Physical improvements to the solar wind reconnection control function for the Earth's magnetosphere

In a 2008 publication a first principles calculation of the dayside reconnection rate expressed in terms of upstream-solar wind parameters led to a rudimentary solar wind coupling function R-1 for the Earth's magnetosphere. Four improvements to that derivation are added in the present paper, resulting in a more correct solar wind control function describing the rate at which solar wind magnetic field lines connect into the Earth's magnetosphere. The first is the inclusion of the effect of -dependent plasma compressibility on the reconnection rate. The second is a corrected calculation of the inflow of magnetic field lines into the reconnection X-line. The third is a more accurate estimate of the orientation of the reconnection X-line for asymmetric reconnection. The fourth correction accounts for the compression ratio of the Earth's bow shock for arbitrary orientation of the solar wind magnetic field. Two solar wind control functions result: one function, R-2CS, is based on the Cassak-Shay equation and another function, R-2CSB, is based on the Cassak-Shay-Birn equation. The control functions are tested using solar wind measurements and geomagnetic indices from 1980 to 2012, and some improved correlation coefficients over the rudimentary function R-1 are found. Simplified approximations to the new control functions are supplied: one is R2CS-approx=1.68x10(-2) sin(2)(/2)n(o)(1/2)v(o)(2)M(A)(-0.3044)exp(-[M-A/3.18](1/2)), where the subscript o denotes the solar wind upstream of the bow shock and M-A is the solar wind Alfven Mach number.