Introduction
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 to
planet Mercury because the Hermean magnetosphere is small and very dynamic.
The average magnetopause distance at Mercury observed by the MESSENGER
(Mercury Surface, Space Environment, Geochemistry and Ranging) mission
is about 1.45RM and the bow shock distance
is 1.89RM, with RM=2440km as planetary
radius . Consequently, the currents of the interaction
influence the magnetic field distribution along the entire orbit of a
spacecraft such as the BepiColombo planetary orbiter at Mercury. The two
BepiColombo spacecraft, the Mercury Planetary Orbiter (MPO) and the Mercury
Magnetospheric Orbiter (MMO), investigate Mercury's environment on polar
orbits (see http://sci.esa.int/). The MPO will travel close to the
surface of Mercury with a period of 2.3 h, a periapsis of
1.2RM, and an apoapsis of 1.6RM with respect to
Mercury's center. The MMO will travel farther out with a period of 9.2 h, a
periapsis of 1.2RM, and an apoapsis of 5.8RM. At the
aphelion of Mercury, the periapsides are the subsolar point, and at
perihelion, the apoapsides are the subsolar point. Thus, especially around
perihelion, the MMO will spend most of the time in the solar wind and crosses the
bow shock twice on each orbit. Note that the MPO will usually not cross the
bow shock and observes the interaction region close to the planet. In this
kind of dynamic environment interspersed with local electrical currents, the
classic Gauss algorithm fails to separate internal and external magnetic
fields in a classical spherical harmonic expansion
. In addition, spacecraft missions
provide only incomplete spatial coverage. For example, the MESSENGER mission
only provided observations close to the Hermean surface in the northern
hemisphere. This leads to a systematic correlation of Gauss coefficients.
Therefore, more elaborate approaches to determine the internal field must be
sought often such as the methods presented here. Accurate internal field
determination is crucial, for example, for the identification of suitable dynamo
models which could explain the enigmatic Hermean planetary magnetic field
e. g.,.
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, used a scaled Tsyganenko-96 model of the Earth's
magnetosphere to include current
systems of the interaction at Mercury. Another empirical model used at
Mercury is the paraboloid magnetosphere model introduced by
. Such a model was also used by to
estimate Mercury's magnetic field with data from MESSENGER. Further
improvement of modeling the magnetosphere was done by ,
including parametric models of magnetopause and cross-tail currents. In
general, empirical models either contain many parameters which also depend on
solar wind conditions or use several assumptions such as the prescription of
shape and location of interaction currents which reduce the number of
parameters to be estimated from data.
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
which is derived from the MHD equations using series
expansion techniques. Further, the reduced model is less sensitive to
numerical errors. The reduced models are suitable for detailed investigations
of the estimation procedure of the planetary magnetic field and the related
errors. Such a reduced model was successfully applied to reconstruct the
solar wind conditions from terrestrial magnetosheath data
. Spacecraft data from the interaction region and the
known planetary magnetic field of the Earth were used to estimate the solar
wind conditions at the subsolar point of the bow shock. Here, this approach
is reversed: the spacecraft data together with solar wind conditions are used
to estimate the planetary magnetic field.
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 . A second approach is to use spacecraft data from the entire
magnetosheath passage along the spacecraft's trajectory with time-dependent
solar wind observations from another spacecraft. Here, we take magnetosheath
data at Earth into account to consider a highly disturbed environment similar
to Mercury. Note that, in general, this approach is not restricted to
magnetosheath data. The magnetospheric magnetic field of the model is
determined by the Biot–Savart law using the magnetosheath current density and
can be compared to magnetospheric observations. To improve precision of such
an extension, further current contributions can be included from different
models as well. The approach can be applied whenever the MPO provides solar
wind data.
The reduced MHD model by Nabert et al. (2013)
The reduced MHD model by provides solutions of the ideal
MHD equations for the mass density ρ, the plasma velocity v, the
gas pressure p, and the magnetic field B assuming a
quasi-stationary situation. In the magnetosheath, the ideal, stationary MHD
equations are
0=∇⋅(ρv),0=ρ(v⋅∇)v+∇p-1μ0∇×B×B,0=∇×v×B,0=∇⋅B,p=kpργ,
with the constant of proportionality kp of the adiabatic law () and the vacuum permeability μ0=4π⋅10-7N/A2.
The solution of the MHD equations can be expressed in vector notation:
u:=ρ,vx,vy,vz,p,Bx,By,BzT.
A Cartesian coordinate system with the solar wind along the x direction and perpendicular directions y and z is considered.
The origin of the coordinate system is located at the bow shock.
The general idea of the approach by is to derive reduced MHD models of the interaction with series expansions of the physical quantities.
Each physical quantity un(x,y,z), where n labels the component of the vector u according to Eq. (),
is expanded by a Taylor series with respect to the y and z directions:
un(x,y,z)=∑j=0NS+1∑k=0NS+1ujkn(x)yjzk.
Here, (NS+1) denotes the expansion order and ujkn(x) the coefficient functions.
To simplify the considerations, we assume a quasi-stationary situation with a
solar wind magnetic field and a planetary dipole moment along the
z direction. Then, the dipole approximation of the Earth's planetary
magnetic field is
BE,x=3zxE-xr5M,
BE,y=3yzr5M,
BE,z=--2z2+xE-x2+y2r5M,
with distance r:=(xE-x2+y2+z2)0.5, dipole
moment M, and the distance of the bow shock to the Earth's center
xE along the x axis. The largest quadrupole contribution to the
magnetic field at about 10RE, an approximation of the subsolar
magnetopause distance at Earth, is less than 1%, where
RE=6371km. Therefore, we neglect any quadrupole
contributions. It should be noted that this is a situation very different
from that one at planet Mercury. However, it seems a valid approximation to
introduce an offset dipole to include Mercury's quadrupole contributions
. The dipole moment representation and solar wind magnetic
field perpendicular to the solar wind direction requires certain symmetries
of the physical quantities, which leads to vanishing coefficient functions
. A symmetric quantity un(x,y,z) to the xy plane
satisfies un(x,y,-z)=un(x,y,z), an anti-symmetric quantity satisfies
un(x,y,-z)=-un(x,y,z) and analogous with respect to the xz plane. In
particular, ρ, vx, p, and Bz are symmetric whereas By is
anti-symmetric with respect to both planes. Further, Bx and vz are
anti-symmetric to the xy plane and symmetric to the xz plane. In
contrast, vy is symmetric to the xy plane and anti-symmetric to the
xz plane.
Similar to the physical quantities, the bow shock and magnetopause geometry
are expanded into Taylor series with respect to the y and z direction. We
restrict our considerations to the situation at Earth and choose
NS=0. The latter assumption restricts the model to the vicinity
of the x axis. The origin of the coordinate system coincides with the
subsolar point of the bow shock. Further, bow shock and magnetopause can be
approximated by series expansions to the y and z direction up to the
second order. Taking into account the symmetries, the shock geometry is given
by
x=cBS,yy2+cBS,zz2,
where the coefficients cBS,y and cBS,z determine curvatures of the shock.
Similarly, the magnetopause parametrization is given by
x=xMS+cMP,yy2+cMP,zz2,
where xMS is the distance of the magnetopause to the bow shock
along the x axis, and cMP,y and cMP,z are
magnetopause curvature parameters. The expansions of the bow shock and the
magnetopause are used to introduce a new coordinate x̃ which adjusts
the coordinate system to the magnetosheath geometry. The new coordinate
x̃ is defined by
x̃=x-cBS,y+Δcyx̃xMSy2-cBS,z+Δczx̃xMSz2,
with Δcy:=cMP,y-cBS,y and Δcz:=cMP,z-cBS,z. Note that x̃=0 gives the bow shock
parametrization () and
x̃=xMS gives the magnetopause
parametrization (). Approximative
analytical expressions for the curvature parameters are determined by
cMP,y=2/(5ΔxMP), cMP,z=1/(2ΔxMP), cBS,y=2/(5ΔxBS), and
cBS,z=1/(2ΔxBS), where ΔxBS=|xE| and ΔxMP denotes the distance of the
magnetopause to the Earth's center .
The coordinate x is replaced by the new coordinate x̃ in expansion
(). Then, the series expansion fits the magnetosheath
geometry which leads to faster convergence of the series expansions. For
NS=0 and taking into account the symmetries of the situation
considered as described above, series expansion ()
simplifies to
ρ(x,y,z)=ρ0(x̃),
v(x,y,z)=(vx0(x̃),vy10(x̃)y,vz01(x̃)z)T,
p(x,y,z)=p0(x̃),
B(x,y,z)=(Bx01(x̃)z,By11(x̃)yz,Bz0(x̃))T,
where the new coordinate x̃ is used and terms of zeroth order in y
and z are labeled by 0 instead of 00.
Substituting this ansatz ()–() into the MHD equations
()–() and equating coefficients of the lowest order,
i.e., y=z=0, a system of ordinary differential equations is obtained:
ρ0vx0′+ρ0vy10+vz01=0,
Bz0vx0′+Bz0vy10=0,
ρ0vx0vx0′+p0′+Bz0Bz0′-Bx01Bz0μ0=0,
p0=kρ0γ.
Derivatives with respect to x̃ are marked by a prime, e.g., p0′=∂x̃p0.
The coefficients of the highest order, i.e., vy10, vz01, and
Bx01, are assumed to be constant to close the system of ordinary
differential Eqs. ()–(). The
values of all coefficients directly behind the bow shock at x̃=0 are
related to the solar wind by solving the Rankine–Hugoniot conditions with
respect to the shock geometry e.g.,. The solar wind
conditions are denoted by NSW for the density, vSW for
the velocity, BSW for the magnetic field, and pSW for the
gas pressure. The expansion coefficients of the bow shock
parametrization () and the magnetopause
parametrization () are determined by
inner boundary conditions related to the planetary magnetic field
()–(). The magnetopause is
considered as a rigid boundary and the plasma flow needs to be tangential to
this boundary. Further, the magnetopause holds a modified pressure balance
according to . A detailed derivation of all boundary
conditions and higher-order equations can be found in .
System ()–() with the
corresponding boundary conditions is referred to as the reduced MHD model of zeroth order.
This model presented is restricted to solar wind magnetic fields along the
z direction. However, a y component can be taken approximately into
account by substituting Bz←(Bz+By)0.5 as shown in
. Thus, the magnetic field within the yz plane is chosen
as a new z component of the magnetic field. Note that effects of a magnetic
field's x component in the solar wind does not contribute to the zeroth-order solution .
Further, we briefly summarize the important relations of an approximative
solution of the zeroth-order model used in this study presented in
as well. The subsolar magnetopause distance to the Earth's
center ΔxMP can be expressed by
ΔxMP=aMPM13,
where aMP depends on the solar wind conditions and is defined by
aMP:=f22μ0kPρSWvSW216.
As shown by f≈2.44 and according to
kP≈0.89 hold for a broad range of
solar wind conditions. The thickness of the magnetosheath along the
x̃ direction is given by
xMS=aMSaMPM13,
with the solar wind dependent factor
aMS:=10.8+mBSgv-1-1.
Here, gv denotes the ratio of the solar wind velocity to the post-shock velocity and mBS measures the solar wind magnetization:
mBS:=1-11+μ0γp0(0)Bz02(0).
Note that gv and mBS are determined using the
Rankine–Hugoniot conditions to obtain an analytical solution for the
situation considered e.g.,. The expression
aMS was derived as part of an analytical solution of the reduced
MHD model in . The factor aMS describes
magnetosheath variations apart from variations in the magnetopause distance.
For example, it contains the magnetosheath broadening due to a magnetic
pile-up of solar wind magnetic field.
Gas pressure and temperature are second-order moments of the velocity
distribution function which are difficult to determine precisely from
spacecraft data . Thus, we use a cold plasma
approximation for our approaches – i.e., the limit of a vanishing solar wind
gas pressure is considered.
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. ()–(). Then, a more
general system of ordinary differential equations is necessary which is
derived using Eq. () instead of ansatz
()–().
Estimating the dipole moment
Using the bow shock location
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 subsolar bow shock distance ΔxBS,
the sum of the magnetopause distance (Eq. ) and the magnetosheath
thickness (Eq. ), is given by
ΔxBS=1+aMSaMPM13.
The coefficients aMP and aMS are determined by the
solar wind conditions at the bow shock as seen in Eqs. ()
and () using Rankine–Hugoniot relations.
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 z axis, and then around the y axis to align the solar
wind velocity vector with the x axis. Further, the origin is transferred
from the Earth's center to the bow shock by x→ΔxBS+x. Although the x axis of the transformed GSM coordinates is aligned
with the x axis of the model coordinates, the z axis is not necessarily
aligned with dipole moment as in the model. Thus, the projection of the
dipole moment onto the z axis is determined. Assuming a solar wind within
the xy plane of the GSM coordinates, the strength of the Earth's dipole
moment in the model varies in the range of its z component and its
magnitude in geographic coordinates. Then, using the IGRF ,
the dipole moment M varies between
-7.74×1015Tm3<M≤-7.63×1015Tm3.
Thus, the accuracy of an estimation of the dipole moment using the MHD model
presented in Sect. is limited to this range.
With respect to the new transformed coordinates, the measured bow shock is
located at xSC/BS in the x direction, at ySC/BS in
the y direction, and at zSC/BS in the z direction. Instead of
xSC/BS, which requires the subsolar bow shock position known to
be calculated from observations in GSM coordinates, the distance along the
x direction from the Earth's center ΔxSC/BS is introduced
which can be easily determined. If the bow shock crossing occurs at a
location off the stagnation streamline that is beside the x axis, the
geometry of the shock according to Eq. () is
taken into account to calculate the subsolar bow shock distance in
Eq. (). Using the analytical expression for the curvature
parameters, the bow shock parametrization ()
is
x=25ΔxBSy2+12ΔxBSz2.
Then, the subsolar bow shock distance is related to the measured location by
ΔxBS=ΔxSC/BS2+ΔxSC/BS24+2ySC/BS25+zSC/BS22.
Solar wind conditions need to be known in addition to the subsolar bow shock
distance in order to calculate the dipole moment with Eq. (). Mean
values of 5 min of THC data in front of the shock transition are used to
determine the solar wind conditions at the shock transition. The data are
taken with a 10 min time gap to the shock because MHD theory is only
approximately valid and solar wind plasma can be affected even in front of
the shock. Ten minutes correspond to a spatial distance of about
600km because of the spacecraft's velocity of about
1kms-1.
Bow shock locations close to the subsolar point and the corresponding solar wind conditions observed by THC.
The z component of solar wind magnetic field is Bz,SW and the magnetic field
magnitude in the yz plane is Bt,SW=(By,SW2+Bz,SW2)0.5.
The computed dipole moments are also presented.
Date
Time
NSW
vSW
Bz,SW
Bt,SW
ΔxSC/BS
ySC/BS
zSC/BS
M/1015
(dd.m.)
(UT)
(1cm-3)
(kms-1)
(nT)
(nT)
(RE)
(RE)
(RE)
(Tm3)
24.8.
00:29
7.61
310.4
1.11
1.74
12.59
-5.52
-2.27
-7.53
25.8.
22:41
6.81
267.8
0.89
1.72
13.64
-4.82
-2.41
-7.25
27.8.
22:20
8.04
272.6
-1.41
3.58
13.41
-4.43
-2.10
-6.94
29.8.
21:06
6.92
324.1
-2.93
3.99
14.37
-3.56
-2.62
-9.05
31.8.
21:49
6.98
334.3
-2.47
2.60
13.63
-3.73
-2.02
-8.48
02.9. (a)
19:48
6.96
250.1
-1.94
1.98
15.22
-2.40
-1.74
-7.92
02.9. (b)
20:58
7.43
276.7
0.93
2.32
14.17
-2.81
-2.57
-8.23
04.9. (a)
21:02
2.05
571.5
1.55
4.10
14.06
-2.33
-1.68
-7.52
04.9. (b)
21:23
2.05
562.1
1.50
4.26
13.76
-2.46
-1.62
-6.87
06.9. (a)
21:26
4.02
546.9
-3.05
3.78
13.50
-2.20
-1.45
-9.42
06.9. (b)
22:30
3.93
560.1
2.92
3.15
12.47
-2.46
-1.49
-7.78
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 . Due to solar wind variations, multiple bow shock
transitions were observed on 2, 4 and 6 September 2008. Note that sufficient
solar wind data are available between two bow shock crossings to obtain
individual solar wind conditions for each crossing. Each observed bow shock
transition is used to calculate the dipole moment M with Eq. ().
Values for multiple shock crossings on an orbit are combined to a single mean
value. The corresponding relative error ΔM of each calculation is
less than 19% with respect to the true value of the Earth's dipole
moment according to Eq. (). The mean value of the computed dipole
moments from the 8 orbits is M=-7.9×1015Tm3 with a
standard deviation of 0.8×1015Tm3. The error of the
mean value differs less than 4% from the true value of the z component
and less than 2% from the magnitude.
During the period considered, the THC spacecraft's distance to the x axis
is less than 5RE, except for the transitions on 24 and 25 August
2008. The subsolar bow shock distance ΔxBS is less than
1.1RE away from the observed shock distance beside the x axis
ΔxSC/BS according to Eq. (). A model error of
the bow shock distance can assumed to be much smaller than this correction; for example, we might choose an error of less than 0.2RE. The
z component of the solar wind magnetic field is always greater than
-3nT. Our model uses the ideal MHD approximation and consequently,
it is not valid for strong southward solar wind magnetic fields, which favor
reconnection and a subsequent departure from an ideal MHD situation. The
corresponding error can be estimated by the earthward shift of the
magnetopause due to the southward magnetic fields using the magnetopause
model by . For a solar wind magnetic field of
-3nT, the magnetopause is about 0.2RE closer to the
Earth than computed with Eq. (). It seems natural to assume a
similar error for the bow shock distance. A further model error can occur due
to the zeroth-order approximation by choosing NS=0 of the reduced
model. This model error for the analytical solution compared to a more
accurate second-order solution was estimated to be about 0.2RE
using the data of the magnetosheath of 24 August 2008 by .
An error of 0.2RE in the determination of the bow shock distance
for typical solar wind conditions at Earth is related to an error of less
than 6% in the calculation of the dipole moment M according to
Eq. (). Typical solar wind conditions at the Earth are a solar wind
ion particle density of about NSW=7cm-3, a velocity
vSW=430kms-1, a solar wind magnetic field
BSW=6nT, and an ion temperature TSW=8×104K .
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 13RE. A variation in the solar
wind density of about 20% compared to typical solar wind conditions
leads to a variation in the bow shock location of about 0.3RE.
This corresponds to a relative error of about 10% for the dipole moment
M computed by Eq. (). Further, a variation in the solar wind
velocity of about 20% corresponds to an error of about 20% for the
estimated dipole moment M. Errors due to solar wind variations are usually
statistical errors which cancel out if a mean value of a large sample size is
considered in contrast to model errors, which can be systematic. The errors
considered here are consistent with the estimations of M using a single bow
shock transition as presented in Table . Further, the
mean value of these estimations is closer to the true value of the Earth's
dipole moment than the individual values as expected.
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.
Using magnetosheath data
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
M. However, the error of the estimation will depend on the level of
knowledge of the solar wind conditions. Such an approach is very suitable for
two-spacecraft missions such as BepiColombo. One spacecraft crosses the
interaction region, while the other spacecraft observes the solar wind.
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 NS=0
presented in Sect. . Similar to , a cost
function can be introduced to quantify the misfit between model solution for
any choice of M and observations. The planetary magnetic field parameter
M in the model is modified until the cost function is minimized. Then, the
model predictions fit to the physical quantities measured by a spacecraft
within the interaction region.
On the orbit of THC across the magnetosheath, the plasma's mass density
ρSC,m, the x component of the velocity vSC,m,
the z component of the magnetic field BSC,m, and the gas
pressure pSC,m are measured in GSM coordinates at locations
(xSC,m,ySC,m,zSC,m) labeled with the
position index m. The corresponding solar wind conditions at the subsolar
point for the mass density ρSW,m, the velocity
vSW,m, the z component of the magnetic field BSW,m,
and the pressure pSW,m at Earth can be determined by OMNI solar
wind data http://omniweb.gsfc.nasa.gov/;
representing measurements from a second spacecraft. Analogously to the
previous considerations using bow shock locations, the data in
GSM coordinates are transferred into the model coordinate system with the
x axis aligned to the solar wind velocity vector.
The physical quantities ρ, vx, p, and Bz are constant for
x̃=const as seen in Eqs. ()–(). Thus, a
location beside the x axis corresponds to a position on the x axis along
the paraboloid geometry with the same values of the physical quantities.
Consequently, a coordinate transformation can be introduced which relates the
location of a data point beside the x axis to a location on the x axis of
the model. Therefore, the magnetosheath geometry needs to be taken into
account according to Eq. (). Together with the
analytical expressions of the curvature parameters, a spacecraft's location
in the model coordinates xSC,ySC,zSC is
related to a location x̃SC on the x axis by
xSC=x̃SC+cBS,y+Δcyx̃SCxMSySC2+cBS,z+Δczx̃SCxMSzSC2.
Measured spacecraft data at the transformed spacecraft coordinate
x̃SC,m of the mass density ρSC,m, the
velocity vSC,m, the magnetic field BSC,m, and the
pressure pSC,m can be compared to the model solution at locations
x̃=x̃SC,m.
The solution of the model ()–() is found by solving the
differential equations ()–() with
the corresponding solar wind conditions and an assumption of the dipole
moment M. A cost function is introduced to quantify the misfit of data and
model solution. Using the method of least-squares regression analysis, we
consider a cost function which includes the THEMIS plasma
as well as the magnetic field data:
K=∑m=1Mdataρ0(x̃SC,m)-ρSC,mρSC,m2+vx0(x̃SC,m)-vSC,mvSC,m2+Bz0(x̃SC,m)-BSC,mBSC,m2,
where Mdata denotes the number of data points in the magnetosheath.
The gas pressure does not contribute to this cost function because it is a
second-order moment of the velocity distribution function which is difficult
to determine precisely from spacecraft data.
Magnetosheath spacecraft data on 31 August 2008 (red) and adjusted
model results (blue) which determined the dipole moment to M using the
time-dependent OMNI solar wind data (green) presented in model coordinates.
The cost function depends on the model solution, which is a function of the
dipole moment M. The cost function is calculated on a grid in the range
-12×1015Tm3≤M≤-6×1015Tm3.
The grid space between two values of M is taken to be 0.004×1015Tm3. The minimum value of the cost function determines the
estimated dipole moment. A value for the dipole moment is estimated from the
magnetosheath data close to the x axis separately for each orbit of THC
between 24 August and 6 September 2008. As an example, the magnetosheath data
on 31 August 2008 in the model coordinates is presented in
Fig. . We choose a 5 min time resolution for the OMNI as well
as THC spacecraft data which is in accord with the quasi-stationary
approximation used in the reduced MHD model. Note that 5 min is the order of
the solar wind transit time across the subsolar magnetosheath. The
time-varying OMNI solar wind data are used as boundary conditions. The reduced
magnetosheath model solution is adjusted to the magnetosheath data via
minimization of cost function (), which determines the dipole
moment M. The corresponding adjusted reduced MHD model solution on 31
August 2008 is also depicted in Fig. . The eight estimated dipole
moments from the eight magnetosheath crossings are shown in Fig. .
Each estimated dipole moment differs not more than 13% from the value of
the Earth's dipole moment. The mean value of the eight computed dipole moments is
M=-7.9×1015Tm3 with a standard deviation of 0.4×1015Tm3. The error is less than 4% with respect to
the z component and less than 2% with respect to the magnitude of the
Earth's 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 z component of the Earth's
dipole moment is depicted as a black line. Dipole moments are normalized to
mnorm=8.0×1015Tm3.
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. . No correlation between the results of
the two different methods is apparent; the correlation coefficient is 0.13.
Thus, one can assume different sources of errors for the two methods. For
example, the estimations with the bow shock locations use solar wind
conditions close to the subsolar point of the shock. Instead, the OMNI solar
wind data take spacecraft data far away from the bow shock to estimate the
conditions at Earth. The latter approach requires a model to transfer the
solar wind conditions to the Earth which is more error-prone. However, using the complete time interval of magnetosheath data instead of
only the bow shock location, more data are taken into account, so that
statistical errors might be reduced.
Additionally to the estimated dipole moments, the value of the cost function
() normalized to the number of data points KDP:=K/Mdata is considered in Fig. . The value of
KDP is in the range of 0.04 and 0.2, which gives a mean deviation
between data and model solution. Smaller values of the normalized cost
function correspond to a better agreement between data and model solution.
Values of the cost function below 0.1, i.e., orbit numbers 1, 3, 4, and 5,
correspond to estimators close to the true value. Consequently, it might be
suitable to give more emphasis to estimates which show a better agreement of
model solution and data.
Estimated dipole moments (red) using cost function () and
the value of the cost function at its minimum normalized to the number of
data points of the respective magnetosheath transition (grey). The dipole
moments are normalized to mnorm=8.0×1015Tm3.
To allow for better comparison of the two different methods, the approach
using magnetosheath data is modified to take only measured bow shock
locations ΔxBS together with the OMNI solar wind data into
account. Then, cost function () is replaced by a new function,
K=ΔxBS-ΔxBS,M2,
where ΔxBS,M denotes the subsolar bow shock location of the
model solution corresponding to a certain dipole moment M. Minimizing this
cost function determines the dipole moment M for any solar wind conditions
and measured bow shock location. The solar wind conditions using OMNI data
can be determined at all shock transitions of THC considered in
Table . The solar wind data are determined as a mean value
of 10 min around a shock crossing. The dipole moments for all 11 shock
transitions are calculated and multiple shock crossings on an orbit are
summed to an average value. Thus, eight dipole moment estimators are obtained
with a mean value of M=-8.2×1015Tm3 with a standard
deviation of 1.2×1015Tm3. This result has a
significantly increased statistical error compared to the previous result
using the entire magnetosheath data. The error is reduced by taking the
entire magnetosheath data into account instead of only 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
<NSW>=6.65cm-3,
the velocity <vSW>=382.2kms-1, and the magnetic
field <BSW>=2.84nT. Using these average values as the
boundary condition of the reduced MHD model, dipole moments M can be
estimated from the eight magnetosheath crossings once again. The results are also
presented in Fig. . The mean value of the dipole moment from the
eight values is M=-8.4×1015Tm3 with a standard deviation
of 1.3×1015Tm3. This corresponds to an error of
about 10% with respect to the Earth's dipole moment.
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.