Abstract
 Top of page
 Abstract
 1. Introduction
 2. Magnetic Field Observations
 3. Paraboloid Magnetospheric Model
 4. Analyses: Estimation of Model Parameters From Orbital Observations
 5. Global Magnetospheric Fields: Results and Discussion
 6. Residual Signatures: Results and Discussion
 7. Conclusions
 Acknowledgments
 References
 Supporting Information
[1] We present a baseline, timeaveraged model for Mercury's magnetosphere, derived from MESSENGER Magnetometer data from 24 March to 12 December 2011, comprising the spacecraft's first three Mercury years in orbit around the innermost planet. The model, constructed under the approximation that the magnetospheric shape can be represented as a paraboloid of revolution, includes two external (magnetopause and magnetotail) current systems and an internal (dipole) field and allows for reconnection. We take advantage of the geometry of the orbital Magnetometer data to estimate all but one of the model parameters, and their ranges, directly from the observations. These parameters are then used as a priori constraints in the paraboloid magnetospheric model, and the sole remaining parameter, the dipole moment, is estimated as 190 nTR_{M}^{3}from a grid search. We verify that the best fit dipole moment is insensitive to changes in the other parameters within their determined ranges. The model provides an excellent firstorder fit to the MESSENGER observations, with a rootmeansquare misfit of less than 20 nT globally. The results show that the magnetopause field strength ranges from 10% to 50% of the dipole field strength at observation locations on the dayside and at nightside latitudes north of 60°N. Globally, the residual signatures observed to date are dominated by the results of magnetospheric processes, confirming the dynamic nature of Mercury's magnetosphere.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Magnetic Field Observations
 3. Paraboloid Magnetospheric Model
 4. Analyses: Estimation of Model Parameters From Orbital Observations
 5. Global Magnetospheric Fields: Results and Discussion
 6. Residual Signatures: Results and Discussion
 7. Conclusions
 Acknowledgments
 References
 Supporting Information
[2] Mercury is the only inner solar system planet other than Earth to possess a largescale magnetic field [Ness et al., 1974, 1975]. Vector magnetic field measurements obtained during the first two flybys of Mercury by the MErcury Surface, Space ENvironment, GEochemistry and Ranging (MESSENGER) spacecraft, together with previous observations from the first and third Mariner 10 flybys, confirmed the presence of a dipolar internal field of dynamo origin [Anderson et al., 2008, 2010; Purucker et al., 2009; Uno et al., 2009]. Although the field is primarily dipolar, and of the same polarity (termed normal polarity) as Earth's field, Mercury's surface magnetic field strength is only about 1% that of Earth's [Ness et al., 1974], corresponding to a dipole moment that is smaller by a factor of about 1000. The underlying mechanism for the weak field is not understood, although dynamo models have been proposed in which the thickness of the liquid outer core [Stanley et al., 2005; Heimpel et al., 2005; Takahashi and Matsushima, 2006], the presence of stably stratified layer(s) [Christensen, 2006; Christensen and Wicht, 2008; Manglik et al., 2010], or the precipitation of solid iron [Vilim et al., 2010] play a role. An alternative dynamo mechanism that produces weak fields invokes feedback between the magnetospheric and core dynamo fields [Grosser et al., 2004; Glassmeier et al., 2007a, 2007b; Heyner et al., 2011]. The combined MESSENGER and Mariner 10 flyby data have resulted in estimates of the dipole moment that vary by up to 40% [Anderson et al., 2010]. This uncertainty reflects tradeoffs in estimating both internal and external fields from the limited observations. In particular, the flyby geometries led to a fundamental ambiguity regarding Mercury's largescale internal field structure: the observations could be represented equally well by the field due to either a centered dipole or a somewhat weaker dipole offset northward along the planetary rotation axis [Connerney and Ness, 1988; Anderson et al., 2010; Alexeev et al., 2010]. Prior to MESSENGER's insertion into orbit around Mercury, it was anticipated that it might not be possible to resolve the ambiguity even with orbital observations [Korth et al., 2004], due to the absence of lowaltitude magnetic field measurements over the southern hemisphere. However, observations from the MESSENGER Magnetometer (MAG) from the first 88 days in orbit around Mercury have enabled the direct identification of the magnetic equator, indicating a dipole offset of ∼0.2R_{M} northward along the rotation axis [Anderson et al., 2011, 2012], where R_{M} (= 2440 km) is Mercury's mean radius.
[3] The high solar wind pressure of the inner heliosphere, combined with Mercury's weak dipole moment, results in a magnetosphere that is about eight times smaller than Earth's relative to the planetary diameter. Thus satellite magnetic field observations, taken within even a few hundred kilometers of the surface, contain substantial contributions from fields produced by magnetospheric current systems [see Anderson et al., 2010]. Temporal changes in the solar wind density and speed, and in the interplanetary magnetic field (IMF), lead to variations in Mercury's magnetospheric conditions on timescales of seconds to a Mercury year (88 Earth days). The location of the magnetopause and magnetotail currents are governed by the origin and tilt of the planetary dipole field, so understanding Mercury's magnetic environment requires knowledge of both the internal and external largescale fields.
[4] During its first Earth year of orbital operations, the MESSENGER spacecraft was in a highly eccentric orbit with a period of 12 h, a periapsis of 200–600 km altitude at ∼60–70°N, and an apoapsis altitude of ∼15,000 km [see Zurbuchen et al., 2011, Figure 1]. These altitude variations present challenges to analyzing the internal field but have the advantage that every orbit transits into, and out of, the magnetosphere. This geometry allows identification of the bow shock and magnetopause on every orbit, as well as measurement of the IMF conditions before, and after, each magnetospheric transit. After one Mercury year, MAG measurements sampled the northern magnetosphere at all local times and sampled the nightside southern lobe of the magnetotail to distances of almost 5 R_{M}. The 88 day annual timescale and associated Mercury solar orbital (MSO) coordinate system govern the geometry of the largescale external current systems at Mercury, such as the magnetopause and magnetotail currents. The origin of the MSO coordinate system is at the planetary center of mass, +X is sunward, +Y lies in the orbital plane, perpendicular to +X and opposite to the direction of planetary orbital motion (i.e., +Y is positive toward dusk), and +Zis normal to the orbital plane and positive northward. With the exception of induction, the Mercury bodyfixed (MBF) coordinate system governs the geometry of the internally generated fields at Mercury. Because of Mercury's small obliquity [Margot et al., 2007, 2012], the MBF and MSO +Zaxes, and hence MSO and MBF latitudes, are nearly identical. Northern hemisphere coverage at all bodyfixed longitudes is achieved once every 59 Earth days (Mercury's sidereal spin period), at spacecraft altitudes from 200 km to over 1000 km. Thus, MAG observations taken in orbit around Mercury afford the first opportunity to examine the global structure of the planet's external and internal fields.
[5] Previous models of Mercury's internal and external fields used a variety of approaches, including spherical harmonic analyses (see reviews by Connerney and Ness [1988] and Anderson et al. [2010]), an empirical magnetospheric model developed for Earth [Tsyganenko, 1995; Tsyganenko and Sitnov, 2005] and scaled to Mercury [Korth et al., 2004; Uno et al., 2009; Anderson et al., 2008, 2010], and a paraboloid magnetospheric model [Alexeev et al., 2008, 2010]. Although physicsbased simulations are desirable and now possible [e.g.,Kabin et al., 2000; Trávníček et al., 2007; Benna et al., 2010], they are still not suited to the practical purpose of parameter searches or inversions involving large quantities of data. Spherical harmonic separation of internal and external fields requires that observations be taken in a currentfree region, a situation that is not the case for MESSENGER Magnetometer data [Alexeev et al., 2008, 2010; Anderson et al., 2010]. In addition, from an estimation perspective, the orbit geometry results in strong covariance among even the coefficients of lowest spherical harmonic degree and order [see, e.g., Connerney and Ness, 1988; Korth et al., 2004]. This covariance is not only among coefficients for the internal and external fields individually but is also between the internal and external fields. Thus, because the fundamental physical assumption of a currentfree region is violated and because the estimation problem results in covariance among the lowest degree and order internal and external fields, we do not pursue the traditional spherical harmonic separation approach here. As we detail below, MESSENGER observations permit direct identification of the geometry of the magnetospheric current systems and hence the fields produced by those currents, circumventing this severe covariance problem. The different average IMF orientation at Mercury than at Earth and the absence of major terrestrial current systems such as the ring current and fieldaligned currents suggest that scaling of Earthbased empirical models [Tsyganenko, 1995; Tsyganenko and Sitnov, 2005] may not be appropriate. Although a scaled version of the Tsyganenko and Sitnov [2005] model was used by us previously [Anderson et al., 2008, 2010; Uno et al., 2009] to try to estimate and remove the external fields from the Mariner 10 and MESSENGER flyby observations, we noted limitations in this approach [see Anderson et al., 2010, section 4.2], in particular in capturing the tail currents. Removing current system modules from Earth magnetospheric models is problematic because the different modules are not independent, in particular the tail and ring current systems [e.g., Tsyganenko and Sitnov, 2005].
[6] Here we use a paraboloid magnetospheric model to represent the internal dipole, magnetotail, and magnetopause fields at Mercury. The paraboloid magnetospheric model (PMM) is based on the assumption, first put forward by Alexeev and Shabansky [1972], that the planetary dipole field is confined within a model magnetopause approximated by a paraboloid of revolution. The model has been further developed for Mercury and has proven successful in modeling the Mariner 10 and MESSENGER flyby data [Alexeev et al., 2008, 2010]. The magnetospheric field is the sum of contributions from the planetary dipole, the magnetopause shielding currents, and the tail current sheet. In addition, reconnection with the IMF is included through a penetration field proportional to a reconnection efficiency parameter. The model formulation is particularly well suited to MESSENGER orbital observations, as these data provide not only measurements inside the magnetosphere but direct observations of the magnetopause and tail current sheet geometry, as well as IMF observations before and after each magnetospheric transit. We do not pursue a comparison of the paraboloid model with other modeling approaches such as a scaled Tsyganenko and Sitnov [2005] model or Gauss separation of internal and external fields. The fundamental limitations of these other approaches outlined above and by Anderson et al. [2010] mean that it is not possible to assess how well they describe the physical system except via an overall misfit of the model to the MAG data away from the regions of current sources. With Mariner 10 and MESSENGER flyby data, Alexeev et al. [2010] demonstrated that the paraboloid model does indeed provide the best fit to the observations. However, more importantly, the validity of the paraboloid model can be assessed in a more powerful way, specifically investigating how well it describes the magnetopause, magnetotail, and internal fields individually (section 4).
[7] In this paper we present a baseline, timeaveraged model for Mercury's magnetosphere, using MAG data from 24 March to 12 December 2011, comprising the first three Mercury years of the MESSENGER spacecraft's orbital observations at Mercury. This approach is philosophically different from that ofAlexeev et al. [2010], who estimated a common internal field but different external fields and reconnection efficiencies for the times of the first two MESSENGER flybys. We do not attempt here to estimate time dependence in the external fields, but solve instead for timeaveraged background internal and external fields. This approach allows us to obtain robust estimates of the external fields from observations during orbits that repeatedly cover the full range of local times and bodyfixed longitudes.Alexeev et al. [2010] used an iterative, linearized minimization approach to solve simultaneously for all the model parameters. Here we take advantage of the orbital MAG data to estimate all but one of the model parameters directly from the observations, thus reducing the model covariance that would be inherent in a multiparameter, nonlinear inversion. We estimate both the mean value and upper and lower bounds for each parameter. From the magnetopause and tail current sheet crossings we constrain the geometry of these currents. From observations during spacecraft transits through the southern lobe of the magnetotail we estimate the flux in a single tail lobe. Observations of the magnetic equator resolve the axial dipole–quadrupole ambiguity otherwise inherent in the orbital geometry [Korth et al., 2004; Anderson et al., 2011, 2012] and provide tight constraints on the dipole tilt and orientation. The remaining model parameter, the average dipole moment, is estimated via a grid search. The baseline model is thus tied to direct constraints from the magnetic field observations and provides a description of the firstorder signal in the data throughout the magnetosphere. Further refinements in current systems, the planetary field, and time variability can then be readily assessed relative to this baseline model.
[8] Central to this study are the orbital MAG observations, which we describe in some detail in section 2. Examples of data collected during orbits at two different local times illustrate the major magnetospheric boundaries and the identification of the magnetic equator. We provide a brief overview of the PMM model in section 3, and we refer the reader to previous publications for model formulation details [Belenkaya et al., 2005; Alexeev et al., 2008, 2010]. A description of how the PMM parameters and their uncertainties are estimated from the MAG observations is given in section 4, and the resulting global model is given in section 5. We discuss possible contributions to the observed residual signatures, including static contributions due to limitations of the model itself, dynamic magnetospheric contributions on a variety of timescales, and contributions related to internal field processes in section 6.
2. Magnetic Field Observations
 Top of page
 Abstract
 1. Introduction
 2. Magnetic Field Observations
 3. Paraboloid Magnetospheric Model
 4. Analyses: Estimation of Model Parameters From Orbital Observations
 5. Global Magnetospheric Fields: Results and Discussion
 6. Residual Signatures: Results and Discussion
 7. Conclusions
 Acknowledgments
 References
 Supporting Information
[9] Vector field observations from orbit about Mercury have been acquired from MESSENGER's MAG since 23 March 2011. Inside the magnetosphere, sampling is typically at 20 samples per second with the exception of a few periods during which operational constraints required that the instrument be turned off [Anderson et al., 2007]. The time that MESSENGER spends inside the magnetosphere during each orbit is on average 1 h when the orbit is close to the terminator and up to 2 h when the orbit is close to the midnight–noon plane. The latitudinal coverage varies with magnetic local time and MBF longitude. Examples of data taken during orbits close to the dawn–dusk and noon–midnight planes are shown in Figures 1 and 2, plotted in the MSO coordinate system. Data from these orbits illustrate how the major magnetospheric boundaries (the bow shock, magnetopause, and tail current sheet) and the magnetic equator may be identified from the MAG data. Our analyses use the reduced data records (RDRs) that contain 1 s averages, and standard deviations, of the magnetic field components.
[10] For each orbit the inbound and outbound bow shock and magnetopause were identified. Diagnostics of the inbound bow shock are an increase in magnetic field magnitude and/or an increase in wave activity just upstream of the shock. Diagnostics of the inbound magnetopause are a rotation in the field direction from the IMF direction toward the planetary field direction, and an accompanying decrease in magnetic field fluctuations [Moldovan et al., 2011; R. M. Winslow et al., Mercury's magnetopause and bow shock from MESSENGER observations, submitted to Journal of Geophysical Research, 2012]. The reverse sequences are seen during outbound bow shock and magnetopause crossings (Figures 1 and 2). Inner and outer limits for each boundary position were identified. The spacecraft is continuously within the magnetosphere between the inner edge inbound and inner edge outbound magnetopause crossings. Inside the magnetosphere the signature of a dipolar field is evident: field magnitudes increase with decreasing spacecraft altitude and increasing northern latitude. For orbits close to the terminator, the field is dominated by the B_{Y} and B_{Z} components. On the ascending (inbound, dawnside) node of the orbit for 1 August 2011 (Figure 1), the B_{Y} component of the field was positive, and on the descending (outbound, duskside) node of the orbit the B_{Y} component was negative. At high northern latitudes the B_{Z}component dominated and was negative. These observations are consistent with the expected field directions in the northern hemisphere for a normalpolarity dipole field.
[11] In contrast, for the midnight–noon orbit on 20 November 2011 (Figure 2), the spacecraft entered the magnetosphere at a downtail distance of X ∼ −4R_{M}. The magnetic field was almost purely antisunward (−B_{X}), diagnostic of the southern magnetotail lobe field. Just before 09:30 UTC the spacecraft passed into the northern lobe, as seen by the rotation of the field into the sunward (+B_{X}) direction. At about 09:50 UTC, the spacecraft started to measure the planetary dipole field. In this orbit geometry, the dipole field is dominated by the B_{X} and B_{Z} components, with B_{X} positive on the nightside and negative on the dayside, and B_{Z}negative at high northern latitudes. In all, the ascending portion of the orbit accounts for over 80% of the 2 h duration of the magnetospheric transit, and the descending dayside portion of the orbit was relatively brief. Highfrequency fluctuations were seen throughout the dayside part of this orbit. Largeamplitude fluctuations, accompanied by depressions in magnetic field magnitude at high dayside latitudes, are diagnostic of the northern cusp region [Winslow et al., 2012].
[12] The magnetic equator was identified on an orbitbyorbit basis with a cylindrical coordinate system defined from standard MSO coordinates (ρ_{MSO}, ϕ_{MSO}, Z_{MSO}), where Z_{MSO} is positive northward, ρ_{MSO} = (X_{MSO}^{2} + Y_{MSO}^{2})^{1/2} is positive outward and parallel to the projection of the radial direction in the X_{MSO}–Y_{MSO} plane, and ϕ_{MSO} is the azimuth angle and is positive eastward and zero at the subsolar point. The magnetic equator on each orbit is the location where the component B_{ρ} passes through zero as it changes sign [Anderson et al., 2011, 2012]. The magnetic equator locations for the orbits with data shown in Figures 1 and 2 are indicated. For orbits that enter the magnetosphere in the deep tail region, such as that shown in Figure 2, the magnetic equator lies within the tail current sheet. The current sheet thickness in the Z direction can be estimated from the time interval (and the associated locations) over which the magnetic field rotates from the antisunward southern lobe direction to the sunward northern lobe direction (compare section 4.2).
[13] Observations spanning all bodyfixed longitudes and local times are obtained every 59 and 88 days, respectively. The magnetic field strength measured inside the magnetosphere (at latitudes north of 30°S) during the second Mercury year of observations is shown inFigure 3. The signature of the dipole field is evident in the increase in field strength with latitude. Orbittoorbit variations in the magnetopause boundary positions are observed. The effects of MESSENGER's highly eccentric orbit are clear: latitudinal coverage depends on local time and spacecraft altitudes range from 200 km to 10,000 km. Observations below 1000 km altitude (the approximate subsolar distance to the inner magnetopause crossings) cover almost all northern latitudes for the descending tracks but cover only latitudes north of ∼60°N for the ascending tracks. Periapsis altitude increases from ∼200 km to ∼600 km during a Mercury year, and the associated decrease in magnetic field strength is clearly seen (Figure 3). For the observations reported here, orbital correction maneuvers were performed every 88 days to lower periapsis altitude to ∼200 km. An example of one orbitcorrection maneuver (OCM) is shown inFigure 3. The overall consistency of the observations over one Mercury year, as well as the consistency from one Mercury year to the next, motivates the generation of a baseline average model for Mercury's magnetosphere.
3. Paraboloid Magnetospheric Model
 Top of page
 Abstract
 1. Introduction
 2. Magnetic Field Observations
 3. Paraboloid Magnetospheric Model
 4. Analyses: Estimation of Model Parameters From Orbital Observations
 5. Global Magnetospheric Fields: Results and Discussion
 6. Residual Signatures: Results and Discussion
 7. Conclusions
 Acknowledgments
 References
 Supporting Information
[14] The PMM [Alexeev and Shabansky, 1972] specifies a model magnetopause that is a paraboloid of revolution and confines the planetary dipole field and the field from the dawntoduskflowing magnetotail current. The full model field is the superposition of the dipole, magnetopause, and tail fields. Reconnection with the IMF,B_{IMF}, can be represented via a reconnection “efficiency” parameter, k, where 0 ≤ k ≤ 1, such that k B_{IMF} is added to the paraboloid magnetospheric model field. The reconnection field requires no change to the magnetopause current system. The full model formulation, as applied here, has been described by Belenkaya et al. [2005] and Alexeev et al. [2008, 2010].
[15] The currents on the magnetopause and in the magnetotail are organized by the dipole origin and orientation – the Mercury solar magnetic (MSM) coordinate system. In general, the MSM coordinate system is related to the MSO system via a displacement (specified by the dipole origin relative to the planetary origin) and a rotation about the X_{MSO} axis (governed by the tilt and azimuth) [Alexeev et al., 2010]. The dipole and magnetopause fields are obtained by solving Laplace's equation in parabolic coordinates (α, β, ϕ), under the assumption that the region between the planetary surface and the magnetopause is current free. In the absence of reconnection of the IMF with the planetary field, the component of the magnetic field normal to the magnetopause is zero. The magnetopause is defined as the parabolic surface on which β = 1. In the X_{MSM}–Z_{MSM} plane, the tail current sheet is confined within the parabola, β = β_{0}.
[16] The PMM is specified by a total of 12 parameters (Table 1). The basic geometry of the model magnetopause and magnetotail is shown in Figure 4. Variable magnetopause flaring (γ ≠ 1) is included in the model formulation [Belenkaya et al., 2005], such that the magnetopause surface (X_{MP}, Y_{MP}, Z_{MP}) is given by the paraboloid β = 1, where
R_{1} is a scale length that determines the magnetopause size. For γ = 1, R_{1} = R_{SS}, the subsolar magnetopause distance, as shown in Figure 4. Values of γ greater than 1 correspond to a less flared magnetopause.
Table 1. Parameters in Magnetospheric Models, Their Average Values, and the Range in Mean Values Derived From MAG ObservationsSymbol  Unit  Mean  Range  Description  Parameter Estimation Procedure (Corresponding Text Section) 

Paraboloid Magnetospheric Model 
R_{SS}  R_{M}  1.45  1.4–1.5  Subsolar magnetopause distance  Paraboloid fit to magnetopause crossings (section 4.2) 
γ  None  1.0  Upper bound of 1.1  Magnetopause paraboloid flaring parameter  Set = 1 (section 4.2) 
R_{2}  R_{M}  1.41  1.36–1.46  Distance to inner edge of crosstail current sheet (closest approach to the planet inXZ plane)  Current sheet was on average tailward of nearplanet equator crossings;R_{2}from radial distance to nearplanet equator crossings within ± 3 h of midnight (section 4.3) 
D_{D}  R_{M}  0.09  0.06–0.14  Half thickness of the crosstail current sheet  Identify rotation in magnetic field associated with the tail current sheet in far tail (section 4.3) 
B_{T}  nT  137  123–152  Model magnetotail field intensity  Estimated from F (average flux in a single tail lobe), R_{2}, and D_{D} (section 4.3) 
X_{0}, Y_{0}, Z_{0}  R_{M}  0, 0, 0.196  0, 0, 0.192–0.200  Center position of planetary dipole relative to planetary center along X_{MSO}, Y_{MSO}, Z_{MSO} axes  Set X_{0} = Y_{0} = 0 on the basis of axisymmetry of field about rotation axis; Z_{0} established from northward offset of magnetic equator (section 4.1) 
Θ, Φ  deg  0,0  Tilt upper bound of 1° at 283°E  Tilt and azimuth of dipole relative to rotation axis.  Magnetic equator analyses and axisymmetry of field about rotation axis (section 4.1) 
m  nT R_{M}^{3}  190  See text  Magnitude of planetary dipole moment  Grid search using paraboloid model to minimize misfit to observations (sections 4.5 and 5) 
k  None  0  Not considered  Magnetopause reconnection efficiency  Set = 0 (section 4.4) 

Shue Magnetopause Model 
R_{SS}  R_{M}  1.45   Subsolar distance  Fit to magnetopause crossings 
α  None  0.5   Tail flaring  
[17] The magnetotail is bounded by the surface β_{0}, given by
where D_{D} and R_{2} are the half thickness and distance (in the X_{MSM}–Z_{MSM}plane) to the inner edge of the crosstail current sheet, respectively, and the flux,F, in one lobe of the tail is
Thus, the tail field parameter, B_{T}, depends linearly on the tail flux, F, and nonlinearly on the other magnetopause and magnetotail parameters, R_{ss}, γ, R_{2}, and D_{D}. We checked that the normal field at the magnetopause boundary (B_{β}) is zero in our numerical implementation of the model, when the reconnection efficiency, k, is set to zero. Specifically, we confirmed that the B_{β} component of the field was less than 2 × 10^{−5} nT along the entire magnetopause from the subsolar point to 6R_{M} downtail. The ratio B_{β}/B, where B is the total field strength, was less than 0.001%. MSO positions at the magnetopause and inside the magnetosphere were corrected for solar wind aberration [Paschmann and Daly, 1998]. An aberration angle was calculated for each MESSENGER orbit using Mercury's instantaneous orbital speed and a mean solar wind speed of 405 km/s from the ENLIL model for the period 24 March to 12 December 2011 [Odstrcil, 2003].
4. Analyses: Estimation of Model Parameters From Orbital Observations
 Top of page
 Abstract
 1. Introduction
 2. Magnetic Field Observations
 3. Paraboloid Magnetospheric Model
 4. Analyses: Estimation of Model Parameters From Orbital Observations
 5. Global Magnetospheric Fields: Results and Discussion
 6. Residual Signatures: Results and Discussion
 7. Conclusions
 Acknowledgments
 References
 Supporting Information
[18] As outlined in section 2, we have used orbital MAG data to identify the magnetopause, the tail current sheet, and the magnetic equator. Here we describe how we used these observations to estimate parameters in the PMM (Table 1). Our procedure differs from the approach of Alexeev et al. [2010], who solved simultaneously for a common internal field but different external fields and reconnection efficiencies for the times of the first two MESSENGER flybys. Although that approach provided a good fit to the observations, there was substantial covariance among the model parameters, as can be seen, for example, for the tail and magnetopause fields from equations (1)–(4) and as discussed by Alexeev et al. [2010]. Data acquired in orbit allowed us to derive the PMM parameters and their uncertainties for the planetary field, magnetopause, and tail current sheet directly from the observations. Importantly, with the exception of the tail flux, which depends on both the tail field and the shape of the magnetopause in the tail region, the parameters derived in this way are independent of each other. These parameters were then used as a priori constraints in the PMM model, and the remaining parameter, the dipole moment, was estimated via a grid search. This approach avoids tradeoffs among the model parameters inherent in a minimization approach, provides quantitative measures of the variability in these parameters, and allows us to assess whether the functional forms implicit in the model may need to be adjusted.
4.1. Dipole Offset and Orientation
[19] We first summarize observations that constrain the dipole offset and orientation, since these parameters (together with the dipole moment; see section 4.4) control the location of the major magnetospheric current systems. MAG observations from MESSENGER's first 88 days in orbit indicated that Mercury's magnetic equator is offset northward from the planetary center [Anderson et al., 2011]. The results of Anderson et al. [2012] confirm this northward offset of the dipole: the mean offset calculated from the first three Mercury years of data is 479 ± 9 km (Table 1). There is no evidence for an offset in the X and Y directions, so we set the dipole origin PMM parameters to be X_{0} = Y_{0} = 0 and Z_{0} = 0.196 R_{M} ± 0.004 R_{M} (Table 1). Anderson et al. [2012] reported a mean value for the offset of 477 ± 7 km on the basis of data that extend into the fourth Mercury year of observations, but we restrict ourselves to the statistic derived from the first 3 years of data for consistency with the remaining analyses discussed here. The magnetic equator observations constrain the tilt in the field to be a maximum of 0.8° with an azimuth of the southern hemisphere pole of 283°E [Anderson et al., 2012, Figure 5], confirming the apparent axisymmetry of the field seen in Figure 3 and observed in the MESSENGER flyby data [Anderson et al., 2010; Uno et al., 2009]. We thus set the tilt and azimuth of the dipole to be zero with an upper bound of 1° tilt at 283°E (Table 1).
4.2. Magnetopause Subsolar Distance and Flaring
[20] Fields due to currents on the magnetopause depend on the internal dipole field and the magnetopause shape. We fit a paraboloid of revolution (Figure 5 and Table 1) to the inbound and outbound magnetopause crossings identified on each magnetosphere pass (Winslow et al., submitted manuscript, 2012), after accounting for the northward offset of the dipole (see Anderson et al. [2011] and section 4.3). To first order the magnetopause is a figure of revolution, as detailed by Winslow et al. (submitted manuscript, 2012). The crossings are fit well on the dayside and on the nightside close to the planet by a paraboloid with R_{SS} = 1.45 R_{M} and γ = 1. Although the nightside magnetopause crossings are better fit with a value for γ greater than 1, values of γ greater than ∼1.1 yield models that provide a poor overall fit to the crossings on the dayside (see, e.g., the curve for γ = 1.3; Figure 5). Thus, we set γ = 1, with an upper bound of 1.1. For this range of γ, a range of R_{SS} values from 1.4 R_{M} to 1.5 R_{M}yield similar root mean square (RMS) misfit values to the nearplanet magnetopause crossings, and so we assign an uncertainty of 0.05R_{M} to the mean R_{SS} values (Table 1). We also fit the magnetopause crossings with the functional form proposed by Shue et al. [1997] for use in our tail flux calculation (section 4.2); this model provides an excellent overall fit to the entire population of magnetopause crossings.
4.3. Tail Current Sheet Geometry and Tail Flux
[21] Orbits that enter the magnetosphere close to the midnight meridian provide information on the thickness and position of the current sheet in the deeptail region (e.g.,Figure 2), whereas orbits that enter the magnetosphere on the dayside close to noon local time sample the neartail region during the descending portion of the transit. We used orbits from the deep tail to constrain the current sheet half thickness (D_{D}) and the tail flux (F). We found that most orbits that sample the neartail region do not typically cross the current sheet, and so these orbits provide only a bound on how close, on average, the current sheet comes to the planet (R_{2}).
[22] For the deeptail observations the current sheet was indicated by a rotation in the field direction from dominantly antisunward in the southern tail lobe to dominantly sunward in the northern tail lobe. The rotation can be seen on individual profiles (Figure 2), and we obtained a mean thickness for the current sheet in this region from a superposed epoch analysis. Orbits from the first three Mercury years of observations, with magnetic equator crossings within 3 h of midnight in the deep tail, were selected. The magnetic equator crossing for each orbit was assigned as the reference time, and the tilt of the field in the ρ–Z plane was calculated from θ_{ρ–Z} = cos^{−1}(B_{ρ}/B_{ρZ}), where B_{ρZ} is the magnitude of B projected onto the ρ–Z plane. Since B_{ρ} is dominated by B_{X} in this region, θ_{ρ–Z} close to 0° and 180° indicate antisunward and sunward field directions, respectively. Note that since the magnetic equator is taken to be the point where B_{ρ} = 0, the center of the current sheet coincides with the magnetic equator by definition. The angle θ_{ρ–Z} and the magnetic field magnitude for the 79 selected deep tail orbits were averaged, and the results are shown in Figures 6a and 6b. The rotation of the field direction from the southern to northern lobe is clear (black curve). After removal of the dipole field (red curve) as given by the best fit dipole moment described in section 4.4, the field is almost purely antisunward in the southern lobe and sunward in the northern lobe. The field rotation is 99% complete within ±220 km of the center of the current sheet or within 0.09 R_{M}. The rotation is 95% complete within 140 km (0.06 R_{M}) of the current sheet center and approaches its asymptotic value at a distance of about 350 km (0.14 R_{M}). The depression in field magnitude associated with the plasma sheet [Korth et al., 2011] is centered on the field reversal, although the magnetic depression is broader than the field rotation, indicating a greater average thickness of the plasma sheet than the current sheet. The minimum field magnitude occurs at the center of the current sheet. From these results we set the PMM parameter D_{D} = 0.09 R_{M}, with a lower bound of 0.06 R_{M} and an upper bound of 0.14 R_{M}.
[23] The neartail region was sampled by MESSENGER on orbits with the descending orbit leg on the nightside (within 3 h of midnight). The magnetic signatures were very different in this region. Inspection of individual neartail orbits showed that although the magnetic equator can be reliably pinpointed, clear localized rotations in the field consistent with current sheet crossings were generally not observed. This observation indicates that these neartail trajectories did not pass from the northern tail lobe to the southern, but rather the spacecraft generally traversed the equator planetward of the crosstail current. Consequently, the MAG data allow us only to place bounds onR_{2}. Depressions in the field magnitude, indicating spacecraft encounters with the plasma sheet, occurred on some but not all crossings of the neartail region. This pattern suggests that the plasma sheet and, hence, possibly, the current sheet were present for only a fraction of the neartail passes. Because in the deep tail the plasma sheet correlated with the current sheet, we selected neartail orbits with stronger plasma sheet signatures by evaluating the minimum value of the magnetic field strength near the equator crossing. Those orbits with the strongest plasma sheet signatures and deepest magnetic field minima should be those that passed closest to the current sheet.
[24] Superposed epoch averages were obtained for the 25% of the orbits with the lowest minimum field magnitudes (Figure 6c). We then subtracted the vector dipole field to assess the field properties due to external currents only, and we took the averages of the residual field magnitude, Z component, and polar angle: B, B_{Z} and θ_{ρ–Z}.
[25] The neartail field, even for these orbits, was quite different from the fartail field. First,θ_{ρ–Z} came only within about 30° of the 180° or 0° direction, indicating that the orbits were, on average, planetward of the current sheet and tail lobes. Second, the magnitude of the external field (after removal of the dipole field) increased rather than decreased near the equator crossing and at the equator was almost entirely in the −B_{Z}direction, that is, southward. This behavior indicates that the external field in this region was dominated by the fringing field of the crosstail current planetward of the tail current sheet. Thus, these orbits passed close to but not through the current sheet, so we estimated a lower bound onR_{2} from the mean radial distance to these equator crossings. On average R_{2} was tailward of 1.41 R_{M}, and we set R_{2} = 1.41 R_{M} in the PMM. The standard deviation in R_{2} is 0.05 R_{M}, and we use this uncertainty to set upper and lower bounds on the mean value of R_{2} (rather than the standard error in the mean) because the spacecraft orbit varied systematically in this region. We note that temporal variability of this distance is likely to occur on timescales less than one orbital period [e.g., Slavin et al., 2012]. In addition, because on average MESSENGER's orbit does not cross the current sheet in the neartail region, we do not have an estimate of the current sheet half width close to the planet. We expect thatD_{D}will be larger in the neartail region than the fartail region, but in the absence of neartail current sheet crossings, we use our fartail estimate ofD_{D} in the PMM (Table 1).
[26] We estimated the tail flux parameter of the PMM model, B_{T}, as follows. To ensure spacecraft residence in the lobe field, we selected only those observations in the southern tail for which the field was oriented to within 10° of the antisunward direction. These locations and the corresponding field magnitudes at each point are shown in Figure 7a. At each location we used the field magnitude, B, together with the radius to the magnetopause boundary from the Shue et al. [1997] model, r_{MP}, measured from the X_{MSM} axis (to account for the offset dipole) parallel to the Y_{MSO}–Z_{MSO} plane at the X_{MSO} coordinate of the spacecraft. We then estimated the southern lobe magnetic flux (Figure 7b) for each data point from
The average tail flux is 2.6 MWb, and the standard deviation is 0.6 MWb (±25%). The tail flux was calculated from data obtained during 63 orbits, and so the standard error of the mean is 0.08 MWb. We used three standard errors to give the upper and lower values of the mean flux (Table 1). With this value of F, together with our estimates for D_{D} and R_{2}, we calculated the tail field parameter B_{T} in the PMM. Note that B_{T} depends on all the other parameters earlier estimated, including γ, R_{1}, R_{2}, and D_{D}. Of these, R_{2} is the least well constrained in an average sense, but for a given flux, B_{T} is relatively insensitive to both D_{D} and R_{2}. Figure 7c shows that for our mean flux, B_{T} varies by less than 6% for D_{D} values between 0.05 R_{M} and 0.25 R_{M}, and for R_{2} values between 1.0 R_{M} (the surface of Mercury) and 2.0 R_{M}. B_{T} scales linearly with the tail lobe flux, which varies from one orbit to another, reflecting different states of the magnetotail. Although the range in the mean flux is small (0.2 MWb), instantaneous tail flux estimates range from 1 to 4.5 MWb, and in section 5 we verify that even a factor of 4 variation in the tail flux or the corresponding B_{T} values does not affect our estimates of the mean dipole moment.
4.4. Reconnection Efficiency
[27] We set the reconnection efficiency, k, to zero for our average baseline magnetospheric model. As the effect of reconnection is modeled simply by adding k B_{IMF} (0 ≤ k ≤ 1) to the predicted fields inside the magnetosphere, reconnection can be examined as one of the possible contributors to the residuals to the baseline model. Reconnection is likely to vary markedly from one orbit to the next and even during a given orbit. The IMF B_{Z} component plays a critical role in reconnection, but at Mercury IMF B_{Z} is small relative to IMF B_{X} and IMF B_{Y}, and it varies on timescales much shorter than a magnetospheric transit time. When MESSENGER is inside the magnetosphere, we do not have simultaneous measurements of the IMF, and so it is unclear how best to estimate the factor k B_{IMF}a priori, even on an orbitbyorbit basis.
4.5. Dipole Moment
[28] We use the values of the PMM parameters estimated as described above as a priori constraints in the paraboloid model. The last remaining model parameter, the dipole moment, m, is constrained by the goodness of fit of the model to the MAG data, globally. We describe the fit process and the resulting global model fields in section 5.
5. Global Magnetospheric Fields: Results and Discussion
 Top of page
 Abstract
 1. Introduction
 2. Magnetic Field Observations
 3. Paraboloid Magnetospheric Model
 4. Analyses: Estimation of Model Parameters From Orbital Observations
 5. Global Magnetospheric Fields: Results and Discussion
 6. Residual Signatures: Results and Discussion
 7. Conclusions
 Acknowledgments
 References
 Supporting Information
[29] The mean values of the magnetospheric parameters estimated by the approaches described above are summarized in Table 1. To complete the PMM model, we conducted a grid search to determine the best fit dipole moment to the observations in an RMS sense. The dipole moment was varied from 170 to 230 nT R_{M}^{3} in 5 nT R_{M}^{3} steps. For each value of the dipole moment, the magnetopause, magnetotail, and dipole fields were predicted at 5 s intervals along every orbit track inside the model magnetopause. The model was compared with the data in the 5 s RDRs, and the RMS misfit of the predicted field to the observations was computed (Figure 8). Both globally inside the entire magnetosphere and for the region restricted to latitudes north of 30°N (where the dipole field dominates), a best fit dipole moment of 190 nT R_{M}^{3} was found. For the latter region, the minimum misfit is 18.9 nT, and the RMS misfit is within 20% of this minimum value for dipole moments within ±10 nT R_{M}^{3} of 190 nT R_{M}^{3}.
[30] Among the PMM parameters estimated from the observations, B_{T}likely deviates the most from its average value on an orbitbyorbit basis (section 4.3). A second grid search confirms that the best fit dipole moment is insensitive to the value of B_{T} for B_{T} values in the range 60 to 260 nT (Figure 8b), or equivalently the observed factor of 4 variations in tail flux (Figure 7b). Orbittoorbit variations in the tail flux result in corresponding orbittoorbit variations in the residuals to the baseline model in the tail region, as discussed further insection 6, but have negligible influence on the planetary moment estimate. We also verified that the best fit dipole moment is insensitive, at the level of the 5 nT R_{M}^{3} increment used in the grid search, to changes in the other parameters within the range of mean values given in Table 1. The robustness of the fit illustrates the advantage of the approach taken here: that of first estimating as many magnetospheric model parameters as possible directly from the observations, and then using these parameters as a priori constraints in the paraboloid model. The alternative approach, that of coestimating all model parameters from nonlinear inversions of the vector data inside the magnetosphere, would yield tradeoffs among model parameters, in particular among the dipole moment, the dipole offset, the magnetopause shape, and the distance from the dipole origin to the magnetopause subsolar point. Furthermore, such an approach would ignore robust a priori observational constraints on model parameters.
[31] Comparison of the predicted model field magnitudes along the orbit tracks for the second year of MESSENGER observations (Figure 9) with the corresponding MAG observations (Figure 3) demonstrates the overall success of the model. Residuals (data minus model) are typically less than 50 nT in magnitude (Figure 9) compared with a maximum signal in the data of ∼500 nT. Model predictions for the aberrated MSO B_{X}, B_{Y}, and B_{Z} components for our example dawn–dusk and midnight–noon orbits provide an excellent fit to the observations (Figures 10 and 11). In addition to the full model prediction, the individual model tail, magnetopause, and dipole fields are shown along each orbit. The importance of the magnetopause fields at high northern latitudes and on the dayside can be seen. For orbit 274 on 1 August 2011 (Figure 10), the dipole field alone would overpredict the amplitudes of the B_{X} and B_{Z} components of the field by ∼40% and ∼10%, respectively. For orbit 497 on 20 November 2011 (Figure 11), the dipole and magnetopause fields contribute almost equally to B_{Z} on the dayside. The magnetopause and dipole fields were essentially zero in the far tail for orbit 497, and the negative B_{X} signal seen in the observations is predicted by the tail field component of the PMM.
[32] The magnitudes of the magnetopause, tail, and dipole fields predicted at the planetary surface are shown in Figure 12. The strong north–south asymmetry in the internal field imposed by the large northward offset of the dipole is evident, with the field strength at high northern latitudes over 700 nT, approximately three times that at the corresponding southern latitudes. Magnetopause fields are strongest on the dayside as expected. In addition, the offset dipole results in a magnetopause surface that is closer to the planetary surface at southern latitudes than at corresponding northern latitudes (see also Figure 4). This north–south difference is greatest at low to middle latitudes at local noon, and thus the strongest magnetopause fields are found at low southern dayside latitudes, where the magnetopause field strength reaches almost 80 nT, about 40% of the dipole field strength at those locations.
[33] Temporal variations in Mercury's globalscale magnetic fields are expected to occur, due to the variation in solar wind ram pressure,P_{ram}, at Mercury that arises from changes in heliocentric distance during Mercury's orbit around the Sun. For a simple balance between the internal magnetic field pressure and the solar wind dynamic pressure, under the assumption of a constant planetary dipole moment, the magnetopause subsolar distance will vary as P_{ram}^{−1/6} [Shue et al., 1997]. Thus, if our mean R_{SS} value of 1.45 R_{M} corresponds to average P_{ram} conditions at Mercury, we can expect a variation in P_{ram} about this average value of ∼50% and a corresponding variation in R_{SS} of about ±7% or ±0.1 R_{M}. This range is consistent with observed variations in R_{SS} (Winslow et al., submitted manuscript, 2012). The variations in R_{SS} result in changes in the magnitude of the magnetopause fields, in particular on the dayside. For R_{SS} variations of ±0.1 R_{M} about the mean value of 1.45 R_{M}, the magnetopause field magnitudes at spacecraft altitudes vary by 25% about their mean values on the dayside and on the nightside at latitudes north of 60°N. In addition, shorter timescale variations in R_{SS}may occur in association with rapid, largeamplitude changes inP_{ram}such as those due to coronal mass ejections or quasiparallel bow shock dynamics.
6. Residual Signatures: Results and Discussion
 Top of page
 Abstract
 1. Introduction
 2. Magnetic Field Observations
 3. Paraboloid Magnetospheric Model
 4. Analyses: Estimation of Model Parameters From Orbital Observations
 5. Global Magnetospheric Fields: Results and Discussion
 6. Residual Signatures: Results and Discussion
 7. Conclusions
 Acknowledgments
 References
 Supporting Information
[34] Residuals for each field component for midnight–noon orbits 497 and 499 on 20 and 21 November 2011 are shown in Figure 13. Orbit 499 occurred 24 h after orbit 497, so these orbits sampled the same largescale static structure in the internal and external magnetic fields. There is some similar structure in the residuals for each orbit. Both orbits exhibit a large dayside negative residual inB_{X} (after 10:10 UTC), accompanied by a positive residual in B_{Z.}However, differences in the residuals are also observed. For example, the largescale structure in theB_{X} residuals is different between about 9:30 and 10:10 UTC, corresponding to northern nightside latitudes (Figure 2) near midnight local time and MBF longitude ∼10°–15°E. The magnetic field during orbit 497 was relatively quiet in the deep tail (08:30–09:10 UTC), whereas the field during orbit 499 was notably more disturbed as evidenced by the highfrequency variability. Orbit 499 shows a persistent positiveB_{Z} residual throughout most of the southern tail lobe until ∼09:10 UTC, or an MSO latitude of ∼10°S. The B_{X}residuals from 09:30 to 09:40 UTC have opposite signs on the two orbits. On the dayside, the magnetic field during orbit 499 was relatively quiet and its inner magnetopause boundary was farther from the planet, compared with orbit 497. Even this cursory comparison of the residuals from these two orbits indicates that timevariable signatures can be comparable to the average residuals but suggests that some systematic residual signatures may be present.
[35] To examine the data for consistent signatures, we first considered the residuals in aberrated MSM coordinates. We transformed the residuals into coordinates organized by the baseline model magnetic field, where B_{par} is parallel to the model field direction, n_{par}, B_{az} is azimuthal and parallel to the ϕ_{MSO} unit vector, n_{ϕ}, and B_{perp} lies in the plane of the model magnetic field and the MSO Z axis such that n_{perp} = n_{ϕ} × n_{par}. Figure 14 shows the residuals in these coordinates for the first three Mercury years of observations. In B_{par}, diamagnetic depressions were seen consistently at dayside latitudes north of 60°N within ±3 h of local noon on both the descending (Figure 14a) and ascending (Figure 14b) tracks, reflecting increased plasma pressure in the northern cusp region [Korth et al., 2011; Winslow et al., 2012]. Depressions in B_{par}associated with the tail plasma sheet were often seen around the deeptail magnetic equator crossings (Figure 14b). Within about 1.5 h of local midnight, nearplanet diamagnetic depressions were seen (Figure 14a); these variations reflect plasma sheet pressures as discussed by Korth et al. [2012]. Consideration of the plasma pressure implied by these neartail magnetic field depressions shows that the plasma pressures are symmetric about the magnetic equator [Korth et al., 2012].
[36] Other major signals that were consistently present are positive residuals parallel to the local field direction at latitudes between 30°N and 60°N and positive (negative) dawn (dusk) residuals at high latitudes in the azimuthal (B_{az}) component. Note that the duskside B_{az}residual signature is also stronger in the higheraltitude data. We suggest that these signatures may in part be due to the difference between the actual magnetopause geometry and that from the paraboloid model. As seen inFigure 5, the paraboloid model magnetopause is more flared than the observations, in particular on the nightside. To assess the major signatures that could be affected by using a different magnetopause shape, we calculated the residuals with respect to a paraboloid magnetopause that is less flared than the γ= 1 baseline model, corresponding to a smallerdiameter magnetotail. We specified a model magnetopause that is less flared, usingγ = 1.3, chosen purely to illustrate the effect that such a magnetopause displacement would have. The γ = 1.3 model provides a better fit to the magnetopause crossings on the nightside for −0.5 R_{M} < X_{MSO} < 2.0 R_{M} but does not fit the dayside crossings as well as the γ = 1 magnetopause shape. Residual signatures larger than 10 nT amplitude are shown in Figure 15 for both our baseline model and the γ = 1.3 model for the second Mercury year of observations, together with the difference between the γ = 1.0 and γ = 1.3 models. The experiment indicates that some, but not all, of the structure in the residual signatures in the B_{par} and B_{perp} components is correlated with structure introduced, or removed, by changing the magnetopause shape. Although signatures similar to those observed in the B_{az} component result from flattening the magnetopause shape, their amplitude is less than 10 nT, and these signatures in the residuals are thus of a different origin.
[37] Steady, nondipolar internal field signatures (of either core or crustal origin) should exhibit residual signatures in the MBF frame that are consistent from one sidereal day to the next. The vector residuals for four sidereal days, starting on 24 March 2011 in MBF spherical coordinates (B_{r}, B_{θ}, B_{ϕ}), are shown in Figure 16 for descending (lower altitude) tracks only, for latitudes north of 30°N (i.e., where spacecraft altitudes are below 1000 km). Signatures such as the cusp region, which are well organized in the MSO frame, are seen rotated in the MBF frame as positive signatures in both B_{r} and B_{θ}. These signatures occur at similar MBF longitudes every third sidereal day because of Mercury's 3:2 spin orbit resonance. A positive anomaly in B_{r} at latitudes north of 60°N, and elongated approximately along 45°E longitude, is seen in all four sidereal days. In contrast, large signals (up to 50 nT amplitude) are seen in the B_{θ} and B_{ϕ} components that are not consistent from one sidereal day to the next. These signatures are better organized in the MSO frame, suggesting an external origin. In a separate study, Purucker et al. [2012] address the question of a possible internal origin for the MBF signature in B_{r} has been addressed.
[38] The residuals to our baseline model in both MSO and MBF coordinates often show large orbittoorbit variations. Such variations, as well as changes in the residuals on timescales shorter than MESSENGER's orbital period, likely reflect a combination of processes. These include variations inR_{SS} (driven by short timescale changes in P_{ram}), as well as possible changes in magnetopause flaring (γ), tail flux (F), and tail current sheet parameters D_{D} and R_{2} associated with reconnection. As discussed earlier, we set the reconnection efficiency, k, in the PMM model to zero in our baseline model. However, observations of the magnetic shear across the dayside magnetopause suggest that the average reconnection efficiency may be substantial (Winslow et al., submitted manuscript, 2012), as expected at Mercury [Slavin et al., 2010, 2012]. Residual signatures also vary markedly in their highfrequency content (e.g.,Figure 13). Investigations of higherorder structure in the internal field should be confined to periods when such highfrequency contributions are absent.
7. Conclusions
 Top of page
 Abstract
 1. Introduction
 2. Magnetic Field Observations
 3. Paraboloid Magnetospheric Model
 4. Analyses: Estimation of Model Parameters From Orbital Observations
 5. Global Magnetospheric Fields: Results and Discussion
 6. Residual Signatures: Results and Discussion
 7. Conclusions
 Acknowledgments
 References
 Supporting Information
[39] Vector magnetic field observations acquired by MESSENGER in orbit around Mercury have allowed, for the first time, robust identification of the planet's globalscale magnetospheric fields. We used observations taken over three Mercury years and 4.5 sidereal days, spanning the period 24 March to 12 December 2011. These measurements provide repeated complete coverage over Mercury's northern hemisphere in local time and in bodyfixed longitude, as well as sampling of Mercury's southern magnetotail region. The repeated coverage has permitted investigations of timeaveraged signatures in the globalscale fields and allowed identification of both steady and timevarying signatures in the residuals to the global fields.
[40] We modeled the largescale external fields from magnetopause and magnetotail currents and the internal dipole field with a paraboloid magnetospheric model [Alexeev and Shabansky, 1972; Belenkaya et al., 2005; Alexeev et al., 2008, 2010]. All but one of the model parameters are specified by direct observations of the magnetopause shape, the tail current sheet geometry, the tail flux, and the dipole origin and tilt. For each MESSENGER orbit we identified the locations of the inbound and outbound magnetopause crossings (Winslow et al., submitted manuscript, 2012). We used these crossing positions to determine the average shape of the magnetopause boundary, assumed for the modeling to be a paraboloid of revolution. The magnetopause crossings indicate an average subsolar standoff distance, R_{SS} of 1.45 R_{M}, and, on the dayside and nearplanet nightside, are consistent with a flaring parameterγ = 1. The orientation and origin of the internal dipole field relative to the planetary center were obtained through identification of the magnetic equator on nearly every orbit [Anderson et al., 2011, 2012], excepting only those orbits on which MESSENGER exited the magnetosphere on the dayside before crossing the magnetic equator. The dipole origin is offset northward from the planet center by 0.196 R_{M}, and to a good approximation the dipole is aligned with the rotation axis. Because the dipole offset from the planet center is so large, the equivalent spherical harmonic description of the field in the MBF frame requires not only an axial quadrupole term that is 40% of the axial dipole term at the planetary surface, but also contributions from axial terms of degree 3 and 4 that are 12% and 3% of the axial dipole term, respectively [see Anderson et al., 2012]. Magnetic field strengths observed in the southern tail lobe, together with a Shue et al. [1997] model for the magnetopause shape, indicate a mean flux in one tail lobe of 2.6 MWb, with a standard deviation of ∼25% of this mean value. Crossings of the tail current sheet indicate a mean half width for the current sheet of 0.1 R_{M}in the deep tail. The absence of current sheet crossings in the neartail region suggests that the closest approach of the current sheet to the planet is on average 1.41R_{M} or greater. The best fit dipole moment was obtained via a grid search and found to be 190 nT R_{M}^{3}. Dipole moments within 10 nT R_{M}^{3} of the 190 nT R_{M}^{3} best fit value yield less than a 20% increase in RMS misfit, and these results were insensitive to variations of the other PMM parameters within the ranges from their mean values given in Table 1.
[41] The PMM provides an excellent firstorder fit to the MESSENGER observations, with an RMS misfit of less than 20 nT globally. Residual field strengths are typically less than 50 nT compared with maximum field strengths in the observations of up to 500 nT. The results show that the magnetopause field strength ranges from 10% to 50% of the dipole field strength at observation locations on the dayside and at nightside latitudes north of 60°N.
[42] Residual signatures include timeinvariant contributions from unmodeled magnetopause fields resulting from the assumed paraboloid shape for the magnetopause, which has too great a flaring on the nightside. The subsolar distance varies with an 88 day period due to changes inP_{ram}during Mercury's highly eccentric orbit. Orbittoorbit changes in the magnetopause boundary position were also observed. Interorbit and intraorbit differences in the observed magnetospheric fields are most obvious in the tail region, likely related to tail loading and unloading [Slavin et al., 2010] and reconnection. Establishment of an empirically derived activity index for the MAG data will greatly aid subselection of orbits for specific studies, such as quiet orbits for internal field modeling and orbits with substantial highfrequency variability for studies of reconnection. Examination of the residuals from the PMM in the bodyfixed frame suggests persistent structure in the radial field at high northern latitudes. Globally, however, the residual signatures observed to date are dominated by magnetospheric processes, confirming the dynamic nature of Mercury's magnetosphere.