University of Amsterdam
MSc Astronomy and Astrophysics
GRAPPA
Master Thesis
Particle Acceleration in Solar System Shocks
A discussion of particle acceleration models in the context of
CME-driven shocks
by
Michiel Bustraan
10097384
August 2015
54 ECTS
September 2014 - July 2015
Supervisor:
Jacco Vink, GRAPPA
Examiner:
Sera Markoff, API
Abstract
Het versnellen van deeltjes in schokgolven is lang bestudeert in de hoop dat dit het spectrum van kosmische stralingen kan verklaren. Hoewel hoge-energie deeltjes in supernova schokken geproduceerd worden, kunnen schokken ook op lagere energie¨en voorkomen. Dit project zal deeltjesversnelling in lage-energie schokken, geproduceerd in de zon, discussi¨eren. Het doel is om deze schokken te analyseren door te kijken naar drie aspecten van schokversnelling: de effici¨entie van deeltjesversnellng, de diffusie van versnelde deeltjes, en de turbulentie in het magnetisch veld in de schok. Het gebruik van ACE, de Advanced Composition Explorer, maakt het mogelijk om deze schokken met hoge precisie te bestuderen. Door het combineren van data verwerking, gebaseerd op metingen van ACE, en voorspellingen die gemaakt zijn door modellen hopen wij een dieper begrip te krijgen van deze lage-energie schokken.
Abstract
The complex subject of particle acceleration in shocks has long been studied in the hopes of explaining the abundance of cosmic rays in the universe. While high-energy cosmic rays are produced in shocks caused by supernovae, shocks can occur at much lower energies. This project will discuss shock acceleration models in the context of low-energy shocks produced by the Sun, known as CME-driven shocks. The goal is to understand these shocks in the context of three different aspects of shock acceleration: the efficiency of acceleration, the diffusion of accelerated particles, and the turbulence of the magnetic field within the shock. By using measurements made by ACE, the Advanced Composition Explorer, we can analyze these shocks with a high resolution that would simply be impossible for supernovae. By combining data analysis, based on ACE measurements, and predictions made by various shock acceleration models, we might be able to gain some deeper understanding of these low-Mach-number, CME-driven shocks.
Contents
I
Introduction and Context
4
1 Coronal Mass Ejections 4
2 Particle Acceleration in Shocks 7
2.1 Shock Structure . . . 7
2.2 Particle Acceleration Mechanism . . . 8
3 Project Goals and Overview 9 4 Shock Model Overview 10 5 Particles in the Solar System 11
II
Sources of Data
14
6 The Advanced Composition Explorer 15 6.1 EPAM . . . 156.1.1 The Low Energy Magnetic Spectrometer . . . 16
6.2 SWEPAM . . . 17
6.2.1 The Solar Wind Ion Instrument . . . 18
6.3 MAG . . . 20
7 Finding Shocks 20
III
Basic Shock Concepts and Methods
20
8 Frames of Reference 21 8.1 The Observer Frame . . . 218.2 The Shock Frame . . . 21
8.3 The Upstream Frame . . . 21
9 Finding the Shock Front 22
10 Characteristic Timescale 22
11 Binning and Errors 23
12 Efficiency and Mach Number 27
12.1 Thermal Pressure . . . 27
12.2 Non-thermal Pressure . . . 28
12.3 Acceleration Efficiency . . . 29
12.4 Compression Ratio and Mach Number . . . 29
13 Efficiency of CME-driven Shocks 30 14 Below the Critical Mach Number 32
V
Diffusion in a Turbulent Magnetic Field
34
15 Diffusion Coefficient and Index 38 15.1 Shock Velocity . . . 3815.2 Diffusion Index . . . 39
16 Diffusion in CME-driven Shocks 41
VI
Shock Gyrofactor and Resonant Waves
43
17 Calculating the Gyrofactor 44 18 Interpreting the Gyrofactor 45 19 Acceleration in a Magnetic Resonance 45 20 The Gyrofactor in CME-driven Shocks 46 20.1 MAG Time Resolution . . . 47VII
Conclusions and Summary
50
21 Further Research 51
Part I
Introduction and Context
Particle acceleration in shocks has always been interesting as a potential explanation for the cosmic ray spectrum1. While high-energy cosmic rays are produced in shocks caused by supernovae, shocks can occur on a much smaller scale. Our Sun can also produce inter-planetary shocks via Coronal Mass Ejections (CMEs), when the Sun releases a large amount of supersonic matter in one instance [29]. Shock produced by these CMEs are known as CME-driven shocks, and they are the focus of this project. While these shocks are not nearly as powerful as supernovae, they are capable of accelerating particles. The purpose of this project is to analyze several CME-driven shocks and determine how their particle acceleration properties match existing shock acceleration models.
These shocks are observed by ACE, the Advanced Composition Explorer, a satellite designed to measure every aspect of the solar wind. The ACE satellite allows us to separately measure the regular solar wind as well as the non-thermal population of accelerated particles. By measuring the density, temperature, and velocity of the solar wind, as well as the intensity of accelerated particles and the interplanetary magnetic field, we are capable of calculating key shock parameters for these CME-driven shocks.
This paper will discuss three shock acceleration models and concepts in the context of calculations made using ACE data of CME-driven shocks. The models that will be discussed are outlined in Section 4, while the discussions themselves take place in Parts IV,V, and VI. Before shock acceleration models are discussed, some fundamental information is provided. Part II describes how ACE measures aspects of the solar wind, and where the shocks analyzed in this project are found. Part III identifies several basic shock concepts and methods that are relevant to all stages of data analysis.
In this part of the paper, the introduction, some basic context is provided. The structure of coronal mass ejections is discussed, as well as the basic principles of shock acceleration. After this, the goal of the project is made clear along with a short summary of the models that will be discussed in the paper. Finally, there is a discussion on the solar environment, and other potential sources of energetic particles.
1
Coronal Mass Ejections
If we are going to analyze these shocks and use them to test shock acceleration models, we should consider the events that cause these shocks.
1For a comprehensive review of the cosmic ray spectrum and the role of shock acceleration in its existence,
Figure 1: A diagram showing the general structure of a coronal mass ejection, as well as the shock that can result. The solid lines represent magnetic field lines. Image obtained from McComas et al., 1998 (Figure 6) [30].
In the simplest terms, a Coronal Mass Ejection (CME) is a large release of gas and plasma into the solar wind [29]. These bursts of charged particles can be powerful enough to cause strong aurora on the Earth’s poles, as well as power outages [32].
During a typical CME, 1015−1016g of plasma is expelled into interplanetary space with a
kinetic energy of order 1031− 1032 ergs [29]. CMEs can be quite gradual events, lasting from
several hours to several days [29]. A paper by Emslie et al. (2012)[12] outlines the results of an analysis of thirty-eight different solar eruptive events, which usually include CMEs. Their
analysis provides estimates of the different energetic components of the events, making it a useful paper to understand the general scale and energy distribution of solar events.
CMEs can vary in composition, particularly at higher energies [38]. A review by Tylka et al. (2005)[38] provides a very comprehensive analysis of the composition of CMEs across a range of energies. Later in this paper, we will see that our calculations of the solar wind only take protons and Helium nuclei into account. While this does discount the higher elements certainly present in the solar wind, the results presented by Tylka et al. also show that these higher elements do not exist in high enough abundances to significantly impact the density or pressure of the shock system.
CMEs can form as closed magnetic loops that are attached to the Sun, or as detached plasmoids [11], as seen in Figure 1. For the former, the plasma is still connected to the Sun through the magnetic field, though reconnection of the magnetic field on the sunward side of the plasmoid could close the loop [11]. If this occurs, the CME propagates as a detached plasmoid.
With velocities typically ranging from 500 to 800 km/s at 1 AU [1], CMEs travel much faster than the typical slow solar wind that we see between the Sun and the Earth [18]. In fact, they travel fast enough to create a shock in the solar wind, known as an Interplanetary Shock. This shock, seen in Figure 1, travels in front of the CME, and can be a source of energetic particles. By accelerating semi-energetic particles2 injected by the CME through the typical process of shock acceleration (discussed in the next section), these CME-driven interplanetary shocks can produce particles with energies much greater than the typical solar wind particle. Particles accelerated this way are referred to as Interplanetary Shock Particles (ISPs), and also fall under the category of Solar Energetic Particles (SEPs). The fact that the charged, dense plasma in the CME bends the Interplanetary Magnetic Field (IMF) such that it is more perpendicular to the shock front also aids in accelerating particles. A model discussed in Dryer, 1994 [11] shows how the shock resulting from a CME propagates through the heliosphere.
Measurements of the solar wind, made by ACE, can be seen in Figure 2. We can see the characteristic jump in velocity for low energy particles that defines the shock front. The corresponding spike in intensity for higher energy particles shows the effect of particle acceleration in CME-driven shocks. A correlation has been observed between the velocity of a CME-driven shock and the intensity of energetic particles detected at the same time [23]. Measurements of all shocks analyzed in this project can be found at the end of this paper, in the Appendix.
2By semi-energetic, we refer to particles above typical solar wind energies that are produced directly in
Figure 2: Measurements made during a CME-driven shock. Left: low-energy measurements of the solar wind velocity, showing the shock front for thermal particles. Right: measurements of SEPs, showing the jump in high-energy particle intensity caused by particle acceleration in a CME-driven shock.
2
Particle Acceleration in Shocks
The theory of particle acceleration in shocks has existed for a long time, and was typically applied to the acceleration of particles in supernova remnant shocks. Particle shock accel-eration was proposed as a way of explaining the presence of high energy cosmic rays, as was done by Axford (1981)[3]. The concept of particle acceleration in shocks, particularly with respect to high energy cosmic rays, was extensively discussed in a pair of papers by Bell, both written in 1978 [5][6]. Currently, there are a number of established theories and equations that are commonly used to calculate the characteristics of a shock.
2.1
Shock Structure
Shocks have been studied for several decades, starting with an article by de Hoffmann & Teller (1950)[19], which introduced the concept of a magneto-hydrodynamic shock. Since then, the theory behind the structure of magneto-hydrodynamic shocks has expanded considerably.
The most established method for calculating the structure of a shock, as well as its jump conditions, is through the Rankine-Hugoniot relations. These relations use the conservation of mass, momentum, and energy across the shock front to describe how the upstream and downstream regions of a shock are related. These relations are critical for calculating some key parameters of a shock, such as Mach number and shock velocity. The first three Rankine-Hugoniot relations are shown below in the shock frame (defined later)[24].
ρ1v1 = ρ2v2,
ρ1v1 e1+ 1 2v 2 1+ P1 ρ1 = ρ2v2 e2+ 1 2v 2 2+ P2 ρ2 .
Where ρ represents fluid density, v is the fluid velocity, P is the pressure, and e is the specific internal energy of the fluid. The subscripts denote whether the value refers to the upstream or downstream region of the shock. These relations are critical for calculating parameters that determine the shock’s physical structure. In the case of a strong magnetic field, the Rankine-Hugoniot relations can be modified by considering the magnetic field in the conservation equations. This leads to the MHD Rankine-Hugoniot relations, seen in Kennel et al., 1989 [24].
Using the Rankine-Hugoniot relations and the concept of a compression ratio, χ, we can define the sonic Mach number of the shock, MS, in terms of shock parameters [39][28].
χ ≡ ρ2 ρ0 = v0 v2 , MS ≡ Vs cs , χ = (γ + 1)M 2 S (γ − 1)M2 S+ 2 , MS = s 2χ γ + 1 − χ(γ − 1).
Where Vs is the shock velocity, cs is the sound velocity in the solar wind, and γ is the
adiabatic index. The Mach number and compression ratio are key parameters that represent the size of the density jump in the shock and the shock velocity, respectively. As such, they have a strong impact on the nature of the shock as a whole.
This model of shock structure, based on Rankine-Hugoniot relations, can be made more complex by including various effects. For example, a paper by Drury & V¨olk (1981)[9] models the influence of cosmic rays on the structure of a hydromagnetic shock.
2.2
Particle Acceleration Mechanism
The method through which particles are accelerated in a shock can vary for specific models, but most models generally incorporate the concept of diffusive shock acceleration. This is the most basic concept to explain shock acceleration. The principle is that particles in the shock are bound to magnetic field lines, and orbit around them with a given gyroradius, though turbulence in the plasma’s magnetic field does allow for the particles to diffuse within the plasma. If the particle is near the shock front and its gyroradius is greater than the thickness of the shock front, the particle can pass freely between the regions upstream and downstream of the shock [5]. Since the thickness of the shock front is comparable to the gyroradius of a thermal proton, a particle must already have a higher energy to be able to cross the shock front at all [10].
The magnetic field near the shock front is assumed to be fairly turbulent. In some papers, such as Bell (1978)[5], the magnetic turbulence is assumed to be caused by energetic particles escaping the shock. Other authors, such as Jokipii (1966)[21], assumed the turbulent magnetic field is caused by the shock itself. Beyond just the source of turbulence, the type of magnetic turbulence has an influence on particle acceleration as well. However, the subject of turbulence in a shock is complex, and is only discussed further in Parts V and VI.
When a particle crosses the shock front, there is a chance that the particle is scattered by the turbulent magnetic field, which might send it back towards the shock [27]. In this way, a particle might cross the shock front multiple times before it escapes the shock entirely. Since the particle has a chance of escaping that is energy dependent [5], particles beyond a certain energy are unlikely to be bound by the shock at all, and simply escape the shock front.
Each time the particle cross the shock front, it gains energy from interactions with the turbulent magnetic field, and occasional collisions with other particles. This process is discussed at length in Bell (1978a)[5] and Drury (1983)[10]. By considering how much energy a particle gains with each shock crossing as well as the velocity of the particle, one can estimate an acceleration rate. This is done in Jokipii (1987)[22]. By considering the escape probability, a maximum particle energy can be calculated, which represents the highest energy that particles can be accelerated to in this shock.
While this is a simplified description of the process, there are a number of sources that describe the process of particle acceleration in shocks in great depth. For instance, a review by Drury, written in 1983 [10], discusses the core principles of diffusive shock acceleration, and can be considered a fundamental paper on the subject of particle acceleration in shocks. A more modern overview of the subject can be found in the review by Malkov & Drury, written in 2001 [28]. While this review is a more accurate representation of the subject of diffusive shock acceleration as it exists today, it also contains more complicated subject matter. As an introduction to the subject, the papers by Bell [5][6] mentioned earlier are a good starting point.
3
Project Goals and Overview
There are many models that expand the general concepts of shock structure and particle acceleration in shocks. However, many of these models have not been tested in, or applied to, low Mach number systems. It would be interesting to know whether low Mach number shocks adhere to certain shock acceleration models, especially if these models were initially designed with higher Mach numbers in mind.
Interplanetary shocks have a number of qualities that make them ideal for testing shock acceleration models. When shocks are applied to physical systems, it is often in the environ-ment of supernova remnant shocks and the cosmic ray spectrum, as in a review by H¨orandel (2008)[20]. While these shocks are certainly relevant as accelerators of cosmic rays, they do not provide a complete picture of shock acceleration. If we are interested in investigat-ing the shock acceleration properties of low-Mach-number shocks, we must look at sources other than supernovae. CME-driven shocks, with Mach numbers below 10, show us how
shocks behave at the lowest Mach numbers possible. This means CME-driven shocks are an important, and irreplaceable, part of the Mach number spectrum of shocks.
CME-driven shocks are also in an ideal position to be measured by us. They can be measured by placing a satellite just beyond the Earth, making them easy to reach. They travel radially between the Sun and the Earth, so angular effects are typically minimal. There is relatively little material between the Sun and the Earth, so the shock is unlikely to interact with matter before reaching our measuring apparatus. Finally, the use of a satellite lets us measure the shocks with precise, local measurements as the shocks pass through. Examples of such satellites are SOHO[17] and ACE[36].
Of course, interplanetary shocks are not perfect. The solar wind that the shock passes through does not have a constant density, and can experience sizable fluctuations, which can interfere with measurements of the shock. Particularly for very low Mach number shocks, where the density jump is small, the noise in the data can be very significant relative to what we want to measure. Even with these imperfections, CME-driven shocks can be used to test shock acceleration models at low Mach numbers, though not all shocks are suitable for analysis.
The goal of this project is to use CME-driven shocks to test three shock acceleration models, outlined in the next section. This is done by identifying, for each model, the relevant shock parameters that are needed to determine whether the shock follows predictions made by the model. Then these shock parameters are calculated using solar wind measurements made by the ACE satellite, discussed more in a later part of the paper. This is done for a number of shocks in order to obtain varied results, as well as results that are statistically more significant. By testing three models that relate to different aspects of shock acceleration, the goal of the project is to determine, to some extent, whether CME-driven shocks agree with shock acceleration models and concepts.
4
Shock Model Overview
In this section, the three models investigated in this project are briefly summarized, as preparation for a more complete discussion in later parts of the paper.
The first model to be discussed, in Part IV, involves the concept of a critical shock Mach number. This model, from a paper by Vink and Yamazaki written in 2014 [39], makes two definite statements for particle acceleration in low Mach number shocks. Firstly, it shows that there should be a minimum Mach number for shock acceleration. Below this Mach number particle acceleration should not be possible. Secondly, the authors show that there is a maximum efficiency for particle acceleration, which depends on the Mach number of the shock. This model is investigated by calculating the Mach number and acceleration efficiency of CME-driven shocks.
Calculating the efficiency of particle acceleration in CME-driven shocks was recently done in a paper by Mewaldt et al. (2008)[31]. In their analysis, Mewaldt et al. estimate the efficiency of particle acceleration in a CME-driven shock by calculating the total kinetic energy of the CME and comparing this to the energy carried by SEPs. This is a simplification,
of course, and I would direct readers to the original article [31] for a more comprehensive discussion. While both Mewaldt et al. and Vink & Yamazaki (2014) [39] refer to the efficiency of particle acceleration, they are referring to different measures of efficiency. In this paper, as well as in Vink & Yamazaki (2014), acceleration efficiency is defined as the ratio between the non-thermal and the total pressure in the shock system. Even so, there could be value in comparing the results of this project to the results obtained by Mewaldt et al. (2008).
The second model, in Part V, is on the subject of diffusion of accelerated particles in shocks. Based on a paper by Giacalone and Jokipii from 1999 [14], the model predicts a particular relationship between the diffusion of particles in the shock and the energy of the diffusing particles. This model is investigated by calculating the diffusion coefficient of the shock precursor at various energies, and observing the dependence of the latter on the former. This can then be compared to the dependence implied by Giacalone and Jokipii.
Finally, a model regarding magnetic resonance in shocks will be discussed, in Part VI. This concept, first introduced by Blandford and Eichler in 1987 [7], states that a resonance can exist between particles rotating around their gyroradius and the magnetic fluctuations that travel through the shock. This could make particle acceleration more efficient in shocks where such a resonance exists.
5
Particles in the Solar System
At any given time, the solar system is filled with a variety of particles from a number of different sources. If we are to discuss results based on measurements of interplanetary shocks, it is important that we understand other potential sources of energetic particles in the solar system. In order to determine whether these other particles can cause a significant amount of uncertainty in our measurements, we have to consider the energies at which they are produced, and how consistent their production is. To do this, we have to consider the mechanism that leads to their production.
The Sun is consistently producing low energy particles in the form of a solar wind. The solar wind exists because these particles, with energies on the order of a few keV, are energetic enough to simply escape the Sun’s corona. The solar wind comes in two forms. The first, known as the slow solar wind, is more common along the heliospheric equator [18], and is hence the form of solar wind that the Earth is typically exposed to. The slow solar wind has a velocity around 400 km/s [18], and a magnetic field on the order of 5 nT, which is shown in ACE measurements in the Appendix at the end of this paper. By contrast, the solar wind being emitted near the Sun’s magnetic poles travels much faster. Dubbed the fast solar wind, it is thought to be caused by holes in the corona [18]. This allows particles to escape directly from the inner regions of the Sun, leading to a solar wind with velocities on the order of 800 km/s [18]. As we will see later in this paper, velocities of this order are high enough to cause a shock in the slow solar wind.
Due to the large difference in velocity between the fast and slow solar wind, regions where they interact or collide can be a source of higher energy particles. Due to the rotation of the Sun, the solar wind streams will form spirals around the Sun. The slow solar wind will
travel along a more curved path than the fast stream. This can cause the fast stream to slam into the rear of the slow stream, as seen in Figure 3. In the overlapping region, the fast stream moving through the slow stream can create a shock in the solar wind. This leads to two shock fronts that propagate away from the interface.
Figure 3: A diagram illustrating how slow and fast streams of the solar wind can interact to create both a forward and reverse shock front. The rotation of the Sun, counter-clockwise in this figure, causes the trajectories of the streams to spiral, with the slow stream following a more bent trajectory. This allows the fast stream to collide with the slow stream. Image obtained from Pizzo, 1978 [34].
The shocks created by the interactions between the slow and fast components of the solar wind are known as Corotating Interaction Regions (CIR). Since this project is focused on particle acceleration in interplanetary shocks, we have to consider whether these CIR
shocks can produce similar measurements. While these shocks are potentially capable of accelerating particles, the fact that they lack an injection of energetic particles to begin with, due to only containing solar wind particles, limits their acceleration capabilities. Because of this, particles accelerated in CIR events will typically be of a lower energy than particles accelerated in CME-driven shocks, on the order of 100 keV per particle [36]. Still, this could mean CIR particles could be measured by ACE and interpreted as particles produced in a CME-driven shock.
However, if we consider the fact that these CIR events take place some distance away from the Sun, and the fact that the measurement apparatus on ACE is directed towards the Sun at all times (discussed in a later section), the measurement of these CIR events will likely not be as noticeable as a measurement of a CME-driven shock. Also, due to the relatively consistent nature of these CIR events, they are more likely to provide a minor background in measurements of higher energy particles, as opposed to leading to any real errors in measuring driven shocks. The differences in composition between a CME-driven shock and a CIR event also make it easy to distinguish one from the other. While CIR events are composed of ordinary solar wind particles, CME-driven shocks typically contain higher abundances of He and heavy ions [33].
There are not many other sources of particles that we seriously have to consider. While there are cosmic rays present in the heliosphere, seen in Figure 4, these are at high enough energies that they will not interfere with the measurements used in this project [36].
Figure 4: This diagram illustrates potential sources of particles in the heliosphere, ranging from the solar wind to galactic cosmic rays. [36]
Part II
Sources of Data
All raw data used in this research project have been obtained from the ACE Science Center (ASC). The ASC [2] handle and process the raw data obtained by the Advanced Composition Explorer (ACE). These data have been used in this research project to calculate relevant shock parameters, for shocks detected via automatic means. This part of the report will explain the source of the data. Both the actual method of data collection by ACE will be
outlined, as well as how shocks were found.
6
The Advanced Composition Explorer
The Advanced Composition Explorer, or ACE, is a satellite launched in August 1997, having been proposed in 1986 [36]. The cylindrical spacecraft, which rotates at 5 rpm, orbits the first Earth-Sun Lagrange point, 240RE sunward of the Earth [36]. Since the spin axis usually faces
the Sun, the scientific equipment is mostly on one side of the cylinder, so that it generally faces the Sun directly.
ACE carries nine independent detecting devices. However, for this research project, only three of these have been used. These detectors, which measure the thermal and non-thermal population as well as magnetic fields, will now be discussed; the data we can obtain from them and other relevant information will be outlined.
6.1
EPAM
The Electron, Proton and Alpha Monitor, also known as EPAM, is used to measure the flux of higher energy particles 3. While EPAM also measures the composition of these particles,
the data of most interest for this project are the flux of ions. This flux is assumed to represent the dynamic behavior of the non-thermal particle population in the solar wind.
EPAM has two devices that measure ion flux, both denoted as Low Energy Magnetic Spectrometers (LEMS) [13]. These detectors, LEMS120 and LEMS30, measure the flux of ions at an angle of 120 degrees and 30 degrees from the spin axis, respectively. Their method of operation is discussed at the end of this section. The result is a measurement of particle flux, in units of 1/(cm**2-s-sr-MeV), where the energy dependence represents a division of the count rate by the size of the energy band. The EPAM measurements used in this research project are at the highest available time resolution of 5 minutes. By taking the measurement from LEMS120 and LEMS30 and averaging them, we are guaranteed to obtain an isotropic particle flux.
The energy bands of the LEMS detectors are defined on the ACE website [2], and given below. The energy bands suggest that particles measured by EPAM will have an energy above that of the typical solar wind, and are therefore non-thermal. Since these energies are below those of galactic cosmic rays, we can assume that particles measured by EPAM were accelerated in the solar system. Because of this, particles accelerated by CME-driven shocks in the solar system would be measured by EPAM, and this is why EPAM is used to measure the behavior of non-thermal accelerated particles in the shocks observed in this project.
LEMS120
Energy Band Mean Energy (keV) Energy Range (keV)
P1’ 56 47-68 P2’ 88 68-115 P3’ 150 115-195 P4’ 250 195-321 P5’ 424 310-580 P6’ 789 587-1060 P7’ 1419 1060-1900 P8’ 3020 1900-4800 LEMS30
Energy Band Mean Energy (keV) Energy Range (keV)
P1 56 46-67 P2 88 67-115 P3 149 115-193 P4 247 193-315 P5 427 315-580 P6 784 580-1060 P7 1412 1060-1880 P8 2973 1880-4700 6.1.1 The Low Energy Magnetic Spectrometer
The EPAM device is primarily composed of two detectors. The Low-Energy Foil Spectrom-eter (LEFS) is used to detect electrons and is not used in this project. The second is the Low-Energy Magnetic Spectrometer (LEMS), which detects ions in the solar wind. Apart from a key difference, discussed later, these two detectors operate in the same way.
Both the LEFS and the LEMS detectors are relatively small detectors encased in a large cylindrical container. This creates an aperture with a look angle of 51 degrees for LEMS, as seen in Figure 5. The detectors themselves are totally depleted, solid-state, silicon surface barrier detectors with a thickness of approximately 200 µm [16]. Incident particles penetrate the detector and create electron-hole pairs in the material, which can be detected. More energetic particles will create more electron-hole pairs, allowing the energy of the particle to be measured. Particles that are energetic enough will pass through the detector entirely. Measurements on these particles will only provide a lower limit for the particle energy, and this also serves as the highest energy the detector can measure, seen in the tables above. The LEMS and LEFS detectors come in pairs, and are placed back-to-back. This allows each detector to serve as the anti-coincidence detector of the other.
The key difference between the LEMS and LEFS detector lies in their structure around the detector, and can be seen in Figure 5. The LEFS detector has a thin aluminized Parylene foil in front of the detector [16]. This foil blocks larger particles, such as protons and alpha particles, while allowing electrons through. In this way, LEFS only detects electrons. The
LEMS detector, on the other hand, has a set of rare-earth magnets in front of its detector [16]. This set of magnets serves to sweep out electrons before they can reach the detectors, while protons and alpha particle are heavy enough to pass through. In this way, LEMS only detects protons and alpha particles. Of course, very slow ions will still be directed away from the detector, leading to the minimum detection energy for LEMS shown in the tables above.
Figure 5: A diagram illustrating the size and structure of the LEFS60 and LEMS120 detec-tors. F’ and M’ are used to point out the solid state detectors for the LEFS60 and LEMS120 systems, respectively. This diagram can be found as Figure 3 in the original paper [16].
6.2
SWEPAM
We will use data obtained from the Solar Wind Electron Proton Alpha Monitor, or SWEPAM, to calculate the behavior of the thermal population. SWEPAM utilizes a spherical section electrostatic analyzer to measure the three-dimensional behavior of electrons and ions [30].
Since the normal to SWEPAM’s aperture is 18.75 degrees from the ACE spin axis, it is capable of measuring ions arriving between 0 degrees and 65 degrees from the spin axis [30]. Since the spin axis faces the Sun, SWEPAM can only measure particles that come directly from the Sun, such as the solar wind. This means that it is unlikely that SWEPAM’s measurements are polluted by particles coming from others sources.
With an energy range of 260 eV to 36 keV, SWEPAM only detects low energy particles that comprise the bulk solar wind. By counting and measuring particles in this energy range, SWEPAM can determine the temperature, density, and velocity of the solar wind. Its time resolution of 64 seconds [2] is significantly greater than that of EPAM, which also leads to greater statistical fluctuations in the data. Because of this, SWEPAM measurements have higher inherent statistical errors than EPAM.
Due to the lower energy range of SWEPAM and its high time resolution, it is the best tool to detect shocks. Since SWEPAM does not usually detect a precursor, as those tend to appear at higher energies, the discontinuity caused by the shock is clearly visible in SWEPAM data. What follows is a more detailed discussion of the SWEPAM device.
6.2.1 The Solar Wind Ion Instrument
The instrument in SWEPAM that is responsible for detections relevant to this project is known as the solar wind ion instrument, or SWEPAM-I. It contains a spherical section electrostatic analyzer (ESA) [30]. Ions enter the instrument through the narrow aperture between the two electrostatic analyzer plates, seen in Figure 6. The inner electrostatic analyzer plate carries a high negative voltage, which only allows particles with a certain energy per charge range and azimuthal angular range to pass through the analyzer. This ensures that only particles in the given energy range are detected. Once the ions pass through the analyzer plates, they encounter 16 channel electron multipliers (CEMs).
The instrument has one enlarged aperture that allows ions to enter from one side of the spherical plates, as seen in the cross sectional side view in Figure 6. The size of the aperture allows particles to enter from polar angles ranging from -21.25 degrees to 59.75 degrees. However, due to the rotating nature of ACE and SWEPAM, this effectively leads to a viewing angle of up to 60 degrees from the ACE spin axis.
Figure 6: A diagram illustrating the solar wind ion instrument SWEPAM-I. This diagram can be found as Figure 10 in the original paper [30]
Particles entering the instrument across this wide range of polar angles are funneled into the 16 CEMs. This effectively provides each CEM with a narrow viewing angle, seen in Figure 6. Hence, a detection by a particular CEM can be traced back to a given polar entrance angle, with a resolution between 5 and 2.5 degrees. This allows ion measurements to be separated into three-dimensional components, by combining an ion’s CEM number with the spin phase of the instrument and ACE.
The CEMs ultimately count the particles that pass through the aperture. Combining the counts from all CEMs allows for solar wind density measurements. Since the voltage on the inner electrostatic analyzer plate controls the energy of the particles that are detected, changing the voltage allows for measuring particles of different energies. By periodically changing the voltage, known as the ESA step level, SWEPAM-I can detect particles across its entire energy range of 260 eV to 36 keV. SWEPAM-I typically only explores the full energy range every 32 minutes, and measures peak energies otherwise [30].
By taking into account the CEM that detects a particle, the spin phase of ACE, and the ESA step level, SWEPAM-I can measure an ion’s energy (and hence velocity and temper-ature) in three dimensions. Averaging these measurements every 64 seconds, this provides velocity and temperature measurements of the solar wind that are separated across three dimensions, in addition to solar wind density measurements.
6.3
MAG
ACE is also equipped with a Magnetometer (MAG), which measure the local interplanetary magnetic field direction and magnitude. the MAG instrument is a twin triaxial fluxgate magnetometer system [2]. This means that it uses two sensors placed on booms that extend slightly beyond the solar panels on ACE. Using this method, MAG is capable of calculating the direction and magnitude of the magnetic field reliably in the range of 4 nT to 65,536 nT, with a resolution ranging from 0.001 nT to 16 nT, respectively [2].
This project will use MAG data at two different time resolutions. When the magnetic field amplitude or direction is needed, a resolution of 16 seconds is used. At a lower resolution of 4 minutes, the variance in the magnetic field can also be obtained directly from ACE. ACE does this by separating the data at 16 second resolution into 4 minute bins, and calculating the variance of each bin. In this way, ACE provides us with the magnetic field amplitude, direction, and variance.
7
Finding Shocks
Since a shock leads to a discontinuity in the density and velocity of the solar wind, predicted by the Rankine-Hugoniot equations, one can detect shocks by looking for sudden changes in the solar wind. While it would be possible for us to simply take the dataset for an entire year and look for discontinuities, this is not what was done. Since the solar wind contains fluctuations even in the absence of a shock, simply looking for sudden changes will lead to many false shock detections.
To reliably find shocks, one must take several aspects of the solar wind into account. By looking for jumps that occur in several datasets at once, a false detection is made less likely. Specifically, we would require a jump to occur in the temperature, velocity, density, and magnetic field amplitude at the same time. For example, a shock detection method by Vorotnikov et al [40] requires a 1.5% jump in velocity, a 15% jump in temperature, and a 20% jump in proton density to occur at the same time during the initial shock selection.
While it would be possible to do this in this research project, it was not necessary to do so. There are various methods, such as the one by Vorotnikov et al, that have already detected a large number of shocks. These detected shocks can be found on a website maintained by the University of New Hampshire [1]. For this project, shocks from this list of detected shocks were selected, and the data for the corresponding dates were downloaded. Within this datafile, the shock is detected by searching for sudden changes in the density.
Part III
Basic Shock Concepts and Methods
8
Frames of Reference
In the following sections, various shocks parameters, such as Mach number, will be defined as functions of velocities and densities. However, we must understand that there are actually three frames of reference that are relevant for these calculations. For clarity, these frames, and the way they will be distinguished, will now be discussed.
Of course, measurements of temperature and density, and hence compression ratio, are frame-independent. It is only that we must be careful when we discuss velocity that we must understand which velocity we are discussing.
8.1
The Observer Frame
First, we have the observer frame. That is to say, the frame that ACE is in. In this frame, the solar wind as well as the shock are traveling towards ACE, with what we define as positive velocity. The particles move with a bulk velocity denoted by ui, where the subscript denotes
the region of the shock. The shock has a velocity defined as Vs, which is not measured
directly by ACE, since ACE only measures the wind before and after the shock.
8.2
The Shock Frame
All data obtained from ACE are in the observer frame, but theoretical models of shocks, such as the Rankine-Hugoniot relations, do not take place in this frame. Equations that describe shocks generally take place in the shock frame, in which the shock is stationary. In this frame, the upstream solar wind is moving towards the shock front with a velocity v0, while the downstream wind is moving away from the shock with velocity v2. Using the
convention above, these would be negative velocities.
Since the shock is stationary in the shock frame and moves with a velocity Vs in the
observer frame, it is clear how one can move between frames in the case of a non-relativistic shock:
ui = vi+ Vs (1)
8.3
The Upstream Frame
While the observer frame is the frame in which measurements are made, and the shock frame is where the Rankine-Hugoniot equations apply, there is one more frame we must consider. This is the frame where the upstream solar wind is stationary. It is in this frame that the upstream sound speed is calculated. In this frame, the precursor is stationary, with the shock
moving towards it with a modified shock velocity Vs∗. Clearly, to move between this frame and the observer frame, we have the following:
Vs = Vs∗+ u0 (2)
9
Finding the Shock Front
Before we can calculate the parameters of the shock, we must first determine accurately where the shock front is for any particular shock. In a previous section it was discussed that shocks are found by looking through a list of previously detected shocks. This provides us with the dates of detected shocks. However, to calculate the parameters of the shock, we need to know exactly where the shock front is in the data set, so that bins can be established relative to the shock front. This can be done by considering the fact that a shock is defined by a jump condition. By simply calculating the difference in density between each data point, we can look for a large gradient in the density. This would represent the shock front.
113.0 113.2 113.4 113.6 113.8 114.0 Day of Year, 2002 0 5 10 15 20 25
Particle Number Density
113.0 113.2 113.4 113.6 113.8 114.0 Day of Year, 2002 0 10 20 30 40 50 60 70 80 90 Density Gradient
Figure 7: Plots showing particle number density (left) and density gradient (right). Notice the peak in the gradient plot at the time of the shock crossing.
In Figure 7, we see what a gradient plot resulting from a density plot looks like. By calculating the density gradient and finding the maximum value, we can easily find the point that defines the shock front. This is critical since the majority of later calculations rely on bins with a position relative to the shock front.
10
Characteristic Timescale
The characteristic timescale can be defined as the timescale on which the non-thermal particle precursor decays in time. This is calculated by fitting an exponential function to the region (20 minutes) just before the shock for the particle intensity data.
This timescale is used to determine the region around the shock front that is considered to be relevant and dominated by shock physics. Specifically, we only consider a region of two characteristic timescales after the shock front to be part of the shock. Also, the characteristic timescale allows us to determine the degree of diffusion in the precursor. From this point on, the characteristic timescale will be denoted as τ .
11
Binning and Errors
In order to calculate more complex shock parameters, such as Mach number, we need to be able to determine key shock values, such as the density of the wind before and after the shock. This is done by taking bins across the necessary regions of the shock. An average is taken of the values in the bin to obtain a value for a basic shock parameter, such as density. This allows us to measure a shock without being heavily affected by the statistical fluctuations that exist in the solar wind. The size of the time bins is typically equal to 2τ .
The Advanced Composition Explorer does not provide instrumental errors for their mea-surements, and so these cannot be taken into account. However, by taking a bin when calculating values, we are able to calculate a statistical error for every value calculated from a shock. The error is determined by calculating the variance of the values in the bin, as is standard for statistical errors.
Part IV
Critical Shock Mach Number
This part of the paper will examine the model discussed by Vink & Yamazaki in their article, published in 2014 [39] . First, the main points and conclusions of the article will be outlined, as well as how the predictions of the model could be tested or confirmed. This is followed by a section that outlines how the relevant shock parameters are calculated (acceleration efficiency and Mach number, in this case). Finally, results from observations of CME-driven shocks are shown and discussed.
The model from Vink & Yamazaki [39] uses the Rankine-Hugoniot relations, as well as the concept of a shock precursor, to predict some interesting shock behaviors at low Mach numbers. By separating the shock into three regions, and applying simple conditions, shown below, the authors were able to show how the acceleration efficiency and escaping energy flux in a shock depends on basic shock parameters. This is done by beginning with standard shock equations, shown below.
χprec = ρ1 ρ0 = v0 v1 , χsub= ρ2 ρ1 = v1 v2
, χtot = χprecχsub =
ρ2 ρ0 = v0 v2 , Mg,0 ≡ s ρ0Vs2 γgPg,0 , Mg,1 ≡ s ρ1v12 γgPg,1 = Mg,0χ−(γprecg+1)/2,
χsub =
(γg+ 1)Mg,12
(γg− 1)Mg,12 + 2
.
Where χ and M represent the compression ratio and Mach number in a given region of the shock, respectively. As you can see, these typically depend on the density ρ, the fluid velocity v, the shock velocity Vs, the adiabatic index γg, and the pressure P . The numbered
subscripts denote a region in the shock, where 0, 1, and 2 refer to the unshocked medium, the precursor, and the region just behind the shock, respectively. The subscript ’g’ refers to the gas in the shock.
These relations, provided in the original paper as equations A2, A3, A4, and A12 [39], lay the groundwork for the derivation of the relations for acceleration efficiency and escaping energy flux provided below. The details of this derivation are not shown here, but are extensively covered in ’Appendix A’ of the original paper [39].
Through the use of these and other relations, the Vink & Yamazaki were able to derive two relations, one for the acceleration efficiency ω and the other for the escaping energy flux , defined below.
ω ≡ Pcr Ptot
≡ 1Fcr
2ρ0Vs3
Where Fcr is the cosmic ray flux. These shock parameters can be rewritten as follows
[39]. ω = 1 − (1 − ω0)χ γg prec+ (1 − ω0)γgMg,02 1 −χ1 prec 1 + (1 − ω0)γgMg,02 1 −χ1 tot , = 1 + 2 γgMg,02 1 1 − ω0 G0− G2 χtot − 2G2 χtot + 1 χ2 tot (2G2− 1), Gi ≡ ωi γcr γcr− 1 + (1 − ωi) γg γg− 1 .
Where G0 and G2 are constants so long as the adiabatic index is the same for the gas and
the cosmic rays [39]. We also see that G0 = G2 = 5/2 in the case where γg = γcr = 5/3, as is
typically assumed for non-relativistic particles. We see that these parameters only depend on the shock Mach number, Mg,0, and the precursor compression ratio, χprec, if we use the
relations given above. One could then calculate the acceleration efficiency and escaping energy flux for a range of compression ratios and Mach numbers. The result of this is shown in Figure 8. As a negative escaping energy flux is nonphysical, the curves are cut off when the escaping energy flux reaches zero. Since they share the same axis, one could plot the acceleration efficiency and escaping energy flux against each other, creating a curve for each individual Mach number. Shocks are expected to exist on this curve if the assumptions made in this model are reasonable. The result of this is shown in Figure 9.
Figure 8: Plots showing acceleration efficiency (top) and escaping energy flux (bottom) as a function of the precursor compression ratio for a discrete range of Mach numbers (shown in the legend). The curves are cut off when the escaping energy flux reaches zero.
Figure 9: Plot showing acceleration efficiency plotted against escaping energy flux. This is based off the plots in Figure 8, where acceleration efficiency and escaping energy flux are calculated as functions of the precursor compression ratio.
The acceleration efficiency and escaping energy flux of a shock are predicted to always follow a curve like the ones shown in Figure 9. The exact nature of this curve depends on the Mach number of the shock. There are, however, two values that exist for each of these curves, independent of Mach number. Firstly, we see that there exists, for every Mach number, an acceleration efficiency for which the escaping energy flux is maximal. While it might be appealing to believe that shocks are more likely to exist near this peak, in order to most effectively release cosmic rays, there is nothing that prevents shocks from existing elsewhere on the curve.
Secondly, we see that each curve ultimately reaches an escaping energy flux of zero as the acceleration efficiency increases. Since a negative escaping energy flux is nonphysical, this effectively places a limit on the acceleration efficiency. This maximum efficiency only seems to depend on the Mach number, in the case of non-relativistic cosmic rays and no preexisting
cosmic ray population4. It should be noted that a shock with this maximum efficiency would
not be considered physical. To obtain this maximum efficiency, the shock would need to be continuous, consisting of only a precursor without a shock jump.
Finally, we see that as the Mach number decreases, the maximum efficiency decreases with it. As one keeps decreasing the Mach number, a point is reached where the escaping energy flux at any non-zero acceleration efficiency will be negative. This means that the shock is nonphysical if particles are accelerated at all. The Mach number where this first occurs is referred to as the Critical Shock Mach Number, and it can be considered the minimum Mach number a shock can have while still accelerating particles. Vink & Yamazaki found this value to be Ms =
√
5 [39]. It should be mentioned that this value is obtained, as shown above, without considering the effects of magnetic fields in the shock. If a shock is magnetically dominated, the critical Mach number would be different, as would the method for obtaining it. This leads to a small modification of the critical Mach number. However, since the CME-driven shocks looked at in this paper do not appear to be heavily magnetically dominated, these considerations are not necessary here.
We see that this model imposes two limitations to low Mach number shocks: a maximum efficiency that depends on Mach number, and a critical shock Mach number. We can test whether the low-Mach-number shocks in the solar systems are limited by these conditions in the way Vink & Yamazaki predict. This is simply done by calculating the acceleration efficiency and sonic Mach number of each shock and checking whether any shock significantly disagrees with the limitations set by Vink & Yamazaki. The next section will outline how these parameters are calculated in this project, using ACE data, and this is followed by a discussion of results.
12
Efficiency and Mach Number
We saw that the acceleration is effectively a ratio of pressures. This means that we need to calculate the components of the pressure in the CME-driven shock in order to calculate the acceleration efficiency. Sonic Mach number can be calculated from the compression ratio using the Rankine-Hugoniot relations, as is shown below. It should be remembered from a previous part of the paper that measurements of the thermal and non-thermal components of the shock are done separately by SWEPAM and EPAM, respectively.
12.1
Thermal Pressure
We calculate the thermal pressure in our system by assuming the simplest equation of state, the ideal gas law. The conditions of the system do not require us to consider a different equation of state, and hence we have the following expression for the thermal pressure:
Pth = nkbT (3)
Where n is the number density, kb is the Boltzmann constant, and T is the average
tem-perature of our thermal particle distribution. Since the number density and temtem-perature are measured directly by SWEPAM as part of ACE, the thermal pressure is simple to calculate.
12.2
Non-thermal Pressure
The non-thermal pressure is more complicated, and relies on integrating over different energy brackets. The non-thermal distribution is measured by EPAM, on ACE, in the form of particle intensities for eight different energy bins. These intensities measure the number of particles passing through an area per second per eV of bin size, per solid angle.
Since the number density of the non-thermal distribution is not measured directly, we need to calculate it from the particles intensities measured by EPAM.
ni =
4πIi
vpart
(4) We see in equation 2 that we can derive the number density from the particle intensity and the velocity of particles. Here, n represents number density, I is the particle intensity, as measured by EPAM, and vpart is the particle velocity, as defined below. The 4π term is
to remove the dependence on solid angle, and the i subscript refers to the energy bin. This means that this only calculates the number density in a specific energy bin.
vpart =
r 2Ei
m
Once we have an energy-dependent number density, we can multiply this by the energy of the bin to obtain an energy density for a specific bin. We also need to multiply this by the size of the bin to account for the fact that the intensity is measured per eV of bin size. By calculating this value for each bin and summing over all bins we can determine the total energy density of the non-thermal population. Then we use the adiabatic index to convert this to a pressure. In the end, the calculation for the total non-thermal pressure is as follows:
Pnth= (γ − 1) 8 X i=1 niEidEi (5) Pnth= (γ − 1) 8 X i=1 4πIi vpart EidEi (6) Pnth = (γ − 1) 8 X i=1 4πIidEi r mEi 2 (7)
Where m, the mass of the particles, is determined from the ratio of Hydrogen to Helium in the system, as measured by ACE. Ei represents the energy of the particles in a bin, and
is determined by the apparatus used to measure the particle intensities (EPAM). dEi is the
can be found on the ACE website, or in a previous part of this paper. Equation 7 lets us calculate the non-thermal pressure using only data obtained from ACE directly. It should be noted that since the bin sizes and energy values are provided in MeV, and the intensity is measured per eV, the non-thermal pressure is initially calculated in units of eVcm-3, and
is then converted to SI units [Jm-3, or Pa].
12.3
Acceleration Efficiency
When the thermal and non-thermal pressure are calculated for a shock, we can calculate the acceleration efficiency of the shock. We define the acceleration efficiency as follows:
ω = Pnth Pnth+ Pth
(8) This efficiency measures how much of the total pressure of the shock comes from the non-thermal population. If we assume that nearly all non-thermal particles in the system come from particle acceleration in the shock5, this efficiency can be seen as a measure of how effectively the shock is accelerating particles, relative to the total energy (pressure) of the shock.
The efficiency can be understood as a percentage, where a value above 0.5 means that thermal pressure is dominant in the system. While there are systems where the non-thermal pressure can dominate, such as supernova remnants, this is unlikely to occur in the solar system.
12.4
Compression Ratio and Mach Number
The compression ratio is a key parameter of a shock, and is related to the base characteristics of the jump conditions of the shock itself. It is denoted by χ, and defined as follows:
χ ≡ n2 n0
= v0 v2
(9) The subscripts 2 and 0 refer to the regions just after the shock and in the unshocked medium, respectively6. The terms n and v represent particle number density and velocity,
as before. As we see, the compression ratio represents the jump in density and velocity, and hence the strength of the shock as a whole. We also see that at this point, compression ratio as defined above is completely model-independent, and is simply a measured quantity of any shock.
In most models that attempt to describe shocks, the compression ratio is strongly related to another basic shock parameter; the Mach number, which is defined as the shock velocity
5More specifically, we must be able to assume that non-thermal particles that do not come from the shock
are insignificant and relatively constant, so that their background does not strongly affect results.
in units of the local sound speed7. In basic shock theory, based on the Rankine-Hugoniot
relations, the compression ratio is related to the Mach number in the following way [39][28]: χ = (γ + 1)M 2 S (γ − 1)M2 S+ 2 (10) Where MS refers to the sonic Mach number and γ is the adiabatic index. If we assume
this relation to hold, which we must do if we wish to consider any existing shock models, we can restructure it as follows:
MS =
s
2χ
γ + 1 − χ(γ − 1) (11) In this way, we can calculate the Mach number of the shock from the compression ratio, which only requires number density data, which can be obtained from ACE directly.
13
Efficiency of CME-driven Shocks
Now that we know how to calculate the acceleration efficiency and Mach number of the CME-driven shocks observed by ACE, we can use data from the solar system to test the model proposed by Vink & Yamazaki. Remembering that this model imposes two limitations on the efficiency of low-Mach-number shocks: a maximum efficiency for each Mach number, and the critical shock Mach number (Macc =
√
5), we can simply plot the results obtained using ACE against the model’s predictions. This is done in Figure 10.
7It should be noted that we are referring to the sound speed in the unshocked medium, as this is the
2
3
4
5
6
7
8
9
10
11
Mach Number
0.0
0.2
0.4
0.6
0.8
1.0
Acceleration Efficiency
Maximum Efficiency
Maximum Escaping Energy Flux
Shock Datapoints
Figure 10: A plot of acceleration efficiency against sonic Mach number. The red data points represent calculations based on CME-driven shocks. The solid black curve is the maximum efficiency as a function of Mach number, while the dashed black line is the efficiency for which the escaping energy flux is maximal.
Shock Date Mach Number Acceleration Efficiency (yyyy, dd) (MS [±]) (ω [±]) 2003, 308 8.75 [1.49] 0.19 [0.03] 2002, 138 3.06 [0.55] 0.20 [0.04] 2002, 113 4.54 [0.53] 0.07 [0.01] 2000, 160 2.93 [0.29] 0.17 [0.02] 2000, 97 4.73 [0.85] 0.16 [0.02] 1999, 49 3.23 [0.59] 0.10 [0.02] 1998, 238 2.60 [0.41] 0.33 [0.05]
The figure above shows the efficiency of solar system shocks, as well as the limitations imposed by Vink & Yamazaki. The red data points are the Mach number and acceleration efficiency for a number of CME-driven shocks, calculated as discussed in the previous section.
The solid black curve is the maximum efficiency as a function of Mach number, while the dashed black curve represents the efficiency for which the escaping energy flux is maximal. These curves are obtained by applying the method shown in Figure 9 to a larger, and more continuous, spectrum of Mach numbers. The points where the escaping energy flux becomes negative form the maximum efficiency curve, while the peaks of each function form the dashed curve. This means that the region above the solid black curve should be unreachable for any shock if the model holds. The data points are represented numerically in the table below Figure 10.
We can see that the majority of the shocks calculated in the project have an efficiency in the range of 10 - 20%. While this shows that a substantial portion of the internal energy in the shock is non-thermal, the thermal component still clearly dominates. None of the results shown above seem to strongly disagree with the limitations imposed by Vink & Yamazaki. While there is one shock with an unusually high efficiency, 32%, it does not seem to disagree with the model in a way that is statistically significant. Taking the errors into account, it is less than a standard deviation away from the maximum efficiency, and cannot be considered a significant disagreement with the proposed model.
There does not seem to be a relation between the Mach number and the actual efficiency of the shock, as the efficiency appears to be fairly constant across a range of Mach numbers. Shocks do not attempt to favor the production of cosmic rays, as the efficiencies calculated are far below those necessary to cause the maximal escaping energy flux. However, this could simply be an injection problem. Since the shock acceleration mechanism requires an input of non-thermal particles, it is possible that these weak shocks simply do not have the initial energy to properly accelerate particles at the efficiencies required for maximal escaping energy flux. Even so, it does not seem like CME-driven shocks tend to efficiencies that maximize the escaping energy flux.
14
Below the Critical Mach Number
You can see in Figure 10 that no shocks below the critical Mach number have been inves-tigated in this project. These shocks are hard to find, as they do not seem to occur often. When they are detected and registered on the ACE list of shocks [1], it is often found that these very weak shocks occur in larger turbulent events, and are not isolated. This makes them difficult to analyze, as the signal-to-noise ratio makes any results insignificant. These shocks typically appear as small jumps in density surrounded by other large variations.
However, we have been able to find one low-Mach-number shock that can serve as a good example of what should occur in a shock below the critical Mach number.
266.0
266.2
266.4
266.6
266.8
267.0
Day of Year, 2004
400
420
440
460
480
500
520
540
560
580
Particle Velocity [km/s]
Solar Wind Velocity
Figure 11: A plot of particle velocity against time for a sub-critical shock. While there is a lot of noise, the shock is fairly clean, and the jump is clearly distinguishable in the turbulence. The red line represents the time that the shock passed by ACE.
In Figure 11, we see the particle velocity as measured in day 266 of the year 2004. This plot, showing the thermal population, clearly depicts a shock moving through the solar wind. With a compression ratio of approximately 2.1, it appears to have a Mach number of around 1.9 [1]. While the turbulence is quite significant, since the velocity jump is quite small, the shock is fairly clear, and unlikely to be anything other than a CME-driven shock. This means that we should expect this shock to agree with the model proposed by Vink & Yamazaki. Since it has a Mach number below the critical Mach number, this shock should not be able to accelerate particles.
266.2
266.4
266.6
266.8
267.0
Day of Year, 2004
10000
20000
30000
40000
50000
Particle Intensity [1/cm^2 s sr MeV]
Intensity in 47 - 66 keV band
Figure 12: A plot of particle intensity in the 47-66 keV band for a sub-critical shock. Again, the red line represents the time of the shock crossing. Notable is the complete absence of energetic particles at the time of the shock crossing.
In Figure 12, we see the data for the non-thermal intensity for the same time period (day 266, 2004), for the first energy band. In contrast to what is typically seen for CME-driven shocks, such as the one in Figure 2, there are not energetic particles measured at notable levels during the shock crossing. There does not appear to be any particle acceleration taking place in this sub-critical shock.
While this shock demonstrates very clearly what should occur below the critical Mach number, it is in itself not a significant endorsement of the model proposed by Vink & Ya-mazaki. For this, more shocks like this one would need to be seen. This makes it difficult to use sub-critical shocks to discuss this model, as these shocks are not typically as clean as the one shown here. However, this shock can serve as an interesting example of a sub-critical shock that does not accelerate particles, and a reminder that not all CME-driven shocks are accompanied by particle acceleration.
Part V
Diffusion in a Turbulent Magnetic
Field
The term diffusive shock acceleration, considered the main mechanism that particles use to accelerate in a shock, is named such because the way the particles escape the shock, as cosmic rays, resembles diffusion. The collective effect of each escaping particles is an exponential decay of energetic particles away from the shock front. This means that the non-thermal population of a shock will have a distribution of the following form [15].
f (x, p) = Af (p) (
exp(v0x/κxx) x ≤ 0
1 x > 0 (12)
Where A is a constant, f (p) is the dependence of the distribution on particle momentum, v0 is the plasma velocity upstream of the shock in the shock frame, κxx is the diffusion
coefficient in the radial direction, and x is the position in the shock system. The shock crossing is at x = 0, while the downstream region exists where x > 1, and the upstream (unshocked) region is where x < 0.
The characteristics of the diffusion coefficient, such as its dependence on particle energy, depend heavily on the nature of the shock. Hence, by investigating the diffusion coefficient, we might be able to learn more about certain aspects of CME-driven shocks. For example, we could use the dependence of diffusion on particle energy, measured by the diffusion index, to distinguish between different types of turbulence in the shock.
A paper by Giacalone & Jokipii, written in 1999 [14], examined how cosmic rays are transported in a turbulent magnetic field. They began by modeling a turbulent magnetic field, based on Kolmogorov-type magnetic turbulence, which is discussed in detail in the original paper. They then took a series of runs where they simulated particles of different energies in the turbulent magnetic field. The parameters of these runs can be found in Table 2 of the original paper. By observing how particles of different energies diffused through the field, they were able to determine the diffusion coefficient at different energies.
Figure 13: Particles of different energies (solid lines) diffusing through a turbulent field, with field lines represented by dotted lines. Taken from [14](Figure 2).
In Figure 13 we see a visual representation of the simulation used by Giacalone & Jokipii. The authors used these particles trajectories to calculate the diffusion coefficients of the particles. This allowed them to calculate how the diffusion coefficient depended on energy.
Figure 14: Calculations of the diffusion coefficient at various particles energies, as done by Giacalone & Jokipii [14]. The different sets of points represent variations in the simulated magnetic field. The x-axis, which is unlabeled due to the source of the plot, represents particle energy, and ranges from 1 to 1000 MeV. The plot comes directly from Giacalone & Jokipii [14].
In Figure 14 we see the result of calculating the diffusion coefficient, both parallel and perpendicular to the magnetic field, across a range of particle energies. From this, the diffusion index can be calculated. Giacalone & Jokipii determined that σ = 0.68, with σ being the diffusion index, as defined below.
κk ≈ κ⊥ ∝ Eσ (13)
κ ∝ E2/3 (14)
In this way, Giacalone & Jokipii predicted the diffusion index of cosmic rays diffusing through a Kolmogorov-type magnetic turbulence. By calculating the diffusion index for CME-driven shocks, we might indirectly learn something about the structure of the magnetic turbulence. Since Giacalone & Jokipii chose conditions similar to the solar wind at 1 AU for their simulation, their results are very applicable to the shocks investigated in this project.
15
Diffusion Coefficient and Index
To calculate the diffusion coefficient, we must make use of the fact that there is an exponential precursor in the non-thermal population measured by ACE. We know that the particle intensity, measured by EPAM, should then measure an exponential increase prior to the shock.
Ii ∝ exp(v0x/κ(E)) (15)
Where I is the particle intensity, as measured by EPAM, i is the EPAM energy band, as defined in a previous section, and E is particle energy. We see that the terms in the exponent could be combined into a characteristic length scale.
∆x = κ(E) v0
(16) However, since ACE is stationary, it cannot measure the shock in terms of distance. Instead, it measures against time, as we have seen in previous figures. Near the shock front, we can make a simple assumption.
∆x = Vsτ =
κ(E) v0
(17) κ(E) = τ Vsv0 (18)
Where Vs is the shock velocity, and τ is the characteristic timescale. By obtaining an
exponential fit to the precursor of the shock, we are able to calculate the characteristic timescale in the system. The fit is typically applied to a 20 minute region before the shock crossing, and is done by fitting a straight line to the logarithm of the intensity data. However, the data rarely form an ideal exponential, and hence the fitting process is far from ideal.
Once we have calculated the characteristic timescale, τ , we only need the shock velocity and the upstream plasma velocity in the shock frame to calculate the diffusion coefficient. However, as we saw in the section on frames of reference, we need the shock velocity to determine the upstream plasma velocity in the shock frame.
15.1
Shock Velocity
In principle, there are two ways to calculate the shock velocity. The first relies on the relation between Mach number and shock velocity, while the second depends on one of the Rankine-Hugoniot relations.
We know that the Mach number is defined as the shock velocity divided by the sound speed in the unshocked plasma. Since we know how to calculate the Mach number using density data, and we know the definition for the sound speed in a plasma, it seems like a simple step to calculate the shock velocity. This was attempted, and initially seemed like a favorable method due to the low statistical errors. However, the calculated values for the shock were typically too low to be feasible in the system. Even after considering the pressure
