## 1. Introduction

[2] While the average properties of the equatorial plasma density in the magnetosphere have been at least approximately described [*Carpenter and Anderson*, 1992; *Gallagher et al.*, 2000], the latitudinal density dependence along field lines is less well known. Methods used to infer the latitudinal density dependence along magnetic field lines include in-situ spacecraft observations, passive remote sensing with whistler waves and ultra low frequency (ULF) toroidal frequencies (see references in [*Goldstein et al.*, 2001] and [*Denton et al.*, 2002]). Another recent technique is active remote sensing using radio waves [*Reinisch et al.*, 2001].

[3] Two previous studies have used the Polar spacecraft plasma wave data to infer the typical dependence of electron density *n*_{e} along field lines. *Denton et al.* [2002] used a method which did not initially assume any particular functional form for the parallel dependence, but then found they could fit their results to the power law form

where *R*_{max} ≈ *LR*_{E} is the maximum geocentric radius *R* to any point on the field line. For a dipole magnetic field, this form becomes *n*_{e} = *n*_{e0} (cos(λ))^{−2α}, which is quite similar to the form recently used by *Reinisch* [2002], *n*_{e} = *n*_{e0} (cos((π/2)λ/(0.8 λ_{inv})))^{−β}, where λ_{inv} is the invariant (Earth's surface) value of the latitude λ. No claim is made that (1) is the optimal functional form for describing parallel density dependence, but for a one parameter fit (α), it appears to do quite well [*Denton et al.*, 2002], especially considering the large spread in the parallel dependence of the data. *Denton et al.* [2002] listed α at a number of *R*_{max} values, finding, for instance, α = 0.8 ± 1.2 at *R*_{max} = 4.4*R*_{E} and α = 2.1 ± 1.4 at *R*_{max} = 7*R*_{E}. *Goldstein et al.* [2001], using a method similar to that of the present paper, found an average value of α = 0.37 ± 0.8 for their plasmasphere data (*n*_{e} ≥ 100 cm^{−3}), and an average value of α = 1.7 ± 1.1 for the plasmatrough (*n*_{e} < 100 cm^{−3}; they discarded outlying points before computing the errors).

[4] There are several improvements in this paper. First of all we use a much larger data set than that of *Goldstein et al.* [2001]. Secondly, we have mapped the magnetic field from the spacecraft position at the time of each measurement to determine *R*_{max} more accurately. Third, we model the statistical average of α as a function of the equatorial density *n*_{e0} and *R*_{max} to find α more accurately than before (average error of 0.65) with a single formula for all types of plasma (plasmasphere and plasmatrough). Furthermore, when we model the average α in this way, there is no remaining dependence of the average α on MLT or Kp (these can be considered to affect *n*_{e0}, from which α can be determined).