*J. Geophys. Res., 91*, 3152 - 3166, 1986.

(Received August 8, 1985;
revised November 8, 1985;
accepted December 2, 1985.)

Copyright 1986 by the American Geophysical Union.

Paper number 5A8825

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**Previous:** 2. Ring Distribution

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Using simple arguments, Strangeway [1985] determined that for distributions like those observed on the polar-orbiting S3-3 spacecraft, the ring distribution given by (1) may be adequate provided there are roughly equal densities of energetic and ambient electrons. Since the discussion was somewhat qualitative, we shall study the effects of temperature in more detail here. The distribution function we shall employ is the spherically symmetric DGH distribution.

where is the Lorentz factor equal to (1 + *p* /
*m ^{2}*

where (*v*) is the gamma function. We therefore
identify with with an inverse temperature,
= 2*m**c* /
*p*,
where *p* is a thermal
momentum, the exponential term in (3) being equivalent to
exp (-*p* /
*p*).

When *j* 0, the DGH
distribution function has a maximum at

As
, the peak occurs at
*p*
*jp*.
In the previous sections we used
*p* / *mc* = 0.1
as a representative ring momentum. For the peak to occur
. For the peak to occur at the same momentum in (3) we require
/ *j* 200.

When *j* = 0 the distribution function given by (2) reduces to
the relativistic Maxwellian. Shkarofsky [1966] has derived the
conductivity tensor for electromagnetic waves for the Maxwellian
. If we denote the conductivity tensor for
*f*(*p*) as
, then

since the derivation of the conductivity tensor required integration over momentum space and the momentum only enters through the Lorentz factor in (2). We can therefore use the results of Shkarofsky to evaluate the conductivity tensor for the distribution given by (2).

One advantage then of using the distribution function (2) is the ability to relate the conductivity tensor (5) to the previous work of Shkarofsky. A second advantage concerns the application to energetic electron distributions as observed on polar-orbiting spacecraft such as S3-3 and DE-1. In general the observed distributions show features reminiscent of both the ring distribution (1) and the spherically symmetric "shell" distribution (2). The electron data often show a peak in the phase space density at 90° pitch angles with the peak momentum in the keV range, similar to a ring distribution. On the other hand, the distributions are much more shell-like at higher momenta. Furthermore, theoretically we expect the combined effect of an accelerating field-aligned potential drop and the magnetic mirror force to produce an oblate spheroidal boundary in momentum space [Chiu and Schulz, 1978]. At low altitudes, the boundary is almost spherical, and so a shell-like distribution may be a reasonable approximation to the theoretically predicted distribution function, apart from the presence of the loss cone. It has been shown [Croley et al., 1978] that the theoretical model of Chiu and Schulz can be related to the observed distribution, bearing in mind the smoothing inherent in the observations due to both plasma instabilities and instrumental effects.

Strangeway [1985] has argued that the dispersion properties
of both ring and shell distributions are likely to be similar.
However, the ring distribution is not easily related to the
observations. For this reason, a shell-like distribution function is
preferable. Unfortunately, including relativistic terms results in
a rather unwieldy form of the conductivity tensor for arbitrary
wave vectors, even for the Maxwellian [Shkarofsky, 1966].
Hence, the ring distribution is of use for studying oblique
propagation. On the other hand, the shell distribution relation
is simplified considerably by assuming **k** = 0. Since we would
like to compare both ring and shell distributions, we shall
therefore assume that **k** = 0, which allows us to identify the
wave modes and explore their dependence on various plasma
parameters. In section 4 we shall address the dependence on
wave vector and justify our identification of the wave modes.

From R. J. Strangeway (unpublished manuscript, 1986)
where the derivation is described in detail, we find that the
dispersion relation for *R-X* polarized waves is given by

:

when **k** = 0, where
indicates summation of electron species,
is the plasma frequency for a particular species,
is the inverse temperature
for that species, *j* is the power of the shell distribution,
= ( -
) /
,
and *Z* is the plasma dispersion
function [Fried and Conte, 1961].

As discussed by R. J. Strangeway (unpublished manuscript,
1986), the introduction of the square root in the argument for
the *Z* function in (6) results in a branch cut from
= 0 to
infinity. If the branch cut is taken to
=
+, then the
analytic continuation of the function is such that the physically
allowable solutions lie on one sheet, which we refer to as
the lower sheet. For the lower sheet Im [(
)] > 0. The value of
*D*() on the upper sheet can be shown to be
simply related to the value on the lower sheet through the analytic
continuation of the *Z* function; hence

except on the branch cut, where the subscripts *u* and *l*
refer to the upper and lower sheets, respectively, and angle brackets
indicate complex conjugation. The relationship (7) indicates
that solutions to (6) come in complex conjugate pairs, except when
Re () > 0 and Im () = 0
. The restriction of solutions to
the lower sheet ensures that a Maxwellian is stable.

Solutions of (6) are plotted in
Figure 3 assuming both hot
and ambient population are present. The hot population can
have different values of *j*, while the ambient population has
*j* = 0, i.e., Maxwellian. Frequency is plotted in the left-hand
column of Figure 3,
with the corresponding growth rate in the
right-hand column for different values of *j*. As indicated by the
heading to the figure, we fix
/
*j* = 200, i.e.,
*p* / *mc*
0.1, the
temperatures of the ambient and hot electron distributions are
the same, as are the number densities. The numbers to the
right of each panel give the *j* parameter for the hot electron
population, with *j* =
corresponding to a delta function.
When *j* = 0, both populations are Maxwellian and we use
ph = 200 for the hot electron population. For reference with
this and subsequent figures, if we define
=
/
*j*, then
=
*j*, *j* 0:
=
, *j* = 0.

Fig. 3. Solutions of thek= 0 shell dispersion relation (6) for variable /. The left-handed column shows frequency while the right column shows the corresponding growth rates. The integers inside and to the right of each panel give thejparameter of the DGH distribution. Forj>1, /j= 200. Whenj= 0, both distributions are Maxwellian and we set = 200. For the prupose of identifying modes, the solid line corresponds to the "trapped" mode, the dashed line corresponds to theR-Xmode, and the dotted line corresponds to a Bernstein mode.

For the delta function we find that there are two modes. The higher-
frequency mode, as indicated by the dashed line, increases in frequency
as
/
is increased. The frequency
of this mode is greater than the gyrofrequency. The
corresponding growth rate, shown in the right-hand column, is
zero, and the mode is marginally stable. We therefore identify
this mode with the *R-X* mode in
Figure 1. The second mode,
shown by the solid line, lies below the gyrofrequency and has
a positive growth rate. We identify this mode with the unstable
trapped mode in Figure 1.
In this and subsequent figures,
we use dashed lines to represent the *R-X* mode, while solid
lines are used to indicate the "trapped" mode. For the high *j*
values this identification is unambiguous, the trapped mode is
the lower-frequency mode, and this mode can be unstable.
However, as we shall show later, the identification of the
modes becomes more difficult for low *j* values.

A third mode is also plotted in Figure 3.
For the purposes of identifying the mode, we consider the solutions shown in
the bottom panels for *j* = 0. These solutions correspond to a
single Maxwellian. At low
/
only one mode
(the *R-X* mode) is shown, and that mode is damped. As
/
is increased, the mode becomes marginally stable and a second
mode is also found. This has been called a "Bernstein" mode
by Pritchett [1984*b*], and
we indicate this mode by dotted
lines. The Bernstein mode shown in
Figure 3 is not present
when *j* = . This is
because there is no branch cut for the
delta function dispersion relation. When *j*
, both the *R-X*
mode and the Bernstein mode lie on the branch cut described
earlier, and therefore are only marginally stable for
>
.
On the lower sheet the *R-X* mode is found as
0 from
above, while the Bernstein mode is found for
0 from
below. As the frequency decreases so that
, both
modes merge, and the solutions are found for complex conjugate
frequencies, one on each sheet. We only show the
solution for the lower sheet and that solution is damped for *j* = 0.

When *j* is large, i.e., low-temperature distributions, the
solutions are very similar to the*j* = case,
apart from the additional Bernstein mode. However, including temperature
restricts the range in instability for the trapped mode. At low
/
the mode is damped as pointed out by Winglee
[1983] for a loss cone type distribution. For lower *j* values, the
trapped mode is damped for all values of
/
.

One last point to be made from
Figure 3 concerns the form of the
solutions for *j* = 1. The *R-X* mode looks very similar.
but the trapped mode behavior is rather different than that
found for *j* > 1. Unlike the other cases, the trapped mode
couples to the Bernstein mode at high
/
. At low
/
.
the mode we have called the trapped mode lies on the branch cut with
>
and
0
from below. It appears then that the labelling
of the modes is not so obvious for *j* = 1.

Before investigating the dependence on other plasma
parameters we shall summarize the properties of the wave
modes. The highest-frequency mode has been identified with
the *R-X* mode, with the implication that this mode
approaches the light cone as wave vector increases. The mode is
either stable or damped and when marginally stable
( >
),
the *R-X* mode lies on the branch cut with
0 from above.
This mode is present even when *j* = . As frequency
decreases, the *R-X* mode couples to a second mode which lies on
the branch cut with
0 from below. We have identified
this mode with a Bernstein mode, which implies that the
frequency of this mode is nearly constant for all wave vectors.
The Bernstein mode and the *R-X* mode merge and are damped for
<
.
The last mode found is the "trapped" mode.
This mode usually has the lowest frequency, and can be
unstable. By implication, this mode cannot propagate for
finite wave vector. In Figure 3,
the trapped mode is separate
from both the *R-X* and Bernstein modes, except for *j* = 1. We
shall verify the inferences made concerning the dispersion
properties of the waves in section 4.

In Figure 4 we fix
/
and vary the number density
ratio. We have again assumed equal temperature electron
populations. At low hot electron densities, the three modes are
very distinct for *j* 2.
The *R-X* mode and Bernstein are both
marginally stable, merging to form a damped mode at sufficiently high
*n* /
*n*
(the Bernstein mode is again not present for
*j* = ). The trapped mode is damped at low
*n* /
*n* and becomes unstable at high
*n* /
*n*.
As might be expected, the required density ratio
for instability decreases as *j* increases.

Fig. 4. Solutions of thek= 0 shell dispersion relation (6) for variablen/n. Similar to Figure 3.

For *j* = 1 the trapped mode behavior shown in
Figure 4 is
again very different. For the particular choice of parameters,
the trapped mode is always marginally stable, lying on the
branch cut. On the other hand, the *R-X* mode merges with the
Bernstein mode, as found for *j* > 1, but the *R-X* mode can
also be unstable for sufficiently high
*n* /
*n*.

The parameter
/
is
varied in Figure 5,
the other parameters being held fixed. If
/
is
large, then the background plasma has a low temperature
in comparison to the hot electrons.
When this is the case, the three modes are very distinct
for 1 *j* < , the
high-frequency *R-X* mode and the
Bernstein mode are both marginally stable, while the low-
frequency trapped mode is unstable provided the energetic
electrons are sufficiently peaked. As the ambient electron
temperature is increased (i.e.,
/
decreases), the three modes approach one another. At sufficiently low energetic
electron temperatures, the low-frequency mode is again unstable, while
the *R-X* mode is damped when *j* 2.
For *j* = 1, the *R-X*
mode also becomes damped. However, the trapped mode
shows a much more complicated structure. Near
/
= 1, the trapped mode changes from being strongly damped (
/
<
-0.005, below the lower limit of the plots) to marginally stable.
At the same time the frequency varies considerably, reaching a maximum as
0
for the trapped mode. When the frequency of the trapped mode maximizes,
both the Bernstein mode and the *R-X* mode merge and form a single
damped mode.

Fig. 5. Solutions of thek=0 shell dispersion relation (6) for variable / . Similar to Figure 3.

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