*J. Geophys. Res., 90*, 9675 - 9687, 1985.

(Received August 24, 1984;
revised June 4, 1985;
accepted June 5, 1985.)

Copyright 1985 by the American Geophysical Union.

Paper number 4A8293.

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**Previous:** 5. Discussion On Group Velocity Variation

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Title and Abstract

One of the principal advantages obtained by using the delta function ring distribution is the simple form of the resultant dispersion relation. This simplicity allows us to carry out extensive parameter studies. While the ring distribution may be considered to be a modification of cold plasma theory which takes into account the presence of hot electrons, it is not clear that the dispersion relation can be arbitrarily applied to the auroral plasma. In this section we shall address this point in more detail, using the results presented in the previous sections as a basis for the discussion.

The two principal results of the analysis are, first, that for
*p*
/
*m*
*c*
/
relativistic corrections are significant, and second, if the electrons can be
considered as two separate species, then the unstable
mode is trapped between the two species
gyrofrequencies. We must hence determine when the
auroral electron distribution function can be said to
satisfy both of these conditions, in which case the
ring distribution may be a reasonable approximation for
the purposes of investigating wave dispersion.

We must reiterate that the main goal for the present analysis is to study wave dispersion. Obviously, the growth rate for instability will depend strongly on the detailed form of the distribution function. However, the wave dispersion is controlled more by the collective properties of particle species, as embodied by the integration over all momentum space when calculating the conductivity tensor for a plasma.

Another point concerning the integration over momentum
space is that at a particular momentum, particles with
pitch angles near 90° occupy a larger solid angle than
particles near 0° or 180°, and so contribute more to the
integrals. For this reason we find that the ring and
shell dispersion relation for **k** = 0 are very similar
[see Pritchett, 1984*b*]. The dispersion relation for the
shell distribution for finite wave vector is not as
tractable as the ring, since the integrals cannot be
readily separated into parts when relativistic
corrections are included. The ring distribution
relation is hence used to extend the analysis to finite
wave vector.

We must now determine whether the ring or shell distributions are applicable to auroral electron distributions. The electron distributions as measured on S3-3 show three basic features, as pointed out by Omidi et al. [1984]. These are a widened loss cone in the upgoing particles, a "hole" at low energies, and a "bump" near 90° pitch angles. We also note that at higher energies the distributions are close to spherical symmetry apart from the loss cone feature. For the purposes of determining wave dispersion the loss cone is probably not significant (again invoking solid angle arguments). We are then left with a spherically symmetric distribution at higher energies, i.e., a shell, with a bump near 90°, i.e., a ring.

It appears then that both a cylindrically symmetric "ring" distribution and a spherically symmetric "shell" distribution can be applied to the auroral electron distribution. However, we have made one further assumption, that is, that the hot electrons have no thermal spread. While the measured distributions show quite large temperatures, the approximation is still reasonable provided we can characterize the ambient and energetic electrons as two species with different gyrofrequencies due to relativistic effects. Unfortunately, the bulk of the low-energy ambient plasma (if present) is presumably below the energy range of an electron detector such as that flown on the S3-3 spacecraft. Nevertheless, we can determine some limits for the applicability of our analysis.

A useful distribution for modeling a spherically symmetric distribution with a peak at some finite momentum is the DGH distribution, which can be generalized as

where *n*
is the number density of the particle species,
*p* is the thermal momentum,
(*v*) is the gamma function, and
*p*
is the momentum at which *f*(*p*) has a maximum. For the
purposes of calculating a dispersion relation,
*p* /
*p*
is usually set to some integer value.

From (8) the average momentum squared for the distribution is given by

where <*w*> is the characteristic energy of the
distribution.

Relativistic effects are important for a particular particle species if

i.e., if

where *n*
is measured in particles per cubic centimeter,
<*w*> is in keV, and
*f*
is the nonrelativistic electron gyrofrequency in megahertz.

We shall use the data published by Mizera and Fennel,
[1977] and Croley et al. [1978] to determine whether or
not (11) is satisfied for measured distributions. From
Croley et al there is a peak in the distribution near
*v* = 1.5 x 10
km/s with *f*(*v*) = 21.5 s
km, while at *v*
= 3 x 10 km/s
*f*(*v*) = 0.0464 s
km. On substitution into (8) this yields
*p*
/
*p*
4, with
*p* /
*m**c*
0.05 and
*n*
0.36 cm.
For this distribution, <*w*> 0.88
keV. At the altitude where the distribution was
measured, 7300 km, the magnetic field was 0.058 G
[Mizera and Fennell, 1977], giving a gyrofrequency of
0.16 MHz. From (11) the density must be
1.1 cm.
Relativistic effects are consequently important for
this distribution.

However, if there is a sufficient amount of low-energy
ambient plasma, these particles will dominate wave
dispersion, and a cold plasma formulation is adequate.
Fortunately, we can obtain a lower limit on the
relative densities for hot and ambient electrons. If
the peak associated with the hot electron distribution
is small in comparison to the phase space density of
the ambient electrons at the peak momentum, the hot
electrons are probably unimportant for wave
dispersion. Noting that (8) reduces to a Maxwellian
when *p*
= 0, we can say that if

then the hot electrons are significant. The subscript a
denotes the ambient Maxwellian plasma. In general we expect
*p* <
*p* , and it can be shown that
*f* /
*f*
is then a monotonically increasing function of
*p* /
*p* provided
*p* /
*p* > 1.5.
The constraint imposed by (12) is consequently less well satisfied when
*p* =
*p* ,
in which case, (12) can be rewritten as

The condition given by (13) tends to zero as
*p* /
*p*
,
i.e., when the temperature tends to zero. This is to be expected,
since at this limit the distribution function is
equivalent to a delta function, and the hot electrons
are distinct from the ambient electrons. On the other
hand, the condition is less easily satisfied as the
temperature increases. The condition given by (13)
therefore reflects the spread in gyrofrequencies
associated with a warmer distribution. For
*p* /
*p*
4,
corresponding to the S3-3 distribution,
*n* /
*n*
0.3.

There is one last constraint which we can apply to the
relative densities. If the ambient particles have too
low a number density, then the hot electrons dominate,
and the electrons are again effectively a
single-particle species. To ensure that there are
sufficient numbers of ambient electrons, we shall
require that the difference in
<*p*>
for each species, and hence the difference in the "average"
gyrofrequency, be comparable to the total
<*p*>, i.e.,

For *p* =
*p* ,
(14) reduces to

Again we find that
*p* /
*p*
must be greater than 1.5.
For *p* /
*p* = 4,
*n* /
*n*
1.67.

As written, the conditions given by (13) and (15) imply that
For *p* /
*p*
2.5,
otherwise the upper limit given by
(15) is lower than the limit given by (13). However, we
note that the limits are flexible, and this requirement
may be relaxed.

To summarize, we have shown that for the S3-3
distribution published by Croley et al. [1978] the
delta function ring distribution is a reasonable
approximation when determining the wave dispersion provided 0.3
*n* /
*n*
1.67.
In obtaining this condition we have made two assumptions
which should be verified. The first is that, provided the
phase space density of 90° pitch angle particles is high,
the asymmetry in parallel momentum associated with the loss
cone is not significant for wave dispersion. The second
is that the dispersion is similar for both a ring and a
shell. Both assumptions essentially reduce to a single
one, that the near-perpendicular particles dominate the
wave dispersion for waves near gyroresonance. The
simulations of Pritchett [1984*a*, *b*] and Pritchett and
Strangeway [1985] indicate that this may be the case.
The latter work included an asymmetric electron
distribution, in which a single-sided loss cone was
included. The growth rates were altered by the use of
different distributions, but the wave modes were
essentially the same. However, a more thorough analysis
would be useful, addressing this issue and also
providing better estimates than those presented here on
the range of plasma parameters for which the dispersion
relation used in the present study is adequate.

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