Magnetopause shape as a bivariate function of interplanetary magnetic field B(sub z) and solar wind dynamic pressure

We present a new method for determining the shape of the magnetopause as a bivariate function of the hourly averaged solar wind dynamic pressure (p) and the north-south component of the interplanetary magnetic field (IMF) B(sub z). We represent the magnetopause (for X(sub GSE) greater than -40 R(sub E)) as an ellipsoid of revolution in solar-wind-aberrated coordinates and express the (p, B(sub z)) dependence of each of the three ellipsoid parameters as a second-order (6-term) bivariate expansion in Inp and B(sub z). We define 12 overlapping bins in a normalized dimensionless (p, B(sub z)) `control space' and fit an ellipsoid to those magnetopause crossings having (p, B(sub z)) values within each bin. We also calculate the bivariate (Inp, B(sub z)) moments to second order over each bin in control space. We can then calculate the six control-space expansion coefficients for each of the three ellipsoid parameters in configuration space. From these coefficients we can derive useful diagnosis of the magnetopause shape as joint functions of p and B(sub z): the aspect ratio of the ellipsoid's minor-to-major axes; the flank distance, radius of curvature, and flaring angle (at X(sub GSE) = 0); and the subsolar distance and radius of curvature. We confirm and quantify previous results that during periods of southward B(sub z) the subsolar magnetopause moves inward, while at X(sub GSE) = 0 the flank magnetopause moves outward and the flaring angle increases.