INTERACTIONS BETWEEN ELECTROMAGNETIC 
ION CYCLOTRON WAVES AND PROTONS IN 
THE MAGNETOSPHERE: SCATHA RESULTS 
 
 
 
Except where reference is made to the work of others, the work described in this 
dissertation is my own or was done in collaboration with my advisory committee.       
This dissertation does not include proprietary or classified information. 
 
 
 
Son Thanh Nguyen 
 
 
 
Certificate of Approval:  
 
 
Yu Lin J. D. Perez, Chair 
Professor Professor  
Physics Physics 
 
 
 
 
Michael Pindzola Edward Thomas, Jr. 
Professor Associate Professor 
Physics Physics 
 
 
 
Allen Landers George T. Flowers  
Assistant Professor        Interim Dean 
Physics Graduate School 
 
 
 
INTERACTIONS BETWEEN ELECTROMAGNETIC 
ION CYCLOTRON WAVES AND PROTONS IN 
THE MAGNETOSPHERE: SCATHA RESULTS 
 
Son Thanh Nguyen 
 
 
 
 
 
 
 
A Dissertation 
 
Submitted to 
 
the Graduate Faculty of 
 
Auburn University 
 
in Partial Fulfillment of the 
 
Requirements for the 
 
Degree of 
 
Doctor of Philosophy 
 
 
 
 
 
 
 
 
 
Auburn, Alabama 
May 10, 2007 
 
 
 
 
INTERACTIONS BETWEEN ELECTROMAGNETIC 
ION CYCLOTRON WAVES AND PROTONS IN 
THE MAGNETOSPHERE: SCATHA RESULTS 
 
 
Son Thanh Nguyen 
 
 
 
Permission is granted to Auburn University to make copies of this dissertation at its 
discretion, upon request of individuals or institutions and at their expense. The au-
thor reserves all publication rights. 
 
 
 
 
 
Signature of Author 
 
 
 
 
Date of Graduation 
 
 
 
 
 
 
 
 
 
 iii
 
 iv
 
 
VITA 
Son Thanh Nguyen, son of Bay Minh Nguyen and Nam Thi Pham, was born on 
January 25, 1956, in Tien Giang Province, Vietnam. He graduated from Auburn Univer-
sity with a Master of Science degree in Physics in August 2005. He also earned a Master 
of Science degree in Physics from the University of Newcastle, Australia in August 1998 
and a Bachelor of Science degree in Physics from the Hochiminh National University, 
Vietnam in August 1979. He worked as a Physics lecturer at the Pre-University School in 
Hochiminh City, Vietnam from 1980 to 1995, and then went to the University of New-
castle, Australia to study. After graduating he was back to Vietnam and continued to 
work at the Pre-University School until July 2002. He joined the Department of Physics, 
Auburn University in August 2002 to pursue a degree of Doctor of Philosophy. 
 
 
 
 
 
 
 
 
 
 
 v
 
 
 
DISSERTATION ABSTRACT 
INTERACTIONS BETWEEN ELECTROMAGNETIC 
ION CYCLOTRON WAVES AND PROTONS IN 
THE MAGNETOSPHERE: SCATHA RESULTS 
 
Son Thanh Nguyen 
 
 
Doctor of Philosophy, May 10, 2007 
(M.S., Auburn University, 2005) 
(M.S., The University of Newcastle, 1997) 
(B.S., The Hochiminh National University, 1979) 
  
 
168 Typed Pages 
Directed by Joe D. Perez 
 
 
 
Electromagnetic ion cyclotron (EMIC) waves and their role in governing proton 
populations in the ring current and radiation belt regions are of importance in understand-
ing the dynamics of the Earth?s magnetosphere. The availability of SCATHA magnetic 
field and proton spectra data allowed us to study the electromagnetic proton cyclotron 
instability as a generation mechanism for EMIC waves and the relationship between 
EMIC waves and protons in the equatorial region of the magnetosphere. To identify 
EMIC waves and obtain wave power spectra from the SCATHA magnetic field data, we 
employed the fast Fourier transform and spectral analysis techniques. 
 
First, we studied the conditions under which the electromagnetic proton cyclotron 
instability acts as a generation mechanism for EMIC waves in the Earth?s magnetosphere. 
The results are consistent with those of previous studies and/or with theoretical expecta-
tions. Especially, the results show an inverse correlation between proton temperature ani-
sotropy and proton parallel beta, that the observed proton temperature anisotropies are 
above the electromagnetic proton cyclotron instability thresholds, and that the observed 
waves actually get energy from energetic and anisotropic proton populations.  
Second, we studied the relationship between EMIC waves and protons in the re-
gion, and found that there were 20 short time intervals showing correlations between 
EMIC waves and proton perpendicular differential fluxes. They indicate that under suit-
able conditions, i.e., the proton distributions are non-gyrotropic or exhibit gyrophase 
bunching, EMIC waves indeed pitch angle scatter protons either toward or away from the 
local magnetic field. To explain the observations, we also established a model that is 
based on resonant interactions between EMIC waves and protons. In this model individ-
ual protons interact resonantly with the whole or a portion of the spectrum of the existing 
EMIC waves and are then pitch angle scattered with respect to the local ambient mag-
netic field. Calculations based on this model are in agreement with the observations and 
suggest that the EMIC waves are responsible for the changes in the proton distribution by 
pitch angle scattering protons with respect to the local ambient magnetic field.  
 
 
 
 
 vi
 
 vii
 
 
 
ACKNOWLEDGEMENTS 
 
The author would like to acknowledge the advice, guidance, continued support, 
and encouragement received from his advisor, Dr. J. D. Perez. Without his, this disserta-
tion would have not been possible. He wishes to express his gratitude to Dr. Y. Lin, Dr. 
M. Pindzola, Dr. Ed. Thomas, and Dr. A. Landers for their servings as advisory commit-
tee members. His sincere appreciations go to all professors in the Department of Physics, 
Auburn University, who taught him during his time at Auburn University, and to Dr. An-
Ban Chen for his help and encouragement. Thanks are also due to Dr. Joseph F. Fennell, 
Space Science Laboratory, the Aerospace Corporation for providing him with SCATHA 
data, to Dr. X. Y. Wang, the Department of Physics, Auburn University, and Dr. T. M. 
Loto?aniu, the Department of Physics, University of Alberta for their help in IDL pro-
gramming, and to others in the Department of Physics, Auburn University, for their help 
in one way or another.  
Finally, he wishes to thank his wife Na T. Tran, his children, his parents, his 
brothers and sister for their encouragement throughout his study in America. 
 
 
 
 
 
 viii
 
 
Style manual or journal used  AGU Style                                                         
 
 
 
Computer software used  Microsoft Word 2003                                               
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ix
 
 
 
TABLE OF CONTENTS 
 
LIST OF TABLES ......................................................................................................... .xiii  
LIST OF FIGURES ...................................................................................................... .. xiv  
1.  INTRODUCTION ..........................................................................................................1  
     1.1  The Earth?s magnetosphere .....................................................................................1  
     1.2  Proton populations in the magnetosphere ...............................................................4 
     1.3  EMIC waves in the equatorial region of the magnetosphere ................................10 
     1.4  Interactions between EMIC waves and protons ....................................................16 
     1.5  Outline of the current dissertation .........................................................................19 
2.  INSTRUMENTATION AND SPECTRAL ANALYSIS TECHNIQUES                   21 
     2.1  Introduction ...........................................................................................................21 
     2.2  The SCATHA satellite ..........................................................................................21 
     2.3  Spectral analysis techniques ................................................................................ 26 
            2.3.1  Fast Fourier transform .................................................................................26 
            2.3.2  Spectral analysis ..........................................................................................28 
                      2.3.2.1  Auto-power ....................................................................................29 
                      2.3.2.2  Cross-power ...................................................................................29 
          2.3.2.3  Problems associated with the fast Fourier transform .....................30 
            2.3.3  Polarization analysis ...................................................................................33 
 
 x
            2.3.4  Frequency-time analysis (dynamic analysis) ..............................................40 
     2.4  SCATHA data processing procedures ..................................................................41 
2.4.1  Introduction ................................................................................................ 41 
2.4.2  Data processing procedures ........................................................................42 
     2.5  Summary ...............................................................................................................46 
3.  ELECTROMAGNETIC PROTON CYCLOTRON INSTABILITY AS 
     A GENERATION MECHANISM FOR EMIC WAVES IN THE  
     EARTH?S MAGNETOSPHERE .................................................................................47 
     3.1  Introduction ...........................................................................................................47 
     3.2  Data preparation and event selection ....................................................................48 
     3.3  Methodology .........................................................................................................50 
            3.3.1  Electromagnetic proton cyclotron instability ..............................................50 
            3.3.2  Estimation of proton temperature anisotropy .............................................51  
            3.3.3  Estimation of proton parallel temperature ..................................................54  
     3.4  Example ................................................................................................................55  
     3.5  Results and discussions .........................................................................................59 
3.5.1  Inverse relation between proton temperature anisotropy and proton 
          parallel beta .................................................................................................59 
3.5.2  Dependence of proton temperature anisotropy on proton perpendicular 
                      temperature .................................................................................................67  
3.5.3  Dependence of proton perpendicular temperature on wave 
                      normalized frequency ..................................................................................71  
     3.6  Conclusions ...........................................................................................................74 
 
 xi
4.  OBSERVATIONS OF PROTON SCATTERINGS BY EMIC WAVES IN  
     THE  MAGNETOSPHERE  ........................................................................................76  
     4.1  Introduction ...........................................................................................................76 
     4.2  Data preparation ....................................................................................................76 
     4.3  Observational results .............................................................................................77 
     4.4  Examples ...............................................................................................................82 
            4.4.1  Scattering of protons away from the local magnetic field ..........................82 
            4.4.2  Scattering of protons toward the local magnetic field ................................89 
 4.4.3  Scattering of protons toward and away from the local 
          magnetic field ..............................................................................................93 
     4.5  Summary ...............................................................................................................98 
5.  A THEORETICAL MODEL OF SCATTERING OF PROTONS BY  
     EMIC WAVES ............................................................................................................99        
     5.1  Introduction ...........................................................................................................99 
     5.2  Cyclotron resonance between EMIC wave and proton .........................................99  
     5.3  Theoretical model of scattering of protons by EMIC waves ..............................100 
     5.4  Pitch angle scattering of protons by EMIC waves ..............................................105 
     5.5  Explanations for examples in Chapter 4 .............................................................108 
5.5.1  Explanations for scattering of protons away from the local  
          magnetic field ............................................................................................108 
5.5.2  Explanations for scattering of protons toward the local  
          magnetic field ............................................................................................114 
 
 
 xii
5.5.3  Explanations for scattering of protons toward and away from the  
          local magnetic field ...................................................................................117 
5.5.3.1 Scattering toward the local magnetic field in the energy bands 
             centered at 36, 71, and 133 keV ...................................................117 
          5.5.3.2  Scattering away from the local magnetic field 
 
in the energy 
                      band centered at 15.6 keV .............................................................118 
     5.6  Discussions .........................................................................................................122 
     5.7  Conclusions .........................................................................................................125 
6.  SUMMARY AND CONCLUSIONS ........................................................................127   
REFERENCES ...............................................................................................................127 
APPENDIX .....................................................................................................................145       
 
 
 
 
 
 
LIST OF TABLES 
 
3.1 Summary of the relationship between A
p
 and ?
||p
 through the coefficients S
p
  
       and ? of the relation A
p
=S
p
?
?
p||
and the correlation coefficients R for different 
       MLT sectors ...............................................................................................................64   
3.2  Summary of the relationship between A
p
 and ?
||p
 through the coefficients S
p 
         
 and ? of the relation A
p
=S
p
?
?
p||
and the correlation coefficients R for different 
       L ranges ......................................................................................................................64 
3.3  Summary of the relationship between A
p
 and T
?
 through the coefficients a  
       and b of the relation A
p
=aT
?
b
 and the correlation coefficients R for different  
       MLT sectors ...............................................................................................................69 
3.4  Summary of the relationship between A
p
 and T
?
 through the coefficients a  
       and b of the relation A
p
=aT
?
b
 and the correlation coefficients R for different 
       L ranges ......................................................................................................................69 
4.1  List of 20 time intervals during which correlations between EMIC waves  
       and proton perpendicular differential fluxes were observed ......................................81 
 
 xiii
 
 xiv
 
 
 
LIST OF FIGURES 
 
1.1  Three-dimensional view of the Earth?s magnetosphere ...............................................2  
2.1  The SCATHA satellite and its instruments ................................................................23 
2.2  The block diagram of the SCATHA magnetic data processing .................................45  
3.1  From top to bottom, the first panel shows the plots of wave percent polarization  
       versus frequency in the FAC (solid pink line) and GSM (dashed brown line)  
       coordinate systems for the 2-minute interval beginning at 23:24 UT, day 048; 
       the second shows those of ellipticity. The third shows the wave spectra for the  
       same event; the solid pink line is P
tran
 in FAC; the dot-dashed black line is P
para
  
       in FAC; and the dashed brown line is P
tran
 in GSM. The fourth shows the 
       plots of the calculated convective growth rate S (dashed brown line) and P
tran
  
       in FAC (solid pink line) versus frequency for the same event ..................................58 
3.2 The upper panel shows a scatter plot of proton temperature anisotropy versus  
       proton parallel beta with the data binned by MLT values and color-coded; the 
       four dotted lines are the fitting curves for four different MLT sectors. The lower 
       panel shows the same plot as the upper panel, but the data are binned by L values;   
       the three dotted lines are the fitting curves for three different L ranges ....................65 
3.3  A scatter plot of proton temperature anisotropy versus proton parallel beta with 
       the data binned by maximum normalized temporal growth rate values, ?
m
/?
p
,  
 
 xv
       and color-coded. The long-dashed blue line depicts the relation given by  
       Gary et al. [1994a] for ?
m
/?
p
=10
-4
; the short-dashed red line depicts that 
       for ?
m
/?
p
=10
-3
; and the dot-dashed green line for ?
m
/?
p
=10
-2
 ..................................66 
3.4  The upper panel shows a scatter plot of proton temperature anisotropy versus 
       proton perpendicular temperature with the data binned by MLT values and  
       color-coded; the four dotted lines are the fitting curves for four different  
       MLT sectors. The lower panel shows the same scatter plot as the upper panel,  
       but the data are binned by L values; the three dotted lines are the fitting  
       curves for three different L ranges .............................................................................70  
3.5  The upper panel shows a scatter plot of proton perpendicular temperature versus 
       wave peak normalized frequency with the data binned by MLT values and  
       color-coded. The lower panel shows the same scatter plot as the upper panel,  
       but the data are binned by L values ...........................................................................73 
4.1  The first, second, third, and fourth panels from the top show the plots 
       of j
perp
 (solid black lines) and j
para
 (dashed red lines), centered at 133.0, 71.0, 
       36.0, and 15.6 keV, versus universal time for two hours beginning at 13:00 UT,  
       day 045; the fifth panel shows the plot of magnetic field strength versus UT; 
       the sixth and the seventh show the dynamic displays of wave ellipticity  
       and transverse power for the same time interval. The white solid line on 
       the seventh panel indicates the local He
+
 gyrofrequency ..........................................86 
4.2  From top to bottom, the first, second, and third panels show the plots of wave  
       percent polarization, ellipticity, and powers versus frequency for 2-minute 
       interval beginning at 14:35 UT, day 045; the fourth shows the plot of wave  
 
 xvi
       magnetic amplitude versus frequency for the same event .........................................87 
4.3  The upper and lower panels show the plots of the observed j
perp
 and j
para
 versus  
       proton energy for one-minute interval at three different instants. The dashed  
       lines are the plots at the event occurrence instant (14:37 UT, day 045); the 
      dotted lines are those at 14:31 UT, six minutes before the event occurs; and 
       the dot-dashed lines are those at 14:43 UT, six minutes after the event ....................88  
4.4 From top to bottom, the first, second, and third panels show the plots of  
       j
perp 
(solid black lines) and j
para
 (dashed red lines), centered at 133, 71, and 36 keV,  
       versus UT for 2 hours beginning at 06:00 UT, day 073; the fourth shows 
       the plot of B
o
 versus UT; the fifth and the sixth show the dynamic displays 
       of wave ellipticity and transverse power for the same time interval .........................91 
4.5  The same format as Figure 4.2 for 2-minute interval from 07:32 UT, day 073 ........92 
4.6  The same format as Figure 4.1 for two hours beginning at 11:00 UT, day 051 ........95 
4.7  The same format as Figure 4.2 for 2-minute interval from 12:23 UT, day 051 ........96 
4.8  The upper and lower panels show the plots of the observed j
perp
 and j
para
 versus  
       proton energy for one-minute interval at three different instants. The dashed lines 
       are the plots at the event occurrence instant (12:24 UT, day 051);  the dotted  
       lines are those at 12:19 UT, five minutes before the event occurs; and the  
       dot-dashed lines are those at 12:29 UT, five minutes after the event ........................97 
5.1  From top to bottom, the first, second, third, and fourth panels show the plots of  
       proton energy, pitch angle, perpendicular velocily, and parallel velocity versus  
       interacting time for the event at 14:36 UT, day 045, and the energy band centered  
       at 36 keV; the fifth shows the plot of wave frequency versus interacting time.  
 
 xvii
       Initial pitch angle of the proton is 40? .....................................................................112 
5.2  From top to bottom, the first and second panels show the plots of proton energy 
       and pitch angle versus interacting time for 15 different values of initial pitch  
       angle ?
o
 from 72? to 10? for the event at 14:36 UT, day 045; the third shows 
       the plots of proton energy versus pitch angle for the same values of ?
o
. The  
       color code is the same for the three panels ..............................................................113 
5.3 From top to bottom, the first and second panels show the plots of proton energy 
       and pitch angle versus interacting time for 15 different values of initial pitch  
       angle ?
o 
from 86? to 80.1? for the event at 07:33 UT, day 073; the third shows  
       the plots of proton energy versus pitch angle for the same values of ?
o
. The 
       color code is the same for the three panels ..............................................................116 
5.4 From top to bottom, the first and second panels show the plots of proton energy  
       and  pitch angle versus interacting time for 11 different values of initial  
       pitch angle ?
o
 from 88.22? to 86? for the event at 12:24 UT, day 051; 
       the third shows the plots of proton energy versus pitch angle for the same  
      values of ?
o
. The color code is the same for the three panels ...................................120 
5.5  From top to bottom, the first and second panels show the plots of proton energy  
       and pitch angle versus interacting time for 7 different values of initial pitch  
       angle ?
o
 from 35? to 5? for the event at 12:24 UT, day 051; the third shows the  
       plots of proton energy versus pitch angle for the same values of ?
o
. The color 
       code is the same for the three panels .......................................................................121 
 
 
 1 
 
 
 
CHAPTER 1 
INTRODUCTION 
  
In this chapter we will give an overview on the Earth?s magnetosphere, its proton 
populations, EMIC waves in its equatorial region and their interactions with protons, and 
the outline of the current dissertation. 
  
1.1  The Earth?s magnetosphere 
The upper atmosphere of the Sun is so hot that materials there can escape the 
Sun?s gravitational attraction, forming a steady streaming outflow called the solar wind. 
Owing to its high temperature and the constant illumination by the Sun, the solar wind is 
a fully ionized plasma, containing mainly protons, helium ions, and electrons. A weak 
magnetic field (about 7 nT near the Earth?s orbit) is embedded in the solar wind plasma 
and called the interplanetary magnetic field (IMF) [e.g., Hundhausen, 1995].  
When the solar wind encounters a magnetized object, such as the Earth with its 
geomagnetic field, it will confine the magnetic field into a limited region around the ob-
ject. This leads to the formation of the Earth's magnetosphere, which is an elongated cav-
ity, fairly blunt in the sunward direction, and nearly cylindrical for a long distance in the 
anti-sunward direction. Figure 1.1 shows a three-dimensional view of the Earth?s magne-
tosphere [after Stern and Ness, 1982].  
 
 2 
 
 
 The boundary of the magnetosphere is called the magnetopause that separates the 
region occupied by the Earth?s magnetic field from the solar wind and the interplanetary 
magnetic field. The location of the magnetopause is determined by the balance between 
the pressure of the geomagnetic field, combined with the thermal pressure of the magne-
tospheric plasma, and the combined pressure of the solar wind and the IMF. Owing to the 
variability of the solar wind and the dynamics of the solar wind-magnetosphere interac-
tion, the magnetosphere is never in a truly steady state, and in fact it is a dynamic and 
constantly changing structure.  
 Figure 1.1  A three-dimensional view of the Earth?s magnetosphere 
[after Stern and Ness, 1982]. 
 
 3 
There are two models of magnetospheres, the open one and the closed one, corre-
sponding to different processes of transporting mass, momentum, and energy across the 
boundary [e.g., Dungey, 1961; Axford, 1964]. According to the open magnetosphere 
model [e.g., Dungey, 1961], the polar cap field lines extend out into the solar wind and 
connect with the IMF, providing access for the solar wind charged particles into the 
Earth's magnetosphere. The plasma sheet plays the role of a storage region for these par-
ticles. When the IMF connects to the Earth?s magnetic field, it produces open field lines 
with only one foot on the Earth. Under the influence of the solar wind, the magnetic field 
lines in the northern and southern hemispheres are stretched into the magnetotail, forming 
the northern and southern tail lobes. These field lines reaching down the magnetotail have 
opposite directions on the northern and southern sides of the equatorial plane. As a result, 
a current sheet exists to separate the northern and southern tail lobes. The open field lines 
in the current sheet region reconnect to form closed field lines that convect back to the 
Earth.                               
The closed magnetosphere is the one in which the geomagnetic field lines close 
on themselves and form a closed system, and there is no merging with the IMF. In the 
closed model of the magnetosphere, all magnetic field lines have two ends on the Earth, 
and the magnetopause is a closed boundary that isolates the Earth?s magnetic field from 
the solar wind and the IMF. According to Axford and Hines [1961], however, in this 
model ?viscous? interactions may occur, allowing plasma inside and near the flanks of the 
magnetosphere to be weakly coupled to the solar wind flow, possibly by waves that 
propagate across the boundary. 
 
 
 4 
1.2  Proton populations in the magnetosphere 
 Protons within the Earth?s magnetosphere belong to different populations with 
different energies and densities. The energies of these protons extend from thermal values 
(order of eV) to several hundred MeV. The primary sources of these protons, except 
those having energies in the MeV range, are the solar wind and the ionosphere. In terms 
of energies, proton populations may be divided into cold ones with energies of a few eV 
or less and hot ones with much higher energies. 
Cold proton populations 
Cold protons can be found in the ionosphere that is an ionized region of the 
Earth?s upper atmosphere. Its lower boundary is located at 50 to 70 km in altitude 
whereas its higher boundary is not well-defined, but is typically at ~500 km in altitude. 
Although the ionosphere is not usually considered as part of the magnetosphere, it plays a 
role of a cold proton source of other regions in the magnetosphere.  
Another region containing cold protons is the plasmasphere. This is the region de-
fined by the magnetic field lines corotating with the Earth, located at the central core of 
the magnetosphere, above the ionosphere, and extending to L~4-6. The plasmasphere?s 
outside boundary is called the plasmapause. Within the plasmasphere cold protons are 
trapped on the corotating field lines and can reach high densities (102 to 104 cm-3).   
Outside the plasmapause, protons do not corotate with the Earth, but are con-
vected to the magnetopause. The corresponding flux tubes expand to large volumes and 
may be connected to the solar wind plasma. As a result, equilibrium cannot be reached. 
This region constitutes the plasma trough whose cold proton density is of several cm-3 
[e.g., Decreau et al., 1982]. In this region, in situ observations have also found rather 
 
 5 
dense, cold plasma structures that have been identified as detached plasma regions and 
plasma tails/plumes [e.g., Chappell, 1974; Carpenter at al., 1993; Moldin et al., 1996, 
2003]. These structures are observed mainly in the noon-dusk quadrant, and their source 
is believed to be the erosion/drainage of the main plasmasphere owing to the effects of 
the convection electric field [e.g., Moldin et al., 1996]. Cold protons can also be found in 
the polar wind that is formed by the cold plasma leaving the ionosphere. 
Hot proton populations 
There are many energetic proton populations inside the magnetosphere, which are 
far from thermal equilibrium. They have free energies associated with their non-
equilibrium state, available for the excitations of waves. 
Hot protons can be found in several boundary layers of the magnetosphere, in-
cluding the polar cusps (or clefts), entry layers, plasma mantle, and low-latitude boundary 
layer. There are two cusps or clefts (one in each hemisphere). These are narrow, funnel-
shaped regions, elongated in longitude, and extending down from the high latitude mag-
netosphere into the polar ionosphere at 70?-80? geomagnetic latitudes. In each hemi-
sphere there also exists an entry layer near the polar cusp region where the magnetic field 
is so weak that the magnetosheath particles can enter the ionosphere. As a result, the 
plasma in the cusps has much the same properties as the magnetosheath plasma, and the 
cusps are thus regarded as extensions of the magnetosheath plasma down to low altitudes, 
allowing a direct access of particles from the entry layers to the ionosphere and vice 
versa. The plasma mantle is a high-latitude boundary layer, extending anti-sunward from 
the cusp to the distant tail. It contains magnetosheath-like plasma with protons having 
 
 6 
energies from several hundreds of eV to about 1 keV. The low-latitude boundary layer 
extends tailward from near the subsolar point. It also contains magnetosheath-like 
plasma.  
 The tail lobes are regions of low-density (less than 0.1 cm-3) plasma outside the 
plasma sheet boundary layer and also contain hot protons. Their plasma source is the so-
lar wind. The plasma sheet is a region around the neutral sheet, consisting of hot particles 
with nearly symmetric velocity distributions. The thickness of the plasma sheet is typi-
cally in the range of 4-6 RE. The plasma sheet ion population is a mixture of particles of 
solar wind (in quiet times) and ionospheric (in active times) origins [e.g., Wolf, 1995]. 
The plasma sheet also contains hot protons. The plasma sheet proton energy ranges from 
1 to 20 keV, with an average of 5 keV, and the proton number density ranges from 0.1 to 
3 cm-3, with an average of 0.5 cm-3.  
Hot protons are also found in the ring current (RC) and the radiation belts. The 
ring current is composed of energetic (up to hundreds of keV) particles in primarily 
closed, Earth-encircling trajectories, in which ions gradient/curvature-drift westward and 
electrons eastward, and generally occupies the spatial region that extends from about L=3 
to L=7, thus overlaps much of the outer plasmasphere and trough region [e.g., Horwitz 
and Moore, 2000]. Therefore, the outer plasmasphere and trough region may be consid-
ered in general as part of the ring current region. Protons are generally the dominant part 
of the ring current over O+ and He+ ions. Protons in the ring current come from either the 
ionosphere or the solar wind. The ionospheric ions could access to the RC region by ei-
ther direct injection from the ionosphere upward along the magnetic field line or earth-
ward injection from the plasma sheet. The radiation belts are the regions lying well inside 
 
 7 
the magnetosphere where electrons and energetic ions including protons are long-term 
magnetically trapped. The energies of these protons extend up to several hundreds of 
MeV, making them the most penetrating of all particles in the magnetosphere. The ex-
tremely large energies, much larger than any electrical potential energy in the magneto-
sphere, suggest an energy source other than electric and magnetic fields associated with 
the Earth [e.g., Walt, 1994]. One source candidate of MeV protons is the cosmic ray al-
bedo neutron decay (CRAND) process. This process is the one in which a neutron, which 
is produced in the Earth?s atmosphere by collisions of cosmic rays with oxygen and ni-
trogen nuclei, decays into a proton, an electron, and an anti-neutrino. If the decay occurs 
while the neutron is moving through the trapping region, the resulting electron and proton 
can become magnetically trapped. The CRAND process dominates in the innermost re-
gion of the magnetosphere. Finally, the very high energy protons can directly penetrate 
the magnetosphere, owing to their very large Larmor radii [e.g., Vacaresse et al., 1999]. 
It is found that the average energy of protons in the radiation belts increases with decreas-
ing L value. 
Precipitating protons are observed in the auroral region [e.g., Cornwall et al., 
1971; Spasojevic et al., 2004]. It is believed that these protons are initially from the ring 
current or the radiation belts, later pitch angle scattered possibly by EMIC waves, then 
falling into the loss cone, and eventually being removed from trapping. When encounter-
ing the auroral ionosphere, those protons can produce detached subauroral proton arcs 
[e.g., Spasojevic et al., 2004; Fuselier et al., 2004]. 
 
 8 
Now we can see that in the outer plasmasphere and plasma trough region there are 
two distinct proton plasma populations: one cold and one hot. These regions are capable 
of supporting the generation/amplification of EMIC waves, as shown in Chapter 3. 
Processes relating to proton populations 
The mechanisms controlling the source, diffusion, and loss of particles in the 
magnetosphere are different in different energy ranges, and the boundaries between these 
different regimes are themselves moving with the level of magnetic activity. For exam-
ple, regarding radial transport, electric field fluctuations dominate for low-energy parti-
cles and essentially explain the build up of the ring current whereas high-energy particles 
are mainly sensitive to major changes in the magnetic field. 
Cold, plasmaspheric protons can be lost owing to the drift to the magnetopause. 
Energetic ring current protons can be lost owing to the drift to the magnetopause, charge 
exchange, and precipitation. The principal processes affecting trapped protons below 500 
keV in the radiation belts are believed to be magnetic storm injection, charge exchange 
with neutrals, Coulomb collision, radial or L shell diffusion, and pitch angle scattering. 
The Earth?s magnetic field can be drastically modified during very active periods. The 
outer L shells can then be opened to interplanetary space, and trapped protons on those 
shells can be lost through the magnetopause. 
Plasmaspheric protons are subject to the radial transport owing to ExB drift. Ring 
current protons are subject to the radial transport owing to ExB drift and gradi-
ent/curvature drift. Radial diffusion of protons across L shells occurs frequently, even in 
magnetically quiet periods, as a result of acceleration and/or injection. It is well estab-
lished that during the active periods, plasmas are injected earthward from the plasma 
 
 9 
sheet from the nightside region. Particle acceleration and heating are known to accom-
pany this process.  
There is also a process called the pitch angle scattering. In this process the pitch 
angles of protons gyrating about the local magnetic field are changed. For those protons 
that are pitch angle scattered toward the local magnetic field, their pitch angles ? are re-
duced. If their ? fall into the loss cone, those protons are then removed from trapping. 
The loss cone is defined as a cone in the velocity space of the particles that are gyrating 
about the local ambient magnetic field line and mirroring between two mirror points such 
that those within this cone will strike the Earth?s atmosphere and thereby be removed 
from trapping. Related to pitch angles of protons, there is another process called the shell 
splitting. This process causes the proton distributions to become more anisotropic as they 
drift from the nightside around the duskside to the dayside, and protons with small pitch 
angles prefer drifting to lower L shells while those with large pitch angles tend to drift to 
higher L shells [e.g., Sibeck et al., 1987].  
Another process also worth mentioning is the one called plasmasphere refilling, 
by which plasma containing protons and other particles is supplied to the plasmaspheric 
flux tube in the first place. If an empty flux tube is connected to the dense ionosphere, 
ionospheric plasma will flow up and accumulate in this flux tube until the plasma back 
pressure is sufficiently high to shut down further upward flow. This process occurs when 
the magnetic activity or the magnetospheric convection intensity decreases from high to 
low degree, making the boundary between the closed and open flux tube trajectories 
move outward. As a consequence, flux tubes that were previously on open trajectories 
 
 10 
now reside on closed ones. Such flux tubes can retain the plasma injected and accumu-
lated from the ionospheric field-aligned flows [e.g., Horwitz and Moore, 2000].  
 
1.3  EMIC waves in the equatorial region of the magnetosphere 
Wave observations  
The Earth?s magnetosphere with a highly structured configuration can sustain the 
generation and propagation of electromagnetic ion cyclotron (EMIC) waves. EMIC 
waves, with frequencies from ?0.1 to 5.0 Hz, have been observed in the equatorial region 
of the magnetosphere by numerous geostationary and elliptically orbiting spacecraft, such 
as ATS-1, ATS-6, DE-1, ISEE-1 and -2, AMPTE/CCE, and CRRES. EMIC waves have 
been observed on the ground as geomagnetic pulsations. Because the frequency range 
(?0.1 to 5.0 Hz) of EMIC waves is the same as that of the Pc 1-2 pulsations, these waves 
are alternatively called Pc 1-2 waves or sometimes simply Pc 1-2 pulsations. 
The studies of Bossen et al. [1976] (ATS-1), Kaye and Kivelson [1979] (OGO-5), 
Fraser [1985] (ATS-6), Ishida et al. [1987] (ATS-6), and Anderson et al. [1990] 
(AMPTE/CCE) showed that EMIC waves occurred in the equatorial region over a wide 
range of L values, from L~3 to the region near the magnetopause. Anderson et al. 
[1992a], using data from the AMPTE/CCE satellite which covered the region from L=3.5 
to 9 and magnetic latitudes between -16? and 16?, found that EMIC waves were more 
common at high L (L>7) than at low L (L<6) and occurred mainly from 03:00 to 18:00 
MLT. A study by Fraser and Nguyen [2001], using data from the CRRES satellite which 
covered the radial region of L=3.5-8 and magnetic latitudes between -30? and 30?, 
 
 11 
showed that EMIC waves occurred both inside and outside the plasmapause in the after-
noon sector with a clear peak around 14:00 to 16:00 MLT and also in the morning sector 
with a minor peak around 05:00 to 07:00 MLT.  
   EMIC waves observed at lower L values are often associated with magnetic ac-
tivities, such as recovery phases of magnetic storms and/or losses of the ring current ions 
[e.g., Kintner and Gurnett, 1977]. On the other hand, those observed at higher L values 
are more likely to be associated with sunward plasma convections [e.g., Kaye and Kivel-
son, 1979] or with plasma sheet ions [e.g., Anderson et al., 1992b]. 
Wave source region  
One of the problems attracting much attention in EMIC wave studies is the loca-
tion of EMIC wave generation region. The early theoretical studies of EMIC wave gen-
eration by Cornwall et al. [1970, 1971], based on a pure proton model of magnetospheric 
plasma, showed that for wave growth to become significant, relatively high plasma num-
ber densities (?100 cm-3) are required. They predicted that the onset of wave growth may 
be correlated with the plasmapause gradient density. As a result, the plasmapause should 
be the preferential region for EMIC waves to be generated because of the enhanced cold 
plasma density at this boundary. The study by Fraser and Nguyen [2001] as mentioned 
earlier in this section, however, showed that the plasmapause is indeed a region of EMIC 
wave generation and propagation, but not necessarily the preferential region, and that 
EMIC waves were observed not only inside but also outside the plasmapause.   
Liemohn et al. [1967] analyzed middle latitude data and measured the wave dis-
persion to determine the magnetic field lines on which the waves were propagating. Their 
 
 12 
results showed that the waves were propagating on magnetic field lines with equatorial 
intercepts between 4 and 9 RE. Their results were in agreement with those of Fraser 
[1968] that showed that the wave generation region crosses the equatorial plane at posi-
tions between 4.5 and 9.5 RE.  
Criswell [1969], using the cyclotron resonance theory, showed that EMIC waves 
are generated in the magnetospheric regions where wave amplifications have maximum 
values. For the waves with f=0.2-5 Hz, the region is inside the plasmasphere, and for 
those with f=0.3-1 Hz, the plasmatrough is a possible candidate. Their results are to some 
extent in agreement with those of Popecki et al. [1993]. Popecki et al. [1993] used data 
obtained from three high-latitude stations, located at South Pole (-75? MLAT), Sondre 
Stromfjord (+74? MLAT), and Siple (-61? MLAT), to examine the source regions of 
EMIC waves. Their results revealed that for EMIC waves with frequencies above 0.4 Hz, 
the source region was the plasmapause region whereas for those with frequencies in the 
range 0.1-0.4 Hz, the source region was between the plasmapause and the magnetopause. 
Mauk [1982] also noted that there is a tendency for ion cyclotron waves to be generated 
in the regions of minimum magnetic field strength along the field lines. This again sug-
gests the equator as the preferred source region. 
  Erlandson et al. [1990], using data from the Viking satellite (59? to 72? MLAT), 
found that the observed EMIC waves propagated nearly parallel to the ambient magnetic 
field lines and the field-aligned Poynting vector components were almost northward, 
away from the equator and downward toward the ionosphere. They suggested that the 
wave source was on approximately the same field lines as Viking, and presumably near 
 
 13 
the equator. Fraser et al. [1992] performed a calculation of the Poynting vector, using 
data from the ISEE-1 satellite, and found that there was a tendency for the wave propaga-
tion to be nearly parallel or anti-parallel to the ambient magnetic field. Labelle and 
Treumann [1992] calculated the field-aligned Poynting vector component, using 
AMPTE/IRM data, and found that the wave propagation direction was away from the 
equator and the source region was located between 1.7? and 6.8? MLAT, north of the 
equator. 
Wave frequency 
In addition to measuring the wave frequency, f, it is often more beneficial to use 
the normalized frequency, X, defined as the ratio of f to the local proton cyclotron fre-
quency, FCH, when considering in situ satellite data. Mauk and McPherron [1980] and 
Fraser [1985] observed that at synchronous orbit X generally are between 0.05 and 0.6, 
corresponding to f from 0.1 to 1 Hz, over a magnetic latitude range of ?10?. The observa-
tion by Kaye and Kivelson [1979], using data from the highly elliptic satellite (OGO-5, 
L=7-14, MLAT=-30? to 30?), showed that the frequencies of observed EMIC waves fell 
in the range from 0.09 to 0.63 Hz. Bossen et al. [1976], using data from ATS-1, found 
that wave frequencies were typically 10-20% of the equatorial proton cyclotron fre-
quency, FCHeq, and most waves observed had frequencies from 0.17 to 0.32 Hz. 
Recently, Anderson et al. [1992a, b] using 7500 hours of data from the 
AMPTE/CCE satellite, which covered an L range from 3.5 to 9 at magnetic latitudes be-
tween ?16?, provided detailed variations in X. From their results, X was shown to be de-
pendent on L value and magnetic local time. Their results showed that the events with 
 
 14 
X<0.25 were observed mainly between 12:00-18:00 MLT while those with X>0.25 were 
seen between 10:00-16:00 MLT, and at the same local time X tended to increase with L 
value. In the magnetic latitude range of ?16?, their results showed that X decreased with 
absolute value of magnetic latitude, |MLAT|. Their results, however, showed no signifi-
cant variation of f with |MLAT|.  
 Wave power 
McPherron et al. [1972] showed that the power of EMIC waves observed by the 
ATS-1 synchronous satellite was much greater than that of their counterparts at the 
ground conjugate station. Perraut et al. [1978] found that the waves observed by   
GEOS-1 had an average amplitude between 0.3 and 1 nT. EMIC waves at high L values 
(7-14) near the equator, which were observed by the OGO-5 satellite, had large ampli-
tudes, ranging from 2 to 8 nT [e.g., Kaye and Kivelson, 1979].  
Recently, Anderson et al. [1992a, b] studied the wave power distribution over the 
L range from 3.5 to 9 and MLAT between ?16?. According to their results, the average 
power spectral density of the observed waves was between 1 and 10 nT2/Hz, correspond-
ing to an average peak to peak amplitude from 1.0 to 3.5 nT. These results showed that 
the wave power tended to increase with local time from morning to afternoon for L=5-9; 
for the afternoon events, the typical peak to peak wave amplitude was about 2.5 nT 
whereas that for the morning events, it was only 1.6 nT. 
Wave polarization 
In the magnetosphere, the polarization plane of EMIC waves is found to be 
mainly perpendicular to the ambient magnetic field, indicating the transverse nature of 
 
 15 
these waves [e.g., Perraut et al., 1978; Kaye and Kivelson, 1979]. It is expected that 
EMIC waves are generated in the left-hand (LH) mode in the equatorial L=4-8 region of 
the magnetosphere by the electromagnetic proton cyclotron instability. Observations of 
EMIC waves by ATS-6, OGO-5, and GEOS-2 near the equator were consistent with this 
understanding, and the observed waves were mainly left-hand polarized [e.g., Perraut et 
al., 1978; Kaye and Kivelson, 1979; Mauk and McPherron, 1980; Perraut, 1982; Fraser, 
1985]. 
Waves observed off the equator, however, can be LH, RH (right-hand), or linearly 
polarized, depending on frequency. Waves with X<0.25 are still mainly LH polarized, but 
some waves with frequencies between 0.1-0.2 Hz are often linearly polarized, possibly 
due to the presence of oxygen ions in the magnetospheric plasma [e.g., Fraser, 1985]. On 
the other hand, waves with X>0.25 can be left-hand, linearly, or right-hand polarized 
[e.g., Mauk and McPherron, 1980; Perraut, 1982; Fraser, 1985]. 
Young et al. [1981] suggested that as the wave frequency passes through a cross-
over frequency introduced by the presence of ions heavier than protons, the wave polari-
zation sense reverses, and the wave is linearly polarized at that frequency. When an 
EMIC wave propagates away from the equatorial region, at a frequency greater than an 
equatorial crossover frequency, into regions where the magnetic field strength is gradu-
ally increasing, the wave frequency will be eventually matched by the local crossover fre-
quency, and the wave polarization changes from LH through linear to RH. Consequently, 
waves with X>0.25 would be expected to be LH, RH, or linearly polarized, depending on 
the local crossover frequency at the location of the satellite when it is off the equator. 
 
 16 
Anderson et al. [1992b] again contributed comprehensive results on the wave po-
larization in the L=3.5-9 region at magnetic latitudes between ?16?. According to their 
results, there was a clear polarization distinction between morning (AM) events and af-
ternoon (PM) events. While the PM events were significantly LH polarized, the AM 
events were significantly linearly polarized. Especially, the AM events for L>7 showed a 
unique linear polarization, requiring a more extensive explanation because this contra-
dicts the conventional theories of wave generation and propagation in a multi-component 
plasma. Their results also showed that the AM events were predominantly linear at all 
latitudes covered (0 to ?16? MLAT). This provided a challenge for theoretical work be-
cause these events had higher normalized frequencies (X=0.4-0.5) when compared to X of 
the waves observed in the other local time sectors. This finding cannot be explained by a 
change of polarization from LH to RH through linear mode when a crossover frequency 
is encountered during the propagation of waves from low to high magnetic field 
strengths. The authors suggested that the AM waves may be generated with linear polari-
zation and the linear characteristic of wave polarization might not be a propagation effect. 
Not surprisingly, their results showed that the PM waves were uniformly LH polarized at 
all |MLAT| below 10?, and became more linearly polarized at |MLAT| above 10?, again 
supporting the conventional theory of EMIC wave generation. 
 
1.4  Interactions between EMIC waves and protons  
The interactions between EMIC waves and protons are of great interest in the 
studies on the dynamics of the magnetopsphere and have been intensively investigated by 
 
 17 
many authors. First, the results show that energetic (ten to several hundred keV) protons 
with an anisotropic (T?>T||) population are believed to be the source of free energy for 
EMIC waves, and it is expected that EMIC waves with frequency below FCH, the local 
proton cyclotron frequency, are generated and amplified favorably in the equatorial re-
gion at the energy expense of such protons through the electromagnetic proton cyclotron 
instability [e.g., Cornwall, 1965; Kennel and Petschek, 1966; Watanabe, 1966; Criswell, 
1969; Gary et al., 1994a], where T? and T|| are proton temperatures perpendicular and 
parallel to the ambient magnetic field Bo, respectively. Our study covers the electromag-
netic proton cyclotron instability, as shown in Chapter 3. 
Second, EMIC waves have capability of pitch angle scattering trapped protons of 
the ring current or radiation belts. It means that the pitch angles of the interacting protons 
can be altered during the time of interaction with EMIC waves. If owing to interactions 
with EMIC waves, the pitch angles of the interacting protons fall into the loss cone, these 
protons will later be lost when encountering the Earth?s atmosphere. Such protons consti-
tute the precipitation populations of protons observed in the auroral regions. Fuselier et 
al. [2004] and Spasojevic et al. [2004] investigated the existence of dayside subauroral 
proton arcs and found those features are caused by proton precipitations possibly as a re-
sult of pitch angle scatterings by EMIC waves that occur in the equatorial region. Simi-
larly, Frey et al. [2004] found that the IMAGE spacecraft occasionally observed subauro-
ral morning proton spots (SAMPS), and they interpreted those spots as the result of the 
interactions between EMIC waves and protons in the equatorial region of the magneto-
sphere. The process of scattering ring current protons into the loss cone occurs on time 
 
 18 
scales of several hours and makes a significant contribution to the loss of protons from 
the trapped population [e.g., Cornwall et al., 1970; Lyons and Thorne, 1972]. In their 
simulation studies, Jordanova et al. [1997] and Kozyra et al. [1997] considered the diffu-
sive pitch angle scatterings of protons in the ring current by EMIC waves and found that 
it would take many hours to have significant losses of the proton populations. Via pitch 
angle scatterings, EMIC waves play a role in transferring energy from the ring current 
protons to ionospheric electrons. These electrons in turn could generate stable auroral red 
(SAR) arc spectra [e.g., Cornwall et al., 1971; Thorne and Horne, 1992]. Our study also 
covers pitch angle scatterings of protons by EMIC waves, as presented in Chapter 4 and 
Chapter 5. 
Erlandson and Ukhorskiy [2001] using wave and particle data from the DE-1 sat-
ellite showed a simultaneous occurrence of EMIC waves and enhanced proton fluxes in 
the loss cone for 3.5<L<5, demonstrating that EMIC waves indeed scatter protons into 
the loss cone. Yahnina et al. [2000] using data from the low-altitude NOAA satellite and 
from the Sodanskyl? Geophysical Observatory, Finland also discovered a direct relation-
ship between proton precipitation and ground Pc1 pulsations.  
To complete the review of the interactions between EMIC waves and protons, it is 
worth mentioning a different approach employed by Guglielmi et al. [1999]. Their study 
is based on the ponderomotive forces exerted by EMIC wave fields on the interacting 
ions including protons and suggests that the ponderomotive effects play a role in forma-
tion of structure and dynamics of the equatorial region of the magnetosphere. 
 
 
 
 19 
1.5  Outline of the current dissertation 
The aim of the current study is (1) to examine the conditions under which the 
electromagnetic proton cyclotron instability acts as a generation mechanism for the 
EMIC waves observed by the SCATHA satellite in the magnetosphere, (2) to present in 
situ observations of proton pitch angle scatterings by EMIC waves in the magnetosphere, 
and (3) to propose a theoretical model for explaining these observations. The dissertation 
is organized as follows. 
Chapter 2 describes the SCATHA satellite, its experiments, and spectral analysis 
techniques employed in the current study. The fast Fourier transform is used to obtain 
power spectral densities and to generate power spectrograms and dynamic displays of 
polarization ellipticity of EMIC waves. The procedures of processing SCATHA data us-
ing spectral analysis techniques are also presented in this chapter. 
 The electromagnetic proton cyclotron instability as a generation mechanism for 
EMIC waves observed by SCATHA is the focus in Chapter 3. This chapter describes 
how SCATHA data are prepared and how wave events are selected. It is also devoted to 
the methodology of investigating the instability and to the implementation of this investi-
gation. A typical example is given in this chapter. The chapter concludes with the results 
and the main conclusions drawn from this investigation. 
Chapter 4 gives details on the observations of proton scatterings by EMIC waves 
in the Earth?s magnetosphere. It starts with data preparation and then shows observational 
results. Three examples will be given to depict three cases of proton scattering by EMIC 
waves. 
 
 20 
 Chapter 5 proposes a theoretical model for pitch angle scattering of protons by 
EMIC waves. Resonant cyclotron interactions between EMIC waves and protons will be 
introduced in detail. The chapter also shows how the pitch angle scattering of protons by 
EMIC waves takes place and gives explanations for the examples shown in Chapter 4. 
 Chapter 6 summarizes the main results and gives the conclusions drawn from the 
current study. 
 
 21 
 
 
 
CHAPTER 2 
INSTRUMENTATION AND SPECTRAL ANALYSIS TECHNIQUES 
 
2.1  Introduction 
The data used in our study are from the experiments onboard the SCATHA (P78-
2) satellite. Therefore, in this chapter we will give descriptions of the satellite and its ex-
periments. To identify EMIC waves from the SCATHA magnetic field data, we employ 
spectral analysis techniques. The calculations of spectra and polarization parameters of 
EMIC waves are performed by using the fast Fourier transform (FFT). The fast Fourier 
transform is an efficient and rapid method to obtain power spectral density estimates, and 
the FFT algorithm was popularized by Cooley and Tukey [1965], although some of the 
principles of this algorithm were first mentioned as early as 1903 [e.g., Otnes and 
Enochson, 1972]. 
In this study, the SCATHA magnetic field data employed take the form of vector 
measurements (x, y, and z components), which are evenly spaced in time. The time series 
of these data are analyzed in the frequency domain. 
  
2.2  The SCATHA satellite 
SCATHA stands for Spacecraft Charging At High Altitude. The SCATHA space-
craft was launched on January 30, 1979 into a near-synchronous orbit with a perigee of 
 
 22 
5.3 RE, an apogee of 7.8 RE, and an inclination to the geographical, equatorial plane of 
7.8?. The primary mission of SCATHA was to obtain information on the processes and 
effects of spacecraft charging. Its specific objectives included (1) obtaining environ-
mental and engineering data to allow the creation of design criteria, materials, techniques, 
tests, and analytical methods to control charging of spacecraft surfaces and (2) collecting 
scientific data about plasma wave interactions, substorms, and the energetic ring current 
[Fennell, 1982]. 
The satellite?s orbit period was ~23.6 hours, resulting in an eastward drift in lon-
gitude at a rate of ~5.3? per day and radial coverage at all MLTs. The spacecraft was spin 
stabilized with a spin period of one minute, and the spin axis was in the orbital plane and 
maintained approximately perpendicular to the sun-earth line. This allowed those ex-
periments with view axes approximately perpendicular to the spin axis to scan a large 
range of angles relative to the local magnetic field, Bo, and also allowed those sensors on 
the end of the spacecraft with fields of view parallel to the spin axis to be viewing nomi-
nally perpendicular to Bo [Fennell, 1982]. 
The satellite was cylindrical and about 1.75 m in length and diameter. On orbit 
five experiment booms were deployed, including SC1, SC2, SC6, SC10, and SC11. The 
other experiments were SC3, SC4, SC5, SC7, SC8, SC9, and ML12. Most instruments 
were contained in the cylinder?s belly band, an area around the middle of the cylinder 
while solar cells covered most of the remaining part of its side [see Figure 2.1]. The 
shorthand notations SC1, SC8, etc., were designated for each of the instruments on board 
SCATHA and have no special significance other than for accounting purposes [Craven et 
 
 23 
al., 1985]. The twelve experiments had a total mass of 87 kg and consumed 110 W. There 
were three 3-m booms, a 2-m boom, and a 4-m boom, all for deployment of experiments. 
In addition, there was a 101.7-m tip-to-tip electric field antenna. 
 The satellite?s mission lasted nearly 12 full years and returned comprehensive 
particle and field measurements of the near-equatorial and near-synchronous space envi-
ronment [Spence et al., 1997]. Through a recovery effort at the Aerospace Corporation, 
these data have recently been made available. According to Fennell et al. [1997], the re-
covery effort produced High Resolution (HR) and Summary data sets that consist of data 
from the experiments: (a) SC1-8 experiment for measurements of VLF (very low fre-
quency) magnetic and electric wave fields, (b) SC2 experiment for measurements of su-
prathermal plasma and energetic protons, (c) SC3 experiment for measurements of high 
 
Figure 2.1  The SCATHA satellite and its instruments [after Olsen 
and Norwood, 1997] 
 
 24 
energy electrons, (d) SC10 experiment for measurements of electric field, and (f) SC11 
experiment for measurements of magnetic field. 
The HR data sets consist of the highest time resolution measurements from these 
experiments. These measurements have been converted to physical units and written onto 
magneto-optical disks in CDF format [Fennell et al., 1997]. The HR data sets have been 
further processed to get Summary data at ?1-minute intervals. In this study we employed 
59 days (year 1986) of HR magnetic field measurements from the SC11 experiment and 
summary data of proton spectra from the SC2 experiment.  
The SC11 experiment  
The Magnetic Field Monitor (SC11) experiment obtained triaxial measurements 
of the geomagnetic field. A triaxial fluxgate magnetometer was used, and the sensors 
were located at the end of a 4-m boom that was extended perpendicular to the satellite 
spin axis [Fennell, 1982], see Figure 2.1. The sensor axes were aligned accurately with 
the corresponding axes of the spacecraft. The magnetic field vectors were measured four 
times per second, resulting in a sampling interval of 0.25 second. Field resolution was 
approximately 0.3 nT with a dynamic range of approximately ?500 nT per axis. Sensor 
response was from DC (direct current) to 70 Hz. A stringent magnetics control program 
was implemented onboard SCATHA to reduce absolute errors in the magnetic field 
measurements [Fennell, 1982].  
The SC2 experiment 
The SC2 experiment was designed to obtain the electron and ion distribution 
functions over a limited range of energy. It also measured the floating potential of the 
 
 25 
graphite-coated spherical probes relative to the spacecraft background. It employed 
plasma sensors that consisted of three miniature analyzers. These sensors measured the 
fluxes of electron and ion distribution functions over a range of energy from several tens 
of eV to about 20 keV. Two of these analyzers (SC2-1 and SC2-2) were contained in the 
boom-mounted spherical probes, and the third (SC2-3) was mounted behind the center 
band of the satellite, see Figure 2.1. The fields of view of these analyzers were parallel to 
each other and to those of the energetic ion sensors. Plasma data obtained by the SC2-3 
sensor, which was mounted on the satellite body, were included in the summary data 
[Fennell et al., 1997].  
The energetic proton (SC2-6) and ion plus background (SC2-3B) sensors pro-
vided measurements of protons, alphas, carbon, nitrogen, and oxygen ions (the CNO 
group) with emphasis on the pitch angle coverage with good resolution [Fennell et al., 
1997]. The experiment recorded the temporal, spatial, and directional variations in posi-
tive ion fluxes. The proton measurements covered the energy range from 17 keV to 3.3 
MeV. The solid-state particle detection technique was used in the experiment to obtain 
instantaneous energy spectra for each one-second interval (?6? angular sampling resolu-
tion). The proton detection system (SC2-6) employed a proton telescope that consisted of 
front and rear solid-state detectors behind a collimator-magnet assembly. After entering 
the collimator, particles passed through a uniform magnetic field that prevented electrons 
from being detected and contaminating the proton measurements. The uniform magnetic 
field had no influences on the proton trajectories. The detector in front was energy ana-
lyzed whereas the detector at rear eliminated penetrating particles from the analysis and 
 
 26 
provided necessary background information. The proton analysis gave differential flux 
measurements in six energy windows, from 0.017 to 0.717 MeV, and two integral flux 
channels. The geometric factor of the proton telescope was 2x10-3 cm2sr. The temporal 
resolution was one second for protons, and 250 ms for alphas and the CNO group 
[Fennell, 1982]. 
The summary proton data were acquired at ten energy channels, seven from the 
SC2-3 sensor (centered at 0.154, 0.360, 0.88, 2.06, 3.60, 8.80, and 15.6 keV) and three 
from the SC2-6 sensor (centered at 36.0, 71.0, and 133.0 keV) [Fennel et al., 1997]. For 
proton differential fluxes, ?perpendicular? refers to those having pitch angles of 90??20? 
for the SC2-3 measurements and 90??10? for the SC2-6 measurements; ?parallel? refers 
to those having pitch angles of 0??30? or 180??30?. There are no measurements of fluxes 
of protons with pitch angles in between those referred to as ?perpendicular? and ?parallel?. 
We also use average energetic (0.15 to 133 keV) proton number density, nh, and 
spacecraft ephemeris data, such as magnetic local times (MLT), magnetic latitudes 
(MLAT), L shell values, and spacecraft geocentric radii (ERAD). These data were in-
cluded in the summary data. Further details are provided by Fennel et al. [1997]. nh are 
needed for calculating EMIC wave convective growth rates, as shown in Chapter 3. 
 
2.3  Spectral analysis techniques  
 2.3.1  Fast Fourier transform 
   The fast Fourier transform (FFT) algorithm can be employed to compute a Dis-
crete Fourier transform (DFT). The DFT can be used to characterize the magnitude and 
 
 27 
phase of a signal. Consider a finite sequence of N signal data points that are sampled at 
intervals of ?t, x(j)=x(j?t), where j=0, 1, 2, ..., N-1, and ?t is the sampling interval. The 
DFT of x(j), denoted as X(n), has the form  
 
          )i2-(exp )(  = )( 1-
0= N
njjxnX g49
g77
pi?                              (2.1) 
 
for n=0, ..., N-1, where i=(-1)1/2. The inverse DFT is defined by 
 
                           )i2(exp )( 1 = )( 1-
0= N
njnX
Njx
g49
g81
pi?                                    (2.2) 
      
Equations (2.1) and (2.2) make up the Fourier transform pair. The value n in X(n) is re-
lated to the frequency by    
      
    f(n)= tNn?                                         (2.3) 
 
where N is the number of elements of the sequence considered and N?t the total length of 
the sample. Using Equation (2.1) implies that the analysis is undertaken in the frequency 
domain, and the sampling interval in the frequency domain is given by 
 
 
 28 
       ?f= tN?1                                         (2.4) 
 
?f defines the frequency resolution, and ?t the time resolution. It is seen from Equation 
(2.4) that the product of ?f and ?t is a constant for a given value of N. It indicates that 
increasing the frequency resolution reduces the time resolution and vice versa. 
Because Equations (2.1) and (2.2) are of the same form, a single FFT algorithm 
can be used to perform both transformations, and we can pass between the time series x(j) 
and its DFT X(n) and back without losing information. The reason for using the FFT is its 
capability of time saving, owing to the efficiency of the FFT algorithm. The number of 
mathematical computations required is proportional to Nlog2N instead of N2 for the direct 
implementation of the DFT if N is a highly composite number such as a power of 2. 
The DFT yields frequency and amplitude information. If x(j) is a sequence in the 
time domain, the sequence X(n) can be regarded as the complex amplitude spectrum of 
the time sequence. At each frequency, the complex components are the amplitudes of a 
sine wave and a cosine wave, and their sum gives a sine wave that has amplitude of 
{X(n)X*(n)}1/2 and phase of tan-1 {Im (X(n))/Re (X(n))}. The asterisk * denotes the com-
plex conjugate. 
 
 2.3.2  Spectral analysis 
The power spectrum of a signal, which is represented by a discrete time series x(j) 
in the time domain, gives information on which frequencies contain the signal?s power in 
 
 29 
the form of a distribution of power values as a function of frequency. In the frequency 
domain, it is the square of X(n), the DFT of the series x(j). 
 
2.3.2.1  Auto-power  
Estimates of auto-power spectral density or the auto-power spectrum, Pxx(n), of a 
time series x(j)=x(j?t), where j=0, 1, 2, ..., N-1, is defined as  
 
Pxx(n)=A|X(n)|2=AX(n)X*(n)                                        (2.5) 
 
where n=0, 1, 2, ..., N-1; X(n) is defined in Equation (2.1), and X*(n) is its complex con-
jugate. A is the normalized factor that is equal to 2?t/N [Otnes and Enochson, 1972]. 
For real data series, the estimates obtained from Equation (2.5) are symmetrical 
about N/2, and the number of independent frequencies is thus an integer equal to N/2+1. 
For N/2+1<n?N-1, Pxx(n) can be regarded as the power spectral density estimates for 
negative frequencies [e.g., Koreda et al., 1977]. Equations (2.1) and (2.5) show that 
power spectral density estimates can be achieved by using the Fourier transform. 
 
2.3.2.2  Cross-power 
Estimates of cross-power spectral density or the cross-power spectrum, Pxy(n), be-
tween two time series x(j) and y(j) is similarly defined as follows 
 
          Pxy(n)=AX(n)Y*(n)                     (2.6) 
 
 30 
where Y(n) is the Fourier transform of the time series y(j) and Y*(n) its complex conju-
gate. 
 
2.3.2.3  Problems associated with the fast Fourier transform 
The power spectrum, obtained from a finite length of data using the FFT, is dis-
torted, and considerable effort must be spent minimizing this effect. There are two poten-
tial problems associated with the power spectrum calculation using the FFT, namely 
aliasing and leakage. These problems exist inherently in the Fourier transformation and 
can give rise to spectrum distortion; consequently they should be well understood. 
Aliasing 
Consider a sinusoidal signal (wave), which is sampled at a uniform interval of 
time, ?t. If ?t is not chosen appropriately, high frequency components of the time series 
can fold back into the lower frequencies leading to distortion of the spectrum. In practice, 
it is impossible to distinguish without ambiguity frequencies above a certain value, called 
the folding frequency or Nyquist frequency, FNYQ. The relationship between the Nyquist 
frequency and the sampling interval, ?t, is given by 
 
  FNYQ= ?t21                        (2.7) 
 
Equation (2.7) shows that the sampling interval should be large enough to sample 
the highest frequency at least twice during each period [e.g., Cooley et al., 1969]. In other 
words, it is necessary to have at least two samples per cycle to detect any frequency  
 
 31 
component. The frequency interval, 0?f?FNYQ, is called the principal interval. No power 
spectral density estimates can be obtained at frequencies above the Nyquist frequency. 
To eliminate aliasing one can choose an appropriate sampling interval and low-
pass filter the time series to filter out frequencies higher than FNYQ before the Fourier 
transformation is performed. The sampling interval should be carefully chosen so that the 
spectrum of the time series is not aliased, and harmonics with frequencies higher than 
FNYQ must be attenuated by a high-cut filter before the spectrum estimation is carried out. 
Alternatively, all powers above the Nyquist are removed before digitizing; this is done by 
applying an anti-alias filter to the analogue signal [e.g., Gubbins, 2004]. 
 Leakage and windowing 
It is implicit that a rectangular window exists for a finite sequence of N sampled 
points. The truncation of the time series can be considered as a multiplication of an infi-
nitely long record by that window. The existence of this window leads to another prob-
lem, called leakage, because the Fourier transform of this window is a function of the 
form sin (x)/x, which has a main peak and subsidiary peaks of decreasing amplitudes at 
nearby frequencies. These subsidiary peaks are called side-lobes, and their appearance 
gives rise to the leakage. Leakage is a problem because it can not only change the peaks 
in the power spectrum but also obscure the structure of the true spectrum. If leakage ex-
ists, the true spectrum of the time series is distorted because the main peak is altered by 
leakage to other frequencies. In other words, the effect of leakage is to smear the power 
spectral density (PSD). Since leakage is an inherent problem in the Fourier transforma-
tion of time series, it cannot be prevented but can be reduced. The procedure of reducing 
 
 32 
leakage involves reducing or suppressing the heights of the side-lobes. To do this a tech-
nique, called windowing, is employed. Owing to the fact that the implicit rectangular 
window, as mentioned earlier, is the source of leakage, the obvious alternative is to mod-
ify the window shape. Before transforming, the time series x(j) is therefore multiplied by 
a data window, W(t), of width T=N?t, which is no longer constant but maximum at the 
center (t=1/(2T)), symmetrical about t=1/(2T), and tapers away toward the ends (t=0, T) 
[e.g., Buckley, 1971]. The applied window will effectively taper the data at the ends of 
the time series to the mean and reduce the heights of the side-lobes.     
There are several windows that can be used for this purpose, such as Parsen, 
Hamming, and Hanning. No window is completely optimum. In this study the Hamming 
window is employed and is described as follows 
 
  W(t)=0.54+0.46cos (2pit/T)                             (2.8) 
           
where t is time. The Hamming window is chosen due to its capability of minimizing the 
heights of the main side-lobes. If the Hamming window is applied to the time series x(j) 
before the Fourier transformation, the modified auto-power spectral density will be  
 
Pxx(n)=(A/U)|C(n)|2                                  (2.9) 
 
 
 33 
where C(n) is the DFT of c(j), defined similarly by Equation (2.1), and c(j) is the multi-
plication of the time series, x(j), and the discrete form, W(j), of the Hamming window 
W(t), and  
     
    )( 1 = 21-
0=
jWNU g49
g77
?                                          (2.10) 
 
A similar process is also applied to obtain the modified cross-power spectral density [e.g., 
Buckley, 1971]. 
 
 2.3.3  Polarization analysis 
The interpretation of EMIC waves requires certain knowledge of the polarization 
of these waves. The contents of this section on wave polarization are based on the studies 
by Fowler et al. [1967], Rankin and Kurtz [1970], Means [1972], Kanasewich [1973], 
and Koreda et al. [1977]. The fundamental assumption is that EMIC waves are quasi-
monochromatic.  
Consider a wave propagating in the z direction with two perpendicular field com-
ponents 
 
     hx(t)=axexp{i(2pift+?1)}                                         (2.11) 
     hy(t)=ayexp{i(2pift+?2)} 
 
 
 34 
In a right-handed Cartesian coordinate system, Equations (2.11) represent an elliptically 
polarized wave propagating in the direction perpendicular to the plane containing the hx 
and hy components. According to Fowler et al. [1967], the polarization of any wave can 
be characterized in terms of the covariance matrix elements of the wave field. The two-
dimensional covariance matrix of the wave described by Equations (2.11) is defined as 
follows  
 
        
)()()()( 
)()()()( 
 = 
*yy*xy
*yx*xx
?
?
?
?
?
?
?
?
?
?
><><
><><
thththth
thththth
J         (2.12) 
 
where the angle brackets represent the time average. 
 According to Means [1972], the covariance matrix is defined in terms of the two 
functions hx(t) and hy(t) that are analytic representations of a measured signal, and, as 
such, are complex. This fact complicates the generation of the analytic representation of 
the covariance matrix in the time domain. In the frequency domain, however, the analytic 
representations of a real signal are easily obtained by multiplying their Fourier transforms 
by the Heaviside step function 
 
       S(f)=0,   f<0                       (2.13) 
        S(f)=1,    f?0 
 
and doubling the resultant values. Then by applying the Parseval?s theorem 
 
 35 
 
      dffHfGdtthtg )()()()( ** ?=? ??????                                  (2.14) 
 
where G(f) and H(f) are the Fourier transforms of g(t) and h(t), respectively, we can ex-
tend the definition of the covariance matrix to the frequency domain. The resulting matrix 
is   
        
)}()({)}()({ 
)}()({)}()({ 
 = 
*yy*xy
*yx*xx
?
?
?
?
?
?
?
?
?
?
fHfHfHfH
fHfHfHfH
J              (2.15) 
 
where the braces represent the average over all the positive frequencies, and the functions 
Hx(f) and Hy(f) are the Fourier transforms of hx(t) and hy(t), respectively.  
Equation (2.15) indicates that the covariance matrix elements are the average 
auto-power and cross-power spectral density estimates of the wave of interest. If the 
wave is monochromatic, the amplitudes (ax and ay) and phases (?1 and ?2) are invariant 
with time. As a result, the wave field is totally polarized, and the covariance matrix J is 
equal to matrix P whose elements are the averaged Pxx, Pxy, Pyx, and Pyy, as defined ear-
lier in Section 2.3.2.  
For a totally polarized wave described by Equations (2.11), the locus of the end 
point of the field vector is an ellipse, called the polarization ellipse, which is contained in 
a rectangle of sides of 2ax and 2ay. The ellipticity, ?, of the wave is defined as the ratio of 
 
 36 
the minor axis to the major axis of the polarization ellipse. Rankin and Kurtz [1970] 
showed that 
 
    ?=tan ?                 (2.16) 
 
where         
 
               sin ?=2Im (Pxy)/(Pxx+Pyy)                 (2.17) 
         
The angle of polarization, ?, between the major axis of the polarization ellipse and x-axis 
of the Cartesian coordinate system can also be obtained by 
 
tan 2?=2Re (Pxy)/(Pxx-Pyy)                 (2.18) 
 
In general, the amplitudes (ax and ay) and phases (?1 and ?2) are functions of time. 
For a quasi-monochromatic wave, the complex amplitudes (ax and ay) remain substan-
tially constant only during a time interval, ??, which is called the coherence time. Gener-
ally, a quasi-monochromatic signal may be considered as the sum of a totally polarized 
signal (monochromatic), whose polarization is characterized by the elements of covari-
ance matrix P, and a completely unpolarized signal, whose polarization is characterized 
by the elements of covariance matrix U. Consequently, the covariance matrix of a quasi-
monochromatic signal is now 
 
 37 
    J=P+U                        (2.19) 
 
For the totally polarized portion, there is a total coherence between the two signal com-
ponents. On the other hand, there is no coherence between two components of the com-
pletely unpolarized portion. Due to the Hermitian property of the matrices J and P, the 
elements of U satisfy the following: Uxy=Uyx=0 and Uxx=Uyy=D?0. Therefore  
 
                    
0 
0  = 
??
?
??
?
D
DU                                 (2.20) 
 
From Equations (2.19) and (2.20), we have  
 
   Jxx=Pxx+D 
   Jxy=Pxy 
Jyx=Pyx=Pxy*                                         (2.21) 
                        Jyy=Pyy+D 
 
By solving for D from Equations (2.21), we obtain  
 
            D=1/2(Jxx+Jyy)?1/2[(Jxx+Jyy)2-4Det |J|]1/2                            (2.22) 
 
 
 38 
where Det |J| is the determinant of the covariance matrix J. Because Pxx and Pyy are posi-
tive, only the negative sign before the radical is accepted [e.g., Fowler et al., 1967]. Sub-
stituting for D in Equations (2.21) leads to  
 
                    Pxx+Pyy=[(Jxx+Jyy)2-4Det |J|]1/2                                 (2.23) 
                        Pxx-Pyy=Jxx-Jyy        
 
The polarization parameters can now be obtained as follows 
a.  Ellipticity  
 Similar to the case of monochromatic wave, for the case of quasi-monochromatic 
wave, we also have 
 
   ?=tan ?                                            (2.24) 
 
From Equations (2.17), (2.21), (2.22), and (2.23), the value of ? is given by 
 
        sin ?=2Im (Jxy)/[(Jxx+Jyy)2-4Det |J|]1/2                           (2.25) 
 
The sense of polarization is given by the sign of ?, ?>0 for clockwise rotation when look-
ing into the propagating wave (right-hand polarized), and ?<0 for counter-clockwise rota-
tion (left-hand polarized).   
 
 
 39 
b.  Angle of polarization 
By using Equations (2.18), (2.21), and (2.23), we obtain 
 
   tan 2?=2Re (Jxy)/(Jxx-Jyy)                                       (2.26) 
 
c.  Degree of polarization 
The degree of polarization of a quasi-monochromatic wave, denoted by R, is de-
fined as the ratio of the polarized power to the total power. By definition we have 
 
                    R=TR |P|/TR |J|=(Pxx+Pyy)/(Jxx+Jyy)                (2.27) 
 
where TR |P| and TR |J| are the traces of the covariance matrices P and J, respectively. 
From Equations (2.21), (2.22), (2.23), and (2.27), we find that  
 
               R=[1-4Det |J|/(Jxx+Jyy)2]1/2                            (2.28) 
 
d.  Coherency 
Coherency between the x and y components, denoted by Cxy, is given by 
 
     Cxy=|Jxy|/[JxxJyy]1/2 =[JxyJyx/JxxJyy]1/2                                (2.29) 
  
Cxy=1 for a totally polarized signal, and Cxy=0 for a completely unpolarized signal. 
 
 40 
 We now see that the elements of the covariance matrix and consequently the po-
larization parameters of a wave propagating in the z direction can be formulated and 
evaluated in terms of auto-power and cross-power spectral density estimates obtained 
from the measurements of x and y components.  
 
2.3.4  Frequency-time analysis (dynamic analysis) 
Dynamic spectral analysis, employing a representation of data in the frequency 
domain with time, has become popular in the field of time series analysis. In the current 
study, this technique is used to generate dynamic power spectra (or spectrograms) and 
dynamic displays of polarization ellipticity of EMIC waves identified from the SCATHA 
magnetic field component measurements. 
The basic idea for this technique is to use a two-dimensional display to represent a 
three-dimensional quantity such as the wave power spectral density as a function of fre-
quency and time. It is performed by sliding an FFT window along the time series record 
in steps of one minute or less and plotting successive spectra along a horizontal time axis. 
Thus, time is plotted on the abscissa and frequency on the ordinate. The third variable, 
which can be the wave power spectral density or polarization parameter at a certain fre-
quency and time, will be represented by intensity modulation using a suitable color or 
grey scale palette. A diagram illustrating this technique is shown below 
 
 
 
 
 
 41 
|-------------------------------------------------------------------   -----------| Ntot 
[-------- NFFT ------------------------] 
<-- Nstep ------->[-------- NFFT ------------------------]   
    <-- Nstep ------->[-------- NFFT ------------------------] 
----------------etc. 
 
where Ntot is the total number of data points of the record, NFFT the FFT length, and 
Nstep the number of data points equivalent to a step of one minute or less. The number of 
FFT blocks (Nblock) calculated for a data record of the length Ntot, is given below 
 
Nblock=(Ntot-NFFT)/Nstep                            (2.30) 
 
The product, ?tNstep, where ?t is the sampling interval, gives the temporal reso-
lution of the spectrogram. Before plotting, time-averaging is performed spectrum by 
spectrum using a certain number of adjacent blocks to reduce the variance and increase 
the statistical reliability. 
 
2.4  SCATHA data processing procedures 
2.4.1  Introduction  
This study employed magnetic field data from the SCATHA satellite. To analyze 
the data, the procedures shown in the block diagram in Figure 2.2 were implemented. The 
aim of the SCATHA data analysis is to obtain dynamic power spectra and ellipticity of 
 
 42 
EMIC waves from the three-component measurements of magnetic field, which will then 
be used to identify EMIC waves and to find their relationship with the concurrently ob-
served proton spectra. The spectral analysis techniques, described in Sections 2.3 and 2.4, 
were adopted to generate those dynamic displays. 
 
2.4.2  Data processing procedures 
 SCATHA data acquisition 
We obtain 59 days of high resolution (HR) and summary SCATHA data con-
tained in CDF (Common Data Format) files from Dr. Fennell (Space Science Laboratory, 
the Aerospace Corporation). From these files we extracted the three-component meas-
urements of magnetic field in the GSM coordinate system, in the form of time series with 
a sampling interval of 0.25 second. By Equation (2.7), the Nyquist frequency is 2 Hz. The 
frequency range from 0 to 2 Hz is expected to capture the majority of EMIC wave events, 
as data from 700 CRRES orbits showed that less than 5% of the total number of EMIC 
wave events observed had frequencies above 2 Hz and no EMIC waves with frequencies 
above 4 Hz were found [e.g., Fraser and Nguyen, 2001]. Note that the CRRES satellite 
covered the radial region of L=3.5-8 and MLAT between -30o and 30o, compatible with 
that covered by SCATHA. 
Data conditioning 
The HR magnetic field time series were first subjected to procedures that remove 
data gaps and spikes, if any. The value of the first data point of a gap was set to be equal 
to the nearest previous true value, and this process continued to the end of the gap. The 
information in the gap under consideration is still not valid, but the information on both 
 
 43 
sides of the gap remains valid, and on dynamic displays the gap corresponds to a blank 
chunk that contains no wave power. This gap removal process can handle as many data 
points as required. A detrending procedure, using a least-square second order polynomial, 
was also included to separate the variation part from the background geomagnetic field.  
Coordinate conversion 
The magnetic field data extracted from the CDF files are represented in the GSM 
coordinate system. Descriptions of the GSM coordinate system are given in the Appen-
dix. Because EMIC wave properties are organized by the local ambient geomagnetic field 
Bo, a transformation from the GSM coordinate system to a field-aligned coordinate 
(FAC) system was then carried out prior to the Fast Fourier transformation and spectral 
analysis. The instantaneous magnetic measurements were resolved into three components 
of a right-handed system: one parallel to Bo, the others perpendicular to Bo.  
Mean removal and windowing 
Before performing the fast Fourier transformation, the mean of every subset of 
data was also removed and the Hamming window applied to reduce leakage, as discussed 
in Section 2.3.  
Fast Fourier transformation 
  The three-component magnetic data sampled at 4 Hz were analyzed day by day. 
To obtain dynamic power spectra and polarization displays, the FFT was used, as dis-
cussed in Section 2.3. The FFT length N was chosen to be 240. Consequently, the fre-
quency resolution is ?f=1/(N?t)=1/60=0.0167 Hz. To perform dynamic spectral analysis, 
the 24-hour time series of magnetic field components were broken into 60-second (240 
 
 44 
data point) overlapping segments. The portion of overlap was set at 2/3, equivalent to a 
sliding step of 80 data points or 20 seconds (Nstep=80). The number of FFT blocks 
(Nblock) performed each day is given by Equation (2.30) with NFFT=240, Nstep=80, and 
the number of data points for each magnetic field component per day equal to 
Ntot=24x3600x(1/?t)=345600. As a result, the number of blocks to be displayed per day 
is 4318.  
Smoothing 
To increase the reliability of the results, time-averaging was also performed be-
fore plotting.  
Dynamic power spectra and ellipticity displays  
For each day, we generated dynamic power spectra of transverse and parallel 
magnetic components in the FAC coordinate system and displays of dynamic polarization 
ellipticity of EMIC waves in the frequency range from 0 to 2 Hz. The calculation of 
power spectral density was based on Equation (2.9), and the calculation of polarization 
parameters on Equations (2.24), (2.25), and (2.28). The obtained dynamic power spectra 
and ellipticity displays were used to identify the wave occurrences. We also produced 
plots of wave powers, ellipticity, and degree of polarization versus frequency for two-
minute intervals of data. These plots were used to obtain values of PSD, wave peak fre-
quency, ellipticity, and degree of polarization at any given time.  
The block diagram of the processing of SCATHA magnetic data is shown in   
Figure 2.2.  
 
 45 
 
SCATHA magnetic field data 
in the GSM coordinate system 
 
Gap removal 
Spike removal 
Polynomial detrending 
 
Coordinate conversion 
from GSM to FAC 
Mean removal 
Hamming window applying 
 
Fast Fourier transformation 
 
Smoothing 
 
Dynamic power display Dynamic ellipticity display 
Figure 2.2  The block diagram of the SCATHA magnetic data 
 processing.  
 
 46 
2.5  Summary 
 The SCATHA magnetic field data in the form of time series with the sampling 
interval of 0.25 second are suitable for investigating EMIC waves in the equatorial region 
of the magnetosphere. Moreover, the availability of SCATHA proton spectra data makes 
possible the finding of the relationship between EMIC waves and protons in this region. 
The fast Fourier transform is a very advanced and rapid technique for calculating 
wave power spectral density and polarization parameters from the measured magnetic 
field time series. For a finite sequence of N data points, the number of mathematical 
computations required for the DFT is proportional to Nlog2N instead of N2. There are two 
inherent problems associated with the FFT, namely aliasing and leakage. Solutions to 
both problems are also discussed. Because EMIC wave properties are organized by the 
local ambient geomagnetic field Bo, a coordinate conversion of the magnetic data from 
the GSM system to the FAC system is carried out prior to the fast Fourier transformation.  
The wave power spectra obtained by employing the spectral analysis techniques 
will be used to study the electromagnetic proton cyclotron instability as a generation 
mechanism for EMIC waves, as described in Chapter 3. The dynamic spectral analysis 
technique is employed due to its capability of representing three-dimensional parameters 
on two-dimensional displays. In the current study, this technique is used to generate dy-
namic displays of wave power and polarization ellipticity. These displays will then be 
used to identify EMIC waves and to find their relationship with proton spectra, and the 
results will be presented in Chapters 4 and 5 along with discussions of these results. 
 
 
 47 
      
 
 
CHAPTER 3 
ELECTROMAGNETIC PROTON CYCLOTRON INSTABILITY 
AS A GENERATION MECHANISM FOR EMIC WAVES 
IN THE EARTH?S MAGNETOSPHERE 
 
3.1  Introduction 
As mentioned in Section 1.3, energetic (ten to several hundred keV) protons with 
an anisotropic (T?>T||) population are believed to be the source of free energy for EMIC 
waves, and it is expected that EMIC waves with frequency below FCH are generated and 
amplified favorably in the equatorial region of the magnetosphere at the energy expense 
of such protons through the electromagnetic proton cyclotron instability. Therefore, in 
this chapter, we will investigate the conditions under which the electromagnetic proton 
cyclotron instability acts as a generation mechanism for the EMIC waves observed by 
SCATHA in the Earth?s magnetosphere. 
Theoretical and experimental studies on electromagnetic proton cyclotron insta-
bility have been carried out by many authors, e.g., Gary et al. [1994a], Gary and Lee 
[1994], Anderson et al. [1994, 1996], and Samsonov et al. [2001, 2006]. These studies 
showed that there is an inverse correlation between proton temperature anisotropy Ap and 
proton parallel beta ?||p for this instability. ?||p is defined as  
 
 
 48 
       ?||p= 2
o
||Bpo k2
B
Tn?                                          (3.1) 
  
where np is the total proton number density, kB the Boltzmann constant, and Bo the mag-
netic field strength.  
Several experimental studies of the local proton cyclotron instability in the mag-
netosphere have been conducted [e.g., Mauk and McPherron, 1980; Anderson et al., 
1996]. Mauk and McPherron [1980] used data from the ATS-6 geostationary satellite to 
show a good matching of predicted and measured frequencies, which is strong evidence 
that the linear, proton cyclotron instability mechanism is responsible for the observed 
EMIC waves. The study by Anderson et al. [1996], using data from the AMPTE/CCE 
satellite, showed that the occurrence of EMIC waves in the magnetosphere is consistent 
with the predictions of the local linear instability theory of proton cyclotron waves, prov-
ing the predictive capability of the linear theory. The study by Anderson et al. [1996], 
however, covered only dawn and noon wave events with L>7. Using data from the 
SCATHA satellite, we extend the range of observations to all local time sectors with L 
values from 5.6 to 8.1 and MLAT from -13.7? to 15.6?. 
 
3.2  Data preparation and event selection  
Following the techniques described in Sections 2.3 and 2.4, the three-component 
magnetic data sampled at 4 Hz (?t=0.25 sec), obtained by SCATHA, were first detrended 
using polynomial curve fitting, and data gaps and spikes were removed. A transformation 
 
 49 
from the GSM coordinate system to the field-aligned coordinate system was then carried 
out. A Hamming window was applied to the time series, and the mean was removed prior 
to Fourier transformation and spectral analysis. The FFT length was chosen to be N=240, 
resulting in a frequency resolution of 0.017 Hz by Equation (2.4). Wave spectra were ob-
tained for transverse ( 2/12y2x ][ BB ?? + ) and parallel components with respect to the local 
ambient magnetic field Bo over the frequency range from 0 to 2 Hz.  
Wave power spectral density, ellipticity, and degree of polarization values are ex-
tracted at 2-minute intervals with 0.25-second resolution to give quantitative measures of 
the power spectral density transverse to Bo, Ptran, and power spectral density parallel to 
Bo, Ppara. These values were generated for both FAC system and GSM coordinate system. 
In the GSM coordinate system, we consider Bz as the ?parallel? component, and the 
?transverse component? means the component perpendicular to Bz. In fact, in the equato-
rial region of interest in this study, the local z direction in the GSM coordinate system 
makes a small angle with respect to that of the local magnetic field line.  
The Ptran spectra obtained in the FAC system show spin effects at 1 Hz and 2 Hz, 
but those in the GSM coordinate system do not. For any particular 2-minute interval, we 
found that, in the neighborhood of the peak, the dependences of Ptran on frequency in both 
coordinate systems are almost the same, and the magnitudes of Ptran in the two coordinate 
systems are different from one another by a nearly constant factor, whose value depends 
on the 2-minute interval considered. Therefore, for frequency and ellipticity in the 
neighborhood of the peak we used Ptran in the GSM coordinate system in place of Ptran in 
the FAC system without losing any information. 
 
 50 
To allow quantitative analysis of a rather large quantity of data, each power spec-
trum of 480 points, corresponding to two minutes of data, was scanned automatically for 
peaks in the wave spectrum. To identify EMIC wave candidates, the following criteria 
were used: (1) The peak spectral power of transverse component Ptran must be at least one 
order of magnitude greater than the corresponding power background. (2) The peak fre-
quency is above the Pc 3 range, i.e., ?0.1 Hz. The Nyquist frequency in this study is 2 Hz 
so the range of peak frequency is from 0.1 to 2 Hz. This covers the range of Pc 2 waves 
(0.1-0.2 Hz) and part of the range of Pc 1 waves (0.2?5 Hz). As mentioned in section 
2.4.2, this frequency range is expected to capture the majority of EMIC wave events.    
(3) In the neighborhood of the peak, the magnitude of Ptran must be significantly greater 
than that of Ppara in the FAC system. Visual inspection of spectra, i.e., the corresponding 
plots of Ptran  and Ppara versus frequency in both coordinate systems for each 2-minute in-
terval, was also performed to make sure only true EMIC waves were identified. In the 59 
days of data examined, there were 1181 wave events defined as a two-minute interval 
during which EMIC waves occur with at least one peak in the transverse power as a func-
tion of frequency. 
In order to focus upon events that were not influenced by the presence of He+ ions, 
we excluded those with two peaks in the spectrum, one above and one below the local 
He+ cyclotron frequency. This criterion eliminated 61 events. We then included only 
events for which SCATHA data provided the hot proton number density nh needed for the 
calculation of the wave convective growth rate S. This resulted in the analysis of 520 
events.  
 
 51 
3.3  Methodology 
 3.3.1  Electromagnetic proton cyclotron instability 
Due to various reasons, such as the presence of the lost cone in the trapped proton 
distributions, the pitch angle diffusion process, and perpendicular energizations of parti-
cles under inward, convective injection processes as a result of the ExB drift, the condi-
tion T?>T|| is often observed in the magnetosphere and can give rise to electromagnetic 
proton cyclotron instability. Standard theories of waves in magnetized plasmas [e.g., 
Swanson, 2003] show that EMIC waves are generated when the proton perpendicular 
temperature T? is greater than the proton parallel temperature T||. As pointed out by Gary 
et al. [1976], when T?>T||, electromagnetic proton cyclotron instability is not the only in-
stability that may arise. But for a pure proton plasma, the fastest growing mode is the 
electromagnetic proton cyclotron instability that has maximum growth at kxBo=0, i.e., 
parallel propagation, and with left-hand circular polarization in that direction of propaga-
tion [Gary, 1993]. Therefore we will use the results from theoretical treatments of EMIC 
waves in the magnetosphere to analyze the events found in the SCATHA data. 
 
 3.3.2  Estimation of proton temperature anisotropy  
For each of the selected events, we obtain the frequencies fpeak, fMax, and fMin. The 
peak frequency fpeak corresponds to the maximum value of Ptran; fMax and fMin are the    
frequencies on either side of the peak frequency at which Ptran reduces to the power back-
ground. We always have fMin<fpeak<fMax for any event. The experiments on board 
SCATHA also give us the local magnetic field strength Bo from which we calculate Xpeak 
 
 52 
and XMax, where Xpeak=fpeak/FCH and XMax=fMax/FCH, and FCH=qBo/(2pimp) is the local pro-
ton cyclotron frequency. The hot proton number density nh is also given in the summary 
data. Using the available values of MLT, MLAT, L, and ERAD (distance from the Earth?s 
center to the location of interest), we estimate the value of the local electron number den-
sity, n, from empirical models. Sheeley et al. [2001] give an empirical formula for the 
total cold electron number density at the magnetic equator valid for 3?L?7, using data 
from CRRES. For L>7, we employ a decay rate of L-4.0 [e.g., Chappell, 1974; Sheeley et 
al., 2001]. The values of n at off-equator locations are estimated using the empirical 
model of field dependence of magnetospheric electron number density established by 
Denton et al. [2002]. Thus from the SCATHA data, we obtain the parameters needed for 
the calculations of linear convective growth rates of EMIC waves. 
As pointed out by Kozyra et al. [1984], the effective amplification of waves in the 
Earth?s magnetosphere depends on the time spent by the waves traveling through the 
growth region, which is of finite extent. Thus, the relevant quantity is the wave convec-
tive growth rate S rather than the temporal growth rate ?. S is the ratio of ? and the wave 
group velocity vg. 
In this investigation to calculate the linear convective growth rates S we use the 
formula established by Gomberoff and Neira [1983] for parallel propagation of EMIC 
waves in a pure proton plasma (ignoring the presence of ions heavier than proton such as 
He+, He++, O+) with the assumption that the distribution of energetic proton population is 
bi-Maxwellian. The formula is  
 
 
 53 
        S=
g74v
? =
||2?
pi [A
p(1-X)-X]exp{ )}1
1()1(
2h||
2
XX
X
?
+?? ?
?
/[X2( X?+11 ? )]              (3.2) 
 
where ?||h=2?onhkBT||/Bo2, nh is the hot proton number density, ?=npc/nh the relative (cold 
to hot) proton number density, npc the cold proton number density, ?||=(2kBT||/mp)1/2 the 
proton parallel thermal velocity, and mp the mass of proton. Ap is the temperature anisot-
ropy of the hot proton population, defined as  
 
              Ap=
||T
T? -1                         (3.3) 
  
For EMIC waves, 0<?<?p, where ?=2pif and ?p=2piFCH=qBo/mp, we are inter-
ested in 0<X<1, and from Equation (3.2), we see that if other parameters are kept con-
stant, the sign of S depends upon the difference 
 
?=Ap- XX?1                              (3.4) 
 
First we see that we only get growth (S>0) when X<1. We also see that the growth stops 
(S=0) when ?=0, so we identify the corresponding normalized frequency as XMax, at 
which the EMIC emissions end, i.e.,  
 
 
 54 
Ap-
Max
Max
1 X
X
? =0                   (3.5) 
 
 As a result, we can determine Ap from the observed XMax. 
 
 3.3.3  Estimation of proton parallel temperature  
As mentioned earlier, with the availability of MLT, MLAT, L, and ERAD values, 
the local total electron number density n was estimated by using the empirical models of 
Chappell [1974], Sheeley et al. [2001], and Denton et al. [2002]. The proton number den-
sity nh was also included in the SCATHA Summary data, as noted in Section 2.2. Having 
obtained n and nh, we deduce the cold proton number density npc by using the charge 
quasi-neutrality condition. 
From Equation (3.2) we can see that given Bo, n, nh, and Ap, the value of wave 
convective growth rate S is a function of X or frequency f with the proton parallel tem-
perature T|| as yet to be determined. Assuming that the peak frequency in the observed 
emission spectrum corresponds to the calculated peak in the convective growth rate al-
lows the determination of T||. With Ap and T|| available, the proton perpendicular tempera-
ture T? is also obtained from Equation (3.3).   
For examining polarization properties of the observed waves, for each event we 
also calculated ?peak and ?ave, where ?peak is the value of ellipticity at the peak frequency, 
and ?ave is the ellipticity weighted by transverse spectral power, 
i.e., ,/ Max
Min
Max
Min
trantranave ??=
g73
g73
g73
g73
PP?? averaged over the frequency range from fMin to fMax.  
 
 55 
For each event, we varied T|| with a resolution of 0.1 keV to match the calculated 
peak frequency with the observed one. T? was calculated using Equation (3.3), and ?||p 
was also calculated using Equation (3.1) with np=n.  
From the observed spectrum for each event, we also calculated the ratio of 
Ptran_Min/Ptran_peak, where Ptran_Min and Ptran_peak are the values of Ptran in the GSM coordi-
nate system at f=fMin and f=fpeak, respectively. For a comparison of the frequency spread 
of the calculated S about the peak frequency with that of the observed Ptran, we also calcu-
lated the ?minimum? frequency of S, fMin_S, at which the corresponding value of the con-
vective wave growth rate, SMin, satisfies SMin/Speak=Ptran_Min/Ptran_peak, where Speak=SMax is 
the value of S at fpeak, i.e., the maximum value of S. The observed frequency range from 
fMin to fMax about fpeak can be considered as the spread of Ptran, and that from the calculated 
fMin_S to fMax_S is considered as the spread of S about its peak frequency. From 
XMax=fMax/FCH, Equation (3.2), and Equation (3.5), we made fMax_S equal to fMax. We 
found that the difference between the observed fMin and the calculated fMin_S was signifi-
cant for only 2% of 520 events considered. The matching of the observed and the calcu-
lated features of the EMIC emission spectra supports the validity of the methodology and 
procedures employed in this investigation. 
 
3.4  Example  
We now consider in detail a typical 2-minute EMIC wave event, centered at 23:25 
UT and occurring at 19.00 MLT, L=6.12, 6.43? MLAT, day 048, 1986. From top to bot-
tom, the first panel of Figure 3.1 shows the plots of the wave percent polarization versus 
 
 56 
frequency in the FAC (solid pink line) and GSM (dashed brown line) coordinate systems 
for this event; the second shows the wave ellipticity. The third shows the wave spectra for 
the same event; the solid pink line is Ptran in the FAC system; the dot-dashed black line is 
Ppara also in the FAC system; and the dashed brown line is Ptran in the GSM coordinate 
system. The fourth panel shows the plots of the calculated wave convective growth rate S 
(dashed brown line) and Ptran in the FAC system (solid pink line) versus frequency for the 
same event. 
From the first and second panels, we see (follow the vertical blue line from the 
top) that in the neighborhood of the peak frequency, the values of percent polarization 
and ellipticity in the FAC (the solid pink lines) and GSM (the dashed brown lines) coor-
dinate systems are almost equal. From the third panel, we also see that in the neighbor-
hood of the peak frequency, the dependences of Ptran on frequency in the two coordinate 
systems are nearly the same (the solid pink line and the dashed brown line), and the mag-
nitudes of Ptran in the two system are different from one another by a nearly constant fac-
tor. From this panel, the spin effects in the FAC system at 1 Hz and 2 Hz are clearly seen 
in Ptran (the solid pink line) and Ppara (the dot-dashed black line), but no spin effects are 
seen in Ptran (the dashed brown line) in the GSM coordinate system. Thus we use Ptran in 
the GSM coordinate system multiplied by the appropriate factor in place of Ptran in the 
FAC system. Also from the third panel, we easily see that in the neighborhood of the 
peak frequency, the value of Ptran (the solid pink line) is approximately an order of mag-
nitude greater than that of Ppara (the dot-dashed black line) in the FAC system, confirming 
the transverse nature of the EMIC waves in this event. 
 
 57 
For this event we found that fpeak=0.28 Hz, fMax=0.35 Hz, fMin=0.17 Hz, the local 
magnetic field strength Bo=101 nT, the measured proton hot density nh=0.22 cm-3 (from 
the SCATHA Summary data), and the total electron number density is taken to be n=13 
cm-3 from the models. The peak ellipticity and averaged ellipticity for this event are  
?peak= -0.45 and ?ave= -0.48, respectively, thus the observed waves are typically left-hand 
polarized.  
Following the procedures outlined in Section 3.2 and Section 3.3, we obtained 
Ap=0.3, T||=13.9 keV, T?=18.0 keV, and ?||p=7.1. We found fMin_S=0.16 Hz, and the dif-
ference between fMin and fMin_S is only 0.01 Hz. The calculated convective growth rate as 
a function of frequency for this event (the dashed brown line in the fourth panel) is com-
pared to Ptran in the FAC system (the solid pink line in the same panel) for T||=13.9 keV in 
the neighborhood of the peak frequency. We see the excellent agreement among the 
minimum, peak, and maximum frequencies, but the observed spectrum is somewhat nar-
rower than the calculated spectrum. 
 
 58 
 
 
Figure 3.1  From top to bottom, the first panel shows the plots of wave 
percent polarization versus frequency in the FAC (solid pink line) and 
GSM (dashed brown line) coordinate systems for the 2-minute interval 
beginning at 23:24 UT, day 048; the second shows those of ellipticity. The 
third shows the wave spectra for the same event; the solid pink line is Ptran 
in FAC; the dot-dashed black line is Ppara in FAC; and the dashed brown 
line is Ptran in GSM. The fourth shows the plots of the calculated convec-
tive growth rate S (dashed brown line) and Ptran in FAC (solid pink line) 
versus frequency for the same event. 
 
 
 
 59 
3.5  Results and discussions 
The 520 events are binned by MLT, ?ave, and L values. The result of binning by 
MLT values shows that 71 events occurred between 03 and 09 MLT, the dawn sector; 
299 between 09 and 15 MLT, the noon sector; 121 between 15 and 21 MLT, the evening 
sector; and 29 from 21 MLT through midnight to 03 MLT, the midnight sector. The num-
ber of events occurring in the noon sector is greater than those in the other local time sec-
tors, in agreement with the result of Anderson et al. [1992a] study using APMTE/CCE 
data that covered nearly the same MLAT range as ours, as shown in their Figure 8. The 
result of binning by ?ave values shows that 311 events occurred with ?ave between -1 and   
-0.1, treated as left-hand polarized; 135 with ?ave between -0.1 and 0.1, linearly polarized; 
and 74 with ?ave between 0.1 and 1, right-hand polarized. This result is consistent with 
that of Anderson et al. [1992b] (see the bottom panel of Figure 6 in their paper) and that 
of Fraser and Nguyen [2001] (see Figure 10 in their paper). Binning the data by L values 
shows that 35 events occurred in the region of L between 5.6 and 6; 196 in that of L be-
tween 6 and 7; and 289 with L between 7 and 8.1. The number of events occurring in the 
region of L between 7 and 8.1 is greater than that in the region between 5.6 and 7, consis-
tent with the finding of Anderson et al. [1992a] that EMIC waves were more common at 
high L (L>7) than at low L (L<6), as mentioned in Section 1.3. 
 
 
 
 
 60 
3.5.1  Inverse correlation between proton temperature anisotropy and proton 
parallel beta 
In a simulation study, Gary et al. [1994a] showed that, in the framework of the 
linear instability theory, the threshold value of the proton temperature anisotropy is re-
lated to the proton parallel beta by the expression 
 
        Ap=Sp pp||??          (3.6) 
 
where Sp is of order unity, but varies as a function of temporal growth rate, and ?p? -0.40 
(?p<0) independent of temporal growth rate for 0.05??||p?5.0. Experimentally, Anderson 
and Fuselier [1993] and Anderson et al. [1994] using the AMPTE/CCE satellite data for 
the subsolar magnetosheath region found the same functional form of the relation be-
tween Ap and ?||p as Equation (6) with Sp=0.85 and ?p= -0.48, showing that the observed 
anisotropy values were higher than the thresholds. In the dawn sector of the magneto-
sphere, Anderson et al. [1996] studied EMIC waves with L>7 using data from the 
AMPTE/CCE satellite and found that the values of Ap were also above the threshold val-
ues as expected. They found the same empirical relationship applied in this region as in 
the magnetosheath for active times with Sp the same and only a slight change in ?p from   
-0.48 to -0.52. In the noon sector of the magnetosphere, however, the Anderson et al. 
[1996] data showed Sp=0.2, much smaller than that in the dawn sector. They attributed 
the lower limit for the noon events to the presence of unmeasured cold plasma. It is to be 
recalled that in this analysis we use an estimate of the cold plasma number density based 
 
 61 
on empirical models and a measured value of the hot proton density obtained from the 
SCATHA data. Kennel and Petsckek [1966] pointed out that a sufficient anisotropy is 
required for EMIC waves to grow and wave particle interactions will then force the parti-
cle distribution toward a marginal linear stability state. The anisotropy causing the wave 
growth is therefore diminished and subsequent wave growths slowed, leading to a final 
state where the plasma remains close to the ion cyclotron instability threshold curve. 
The current investigation shows that Equation (3.6) holds for the SCATHA data 
and also for different MLT sectors and L ranges with corresponding values of Sp and ?p. 
A fit to our full data of Ap and ?||p yields Sp=0.70 and ?p= -0.34 with a correlation coeffi-
cient R= -0.63. The results for binned data are shown in Figure 3.2. The upper panel of 
this figure shows a scatter plot of Ap versus ?||p with the data binned by MLT values and 
color-coded; the four dotted lines are the fitting curves for four different MLT sectors. 
From this panel, we see that the events occurring in the dawn sector (blue diamonds) 
have rather high Ap and mainly low ?||p (<3) except for two with ?||p>10. The events in the 
noon sector (red triangles) have a bigger spread in values of Ap with a concentration at 
high Ap and span nearly the entire range of ?||p with a concentration at low ?||p (<4)    
Those in the evening sector (green squares) concentrate at low values of Ap and high val-
ues of ?||p. It is interesting to see that the events in the midnight sector (black crosses) are 
limited to a rather narrow region of low values of both Ap and ?||p. A fit to the data for 
each sector results in different values of Sp, ?p, and R, as shown in Table 2.1. From the 
table, for the dawn sector with L>7, we have Sp=0.63 and ?p= -0.37; these values are dif-
ferent from those of Anderson et al. [1996] (Sp=0.85 and ?p= -0.52); this difference may 
 
 62 
be due to the fact that we used more events (69) than they did (12) and they considered  
events with the averaged value of maximum normalized temporal growth rate, ?m/?p, of 
2.5x10-2, much greater than the value of 0.62x10-2 for the 69 events considered here. 
Higher ?m/?p correspond to more energy the waves can get from the interacting proton 
populations and thus higher Ap are expected. 
The lower panel shows the same scatter plot as the upper panel, but the data are 
binned by L values; the three dotted lines are the fitting curves for three different L 
ranges. From this panel, we see that the events in the region of L between 7 and 8.1 (red 
squares) clearly have higher values of Ap than those in the region of L between 5.6 and 7 
(brown diamonds and blue triangles). It appears that higher L corresponds to higher Ap. 
This is consistent with the consequences of the drift shell splitting process, as described 
in Section 1.2. Similarly, a fit to the data for each L range leads to different sets of Sp, ?p, 
and R. Table 3.2 summarizes the radial variation of the Ap-?||p relation.  
Gary et al. [1994a] denoted ?m as maximum temporal growth rate, used the linear 
Vlasov theory, and found that with the value of T||e/T||p=0.25, where T||e is the electron 
parallel temperature, and the normalized Alfven velocity of vA/c=10-4, Sp=0.35 and      
?p= -0.42 for ?m/?p=10-4; Sp=0.43 and ?p= -0.42 for ?m/?p=10-3; Sp=0.65 and ?p= -0.40 
for ?m/?p=10-2. Samsonov et al. [2001] solved the linear Vlasov dispersion equation as a 
function of ?||p for a pure proton plasma to find the threshold conditions for the electro-
magnetic cyclotron instability and the mirror instability assuming fixed ?m/?p for each 
instability, and found that the relation (3.6) holds for both instabilities. For the electro-
magnetic cyclotron instability, their results gave the values of Sp and ?p that are almost 
 
 63 
equal to those found by Gary et al. [1994a], for 0.01??||p?10.0 and the same three values 
of ?m/?p, see Figure 1 in their paper. Recently, Samsonov et al. [2006], using magne-
tosheath data from Cluster, found that for low ?||p (?||p<1.0) the observed proton tempera-
ture anisotropy Ap is in agreement with the proton cyclotron threshold and for higher ?||p 
(?||p?1.0) the observed Ap is close to both proton cyclotron threshold and mirror threshold, 
as shown in their Figure 10. To compare our results with those of Gary et al. [1994a], we 
also calculated ?m, by using ?m=SMaxvg, and ?m/?p that is the value of ?m normalized to the 
local proton cyclotron frequency. Figure 3.3 shows a scatter plot of Ap versus ?||p with the 
data binned by ?m/?p values and color-coded. The long-dashed blue line depicts the rela-
tion found theoretically by Gary et al. [1994a] for ?m/?p=10-4; the short-dashed red line 
depicts that for ?m/?p=10-3; and the dot-dashed green line for ?m/?p=10-2. From this fig-
ure, we see that the Ap values found are all above the corresponding thresholds for differ-
ent ranges of ?m/?p, and that in general for a given value of ?||p, higher ?m/?p corresponds 
to higher Ap. This supports the finding that EMIC waves get free energy from energetic 
protons with an anisotropic distribution; proton populations with higher Ap provide more 
energy for the waves, leading to higher values of ?m/?p. 
We found that there is no clear dependence on ellipticity; thus we did not show 
parameters as a function of ellipticity.  
The finding that Sp varies with MLT sector and L range is in agreement with the 
argument by Gary et al. [1994b] that this coefficient depends on macroscopic factors, 
particularly how hard external forces drive the energetic proton temperature anisotropy. 
 
 64 
Table 3.1 Summary of the relationship between Ap and ?||p through the coefficients Sp and 
?p of the relation Ap=Sp pp||?? and the correlation coefficients R for different MLT ranges. 
 
MLT 
 
   Sp    ?p     R No. of events 
03<MLT<09 
(Dawn sector) 
 
0.63 -0.37 -0.82 69 
09<MLT<15 
(Noon sector) 
 
0.75 -0.34 -0.45 299 
15<MLT<31 
(Evening sector) 
 
0.65 -0.32 -0.89 121 
21<MLT<03 
(Midnight sector) 
 
0.61 -0.20 -0.73 29 
All 
 
0.70 -0.34 -0.63 518 
 
 
 
Table 3.2  Summary of the relationship between Ap and ?||p through the coefficients Sp 
and ?p of the relation Ap=Sp pp||?? and the correlation coefficients R for different L ranges. 
 
 
 
L values 
 
Sp ?p R No. of events 
L<6 0.63 -0.32 -0.90 35 
 
6<L<7 0.57 -0.28 -0.81 196 
 
L>7 
 
0.81 -0.27 -0.43 289 
All 
 
0.70 -0.34 -0.63 520 
 
 
 
 
 65 
 
 
Figure 3.2  The upper panel shows a scatter plot of proton temperature anisot-
ropy versus proton parallel beta with the data binned by MLT values and 
color-coded; the four dotted lines are the fitting curves for four different MLT 
sectors. The lower panel shows the same scatter plot as the upper panel, but 
the data are binned by L values; the three dotted lines are the fitting curves for 
three different L ranges.  
 
 
 66 
 
 
Figure 3.3  A scatter plot of proton temperature anisotropy versus pro-
ton parallel beta with the data binned by maximum normalized tempo-
ral growth rate values, ?m/?p, and color-coded. The long-dashed blue 
line depicts the relation given by Gary et al. [1994a] for ?m/?p=10-4; 
the short-dashed red line depicts that for ?m/?p=10-3; and the dot-
dashed green line for ?m/?p=10-2. 
 
 67 
3.5.2  Dependence of proton temperature anisotropy on proton perpendicu-
lar temperature  
  Each panel of Figure 3.4 shows a scatter plot of Ap versus T? with the same for-
mat as Figure 3.2. From either panel, we see that for EMIC waves to be destabilized in 
general lower T? requires higher Ap. This is consistent with a finding that the free energy 
source of EMIC waves is energetic protons with an anisotropic distribution. Consider two 
energetic and anisotropic (T?>T||) proton populations with the same T?, and we expect 
that the one with higher Ap can give more energy to the waves than the other. As a result, 
given a low value of T? the corresponding value of T|| must be small enough so that Ap is 
high enough to give sufficient energy for the waves to grow or to be amplified. From 
each panel, we also see that Ap and T? follow the relation  
 
                         Ap=aT?b                             (3.7) 
 
where a and b are constants. When all 520 events are included, a fit to the data gives 
a=1.36 and b= -0.34 with R= -0.66. 
 From the upper panel of Figure 3.4, we see that the events occurring in the dawn 
sector (blue diamonds) and in the noon sector (red triangles) are more likely to have 
lower values of T? while those in the evening sector (green squares) are more likely to 
have higher values. Again we see that the events in the midnight sector (black crosses) 
have a rather narrow range of low values of T? from 2 to 20 keV. Table 3.3 summarizes 
the MLT variation of the Ap-T? relation in Equation (3.7). 
 
 68 
From the lower panel of Figure 3.4, we see that the events in the region of L be-
tween 5.6 and 6 (brown diamonds) are more likely to have high values of T? while those 
in the region of L between 7 and 8.1 (red squares) concentrate at lower values of T? and 
those with of L between 6 and 7 (blue triangles) are fairly evenly distributed over the ob-
served range of T?. Table 3.4 summarizes the radial variation of the Ap-T? relation.  
We again see that the coefficients a and b in Equation (3.7) vary with local time 
and L value. This is what is expected from the argument presented by Gary et al. 
[1994b], as described in the last paragraph of Section 3.5.1. 
 
 
 
 
 69 
Table 3.3  Summary of the relationship between Ap and T? through the coefficients a and 
b of the relation Ap=aT?b and the correlation coefficients R for different MLT sectors. 
 
 
 
MLT 
 
a b R No. of events 
03<MLT<09 
(Dawn sector) 
 
1.82 -0.46 -0.81 71 
09<MLT<15 
(Noon sector) 
 
1.06 -0.13 -0.35 299 
15<MLT<31 
(Evening sector) 
 
0.98 -0.31 -0.86 121 
21<MLT<03 
(Midnight sector) 
 
0.91 -0.17 -0.73 29 
All 
 
1.36 -0.34 -0.66 520 
 
 
 
 Table 3.4  Summary of the relationship between Ap and T? through the coefficients a and 
b of the relation Ap=aT?b and the correlation coefficients R for different L regions. 
 
 
 
L values 
 
a b R No. of events 
L<6 1.02 -0.30 -0.94 35 
 
6<L<7 0.86 -0.26 -0.78 196 
 
L>7 
 
1.32 -0.18 -0.43 289 
All 
 
1.36 -0.34 -0.66 520 
 
 70 
 
 
 
Figure 3.4  The upper panel shows a scatter plot of proton anisotropy versus 
proton perpendicular temperature with the data binned by MLT values and 
color-coded; the four dotted lines are the fitting curves for four different 
MLT sectors. The lower panel shows the same scatter plot as the upper 
panel, but the data are binned by L values; the three dotted lines are the fit-
ting curves for three different L ranges.  
 
 
 71 
3.5.3  Dependence of proton perpendicular temperature on wave normalized 
frequency  
Each panel of Figure 3.5 shows a scatter plot of proton perpendicular temperature 
T? versus wave peak normalized frequency X; the data were also binned and color-coded 
in the same ways as those used in Figure 3.2. From either panel, we see that T? decreases 
strongly with increasing X, approaching very small values when X is near to ?0.70; this is 
in agreement with the theoretical finding of Gendrin et al. [1971]. They used the kinetic 
theory, assumed a bi-Maxwellian distribution of hot protons with T?>T|| and parallel 
propagation of EMIC waves, and found that for the range of Ap between 0.5 and 3.0 that 
is comparable with ours, EMIC waves can be amplified when X is between 0.2 and 0.7, 
see Figure 5 in their paper. Similarly, X for EIMC waves were observed in the range from 
0.05 to 0.50 in the study of Mauk and McPherron [1980] using data from ATS-6. Be-
cause there is no theoretical basis for a functional form, we did not do a fit to the data, 
and thus there is no fitting curve in each panel of Figure 3.5. 
From the upper panel of Figure 3.5, we see that the events in the dawn sector (blue 
diamonds) occur with X mainly from 0.2 to 0.6; those in the noon sector (red triangles) 
span nearly the observed range of X, from 0.1 to 0.7, but concentrate in the range between 
0.27 and 0.55. It is clear that the events in the evening sector (green squares) concentrate 
at X<0.22, and we see that those in the midnight sector (black crosses) have X falling in 
the narrow range between 0.27 and 0.40. The variation of X with MLT is in agreement 
with that found by Anderson et al. [1992b], see the top panel of Figure 8 in their paper.  
 
 72 
From the lower panel of Figure 3.5, we see that the events in the region of L be-
tween 5.6 and 6 (brown diamonds) occur with X<0.58. The events with L between 6 and 
7 (blue triangles) cover mainly the range from ~0.10 to 0.46 while those with L between 
7 and 8.1 (red squares) cover nearly the observed range X from ~0.1 to 0.7 and concen-
trate at high X. It appears that in general X increases with increasing L. This is consistent 
with the result of Fraser and Nguyen [2001], see Figure 8 in their paper. 
 
 
 
    
 
 73 
 
 
 
Figure 3.5  The upper panel shows a scatter plot of proton perpendicular 
temperature versus wave peak normalized frequency with the data binned 
by MLT values and color-coded. The lower panel shows the same scatter 
plot as the upper panel, but the data are binned by L values.  
 
 
 74 
3.6  Conclusions 
In summary, in this investigation, data from 59 days in 1986 obtained by the 
SCATHA satellite, have been analyzed to identify the electromagnetic ion cyclotron 
waves and to find the conditions under which the electromagnetic proton cyclotron insta-
bility acts as a generation mechanism for EMIC waves in the Earth?s magnetosphere. Fol-
lowing the procedures outlined in Sections 3.2 and 3.3, we selected 520 two-minute in-
terval events for analysis. These events occurred at all local time sectors, with L values 
from 5.6 to 8.1 and magnetic latitudes between -13.7? and 15.6?. The wave peak frequen-
cies ranged from 0.13 to 1.08 Hz, i.e., ~0.1 to 0.7 normalized to the local proton cyclo-
tron frequency. For each event, in the FAC system the values of Ptran are significantly 
greater than those of corresponding Ppara, consistent with the transverse nature of the 
EMIC waves. 
 From this investigation several conclusions can be drawn: 
 1) There are several findings that are consistent with those of previous studies 
and/or with theoretical expectations. They include (a) the inverse correlation between 
proton temperature anisotropy and proton parallel beta and that the observed Ap are above 
the electromagnetic proton cyclotron instability thresholds, (b) that the observed waves 
actually get energy from energetic and anisotropic proton populations, and (c) that the 
peak normalized frequencies of the observed waves are less than 0.7 for the observed 
range of Ap. 
2) Owing to the large number of events and the complete set of data accompany-
ing each, we are able to study predicted theoretical relationships as a function of MLT 
 
 75 
and L. We found that the inverse correlation Ap=Sp pp||??  holds not only for all events con-
sidered but also for those in different local time sectors and L ranges. More interestingly, 
we established a similar relation between Ap and T?, given by Equation Ap=aT?b; this re-
lation again applies for all events and for those in different MLT sectors and L ranges. 
The finding that the coefficients depend on macroscopic factors such as local time and L 
value reflects the degree of strength of external forces driving the energetic proton tem-
perature anisotropy. We found the correlation coefficients R are greater than 0.6 for all 
MLT sectors and L ranges, except for the local noon sector and the region with L from 7 
to 8.1.  
3) The longitudinal or MLT variations are outstanding. The events in the dawn 
sector have rather high Ap, mainly low ?||p, rather low T?, and X mainly between 0.2 and 
0.6. Those in the noon sector have mainly high Ap, mainly low ?||p, low T?, and X concen-
trating in the range between 0.27 and 0.55. Those in the evening sector have lowest Ap, 
highest ?||p, highest T?, and X concentrating at X<0.22. Special results are found in the 
midnight sector; the events in this sector occur with limited ranges of Ap, ?||p, and T?, and 
with X just between 0.27 and 0.40. 
4) The radial variations are also notable. We found that moving outward, or in-
creasing L value, corresponds to increased Ap, in agreement with the drift shell splitting 
process, and also to lower ?||p and higher X.  
 
 76 
 
 
 
CHAPTER 4 
OBSERVATIONS OF PROTON SCATTERINGS  
BY EMIC WAVES IN THE MAGNETOSPHERE 
 
4.1  Introduction 
 As mentioned in Section 1.3, EMIC waves have been observed in the equatorial 
region of the Earth?s magnetosphere, and these waves are believed to have the capability 
of pitch angle scattering trapped protons in the ring current or radiation belts. To our 
knowledge, however, there have not been in situ observations of proton pitch angle scat-
tering by EMIC waves in the equatorial region of the magnetosphere other than a report 
by Erlandson and Ukhorskiy [2001] of simultaneous occurrences of EMIC waves and 
enhanced proton fluxes in the loss cone for 3.5<L<5, indicating that EMIC waves indeed 
scattered protons into the loss cone. Taking advantage of the wealth of SCATHA satellite 
data, in this chapter we will show that under certain conditions EMIC waves scattered 
protons either away from or toward the magnetic field line. 
 
4.2  Data preparation 
Because there is one day on which we have no data on proton differential fluxes, 
in this investigation we use 58 days of SCATHA magnetic field and proton spectra data. 
Employing dynamic spectral analysis techniques described in Sections 2.3 and 2.4, for 
 
 77 
each day we produced 24-hour dynamic displays of wave powers and ellipticity in the 
field-aligned coordinate (FAC) system. Inspection of these displays allows the identifica-
tion of EMIC wave occurrences on that day. The occurrences of EMIC waves on each 
day considered are indicated by the enhancements on the corresponding dynamic display 
of wave transverse power. Then for the days identified with EMIC waves, those dynamic 
displays for any 2-hour interval during which EMIC waves occurred intermittently were 
generated, giving more details on the previously identified waves and their ellipticities 
along with other information such as UT, MLT, MLAT, and L value.  
For investigating the relationship between EMIC waves and protons, the plots of 
proton differential fluxes, perpendicular and parallel to Bo, for 8 different energy bands 
(centered at 0.88, 2.06, 3.60, 8.80, 15.60, 36.00, 71.00, and 133.00 keV), versus universal 
time were also produced for the same two-hour intervals, just mentioned. Visual inspec-
tions of both wave dynamic displays and proton differential flux versus UT plots for 
these two-hour intervals were then performed to find time intervals during which both 
strong EMIC emission occurred and increases or decreases in proton differential fluxes 
were observed. 
 
4.3  Observational results 
As mentioned above, based on the 24-hour dynamic displays of wave powers and 
ellipticity, obtained by using data from the SCATHA satellite and dynamic spectral 
analysis techniques described in Sections 2.3 and 2.4, we identified 37 out of 58 days on 
which EMIC waves occurred intermittently. For the observed waves, the transverse 
 
 78 
power was much greater than the parallel power, indicating the transverse nature of the 
EMIC waves.  
By matching the plots of proton differential fluxes versus universal time and the 
dynamic displays of EMIC waves for the same two-hour intervals during which EMIC 
waves were found, we were able to identify 20 short time intervals showing correlations 
between EMIC waves and proton perpendicular differential fluxes jperp in the energy 
bands centered at 15.6, 36.0, 71.0, and 133.0 keV. By correlation we mean that there is a 
sudden, strong increase (a peak) or strong decrease (a dip or a valley) in proton perpen-
dicular differential flux, occurring simultaneously with the observed EMIC waves. Only 
events in which (1) the changes in proton perpendicular flux had the same time duration 
as that of the EMIC wave emissions, and (2) the peaks (or dips) were clearly above (or 
below) background and/or similar variations in that time frame were recorded. This pro-
cedure may have omitted some weak correlations as only those that were unambiguous 
were included. Each of these 20 intervals lasts only several minutes, and these intervals 
account for only approximately 3.5% of the total time of wave observation. Therefore 
they may be indicative of special conditions that require that the proton pitch angle popu-
lations are both anisotropic and non-gyrotropic, as explained later in Chapter 5. They 
were found in the magnetospheric region of L values from 5.6 to 7.5 and magnetic lati-
tudes from -15.6? to 15.4?, mainly near local noon and in the evening sector.  
For these 20 intervals, the wave frequencies fell in the range 0.16-1.06 Hz, which 
corresponded to 0.06-0.56 normalized to the local proton cyclotron frequency. Inspection 
of all spectra of these waves showed that the peak frequencies were either above or below 
 
 79 
the local He+ gyrofrequency, not both, and these waves had rather broad bands. This 
means that for these cases He+ ions did not make a major contribution. Nineteen out of 
these 20 intervals had ?<0.0, and only one had ? near 0.0. As mentioned in Section 1.4 
and shown in Chapter 3, EMIC waves are expected to be generated in the left-hand mode 
by the electromagnetic proton cyclotron instability. While in the simple linear theory, 
EMIC waves propagating along the ambient magnetic field line are expected to be purely 
left-handed, i.e., ?=-1, this is not what is typically observed. Examples of the spread of ? 
of EMIC waves observed in the equatorial region of the magnetosphere can be found in 
the study of Anderson et al. [1992b], Figures 1, 6, 7, 8, and 9. 
The summary of the observations of these intervals is given in Table 4.1. In this 
table, for each time interval, the date (day of year 1986), the middle values of UT, MLT, 
MLAT, L, and Bo, the value of Dst (as an indication of the magnetic activity), the peak 
frequency of the observed waves, and the energy bands in which dips or peaks in jperp 
were observed are given.  
We found no correlations between EMIC waves and ?lower energy? (energy 
bands centered at 0.88, 2.06, 3.60, and 8.80 keV) proton differential fluxes either parallel 
or perpendicular to the ambient magnetic field. For ?high energy? (energy bands centered 
at 15.6, 36.0, 71.0, and 133.0 keV), we also found no correlations between proton parallel 
differential fluxes jpara and EMIC waves. This is despite the fact that EMIC waves are ex-
pected to scatter protons into the loss cone, as mentioned in Section 1.4. We think this is 
because jpara consists of protons with pitch angles between -30? and 30?, whereas the loss 
cone covers a much smaller pitch angle range. 
 
 80 
The occurrences of strong peaks (dips) in jperp during the same time intervals of 
EMIC wave existence can be attributed to the pitch angle scatterings of protons away 
from (toward) the local magnetic field line as a result of resonant interactions between 
those protons with the present EMIC waves. It should be noted that interaction with 
EMIC waves is not the only possible cause for the changes in proton perpendicular flux. 
For example, magnetic compressions are known to energize protons in the perpendicular 
direction [e.g., Anderson and Hamilton, 1993; Arnoldy et al., 2005]. In this investigation, 
however, we have found many EMIC wave periods together with increases or decreases 
in ambient magnetic field strength but with no changes in proton perpendicular flux.   
 
 
 
 81 
Table 4.1  List of 20 time intervals during which correlations between EMIC waves and 
proton perpendicular differential fluxes were observed 
Day UT MLT    MLAT(?)  L  f (Hz)   Bo(nT)  Dst(nT)   Correlations (Peak/Dip in jperp   
        and centered energy, keV) 
86037 14:15 8.56 15.42 7.47 1.06 124.3 -10 Peaks at 36 and 71   
86037 22:48 14.75 6.84 7.03 0.54 94.46 -10 Peak at 133  
86041 21:02 14.36 9.16 7.11 0.58 115.16 -130 Peaks at 36, 71 and 133 
86045 14:16 10.49 13.89 7.51 0.57 95.61 -65 Peaks at 36, 71 and 133 
86045e  14:37 10.72 13.75 7.51 0.66 101.36 -65  Peaks at 36, 71 and 133 
86048 21:32 16.87 7.246 6.64 0.2 109.40 -30 Peaks at 36, and 71  
86051e 12:24 10.48 11.25 7.38 0.7 107.73 -38     Dips at 36, 71, 133; Peak at 15.6 
86054 12:30 11.18 8.65 7.28 0.8 93.63 -58  Dips at 71 and 133; Peak at 36  
86054 14:23 12.50 8.20 7.26 0.49 91.87 -58 Peaks at 36, 71 and 133  
86054 14:45 12.76 8.08 7.19 0.53 91.94 -58 Peaks at 36, 71 and 133 
86054 19:44 17.09 4.90 6.46 0.28 95.34 -58      Dips at 71, 133; Peaks at 36, 15.6 
86073*   6:31 10.87 -7.68 7.31 0.8 98.37 -50 Peaks at 36, 71 and 133 
86073*  6:56 11.18 -7.83 7.28 0.74 99.66 -50 Peaks at 36, 71 and 133 
86073*e 7:33 11.64 -8.07 7.22 0.58 105.17 -50 Dips at 36, 71 and 133  
86077* 14:33 20.47 -13.66 5.9 0.16 177.72 -16 Dips at 36, 71 and 133 
86080*  7:23 13.28 -13.18 6.98 0.92 149.83 -7 Peaks at 36, 71 and 133 
86083l 10:34 17.86 -15.6 6.15 0.16 175.52 -20 Dips at 36, 71 and 133 
86086*  9:47 18.24 -14.77 6.04 0.23 171.45 -30 Peaks at 36, 71 and 133 
86099  5:43 18.04 -6.7 5.78 0.22 128.66 -10 Dips at 71 and 133 
86099  6:49 19.56 -4.20 5.65 0.22 147.52 -10 Dips at 71, 133; Peak at 36 
 
 82 
*For days 073, 077, 080, 083, and 086, SCATHA proton differential flux data are avail-
able for the energy bands centered at 36, 71, and 133 keV only. The three intervals to be used as 
examples in this chapter are highlighted by the subscript ?e?. The interval at 10:34 UT, day 083, is 
the one with the wave ellipticities near 0, marked by the subscript ?l?. 
 
4.4  Examples 
 4.4.1  Scattering of protons away from the local magnetic field 
We consider an event in which protons are scattered away from Bo. From top to 
bottom of Figure 4.1, the first, second, third, and fourth panels show the plots of proton 
perpendicular differential fluxes (solid black lines) and proton parallel differential fluxes 
(dashed red lines), in the energy bands centered at 133.0, 71.0, 36.0, and 15.6 keV, versus 
universal time for two hours beginning at 13:00 UT, on day 045. The fifth shows the plot 
of magnetic field strength Bo versus UT for the same time interval. The sixth and seventh 
show the dynamic displays of wave ellipticity and transverse power in the field-aligned 
coordinate system. The frequency-time (dynamic) displays of ellipticity include only el-
lipticity values where the wave transverse power exceeds a certain power threshold that is 
appropriate to the digitalization level of the instrument range, and a degree of polarization 
?0.7. The dark horizontal bars at 1 Hz and 2 Hz on the wave power dynamic display and 
their corresponding counterparts on the wave ellipticity dynamic display are due to the 
satellite spin effects and are not real. The white solid line on the wave power dynamic 
display indicates the local He+ gyrofrequency.  
From the dynamic displays of ellipticity and transverse power, we see that during 
this 2-hour interval, EMIC waves occurred at 5 different UT intervals, lasting several 
 
 83 
minutes. By matching them with the plots of proton differential fluxes versus UT also for 
this 2-hour interval (the first, second, third, and fourth panels from the top), we see that 
there are correlations between EMIC waves and jperp in two intervals, one at 14:16 UT 
with a single peak in jperp, marked by the left vertical line in Figure 4.1, and the other with 
a double peak in jperp at 14:37 UT, marked by the right vertical line in Figure 4.1. These 
correlations occur in the energy bands centered at 36.0, 71.0, and 133.0 keV, but not at 
15.6 keV. We see that the durations of the wave occurrences correspond well with those 
of the peaks in the observed jperp.  
There are some weak variations in proton perpendicular flux that occur with the 
strong EMIC emissions between 13:25-13:30, 14:45-14:50, and 14:55-15:00 UT. We did 
not classify these as correlations because for the former it is not clear that the time dura-
tions are the same, and the latter two lack the unambiguous rise of the event at 14:37 UT. 
The proton parallel fluxes are also plotted in Figure 4.1. They are small and do not show 
strong correlations with the observed EMIC waves. So they are also not classified as cor-
relation events. 
The correlation time interval that is centered at approximately 14:37 UT shows a 
double peak in jperp, and the peak is stronger than that of the correlation interval at 14:16 
UT. This may be due to the fact that the frequency bandwidth for the former is much 
broader than that for the latter. 
 We now consider in detail a two-minute wave event occurring at 14:36 UT, 10.72 
MLT, L=7.5, MLAT=13.8?, day 045. The local magnetic field strength was Bo=101 nT 
(from the SCATHA Summary data). In Figure 4.2, the first, second, and third panels 
 
 84 
show the plots of wave percent polarization, ellipticity, and powers versus frequency for 
the two minute; the fourth shows the plot of wave magnetic field amplitude |Bw| versus 
frequency for the same event. To generate the plots of wave percent polarization, elliptic-
ity, and powers versus frequency for this two minute, we used the spectral analysis tech-
niques described in Sections 2.3 and 2.4. To determine |Bw| we used the formula 
|Bw|=(Ptran?f)1/2, where ?f=0.017 Hz is the frequency resolution; this formula is similar to 
that employed by Anderson et al. [1992a]. To eliminate the satellite spin effects at fre-
quencies around 1 Hz and 2 Hz, we used the corresponding 2-min wave spectrum ob-
tained in the GSM coordinate system. We found that Ptran spectra obtained in the GSM 
coordinate system showed no spin effects and had the same dependences of Ptran on fre-
quency as those obtained in the FAC system, and the magnitudes of the two were differ-
ent from one another by a nearly constant factor, whose value depends on the 2-min in-
terval considered. Therefore, we obtained the values of Ptran in the FAC system at fre-
quencies around 1 Hz and 2 Hz by using those in the GSM coordinate system for the 
same event and multiplying them by the approximate factor. From the third and second 
panels of Figure 4.2, we see that at frequencies with strong EMIC waves the transverse 
power is at least an order of magnitude larger than the parallel power and mainly ?<0.0; 
the waves were thus mainly left-hand polarized. 
The change of the proton populations relating to the event shown in this example 
is illustrated in Figure 4.3. The upper and lower panels of this figure show the plots of the 
observed jperp and jpara versus proton energy for one-minute interval at three different in-
stants. The dashed lines show the dependences of the observed jperp and jpara on proton 
 
 85 
energy at the event occurrence instant (14:37 UT, day 045); the dotted lines show those at 
14:31 UT, six minutes before the event occurs; and the dot-dashed lines show those at 
14:43 UT, six minutes after the event. Comparing the observed jperp and jpara at each in-
stant, we see that at all three instants the jperp are clearly greater than their counterpart jpara 
for high energies. Thus the three corresponding populations are anisotropic with T?>T||. 
The significant increases in the observed jperp in the energy bands centered at 36, 71, and 
133 keV (see the upper panel of Figure 4.3) are clear. This, however, does not lead to as 
large a change in the counterpart jpara, as clearly seen in the lower panel of Figure 4.3.  
The sudden increases in jperp occurring at the same time as the existence of EMIC 
waves indicate that protons in the corresponding energy bands were pitch angle scattered 
away from the local magnetic field due to the interactions with these waves. Explanations 
for these scatterings will be given in Section 5.5.1, Chapter 5. 
 
 
 
 86 
 
 
 
Figure 4.1  The first, second, third, and fourth panels from the top show the plots of jperp 
(solid black lines) and jpara (dashed red lines), centered at 133.0, 71.0, 36.0, and 15.6 keV, 
versus UT for 2 hours beginning at 13:00 UT, day 045; the fifth shows the plot of Bo versus 
UT; the sixth and the seventh show the dynamic displays of wave ellipticity and transverse 
power. The white solid line on the seventh panel indicates the local He+ gyrofrequency. 
 
 
 87 
  
 
 
 
 
 
 
 
Figure 4.2  From top to bottom, the first, second, and third panels 
show the plots of wave percent polarization, ellipticity, and powers 
versus frequency for the 2-minute interval beginning at 14:35 UT, day 
045; the fourth shows the plot of wave magnetic field amplitude ver-
sus frequency for this event. 
 
 
 
 88 
 
 
 
 
 
Figure 4.3  The upper and lower panels show the plots of the 
observed jperp and jpara versus proton energy for one-minute in-
terval at three different instants. The dashed lines are the plots 
at the event occurrence instant (14:37 UT, day 045); the dotted 
lines are those at 14:31 UT, six minutes before the event oc-
curs; and the dot-dashed lines are those at 14:43 UT, six min-
utes after the event.  
 
 
 89 
4.4.2  Scattering of protons toward the local magnetic field 
We now consider an event in which protons are scattered toward Bo. Figure 4.4 
has the same format as Figure 4.1 for two hours beginning at 06:00 UT, on day 073; ex-
cept that there are no plots of proton differential fluxes in the energy band centered at 
15.6 keV because no SCATHA data on jperp and jpara are available in this energy band for 
this time. 
From Figure 4.4 (the fifth and sixth panels from the top), we see that during this 
2-hour interval EMIC waves occurred at 4 different time intervals, lasting several min-
utes. By matching them with the plots of proton differential fluxes versus UT also for this 
2-hour interval (the first, second, and third panels from the top), we see that there are cor-
relations between EMIC waves and proton perpendicular differential fluxes jperp in three 
intervals at 06:32 UT, 06:56 UT, and 07:33 UT, marked by three corresponding vertical 
lines in this figure. The correlations at 06:32 UT and 06:56 UT are indicated by the sud-
den increases (peaks) in the observed jperp, and the one at 07:33 UT by sudden decreases 
(dips) in the observed jperp, in the energy bands centered at 36, 71, and 133 keV. We again 
see that the durations of the wave occurrence also correspond well with those of the 
peaks or dips in the observed jperp.  
There are weak EMIC emissions and fluctuations in jperp between 7:10 and 7:18 
UT. But these peaks and dips are typical of those seen throughout this two-hour period 
and are not classified as correlation events. There is also a time period near 6:40 UT in 
which the proton perpendicular fluxes are very low, but this is between two peaks and not 
during a period of strong EMIC waves. So it is not classified as a correlation event either. 
 
 90 
The proton parallel fluxes are again much weaker than the perpendicular fluxes and show 
no significant variations so we do not include any of these as correlation events. 
We focus on the correlation interval, marked by the rightmost vertical line in Fig-
ure 4.4, which occurred at 07:33 UT, 11.64 MLT, L=7.2, MLAT=-8.07?. Figure 4.5 has 
the same format as Figure 4.2 for the 2-minute interval from 07:32 to 07:34 UT, day 073. 
The local magnetic field strength was Bo=105 nT. From the third and second panels of 
Figure 4.5, we again see that at frequencies with strong EMIC waves the transverse 
power is at least an order of magnitude larger than the parallel power and ?<0.0; the 
waves were thus left-hand polarized. 
Because for this time interval the measurements of jperp and jpara in the energy 
bands centered at 15.6, 8.80, 3.60, 2.06, and 0.88 keV are not included in the SACTHA 
data available to us, we do not produce plots similar to those in Figure 4.3 for the event 
shown in this example. We, however, also found that the sudden decreases in the ob-
served jperp in the 36, 71, and 133 keV bands do not lead to sudden increases in the coun-
terpart jpara.  
The sudden decreases in jperp occurring at the same time as the existence of EMIC 
waves indicate that protons in the corresponding energy bands were pitch angle scattered 
toward the local magnetic field due to the interactions with these waves. Explanations for 
these scatterings will be given in Section 5.5.2, Chapter 5. 
 
 
 
 91 
 
Figure 4.4  From top to bottom, the first, second, and third panels show the plots of jperp 
(solid black lines) and jpara (dashed red lines), centered at 133, 71, and 36 keV, versus UT 
for 2 hours beginning at 06:00 UT, day 073; the fourth shows the plot of Bo versus UT; the 
fifth and the sixth show the dynamic displays of wave ellipticity and transverse power. 
 
 
 
 
 92 
 
 
Figure 4.5  The same format as Figure 4.2 for the 2-minute 
interval from 07:32 UT, day 073. 
 
 93 
4.4.3  Scattering of protons toward and away from the local magnetic field 
We now consider an event in which a peak in one energy band and dips in other 
energies bands in jperp were observed. Figure 4.6 has the same format as Figure 4.1 for 
two hours beginning at 11:00 UT, on day 051. From the sixth and seventh panels of this 
figure, we see that during this 2-hour interval EMIC waves occurred at 4 different time 
intervals, lasting several minutes. By matching them with the plots of proton differential 
fluxes versus UT also for this 2-hour interval (the first, second, third, and fourth panels 
from the top), we see that there are correlations between EMIC waves and proton per-
pendicular differential fluxes jperp in one interval at 12:24 UT, marked by a vertical line in 
this figure. These correlations are indicated by sudden decreases (dips) in jperp in the en-
ergy bands centered at 36.0, 71.0, and 133.0 keV and a sudden increase (peak) in jperp in 
the energy band centered at 15.6 keV. We see that the duration of the wave occurrence 
corresponds well with those of the peak and dips in the observed jperp.  
There is a period of strong EMIC wave emission from 11:50 to 11:55 UT. But 
there are no significant changes in jperp, so we do not classify this as a correlation event. 
The parallel fluxes are again smaller than the perpendicular fluxes. They do show varia-
tions but no unambiguous correlation with the EMIC waves. 
We concentrate on the two-minute wave event occurring at 12:24 UT, 10.48 
MLT, L=7.4, and MLAT=11.25? with Bo=108 nT, as shown in Figure 4.7. Figure 4.7 has 
the same format as Figure 2 for this two minute. From the third and second panels of this 
figure, we also see that at frequencies with strong EMIC waves the transverse power is at 
 
 94 
least an order of magnitude larger than the parallel power and mainly ?<0.0; the waves 
were thus mainly left-hand polarized. 
The change of the proton populations relating to the event shown in this example 
is illustrated in Figure 4.8. Similarly to Figure 4.3, the dashed lines in both panels of Fig-
ure 4.8 show the dependences of the observed jperp and jpara on proton energy at the event 
occurrence instant (12:24 UT, day 051); the dotted lines show those at 12:19 UT, five 
minutes before the event occurs; and the dot-dashed lines show those at 12:29 UT, five 
minutes after the event. Comparing the observed jperp and jpara at each instant, we see 
again that at all three instants the jperp are greater than the counterpart jpara for all energy 
bands. Thus the three corresponding populations are also anisotropic with T?>T||. The 
presence of the EMIC waves at 12:24 UT clearly causes a significant increase in the ob-
served jperp in the energy band centered at 15.6 keV and significant decreases in the ob-
served jperp in the energy bands centered at 36, 73, and 133 keV (see the upper panel of 
Figure 4.8). The sudden increase/decreases in jperp, however, do not lead to sudden de-
crease/increases in jpara, as clearly seen in the lower panel of Figure 4.8.  
The sudden decreases and increase in jperp occurring at the same time as the exis-
tence of EMIC waves indicate that protons in the corresponding energy bands were pitch 
angle scattered toward (for the bands centered at 36.0, 71.0, and 133.0 keV) and away 
from (for the band centered at 15.6 keV) the local magnetic field due to the interactions 
with these waves. Explanations for these scatterings will be given in Section 5.5.3,   
Chapter 5. 
 
 95 
 
 
Figure 4.6  The same format as Figure 4.1 for two hours beginning 
at 11:00 UT, day 051. 
 
 
 
 96 
 
 
 
 
 
 
 
 
 
Figure 4.7 The same format as Figure 4.2 for the 2-minute 
interval from 12:23 UT, day 051. 
 
 
 
 97 
 
 
Figure 4.8  The upper and lower panels show the plots of the 
observed jperp and jpara versus proton energy for one-minute in-
terval at three different instants. The dashed lines are the plots 
at the event occurrence instant (12:24 UT, day 051); the dotted 
lines are those at 12:19 UT, five minutes before the event oc-
curs; and the dot-dashed lines are those at 12:29 UT, five min-
utes after the event.  
 
 
 98 
4.5  Summary 
 Using the SCATHA data and dynamic spectral techniques, we found that EMIC 
waves occurred intermittently on 37 out of 58 days considered and there were 20 time 
intervals showing correlations between EMIC waves and proton perpendicular differen-
tial fluxes. Each of these 20 intervals lasted only a few minutes. They were mainly near 
local noon and in the evening sector with L values from 5.6 to 7.5 and magnetic latitudes 
within 15 degrees of the equator. The wave frequencies fell in the range 0.16-1.06 Hz, 
which corresponds to 0.06-0.56 normalized to the local proton gyro-frequency. Although 
these intervals account for approximately 3.5% of the total time of wave observation, 
they indicate that EMIC waves indeed pitch angle scatter protons either toward or away 
from the local magnetic field. In the next chapter we will consider a mechanism for this 
scattering and give explanations for the examples given in this chapter.  
 
 
 99 
 
 
 
CHAPTER 5 
A THEORETICAL MODEL FOR SCATTERING  
OF PROTONS BY EMIC WAVES 
 
5.1  Introduction 
As shown in Chapter 4, we have found that in a small percentage of the time in-
tervals in which EMIC wave emissions were observed by SCATHA, there are correla-
tions with the proton perpendicular differential fluxes jperp. They indicate that EMIC 
waves indeed pitch angle scatter protons either toward or away from the local magnetic 
field, Bo. In this chapter we present a linear analysis of the interactions of protons with 
the observed spectrum of EMIC waves that suggests that it is reasonable to attribute the 
changes in jperp to resonant interactions with the EMIC waves if the proton pitch angle 
distribution is non-gyrotropic or exhibits gyrophase bunching in these unusual cases. 
These interactions result in pitch angle scatterings of protons with respect to Bo. 
 
5.2  Cyclotron resonance between EMIC wave and proton  
Cyclotron (or gyro) resonance is the term used in a process in which the wave 
electric field vector of a circularly polarized wave and a particle both rotate about a mag-
netic field line at the same frequency and can exchange energy between one and the 
other. Consider a proton with a finite perpendicular velocity, spiraling along the local 
 
 100 
ambient magnetic field Bo and an EMIC wave propagating parallel to Bo, but in the direc-
tion opposite to that of the proton. If the Doppler-shifted wave frequency is equal to the 
local proton gyrofrequency, the wave and proton are in cyclotron resonance. When this 
happens, the wave electric field vector Ew and v? of the proton maintain a constant phase 
angle with respect to each other, allowing Ew and Bw, which is the wave magnetic field 
vector, to exert forces on the proton for a reasonable time interval, leading to energy 
transfer [e.g., Brice, 1964]. 
 
5.3  Theoretical model for scattering of protons by EMIC waves 
For a proton to interact resonantly with an EMIC wave with angular frequency, ?, 
the following condition must be satisfied [e.g., Kennel and Engelmann, 1966] 
 
?-k||v||R=m?p                              (5.1) 
 
where m=0, ?1, ?2, ?3, ?, and ?p=qBo/mp is the proton cyclotron angular frequency, k|| 
the component parallel to Bo of the wave vector k, v||R the component parallel to Bo of the 
resonant proton?s velocity, q the proton electric charge, and mp the proton mass. The val-
ues m?0 give cyclotron harmonic resonances where the wave angular frequency in the 
particle?s frame of reference is equal to some harmonic of the particle?s cyclotron angular 
frequency. The value m=1 gives the principal harmonic resonance. 
According to Kennel and Petschek [1966] and Ginet and Albert [1991], for the 
values of m in Equation (5.1) such that |m|>1 the wave must propagate at an angle (??0?) 
 
 101 
with respect to Bo. Therefore, for EMIC waves propagating parallel to Bo and for the 
analysis presented in this chapter, the value m=1 is assumed. According to Somov [2000], 
the strongest interaction usually occurs when the Doppler-shifted wave frequency exactly 
matches the particle gyrofrequency. Substituting m=1 into Equation (5.1) leads to (for 
parallel propagation we set k||=k) 
 
v||R= k?? p?                              (5.2) 
 
For an EMIC wave with 0<?<?p, from Equation (5.2) we can see that the prod-
uct, kv||R, must be negative, and thus the parallel velocity of the resonant proton must be 
opposite to the wave phase velocity [e.g., Helliwell, 1970]. From Equation (5.2) we easily 
see that if v|| of the interacting proton changes, the current resonance condition is broken. 
For a strong interaction between the particle and the wave to continue, the particle must 
resonate with a different frequency in the emission spectrum, establishing a new reso-
nance condition.  
Since the observed EMIC waves are traveling at small angle with respect to Bo, 
we will treat the case of parallel propagation. Then the wave electric field vector Ew is 
perpendicular to Bo and corotating with respect to the gyromotion of the proton that is 
also spiraling along Bo. The corotation angle or relative phase angle, ?c, is defined as the 
angle between v? of the proton and Ew such that ?c=cos-1(v?.Ew/|v?||Ew|), where v? is the 
vector component perpendicular to Bo of the proton?s velocity. 
 
 102 
As mentioned earlier, for the energy transfer between the wave and the proton to 
be efficient, the particle and the wave must maintain the same phase relation for a suffi-
cient period of time [e.g., Hargreaves, 1979], so that the proton in resonance sees the 
wave fields Ew and Bw as static ones. The wave electric force acting on the proton is qEw 
and in the perpendicular direction. The wave magnetic force acting on the proton is 
q(vxBw)=q(v||xBw)+q(v?xBw). The sign ?x? here denotes the cross product of two vectors. 
We have used v=v||+v?.  
Because Ew and Bw are both perpendicular to Bo, and the wave propagates along 
Bo, opposite to the spiraling proton (v|| and k are antiparallel while the resonance condi-
tion holds), and the three vectors, Ew, Bw, and k, make a right-handed system, we can see 
that q(v||xBw) is always parallel to Ew. As a result, the total force due to the wave acting 
on the proton in the perpendicular direction is q(Ew+v||xBw) and the equation of motion of 
the proton in the perpendicular direction is 
 
    mp dtdv ? =q|Ew+v||xBw|cos ?c                              (5.3) 
 
or                        dtdv ? =
pm
q (v
||-vphase)|Bw|cos ?c                              (5.4) 
 
where |Bw| is the amplitude of the wave magnetic field. We have used |vphase|=|Ew|/|Bw|, 
where vphase is the wave phase velocity. 
 
 103 
The vector component q(v?xBw) acts on the proton in the parallel direction, lead-
ing to the equation of motion of the proton in this direction 
 
        dtdv|| = -
pm
q v
?|Bw|cos ?c                    (5.5) 
 
If we multiply both sides of Equation (5.4) by v?dt and both sides of Equation 
(5.5) by (v||-vphase)dt and then add the results together, we obtain 
 
                  v?dv?+(v||-vphase)dv||=0                               (5.6)  
 
Integrating both sides of Equation (5.6) leads to [Brice, 1964] 
 
    2?v +(v||-vphase)2=constant                    (5.7) 
 
Multiplying both sides of Equation (5.7) by mp/2, the left hand side becomes the proton 
energy in the rest frame of the wave. Thus in this frame the energy of the particle is con-
stant so that if v|| increases (decreases), v? decreases (increases), and the particle pitch an-
gle decreases (increases). 
For an EMIC wave propagating parallel to Bo in a pure proton plasma, we have 
the following approximate dispersion relation [e.g., Lyons and Williams, 1984; Baumjo-
hann and Treumann, 1996; Boyd and Sanderson, 2003] 
 
 104 
 
          ?2= )1(
p
2A2
?
?vk ?                 (5.8) 
 
where vA=Bo(?ompn)-1/2  is the local Alfven velocity and n the local total electron number 
density. From Equation (5.8) we have  
 
       |k|=
2/1
p
A )1( ?v
?
?
?
                                 (5.9)  
 
Combining Equations (5.2) and (5.9) leads to 
 
 v||R=vA 2/1
p
p )1)(1(
?
? ?
? ??               (5.10) 
 
 and from Equation (5.9) and |vphase|=? /|k|, we have 
 
       |vphase|=vA 2/1)p1( ???                  (5.11) 
 
Now combining Equations (5.4), (5.10), and (5.11) yields 
 
 
 105 
  dtdv ? =vA
pm
q |B
w| 2/1
p
p )1)((
?
? ?
? ? cos ?c     (5.12) 
 
while the resonance condition holds. So that all the terms on the right hand side of Equa-
tion (5.12) are available from the SCATHA data except the Alfven velocity. To estimate 
the Alfven velocity vA, we assume charge neutrality and use the empirical models, as de-
scribed in Section 3.3.2, for the local cold electron number density n because values of n 
were not included in the SCATHA data. 
 
5.4  Pitch angle scattering of protons by EMIC waves 
Assuming a value of relative phase angle between the wave electric field and the 
gyro-motion of the proton and that the proton stays in resonance as its parallel velocity 
changes by interacting with the observed strength of the waves at new resonant frequen-
cies, we now solve the equations of motion for the particle, i.e., Equations (5.5) and 
(5.12). We then obtain v||, v?, ?, and E of the proton as functions of time over the period 
during which the resonant interaction is maintained, i.e., tinteract. Here ? is the proton pitch 
angle given by  
 
                 tan ?=
||v
v ?                   (5.13) 
 
and E is the total energy of the interacting proton. Note that during the interacting period, 
Equation (5.10) is always met, meaning the proton and the wave are always in resonance. 
 
 106 
While v|| changes, the proton interacts resonantly with other waves with different frequen-
cies in the spectrum, such that Equation (5.10) is satisfied. 
We consider two cases. In the first case, for the relative phase angle ?c is an acute 
angle such that 0?|?c|<pi/2 and cos ?c>0, solutions to Equations (5.5) and (5.12) show that 
v|| decreases and v? increases, i.e., ? increases over time, and the particle is pitch angle 
scattered away from Bo. In this case the effect of perpendicular acceleration on the proton 
energy is greater than that of parallel deceleration so E also increases. Such a particle in-
creases its pitch angle and energy so that it appears in one of the perpendicular differen-
tial flux bands observed by SCATHA. If more particles are scattered into a particular 
band than out of the band, a peak in jperp will be observed in that band as a result of the 
resonant interactions of the protons with the EMIC waves.  
In the second case, for the relative phase angle ?c is an obtuse angle such that 
pi/2<|?c|?pi and cos ?c<0, solutions to Equations (5.5) and (5.12) show that v|| increases 
and v? decreases, i.e., ? decreases over time, and the particle is pitch angle scattered to-
ward Bo. In this case the effect of perpendicular deceleration on the proton energy is 
greater than that of parallel acceleration so E also decreases. If more particles are scat-
tered out of a particular band than into that band, a dip in the proton perpendicular differ-
ential flux in that band is observed. 
In the interaction region, individual protons respond to the waves. Protons whose 
parallel velocities satisfy Equation (5.10) interact strongly with the waves. The waves can 
get energy from the protons leaving the proton distribution less anisotropic, and the jperp 
decreases; or the protons can get energy from the waves, and the jperp increases. For a 
 
 107 
whole distribution of interacting protons in the Earth?s radiation belts or ring current re-
gion, the two processes can occur simultaneously and compete with each other. One ex-
pects to observe correlations between EMIC wave occurrences and proton perpendicular 
differential fluxes only when the plasma conditions are favorable, i.e., when 
a) a significant amount of protons in the interaction region interact resonantly 
with the EMIC waves, and 
b) the number of resonant protons whose relative phase angles satisfy 0?|?c|<pi/2 
is significantly different from that of resonant protons whose relative phase angles satisfy 
pi/2<|?c|?pi.  
We thus see that the effects of resonant interactions depend very much on the val-
ues of gyrophase of the individual protons of the interacting population. If the gyrophases 
of the interacting protons are initially distributed in a random fashion, i.e., without any 
preferences in the azimuthal direction in the plane perpendicular to Bo, the values of the 
relative phase angles |?c| will be equally distributed between 0 and pi, and the number of 
protons that are pitch angle scattered away from Bo will not be significantly different 
from that of protons that are pitch angle scattered toward Bo, and the net result will lead 
to no significant increases or decreases in jperp. As mentioned in Chapter 4, the 20 time 
intervals of correlation account for only about 3.5% of the total time of wave observation 
and may be indicative of special conditions. It appears that the gyrophases of the interact-
ing protons are initially so distributed that the number of protons with |?c|<pi/2 is signifi-
cantly greater than that of those with |?c|>pi/2 in the case of observations of sudden in-
creases or peaks in jperp, and the number of protons with |?c|<pi/2 is significantly smaller 
 
 108 
than that of those with |?c|>pi/2 in the case of observations of sudden decreases or dips in 
jperp. In other words, the pitch angle distributions of interacting proton populations are 
non-gyrotropic or exhibit phase bunching.  
 
5.5  Explanations for the examples in Chapter 4 
5.5.1 Explanations for scattering of protons away from the local magnetic 
field 
We now consider explicitly the example described in Section 4.4.1. We assume in 
this case that the interaction between the waves and proton is of cyclotron resonance. As 
described above, if resonant protons have a corotation angle near zero they will be accel-
erated in the perpendicular direction and their pitch angles will be increased. For there to 
be a net effect on the proton distribution, there must be more protons being accelerated 
than being decelerated in the perpendicular direction, i.e., the proton distribution must be 
non-gyrotropic or exhibit gyrophase bunching. For the strongest effect of this interaction 
we chose the corotation angle to be ?c=0 (Ew and v? are parallel), and then using the ob-
served |Bw| (the fourth panel from the top of Figure 4.2) and allowing the proton of inter-
est to interact resonantly with the observed wave spectrum, we solved Equations (5.5) 
and (5.12) with different initial values of proton v?, ?, and E. The results for an interact-
ing proton with the initial pitch angle ?o=40? and parallel velocity v||o=1.8x106 m/s are 
shown in Figure 5.1. From this figure, we see that during the course of interaction v|| of 
the proton decreased (the fourth panel from the top) and matched the resonance condition 
(Equation 5.10) with different frequencies of the wave spectrum (the fifth panel from the 
 
 109 
top), and its v?, ?, and E all increased with time (the third, second, and first panels from 
the top). The proton pitch angle ? changed from 40? to 83?, and E changed from 30 to 41 
keV, staying in the SCATHA energy band centered at 36 keV (for this energy band 
24?E?48 keV) but moving into the pitch angle band included in jperp. The effective time 
of interaction tinteract was about 25 seconds for covering the whole spectrum with fre-
quency from 0.5 to 1.0 Hz (the fifth panel from the top), corresponding to the v||R range 
from 1.8x106 to 0.4x106 m/s.  
We considered other protons with the same v||o but different ?o and let them inter-
act resonantly with the waves in the same manner just described, and the results are 
shown in Figure 5.2. In Figure 5.2, from top to bottom, the first and second panels show 
the plots of proton energy and pitch angle versus interacting time for 15 different values 
of ?o from 72? to 10? for this event; the third shows the plots of proton energy versus 
pitch angle for the same values of ?o. The color code is the same for all three panels. 
Each color line coresponds to a value of ?o and covers the whole spectrum of the waves.  
We have 15 different values of ?o, corresponding to 15 interacting protons, and 
the same value of v||o=1.8x106 m/s that corresponds to the lower edge of the wave spec-
trum, fo=0.5 Hz; v||o and fo satisfy Equation (5.10). After letting these 15 protons interact 
resonantly with the whole spectrum of the waves for suitable tinteract, their pitch angles and 
energies will increase by significant amounts, as explained above; thus these protons are 
pitch angle scattered away from Bo. As shown in Figure 5.2, depending on the value of 
?o, the energy of each proton will fall in a particular energy band centered either at 36 or 
 
 110 
71 or 133 keV. When its ? changes from below 80? to above 80?, that proton begins to 
contribute to the observed jperp in the corresponding energy band.  
From the first and second panels of Figure 5.2, we see that the scattering of pro-
tons away from the magnetic field occurred as ? increased and the boundary value of 
?boundary=80? was crossed for the energy bands centered at 36, 71, and 133 keV. This is 
consistent with the observation of the peaks in jperp in these 3 energy bands. We also see 
that for a proton interacting with the whole wave spectrum, a higher energy corresponds 
to a shorter tinteract; tinteract ranges from about 45 seconds for a proton with energy less than 
30 keV to less than 10 seconds for a proton with energy greater than 170 keV (for the en-
ergy band centered at 133 keV, 94?E?172 keV). 
It should be emphasized that it is not necessary that a proton have all successive 
values of v|| in the v||R range, which corresponds to the whole spectrum, during the course 
of interaction. In fact, with any initial value of v|| that falls in that range, the proton can 
start interacting resonantly with the corresponding frequency in the spectrum such that 
Equation (5.10) is met, and its ? and E will increase over time when a portion of the 
whole spectrum is covered; thus the proton is also pitch angle scattered away from Bo. If 
its ? crosses ?boundary and its E falls in one of the three energy bands, the proton will con-
tribute to the observed jperp in that energy band. For a proton interacting with only a por-
tion of the whole spectrum, tinteract is clearly shorter than that for one interacting with the 
whole spectrum. The nearer to 80? from below, the shorter tinteract required to have ? 
above 80?. The observed significant increases in jperp in these energy bands imply that the 
 
 111 
plasma conditions were such that protons interacted resonantly with the EMIC waves for 
suitable times so that their pitch angles and energies increased so as to contribute to jperp. 
For the value ?c=0, corresponding to the strongest interaction effect, depending on 
the proton energy and the portion of the wave spectrum covered, the values of tinteract re-
quired for significant changes in proton pitch angles for different energy bands range 
from less than 50 seconds to less than 10 seconds. These values are shorter than the sev-
eral minutes of the correlation event considered here. If we chose the value of ?c such that 
0<|?c|<pi/2, not corresponding to the strongest effect, the total time of interaction required 
would be higher but may still fall within the timescale of several minutes of the correla-
tion event. For example, if we set |?c| equal to pi/4 instead of 0, tinteract would be 1.4 times 
those obtained with the value of 0 and still be within the timescale of observation. 
For protons in the energy band centered at 15.6 keV, the last lines, i.e., 15th Lines 
in the three panels of Figure 5.2 show that protons in this energy band must have a small 
pitch angle to satisfy the resonant condition. Therefore it takes more time for them to be 
scattered into the perpendicular direction. More importantly, the results show that many 
of the particles that might eventually have pitch angles above 70? would also have their 
energy increased to values high enough to remove them from the energy band centered at 
15.6 keV (for the energy band centered at 36 keV, 24?E?48 keV). Thus the fact that 
there is no observed peak in jperp in this energy band for this event is consistent with the 
idea that the operating mechanism is the resonant interaction of the protons with the 
EMIC waves. 
 
 112 
 For protons in the energy bands centered at 71 and 133 keV, those observed in 
jpara have parallel velocities too high to be in resonance with the EMIC waves. Some pro-
tons in the band centered at 36 keV may be in resonance with the waves, but it is only a 
small fraction of those observed in jpara. Thus the fact that there are no significant changes 
in jpara in these energy bands is consistent with the calculations presented here. 
 
 
 
 
Figure 5.1 From top to bottom, the first, second, third, and fourth 
panels show the plots of proton energy, pitch angle, perpendicular 
velocily, and parallel velocity versus interacting time for the event 
at 14:36 UT, day 045, and the energy band centered at 36 keV; the 
fifth shows the plot of wave frequency versus interacting time. 
Initial pitch angle of the proton is 40?. 
 
 
 113 
 
 
Figure 5.2 From top to bottom, the first and second panels show 
the plots of proton energy and pitch angle versus interacting 
time for 15 different values of initial pitch angle ?o from 72? to 
10? for the event at 14:36 UT, day 045; the third shows the plots 
of proton energy versus pitch angle for the same values of ?o. 
The color code is the same for the three panels. 
 
 
 
 114 
5.5.2  Explanations for scattering of protons toward the local magnetic field 
 We now consider explicitly the second example described in Section 4.4.2. We 
again assume that the interaction between the EMIC waves and protons is of cyclotron 
resonance. For the strongest effect of this interaction, we chose the corotation angle to be 
|?c|=pi (Ew and v? are antiparallel). Then using the observed |Bw| (the fourth panel from 
the top of Figure 4.5) and allowing the protons of interest to interact resonantly with the 
observed wave spectrum, we solved Equations (5.5) and (5.12) with different initial val-
ues of proton v?, ?, and E.  
The results for 15 interacting protons with different initial pitch angles ?o from 
86.0? to 80.1? and the same v||o=0.4x106 m/s, corresponding to the higher edge of the 
wave spectrum fo=0.98 Hz, are shown in Figure 5.3; v||o and fo satisfy Equation (5.10). In 
Figure 5.3, from top to bottom, the first and second panels show the plots of proton 
energy and pitch angle versus interacting time for different values of ?o for this event; the 
third shows the plots of proton energy versus pitch angle for the same values of ?o. Each 
color line coresponds to a value of ?o and covers the whole spectrum of the waves.  
After letting these protons interact resonantly with the whole spectrum of the 
waves with frequency from 0.98 to 0.50 Hz, corresponding to the v||R range from 0.4x106 
to 1.9x106 m/s, for suitable tinteract, their pitch angles and energies will decrease by sig-
nificant amounts; thus these protons are pitch angle scattered toward Bo. As shown in 
Figure 5.3, depending on the value of ?o, the energy of each proton will fall in a particular 
energy band centered either at 36 or 71 or 133 keV. When its ? changes from above 80? 
 
 115 
to below 80?, that proton no longer contributes to the observed jperp in the corresponding 
energy band.  
From the first and second panels of Figure 5.3, we see that the scattering of pro-
tons toward the magnetic field occurred as ? decreased and the boundary value of ?bound-
ary=80? was crossed for protons in the energy bands centered at 36, 71, and 133 keV. This 
is consistent the observation of the dips in jperp in these 3 energy bands. We also see that 
for a proton interacting with the whole wave spectrum, a higher energy corresponds to a 
shorter tinteract; tinteract ranges from less than 40 seconds for a proton with energy less than 
30 keV to less than 10 seconds for a proton with energy greater than 170 keV. These val-
ues are again shorter than the timescale in which the proton perpendicular flux is corre-
lated with the strength of the EMIC waves. This is consistent with the presumption that 
the observed dips in jperp in these 3 energy bands are due to resonant interactions of pro-
tons with EMIC waves. 
 If we chose the value of ?c such that pi/2<|?c|<pi, not corresponding to the strong-
est effect, the total time of interaction would be higher and may still fall within the time-
scale of several minutes of this correlation event. For example, if we set |?c| to be equal to 
3pi/4 instead of pi, tinteract would be 1.4 times those obtained with the value of pi but still be 
within the timescale of observation. 
 
 116 
 
 
 
 
 
 
Figure 5.3 From top to bottom, the first and second panels 
show the plots of proton energy and pitch angle versus 
interacting time for 15 different values of initial pitch angle ?o 
from 86? to 80.1? for the event at 07:33 UT, day 073; the third 
shows the plots of proton  energy versus pitch angle for the 
same values of ?o. The color code is the same for the three 
panels. 
 
 
 117 
5.5.3  Explanations for scattering of protons toward and away from the local 
magnetic field 
For the third correlation event presented in Section 4.4.3, there are dips in jperp in 
the energy bands centered at 36, 71, and 133 keV, and a peak in jperp in the energy band 
centered at 15.6 keV. The existence of these extremes may also be attributed to the pitch 
angle scatterings of protons by the present EMIC waves. 
 
5.5.3.1  Scattering toward Bo in the bands centered at 36, 71, and 133 keV 
For the dips in the energy bands centered at 36, 71, and 133 keV, we chose the 
corotation angle to be |?c|=pi (Ew and v? are antiparallel). Then using the observed |Bw| 
(the fourth panel from the top of Figure 4.7) and allowing the proton of interest to interact 
resonantly with the observed wave spectrum, we solved Equations (5.5) and (5.12) with 
different initial values of proton v?, ?, and E. The results for 11 interacting protons with 
different initial pitch angles ?o from 88.22? to 86? and the same v||o=0.2x106 m/s, corre-
sponding to the higher edge of the wave spectrum fo=1.25 Hz, are shown in Figure 5.4. 
v||o and fo satisfy Equation (5.10). After letting these protons interact resonantly with the 
whole spectrum of the waves with frequency from 1.25 to 0.54 Hz, corresponding to the 
v||R range from 0.2x106 to 1.9x106 m/s, for suitable tinteract, their pitch angles and energies 
will decrease by significant amounts; thus these protons are pitch angle scattered toward 
Bo. As shown in Figure 5.4, depending on the value of ?o, the energy of each proton will 
fall in a particular energy band centered either at 36 or 71 or 133 keV. When its ? 
 
 118 
changes from above 80? to below 80?, that proton no longer contributes to the observed 
jperp in the corresponding energy band.  
From the first and second panels of Figure 5.4, we see that the scattering of pro-
tons toward the magnetic field occurred as ? decreased and the boundary value of ?bound-
ary=80? was crossed for every energy band centered at 36 or 71 or 133 keV. This is again 
consistent with the observation of the dips in jperp in these 3 energy bands. We also see 
that for a proton interacting with the whole wave spectrum, a higher energy corresponds 
to a shorter tinteract; tinteract ranges from less than 52 seconds for a proton with energy less 
than 30 keV to less than 20 seconds for a proton with energy greater than 170 keV. 
 
5.5.3.2  Scattering away from Bo in the band centered at 15.6 keV 
For the peak in jperp in the energy band centered at 15.6 keV, we chose the corota-
tion angle to be ?c=0 (Ew and v? are parallel). Then using the observed |Bw| (the fourth 
panel from the top of Figure 4.7) and allowing the proton of interest to interact resonantly 
with a portion of the wave spectrum with frequency from 0.62 to 0.93 Hz, we solved 
Equations (5.5) and (5.12) with initial values of proton v?, ?, and E. The results for 7 pro-
tons with different values of ?o from 35? to 5? and the same value of v||o=1.5x106 m/s that 
corresponds to fo=0.62 Hz, are shown in Figure 5.5. v||o and fo satisfy Equation (5.10). Af-
ter letting these 7 protons interact resonantly with the waves with frequency in the range 
from 0.62 to 0.93 Hz, corresponding to the v||R range from 1.5x106 to 0.6x106 m/s, for 
suitable tinteract, their pitch angles and energies will increase by significant amounts; thus 
these protons are pitch angle scattered away from Bo. As shown in Figure 5.5, the times 
 
 119 
of interaction are less than 32 seconds, and when the pitch angles are above 70?, the en-
ergy of most particles remains in the 15.6 keV band. Thus the observation of the peak in 
this energy band is also in agreement with the calculations presented here. 
For |?c|=0, corresponding to the strongest interaction effect, again depending on 
the proton energy and the portion of the wave spectrum covered, the values of tinteract re-
quired for significant changes in proton pitch angle for this energy band are less than 32 
seconds. These values are also shorter than the timescale of several minutes of the corre-
lation event considered here. If we chose the value of ?c such that 0<|?c|<pi/2, not corre-
sponding to the strongest effect, the total time of interaction would be higher but still fall 
within the timescale of this correlation event. 
The simple analysis presented here gives no explanations for why the correlation 
angle is 180 degrees at high energies and 0 degree at low energies. The observations, 
however, seem to require this unusual pitch angle distribution, and the calculations pre-
sented in this section show that under the proper conditions the existing EMIC waves 
could be responsible for the observed changes in proton perpendicular flux. 
 
 
 120 
 
Figure 5.4 From top to bottom, the first and second panels 
show the plots of proton energy and pitch angle versus 
interacting time for 11 different values of initial pitch angle 
?o from 88.22? to 86? for the event at 12:24 UT, day 051; 
the third shows the plots of proton energy versus pitch 
angle for the same values of ?o. The color code is the same 
for the three panels. 
 
 
 
 
 121 
 
   
Figure 5.5 From top to bottom, the first and second panels 
show the plots of proton energy and pitch angle versus 
interacting time for 7 different values of initial pitch angle 
?o from 35? to 5? for the event at 12:24 UT, day 051; the 
third shows the plots of proton and energy versus pitch 
angle for the same values of ?o. The color code is the same 
for the three panels. 
 
 
 
 
 122 
5.6  Discussions  
 To our knowledge there have not been in situ observations of correlations be-
tween EMIC waves and proton perpendicular differential fluxes in the magnetosphere. 
The theoretical studies of the role of energetic and anisotropic proton populations in gen-
erating EMIC waves in the magnetosphere found that EMIC waves get free energy from 
such populations and scatter protons toward the local magnetic field. This is supported by 
the observations of proton precipitations in the auroral or polar regions, as mentioned in 
Section 1.4. We for the first time report in situ observations of correlations between 
EMIC waves and proton perpendicular differential fluxes, possibly showing that protons 
are pitch angle scattered toward or away from Bo due to interactions with EMIC waves in 
the magnetosphere. 
Resonant wave growth/amplification and particle scattering in general requires a 
nonlinear treatment. Wave energy and particle energy are conserved together; the waves 
gain energy at the expense of the interacting particles and vice versa. If we assume, how-
ever, that the wave amplitudes are sufficiently small, we can treat the wave 
growth/amplification and particle scattering separately. In this chapter we show that this 
treatment gives the results in agreement with the observed features. 
Our approach is to allow individual protons to interact resonantly with the whole 
or a portion of the spectrum of EMIC waves and to assume a non-gyrotropy feature of the 
distributions of the interacting proton populations. Depending on the frequency spectrum 
of the waves and the initial value of v|| of the interacting proton, its pitch angle and energy 
can change significantly owing to the resonant interaction with the present waves, and the 
proton is pitch angle scattered either away from or toward the local magnetic field. If the 
 
 123 
value of ?boundary is crossed, the contributing status of the proton to the observed jperp also 
changes. If the local plasma conditions are favorable, i.e., existing of non-gyrotropic pro-
ton distributions or gyrophase bunching, there are many protons that interact resonantly 
with the same waves in the same fashion and are then pitch angle scattered either away 
from or toward Bo, and the number of protons scattered away from Bo is significantly 
greater/smaller than that of protons scattered toward Bo, leading to a peak/dip in jperp, as 
observed. For |?c|=0 or pi, that corresponds to the strongest effect of the resonant interac-
tions, depending on the energies of the interacting protons, tinteract falls in the range from 
less than 10 seconds to less than 50 seconds, shorter than the timescales of several min-
utes of the observed correlations. 
To show that the changes in proton perpendicular flux may be indeed caused by 
resonant interactions with the EMIC waves, we presented calculations of single protons 
interacting with the observed EMIC spectrum. Our approach is similar to that of Gendrin 
[1968] in allowing protons whose v|| are changing to interact resonantly with different 
frequencies of electromagnetic waves, satisfying the resonance condition. The energies of 
the particles also change during the course of resonant interaction. Gendrin [1968], how-
ever, put emphasis on calculations of pitch angle and energy diffusion coefficients, as-
suming a diffusive process of wave particle interaction with durations of many hours. In 
our investigation, it was observed clearly that the timescales of the correlations were only 
a few minutes, and so the observed peaks/dips in jperp may result from the deterministic 
pitch angle scatterings of protons with respect to Bo due to resonant interactions with 
EMIC waves, i.e., not from diffusive processes. For diffusive pitch angle scatterings of 
 
 124 
protons by EMIC waves, the results of the studies by Cornwall et al. [1970], Lyon and 
Thorne [1972], Jordanova et al. [1997], and Kozyra et al. [1997] showed that it would 
take hours to have significant effects on the proton populations, as mentioned in Sect. 1.4. 
Our calculations were based on the resonant interactions of single protons with se-
lected phase angles with respect to the EMIC waves, while the observed jperp were the 
result of the behavior of a full distribution of protons. In reality, the processes of scatter-
ing protons toward and away from Bo compete with each other. Only when the local 
plasma conditions favor one process over the other, does a strong correlation result in and 
a clear peak or dip in jperp is generated. This implies that in the special cases reported here 
there were proton pitch angle distributions that were not only anisotropic but also non-
gyrotropic. 
We emphasize that if the observations are indeed indicative of non-gyrotropic 
proton distribution functions, they are not the norm. The suggestive events are only a 
small fraction of the observed EMIC emissions. This result is not totally unexpected.  
Non-gyrotropic distributions in the geosphere have been observed experimentally and 
predicted theoretically. The simulation study of Hoshino and Teresawa [1985] suggested 
that an electromagnetic ion beam instability can be a direct source of gyrophase bunch-
ing. Pantellini et al. [1994] used the linearized Vlasov theory and the Liouville mapping 
method to investigate the linear evolution of particle distribution functions in kinetic in-
stabilities in a homogeneous collisionlesss plasma. For the electromagnetic ion cyclotron 
instability with parallel propagation, they found that the instability is driven by ions un-
dergoing cyclotron resonance and that the ion velocity distributions are non-gyrotropic 
during the linear phase of the growth. The simulation study of Gary et al. [1996] showed 
 
 125 
that the enhanced fluctuations from the electromagnetic proton cyclotron instability drive 
a hot proton distribution function that is initially non-gyrotropic toward the gyrotropic 
condition in the postsaturation phase. Thomsen et al. [1985] observed gyrophase bunched 
ions associated with nearly monochromatic, low frequency hydromagnetic waves in the 
upstream of the Earth?s bow shock using data from ISEE-1 and -2. Eastman et al. [1981] 
also using data from ISEE-1 and -2 found correlations between enhanced broadband (100 
Hz to ?10 kHz) emissions and non-gyrotropic distributions of energetic ions in the 
Earth?s foreshock region. The existence of non-gyrotropic distribution of alpha particles 
and protons in the solar wind plasma and a close relation between the non-gyrotropy of 
the particle distributions and the large temperature anisotropies were also found by As-
tudillo et al. [1996] at heliocentric distances near 1.3 AU, using data obtained in the ex-
periment TAUS on the PHOBOS-2 mission to planet Mars. Thus the existence of non-
gyrotropic distributions under unusual conditions is not unreasonable.  
 
5.7  Conclusions 
The main conclusions drawn from the investigation presented in Chapter 4 and 
Chapter 5 are as follows 
 1) Using SCATHA data, we found correlations between the EMIC wave occur-
rences and the observed proton perpendicular differential fluxes in 20 time intervals.  
These intervals were mainly near local noon and in the evening sector with L values from 
5.6 to 7.5 and magnetic latitudes within 15 degrees of the equator, and accounted for only 
a small fraction, approximately 3.5%, of the total time of wave observation. The wave 
 
 126 
frequencies fell in the range 0.16-1.06 Hz corresponding to 0.06-0.56 normalized to the 
local proton gyrofrequency. The waves observed in these time intervals had rather broad 
bands, allowing effective scatterings of protons. 
2) Linear analysis of the resonant interaction of individual protons with the ob-
served EMIC waves shows that it is reasonable to attribute the changes in proton perpen-
dicular flux to the deterministic pitch angle scatterings of protons with respect to the local 
magnetic field, as a result of their interactions with the existing EMIC waves. 
3) Based on cyclotron resonance and the parallel propagation of EMIC waves and 
the assumption of a non-gyrotropy feature of the pitch angle distributions of the interact-
ing proton populations, the calculation results are consistent with the observations. 
 
 127 
 
 
CHAPTER 6 
SUMMARY AND CONCLUSIONS 
 
The role of electromagnetic ion cyclotron waves in governing proton populations 
in the ring current and radiation belt regions is very important in understanding the dy-
namics of the magnetosphere. Magnetic field data obtained by SCATHA in the form of 
time series with the sampling interval of 0.25 second, corresponding to the Nyquist fre-
quency of 2 Hz, are suitable for identifying and obtaining power spectra of EMIC waves 
in the equatorial region of the magnetosphere. To identify and obtain power spectra of 
EMIC waves from these data, we employed the fast Fourier transform and spectral analy-
sis techniques. The fast Fourier transform is used because it is a very advanced and rapid 
technique for calculating wave power spectra and polarization parameters from the meas-
ured magnetic field time series.  
First, the obtained wave power spectra were used to investigate the conditions un-
der which the electromagnetic proton cyclotron instability acts as a generation mecha-
nism for EMIC waves in the region. Based upon the event selection criteria, 520 were 
selected out of 1181 events identified from 59 days of SCATHA magnetic field data for 
this investigation. The results of the analysis of these 520 events can be summarized as 
follows 
 
 128 
1) There are several findings that are consistent with those of previous studies 
and/or with theoretical expectations, including (a) the inverse correlation between proton 
temperature anisotropy and proton parallel beta and that the observed Ap are above the 
electromagnetic proton cyclotron instability thresholds, (b) that the observed waves actu-
ally get energy from energetic and anisotropic proton populations, and (c) that the peak 
normalized frequencies of the observed waves are less than 0.7 for the observed range of 
Ap. 
2) The inverse correlation Ap=Sp pp||??  holds not only for all events considered but 
also for those in different local time sectors and L ranges. More interestingly, a similar 
relation between Ap and T?, given by Equation Ap=aT?b, is also established. This relation 
again applies for all events and for those in different MLT sectors and L ranges. The co-
efficients of the two relations are found to depend on macroscopic factors such as MLT 
and L value, reflecting the degree of strength of external forces driving the energetic pro-
ton temperature anisotropy.  
3) The longitudinal or MLT variations are outstanding. The events in the dawn 
sector have rather high Ap, mainly low ?||p, rather low T?, and X mainly between 0.2 and 
0.6. Those in the noon sector have mainly high Ap, mainly low ?||p, low T?, and X concen-
trating in the range between 0.27 and 0.55. Those in the evening sector have lowest Ap, 
highest ?||p, highest T?, and X concentrating at X<0.22. The events in the midnight sector 
occur with limited ranges of Ap, ?||p, and T?, and with X just between 0.27 and 0.40. 
 
 129 
4) The radial variations are also notable. It is found that moving outward, or in-
creasing L value, corresponds to increased Ap, in agreement with the drift shell splitting 
process, and also to lower ?||p and higher X.  
Second, besides magnetic field data, the experiments on board SCATHA also 
provide proton spectra data, allowing us to study the relationship between EMIC waves 
and protons. From 58 days of magnetic field and proton spectra data, we found that there 
were 20 short time intervals showing correlations between EMIC waves and proton per-
pendicular differential fluxes. They indicate that under suitable conditions, EMIC waves 
indeed pitch angle scatter protons either toward or away from the local magnetic field Bo.  
These observations required theoretical explanations. Therefore, we also estab-
lished a model that is based on resonant interactions between EMIC waves and protons. 
In this model individual protons interact resonantly with the whole or a portion of the 
spectrum of the existing EMIC waves and are then pitch angle scattered with respect to 
Bo. Calculations based on this model are in agreement with the observations and suggest 
that the EMIC waves are producing the changes in proton distributions by pitch angle 
scattering protons with respect to the local ambient magnetic field. Depending on the fre-
quency spectrum of the waves and the initial value of v|| of the interacting proton, the pro-
ton pitch angle and energy can change significantly owing to resonant interactions with 
the present EMIC waves. If the local plasma conditions are favorable, i.e., existing of 
non-gyrotropic proton distributions, there are many protons that interact resonantly with 
the same waves and are then pitch angle scattered either away from or toward Bo, and the 
 
 130 
number of protons scattered away from Bo is significantly greater/smaller than that of 
protons scattered toward Bo, leading to peaks/dips in jperp, as observed.  
In our investigation, it was observed clearly that the timescales of correlations 
were only a few minutes, and so the observed peaks/dips in jperp may result from the de-
terministic pitch angle scatterings of protons with respect to Bo due to the resonant inter-
actions with EMIC waves, not from diffusive processes.  
The main results of this investigation are given below  
 1) Correlations between the EMIC wave occurrences and the observed proton per-
pendicular differential fluxes were found in 20 time intervals. These intervals were 
mainly near local noon and in the evening sector with L values from 5.6 to 7.5 and mag-
netic latitudes within 15 degrees of the equator, and accounted for approximately 3.5% of 
the total time of wave observation. The waves observed in these time intervals had rather 
broad bands, allowing effective scatterings of protons. 
2) Linear analysis of the resonant interaction of individual protons with the ob-
served EMIC waves shows that it is reasonable to attribute the changes in proton perpen-
dicular flux to the pitch angle scatterings of protons with respect to the local magnetic 
field, as a result of the resonant interactions with the existing EMIC waves. 
3) The calculation results, based on proton cyclotron resonance and the parallel 
propagation of EMIC waves and the assumption of a non-gyrotropy feature of the proton 
pitch angle distributions, are in agreement with the observations. 
 
 
 
 131 
 
 
 
REFERENCES 
 
Albert, J. M. (2003), Evaluation of quasi-linear diffusion coefficients for EMIC waves in 
    a multispecies plasma, J. Geophys. Res., 108(A6), 1249, doi:10.1029/2002JA009792. 
Anderson, B. J., and S. A. Fuselier (1993), Magnetic pulsations from 0.2 to 4 Hz and  
    associated plasma properties in the Earth?s subsolar magnetosheath and plasma  
    depletion layer, J. Geophys. Res., 98, 1461. 
Anderson, B. J., and D. C. Hamilton (1993), Electromagnetic Ion Cyclotron Waves  
    Stimulated by Modest Magnetospheric Compressions, J. Geophys. Res., 98, 11369- 
    11382. 
Anderson, B. J., K. Takahashi, R. E. Erlandson, and L. J. Zanetti (1990), Pc1 pulsations 
    Observed by AMPTE/CCE in the Earth?s outer magnetosphere, Geophys. Res. Lett.,  
    17, 1853. 
Anderson, B. J., R. E. Erlandson, and L. J. Zanetti (1992a), A statistical study of Pc1-2  
    magnetic pulsations in the equatorial magnetosphere: 1. Equatorial occurrence  
    distribution, J. Geophys. Res., 97, 3075. 
Anderson, B. J., R. E. Erlandson, and L. J. Zanetti (1992b), A statistical study of Pc1-2  
    magnetic pulsations in the equatorial magnetosphere: 2. Wave properties, J. Geophys.  
    Res., 97, 3089. 
Anderson, B. J., S. A. Fuselier, S. P. Gary, and R. E. Denton (1994), Magnetic spectral 
 
 132 
    signatures in the Earth?s subsolar magnetosheath and plasma depletion layer, J.  
    Geophys. Res., 99, 5877. 
Anderson, B. J., R. E. Denton, G. Ho, D. C. Hamilton, S. A. Fuselier, and R. J.  
    Strangeway (1996), Observational test of local proton cyclotron instability in  
    the Earth?s magnetosphere, J. Geophys. Res., 101, 21527-21543. 
Arnoldy, R. L., M. J. Engebretson, R. E. Denton, J. L. Posch, M. R. Lessard, N. C. 
    Maynard, D. M. Ober, C. J. Farrugia, C. T. Russell, J. D. Scudder, R. B. Torbert,  
    S.-H. Chen, and T. E. Moore (2005), Pc 1 waves and associated unstable distributions 
    of magnetospheric protons observed during a solar wind pressure pulse, J. Geophys. 
    Res., 110, A07229, doi:10.1029/2005JA011041. 
Astudillo, H. F., S. Livi, E. Marsch, and H. Rosenbauer (1996), Evidence for  
    nongyrotropic alpha particle and proton distribution functions: TAUS solar wind 
    measurements J. Geophys. Res., 101, 24423-24432. 
Axford, W. I. (1964), Viscous interaction between the solar wind and the Earth?s  
    magnetosphere,  Planet. Space Sci., 12, 45. 
Axford, W.I., and C. O. Hines (1961), A unifying theory of high-latitude geophysical 
    phenomena and geomagnetic storms, Can. J. Phys, 39, 1433. 
Baumjohann, W., and R. A. Treumann (1996), Basic Space Plasma Physics, Imperial  
    College Press, p. 288. 
Bossen, M., R. L. McPherron, and C. T. Russell (1976), A statistical study of Pc1  
    magnetic pulsations at synchronous orbit, J. Geophys. Res., 81, 6083. 
Boyd, T. J. M., and J. J. Sanderson (2003), The Physics of Plasmas, Cambridge 
    University Press, p. 201. 
 
 133 
Brice, N. (1964), Fundamentals of Very Low Frequency Emission and Generating  
    Mechanism, J. Geophys. Res., 69, 4515. 
Buckley, R., Some notes on practical digital power spectral, auto- and cross correlation 
    analysis using the fast Fourier transform (1971), Department of Physics, The 
    University of Adelaide, Australia, 1971. 
Carpenter, D. L., A. J. Smith, B. L. Giles, C. R. Chappell, P. M. E. Decreau, R. R. 
    Anderson, A. M. Persoon, A. J. Smith, Y. Corcuff, and P. Canu (1993), Plasmasphere  
    dynamics in the duskside bulge region: A new look at an old topic, J. Geophys. Res., 
    98, 19243.  
Chappell, C. R. (1974), Detached plasma Regions in the magnetosphere, J. Geophys.  
    Res., 79, 1861. 
Chaston, C. C., Y. D. Hu, and B. J. Fraser (2000), Quasi-linear ion cyclotron heating in 
    the near-Earth magnetotail, J. Geophys. Res., 105(A3), 5507-5516. 
Cooley, J. W., and J. W. Tukey (1965), An algorithm for the machine calculation of  
    complex Fourier series, Math. Computation, 19, 297. 
Cooley, J. W., P. A. W. Lewis, and P. D. Welch (1969), The fast Fourier transform and 
    its applications, IEEE Trans. Ed., 12, 27. 
Cornwall, J. M. (1965), Cyclotron instabilities and electromagnetic emissions in the ultra- 
    low frequency and very-low frequency ranges, J. Geophys. Res., 70, 61. 
Cornwall, J. M., F. V. Coroniti, and R. M. Thorne (1971), Unified theory of SAR arc  
    formation at the plasmapause, J. Geophys. Res., 76, 4428. 
Cornwall, J. M., F. V. Coroniti, and R.M. Thorne (1970), Turbulent loss of ring current 
    protons, J. Geophys. Res., 75, 4699. 
 
 134 
Craven, P. D., R. C. Olsen, J. Fennell, D. Croley, and T. Aggson (1985), Potential 
    Modulation on the SCATHA Spacecraft,  J. Spacecraft, Vol. 24, No. 2, 150-157. 
Criswell, D. R. (1969), Pc1 micropulsation activity and magnetospheric application of 0.2 
    to 5.0 Hz hydromagnetic waves, J. Geophys. Res., 74, 205. 
Decreau, P. M. E., C. Beghin, and M. Parrot (1982), Global characteristics of the 
    cold plasma in the equatorial plasmapause region as ducted from the GEOS-1 mutual 
    impedance probe, J. Geophys. Res., 87, 695. 
Denton, R. E., J. Goldstein, and J. D. Menietti (2002), Field line dependence of 
    magnetospheric electron density, Geophys. Res. Lett., 29, No 24, 58. 
Dungey, J. W. (1961), Interplanetary magnetic field and the auroral zones, Phys.  
    Rev. Lett., 6, 47.  
Eastman, T. E., R. R. Anderson, and L. A. Frank (1981), Upstream Particles Observed  
    in the Earth?s Foreshock Region, J. Geophys. Res., 86, 4379-4395. 
Erlandson, R. E., L. J. Zanetti, T. A. Poterma, L. P. Block, and G. Homgren (1990), 
    Viking magnetic and electric field observation of Pc1 waves and high latitudes, 
   J. Geophys. Res., 95, 5941. 
Erlandson, R. E., and A. J. Ukhorskiy (2001), Observations of electromagnetic ion  
    cyclotron waves during geomagnetic storms: Wave occurrence and pitch angle  
    scattering, J. Geophys. Res., 106(A3), 3883-3995. 
Fennell, J. F. (1982), Description of P78-2 (SCATHA) satellite and experiment, pages 
    68-81, in The IMS Source book, Guide to the International Magnetospheric Study data  
    Analysis, edited by C. T. Russell and D. J. Southwood, American Geophysical Union, 
    Wash. DC. 
 
 135 
Fennell, J. F., G. M. Boyd, M. T. Redding, and M. C. McNab (1997), Data Recovery  
    from SCATHA satellite, Prepared for European Space Research and Technology 
    Centre, Noordwijk, The Netherlands, Contract No. 11006/94/NL/CC and National  
    Aeronautics and Space Administration, Washington, DC 20546, Grant No. NAGW- 
    4141, Space and Environment Technology Center, the Aerospace Corporation. 
Fowler, R. A., B.  J. Kotick, and R. D. Elliott (1967), Polarization analysis of natural and 
    artificially induced geomagnetic pulsations, J. Geophys. Res., 72, 2871. 
Fraser, B. J. (1968), Temporal variation in Pc1 geomagnetic micropulsations, Planet.  
    Space Sci., 16, 111. 
Fraser, B. J. (1985), Observations of ion cyclotron waves near synchronous orbit and on 
    the ground, Space Sci. Rev., 42, 357. 
Fraser, B. J., J. C. Samson, Y. D. Hu, R .L. McPherron, and C . T. Russell (1992), 
    Electromagnetic ion cyclotron waves observed near the oxygen cyclotron frequency  
    by ISEE 1 and 2, J. Geophys. Res., 97, 3063. 
Fraser, B. J., and T. S. Nguyen (2001), Is the Plasmapause a Preferred Source Region of 
    Electromagnetic Ion Cyclotron Waves in the Magnetosphere, Journal of Atmospheric  
    and Solar-Terrestrial Physics, 63, 1225. 
Fraser, B. J., H. J. Singer, W. J. Hughes, J. R. Wygant, R. R. Anderson, and Y. D. Hu  
    (1996), CRRES Poynting vector observation of electromagnetic ion cyclotron waves  
    near the plasmapause, J. Geophys. Res., 101, 15331. 
Frey, H. U., G. Haerendel, S. B. Mendel, W. T. Forrester, T. J. Immel, and N. ?stgaard  
    (2004), Subauroral morning proton spots (SAMPS) as a result of plasmapause-ring 
    -current interaction, J. Geophys. Res., 109, A10305, doi:10.1029/2004JA010516,  
 
 136 
Fuselier, S. A., S. P. Gary, M. F. Thomsen, E. S. Claflin, B. Hubert, B. R. Sandel, and 
    T. Immel (2004), Generation of transient dayside subauroral proton precipitation, 
    J. Geophys. Res., 109, A12227, doi:10.1029/2004JA010393.  
Gary, S. P., M. D. Montgomery, D. W. Feldman, and D. W. Forslund (1976), Proton 
    temperature anisotropy instability in the solar wind, J. Geophys. Res., 81, 1241. 
Gary, S. P. (1993), Theory of space plasma microinstabilities, Cambridge Atmospheric 
    and Space Science Series, p. 128, Cambridge University Press. 
Gary, S. P., and M. A. Lee (1994), The ion cyclotron instability and the inverse beta  
    correlation between proton anisotropy and proton beta, J. Geophys. Res., 99, 11297. 
Gary, S. P., M. E. McKean, D. Winske, B. J. Anderson, R. E. Denton, and S. A. Fuselier 
    (1994a), Proton cyclotron anisotropy and the anisotropy/beta inverse correlation, J. 
    Geophys. Res., 99, 5903. 
Gary, S. P., M. B. Moldwin, M. F. Thomsen, D. Winske, and D. J. McComas (1994b),  
    Hot proton anisotropies and cool proton temperatures in the magnetosphere, J.  
   Geophys. Res., 99, 23603. 
Gary, S. P., V. M. Vazquez, and D. Winske (1996), Electromagnetic ion cyclotron  
    instability: Proton velocity distributions, J. Geophys. Res., 101, 13327-13333. 
Gendrin, R. (1968), Pitch Angle Diffusion of Low Energy Protons due to Gyroresonant 
    Interaction with Hydromagnetic Waves, J. of Atmos. and Terr. Physics, 30, 1313- 
    1330.  
Gendrin, R., S. Lacourly, A. Roux, J. Solomon, F. Z. Feigin, M. V. Gokhberg, V. A. 
    Troitskaya, and V. L. Yakimenko (1971), Wave packet propagation in an amplifying 
    medium and its application to the dispersion characteristics and to the generation 
 
 137 
    mechanisms of Pc1 events, Planet. Space Sci., 19, 165. 
Gendrin, R., and A. Roux (1980), Energization of Helium Ions by Proton-Induced  
    Hydromagnetic Waves, J. Geophys. Res., 85, 4577-4586. 
Ginet, G. P., and J. M. Albert (1991), Test particle motion in the cyclotron resonance  
    regime, Phys. Fluids B, 3(11), 2994. 
Gomberoff, L., G. Gnavi, and F. T. Gratton (1995), Parametric decays of electromagnetic 
    ion cyclotron waves in a H+-He+-O+ magnetospheric plasma, J. Geophys. Res., 10, 
    17220-17229. 
Gomberoff, L., and R. Neira (1983), Convective growth rate of ion cyclotron waves in a 
    H+-He+ and H+-He+-O+ plasma, J. Geophys. Res., 88, 2170. 
Gubbins, D. (2004), Time Series Analysis and Inverse Theory for Geophysicists, 
    published by the Press Syndicate of the University of Cambridge, Cambridge 
    University  Press.  
Guglielmi, A. V., K. Hayashi, R. Lundin, and A. Potapov (1999), Ponderomotive impacts 
    in the equatorial zone of the magnetosphere, Earth Planet Space, 51, 1297-1308. 
Hargreaves, J. K. (1979), The Upper Atmosphere and Solar-Terrestrial Relation: An  
    Introduction to the Aerospace Environment, Chapter 9: Waves in the Magnetosphere, 
    Van Nostrand Reinhold Co., Ltd. 
Helliwell, R. A. (1970), Particles and Fields in the Magnetosphere: Intensity of Discrete 
    VLF Emissions, P. 292, Vol. 17, Astrophysics and Space Science Library (edited by B. 
    M. McCormac), D. Reidel Publishing Company, Dordrecht-Holland. 
Horne, R. B., and R. M. Thorne (1990), Ion Cyclotron Absorption at the Second  
    Harmonic of the Oxygen Gyrofrequency, Geophys. Res. Let., 17, 2225-2228. 
 
 138 
Horwitz J. L., and T. E. Moore (2000), Core Plasmas in Space Weather, IEEE 
    Transactions on Plasma Science, vol. 28, No. 6, p. 1840-1853. 
Hoshino, M., and T. Terasawa (1985), Numerical Study of the Upstream Wave  
    Excitation Mechanism, 1.  Nonlinear Phase Bunching of Beam Ions (1985), J.  
    Geophys.  Res., 90, 57-64. 
Hundhausen, A. J. (1995), The solar wind, in Introduction to space physics, ed. by M.  
    G. Kivelson and C. T. Russell, Cambridge University Press, New York. 
Ishida, J., S. Kokubun, and R. L. McPherron (1987), Substorm effects on spectral  
    structures of Pc1 waves at synchronous orbit, J. Geophys. Res., 92, 143. 
Jordanova, V. K., C. J. Farrugia, R. M. Thorne, G. V. Khazanov , G. D. Reeves, and M. 
    F. Thomsen (2001), Model of the ring current proton precipitation by electromagnetic  
    ion cyclotron waves during the May 14-16, 1997, storm, J. Geophys. Res., 106, 7. 
Jordanova, V. K., J. U. Koryza, A. F. Nagy, and G. V. Khazanov (1997), Kinetic model  
    of the ring current-atmosphere interactions, J. Geophys. Res., 102, 14279. 
Kanasewich, E. R. (1973), Time sequence analysis in geophysics, Uni. Alberta Press,  
    Edmonton. 
Kaye, S. M., and M. G. Kivelson (1979), Observations of Pc1-2 waves in the outer  
    magnetosphere, J.  Geophys. Res., 84, 4267. 
Kennel C. F., and F. Engelmann (1966), Velocity Space Diffusion from Weak Plasma  
    Turbulence in a Magnetic Field, Phys. Fluids, 9, 2377. 
Kennel, C. F., and H. E. Petschek (1966), Limit on stably trapped particle fluxes, J.  
    Geophys. Res., 71, 1. 
Khazanov, G. V., K. V. Gamayunov, and V. K. Jordanova (2003), Self-consistent model 
 
 139 
    of magnetospheric ring current and electromagnetic ion cyclotron waves: The 2-7 May 
    1998 storm, J. Geophys. Res., 108(A12), 1419. 
Kintner, P. M., and D. A. Gurnett (1977), Observations of ion cyclotron waves within the  
    plasmasphere by Hawkeye 1, J. Geophys. Res., 82, 2314. 
Kodera, K., R. Gendrin, and C. de Villedary (1977), Complex representation of a 
    polarized signal and its application to the analysis of ULF waves, J. Geophys. Res.,  
    82, 1245. 
Kozyra, J. U., T. E. Cravens, A. F. Nagy, E. G. Fontheim, and R. S. B. Ong (1984),  
    Effects of energetic heavy ions on electromagnetic ion cyclotron wave generation in 
    the plasmapause region, J. Geophys. Res., 99, 2217-2233. 
Kozyra, J. U., V. K. Jordanova, R. B. Horne, and R. M. Thorne (1997), Modeling of the 
    Contribution of Electromagnetic Ion Cyclotron (EMIC) Waves to Stormtime Ring  
    Current Erosion, in Magnetic Storms, Geophysical Monograph 98, 187-202. 
LaBelle, J., and R. A. Treumann (1992), Poynting vector measurement of  
    electromagnetic ion cyclotron waves in the plasmasphere,  J. Geophys. Res., 97,    
    13789. 
Liemohn, H. B., J. F. Kenney, and H. B. Knaflich (1967), Proton densities in the  
    magnetosphere from pearl dispersion measurement, Earth Planet. Sci. Lett., 2, 360. 
Lorentzen, K. R., M. P. McCarthy, G. K. Parks, J. E. Foat, R. M. Milan, D. M. Smith, 
    and R. P. Lin (2000), Precipitation of relativistic Electrons by Interaction with  
    electromagnetic ion cyclotron waves, J. Geophys. Res., 105(A3), 5381-5389. 
Lyons, L. R., and D. J. Williams (1984), Quantitative aspects of Magnetospheric Physics, 
    D. Reidel, Dordrecht, Holland. 
 
 140 
Lyons, L. R., and R. M. Thorne (1972), Parasitic pitch angle diffusion of radiation belt 
    particles by ion cyclotron waves, J. Geophys. Res., 77, 5608-5616. 
Mauk, B. H. (1982), Helium resonance and dispersion effects on geostationary 
    Alfven/ion cyclotron waves, J. Geophys. Res., 87, 9107. 
Mauk, B. H., and R. L. McPherron (1980), An experimental test of the electromagnetic 
    ion cyclotron instability within the Earth?s magnetosphere, Phys. Fluid, 23, 2111. 
McPherron, R. L., C. T. Russell, and P. J. Coleman, Jr. (1972), Fluctuating magnetic 
    fields in the magnetosphere, II, ULF waves, Space Sci. Rev., 13, 411. 
Means, J. D. (1972), Use of the three-dimensional covariance matrix in analyzing the 
    polarization properties of plane waves, J. Geophys. Res., 77, 5551. 
Meredith, N. P., R. M. Thorne, R. B. Horne, D. Summers, B. J. Fraser, and R. R.  
    Anderson, Statistical analysis of relativistic electron energies of cyclotron resonance 
    with EMIC waves observed on CRRES, J. Geophys. Res., 108, pp. SMP 17-1, 
    doi:10.1029/2002JA009700. 
Moldwin, M. B., M. F. Thomsen, S. J. Bame, D. J. McComas, L. A. Weiss, G. D. Reeves, 
    and R. D. Belian (1996), The appearance of plasmaspheric plasma in the outer  
    magnetosphere in association with substorm growth phase, Geophys. Res. Lett., 23,  
    No. 8, 801-804. 
Moldwin, M. B., B. R. Sandel, M. F. Thompson, and R. C. Elphic (2003), Quantifying 
    global plasmaspheric images with in situ observations, Space Science Reviews, 109,  
    47-61. 
Moukikis C. G., L. M. Kistler, W. Baumjohann, E. J. Lund, A. Korth, B. Klecker, M. A.  
    Popecki, J. A. Sauvaud, H. R?me, A. M. Di Lellis, M. McCarthy, and C.W. Carlson 
 
 141 
    (2002), Equator-S observations of He+ Energization by EMIC waves in the dawnside  
    equatorial magnetosphere, Geophys. Res. Lett., 29, 1432. 
Olsen, R. C., and C. W. Norwood (1992), SPACE-GENERATED IONS, J. Geophys.  
    Res., 96, 15951-15962. 
Otnes, R. K., and L. Enochson (1972), Digital time series analysis, John Wiley & Sons, 
   New York. 
Pantellini, F. G. E., D. Burgess, and S. J. Schwartz (1994), Phase space evolution in  
    linear instabilities (1994), Phys. Plasmas, 1, 3784-3791. 
Perraut, S. (1982), Wave-particle interactions in the ULF range: GEOS-1 and -2 results,   
   Planet  Space Sci., 30, 1219. 
Perraut, S., R. Gendrin, P. Robert, A. Roux, and C. De Villedary (1978), Ultra-low 
    frequency waves observed with magnetic and electric sensors on GEOS-1, Space Sci.  
   Rev., 22, 347. 
Popecki, M., R. Arnoldy, M. J. Engebretson, and L. J. Cahill, Jr. (1993), High-latitude 
    ground observations of Pc1/2 micropulsations, J. Geophys. Res., 98, 21481. 
Rankin, D. and R. Kurtz (1970), Statistical study of micropulsation polarization, J. 
    Geophys. Res., 75, 5444. 
Samsomov, A. A., M. I. Pudovkin, S. P. Gary, and D. Hubert (2001), Anisotropic MHD 
    model of the dayside magnetosheath downstream of the oblique bow shock, J. 
    Geophys. Res., 106, 21689-21699. 
Samsomov, A. A., O. Alexandrova, C. Lacombe, M. Maksimovic, and S. P. Gary (2006),  
    Proton temperature anisotropy in the magnetosheath: comparison of 3-D MHD  
    modelling with Cluster data, submitted to Ann. Geophys. 
 
 142 
Sheeley, B. W., M. B. Moldwin, and H. K. Rassoul (2001), An empirical plasmasphere 
    and trough density model: CRRES observations, J. Geophys. Res., 106, NO. A11,  
    25631. 
Sibeck, D. G., R. W. McEntire, A. T. Y. Lui, R. E. Lopez, and S. M. Krimigis (1987), 
    Magnetic field drift shell splitting: Cause of unusual dayside particle pitch angle  
    distributions during storms and substorms, J. Geophys. Res., 92, 13485. 
Somov, V. Boris (2000), Cosmic Plasma Physics, P. 122, Vol. 251, Astrophysics and 
    Space Science Library, Kluwer Academic Publishers. 
Spasojevic, M., H. U. Frey, M. F. Thomsen, S. A. Fuselier, S. P. Gary, B. R. Sandel, and 
    U. S. Inan (2004), The link between a detached subauroral proton arc and a  
    plasmaspheric plume, Geophys. Res. Lett., 31, L04803, doi:10.1029/2003GL018389. 
Spence, H. E., A. M. Jorgensen, W. J. Hughes, J. F. Fennell, and J. L. Roeder(1997),  
    Towards Inner Magnetosphere Particle and Field Models, Adv. Space Res., Vol. 20, 
    No. 3, 427-430, Published by Elsevier Ltd, Pergamon. 
Stern, D. P., and N. F. Ness (1982), Planetary Magnetospheres, Ann. Rev. Astro.  
    Astrophys., 20, 139-161. 
Stevens, J. R., and A. L. Vampola (1978), Description of the space test program P78-2  
    spacecraft and payloads, Air Force Space and Missile Systems Organization (now  
    Space Division), report SAMSO TR-78-24. 
Summers, D., and R. B. Thorne (2003), Relativistic electron pitch-angle scattering by 
    electromagnetic ion cyclotron waves during geomagnetic storms, J. Geophys. Res., 
    108(A14), 1143. 
Summers, D., R. B. Thorne, and F. Xiao (1998), Relativistic theory of wave-particle  
 
 143 
    resonant diffusion with application to electron acceleration in the magnetosphere, J.  
    Geophys. Res., 103, 20487. 
Swanson, D. G. (2003), Plasma waves (Second edition), Institute of Physics, Series in 
    Plasma Physics, Taylor and Francis Publisher. 
Thomsen, M. F., J. T. Gosling, and S. J. Bame (1985), Gyrating Ions and Large- 
    Amplitude Monochromatic MHD Waves Upstream of the Earth?s Bow Shock, J.  
    Geophys. Res., 90, 267-273. 
Thorne, R. M., and C. F. Kennel (1971), Relativistic Electron Precipitation During  
    Magnetic Storms Main Phase, J. Geophys. Res., 76, 4446-4453. 
Thorne, R. M., and R. B. Horne (1992), The Contribution of Ion-Cyclotron Waves to 
    Electron Heating and SAR-Arc Excitation near the Storm-time Plasmapause, Geophys. 
    Res. Lett., 19, 417-420. 
Thorne, R. M., and R. B. Horne (1993), Cyclotron absorption of ion-cyclotron waves at 
    bi-ion frequency, Geophys. Res. Lett., 20, 317-320.  
Thorne, R. M., and R. B. Horne (1994), Energy transfer between energetic ring current 
    H+ and O+ by electromagnetic ion cyclotron waves, J. Geophys. Res., 99, 17,275- 
    17,282. 
Thorne, R. M., and R. B. Horne (1997), Modulation of electromagnetic ion cyclotron  
    instability due to interaction with ring current O+ during magnetic storms, J. Geophys.  
    Res., 102, 14,155-14,163. 
Vacaresse, A., D. Boscher, and S.  Bourdarie (1999), Modelling the high-energy proton 
    belt, J. Geophys. Res., 104, 28,601-28,613. 
Walt, M. (1994), Introduction to Geomagnetically Trapped Radiation, in Cambridge  
 
 144 
    atmospheric and space science series, Cambridge University Press. 
Watanabe, T. (1966), Quasi-linear theory of transverse plasma instabilities with 
    applications to hydromagnetic emissions from the magnetosphere, Can. J. Phys, 44,  
    815. 
Wolf, R. A. (1995), Magnetospheric configuration, in Introduction to Space Physics, 
    ed. by M.G. Kivelson and C. T. Russell, Cambridge University Press, New York. 
Yahnina, T. A., A. G. Yahnin, J. Kangas, and J. Manninen (2000), Proton precipitation 
    related to Pc1 pulsations, Geophys. Res. Lett., 27, 3575-3578.  
Young, D. T., S. Perraut, A. Roux, C. de Villedary, R. Gendrin, A. Korth, G. Kremser,  
    and D. Jones (1981),  Wave-particle interactions near ?He+ observed on GEOS 1 and  
    2, 1. Propagation of ion cyclotron waves in He+ -rich plasma, J. Geophys. Res., 86,  
    6755. 
 
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APPENDIX 
 
AMPTE/CCE and AMPTE/IRM: AMPTE stands for Active Magnetospheric Particle 
Tracer Explorers. The AMPTE program was a three-nation, three-satellite mission to 
study the sources, transport, and acceleration of energetic magnetospheric ions, and to 
investigate the interactions between clouds of cool, dense, artificially injected plasma and 
the hot, magnetized, rapidly flowing natural plasmas of the magnetosphere and solar 
wind. The three AMPTE satellites are the NASA Charge Composition Explorer (CCE), 
the Federal Republic of Germany's Ion Release Module (IRM), and the United Kingdom 
Satellite (UKS). All three were launched together on August 16, 1984 into near-
equatorial elliptical orbits and contained extensive instrumentation. The CCE and IRM 
satellites provided the complete data set on energetic ion spectra, composition, and 
charge state throughout the near-earth magnetosphere.  
 ATS-1 and ATS-6: These two satellites were part of the Applications Technology Satel-
lite (ATS) program. The ATS series was a set of 6 NASA satellites created to explore and 
test new technologies and techniques for communications and meteorological and naviga-
tion satellites. The areas investigated during the program included spin stabilization, 
gravity gradient stabilization, complex synchronous orbit maneuvers, a number of com-
munications experiments, and the geostationary orbit environment. The ATS flights also 
 
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collected and transmitted meteorological data, and functioned at times as communica-
tions satellites. ATS-1 was launched on December 07, 1966, and its mission lasted 18 
years. ATS-6 was launched on May 30, 1974, and its mission lasted 5 years.  
Cluster: The Cluster II mission is an European Space Agency (ESA) unmanned space 
mission devoted to studying the Earth's magnetosphere, using four identical spacecraft 
flying in a tetrahedral formation. Their spacing can vary between 200 km and 18000 km. 
The four spacecraft give three-dimensional views of the near-Earth particle, field, and 
plasma environments. The four were put into a polar orbit with a perigee of 19000 km, an 
apogee of 119000 km, an inclination of 98.6?, and a period of 57 hours. The first two, 
named Salsa and Samba, were launched on July 16, 2000; the second two, named Tango 
and Rumba, were launched on August 20, 2000. Each satellite carries a set of 11 instru-
ments that measure electric and magnetic fields, electrons, protons, ions, and plasma 
waves. The Cluster mission is currently investigating small-scale structures (in 3 D) of 
the Earth's plasma environment, such as those involved in the interaction between the so-
lar wind and the magnetospheric plasma, in global magnetotail dynamics, in cross-tail 
currents, and in the formation and dynamics of the neutral line and of plasmoids. This 
mission makes it possible, for the first time, to distinguish spatial and temporal variations 
of magnetospheric phenomena. The mission is set to end in December 2009. 
CRRES: The Combined Release and Radiation Effects Satellite (CRRES) was launched 
on July 25, 1990 into a geosynchronous transfer orbit, with a perigee of 350 km and an 
apogee of 6.3 RE. The satellite?s orbit period was 9 hours 52 minutes. Due to its orbital 
 
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inclination of 18.2?, the satellite covered the magnetic latitude range from -30? to 30?. 
The local time at apogee was 08 MLT at launch and decreased at a rate of 1.3 hours per 
month to about 14 MLT when the spacecraft ceased its operation on October 21, 1991. 
The main objectives of the CRRES mission included studies of the natural radiation belt 
environment and its effects on a variety of microelectric components, studies of new 
static and dynamic models, descriptions of processes in the radiation belts, performance 
of active chemical release experiments in the ionosphere and magnetosphere, and low-
latitude studies of ionospheric irregularities. 
DE-1: DE stands for Dynamics Explorer. The DE-1 satellite was launched on August 3, 
1981 into an elliptical polar orbit, with a perigee of 675 km, an apogee of 24875 km, and 
an inclination of 90?. Its orbit period was about 7.5 hours. The satellite carried an auroral 
imager and detectors designed primarily for field and particle measurements. DE-1 
ceased its operation in March 1991. 
Dst Index: The Dst or disturbance storm time index is a measure of geomagnetic activity 
and used to assess the severity of geomagnetic storms. It is expressed in nanoteslas and is 
based on the average value of the horizontal component of the Earth's magnetic field ob-
tained hourly at four near-equatorial geomagnetic observatories. Negative Dst values in-
dicate that a magnetic storm is in progress; the more negative Dst the more intense the 
magnetic storm. The negative deflections in the Dst index are caused by the storm time 
ring current that flows around the Earth from east to west in the equatorial plane. The use 
of the Dst as an index of storm strength is possible because the strength of the surface 
 
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magnetic field at low latitudes is inversely proportional to the energy content of the ring 
current, which increases during geomagnetic storms. 
GEOS-1 and GEOS-2: GEOS-1 and GEOS-2, two identical satellites of the European 
Space Agency (ESA), were devoted to the study of the Earth's magnetosphere. The two 
played the roles of reference spacecraft for the International Magnetospheric Study (IMS) 
and carried out correlative measurements with extensive ground-based networks in Scan-
dinavia. The instruments on board the two were to measure DC and AC electric and 
magnetic fields; gradient of the magnetic field; thermal and suprathermal plasma parallel 
and perpendicular to the magnetic field; energy spectra, angular distribution, and compo-
sition of positive ions; and angular distribution and energy spectra of energetic electrons 
and protons. GEOS-1 was launched on April 20, 1977, and initially aimed to be a geo-
synchronous satellite, but ended up in the wrong orbit. As a result, it had a perigee of 
2767 km, an apogee of 38389 km, an inclination of 26.8?, and an orbit period of 734 
minutes. GEOS-2 was launched on July 14, 1978 into a truly geostationary orbit with a 
perigee of 36020 km, an apogee of 36061 km, an inclination of 11.8?, and a period of 
1449 minutes. 
GSM coordinate system: GSM stands for Geocentric Solar Magnetospheric. This coor-
dinate system is Earth centered and has its x-axis pointing from the Earth toward the Sun. 
The positive z-axis is perpendicular to the x-axis and parallel to the projection of the 
negative moment of the Earth?s magnetic field on a plane perpendicular to the x-axis. 
Thus the magnetic dipole axis lies within the xz plane. The y-axis is directed toward the 
dusk to produce a right-handed orthogonal coordinate system.   
 
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IMAGE: IMAGE stands for Imager for Magnetopause to Aurora Global Exploration. 
The IMAGE spacecraft was launched on March 25, 2000 into an elliptical polar orbit, 
with an apogee altitude of 45922 km and a perigee altitude of 1000 km. It has a nominal 
spin period of 2 minutes, and its spin axis is perpendicular to the orbital plane. IMAGE 
was the first satellite devoted to imaging the Earth's magnetosphere. IMAGE used neutral 
atom, ultraviolet, and radio imaging techniques to obtain the first comprehensive global 
images of the plasma populations in the inner magnetosphere, enabling space scientists to 
observe the large-scale dynamics of the magnetosphere and the interactions among its 
constituent plasma populations. On December 18, 2005, after 5.8 years of successful op-
erations, IMAGE's telemetry signals were not received during a routine pass. Currently, 
IMAGE has not responded to commands. Although the IMAGE mission was designed as 
a two-year one, it has exceeded all its scientific goals and produced many stunning im-
ages of the previously invisible region of space in the inner magnetosphere.  
ISEE: ISEE stands for International Sun-Earth Explorers. The ISEE program consisted of 
three spacecraft, ISEE-1, ISEE-2, and ISEE-3. ISEE-1 and ISEE-2 were launched on Oc-
tober 22, 1977 into almost coincident orbits around the Earth with periods of approxi-
mately 57 hours and inclinations of 23?, and their time separation in this orbit could be 
altered by maneuvering ISEE-2. These two spacecraft, separated by a variable distance 
and with similar instrument complements, were able to break the space-time ambiguity 
inevitably associated with measurements by a single spacecraft on thin boundaries that 
may be in motion, such as the bow shock and the magnetopause. The scientific objective 
of the ISEE mission was to study magnetosphere and its relation to solar activities. With 
 
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an apogee of 23 RE, ISEE-1 and ISEE-2 penetrated into the interplanetary medium for up 
to 3/4 of their periods depending upon the time of year. The ISEE-1 and ISEE-2 space-
craft reentered the Earth's atmosphere in September 1987 after covering 1517 orbits.  
L value: When considering the radiation belts, it is often useful to use the B-L coordinate 
system. The L value of a particular point in the magnetosphere is just the distance in RE 
from the Earth?s center to the position at which the field line through that point crosses 
the equatorial plane.  
OGO-5: OGO stands for Orbiting Geophysical Observatories. A series of six Orbiting 
Geophysical Observatories were put into orbit by NASA between 1964 and 1969. They 
were designed to study the Earth's atmosphere and magnetosphere, and the space between 
the Earth and the Moon. The fifth, OGO-5, was launched on March 4, 1968 into a highly 
elliptical orbit, with a perigee of 272 km and an apogee of 148228 km, and was primarily 
devoted to the Earth observations. Its orbital inclination was 31.1?, and the satellite took 
3796 minutes to complete one orbit. The operation of OGO-5 terminated on July 14, 
1972. 
PHOBOS: PHOBOS was an international project that was devoted to the investigations 
of Mars and its satellites. The objectives of the PHOBOS mission were to conduct studies 
of the interplanetary environment, perform observations of the Sun, characterize the 
plasma environment in the Martian vicinity, and study the surface and atmosphere of 
Mars, and the surface composition of the Martian satellite Phobos. The project included 
two unmanned interplanetary probes, PHOBOS-1 and PHOBOS-2. PHOBOS-1 was 
launched on July 07, 1988, and PHOBOS-2 on July 12, 1988. PHOBOS-2 operated 
 
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nominally throughout its cruise and Mars orbital insertion phases, obtaining data on the 
Sun, interplanetary medium, Mars, and Phobos. Contact with PHOBOS-2 was lost 
shortly before the final phase of the mission, during which the spacecraft was to approach 
within 50 meters of Phobos' surface and release two landers, one a mobile hopper and the 
other a stationary platform. The mission of PHOBOS-2 ended on March 27, 1989, when 
the spacecraft signals failed to be successfully reacquired. A malfunction of the on board 
computer was the cause of the failure of PHOBOS-2. On board PHOBOS-2, the TAUS 
experiment was conducted to collect plasma data. The TAUS instrument was a mass-
resolving ion spectrometer that, with its hemispherical electrostatic and electrostatic-
magnetic analyzers, provided separate measurements of three-dimensional velocity dis-
tributions for protons and alphas. 
Viking: Sweden's first satellite, Viking, was launched on February 22, 1986, and placed 
into a final 817 km-13,530 km polar orbit with an inclination of 98.6? and a period of  
261.2 minutes, where it conducted scientific observations of complex plasma processes in 
the magnetosphere and ionosphere of the Earth. The experiments on board the satellite 
were designed to measure electric fields, magnetic fields, charged particles, waves, and 
auroral images. The satellite ceased its operations on May 12, 1987.