Originally Published In: Advances in Space Research, 18, (8)207-(8)212, 1996.
Copyright © 1995 COSPAR
C. T. Russell
Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA
Magnetopause crossings by the ISEE spacecraft have been mapped to the subsolar region along conic sections derived from average fits to northward and southward IMF crossing sets. These mapped positions have been normalized by the solar wind pressure, which is derived from observations by ISEE-3 and IMP-8. Some of the scatter in the extrapolated positions is due to the assumption that a simple conic section can be used to describe the shape of the dayside magnetopause. The existence of the cusps causes deviations from this simple shape in the meridian plane. Crossings of the dayside magnetopause by the ISEE spacecraft when the interplanetary magnetic field is northward are used to examine the influence of the angle of tilt of the Earth's dipole on the magnetopause shape, and are compared with a numerical magnetopause model.
The shape of the magnetopause and size of the magnetosphere have been topics of interest for several decades. Research in this area is essential for understanding the interaction between the solar wind and the magnetosphere. The first suggestion of the existence of a boundary between the interplanetary environment and the near-Earth region was put forth by Chapman and Ferraro /1/. Since that time, more rigorous treatments of the magnetopause shape have been performed. Explicit two- dimensional solutions were later derived /2, 3/. More detailed computational models of the magnetopause were soon determined by others /4, 5, 6, 7, 8, 9/. All of these models begin with the basic assumption that the component of solar wind dynamic pressure normal to the boundary is balanced by the magnetospheric magnetic field at the inner edge of the magnetopause (although the exact nature of the interaction (elastic or inelastic), which scales the size of the magnetosphere, was not properly understood until the work of Spreiter et al. /10/). The superconducting property of the magnetospheric boundary and the nature of the dipole field combine to produce magnetic nulls, portrayed in these computational models as indentations of the magnetopause surface.
The first empirical derivation of an average shape and size of the magnetopause was undertaken in 1971 /11/. In that study, crossing positions of the magnetopause by several different spacecraft were assembled, and a fit to these crossings was performed. The magnetopause was assumed to be well approximated by a conic section with five free parameters, and was assumed to be cylindrically symmetric about the average solar wind flow direction. From the fit parameters, it was found that the magnetopause shape could be adequately represented by an ellipse with one focus positioned at the center of the Earth ( = 0.4, ro = 11.0 Re). Fits were also performed for subsets of northward and southward IMF crossings (ro = 11.6±2.0 Re and 10.5±1.3 Re, respectively). Similar empirical fits to magnetopause crossings have been estimated by other authors, using additional or other magnetopause crossing sets from later missions /12, 13, 14, 15, 16/. The magnetopause shapes derived in these studies are also fits to conic sections, though the number of free parameters varies between two and nine, depending on the particular study. However, evidence for the indentation of the magnetopause at the cusps has been noted /17/, though the number of crossings is very small.
In the work of Petrinec et al. /15/, average ellipsoidal dayside magnetopause shapes were found from traversals of the magnetopause by the ISEE spacecraft, for strongly northward and southward IMF conditions. Crossings of the magnetopause during periods of northward IMF are mapped to the subsolar region along an ellipsoid of eccentricity 0.42. The variation with the total pressure of the solar wind is displayed in Figure 1a. The dashed line represents the theoretical variation, while the solid line is a linear regression to the data. The close agreement between theory and observation indicates that the size of the magnetosphere responds to the solar wind pressure as expected. However, a significant amount of scatter of mapped positions appears about the regression line. When the IMF is southward, one expects that time integrated effects of magnetospheric reconnection at the dayside magnetopause may increase the level of scatter of the mapped crossing positions. The magnetopause has also been observed to oscillate about its equilibrium, and the level of oscillation is greater for southward than for northward IMF conditions /18/. Even for the northward IMF crossings which have been normalized to the solar wind pressure, however, the standard deviation of the mapped stand-off positions is found to be 0.81 Re. Figure 1b illustrates fits of magnetopause crossings for both northward and southward IMF, normalized by the solar wind pressure. The data has been binned according to the value of IMF Bz, and linear regressions performed for northward and southward IMF subsets separately, with the constraint that the two curves meet at IMF Bz = 0 nT. The small slope obtained for northward IMF indicates that the size of the magnetosphere is independent of the value of northward IMF Bz.
In this study we consider in greater detail the ellipsoidal approximation to the shape of the magnetopause, and its contribution to the scatter of stand-off positions about the median value. An ellipsoid of revolution may be adequate in the equatorial plane, but as is evident from theoretical models of the magnetopause shape, the existence of the cusps can cause significant deviations from this simple shape. We can examine the magnitude of this effect with actual crossing positions from ISEE- 1 and -2. Although the orbit of these spacecraft was mainly in the equatorial plane, they were occasionally able to reach high enough magnetic latitudes to be affected by the cusp regions.
Fig. 1. Influence of solar wind parameters on the position of the magnetopause. a) Northward IMF crossings, as a function of solar wind pressure. b) Binned medians of normalized crossings, as a function of IMF Bz.
Fig. 2. Normalized magnetopause crossings taken from the northward IMF data set and mapped into the meridian plane. a) Positive values of dipole tilt. b) Negative values of dipole tilt. The solid line is the average northward IMF fit, and the dashed line is the shape from the Spreiter and Briggs model, for a dipole tilt angle of ±10 degrees, for panels a and b respectively.
Fig. 3. The normalized stand-off distance from the crossings displayed in Figure 2, versus the sum of magnetic latitude and the dipole tilt angle. The solid line represents a polynomial regression to the median values (solid squares), which have been folded about the y=0 axis, and the dashed line is calculated from the Spreiter and Briggs  model.
To examine the effect of the magnetic cusps on the shape of the magnetopause, we only examine those crossings which occur for northward IMF and have been normalized by the solar wind pressure. Since the shape of the magnetopause is most affected by the cusp regions in the noon-midnight meridian plane, our set is further restricted such that 5 Re. These points are separated according to the sign of the dipole tilt angle, and mapped into the meridian plane along the average shape, keeping the latitude of the crossing constant . The crossing positions are displayed in Figure 2, with the solid curves representing the average fit for all strongly northward IMF crossings ( = 0.42, ro = 10.3 Re) /15/, while the dashed line represents the shape determined by a theoretical model /4/ for dipole tilt angles of ±10 degrees dipole tilt, respectively. The effect of the cusp region on the magnetopause shape is tested by examining the sum of the magnetic latitude ( = Arcsin(z/r)) and the dipole tilt angle () for each crossing, for each normalized stand-off distance, ro. For those points closest to one of the cusp regions, the magnitude of + will be largest, and values near zero indicate that the crossing position is far from either cusp. Figure 3 shows the result of this calculation. The average stand-off distance is largest for those points far from either cusp ( + near zero), and decreases as this value increases in magnitude. Two extrapolated normalized crossings with angles near 40 degrees are believed to have crossed the magnetopause through the cusp region, because of the very small values obtained for the normalized stand-off distance. Median values of ro are calculated (in 10 degree bins - represented by solid squares), after folding the data set about the + = 0 degree axis. A parabolic fit (solid line) to these points is calculated. In addition, we have used a theoretical model /4/ to calculate stand-off positions as a function of magnetic latitude, performing the same ellipsoidal model mapping that was performed on the actual crossings (dashed line). The decrease in the normalized stand-off position with increasing magnitude of magnetic dipole latitude is found to be identical to the fit for the actual crossings. This result confirms our expectation that the cusps of the magnetosphere can still on occasion strongly affect where the magnetopause is observed and how crossing positions are mapped, even for a near equatorial orbiting spacecraft such as ISEE.
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18. P. Song, R. C. Elphic, and C. T. Russell,
ISEE 1 & 2 observations of the oscillating magnetopause,
Geophys. Res. Letts., 15, 744-747, 1988.
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18. P. Song, R. C. Elphic, and C. T. Russell, ISEE 1 & 2 observations of the oscillating magnetopause, Geophys. Res. Letts., 15, 744-747, 1988.