Dipole model of the Earth's magnetic field

Equations

The following equations describe the dipole magnetic field.

First, define B 0 as the mean value of the magnetic field at the magnetic equator on the Earth's surface. Typically B 0 = 3.12 × 10 − 5   T .

Then, the radial and azimuthal fields can be described as

B r = − 2 B 0 ( R E r ) 3 cos ⁡ θ

B θ = − B 0 ( R E r ) 3 sin ⁡ θ

| B | = B 0 ( R E r ) 3 1 + 3 cos 2 ⁡ θ

where R E is the mean radius of the Earth (approximately 6370 km), r is the radial distance from the center of the Earth (using the same units as used for R E ), and θ is the azimuth measured from the north magnetic pole.

It is sometimes more convenient to express the magnetic field in terms of magnetic latitude and distance in earth radii. The magnetic latitude λ is measured northwards from the equator (analogous to geographic latitude) and is related to θ by λ = π / 2 − θ . In this case, the radial and azimuthal components of the magnetic field (the latter still in the θ direction, measured from the axis of the north pole) are given by

B r = − 2 B 0 R 3 sin ⁡ λ

B θ = B 0 R 3 cos ⁡ λ

| B | = B 0 R 3 1 + 3 sin 2 ⁡ λ

where R in this case has units of Earth radii ( R = r / R E ).

Invariant latitude

Invariant latitude is a parameter that describes where a particular magnetic field line touches the surface of the Earth. It is given by

Λ = arccos ⁡ ( 1 / L )

or

L = 1 / cos 2 ⁡ ( Λ )

where Λ is the invariant latitude and L is the L-shell describing the magnetic field line in question.

On the surface of the earth, the invariant latitude ( Λ ) is equal to the magnetic latitude ( λ ).