U
NIVERSITY OF
A
MSTERDAM
MSc Physics and Astronomy
Track: Astronomy & Astrophysics
M
ASTER
T
HESIS
Particle Acceleration in CME-driven Shocks
Study of Solar Energetic Particle Events
as a Function of Shock Mach Number
by
Monika Pikhartov´a
12301469 (UVA)
60 ECTS
2nd of September 2019 - 31st of July 2020
Supervisors:
Jacco Vink
Dimitrios Kantzas
Examiners:
Jacco Vink
Shin’ichiro Ando
Anton Pannekoek
Institute
Contents
Popular Abstract
1
Abstract
3
Acknowledgements
5
1
Introduction
7
1.1
Cosmic Rays
. . . .
7
1.1.1
Solar Energetic Particles . . . .
9
1.2
Coronal Mass Ejections . . . .
10
1.3
Collisionless Shocks . . . .
13
1.4
Particle Acceleration . . . .
15
1.4.1
Second Order Fermi Acceleration Theory . . . .
15
1.4.2
Diffusive Shock Acceleration
. . . .
17
1.4.3
DSA Theory . . . .
17
1.4.4
Non-Linear Diffusive Shock Acceleration . . . .
20
1.4.5
Low Mach Number Shocks
. . . .
20
1.5
Project Goals . . . .
21
2
Advanced Composition Explorer
22
2.1
EPAM . . . .
25
2.2
SWEPAM . . . .
26
2.3
MAG
. . . .
26
3
Analysis
27
3.1
Data Selection and Preparation . . . .
27
3.2
Shock Notation . . . .
28
3.3
Compression Ratios and Mach Numbers . . . .
29
3.4
Power-Law Slopes of ACE Data . . . .
30
3.5
Thermal and Non-thermal Pressures . . . .
30
3.6
Acceleration Efficiency and Theoretical Non-Thermal Pressure . . . .
31
3.7
Theoretical Non-Thermal Pressure and Magnetic Field Pressure
. . . .
32
4
Results
34
4.1
Classification . . . .
35
Contents
4.2
Particle distributions
. . . .
41
4.2.1
Electron contribution . . . .
45
4.3
Acceleration efficiency . . . .
46
4.4
Pre-Existing Particles in Momentum Conservation Law . . . .
48
4.5
Magnetic Field Importance - Plasma Beta . . . .
51
4.6
Magnetic Field in Momentum Conservation Law . . . .
53
5
Discussion and Conclusions
56
5.1
General Remarks . . . .
56
5.2
Caveats . . . .
59
5.3
Highlights . . . .
60
Appendix
65
A Supplemental Figures
65
A.1 Slopes of Particle Distributions . . . .
65
A.2 Including Pre-Existing Particles and Magnetic Field
in Momentum Conservation Law . . . .
68
Popular Abstract
The nearest star to Earth, the Sun, affects us directly in various manners. It influences the life
on Earth, as well as in space. Despite the detailed studies for decades, many properties of
the Sun are still poorly understood. In particular, processes that are related to the activity of
the Sun, such as the coronal-mass ejections (CMEs; ejections of material that create shock
waves, disturbances, traveling through space), their launching mechanism, their influence
on the medium CME travel through, the acceleration of solar energetic particles (SEPs)
in the aforementioned shock waves and others, are complex and the basic models do not
sufficiently explain the underlying physics. SEPs arising from solar activity play a crucial
role in space technology and influence space missions. If these particles have very high
energy and penetrate into Earth’s atmosphere, they can even disable navigation systems or
cause storms on Earth that drop power grids. In this work, we concentrate on SEP events
and in particular, how are SEPs accelerated by the shock waves coming from the Sun. There
are two acceleration mechanisms that are generally considered. We study 19 years of data
from a spacecraft that orbits between the Sun and the Earth to distinguish which of these two
fundamental mechanisms dominates. We find that the data shows no preference for either
of the acceleration mechanisms we consider. We suggest including pre-existing accelerated
particles and magnetic field in the equations of conservation of mass and momentum that
describe the shock. We see that including pre-existing particles in the conservation equations
could improve the theoretical predictions of our data while including magnetic field would
actually diminish them.
Abstract
The nearest star to Earth, the Sun, affects us directly in various manners. It influences the life
on Earth, as well as in space. Despite the detailed studies for decades, many properties of the
Sun are still poorly understood. In particular, processes that are related to the activity of the
Sun, such as the coronal-mass ejections (CMEs), their launching mechanism, their influence
on the interplanetary medium, the acceleration of solar energetic particles (SEPs) in
CME-driven shocks and others, are complex and the basic models do not sufficiently explain the
underlying physics. SEP events play a crucial role in space technology and influence space
missions. In this work, we focus on the SEP events and in particular, on the acceleration
mechanisms of SEPs in CME-driven shocks. We consider the particle acceleration to occur
due to multiple shock crossings as explained by the diffusive shock acceleration (DSA). We
focus on the non-linear regime of this mechanism (NLDSA) and discuss its feasibility based
on data obtained by the Advanced Composition Explorer (ACE) spacecraft. We examine 19
years of data and discuss the connection between the macroscopic quantities of the
CME-driven shocks (e.g., flow temperature, velocity, pressure and magnetic field) and the
micro-scopical ones (e.g., the accelerated particle distributions and the acceleration efficiency). We
suggest that models of (NL)DSA should take into account macroscopic quantities like the
pressure of the pre-existing particles but it is a valid assumption to neglect the magnetic field.
Our data do not show any preference for DSA or NLDSA. We see no propensity of shock
acceleration efficiency or particle distribution on the velocity of the shock.
Acknowledgements
First and foremost, I would like to express my sincere gratitude to my advisor, Dr. Jacco
Vink. Throughout the whole year his advice, knowledge and ardor about the topic bolstered
me continuously.
I would like to offer a very special thank you to my daily supervisor, Dimitrios Kantzas
for always being there for me, supporting me when I needed or wanted it and even when I
did not. His endless enthusiasm and encouragement was what kept me going no matter the
hardships I was facing.
Thank you to my fellow classmates and Master room office mates for sharing your
knowl-edge and struggles with me and for the comic relief that was solely needed and appreciated.
I would like to specifically thank Bart van Baal for being my coding hero.
Lastly, I would like to thank to the rest of the Supernova Remnant research group for their
input and assistance throughout my research project.
Chapter 1
Introduction
1.1
Cosmic Rays
Cosmic rays (CRs) have been thoroughly studied ever since their discovery in 1912 by Victor
Hess. Victor Hess used balloon experiments of the atmosphere and found that the penetrating
radiation, whose origin has been long debated, had to be of an extraterrestrial origin. He
concluded that the Sun could not be its source due to lack of radiation decrease during night
and solar eclipses (
Rigden 2003
;
Hess 2018
).
CRs are a population of elementary particles and nuclei with energies that range from a
few MeV up to more than ∼ 10
10
GeV. The CR energy spectrum is a steep broken
power-law (
Tanabashi et al. 2018
). In Figure
1.1
adopted from
Evoli
(
2018
), we show the CR
energy flux as a function of the total energy of the particles. The gray shaded region
with a turnover shows the solar contribution to CRs. Protons dominate the low GeV
en-ergies. The contributions of other CR particles, such as protons, antiprotons, electrons, etc.,
are shown as well for comparison. The flux of these particles ranges between 10
−7
and
∼ 10
3
GeV m
−2
s
−1
sr
−1
.The proton flux is of the order of 1 cm
−2
s
−1
around GeV range.
Particles with energy more than ∼ 1 GeV are of extrasolar but still Galactic origin. These
Galactic CRs go up to the so called ‘knee’, which lies at ∼ 3 PeV. Up to this energy, the
spectrum follows a power-law with a spectral index of -2.7. Around the ‘knee’, the flux is
of the order of 1 m
−2
yr
−1
. Above the knee, the origin of the CRs is unclear but most likely
extragalactic. The spectral index of the power-law in this energy range is -3.1. The energy
range of the Large Hadron Collider is in this region. Around the so called ‘ankle’ (from ∼
3 PeV to ∼ 5 EeV), the flux is of the order of 1 km
−2
yr
−1
and the spectrum hardens
(flat-tens). The spectrum shows a cutoff at around ∼ 10
20
eV. The data were was collected by
various facilities both on the surface of Earth and in orbit (see
Evoli 2018
, and references
therein).
Cosmic Rays
F
IG
. 1.1
–
The CR energy spectrum as a function of the CRs’ total energy. The famous CR-spectrum features,
the ‘knee’ and the ‘ankle’ are indicated. The different colors indicate the different instruments collecting these
data in various energy ranges as shown in the legend. Adopted from
Evoli
(
2018
).
the CR spectrum (
Forbush 1946
). Supernovae (SN) and Supernova Remnants (SNRs;
Baade
& Zwicky 1934
), Pulsar Wind Nebulae (
Manconi et al. 2020
), X-Ray Binaries (
Cooper et al.
2020
) and clusters of massive stars (
Aharonian et al. 2018
) are considered the dominant
candidate sources of Galactic CRs. Active Galactic Nuclei, Gamma-Ray Bursts, clusters of
galaxies are among the dominant sources for extragalactic CRs (see e.g.,
Hillas 1984
;
Kotera
& Olinto 2011
;
Moskalenko & Seo 2018
;
Boezio et al. 2020
, and references therein).
Further open questions regarding CRs are: the CR origin; the CR composition, especially in
the high-energy regime where one experiment favors protons (
Hanlon 2019a
,
b
) while other
favors heavier elements (
Deligny 2019
;
The Pierre Auger Collaboration et al. 2019
;
Perrone
& the Pierre Auger Collaboration 2020
); the potential role that CRs play in the evolution
and the dynamics of stars and galaxies. The mechanism that drives the CR acceleration
Cosmic Rays
up to more than ∼ 10
10
GeV is another open question (
Drury 1983
). In this work we are
investigating the mechanism behind the acceleration of solar cosmic rays.
1.1.1
Solar Energetic Particles
Solar Energetic Particles (SEPs) are electrons, protons, and heavier nuclei associated with
Coronal Mass Ejections (CMEs; see below) and solar flares (see
Cane & Lario 2006
, and
references therein). In the CR spectrum, SEPs lay in the low energy region (up to GeV).
SEPs were discovered by
Forbush
(
1946
) in 1942 who noticed an increase of CR intensity
related to solar activity. SEP events are either impulsive or gradual. The impulsive events
are associated with solar flares. The gradual events are associated with CMEs and can last
up to days (
Li et al. 2005
).
We show the typical particle intensity of protons and electrons over time in Figure
1.2
adopted from (
Reames 1999
). The left panel shows a gradual event on 1981 December 5
in which a filament erupts from the Sun as a CME, without any accompanying solar flare.
The right panel shows events on 1982 August 13 and 14 that are associated with
impul-sive solar flares with no accompanying CMEs. The difference in the time scales of these
SEP events are clearly visible in the Figure. We see that the gradual event is dominated by
protons while the impulsive events are dominated by electrons.
F
IG
. 1.2
–
Particle intensity profile over time for a gradual SEP event on 1981 December 5 (left) and an
impulsive SEP event on 1982 August 13, 14 (right). The different points correspond to different particles as
depicted in the legend. Figure adopted from
Reames
(
1999
).
Coronal Mass Ejections
SEPs play a crucial role to the physical processes related to the space weather as they have
been considered a notorious space hazard. If these energetic particles hit a spacecraft or
penetrate astronauts’ spacesuit, they damage the equipment and can be hazardous to human
health (
Feynman & Gabriel 2000
;
Reames et al. 2001
). Shielding needs to be incorporated
into spacesuits and instrumental setups in order to safeguard against SEPs. SEPs can cause
storms in the Earth’s atmosphere that can disrupt power grids (see e.g.,
Stauning 2002
). All
these reasons are why we need to study, in particular: their origin, their composition, their
connection to CME-driven shocks and how they are accelerated.
1.2
Coronal Mass Ejections
Most strong SEP events are accelerated by CME shock waves (see e.g.,
Gargat´e et al. 2014
,
and references therein). During a CME, plasma carrying magnetic field is ejected from the
Sun’s corona into the interplanetary medium (IPM). About 10
15
g of plasma gets expelled
during a typical CME. The kinetic energy of this plasma can reach up to ∼ 10
32
erg with
plasma velocities in the range of 20–2500 km s
−1
(
Manchester et al. 2005
). CME-driven
shocks are considered efficient SEP accelerators. Type II radio bursts that often accompany
a CME shock are regarded as a sign for particle acceleration (
Gopalswamy et al. 2010
). Up
to 10% of the kinetic energy of a CME could go into energetic particles (
Mewaldt 2006
).
CMEs can significantly vary in time and can last from hours up to days (
Manchester et al.
2005
;
Webb & Howard 2012
). The frequency of CME occurrence follows the solar cycle
(
Webb & Howard 1994
; more CMEs during maximum solar activity and vice versa).
CMEs form as a result of an eruption of a magnetic flux rope through magnetic reconnection
in the Sun’s corona. A magnetic flux rope is a coherent magnetic structure with all magnetic
field lines twisting around a central, usually helical, axis. This magnetic flux rope can be
established during or even before to a solar eruption (see
Song & Yao 2020
;
Xing et al.
2020
, and references therein). We show a schematic of a magnetic flux rope in Figure
1.3
that was adopted from Professor Lang of the Tufts University. The CME ejection starts
as a prominence-like loop that stretches further and further away from the Sun. A shock is
formed at the front and is driven by the CME that carries the trapped hot plasma (grey shaded
region). As the magnetic field lines stretch with the shock travelling further away from the
Sun, lines going in opposite directions come closer and closer until finally they snap and
magnetic field reconnects. Magnetic reconnection takes place at the thick black vertical line.
The closed field region above the prominence (black shaded region) becomes the flux rope.
We show two images of a CME in Figure
1.4
. The images were taken by two of the
corona-graphs (C2 and C3) of the Large Angle and Spectrometric Coronograph (LASCO) on board
of the Solar and Heliospheric Observatory (SOHO). The images in the visible spectrum were
taken roughly ∼ 6 hours apart. We can see a “lightbulb” shape of the the blast in white color.
The field of view of LASCO C2 ranges from 1.5 to 6 solar radii, while LASCO C3 ranges
1
Coronal Mass Ejections
F
IG
. 1.3
–
An illustration of a CME formation adopted from Lang (2010)
1.
from 3.5 to 30 solar radii (
Domingo et al. 1995
).
CME shock waves may interact with the Earth’s magnetosphere when travelling through
IPM towards Earth. When the CME shock carries high energetic SEPs, the interaction with
Earth’s magnetosphere could cause geomagnetic storms that affect for example electricity
grids, e.g. transportation, satellite navigation and others (see e.g.
Gosling et al. 1991
). In
Figure
1.5
we show an illustration of a CME and its subsequent interaction with the Earth’s
magnetic field. The left side of the Figure is a superimposed Extreme ultraviolet Imaging
Telescope (EIT) image, taken at the 30.4 nm wavelength, on a LASCO C2 optical image.
Both are on board of SOHO. The right side of the Figure is an artist’s impression of the
shock’s interaction with the Earth’s magnetic field. Magnetic field lines are shown in blue
color. Due to the impact these CME-driven shock waves may have on Earth, we are
moti-vated to study the acceleration of the SEPs that are driven by CME shocks.
2
https://sohowww.nascom.nasa.gov/gallery/images/las02.html
3
https://www.esa.int/ESA Multimedia/Images/2003/04/Coronal mass ejection CME blast
Coronal Mass Ejections
F
IG
. 1.4
–
CME observed on 2000 February 27 by LASCO C2 and C3 ∼ 6 hours apart. The white circle
denotes the optical sun, while the larger filled circle blocks the direct light. Credits: SOHO/LASCO consortium
2.
Collisionless Shocks
1.3
Collisionless Shocks
Shocks are ubiquitous phenomena in the Universe. They can be described as transition layers
in which the flow and thermodynamic properties of the plasma, such as the bulk velocity or
the number density, rapidly change from one moment to another. They form when a
distur-bance propagates through a medium at a supersonic speed; i.e. supernova explosions, stellar
bow shock, Gamma Ray Bursts, Active Galactic Nuclei and other explosive astrophysical
phenomena.
The CME-driven shocks described above are collisionless shocks. Collisionless means that
energy is not dissipated through particle-particle collisions but by wave-particle interactions
instead (
Treumann 2009
;
Burgess & Scholer 2015
). The interaction cross-section between
particles is so low that collisions do not have an effect on the system.
In this work, we study non-relativistic collisionless CME-driven shocks. In Figure
1.6
, we
show six examples of CME-driven collisionless shocks detected by the Advanced
Composi-tion Explorer (ACE) spacecraft and studied by
Giacalone
(
2012
). In each of these six plots
the panels from top to bottom are: ion flux, magnetic field, flow velocity and bulk number
density. The vertical line indicates the shock crossing the spacecraft and clearly separates
the unshocked medium on the left side from the shocked medium on the right. The magnetic
field, flow velocity and number density of each of the shocks have the same profiles. Notable
is, however, how varied the shocks are in the flux (top panel). All of the shocks have an
in-crease in flux already before the shock, but the inin-crease can vary in length. After the shock
crossing, the flux either stays high or drops down. These drops in flux also vary in length.
Collisionless Shocks
F
IG
. 1.6
–
Examples of ion flux at two different energies (47 – 65 keV solid line, 65 – 112 keV dashed line),
magnetic field, flow velocity and number density (each panel from top to bottom) of CME-driven shocks. Figure
adopted from
Giacalone
(
2012
), data taken by ACE.
Particle Acceleration
1.4
Particle Acceleration
CRs gain high energies that reach up to 10
20
eV via various mechanisms.
Fermi
(
1949
) first
postulated a way of accelerating charged particles to such high energies. Particle acceleration
is the inevitable outcome of charged particles interacting with turbulent motions. This initial
idea, nowadays known as the second order Fermi acceleration, states that the net particle
en-ergy gain scales as (v/c)
2
, where v is the velocity of the perturbations and c is approximately
the velocity of the accelerated particles. The value of the exponent drives the name of this
acceleration mechanism. The resulting particle energy spectrum follows a power-law. Since
the velocity of the shock is much less than the speed of light (v c), this mechanism is not
very efficient.
1.4.1
Second Order Fermi Acceleration Theory
Let us first discuss the second order Fermi acceleration, which was also historically presented
first.
Fermi
(
1949
) postulated that CRs are accelerated primarily in the interstellar space by
collisions against moving magnetic fields. These moving magnetic fields are interstellar
clouds. Nowadays, these clouds are interpreted as shock waves.
Following
Longair
(
2011
) and
Uroˇsevi´c et al.
(
2019
), we consider a particle with mass m,
kinetic energy E and total energy ε = m c
2
+E moving through a medium.When the particle
collides with a cloud that moves along the x-axis with speed v
s
the particle is reflected on
the pitch angle θ. Right before the collision, the energy and the momentum of the particle in
the shock (moving) frame of reference are:
ε
0
= Γ (ε + v
s
p
x
) ,
(1.1)
p
0
x
= Γ (p
x
+ v
s
ε/c
2
) ,
(1.2)
where Γ ≡ (1 − v
s
2
/c
2
)
−1/2
is the Lorentz factor of the shock. The collision is elastic
and therefore, the initial total energy of the particle in the shock frame is conserved (ε
0
i
=
ε
0
f
). The particle moves in the opposite direction, thus p
0
x,i
= −p
0
x,f
. Using the Lorentz
transformation, the total energy of the particle after the collision in the lab frame is:
ε
f
= Γ (ε
0
+ v
s
p
0
x
) = Γ
2
(ε + 2 v
s
p
x
+ v
2
s
ε/c
2
) .
(1.3)
And since p = Γ m v and ε = Γ m c
2
, we can write that:
p
x
ε
=
p cos θ
ε
=
v cos θ
c
2
.
(1.4)
Particle Acceleration
Γ = (1 − v
s
2
/c
2
)
−1/2
≈ 1 + (v
s
/c)
2
, we can write:
ε
f
ε
=
1 + 2 (v
s
/c) (v/c) cos θ + (v
s
/c)
2
1 − (v
s
/c)
2
≈ 1 + 2 (v
s
/c) (v/c) cos θ + (v
s
/c)
2
.
(1.5)
Assuming a random distribution of angles θ and neglecting the rest energy in the relativistic
regime of accelerated particles, the average kinetic energy gain per collision is:
∆E
E
=
E
f
− E
E
'
8
3
v
s
c
2
.
(1.6)
The energy gain of the particle scales as the square of the velocity of the shock (v
s
/c)
2
and
that is why this process is called “second-order” Fermi acceleration. If the average time
between collisions is 2L/c, where L is the mean free path of the particle, the rate of energy
increase is:
dE
dt
=
4
3
v
2
s
cL
E ≡ α E .
(1.7)
Since the particle propagates away from the shock-front, the actual acceleration can only
continue for time τ , while the particle stays within the accelerating region.
To derive the resulting energy spectrum of the accelerated particles, we solve the kinetic
equation (for derivation see Chapter 17.3 of
Longair 2011
):
dN (E)
dE
= −
1 +
1
α τ
N (E)
E
,
(1.8)
which leads to
n(E) = C
2
× E
−
α τ + 1
α τ
,
(1.9)
where n is the number of particles at a certain energy and C
2
is a constant. The particle
energy spectrum of the second order Fermi acceleration mechanism follows a power-law.
The second order Fermi acceleration mechanism is inadequate for explaining multiple
as-pects. The theory does not properly explain why the energy spectrum has the observed slope
of around -2 for strong shocks (see e.g.,
Vink 2012
, and references therein). The random
velocities of the postulated clouds and the mean free path for CR scatterings are
compara-tively low and that leads to a slow, almost impossible particle acceleration (
Longair 2011
).
This acceleration mechanism served instead as a foundation for later studies of particle
ac-celeration, and for establishing the first order Fermi acceleration theory (
Axford et al. 1977
;
Krymskii 1977
;
Bell 1978a
,
b
;
Blandford & Ostriker 1978
;
Drury 1983
).
Particle Acceleration
1.4.2
Diffusive Shock Acceleration
The essence of first order Fermi acceleration, or Diffusive Shock Acceleration (DSA), is that
charged particles of the shock-heated plasma move diffusively and consecutively cross the
shock front head-on due to magnetic field irregularities. This repeating process ultimately
results in an energy gain that scales as v/c, making DSA more efficient than the second order
Fermi acceleration (
Bell 1978a
). The resulting particle energy spectrum follows a power-law
that scales as E
−q
.
1.4.3
DSA Theory
A more efficient mechanism for acceleration of CRs is DSA or first order Fermi acceleration.
Essential to further understanding of DSA is to describe the conditions around the
CME-driven shock. The physical properties and the structure of the shock, both in the unshocked
and shocked region, can be described by the Rankine-Hugoniot relations (
Landau & Lifshitz
1987
;
Landau & Sykes 1987
;
Shu 1992
;
Nieuwenhuijzen et al. 1993
). These relations treat
the shock macroscopically as a single discontinuous jump and are:
ρ
0
v
0
= ρ
2
v
2
,
(1.10)
ρ
0
v
0
2
+ P
0
= ρ
2
v
2
2
+ P
2
,
(1.11)
h
0
+
1
2
v
2
0
= h
2
+
1
2
v
2
2
.
(1.12)
where ρ is the mass density of the fluid, v is the velocity of the flow, P is the total pressure,
and h is the enthalpy. The equations are in the frame co-moving with the shock. Plasma
enters the shock with v
0
= v
sh
and moves away from the shock with v
2
(shocked). These
equations describe the conservation of mass, momentum and energy across the shock,
re-spectively. The equations show that no mass accumulates in the shock layer and the kinetic
energy converts to enthalpy. The enthalpy for a perfect gas satisfies the following relation:
h =
γ
γ − 1
P
ρ
=
γ
γ − 1
kT
m
= E +
P
ρ
.
(1.13)
γ is the adiabatic index, k is the Boltzmann constant, T is the temperature, m is the
elemen-tary mass of the crossing particles, and E is the internal energy. By taking into account the
magnetic field of the upstream and the downstream regions, we can include extra terms for
the magnetic field pressure and energy density in the above equations (
Kennel et al. 1989
;
see Section
3.7
for more detail).
A schematic diagram from
Longair
(
2011
) shows the dynamics of the high energy particles
in proximity of a shock (see Figure
1.7
). The top left depiction shows a strong shock wave
propagating at a velocity U that is faster than the speed of the medium. The density, pressure
and temperature (ρ, P, and T, respectively) differ between the unshocked (subscript 1) and the
Particle Acceleration
shocked medium (subscript 2). The conditions can be described by the Rankine-Hugoniot
re-lations (see Equations
1.10
–
1.12
). The top right depiction shows the shock reference frame.
Equation
1.10
of the Rankine-Hugoniot relations, the ratio of unshocked to shocked region
velocity is (γ + 1)/(γ − 1). For a fully ionized non-relativistic plasma, γ = 5/3, that gives
a ratio of 4. In the bottom left depiction is the frame of reference where the upstream gas
is stationary, showing how the velocity in the shocked region equals to 3U /4 (we later call
that v
s
). The bottom right depiction is the frame of reference where the shocked plasma is
stationary.
F
IG
. 1.7
–
Top left: A shock moving with a speed U . Pressure, density and temperature in the unshocked
medium (p
1, ρ
1, T
1) and in the shocked plasma (p
2, ρ
2, T
2). Top right: Using the Rankine-Hugoniot relations,
the ratio between the unshocked and shocked region velocities is v
1/v
2= 4 because γ = 5/3. Bottom left:
Reference frame where the unshocked gas is stationary. The velocity of the downstream flow is 3/4 U. Bottom
right: Reference frame where the shocked gas is stationary. The velocity of the upstream flow is then 3/4 U.
Figure adopted from
Longair
(
2011
).
We assume that high energy particles are in both unshocked and shocked regions of the
shock. When such a particle crosses the shock front from the unshocked to the shocked
region, it encounter the gas behind the shock that moves at speed of 3v
s
/4. The particle
scatters due to magnetic field irregularities and gains energy. This gain scales as ∝ v
s
/c.
When on the other hand, a particle crosses from the shocked to the unshocked region, it
again encounters a gas there that moves at speed of V = 3v
s
/4 and again gains the same
amount of energy. This process can repeat multiple times allowing for the particles to attain
high enough energies.
Particle Acceleration
crossing into the shocked region in the shock frame is:
ε
0
= Γ (ε + V p
x
) ,
(1.14)
where V is the shocked plasma approaching the particle (V = 3/4v
s
), the x-axis is
perpen-dicular to the shock front, the shock is non-relativistic (V c, Γ = 1). The particles are
relativistic, therefore, E = p c, p
x
= (E/c) cos θ and:
∆E = p V cos θ and
∆E
E
=
V
c
cos θ ,
(1.15)
where ∆E is the difference in energy in each shock crossing. The average increase in energy
can be written as:
∆E
E
=
2
3
V
c
.
(1.16)
From classical kinetic theory (see Chapter 17.4 of
Longair
(
2011
) for details), n c/4 is the
flux of particles crossing the shock in either direction, where n is the number density of the
particles. Particles in the shocked region are advected or swept away from the shock at a rate
nV = nv
s
/4 creating a fraction of particles that escape per unit time, v
s
/c. Since only a
small fraction of particles is lost per cycle, the probability that a particle remains within the
accelerating region after one collision is P
e
= 1 − (v/c). After some number of collisions
k, there are n
f
= n
i
P
e
k
particles with energies E
k
= E
i
β
k
, where β is a constant connected
to the energy gain (β = 1 + 4V /(3c) in one round trip). We can eliminate k by taking the
logarithm of both n
f
/n
i
and E
k
/E
i
:
ln(n
f
/n
i
)
ln(E
k
/E
i
)
=
ln P
e
ln β
,
(1.17)
which results in:
n
f
n
i
=
E
k
E
i
ln P
e/ ln β
.
(1.18)
For infinite number of collisions (k → ∞) we neglect the subscript and write E
k
= E. From
the above equation we can obtain the energy spectrum of the accelerated particles:
n(E) dE = C
1
× E
−1+(ln P
e/ ln β)
dE .
(1.19)
If we now use the values of P
e
and E
f
we derived earlier, we can get the differential
power-law spectrum of:
n(E) dE = E
−q
dE ,
(1.20)
where q = (χ + 2)/(χ − 1) (q = 2 for a strong shock χ = 4) and we considered that v
is also a function of compression ratio. We have to stress that the above derivation is for
relativistic particles (
Bell 1978a
). For the non-relativistic particle limit the spectrum gets as
flat as q = 1.5 (
Asvarov et al. 1990
;
Caprioli 2014
).
Particle Acceleration
1.4.4
Non-Linear Diffusive Shock Acceleration
When the accelerated particles carry a sizable fractions of pressure and energy density of
the shock they start affecting the dynamics of the system. This is the non-linear regime of
DSA (NLDSA;
Drury 1983
;
Caprioli & Haggerty 2019
). This theory suggests that particles
with higher energies diffuse further away from the shock forming the so called precursor
and slowing down the upstream plasma. This precursor results in an additional pressure in
the unshocked region which compresses the plasma and makes the shock-jump smaller (less
strong). Accelerated particles also enhance the magnetic field strength and turbulence that
have a back reaction to the accelerated particles.The spectrum of the accelerated particles
does not follow a power-law anymore but is concave instead. In particular, at low energies,
the spectrum becomes steeper and at higher energies the spectrum becomes flatter than the
DSA’s prediction of q = 2 (for strong shocks and relativistic particles).
Nevertheless, the local interstellar spectra (Galactic CRs) measured by Voyager 1 shows a
power-law with ∼ E
−1.45
at low energies (kinetic energy below GeV; see e.g.,
Stone et al.
2013
;
Potgieter 2014
, and references therein). This is in agreement with simulations of
ac-celerated non-relativistic Galactic CRs that follow a distribution of E
−1.5
(see e.g.,
Caprioli
& Spitkovsky 2014
). Observations of electrons (see e.g.,
Lin et al. 1982
) and simulations
of solar CR spectra (see e.g.,
Kong et al. 2019
, and references therein) show a variability in
the power-law spectrum between ∼ E
−0.6
and ∼ E
−4.3
. The above-mentioned power-law
slopes are a hint for NLDSA.
When magnetic field amplification is not important, the high energy particles stream further
upstream, experiencing larger compression, thus having a flatter distribution than DSA. On
the other hand, when magnetic field amplification becomes important, the particle energy
spectrum experiences the opposite effect. The lower energies become flatter, while higher
energies become steeper (
Caprioli 2012
).
1.4.5
Low Mach Number Shocks
Shock acceleration is tightly connected to the properties of the shock where particles
accel-erate. A fundamental quantity that describes the shock is the Mach number that is defined
as the ratio of the flow velocity v with respect to the local speed of sound c
s
. The larger
the value of Mach number, the “stronger” the shock. When studying interstellar shocks
de-tected in situ, it is useful to derive the Mach number from measurable quantities. We show
in Section
3.3
that a good approximation for the Mach number is
M =
s
2γ
γ + 1 − χ(γ − 1)
,
(1.21)
where γ is the adiabatic index (5/3). The compression ratio χ is the ratio of the number
density in the shocked over the unshocked medium.
Project Goals
An open question is whether shocks with low Mach number are associated with efficient
par-ticle acceleration.
Gopalswamy et al.
(
2010
) indicates that there is a possible critical Mach
number under which particle acceleration does not happen. It is the Alfv´en Mach number
M
A
≈ 2.7, where M
A
is defined as the ratio of the flow velocity with respect to the Alfv´en
velocity (v
A
= B (4 π ρ)
−1/2
, where B is magnetic field and ρ is the bulk density).
Lee et al.
(
2009
) found a critical Mach number of 2.46 with particle-in-cell simulations for the solar
wind termination shock (at the point in the heliosphere where the solar wind slows down
to subsonic speed because of its interactions with the local ISM). The termination shock is
also associated with particle acceleration (see e.g.,
Florinski et al. 2013
).
Giacalone
(
2012
)
presented that, from observations, CME-driven shocks with χ > 2.5 show evidence of
parti-cle acceleration.
Vink & Yamazaki
(
2014
) analytically derived a similar critical lower Mach
number that should exist for DSA, below which shocks do not accelerate particles. This
critical value of Mach number is
√
5 (that corresponds to χ = 2.5) if magnetic fields are not
dynamically important. If magnetic field pressure dominates the pressure in the unshocked
medium, the critical Mach number would then be 2.5. More recently,
van Marle
(
2020
)
performed a series of hybrid simulations (magnetohydrodynamics and particle-in-cell) for
collisions of two galaxy clusters and the subsequent collisionless shock that is formed.
van
Marle
(
2020
) found that no DSA occurs in shocks with a Mach number below 2.25.
1.5
Project Goals
Life on Earth is directly affected by the Sun. Accelerated SEPs propagating towards Earth
can depreciate equipment or instruments in space. They can also cause geomagnetic storms
on Earth that can disrupt global electrical power grids or navigation systems. The physical
processes that are related to these particles, e.g. the CME launching mechanism, the
influ-ence of CME-driven shocks to the ISM, the acceleration of SEPs in these shocks and other,
are complex and not yet fully understood. Nevertheless, it is crucial to study them further
so as to be able to tackle SEP events better in the future. The analytical models describing
these processes are still fairly basic using many assumptions and approximations.
The main objective of this work is to investigate the mechanisms behind acceleration of SEPs
due to CME-induced non-relativistic collisionless shocks using real data from the Advanced
Composition Explorer (ACE) spacecraft (see Chapter
2
for details). We investigate SEP
ac-celeration, its relationship to the shock Mach number and the disposition toward (NL)DSA.
We consider the connection between the microscopical and macroscopical quantities of the
shock and test whether including microscopic quantities into analytical models could
im-prove their reliability. We specifically examine whether including pre-existing particles and
magnetic field into the shock conservation relations (Equations
1.10
–
1.12
) increases the
ac-curacy of theoretical predictions. We inspect the data for a “critical” Mach number as we
introduced in the above section. In the absence of such a “critical” Mach number, we
inves-tigate whether shocks of smaller Mach number accelerate particles less efficiently.
Chapter 2
Advanced Composition Explorer
F
IG
. 2.1
–
Illustration of the ACE spacecraft. Figure adopted from the ACE Photo Gallery
1.
The Advanced Composition Explorer (ACE) (see Figure
2.1
), being part of the NASA
Ex-plorer program, was launched in 1997 and is still active in 2020, exceeding the initial
per-spective of a 5 year mission. ACE is in orbit at the L1 Lagrangian point of the Sun-Earth
sys-tem, ∼ 0.01 au sun-ward of the Earth. The instruments on board the spacecraft include six
high-resolution spectrometers and three additional monitoring instruments. Together they
study the elemental, isotopic, and ionic composition of energetic nuclei in interplanetary
space, at energies ranging from ∼ 1 keV/nucleon (solar wind) to ∼ 0.5 GeV/nucleon
(Cos-mic Radiation). This includes ions accelerated at different places: the Sun, ISM, the edge of
the heliosphere (bow shock), the Galaxy (for Galactic sources see Section
1.1
). The list of
all the instruments and their positions on board ACE is in Table
2.1
and Figure
2.2
.
We use data from ACE because the spacecraft is in a unique position for us to study particles
in situ. ACE also measures in a uniquely low energy regime which is what our work
con-centrates on. For this work, we combine data from three of the instruments on board ACE:
1
http://www.srl.caltech.edu/ACE/Gallery/gallery.html
2http://www.srl.caltech.edu/ACE/ASC/level2/index.html
3
Chapter 2. Advanced Composition Explorer
Acronym
Full Name
Type of Investigation
Energy
Range
CRIS
Cosmic Ray Isotope
Spectrometer
Isotope composition of cosmic rays
100–500
MeV/nuc.
EPAM
Electron, Proton and
Alpha Monitor
Energetic particles across a broad
range of energies
0.04-93
MeV
MAG
Magnetic Field
Exper-iment
Local magnetic field
SEPICA
Solar Energetic
Parti-cle Charge Analyser
Ionic charge states of energetic
parti-cles
0.1–5
MeV/nuc.
SIS
Solar
Isotope
Spec-trometer
Isotopic composition of energetic
nuclei from He to Ni
5–150
MeV/nuc.
SWICS
Solar Wind Ion
Com-position Spectrometer
Chemical and ionic charge state
composition of the solar wind, H and
He isotopes only
0.5–100
keV/Q
SWIMS
Solar Wind Ion Mass
Spectrometer
Chemical and isotopic composition
of the solar wind from He to Ni
0.5–20
keV/Q
SWEPAM
Solar Wind Electron,
Proton,
and
Alpha
Monitor
Solar wind plasma electron and ion
fluxes
0.001–36
keV
ULEIS
Ultra Low Energy
Iso-tope Spectrometer
Ion fluxes of suprathermal and
ener-getic particle ranges of H through Ni
0.02–14
MeV/nuc.
T
ABLE
2.1 List of scientific payload on board the ACE spacecraft, their purpose and the energy range where
they operate. Q is the charge and ‘nuc.” stands for nucleon. For this project, we use EPAM, SWEPAM and
MAG. Information taken from the ACE Science Center (ASC) website
2.
EPAM, SWEPAM and MAG. All the data have been obtained from the ACE Science Center
(ASC) public archives (
Garrard 1997
;
Garrard et al. 1998
). In this Chapter, we discuss the
the instruments we use in more depth.
Chapter 2. Advanced Composition Explorer
F
IG
. 2.2
–
Exploded model of the ACE spacecraft and all its instruments on board. Figure adopted from the
ACE Photo Gallery
3.
EPAM
2.1
EPAM
The Electron, Proton and Alpha Monitor (EPAM) is composed of two Low Energy Foil
Spec-trometers (LEFS), LEFS60/LEFS150, two Low Energy Magnetic SpecSpec-trometers (LEMS),
LEMS30/LEMS120, and Composition Aperture (CA), CA60. Each of the numbers after the
acronyms denote the orientation of the instrument with respect to the spacecraft spin-axis in
degrees. As the spacecraft spins, these five instruments sweep out the area around the
tele-scope, presenting nearly full coverage for ions and approximately 40% coverage for
elec-trons, where the electronics automatically break out the sample evenly (
Gold et al. 1998
).
The different time resolutions of EPAM are 12 s, 5 min, hourly and daily averages. LEFS
measure the flux and directions of electrons between ∼44 keV and ∼ 4.9 MeV. LEMS
mea-sure the flux and directions of ions between approximately 46 keV and 4.8 MeV. There is
also a Deflected Electrons (DE) detector, or DE30, measuring electrons that are being
de-flected from the LEMS30 detector by a rare-earth magnet. CA measures ion composition
between ∼ 0.4 and 92.7 MeV. Each of these energy ranges is split up between two or more
energy channels (
Stone et al. 1998
;
Hawkins 1999
). The large dynamic range is crucial for
understanding the dynamics of solar flares, interplanetary shock acceleration and
propaga-tion through heliosphere, and helps support the other instruments on board of ACE as well
(
Gold et al. 1998
).
LEMS30
LEMS120
Channel
Energy Range (keV)
Mean Energy (keV)
Energy Range (keV)
Mean Energy (keV)
1
46-67
56
47-86
56
2
67-115
88
68-115
88
3
115-193
149
115-195
150
4
193-315
247
195-321
250
5
315-580
427
310-580
424
6
580-1060
784
587-1060
789
7
1060-1880
1412
1060-1900
1419
8
1880-4700
2973
1900-4800
3020
T
ABLE
2.2 EPAM channels and their specific energy ranges. Information from
Hawkins
(
1999
).
DE30
Channel
Energy Range (keV)
Mean Energy (keV)
1
38-53
45
2
53-103
74
3
103-175
134
4
175-315
235
T
ABLE
2.3 DE30 channels and their specific energy ranges. Information from
Hawkins
(
1999
).
For this work, we use all the 16 LEMS30/LEMS120 and all the 4 DE30 channels (see Tables
2.2
–
2.3
for specific energy ranges) at the best possible resolution the instrument provides,
which is 12 seconds. The data is given in average particle intensities.
MAG
2.2
SWEPAM
The Solar Wind Electron, Proton, and Alpha Monitor (SWEPAM) measures the bulk flow
and kinetic properties of the solar wind. Some of the primary objectives of SWEPAM are
determining coronal composition, studying solar plasma conditions and solar wind
accelera-tion, and constraining particle acceleration models. SWEPAM has two electrostatic
analyz-ers that measure electrons and ions for studying these phenomena. Both analyzanalyz-ers measure
the energy per charge of a particle by bending their flight paths, where the ions are detected
between 0.26 and 36 keV and electrons between 0.001 and 1.35 keV (
McComas et al. 1998
;
Stone et al. 1998
).
SWEPAM data is available in 64 s, hourly or daily averages. We select the best possible
resolution, which is 64 s. We use the provided bulk proton number density, velocity and
temperature. The proton velocity SWEPAM measures is also determined with respect to
the GSE (Geocentric Solar Ecliptic)
4
, GSM (Geocentric Solar Magnetospheric)
5
, and RTN
(Radial Tangential Normal)
6
coordinates (
Stone et al. 1998
). We neglect the direction
infor-mation in lieu of making the simplest model possible.
2.3
MAG
The Magnetic Field Experiment (MAG) measures the local magnetic field in the IPM. MAG’s
observations facilitate studies of propagation and evolution of thermal and energetic particle
populations because MAG measures the large-scale structure of the interplanetary magnetic
field (IMF), its fluctuation and turbulence. MAG consists of two duplicate fluxgate
magne-tometers that are attached to booms extending outwards from opposite sides of the spacecraft
(
Smith et al. 1998
).
The time resolution of the data is in 1 s, 16 s, 64 s, 4 min, hourly, and daily averages. MAG
also measures the magnetic field with respect to the GSE, GSM and RTN coordinates. We
again neglect these measurements. Since we combine the data with those of SWEPAM,
which has a 64-second resolution, we select 64 s for MAG as well. We download the data
in bulk from the ACE Science Center website as MAGSWE merged datasets of MAG and
SWEPAM(
Stone et al. 1998
). From the MAG part of the dataset we use the average magnetic
field estimates.
4
GSE coordinate system is defined as having an origin at the Earth, positively increasing towards the Sun
(x-axis). The z-axis has a direction of the North Ecliptic Pole. The y-axis is the cross product of the x- and
y-axes.
5
GSM coordinate system has the same x-axis as the GSE coordinate system. The z-axis is the projection of
the Earth’s magnetic dipole axis onto the yz plane. The y-axis is again the cross product of the x- and y-axes.
6
RTN is the spacecraft centered coordinate system. The r-vector (x-axis) goes from the Sun towards the
spacecraft. The y-axis (t-vector) is the cross product of the solar rotational axis and the x-axis (r-vector). The
z-axis (n-vector) is once more the cross product of the x- and y-axes
Chapter 3
Analysis
3.1
Data Selection and Preparation
The selection of CME-driven shocks we survey in this work is based on the ACE Lists of
Disturbances and Transients
observed by ACE (
Smith 2016
)
1
. The MAG team has reported
the detection of 609 shocks between 1997 and 2016. We use all the identified 609 shocks
and the identified times of shock crossings that are when the ACE spacecraft encountered
each individual shock. The data are acquired from the ACE Science Center (ASC) public
archives (
Garrard 1997
;
Garrard et al. 1998
)
2
in 24-hour blocks of EPAM and MAGSWE
(merged MAG and SWEPAM data) in the time resolution of 12 s for EPAM and 64 s for
MAGSWE.
Whenever ACE was encountering difficulties on taking measurements that led to missing
or invalid data, the collaboration set the values to -9999.9 or -999.9 to distinguish from the
proper data. We replace these values by a python value describing empty measurement,
numpy.nan
or NaN (Not A Number), so as to be able to perform numerical mathematical
analysis with these data.
ACE does not provide instrumental errors for their measurements. All the uncertainties we
provide are statistical errors derived from error propagation or 1σ errors when discussing
mean values.
1
http://www.ssg.sr.unh.edu/mag/ace/ACElists/obs list.html
2
