Investigation of the Magnetosphere of Ganymede with Galileo's Energetic Particle Detector

Ph.D. dissertation by Shawn M. Stone, University of Kansas, 1999.

Copyright 1999 by Shawn M. Stone.  Used with permission.

4.3 Modeling of External Sources: Magnetopause and Tail Fields

As was discussed in Chapter 1, the solar wind plasma impacts the magnetic field of the Earth, compressing it until a balance in pressure is achieved between the plasma pressure of the solar wind and the magnetic pressure of the Earth’s magnetic field. A magnetopause boundary forms, confining the magnetic field of the Earth from the interplanetary magnetic field. A tail field also results from currents that are induced when the solar wind plasma sweeps down the magnetosphere of the Earth.

The modeling of the Earth’s dayside magnetosphere has been approached in two ways. The first, and least desirable, is the method of an image dipole with variable position and strength to model the magnetopause boundary [Hones, 1963]. The second, more self-consistent, method is the boundary surface model which uses the external expansion in Equation [4.4] to construct a magnetopause field configuration [Mead, 1964; Choe et al., 1973; Choe and Beard,1974a]. The second method has also been applied to the tail field of the Earth [Choe and Beard, 1974b; Beard 1979].

4.3.1 Multipole Model of the Earth's Magnetopause Field

The magnetic field of the Earth is assumed to be the vector sum of Earth’s internal dipole B_d and the external field caused by the surface currents on the magnetopause boundary B_s. The total field is, then, in a system where the Z axis is the geographic north pole of the Earth and the X axis points in the anti-solar wind direction (towards the sun), in spherical coordinates

[4.7]

where λ is the dipole tilt angle [Choe and Beard, 1974a].  

The dipole field expressed in gauss and r measured in Earth radii is

The external field contribution depends on the surface current and the shape of the boundary. The currents on these surfaces are integrated using the Biot-Savart integral to obtain their contribution to the field at r:

[4.9]

Choe and Beard, 1974a], as illustrated in Figure [4.2].

Figure 4.2 Schematic of the geometry required to integrate Equation [4.9].  The magnetic field contribution at r, from the surface currents at ρ, is obtained from the integration of the Biot-Savart integral along the surface S.

The solution to Equation [4.9] is rather computationally complex. It begins by specifying an initial surface according to Mead and Beard [1964], and the field from the currents flowing on this first surface is calculated. This field is then used to improve the first surface configuration, and the process is repeated iteratively. It was found that after 9 iterations there was no more significant improvement in B_s [Choe et al., 1973; Choe and Beard, 1974a]. Since the magnetic field is assumed in this calculation to be irrotational inside the magnetosphere, the surface current field B_s can be expressed in terms of the external scalar potential expansion of Equation [4.4].

Using arguments of symmetry, the sin(mΦ) terms drop out [Choe et al., 1973], and B_s becomes

[4.10]

where i=(1,2,3) corresponding to the three spherical coordinates r, θ, and Φ. The T_n^mi terms depend on component

[4.11]     [4.12]

[4.13]

Equation [4.10] can be expressed in terms of the sub-solar point r_o (or offset distance) which is the point of pressure balance:

[4.14]

where a_j=0 if n+m+j is odd from symmetry of the spherical harmonics. The coefficients K_n^m(λ) are calculated by a least squares fitting routine using 688 magnetic field values measured in the dayside magnetosphere [Choe et al., 1973; Choe and Beard, 1974a]. Table 4.2 shows these values up to n_max=2, but the coefficients up to n_max =7 can be found in Choe and Beard [1975].

Table 4.2 The coefficients a_j in the series expansion of the spherical harmonic coefficients

K_n^m(λ), of the scalar magnetic potential for the magnetopause field up to n_max=2 [Choe and Beard, 1975].

Next: 4.3.2 Multipole Model of the Earth's Magnetotail Field 

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