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Theory of Horseshoe Maser Instability

I Vorgul1, R A Cairns1, R Bingham2,3, K. Ronald3, A.D.R. Phelps3, A.W. Cross3,
D.C. Speirs3 and S.L. McConville3
1.School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK.
2.CCLRC Rutherford Appleton Laboratory, Chilton, Didcot, Oxon. OX11 0QX,UK.
3.SUPA Department of Physics, University of Strathclyde, Glasgow, G4 0NG,UK.
Abstract. Horseshoe- or crescent-shaped electron velocity distribution function is investigated as a cause of cyclotron
maser instability. The horseshoe distribution function is formed when an electron beam propagates in a convergent
magnetic field. The instability is analyzed in cylindrical geometry relevant to auroral kilometric radiation and to a
laboratory experiment. Exact modelling of coupled TE and TM modes radiation shows high growth rate for modes with
almost perpendicular propagation.
Keywords: Plasma maser instability, auroral kilometric radiation
PACS: 95.30.Qd, 94.20.wc, wf, 94.30.cq
When a beam of electrons with a thermal spread moves along converging magnetic field lines, the distribution
function evolves into a horseshoe shape in (p|| , p⊥) space. In such a distribution there is a population inversion of
particles in perpendicular momentum which results in the plasma being subject to a cyclotron maser instability.
There is strong evidence that this type of instability is the source of auroral kilometric radiation1.

A satellite passing through the acceleration region (long-dashed line in Fig.1) observes precipitating electrons,
anti-Earthward ion fluxes, and large-amplitude electric fields at the boundaries. Satellite data also contain an evident
horseshoe-shape distribution function.
The theory of the instability2 indicates that the phenomenon can be scaled to laboratory dimensions, with
centimetre rather than kilometre wavelengths, and an experiment to do this has been constructed at the University of
Strathclyde3 (see the paper of K.Ronald and D.C. Speirs et al at this conference). Here we present an analysis of the
instability in cylindrical geometry, with a circular beam partially filling a circular conducting waveguide. Plasma
frequency is small compared to cyclotron frequency for aurora, the experiment and modeling.
We consider an electron beam moving into convergent magnetic field. Magnetic moment is an adiabatic
invariant, i.e. approximately constant so long as the field varies over a scale >> Larmor radius. Energy
conservation means .
For a drifting Maxwellian initial distribution the laws of conservation of magnetic moment and particle energy result
in the transformed distribution with a shape of a horseshoe in velocity space, as shown in Fig.2

Such a model gives us about 2% energy efficiency estimate, which is in line with results of the experiment and
PIC code simulations.
This work was supported by UK Science and Engineering Research Council
1. R.E.Ergun, C.W.Carlson, C.W.McFadden et al., Astrophys. J. 538, 456 (2000).
2. R.Bingham and R.A.Cairns, Phys. Plasmas 7, 3089 (2000)
3. R.A.Cairns, D.C.Speirs, K.Ronald et al., Phys.Scr., T T116, 23 (2005).
4. I.Vorgul, R.A.Cairns and R.Bingham, Phys.Plasmas 12, 122903 (2005).

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