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Chapter 2
GUIDING CENTER EQUATIONS
The basis for the test-particle calculations described in this thesis are the guiding
center equations governing single particle motion in the presence of electromagnetic
fields which vary slowly in both time and space. These equations are derived from the
exact equations of motion by means of perturbation analyses, where only the firstorder terms with respect to the parameters, ρ   and   ge are retained [Northrop,
1961, 1966; Schmidt, 1979]. Here,  refers to the particle gyroradius,   is the length
scale of the transverse gradient in ambient geomagnetic and wave electromagnetic
fields,  represents a characteristic frequency of oscillation of the electromagnetic
field, and  ge is the gyrofrequency of the particle. The guiding center equations,
which the test-particle equations are based upon, are presented in this chapter. Details
of the derivations may be found in Appendix A, B and C.
The first section begins with a general statement of charged particle motion. The
principals of guiding center approximation and its application to the particle equations
of motion are also discussed. In Sections 2.2 and 2.3, the guiding center equations are
projected into perpendicular and parallel components with respect to the geomagnetic
field. Both sections include an evaluation of neglected terms. The resulting equations
of motion are presented at the end of this chapter.
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2.1 BASIC EQUATIONS OF MOTION
Newton’s second law of motion applied to a charged particle in electromagnetic
E, Band gravitational  g  fields is,
mr  qEr, t   r  Br, t   mgr, t  .
2.1
The particle mass and charge are designated by m and q, respectively. The
spatiotemporal properties of the fields, r and t, are unrestricted in the equation. The
guiding center approximation introduced by Northrop [1961, 1966] transforms the
system of equations (2.1) into a reduced set of equations describing the particle drift
velocity and its magnetic field-aligned acceleration in general curvilinear orthogonal
coordinates. It is assumed that a locally orthogonal coordinate system can be defined
with one coordinate parallel to a presumably static ambient magnetic field.
The gravitational force exerted on the particles in the near-Earth space environment is
proportional to the inverse square of the particle distance from the Earth, which is very
small at the distance of interest, approximately 7.5RE. Usual LLBL locations are 10RE
or more. The gravitational potential energy of the particle, calculated at the lowest
altitude (320 km) relevant for this study, is 3.3810 4 eV. The ratio of the gravitational
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potential energy to the average magnetosheath particle energy (100 eV), considered in
the numerical examples of Chapters 5 and 6, is 3.3810 6 eV. This ratio is sufficiently
-
less than one so that the gravitational force and associated drift term may be neglected.
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2.2 GUIDING CENTER APPROXIMATION
Figure 2-1 illustrates the orthogonal coordinate system with respect to the ambient
magnetic field lines established by Northrop [1961]. A vector summation, r = R + ρ,
represents the explicit particle position, r, with respect to the guiding center position,
R, and the gyroradius, ρ. The position vectors themselves are functions of time.
Figure 2-1. A charged particle gyrating about its guiding center along the magnetic field B is the
definition of gyro motion of particles. This figure also illustrates the vector addition r = R + ρ
[Northrop, 1961] associated with gyro motion and the guiding center approximation.
The derivation proceeds by expanding equation (2.1) about ρ  0 and time averaging
over the gyroperiod so the results are independent of gyroradius. The system is now
represented as a function only of the guiding center position. The result of this
expansion and averaging is,
  q E  R
  B  M  B ,
R
m
m
2.2
where E and B are now regarded as representations of the electromagnetic fields as
functions of the guiding center position vector, R, and time. The nominal particle
magnetic moment, M, is defined as, M  W B , the amount of perpendicular particle
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energy, W , per magnetic field strength, B. The magnetic moment is assumed to be an
adiabatic invariant [Northrop, 1961; Schmidt, 1979].
2.3 PARTICLE DRIFT VELOCITY
The particle motion in this model is resolved into perpendicular and parallel degrees of
freedom relative to the ambient magnetic field. Equation (2.2) can be expressed in
perpendicular and parallel components of the guiding center system.
The particle motion perpendicular to the total magnetic field, B , is obtained from the
vector product of B with equation (2.2),
  B    E  M  B  m R
  .
R


2
q
q 
B

2.3
The total magnetic field is defined as B  B 0  δB , where B 0 is the ambient magnetic
field and δB is the perturbed magnetic wave field. The first bracketed term denotes
the electric field drift velocity, which dominates the drift motion when compared to
the gradient and inertial drift velocities, the middle and last terms of equation (2.3),
respectively. The perpendicular drift velocity components given below are defined in
terms of the ambient magnetic field, three wave field parameters, E∥, E⊥ and δB, used
in the simulation specified by Streltsov et al. [1998], and the gradient length scales,  ||
and   . It is assumed that the following inequalities exist between the parallel and
perpendicular components of the wave electric field and the parallel and perpendicular
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gradient scale lengths characterizing the wave variations: E∥ << E⊥ and   <<  || . The
ambient geomagnetic and wave magnetic fields satisfy the inequality δB << B0 .
When can the gradient and inertial drifts be neglected? The electric drift is
approximated as v E  E B0 for the comparison to the drift terms below.
Gradient Drift. The gradient velocity, v  , is produced by transverse variations in the
total magnetic field and is defined as,
v 
m B M
 B .
q B2 m
2.4
Assuming the background magnetic field is large and its variations are small, the ratio
of the gradient drift to the perpendicular electric drift is of order,
v
v2 1
E B0 1
~ 
vE
2   ge
~
v
   E B0 1 .
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This expression is first order in the scale factor ρ   . The gradient drift is assumed a
small quantity. The ratio v  v E is evaluated at an equatorial radial distance of 7.5RE,
an electron of thermal speed 100eV, a perpendicular length scale of 2.5RE and a wave
perpendicular electric field of 1mV/m. Note that the perpendicular length scale,
1    B0 B0  3 LRE  1 2.5 RE for L=7.5. The resulting ratio, v  v E  0.006 , is
much less than one and the gradient drift term is neglected in the test-particle
calculations.
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Polarization Drift. The polarization drift velocity, v p , depends on the time rate of
change of the electric field, d E dt , along the particle trajectory. The wave
frequency, , for variation of E⊥ is assumed to be much less than the gyrofrequency.
The polarization drift velocity is defined as,
vp 
m B  vE

.
q B2  t
2.5
Its ratio to the electric drift velocity ( v E ) is,
vp
vE
~
m B0 E 
E  B0 1
2
q B0 B0
~   ge .
The polarization drift is negligible compared to the electric drift because   ge << 1.
Centrifugal Drift. The centrifugal or curvature drift velocity results from particle
parallel motion along a curved magnetic field line and is given by,
v CD 
B m 2 1
 v|| B .
B
B2 q
2.6
The ratio of centrifugal drift to the electric drift term is,
2
2
v CD
v  2v||
m v||
1
E B0  
~
.
vE
qB0  
v E v 2
Since v  v E has been shown to be small and v||2 v2 <10, resulting from the
calculations presented in Chapter 6, the centrifugal drift term is neglected.
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The remaining nonlinear drift terms are of the same order or smaller than the
neglected drift terms considered above and therefore are neglected in the test-particle
calculations (see Appendix C).
2.4 PARALLEL PARTICLE ACCELERATION
Returning to the particle motion along the field line, the basic vector definitions are
again exploited in this section. The terms representing parallel acceleration are
isolated. Total particle acceleration along magnetic field lines,
dv ||
dt

q
M B0  dê 1
,
E || 
R
m
m s
dt
2.7
results from the scalar product of equation (2.2) with the tangential unit vector.
Acceleration by the parallel electric field and the magnetic mirror force (second term)
are assumed to dominate the particle acceleration (see Appendix C).
The reduced guiding center equations, including the approximations and assumptions
described in Sections 2.2 and 2.3 are:
vD 
EB
,
B2
v||  
q
M B
E || 
.
m
m s
2.8
2.9
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