spatial zones of resonance for electron...

SPATIAL ZONES OF RESONANCE FOR ELECTRON INTERACTION WITH

MAGNETOSPHERIC PLASMA WAVES

Danny Summers1School of Space Research, Kyung Hee University,

Yongin, Gyeonggi, Korea2Memorial University of Newfoundland

St.John’s, Newfoundland, Canada

Collaborators:Binbin Ni, UCLA, USA

Rongxin Tang, MUN, St.John’s, Canada

1,2

OUTLINE1. Introduction

Wave-particle interactions and radiation belt dynamicsPlasma wave modes in the inner magnetosphere

2. TheoryCyclotron resonanceWave dispersion relations

3. Distributions in ( ) space:(1) electron density(2) cold plasma parameter

4. Resonance zones for field-aligned waves5. Resonance zones for oblique waves6. Conclusions

22pee ωΩ

λ,L

WAVE-PARTICLE INTERACTIONS1. Influence the transport, acceleration and loss of

radiation belt electrons2. Generate relativistic (~MeV) electrons in the outer zone3. Cause microburst precipitation of electrons in the energy

range 100keV ~ several MeV4. Contribute to diffuse auroral precipitation of

100eV~20keV plasmasheet electrons5. Scatter energetic electrons from the ‘slot region’

WAVE MODES IN THE INNER MAGNETOSPHERE1. Outside the plasmasphere:

Whistler-mode (VLF) chorus, magnetosonic waves (equatorial noise)

2. Inside the plasmasphere:Whistler-mode (ELF) hiss, EMIC waves, magnetosonic waves

DOPPLER GYRORESONANCE CONDITIONγω /|||| eNvk Ω=−

=ωθcos|| kk =

=k=θ

=±±= K,2,1N

wave frequency

wave numberwave normal angle

cyclotron harmonic

=−=Ω )/(0 cmBe ee

αcos|| vv ==v=α

=−= − 2/122 )/1( cvγ

electron gyrofrequency

particle speedpitch-angle

Lorentz factor

Field-aligned waves: 1,1,0 +=−== NNθ (R-mode waves) (L-mode waves)

DISPERSION RELATIONS FOR FIELD-ALIGNED EM WAVES

)/)(/()1(1 *

2

ee ssck

Ω−+Ω+

+=⎟⎠⎞

⎜⎝⎛

ωεωαε

ω=Ω= 22* / pee ωα

== 2/120 )/4( epe meNπω

=0N1=s

1−=s

cold plasma parameter

electron number density

(R-mode chorus, R-mode hiss)

(L-mode EMIC waves)

=cpe mm /=ε

=−= 2)1( cmE ek γ

speed of light

kinetic energy

electron plasma frequency

CALCULATION OF RESONANT WAVE FREQUENCIESWave frequencies resonant with a particle of given kinetic energy

and pitch-angle are given by the simultaneous solution of the gyroresonance condition and the dispersion relation.

Resonant wave frequencies satisfy

where the coefficients depend on .

MODEL ELECTRON NUMBER DENSITY DISTRIBUTIONInside the plasmasphere,Outside the plasmasphere,Plasmapause boundary

kEα

0432

23

14 =++++ axaxaxax

ex Ω= /ω

4321 ,,, aaaa 00 ,,, NBEk αEARTH’S DIPOLE MAGNETIC FIELD

,cos/)sin31()(,/),( 62/1230 λλλλ +=== fLBBfBB Eeqeq

equatorial pitch-angle,== eqeqf ααλα ,sin)]([sin 2/1

magnetic shell,=L magnetic latitude,=λ gauss311.0=EB

PPP LLfNLN <<= 1,)]([),(0σλλ

PPT LLfNLN >= ,)]([),(0κλλ

5.3=PPL34383.4 )/3(124,)/3(1390 −− == cmLNcmLN TP (Sheeley et al., 2001)

RESONANCE ZONES

Conditions for gyroresonant wave-particle interaction depend on wave mode, band width, particle kinetic energy , equatorial pitch-angle , L-shell, and magnetic latitude .

For a particle of given energy and given L-shell, over what regions in ( ) space does gyroresonance occur?

For a particle of given energy and given equatorial pitch-angle, over what regions in ( ) space does gyroresonance occur?

kE eqα

λ,L

λ

λα ,eq

Figure 1. Dispersion curves

for field-aligned whistler-

mode chorus, whistler-mode

hiss and EMIC waves.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40Dispersion Curves

ω /|Ωe|

kc/|

e|CHORUSL = 4

* = 0.06

0 0.01 0.02 0.03 0.04 0.05 0.060

1

2

3

kc/|

e|

HISSL = 3

*= 0.009

10 10 10 10 010

10

10

10

10 0

10 1

/ p

kc/|

e|

EMICL = 3

* = 0.009

/| e|

Electron Density Distribution

pp = 3.5

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

(σ, κ) = (0, 0)

Z(R

e)

X (Re)

L

pp = 3.5(σ, κ) = (1, 1)

L

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Z(R

e)

N (cm )

10

100

1000

10000

100000

0

Z(R

e)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

0.01

0.1

1

10

100

1000

10000

0.01

0.1

1

10

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

X (R e)Z

(Re

)

pp = 3.5(σ, κ) = (0, 0)L

pp = 3.5(σ, κ) = (1, 1)L

α *

Figure 2. (left) Model electron density distribution in the inner magnetosphere correspond -ing to the plasmapause location . Inside the plasmasphere ( ) we set the electron density where and outside the plasmasphere we set , where the equatorial plasmasphericdensity and trough density are due to Sheeley et al. (2001). (right) 2D-plot of

corresponding to the electron number density distribution in the left panel.)(LNP

κλλ )]([)(),(0 fLNLN T ×=

5.3=PPL 5.31 << L=)(λfσλλ )]([)(),(0 fLNLN P ×= ,cos/)sin31( 62/12 λλ+

)(LNT22* / pee ωα Ω=

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Z(R

e)

Minimum Resonant Energy (E )mink

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Z(R

e)

X (Re)

CHORUSL = 3.5pp

κ = 0

κ = 1

(keV) 1 10 100 1000 10000 100000

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

100 1000 10000

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

X (Re)

HISSL = 3.5pp

σ = 0

σ = 1

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

(keV) (MeV) 10 100 1000 10000

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

X (Re)

EMICL = 3.5pp

σ = 0

σ = 1

Figure 3. 2D-plot of the minimum electron resonant energy for the given wave modes. The upper cut-off frequency is for chorus, for hiss, and for EMIC waves.

65.0/ =Ωeqeucω Hzuc 20002/ =πω

5.0)/( =Ω eqPucω

Ma

gn

eti

cL

ati

tud

eλ

(de

g)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90Ek = 200 keV, L = 4

90

Equatorial Pitch Angle α eq (deg)0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

60

70

80

90

90

Ma

gn

eti

cL

ati

tud

eλ

(de

g)

CHORUS κ = 0

κ = 1

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Ek = 1 MeV, L = 4CHORUS κ = 0

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

κ = 1

Equatorial Pitch Angle α eq (deg)

0.05 < ω / |Ωe|eq < 0.65(a)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Ek = 200 keV, L = 6

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

0 10 20 30 40 50 60 70 80 90

Ek = 1 MeV, L = 6

Equatorial Pitch Angle α eq (deg) Equatorial Pitch Angle α eq (deg)

CHORUS κ = 0

κ = 1 κ = 1

CHORUS κ = 0

Figure 4. Regions of resonance in ( ) space for chorus-electron interaction at L=4 and L=6, at the given energies, for the wave band

.

λα ,eq

65.0/05.0 <Ω<eqeω

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Ma

gn

eti

cL

ati

tud

eλ

(de

g)

Ek = 200 keV, L = 4

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Ma

gn

eti

cL

ati

tud

eλ

(de

g)

Equatorial Pitch Angle α eq (deg) Equatorial Pitch Angle α eq (deg)

0.05 < ω / |Ωe|eq < 0.4(b)

CHORUS κ = 0

κ = 1

0

10

20

30

40

50

60

70

80

90

κ = 1

0 10 20 30 40 50 60 70 80 90

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Ek = 1 MeV, L = 4

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Ek = 200 keV, L = 6

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90


0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90


0 10 20 30 40 50 60 70 80 90

Ek = 1 MeV, L = 6

CHORUS κ = 0

κ = 1

CHORUS κ = 0

κ = 1

CHORUS κ = 0

Figure 5. Regions of resonance in ( ) space for chorus-electron interaction at L=4 and L=6, at the given energies, for the wave band

.

λα ,eq

4.0/05.0 <Ω<eqeω

0 1 2 3 4 5 6 7 801

234

56

78

0 1 2 3 4 5 6 7 801

23

45

67

8

0 1 2 3 4 5 6 7 801

23

456

78

0 1 2 3 4 5 6 7 801

23

45

67

8

αeq = 100 αeq = 750

X (Re) X (Re)

CHORUSEk = 1 MeVL = 3.5pp

0 1 2 3 4 5 6 7 8012

34

56

78

0 1 2 3 4 5 6 7 801

23

45

678

0 1 2 3 4 5 6 7 801

234

56

78

0 1 2 3 4 5 6 7 801

23

456

78

αeq = 100 αeq = 750Z

(Re)

κ=0

Z(R

e)

κ=1

X (Re) X (Re)

0.05 < ω / |Ωe|eq < 0.65(a)

Ek = 1 MeVCHORUS

L = 3.5pp

CHORUSEk = 200 keVL = 3.5pp

Ek = 200 keVCHORUS

L = 3.5pp

Figure 6. Resonance zones in ( ) space for chorus-electron interaction at the given equatorial pitch-angles and energies, for the wave band

.

λ,L

65.0/05.0 <Ω<eqeω

0 1 2 3 4 5 6 7 8012345678

0 1 2 3 4 5 6 7 8012345678

0 1 2 3 4 5 6 7 8012345678

0 1 2 3 4 5 6 7 8012345678

αeq = 100 αeq = 750

X (Re) X (Re)

0 1 2 3 4 5 6 7 8012345678

0 1 2 3 4 5 6 7 8012345678

0 1 2 3 4 5 6 7 8012345678

0 1 2 3 4 5 6 7 8012345678

αeq = 100 αeq = 750

Z(R

e)κ=0

Z(R

e)κ=1

X (Re) X (Re)

0.05 < ω / |Ωe|eq < 0.4(b)





Figure 7. Resonance zones in ( ) space for chorus-electron interaction at the given equatorial pitch-angles and energies, for the wave band

.

λ,L

4.0/05.0 <Ω<eqeω

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90HISS100 Hz < ω / 2π < 2 kHz

90

Ek = 200 keV, L = 3

σ = 0

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

σ = 1


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

HISS100 Hz < ω / 2π < 2 kHz

σ = 0

Ek = 1 MeV, L = 3

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90


σ = 1

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Ma

gn

eti

cL

ati

tud

eλ

(de

g)

Ek = 200 keV, L = 2

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Ma

gn

eti

cL

ati

tud

eλ

(de

g)


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Ek = 1 MeV, L = 2

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90



σ = 0

σ = 1


σ = 0

σ = 1

Figure 8. Regions of resonance in ( ) space for hiss-electron interaction at L=2 and L=3, at the given energies, for the wave band

.

λα ,eq

HzHz 20002/100 << πω

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

αeq = 100 αeq = 600

X (Re) X (Re)

HISS HISSEk = 1 MeV100 Hz < ω / 2π < 2 kHzL = 3.5pp

Ek = 1 MeV100 Hz < ω / 2π < 2 kHzL = 3.5pp

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

αeq = 100 αeq = 600Z

(Re)

σ=0

Z(R

e)σ=1

X (Re) X (Re)

HISSEk = 200 keV100 Hz < ω / 2π < 2 kHzL = 3.5pp

HISSEk = 200 keV100 Hz < ω / 2π < 2 kHzL = 3.5pp

Figure 9. Resonance zones in ( ) space for hiss-electron interaction at the given equatorial pitch-angles and energies, for the wave band

.

λ,L

HzHz 20002/100 << πω

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Ek = 3 MeV, L = 3

EMIC1/6 < ω / (Ω p )eq < 1/2

σ = 0

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90


σ = 1

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Ek = 10 MeV, L = 3

EMIC1/6 < ω / (Ω p )eq < 1/2

σ = 0

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

σ = 1


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Mag

net

icL

atit

ud

eλ

(deg

)

Ek = 3 MeV, L = 2

EMIC1/6 < ω / (Ω p )eq < 1/2

σ = 0

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Mag

net

icL

atit

ud

eλ

(deg

)


σ = 1

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90

Ek = 10 MeV, L = 2

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

90


EMIC1/6 < ω / (Ω p )eq < 1/2

σ = 0

σ = 1

Figure 10. Regions of resonance in ( ) space for EMIC-electron interaction at L=2 and L=3, at the given energies, for the wave band

.

λα ,eq

2/1)/(6/1 <Ω< eqPω

0 1 2 3 4 5 6 7 8012345678

0 1 2 3 4 5 6 7 8012345678

0 1 2 3 4 5 6 7 8012345678

0 1 2 3 4 5 6 7 8012345678

X (Re) X (Re)

Z(R

e)

σ=0

Z(R

e)σ=1

αeq = 100 αeq = 600

EMIC EMIC

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Z(R

e)σ=0

Z(R

e)σ=1

X (Re) X (Re)

αeq = 100 αeq = 600

CIMECIME

(a)Ek= 3 MeV (b)Ek

= 10 MeV

L = 3.5pp

1/6 < ω / (Ω p )eq < 1/2L = 3.5pp

1/6 < ω / (Ω p )eq < 1/2L = 3.5pp

1/6 < ω / (Ω p )eq < 1/2L = 3.5pp

1/6 < ω / (Ω p )eq < 1/2

Figure 11. Resonance zones in ( ) space for EMIC-electron interaction at the given equatorial pitch-angles and energies, for the wave band .

λ,L

2/1)/(6/1 <Ω< eqPω

resonant wave frequencies satisfy

Whistler-mode waves

)(,]cos)(4[sinsin2),( 02/122242 cmBe PPP =ΩΩ+−−=Φ θωθθθωwhere

DISPERSION RELATIONS FOR OBLIQUE EM WAVES

Coefficients depend oniii dcb ,,

)/)(cos/(11 *

2

ee

ckΩ−Ω

+=⎟⎠⎞

⎜⎝⎛

ωθωαω

EMIC waves ),(2

*

2

θωεαω Φ=⎟

⎠⎞

⎜⎝⎛ ck

Magnetosonic waves )/)(cos/(11 *

2

ee

ckΩ−+Ω

+=⎟⎠⎞

⎜⎝⎛

ωθεωαω

resonant wave frequencies satisfy

ex Ω= /ω

0322

13 =+++ bxbxbx

0652

43

34

25

16 =++++++ cxcxcxcxcxcx

ex Ω= /ω

0432

23

14 =++++ dxdxdxdx

resonant wave frequencies satisfyex Ω= /ω

)2,1,0(,,,,, K±±=NLE eqk θλα

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40Dispersion Curves

ω/|Ωe|

kc/|Ω

e| CHORUSL = 4α* = 0.06

0 0.01 0.02 0.03 0.04 0.05 0.060

2

4

6

8

kc/|Ω

e|

HISSL = 3

α* = 0.009

10 10 10 10010101010100101

ω/Ωp

kc/|Ω

e|

EMICL = 3

α* = 0.009

10 10 1010

100

101

kc/|Ω

e|

MAGNETOSONICθ = 89°

θ = 10°

θ = 50°

θ = 10°

θ = 80°

θ = 10°

θ = 80°

L=4, α*=0.06L=3, α*=0.009

ω/|Ωe|

ω/|Ωe|

Figure 12. Dispersion curves

for obliquely propagating

whistler-mode chorus,

whistler-mode hiss, EMIC

and magnetosonic waves.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11010100101102103104

Minimum Electron Resonant Energies

E min

(keV

) CHORUSL = 4

α* = 0.06θ = 10°

0 0.01 0.02 0.03 0.04 0.05 0.06101

102

103

104

E min

(keV

) HISSL = 3

α* = 0.009θ = 10°

10 10 10 100102103104105106107

ω/Ωp

E min

(keV

) EMICL = 3

α* = 0.009θ = 10°

10 10 10103

104

105

106

E min

(keV

) MAGNETOSONICL = 4

α* = 0.06θ = 89°

N = 1 N = 2

ω/|Ωe|

ω/|Ωe|

ω/|Ωe|

Figure 13. Minimum electron

resonant energy as a function

of normalized wave frequency

for the given wave modes and

cyclotron resonances

.2,1 ±±=N

Ek = 200 keV, L = 4

N=-2

N=-1

N=0

N=1

N=2

λ(deg)

θ = 100κ = 0, θ = 100κ = 1, θ = 500κ = 0, θ = 500κ = 1,

0102030405060708090

λ(deg)

0102030405060708090

0102030405060708090

λ(deg)

0102030405060708090

λ(deg)

0102030405060708090

λ(deg)

α (deg)eq α (deg)eq α (deg)eq α (deg)eq0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80

CHORUS CHORUS CHORUS CHORUS

Figure 14. Regions of resonance in ( ) space for chorus-electron interaction at L=4 and at the given wave normal angles, for the wave band , corresponding to .

λα ,eqkeVEk 200= 65.0/05.0 <Ω<

eqeω2,1,0 ±±=N

Ek = 1 MeV, L = 4

N=-2

N=-1

N=0

N=1

N=2

λ(deg)

θ = 100κ = 0, θ = 100κ = 1, θ = 500κ = 0, θ = 500κ = 1,

0102030405060708090

λ(deg)

0102030405060708090

0102030405060708090

λ(deg)

0102030405060708090

λ(deg)

0102030405060708090

λ(deg)



Figure 15. Regions of resonance in ( ) space for chorus-electron interaction at L=4 and at the given wave normal angles, for the wave band , corresponding to .

λα ,eqMeVEk 1= 65.0/05.0 <Ω<

eqeω2,1,0 ±±=N

Z(R

e)

012345678

Z(R

e)012345678

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Ek = 200 keV, θ = 100

ακ = 0, ακ = 0, ακ = 1, ακ = 1,

N=-2

N=-1

N=0

N=1

N=2

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

CHORUSCHORUS CHORUS CHORUS CHORUS

Figure 16. Resonance zones in ( ) space for chorus-electron interaction at the given equatorial pitch-angles for and the wave normal angle , for

and the wave band .

λ,LkeVEk 200=

65.0/05.0 <Ω<eqeω2,1,0 ±±=N

o10=θ

Z(R

e)

012345678

Z(R

e)012345678

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Ek = 1 MeV, θ = 100

ακ = 0, ακ = 0, ακ = 1, ακ = 1,

N=-2

N=-1

N=0

N=1

N=2

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)



and the wave band .

λ,LMeVEk 1=

65.0/05.0 <Ω<eqeω2,1,0 ±±=N

o10=θ

Z(R

e)

012345678

Z(R

e)012345678

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Ek = 200 keV, θ = 500

ακ = 0, ακ = 0, ακ = 1, ακ = 1,

N=-2

N=-1

N=0

N=1

N=2

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

CHORUSCHORUS CHORUS CHORUS


and the wave band .

λ,LkeVEk 200=

65.0/05.0 <Ω<eqeω2,1,0 ±±=N

o50=θ

Z(R

e)

012345678

Z(R

e)012345678

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Ek = 1 MeV, θ = 500

ακ = 0, ακ = 0, ακ = 1, ακ = 1,

N=-2

N=-1

N=0

N=1

N=2

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

CHORUS CHORUS CHORUS CHORUSCHORUS


and the wave band .

λ,LMeVEk 1=

65.0/05.0 <Ω<eqeω2,1,0 ±±=N

o50=θ

Ek = 200 keV, L = 3

N=-2

N=-1

N=0

N=1

N=2

λ(deg)

θ = 100σ = 0, θ = 100σ = 1, θ = 800σ = 0, θ = 800σ = 1,

0102030405060708090

HISS HISS HISS

λ(deg)

0102030405060708090

0102030405060708090

λ(deg)

0102030405060708090

λ(deg)

0102030405060708090

λ(deg)


HISS

Figure 20. Regions of resonance in ( ) space for hiss-electron interaction at L=3 andat the given wave normal angles, for the wave band ,

corresponding to the cyclotron resonances and Landau resonance .

λα ,eq

keVEk 200= HzHz 20002/100 << πω2,1 ±±=N 0=N

Ek = 1 MeV, L = 3

N=-2

N=-1

N=0

N=1

N=2

λ(deg)

θ = 100σ = 0, θ = 100σ = 1, θ = 800σ = 0, θ = 800σ = 1,

0102030405060708090

λ(deg)

0102030405060708090

0102030405060708090

λ(deg)

0102030405060708090

λ(deg)

0102030405060708090

λ(deg)


HISS HISS HISS HISS

Figure 21. Regions of resonance in ( ) space for hiss-electron interaction at L=3 andat the given wave normal angles, for the wave band ,

corresponding to the cyclotron resonances and Landau resonance .

λα ,eq

MeVEk 1= HzHz 20002/100 << πω2,1 ±±=N 0=N

Z(R

e)

012345678

Z(R

e)012345678

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Ek = 200 keV, θ = 100

αeq = 100

σ = 0, αeq = 600

σ = 0, αeq = 100

σ = 1, αeq = 600

σ = 1,

N=-2

N=-1

N=0

N=1

N=2

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

HISS HISS HISS HISS

Figure 22. Resonance zones in ( ) space for hiss-electron interaction at the given equatorial pitch-angles for and the wave normal angle , for

and the wave band .

λ,LkeVEk 200=

2,1,0 ±±=N

o10=θHzHz 20002/100 << πω

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Ek = 1 MeV, θ = 100

ασ = 0, ασ = 0, ασ = 1, ασ = 1,

N=-2

N=-1

N=0

N=1

N=2

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

HISS HISS HISS HISS


and the wave band .

λ,LMeVEk 1=

2,1,0 ±±=N

o10=θHzHz 20002/100 << πω

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Ek = 200 keV, θ = 800

αeq = 100

σ = 0, αeq = 600

σ = 0, αeq = 100

σ = 1, αeq = 600

σ = 1,

N=-2

N=-1

N=0

N=1

N=2

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

HISS HISSHISSHISS


and the wave band .

λ,LkeVEk 200=

2,1,0 ±±=N

o80=θHzHz 20002/100 << πω

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Ek = 1 MeV, θ = 800

ασ = 0, ασ = 1,ασ = 0, ασ = 1,

N=-2

N=-1

N=0

N=1

N=2

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

HISS HISS HISS HISS


and the wave band .

λ,LMeVEk 1=

2,1,0 ±±=N

o80=θHzHz 20002/100 << πω

Ek = 10 MeV, L = 3

N=-2

N=-1

N=1

N=2

λ(deg)

θ = 100σ = 0, θ = 100σ = 1, θ = 800σ = 0, θ = 800σ = 1,

0102030405060708090

λ(deg)

0102030405060708090

0102030405060708090

λ(deg)

0102030405060708090

λ(deg)


EMIC EMIC EMIC EMICEMIC

Figure 26. Regions of resonance in ( ) space for EMIC wave-electron interaction at L=3 and at the given wave normal angles, for the wave band ,corresponding to the cyclotron resonances .

λα ,eqMeVEk 10= 2/1)/(6/1 <Ω< eqPω

2,1 ±±=N

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

Ek = 10 MeV, θ = 100

ασ = 0, ασ = 1,ασ = 0, ασ = 1,

N=-2

N=-1

N=1

N=2

0 1 2 3 4 5 6 7 8X (R )

0 1 2 3 4 5 6 7 8X (Re )

0 1 2 3 4 5 6 7 8X (R )

0 1 2 3 4 5 6 7 8X (Re)

EMIC EMICEMIC EMIC

Figure 27. Resonance zones in ( ) space for EMIC wave-electron interaction at the given equatorial pitch-angles for and the wave normal angle , for

and the wave band .

λ,LMeVEk 10=

2,1 ±±=N

o10=θ2/1)/(6/1 <Ω< eqPω

E k=

200

keV

E k=

1M

eVE k

=10

MeV

N = 0θ = 890

σ = 0, L = 3 σ = 1, L = 3 κ = 0, L = 6 κ = 1, L = 6

0102030405060708090

λ(deg)

0102030405060708090

λ(deg)

0102030405060708090

λ(deg)


,

MAGNETOSONIC MAGNETOSONICMAGNETOSONICMAGNETOSONIC

Figure 28. Regions of resonance in ( ) space for magnetosonic wave-electron interaction at L=3 and L=6, for the wave normal angle and the given energies, for the wave band , corresponding to the Landau resonance .

λα ,eq

0=No89=θ

0044.0/0026.0 <Ω<eqeω

Z(R

e)

012345678

Z(R

e)

012345678

Z(R

e)

012345678

θ = 890

(σ,κ) = (0, 0)N

=0

N=

0N

=0

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

(σ,κ) = (1, 1)

αeq = 800,

Ek = 10 MeVEk = 10 MeV

Ek = 1 MeV Ek = 1 MeV

Ek = 200 keVMAGNETOSONIC MAGNETOSONICEk = 200 keV

Figure 29. Resonance zones in ( ) space for magnetosonicwave-electron interaction for the equatorial pitch-angle , the wave normal angle and the given energies, and for the wave band , corresponding to the Landau resonance .

λ,L

0=N

o89=θ

o80=eqα

0044.0/0026.0 <Ω<eqeω

Ek = 10 MeV,N=-1

N=1

θ = 890

σ = 0, L = 3 σ = 1, L = 3 κ = 0, L = 6 κ = 1, L = 6λ(deg)

0102030405060708090

0102030405060708090

λ(deg)


MAGNETOSONIC MAGNETOSONIC MAGNETOSONIC MAGNETOSONIC

Figure 30. Regions of resonance in ( ) space for magnetosonic wave-electron interaction at L=3 and L=6, for and the wave normal angle , for the wave band , corresponding to .

λα ,eq

0044.0/0026.0 <Ω<eqeω 1±=N

o89=θMeVEk 10=

Z(R

e)

012345678

Z(R

e)

012345678

Ek = 10 MeV, θ = 890

(σ,κ) = (0, 0)N=-1

N=1

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

0 1 2 3 4 5 6 7 8X (Re)

(σ,κ) = (0, 0) (σ,κ) = (1, 1) (σ,κ) = (1, 1)

αeq = 600αeq = 10

0 αeq = 100 αeq = 60

0MAGNETOSONICMAGNETOSONIC MAGNETOSONICMAGNETOSONIC

Figure 31. Resonance zones in ( ) space for magnetosonic wave-electron interaction at the given equatorial pitch-angles, for and the wave normal angle , for and the wave band .

λ,L

0044.0/0026.0 <Ω<eqeω1±=No89=θ

MeVEk 10=

CONCLUSIONS1. Resonant wave-particle interactions play an essential role in

realistic models that seek to describe the transport, acceleration and loss of radiation belt electrons.

2. For VLF chorus, ELF hiss, EMIC and magnetosonic waves in the Earth’s inner magnetosphere, we have constructed two types of region for electron cyclotron resonance: regions in ( ) space for electrons of a given energy at a given L-shell, and regions in ( ) space for electrons of a given energy and given equatorial pitch-angle.

3. Both types of resonance region are useful in determining where particular wave modes can contribute to the pitch-angle scattering and acceleration of radiation belt electrons.

4. Determination of resonance zones can aid in the modeling of radiation belt electron dynamics as well as in the interpretation of particle and wave observational data.

λ,L

λα ,eq

spatial zones of resonance for electron...

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