Oj

terms, are also given in Table H-1. With these coefficients and a definition of the associated Legendre functions, P™, it is possible to calculate the magnetic field at any point in space via Eqs. (H-l) and (H-2).

The coeffients of the IGRF assume that the P™ are Schmidt normalized [Chapman and Bartels, 1940], i.e., fr[/>nm(0)]2sin0d0 = Jo

where the Kronecker delta, 8j =1 if i = j and 0 otherwise. This normalization, which is nearly independent of m, is chosen so that the relative strength of terms of the same degree (n) but different order (m) can be gauged by simply comparing the respective Gaussian coefficients. For Schmidt normalization, the P™ (0) have the form

sinm0

(n - m)(n - m - I )(« - m - 2)(n - m - 3) 2-4(2«— l)(2/i —3)

where (2n — 1)!!s 1 - 3-5 - - • (2«- 1). The square root term in Eq. (H-4) is the only difference between the Schmidt normalization and the common Neumann normalization described in Appendix G. The computation time required for the field models can be significantly reduced by calculating the terms in Eq. (H-4) recursively, i.e., expressing the «th term as a function of the (« — l)th term. The first step is to convert the coefficients in Table H-l from Schmidt to Gauss normalization, which saves about 7% in computation time [Trombka and Cain, 1974]. The Gauss functions, are related to the Schmidt functions, P™, by

0 0

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