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) Bz. We represent the magnetopause (for XGSE>−40RE) as an ellipsoid of revolution in solar-wind-aberrated coordinates and express the (p, Bz) dependence of each of the three ellipsoid parameters as a second-order (6-term) bivariate expansion in lnp and Bz. We define 12 overlapping bins in a normalized dimensionless (p, Bz) “control space” and fit an ellipsoid to those magnetopause crossings having (p, Bz) values within each bin. We also calculate the bivariate (lnp, Bz) 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 diagnostics of the magnetopause shape as joint functions of p and Bz: the aspect ratio of the ellipsoid's minor-to-major axes; the flank distance, radius of curvature, and flaring angle (at XGSE = 0); and the subsolar distance and radius of curvature. We confirm and quantify previous results that during periods of southward Bz the subsolar magnetopause moves inward, while at XGSE = 0 the flank magnetopause moves outward and the flaring angle increases. These changes are most pronounced during periods of low pressure, wherein all have a dependence on Bz that is stronger and functionally different for Bz southward as compared to Bz northward (i.e., the behavior of a “half-wave rectifier”). In contrast, all these changes are much less sensitive to IMF Bz at the highest pressures. As an application of these new results, we use a pressure balance relationship to estimate the difference between the magnetic field strength just inside the subsolar magnetopause and that of the dipole field, and we find that this difference decreases rapidly as Bz becomes more negative (although it is relatively insensitive to northward changes in Bz). Quantitative comparison shows that Region 1 Birkeland currents could make the dominant contribution to this depression in the inferred magnetic field at the subsolar point.
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