Field line distribution of mass density at geostationary orbit

The distribution of mass density along the field lines affects the ratios of toroidal (azimuthally oscillating) Alfv\u27{e}n frequencies, and given the ratios of these frequencies we can get information about that distribution. Here we assume the commonly used power law form for the field line distribution, rho_{m} = rho_{m,eq} ( L R_{E} /R )^alpha, where rho_{m,eq} is the value of the mass density rho_{m} at the magnetic equator, L is the L shell, R_{E} is the Earth\u27s radius, R is the geocentric distance to a point on the field line, and alpha is the power law coefficient. Positive values of alpha indicate that rho_{m} increases away from the magnetic equator, zero value indicates that rho_{m} is constant along the magnetic field line, and negative alpha indicates that there is a local peak in rho_{m} at the magnetic equator. Using 12 years of observations of toroidal Alfven frequencies by the Geostationary Operational Environmental Satellites (GOES), we study the typical dependence of inferred values of alpha on the magnetic local time (MLT), the phase of the solar cycle as specified by the F10.7 extreme ultraviolet solar flux, and geomagnetic activity as specified by the auroral electrojet (AE) index. Over the mostly dayside range of the observations, we find that alpha decreases with respect to increasing MLT and F10.7, but increases with respect to increasing AE. We develop a formula that depends on all three parameters, alpha_{3Dmodel} = 2.2 + 1.3 cos(MLT 15 degrees) + 0.0026 {AE} cos((MLT-0.8) 15 degrees) + 2.1 10^{-5} {AE} {F10.7} - 0.010 {F10.7},\r\nthat models the binned values of alpha within a standard deviation of 0.3. While we do not yet have a complete theoretical understanding of why alpha should depend on these parameters in such a way, we do make some observations and speculations about the causes. At least part of the dependence is related to that of rho_{m,eq}; higher alpha, corresponding to steeper variation with respect to MLAT, occurs when rho_{m,eq} is lower