

As the MP is almost parallel to the *x* axis at lunar distances, we
concentrate on differences between position prediction and actual position
along the previously described *d* axis, or, as we rotated all positions into
the equatorial *x**y* plane, along the *y* axis. Rotating into the equatorial
plane or using the defined *d* axis are equal to each other.

The standard deviation, which is the distance between minimum and maximum
location along the *y* axis, is called error range *δ**y* by us; see
Fig. 3. To quantify the actual MP position in relation to
the model, its distance Δ*y* to the mean model location is normalized
to *δ**y*∕2. We call this the normalized error of MP distance. Using
this definition a MP laying exactly at the position predicted by the model
has a normalized distance 2Δ*y*∕*δ**y* of zero. A MP laying
exactly at the model MP with error has the normalized distance 2Δ*y*∕*δ**y* of ±1.

Figure 4 shows the distribution of normalized
positions for the innermost crossing and Fig. 5
for the outermost crossing. In the case of the innermost crossing the mean
distance is at −1.12 with a standard deviation of 1.94 and a skewness of
0.76. This means that the MP is usually found more close to the
magnetotail, as predicted by the model. In about 54 *%* of the crossings the model
overestimates the location of the MP.

The situation is different with an outermost crossing; see Fig. 5. Here the mean distance is 2.48 with a standard deviation of 2.37 and a skewness of 0.89. Accordingly the location of the MP is underestimated by the model.

The normalized MP distance does not show any strong correlation to the MP
position along the *x*_{GSM} axis, the strength of the SW
*B*_{z} component, or the SW speed. Each of the respective correlation
coefficients is below 0.6. As an example,
Figs. 6 and 7
display the scattering of the *x* position against the normalized distance.
Because of that, we conclude that there is no systematic deviation between
modelled and actually observed MP distance with respect to these parameters.

Besides its radial distance, the direction of the magnetopause normal can
also be deduced from the Shue model and compared with the observations at
lunar distances. For this purpose model and observed normal directions are
projected onto the *y**z* planes (polar plane) and *x**y* planes (equatorial),
respectively, afterwards the deviation angles *α*, respectively *β*,
between model and observed normal directions are determined. For deviation
angles in the *y**z* plane (*x**y* plane) the sign of the angle is defined
positive for situations in which the actual direction is pointing towards the
positive *z* (*x*) direction in relation to the model direction.
Figure 8 illustrates this angle convention.

The thus-defined deviation angles allow deviations of the
magnetopause's opening angle to be highlighted, in the case of the angle laying in the *x**y* plane,
which corresponds to the Shue flaring parameter, as well as deviations from
the ideal axial symmetry, in the case of the angle laying in the *y**z* plane. For
each identified crossing SW data are used to calculate the model
magnetopause. The expected distribution of angles *γ* between the model
normal direction and the *y*_{GSM} axis is shown in
Fig. 9 for the innermost and
Fig. 10 for the outermost crossings. Angles with a mean of 4.6^{∘} (5.1^{∘}) are expected for the
innermost (outermost) crossing directed sunwards, or positive direction,
following our convention. This reinforces the assumption of a MP almost
parallel to the *x* axis; see Sect. 3.

Figures 11 and 12 display the deviation angle distributions.

For the innermost crossings we get the following results. The median
deviation angles *α* for the *y**z*_{GSM} plane are 1.9
(44.6)^{∘} for inbound and −7.1 (45.9)^{∘} for
outbound crossings. Values in parenthesis denote the respective standard deviation. For
the angles *β*, the *x**y*_{GSM} plane values are 8.0
(38.3)^{∘} for inbound and 5.0 (42.7)^{∘} for outbound passes. And results
for the outermost crossings are as follows. The median deviation angles
*α* for the *y**z*_{GSM} plane are −3.3 (37.2)^{∘} for
inbound and 6.1 (42.7)^{∘} for outbound crossings. For the angles
*β*, the *x**y*_{GSM} plane values are 9.1 (33.9)^{∘} for
inbound and 8.3 (40.2)^{∘} for outbound passes.

Both angles *α* and *β* show median values near zero for all cases
but come along with high scattering of more than 30^{∘}. Since we only
observe one single crossing event per spacecraft and month, due to the
spacecraft orbit, the high scattering is not surprising. But, with some
caution, we conclude that the predicted directions agree well the actual
directions.

The location of the magnetopause at lunar distances shows systematic differences to the model prediction. When choosing the innermost crossing of the MP, which is the same methodology as in Shue et al. (1997), the location is overestimated. In that case the MP is on average found much closer to the centre of the magnetotail. On the other hand, when choosing the outermost crossing, Shue et al. (1997) underestimates the location and the MP is found in much greater distance to the magnetotail centre than expected.

Different to this are predictions about the normal direction of the MP. These scatter over a wider range of angles, but show a clear tendency to conform to the model-predicted directions. Since the standard deviation is very large, it is not possible to make a well-founded statement about differences in in- and outbound traversals. Due to the high variability of the MP location caused by constantly changing SW conditions, the scattering in the normal direction is as expected, since the SW directly changes the MP standoff distance and indirectly induces surface waves such as Kelvin–Helmholtz instabilities due to differences in the plasma flow velocity. Essentially, the axial symmetry of the model can be confirmed for lunar distances in the magnetotail and near to the equatorial plane.

We conclude that the uncertainty in determination of the MP location increases with greater distance to the Earth. This implies that the statistical width of the MP is larger than it is closer to Earth.

THEMIS data and the latest calibration files are publicly available at http://themis.ssl.berkeley.edu/ (last access: 15 November 2018) or via the SPEDAS software.

The authors declare that they have no conflict of interest.

We acknowledge use of NASA/GSFC's Space Physics Data Facility's OMNIWeb
service, and OMNI data. We acknowledge NASA contract NAS5-02099 and V. Angelopoulos for use
of data from the THEMIS Mission – specifically,
C. W. Carlson and J. P. McFadden for use of ESA data. This study is
financially supported by the German Ministerium für Wirtschaft und Energie
and the Deutsches Zentrum für Luft- und Raumfahrt under contract 50 OC 1403.

Edited by: Christopher Owen

Reviewed by: one anonymous referee

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