ION KINETICS AT THE HELIOSPHERIC TERMINATION SHOCK PIN ...

BOSTON UNIVERSITY
GRADUATE SCHOOL OF ARTS AND SCIENCES
Dissertation
ION KINETICS AT THE HELIOSPHERIC TERMINATION SHOCK
by
PIN WU
B.S., University of Science and Technology of China, 2001
M.A., Boston University, 2005
Submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
2010

© Copyright by
PIN WU
2010

First Reader
Second Reader
Third Reader
Approved by
Nathan A. Schwadron, Ph.D.
Associate Professor of Astronomy
Boston University
W. Jeffrey Hughes, Ph.D.
Professor of Astronomy
Boston University
Dan Winske, Ph.D.
Laboratory Fellow
Los Alamos National Laboratory

Men encourage her to accomplishments yet she remains eternally feminine.
—Chien-Shiung Wu

For my grandparents

Acknowledgements
I am deeply grateful to my advisor Professor Nathan Schwadron, who trains me; to my
mentors Drs. S. Peter Gary and Dan Winske, who coach me; to my advisory committee
Professors W. Jeffrey Hughes, Harlan Spence and Alan Marscher, who guide me; to
everyone in the Astronomy department and everyone in the ISR-1 division of the Los
Alamos National Laboratory, who accompany me; to my friends at home and abroad,
who believe in me; and mostly, to my family, who love me.
My thanks extend to the National Aeronautics and Space Administration (NASA)
Interstellar Boundary Explorer (IBEX) mission, which funds my dissertation at Boston
University; and to the NASA Solar and Heliospheric Physics SR&T Program, which
funds the Los Alamos portion of this dissertation, performed under the auspices of the
U.S. Department of Energy (DOE).
vi

ION KINETICS AT THE HELIOSPHERIC TERMINATION SHOCK
(Order No. )
PIN WU
Boston University Graduate School of Arts and Sciences, 2010
Major Professor: Nathan A. Schwadron, Associate Professor of Astronomy
ABSTRACT
A number of interesting phenomena were revealed after the two Voyager spacecraft
crossed the termination shock in 2004 and 2007, respectively. This dissertation presents
theoretical and computational studies to address ion heating, energy dissipation and ion
speed distributions at the quasi-perpendicular termination shock. In our model, the
termination shock is distinguished by the presence of a significant fraction of pickup ions,
interstellar atoms which have become ionized and subsequently accelerated up to solar
wind flow speeds. Using the Los Alamos hybrid simulation code, we demonstrate that
the heating of pickup ions is dependent on the phase of gyration about the local magnetic
field when they encounter the termination shock. The temperature of solar wind ions is
raised by a larger factor than that of the pickup ions because some solar wind ions are
specularly reflected. An analytic model for energy partition is developed based on the
Rankine–Hugoniot relations and a polytropic energy equation (the Multicomponent
Rankine–Hugoniot model). The polytropic index γ is varied to improve agreement
between the model and the simulations. When the pickup ion relative density is 0%, the
polytropic index is 5/3. As the pickup ion relative density increases toward 40%, the
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polytropic index increases toward 2.2, suggesting a fundamental change in the character
of the shock. We infer that the pickup ion relative density is about 25%, and that pickup
ions gain the larger share (~90%) of the dissipated energy near the nose of the
termination shock, consistent with Voyager 2 observations. Further, we explore the
consequences of four different assumptions regarding upstream pickup ion velocities,
each corresponding to a different thermalization level. The downstream ion speed
distribution is found to be almost identical for each case, with two Maxwellian
components providing a good fit. In addition, the downstream heated ion spectrum scales
with the solar wind speed, with smaller spectral indexes corresponding to faster solar
wind speeds, consistent with the Interstellar Boundary EXplorer (IBEX) inference. The
IBEX team plans to use the constructed spectra to help interpret their measurements.
Finally, we validate our simulations with numerical tests and observations.
viii

Contents
1 Introduction............................................................................................................... 1
1.1 Motivation ........................................................................................................ 1
1.2 Upstream Conditions........................................................................................ 8
1.2.1 Solar Wind .......................................................................................... 8
1.2.2 Pickup ions.......................................................................................... 9
1.3 Observations of the Termination Shock......................................................... 14
1.4 Kinetic Physics of the Termination Shock ..................................................... 20
1.5 Research Background and Goals of this Dissertation .................................... 26
2 Hybrid Simulations for an Idealized Termination Shock ................................... 32
2.1 The Hybrid Simulations ................................................................................. 32
2.1.1 Computer Simulation Methods......................................................... 32
2.1.2 Previous Simulations of the Termination Shock .............................. 35
2.1.3 Methodology of Hybrid Simulations ................................................ 38
2.1.4 The Los Alamos Hybrid Simulation Model...................................... 41
2.2 Simulation Results Overview......................................................................... 47
2.3 Kinetic Structure of the Termination Shock................................................... 51
2.4 Qualitative Examination of Heating via Phase-Space Plots........................... 58
2.5 Heating and Energy Partition ......................................................................... 65
2.6 Pickup Ion Energization ................................................................................. 70
2.6.1 Simulated Pickup Ion Kinetics.......................................................... 71
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2.6.2 Gyrophase-Dependent Acceleration Model for Pickup Ions............ 76
2.7 Discussion ...................................................................................................... 82
3 Analytic Model for an Idealized Termination Shock .......................................... 85
3.1 Baseline Analytic Formula............................................................................. 85
3.1.1 Heating of a Transmitted Ion ............................................................ 85
3.1.2 Analysis of Upstream Mach Numbers.............................................. 86
3.2 Multicomponent Rankine–Hugoniot Model .................................................. 87
3.2.1 Modified Compression Ratio............................................................ 87
3.2.2 Component Pressures........................................................................ 88
3.2.3 Reflection Efficiency ........................................................................ 91
3.2.4 Energy Partition ................................................................................ 96
3.2.5 Downstream Mach numbers ............................................................. 99
3.3 Downstream Multi-Ion Speed Distributions ................................................ 101
3.3.1 Dimensional Analysis ..................................................................... 102
3.3.2 Downstream Two-Maxwellian Speed Distribution ........................ 103
3.3.3 Refined Downstream Two-Maxwellian Speed Distribution........... 108
3.4 Discussion .................................................................................................... 110
4 Hybrid Simulations for a More Realistic Termination Shock.......................... 115
4.1 Variation of Pickup Ion Velocity Distributions ........................................... 115
4.1.1 Analytic Derivation of Pickup ion Beta and Speed Range............. 116
4.1.2 Simulated Downstream Speed Distribution.................................... 123
4.2 Variation of Alfvén Mach Number .............................................................. 132
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4.3 Variation of Shock Normals......................................................................... 134
4.4 Discussion .................................................................................................... 136
5 Accuracy and Convergence.................................................................................. 139
5.1 Cell Size ....................................................................................................... 139
5.1.1 Cases without Pickup Ions.............................................................. 140
5.1.2 Cases with Pickup Ions ................................................................... 143
5.2 Super-Particles per Cell................................................................................ 146
5.3 Discussion .................................................................................................... 149
6 Theory/Model Connection to Observations........................................................ 154
6.1 Comparison with Voyager 2......................................................................... 154
6.1.1 Structure of the Termination Shock................................................ 154
6.1.2 Core Ions......................................................................................... 156
6.2 Implication for IBEX.................................................................................... 158
7 Summary................................................................................................................ 162
7.1 General Overview......................................................................................... 162
7.2 Review of Scientific Context ....................................................................... 163
7.3 Future Work ................................................................................................. 167
List of Journal Abbreviations...................................................................................... 172
References...................................................................................................................... 173
Curriculum Vitae.......................................................................................................... 188
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List of Tables
Table 1-1. Voyager 2 Observations in the Upstream Region. .......................................... 15
Table 1-2. Voyager 2 Observations in the Downstream Region. ..................................... 16
Table 1-3. Voyager 2 Observations of the Termination Shock. ....................................... 16
Table 1-4. Voyager 2 Observations of the Foreshock Region.......................................... 16
Table 1-5 Voyager 1 (V1) Observations of the Termination Shock................................. 19
Table 2-1 Results Calculated from the Hybrid Simulations (MA = 8, β sw = 0.05). ......... 67
Table 4-1 Results Calculated from the Hybrid Simulations (MA=8, β sw =0.05, β PUI =8.53)
...................................................................................................................... 125
Table 4-2 Properties of the two downstream Maxwellians from the four simulations
(MA=8, β sw =0.05, β PUI =8.53)..................................................................... 128
Table 4-3 The goodness (R) of the two-Maxwellian and the refined two-Maxwellian fits
for the four simulations (MA=8, β sw =0.05, β PUI =8.53) .............................. 131
Table 4-4 The downstream temperatures of the simulations ( sw
β =0.05, φ =20%
Vasyliunas-Siscoe distributed upstream pickup ions).................................. 133
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List of Figures
Figure 1-1 An artist's rendition of the termination shock and its relation to our
heliosphere and the interstellar medium. Adapted from Walt Feimer, National
Aeronautics and Space Administration (NASA).............................................. 2
Figure 1-2 The heliosphere inflated by the solar wind in the region surrounding the Sun
in the interstellar medium. The heliosphere is formed by the supersonic solar
wind as it expands radially from the Sun. At the termination shock, the solar
wind abruptly slows and heats as it turns toward the tail of the heliosphere.
The heliopause is boundary that separates the solar wind flow and the
interstellar flow, where the pressure of the solar ions balances the pressure of
the interstellar ions. In this model, the interstellar wind is also supersonic and
a bow shock forms upstream of the nose of the heliosphere. Adapted by Stone
[2001] from Zank [1999].................................................................................. 4
Figure 1-3 The Parker Spiral. Beyond 20 AU, the interplanetary magnetic field is nearly
toroidal as the solar rotation wraps up the field. Hence, this magnetic field is
nearly perpendicular to the solar wind flow and the termination shock normal.
Figure adapted from Steve Suess, NASA. ..................................................... 11
Figure 1-4 A conceptual illustration of the pickup process, and the subsequent velocity
distribution in the solar wind frame (upper panel) and the spacecraft frame
(bottom panel). The pickup process ends when the pickup ion’s guiding
center moves with the solar wind flow velocity............................................. 12
Figure 1-5 Magnetic field profile from one of the Voyager 2 observed termination shock
crossing. Adapted from: Bulaga et al. [2008]................................................. 15
Figure 1-6 The magnetic field profile of a perpendicular supercritical shock from a hybrid
simulation. The foot, ramp, and overshoot are indicated. From Wu et al.
[1984]. ............................................................................................................ 24
Figure 2-1 A one-dimensional representation of the simulation cells, the mesh points
(center of each cell), and the simulation domain. Cell 1 and Cell NX2 are
ghost cells that are necessary for the specification of boundary conditions.
The simulation domain ranges from Cell 2 to Cell NX1................................ 40
Figure 2-2 The upper panel is a representative setup of the one-dimensional simulation
for the termination shock. The bottom panel shows the time evolution of the
magnetic field B z profile from a 0% pickup ion simulation. Since the
simulation is run in the downstream rest frame, the shock propagates to the
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left. In the bottom panel, x is normalized by c / ω pi where ω pi is the ion
plasma frequency and time t is normalized by i Ω where Ω i the ion cyclotron
frequency based on the upstream magnetic field. From Wu et al. [2009]...... 43
Figure 2-3 Density profile (upper panel) and magnetic field profile (bottom panel) from a
0% pickup ion simulation. The ion density n i is normalized by the upstream
ion density u n ; the magnetic field B z is normalized by the upstream magnetic
field u B ; and x is normalized by / c ω pi . From Wu et al. [2009]................... 49
Figure 2-4 The shock strength (bottom panel) and the downstream flow speed (upper
panel) as a function of pickup ion relative density......................................... 50
Figure 2-5 By and Ez from the 0% (black) and the 20% (blue) pickup ion simulations. By
is normalized to the upstream magnetic field Bu and Ez is normalized by
vABu/c. The simulation dimension x is normalized by ion inertial length
c/ω . ............................................................................................................. 51
pi
Figure 2-6 Profiles of Bz (top panel), Ex (middle panel) and Ey (bottom panel) from the
0% (black) and 20% (blue) pickup ions simulations. The magnetic field is
normalized by the upstream magnetic field Bu; the electric fields are
normalized by vABu/c, and the spatial length x is normalized to the ion inertial
length c/ωpi. .................................................................................................... 53
Figure 2-7 Pickup ion (red), solar wind ion (green) and all physical ions (black) density
and velocity profiles. Panel a, b, c, and d are from the 0% pickup ion
simulation (where pickup ions are test particles). Panel a’, b’, c’ and d’ are
from the 20% pickup ion simulation. In Panel a and a’, all densities are
normalized to the upstream density (nu). In Panel b and b’, each density is
normalized to its upstream value (red: nPUI / n u, PUI , green: nsw / n u, sw,
black:
n/ n u ). The spatial length x is normalized to the ion inertial length c/ωpi. .... 55
Figure 2-8 The phase-space plots of solar wind ions in the 0% pickup ion simulation. The
panels on the top are the x v – x , y v – x and v z – x plots for a) transmitted solar
wind ions and b) reflected solar wind ions. The panels on the bottom are v x
velocity distributions and x v – v y phase-space plots for c) upstream solar wind
ions, d) downstream solar wind ions, e) downstream transmitted solar wind
ions, and f) downstream reflected solar wind ions. The shock is marked by a
dashed line in both Panel a) and b). The solar wind phase-space plot evolves
from the upstream Panel c) into the downstream Panel d) passing the shock.
Empirically, we can separate the downstream population d) into the
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transmitted ions in Panel e) and the reflected ions in Panel f). The velocities
are all normalized by the upstream Alfvén speed v A and x is normalized by
c / ω pi . From Wu et al. [2009]........................................................................ 60
Figure 2-9 The phase-space plots of the 0% pickup ion simulation where pickup ions are
treated as test particles. Solar wind ions are plotted in the panels (a,c,d) on the
left and pickup ions are plotted in the panels (b,e,f) on the right. Upper panels
are magnetic field profile B z –x, x v –x phase-space plots, and v y –x phasespace
plots for a) solar wind ions and b) pickup ions. Lower panels are v x
velocity distributions and x v – v y phase-space plots for c) upstream solar wind
ions, d) downstream solar wind ions, e) upstream pickup ions, and f)
downstream pickup ions. The velocities are all normalized by the upstream
Alfvén speed v A and x is normalized by c / ω pi . From Wu et al. [2009]. ..... 62
Figure 2-10 The phase-space plots of the 20% pickup ions simulation. Solar wind ions
are plotted in the panels (a,c,d) on the left and pickup ions are plotted in the
panels (b,e,f) on the right. Upper panels are magnetic field profile z B –x, v x –x
phase-space plots, and v y –x phase-space plots for a) solar wind ions and b)
pickup ions. Lower panels are v x velocity distributions and x v – v y phasespace
plots for c) upstream solar wind ions, d) downstream solar wind ions, e)
upstream pickup ions, and f) downstream pickup ions. The velocities are all
normalized by the upstream Alfvén speed v A and x is normalized by c / ω pi .
From Wu et al. [2009]. ................................................................................... 64
Figure 2-11 The simulated temperature jumps (blue: solar wind; red: pickup ion) as
compared with adiabatic temperature jumps. The polytropic indexes are
calculated from Equation (2.10). In the 0% pickup ion simulation, pickup ions
are treated as test particles. They are subject to the electricmagnetic fields but
make no contributions to the source terms..................................................... 67
Figure 2-12 Velocity phase-space plots (vx–vy) for pickup ions (left panels) and solar
wind ions (right panels) within 20 c/ωi of the shock front from the 0% pickup
ion simulation. All velocities are normalized by vA....................................... 72
Figure 2-13 Phase space evolution of the pickup ions (first row of the vx–vy panels) and
of the solar wind ions (second row of the vx–vy panels) as they cross the 20%
pickup ion shock. The kinetic shock structure from Section 2.3 is zoomed in
for comparison. Each panel of the evolution series corresponds to the color
marked spatial region of 20 c/ωpi. The center of each adjacent panel is 5 c/ωpi
apart................................................................................................................ 74
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Figure 2-14 Pickup ion trajectories from the 20% pickup ions simulation in velocity
space perpendicular to the background magnetic field. Both the x v and v y are
normalized to the upstream Alfvén speed v A . Each panel shows a
representative ion trajectory from time zero (indicated by an asterisk) to the
end of the simulation. The arrows indicate the sense of temporal progress of
each trajectory. From time zero, the color of each trajectory changes from
black through blue, green, yellow, orange and, near the end of the simulation,
to red. From Wu et al. [2009]......................................................................... 77
Figure 2-15 Schematic interpretation of pickup ion energy gain (downstream rest frame).
Left side of the figure corresponds to a representative trajectory of a
“reversed” ion. Right side of the figure corresponds to a representative
trajectory of a “crossing” ion. Both curves correspond to gyromotions with
increasing Lamour radii. The electric field Ex and the electric field Ey in the
shock region (foot, overshoot) are also shown. The downstream oscillation
these fields are not shown. Instread, thick dash lines mark the average
downstream fields........................................................................................... 79
Figure 2-16 Energy distribution histogram of the downstream pickup ions. Blue line: the
distribution for “reversed” ions. Black line: the distribution for “crossing”
ions. The pickup ion energy is normalized to the upstream solar wind
dynamic energy. ............................................................................................. 82
Figure 3-1 Compression ratio r S , pressure jump Pd / P u and downstream pickup ions
PUI
thermal pressure ratio Pd / P d as a function of the pickup ion relative
density φ . Dashed lines are theoretical predictions when γ is set to be 5/3,
the solid lines are theoretical predictions when γ is 2.2. The diamond
symbols mark values from simulations. From Wu et al. [ 2009]. .................. 94
Figure 3-2 Solar wind reflection efficiency ε ref (%) as a function of the pickup ion
relative density φ . The dashed lines correspond to γ =5/3 and the solid lines
correspond to γ =2.2. Adapted from Wu et al. [2009]. ................................. 96
Figure 3-3 Termination shock energy partition η (%) as a function of the pickup ion
relative density φ . The dashed lines correspond to γ =5/3 and the solid lines
correspond to γ =2.2. Red diamond symbols mark the percentage of heating
the pickup ions gain from our simulations. Adapted from Wu et al. [2009].. 98
Figure 3-4 Upstream and downstream Mach numbers as functions of pickup ion relative
density. The three panels on the right are subsets of the same quantities of the
three panels on the left: Alfvén Mach number M A (top panel), sonic Mach
xvi

number cs M (middle panel), Magnetosonic Mach number M MS (bottom
panel). In the right panels, we have a better view of how the solid red lines
differ from the dashed lines. The black lines are the upstream Mach numbers
as a function of the pickup ion relative density φ ; the red lines are the
analytically calculated downstream Mach numbers at γ =2.2 and the red
dashed lines are the analytically calculated downstream Mach numbers at
γ =5/3. From Wu et al. [2009]. .................................................................... 100
Figure 3-5 Downstream solar wind (top panel) and pickup ion (bottom panel) thermal
speeds as functions of the pickup ion relative density and the polytropic index
γ for a MA=8, β sw =0.05 perpendicular termination shock. The vertical line
marks the 20% shell distributed pickup ion case, in which the γ =1.95
solutions predict that the downstream solar wind thermal speed is 1.93 vA and
the downstream pickup ion thermal speed is 12.20 vA, consistent with the
simulation fitted thermal speeds (Chapter 4). .............................................. 106
Figure 3-6 Downstream Maxwellian-1 (top panel) and Maxwellian-2 (bottom panel)
thermals speeds as functions of the shell distributed pickup ion relative
density and the polytropic index γ for a MA=8, β sw =0.05 perpendicular
termination shock. For the 20% pickup ion case, the γ =1.95 solutions predict
that the downstream Maxwellian-1 thermal speed is 1.39 vA and the
downstream Maxwellian-2 thermal speed is 12.26 vA, consistent with the
simulation fitted thermal speeds (Chapter 4). .............................................. 109
Figure 4-1 The distribution function fPUI ( v⊥ ) versus v⊥ for the pickup ion shell (black),
the pickup ion sphere (green), the Vasyliunas–Siscoe distributed pickup ions
(orange), and the Maxwellian pickup ions (red) at the same β PUI =8.53. The
panel on the left is ploted in linear-linear scale and the panel on the right is the
same plot in linear-log scale. The pickup ion’s perpendicular velocity vperp is
normalized to the upstream velocity uu . From Wu et al. [in preparation]... 121
Figure 4-2 The magnetic field profile (Bz) of the four simulations. Black: the simulation
with a pickup ion velocity shell. Green: the simulation with a pickup ion
velocity sphere. Orange: the simulation with the Vasyliunas–Siscoe
distributed pickup ions. Red: the simulation with the Maxwellian distributed
pickup ions. From Wu et al. [in preparation]. .............................................. 125
Figure 4-3 The upstream and the downstream speed distributions in their respective
plasma frames. Black: the simulation with a pickup ion shell. Green: the
simulation with a pickup ion sphere. Orange: the simulation with the
Vasyliunas–Siscoe distributed pickup ions. Red: the simulation with the
xvii

Maxwellian distributed pickup ions. Upper left panel: upstream speed
distribution in log-log scale. Upper right panel: upstream speed distribution in
log-linear scale. Lower left panel: downstream speed distribution in log-log
scale. Lower right panel: downstream speed distribution in log-linear scale.
From Wu et al. [in preparation].................................................................... 127
Figure 4-4 The fitting (orange line) of the downstream distributions (in their respective
plasma frames) with the two maxwellian components as derived from Section
3.3.3. The left panel is plotted in log-log scale and the right panel is plotted in
the log-linear scale. The dotted lines are downstream speed distributions from
the simulations. Black dots: the simulation with a pickup ion ring shell. Green:
the simulation with a pickup ion sphere. Orange: the simulation with the
Vasyliunas–Siscoe distributed pickup ions. Red: the simulation with the
Maxwellian distributed pickup ions. Right panel: log-linear scale. Left panel:
log-log scale. The perpendicular speed vperp is in the unit of the upstream
Alfvén speed vA. From Wu et al. [in preparation]........................................ 130
Figure 4-5 The fitting of the downstream distributions (in their respective plasma frames)
with the two maxwellian components as derived from Section 3.3.2. The left
panel is plotted in log-log scale and the right panel is plotted in the log-linear
scale. The dotted lines are downstream speed distributions from the
simulations. Black dots: the simulation with a pickup ion ring shell. Green:
the simulation with a pickup ion sphere. Orange: the simulation with the
Vasyliunas–Siscoe distributed pickup ions. Red: the simulation with the
Maxwellian distributed pickup ions. Right panel: log-linear scale. Left panel:
log-log scale. The perpendicular speed vperp is in the unit of the upstream
Alfvén speed vA. From Wu et al. [in preparation]........................................ 131
Figure 4-6 The downstream ion speed distributions (in their respective plasma frames)
from the simulations: MA=4 (blue), MA=6 (green), MA=8 (black), MA=12
(orange) and MA=16 (red). The left panel is in the log-log scale and the right
panel is in the log-linear scale. From Wu et al. [in preparation].................. 133
Figure 4-7 The magnetic field profile (Bz, Bx) of the three simulations with different
shock nomal. Black: the simulation with θBN = 90°. Orange: the simulation
with θBN = 80°. Blue: the simulation with θBN = 70°. ................................... 135
Figure 4-8 The downstream ion speed distributions (in their respective plasma frames)
from the three simulations: θBN = 90° (black), θBN = 80° (orange) and θBN =
70° (blue). The left panel is in the log-log scale and the right panel is in the
log-linear scale. ............................................................................................ 136
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Figure 5-1 Magnetic field profiles of the three simulations without pickup ions: a) H=
c/ω pi (top panel), b) H=0.5 c/ω pi (middle panel), and c) H=0.25 c/ω pi
(bottom panel). ............................................................................................. 142
Figure 5-2 The compression ratios as a function of time calculated from the average
density of a 50 c/ω pi wide region that is 10 c/ω pi away from the overshoot in
the downstream. The three simulations are presented: a) H= c/ω pi (blue), b)
H=0.5 c/ω pi (black), and c) H=0.25 c/ω pi ( red curve).............................. 143
Figure 5-3 Magnetic field profiles of the three simulations at the same resistivity but
different cell sizes: a) H= c/ω pi , b) H=0.5 c/ω pi , and c) H=0.25 c/ω pi .
Panel c) shows an unphysical fast moving shock caused by the small cell size.
This effect can be eliminated with a larger resistivity as shown in Panel c’),
where the cell size is as small as Panel c) but the resistivity is 5 times bigger
the previous 3 panels. ................................................................................... 144
Figure 5-4 The compression ratios as a function of time calculated from the average
density of a 50 c/ω pi wide region that is 10 c/ω pi away from the overshoot in
the downstream. The four simulations are presented: a) H= c/ω pi (blue), b)
H=0.5 c/ω pi (black), c) H=0.25 c/ω pi (dashed red curve) and c’) H=0.25
−4
c/ω with higher resistivity (red curve) η = 10 (4 π / ω ) ....................... 146
pi
Figure 5-5 Magnetic field profiles of three simulations with different numbers of particles
per cell. Top panel: simulation with 25 particles per cell; middle panel:
simulation with 100 particles per cell; bottom panel: simulation with 400
particles per cell............................................................................................ 148
Figure 5-6 Entropy (black) of the system at the end of the simulation run: 20% pickup ion
shell, MA=8, β sw =0.05, β PUI = 8.53. The temperature profile (blue) is scaled
and overplotted. ............................................................................................ 152
Figure 6-1 Comparison of our simulated magnetic field profile (orange) with the Voyager
observed TS-3 magnetic field profile (black). Voyager 2 observing time is
label on the x axis on the bottom and the simulation length is marked on the x
axis on the top. Voyager 2 Data are provided by Leonard F. Burlaga......... 155
Figure 6-2 The fitting of the simulated downstream distribution (MA=8, β sw =0.05, and
β PUI =8.53) with one maxwellian component. The left panel is plotted in loglog
scale and the right panel is plotted in the log-linear scale. The circles are
xix
pi

downstream speed distributions from the simulations. The lines are the fitted
function. The perpendicular speed vperp is in the unit of the upstream Alfvén
speed vA........................................................................................................ 158
Figure 6-3 The modeled x direction (r direction in the RTN coordinate system) energy
spectra of the sunward directing ions in the shock frame (left). Partial spectra
of Figure 6-3 that fall into the IBEX-Hi detector limit (right). We fit the
spectra with power law functions (solid lines). Blue: MA=4, black: MA=8, and
red: MA=16. .................................................................................................. 159
Figure 7-1 The three upstream components in the solar wind frame. The solid curve
represents the Maxwellian solar wind; the dotted curve represents the
Vasyliunas–Siscoe pickup ions; the dashed curve represents the powerlaw tail.
...................................................................................................................... 169
xx

List of Abbreviations
ACR Anomalous Cosmic Ray
AU Astronomical Unit
BU Boston University
ENA Energetic Neutral Atoms
GPDA Gyrophase-dependent Acceleration
IBEX Interstellar Boundary Explorer
MHD Magnetohydrodynamics
NASA National Aeronautics and Space Administration
PIC Particle-in-cell
RTN Radial-Tangential-Normal
SDA Shock Drift Acceleration
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List of Symbols
B magnetic field
Bj the jth component of magnetic field
c speed of light
d as subscript or superscript, denotes downstream
DT simulation time step
E electric field
Ej the jth component of electric field
e elementary charge
19
1.6022 10 −
× C
e as subscript, denotes electron species
f distribution function
fi distribution function of the jth particle/species/component
j as superscript and subscript, denotes ion species/component
kB Boltzmann constant
MA Alfvén Mach number
Mcs sonic Mach number
MMS Magnetosonic Mach number
mj mass of a jth species particle
n number density
23
1.38 10 −
× J･K -1
nj number density of the jth species/component
P pressure
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Pj pressure of the jth component
rL Larmor radius (gyroradius)
rS compression ratio (shock strength)
T temperature
Tj temperature of the jth species/component
u flow velocity
u as subscript or superscript, denotes upstream
ud downstream flow velocity
uu upstream flow velocity
v individual ion velocity
vA Alfvén speed
vcs sound speed
vth thermal speed
β plasma beta
β j plasma beta for the jth component
γ polytropic index
γ j
polytropic index for the jth component
ε ref solar wind ion reflection efficiency
ε 0
permittivity of free space
xxiii
12
8.8542 10 −
× F･m -1
η when expressed without subscript, denotes resistivity
η j
the percentage of dissipated energy that heats the jth component

θ BN shock normal angle
κ power law index
λ D Debye length
μ 0 permeability of free space
τ temperature jump
xxiv
4π10 −7
× H･m -1
τ j
temperature jump of the jth component
ϕ pickup ion relative density
Ω cyclotron frequency
Ωj cyclotron frequency of the jth species/component
χ d downstream pickup ion relative pressure
ω p wave frequency
ω pj plasma frequency of the jth species

1 Introduction
1.1 Motivation
Our solar system lies within the heliosphere, an asymmetric bubble inflated into the
interstellar medium by the outflowing solar wind. The boundary of the heliosphere is the
heliopause (located at ~120 astronomical units 1 ), where the solar wind pressure balances
the interstellar wind pressure. The heliopause is therefore a separator between the solar
wind and the instellar medium. Inside the heliopause, there is another heliospheric
boundary: the heliospheric termination shock (located at ~90 AU). This shock, like all
collisionless heliospheric shocks, is relatively thin, about 10 -5 heliosphere radius (10 7 km)
in thickness. At the shock, the interstellar medium starts to have an effect on the solar
wind. The solar wind is slowed from a supersonic flow in the upstream region to a
subsonic flow in the downstream region and then deflected back to the tail.
The relationship between the termination shock, the heliosphere and the interstellar
medium is illustrated in Figure 1-1. In the figure, upstream is sunward of the termination
shock; the heliosheath lies downstream, beyond the termination shock. The nose of the
termination shock is in the direction of the solar system's motion through the local
interstellar medium. If this relative motion is supersonic, a bow shock will form in the
interstellar medium. In this case, the heliosheath is divided into two regions: the inner
heliosheath (the region between the termination shock and the heliopause) and the outer
heliosheath (the region between the bow shock and the heliopause). The inner heliosheath
1 1 astronomical unit (AU) =1.49×10 13 cm, approximately the Sun–Earth distance.

2
is the sheath of primarily solar wind ions and the outer heliosheath is the sheath of
primarily interstellar material.
Outer
Heliosheath
nose
Inner
Upstream Downstream
Figure 1-1 An artist's rendition of the termination shock and its relation to our
heliosphere and the interstellar medium. Adapted from Walt Feimer, National
Aeronautics and Space Administration (NASA).

3
A more physical image of the termination shock and the heliospheric environment
was produced by Zank [1999] through a Magnetohydrodynamic (MHD) simulation and
was later adapted by Stone [2001], as shown in Figure 1-2. This model assumes the
existence of a bow shock in front of the heliosphere. The interstellar wind is relatively
cool upstream of the bow shock (dark blue), just as the solar wind is relatively warm
upstream of the termination shock (yellow). Both plasmas are heated at their respective
shocks, although the heating of the interstellar wind (light blue) is less than that of the
solar wind (red) because it is expected that the bow shock is considerably weaker than the
termination shock. The heated interstellar wind and the heated solar wind encounter at the
heliopause (green region), where the two flows mix and where the temperature falls in-
between the temperatures of both flows.
A closer examination of Figure 1-2 shows that the termination shock forms a comet-
like oval instead of a perfect sphere. The nose of the termination shock is compressed as
the solar system moves into the local interstellar medium. On the opposite site, the tail of
termination shock extends into the local interstellar medium. This tail feature is more
pronounced for the inner heliosheath, as the downstream solar wind flows deflect and are
dragged tailward (the flow directions are indicated with black arrows). The solar wind
plasma inside the termination shock is generally cooler than the solar wind plasma in the
inner heliosheath, because the termination shock slows the flow and heats the plasma.
The Voyager spacecraft are currently in the inner heliosheath. Their trajectories are
marked by the blue arrow (Voyager 1) and red arrow (Voyager 2). Both Voyagers
encounter the shock in the upwind direction near the nose of the termination shock. The

4
Voyager measurements of the termination shock will be summarized in Section 1.3 of
this chapter.
Figure 1-2 The heliosphere inflated by the solar wind in the region surrounding the Sun
in the interstellar medium. The heliosphere is formed by the supersonic solar wind as it
expands radially from the Sun. At the termination shock, the solar wind abruptly slows
and heats as it turns toward the tail of the heliosphere. The heliopause is boundary that
separates the solar wind flow and the interstellar flow, where the pressure of the solar
ions balances the pressure of the interstellar ions. In this model, the interstellar wind is
also supersonic and a bow shock forms upstream of the nose of the heliosphere. Adapted
by Stone [2001] from Zank [1999].

5
Unlike the Earth’s bow shock which has been extensively measured in the past 50
years with satellite observations, the termination shock is not well explored or understood
due to its distant location and large size. Voyager 1 and Voyager 2, launched in 1977, are
the only spacecraft to reach the termination shock in 2004 and 2007, respectively.
However, the Voyagers are not able to measure the suprathermal ions (~keV). Voyager
1’s plasma instrument failed early before the spacecraft crossed the termination shock.
Voyager 2 has a working plasma instrument, but the resolution of this instrument
prohibits the measurement of the suprathermal ions, because the instrument is not able to
distinguish signals from background noise in this energy range. The first indirect
measurement of the suprathermal ions was made by the Interstellar Boundary EXplorer
(IBEX) [McComas et al., 2009], through the remote measurement of energetic neutral
atoms (ENAs).
The IBEX spacecraft was launched on October 19, 2008 as an Earth orbiting satellite.
The spacecraft’s low energy sensor IBEX-Lo measures neutral atoms ranging from 2 eV
to 2 keV. The spacecraft’s high energy sensor IBEX-Hi measures energetic neutral atoms
(ENAs) ranging from 300 eV to 6 keV. When IBEX’s orbit is outside the Earth’s
magnetosphere, the spacecraft measures ENAs originated behind the termination shock in
the heliosheath. The important fact about these ENAs is that they are originally
termination shock heated ions. The ions undergo charge exchange with interstellar neutral
atoms in the inner heliosheath. As a result of the charge exchange, the termination shock
heated ions are converted into ENAs. As neutrals, the ENAs are unconfined by magnetic
fields. The ENAs which have a sunward directed velocity will propagate into the

6
heliosphere toward IBEX’s location. Therefore, IBEX’s measurement is a line-of-sight
integration of the ENAs. To interpret the ENA measurements from IBEX, one needs to
understand the termination shock heated ions. Our theoretical study of the ion kinetics at
the termination shock is thus motivated by the IBEX mission to take important first steps
toward understanding these ions.
Collisionless shocks heat plasmas, both electrons and ions. Unlike the Earth’s bow
shock, which is dominated by the thermal solar wind ion, the termination shock has an
additional ion component, the suprathermal pickup ions. The presence of these ions
implies that the physics of the termination shock is significantly different from that of the
Earth’s bow shock. The important physical questions then are: How is the shock affected
by this suprathermal ion component? What is the partition of dissipated energy between
the solar wind component and the pickup ion component? How is each ion component
energized? The related processes that provide answers to these questions are not only
important to the termination shock but also generic to plasma physics. For example, other
astrospheres 2 may also have more than one ion component at astrospheric termination
shocks. Although the ions of these astrospheres operate at different energies, some of the
scaling concepts and physical insights we can gain from studying the heliospheric
termination shock are extendable to astrospheric termination shocks, particularly for the
astrospheres that are similar to the heliosphere.
2 Like the heliosphere, an astrosphere is a bubble blown by hot plasmas (stellar wind) originating from a
single massive Sun-like star into the interstellar medium. Analagous to the heliospheric termination shock,
an astrospheric termination shock is where the stellar wind becomes subsonic.

7
We emphasize here that in this dissertation we do not consider the role of energetic
particles (anomalous cosmic rays with energy ranging from ~100 keV to the MeV level
and galactic cosmic rays with energy ranging from the MeV level to the TeV level),
which are widely believed to be accelerated by diffusive shock acceleration at various
astrophysical shocks (e.g., supernova remnants) [Kirk and Dendy, 2001]. Rather, to
explain Voyager observations and to complement the recent IBEX observations at 2 eV
to 6 keV [McComas et al., 2009, Funsten et al., 2009, Schwadron et al., 2009, Fuselier et
al., 2009], we focus on the modeling aspects of the thermal and suprathermal ions (~95%
protons) at the termination shock. These ions and the associated magnetic fields, when
extended to three spatial dimensions, will provide the groundwork upon which to build
energetic particle models. Similar work has been done from different perspectives. Lucek
and Bell [2000] perform kinetic cosmic ray simulations coupled to a single fluid
magnetohydrodynamic code. Giacalone and Ellison [2000] investigate the behavior of
energetic particles using hybrid simulations that treat pickup ions as test particles.
Although we have not considered or included energetic particles, our treatment of the
suprathermal pickup ions throughout this work is self-consistent (meaning that pickup
ions are not treated as test particles but as particles fully participating in the dynamics).
With energetic particles in mind for the future, we focus here on understanding the
fundamental scaling of the termination shock and the relative importance of the ion
components.

1.2 Upstream Conditions
8
Upstream of the termination shock, the primary charged particles are electrons, the solar
wind ions and the pickup ions. In this work, we assume all ions are protons. This
assumption is valid for most quiet time solar wind conditions, when helium 3 and other
heavy ions constitute only a very small fraction of the upstream solar wind population,
e.g., the Helium to proton number density ratio is He/H

9
articles predict that the termination shock location falls between 80 AU and 100 AU. This
prediction was confirmed by the Voyagers’ crossings.
1.2.2 Pickup ions
In contrast to the terrestrial bow shock, the termination shock contains a significant
component of very hot pickup ions. Pickup ions are created from interstellar neutral
atoms that drift in through the heliosphere, unaffected by the solar wind's electromagnetic
fields. These atoms become ionized through charge-exchange (electron transfer) with
solar wind ions or by photo ionization. The ionization process is very fast. The newborn
proton maintains its initial velocity, since the interstellar medium velocity uISM~25 km/s
(very small compared with the solar wind bulk velocity usw~400km/s). After ionization,
this new proton is picked up by the solar wind flow through interactions with the
interplanetary magnetic field, as described in the pickup process detailed below.
Consider a newly ionized pickup ion. This new proton starts to gyrate around the
magnetic field. In the simplest case when the solar wind flow is perpendicular to the
magnetic field (occurs at a large distance from the Sun, as implied by the Parker spiral as
shown in Figure 1-3), the proton gyrates around the interplanetary magnetic field with an
initial velocity ~usw as seen in the solar wind frame (top panel of Figure 1-4). In the
spacecraft frame (bottom panel of Figure 1-4), the proton starts with a very small velocity
(~25 km/s) as compared with the solar wind flow. Therefore, the proton “sees” a
motional electric field caused by the moving solar wind plasma

10
Esw = − usw × B.
(1.1)
A new proton gains energy from the motional electric field until its guiding center
reaches a velocity that is equivalent to the solar wind flow velocity so that it becomes part
of the bulk flow and no longer “sees” the motional electric field. In the spacecraft frame,
this process appears as if those new protons are picked up by the solar wind flow. Thus,
the protons received the name “pickup ions”. As a result, the pickup ions form a velocity
ring by the end of the pickup process (in a gyroperiod), as shown in Figure 1-4. It should
be noted that in reality, the magnetic field is not strictly perpendicular to the solar wind
flow. The pickup ion’s gyrovelocity in the solar wind frame should be modified by a
factor “sinθ”, where θ is the angle between the magnetic field and the solar wind flow.
This factor can be neglected at large heliospheric distances because of the Parker spiral.
Due to the initial deviation of the ring distribution from a Maxwellian, hydromagnetic
waves in the solar wind are rapidly amplified by various anisotropic phase-space
instabilities. These waves mediate the coupling of pickup ions parallel velocity to the
solar wind flow [Burgess, 1997]. In addition, pre-existing solar wind
turbulence/fluctuations enhance this coupling too [e.g., Burgess, 1997; Intriligator et al.,
1996]. Upstream of the termination shock, it is widely believed that the pickup ions at
least form a shell distribution due to the strong pitch angle scattering [Burgess, 1997]. A
ring distribution is unlikely unless the mass loading is extremely strong within a few
pickup ion gyro radii upstream of the termination shock.

Shock normal
11
Figure 1-3 The Parker Spiral. Beyond 20 AU, the interplanetary magnetic field is nearly
toroidal as the solar rotation wraps up the field. Hence, this magnetic field is nearly
perpendicular to the solar wind flow and the termination shock normal. Figure adapted
from Steve Suess, NASA.

u
B
Esw
usw
Solar wind frame
sw
vy
vy
usw
Spacecraft frame
12
pickup ring
solar wind
Figure 1-4 A conceptual illustration of the pickup process, and the subsequent velocity
distribution in the solar wind frame (upper panel) and the spacecraft frame (bottom
panel). The pickup process ends when the pickup ion’s guiding center moves with the
solar wind flow velocity.
usw
solar wind
usw
vx
pickup ring
2usw
vx

13
While the solar wind ions have a relatively low temperature (a few eV upstream of
the termination shock), pickup ions are created with a high effective temperature (~1
keV). Hence, pickup ions are crucial to this study, because they dramatically increase the
effective temperature of the upstream flow. It is also important to note that the increase of
upstream flow speed (fast solar wind) significantly increases pickup ion energies. The
effect of this on the shock will be discussed in Chapter 4.
Although the pickup process upstream of the shock creates a ring-shaped ion velocity
distribution initially, this anisotropic distribution is unstable to the growth of waves.
These waves pitch-angle scatter the pickup ions into a shell distribution. From there the
pickup ions are further scattered into broader distributions: first a velocity sphere, then a
velocity distribution that described by Vasyliunas and Siscoe [1976] (which we refer to as
the Vasyliunas–Siscoe distribution), and finally a Maxwellian distribution. The exact
pickup ion velocity distribution has not been measured upstream of the shock, due to the
Voyager spacecraft’s inability to resolve pickup ions. For the simulations described in
later chapters, we assume that the upstream pickup ion distributions are a velocity shell, a
velocity sphere, a Vasyliunas–Siscoe distribution and a Maxwellian distribution,
respectively. Each velocity assumption corresponds to a successive level of
thermalization experienced by the pickup ions as they flow away from the Sun, as well as
a successive level of the pickup mass loading process along the path. By exploring the
outcomes of these different assumptions, we can determine the importance of this
unknown parameter and understand the kinetics.

14
1.3 Observations of the Termination Shock
To date, the only in-situ measurements of the termination shock have been made by the
Voyager spacecraft. Voyager 1 crossed the termination shock in December 2004 at a
heliocentric distance of 94 AU and a heliospheric latitude of 34.1 ° [Stone et al., 2005]. In
August 2007 Voyager 2 crossed the termination shock at 84 AU and -27.5 ° in
heliospheric latitude [Decker et al., 2008]. At the times of their respective crossings, both
Voyager 1 and Voyager 2 carried operating magnetometers and instruments to measure
energetic charged particles, but only Voyager 2 carried a functional plasma instrument
(energy range 10-5950 eV). However, because of the plasma instrument’s low energy
resolution (29%), Voyager 2 was not able to measure the relatively low flux of
suprathermal ions (~keV ions). Therefore, there are no direct observations of the partition
of dissipated energy at the termination shock.
From the Voyager measurements, Richardson et al. [2008] conclude that the
termination shock is a quasi-perpendicular, supercritical shock. Burlaga et al. [2008]
further show that the shock’s magnetic profile has the typical supercritical shock feature 4 :
a foot, a ramp, an overshoot, and an undershoot, as illustrated in Figure 1-5.
Richardson [2008] provides a detailed analysis of the Voyager 2 plasma data at the
shock and into the heliosheath. Both the Richardson et al. [2008] and the Richardson
[2008] papers demonstrate that there were substantial fluctuations in the plasma
properties, as shown in the following tables that summarize Voyager 2 observation from
4
The physical importance of these structures as to supercritical shocks will be explained in Section 1.4
when we describe the kinetic physics.

15
Richardson et al. [2008], Richardson [2008], Li et al. [2008], and Burlaga et al. [2008].
In the tables, TS-2 is the second crossing when the termination shock is moving
outwards, and TS-3 is the third crossing when the termination shock is moving inward.
Figure 1-5 Magnetic field profile from one of the Voyager 2 observed termination shock
crossing. Adapted from: Bulaga et al. [2008]
Table 1-1. Voyager 2 Observations in the Upstream Region.
Crossings nu (cc) uu (km/s) MMS,sw-core ≈ MA β Bu (nT)
u
TS-2 0.0013 (321.0, 11.3, 1.1)
TS-3 0.0013
4.9±0.1 0.053 (-0.012, -0.063, -0.022)
(321.0, 11.3, 1.1) 8.8±1.2 0.053 (-0.012, -0.063, -0.022)

16
Table 1-2. Voyager 2 Observations in the Downstream Region.
Crossings nd (cc) ud (km/s) MMS, sw-core, d ≈MA, d β Bd (nT)
d
TS-2 0.0033 (184.7, 11.9, 5.8) 1.1±0.1
TS-3 0.0022 (167.6, 29.7, 8.9) 2.8±0.4
Table 1-3. Voyager 2 Observations of the Termination Shock.
1.143 (-0.003, -0.132, -0.022) )
0.508 (-0.006, -0.122, -0.013)
Crossings θBN rS=nd/nu uS (km/s) wS (km) wramp (km) η τ
sw
TS-2 82.8°±3.9° 2.38±0.14 94.0±3.4 300,000
TS-3 74.3°±11.2° 1.58±0.71 -67.9±17.3 100,000
Table 1-4. Voyager 2 Observations of the Foreshock Region.
- ~15% ~10 – 20
6000 ~15% ~10 – 20
rforeshock (AU) uu,foreshock(km/s) |ΔEsw,ram|/ Esw,ram
75 400-300 (in three steps) 40%
In the tables, n denotes the plasma density; u denotes flow velocity, subscripts “u”
and “d” denote upstream and downstream; MMS is the magnetosonic Mach number
defined as MMS=uu/(vA 2 +vcs 2 ) 1/2 , where vA is the Alfvén speed and vcs is the sound speed
(caution: in the observations, vcs can only be inferred from the core solar wind, which in
reality is expected to be much smaller than its true value.); MA is the Alfvén Mach
number defined as MA=uu/vA; β is the plasma beta of the observed core solar wind ions;
B is the magnetic field; θBN is the shock normal (defined as the angle between the
magnetic field and the shock normal); rS is the shock strength (compression ratio); uS is
the shock velocity (where a negative value means that the shock is moving inward); τ is
the temperature jump of the measure core solar wind ions; wS is the width of the shock;

17
wramp is the width of the ramp (notably for TS-3, w ramp ~1 c/ ω pi, u [Burlaga et al., 2008]);
η represents energy partition and η sw is the amount of ram energy that is converted into
solar wind heating; rforeshock (AU) is the location from the Sun of the foreshock
measurements; uu,foreshock is the bulk speed of the upstream solar wind in the foreshock
region which drops from 400 km/s to 300 km/s in three steps; |ΔEsw,ram|/ Esw,ram is the
amount of solar wind ram energy that has been lost in the foreshock region. The vectors
in the table are expressed in the Radial-Tangential-Normal (RTN) coordinate system (R,
points from the Sun to the spacecraft; T, the Sun’s rotation vector crossed into R; N,
completes the right-handed triad). For Li et al. [2008], we cite the mean values from their
Table 1 and 2 in order to be consistent with Richardson et al. [2008], Richardson [2008],
and Burlaga et al. [2008]. The resolutions of the Voyager 2 measurements are:
• Magnetic field: 48 s [Burlaga et al., 2008],
• Plasma: 192 s [Richardson, 2008],
• Energy: 29% [Richardson, 2008].
There are some inconsistencies between those published values, due to fluctuations
and the limited resolution of the instruments onboard Voyager 2. For example, the
downstream flow speeds of TS-2 and TS-3 inferred by Li et al. [2008] seem at odds with
each other. As TS-2 is both more perpendicular and stronger than TS-3, the downstream
flow speed of TS-2 should be smaller than that of TS-3. However, this is not the case in
the tables. Also, the inferred temperatures
2
Td = βdBd /(2 μ0kBnd)
are 174,000 K
(downstream) and 5000 K (upstream) for TS-2 and 100,000 K (downstream) and 5000 K

18
(upstream) for TS-3. However, Richardson [2008] measured 11,000 K over a wider range
upstream and 181,000 K over a wider range downstream. For the above reasons, we
adopt an estimate that the temperature jump of the core solar wind is on the order of ~10
– 20 as listed in the last column of Table 1-3.
In a plasma, the total sound speed is dominated by hot ions. The magnetosonic Mach
number is the coupled Mach number of the sonic Mach number and the Alfvén Mach
number. Excluding the hot ions, the Voyager-measured magnetosonic Mach number
actually approximates the entire flow’s Alfvén Mach number (MMS,sw-core≈ MA), as
marked in Table 1-1 and Table 1-2. The reason is that the Voyager measured core solar
wind has a very small plasma beta and the sound speed calculated from these low energy
core solar wind ions is only a small fraction of the total sound speed of the flow.
The Alfvén Mach number M A = u miμ0 n / B,
for a strictly perpendicular shock, is
MA,d=MA,u/rS 1.5 (see Section 3.2.5 for derivation). Taking TS-3, if rS=1.58 (as in Table 3),
then MA,d=4.35 (does not agree with Table 2); if MA,d=2.8 (as in Table 2), then rS=2.14
(upper limit of the compression ratio in Table 3). If TS-3 has a MA twice that of TS-2’s as
Table 2 shows, assuming that in the upstream TS-2 and TS-3 has the same conditions
(which is not true for MA in Table 1), then nd,TS-2/nd,TS-3≈ Bd,TS-2/Bd,TS-3≈2 (1/1.5) =1.58.
Comparing with the observations (Table 2), the density ratio (0.0033/0.0022=1.5) is in
agreement but the magnetic field ratio (0.132/0.122=1.1) seems insufficient. Another
observational inconsistency is that the upstream MA of TS-3 is about twice the upstream

19
MA of TS-2, while these two upstream regions have similar magnetic fields, densities,
and upstream velocities.
The
B
B
nd ,
nu ,
(downstream normal component of magnetic field over upstream normal
component of magnetic field) is 2.10 for TS-2 and 1.94 for TS-3, as can be calculated
from Table 1-1 and Table 1-2. This shows that the shocks should be very similar, despite
the fluctuations and different versions of calculations, inferences and interpretations from
different publications.
Voyager 1 observations at the termination shock are listed in Table 1-5.
Measurements from Voyager 1 are much more limited than those from Voyager 2, due to
the instrument limitations. To start with, we are concerned with the “average” properties
of the shock Voyager 2 observed: compression ratio ~2, plasma density np ≅ 0.0013 cm -3 ,
magnetic field B ≅ 0.063 nT (upstream), upstream flow speed uu=300 km/s and upstream
temperature 5000 K, which corresponds to Alfvén speed vA= 38 km/s (calculated from
v
A
2
Bu
= , ignoring pickup ion density contribution to the upstream density nu,),
μ nm
0
u i
Alfvén Mach number of MA ≅ 8, βsw ≅ 0.05 and solar wind proton temperature Tp ≅ 0.43
eV (upstream).
Table 1-5 Voyager 1 (V1) Observations of the Termination Shock.
encounters r (AU) rS uu (km/s) ud (km/s) θBn wS (km) τ Td (k)
V1 94 0.4
2.6 +
−0.2
200 100 - -
-
-

20
There are three important puzzles posed by the Voyager 2 measurements. First,
Richardson [2008] found that solar wind ion heating accounts for only ~15% of the flow
dissipation. Richardson [2008] postulated that pickup ions account for most of the
dissipation. Second, Richardson et al. [2008] and Li et al. [2008] concluded that
downstream of the termination shock, the heliosheath flow speed remains super-
magnetosonic. Third, the peak of anomalous cosmic rays (ACR, ~100keV – MeV)
predicted by diffusive shock acceleration [e.g., Pesses et al., 1981; Jokipii 1986] is not
observed [Stone et al., 2005] and the ACR enhancement is anisotropic in two opposite
directions for Voyager 1 and Voyager 2 observations, respectively.
We emphasize that the density jump and magnetic field jump are ~2, indicating that
the shock crossed by Voyager 2 is relatively weak by comparison with the strong shock
limit of 4. Although suprathermal ions are too tenuous to be measured by the plasma
instrument, Richardson [2008] infers that pickup ions make up roughly 20% of the sheath
plasma and have energies of about 6 keV in the heliosheath.
1.4 Kinetic Physics of the Termination Shock
Because this dissertation focuses on kinetics, we group shocks, based on the nature of the
collision process, into gas kinetic shocks and collisionless plasma shocks.
In general, a shock forms when the flow encounters an obstacle with a speed that is
faster than the fastest wave in the flow medium. For a gas, this speed is the sound speed;

21
for a plasma, this speed may be the magnetosonic speed (fast mode wave speed), the
Alfvén speed, or the slow mode wave speed.
Pure gas kinetic shocks, involving no electromagnetic forces, are dissipated by
collisions between gaseous particles. Because those particles are neutral molecules or
atoms, the Rankine–Hugoniot relations (the conservation rules of mass, momentum and
energy) can be formulated within hydrodynamics [Gombosi, 1994].
Collisionless plasma shocks such as the Earth’s bow shock, however, are not
dissipated by binary collisions. Plasmas of interplanetary space are considered
collisionless, with a mean free path around 1 AU or more (much larger than their average
Coulomb interaction range). In fact, a half century ago, it was hotly debated whether
collisionless shocks even existed or not. The debate was not resolved until the
observation of the Earth’s bow shock [Sonett and Abrams, 1963].
For shocks in collisionless plasmas, energy dissipation is provided by mechanisms
other than collisions. Here, we only address the quasi-perpendicular shocks, which are
relatively thin shocks with a shock normal nearly perpendicular to the magnetic field.
Consider such a quasi-perpendicular shock with a single, relatively cold, upstream ion
component. At the shock, the dynamic energy of the plasma is dissipated mainly by
cross-field current driven instabilities [Winske, 1984] if the shock is subcritical (i.e., the
Alfvén Mach number is below a critical number). If this shock is a supercritical shock, it
is instead dissipated mainly by ion reflection [e.g., Paschmann et al., 1982], as is the case
in the Earth’s bow shock. The first study of such supercritical shocks was made by

22
Marshall [1955] in which he found that a flow of fast Mach number 2.76 into a cold
MHD fluid required more dissipation than the maximum possible from resistivity.
Further studies have since concluded that the critical Mach number M c [Woods et al.,
1969] that divides subcritical shocks and supercritical shocks is a strong function of the
upstream plasma beta and shock normal angle [Kennel et al., 1985]. The critical Mach
number increases with decreasing plasma beta or increasing shock normal θBN (defined as
the angle between the magnetic field and the shock normal) [Kennel et al., 1985]. This is
because a high beta plasma has a large upstream thermal velocity to start with and thus
can account for more of the net amount of dissipation by transmitted heating (like pickup
ions in Chapter 2 and 3). For typical solar wind parameters, the critical Mach number is
less than 2 [Kennel et al., 1985].
The early magnetohydrodynamic (MHD) treatments of quasi-perpendicular shocks
involved artificial collision frequencies in the fluid equations that automatically assume a
picture of local and diffusive dissipation. These processes saturate when the shock is so
fast (supercritical) that adiabatic heating and anomalous resistivity are not sufficient to
account for energy dissipation. Kinetic effects are necessary to go beyond fluid approach.
Ion reflection, a phenomenon confirmed by hybrid simulations [e.g., Quest, 1985;
Gosling and Robson, 1985; Goodrich, 1985], laboratory studies [e.g., Phillips and
Robson, 1972] and spacecraft observations of Earth's bow shock [e.g., Paschmann et al.,
1982, Sckopke et al., 1983], has been introduced as a means to provide the extra
dissipation necessary for supercritical shocks.

23
The reflection process causes some ions to move back upstream of the shock. Those
reflected ions are heated by the conversion of some of their ram energy into the energy of
ion gyration. They are then convected downstream. Because the ion reflection process is
nearly specular [Gosling and Robson, 1985], the gyrovelocities of reflected ions
approximate the upstream bulk speed [Burgess, 1995; Gosling and Robson, 1985], which
in this case is the upstream solar wind speed. An illustration of how the ion reflection
process leads to a standard supercritical quasi-perpendicular shock structure is shown in
Figure 1-6 [Wu et al., 1984]. Although Figure 1-6 is a result from a one-dimensional
hybrid simulation, the structure presented in the figure is confirmed by early ISEE
observations [Sckopke et al., 1983]. As Wu et al. [1984] and Kennel et al. [1985]
originally discussed, the reflected ions, usually ~20%, upon reflection, enhance the
upstream magnetic field as the ions compress the flow, thus creating an extended foot
region as marked in Figure 1-6. As those ions gyrate around the upstream magnetic field,
their Lorentz force adds to the longitudinal electric field gradually and lead to the
formation of the overshoot, also marked in Figure 1-6. The overshoot effectively
insulates the downstream region from the upstream region [Kennel et al., 1985]. In
Chapter 2, we will re-examine specular reflection in relation to the shock structure. Our
emphasis will be to provide a conceptual picture that captures the essential physical
processes, which are critical to the understanding of ion kinetics. On the other hand, a
complex analytic theory of ion reflection has already been developed by Leroy [1983].
Leroy [1983] also offers a quantitative explanation of the magnetic field and the potential
overshoot.

24
Figure 1-6 The magnetic field profile of a perpendicular supercritical shock from a hybrid
simulation. The foot, ramp, and overshoot are indicated. From Wu et al. [1984].
In terms of the macroscopic conservation laws, since collisionless shocks largely
involve magnetic forces and energies, classical hydrodynamics is insufficient to describe
such shocks. Magnetodydrodynamics must be applied to derive the modified Rankine–
Hugoniot relations [Burgess, 1995]. In the frame of a steady shock, both upstream and
downstream plasma are assumed to satisfy the equations of MHD. Ideally, the MHD
equations should be integrated from negative infinity to the shock for the upstream and
from the shock to positive infinity for the downstream. Often, for a wide enough
upstream and downstream region, the reasonable steady state approximations are
[Burgess, 1995]:

25
[ ρ u]
= 0,
(1.2)
[ uB ] = 0,
(1.3)
B
+ + = (1.4)
2
[ ρu
P
2
]
2μ0
0,
1
γ
P B
2
2
[ ρu(
u + ) + u ] = 0
2 γ −1ρ
μ0
(1.5)
for the conservation of mass, the continuity of tangential electric field, the conservation
of momentum, and the conservation of energy, respectively. Here, “[X]” represents the
upstream and downstream difference of the physical parameter X across the shock
[X]=Xu-Xd. In order to close the system of equations (also known as the closure problem),
we take the equation that describes the polytropic process P
non-adiabatic situations, γ should be interpreted as a polytropic index.
γ
∝ ρ in Equation (1.5). For
For a perpendicular shock in which the flow is perpendicular to the magnetic field,
r
S
ρd
uu Bd
= = = .
(1.6)
ρ u B
u d u
Combining (1.2)–(1.5), we find [Burgess, 1995]:
2−γγ2 − + + + − − + = (1.7)
2
( rS 1)[ rS r ( 1) ( 1)] 0,
2 S
γ γ
2 2
M A MA Mcs

where
1/2
A u( μρ 0 u) / u
26
1/2
M = u B is the Alfvén Mach number and M = u ( ρ / γP
) is the
cs u u u
sonic Mach number. It can demonstrated with algebra that the largest compression ratio
one can achieve with a γ = 5 / 3 shock is rS=4 [Burgess, 1995].
The Rankine–Hugoniot relation derived from MHD theory applies to shocks with a
single ion component upstream. However, the termination shock has two ion components.
As discussed earlier, the presence of hot (1~4 keV) pickup ions upstream changes the
character of the shock by greatly increasing the effective plasma beta. The large effective
plasma beta implies that the termination shock can not be treated as a classical strong
shock [Wu et al, 2009], because the supercritical Mach number for such a shock is larger
than a low beta shock [Kennel et al., 1985]. New models to describe as well as quantify
the physical processes are thus necessary.
1.5 Research Background and Goals of this Dissertation
This section presents the goals of this work, in the context of previous studies of the
termination shock. As discussed, the termination shock resembles some Alfvénic shock
properties but is complicated by the presence of pickup ions.
Various analytical models and computational simulations have reached different
conclusions on the partition of dissipated energy between the solar wind ions and the
pickup ions, as well as the mechanism for pickup ion energization. Zank et al. [1996] and
Lee et al. [1996] independently proposed that pickup ions gain a large amount of energy
through repeated reflections from a very thin–-electron Debye length scale–-cross shock

27
potential layer. With a one-dimensional hybrid kinetic simulation that includes electron
inertia terms (electron resistive heating), Lipatov and Zank [1999] demonstrated that
pickup ions can gain more energy than the solar wind ions through multiple reflections.
As a result, a few pickup ions gain substantial energy. If these ions have a sufficiently
long mean free path they can, in principle, be further accelerated via diffusive shock
acceleration process. However, earlier Liewer et al. [1993] reported that even with a 20%
pickup ion relative density, their one-dimensional hybrid simulations showed that solar
wind ions provide most of the termination shock dissipation. They also found that
“pickup ions lead to the formation of an extended foot in front of the shock ramp of
length approximately the gyroradius of the energetic pickup ion” [Liewer et al., 1993].
The fundamental differences between the Lipatov and Zank [1999] simulations and the
Liewer et al. [1993] simulations are: 1) the length scale of the shock ramp where pickup
ions are most likely to be accelerated, and 2) whether the assumption of electron resistive
heating is physical or not.
We argue that: 1) the ramp of the termination shock scales as the ion inertial length,
not the electron inertial length, and 2) the electrons are adiabatically heated. Based on
early ISEE observations of strong shocks such as the Earth's bow shock (compression
ratio ~4), Scudder et al. [1986a] showed that the thickness of such a shock ramp is some
fraction of the upstream proton inertial length, that is, roughly the ion gyroradius. More
recently, Bale et al. [2005] showed similar ramp thicknesses based on Cluster
observations. Our simulations with zero percent pickup ions are consistent with these
observations. Intuitively, we expect that with the addition of pickup ions at the

28
termination shock, the shock will weaken (compression ratio ~2 as Voyager observed)
and the ramp should widen. More directly, Burlaga et al. [2008] reported the direct
observation from Voyager 2 that the ramp of the termination shock is about one ion
inertial length. As to the electrons, Goodrich and Scudder [1984] demonstrate
convincingly that resistive heating of electrons via various cross-field instabilities was
nearly canceled due to the energy lost by electrons drifting along the shock in the ramp,
so that the actual amount of energy gained by electrons was nearly adiabatic. Assuming
resistive heating of electrons in a simulation causes the electrons to be overheated, this
will result in a very large electron pressure gradient and hence a huge shock potential
which further compresses the shock ramp to the electron scale length. This is not
consistent with observations. As Lembege et al. [2004] concludes, the electron
temperature jump on the order of ~50 – 100 from the Lipatov and Zank [1999] simulation
is unphysical. Therefore, hybrid simulations with zero electron mass (the same model
used by Liewer et al. [1993]) are a valid tool to examine the termination shock. Our study
utilizes the Los Alamos hybrid simulation code to model the relative heating of the solar
wind and pickup ions at the termination shock. Our simulation has fundamentally the
same physics as the Liewer et al. [1993] simulation. However, we apply a more accurate
set of input parameters (e.g., solar wind beta) derived from the recent Voyager
observations.
The presence of a substantial number of pickup ions at large heliospheric distances
suggests that the environment of the termination shock is very different from that of the
terrestrial bow shock. The processes by which the relatively hot pickup ions gain energy

29
at the shock are fundamentally different from the specular reflection of the relatively cold
solar wind ions. Using the Los Alamos hybrid simulations, we start with the simplest case
of a strictly perpendicular termination shock and gradually include more realistic
conditions. The physical processes involved are also examined layer by layer as we
poceed.
In Chapter 2, simulation methods are introduced with an emphasis on hybrid
simulations—the tool we have chosen. This chapter models the termination shock in a
generic sense, rather than trying to reproduce explicit features of any of the Voyager
shock crossings. As such, this chapter concerns two major findings from Voyagers: that
the average fraction of energy gain by the pickup ions at the shock is about 85% and that
the average shock strength is about 2. The physical processes of ion energization are
examined and ions are grouped according to the mechanisms by which they are energized.
In Chapter 3, we develop an analytic description, the multicomponent Rankine–Hugoniot
model of shock dissipation, solar wind heating and pickup ion heating. The analytic
results are compared with simulations from Chapter 2. Both the analytic solutions and the
simulations are consistent with Voyager observations. One puzzle reported by the
Voyager team is that the measured downstream solar wind flow remains super-
magnetosonic [Li et al., 2008]. We will address this issue with our analytic model as well.
For both Chapter 2 and Chapter 3, most upstream parameters are fixed with Voyagers
observed average values. Only the pickup ion relative density is varied to examine this
most critical component’s effect on shock kinetics. Therefore, results from these two
chapters are characterized as functions of the pickup ion relative density, φ , which is

30
defined as the upstream pickup ion number density over the total upstream number
PUI
density, n / n .
u u
In Chapter 4, we work toward simulating a more realistic termination shock step by
step. We describe the results of simulations in which we vary the pickup ion velocity
distribution, the shock normal angle, and the upstream Alfvén Mach number to span a
wider range of Voyager observations. Scattering at the shock is found to be so strong that
different upstream pickup ion distributions result in nearly identical downstream speed
distributions. The upstream Alfvén Mach number is found to be correlated with the
downstream ion energies. In Chapter 5, the accuracy and convergence of our simulations
are examined. In Chapter 6, we compare results from our simulations with Voyager
observations and discuss the implications of our results for the IBEX mission. In Chapter
7, we summarize the scientific findings, discuss the limitations of this work, and point to
possible future directions of research.
Throughout this work, the flow velocity and the velocity of individual ions are
denoted by “u” and “v”, respectively. The subscript “u” represents upstream and the
subscript “d” represents downstream. For example, the quantities u u and u d are the bulk
velocities upstream and downstream, respectively. The quantity r S denotes the shock
strength (or compression ratio), defined by the density jump nd / n u (downstream density
over upstream density) or the jump of magnetic field transverse component. The quantity

31
θ BN is the angle between the shock normal and the local magnetic field. The quantity T
is the temperature. The temperature jump is denoted by τ , where τ = T / T .
d u
To conclude, the goal of this dissertation is to address the following questions:
• How do pickup ions modify the termination shock?
• How much is each ion component heated across the termination shock?
• What is the partition of dissipated energy among the ion components?
• How are ions (solar wind and pickup ions) energized?
• What are the downstream ion velocity distributions?

2 Hybrid Simulations for an Idealized Termination Shock
The termination shock is believed to be quasi-perpendicular at most heliospheric latitudes
because of the Parker spiral structure of the heliospheric magnetic field and the shock's
great distance from the Sun. In this chapter’s simulations, we assume an idealized
termination shock: the shock is strictly perpendicular, the upstream ions are all protons;
the cold solar wind ions are Maxwellian distributed, and the hot pickup ions form a shell
distribution. This idealization allows us to gain physical insights of how hot pickup ions
and cold solar wind ions are energized differently at the termination shock. The pickup
ion relative density is varied to investigate the consequent shock response.
2.1 The Hybrid Simulations
In this section, the basic principles of hybrid simulations will be explained. But first, we
will introduce basic plasma computational methods and review previous simulations of
the termination shock. The advantages and disadvantages of each approach will be
discussed. We will argue that for our goals here, the hybrid simulation is the ideal tool.
2.1.1 Computer Simulation Methods
When a physical system is not subject to analytic description, or is inaccessible to
experimental discovery, computer simulations become necessary. The history of
computer simulations can be traced back to the Manhattan project when scientists used
the Monte Carlo algorithm to understand nuclear reactions. Computer simulations not
only save us from performing expensive and dangerous experiments but also allow us to
resolve the fast temporal and spatial variations that are difficult to track down in a

33
laboratory experiment. For large scale space plasma structures, such as the termination
shock, it is simply not possible to measure the whole physical domain of interest with
satellites. In these circumstances, we resort to computer simulations to bridge the gap
between analytic theories and experimentally unrealizable areas, hoping that this method
puts the physical picture into readable form. The immediate advantage is that computer
simulations are inexpensive and controllable. What makes this method attractive is that
we can make as many measurements as we want. None of the measurements made
perturb the system (unlike real experiments). A successful computer simulation should
obey fundamental physical principles and be consistent with the limited experiments.
Further, it should provide physical insight to the system studied.
A typical computer simulation solves a set of mathematical equations to model the
physical phenomenon of interest. The equations are cast into discrete algebraic form
which is amenable to numerical solution. As the equations evolve in time with time step
DT, the modeled system can be investigated as in a lab experiment. In a sense, a
computer simulation is a numeric experiment—an experiment that is carried out through
mathematical algorithms instead of hardware.
Before we summarize previous numerical studies of the termination shock, we should
briefly introduce plasma simulation methods, pioneered by Oscar Buneman (1913 –1993)
and John M. Dawson (1930 – 2001). Based on the way particles are treated with respect
to the electromagnetic fields, plasma simulation methods can be classified into three
major categories: 1. particle orbit theory, 2. fluid simulations, and 3. kinetic simulations.

34
Category 1 (particle orbit theory) is the test particle method. This method investigates
plasma ion/electron behavior by tracking them as test particles in prescribed electric and
magnetic fields. Hence, this method is not self-consistent, as it does not allow the
plasmas to contribute back to the electric and magnetic fields. Method 2 (fluid
simulations), e.g., Magnetodydrodynamic (MHD) simulation, solves a set of partial
differential equations (transport equations and Maxwell’s equations) assuming that
plasmas can be represented as fluids with approximate transport coefficients. This
method is typically applied to large scale, low frequency phenomena where kinetic
effects are not important. However, when plasma velocity distributions deviate
significantly from a local Maxwellian, the fluid approximations often fail and we have to
apply method 3 (kinetic simulations). Kinetic descriptions are by far the most
sophisticated and complex method that allows the microscopic physical properties of
plasmas to be studied.
There are two distinct techniques within method 3 (kinetic description), the Vlasov
method and the particle-in-cell (PIC) method. The Vlasov method treats phase-space as a
continuum and integrates the distribution function f ( r , v , t) forward in time as coupled
to Maxwell’s equations. It is very challenging to extend this method to multi-dimensional
simulations. The PIC method represents the plasma as a large number of computational
super-particles that move according to single particle equations of motion in self-
consistent electromagnetic fields. In this technique: the particles are advanced in time
using the equations of motion for charged particles in an electromagnetic field; and the
electromagnetic fields are advanced through calculations of current densities from the

35
particles. Although the particles’ motions are fully resolved, the fields are interpolated
between the simulation cells. The introduction of the “cell” concept is to limit the
numerical operations of this many-body system, as discussed later. Nearly all plasma
physics of interest comes down to some scale lengths, within which physical quantities
(density, current, etc.) can be considered/approximated as continuous. Therefore, the
introduction of the simulation cell concept is not just a trick to avoid enormous amount of
numerical operations, but also a physically valid and intelligent choice for describing a
massively complicated system. The cell size should be carefully chosen, as will be
discussed briefly later in this chapter and more extensively in Chapter 5.
In hybrid simulations, fluid and particle-in-cell treatments are given to different
components of a plasma.
2.1.2 Previous Simulations of the Termination Shock
Fluid models such as MHD simulations [e.g., Opher et al., 2006; Pogorelov et al., 2008]
are the simplest way to represent the termination shock. In ideal MHD simulations, there
are three forces: magnetic tension, magnetic pressure, and gas pressure. Kinetic (i.e.,
particle) effects are absent. There is no physical dissipation, although there is unavoidable
numerical dissipation/diffusion due to the Lax Scheme [Hockney and Eastwood, 1988]
(which allows the simulation to discretize the convection equation with stability). To
approximate the consequence of specular reflection, a MHD simulation has to include
anomalous viscosity. Hence, MHD simulations ignore the microphysics of the shock and
are not sufficient to give kinetic information on the plasmas. Without kinetic physics,

36
MHD models assume that the velocity distributions of plasmas species remain near-
Maxwellian, thus do not provide information about the non-thermal ion heating and
thermal ion distributions which we seek. Therefore, although MHD models are useful to
study the global structure as well as the shock’s relation to the solar wind, these models
are too simplified to address the issues of concern to us.
Kinetic Monte Carlo simulations calculate the effects of Coulomb collisions and
charge-exchange processes for neutral interstellar atoms. Such calculations, when
coupled directly to MHD codes, can model the interaction between the solar wind and the
interstellar medium [Müller et al., 2000; Heerikhuisen et al., 2006], and provide some
information about ion velocity distributions in the heliosheath. However, Monte Carlo
codes typically do not include the effects of electric and magnetic fields that are critical
in understanding how ions gain energy at the termination shock.
The most complete representation of plasma physics is provided by PIC simulations
which yield a self-consistent, kinetic description of plasma ions and electrons as well as
the associated electric and magnetic fields. Although PIC simulations have been applied
to the termination shock [Lee et al., 2005; Chapman et al., 2005], such computations
require that the electron dynamics be followed in detail. This typically restricts the
calculations to either relatively short times or to small spatial ranges and dimensions,
preventing a full physical representation of the ion distributions at and downstream of
this shock. Also, with a full PIC code, one typically needs to use artificial parameters,
such as a small ion to electron mass ratio to reduce the difference between the electron

37
and ion spatial and time scales. In addition, both the fundamental ion parameters (ion
2
plasma frequency ω = ne ( mε
) and ion cyclotron frequency Ω = eB / m ) and the
pi i i
fundamental electron parameters (electron plasma frequency
0
i i
ω = ne ( mε
) ,
2
pe e e 0
electron cyclotron frequency Ω e = eB / me
, and Debye length λD) need to be included in
full PIC codes. Another disadvantage of using the PIC model is that the discrete character
of the super-particle electrons leads to additional numerical noise and increases the risk
of the accumulation of numerical errors. One should be very careful in checking the
stability as well as the energy conservation of the system. In most situations, smoothing
over space and time is necessary to reduce noise. The question then, is, whether this extra
effort is necessary. Both Hockney and Eastwood [1988] and Birdsall and Langdon [1985]
stress that one only needs to do well enough to present the correct physics and minimize
the chance of amplifying numerical noise.
The middle road between MHD and PIC simulations is the hybrid simulation. Most
hybrid simulations [e.g., Winske and Leroy, 1984] that have been successfully applied to
shocks treat ion species as particles (as in PIC simulations) and electrons as a massless
fluid (as in MHD simulations), and thus provide a fully self-consistent treatment of the
electric and magnetic fields up to the ion cyclotron frequency (or gyrofrequency). It was
convincingly demonstrated and validated in the past two to three decades, through
comparison with observations [e.g., Winske and Leroy, 1984], that collisionless shocks
are ion-inertial-length plasma phenomena which can be effectively described by hybrid
simulations. While PIC codes provide a more complete representation of the physics,

38
hybrid codes reduce numerical noise and extract the essential physics at frequencies up to
the ion cyclotron frequency. With the hybrid code, longer run-times (in order to create a
well-formed shock) and wider spatial ranges are easily achievable, neglecting the small
amplitude high frequency fluctuations that are not expected to make a significant
contribution to the low frequency ion physics.
Liewer et al. [1993] used a one-dimensional hybrid code with a zero-mass electron
fluid model to provide early, seminal results for high Mach number quasi-perpendicular
termination shocks. Their simulations showed a shock transition length of about an ion
inertial length and a moderate energization for pickup ions. Lipatov and Zank [1999] also
used a one-dimensional hybrid code, but with a finite electron mass and a finer cell size
(~0.006 c/ω pi ), to simulate the quasi-perpendicular termination shock. (Hybrid codes
with finite electron mass, like PIC codes, include all the short wavelength electron scales.)
In contrast to zero-electron-mass computations, Lipatov and Zank [1999] show a very
narrow shock transition, on a scale less than 5% of the ion gyroradius and a pickup ion
energy gain up to 0.5 MeV. It is our goal to re-address those issues, in the context of the
recent Voyager 2 observations.
2.1.3 Methodology of Hybrid Simulations
In a hybrid model, some plasma components are treated kinetically as super-particles
(like in PIC), while some plasma components are treated as fluid. First, we need to
introduce the PIC concept.

39
The concept of super-particle in the PIC method is constructed on the basis that the
plasma is collisionless 5 . This basis is valid for the solar wind plasmas in most situations.
By “super-particle”, one means that the particles are uncorrelated (also called
collisionless, or weakly coupled), as their distribution function satisfies
f ( xi, vi, x j, v j, t) f ( xi, vi, t) ⋅ f ( x j, v j,
t)
(2.1)
ij i j
where i and j are any two particles in the system, fij is the two-particle distribution
function, and fi and fj are one-particle distribution functions. Although the super-particles
do not correspond to particles of real plasmas on a one-to-one basis, they retain much of
the collective behavior of the real plasma. The immediate advantage is that we can avoid
solving N 2 (N is the total number of particles in the system) equations of particle-particle
qq i j
Coulomb force F
ij = r at every time step. The forces will instead be described by the
2 ij
r
ij
continuous electric field E interpolated between mesh points (center of simulation cells,
as illustrated in Figure 2-1) over simulation cells: i = i
computation cycle from N 2 to N logN .
F q E.
This way one can reduce the
Whether it is in a PIC simulation or a hybrid simulation, the fluid source terms
(number density, charge density, current, electric field, and magnetic field) are specified
at mesh points of computational cells. In between these mesh points at a particle’s
5 Collisionless plasma satisfy nλD 3 >>1. This means that the average Coulomb interaction range a=e 2 /(4π
ε 0kBT) is much smaller than the average particle spacing 1/n 1/3 (kinetic energy>>Coulomb type of
potential energy).

40
location, the source terms are calculated through interpolation/weighting. The most
straightforward weighting method is linear weighting, which we use for our simulations
of the termination shock. In the linear weighting, a source term is evenly distributed
between two cells in a linear manner (one dimension: cell length; two dimensions: cell
area; three dimensions: cell volume). The noise level of linear interpolation may be
intolerable in some plasma problems; in such cases higher order weighting such as
quadratic weighting (second order interpolation) must be applied.
x= 0 H Simulation domain in one dimension
XMAX
1 2 3 Mesh points (cell numbers) NX NX1 NX2
Figure 2-1 A one-dimensional representation of the simulation cells, the mesh points
(center of each cell), and the simulation domain. Cell 1 and Cell NX2 are ghost cells that
are necessary for the specification of boundary conditions. The simulation domain ranges
from Cell 2 to Cell NX1.
In Figure 2-1, there are two cells at the boundaries of a simulation dimension. These
cells (Cell 1 and Cell NX2) are called ghost cells. They do not provide output for the
simulation. The simulation domain ranges from Cell 2 to Cell NX2. The ghost cells allow
easy manipulations of various boundary conditions to ensure that there is no sink/lost in
the system. The particles as well as the source terms that enter Cell NX2 are either put
back into cell NX1 (reflecting wall boundary) or cell 2 (periodic boundary).

41
After identifying the species (some fluid, some PIC) and the cells, one can apply
different numerical schemes to solve the Maxwell’s equations, the fluid equations, and
the equations of particle motions (Newton’s law) under various approximations (full
Maxwell, Darwin approximation, electrostatic, etc.), initial conditions, and boundary
conditions that are tailored to the specific problem one intends to solve.
For the simulation to run smoothly with minimum errors, natural units are often
applied such that the important physical quantities are presented with values at least
initially of order unity. Stability analysis should also be performed to ensure that there are
no growing numerical instabilities. This can be done by applying the von Neunmann
analysis to the numerical algorithm. One should check conservation laws with test runs
on simple examples before going into complex physical systems. At times, smoothing is
necessary in order to reduce noise.
2.1.4 The Los Alamos Hybrid Simulation Model
The Los Alamos hybrid code for collisionless shock was developed by Dan Winske
and collaborators. In the Los Alamos Hybrid code, ions are treated kinetically as super-
particles (PIC method) and electrons are regarded as a massless fluid [Winske and Omidi,
1993, Winske et al., 2003]. The code is ideal for computing ion responses to plasma
phenomena (such as the termination shock) at ion inertial length and time scales. The
simulated plasma is quasi-neutral. The electron densities are set equal to ion densities
everywhere and the electrons respond to the changes of the system instantaneously
(inertia-less). The electromagnetic fields are treated in the magnetostatic condition (the

42
1 ∂ E
displacement current term in Ampere’s law is neglected = ∇× B− μ
2
0 J = 0 ) in the
c ∂t
low-frequency approximation. We restrict the discussion to one-dimensional simulations,
which we use. This hybrid shock model has been widely used in space physics for more
than two decades and evolved through several versions of numerical schemes [Winske et
al., 2001]. The present form of the model has been tested to be robust and is well
validated against data from quasi-perpendicular shocks, quasi-parallel shocks, and
cometary shocks [Winske et al., 2001].
To apply the hybrid code to the termination shock, the simulation is run in the
downstream rest frame in which the shock propagates to the left, as illustrated in Figure
2-2 (upper panel). (Throughtout the text, except where specified, figures are shown in the
downstream rest frame.) As shown in Figure 2-2, the simulated dimension is the x
direction. A perpendicular magnetic field is in the z direction. Although the simulation is
one-dimensional, the ion velocities, fluid velocities and electromagnetic fields are fully
three-dimensional. The one-dimensional concept implies that ∂ / ∂ y and ∂/ ∂ z are zero
for all physical parameters, and that particle motions in y and z directions are not
followed.

43
Figure 2-2 The upper panel is a representative setup of the one-dimensional simulation
for the termination shock. The bottom panel shows the time evolution of the magnetic
field B z profile from a 0% pickup ion simulation. Since the simulation is run in the
downstream rest frame, the shock propagates to the left. In the bottom panel, x is
normalized by c / ω pi where ω pi is the ion plasma frequency and time t is normalized by
Ω i where Ω i the ion cyclotron frequency based on the upstream magnetic field. From
Wu et al. [2009].

and
44
The hybrid code solves the following equations
dvi
mi = e( E+ vi× B) −eηJ,
dt
dxi
dt =
(2.2)
vi
(2.3)
for each ion super-particle. Equation (2.2) and (2.3) are solved using the leap-frog
scheme: velocities are advanced at time step N+1/2; and positions are advanced at time
step N. This method is second order accurate and is widely used to replace the first order
Euler method.
The transverse components (divergence free) of magnetic fields (By and Bz) are
explicitly solved from Faraday’s law
∂ B
= −∇× E
(2.4)
∂t
using the fourth order Runge–Kutta scheme and subcycling [Winske and Quest, 1988].
The fourth order Runge–Kutta solver ensures simulation accuracy and the subcycling
ensures the convergence of the calculation, as will be discussed in Chapter 5. We apply
10 subcycles for the magnetic field solver, each performing a fourth order Runge-Kutta
integration of Equation (2.4) within a time step DT/10. In contrast, solving the normal
component of magnetic field (Bx) is relatively straightforward. The zero divergence of

45
the magnetic field ∇⋅ B = 0 , together with this simulation’s one-dimensional setting in
the x direction, imply that Bx=constant. In Equation (2.4), the electric fields are
calculated from electron momentum equation
∂
nmu e e e = 0 =− ene( E+ ue× B) −∇⋅ Pe + eneη⋅J, (2.5)
∂t
where ue is the electron bulk velocity, ui is the ion bulk velocity, η is the
phenomenological resistivity (set to be a constant) to approximate the electron-ion
interaction. Equation (2.5) can be rewritten as the Ohm’s law
∇⋅P
e E = − ue× B− + η ⋅J.
(2.6)
en
This equation for the electric field does not require a time advance, unlike Equation (2.4)
for the magnetic field. The electron pressure term Pe is advanced adiabatically. Ampere’s
law
is used to eliminate ue in Equation (2.5).
∇× B = 4πμ J = 4 π enμ( ui −ue)
0 i 0
e
(2.7)
The last of Maxwell’s equations—the Poisson’s equation ∇⋅ E = 4 πe( n −n
)/ ε0
, is
replaced by the quasi-neutral approximation (plasma approximation)
i e
ne = ni.
(2.8)

46
Now the simulation is self-consistent and self-contained. With appropriate choices of
grid size, time step, particles per cell etc. (as will be discussed in Chapter 5) to ensure
acceptable (optimal) accuracy and convergence, we are ready to study the physics of the
system.
In our simulations, the system is initiated with particles and fields that satisfy the
Rankine–Hugoniot jump conditions and subsequently evolve into a self-sustaining shock
with width of the order of an ion inertial length. As shown in Figure 2-2 (bottom panel),
the shock jump is initially a step function at time zero, like in MHD simulations. As time
advances, the shock gradually develops into a finite width structure that has the same
features as the Leroy et al. [1982] simulation (Figure 1-6). Upstream, particles are
initiated with appropriate distribution functions, depending on the particle species.
Particles are continuously injected from the left wall with velocity ( uu − ud
) as the
simulation advances. Downstream, we have initially included a narrow region of heated
plasma with plasma conditions determined by the Rankine–Hugoniot relations. As we
continue to inject particles from the left hand side, the right wall boundary is set to reflect
particles to ensure the balance of flux. This reflecting wall can be conveniently set up
because the simulations are run in the downstream rest frame.

2.2 Simulation Results Overview
47
This section examines the hybrid simulation results in a general sense. We investigate the
effect of changes of basic macroscopic parameters of the shock as the pickup ion relative
density increases.
The length of the simulated system is 400 ion inertial lengths (400 ω / ). The ion
plasma frequency here is calculated from the total upstream ion density. The system is
divided into 800 cells and the initial number of particles is 80,000 (about 100 particles
−5
per cell) for each ion component. The resistivity is set to a small value: 2× 10 (4 π / ω ) ,
corresponding to a resistive length on the order 1% of the cell width. The time step is
DT=0.02Ωi -1 . Here Ω i is the ion cyclotron frequency based on the upstream magnetic
field. We assume that all ions are protons, and that the shock is strictly perpendicular
with a shock normal θ Bn = 90 ° . The upstream parameters are chosen to be consistent with
the Voyager 2 observations. Here the plasma beta β sw =0.05, and uu = 8vA
(or
equivalently M A = 8)
in the shock frame. The solar wind's upstream velocity distribution
is assumed to be Maxwellian. Furthermore, the upstream pickup ion velocity distribution
is taken to be a spherical shell with the shell radius equal to the upstream bulk speed u u ,
as done by Liewer et al. [1993] and Lipatov and Zank [1999]. We vary the pickup ion
relative density from 0% to 40% to study different scenarios. As stated in Chapter 1, we
only consider the conditions near the shock. We do not simulate the foreshock region
which extends about 10–15 AU upstream of the shock, nor do we extend our simulations
pi c
pi

48
deep into the downstream heliosheath. Both these regions are more appropriately
modeled by MHD simulations, considering the scale lengths.
The 0% pickup ion simulation provides a baseline computation for comparison with
the more realistic simulations (with pickup ions) to follow. The bottom panel of Figure
2-2 shows the spatial profiles of the magnetic field B z from the 0% pickup ion simulation
at constant time intervals, stacked with the t=0 profile at the bottom, extending to the
−1
t=48 Ω profile on the top. This plot demonstrates a well-formed shock propagating to
i
the left at a nearly constant speed, u = − 1.98v
. The speed of the solar wind relative to
the shock is u − u + u 8v
.
u d sh A
sh A
Figure 2-3 shows the spatial profiles of the magnetic field and the ion density. The
density and magnetic field profiles are similar, as expected for a perpendicular shock.
The shock jump is clearly evident at x=259 c / ω pi . The upstream values are nearly
constant, and the downstream average values of the density and magnetic field are ~ 4.
The density and magnetic field jumps, as well as the amount of downstream heating, is
consistent (to within ~ 5%) with the values based on the Rankine–Hugoniot relations
[Burgess, 1995].
From the simulations with varying pickup ion relative density, we calculate the
downstream flow speed as a function of pickup ion relative density, as shown in Figure
2-4 (top panel). The shock strength rS (bottom panel) is calculated from the density jump
rS=nd/nu. In the figure, the downstream flow speed increases with increasing pickup ion

49
relative density whereas the shock strength decreases. As the presence of pickup ions
increase the upstream effective beta (sound speed), the magnetosonic speed is increased
and the steepening (compression) of the shock is reduced. Particularly, the shock strength
rS is reduced to 1.85 – 2.6 (Voyager measured range) when the pickup ion relative density
falls within 20 – 30%.
Figure 2-3 Density profile (upper panel) and magnetic field profile (bottom panel) from a
0% pickup ion simulation. The ion density n i is normalized by the upstream ion density
n u ; the magnetic field B z is normalized by the upstream magnetic field B u ; and x is
normalized by c / ω pi . From Wu et al. [2009]

Figure 2-4 The shock strength (bottom panel) and the downstream flow speed (upper
panel) as a function of pickup ion relative density.
50

51
2.3 Kinetic Structure of the Termination Shock
From the previous section, we know that the scaling is monotonic. For simplicity, we
compare the 0% pickup ion simulation with the 20% (Voyager inference) pickup ion
simulation in this section to study the change of the kinetic structure of the shock with
and without pickup ions. These kinetic structures of the shocks, as revealed by such fluid
terms as E , B , n, and u , are the “sums” of the kinetic behaviors of the ions involved in
the termination shock, and thus offer a first order macroscopic examination of the
relevant microscopic processes.
In the one-dimensional simulation, Bx=0 for perpendicular shock, as discussed before.
We show in Figure 2-5 that By is trivial (nearly zero). As Ez=-uxBy, Ez is also a very
small quantity (same plot). To compare the kinetic structure of the strictly perpendicular
shock profiles via electric and magnetic fields, we only need to compare Bz, Ex, and Ey.
Figure 2-5 By and Ez from the 0% (black) and the 20% (blue) pickup ion simulations. By
is normalized to the upstream magnetic field Bu and Ez is normalized by vABu/c. The
simulation dimension x is normalized by ion inertial length c/ω pi .

52
The profile of Bz, Ex, and Ey from the 20% pickup ion simulation are shifted such that
the shock lines up with the 0% pickup ions shock, as shown in Figure 2-6. In the top
panel, one sees the well established characteristics (foot, ramp, overshoot, undershoot) of
a supercritical shock from both simulations. There is a sharp peak (overshoot) of the
magnetic field at the shock front (x ~262) that is much larger than the average
downstream field. The magnitude of the overshoot from the 20% pickup ion simulation is
significantly smaller than that of the 0% pickup ion simulation. There is an extended foot
just upstream of the shock front. The length and magnitude of the foot from the 20%
pickup ion simulation is larger than those of the 0% pickup ion simulation. The
steepening at the shock front (where the ramp is) is weaker for the 20% pickup ion
simulation than for the 0% pickup ion simulation. The upstream magnetic field Bz
fluctuates as we include pickup ions, because the pickup ions increase the upstream
velocity anisotropy and induce more upstream waves. In the downstream, both
simulations show a number of large, quasi-periodic oscillations. The average frequency
of these oscillations from the 20% pickup ion simulation is much smaller than that of the
0% pickup ion simulation. Hence, the oscillating features downstream of 20% pickup ion
simulation have longer wavelengths as compared with the 0% pickup ion simulation.
In calculating Ex, we subtract the non-physical contributions from the hot plasma near
the reflecting boundary of the wall. One sees that the average Ex upstream is
approximately zero, as is the average Ex downstream. There is a decrease of Ex in the foot
due to the increase in the electrostatic potential, Ex = - ∇ϕ. This Ex slows the upstream
flow, which creates the compressed foot region as seen in the Bz profile. There is a sharp

53
dip and peak in Ex at the shock front that is due to and
∇pe
E
x = − uy× Bz − ⋅ x.
Similarly,
en
the magnitude of Ex is reduced with the inclusion of pickup ions, as well as the oscillating
features. And the extended foot of shock is more prounced in the 20% pickup ion
simulation.
Figure 2-6 Profiles of Bz (top panel), Ex (middle panel) and Ey (bottom panel) from the
0% (black) and 20% (blue) pickup ions simulations. The magnetic field is normalized by
the upstream magnetic field Bu; the electric fields are normalized by vABu/c, and the
spatial length x is normalized to the ion inertial length c/ωpi.

54
Profiles in Ey show the expected upstream motional electric field
( u )
E = − u B (recall that we are working in the downstream frame). This motional
y u d z
electric field is reduced significantly at the foot as the flow is decelerated by the reflected
ions. Ey has a large negative spike at the shock front that corresponds to both the
motional electric field and the induced electric field
2 ∂BZ
E E
y = Ey−mot + y−ind =− ux× Bz −c ⋅ ⋅DT⋅y that lead to the large overshoot in Bz. In
∂x
the downstream region, the average value of Ey is zero, because the flow speed is zero
(downstream rest frame) and therefore the motional electric field Ey-mot becomes zero.
The downstream Ey oscillates at the same frequency but the opposite direction as Bz due
to the induced electric field (Ey-ind).
Figure 2-7 shows additional results of the kinetic structure of the two shocks (0% and
20% pickup ion shocks) via the fluid densities and velocities of pickup ions, solar wind
ions and all ions. Again, the z component flow velocities are not shown because these
velocities have very small values for the one-dimensional simulation. Notice that Panel a
(a’) and Panel b (b’) both depict the same three densities (pickup ion density: red; solar
wind ion density: green; all ions density: black) however differ in the normalizations of
these densities. Panel a (a’) densities are all normalized to the upstream density nu such
that the magnitudes of the plotted densities can be compared. Panel b (b’) densities are
normalized to each component’s upstream value such that the compression of each
component is compared in the same scale. Because the unprimed panels are from the 0%

a)
b)
c)
d)
a’)
b’)
c’)
d’)
u
u
u
u
55
Figure 2-7 Pickup ion (red), solar wind ion (green) and all physical ions (black) density
and velocity profiles. Panel a, b, c, and d are from the 0% pickup ion simulation (where
pickup ions are test particles). Panel a’, b’, c’ and d’ are from the 20% pickup ion
simulation. In Panel a and a’, all densities are normalized to the upstream density (nu). In
Panel b and b’, each density is normalized to its upstream value (red: nPUI / n u, PUI , green:
nsw / n u sw,
black: n/ n u ). The spatial length x is normalized to the ion inertial length c/ωpi.
,
Overshoot
Dip
Overshoot
Dip

56
pickup ion simulation where pickup ions are test particles; pickup ions make no
contributions to the source terms, as can be seen in Panel a that the red line is essentially
zero, and that all the green curve (solar wind ions) in the unprimed panels overlap the
black curves (all ions) precisely. This is not the case in the primed panels (the 20%
pickup ion simulation), where the green curves and the red curves show visible deviations
and sometimes substantial local differences.
In Panel a Figure 2-7, the density of the solar wind ions shows a large enhancement
(overshoot) at the shock front. There is also an extended foot in the upstream shock
transition region and there are downstream oscillations. These features persist in Panel a’
(where there are 20% pickup ions) and are consistent with the Wu et al. [1984] specular
reflection magnetic field structure for a supercritical shock (Figure 1-6). This indicates
that some solar wind ions are specularly reflected to account for some of the dissipation
of the shock, whether there are pickup ions or not. The difference is that with pickup ions,
the length scale of the supercritical shock structure is broadened and the magnitude of the
overshoot is reduced, suggesting that the solar wind reflection efficiency is reduced. Also,
in Panel a’, since solar wind ions make up 80% of the density, the total density profile of
the shock is determined by the solar wind density and the pickup ion density is only a
small fraction of the total density. We compare pickup ion density compression at the
same scale with the solar wind density compression in Panel b and b’ of Figure 2-7. In
both panels, the pickup ion density rises earlier before the shock. This suggests that the
extended foot (Panel a’ and b’, black curves) of the 20% pickup ion shock is caused by
the pickup ions. Such an extended foot is not seen in the 0% pickup ion shock (Panel a

57
and b, black curves), because as test particles pickup ions make no contribution to the
source terms in this simulation. Behind the foot, the pickup ion density is nearly flat
through the shock front and into the downstream, suggesting that pickup ions are not
energized in the same manner as the solar wind ions for either shock [Winske et al., 2009].
The small amplitude fluctuations among the nearly flat density suggest that pickup ions
are mostly gyrating and drifting with almost unperturbed orbits such as those from
specular reflection.
The flow velocity ux (Panels c and c’) show that both the pickup ion and the solar
wind ion have the same average upstream velocity (uu-ud) and the same average
downstream velocity ~zero (reminder: the simulations are run in the downstream rest
frame). The pickup ion upstream ux fluctuates more due to the pickup ions’ larger
gyroradii. At the shock, the pickup ion ux has a rather gradual transition from (uu-ud) to
zero. In contrast, the solar wind ion ux has a sharp dip (negative ux) that corresponds to
the overshoot and the solar wind ion reflection. As the reflected solar wind ions are
carried further downstream, after 1-2 gyroradii, the solar wind ion ux recovers from the
sharp dip and displays low amplitude periodic oscillations due to the downstream solar
wind ion gyrations. Again, both the dip and the oscillation magnitudes are reduced with
the inclusion of the pickup ions.
For the flow velocity uy (Panel d and d’), we subtract the non-physical drifting
resulting from the potential created by the hot plasmas near the reflecting boundary of the
wall. Therefore, in both the upstream and the downstream region, uy averages to zero.

58
The fluctuations outside the shock region are caused by ion gyrations. In the shock
region, there is a wide peak of positive uy that spans the entire foot region of the shock,
suggesting an energization mechanism preferentially accelerating pickup ions in the y
direction in the shock front. Of course the 0% pickup ion shock does not “see” this
enhancement and the broadened foot, because pickup ions are test particles in this case.
But pickup ions “see” the shock and display a similar response to that in the 20% pickup
ion shock.
The important result of Figure 2-7 is: whether we examine the 0% pickup ion
simulation or the 20% pickup ion simulation, pickup ions display similar behavior that
suggest a similar energization mechanism for pickup ions (will be discussed in later
sections) relatively independent of the pickup ion density, as evident in the similar pickup
ion features in panels b and b’, c and c’, and d and d’. Both Figure 2-6 and Figure 2-7
indicate that the kinetic supercritical perpendicular shock structure is weaker and wider
with the inclusion of pickup ions.
2.4 Qualitative Examination of Heating via Phase-Space Plots
In this section, we compare phase-space plots obtained from simulations of different
pickup ion relative densities. We draw qualitative conclusions on the solar wind ion
heating and reflection [Wu et al., 2009], as well as the pickup ion heating and gyration
[Winske et al., 2009].

59
Figure 2-8 shows a series of phase-space plots from the 0% pickup ion simulation.
The shock is marked by a dashed line in Figure 2-8a and Figure 2-8b. In the figures, each
dot corresponds to a simulation ion. We identify reflected solar wind ions as the particles
that are streaming upstream ( v < − 0.4u
and v > 0.4u
) within a finite distance
x u
y u
( 2 c / ω pi ) of the shock center. The limitation is that this method can not separate reflected
ions from transmitted ions unambiguously. Because the transmitted ions can also gyrate
back, a backward motion inside the shock can be due to gyration or reflection.
Nevertheless, this method provides a qualitative picture of ion reflection. The transmitted
solar wind ions are shown in Figure 2-8a, and the reflected solar wind ions are shown in
Figure 2-8b. The v x – x and v y –x plots indicate that the gyrophases of reflected ions are
∼ 180 ° off those of the transmitted solar wind ions. Further, the gyrovelocities of the
reflected ions approximate the upstream flow velocity (in the shock frame). The v z – z
plots show that there is no heating in the z- direction for both the transmitted and
reflected solar wind ions, a consequence of the one-dimensional perpendicular shock.
The vx–vy plots confirm that the upstream solar wind ions (Figure 2-8c) form a
Maxwellian distribution as assumed. The velocity (vx–vy) distributions of the downstream
ions (Figure 2-8d) are divided into two populations: a heated transmitted population
(Figure 2-8e) and a suprathermal tail formed by the reflected ions (Figure 2-8f). Both
downstream populations are gyrotropic.

60
Figure 2-8 The phase-space plots of solar wind ions in the 0% pickup ion simulation. The
panels on the top are the v x – x , y v – x and v z – x plots for a) transmitted solar wind ions
and b) reflected solar wind ions. The panels on the bottom are v x velocity distributions
and x v – v y phase-space plots for c) upstream solar wind ions, d) downstream solar wind
ions, e) downstream transmitted solar wind ions, and f) downstream reflected solar wind
ions. The shock is marked by a dashed line in both Panel a) and b). The solar wind phasespace
plot evolves from the upstream Panel c) into the downstream Panel d) passing the
shock. Empirically, we can separate the downstream population d) into the transmitted
ions in Panel e) and the reflected ions in Panel f). The velocities are all normalized by the
upstream Alfvén speed v A and x is normalized by c / ω pi . From Wu et al. [2009].

61
Figure 2-9 presents the phase-space plots of both the solar wind ions and the pickup
ions for the 0% pickup ion simulation. The pickup ions are test particles here—they are
subject to the electromagnetic fields as usual, but they make no contribution to the source
terms such as density and current. In other words, the test particles are acted on by the
plasma but they do not react back to modify the plasma flow or shock in any way. In the
figure, solar wind ions are plotted in the panels on the left: a), c) (upstream), and d)
(downstream). Pickup ions are plotted in the panels on the right: b), e) (upstream), and f)
(downstream). The upper panels of Figure 2-9a and Figure 2-9b display the same
magnetic field profile B z from the simulation. The profile is plotted on top of both the
solar wind ion panels and the pickup ion panels so that we can compare the scale length
of B z with the scale lengths of both populations. As Figure 2-9f illustrates, some pickup
ions are energized. However, the average heating of the pickup ions is much weaker than
that of the solar wind ions. In Figure 2-9b, the energized pickup ions upstream of the
shock form an extended region in the v x –x and the v y –x spaces (middle and bottom
panels) as marked by the two vertical grey lines. However, as test particles, pickup ions
do not affect the foot of the magnetic profile (top panel of the same figure).
Figure 2-10 presents the same plot as Figure 2-9 for the 20% pickup ion simulation.
In this simulation, the shock is much weaker ( r S ~ 2.5 ). There is a broader foot in front of
the shock front in the B z profile, which is roughly equal to the width ( ∼ 8 pi / c ω ) of the
upstream pickup ion gyroradius [Liewer et al., 1993]. This wider shock foot, due to the
presence of pickup ions in front of the shock, is marked by the two vertical grey lines in

62
Figure 2-9 The phase-space plots of the 0% pickup ion simulation where pickup ions are
treated as test particles. Solar wind ions are plotted in the panels (a,c,d) on the left and
pickup ions are plotted in the panels (b,e,f) on the right. Upper panels are magnetic field
profile z B –x, x v –x phase-space plots, and v y –x phase-space plots for a) solar wind ions
and b) pickup ions. Lower panels are v x velocity distributions and v x – v y phase-space
plots for c) upstream solar wind ions, d) downstream solar wind ions, e) upstream pickup
ions, and f) downstream pickup ions. The velocities are all normalized by the upstream
Alfvén speed v A and x is normalized by c / ω pi . From Wu et al. [2009].

63
Figure 2-10b. Recall that in the top panel of Figure 2-9b (the 0% pickup ion simulation),
the foot is much shorter and not as visible.
Figure 2-10a shows that for the 20% pickup ion simulation, solar wind ions are
heated less than in the 0% pickup ion simulation (Figure 2-9a). When the pickup ion
relative density is 20%, fewer solar wind ions are reflected. Assuming that the wings of
the downstream solar wind distributions correspond to the more energetic reflected ions,
the reflection efficiency of the solar wind in the simulation with 20% pickup ions (Figure
2-10d) is significantly reduced in comparison with the reflection efficiency in the 0%
pickup ion (Figure 2-9d) simulation. The average heating of the solar wind ions,
however, is still stronger than that of the pickup ions due to reflection. The downstream
thermal velocity of the solar wind (Figure 2-10d) is roughly 3–4 times its upstream
velocity (Figure 2-10c). Some pickup ions are energized to form a heated ring
distribution (Figure 2-10f), as in Figure 2-9f. The downstream pickup ions (Figure 2-10f)
are heated only ~ 2 times the upstream pickup ion velocity (Figure 2-10e) because the
shock is much weaker [Wu et al., 2009]. More precisely quantified heating ratios will be
discussed in the next section.

64
Figure 2-10 The phase-space plots of the 20% pickup ions simulation. Solar wind ions
are plotted in the panels (a,c,d) on the left and pickup ions are plotted in the panels (b,e,f)
on the right. Upper panels are magnetic field profile z B –x, v x –x phase-space plots, and
v y –x phase-space plots for a) solar wind ions and b) pickup ions. Lower panels are v x
velocity distributions and v x – v y phase-space plots for c) upstream solar wind ions, d)
downstream solar wind ions, e) upstream pickup ions, and f) downstream pickup ions.
The velocities are all normalized by the upstream Alfvén speed v A and x is normalized
by c / ω pi . From Wu et al. [2009].

2.5 Heating and Energy Partition
65
In this section, we use temperature to quantify the average microscopic particle
energization/heating, and pressure to quantify the macroscopic partition of energy. We
conclude that the temperature of solar wind ions is raised by a larger factor than the
temperature of the pickup ions; however, the majority of the dissipated energy is
deposited on the pickup ions [Wu et al., 2009].
Generally, the thermal pressure can be calculated from P= nkBT . The temperature, T,
has the dimension of energy per unit particle and represents the average kinetic energy of
these particles. The pressure, P, has the dimension of energy per unit volume. For our
simulations, the temperatures are defined as follows
m
T = < v −< v > >+< v −< v > >+< v −< v > > (2.9)
j 2 2 2 2 2 2
j ( x x y y z z ).
3kB
Here the angular brackets denote the average value (ensemble average) of the
enclosed parameter over all the ions of component j (j could be the pickup ion or the solar
wind ion) and over some finite spatial region within the simulation domain; m j is the
mass of ions of component j; kB is the Boltzmann constant. In the one-dimensional
simulation of perpendicular shocks, as shown in Figure 2-8, there is nearly no heating in
v z (the velocity component parallel to the background magnetic field). Therefore, it is
more accurate to calculate the temperatures inferred from the simulation using only the
v x and y
v terms (velocity components perpendicular to the background magnetic field)

66
in Equation (2.9). The downstream temperature calculated in this manner from the 0%
pickup ion simulation satisfies the Rankine-Hugoniot relations [Wu et al., 2009].
Applying the temperatures computed over a spatial extent of 60 c / ω pi upstream and
downstream of the shock, we calculate the temperature jump of the solar wind ions τ sw
and the temperature jump of the pickup ions τ PUI , both of which are listed in Table 2-1
and plotted in Figure 2-11. In the table, nPUI / n (input column) is the relative density of
upstream pickup ions. Ideally, if all the particles are transmitted, a polytropic energy
equation predicts that the temperature jump across a shock should be given by (see
Section 3.1.1 for derivation)
τ
T
r
d γ −1
= = S .
(2.10)
Tu
Here the polytropic index is to be decided and should not be mistaken as the adiabatic
value of 5/3.
In Table 2-1 and Figure 2-11, an adiabatic temperature jump corresponding to γ =5/3
is presented by the τ adiabat column. An adiabatic transition involves no change in entropy
across the shock. However, by definition, a shock is a transition across which there is an
entropy increase. Because of this non-adiabatic nature of shock crossings, both the solar
wind temperature jump τ sw and the pickup ion temperature jump τ PUI are larger
thanτ adiabat . In particular, the solar wind ions go through a much larger temperature jump

67
Table 2-1 Results Calculated from the Hybrid Simulations (MA = 8, β sw = 0.05).
PUI
nu
nu
τ adiabat τ sw τ PUI γ PUI η PUI
Pd
P u
PUI
Pd
χ d =
Pd
0% 2.47 433.64 6.91 2.43 - 1632.68 -
5% 2.18 271.24 5.86 2.51 0.42 43.32 0.46
10% 2.04 181.30 4.75 2.45 0.59 23.82 0.65
15% 1.95 88.79 4.42 2.49 0.80 15.71 0.84
20% 1.86 49.07 3.85 2.44 0.87 12.13 0.90
25% 1.75 48.85 3.43 2.45 0.87 9.97 0.90
30% 1.67 42.69 3.04 2.44 0.84 8.14 0.89
35% 1.59 35.39 2.87 2.52 0.88 7.05 0.92
40% 1.50 31.51 2.59 2.56 0.89 5.76 0.93
Here energy partition η j =
j j
j
φjTd −φjTu
Td
[see Equation (2.12)] and temperature jump τ i i
j = ,
j
( φT −φT
)
T
∑
i= sw, PUI
i d i u
where j denotes an ion component. The polytropic index γ PUI is derived from columns τ PUI and S
u
r using
PUI 1
τ PUI = rS γ − . In the 0% pickup ion simulation, pickup ions are treated as test particles. They are subject
to the electricmagnetic fields but make no contributions to the source terms. Adapted from Wu et al. [2009].
γ PUI =2.48
γ =5/3
adiabatic
Figure 2-11 The simulated temperature jumps (blue: solar wind; red: pickup ion) as
compared with adiabatic temperature jumps. The polytropic indexes are calculated from
Equation (2.10). In the 0% pickup ion simulation, pickup ions are treated as test particles.
They are subject to the electricmagnetic fields but make no contributions to the source
terms.

68
than the pickup ions: τ sw >> τPUI > τadiabat
(Figure 2-11). Recall that the Rankine–
Hugoniot relations determine the temperature jump across the shock, but do not indicate
the nature of the heating, nor the partition of the heating if there is more than one plasma
component. A polytropic law of the form of Equation (2.10) has often been used to
characterize the heating at shocks [e.g., Feldman, 1985], where γ becomes a parameter
determined from observations or simulations. We thus define γ PUI as such a parameter
for characterizing pickup ion heating. The values of γ PUI from the simulations can be
obtained from Equation (2.10), as shown in Table 2-1. It is apparent that γ PUI varies
between 2.4 and 2.6 and does not depend on the pickup ion relative density [Wu et al.,
2009]. This result will be applied in Chapter 3 to analytically describe heating and energy
partition.
Compared with the observed solar wind temperature jump in Table 1-3, the simulated
values of τ sw in Table 2-1 are larger than the observed ones. This is because the plasma
instrument on Voyager 2 does not measure reflected solar ions in the suprathermal range
(~keV) of the velocity distributions and therefore underestimatesτ sw , in comparison with
the simulation values [Wu et al., 2009].
We define η j as the percentage of thermal energy gained by each ion component
η
PV −P( rV) P −Pr
,
( ) ( )
j j j j
j = d d u S d
PV d d −Pu rV S d
= d u S
i i
∑ Pd −Pr
u S
i= sw, PUI
(2.11)

69
where j denotes the ion component (the pickup ion or the solar wind ion); V d is the
volume of the downstream ions; and so ( rV S d)
is the ions’ upstream volume before they
j j j
cross the shock. As P = n kT , Equation (2.11) can be re-written as
η
φ T −φ
T
j j
j =
j d
∑
i= sw, PUI
i
j
i
d
u
i
i
u
( φT −φT
)
(2.12)
for a simulation. Here φ j is the relative density of the ion component j. The calculated
η PUI (Table 2-1) from simulations shows that the net energy gain of pickup ions
increases with the increasing pickup ion relative density. When the pickup ion relative
density is greater than 15%, pickup ions account for more than 80% of the dissipation.
This is consistent with the postulation that most energy goes to pickup ions, which
Richardson et al. [2008] draws from the Voyager 2 observations [Wu et al., 2009]. This
does not contradict the results that τ sw >> τ PUI.
Because pickup ion starts with an initial
energy that is ~1000 times that of the solar wind energy, even a relatively small jump in
the temperature implies a large value of net energy gain, as illustrated in the following
algebraic (unitless) analogy for a single representative pickup ion and a single
representative solar wind ion:
τ sw 40 >> τ pui
4
Δ Esw = Esw(
τ sw − 1) = 1 × (40− 1) = 39,
Δ EPUI = Epui(
τ pui − 1) = 1000 × (4 − 1) = 4000,
Δ E = 4000 >>Δ E = 39
PUI sw

70
The shock has more than a single pickup ion and a single solar wind ion. The complete
calculation is more complex than the simple analogy, as will be analytically formulated in
Chapter 3.
Another way of characterizing the energy partition, for easy comparison with
Voyager observations, is through the downstream thermal pressure ratio defined as
PUI
Pd
χ d = .
(2.13)
P
The pressure ratio χ d is displayed in Table 2-1. For the most cases, χ d
approximates η PUI . Both χ d and η PUI increase as the pickup ion relative density
increases. With a pickup ion ratio of 15%, χ d is about 84%—the fraction gain of
dissipated energy for pickup ions, as inferred from the Voyager 2 observations by
Richardson et al. [2008].
2.6 Pickup Ion Energization
The primary mechanisms for heating the solar wind ions are compression of the
transmitted ions and shock reflection of the relatively small number of ions which are
turned back upstream. Both processes have been analyzed in great detail elsewhere [e.g.,
Leroy et al., 1982; Leroy, 1983; Burgess et al., 1989] and are not the emphasis here. The
pickup ions, however, are much more energetic than the cool solar wind ions upstream;
as a result, they gain energy at the shock in a completely different manner. Several
different studies have suggested energy gains by orders of magnitude larger than the
d

71
upstream energy [e.g., Zank et al., 1996; Lee et al., 1996], e.g., 0.5 MeV. In contrast, we
address the initial heating of the pickup ions.
2.6.1 Simulated Pickup Ion Kinetics
Early on in Section 2.3, we demonstrated that at the level of a fluid description, the
supercritical characteristics of the termination shock are a result of solar wind ion
specular reflection. We have also shown that the collective behavior of pickup ions
extends the length of the termination shock foot and the spatial scales of the downstream
oscillations (Figure 2-6 and Figure 2-7). This section goes beyond a fluid parameter
analysis and examines the response of individual pickup ions.
The simplist case is one in which that pickup ions are treated as test particles, the 0%
pickup ion simulations. Figure 2-12 shows individual ion’s vx – vy phase (pickup ions:
left panel; solar wind ions: right panel) from such a simulation. In the figures, each dot
corresponds to a simulation ion. These ions are the ions from the vicinity of 20 c/ω pi that
covers the range of the foot, the ramp, the overshoot and the undershoot of the shock as
well as some partial upstream adjacent to the foot.
Solar wind ion reflection is apparent in Figure 2-12, where some solar wind ions
stream upstream from a collection of cold upstream ions concentrated at vx/vA ~ 5. These
ions acquire significant energy through vy as they return toward the shock and develop a
ring-like character. In contrast, there seem to be two streams of ions that are pulled out
from the pickup ion shell at two particular gyro-phase angles, 1. vx< 0 and vy>0 (later

72
Figure 2-12 Velocity phase-space plots (vx–vy) for pickup ions (left panels) and solar
wind ions (right panels) within 20 c/ωi of the shock front from the 0% pickup ion
simulation. All velocities are normalized by vA.
will be referred to as “reversed” ions as the backward gyration that leads to energization
starts in the foot region upstream of the shock, therefore the ions encounter the shock
front from upstream), 2. vx>0 and vy

73
If we examine a spatial series of the above phase space plots, we obtain the evolution
of the ions’ velocities as they cross the shock. Such an analysis for the 20% pickup ion
simulation is shown in Figure 2-13. Again, we include both pickup ions and solar wind
ions, respectively. Because the shock compression is reduced with the inclusion of pickup
ions in this simulation, the energization level is reduced compared with Figure 2-12.
Nevertheless, the same features of the energization processes persist, indicating that the
density of pickup ions does not affect the basic energization mechanism.
In Figure 2-13, the red labeled pickup ion phase space corresponds to the red labeled
upstream and partial foot region of the kinetic shock profile (detail description and
explanation of the profile can be found in Section 2.3 and thus will not be repeated here).
There is a small stream of “reversed” pickup ions in the vx0 region. As the
pickup ions enter further into the foot, more “reversed” pickup ions are seen (as shown in
the orange labeled phase space plot), indicating that the “reversed” ions originate from
the foot region upstream of the shock. Up to this point, there is no evidence of the
“crossing” ions in both the red and the orange labeled regions which include the foot but
exclude the other shock structures. Interacting with the negative Ex in this upstream foot
region, the “reversed” pickup ions gain energy because vxEx>0. As we move to the
yellow labeled region which covers the ramp, the overshoot, and the undershoot as well
as the foot, the “crossing” ions appear in the vx>0 and vy0. Both groups of pickup ions continue to gain energy as they
gyrate around the local magnetic fields, as shown in the green labeled phase space. As we

74
Figure 2-13 Phase space evolution of the pickup ions (first row of the vx–vy panels) and
of the solar wind ions (second row of the vx–vy panels) as they cross the 20% pickup ion
shock. The kinetic shock structure from Section 2.3 is zoomed in for comparison. Each
panel of the evolution series corresponds to the color marked spatial region of 20 c/ωpi.
The center of each adjacent panel is 5 c/ωpi apart.

75
continue to move downstream, the energized pickup ions form a gyrotropic distribution
that is nearly uniform in the covered velocity range, suggesting that both energization
processes are continuous, just as indicated by the constant value of the pickup ion density
across the shock.
In comparison, some solar wind ions show an initial reversal of vx that later develops
into the low density halo around the ions that are heated through transmission. The
downstream gyrotropic distribution of the solar wind (purple labeled phase space plot) is
rather non-uniform with the reflected ion ring bounding the denser core. This is
consistent with the fact that specular reflection creates an energization gap between
transmitted ions and reflected ions.
Phase-space evolutions of pickup ions in a similar format as Figure 2-13 also appear
in Lipatov and Zank [1999] and Chapman et al. [2005], but look significantly different.
In the former case, the simulation contains strong resistive electron heating and very
small spatial scales (~ electron Debye length) that generating a very large spike in Ex that
accelerates pickup ions significantly through multiple reflections (surfing). In the latter
case, a reformatting shock that involves large, time-varying electric fields is quoted as the
reason for the acceleration of the pickup ions. Our hybrid simulations, instead, gives
steady state solutions.
Figure 2-13 suggests, and Figure 2-14 confirms, that there is a gyrophase-dependent
energy gain process initially. The panels of Figure 2-14 illustrate the temporal evolution
of pickup ions that originate from the same upstream location at the start of the

76
simulation, with one ion per panel. The start time in each panel is marked with an asterisk,
and the color changes from black through blue, green, yellow, orange and red; the last
color corresponds to late-time downstream conditions. Except for the last panel, these
trajectories are characteristic of the fraction of pickup ions that gain substantial energy at
the shock. The black to green circles with relatively large average v x (~uu-ud) are
upstream velocity trajectories; the transition region from green to yellow corresponds to
parts of the trajectories near the shock; and the large red circles with relatively small
average v x (~0, recall that the simulations are run in the downstream frame) are from
downstream. These trajectories suggest that some pickup ions can achieve a significant
energy gain at the shock, and the process involves both v x and v y . The mechanism by
which these ions are energized [Winske et al., 2009] is the gyrophase-dependent
acceleration (GPDA), discussed in the next section.
2.6.2 Gyrophase-Dependent Acceleration Model for Pickup Ions
To explain how the two streams of pickup ions are energized across the shock, consider
the Figure 2-15 diagram. The dashed circles mark the upstream pickup ion trajectories in
x–y space and vx–vy space. In the context of this figure, we consider two types of pickup
ions.
In the downstream rest frame, consider the first type of pickup ion well upstream of
the shock (in the left side of the figure), as it gyrates in the upper right quarter of its
upstream orbit in x–y space (from Point P to Point B). By nature of gyration, the ion has a

77
Figure 2-14 Pickup ion trajectories from the 20% pickup ions simulation in velocity
space perpendicular to the background magnetic field. Both the x v and v y are normalized
to the upstream Alfvén speed v A . Each panel shows a representative ion trajectory from
time zero (indicated by an asterisk) to the end of the simulation. The arrows indicate the
sense of temporal progress of each trajectory. From time zero, the color of each trajectory
changes from black through blue, green, yellow, orange and, near the end of the
simulation, to red. From Wu et al. [2009].

78
positive vx and negative vy. The ion loses energy in the foot region because ∫ vE x x 0 and Ex

“Reversed” ion
B
D x
Ex
0
Ey
0
C
vy
y
C
D
Shock front
P
vx
79
“Crossing” ion
B
P
P
B
Figure 2-15 Schematic interpretation of pickup ion energy gain (downstream rest frame).
Left side of the figure corresponds to a representative trajectory of a “reversed” ion.
Right side of the figure corresponds to a representative trajectory of a “crossing” ion.
Both curves correspond to gyromotions with increasing Lamour radii. The electric field
Ex and the electric field Ey in the shock region (foot, overshoot) are also shown. The
downstream oscillation these fields are not shown. Instread, thick dash lines mark the
average downstream fields.
Ex
0
Ey
0
D
C
vy
Shock front
y
P
C
D
B
x
vx

80
The ion gains some energy initially by∫ vE y y >0 (for vy0) and vE x x
∫ >0 (for vx

81
loss), while the “crossing” ions gain some initial energy as they cross back over the shock
front in a narrow region (the width of the overshoot, ~ 1–2 c/ωpi) with a large |Ey|.
For both types of ion, the major share of the energy gain occurs when the ions have
returned upstream of the shock front (from Point C to Point D), through the usual shock
drift acceleration (SDA) mechanism (vy>0, Ey>0). That is, the drift of the ions along the
motional electric field upstream of the shock. The important difference is, however, that
the “crossing” ions receive an initial kick for a minor energy gain before this major
energy gain, while the “reversed” ions receive an initial kick for a minor energy loss
before the major energy gain. This initial energy gain/loss is a result of the ion’s first
interaction with the shock at different physical locations (in x–y space), or the shock
front’s first interaction with the ions at different phases (in vx–vy space). Therefore, we
name our model the gyrophase-dependent acceleration (GPDA), to distinguish it from
SDA in our emphasis of the initial minor energy gain/loss, as seen in velocity phase space
(recall Figure 2-14).
In order to examine the relative efficiency of these two processes, we have plotted
Figure 2-16 from a simulation with 20% pickup ions. The “reversed” ions and the
“crossing” ions are tagged and followed, respectively, as they cross the shock in the
simulation. Their “final” energy is then computed some distance (~ 50 c/ ω pi )
downstream of the shock front. The maximum energy achieved of the “crossing” ions is
clearly larger than that gained by the “reversed” ions, as expected. More importantly,
there are many more “crossing” ions with higher energy gain than “reversed” ions, due to

82
the different initial kicks the ions receive from the shock that result in minor energy
gain/loss as predicted by GPDA.
Figure 2-16 Energy distribution histogram of the downstream pickup ions. Blue line: the
distribution for “reversed” ions. Black line: the distribution for “crossing” ions. The
pickup ion energy is normalized to the upstream solar wind dynamic energy.
2.7 Discussion
“Reversed” ions
“Crossing” ions
We have used the one-dimensional Los Alamos hybrid code to carry out a series of
simulations of the idealized perpedicular termination shock in the presence of solar wind
ions and pickup ions. Our simulations show that, although the presence of the pickup ions
weakens the shock, some of the upstream solar wind ions are still reflected, helping to
dissipate the energy required for the shock transition. The reflected solar wind ions gain a

83
gyrotropic speed of the order of the upstream flow speed, and are swept downstream.
This process gives some solar wind ions the same order of magnitude of kinetic energy as
the pickup ions, so that it may be difficult to observationally separate reflected solar wind
ions from pickup ions downstream. The simulations also show that the presence of the
pickup ions significantly raises the magnetosonic wave speed so that the termination
shock is significantly weakened. The shock strength reduces to between 2 and 2.5 when
the pickup ion relative density is 20–30%. Recall that the compression ratio is about 4 for
strong supercritical shocks.
The simulations also show that the temperature jump of solar wind ions across the
shock is significantly larger than that of the pickup ions. This does not contradict the
finding (also from the simulations) that pickup ions account for more than ~85% of the
dissipated energy when the pickup ion relative density is larger than 15%. The reason is
that the pickup ions start with initial energies that are ~1000 times those of the solar wind
ions. A small fraction of energy gain results in a large portion of the dissipated energy
being transferred to pickup ions.
We demonstrate that there are two types of energized pickup ions, consistent with the
Lever et al. [2001] test particle calculation. Both types gain moderate energies but on
average the “crossing” ions gain more energy than the “reversed” ions, consistent with
the finding of Lever et al. [2001]. The acceleration efficiency of the “crossing” ions is
independent of the pickup ion relative density, or the thickness of the shock ramp. Most
importantly, we propose the gyrophase-dependent acceleration (GPDA) to explain the

84
physical difference in the initial kick of the minor energy gain/loss for the
“crossing”/”reversed” ions, a mechanism not included in the Lever et al. [2001] model.
The primary method for the initial energy gain by the pickup ions at the termination
shock is through the gyrophase-dependent acceleration (GPDA) process described by
Winske et al. [2009]. This GPDA mechanism demonstrates that, rather than a simple
reversal of the radial velocity ( v x ) as in reflection, both the radial and the transverse ( v y )
velocity components play a role in the transfer of energy to the pickup ions [Winske et al.,
2009]. GPDA also explains why the pickup ion energy gain is greater than the value
predicted by simple compression. Only pickup ions with sizeable components of vx and
vy can sample the region near the shock front where the motional electric field (Ey) has a
large negative spike on the spatial scale of the width of the magnetic field overshoot.
When the pick-up ions dominate the downstream pressure, the upstream foot and
downstream oscillations of Bz have a longer spatial scale length, characterized in terms of
the gyro-radius of the energized pickup ions. Thus, The GPDA includes SDA but
emphasizes the pickup ion’s very first moment of encounter with the shock.
In conclusion, from the ion energization point of view, there are three primary
mechanisms for ion heating at the termination shock: 1) the compression of transmitted
solar wind ions; 2) the specular reflection of some solar wind ions; and 3) the GPDA
mechanism operating on the pickup ions. We can use the polytropic index γ PUI (as
shown in Table 2-1) to characterize the pickup ions. The next chapter will thus be built
upon the above findings.

3 Analytic Model for an Idealized Termination Shock
This chapter develops an analytic multicomponent Rankine–Hugoniot model to address
the partition of dissipated energy at the perpendicular termination shock. A polytropic
index is introduced to characterize the shock. Based on the model, we analytically
formulate the downstream multi-ion speed distribution as a superposition of Maxwellian
distributions.
3.1 Baseline Analytic Formula
3.1.1 Heating of a Transmitted Ion
To derive the heating of an ion for a single transmission across the shock, we start with
the polytropic energy equation
where
or
P nk T nk v
V ∝ n,
so
2
= B ∝ B and 1/
γ
PV = constant,
(3.1)
2 γ 2 γ−1
nv n v n constant
(1 / ) = / = ,
(3.2)
v / n = v / n .
(3.3)
2 γ−1 2 γ−1
u u d d
If we substitute shock strength r = n / n into Equation (3.3), we arrive at
S d u
v = v r
(3.4)
( γ −1)/2
d u( S)
.

Defining the temperature jumpτ ,
τ
T
86
v v r
d
2 γ −1
= = ( d / u) = ( S)
.
(3.5)
Tu
Here the derivation is very general and the polytropic index should not be mistaken for
the adiabatic value of 5/3. It should be determined by the shock’s characteristics, as we
discuss below.
3.1.2 Analysis of Upstream Mach Numbers
We analyze the upstream Mach numbers to gain preliminary insights into the termination
shock. The magnetosonic Mach number M MS , as the coupled Mach number of the Alfvén
Mach number A M and the sonic Mach number M cs , can be written as
M
MS
uuM AMcs M A
Mcs
= = = =
2 2 2 2
v + v M + M β γ /2+ 1 2/( β γ)
+ 1
A cs A cs
eff eff
(3.6)
where β eff is the effective plasma beta (calculated from the sum of solar wind beta and
the pickup ion beta). Assuming γ =5/3, MA=8 (Voyager observations), β sw =0.05
(Voyager observations) and β PUI =8.53 6 , we find that MMS=2.80~Mcs=2.99. Both Mach
numbers are much smaller than the Alfvén Mach number (MMS~Mcs

87
the effective beta dramatically and change the shock’s characteristic from the MA-
dominant to the Mcs-dominant form.
3.2 Multicomponent Rankine–Hugoniot Model
Because of the importance of pickup ions, we here develop an analytic model for the
perpendicular termination shock, based on the Rankine–Hugoniot jump conditions, a
polytropic energy equation, and specular reflection (for solar wind ions). The model
computes the relative energy gain at a perpendicular shock for three proton components:
transmitted solar wind ions, reflected solar wind ions, and pickup ions.
3.2.1 Modified Compression Ratio
To apply the Rankine–Hugoniot relations, we first need to define the upstream conditions.
As noted earlier, we consider a perpendicular shock, with an upstream velocity u u and a
corresponding Alfvén Mach number M A = uu / uA
, where u A is the upstream Alfvén
speed
u = B /( μ ρ ) . We also specify the upstream solar wind temperature sw
T
2 2
A u 0 u
sw sw 2
using β = 2 μ kT ρ /( mB)
. For the pickup ions, we assume the idealized spherical
sw 0 B u u i u
shell distribution (as done in Chapter 2), with speed vshell = uu.
It then follows from
Equation (2.9) that
PUI
2 2
Pu = ρPUIvPUI, u /3 = φρuuu
/3,
(3.7)
where φ is the upstream pickup ion relative density. Therefore, the upstream thermal
pressure is given by
u

It is then useful to define
88
β φ
P = − P + P = − + u
(3.8)
sw PUI sw
2
u (1 φ) u φ u [(1 φ) ] ρ .
2 u u
2MA3 P β φ
δ = = (1 − φ)
+ .
(3.9)
ρ
2 3
u sw
2 2
uuu MA
With given upstream solar wind beta β sw , Alfvén Mach number M A and heating
parameters (γ , γ PUI ), for any chosen pickup ion relative density φ we can calculate δ
and the sonic Mach number
Then Equation (1.7) can be rewritten as
M = 1/( γδ ).
(3.10)
2
cs
2 −γ
γ
r − r + r + + − − + = (3.11)
2
( S 1)[ S ( 2 1) ( 1)] 0
2 S γδ γ γ
2
MA MA
and solved for r S . Hence, we modified the original Rankine-Hugoniot compression ratio
equation with the inclusion of the pickup ion relative density [Wu et al., 2009].
3.2.2 Component Pressures
Both the Voyager 2 observations and our hybrid simulations illustrate that downstream
pickup ions gain more energy than would be described by γ = 5 / 3 in Equation (2.10). To
represent this concept in the analytic model, we assume that the thermalization of the

89
pickup ions follows a polytropic energy equation on the average sense, as characterized
in Chapter 2. Using γ PUI to represent the amount of pickup ion heating,
Equation (3.7) and (3.12) together give
PUI γ PUI PUI
P r P .
(3.12)
d S u
P = r u
(3.13)
PUI γ PUI 2
d S φρu
u /3.
Our simulations further show that, even with the inclusion of pickup ions, some solar
wind ions are specularly reflected at the shock. Therefore, we assume that the solar wind
ions are divided into two components: a transmitted component and a reflected
component. Let ε ref be the reflection efficiency of the solar wind ions: the number
density of reflected solar wind ions divided by the number density of the solar wind ions.
We further assume that the heating of transmitted solar wind ions obeys the polytropic
energy equation, with the same γ as used in Equation (3.11). Then the transmitted solar
wind population has a downstream pressure of
sw−trans γ sw−transγ sw
P = r P = r (1 −ε ) P ,
(3.14)
d S u S ref u
where the superscripts “ref” and “sw-trans" represent “reflected solar wind ions” and
“transmitted solar wind ions”, respectively. In addition, we can express the upstream
solar wind pressure as

90
P = P − P = P − u
(3.15)
sw PUI
2
u u u u φρu
u /3,
where we made use of equation (3.7). Equations (3.14) and (3.15) give
P r ε P u
(3.16)
sw−trans γ
2
d = S (1 − ref )( u −φρu
u / 3).
For the reflected solar wind ions, because of specular reflection and the subsequent
sw−ref energy gain in the upstream region, their downstream thermal speed is v ~ u
d u
sw−ref [Gosling and Robson, 1985]. In fact, our 0% pickup ion simulation gives v = 1.5u
.
d u
The additional energy gain comes from the shock compression and other non-linear
processes in the shock. To simplify the calculation, we adapt a value of 2 and rewrite
sw−ref v 2u
(3.17)
d u
for the reflected solar wind ions. It then follows that
P = ρ v = ε − φ ρ u = r ε −φ
ρ u (3.18)
sw−ref sw−ref sw−ref 2 2 2
d d ( d ) /3 ( ref (1 ) d )2 u /3 2 S ref (1 ) u u /3.
The total downstream thermal pressure can then be expressed as the sum of the
transmitted solar wind thermal pressure, the reflected solar wind thermal pressure and the
transmitted pickup ion thermal pressure
sw−trans sw−ref PUI
P = P + P + P .
(3.19)
d d d d

91
Using Equation (3.14), (3.16) and (3.18), we can rewrite the above equation as [Wu et al.,
2009]
P = r −ε P − φρ u + r ε − φ ρ u + r φρ u (3.20)
γ
2 2 γ PUI 2
d S (1 ref )( u u u / 3) 2 S ref (1 ) u u / 3 S u u / 3.
3.2.3 Reflection Efficiency
The Rankine–Hugoniot conservation of momentum requires that
B B
ρ + + = ρ + + .
(3.21)
2 2
2
uuu Pu u
2μ0 2
dud Pd
d
2μ0
Substituting d P with equation (3.20) and B d with Equation (1.6), we obtain
B rB
ρ ρ (1 )( φρ / 3) 2 (1 φ) ρ / 3 φρ / 3 .
2 2 2
2 u
uuu + Pu + =
2μ0 2 γ
dud + rS −εref Pu −
2
uuu + rSε ref −
2
uuu γ PUI + rS 2
uuu S u +
2μ0
2
Dividing Equation (3.22) by u u u ρ and solving forε ref , we find
(3.22)
2
1 1−
rS
φ γ γ PUI
1 − + + (1 − r ) ( )
2
S δ + rS −rS
rS 2M A
3
ε ref =
.
(3.23)
γ
2 rS(1 −φ)
rSφ
γ
+ −rSδ
3 3

92
Because the compression ratio r S has been obtained from Equation (3.11) in Section
3.2.1, we can solve for the solar wind reflection efficiency ε ref from Equation (3.23). In
addition, the thermal pressure jump can be derived as
γ
2 (1 ) PUI
P r
d γ φ Sεref−φ rS
φ
= rS
(1 −ε ref )(1 − ) + +
(3.24)
P
3δ3δ 3δ
u
and the downstream pickup ion thermal pressure ratio [defined in Equation (2.13)] is [Wu
et al., 2009]
PUI
γ PUI
Pd rS
φ
χ = =
.
γ
γ PUI
P r (1 −ε )(3 δ − φ) + 2 r ε (1 − φ) + r φ
d S ref S ref S
(3.25)
In order to compare model results with Voyager observations and hybrid simulations,
we choose the same input values of M A = 8 and β sw = 0.05 (as in our simulations) for our
evaluation of the analytic model. In addition, Table 2-1 shows that γ PUI varies between
2.4 and 2.6. We assume an average value, γ PUI =2.48, independent of the pickup ion
relative density, in our analytic formula. Figure 3-1 then shows model results for
PUI
r , P / P , and P / P as functions of the pickup ion relative density for the choice of
S
d u
d d
two different values of γ : the usual adiabatic value γ =5/3 and an empirical choice of
γ =2.2; the simulation results are indicated by the diamond symbols [Wu et al., 2009].
Our multicomponent Rankine–Hugoniot analysis with these input conditions shows
that, when the pickup ion relative density φ increases, the compression ratio r S (see the

93
top panel of Figure 3-1) and the pressure jump Pd/ P u (as in the middle panel of Figure
3-1) drop as the shock weakens. In contrast, the downstream partition of pickup ion
PUI
energy density P / P (bottom panel) increases as the pickup ion number density
d d
increases. The reason is that the increase of upstream pickup ion number density raises
the partition of upstream pickup ion energy density significantly. For a fixed value of the
pickup ion relative density, an increase in the polytropic index γ corresponds to a
PUI
decrease in the pressure jump and a smaller compression ratio. Both the r S and Pd / P d
panels demonstrate that with the increase of the pickup ion relative density, the simulated
values trend from the γ =5/3 curve toward the γ =2.2 curve [Wu et al., 2009].
Overall, as seen from the r S panel, above ~15% pickup ion relative density, the shock
tends toward the γ =2.2 solution. With the γ =5/3 solution, the downstream partition of
PUI
pickup ion energy density P / P reaches ~100% abruptly when the pickup ion relative
d d
density reaches 11%. This solution is not consistent with Voyager result. So a γ =5/3
solution is unlikely to be true even when pickup ion density is below 15% but large
enough to change the shock’s character. Approximately when pickup ion relative density
approaches 5%, a different γ should be applied. In fact, the transition from γ =5/3 to
γ =2.2 should be gradual and monotonic when we raise pickup ion relative density from
0% to 40%. There are many intermedium values of γ between 5/3 and 2.2. In particular,
the solution of γ = 1.95 fits the simulation with 20% pickup ions.

94
Figure 3-1 Compression ratio r S , pressure jump Pd / P u and downstream pickup ions
PUI
thermal pressure ratio Pd / P d as a function of the pickup ion relative density φ .
Dashed lines are theoretical predictions when γ is set to be 5/3, the solid lines are
theoretical predictions when γ is 2.2. The diamond symbols mark values from
simulations. From Wu et al. [ 2009].

95
PUI
The downstream fraction of pickup ion energy density ( P / P ) inferred from
d d
Voyager 2 observations [Richardson, 2008] is approximately 85%. The values from our
PUI
simulations in Chapter 2 show that, for φ > 15% , P / P approaches a constant value
d d
of approximately 90%, quite close to the Richardson [2008] inference. If we combine this
with the results of the top panel of Figure 3-1 and apply the observed average
compression ratio of ~ 2, the simulations and the model at γ = 2.2 both indicate that the
pickup ion relative density near the nose of the termination shock is around 25%, which
is consistent with the Richardson [2008] estimate of 20% [Wu et al., 2009].
The solar wind reflection efficiency ε ref is plotted in black as a function of the pickup
ion relative density φ in Figure 3-2. The dashed line corresponds to γ =5/3 and the solid
line corresponds to γ =2.2. The modeled reflection efficiency for the 0% pickup ion case
is 25% and is consistent with earlier hybrid simulations [e.g., Leroy et al.; 1982]. The
value of the reflection efficiency drops dramatically to zero when the pickup ion relative
density arrives at φ =11% for the γ =5/3 solution. For the γ =2.2 solution, the reflection
efficiency of solar wind ions remains notable for all values of the pickup ion relative
density. When the pickup ion relative density is 20%, the model predicts that there are
still a significant number of reflected solar wind ions, consistent with the phase-space
plots in Capter 2 (Figure 2-10) [Wu et al., 2009].

96
Figure 3-2 Solar wind reflection efficiency ε ref (%) as a function of the pickup ion
relative density φ . The dashed lines correspond to γ =5/3 and the solid lines correspond
to γ =2.2. Adapted from Wu et al. [2009].
3.2.4 Energy Partition
Applying the results from the last section, the percentage of thermal energy gain by the
transmitted solar wind ions can be derived from Equation (2.11)
η
sw−trans γ
(1 −εref )( rS−rS)( δ −φ
/ 3)
=
.
γ PUI
γ
φ( r − r )/3 + ( δ −φ /3)[ r (1 −ε ) − r ] + 2 ε r (1 −φ)/3
S S S ref S ref S
(3.26)

97
Similarly, the percentages of thermal energy gain by reflected solar wind ions and pickup
ions are
and
η
sw−ref η
PUI
2 εref rS(1 −φ )/3 −εref rS(
δ −φ
/3)
=
,
γ PUI
γ
φ( r − r )/3 + ( δ −φ /3)[ r (1 −ε ) − r ] + 2 ε r (1 −φ)/3
S S S ref S ref S
γ PUI φ(
rS − rS)/3
=
,
γ PUI
γ
φ( r − r )/3 + ( δ −φ /3)[ r (1 −ε ) − r ] + 2 ε r (1 −φ)/3
S S S ref S ref S
(3.27)
(3.28)
respectively. The fractions of dissipation energies acquired by the reflected solar wind
( ηsw− ref
, blue) and the pickup ions ( η PUI , in red) are plotted in Figure 3-3. The dashed
lines correspond to γ =5/3, the solid line corresponds to γ =2.2, and the simulation results
are indicated with diamond symbols. The percentage of thermal energy gain by the
reflected solar wind ions decreases with increasing pickup ion relative density φ . The
percentage of thermal energy gain by the transmitted solar wind ions ( ηsw− trans
negligibly small and is not shown. The percentage of thermal energy gain by the pickup
ions η PUI increases with an increasing pickup ion relative density. Above φ ~ 15% , the
simulated η PUI is nearly constant. At relative small values of pickup ion relative density,
the simulated values are more in agreement with the Multicomponent Rankine–Hugoniot
model based on a γ = 5/3 shock. These values tend toward the γ = 2.2 shock solution at
higher pickup ion relative densities as Figure 3-2 shown [Wu et al., 2009]. The solutions
) is

98
with intermedium values of γ are not shown but can be infered to fall monotonically
within the range of the γ = 5 / 3 solutions and the γ = 2.2 solutions.
Figure 3-3 Termination shock energy partition η (%) as a function of the pickup ion
relative density φ . The dashed lines correspond to γ =5/3 and the solid lines correspond
to γ =2.2. Red diamond symbols mark the percentage of heating the pickup ions gain
from our simulations. Adapted from Wu et al. [2009].

3.2.5 Downstream Mach numbers
The downstream Alfvén Mach number is obtained with the aid of Equation (1.6)
99
u μ ρ μ ρ
M = = u ( ) = r u ( ) = r M . (3.29)
d 0 d 1/2 −1.50 u 1/2 −1.5
A, d
vAd ,
d 2
Bd S u 2
Bu
S A
Let the upstream Alfvén Mach number be M A = 8,
for the pickup ion relative density
range φ =[0, 40%]. Then, M A, d =[1.11, 2.26] (for γ =5/3) or A, d
M =[1.90, 3.43] (for
γ =2.2). This indicates that downstream of the termination shock, the flow is still super-
Alfvénic.
The downstream sonic Mach number can also be obtained analytically
u ρ M
M = = u = (3.30)
d d 1/2
cs
cs, d d ( ) ,
vcs γ PdrP S d / Pu
where the upstream sonic Mach number M = 1/ ( γδ ) (as discussed in Section 3.2.1). The
cs
pressure jump Pd / P u is known from Equation (3.24). For the pickup ion relative density
range φ = [0, 40%], we obtain that the downstream sonic Mach number M cs, d = [0.50,
0.62] (γ =5/3) or M cs, d = [0.55, 0.69] (for γ =2.2).

100
Figure 3-4 Upstream and downstream Mach numbers as functions of pickup ion relative
density. The three panels on the right are subsets of the same quantities of the three
panels on the left: Alfvén Mach number M A (top panel), sonic Mach number M cs
(middle panel), Magnetosonic Mach number M MS (bottom panel). In the right panels, we
have a better view of how the solid red lines differ from the dashed lines. The black lines
are the upstream Mach numbers as a function of the pickup ion relative density φ ; the red
lines are the analytically calculated downstream Mach numbers at γ =2.2 and the red
dashed lines are the analytically calculated downstream Mach numbers at γ =5/3. From
Wu et al. [2009].

101
The magnetosonic Mach number M MS can be calculated from
M = M M M + M For all of our simulations, the downstream magnetosonic
2 2
MS A cs A cs .
Mach numbers fall within the range of [0.46, 0.60] (for γ =5/3) or [0.53, 0.67] (for
γ =2.2).
The analytically calculated Mach numbers are plotted in Figure 3-4 as a function of
the pickup ion relative density φ . The polytropic index of γ =2.2 predicts slightly larger
downstream Mach numbers than the Mach numbers predicted by γ =5/3. This is because
a larger γ increases the heating, and thus lowers the compression ratio, which results in a
faster downstream flow with larger downstream Mach numbers. Although the
downstream flow is super-Alfvénic (or measured to be super-magnetosonic due to
instrument limitations), it is still subsonic and strictly sub-magnetosonic. The termination
shock has the character resembling a gas kinetic shock as opposed to the terrestrial bow
shocks. The participation of pickup ions in the shock dynamics give rise to the gas kinetic
nature of the termination shock [Fisk, 1996]. As indicated in Section 3.2.1, the
termination shock is a sonic Mach number dominant shock.
3.3 Downstream Multi-Ion Speed Distributions
In this section, we build on the Multicomponent Rankine–Hugoniot model to derive
functions that describe the ion speed distributions downstream of the termination shock.
Recall from Section 2.5 and Figure 2-8 that the parallel velocity of a perpendicular shock

102
d u
from a one-dimensional simulation remains nearly unchanged f ( v ) = f ( v )
consider the perpendicular speed distribution f ( v⊥ ) in this work.
3.3.1 Dimensional Analysis
, we only
In this section, we examine if the one-dimensional Maxwellian can be generalized to a
higher dimension.
For the purposes of comparing with our one-dimensional simulations (where only
perpendicular heating is achieved), we restrict the discussion to the perpendicular
velocities vx and vy. Assuming the distribution is gyrotropic, we write
where the perpendicular speed
f ( v ) dv ⋅ f( v ) dv = f( v , θ ) ⋅ v dv dθ,
(3.31)
x x y y
v = v + v .
⊥
2 2
x y
Consider the Maxwellian distribution,
⊥ ⊥ ⊥
2
2 2 2
v vy vx+ v
x
y
− − −
2 2 2
vthvth vth
x θ x⋅ y y = 1 ⋅ 2 x y = 1 2
x y
f ( v , ) dv f( v ) dv ae a e dv dv aa e dv dv
So ( , )
f v θ
⊥ is still Maxwellian in form.
2
v⊥
−
2
vth
= ae vdvdθ = f( v) ⋅vdvdθ.
⊥ ⊥ ⊥ ⊥ ⊥
(3.32)

103
3.3.2 Downstream Two-Maxwellian Speed Distribution
In this section, we characterize the downstream ion speed distribution as a superposition
of two Maxwellian distributions.
Although the reflected solar wind ions are heated a lot more than the transmitted solar
wind ions, their energies still fall to the lower end of the downstream pickup ion energy
range. We make an assumption (to be tested in Chapter 4 simulations) that downstream
of the shock, the solar wind ions form one Maxwellian and the pickup ions form another
Maxwellian. This assumption is only valid when there are not too many reflected solar
wind ions, i.e., when the pickup ion relative density is sufficiently large (as will be
discussed in Section 3.4). So the downstream multi-ion speed distribution can be written
as
v v
f( v ) = a exp[ − ( ) ] + a exp[ − ( ) ]
(3.33)
⊥
2 2
⊥ 2 ⊥ 2
1 sw
vth, d
2
PUI
vth,
d
in the plasma frame. Here a1 and a2 are normalization constants and vary from case to
sw
PUI
case, v th, d is the downstream thermal speed of the solar wind Maxwellian, and th, d
sw
downstream thermal speed of the pickup ion Maxwellian. To specify ,
have to apply the results from the multicomponent Rankine-Hugoniot model.
sw
First, we calculate the upstream solar wind thermal speed v th, u from
v is the
PUI
v th d and th, d
v , we

β
sw
104
2kT 2kT
n k T
= =
m
=
m
=
B sw B sw
sw B
2
B
sw i
2
B
i
2
B
sw 2
(1 −φ
)( vth,
u )
, 2
vA
2 μ μ n m μ (1 −φ)
n m
0 0 sw i 0
u i
(3.34)
where φ , as defined before, is the pickup ion relative density. Equation (3.34) can be
rewritten as
β
v v
(1 −φ
)
sw sw 1/2
th, u = [ ] ⋅ A.
PUI
Similarly, the upstream pickup ion effective thermal speed v , is
Then the downstream pickup ion thermal speed is
th u
(3.35)
PUI βPUI
1/2
vth, u = ( ) vA.
(3.36)
φ
γ PUI −1
PUI PUI 2
th, d th, u S ,
v = v ⋅ r
(3.37)
where we apply Equation (3.4). Equations (3.36) and (3.37) together specify
For the downstream solar wind thermal speed,
γ PUI −1
PUI βPUI
1/2 2
vth, d = ( ) ⋅rS⋅ vA.
(3.38)
φ

105
v = [ ε ( v ) + (1 −ε
)( v ) ]
sw sw ref 2 sw trans 2 1/2
th, d ref th, d ref th, d
ref
2 2
M AvA ref
sw
vth, u
2 1 1/2
rS γ
− −
= ε + −ε
−
[ (2 ) (1 )( ) ] ,
(3.39)
where we apply Equation (3.17) and (3.4). Equations (3.36) and (3.37) together specify
β
v ε M ε r v
(1 −φ
)
sw 2 sw γ −1
1/2
th, d = [2 ref ⋅ A + (1 − ref ) ⋅ ⋅ S ] A.
(3.40)
In Equation (3.38) and Equation (3.40), the compression ratio rS and the solar wind
reflection efficiencyεref can be obtained analytically from Equations (3.11) and (3.23).
The pickup ion polytropic index γ PUI is set to be 2.48 as done before. The other
polytropic index γ is varied from 5/3 to 2.2 with an intermediate value γ =1.95 which
corresponds to the 20% pickup ion simulation (Section 3.2.2). With given upstream
parameters β sw , β PUI , and MA, the downstream thermal velocities of the two
sw
Maxwellians ( ,
PUI
v th d and th, d
v ) can be analytically solved as functions of the pickup ion
relative density and the polytropic index γ . Figure 3-5 shows such solutions of the two
downstream thermal speeds for the upstream parameters: β sw =0.05, β PUI =8.53, and
MA=8. One should be very careful in applying this figure. As stated, the assumptions of
this section’s derivation only apply when the pickup ion relative density is sufficiently
large that the reflected solar wind ions are a very small percentage of total density.
Figure 3-5 shows that as the pickup ion relative density φ increases, both the
downstream solar wind thermal speed (top panel) and the downstream pickup ion thermal

106
Figure 3-5 Downstream solar wind (top panel) and pickup ion (bottom panel) thermal
speeds as functions of the pickup ion relative density and the polytropic index γ for a
MA=8, sw
β =0.05 perpendicular termination shock. The vertical line marks the 20% shell
distributed pickup ion case, in which the γ =1.95 solutions predict that the downstream
solar wind thermal speed is 1.93 vA and the downstream pickup ion thermal speed is
12.20 vA, consistent with the simulation fitted thermal speeds (Chapter 4).

107
speed (bottom panel) decrease, in response to the weakening of the shock. The decreasing
of the solar wind thermal velocity is so fast that the γ =5/3 solution is incapable of
capturing the right physics at φ >11% (consistent with Section 3.2 result). The γ =2.2
solution of the downstream solar wind thermal speed is very different from the γ =5/3
solution. It is then necessary to find the exact value of γ (which is 1.95) for the 20%
pickup ion case that we are particularly interested in. The γ =1.95 solution (dotted
curves) indicates that with 20% pickup ions (marked by the grey vertical line), the solar
wind thermal speed is 1.93 vA and the pickup ion thermal speed is 12.20 vA, which imply
that downstream the multi-ion speed distribution for the 20% shell distributed pickup ion
case is
2 2
v⊥ v⊥
f( v⊥ , 20%, M A = 8) = 0.54nd exp( − ) + 0.0027n exp( ).
2 d − (3.41)
2
1.93 12.20
where nd is the downstream density, the normalization factors 0.54 and 0.0027 are
derived from the integration of the distribution functions and the relative densities (0.8
0.8
and 0.2) of the two Maxwellian components: = 0.54
2
v⊥ exp( − ) 2
1.93
0.2
= 0.0027 . This relationship will be tested in Chapter 4 in simulations.
2
v⊥ exp( − ) 2
12.20
In addition, the downstream pickup ion thermal speed (bottom panel of Figure 3-5) is
not as a sensitive function of γ as the downstream solar wind thermal speed. The reason
and

108
is that the gyrophase-dependent acceleration mechanism [Winske et al., 2009] for pickup
ions is not a sensitive function of γ .
3.3.3 Refined Downstream Two-Maxwellian Speed Distribution
In the previous section, we make the assumption to characterize the downstream into a
solar wind Maxwellian and a pickup ion Maxwellian. However, as the simulations show,
the reflected solar wind ions have gained slightly more energy than those pickup ions
which gain no energy (as shown in the last panel of Figure 2-14). A better description of
the downstream components would be to include the low energy pickup ions into the
relatively cold Maxwellian (which we refer to as Maxwellian-1) and to include the
reflected solar wind ions into the relatively hot Maxwellian (which we refer to as
Maxwellian-2). This method, again, only applies when the pickup ion relative density is
large enough so that there are sufficient low energy pickup ions to thermalize fully with
the transmitted solar wind ions.
Using this prescription with Equations (3.37) and (3.39), the thermal speeds for
Maxwellian-1 and Maxwellian-2 are
and
γ PUI −1
PUI 2
2 2
th d th u S εrefφ εref φ
v 1, = v , ⋅r − ⋅(1 − ) ⋅ M + ⋅(1 − ) ⋅ 2M
(3.42)
A A
v [ ( M v ) (1 )( v ) r ] ,
γ
= ε + − ε
(3.43)
2 2 sw 2 −1
1/2
th2, d ref A A ref th, u S

109
respectively. For the same parameters as used in the previous section, the modified
and
Max2
th, d
v
Max1
th, d
v are plotted in Figure 3-6. This modified method of decomposing the
Figure 3-6 Downstream Maxwellian-1 (top panel) and Maxwellian-2 (bottom panel)
thermals speeds as functions of the shell distributed pickup ion relative density and the
polytropic index γ for a MA=8, sw
β =0.05 perpendicular termination shock. For the 20%
pickup ion case, the γ =1.95 solutions predict that the downstream Maxwellian-1 thermal
speed is 1.39 vA and the downstream Maxwellian-2 thermal speed is 12.26 vA, consistent
with the simulation fitted thermal speeds (Chapter 4).

110
distribution into two Maxwellian components reduces the temperature of the relatively
cold Maxwellian notably and increases the temperature of the relatively hot Maxwellian
slightly. The method also increases the normalization factor of the relatively cold
Maxwellian. So the downstream distribution for the MA=8, φ =20% shell distributed
pickup ion case is predicted to be
2 2
v⊥ v⊥
f( v⊥ ,20%, MA = 8) = 0.83nd exp( − ) + 0.0027n exp( ).
2 d − (3.44)
2
1.39 12.26
This relation will also be tested with the simulations, as compared to the relation in
Equation (3.41).
3.4 Discussion
Our Multi-Component Rankine–Hugoniot model is based on the Rankine–Hugoniot jump
conditions, a polytropic energy equation, and specular reflection (for a few solar wind
ions). The model computes the relative energy gain at a perpendicular shock for three
proton components: transmitted solar wind ions, reflected solar wind ions, and pickup
ions. Applying fitting parameters derived from our hybrid simulations to the polytropic
energy equation, we can characterize the energy gains for transmitted solar wind ions and
pickup ions. The reflected solar wind ion energy gain is characterized by a different
parameter (which represents specular reflection) derived from the 0% pickup ion
simulation. The model results are in good agreement with our simulations and are
consistent with the plasma observations made by Voyager 2 at the termination shock [Wu
et al., 2009].

111
Our model and our simulations predict that the pickup ion relative density at the
Voyager 2 crossing of the termination shock is about 25%, similar to the value of ~ 20%
inferred from the Voyager data by Richardson [2008]. Comparison between the analytic
results and the simulations shows that, as pickup ion relative density increases, the
polytropic index increases. As the pickup ion relative density approaches 40% (from 0%),
the polytropic index approaches 2.2 (from 5/3). The reason is that the pickup ions widen
the shock transition region (as shown in Chapter 2 Figure 2-10). A wider shock has a
longer compression region and thus enhanced heating. This increased heating,
represented by γ >5/3 and γ PUI >5/3, reduces the shock strength as can be sensed from
the polytropic equation that the temperature jumps scale with
1
rS γ − . When γ increases, rS
is allowed to “relax” without having to sacrifice too much on the temperature jump. For
example, the strongest shock with γ =5/3 has a temperature jump of 4 5/3-1 =2.52 for
transmitted solar wind ions. To achieve the same temperature jump at γ =2.2, the shock
strength only has to be rS=2.2 (2.2 2.2-1 =2.52). Essentially this change of shock
characteristic is an unavoidable result of the Winske et al [2009]’s GPDA mechanism. It
is the GPDA gyrating motion of pickup ion that widens the shock transition region (e.g.,
the foot of the shock is determined by pickup ion gyroradius) and relaxes the shock to a
state that the compression ratio can be lowered to a value ~2.
The simulations and the model further show that the relative energy gains of the two
upstream components are sensitive functions of the relative pickup ion density. Only
when the pickup ion relative density is less than ~ 5% does more energy go into the solar

112
wind reflection than the pickup ion heating; in this case, the solar wind component
dominates the shock dissipation [Liewer et al., 1993]. However, for larger pickup ion
relative densities, the energy gain of pickup ions becomes much greater than that of the
solar wind ions, which is the case of the termination shock. Our results support the
inference by Richardson [2008] that most ( ~ 90%) of the dissipation is caused by pickup
ions.
Further, our multicomponent Rankine–Hugoniot model allows the downstream Mach
number to be formulated analytically. We find that downstream of the shock the flow is
still super-Alfvénic; however, it is subsonic and sub-Magnetosonic. There is no mystery
to the observation that the downstream appears to be super-magnetosonic, because the
observation inferred magnetosonic Mach number excludes pickup ions and possibly
reflected solar wind ions. We infer that instead of a strong Alfvénic shock such as the
familiar terrestrial bow shock, the presence of the energetic pickup ions raises the
magnetosonic wave speed so that the termination shock has the dual identity of a
Alfvénic shock and a gas kinetic shock [Wu et al., 2009].
Using the multicomponent Rankine–Hugoniot model and the general notion that the
downstream distributions are better described by the superposition of at least two
Maxwellians, we formulate the analytic two-Maxwellian speed distribution for the
downstream. The two components approximately correspond to heated solar wind ions
and heated pickup ions, assuming that each ion population is fully thermalized within
itself by the shock (as will be tested by simulations in the next Chapter). The thermal

113
speeds of the two groups are described as functions of the pickup ion relative density and
the polytropic index γ . It is found that the downstream solar wind thermal velocity has a
stronger dependence on γ while the downstream pickup ion thermal velocity is weakly
dependent on γ . The reason lies in the pickup ion energization mechanism proposed by
Winske et al. [2009] (Chapter 2). Because the solar wind ions are energized through the
combination of specular reflection and transmission, and because specular reflection
(which allows significantly more energy gain than the simple transmission) strongly
depends on γ (Figure 3-2), a small change in γ greatly alters the downstream thermal
velocity. For the pickup ions, the mechanism is rather different. The gyrophase-
dependent acceleration (Winske et al. [2009]) does not show significant dependence on
either γ , or γ PUI .
As reflected solar wind ions have a speed that is slightly higher than un-energized
pickup ions, the real situation is more complex. A refined two-Maxwellian speed
distribution is formulated to account for the effect. The relatively cold Maxwellian is
assumed to be a mixture of transmitted solar wind ions and low energy pickup ions. Both
the two-Maxwellian model and the refined version only apply to situations where the
pickup ion relative density is large (>=20%). When the pickup ion relative density is very
small (

114
observation and simulation. Our analytic approximation will still hold for the range of
parameters we are concerned with.

4 Hybrid Simulations for a More Realistic Termination Shock
Our simulations of an idealized termination shock (Chapter 2) use most of Voyager 2
observed upstream values as inputs, but assume a perpendicular shock and vary pickup
ion relative density to gain physical insight into the ion energization processes and the
energy partition. In this chapter’s simulations, the pickup ion relative density will be
fixed at 20%. First, we vary the upstream pickup ion velocity distribution to understand
the physical consequences of such variations. Second, we vary the upstream Alfvén Mach
number to investigate the associated downstream ion energy dependence. Third, we will
vary the shock normal in our simulations to span the observed values, thereby
investigating the physics of quasi-perpendicular shocks.
4.1 Variation of Pickup Ion Velocity Distributions
The pickup ions initially form a ring velocity distribution initially by virtue of the pickup
process. Subsequent scattering produces broader velocity distributions such as the
Vasyliunas–Siscoe distribution. In the simulations described here, we assume that the
upstream pickup ion distributions may be presented as a velocity shell, a velocity sphere,
a Vasyliunas–Siscoe distribution and a Maxwellian distribution, corresponding to
successive levels of thermalization the pickup ions go through as they flow away from
the Sun. For each of these simulations, we demonstrate that scattering near the shock is
strong enough to yield almost identical distributions downstream.

116
4.1.1 Analytic Derivation of Pickup ion Beta and Speed Range
This section provides an analytic description of the assumed velocity specifications for
the hybrid simulations of a more realistic termination shock. We assume that pickup ions
gain an initial gyrovelocity that is equal to the solar wind upstream speed uu. Hence,
newly born pickup ions form a ring-shaped velocity distribution with an average flow
speed equal to that of the solar wind. As they flow outward toward the termination shock,
the pickup ions are scattered by both particle-particle and wave-particle interactions,
broadening this distribution in both pitch-angle and in speed. The upstream pickup ion
distribution also depends on the interstellar neutral ion density, the charge-exchange rate,
and the ionization rate along the ion trajectories as they flow away from the Sun
[Vasyliunas and Siscoe, 1976].
Here we consider five distinct velocity distributions corresponding to successive level
of scattering which follow the initial pickup:
• A ring distribution with perpendicular speed equal to the upstream flow speed,
• A shell distribution with speed equal to the upstream flow speed;
• A velocity sphere with maximum speed determined by the pickup ion beta;
• The Vasyliunas–Siscoe distribution with maximum speed determined by the
pickup ion beta;
• A fully thermalized Maxwellian distribution.

117
For each of these upstream distributions, we derive and calculate the effective pickup ion
beta,
ring
β PUI ,
shell
β PUI ,
sphere
β PUI ,
β , and Max
β , respectively. Here VS denotes the Vasyliunas–
VS
PUI
Siscoe distribution and “Max” denotes the Maxwellian.
PUI
We assume that for each distribution, the pickup ion relative density is φ and an
upstream Alfvén Mach number MA is maintained constant. For a pickup ion distribution
f(v), the corresponding β PUI in the solar wind frame is
β
PUI
φ
m
∫
∫
= = ⋅ ,
i nk u B
3kB 2
f( vvdv )
2φ
2
f( vvdv )
2
Bu /(2 μ0)
f( v) dv
2
3vA
f( v) dv
∫ ∫
(4.1)
where kB is the Boltzmann constant; μ0 is the permeability; nPUI,u is the upstream
pickup ion number density, nu is the total upstream number density; Bu is the upstream
magnetic field; mi is the pickup ion (here proton) mass.
ring
Starting with the ring distribution, f ( v , v ) = δ( v −u
) δ(
v )
PUI ⊥ ⊥ u ,
∞
2 2
δ( v uu ) δ(
v )( v v ) v dv dv
ring 2φ ∫ ⊥ − + ⊥ ⊥ ⊥
0
2φ 2 2 2
PUI = ⋅ = u 2 2 u = M
∞
A
3vA 3vA 3
∫δ(
v⊥ − uu ) δ(
v) v⊥dvdv⊥ 0
β φ
shell
For a shell distribution, f ( v) = δ ( v − u ) ,
PUI u
,
(4.2)

118
∞
4
δ ( v uu) v dv
shell 2φ ∫ −
0
2φ
2 ring
PUI = ⋅ = ⋅ u
2 2 u =
∞
PUI
3vA 2 3vA
∫δ
( v − uu) v dv
0
β β
(4.3)
For the other three distributions, our procedure is to choose a velocity cutoff or a
thermal speed so that the corresponding pickup ion beta matches those of Equations (4.2)
and (4.3). This will provide us with pickup ion distributions with the same upstream
kinetic energy in the simulations described in the next section.
sphere
sphere
For a velocity sphere, f ( v ) = 1 at 0 < v< v and zero otherwise,
β
PUI
sphere
vu
4
1 vdv
sphere 2φ ∫ ⋅
0 2φ 3 sphere 2
PUI = ⋅ ( ) ,
2 sphere = ⋅ v
v
2 u
3v u
A 3v 5
2
A
∫
0
1⋅vdv
In order for Equation (4.4) to be equal to ring
β PUI , the upper limit of the sphere has to be
u
(4.4)
sphere 5
vu = uu.
(4.5)
3
For a velocity distribution described by Vasyliunas and Siscoe [1976],
⎧ 3n
β r
λ
f v v v
2
PUI H 11
VS ( ) = ⎪
exp[ − ],0 ≤ 4 3/2 3/2
⎨8
πur
u TS( v/ uu) rTS( v/ uu)
⎪
VS
0, v> vu
≤
VS
u
⎩
(4.6)

119
where nH is the density of the interstellar neutrals (~0.1cm -3 ); β 1 is the ionization rate at 1
astronomical unit (AU); r1 is the length of 1 AU, rts is the location of the termination
shock (~94 AU); uu is the upstream solar wind speed (~300 km/s); λ is where neutrals get
PUI
fully ionized and fully picked up (~4 AU). The complete integration of f ( v ) in three
dimensions (v 2 λ uu
3/2
dv) involves the exponential integral function Ei
( − ( ) ) , where
r v
∞ −
t
e dt
Ei( x)
=−∫ (van Heemert, 1957). The exponential integral function is not
t
− x
PUI
analytically solvable. Notice that because the exponential term in f () v does not affect
f () v much when v is larger than 0.1uu, we simplify Equation (4.7) to
PUI
VS
3n
β r
f () v = ,0.01 ≤v≤ v .
(4.7)
PUI H
2
11
VS
VS 4 3/2
8 πur
u TS( v/ uu)
u
Therefore, the pickup ion beta can be analytically integrated,
β
VS VS
vu vu
PUI 4 5/2
∫ fVS () v ⋅v
dv ∫ v dv
VS φ 0.1u φ u 0.1uu PUI = ⋅ 2 VS = ⋅ 2 VS
v 3 u v
v 3 u
A v
PUI 2 A 1/2
∫ fVS () v ⋅v
dv ∫ v dv
φ
⋅ 2
3vA 7
VS
vu
2
0.1u 0.1u
2 2 2 3
( ) . (4.8)
u u
In order for Equation (4.8) to be equal to ring
β PUI , the upper limit of the Vasyliunas–Siscoe
distribution has to be
VS
TS
VS

120
VS 7
vu = uu.
(4.9)
3
v / vth
For a Maxwellian distribution, f () v ce −
= , where c is a constant. Hence,
β
Max
PUI
∞
∫
2 2
−v
/ vth
4
2 2
2φ = ⋅
e ⋅v ⋅dv
2φ 5
= ⋅ v ,
e ⋅v ⋅dv
Max
0
2
PUI 2
3vA ∞
2 2
−v
/ vth
2
2
3vA 2
th
In order for Equation (4.10) to be equal to ring
β PUI , vth has to satisfy
∫
0
(4.10)
2
vth = uu.
(4.11)
5
If we set φ =20% and MA=8, then β PUI =8.53 for all the above five pickup ion
distributions.
Below, we show the analytic solutions of the perpendicular distribution function
v
u
( fPUI ( v⊥) = ∫ f( v) ⋅dv
) for β PUI =8.53 pickup ions, applying the above prescriptions for
v
− u
four cases we will examine soon in the simulations (next section).

121
Figure 4-1 The distribution function fPUI ( v⊥ ) versus v⊥ for the pickup ion shell (black),
the pickup ion sphere (green), the Vasyliunas–Siscoe distributed pickup ions (orange),
and the Maxwellian pickup ions (red) at the same β PUI =8.53. The panel on the left is
ploted in linear-linear scale and the panel on the right is the same plot in linear-log scale.
The pickup ion’s perpendicular velocity vperp is normalized to the upstream velocity uu .
From Wu et al. [in preparation].
For the pickup ion shell case,
∞
∫ δ (
2 2
v + v⊥−uu) ⋅dv
PUI −shell ⊥
−∞ = ∞
2
∫ v ⋅δ( v−uu) ⋅dv
−∞
f ( v ) .
(4.12)
To simplify the analytic solution, we normalized all velocities to uu, the integration
becomes
∞
v
2
fPUI −shell ( v⊥) = ∫ δ ( v−1) ⋅ ⋅ dv=
.
(4.13)
v −v 1−v
2 2 2
−∞ ⊥ ⊥

122
As the black line (Figure 4-1) shows, the shell pickup ion distribution peaks at v ⊥ =uu,
drops sharply below uu, has a small value at smaller v ⊥ , and is zero when v ⊥ is larger
than uu.
For the pickup ion sphere
0
5/3
dv
fPUI −sphere( v⊥
) =
− 5/3
5/3
2
v ⋅ dv
= 18/5.
∫
∫
(4.14)
As the green line (Figure 4-1) shows, the sphere pickup ion distribution is flat and has a
wider range with the v ⊥ upper limit 5/3uu.
For the Vasyliunas–Siscoe distributed pickup ions,
f ( v ) =
PUI −VS ⊥
2
7/3
∫
0.1
2 2 3/2
⊥
7/3 2
0.1
1
( v + v )
∫
v
v
3/2
⋅dv
⋅dv
1 3 5 2 1 3 5
2
= 10.222 2F1[ , , , −100 v⊥] −2.615 2F1[ , , , −0.429
v⊥].
4 4 4 4 4 4
(4.15)
The integration of Equation (4.15) results in Gauss’s hypergeometric function 2F1, which
can be solved with the Mathematica program numerically. As the orange line (Figure 4-1)
shows, the Vasyliunas–Siscoe pickup ion distribution has a wider range than the pick up
ion sphere, with the v ⊥ upper limit 7/3uu.

For the Maxwellian pickup ions,
123
∞ 5 2 2
− ( v⊥+ v)
2
∫
e ⋅dv
5 2
− v⊥
2
−∞
fPUI −Max ( v⊥) = = 10 e .
∞ 5 2
− ( v )
2 2 v ⋅e ⋅dv
∫
−∞
(4.16)
As the red line (Figure 4-1) shows, the Maxwellian pickup ion distribution has the widest
range but as it drops off very fast it is not obvious at large v ⊥ .
4.1.2 Simulated Downstream Speed Distribution
The hybrid simulations described in this section utilize the isotropic shell, sphere,
uniform sphere and the Vasyliunas–Siscoe pickup ion distributions as well as the velocity
range discussed in the last section as input to study the ion speed distribution downstream
of the perpendicular termination shock. The ring pickup ion distribution is not simulated
here because the distribution is highly anisotropic and should be simulated using a three-
dimensional hybrid code as will be discussed in Chapter 7.
The Los Alamos hybrid code used here is the same as that used in Chapter 2 with the
ions represented as superparticles and the electrons described by a zero-mass adiabatic
fluid model with γ =5/3. The simulations are run on a system of length 400 ω / and
−1
800 cells in the x-direction; the integration time step is DT=0.02 Ω . The magnetic field
is in the z-direction; spatial variations are allowed only in the x-direction, but the full
three-dimensional variations of the fields and the superparticle velocities are included.
ci
pi c

124
The simulation parameters used here are similar to the ones described in Chapter 2. The
initial number of particles is 80,000 (about 100 particles per cell) for each ion component.
The resistivity is set to a small value:
−5
2 10 (4 π / ω pi )
× . We assume that all ions are
protons, and that the shock is perpendicular (shock normal θ Bn = 90 ° ).
The upstream Alfvén Mach number is set to be MA=8, the upstream pickup ion
relative density is set to be 20%; and the upstream solar wind velocity distribution is
assumed to be a Maxwellian with β sw =0.05 as inferred from Voyager 2 observations
[e.g., Richardson et al. 2008; Li et al., 2008]. The upstream β PUI pickup ion beta is set to
be 8.53 for all four of the upstream distributions used here.
The results of the simulations are listed in Table 4-1. Both the compression ratio rS
and the downstream bulk speeds are relatively independent of the upstream pickup ion
velocity distribution. The magnetic field profiles, as shown in Figure 4-2, illustrate that
the simulations show very similar shock profiles in magnitude and oscillations, also
independent of the upstream pickup ion distributions.

125
Table 4-1 Results Calculated from the Hybrid Simulations (MA=8, β sw =0.05, β PUI =8.53)
Pickup ion
velocity distribution
Shell Sphere Vasyliunas–Siscoe Maxwellian
compression ratio rS 2.47 2.38 2.38 2.39
ud/vA 2.93 2.97 3.02 3.04
nd,sw /nu 1.98 1.95 1.99 1.97
nd,PUI /nu 0.50 0.49 0.49 0.50
Figure 4-2 The magnetic field profile (Bz) of the four simulations. Black: the simulation
with a pickup ion velocity shell. Green: the simulation with a pickup ion velocity sphere.
Orange: the simulation with the Vasyliunas–Siscoe distributed pickup ions. Red: the
simulation with the Maxwellian distributed pickup ions. From Wu et al. [in preparation].

126
The perpendicular speed distributions (un-normalized) upstream and downstream of
the shock are plotted in their respective plasma frames, as shown in Figure 4-3. Both
distributions are displayed in the log scale and the linear scale for the x axis. For the y
axis, all the four panels are plotted in log scale. Upstream of the shock, the solar wind
forms a Maxwellian close to the y axis for all four simulations. The velocity shell (black)
have a speed peak at ~8 vA while the velocity sphere (green) forms a flat plateau beyond
the solar wind Maxwellian. Despite these differences, all four downstream ion speed
distributions have a two-Maxwellian character: a relatively cold Maxwellian (which we
refer to as Maxwellian-1) and a relatively hot Maxwellian (which we refer to as
Maxwellian-2). Maxwellian-1 resulted from all the four simulations are nearly identical,
while Maxwellian-2 differentiates slightly at the higher energy end for the four
simulations. Although the macroscopic electric fields and magnetic fields in a one-
dimensional simulation are unable to isotropize downstream ions, they strongly scatter
the ions into a downstream distribution that is relatively independent of the upstream
pickup ion distribution.

127
Figure 4-3 The upstream and the downstream speed distributions in their respective
plasma frames. Black: the simulation with a pickup ion shell. Green: the simulation with
a pickup ion sphere. Orange: the simulation with the Vasyliunas–Siscoe distributed
pickup ions. Red: the simulation with the Maxwellian distributed pickup ions. Upper left
panel: upstream speed distribution in log-log scale. Upper right panel: upstream speed
distribution in log-linear scale. Lower left panel: downstream speed distribution in loglog
scale. Lower right panel: downstream speed distribution in log-linear scale. From Wu
et al. [in preparation].

128
Table 4-2 Properties of the two downstream Maxwellians from the four simulations
(MA=8, β sw =0.05, β PUI =8.53)
Pickup ion
velocity distribution
Shell Sphere Vasyliunas– Siscoe Maxwellian
nmax1/nd 0.79 0.80 0.80 0.79
nmax2/nd 0.21 0.21 0.20 0.21
nmax1 /nu 1.95 1.94 1.99 1.96
nmax2 /nu 0.53 0.50 0.49 0.51
T 45.11 41.59 43.25 49.00
Tmax1 /
Tmax2 /
sw
u
T 2266.56 2394.77 2407.63 2252.26
sw
u
vth1 /vA 1.68 1.61 1.64 1.75
vth2 /vA 11.90 12.23 12.27 11.86
T 61.52 53.56 56.85 53.56
Tsw,d /
TPUI,d /
sw
th, d
sw
u
T 2239.18 2394.41 2353.45 2394.42
sw
u
v /vA 1.96 1.83 1.89 1.83
v /vA 11.83 12.23 12.13 12.23
PUI
th, d
Td/ T 497.05 521.73 516.17 521.73
sw
u
Note: As discussed in Chapter 2, we calculate the downstream temperature using only the perpendicular
velocity components for the two Maxwellians.

129
We divide the distributions into two Maxwellian components at '
v =4.4 vA. This
⊥
value of
'
v is empirically chosen such that the relatively cold Maxwellian has 20% of the
⊥
downstream density, and such that we can test Sections 3.3.2 and 3.3.3 predictions. The
relative densities of the two simulated Maxwellian components are listed in Table 2 for
the MA=8 simulations: nmax1 for Maxwellian-1 and nmax2 for Maxwellian-2, both in the
units of the downstream density nd and the upstream density nu. Compared with the
downstream solar wind and pickup ion densities nd,sw and nd,PUI in Table 4-1, we find that
nmax1 ≈ nd,sw and nmax2 ≈ nd,PUI as expected, suggesting that most of Maxwellian-1 ions are
heated solar wind ions and most of Maxwellian-2 ions are heated pickup ions. However,
an examination of the temperatures (or effective thermal speed) shows that although
Tmax2 ≈ TPUI,d (or vth2 ≈ ,
PUI
sw
v th d ) for each simulation, Tmax1 < Tsw, d (or vth1 < th, d
v ) notably.
This is because the relatively low energy pickup ions (as suggested by Section 3.3.3)
instead of the reflected solar wind ions (as suggested by Section 3.3.2) participate in the
thermalization process with the transmitted solar wind ions to form Maxwellian-1. The
reflected solar wind ions (which have more energy than the low energy pickup ions)
instead fall into Maxwellian-2. In addition, the four simulated speed distributions have
nearly the same effective downstream temperatures, as required by conservation of
energy. Given that Maxwellian-1 has nearly the same density as the solar wind ions, we
can apply section 3.3.3 result (refined two-Maxwellian) multiplied by a normalization
factor to fit the distributions (Figure 4-4). We also apply the two-Maxwellian result of

130
section 3.3.2 multiplied by a normalization factor to fit the distributions as a comparison
(Figure 4-5).
Section 3.3.4 fit Section 3.3.4 fit
Figure 4-4 The fitting (orange line) of the downstream distributions (in their respective
plasma frames) with the two maxwellian components as derived from Section 3.3.3. The
left panel is plotted in log-log scale and the right panel is plotted in the log-linear scale.
The dotted lines are downstream speed distributions from the simulations. Black dots: the
simulation with a pickup ion ring shell. Green: the simulation with a pickup ion sphere.
Orange: the simulation with the Vasyliunas–Siscoe distributed pickup ions. Red: the
simulation with the Maxwellian distributed pickup ions. Right panel: log-linear scale.
Left panel: log-log scale. The perpendicular speed vperp is in the unit of the upstream
Alfvén speed vA. From Wu et al. [in preparation].

Section 3.3.3 fit
131
Section 3.3.3 fit
Figure 4-5 The fitting of the downstream distributions (in their respective plasma frames)
with the two maxwellian components as derived from Section 3.3.2. The left panel is
plotted in log-log scale and the right panel is plotted in the log-linear scale. The dotted
lines are downstream speed distributions from the simulations. Black dots: the simulation
with a pickup ion ring shell. Green: the simulation with a pickup ion sphere. Orange: the
simulation with the Vasyliunas–Siscoe distributed pickup ions. Red: the simulation with
the Maxwellian distributed pickup ions. Right panel: log-linear scale. Left panel: log-log
scale. The perpendicular speed vperp is in the unit of the upstream Alfvén speed vA. From
Wu et al. [in preparation].
Table 4-3 The goodness (R) of the two-Maxwellian and the refined two-Maxwellian fits
for the four simulations (MA=8, β sw =0.05, β PUI =8.53)
Pickup ion
velocity distribution
Shell Sphere Vasyliunas– Siscoe Maxwellian
Refined two-Maxwellian 0.988 0.993 0.994 0.988
Two-Maxwellian 0.954 0.961 0.950 0.958

132
2
The goodnesses ( κ =R) of the fits to the simulations are listed in Table 4-3; the
refined two-Maxwellian formula fits the MA=8 and φ =20% simulations much better as
can be inferred from the goodness of the fits and the figures. The refined formula
describes the mixing region of the two Maxwellian components rather poorly, as can be
seen from the figure. This will be discussed at the end of this chapter’s discussion section.
4.2 Variation of Alfvén Mach Number
In this section, the upstream pickup ion velocity distribution is chosen to be a Vasyliunas-
Siscoe population with 20% relative density. The upstream Alfvén Mach number is
varied to investigate how this parameter changes the downstream ion properties.
Assuming that the upstream magnetic field variation is very small, the variation of the
upstream Alfvén Mach number corresponds to the variation of the upstream solar wind
speed as well as the variation of the upstream pickup ion beta (as implied by Equation
(4.2)). As shown in Figure 4-6, when the Alfvén Mach number increases, the speed
distribution broadens because the upstream flow has more energy.
The temperatures of the solar wind ions and pickup ions are listed in Table 4-4. As
the Alfvén Mach number ranges from MA=[6,16], the pickup ion effective thermal energy
falls within the range=[600 eV, 6 keV], the energy range of interest to IBEX. We will
return to these results in Chapter 6.

MA=4
MA=6
MA=8
MA=12
MA=16
133
Figure 4-6 The downstream ion speed distributions (in their respective plasma frames)
from the simulations: MA=4 (blue), MA=6 (green), MA=8 (black), MA=12 (orange) and
MA=16 (red). The left panel is in the log-log scale and the right panel is in the log-linear
scale. From Wu et al. [in preparation].
Table 4-4 The downstream temperatures of the simulations ( β sw =0.05, φ =20%
Vasyliunas-Siscoe distributed upstream pickup ions)
sw
Tsw,d/ T Tsw,d (eV) sw
u
TPUI,d/ T u
TPUI,d (keV)
MA=4 5.42 2.33 467.56 0.20
MA=6 20.54 8.83 1256.07 0.54
MA=8 56.85 24.45 2353.45 1.01
MA=12 196.20 84.366 5093.85 2.19
MA=16 460.38 197.96 8680.65 3.73
As calculated in Section 1.3, at the Voyager 2 observed average upstream conditions
(vA=38 km/s), the upstream solar wind temperature is about T ≅ 0.43 eV.
sw
u
MA=4
MA=6
MA=8
MA=12
MA=16

4.3 Variation of Shock Normals
134
Due to the Parker spiral (Chapter 1), the idealistic termination shock has been assumed to
have a shock normal of 90°. In reality, Voyager 2 observed termination shock normals (to
the magnetic field) to be 74° and 82°. It is safe to assume that the realistic quasi-
perpendicular termination shock near the ecliptic plane has a shock normal in the range of
[70°, 90°].
For the simulations in this section, we chose the upstream pickup ion distribution to
be of the Vasyliunas–Siscoe type with a relative density of 20%, the Alfvén Mach
number MA=8, β sw =0.05 and β PUI =0.05. The shock normal θBN is set to be 70°, 80°, 90°
in successive simulations. Figure 4-7 shows the magnetic field profiles of the simulations
after the same run time. The Bz components of the simulations have the similar
oscillating features. As the shock normal decreases, it appears that the shock propagates
slightly slowly. This is caused by the fact that it takes slightly longer simulation time to
form a self-sustaining shock when the shock normal decreases. Actually, the shock speed
increases slightly when the shock normal reduces. The shock is slightly weaker than the
strictly perpendicular shock. The magnitude of Bz also decreases slightly with decreasing
shock normal because Bz = Bu⋅ CosθBN.
The other transverse component By is found to
increase with decreasing shock normal. In the quasi-perpendicular shock, the non-zero
parallel component of magnetic field Bx introduces Ez uy x B = − × . This oscillating (uy

135
oscillates due to gyration) Ez gives rise to the oscillating By component, as required by
z By= E ∂
⋅ DT
∂x
. When θBN = 90°, Bx is zero and so is By, as seen in Chapter 2.
θBN = 70°
θBN = 80°
θBN = 90°
θBN = 70°
θBN = 80°
θBN = 90°
Figure 4-7 The magnetic field profile (Bz, Bx) of the three simulations with different
shock nomal. Black: the simulation with θBN = 90°. Orange: the simulation with θBN = 80°.
Blue: the simulation with θBN = 70°.

136
The downstream ion speed distributions, as shown in Figure 4-8, tend toward slightly
lower energy when the shock normal decreases. However, the change, again, is very
small, unlike the changes caused by the variation of Alfvén Mach numbers shown in the
previous section.
Figure 4-8 The downstream ion speed distributions (in their respective plasma frames)
from the three simulations: θBN = 90° (black), θBN = 80° (orange) and θBN = 70° (blue).
The left panel is in the log-log scale and the right panel is in the log-linear scale.
4.4 Discussion
θBN = 70°
θBN = 80°
θBN = 90°
θBN = 70°
θBN = 80°
θBN = 90°
We first carry out four different simulations with four distinct upstream pickup ion
distributions (shell, sphere, Vasyliunas–Siscoe, Maxwellian) with the same temperature.
The simulated compression ratio, the magnetic field profile and the speed profile all
indicate that all four shocks are nearly identical, despite the different prescription for the
upstream pickup ions. The pickup ions’ interaction with the electric and magnetic fields

137
inside the shock allows them to thermalize into nearly identical downstream speed
distributions, independent of the upstream distributions. This result is not unexpected,
given the pickup ion heating mechanism (GPDA) described in Section 2.6. Therefore, we
can not use the downstream information to infer the upstream pickup ion velocity
distribution.
The two-Maxwellian distribution and refined two-Maxwellian distribution provide
good fits downstream in the MA=8 simulations. Particularly, the refined formula is more
successful with an average goodness of fit equal to 0.99. However, the two-Maxwellian
fit describes the mixing region of the two Maxwellian components poorly. This implies
that an analytic formula derived from the simple assumption of two Maxwellian
components does not capture all the correct physics of the shock. Particularly, the
thermalization processes inside the shock mixes the three ion components (transmitted
solar wind ions, reflected solar wind ions, and pickup ions) in a non-linear manner. This
mixing should be a function of the shock strength, the upstream Mach numbers, the
pickup ion relative density etc. An analytic model is unlikely to capture the full
complexity of the processes. A rigorous multi-Maxwellian decomposition of the speed
distribution is unlikely to be useful for practical purposes. In data analysis, it is better to
fit the tail part of the distribution with a power law, as will be done in Chapter 6.
To summarize, this chapter emphasizes the construction of the various downstream
heated ion speed distributions from the more realistic simulations. The upstream pickup
ion velocity distributions, the upstream Alfvén Mach numbers, and the upstream shock

138
normals are varied independently to examine the relative importance of these three
factors. We found that the upstream pickup ion velocity distribution and the shock normal
have very little influence on the downstream speed distributions and the downstream
temperatures. In contrast, the upstream Alfvén Mach number affects the downstream
speed distribution and the downstream temperatures significantly. A large upstream
Alfvén Mach number implies a large ion energy input to the shock. The downstream
temperatures calculated from the MA = [6, 16] simulations fall within the range [600 eV –
6 keV], the range of interest to IBEX observations, as discussed later in Chapter 6.

5 Accuracy and Convergence
In Chapter 4, we studied the sensitivity of the simulation results to variations in physical
parameters; in this chapter, we study the sensitivity of the simulation results to variations
in numerical parameters. Therefore, the goal here is to explore the validity of our
simulations. We vary the cell size and the number of super-particles per simulation cell to
investigate how these parameters affect the stability as well as the physical interpretations
of the simulations.
5.1 Cell Size
A van Neumann stability analysis shows that a simulation is stable if it satisfies the
Courant condition (also called the Courant–Friedrichs–Levy condition)
v⋅ DT < H
(5.1)
where v is the phase speed of the fastest wave in the system, DT is the computational
time step and H is the cell size. This relation means that the cell size must be large
enough so no signal can travel beyond one cell within one computational time step DT. If
this condition is not satisfied, causality is violated: the source terms in the cell change
faster than the physics such that the simulation either crashes or produces unphysical
results. For a hybrid simulation, we further require that the fastest ion should not move
more than one cell length in one time step. Actually it is better if the fastest ion does not
move more than ½ of a one cell length in one time step. This is because the
electromagnetic forces that the ion experiences are interpolated from 2 adjacent grid

140
points. At the same time, the hybrid simulation time step is limited by the condition that
the ion gyro-motion must be well resolved. This means that DT should be small
compared with the inverse ion cyclotron frequency. A typical rule of thumb is to set
DT0.1 c/ω pi [Winske and Omidi, 1993].
Even when the Courant condition is satisfied, the cell size still needs to be carefully
chosen. If the cell size is too big, the resolution is too low to resolve the physical
processes. If the cell size is too small, DT has to be reduced to satisfied Courant condition,
and the total number of super-particles has to be increased to keep the same thermal noise
level (discussed in the next section). Both changes result in longer run times. In addition,
numerical whistler waves can be introduced. Recall that the simulations are zero electron
mass simulations. Under such an approximation, whistler waves have a strong dispersion
ω ~ k 2 at short wavelengths. These fast high frequency whistler waves are artificial in a
sense that when the electron mass is taken into account, as is the case in reality, the
speeds of high frequency whistler waves are reduced. To suppress the artificial fast
whistler waves in zero electron mass models such as the hybrid simulations, a medium
cell size is preferred, as can be seen in the following examples.
5.1.1 Cases without Pickup Ions
To study the effect of simulation cell size, we first show results from simulations without
pickup ions. The shock is therefore similar to the Earth’s bow shock, with structure that
has been observationally verified. Three simulations with different cell sizes are studied:
a) H= c/ω pi , b) H=0.5 c/ω pi (the cell size generally used in other chapters of this

141
dissertation), c) H=0.25 c/ω pi . For case c), the time step DT is also reduced. The physical
input parameters are all the same for the three cases: MA=8, θBN=90°, and β sw =0.05. This
is evidently a supercritical shock that should yield a compression ratio of 4.
As shown in the upper panel of Figure 5-1, a coarse spatial resolution of c/ω pi gives
an incorrect downstream magnetic field profile. There is no clearly defined overshoot,
because some of the downstream peaks are larger than the first peak. This simulation
does not represent the downstream situation well. The compression ratio of this case
fluctuates and is much larger than 4 (Figure 5-2 blue curve); this is unphysical because
the maximum compression value for a supercritical perpendicular shock is rS=4 if γ =5/3
[Burgess, 1995].
When the cell size is reduced to 0.5 c/ω pi (middle panel of Figure 5-1), a steady
shock forms. There is an overshoot at the shock front and there are downstream
oscillations, in agreement with typical bow shock observations [e.g., Sckopke et al., 1983,
Leroy et al., 1982]. The compression ratio is ~4 (Figure 5-2 black curve) as expected. If
the cell size is further reduced to 0.25 c/ω pi (bottom panel of Figure 5-1), the
downstream oscillation near the overshoot is very regular, which is not typically observed.
In addition, the compression ratio is a bit small (Figure 5-2 red curve), because of more
time variation of the shock structure that gives rise to slightly more heating. At even
smaller cell size, artificial fast whistler waves set in. These waves increase heating and
reduce the compression ratio even further. Our choice is either to adapt the medium cell

142
size or to increase resistivity to suppress these unwanted numerical waves. We have
chosen the former in the thesis, because it is better not to include the unphysical waves
and artificially large resistivity for compensation.
a)
b)
c)
Figure 5-1 Magnetic field profiles of the three simulations without pickup ions: a) H=
c/ω pi (top panel), b) H=0.5 c/ω pi (middle panel), and c) H=0.25 c/ω pi (bottom panel).

143
Figure 5-2 The compression ratios as a function of time calculated from the average
density of a 50 c/ω pi wide region that is 10 c/ω pi away from the overshoot in the
downstream. The three simulations are presented: a) H= c/ω pi (blue), b) H=0.5 c/ω pi
(black), and c) H=0.25 c/ω pi ( red curve).
5.1.2 Cases with Pickup Ions
In this section, shell distributed pickup ions with 20% relative density are added to the
simulations described in the previous section: a). H= c/ω pi , b). H= 0.5 c/ω pi (our
choice), and c). H= 0.25 c/ω pi . We add case c’) (H= 0.25 c/ω pi with higher resistivity)
to show that a higher resistivity helps to suppress some numerical waves. The physical
input parameters are all the same for the four cases: MA=8, θBN=90°, β sw =0.05, and
β PUI =8.32.

a)
b)
c)
c’)
144
Resistivity
Resistivity
Resistivity
Resistivity
η = ×
π ω
−5
2 10 (4 / pi )
η = × π ω
−5
2 10 (4 / pi )
η = × π ω
−5
2 10 (4 / pi )
η = π ω
−4
10 (4 / pi )
Figure 5-3 Magnetic field profiles of the three simulations at the same resistivity but
different cell sizes: a) H= c/ω pi , b) H=0.5 c/ω pi , and c) H=0.25 c/ω pi . Panel c) shows
an unphysical fast moving shock caused by the small cell size. This effect can be
eliminated with a larger resistivity as shown in Panel c’), where the cell size is as small as
Panel c) but the resistivity is 5 times bigger the previous 3 panels.

145
Figure 5-3 illustrates the magnetic field profiles of the three runs at the same real time
(50 Ωi -1 ) when the shock is well formed. For the coarse cell size case (H= c/ω pi ) in Panel
a), the downstream is not well resolved, as is the case a) in the previous section. For the
medium cell size case (H=0.5 c/ω pi , our choice in this work) in Panel b), the supercritical
shock features are well presented. For the smaller cell size case (H=0.25 c/ω pi ) in Panel
c), the magnitude of the overshoot is too large and the downstream oscillation is too
regular, even after a few pickup ion gyrating periods. (Here, the oscillation is associated
with pickup ions, as discussed in Chapter 2.) Most disturbingly, the shock propagates too
fast toward upstream, which indicates that there are unphysical (numerical) waves that
over-heat the ions. This shock speed can be corrected if we increase the resistivity as
shown in Panel c’). Still, the downstream oscillation is too regular and unlikely to be real.
The compression ratios of the four simulations are plotted in Figure 5-4. Case a) and
case b) have the same overall compression ratios. Case c) (dashed red curve) predicts a
compression ratio that is too small. However, with a larger resistivity, case c’) (solid red
curve) achieves the correct compression ratio as the other cases. In terms of resolution,
the noise is smaller when we reduce the cell size.

146
Figure 5-4 The compression ratios as a function of time calculated from the average
density of a 50 c/ω pi wide region that is 10 c/ω pi away from the overshoot in the
downstream. The four simulations are presented: a) H= c/ω pi (blue), b) H=0.5 c/ω pi
(black), c) H=0.25 c/ω pi (dashed red curve) and c’) H=0.25 c/ω pi with higher resistivity
−4
(red curve) η = 10 (4 π / ω ) .
5.2 Super-Particles per Cell
pi
The number of ions per simulation cell is varied in the section. For a simulation, the
thermal noise is proportional to 1/ N cell , where Ncell is the number of super-particles per
cell. Ideally if we increase Ncell, the thermal noise will decrease and the result will be
more accurate. The tradeoff is that the computing time will increase significantly. For this
thesis, we adopt a value of Ncell=100, as typically chosen in most codes that apply the PIC
techniques.
Black: H=0.5 c/ω pi
Blue: H= c/ω pi
Dashed Red: H=0.25 c/ω pi
Solid Red: H=0.25 c/ω pi with higher resistivity
η =
π ω
−4
10 (4 / pi )

147
We compare our simulations to simulations with Ncell=25, and Ncell=400. This small
variation of noise level (a factor of 1/2 and of 2) hardly affects the results or the overall
conclusions for the shock simulation. The shock speed and the compression ratios are
nearly identical.
Figure 5-5 show the magnetic field profile of the simulations. In the upstream, it is
apparent that with more particles per cell, the simulation is less noisy. The magnitudes of
the overshoots from the three simulations are very similar. However, for the Ncell=25
simulation, the overshoot bifurcates and is not well resolved. This bifurcation is reduced
when the particles per cell is increased to 100. When there are 400 particles per cell, the
bifurcation disappear and there is a well resolved supercritical shock structure. The
Ncell=100 solution, although not as perfect as the Ncell=400 simulation, is still appropriate
for our problem, because it preserves the physics without requiring a longer run time the
Ncell=400 simulation demands. In fact, even the 25 particles per cell simulation gives
correct shock jump condition, despite the noisy magnetic field profile. Recall that the
simulated super-particles do not correspond to real particles on a one-on-one basis. In fact,
all published hybrid or PIC simulations contain orders of magnitude fewer particles than
are found in real plasmas. What is preserved in those simulations is the general character
of the plasma behavior, the overall physics and the scaling of the systems. Simulations
with realistic particle density will not be available any time in the near future.

148
Figure 5-5 Magnetic field profiles of three simulations with different numbers of particles
per cell. Top panel: simulation with 25 particles per cell; middle panel: simulation with
100 particles per cell; bottom panel: simulation with 400 particles per cell.

5.3 Discussion
149
In this chapter, we have demonstrated that the choice of cell size is crucial for the proper
representation of shock physics. If the cell size is too big, the simulation resolution is
inadequate and the small scale physics is not properly represented. However, if the cell
size is too small, unwanted numerical errors increase. The ideal cell size should optimize
the system, demonstrate the important physical processes, and produce results that are in
agreement with observations. The other issue with small cell sizes is that the total amount
of cells in the simulation becomes very large, thus increasing the simulation time. Often,
we also need to reduce the time step. The larger number of particles and smaller time
steps result in very time-consuming and costly simulations. The important issue is that in
the zero electron mass hybrid simulations, whistler waves become unphysically fast if the
cell size is too small. Increasing the resistivity tends to suppress short wavelength modes
and makes the downstream look much too regular. It’s still a solution of the physical
equations, but may not look as “real”. The cell size we chose for this thesis is
H=0.5 c/ω pi .
Unlike the cell size, we find that the number of super-particles per cell Ncell does not
significantly affect the physical results, e.g., the compression ratio stays unchanged.
However, the detail magnetic field structure is less noisy when there are more particles
per cell due to the decrease of thermal noise. A typical thumb of rule is to choose 100
particles per cell, as we did in this work.

150
Numerical algorithms are also directly related to the accuracy and convergence of the
simulation. Particularly, the solver for magnetic field has gone through several
generations of change. Originally, in the early 1980’s, the magnetic field solver was
implicit [Winske and Leroy, 1984]. One solves the inverse matrix for the vector potential
at a time step. This method is not as accurate as explicit methods, which are more
commonly used nowadays. The fourth order Runge–Kutta method is a fairly accurate (to
the fourth order) explicit method. However, this method lacks convergence at large time
steps. Winske and Quest [1988] presented a new method for solving the magnetic field in
perpendicular shock simulations. This method divides the simulation time step DT into
several sub time steps DDT=DT/Ncycle. Here Ncycle is the number of sub-cycles within a
time step. Within each of these sub-cycles, the magnetic field is calculated using the
fourth order Runge–Kutta algorithm. This method has achieved much success as a robust
solver and is applied in our simulations. Of course, there is still room for further
numerical improvement. One may try different weighting algorithms for interpolating the
source terms, or include several smoothing cycles using the Hamming filter [Hamming,
1977].
In addition, we have checked conservation laws (the Rankine–Hugoniot relations, as
discussed in detail in Chapter 3) to ensure the validity of the code before analyzing the
simulations. Finally, we compute the entropy of the simulation system to make a
connection with gas-dynamic and MHD shocks. Take the 20% pickup ion (Vasyliunas–

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Siscoe distribution) simulation with H=0.5 c/ω pi and 100 particles per cell. The values of
entropy are calculated in each cell using
S =−kB∫ dv⊥⋅v⊥f( v⊥)ln f( v⊥)
(5.2)
for all ions at the end of the simulation. That is, we computer an entropy value every half
ion inertial length for ~100 ions. In calculating each entropy value, the velocity bin is
0.125 vA wide and apart. A different bin size will not change the overall result much but
will alter the details of the values. The entropy values are very noisy, calculated from
binning only ~100 ions in each cell. Therefore, we apply the Hamming filter [Hamming,
1977] once to smooth the values.
As shown in Figure 5-6, entropy increases across the shock. There are localized
decrease due to fluctuactions, similar to the local decrease shown in Gombosi [1994].
This does not violate the second law of thermaldynamics, because the system’s total
entropy still increases with time as ions move from the low average entropy upstream
region to the high average entropy downstream region. Physically, this is what we
expected because the shock crossing is an irreversible process that thermalizes these ions.
The thermalization turns the ion’s distributions from the relatively cold distributions
(upstream) into the relatively hot distributions (downstream). A cold distribution is
perfectly ordered; all particles have similar velocities. As temperature increases, there is
a greater thermal speed with greater thermal spread, which corresponds to more disorder
and larger entropy. Put simply, the number of microstates increases across the shock.

152
Notice that if the particle distribution is a single Maxwellian, Equation (5.2) becomes
S =−k dv ⋅v
f ( v )ln f ( v )
Max B ⊥ ⊥ Max ⊥ Max ⊥
2 2 2 2
⊥/ th ⊥/
th
∝−k dv ⋅v
e ln e
B
B
v
= k dv ⋅v e ⋅
∝ T
∫
Max
∫
∫
.
−v v −v
v
⊥ ⊥
2
⊥ ⊥
2 2
−v⊥/ vth
⊥
2
vth
(5.3)
Figure 5-6 Entropy (black) of the system at the end of the simulation run: 20% pickup ion
shell, MA=8,
overplotted.
β sw =0.05, β PUI = 8.53. The temperature profile (blue) is scaled and

153
Equation (5.3) says that the entropy of a Maxwellian distribution is proportional to its
temperature. In Figure 5-6, the temperature (blue) calculated from each simulation cell is
re-scaled such that the average temperatures upstream and downstream overlap the
average entropies upstream and downstream, respectively. Like for the entropy, we also
smooth the temperature by applying the Hamming filter once. In the upstream, there are
80% of the ions (solar wind ions) in a Maxwellian that make little contribution to the
upstream temperature. The upstream temperature shape deviates from the upstream
entropy shape significantly, as the upstream temperature is dominant by the non-
Maxwellian pickup ions. Therefore, in the figure the upstream temperature is nearly flat
while the upstream entropy fluctuates. In contrast, the shape of the downstream
temperature shows more agreement with the shape of the downstream entropy. This is
due to Maxwellian-1 (discussed in Chapter 4) that includes both the transmitted solar
wind ions and some pickup ions.
I want to end this Chapter by quoting Charles K. Birdsall and A. Bruce Langdon who
have contributing immeasurably to the subject of numerical simulations of plasmas and
whom I greatly admire, “Practitioners must exercise a great deal of care, enough to obtain
the essence of the problem, but not so much as to inhibit achieving any result.”

6 Theory/Model Connection to Observations
6.1 Comparison with Voyager 2
In Chapter 2, we found agreement between Voyager observations and several simulation
results including the energy partition among several downstream ion components, and the
pickup ion relative density. This section further compares observed termination shock
properties with our simulated shock properties.
6.1.1 Structure of the Termination Shock
Figure 6-1 shows a magnetic field profile from a simulation with the following upstream
inputs that are consistent with Voyager TS-3 crossing: θBN=74°, MA=8.8 and pickup ion
relative density φ =20%. The pickup ion velocity distribution upstream follows the
Vasyliunas–Siscoe prescription, which is widely believed to be the pickup ions velocity
distribution at the termination shock [e.g., Schwadron, 1998]. In addition, our analysis in
Chapter 4 shows that the pickup ion velocity assumption does not affect the magnetic
field profile of the shock much. The choice of the Vasyliunas–Siscoe pickup ion for the
purpose of comparison with Voyager 2 observation is therefore a valid choice.
The structure (foot, ramp, overshoot, undershoot) of the termination shock observed
by Burlaga et al. [2008] is compared side by side with our simulated shock structure, as
shown in the figure. The two magnetic field profiles are superimposed in a way that 1. x-
axis: the ramps from both profiles overlap and the scale length of the feet of both profiles
overlap; 2. y-axis: the average upstream magnetic fields from both profiles overlap and
the average downstream magnetic field from both profile overlap.

Ramp
Foot
155
Overshoot
Undershoot
Figure 6-1 Comparison of our simulated magnetic field profile (orange) with the Voyager
observed TS-3 magnetic field profile (black). Voyager 2 observing time is label on the x
axis on the bottom and the simulation length is marked on the x axis on the top. Voyager
2 Data are provided by Leonard F. Burlaga.
Figure 6-1 demonstrates that the scaling of the simulation is very similar to the
scaling of the observation. In particular, the simulated ramp is about one ion inertial
length, just as Burlaga et al. [2008] conclude from Voyager 2 observations. The
simulated lengths of the foot is about 8.8 ion inertial length, that is, the pickup ion
gyroradius, also consistent with observations. The magnitudes and scale lengths of the
overshoot and the undershoot are also in good agreement with the observations. The
simulated downstream oscillation has similar magnitude as the observed oscillations.

156
However, the exact phases of the oscillations are not in sync. In addition, the magnitude
of the simulated foot is slightly smaller than the observed foot; the magnitude of the
simulated overshoot is slightly larger than the observed overshoot; and the simulated
overshoot bifurcates (note that as shown in Figure 5-5, the 400 particles per cell
simulation has a better resolved shock structure without bifurcation). These differences
are unavoidable results of comparing space plasma observations which have substantial
temporal and spatial fluctuations as well as variations in the upstream solar wind
conditions with a limited one-dimensional simulation with limited particles per cell (100
particles per cell). Nevertheless, the qualitative agreement between the two magnetic
field profiles provides convincing evidence that the simulation has captured the basic
properties (foot, ramp, overshoot, undershoot) of the observed supercritical shock.
6.1.2 Core Ions
In Chapters 3 and 4, we fit the downstream ions with two Maxwellian components for
theoretical purposes. In this section, for comparison with Voyager 2 observations, we fit
the downstream ions with a single Maxwellian and ignore the high velocity “tail” of the
distribution. As Bridge et al. [1977] originally presented, Voyager 2’s plasma instrument
measures from 10 eV to 5950 eV. However, the instrument’s minimum measurable flux
is above the flux of the suprathermal ions. Therefore, Voyager 2 can not distinguish
suprathermal ion flux from the background. In fitting a single Maxwellian to the
simulated downstream ions, we ignore the low flux suprathermal ions and bring out the
core ions measured by Voyager 2 measures.

157
Consider the 20% pickup ion simulation of a perpendicular shock with a Vasyliunas–
Siscoe distribution upstream, MA=8, β sw =0.05, and β PUI =8.53. Figure 6-2 shows the
single Maxwellian fit to the downstream ions in log and linear scales. The goodness of
this fit is R=0.995, despite the fact that we ignore the tenuous suprathermal part of the
distribution (as clearly evident in the log scale figure). The temperature of this fitted
Maxwellian is Tcore 19 Tu,sw. Similarly, if we fit the MA=6 simulation with a single
Maxwellian, the temperature is Tcore 10 Tu,sw. Both temperature jumps with respect to
the upstream solar wind temperature fall within Voyager 2 observed temperature jump
range ~10–20.
These results suggest that the Voyager observed downstream core ion is a thermalized
population that includes both the transmitted solar wind ions (whose temperature jump is
1
rS γ − ~2 as describe in Chapter 3) and some low energy pickup ions whose effective
temperature is more than 10 times of the upstream solar wind temperature. The mixing of
these two populations generate the core Maxwellian with a temperature that is ~10–20
times of the upstream solar wind temperature when MA= 6–8, consistent with Voyager 2
measurements.

( )
f v ⊥
10 4
1000
100
10
1
0.1
y = m1*exp(-x^ 2/1.1^ 2)
m1
Chisq
Value
3723.7
7.1089e+5
Error
24.835
NA
158
R 0.99506 NA
0.01
0.01 0.1 1 10 100
4000
3500
3000
2500
2000
1500
1000
500
y = m1*exp(-x^ 2/1.1^ 2)
m1
Chisq
Value
3723.7
7.1089e+5
Error
24.835
NA
R 0.99506 NA
0
0.01 0.1 1 10 100
Figure 6-2 The fitting of the simulated downstream distribution (MA=8, β sw =0.05, and
β PUI =8.53) with one maxwellian component. The left panel is plotted in log-log scale
and the right panel is plotted in the log-linear scale. The circles are downstream speed
distributions from the simulations. The lines are the fitted function. The perpendicular
speed vperp is in the unit of the upstream Alfvén speed vA.
6.2 Implication for IBEX
In this section, we present the x direction (r direction in the RTN coordinate system)
energy spectra of the sunward moving downstream ions (in the shock frame) and discuss
implications for IBEX.
v⊥ / vA
v⊥ / vA
Figure 6-3 shows the complete spectra of three simulations with MA=4, 8, 16. These
Mach numbers correspond to upstream solar wind speeds uu=128 km/s, 304 km/s and 608
km/s, respectively, assuming that the upstream Alfvén speed is 38km/s (Chapter 1). The
horizontal axis represents the x direction energy in the shock frame. The vertical axis
represents the counts of ions as a function of the x direction energy. As expected from the

159
discussion in Chapter 4, when the Alfvén Mach number increases, the spectra broaden
and more suprathermal ions are produced. Using the Alfvén speed observed by Voyager
2 vA=38 km/s (Chapter 1), IBEX-Hi detector’s energy range (~0.6–6 keV) [Funsten et al.,
2009] corresponds to vx 2 /vA 2 = [90, 804], as shown in the right panel. If we fit these
spectra with power law functions, the corresponding power law indexes are κ =5.89
(MA=4), κ =2.76 (MA=8) and κ =1.22 (MA=16). A higher Alfvén Mach number leads to
a gentler slope for the suprathermal ions. The reason is that a higher Mach number
implies a hotter upstream population (larger pickup ion speeds) as well as more strongly
heated downstream ions.
10 4
1000
100
10
1
1 10 100 1000 10 4
MA=4
MA=8
MA=16
2 2
v /vA
x
1000
100
10
y = 6.4518e+12 * x^(-5.8907) R= 0.99048
y = 8.7629e+7 * x^(-2.7604) R= 0.94592
y = 49504 * x^(-1.222) R= 0.97503
MA=4
MA=8
MA=16
1
10 100 1000
2 2
v /vA
x
Figure 6-3 The modeled x direction (r direction in the RTN coordinate system) energy
spectra of the sunward directing ions in the shock frame (left). Partial spectra of Figure
6-3 that fall into the IBEX-Hi detector limit (right). We fit the spectra with power law
functions (solid lines). Blue: MA=4, black: MA=8, and red: MA=16.

160
Ions heated at the termination shock that undergo charge exchange with interstellar
neutrals are converted into neutrals, which may subsequently be detected by IBEX. Our
result implies that for IBEX high energy detector observed spectra, the decreases in the
power law index scales with the increasing solar wind speeds. This scaling is consistent
with the new results returned from the IBEX measurements that the spectral index is
relatively large ( κ =1.95±0.09 at 1.7 keV and κ =1.91±0.07 at 2.7 keV) at low
heliospheric latitudes where the solar wind is relatively slow and relatively small
(κ =1.49±0.05 at 1.7 keV and κ =1.39±0.08 at 2.7 keV) at high heliospheric latitudes
where the solar wind is relatively fast [Funsten et al., 2009].
The predicted κ range of the downstream ions is however slightly larger than the
IBEX measured κ range of the energetic neutral atoms (ENAs), as McComas et al.
[2009]’s Figure 2 illustrated sky map of κ shows that κ has a range of approximately [1,
4]. A more complete comparison of IBEX observations and simulation results is beyond
the scope of this work. Recall that IBEX globally observes the line-of-sight integration of
the sunward propagating heliosheath neutrals. Heating processes beyond the termination
shock and into the heliosheath [e.g., Schwadron and McComas, 2006; Fisk and Gloeckler,
2006; Drake et al., 2009] as well as the dynamical properties of the heliopause must be
included. The deflection of flow in the heliosheath can be described by MHD models
[e.g., Opher et al., 2006; Pogorelov et al., 2008]. The charge exchange processes to
convert termination shock heated ions into ENAs can be modeled with a code developed
in Boston University (BU) [Prested et al., 2008, Schwadron et al., 2009]. In addition, the
BU model can take into account the ENA spectral’s energy shift as they are deflected and

161
by the Sun’s gravitational field and radiation pressure [Schwadron et al., 2009, Möbius et
al., 2009]. Nevertheless, our result provides physical insights for the scaling of IBEX
measurements, as well as some quantitative estimates of how much heating the
termination shock gives to the ions and how that amount of heating compares to the range
that IBEX measures.
Finally, it is worth mentioning that the newly returned IBEX measurements discover
a “ribbon” of higher flux ENAs across the global sky map, with spectral indexes not
significantly different from adjacent areas at the same heliospheric latitude [McComas et
al., 2009, Funsten et al., 2009, Schwadron et al., 2009, Fuselier et al., 2009]. This
suggests that the ribbon is caused by merely local confinement of higher density plasmas,
other than additional dynamic heating at the termination shock [McComas et al., 2009,
Funsten et al., 2009, Schwadron et al., 2009]. Many ideas are suggested to account for
the ribbon, among them there is the concept of a strong interstellar magnetic field that
distorts the heliopause and possibly the heliosheath [McComas et al., 2009, Funsten et al.,
2009, Schwadron et al., 2009]. Given the interstellar magnetic field is inferred to be more
important than the community has previously anticipated, it may be important to include
this magnetic field in future modeling efforts of the heliopause, which can also be done
with hybrid simulations.

7 Summary
7.1 General Overview
The study of ion kinetics at the helispheric termination shock is driven by scientific
curiosity and practical necessity. As the Voyager spacecraft return in situ measurements
from their crossings of the termination shock and traversal of the heliosheath, and as the
IBEX spacecraft starts to make remote measurements of this region, space scientists are
using theoretical studies and numerical simulations to interpret these observations and
increase our physical understanding of this most remote region of our solar system. This
dissertation describes our research using hybrid simulations to self-consistently model the
ion scattering and energization processes at the shock. We quantify ion heating, ion
energy dissipation and the ion speed distribution with the aid of analytic methods.
The organization of the thesis is summarized as follows. Chapter 1 describes the
motivation, summarizes what known and unknown about the termination shock in both
theory and observations, and sets the research background of the topic. Chapter 2 focuses
on understanding the fundamental ion energization processes at the termination shock,
simulating an idealized termination shock with varying pickup ion relative density.
Chapter 3 builds on the knowledge from Chapter 2, providing an analytic multi-
component Rankine–Hugoniot model to address the heating, energy partition and
velocity distributions. Chapter 4 examines the sensitivity of the resulting downstream ion
speed distribution to the upstream pickup ion velocity distribution, the Alfvén Mach
number, and the shock normal angle relative to the magnetic field. Chapter 5 investigates

163
the sensitivity of the simulations to simulation cell size and numbers of particles per cell.
Chapter 6 further validates the simulation model with direct comparison with Voyager
observations and discusses the implication for the IBEX mission.
It is important to clarify two important aspects of this work. First, we do not model
the foreshock region, which is on the scale of ~10 AU, nor do we model the inner
heliosheath, which is on the scale of ~30 AU. Both regions represent large scale
structures beyond the scope we consider. The foreshock involves instabilities that scatter
the upstream ion distributions and the heliosheath involves instabilities that isotropize the
downstream ions. Those physical processes are not addressed here. Second, we do not
model the energetic particles at energies greater than 10 keV or their acceleration
mechanisms. Instead, we address the acceleration of thermal (eV) and suprathermal (few
keV) ions within IBEX’s range of measurements, as discussed in Chapter 1 (end of
Section 1.1) and Chapter 6 (Section 6.2). The issue with energetic particles is future work
and should be carefully modeled with three-dimensional simulations in large simulation
domains.
7.2 Review of Scientific Context
To summarize the scientific content, we re-visit the goals of this dissertation (end of
Chapter 1), answering the following essential questions.
How do pickup ions modify the termination shock? Pickup ions, typically ~1000
times hotter than the solar wind ions, significantly increase the effective beta of the

164
upstream plasma. As a result, both the upstream sound speed and the fast wave speed are
increased with increasing pickup ion relative density. When 20% of the ions are pickup
ions, the upstream magnetosonic Mach number is reduced. The shock becomes much
weaker than the terrestrial bow shock. Still, the shock remains supercritical, as is clearly
evident in the simulations with the formation of the overshoot in the magnetic field
profile and through the solar wind ion specular reflection. The difference here is that the
foot of the shock is determined by the pickup ion gyroradius, as a result of the gyro-phase
dependent acceleration. This foot is notably broader than that resulted from solar wind
ion gyrations.
How much is each ion component heated across the termination shock? The
heating rate of the ions depends on the pickup ion relative density, which modifies the
compression ratio (strength) of the shock. The heating is quantified by our analytic multi-
component Rankine–Hugoniot model. For transmitted solar wind ions, the heating is very
1
small and the temperature jump is about τ = r γ − , which is about τ ~2 for a compression
ratio rS~2 shock. This polytropic index increases with pickup ion relative density. When
there are no pickup ions, γ ≈5/3; when there are 20% pickup ions, γ ≈2; and when there
are 40% pickup ions, γ ≈2.2. For reflected solar wind ions, their dynamic energy is
converted into thermal energy. The temperature jump of reflection solar wind ions is on
the order of a thousand. Because the reflected solar wind ions are a small portion of the
solar wind population (reflection efficiency is less than 10% when pickup ion relative
density is larger than 20%), the temperature jump of all solar wind ions are on the order
S

165
of ten. For pickup ions, the average temperature jump is described by
is roughly a constant (~2.5), independent of the pickup ion relative density.
γ
1
rS PUI γ −
, where PUI
What is the partition of dissipated energy among the ion components? The
partition of dissipated energy is also a function of the pickup ion relative density. When
this density is smaller than 5%, solar wind ion reflection is the dominant dissipation
process. Most of the dissipated energy is deposited into solar wind ions. When the pickup
ion relative density is larger than 5%, pickup ions account for most of the shock
dissipation. Particularly when pickup ion relative density is equal or larger than 20%,
more than 85% of the dissipated energy is converted into pickup ion heating. This result
is consistent with inference from Voyager 2 measurements. In addition, this result does
not contradict the finding that the temperature jump of the pickup ion is much smaller
than that of the solar wind ions. Because pickup ions started with an initial temperature
that is three orders of magnitude larger the initial temperature of the solar wind ions, a
small temperature jump still enables a much larger net dissipated energy gain for the
pickup ions than for the solar wind ions.
How are ions (solar wind and pickup ions) energized? The ion components are
heated through different mechanisms. The transmitted solar wind ions are heated by
shock compression. The reflected solar wind ions are heated through specular reflection,
which convert these ions’ incident velocity into gyrovelocity. These mechanisms are the
same as the ion energization mechanisms of the terrestrial bow shocks. What is different
is the energization process of the pickup ions. The pickup ions already have very high

166
upstream temperatures. Therefore, they encounter the termination shock in a completely
different manner compared with the cold solar wind ions. This encounter involves both
the guiding center motions and the gyro motions of the pickup ions. We found that
pickup ions’ energization is a gyrophase-dependent acceleration (GPDA) process, with
the “crossing” ions gaining more energy than the “reversed” ions. This is because the
“crossing” ions’ initial encounter with the shock results in some initial minor energy gain,
while the “reversed” ions’ initial encounter with the shock results in some initial minor
energy loss. While both types of pickup ions’ major energy gain is due to the well-known
shock drift acceleration (SDA) during a single encounter with the shock, GPDA causes
differences in the ions’ final energy, according to the gyrophases of the ions as they
encounter the shock. Those pickup ions with favorable phase angles (positive vx, negative
vy) will be preferentially accelerated by the inhomogeneous electric field Ey (

167
In addition, results from our one-dimensional hybrid simulation are in agreement with
Voyager 2 observed termination shock magnetic field profile, Voyager 2 inferred pickup
ion relative density, and Voyager 2 inferred energy partition of dissipated energy. We
also validate our simulations by checking accuracy and convergence.
7.3 Future Work
This dissertation makes a fundamental contribution to the understanding of the physical
processes associated with scattering and heating of thermal and suprathermal ions at the
termination shock. Along the way, we identify directions for future work:
1. From the perspective of hybrid simulations, the following projects can be
considered in the future:
I. Dimension. An important limitation of our work is that ions are tied to magnetic
fields in the one-dimensional simulation. The one-dimensional code precludes the
excitation of some downstream waves as well as cross-field diffusion. For example,
quasi-perpendicular shocks often generate strong T⊥ > T|| anisotropies which can lead to
ion cyclotron instabilities [e.g., Scholer et al., 2000]. These waves scatter the heated solar
wind and pickup ions in the direction of the magnetic field in two-dimensional or three-
dimensional simulations. The scattering relaxes the downstream plasma [e.g., Winske and
Quest, 1986; Thomas, 1989; McKean et al. JGR, 1995], which in one-dimensional
simlations remains in a highly anisotropic state. With three-dimensional simulations, one

168
can fully resolve the downstream ion velocity distribution as well as investigating the
additional accelerations (if any) caused by various instabilities.
II. Energetic Particles. To describe energetic particles, one can include the energetic
particle population into the upstream input, along with the solar wind population and the
pickup ion population. The ubiquitous co-existence of these three populations is observed
in the heliosphere all the time, even during quiet-time solar wind conditions [Fisk and
Gloeckler, 2006]. Assuming that those ions are isotropic protons, Figure 7-1 depicts our
illustration of the three populations in the upstream region. In the figure, the solar wind is
a Maxwellian, the pickup ions form a Vasyliunas–Siscoe distribution, and the energetic
particles form the power law tail of the spectra.
In addition, the simulation box should be extended significantly when energetic
particles are included. A MeV energetic particle will have a gyrospeed that is more than
1000 times of the Alfvén speed. Then this particle’s gyroradius is more than 1000 times
the ion inertial length 7 . To describe energetic particles properly, one should extend the
simulation box to resolve energetic particle gyrations. The appropriate length of the
simulation box is expensive to achieve.
Further, Giacalone and Ellison [2000] argue that energetic particles can be generated
directly from thermal particles. Treating pickup ions as test particles, they apply
7 Energetic particle larmor radius (gyroradius) ,
here λi is the ion inertial length.
v v v c ω v c v
EP EP A pi EP EP
LEP= = ⋅ ⋅ ⋅ = ⋅ = i
Ωc vA c ωpi Ωc
vA ωpi
vA
r
λ ,

169
Figure 7-1 The three upstream components in the solar wind frame. The solid curve
represents the Maxwellian solar wind; the dotted curve represents the Vasyliunas–Siscoe
pickup ions; the dashed curve represents the powerlaw tail.
prescribed Alfvén waves to their three-dimensional hybrid simulations. They find that the
downstream spectra broaden when the simulation box size increases, thereby suggesting
that the long wavelength waves accelerate particles by enabling cross-field diffusion and
magnetic field random walks. To test their acceleration mechanism, one needs the system
length to be on the order of 20,000 c/ω pi (~1 AU) in the direction of solar wind
propagation (x direction in our simulation). The requirement is unattainable in present
three-dimensional simulations but may be possible with more powerful parallel
computatation.
Solar wind
Pickup ions
Energetic particles

170
III. Heavy Ions. Although to first order, heavy ions are not included in this work, it
would be a useful addition to carry out more realistic simulations in the future. Helium
ions, which usually make up

171
solar system boundaries. Future projects can be built upon this work to continue the
exploration of the termination shock, other astrophysical shocks, and the universe.

List of Journal Abbreviations
Adv. Space Res. Advance in Space Research
Astrophys. J. Astrophysical Journal
Geophys. Monogr. Ser. Geophysical Monograph Series
Geophys. Res. Lett. Geophysical Research Letters
J. Geophys. Res. Journal of Geophysical Research
J. Phys. G: Nucl. Part. Phys. Journal of Physics G: Nuclear and Particle
Physics
J. reine angew. Math. Journal für die reine und angewandte
Mathematik
Mon. Not. R. Astron. Soc. Monthly Notices of the Royal Astronomical
Society
Phys. Fluids Physics of Fluids
Phys. Rev. Lett. Physical Review Letters
Phys. Plasmas Physics of Plasmas
Proc. R. Soc. Proceedings of the Royal Society
Space Sci. Rev. Space Science Reviews

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Curriculum Vitae
EDUCATION
Pin Wu
Ph.D. Candidate, Department of Astronomy, Boston University
Affiliate, ISR-1, Los Alamos National Laboratory
Department of Astronomy, 725 Commonwealth Ave, Boston, MA 02215
pwu@bu.edu | (857)753-7725 (cell) | http://people.bu.edu/pwu
B.S., Modern Physics, University of Science and Technology of China (USTC), 2001
M.A., Astronomy, Boston University, 2003–2005
Ph.D., Astronomy, Boston University, 2003–2009
Dissertation: Ion Kinetics at the Heliospheric Termination Shock
Advisor: Nathan A. Schwadron, Boston University
Mentors: S. Peter Gary and Dan Winske, Los Alamos National Laboratory
JOURNAL PUBLICATIONS
As the First Author (Chronologically)
1. Wu, P., T. A. Fritz, B. Larvaud, and E. Lucek (2006), Substorm associated
magnetotail energetic electrons pitch angle evolutions and flow reversals: Cluster
observation, Geophys. Res. Lett., 33, L17101, doi:10.1029/2006GL026595.
2. Wu, P., N. A. Schwadron, G. L. Siscoe, and P. Riley (2008), Initial condition
influence on coronal mass ejection propagation, J. Geophys. Res., 113, A00B05,
doi:10.1029/2008JA013082.
3. Wu, P., D. Winske, S. P. Gary, N. A. Schwadron and M. A. Lee (2009), Energy
dissipation and ion heating at the heliospheric termination shock, J. Geophys. Res.,
114, A08103, doi:10.1029/2009JA014240.
4. Wu, P., D. Winske, S. P. Gary, and N. A. Schwadron (2009), Hybrid simulations of
perpendicular termination shocks: consequences of variations in pickup ion
distributions, in preparation.
As a Contributing Author
5. Winske, D., S. P. Gary, P. Wu and N. A. Schwadron (2009), Pickup ion energization
at the termination shock, in preparation.

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6. Lee, M. A., H. J. Fahr, H. Kucharek, E. Moebius, C. Prested, N. Schwadron and P.
Wu (2009), Physical processes in the outer heliosphere, Space Science Reviews, doi:
10.1007/s11214-009-9522-9.
7. Schwadron, N. A., G. Crew, R. Vanderspek, F. Allegrini, M. Bzowski, R. DeMagistre,
G. Dunn, H. Funsten, S. A. Fuselier, K. Goodrich, M. Gruntman, J. Hanley, J.
Heerikuisen, D. Heirtlzer, P. Janzen, H. Kucharek, G. Loeffler, K. Mashburn, K.
Maynard, D. J. McComas, E. Moebius, C. Prested, B. Randol, D. Reisenfeld, M.
Reno, E. Roelof and P. Wu (2009), The Interstellar Boundary Explorer Science
Operations Center, Space Science Reviews, doi:10.1007/s11214-009-9513-x.
INVITED TALKS (Chronologically)
1. Substorm associated magnetotail energetic electrons pitch angle evolutions and flow
reversals: Cluster observation, Los Alamos National Laboratory, ISR-1
magnetospheric physics group Seminar, July 6, 2005, Los Alamos, New Mexico.
2. Hybrid simulation of the heliospheric termination shock, Los Alamos National
Laboratory ISR-1 solar & heliospheric physics group Seminar, September 26, 2007,
Los Alamos, New Mexico.
3. Energy dissipation and ion heating at the heliospheric termination Shock, New
England Space Consortium, MIT, January 26, 2009, Boston, Masschusetts.
4. Energy dissipation and ion heating at the heliospheric termination Shock, Los Alamos
National Laboratory ISR-1 solar & heliospheric physics group Seminar, April 22,
2009, Los Alamos, New Mexico.
5. Energy dissipation and ion heating at the heliospheric termination Shock, 8 th Annual
International Astrophysics Conference, May 1–7, 2009, Big Island, Hawaii
APPOINTMENTS
1. Teaching Fellow, Boston University, 2003–2004
2. Graduate Research Assistant, Boston University, 2004–2009
3. Visitor, sponsors/mentors: Reiner Friedel and Geoff Reeves, ISR-1, Los Alamos
National Laboratory, July 5, 2005–July 15, 2005
4. Staff Research Assistant (Intern), sponsors/mentors: S. Peter Gary and Dan Winske,
ISR-1, Los Alamos National Laboratory, July 2, 2007–September 28, 2007
5. Visitor, sponsors/mentors: S. Peter Gary and Dan Winske, ISR-1, Los Alamos
National Laboratory, April 20, 2009 - May 15, 2009
6. Visitor, sponsors/mentors: S. Peter Gary and Dan Winske, ISR-1, Los Alamos
National Laboratory, September 21, 2009 – October 2, 2009

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CONFERENCE/WORKSHOP EXPERIENCES
1. Center for Integrated Space Weather Modeling (CISM) Summer School (2003:
attended)
2. Geospace Environment Modeling (GEM) workshop (2004: gave a poster; 2005: gave
a student tutorial talk and a poster)
3. Solar Heliospheric & INterplanetary Environment (SHINE) workshop (2008: gave a
poster, chaired an invited talk section; 2009: gave a student tutorial and a poster)
4. AGU Fall Meeting (2004, 2005, 2007, 2008: gave a poster; 2005: gave a talk)
5. IBEX Science Working Team meeting (2007: gave a talk at Orbital Science
Corporation; 2008: gave a talk at BU)
6. New England Space Consortium (2005–now: attended every season)
7. Solar Physics Summer School (2009: attended)
TITLES/AWARDS
Student ambassador, Boston Ballet, 2007–2009
Outstanding student scholarship, USTC, 2000
2 nd prize, dance, College student art festival, Anhui TV, 1999
Outstanding student scholarship, USTC, 1998
DiAo outstanding student scholarship, USTC, 1997
AFFILIATIONS
Los Alamos National Laboratory, NASA COSMICOPIA, Boston Ballet
PROFESSIONAL MEMBERSHIP
Student Member, American Geophysics Union (AGU), 2004–present
Student Member, American Physics Society (APS), 2005–2006
SKILLS U
Operating System: Linux, MAC, Windows
Programming: IDL, FORTRAN, OpenDX, CismDX, C, HTML, LaTex
Language: English (fluent), Mandarin (native), Taiwanese (native)