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Generation of Radio Emission from Energetic Electron Beams

Generation of Radio Emission from Energetic Electron Beams

Robert Bingham,
LOFAR and the Transient Radio Sky, Amsterdam, December 2008

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transientskp

June 18, 2012
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Generation of Radio Emission from Energetic Electron Beams Robert Bingham Rutherford Appleton Laboratory, Space Science & Technology Department B. J. Kellett1, V. Graffagnino1, T.W.B. Muxlow2, A.G. Gunn2, D.C. Speirs3, I. Vorgul4, R.A. Cairns4, K. Ronald3, S.L. McConville3, A.D.R. Phelps3, A.W. Cross3, C.W. Robertson3, C.G. Whyte3, W. He3 and K. Gillespie3. 1.  STFC Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX, UK. 2.  Merlin/VLBI National Facility, Jodrell Bank Observatory, The University of Manchester, U.K. 3.  Department of Physics, University of Strathclyde, Glasgow, G4 0NG, UK. 4.  School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife, KY16 9SS, UK. LOFAR and the Transient Radio Sky Amsterdam, December, 2008.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Relevance •  Various processes in space physics and astrophysics:- –  Auroral Kilometric Radiation (AKR) – emission from Earth’s aurora –  Low frequency radio emission from extra-solar planets –  Stellar maser radiation • UV Ceti –  100% polarised radio flares • CU Vir –  100% polarised coherent and periodic radio enhancements –  First stellar pulsar – “pulstar”!! •  Regions in narrow frequency band close to local cyclotron frequency. •  Microwave generation – gyrotrons, etc. [1] Bingham R. and Cairns R.A., Physics of Plasmas, 7 (2000) 3089. [2] Bingham R. et al., Astron. and Astrophys., 370 (2001) 1000. [3] Kellett B.J. et al., MNRAS, 329, 102, (2002). [4] Kellett et al., 2008.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. All solar system planets with strong magnetic fields (Jupiter, Saturn, Uranus, Neptune, and Earth) also produce intense radio emission – with frequencies close to the cyclotron frequency. Planetary Magnetospheres Planetary Aurora Animation courtesy of NASA Jupiter’s aurora Solar wind electron beams Radio emission region
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Planetary Radio Emission •  (a) Initial radio Bode’s law for the auroral radio emissions of the five radio planets (Earth, Jupiter, Saturn, Uranus and Neptune) (Desch and Kaiser, 1984; Zarka, 1992). JD and JH correspond to the decameter and hectometer Jovian components, respectively. The dashed line has a slope of 1 with a proportionality constant of 7.10-6. Error bars correspond to the typical uncertainties in the determination of average auroral radio powers. (b) Magnetic radio Bode’s law with auroral and Io- induced emissions. The dotted line has a slope of 1 with a constant of 3.10-3. i.e. due to solar wind ram pressure
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. CU Vir at Jodrell Bank (Kellett et al., ApJ, 2008)
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. CU Vir at Jodrell Bank (Kellett et al., ApJ, 2008) Right Hand Circular Left Hand Circular ~150o
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Stellar Radio Emission Artists impression of the MCP star CU Virginis
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Electron acceleration in the aurora •  DE-1 at 11000 km over the polar cap [Menietti & Burch, JGR, 90, 5345, 1985] ! ! Electron distribution with a crescent shaped peak in the downward direction A crescent-shaped peak (p) with the addition of a field-aligned hollow (h).
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. FAST Observations - Delory et al. GRL, 25(12), 2069, 1998
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Modelling the Growth Rate of electron cyclotron l Model Input ! !
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Horse-shoe Formation Process Diagrammatic representation of horseshoe distribution formation due to conservation of magnetic moment µ.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Converging Magnetic Field •  Imagine a converging magnetic field geometry … •  … now imagine an electron beam moving down … •  … the electrons will become increasing aware of the magnetic field “squeezing” them … •  … this will result in the conversion of parallel velocity into perpendicular velocity. •  This is a result of the conservation of the first magnetic invariant.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Horseshoe Formation Field aligned electron beams naturally form a horseshoe distribution as they move into stronger magnetic field regions. The adiabatic invariance v ⊥ 2 /B = constant causes the electrons to lose parallel energy and increase their perpendicular energy producing the characteristic horseshoe distribution with ∂fe / ∂v ⊥ > 0. Requirements ! ! ÷ DOWNWARD ACCELERATED ELECTRONS AURORAL DENSITY CAVITY AURORAL KILOMETRIC RADIATION 0 v > ∂ ∂ ⊥ e f pe c ω > Ω 2 1 0 2 0 pe , where ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = Ω ε ω e e c m e n m eB Low density cold background such that nH > nC
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Laboratory Experiment •  So, in order to perform an experiment, we simply need to construct a converging magnetic field … •  … and then fire in an electron beam! •  (couldn’t be simpler – at least for a theoretician ! – it might be a little more difficult to actually build … )
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Experimental Progress •  Experiments now underway in Lab. at University of Strathclyde!
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Modelling the Laboratory Experiment Bz ~ 0.5 T Bz ~ 0.03 T Distribution Functions v⊥ v⊥ v⊥ v|| v|| v|| Electron gun B field Im(n) Im(n) Im(n) ω ω ω
  17. The image cannot be displayed. Your computer may not have

    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Karat Simulations Evolution of the electron beam velocity distribution as a function of axial position. (a)  At z = 6cm a well defined horseshoe-shaped velocity distribution is present. (b)  At z = 130cm there is evidence of a cyclotron-maser instability in the Vtransverse vs Vz plot, with smearing of the transverse velocity profile at high pitch factors. In the corresponding Vtheta vs Vradial plot there is also evidence of azimuthal phase bunching. (c)  Finally, at z = 200cm the smearing in transverse velocity is apparent across most of the pitch range. There is also evidence of phase trapping in the corresponding Vtheta vs Vradial plot, indicative of the instability having reached a saturated state.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Microwave output frequency Observed 11.6 GHz maser radiation from experiment!
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Basic Physics •  Instability is driven by interaction between cyclotron motion of electrons and right circularly polarized component of electric field. This component rotates in the same sense as the electrons. •  Electrons uniformly distributed in phase around orbit – as many gaining energy as losing energy. •  Need some sort of bunching to get instability. •  Important frequencies: -  cyclotron frequency corrected for relativistic mass shift. - wave frequency Doppler shifted by parallel motion of electron. γ γ Ω = e m eB 0 || || v k − ω
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Modelling the Growth Rate of electron cyclotron l The spatial growth rate can be obtained by solving where n is the refractive index and in spherical polar co-ordinates (p, µ, φ) θ is r eplaced by µ = cos θ = p|| / p and the resonant momentum p0 = mc (2(Ωc0 -ω)/ Ωc0 )1/2 The horseshoe distribution f(p, µ) = F(p) g(µ) destabilizing stabilizing ! ( ) ( ) 2 0 4 2 2 2 0 2 0 0 2 Im c p pe c c e k c n Ω − Ω Ω − − = Ω ω ω ω α ( ) 0 2 2 1 1 2 2 2 2 0 2 1 2 4 1 p p e e c pe f p p f p d c m = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ − Ω = ∫ µ µ µ µ π ω α 0 p p p F Q p F P = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ∂ ∂ Γ = α
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Modelling the Growth Rate of electron cyclotron where The first term in α results in emission of the waves if ∂F/∂p is +ve at the resonant momentum. The second term is -ve and goes to zero if g becomes uniform on the interval [ -1, 1 ] •  The beam requires the correct ⊥ spread to trigger the emission of AKR ( ) ( ) µ µ µ d g P ∫ − − = 1 1 2 1 ( ) ( ) µ µ µ d g Q ∫ − − = 1 1 2 3 1
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Growth Rate of Electron Cyclotron •  AKR in Auroral Zone Consider a horseshoe centred on p|| = 0.1 me c - i.e. a 5 keV beam, with a thermal width of 0.02 me c and an opening angle of µ0 = 0.5 moving in a low density Maxwellian plasma with Te = 312 eV, ! ! ωp /Ωce = 1/40. A typical convective growth length across B Lc = 2π/Im k⊥ is 10 λ. For a cyclotron frequency of 440 kHz the convective growth distance is of order 5 km allowing many e-foldings within the auroral cavity which has a latitudinal width of about 100 km. The growth rate decreases for increasing µ0 and increasing thermal width of the horseshoe distribution.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Electron Cyclotron Maser as a plasma instability •  Chu and Hirshfield (1987) and Pritchett (1984) considered this process as a plasma instability, starting from the equilibrium distribution function - monoenergetic electrons, in frame moving with parallel velocity. •  Put this into standard dielectric tensor (from Stix for example) and get dispersion relation for right hand polarized wave propagating along B0 . •  Result ( ) ( ) || 0 0 2 1 p p p p f δ δ π ⊥ ⊥ ⊥ − = ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Ω − − + Ω − = − 2 2 2 2 2 2 2 2 2 2 2 1 γ ω ω γ ω ω γ ω ω c k V k c k p γ e m p V 0 ⊥ =
  24. The image cannot be displayed. Your computer may not have

    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Electron Cyclotron Maser (From Chu and Hirshfield, 1978) Stable branches join the two unstable branches. Note: Dispersion relation has real coefficients – expect complex conjugate roots. Where imaginary part disappears would expect real part to split into two branches.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Perpendicular Propagation •  This is of interest to the AKR and stellar maser problems where observations suggest waves are generated in the X mode, propagating across the field. •  Look at similar ring distribution for this problem. •  Dispersion relation contains Bessel functions Jn (k⊥ p⊥ / me Ω). •  Assume perpendicular wavelength >> Larmor radius – take lowest order J1 (x) / x ~ 1/2 to get rid of these and k dependence coming from them. •  Result for ω ∼ Ω :- ⊥ ⊥ − = ε ε ε ω 2 2 2 2 xy c k ( ) ( ) ( ) ( )2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 1 1 γ ω ω γ ω γ ω γω ω ε γ ω ω γ ω γ ω γω ω ε Ω − Ω + Ω − − = Ω − Ω + Ω − − = ⊥ c V c V p p xy p p
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Perpendicular Propagation •  Again there is a stable branch connecting the unstable branches. •  Compare cold plasma dispersion curves. •  Question on AKR – instability on lower branch below UH frequency. Radiation escapes magnetosphere – how does it get onto upper branch which connects to vacuum radiation? Possible answer – the beam changes topology of dispersion curves.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Real & Imaginary Frequency vs. Wavenumber in Slab
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Spread in Particle Energies •  In laboratory device can produce almost monoenergetic electron beams. •  In space applications expect spread in energies and instability driven by resonant particles. •  Cyclotron instability produces particle diffusion in perpendicular degree of freedom – instability needs population inversion along p ⊥ axis. •  Loss cone – electrons moving along B field lost through magnetic mirror. •  Horse-shoe distribution – produced when beam with thermal spread moves into increasing magnetic field. •  This appears to be a good candidate for the source of AKR. •  Resonant condition:- ω = Ω / γ •  Expect instability if resonant particles lie around inside of horseshoe.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Laboratory Experiment •  Analysis of horseshoe instability - Bingham and Cairns (2000), Vorgul, Cairns and Bingham (2005) - shows strong instability. •  Experiment at University of Strathclyde - send electron beam with velocity spread into increasing magnetic field. •  Results show emission in narrow frequency band below cold plasma cyclotron frequency. Efficiency of energy conversion from beam 1-2%. Consistent with AKR observations. •  Some future questions:- –  Look at dispersion properties of plasma with horseshoe distribution in more detail - how does energy produced in unstable region in an inhomogeneous plasma propagate away from this region? –  Can experiment be modified to look at other beam instabilities?
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Slab Geometry •  What happens in an inhomogeneous magnetic field where the wave frequency and the cyclotron frequency are only close to each other in a limited spatial range. •  Consider the simple problem of a uniform plasma slab within which the wave is unstable, and on each side of it regions where the wave propagates stably. •  From the dispersion curves it can be seen that for the low density which we consider the wave speed on each side of the cyclotron resonance layer is very close to that of vacuum propagation. •  Take three layers:- 1. x < 0 propagating wave with wavevector k0 ≈ ω / c 2.  0 < x < L layer in which the system is described by the dispersion relation which we denote for convenience by the general form F(ω, k) = 0 3. x > L as for 1.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Conclusion •  We have shown that the existence of this branch means that radiation generated by an instability below the cyclotron frequency can produce radiation propagating freely into a region where the wave frequency is above the cyclotron frequency and onto the branch of the dispersion relation which connects to vacuum propagation. •  A long standing problem in the theory of such emissions is how radiation which is generated below the local cyclotron frequency escapes into the vacuum. •  The present analysis, shows that in the presence of energetic particle distributions, the dispersion curves below the cyclotron frequency connect to those in the vacuum, and could be relevant to this problem.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. SLAB Geometry (2) •  We look for a solution in which there are only waves propagating away from the layer on each side, so that the amplitude is given by where we can choose the amplitude of the wave in x < 0 to be unity. •  Imposing conditions of continuity of the amplitude and its derivative at the boundaries gives •  If we put K = k/k0 and l = k0 L, then from the above we obtain the relation which determines K. ( ) 1 2 tan 2 + = K K KL i (1)
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Solutions •  Possible solutions representing a locally unstable region emitting waves in both directions are found by first finding a value of K which satisfies (1). •  We must than look for a non-trivial solution, if such exists, of •  In particular we look for solutions with positive imaginary part, whose existence shows that there is an unstable layer which can emit growing waves in both directions. Equation (1) has multiple solutions, the first few of which are plotted here for L = 20. K versus Mode No. Real (x) and imaginary (+) parts of K for the first 9 modes.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. •  In one direction the wave will get into a region of decreasing magnetic field and density and in a suitable geometry the wave will connect smoothly onto the vacuum wave. •  In the other direction the wave will, in general, encounter a cut-off and be reflected, so we ask what happens if it re-enters the cyclotron resonance region. •  For the ring distribution, the dispersion relation yields one value of k2 at each point and this value is positive, as shown here. •  The wave just propagates through the region with no absorption or reflection by solving the equation the x dependence in F coming from the x dependent cyclotron frequency. Inhomogeneous System – Varying Magnetic Field Ω k ( ) 0 , 2 2 = + φ ω φ x F dx d
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Introduction •  Cyclotron maser instabilities are important in laboratory devices like gyrotrons for generation of high power radiation and are thought to play a role in radio emission from planets and stars. •  Recently we and others [1] have argued that a cycloton instability driven by a horseshoe shaped distribution in velocity space may be responsible for auroral kilometric radiation from the Earth and other planets, as well as for emission from a variety of stellar objects. •  One long standing problem is that of how the radiation, generated at frequencies below the upper hybrid resonance, gets on to the higher frequency branch of the dispersion relation which connects to the regime of vacuum propagation. •  We will look at some of the dispersion properties of waves in the presence of energetic particles populations and show that they have properties which suggest a solution to this problem. •  We begin with a simpler problem than the horseshoe distribution, namely a monoenergetic ring distribution with zero parallel momentum and a fixed perpendicular energy.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Observations of auroral electrons Mountain-like surface plot of an auroral electron distribution exhibiting a distinct beam at the edge of a relatively broad plateau.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Evolution of an auroral electron beam distribution (Bryant and Perry, JGR, 100, 23711, 1995) A - a cold, background plasma B – inject an electron beam parallel to the magnetic field. C-G – electron beam moves down into the converging field and slowly swaps parallel velocity for perpendicular velocity H – Form complete “horseshoe” distribution
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. FAST Observations of electron distributions in the AKR source region •  Delory et al. - GRL 25 (12), 2069-2072, 1998. ! ! Delory et al. reported on high time-resolution 3-D observations of electron distributions recorded when FAST was actually within the AKR source region. In general, the electron distributions show a broad plateau over a wide range of pitch angles. They presented computer simulations of the evolution of the electron distribution which assumed plasma conditions similar to those observed by FAST and which show similar results to those observed.
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. AKR Laboratory Analogy
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Energy Gain and Loss E Electrons gain energy Electrons loss energy When electrons gain energy – their mass increases and cyclotron frequency decreases. When electrons loss energy – their mass decreases and cyclotron frequency increases. This will lead to bunching around the orbit. If - field slips in phase so as to gain energy from particles in bunch. || || v k − < Ω ω γ E B0 B v x B force along axis, opposite directions on opposite sides of orbit. Particles gaining and losing energy have opposite axial shifts - opposite Doppler shifts - bunching on orbit. If the relative shift in the particle and wave phase is such as to give a net loss of energy from the particles to the wave. || || v k − > Ω ω γ
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    enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Strangeway et al. 2001 – FAST Data “Cartoon” The figure shows an electron distribution function acquired by FAST within the auroral density cavity (see later). This is the region where the auroral kilometric radiation (AKR) is generated. The figure also shows the envisaged flow of energy. Parallel energy gained from the electric field (stage 1) is converted to perpendicular energy by the mirror force (stage 2). This energy is then available for the generation of AKR and diffusion to lower perpendicular energy (stage 3).