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Modelling the Velocity Evolution of an Electron Beam with
Stochastic Simulations
JA Pollock, L Fletcher
Physics & Astronomy Dept. University of Glasgow, Glasgow, G12 8QQ
Introduction
Results
In a solar flare, charged particles (for example fast electrons) are accelerated
along magnetic field lines in the solar atmosphere, travelling through the
ambient coronal plasma. As they pass through the plasma, collisions generate
a positive slope in the beam velocity space, which leads to the generation of
Langmuir waves. These in turn interact further to produce Type III radio bursts
(the wavelength of which is directly related to the plasma frequency and hence
the ambient density and the velocity of the particles along the magnetic field).
The growth rate of Langmuir waves is a function of the wavenumber and the
beam electron distribution function.
We have simulated the evolution of the distribution function of an electron beam
passing through the ambient coronal plasma using Eqs. 6 in different situations.
The velocities (energies) predicted by the simulation at given distances for given
initial energies agree with those predicted by the equation for energy as a function
of column depth and initial angle in Emslie, 1978 [5], thus verifying the simulation.
We study the development of an electron beam under the influence of Coulomb
collisions, and investigate conditions for reverse-drift Type III radio emission.
We have assumed non-relativistic electrons, and consider only Coulomb collisions
as a source of scattering. It should be straightforward in the future to add further
forms of scattering to the simulation. We assume Type III radio emission of
wavelength greater than 10cm, which corresponds to a minimum velocity (energy)
in the direction of the field (via Eq. 8) of 2.839×109 cms-1 .
Type III Radio Bursts
The electrons in the atmosphere are primarily thermal, and therefore have a
thermal distribution function, however the accelerated component has higher
energy and a different distribution – we shall consider a power law
distribution. Coulomb collisions affect the low energy part of the distribution
(see Eq. 6) and this leads to a ‘bump-in-tail’ instability in the distribution
function of the electrons. The growth rate of this instability depends on the
rate of change of the distribution function with velocity along the field line
according to the following equation[3]:
Fig. 2: Log-log plot of electron number density (≡
distribution function) against component of velocity along
the field (vz) line at different distances along the field line (z).
Initial angle of injection constant (60°), initial velocities
randomly chosen from power law distribution in the range
5.3 - 8.3×109 cms-1.
(1)
F ( v z )
f (v z )
where the sign of
v z hence a positive
v z is the same as the sign of
slope in a histogram of vz, the component of the velocity along the field
line, results in the growth of waves.
These are Langmuir waves which, through a series of further
interactions, result in Type III radio emission.
The
wavelength of the resultant emission is related to both vz and
n, the coronal number density, by the following equation (in
cgs units):
These graphs allow us to see where a positive slope in velocity space develops in
relation to the required minimum velocity for Type III radio emission and hence at
what distance along the field line a wave resulting in this sort of emission is
created. Thus we can postulate where Type III emission originates. The y-axis is
number density (the data were binned into histograms) and represents the electron
distribution function.
Figure 2 shows that for electrons of injection angle 60° and initial energies chosen
randomly from a power law distribution between 7.997keV and 19.612keV, a
positive slope at the velocity threshold first appears after a distance of
approximately 1×107 cm (~100km) for a coronal density of 109cm-3. Varying the
initial energies very soon results in all positive gradients appearing before or after
the minimum threshold velocity, in which cases there is the possibility of Type III
emission at no or all distances along the field line respectively. The same is true of
varying the initial injection angle.
(2)
Hence for Type III radio emission of minimum wavelength
λ=10cm, the minimum velocity at which the slope must
appear positive is vz=2.389×109 cms-1.
Fig. 3: Log-log plot of electron number density (≡
distribution function) against component of velocity along
the field (vz) line at different distances along the field line (z).
Initial angle of injection chosen from uniform distribution
between 0° and 90°, initial velocities randomly chosen from
power law distribution in the range 1.0 - 1.5×1010 cms-1.
Fig. 1: Reverse-drift Type
III
radio
bursts.
[Aschwanden & Benz, ApJ
480, 1997]
Stochastic Simulations
The general distribution function of electrons in a beam can be described by[1]:
(3)
df
which is equivalent to the Boltzman Equation if dt  0 . When electromagnetic
forces are introduced, this can be written:
Figure 3 shows the results of a simulation with the initial injection angles chosen
randomly from a uniform distribution between 0° and 90°, and the initial energies
chosen randomly from a power law distribution between 28.469keV and
128.109keV. In this situation, increasing the initial energies results in much less
change in the evolution of the distribution function over time compared to the
situation presented in Fig. 2. In fact, even at these near relativistic energies, the
positive slopes in velocity space do not reach the minimum velocity required for
Type III emission of λ > 10cm.
The results so far would suggest that Type III emission only results from beams,
since for initial injection angles chosen randomly from a uniform distribution,
positives gradients in velocity space do not appear at the threshold minimum
velocity even for near-relativistic particles. Further work into relativistic particles
will investigate this further.
(4)
which is the Vlasov Equation. This equation, when collisions are introduced,
provides the basis of the evolution of the electron distribution function, which
can be described thus (e.g. Kovalev & Korolev, 1981):
(5)
4e n
where μ=cosθ, the angle of particle injection, and D 
me2
4
(Λ≈20 is the
Further Work
We intend to consider further the effects of different initial velocity and injection
angle distributions with the existing simulations, and later to include further
scattering terms, curved and varying magnetic fields and relativistic particles. We
also hope to be able to incorporate different atmospheric models rather than
assume constant density as we have here.
Coloumb logarithm and n≈109cm-3 is the coronal number density). We assume
a constant magnetic field. This is a Fokker-Plank Equation, which has been
shown (in the normalised situation) to be equivalent to the set of stochastic
differential
“Physics of Space Plasmas” GK Parks
[2] “Stochastic simulation of fast particle diffusive transport.” AL McKinnon & IJD Craig, Astron.
& Astrophys. 251, 1991
[3] “Instabilities In Space & Laboratory Plasmas” DB Melrose
[4] “On the generation of loop-top impulsive hard X-ray sources.” L Fletcher, Astron.& Astrophys.
303, 1995
[5] “Collisional interaction of a beam of charged particles with a hydrogen target of arbitrary
ionization level.” AG Emslie, Astrophysical Journal, 224, 1978
[6] “Plasma Astrophysics” 2nd Edition, AO Benz, ASSL / Kluwer 2002
[7] “Physics of the Solar Corona - An Introduction” M Aschwanden, Springer / Praxis 2004
[1]
equations[2]:
(6)
where r(t) is a Gaussian random noise process of mean 0 and variance 2.
We have used these stochastic differential equations to calculate v, z and µ
incrementally in a Java program, and from this produced histograms of the
resultant velocities of particles in the z direction (along the field lines). We
have assumed straight field lines.
Acknowledgements: Iain Hannah, Ross Galloway, Dr Norman Grey & Dr Graeme Stewart.