Generation of Radio Emission from Energetic Electron Beams

1
Generation of Radio Emission from Energetic
Electron Beams
LOFAR and the Transient Radio Sky Amsterdam,
December, 2008.

• 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.

• STFC Rutherford Appleton Laboratory, Chilton,
  Didcot, Oxfordshire, OX11 0QX, UK.
• Merlin/VLBI National Facility, Jodrell Bank
  Observatory, The University of Manchester, U.K.
• Department of Physics, University of Strathclyde,
  Glasgow, G4 0NG, UK.
• School of Mathematics and Statistics, University
  of St Andrews, St Andrews, Fife, KY16 9SS, UK.

2
Relevance

• Various processes in space physics and
  astrophysics-
• Auroral Kilometric Radiation (AKR) emission
  from Earths 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.
3
Planetary Magnetospheres

• 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 Aurora
Radio emission region
Solar wind
electron beams
Animation courtesy of NASA
Jupiters aurora
4
Planetary Radio Emission
i.e. due to solar wind ram pressure

• (a) Initial radio Bodes 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 Bodes
  law with auroral and Io-induced emissions. The
  dotted line has a slope of 1 with a constant of
  3.10-3.

5
CU Vir at Jodrell Bank(Kellett et al., ApJ, 2008)

6
CU Vir at Jodrell Bank(Kellett et al., ApJ, 2008)

Right Hand Circular Left Hand Circular
150o
7
Stellar Radio Emission

Artists impression of the MCP star CU Virginis
8
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).
9
FAST Observations - Delory et al. GRL, 25(12),
2069, 1998
10
Modelling the Growth Rate of electron cyclotron

• ? Model Input

11
Horse-shoe Formation Process

Diagrammatic representation of horseshoe
distribution formation due to conservation of
magnetic moment ?.
12
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.

13
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? gt 0.
• Requirements

Low density cold background such that nH gt nC
14
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!
• (couldnt be simpler at least for a
  theoretician ! it might be a little more
  difficult to actually build )

15
Experimental Progress

• Experiments now underway in Lab. at University of
  Strathclyde!

16
Modelling the Laboratory Experiment

Distribution Functions
v
v
v
v?
v?
v?
Bz 0.03 T
B field
Electron gun
Bz 0.5 T
Im(n)
Im(n)
Im(n)
?
?
?
17
Karat Simulations

• Evolution of the electron beam velocity
  distribution as a function of axial position.
• At z 6cm a well defined horseshoe-shaped
  velocity distribution is present.
• 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.
• 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.

18

Microwave output frequency
Observed 11.6 GHz maser radiation from experiment!

19
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.

20
Modelling the Growth Rate of electron cyclotron

• ? The spatial growth rate can be obtained by
  solving
• where n is the refractive index and
• in spherical polar co-ordinates (p, µ, f) ?
  is r eplaced by µ cos ? p / p
• and the resonant momentum p0 mc (2(Oc0-?)/
  Oc0)1/2
• The horseshoe distribution f(p, µ)
  F(p) g(µ)

21
Modelling the Growth Rate of electron cyclotron

• where
• The first term in a 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

22
Growth Rate of Electron Cyclotron

• AKR in Auroral Zone
• Consider a horseshoe centred on p 0.1
  mec - i.e. a 5 keV beam, with a thermal width of
  0.02 mec and an opening angle of µ0 0.5 moving
  in a low density Maxwellian plasma with Te 312
  eV,

?p/Oce 1/40. A typical convective growth
length across B Lc 2p/Im k? is 10 ?. 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.
23
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

24
Electron Cyclotron Maser
Note Dispersion relation has real coefficients
expect complex conjugate roots. Where imaginary
part disappears would expect real part to split
into two branches.

(From Chu and Hirshfield, 1978)
Stable branches join the two unstable branches.
25
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? / meO).
• Assume perpendicular wavelength gtgt Larmor radius
  take lowest order J1(x) / x 1/2 to get rid
  of these and k dependence coming from them.
• Result for ? ? ? -

26
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.
27
Real Imaginary Frequency vs. Wavenumber in Slab

28
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-
• ? ? /
  ?

• Expect instability if resonant particles lie
  around inside of horseshoe.

29
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?

30
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-
• x lt 0 propagating wave with wavevector k0 ? ?
  / c
• 0 lt x lt L layer in which the system is described
  by the dispersion relation which we denote for
  convenience by the general form F(?, k) 0
• x gt L as for 1.

31
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.

32

The End!
33
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 lt 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 k0L, then from the
  above we obtain the relation
• which determines K.

(1)
34
Solutions
K versus Mode No.

• 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.

Real (x) and imaginary () parts of K for the
first 9 modes.
35
Inhomogeneous System Varying Magnetic Field
k

• 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.

?
36
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37
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.

38
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.

39
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

40
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.
41
AKR Laboratory Analogy

42
Energy Gain and Loss
E
Electrons gain energy

When electrons gain energy their mass increases
and cyclotron frequency decreases.
When electrons loss energy their mass decreases
and cyclotron frequency increases.
Electrons loss energy
This will lead to bunching around the
orbit. If - field slips in phase so as to gain
energy from particles in bunch.
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.
E
B0
B
43
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).