UNCERTAINTIES IN FIELD-LINE TRACING IN THE MAGNETOSPHERE .1. THE AXISYMMETRICAL PART OF THE INTERNAL GEOMAGNETIC-FIELD

The technique of tracing along magnetic field lines is widely used in magnetospheric physics to provide a ''magnetic frame of reference'' th at facilitates both the planning of experiments and the interpretation of observations. The precision of any such magnetic frame of referenc e depends critically on the accurate representation of the various sou rces of magnetic held in the magnetosphere. In order to consider this important problem systematically, a study is initiated to estimate fir st the uncertainties in magnetic-held-line tracing in the magnetospher e that arise solely from the published (standard) errors in the specif ication of the geomagnetic field of internal origin. Because of the co mplexity in computing these uncertainties for the complete geomagnetic field of internal origin, attention is focused in this preliminary pa per on the uncertainties in magnetic-field-line tracing that result fr om the standard errors in just the axisymmetric part of the internal g eomagnetic held. An exact analytic equation exists for the magnetic fi eld lines of an arbitrary linear combination of axisymmetric multipole s. This equation is used to derive numerical estimates of the uncertai nties in magnetic-field-line tracing that are due to the published sta ndard errors in the axisymmetric spherical harmonic coefficients (i.e. g(n)(0) +/- delta g(n)(0)) Numerical results determined from the anal ytic equation are compared with computational results based on stepwis e numerical integration along magnetic held lines. Excellent agreement is obtained between the analytical and computational methods in the a xisymmetric case, which provides great confidence in the accuracy of t he computer program used for stepwise numerical integration along magn etic held lines. This computer program is then used in the following p aper to estimate the uncertainties in magnetic-held-line tracing in th e magnetosphere that arise from the published standard errors in the f ull set of spherical harmonic coefficients, which de fine the complete (non-axisymmetric) geomagnetic field of internal origin. Numerical es timates of the uncertainties in magnetic-held-line tracing in the magn etosphere, calculated here for the axisymmetric part of the internal g eomagnetic field, should be regarded as ''first approximations'' in th e sense that such estimates are only as accurate as the published stan dard errors in the set of axisymmetric spherical harmonic coefficients . However, all procedures developed in this preliminary paper can be a pplied to the derivation of more realistic estimates of the uncertaint ies in magnetic-held-line tracing in the magnetosphere, following furt her progress in the determination of more accurate standard errors in the spherical harmonic coefficients.