Observations of the electron density ne based on measurement of the upper hybrid resonance frequency by the Polar spacecraft Plasma Wave Instrument (PWI) are available for March, 1996 to September, 1997, during which time the Polar orbit sampled all MLT values three times. Using this data set, we assume a power law form for the electron density dependence along field lines ne = ne0 (Rmax/R)α, where ne0 is the equatorial electron density and Rmax ≈ LRE is the maximum geocentric radius R to any point on the field line, and model the statistical average of α as αmodel = 8.0 − 3.0 log10ne0 + 0.28 (log10ne0)2 − 0.43(Rmax/RE) for all categories of plasma (plasmasphere and plasmatrough), with an average error of 0.65. The data set on which this result is based is limited to 2.5RE ≤ Rmax ≤ 8.5RE, 2RE ≤ R ≤ Rmax, and 2 ≤ ne0 ≤ 1500 cm−3. There is no remaining dependence of the average α – αmodel on MLT or Kp.
 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].
 Two previous studies have used the Polar spacecraft plasma wave data to infer the typical dependence of electron density ne along field lines. Denton et al.  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 Rmax ≈ LRE is the maximum geocentric radius R to any point on the field line. For a dipole magnetic field, this form becomes ne = ne0 (cos(λ))−2α, which is quite similar to the form recently used by Reinisch , ne = ne0 (cos((π/2)λ/(0.8 λinv)))−β, where λinv 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.  listed α at a number of Rmax values, finding, for instance, α = 0.8 ± 1.2 at Rmax = 4.4RE and α = 2.1 ± 1.4 at Rmax = 7RE. Goldstein et al. , using a method similar to that of the present paper, found an average value of α = 0.37 ± 0.8 for their plasmasphere data (ne ≥ 100 cm−3), and an average value of α = 1.7 ± 1.1 for the plasmatrough (ne < 100 cm−3; they discarded outlying points before computing the errors).
 There are several improvements in this paper. First of all we use a much larger data set than that of Goldstein et al. . Secondly, we have mapped the magnetic field from the spacecraft position at the time of each measurement to determine Rmax more accurately. Third, we model the statistical average of α as a function of the equatorial density ne0 and Rmax 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 ne0, from which α can be determined).
2. Example of Density Data
 The electron density values used in this paper are obtained using the Polar Plasma Wave Instrument (PWI) [Gurnett et al., 1995]. The electron number density can be determined from noise emission which has an upper edge in frequency at the upper hybrid resonance (UHR) frequency [Goldstein et al., 2001]. For each data point, a field line mapping program was used to map the spacecraft location to the position along the field line with maximum radius, Rmax (= LRE for a dipole field). A Tsyganenko magnetic field model was used [Tsyganenko, 1995] as described by Denton et al. .
Figure 1a shows the electron density ne as determined by the PWI on February 3, 1997, 1322–1453 UT. Due to the nature of the Polar orbit [Goldstein et al., 2001], the trajectory of the spacecraft crosses Rmax values at two different radii (Figure 1b). In Figure 1a, ne is plotted versus Rmax for the outer (thin curve) and inner (bold curve) portions of the orbit. The PWI data points were extracted by hand; the highest possible resolution was about 2.4 s, but the actual temporal separation of data points may range from this value up to a couple of minutes. The inner curve in Figure 1a has 31 data points, and a typical Rmax separation between these points is 0.1. While it is clear that there is some azimuthal or temporal dependence (the plasmapause position is not exactly the same for the large and small radius portions of the orbit), in this particular case that dependence appears to be small. Furthermore, the smooth variation of density on both portions of the orbit gives evidence that in this case there is not a great amount of structure in the azimuthal direction.
 Now we determine the parallel dependence of density for this data (α values) at a number of Rmax values spaced approximately 0.1 apart and only within the range represented by the data on both inner and outer radius segments (here, Rmax = 3.0 − 6.0 RE). At each of these data points, we interpolate the ne and R values from each of the two segments, then calculate α at these data points using (from (1)) α = log(ne2/ne1)/log(R1/R2), where the subscripts 1 and 2 indicate data on the inner and outer segments of the Polar trajectory. The α values are plotted in Figure 1c. For our statistical survey, we will exclude data with sharp gradients such as the region around the plasmapause in Figure 1, since a slight error in Rmax value for either the inner or outer radius portion of the orbit would result in a large error in α. We also discard from the survey data for which the large and small radius values differed by less than 5% (at R1 = R2, our formula blows up). Finally, we also exclude data with a large amount of irregular variation, which can result from azimuthal structure. Undoubtably there is still some error due to time dependence and azimuthal structure, but our selection procedure should reduce this considerably, and we expect that the statistical results will yield averages which are more reliable than any one particular measurement. Once we know α, we can also solve for the equatorial electron density ne0 using ne0 = ne1 (R1/Rmax)α.
 Note that the value of α is small within the plasmasphere where the density on the inner and outer radius portions of the orbit are nearly the same. A value of α = 0 corresponds to a flat, or constant, density dependence (Equation (1)). For this particular event, α is also small just outside the plasmapause, but it increases farther out in the plasmatrough where the density on the inner and outer radius portions of the orbit differ. While the low α values within the plasmasphere are consistent with diffusive equlibrium (which leads roughly to ne ∝ 0–1), the larger α values characteristic of the plasmatrough are more consistent with a collisionless model (roughly ne ∝ R−4) or to an intermediate dependence [Lemaire and Gringauz, 1998].
3. Statistical Study
 We now make use of a large set of the Polar PWI data in order to characterize the typical dependence of α on various parameters. For each Polar orbit used in the survey, we divide the density data into as many as three categories. If there is a plasmapause (drop in ne of at least a factor of 3 within ΔRmax = 0.4), we categorized the density data inside (outside) the plasmapause as plasmasphere (plasmatrough) data. If the orbit had only ne values >300 cm−3, we categorized it as plasmasphere data, while if the orbit had only ne < 30cm −3 with some values decreasing down to ∼10 cm−3, we categorized it as plasmatrough data. Sometimes ne decreased gradually with respect to Rmax from values >300 cm−3 to values ∼10 cm−3 with no clear plasmapause, and we counted this data as a third “gradually decreasing” (no plasmapause) category.
 The PWI data set has been described by Denton et al. . It spans the parameters 2.5 RE ≤ Rmax ≤ 8.5 RE, 2 RE ≤ R ≤ Rmax, and 2 ≤ ne0 ≤ 1500 cm−3. Because of its polar orbit (sampling two opposite MLT values in any single orbit), the Polar spacecraft sampled all MLT values in 1/2 year (owing to the Earth's revolution). Since data in the plasmasphere category were so abundant, we used data in that category only in the first 1/2 year of PWI operation from March 26, 1996 to September 26, 1996. However, for the other two categories we used data from the entire time period for which the PWI was in operation, from March 26, 1996, to September 16, 1997 (sampling all MLT values approximately three times).
Figure 2 shows the α values as a function of ne0 for the three categories of plasma data. It is apparent from this figure that the α values are ordered with respect to ne0 in such a way as is roughly independent of their plasma category (the same polynomial fit is plotted in each panel of Figure 2). This suggests that we might model α as a function of ne0.
 A better fit can be found, however, by using a functional dependence on ne0 and Rmax. By minimizing the standard error, we have fit the α values in the entire data set (all three categories) to the following form
Figure 3 shows a plot of α – versus ne0 (α value with the Rmax dependence subtracted out) versus ne0 for all three categories of plasma data combined. The solid curve is which does a good job of fitting the ne0-dependent part of the α dependence. Similarly, Figure 4 shows that does a good job of fitting the Rmax-dependent part of the α dependence.
 The range of α values found in the magnetosphere will not be as large as Figure 3 by itself would seem to indicate because large (small) ne0 tends to occur at small (large) Rmax, at which will lead to increased (decreased) α. Using Equation (4), we find that the average error between the α values and αmodel, .
 We also examined the α dependence as a function of MLT and the averaged Kp, 〈Kp〉 (averaged over the preceding time t using the weighting factor exp(−t/t0), where t0 = 1.5 days). There is no dependence of the average α – αmodel on MLT or 〈Kp〉 (not shown). This is despite the fact that there is a significant dependence of α on MLT and 〈Kp〉 (as shown by Denton et al. ). Thus the MLT and Kp α dependence can be accounted for by changes in ne0.
 Using the Polar PWI data set, which spans the parameters 2.5 RE ≤ Rmax ≤ 8.5 RE, 2RE ≤ R ≤ Rmax, and 2 ≤ ne0 ≤ 1500 cm−3, we have found that the statistical average parallel dependence for the electron density ne can be described by a function of equatorial density ne0 and Rmax(4), resulting in an average error for α of 0.65. Variations of the average α with respect to MLT or Kp [Denton et al., 2002] can be attributed to changes in ne0. It must be stressed that our results describe only the average field line dependence; there will surely be deviations from the average behavior, some of which will be related to time-dependent effects.
 One benefit to having a model for α which depends on ne0 is that the equatorial density is relatively well known, so that a magnetospheric density model can easilly be constructed using our (4). One disadvantage to using a functional dependence on ne0 is that the calculated values of α are derived from the two density values (one of which is at high altitude), so α is not entirely independent of ne0. For instance, random errors (or mismatches) in the low and high radius density values could lead to a correlation between large α and small ne0. This problem is ameliorated, however, by the fact the Rmax dependence of ne on the low radius portion of the Polar orbit is usually small where α is large (Figure 1). Because of this, there will not generally be a great error choosing a density measurement on the low radius portion of the orbit to match one on the higher portion.
 We are grateful to Iowa PWI team, including D. Gurnett and J. Pickett for supplying the upper hybrid resonance data (supported by NASA contract NAS5-30371 and grant NAG5-11942), the Polar spacecraft science data team for supplying spacecraft ephemeris data, and NSSDC for making the IMP-8 (GSFC and LANL) and WIND (courtesy of R. P. Lepping and K. W. Ogilvie) solar wind data, Dst (Kyoto University), and Kp (NOAA) data available on OMNIWeb. We thank Dennis Gallagher for providing averaged Kp data. Work at Dartmouth was supported by NASA grants NAG5-11712 and NSF grant ATM-9911975. Work at the University of Iowa was supported by NASA grants NAG5-11942 and NAG5-9561.