* 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, *r _{o}*
= 11.0

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 1*a*. 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 *R _{e}*. Figure 1

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 B_{z}.

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 [1961] 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
*R _{e}*. 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,

8. W. P. Olson, The shape of the tilted
magnetopause, *J. Geophys. Res., 74*, 5642-5651, 1969.

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