3 Linear Dispersion Relation
To obtained the linear dispersion relation, we substitute linearized forms of
Eqs. (10)(15) to the Poisson's equation (16) and restrict up to the first order, which yield:

(17) 
where the appearance of a normalized dependent screening factor
(scaled Debye wavenumber)
is defined by

(18) 
Eq. (17) contains a suprathermality term, a Langmuir wave mode, and a positron wave mode.
In this equation,
corresponds to the normalized cool electron thermal velocity, and
is associated with the normalized positron thermal velocity. In the absence of the hot electrons, a linear dispersion is derived in agreement with Eq. 4 im Ref. [63] and Eq. 8 im Ref. [48]. It is seen that the phase speed increases
with higher cooltohot electron temperature ratio
, in
agreement with what found previously [31]. In the limit
(in the absence of the positrons), we obtain
Eq. (14) of [31]. From Eq. (17), we
see that the frequency , and hence the phase speed, increases
with higher positrontohot electron temperature ratio
.
However, this linear thermal effect may not be noticeable due to the range of
parameters adopted here, i.e., low values of the positrontocool electron
density ratio (
).
Figure 1:
Variation of the dispersion curve for different values
of the positrontocool electron density ratio .
Curves from bottom to top: (solid), (dashed),
(dotdashed curve), and (dotted curve). Here, ,
,
and .

In Figure 1, we plot the dispersion curve (17) in the
electron cold limit (), showing the effect of varying the values of
the positrontocool electron charge density ratio . It can be seen that an
increase in the positron parameter weakly increases the phase speed
(). Noting that the positron fraction in astrophysical plasmas, for
example in the solar wind [5], is very small, its effect on the smallamplitude wave solutions is
negligible. Otherwise, a large fraction of the positron fraction can
significantly modify the dispersion relation. Previously, we found that the
dispersion relation and the condition for Landau damping are considerably
changed, when the plasma is dominated by hot distributed electrons (see Figure 1 in Ref. [31]). Similarly,
increasing the number density of suprathermal hot electrons or/and the
suprathermality (decreasing ) also decreases the phase speed.
Ashkbiz Danehkar
20180328