Knowledge of planetary magnetic fields provides deep insights into the structure and dynamics of planets. Due to the interaction of a planet with the solar wind plasma, a rather complex magnetic environment is generated. The situation at planet Mercury is an example of the complexities occurring as this planet's field is rather weak and the magnetosphere rather small. New methods are presented to separate interior and exterior magnetic field contributions which are based on a dynamic inversion approach using a reduced magnetohydrodynamic (MHD) model and time-varying spacecraft observations. The methods select different data such as bow shock location information or magnetosheath magnetic field data. Our investigations are carried out in preparation for the upcoming dual-spacecraft BepiColombo mission set out to precisely estimate Mercury's intrinsic magnetic field. To validate our new approaches, we use THEMIS magnetosheath observations to estimate the known terrestrial dipole moment. The terrestrial magnetosheath provides observations from a strongly disturbed magnetic environment, comparable to the situation at Mercury. Statistical and systematic errors are considered and their dependence on the selected data sets are examined. Including time-dependent upstream solar wind variations rather than averaged conditions significantly reduces the statistical error of the estimation. Taking the entire magnetosheath data along the spacecraft's trajectory instead of only the bow shock location into account further improves accuracy of the estimated dipole moment.

The interaction of a planetary magnetic field with the solar wind strongly
modifies the magnetic field environment around the planet. If in situ
spacecraft data are used to estimate the planetary magnetic field, this
interaction needs to be taken into account. This is of particular importance
for the upcoming two-spacecraft mission BepiColombo

We investigate various methods to estimate a planetary magnetic field from
in situ spacecraft observations. Previous approaches consider the interaction
of Mercury's magnetic field with the solar wind using empirical models. For
example,

Here, we consider two different approaches employing an magnetohydrodynamic (MHD) model to compute
the interaction depending on the varying solar wind conditions. For an
efficient calculation, we use a reduced MHD model presented by

We consider terrestrial THEMIS data to reconstruct the well-known planetary magnetic field of the Earth as a test case for the more challenging situation at Mercury in preparation for the BepiColombo mission. However, with respect to the strongly modified magnetic field environment due to the solar wind at Mercury, we choose THEMIS data from the magnetosheath. In this region, the measured magnetic field at the Earth is strongly influenced by the interaction with the solar wind comparable to the situation at Mercury.

With the reduced MHD model, different procedures can be applied to obtain the
planetary magnetic moment and are investigated with respect to systematic and
statistical errors. A first method, suitable also for single-spacecraft
missions, considers observations from a spacecraft crossing the bow shock
which measures the solar wind conditions on the sunward side of the shock.
Then, Earth's dipole moment can be directly calculated with analytical
expressions of the shock's distance in the reduced MHD model. At Mercury,
this approach is applicable for the BepiColombo mission using the bow shock
observations of the MMO around perihelion. Note that, in general, the quality
of solar wind data of the MESSENGER mission at Mercury is not
sufficient to apply this method presented here

The reduced MHD model by

To simplify the considerations, we assume a quasi-stationary situation with a
solar wind magnetic field and a planetary dipole moment along the

Similar to the physical quantities, the bow shock and magnetopause geometry
are expanded into Taylor series with respect to the

The coordinate

Substituting this ansatz (

The coefficients of the highest order, i.e.,

This model presented is restricted to solar wind magnetic fields along the

Further, we briefly summarize the important relations of an approximative
solution of the zeroth-order model used in this study presented in

Gas pressure and temperature are second-order moments of the velocity
distribution function which are difficult to determine precisely from
spacecraft data

Although the assumptions of the model are usually valid at Earth and Mercury,
our approach can be generalized. Instead of considering only a dipole moment,
higher-order moments can be included as well, modifying
Eqs. (

For a single-spacecraft mission such as MESSENGER, usually no solar wind data are available while the spacecraft is crossing the interaction region. Only at the bow shock can the solar wind conditions often be extracted from the spacecraft data directly in front of the shock. Within the scope of ideal MHD, unperturbed solar wind reaches the bow shock and is decelerated at the infinitesimal thin shock. This approximation is usually valid at the bow shock of Earth and Mercury because the shock's thickness is much smaller than the shock's subsolar distance. Then, the solar wind information and bow shock location can be used to estimate planetary magnetic field parameters with an MHD model of the interaction.

We consider the subsolar bow shock distance resulting from the analytical
approximation of the reduced MHD model of the magnetosheath solution by

The terrestrial dipole moment is calculated with THEMIS data of the THC
spacecraft in GSM coordinates. The data need to be transferred into the
coordinate system of the model. Therefore, the GSM coordinates are first
rotated around the

With respect to the new transformed coordinates, the measured bow shock is
located at

Bow shock locations close to the subsolar point and the corresponding solar wind conditions observed by THC.
The

We investigate 11 bow shock transitions close to the subsolar point on 8
orbits of the THC spacecraft between 24 August and 6 September 2008. During
this time interval, magnetosheath transitions near the stagnation streamline
can be observed as discussed below. The data are presented in
Table

During the period considered, the THC spacecraft's distance to the

Additionally, errors of estimating the dipole moment can occur due to a bow
shock motion caused by varying solar wind conditions or data errors. The
subsolar bow shock distance to the Earth's center for typical solar wind
conditions at Earth is about

The method presented takes bow shock locations into account. In contrast to magnetopause observations, the solar wind conditions can be determined at bow shock crossings, even for single-spacecraft missions. The method provides a valid estimator for the planetary dipole moment, in which a larger sample size might reduce the statistical error further.

The previous considerations took into account the location of the bow shock
only. From a statistical point of view, it seems advantageous to include more
data points on an orbit for the estimation of the dipole moment. If the solar
wind conditions during the magnetosheath crossings of THC are known, each
data point within the magnetosheath can be used to estimate dipole moment

A model of the interaction relates the observations within the interaction
region together with the solar wind data to the planetary magnetic field.
Here, we use the reduced MHD model with the approximation

On the orbit of THC across the magnetosheath, the plasma's mass density

The physical quantities

The solution of the model (

Magnetosheath spacecraft data on 31 August 2008 (red) and adjusted
model results (blue) which determined the dipole moment to

The cost function depends on the model solution, which is a function of the
dipole moment

Estimated dipole moments using THC data at orbits from 24.8, 25.8,
27.8, 29.8, 31.8, 2.9, 4.9, and 6.9, labeled by orbit numbers 1 to 8. The
dipole moment was estimated by THC bow shock observations together with
pre-shock solar wind conditions (blue), by THC magnetosheath data using the
time-dependent OMNI solar wind observations, and by THC magnetosheath data
with average solar wind conditions (green). The

The value of the estimated dipole moment using the entire magnetosheath data
seems comparable to the estimate using bow shock location information.
However, the standard deviation, related to the statistical error, is halved.
The estimated dipole moments taking only bow shock locations into account are
also displayed in Fig.

Additionally to the estimated dipole moments, the value of the cost function
(

Estimated dipole moments (red) using cost function (

To allow for better comparison of the two different methods, the approach
using magnetosheath data is modified to take only measured bow shock
locations

For single-spacecraft missions, the solar wind conditions during a
magnetosheath transition are usually not well known and average values need
to be assumed. To investigate the advantage of time-dependent considerations,
the Earth dipole moment is estimated using the mean values of the solar wind
conditions of all transitions considered here. These average conditions of
all eight magnetosheath transitions are for the ion particle density

The use of average solar wind conditions leads to a result that is worse compared to the use of time-dependent solar wind conditions. The dipole moment is significantly overestimated and standard deviation is 3 times larger. The reason is the nonlinear dependence of the MHD model solution on the solar wind conditions. Thus, for a precise estimation of the planetary magnetic field from spacecraft data obtained within a region strongly influenced by the interaction with the solar wind, it is an advantage to include the actual solar wind conditions instead of using average conditions.

We examined two different methods to estimate the planetary magnetic dipole
field using a reduced MHD model to take the solar wind interaction into
account. The methods presented were investigated in preparation for the
analysis and interpretation of measurements from the BepiColombo mission to
Mercury

In this study, we used a simple reduced MHD model of the interaction. In a
next subsequent step, more data need to be included to reduce the
statistical error. Therefore, the reduced MHD model, which is valid close to
the

Data from the THEMIS mission are publicly available and can
be obtained from

The authors declare that they have no conflict of interest.

This work was financially supported by the German Ministerium für Wirtschaft und Technologie and the Deutsches Zentrum für Luft- und Raumfahrt under contracts 50OC1403 and 50QW1501. 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. We acknowledge use of NASA/GSFC's Space Physics Data Facility's OMNIWeb service, and OMNI data. The topical editor, E. Roussos, thanks V. M. Vasyliunas and one anonymous referee for help in evaluating this paper.