International Journal of Geomagnetism and Aeronomy
Vol 2, No. 1, August 2000

Simple analytical model of the magnetosphere

Yu. A. Romashchenko and P. D. Reshetnikov

Institute of Space Physics and Aeronomy, Yakutsk, Sakha Republic, Russia



The analytical solution of the problem on an inclined displaced dipole in a closed sphere is presented. The solution obtained may be used as a first approximation for description of the magnetic field in the inner magnetosphere. In this a dipole axis may precess with some angular velocity around an immovable axis. The suggested model describes fairly well analytically the Earth's magnetic field in the cusps and inner magnetosphere.


For solution of theoretical and experimental problems of magnetospheric physics, one should take a magnetic field model of the Earth's magnetosphere. Currently, there exist very reliable empirical magnetospheric models of which we would like to mention the Tsyganenko models [Sergeev and Tsyganenko, 1980; Tsyganenko, 1990]. The only defect of these models is their multiparameterization that is not always convenient in consideration of some theoretical problems. Therefore we suggest a very simple model of the inner magnetosphere based on a solution of the problem of a closed displaced dipole in a spherical cavity. We solved this problem a long time ago [Krymskiy and Romashchenko, 1975], but we believe that this paper is of some interest, again owing to the new attention given to magnetospheric cusp physics. Moreover, the early paper was published in a peripheral journal and has an error (the integration constant was determined incorrectly).

Formulation of the Problem

The problem is to determine the magnetic field of an inclined dipole placed in a sphere with radius a, the sphere center, and dipole not coinciding. The dipole axis may precess with a constant angular velocity W. The magnetic field normal component at the sphere boundary is zero.

Problem Solution

For solving of this internal problem of magnetostatics we use the results of the Veiss theorem for a sphere [Milne-Thomson, 1964]. Since this theorem is seldom used in the physics of magnetic phenomena related to magnetospheric process modeling, we formulate this theorem in full. It is necessary to take into account that the Veiss theorem was formulated in hydrodynamical terms, but since the equations of hydrodynamics and nondissipative electrodynamics (in particular, magnetostatics) coincide apart from constants (the velocity v is similar to the magnetic field B, rot v Rightarrow the current density j, the velocity potential is similar to the magnetic field potential, and current velocity function Rightarrow the magnetic flux), the hydrodynamical results are automatically transposed to magnetostatics.

Veiss Theorem for a Sphere

In a boundless space, let there be an irrotational flow of an ideal incompressible liquid with the velocity potential j(r,q,w) in spherical coordinates j(r,q,w), all singular points of this function being placed at a distance from the origin larger than a. If one places a sphere with r = a into the region of this flow, then the velocity potential that causes the normal component of velocity at the surface of the sphere to vanish may be expressed as


The proof of this theorem was presented by Milne-Thomson [1964], so we do not present it here.

fig01 Use the results of the above mentioned theorem for solution of the problem on a radial dipole placed outside the sphere. Thus let there be a sphere with r = a placed into a spherical coordinate system with the center at the origin. At a distance f from the sphere center (f > a) we place a dipole with the magnetic moment M (Figure 1).

It is known that the dipole potential in "its own" coordinate system is


Being displaced at a distance f from the origin, this dipole will have the potential


We use the fact that r cos q = r' cos q' and g is the angle between the radius vector r and vector OA= f. If the azimuth w is measured from the line OA, then


Using (1), we write (omitting the implicit arguments q, w of j for simplicity)


Substituting (3) into this expression and making a simple integration, we obtain


(We introduced an inversion point relative the d = a2/f sphere.)

The analysis of this formula shows that the sphere influence leads to an appearance within the sphere in the inversion point d of the mapped dipole with the power Ma3/f3 < M and some additional source within the sphere described by the third term.

Finally, we consider our main problem of a dipole inside the sphere placed at the distance d from the sphere center (d < a). Solving the external problem, we saw that the dipole outside the sphere induces within the sphere a mapped dipole and an additional field. In order to compensate this additional field we have to assume that outside the sphere there should exist some additional field source besides the dipole. Thus let there be a dipole with the power M1 described by the potential j0(r,q,w) outside the sphere at the distance f and some source described by the potential j1(r,q,w). (As earlier, we use fd = a2.) Then from the Veiss theorem,

F(r, q, w)= j0(r, q, w)+ j1(r, q, w)


The first integral on the right-hand side can be broken down into


To find the potential j1, we require that


for all values of r>a, meaning that the integrand must equal zero and hence


The integral obtained is indefinite; therefore one can replace R by r.



We chose the constant C(q,w) in such a form that there is no singularity at grightarrow 0.

fig02 Thus, if a single dipole with the moment M [M = M1(a/f)3] is placed inside the sphere, then the total potential is equal to


The solutions obtained are easily generalized for the case when the inner dipole axis is inclined from the normal to the equatorial plane by an angle q0 and rotates around the vertical axis with an angular velocity W.










Here cos g = sin q cos w and cos g1 = cos q cos q0 + sin q sin q0 cos (Wt - w). Figure 2 shows the cross section of the sphere with the locked dipole calculated by (13) with 2t = p. At q0 = 0 (when the inclination is absent), (13) transforms into (12). For the sake of simplicity, we consider again the expression (12). Two free parameters are present there: a and d (or f ).

The parameter d is unambiguously connected with the cusp latitude. This connection is easily obtained if the Bq component of the magnetic field at the boundary of the spherical cover in the cusp point is taken to be zero.

For the central and frontal parts of the Earth's magnetosphere one can use this analytical approximation. In reality, the magnetosphere is aligned in the antisolar direction; therefore for distant magnetospheric regions this model does not work.


In this paper a simple analytical model of the Earth's magnetosphere which may be used for description of the inner magnetosphere and polar cusps is suggested. A formula for the distance Rs from the dipole center to the subsolar critical point is also suggested.


Fairfield, D. H., Average and unusual locations of the Earth's magnetopause and bow shock, J. Geophys. Res., 76, 6700, 1971.

Krymskiy, G. F., and Yu. A. Romashchenko, Magnetohydrodynamical model of magnetosphere, Issled. Geomagn. Aeron. Fiz. Solntsa (in Russian), 36, 174, 1975.

Milne-Thomson, L. M., Theoretical Hydrodynamics, St. Martin's Press, London, 1964.

Pudovkin, M. I., et al., Physical Basis of Magnetospheric Disturbance Prognosis (in Russian), p. 312, Nauka, Leningrad, 1977.

Sergeev, V. A., and N. A. Tsyganenko, The Earth's Magnetosphere, (in Russian) Nauka, Moscow, 1980.

Tsyganenko, N. A., Quantitative models of the magnetospheric magnetic field: Method and results, Space Sci. Rev., 54, 75, 1990.

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