## Geomagnetic field analysis

### Geomagnetic Coordinates

At low altitude, the Earth's field is approximately that of a magnetic dipole, while at high altitude it is strongly distorted. Models of the Earth's geomagnetic field are required for trapped particle, solar event and cosmic-ray environment modelling. Trapped particle morphologies are described in terms of location in idealised geomagnetic dipole space, while the field is needed in order to include geomagnetic shielding effects for solar particle events and cosmic-ray environments. Field models are also used for magnetospheric physics studies, such as tracing the trajectories of particles through the magnetosphere.

Trapped radiation environment models give energetic particle fluxes as functions of energy and of the geomagnetic coordinates L and B/B_{0}. L is the radial distance of the field line from the axis at the geomagnetic equator in an ideal dipole field, and B is the magnetic field strength, determining the position along a field line from the minimum (B_{0}) at the geomagnetic equator. For many applications the pair B and L are sufficient to define a location in the field due to the azimuthal symmetry. In the geomagnetic field, which is only quasi-dipolar, L is defined by means of a function of the adiabatic integral invariant I. I is a constant on a field line or drift shell, and L is found to be nearly constant along a drift shell.

By transforming orbital locations into the B, L coordinate system and accessing the radiation environment models throughout the orbit, predictions can be made of satellite radiation exposures.

Although to a first approximation the Earth's magnetic field is dipolar, the non-dipolar contributions are important and are best described by numerical models of the field which also account for the offset and tilt of the geomagnetic axis with respect to the Earth's rotation axis. Despite the fact that the field is not a true dipole, the B and L coordinates remain valuable for describing locations within it and their convenience has led to their adoption as the system for radiation environment analyses.

### Geomagnetic Field Models

The numerical models of the geomagnetic field in general use describe the internal field and its secular variations by spherical harmonic expansions of the scalar potential V. Although a number of geomagnetic field models are available, the principle ones are those in the International Geomagnetic Reference Field (IGRF) series. The potential expansion is:

Where a is the radius of a reference sphere and has a value of 6371.2 km for the IGRF models, corresponding to the mean Earth radius. The position of a point of interest is specified with r, theta and phi (the geocentric distance, coelevation and longitude respectively); g(m,n) and h(m,n) are the model coefficients and P(m,n) are Schmitt-normalised Legendre functions.

The IGRF models released since 1960 have had 120 spherical harmonic coefficients (to degree and order 10) and a further 80 (to degree and order 8) describing the secular variations of the corresponding main field coefficients in a linear fashion. The most important harmonic coefficients can be found in the figure below:

The term g(2,0) represents the flattening of Earth, which means that there is an additional ring of mass around the equator. g(3,0) is presenting the "pear" shape of Earth, meaning additional mass at the South pole and the northern mid-latitudes.

### External Field Models

The models described above only describe the field generated by processes within Earth. At high altitudes most of the higher-order terms become negligible and the dipole approximation is often adequate to describe this contribution. However, the solar wind causes large diurnal distortions of the field at high altitude. This effect, together with the ring current from azimuthally drifting particles, and other current systems means that the internal field is a poor representation of the total field. Various models for the external contributions to the field have been developed by: Tsyganenko; Olsen and Pfitzer; and Mead and Fairfield. Since these models include the diurnal asymmetry of the field and depend on geomagnetic activity indices, they are attractive for use in mapping energetic particle fluxes. Models have yet to be produced using them but some studies have been undertaken. The CRRES programme is also planning to use external field models in its analysis of energetic particle data.

Last update: 6 May 2014

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