Model magnetosphere of Mercury

218

Physics o/the Earth and Planetary Interiors, 20 (1979) 218-230 © Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

MODEL MAGNETOSPHERE OF MERCURY Y.C. WHANG Catholic University o f America, Washington, DC 20064 (U.S.A.)

(Accepted for publication in revised form March 21, 1979)

Whang, Y.C., 1979. Model magnetosphere of Mercury. Phys. Earth Planet. Inter., 20: 218-230. The paper presents a three-dimensional quantitative model of Mercury's magnetosphere based on the entire combined set of observational data obtained from the first and third encounters of Mariner 10 with Mercury. The model assumes that the surface magnetic field of the planet Mercury consists of a dipole, a quadrupole and an octupole. The dipole moment of Mercury is 2.4 X 1022 G em 3, tilted 2.3° from the normal to the planetary orbital plane and having the same directional sense as that of the Earth. The intensity of the quadrupole moment is approximately 45% of the dipole, and that of the octupole moment 29% of the dipole. The model meets four critical tests: (1) it produces the smallest residuals among all existing models, (2) it can reproduce the crossing of a tail current sheet by Mariner 10, (3) all planetary field lines are confined inside the model magnetosphere, and (4) the size of the model magnetosphere agrees well with the magnetopause crossings directly observed from Mariner 10. The model can also be used to explain two observational features: (1) the plasma characteristics observed in different regions of the magnetosphere, and (2) the regions of quiet and disturbed signatures directly observed from Mariner 10.

1. Mariner 10 observations This paper discusses a three-dimensional quantitative model based on direct observation of Mercury's magnetosphere from Mariner 10. The first encounter of Mariner 10 with the planet Mercury (hereafter referred to as Mercury 1) occurred on March 29, 1974. The magnetic-field measurements of Mercury 1 unexpectedly revealed that the planet Mercury has a modest but significant intrinsic field (Ness et al., 1974). The planetary field is compressed and confined by the dynamic pressure of the solar wind to form a magnetosphere. The third, and final, encounter of Mariner 10 with Mercury (hereafter referred to as Mercury 3) took place on March 16, 1975. Magnetic-field data obtained from Mercury 3 dramatically confirmed the earlier interpretation that there exists an intrinsic planetary magnetic field and yielded further data on the nature of its interaction with the solar wind (Ness et al., 1975a). The magnetic-field experiment on Mariner 10 has collected 17 min of magnetospheric magnetic-field

data from Mercury 1 and 12.5 min of such data from Mercury 3. The heliocentric orbital period of Mariner 10 was 176 days, exactly twice the orbital period of Mercury and three times the rotation period of Mercury. Thus, with reference to a planet-centered Mercury orbit coordinate system (MO coordinates), the contribution of planetary intrinsic field at any given point should remain unchanged at successive encounters, and the combined set of Mercury 1 and Mercury 3 data may be used to study the model magnetosphere of Mercury. The MO coordinates are used here as the frame of reference. The Mercury's orbital plane makes an angle of 7 ° with the solar ecliptic. The x-axis is directed from Mercury to the sun, the z-axis is normal to the orbital plane of Mercury and directed northward, and x y z forms a right-handed coordinate system. The planetary radius (Rm = 2439 km) is used as a unit length. The combined set of 30 min of magnetospheric magnetic-field data is invaluable and unique in studying the planetary magnetic field of Mercury and its interaction with the solar wind. In addition, Mariner 10

219 angle of approximately 69 ° with the normal to the orbital plane near the point of closest approach where the observed field magnitude reached a maximum of 98 3'- Figure 2 presents an enlarged plot of 1.2-s averages of the x-component of magnetic-field data obtained within the magnetopause crossings of Mercury 1. Immediately following the closest approach a very sharp change in Bx (the x-component of the magnetic field) was observed. In about 40 s, Bx changed by approximately 80 % which is a substantial fraction of the maximum field magnitude measured from Mercury 1. We interpret this sudden change in Bx as the crossing of the tail current sheet by Mariner 10. From the trajectory data, we can identify that the spacecraft entered the tail sheet at (x,y,z) = (-1.23, -0.41, 0.00) and exited the sheet at (x,y,z) = (-1.19, -0.56, 0.06). Thus the location of the sheet's inner edge xt > -1.2, the tail current sheet is immediately above the Mercury's orbital plane and the thickness of the sheet is

has observed several important features of Mercury's magnetosphere. They include (1) the tail current sheet, (2) the size of the magnetosphere, (3) plasma characteristics, (4) quiet and disturbed regions of the magnetospheric magnetic field, and others. A successful quantitative model is expected to reproduce or to explain at least some of these features. The trajectory of Mercury 1 was a dark-side pass with a closest approach distance from the surface of 723 km. Figure 1 shows the three vector components of the observed magnetic field from Mercury 1, from which one can identify the crossings of the bow shock, magnetopause, and tail current sheet. Immediately before entry and after exit of the bow shock, the interplanetary magnetic field was observed to have a northward component. The trajectory of Mercury 1 was approximately perpendicular to the Mercury-Sun line on the dark side of the planet. The spacecraft crossed the orbital plane of Mercury from below, making an

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ca. 0.06R m (ca. 150 km). Mercury 3 did not fly close to the tail current sheet. During the Mercury 1 encounter the inbound magnetopause crossing was located at x = - 1 . 7 1 , r = (y2 + z2) ~rz = 2.23; and the outbound crossing at x = - 0 . 7 5 , r = 2.23. The inbound and outbound magnetopause crossings of Mercury 3 occurred at x = - 0 . 9 2 , r = 2.14, and x = 0.02, r = 1 3 0 , respectively. From these directly observed magnetopause crossings we can estimate that the magnetosphere has a radius of approximately 2.2 planetary radii for - 1 . 7 < x < - 0 . 9 . During Mercury 1 and Mercury 3, the solar wind electron spectrometer observed magnetospheric elec-

tron populations with three distinct characteristics: polar low flux electrons, hot electrons in the keV range and cool electrons of energy ca. 1 0 0 - 2 0 0 eV. Before the closest encounter of Mercury 1, the magnetospheric magnetic fields were relatively quiet. As shown in Fig. 1, large variations in the magnetic field vector were observed during the second half of the magnetosphere residence time of Mercury 1. The magnetospheric magnetic fields observed during Mercury 3 were relatively quiet except in the first few minutes of its magnetosphere residence time. The maximum field observed at Mercury 3 encounter was 400 %

221 2. Model study

The model for Mercury's magnetospheric magnetic field by Whang (1977) assumes that '(1) the total magnetospheric magnetic field is the vector sum of an intrinsic planetary field and an external field, (2) the magnetic field is divergence-free and curl-free everywhere inside the magnetosphere except at the tail current sheet, (3) the intrinsic field Bi is symmetrical about an axis passing through the planetary center and represented by the sum of a dipole, a quadrupole and an octupole field, and (4) the major sources of the external field are the field due to currents in the magnetopause Bp, and the field due to currents in the tail sheet Bt. Let the intrinsic magnetic field = -VV

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at a distance x i upstream from the center of the apparent planetary dipole. The image dipole is perpendicular to the Mercury-Sun line in the same plane with the planetary dipole. The presence of an intrinsic quadrupole field (G2 :/= 0) is equivalent to an axial offset of the planetary magnetic center by a distance o f d = 0.5 G2 Rm from the planetary gravitational center.

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Whang (1977) shows that at a point (x,y, z) the magnetic field due to a tail current sheet

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Therefore, the center of the image dipole is also placed at a distance d from the Mercury-Sun line. The two dimensionless parameters Pi and xi control the shape and size of the magnetosphere. We assume that the tail current sheet of Mercury is a sheet of finite thickness and is parallel to the orbital plane of the planet on its dark side, as described by the region

where the integration is to be carried out over the region given in (5) and the two sets of variables of integration ~ , ~ ) and (xe, Ze) are related by x = x e + p cos co and y = Ye + P sin co Equation (6) is analytically integrable and yields a solution in closed form for the tail field. This field remains curl-free everywhere except at the tail current sheet of f'mite thickness. The magnetic field inside Mercury's magnetosphere and its magnetic tail region is expressed as the vector sum of a planetary intrinsic field Bi, a magnetopause field Bp, and a tail field Bt B = B i + Bp + Bt

Our study used the entire combined set of 6-s average data measured inside the planetary magnetosphere during Mercury 1 and Mercury 3. The Mercury 1 magnetosphere magnetic field data set consists of 170 points, the Mercury 3 data set consists of 125 points. Let Bobs denote the 6-s average of the observational data. We can write

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222 ing value of o. A minimum of o is reached when G2 = 0.45 and Ga = 0.29. We use this combination to represent the best-fit solution of the model magnetosphere. The values of the two parameters representing the tail field for the best-fit solution are xt = - 1 . 1 0 and F t = 38.9 7. The values of the three parameters representing the planetary dipole field (in units of 3') are

The values of ol and oa are functions of nine independent parameters: g], hi, gO, G2, G3, I'i, xi, xt, and F t . The first five parameters are related to the intrinsic field, and the other four unknown parameters are used to describe the external source field. The shape and size of the magnetosphere are dominated by the relative strength of the magnetopause field which is characterized by the two dimensionless parameters Iai and xi. Very reasonable topological features of Mercury's magnetosphere are obtained when the magnetopause field is represented by an image dipole placed at a distance of approximately 7.4 planetary radii upstream of the planetary dipole and having a moment of approximately 160 times the dipole moment. For any given combination of G2 and G3 we can determine the value o f g L h~, gO, Ft ' and x t using the method of least squares and calculate the correspond40

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g~ = 0.0, h~ = - 6 . 8 , g ° = -166.3 The dipole moment of Mercury, g l, deduced from this model study is 166.4 3'R~, or 2.41 X 1022 G cm 3, tilted 2.3 ° from the normal to the planetary orbital plane and having the same directional sense as that of the Earth. The intensity of the q u a d r u p l e moment is approximately 45% of gl and that of the octupole mo T ment 29% o f g l . The presence of a quadrupole moment indicates an

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axial offset of the apparent dipole center by a distance, d ~ 0.5 X G2 X planetary radius, northward from the planetary gravitational center. The presence of an octupole moment indicates that the average radius of the source current system inside the planet is a significant fraction of the planetary radius. The planet Mercury is believed to have a large dense solid core with a radius of approximately 3/4 of the planetary radius. A large octupole moment for the surface field of Mercury is consistent with our present understanding of the internal structural model of Mercury (Whang, 1977; Elphic and Russell, 1978).

The least-squares fit of the model magnetosphere yields a residual of 15.1 7 for the entire set of Mercury 1 data. This residual amounts to 15% of the maximum field at Mercury 1 encounter. For the entire set of Mercury 3 data the residual is 24 % which is 6% of the maximum field at Mercury 3 encounter. The average ' deviation is 20 7 for the entire combined set of 295 data points. A comparison of the observation data with the magnetic field calculated from the model magnetosphere is shown in Figs. 3 and 4 in which the solid lines represent the model field.

224 3. Estimations of the planetary magnetic field

Our model magnetosphere attempts to determine a magnetic-field representation valid throughout the entire magnetosphere of Mercury based on the entire combined set of Mercury 1 and Mercury 3 data. Because the planet Mercury occupies a very large fraction of its magnetosphere, when measurements are performed relatively close to the surface of Mercury, the total magnetic field includes a substantial contribution attributable to external field sources. A proper representation for the external field becomes very important in the analysis and interpretation of the magnetic field data from Mariner 10. In the first phase of data analysis, the external fields were represented by harmonics of order 1 or 2 and the studies were based on some selected quiet subsets of magnetometer data. The dipole moment of Mercury was first estimated by Ness et al. (1975b) to be approximately 5.1 X 1022 G cm a using the leastsquares fit of a spherical harmonic model to a subset of quiet period of Mercury 1 data which covers 40% of the magnetosphere residence time of Mercury 1 . Using the same model, an estimated dipole moment of 4.7 X 1022 G cm 3 was obtained based on a subset of Mercury 3 data which comprised 56% of the data set obtained during the magnetosphere residence time of Mercury 3 (Ness et al., 1976). The simple external field used in these analyses cannot produce a magnetospheric magnetic field to satisfy the boundary condition that all planetary magnetic field lines are confined in a magnetosphere-like region. This boundary condition is satisfied in phase two analyses of Mercury's magnetic field. However, the results of phase two study have substantially revised the phase one estimate of Mercury's dipole moment. Prior to Mercury 3 encounter, the entire set of

Mercury 1 data was used by Whang and Ness (1975) to study the model magnetosphere of Mercury using an image dipole representation for the external field. The inclusion of a magnetosphere-like boundary revised the estimated planetary dipole moment of phase one analyses by a factor of 2. After Mercury 3 encounter, further studies of Mercury's model magnetosphere using the entire combined set of Mercury 1 and 3 data revealed that the planetary field is very much distorted from a simple centered dipole; the planetary quadrupole and octupole are quite significant in magnitude compared with its dipole moment (Whang, 1977). A summary of results obtained from these two image dipole models is given in Table I. When the entire combined set of data from Mercury 1 and 3 were used, the inclusion of a quadrupole and an octupole is very effective in reducing the model residuals. The residual from the multipole model amounts to 50% of that from a dipole model. A spherical harmonic representation for the external field has also been used to study the model magnetosphere of Mercury to include magnetosphere-like boundaries (Jackson and Beard, 1977; Ng and Beard, 1979). Table II summarizes various phase two models based on the entire combined set of Mercury 1 and 3 data, and their calculated deviations between model field and observed field. These models are not capable of further reducing the residuals to below 20 3'. All results of phase two support the findings of Whang's model that the dipole moment of Mercury is approximately 2.4 X 1022 G cm 3. It should be noted that the reduction of residuals is an essential but not the only criterion from which a comparison of various models should be made. A fair comparison must include many other qualities of the various models. The next section will discuss some fine qualities of whang's three-dimensional quantita-

TABLE I Planetary magnetic field calculated from image dipole models Model reference

Whang and Ness (1975)

whang (1977)

Planetary field Data base

Centered dipole Mercury 1

Dipole, quadrupole, and octupole

Dipole moment al, residuals of Mercury I

2.3 X 1022 G cma

2.4 X 1022 Gcm 3

16.5 3"

15.1 3'

Mercury 1 and 3

225 TABLE II Models based on all Mercury 1 and 3 data Model reference

Whang (1977)

Jackson and Beard (1977)

Ng and Beard (1979)

Dipole moment

2.4 X 1022 Gcm 3

2.44 X 1022 Gcm 3

Quadrupole/dipole Octupole/dipole Deviations:

0.45 0.29

0.66 0

2.37 × 1022 G em 3 (offset = 0.24R m) 0

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tive model in the light of many main features of the magnetosphere of Mercury observed from Mariner 10.

4. Model magnetosphere

As shown in Fig. 3, near the location o f tail crossing the model magnetosphere agrees very well with observations in both the location and the thickness of the tail current sheet. The presence of a non-zero quadrupole means that the center of the apparent planetary dipole is offset approximately 0.2 planetary radius northward from the planetary center. This causes the asymmetry of the planetary intrinsic field and the magnetosphere magnetic field about the orbital plane of Mercury. On the midplane of the tail current sheet, Bx is negative instead of zero. This means the midplane of the current sheet is located south of the planetary magnetospheric equator. Figure 5 shows the field lines and the magnetopause of the model magnetosphere in the n o o n - m i d night meridian plane. Field lines are tangential to a magnetopause-like surface at the boundary. The size of the model magnetosphere agrees well with the magnetopause crossings directly observed from Mariner 10. On the surface of the planet, each magnetic-field line may be identified by two surface variables: the local time and the planetary latitude. Since all field lines inside the model magnetosphere originate at the planetary surface, all field lines on a cross-section of the tail may be identified with their surface variables. Figure 6 shows that the cross-sections o f the magneto-

spheric tail on planes normal to the Mercury-Sun line are nearly circular; it also shows the identification o f all field lines. Figure 7 shows the field isointensity con. tours on a cross-section o f the tail calculated from the model magnetosphere. Since the field intensity varies by a factor greater than two over the cross-section, a quantitative model of the magnetosphere can provide new information in studying the dynamical process of the tail plasma. Figure 8 shows the field isointensity contours in

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the noon-midnight meridian plane of the model magnetosphere. The solar wind stagnation point was located at an altitude of 0.32 planetary radius above the planetary surface, and the field magnitude at the stagnation point of the model magnetosphere is 160 7. The standoff distance to the subsolar point of the magnetopause was not directly observed. However, from the solar wind velocity and density measured by the plasma science experiment (Ogilvie et al., 1974) we can estimate the magnetic field at the stagnation point to be ca. 160 7Since all field lines on the planetary surface may be identified by the planetary latitude and the local time, the model magnetosphere can identify the region of the magnetosphere where all field lines are connected with the tail current sheet, and the region of closed field lines as shown in Fig. 9. The surface area connected with the tail current sheet is larger in the southern hemisphere than in the northern hemisphere. Those field lines crossed by the trajectories of Mercury 1 and 3 are also shown in Fig. 9, from which

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we can see that Mariner 10 did not make any observation in the region of closed field lines inside Mercury's magnetosphere. The magnetospheric plasma should exhibit different characteristics in different regions of the model magnetosphere. Plasma experiments of Ogilvie et al. (1977) have observed various electron populations from Mariner 10. A polar, low-flux region was observed along the field lines connected with the north pole. Hot electrons in the keV range were observed along the field lines connected with the inner edge of the tail current sheet. Cool electrons of energy ca. 100-200 eV were observed in the remainder of Mercury 1 and Mercury 3 trajectories inside the magnetosphere. In addition, intense bursts of high-energy charged particles, identified as A, B, or C events (Simpson et al., 1974) were observed along some field lines connected with the inner edge of the tail current sheet. In Mercury's magnetosphere, the external field represents a significant fraction of the total field. In Fig. 10 are plotted the constant contour lines for the magnitude of external field on the two trajectory

planes of Mariner 10. The trajectory of Mercury 1 was approximately perpendicular to the Mercury-Sun line on the dark side of the planet. The spacecraft crossed the tail current sheet from below. During the first half of magnetosphere residence time of Mercury 1, the magnitude of the observed field increased from 40 to 98 7- The external field, which has a magnitude of about 30 ~,, represents a significant fraction of the observed field. Therefore, a proper representation for the external field becomes very important in data inter. pretation. Transmission of unsteady-state features of the solar wind into the magnetosphere is modulated by the gross size of a magnetosphere. Owing to the small size of Mercury's magnetosphere, the magnetopause and the tail sheet of Mercury's magnetosphere respond to unsteady-state solar wind conditions with very little damping. Since the current sources for the external field are distributed in the magnetopause and the tail current sheet, the location and strength of the surface of the two sources vary all the time, responding to conditions of the solar wind and the magnetosheath.

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Thus, unlike the planetary intrinsic field, the external field is not expected to be a steady-state field. We suggest that this temporal variation is also responsible for the fluctuation of the external field in the magnetosphere, particularly in the region where the gradient of the external field is large. The gradient of the external field BE = elB1 + e2B2 + e3B3

is a dyadic quantity and consists of nine components

aB/ (i,/= axi

fluctuations in magnetic field observed on Mercury 1 and 3. This means that when the spacecraft was in the region with large gradient of external field, the motion of current-source surface causes large fluctuations in the observed magnetospheric magnetic field. This offers a new possible explanation for the observed large variations in magnetic field of Mercury 1 which was interpreted by Siscoe et al. (1975) as temporal events resembling substorm phenomena in the Earth's magnetosphere.

1,3)

Its magnitude may be represented by

Making use of the model magnetosphere, the magnitude for the gradient of the external field is calculated on the trajectory plane of Mercury 1 and Mercury 3 as shown in Fig. 11. The regions with large external field gradient are indicated by shaded areas. The area of large gradient correlates positively with the large

Acknowledgements

The author wishes to thank N.F. Ness, R.P. Lepping, and K.W. Behannon for their assistance throughout the course of this research. The work was supported by NASA grant NGR 09-005-063. References Elphic, R.C. and Russel, C.T., 1978. On the apparent source

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