## Electrostatic waves and solitons in electron-positron plasmas.

##### Abstract

The magnetosphere of pulsars is thought to consist of an electron-positron
plasma rotating in the pulsar magnetic field (Beskin, Gurevich & Istomin
1983; Lominadze, Melikidze & Pataraya 1984; Gurevich & Istomin 1985). A
finite, and indeed large, longitudinal electric field exists outside the star, and
may accelerate particles, stripped from the surface, to high energies (Goldreich
& Julian 1969; Beskin 1993). These particles may leave the magnetosphere
via open magnetic field lines at the poles of the pulsar. This depletion
of particles causes a vacuum gap to arise, a double layer of substantial potential
difference. The primary particles, extracted from the star's surface,
are accelerated in the double layer, along the pulsar magnetic field lines,
and so produce curvature radiation. The curvature photons, having travelled
the distance of the double layer may produce electron-positron pairs
above the vacuum gap. These first-generation secondary particles, although
no longer accelerating, may synchroradiate, generating photons which may
then produce further electron-positron pairs. These synchrophoton produced
pairs will be at energies lower than curvature photon produced pairs, since
synchrophoton energies are approximately an order of magnitude less than
that of the parent curvature photon.
An attempt to model the electron-positron pulsar magnetosphere is made.
A four component fluid electron-positron plasma is considered, consisting of a
hot electron and positron species, at temperature Th , and a cool electron and
positron species at temperature Tc . The hot components represent the parent
first-generation curvature-born pairs, and the cooler components represent
the second-generation pairs, born of synchrophotons. The hot components
are assumed to be highly mobile, and are thus described by a Boltzmann
density distribution. The cool components are more sluggish and are thus
described as adiabatic fluids. The model is symmetric in accordance with
pair production mechanisms, so that both species of hot(cool) electrons and
positrons have the same temperature Th(Tc, and number density Nh(Nc ) .
In the interests of completeness, linear electrostatic waves in five different
types of electron-positron plasmas are considered. The dispersion relations
for electrostatic waves arising in these unmagnetized plasmas are derived.
Single species electron-positron plasmas are investigated, considering
the constituents to be: both Boltzmann distributed; both adiabatic fluids;
and finally, one species of each type. Linear electrostatic acoustic waves in
multi-component electron-positron plasmas are then considered, under the
four component model and a three component model (Srinivas, Popel &
Shukla 1996).
Small amplitude nonlinear electron-positron acoustic waves are investigated,
under the four component electron-positron plasma model. Reductive
perturbation techniques (Washimi & Taniuti 1966) and a derivation of the
Korteweg-de Vries equation result in a zero nonlinear coefficient, and a purely
dispersive governing wave equation. Higher order nonlinearity is included,
leading to a modified Korteweg-de Vries equation (Watanabe 1984; Verheest
1988), which yields stationary soliton solutions with a sech dependence rather
than the more familiar sech2.
Arbitrary amplitude solitons are then considered via both numerical and
analytical (Chatterjee & Roychoudhury 1995) analysis of the Sagdeev potential.
The symmetric nature of the model leads to the existence of purely
symmetrical compressive and rarefactive soliton solutions. Small and arbitrary
amplitude soliton solutions are compared, and show good correlation.
Under the assumption of Boltzmann distributed hot particles, severe restrictions
are imposed on the existence domains of arbitrary amplitude soliton
solutions. The Boltzmann assumption places a stringent upper limit on the
cool species number density, in order for the solutions to be physical.
An investigation is made of results obtained for an asymmetric electronpositron
plasma (Pillay & Bharuthram 1992), consisting of cold electrons
and positrons, and hot Boltzmann electrons and positrons at different temperatures
Teh and Tph , and number density Neh and Nph . It is found that
the assumption of Boltzmann particles again places restrictions on the acoustic
soliton existence space, and that the results obtained may be physically
invalid. Valid solutions are obtained numerically, within the boundaries of
allowed cool species density values.