J. Geophys. Res., 90, 9650-9662, 1985
(Received January 14, 1985; revised June 11, 1985; accepted June 12, 1985)
Copyright 1985 by the American Geophysical Union
Paper number 5A8480
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There are three main features to be determine in the auroral zone model. These are the total number density as a function of altitude, the choice of electron populations and the velocity-space distribution function for each, and the relative abundance of each component as a function of altitude. While measurements exist regarding the first two questions, there are no direct observations of the low-energy electron population on auroral field lines, and thus the relative abundances are not well determined.
We shall use the results of Calvert  for the total number density as a function of altitude along an auroral zone field line. In his work, Calvert deduced the presence of an auroral density-cavity using wave data from the Hawkeye spacecraft. This density depletion produces a minimum in the plasma frequency/gyrofrequency ratio (/ ) at a geocentric distance of 1.8 R. The variation of / with altitude is shown in Figure 1 [after Strangeway, 1985]. In addition, we have shown the variation of the gyrofrequency along the field line as a function of altitude, assuming a dipole field with an invariant latitude of 70°.
Fig. 1. Variation of the electron gyrofrequency , the plasma frequency/gyrofrequency ratio / , and the characteristic perpendicular momentum p as a function of geocentric distance along an auroral field line [from Strangeway, 1985].
To determine the electron distributions and abundances, we follow the work of Chiu and Schulz . We include two distinct electron populations in our model. The first is the primary energetic component, whose source is taken to be a downward propagating half-Maxwellian distribution at the magnetospheric (high altitude) end of the field line. This is refered to as the "magnetopheric" component. The other is a secondary component which is assumed to evolve from a Maxwellian distribution at the ionospheric (low altitude) end of the field line. This component arises as a result of backscattering of the primary electrons, and so it can have relatively high temperature. While it is not truly ionospheric in origin, we shall for purposes of distinction refer to it as the "ionospheric" component. We shall ignore any low-temperature (~ 1 eV) electrons which come from the ionosphere, as these electrons are excluded from the auroral field line by the parallel electric field. We assume the electric field to be present over a range from 1.3 R to 3.3 R geocentric distance. We do not consider explicit ion distributions but simply assume that quasi-neutrality is maintained.
As discussed in detail by Chiu and Schulz , these electron distributions are restricted to different regions of momentum space because of the effects of electric field acceleration and magnetic mirroring. We shall briefly review the work of Chiu and Schulz in the context of the present study.
The particles conserve energy and magnetic moment:
where p is the electron momentum, m is the electron mass, e is the magnitude of the electron charge, is the local electrostatic potential, and B is the local magnetic field strangeth. The subscripts and || are used to denote perpendicular and parallel with respect to the ambient magnetic field, p = p + p . These conservation equations can be rewritten to give the characteristic boundaries in momentum space due to the accelerating electric field and the atmosphere loss cone. The electric field results in an elliptical boundary
where B denotes the magnetic field strength at the magnetospheric end of the field line where = 0. Only particles with momenta greater than the momentum given by this curve can reached an altitude characterized by B and and have originated in the magnetosphere. All other electrons are trapped by the electric field. Loss at the atmosphere results in a loss cone hyperbola
where the subscript I denotes the ionospheric end of the field line. All electrons with a momentum less than this boundary curve come from the ionosphere, although the actual source may be backscatter of the primary auroral electrons.
Chiu and Schulz pointed out that the curves given above only yield the particle accessibility boundaries provided d / dB 0. If this is not the case, it is possible that local electrostatic mirrors exist. For this purpose of simplicity we shall assume that d / dB = 0, i.e.,
With this assumption the p = 0 intercepts of the loss cone hyperbola and the acceleration ellipse are both given by
We can rewrite (5) as p = p I B/ BI, where p I is p at the ionospheric end of the field line. The momentum plotted in Figure 1 corresponds to the p = 0 intercept given in (5). The momentum has been normalized to mc. we have chosen p I= 0.2mc, corresponding to e = 10 keV.
Having determine the boundaries that result from mapping of particle momentum from one location to another along a field line, we must also determine how the particle distribution function is mapped along the field line. Liouville's theorem states that the phase space density is a constant along a particle trajectory, i.e.,
The subscripts 1 and 2 denote different locations on the field line. We shall assume that the magnetospheric electrons have a Maxwellian distribution at B = B. Consequently, the distribution function as a function of position along the field line is given by
where p is the magnetospheric electron thermal momentum. Similarly, an ionospheric Maxwellian distribution when mapped up the field line results in a local distribution given by
where p I is the ionospheric electron thermal momentum. The expressions (8) and (9) for the electron distribution functions are valid only within the kinematically allowed regions for each species as determined from the boundary curves (2) and (3). Outside the allowed regions the distribution functions vanish. Note that because of these kinematically forbidden regions the constants n0 and n0I, in (8) and (9) do not represent the actual total number densities of magnetospheric electrons at B = B or of ionospheric electrons at B = BI . The expressions for the number densities will be given in (10) and (11).
The mapping of the electron distribution functions along an auroral field line is illustrated in Figure 2. The figure shows contours of constant phase space density, together with the boundaries in momentum space due to the loss cone and the parallel electric field. The boundaries are given by the thicker lines. We have shown three locations, one corresponding to the ionospheric end of the field line where = I, an intermediate location where = 0.2I and the magnetospheric end of the field line where = 0. We have assumed that the magnetic field can be modeled by a dipole field with an invariant latitude of 70°, that the electric field lies between 1.3 and 3.3 R, and that the potential scales linearly with the magnetic field magnitude.
Fig. 2. The electron distribution function at three altitudes along an auroral field line. Contours of constant phase density are shown for the magnetospheric electron population (solid circles) and for the ionospheric electron population (dotted circles).
The solid circles in Figure 2 show the first 15 e-foldings of the phase density for the magnetospheric electron population. The contours are calculated with respect to the p = 0 value of the distribution function at R = 3.3 R. As a consequence of Liouville's theorem the phase density is also equal to this value at the p = 0 intercept of the acceleration ellipse for all locations along the field line. We have chosen a thermal energy of 1 keV for these electrons, which is 0.1 of the total parallel potential, and it is apparent that the total number density of the magnetospheric electrons will not vary greatly as a function of position. The number density will only show a marked decrease at lower altitudes for much lower temperatures.
The dotted circles show the ionospheric electron phase space density. In this case the contours are normalized to the p = 0 value at 1.3 R. We have again assumed a temperature of 1-keV for these electrons, indicating that this distribution is mainly produced by the backscattering of the primary auroral electrons. It is apparent that even for a 1-keV temperature the electric field excludes a large amount of these particles. At R = 2.054 R the phase space density has descreased by eight e-foldings at p = 0.
Having determined how the phase space density maps along an auroral zone field line, we can calculate the local number densities associated with the magnetospheric and ionospheric electron distributions. First, we define
Apart from normalization, these are the same as the integrals defined by Chiu and Schulz . We have normalized the integrals so that E(0) = 1. In addition, E(x) and E(x) are directly related to the complex complementary error function w(z) as defined, for example, by Abramowitz and Stegun :
Next, we let
where p, is a thermal momentum, which can either be p or pI depending on the particle species. It should be noted that and * are constant; the altitude dependence is incorporated into b and b*. Another important point concerning the variables defined above is that both and * depend only on the ratio eI/ p; neither temperature nor potential appears separately. This is due to the assumption that the potential scales linearly with the magnetic field strength and that the distribution functions are Maxwellian.
For the magnetospheric electrons,
where denotes evaluated with p. As b increases, correponding to a movement to lower altitudes, the first term in (10) increases, whereas the other two terms decrease. In general, the magnetospheric electron population is expected to show an initial decrease followed by a subsequent increase as a function of decreasing altitude. Furthermore, n will only show a large increase at lower altitudes if b is large. For the parameters used in Figure 2, we expect an increase in number density of about a factor of 2 at the ionospheric end of the field line.
In contrast, the number density of the ionosphere electrons is expected to be dominated by the deceleration due to the parallel electric field,
with computed using p. It should be noted that * - = 2me/ p. We have written (11) so as to bring out the relationship between n and n. The term multiplied by exp (-* + ) in (11) is functionally the same as (10). This term corresponds to that part of the ionosphere electron population that would lie outside the loss cone at the magnetospheric end of the field line. The dominant term (11) is the first term, as expected. As previously deduced, the ionosphere electron population will only be dominant at the lower altitudes. For these secondary electrons to be dominant at the magnetospheric end of the field line,
Having determined the functional dependence of the electron number density, as given as given by (10) and (11), we can compare different predictions with the observed number density [Calvert, 1981]. There are four free parameters we can vary when attempting to fit the data. These are the temperatures of the two electrons populations, together with the factors n0 and n0 in equations (10) and (11). The latter two parameters can also be specified by fixing the number densities of the ionospheric and magnetospheric electrons at their respective ends of the field line.
One such fit is shown in Figure 3. The data of Calvert  show a rapid decrease in number density with increasing altitude down to a value of 1 cm at 1.8 R. Above this altitude "the plasma density within the cavity is roughly uniform with altitude." We have chosen a density of 0.5 cm in the cavity to produce the curve in Figure 3, but this value could reasonably be changed by 50%. The rapid falloff in total density at the low altitude arises from the sharp decrease of the ionospheric population with increasing altitude. The variation in the density of the magnetospheric component is much less dramatic, and this component by itself provides a very good fit to the constant in the cavity. The transition from a majority ionospheric component to a majority magnetosphreic component occurs at 1.7 R. If the cavity density were taken to be as low as 0.25 cm . The transition point would increase only to 1.8 R. For the fit shown in Figure 3 the temperatures of the magnetospheric and ionospheric electrons were 0.92 and 0.82 keV, respectively, whereas the total potential drop was assumed to be 10 kV.
Fig. 3. The electron density as a function of geocentric distance along an auroral field line. The curve labeled "data" gives the observed variation [Calvert, 1981]. The curve "fit," which is the sum of n and n given by equations (10) and (11), is best fit to this observed density.
As stated previously, we have assumed that the region of parallel electric field is confined to the range 1.3-3.3 R and furthermore that the potential varies linearly with magnetic field. Given the uncertainty in the electron densities and distribution functions measured on the auroral zone field line, these assumptions are probably not unreasonable. Figure 2 and equation (11) both show that the dominant factor affecting the ionospheric electron density is the exponential decrease due to the parallel electric field. Since the ionospheric distribution remains Maxwellian with constant temperature as a function of altitude and Calvert's data give us an estimate for the number density at 1.3 R, the presence of accelerating electric fields below this altitude need not be included. Additionally, while the assumption of a 10-kV total potential drop may overestimate typical auroral zone potential drops, the decrease with altitude is only dependent on the ratio 2me/ p (~ 10). This ratio was obtained by fitting the densities given by (10) and (11) to Calvert's data and is not a free parameter. If we decrease the assumed total potential drop, we must decrease the temperature by the same factor. Relaxing the constraint that the potential varies linearly with the magnetic field strength may allow us to change the temperature of the ionospheric distribution, but the large decrease in density present in Calvert's data will still result in a decrease in the ionospheric population similar to that shown Figure 3.
Similarly, changing the assumptions concerning the parallel potential drop will not result in large changes in the magnetospheric electron density. However, it should be noted that while the number densities will not change markedly, the boundaries in momentum space may well change significantly on altering the assumptions concerning the parallel potential. For example, if we assume that the losses to the atmosphere occur at a lower altitude, the loss cone will be narrower. In general, changes of this nature will affect the growth rates of instabilities, not the modifications of the wave dispersion introduced by the presence of hot electrons.
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