Vol 2, No. 1, August 2000

*Yu. A. Romashchenko and P. D. Reshetnikov*

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

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).

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.

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** the current density
** j**,
the velocity potential
is similar to the magnetic field potential, and current velocity
function
the magnetic flux), the hydrodynamical
results
are automatically transposed to magnetostatics.

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

(1) |

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

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

(2) |

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

(3) |

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

(4) |

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

(5) |

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

(6) |

(We introduced an inversion point relative the
*d* = *a*^{2}/*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
*Ma*^{3}/*f*^{3} < 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
*M*_{1} described by the
potential
*j*_{0}(*r*,*q*,*w*) outside the sphere at the distance
*f* and some source described by
the potential
*j*_{1}(*r*,*q*,*w*). (As earlier, we use
*fd* = *a*^{2}.) Then from the Veiss theorem,

(7) |

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

(8) |

To find the potential
*j*_{1}, we require that

(9) |

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

(10) |

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

Therefore

(11) |

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

Thus, if a single dipole with the moment
*M*
[*M* = *M*_{1}(*a*/*f*)^{3}] is
placed inside the sphere, then the total potential is equal to

(12) |

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
*q*_{0} and rotates around the vertical
axis with an angular
velocity
*W*.

(13) |

Here
cos *g* = sin *q*
cos *w* and
cos *g*_{1} = cos *q*
cos *q*_{0} + sin *q*
sin *q*_{0} cos (*W**t*
- *w*).
Figure 2
shows
the cross section of the sphere with the locked dipole
calculated by (13) with
2*t* = *p*.
At
*q*_{0} = 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
*B*_{q} 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
*R*_{s} 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.