# HESPERIA Forecasting Tools: Real-Time and Post-Event

Marlon Núñez, Karl-Ludwig Klein, Bernd Heber, Olga E. Malandraki, Pietro Zucca, Johannes Labrens, Pedro Reyes-Santiago, Patrick Kuehl, Evgenios Pavlos

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Marlon Núñez, Karl-Ludwig Klein, Bernd Heber, Olga E. Malandraki, Pietro Zucca, et al., HESPERIA Forecasting Tools: Real-Time and Post-Event. Malandraki, Olga E.; Crosby, Norma B. Solar Particle Radiation Storms Forecasting and Analysis, 444, pp.113-131, 2018, 10.1007/978-3-319-60051-2_7. [https://doi.org/10.1007/978-3-319-60051-2_7](https://doi.org/10.1007/978-3-319-60051-2_7) [hal-03728309]

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Olga E. Malandraki
Norma B. Crosby
*Editors*

Solar Particle
Radiation Storms
Forecasting
and Analysis

The HESPERIA HORIZON 2020 Project
and Beyond

AS
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Solar Particle Radiation Storms Forecasting
and Analysis
Astrophysics and Space Science Library

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W. B. BURTON, *National Radio Astronomy Observatory, Charlottesville, Virginia, U.S.A.* (bburton@nrao.edu); *University of Leiden, The Netherlands*
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F. BERTOLA, University of Padua, Italy

C. J. CESARSKY, *Commission for Atomic Energy, Saclay, France*

P. EHRENFREUND, Leiden University, The Netherlands

O. ENGVOLD, University of Oslo, Norway

E. P. J. VAN DEN HEUVEL, University of Amsterdam, The Netherlands

V. M. KASPI, McGill University, Montreal, Canada

J. M. E. KUIJPERS, University of Nijmegen, The Netherlands

H. VAN DER LAAN, University of Utrecht, The Netherlands

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Olga E. Malandraki • Norma B. Crosby
Editors

# Solar Particle Radiation Storms Forecasting and Analysis

## The HESPERIA HORIZON 2020 Project and Beyond

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OPEN

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Editors

Olga E. Malandraki
National Observatory of Athens
IAASARS
Athens, Greece

Norma B. Crosby
Space Physics Division - Space Weather
Royal Belgian Institute for Space Aeronomy
Brussels, Belgium

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ISSN 0067-0057

ISSN 2214-7985 (electronic)

Astrophysics and Space Science Library

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DOI 10.1007/978-3-319-60051-2

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# Preface

Ranging in energy from tens of keV to a few GeV solar energetic particles (SEPs) are an important contributor to the characterization of the space environment. Emitted from the Sun they are associated with solar flares and shock waves driven by coronal mass ejections (CMEs). SEP radiation storms may last from a period of hours to days or even weeks and have a large range of energy spectrum profiles. These events pose a threat to modern technology relying on spacecraft and humans in space as they are a serious radiation hazard. Though our understanding of the underlying physics behind the generation mechanism of SEP events and their propagation from the Sun to Earth has improved during the last decades, to be able to successfully predict a SEP event is still not a straightforward process.

The motivation behind the 2-year HESPERIA (High Energy Solar Particle Events forecasting and Analysis) project of the EU HORIZON 2020 programme, successfully completed in April 2017, was indeed to further our scientific understanding and prediction capability of high-energy SEP events by building new forecasting tools while exploiting novel as well as already existing datasets. HESPERIA, led by the National Observatory of Athens, with Project Coordinator Dr. Olga E. Malandraki, was a consortium of nine European teams that also collaborated during the project with a number of institutes and individuals from the international community. The complementary expertise of the teams made it possible to achieve the main objectives of the HESPERIA project:

* To develop two novel real-time SEP forecasting systems based upon proven concepts.

* To develop SEP forecasting tools searching for electromagnetic proxies of the gamma-ray emission in order to predict large SEP events.

* To perform systematic exploitation of novel high-energy gamma-ray observations of the FERMI mission together with in situ SEP measurements near 1 AU.

* To provide for the first time publicly available software to invert neutron monitor observations of relativistic SEPs to physical parameters that can be compared with space-borne measurements at lower energies.
* To perform examination of currently unexploited tools (e.g. radio emission).

* To design recommendations for future SEP forecasting systems.

This book reviews our current understanding of SEP physics and presents the results of the HESPERIA project. In Chap. 1 the book provides a historical overview on how SEPs were discovered back in the 1940s and how our understanding has increased and evolved since then. Current state of the art based on the unique measurements analysed in the three-dimensional heliosphere and the key SEP questions that remain to be answered in view of the future missions Solar Orbiter and Parker Solar Probe that will explore the solar corona and inner heliosphere are also presented. This is followed by an introduction to why SEPs are studied in the first place describing the risks that SEP events pose on technology and human health. Chapters 2 through 6 serve as background material covering solar activity related to SEP events such as solar flares and coronal mass ejections, particle acceleration mechanisms, and transport of particles through the interplanetary medium, Earth's magnetosphere and atmosphere. Furthermore, ground-based neutron monitors are described. The last four chapters of the book are dedicated to and present the main results of the HESPERIA project. This includes the two real-time HESPERIA SEP forecasting tools that were developed, relativistic SEP related gamma-ray and radio data comparison studies, modelling of SEP events associated with gamma-rays and the inversion methodology for neutron monitor observations that infers the release timescales of relativistic SEPs at or near the Sun.

With emphasis on SEP forecasting and data analysis, this book can both serve as a reference book and be used for space physics and space weather courses addressed to graduate and advanced undergraduate students. We hope the reader of this book will find the world of SEP events just as fascinating as we do ourselves.

Olga E. Malandraki
Norma B. Crosby
Acknowledgements

The HESPERIA project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 637324. The authors thank the EU for this support making it possible to further our knowledge in solar energetic particle research and forecasting, as well as write this book.

The authors of Chaps. 4, 9 and 10 acknowledge the use of ERNE data from the Space Research Laboratory of the University of Turku and of the SEPEM Reference Data Set version 2.00, European Space Agency (2016). They thank the ACE/EPAM, SWEPAM and MAG instrument teams and the ACE Science Center for providing the ACE data. They acknowledge the use of publicly available data products from WIND/SWE and 3DP, GOES13/HEPAD and the CME catalogues from SoHO/LASCO and STEREO/COR1. SoHO is a project of international cooperation between ESA and NASA. They acknowledge also the use of the Harvard-Smithsonian Interplanetary shock Database maintained by M. L. Stevens and J. C. Kasper and of the Heliospheric Shock Database, generated and maintained at the University of Helsinki.

Rolf Bütikofer thanks Erwin Flückiger and Claudine Frieden for their suggestions and assistance in writing Chaps. 5 and 6. This work was supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under the contract number 15.0233 and by the International Foundation High Altitude Research Stations Jungfraujoch and Gornergrat.

The authors of Chap. 7 thank the National Oceanic Atmospheric Administration (NOAA) for providing GOES data files which were used to calibrate and evaluate the HESPERIA UMASEP-500 tool. They acknowledge the NMDB database (www.nmdb.eu) funded under the European Union's FP7 programme (contract No. 213007). They also acknowledge Dr. Juan Rodriguez from NOAA for his support on the estimation of >500 MeV integral proton flux and expert advice on the GOES/HEPAD data.

The authors of Chap. 8 acknowledge STEREO/HET/LET/SEPT, ACE/EPAM, ACE/SIS, GOES/HEPAD, WIND/3DP and SoHO/ERNE/EPHIN teams as well as the SEPServer team for the availability of the energetic particle data. The STEREO/SEPT and the SoHO/EPHIN projects are supported under grant
50OC1702 by the Federal Ministry of Economics and Technology on the basis of a decision by the German Bundestag. Gerald H. Share and Ronald J. Murphy (Department of Astronomy, University of Maryland, College Park MD 20742 and National Observatory of Athens; Naval Research Laboratory, Washington DC 20375) are acknowledged for making Fermi/LAT data available to the project prior to their publication. Specifically, the authors of Chap. 9 thank G. Share for providing the data on interacting proton spectra derived from the Fermi/LAT γ-ray observations.

Alexandr Afanasiev and Rami Vainio acknowledge the financial support of the Academy of Finland (project 267186) and the computing resources of the Finnish Grid and Cloud Infrastructure maintained by CSC—IT Centre for Science Ltd. (Espoo, Finland) and co-funded by the Academy of Finland and 13 Finnish research institutions. The team of the University of Barcelona has been also partially supported by the Spanish Ministerio de Economía, Industria y Competitividad, under the project AYA2013-42614-P and MDM-2014-0369 of ICCUB (Unidad de Excelencia 'María de Maeztu'). Computational support was provided by the Consorci de Serveis Universtiaris de Catalunya (CSUC).

Alexis P. Rouillard (external collaborator of the HESPERIA project) acknowledges support from the plasma physics data center (Centre de Données de la Physique des Plasmas; CDPP; http://cdpp.eu/), the Virtual Solar Observatory (VSO; http://sdac.virtualsolar.org), the Multi Experiment Data & Operation Center (MEDOC; https://idoc.ias.u-psud.fr/MEDOC), the French space agency (Centre National des Etudes Spatiales; CNES; https://cnes.fr/fr) and the space weather team in Toulouse (Solar-Terrestrial Observations and Modelling Service; STORMS; https://stormsweb.irap.omp.eu/). This includes the data mining tools AMDA (http://amda.cdpp.eu/) and the propagation tool (http://propagationtool.cdpp.eu). He also acknowledges financial support from the HELCATS project under the FP7 EU contract number 606692. The STEREO SECCHI data are produced by a consortium of RAL (UK), NRL (USA), LMSAL (USA), GSFC (USA), MPS (Germany), CSL (Belgium), IOTA (France) and IAS (France).

The authors thank Springer for their interest in the HESPERIA project and the opportunity for the publication of its results.
# Contents

<table><tr><td>1</td><td>Solar Energetic Particles and Space Weather: Science and Applications</td><td>1</td></tr><tr><td></td><td>Olga E. Malandraki and Norma B. Crosby</td><td></td></tr><tr><td>2</td><td>Eruptive Activity Related to Solar Energetic Particle Events</td><td>27</td></tr><tr><td></td><td>Karl-Ludwig Klein</td><td></td></tr><tr><td>3</td><td>Particle Acceleration Mechanisms</td><td>45</td></tr><tr><td></td><td>Rami Vainio and Alexandr Afanasiev</td><td></td></tr><tr><td>4</td><td>Charged Particle Transport in the Interplanetary Medium</td><td>63</td></tr><tr><td></td><td>Angels Aran, Neus Agueda, Alexandr Afanasiev, and Blai Sanahuja</td><td></td></tr><tr><td>5</td><td>Cosmic Ray Particle Transport in the Earth's Magnetosphere</td><td>79</td></tr><tr><td></td><td>R. Bütikofer</td><td></td></tr><tr><td>6</td><td>Ground-Based Measurements of Energetic Particles by Neutron Monitors</td><td>95</td></tr><tr><td></td><td>R. Bütikofer</td><td></td></tr><tr><td>7</td><td>HESPERIA Forecasting Tools: Real-Time and Post-Event</td><td>113</td></tr><tr><td></td><td>Marlon Núñez, Karl-Ludwig Klein, Bernd Heber, Olga E. Malandraki, Pietro Zucca, Johannes Labrens, Pedro Reyes-Santiago, Patrick Kuehl, and Evgenios Pavlos</td><td></td></tr><tr><td>8</td><td>X-Ray, Radio and SEP Observations of Relativistic Gamma-Ray Events</td><td>133</td></tr><tr><td></td><td>Karl-Ludwig Klein, Kostas Tziotziou, Pietro Zucca, Eino Valtonen, Nicole Vilmer, Olga E. Malandraki, Clarisse Hamadache, Bernd Heber, and Jürgen Kiener</td><td></td></tr><tr><td>9</td><td>Modelling of Shock-Accelerated Gamma-Ray Events</td><td>157</td></tr><tr><td></td><td>Alexandr Afanasiev, Angels Aran, Rami Vainio, Alexis Rouillard, Pietro Zucca, David Lario, Suvi Barcewicz, Robert Siipola, Jens Pomoell, Blai Sanahuja, and Olga E. Malandraki</td><td></td></tr></table>
<table><tr><td>10</td><td>Inversion Methodology of Ground Level Enhancements</td><td>179</td></tr><tr><td></td><td>B. Heber, N. Agueda, R. Bütikofer, D. Galsdorf, K. Herbst, P. Kühl, J. Labrenz, and R. Vainio</td><td></td></tr><tr><td></td><td>Index</td><td>201</td></tr></table>
List of Abbreviations

<table><tr><td>ACE</td><td>Advanced Composition Explorer</td></tr><tr><td>AEPE</td><td>Atypical Energetic Particle Event</td></tr><tr><td>AIA</td><td>Atmospheric Imaging Assembly</td></tr><tr><td>AMS</td><td>Alpha Magnetic Spectrometer</td></tr><tr><td>AU</td><td>Astronomical Unit</td></tr><tr><td>AWT</td><td>Average Warning Time</td></tr><tr><td>CDAW</td><td>Coordinated Data Analysis Workshops</td></tr><tr><td>CGRO</td><td>Compton Gamma-Ray Observatory</td></tr><tr><td>CME</td><td>Coronal Mass Ejection</td></tr><tr><td>CORONAS</td><td>Complex Orbital near-Earth Observations of Solar Activity</td></tr><tr><td>CIR</td><td>Corotating Interaction Region</td></tr><tr><td>CSA</td><td>Coronal Shock Acceleration</td></tr><tr><td>CS</td><td>Current Sheet</td></tr><tr><td>DSA</td><td>Diffusive Shock Acceleration</td></tr><tr><td>DSP</td><td>Downstream Propagation</td></tr><tr><td>EGRET</td><td>Energetic Gamma Ray Experiment Telescope</td></tr><tr><td>EPAM</td><td>Electron, Proton, and Alpha Monitor</td></tr><tr><td>EPD</td><td>Energetic Particle Detector</td></tr><tr><td>EPHIN</td><td>Electron Proton Helium Instrument</td></tr><tr><td>ESA</td><td>European Space Agency</td></tr><tr><td>ESP</td><td>Energetic Storm Particle</td></tr><tr><td>FAR</td><td>False Alarm Ratio</td></tr><tr><td>GBM</td><td>Gamma-ray Burst Monitor</td></tr><tr><td>GCR</td><td>Galactic Cosmic Rays</td></tr><tr><td>GLE</td><td>Ground Level Enhancement</td></tr><tr><td>GME</td><td>Goddard Medium Energy</td></tr><tr><td>GOES</td><td>Geostationary Operational Environmental Satellites</td></tr><tr><td>GSE</td><td>Geocentric Solar Ecliptic</td></tr><tr><td>HCS</td><td>Heliospheric Current Sheet</td></tr><tr><td>HEPAD</td><td>High Energy Proton and Alpha Detector</td></tr></table>
<table><tr><td>HESPERIA</td><td>High Energy Solar Particle Events forecasting and Analysis</td></tr><tr><td>ICME</td><td>Interplanetary Coronal Mass Ejection</td></tr><tr><td>IMF</td><td>Interplanetary Magnetic Field</td></tr><tr><td>IMP</td><td>International Monitoring Platform</td></tr><tr><td>INTEGRAL</td><td>INTERNational Gamma-Ray Astrophysics Laboratory</td></tr><tr><td>ISIS</td><td>Integrated Science Investigation of the Sun</td></tr><tr><td>ISS</td><td>International Space Station</td></tr><tr><td>L1</td><td>first Lagrangian point</td></tr><tr><td>LAT</td><td>Large Area Telescope</td></tr><tr><td>LMA</td><td>Levenberg-Marquardt algorithm</td></tr><tr><td>MED</td><td>Medium Energy Detector</td></tr><tr><td>MHD</td><td>MagnetoHydroDynamics</td></tr><tr><td>NASA</td><td>National Aeronautics and Space Administration</td></tr><tr><td>NGDC</td><td>National Geophysical Data Center</td></tr><tr><td>NNLS</td><td>Non-Negative Least Squares</td></tr><tr><td>NOAA</td><td>National Oceanic and Atmospheric Administration</td></tr><tr><td>NoRP</td><td>Nobeyama Radio Polarimeters</td></tr><tr><td>NM</td><td>Neutron Monitor</td></tr><tr><td>PAD</td><td>Pitch Angle Distribution</td></tr><tr><td>PAMELA</td><td>Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics</td></tr><tr><td>PFSS</td><td>Potential Field Source Surface</td></tr><tr><td>POD</td><td>Probability of Detection</td></tr><tr><td>QLT</td><td>Quasi-Linear Theory</td></tr><tr><td>PROBA</td><td>Project for On-Board Autonomy</td></tr><tr><td>REleASE</td><td>Relativistic Electron Alert System for Exploration</td></tr><tr><td>RHESSI</td><td>Ramaty High Energy Solar Spectroscopic Imager</td></tr><tr><td>RICH</td><td>Ring Imaging CHerenkov</td></tr><tr><td>RSTN</td><td>Radio Solar Telescope Network</td></tr><tr><td>SaP</td><td>Shock and Particle</td></tr><tr><td>SCR</td><td>Solar Cosmic Rays</td></tr><tr><td>SDA</td><td>Shock Drift Acceleration</td></tr><tr><td>SDO</td><td>Solar Dynamics Observatory</td></tr><tr><td>SEP</td><td>Solar Energetic Particle</td></tr><tr><td>SEPEM</td><td>Solar Energetic Particle Environment Modelling</td></tr><tr><td>SMM</td><td>Solar Maximum Mission</td></tr><tr><td>SOHO</td><td>SOlar and Heliospheric Observatory</td></tr><tr><td>SoIO</td><td>Solar Orbiter</td></tr><tr><td>SSD</td><td>Solid State Detector</td></tr><tr><td>STEREO</td><td>Solar Terrestrial Relations Observatory</td></tr><tr><td>SWAP</td><td>Sun Watcher using Active pixel system detector and image Processing</td></tr></table>
<table><tr><td>SXR</td><td>Soft X-Ray</td></tr><tr><td>UMASEP</td><td>University of MAlaga Solar particle Event Predictor</td></tr><tr><td>WCP</td><td>Well-Connected Prediction model</td></tr><tr><td>WL</td><td>White Light</td></tr></table>
# Chapter 1
## Solar Energetic Particles and Space Weather: Science and Applications

Olga E. Malandraki and Norma B. Crosby

**Abstract** This chapter provides an overview on solar energetic particles (SEPs) and their association to space weather, both from the scientific as well as from the applications perspective. A historical overview is presented on how SEPs were discovered in the 1940s and how our understanding has increased and evolved since then. Current state-of-the-art based on unique measurements obtained in the 3-dimensional heliosphere (e.g. by the Ulysses, ACE, STEREO spacecraft) and their analysis is also presented. Key open questions on SEP research expected to be answered in view of future missions that will explore the solar corona and inner heliosphere are highlighted. This is followed by an introduction to why SEPs are studied, describing the risks that SEP events pose on technology and human health. Mitigation strategies for solar radiation storms as well as examples of current SEP forecasting systems are reviewed, in context of the two novel real-time SEP forecasting tools developed within the EU H2020 HESPERIA project.

### 1.1 Science

#### 1.1.1 Historical Perspective of Solar Energetic Particle (SEP) Events

It is widely accepted that protons, electrons, and heavier nuclei such as He-Fe are accelerated from a few keV up to GeV energies in at least two distinct locations, namely the solar flare and the coronal mass ejection (CME)-driven interplanetary (IP) shock. The particles observed in IP space and near Earth are commonly referred to as solar energetic particles (SEPs). Those accelerated at flares are known as

<footnote>O.E. Malandraki (✉)
National Observatory of Athens, IAASARS, Athens, Greece
e-mail: omaland@noa.gr</footnote>

<footnote>N.B. Crosby
Royal Belgian Institute for Space Aeronomy, Brussels, Belgium
e-mail: Norma.Crosby@aeronomie.be</footnote>
<figure><img src="image_8.png" /><figcaption>**Fig. 1.1** Early observation of a solar energetic particle event (Reproduced from Meyer et al. 1956, permission for reuse from publisher American Physical Society for both print and electronic publication)</figcaption></figure>

impulsive SEP events, particle populations accelerated by near-Sun CME-shocks are termed as gradual SEP events, and those associated with CME shocks observed near Earth are known as energetic storm particle (ESP) events (Desai and Giacalone 2016).

The first SEP event observations extending up to GeV energies were made with ground-based ionization chambers and neutron monitors in the mid 1940s (Forbush 1946). One early event is shown in Fig. 1.1. Until the mid-1990s the so-called ‘solar flare myth’ scenario was prevalent, in which large solar flares were considered to be the primary cause of major energetic particle events observed at 1 AU (Gosling 1993). However, Wild et al. (1963) had reviewed radio observations and on the basis of the slow-drifting type II bursts observed in close association with the SEP events, proposed an alternative view for the particle acceleration at magnetohydrodynamic shock waves, typically accompanying the flares.

By the end of the 1990s a two-class paradigm of SEP events (see Fig. 1.2) had been generally accepted (e.g. Reames 1999). The flare-related impulsive events lasted a few hours and were typically observed when the observer was magnetically connected to the flare site, were electron-rich and associated with type III radio bursts. These events also had $ ^3He / ^4He $ ratios enhanced by factors $ 10^3–10^4 $, enhanced Fe/O ratios by a factor of 10 over the nominal coronal values, and Fe ionization states of up to ~2. On the other hand, the gradual events lasted several days, had larger fluences, and were attributed to be a result of diffusive acceleration at CME-driven coronal and IP shocks. They were proton-rich, had average Fe/O ratios of 0.1 and Fe ionization states of 14 and were associated with type II radio bursts (e.g. Cliver 2000; Reames 2013).
<figure><img src="image_9.png" /><figcaption>**Fig. 1.2** The two-class paradigm of SEP events is presented (a) the gradual SEP events occur as a result of diffusive acceleration at CME-driven coronal and IP shocks and populate interplanetary magnetic field (IMF) lines over a large longitudinal extent (b) the impulsive SEP events which are produced by solar flares and which populate only those IMF lines well-connected to the flare site. Comparison of intensity-time profiles of electrons and protons in ‘pure’, (c) gradual and (d) impulsive SEP events. The gradual event is a disappearing—filament event with a CME but no impulsive flare. The impulsive events come from a series of flares with no CMEs (Reproduced from Desai and Giacalone 2016, permission for reuse from publisher Springer for both print and electronic publication)</figcaption></figure>

Since then, observations have indicated that there are ‘hybrid’ or mixed event cases, where both mechanisms appear to contribute, with one accelerating mechanism operating in the flare while the other operates at the CME-driven shock (Kallenrode 2003). Such hybrid events may result from the re-acceleration of remnant flare suprathermals by shock waves (Mason et al. 1999; Desai et al. 2006) or from the interaction of CMEs (Gopalswamy et al. 2002). It is noteworthy however, that based on large enhancements in the Fe/O during the initial phases of two large SEP events observed by Wind and Ulysses when the two spacecraft (s/c) were separated by 60° in longitude (Tylka et al. 2013) argued that the initial Fe/O enhancements cannot be cited as evidence for a direct flare component, but instead
<figure><img src="image_10.png" /><figcaption>**Fig. 1.3** Proton spectra of the SEP events of April 1998 (green, Tylka et al. 2000) and September 1989 (blue, Lovell et al. 1998) are compared. In *yellow* the hazardous portion of the spectrum during the April 1998 event is highlighted. The region of additional hazardous radiation from the September 1989 event is shaded *red* (Reproduced from Reames 2013, permission for reuse from publisher Springer for both print and electronic publication)</figcaption></figure>

they are better understood as a transport effect, driven by the different mass-to-charge ratios of Fe and O.

High-energy protons in the largest SEP events can pose significant radiation hazards for astronauts and technological systems in space, particularly beyond the Earth's protective magnetic field (National Research Council (NCR) 2008; Cucinotta et al. 2010; Xapsos et al. 2012) (see Sect. 1.2 for more details).

Protons of ~150 MeV are considered as 'hard' radiation since they are very difficult to shield against. Essentially most of the radiation risk of humans in space from SEPs is due to proton intense fluxes of above ~50 MeV, i.e. 'soft' radiation, the energy at which protons begin to penetrate spacesuits and s/c housing. Figure 1.3 compares the proton energy spectra for two large SEP events, presenting typical knee energies of soft and hard radiation SEP events. The most important factor in the radiation dose and depth of penetration of the ions is the location of the energy spectral knee. In yellow in Fig. 1.3, the hazardous part of the spectrum for the April 1998 event is shown, whereas the red shaded area denotes the region of the additional hazardous radiation from the September 1989 event. In the April 1998 event the spectrum rolls over much more steeply at high energies, whereas in the September 1989 event the spectral knee occurred between ~200 and 300 MeV. Events with higher roll-over energies have significantly higher proton intensities above ~100 MeV and can constitute a severe radiation hazard to astronauts (Reames 2013). In fact during the September 1989 event, even an astronaut behind 10 g cm⁻² of material would receive a dose of ~40 mSv. The annual dose limit for a radiation worker in the United States is 20 mSv (Zeitlin et al. 2013; Kerr 2013). In each solar
<figure><img src="image_11.png" /><figcaption>**Fig. 1.4** Typical intensity-time profiles of 1–30 MeV protons for gradual SEP events observed at three different solar longitudes relative to the parent solar event. *Dashed lines* indicate the passage of shocks (Reproduced from Reames 2013, permission for reuse from publisher Springer for both print and electronic publication)</figcaption></figure>

cycle several events of this intensity occur, thus, knowledge of the spectral knee energies is essential.

1.1.2 Large Gradual SEP Events

Early multi-spacecraft SEP observations revealed that 1–30 MeV proton time-
intensity profiles in large gradual SEP events observed in the ecliptic plane at 1
AU are organized in terms of the longitude of the observer with respect to the
traveling CME-driven shock and can be understood if the strongest acceleration
occurs near the ‘nose’ of a CME-driven shock radially expanding outward from the
Sun (see Fig. 1.4, Reames 2013; Cane et al. 1988; Cane and Lario 2006).<fnref n="1" /> Figure
1.4 shows proton intensity profiles of several SEP events observed by the IMP-8 s/c
as a function of longitude of the parent solar event. For observers at solar longitudes
to the east of the source, the intensities have rapid rises peaking relatively earlier
during the event when there is magnetic connection to the nose of the CME-shock

<footnote n="1">¹When observing images of the Sun east and west are reversed.</footnote>
<figure><img src="image_12.png" /><figcaption>**Fig. 1.5** Peak proton intensity in SEP events at 2 and 20 MeV as a function of CME speed. The different symbols denote two combinations of SEP instruments (Wind, Helios) and coronagraphs (LASCO, SOLWIND). Linear least-squares fits as well as the corresponding correlation coefficients are shown for each proton energy (Reproduced from Kahler 2001, permission for reuse from publisher John Wiley and Sons for both print and electronic publication)</figcaption></figure>

near the Sun. Gradual decreasing intensities are observed subsequently, as the shock moves outward and the *s/c* becomes magnetically connected to the eastern flanks of the shock. For sources near the central meridian the proton intensities peak when the nose of the shock itself arrives at the *s/c* location. Observers located to the west of the source observe slowly rising intensities that peak after the local passage of the shock.

Comparison between the SEP and the CME or IP shock properties have shown no evidence of a clear correlation. In Fig. 1.5 (Kahler 2001) it is shown that CMEs with similar speeds are associated with a significant spread (~3–4 orders of magnitude) in the peak proton intensities at 2 and 20 MeV of the associated SEPs at 1 AU. This study subsequently constituted the basis for comparison of a more recent multi-spacecraft study by (Rouillard et al. 2012) in which shock speeds could be measured where the shock intersected the field lines to each *s/c* in the heliosphere (see Sect. 1.1.4). Using a large number of SEP events, Kahler (2013a) examined the SEP-CME relationship calculating three different SEP event timescales: the onset time from CME launch to the 20 MeV SEP onset time, the rise time from
SEP onset to half the peak intensity and the duration of the SEP intensity above
half the peak value. Comparison of these timescales with the CME properties such
as speed, acceleration, width and location confirmed that faster (and wider) CMEs
drive shocks, and accelerated SEPs over longer periods of time produce SEP events
with longer timescales and larger fluences.

A flatter size distribution of SEP events relative to that of flare soft X-ray (SXR) events has been previously reported, with the power-law characterizing SEP size being significantly flatter than that of the SXR flux (e.g. Hudson 1978; Belov et al. 2007; Cliver et al. 2012). Cliver et al. (2012) have shown that this difference is primarily due to the fact that flares associated with large gradual SEP events are an energetic subset of all flares also characteristically accompanied by fast (>1000 km/s) CMEs that drive coronal/IP shock waves. They also concluded that the difference of ~0.15 between the slopes of the SEP event distributions and SEP SXR flares is consistent with the observed variation of SEP event peak flux with SXR peak flux. Kahler (2013b) presented arguments against using scaling laws for the description of the relationship between the size distributions of SXR flares and SEP events. They suggested an alternative explanation for flatter SEP power-law distributions in terms of the recent model of fractal-diffusive self-organized criticality proposed by Aschwanden (2012), providing evidence against a close physical connection of flares with SEP production. Trottet et al. (2015), although based on a limited SEP event sample, have recently studied the statistical relationships between SEP peak intensities of deka-MeV and near-relativistic electrons and characteristic parameters of CME and solar flares: the CME speed as well as the peak flux and fluence of SXR emission and the fluence of microwave emission. Via a partial correlation analysis they showed that the CME speed and SXR fluence are the only parameters that significantly affect the SEP intensity and concluded that both flare acceleration and CME shock acceleration contribute to the deka-MeV proton and near-relativistic electron populations in large SEP events.

Above a few tens of MeV per nucleon, large gradual SEP events are highly variable in their spectral characteristics and elemental composition. As an example, Fig. 1.6 (left) shows the event-integrated Fe/C ratio as a function of energy for the SEP events of April 21, 2002 and August 24, 2002 (Tylka et al. 2005). Both events were associated with flares nearly identical in terms of their sizes and solar locations (~W80), as well as with CMEs with similar speeds of ~2000 km/s, however, there were remarkable differences observed in their associated heavy ion spectral behaviour. To explain these differences, Cane et al. (2003) and Cane et al. (2006) proposed a direct flare particle component above ~10 MeV/nuc and that large SEP events are a mixture of flare-accelerated and shock-accelerated populations. According to this scenario, well-connected western hemisphere events are dominated by flare-accelerated particles above ~10 MeV/nuc, causing the significant increase of Fe/O, and could also account for the increasing energy dependence of the Fe/O ratios observed e.g. during the August 24, 2002 event. On the other hand the CME shock during the April 21, 2002 event is strong enough to accelerate >10 MeV/nucleon particles at 1 AU and lead to the observed Fe/O decrease with increasing energy.
<figure><img src="image_13.png" /><figcaption>**Fig. 1.6** The *left panel* shows a comparison of the energy dependence of the event-integrated Fe/C for the two SEP events of 21 April 2002 (blue) and 24 August 2002 (red) which are otherwise similar in their properties (Tylka et al. 2005). The *right panel* shows hypothetical spectra of the suprathermal seed populations for shock-accelerated SEPs, comprising both solar wind and flare-accelerated ions. Different injection thresholds will yield different abundance ratios (Reproduced from Reames 2013, permission for reuse from publisher Springer for both print and electronic publication)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 1.7** (*left*) The 90 large SEP events defined as events with >12 MeV/nucleon Fe fluences > 0.1/(cm² sr) from 1998 to 2005. Days with high fluence only occur when the density of pre-existing suprathermal Fe was >0.3 Dm⁻³. (*right*) Histogram of daily averaged suprathermal Fe densities for all days from March 1998 to December 2005 (Reproduced from Mewaldt et al. 2012a, permission for reuse from publisher AIP Publishing LLC for both print and electronic publication)</figcaption></figure>

Re-acceleration of remnant flare suprathermals or from accompanying flares has been another plausible idea to account for the observed elemental composition variability in SEP events. Mewaldt et al. (2012a) examined the dependence of SEP fluences on suprathermal seed-particle densities. In Fig. 1.7 (left) the Fe fluence in 90 large SEP events is compared with the pre-existing number density of suprathermal Fe at 1 AU 1 day before the occurrence of the SEP event. They
found that the maximum Fe daily-average SEP fluences measured by ACE/SIS are apparently limited by the pre-existing suprathermal number density. In Fig. 1.7 (right) it is shown that the suprathermal Fe densities are significantly greater before the occurrence of these large SEP events with respect to all other days, strongly suggesting that the large fluences of Fe in SEP events only occurred when there was a pre-existing high density of suprathermal Fe. According to these authors remnant flare suprathermal ions, as well as suprathermal material accelerated at previous CME shocks, existed in the heliosphere and served as seed particles subsequently re-accelerated by the CME shock that produced the large CME event (Mason et al. 1999; Desai et al. 2006).

An alternative scenario that (Tylka et al. 2005) proposed is that the observed variability in the energy dependence of the Fe/O ratio could be due to the interplay of two factors namely the evolution in the shock-normal angle as the shock moves outward from the Sun and a compound seed population, typically comprising at least suprathermals from the corona (or solar wind) and flare suprathermals. In this scenario, (Fig. 1.6, right), since the quasi-perpendicular (Q-Perp) shock needs higher injection energy, it may only effectively accelerate impulsive suprathermals originating from the flare acceleration process to high energy, producing the Fe-rich events. On the other hand, since quasi-parallel (Q-Par) shocks have lower injection thresholds they can accelerate the ambient solar wind (or coronal suprathermal ions) producing the Fe-poor events at higher energies. Tylka and Lee (2006) formalized the ideas put forward by Tylka et al. (2005) in an analytical model which above ~1 MeV/nucleon the Tylka and Lee (2006) model reproduced key features of the SEP variability observed in terms of the energy dependence of Fe/O, the ³He/⁴He ratio and the mean ionic charge state of Fe. Schwadron et al. (2015) further improved the model of coronal shock acceleration. In the left panel of Fig. 1.8, the injection energy of shock-accelerated particles is shown as a function of θ<sub>Bn</sub> for a range of the perpendicular to parallel diffusion coefficient ratios, whereas in the right panel, the time profiles of the shock or compression radial position (top panel) and θ<sub>Bn</sub> (bottom

<figure><img src="image_15.png" /><figcaption>**Fig. 1.8** Left panel: the injection energy of shock-accelerated particles as a function of θ<sub>Bn</sub> for a range of the perpendicular and parallel diffusion coefficient ratio. Right panel: Time profiles of the shock radial position θ<sub>Bn</sub> relative to 01:28:57 in the simulation time (see text) (© AAS. Reproduced with permission from Schwadron et al. 2015)</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Fig. 1.9** Energy dependence of (Fe/O)ₙ predicted by the Tylka and Lee (2006) model shown for different values of the parameter R, which reflects the relative strengths of the remnant flare and coronal source contributions at a parallel shock, where seed ions from both populations are injected with equal efficiency (Reproduced from Reames 2013, permission for reuse from publisher Springer for both print and electronic publication)</figcaption></figure>

panel) relative to 01:28:57 in the simulation time are shown. Apparently as the shock moves outward, θ<sub>Bn</sub> decreases, and the geometry of the shock would change from Q-Perp to Q-Par. Schwadron et al. (2015) noted that the CME expansion and acceleration in the low corona may naturally give rise to rapid particle acceleration and broken power-law distributions in large SEP events. Figure 1.9 shows the results
of the (Tylka and Lee 2006) model for the case in which the injection of coronal seed
ions at Q-Perp shocks is suppressed. The energy dependence of the normalized Fe/O
ratio i.e. (Fe/O)$_n$ is shown for different values of the impulsive suprathermal fraction
R in the seed population. In the Q-Par shock event (Fe/O)n ~1 at lower energies
(E < 2 MeV/nucleon), while (Fe/O)$_n$ monotonically decreases with increasing E.
In contrast, in the Q-Perp shock (Fe/O)$_n$ is between 1 and 8 at lower energies,
depending on the impulsive suprathermal fractions. With increasing energy the
normalized ratio exhibits a complex variation e.g. approaching a plateau or reaching
a minimum and further increasing afterwards. Tylka et al. (2005) hence assumed that
the high-energy Fe/O ratio could be used as a crude proxy for shock geometry, with
Fe-poor and Fe-rich events corresponding to Q-Par and Q-Perp shock geometries,
respectively.

It should be noted that these explanations have not taken into account the IP
transport effect, which could further distort the Fe/O ratio that emerged from the
CME-shock acceleration process (e.g. Tylka et al. 2013). Recently, (Tan et al. 2017)
examined 29 large SEP events with peak proton intensity Jpp (>60 MeV) > 1 pfu
during solar cycle 23. The emphasis of their examination was put on a joint analysis
of the Ne/O and Fe/O data in the 3–40 MeV/nucleon energy range as covered
by the Wind/LEMT and ACE/SIS sensors in order to differentiate between the
Fe-poor and Fe-rich events at higher energies that emerged from the CME-driven
shock acceleration process, after correcting the IP transport effect. One of the main
findings of this work is presented in Fig. 1.10 in which the plot of the source plasma
temperature T as very recently reported by Reames (2016) versus the normalized
Ne/O ratio i.e. (Ne/O)$_n$ at E = 30 MeV/nucleon is shown. T is well correlated
with (Ne/O)$_n$ with the linear correlation coefficient (CC) = 0.82. Therefore, the
(Ne/O)$_n$ value at high energies should be a proxy of the injection energy in the shock
acceleration process, and hence the shock $\theta_{Bn}$ according to the models of Tylka and
Lee (2006) and Schwadron et al. (2015).

1.1.3 Ground Level Enhancement (GLE) Events

Ground Level Enhancement (GLE) events form a particular case of high-energy SEP
events associated with ~GeV protons. These events pose severe radiation hazards
to astronauts and technological assets in space and disrupt airline communications
(Shea and Smart 2012). GLEs are nowadays measured with better coverage from
space than at ground level, including ~80 MeV/amu to ~3 GeV/amu H and He
spectra (Adriani et al. 2011), onsets (Reames 2009a, b), energy spectral shapes
and abundances (Mewaldt et al. 2012b), electrons (Kahler 2007, 2012; Tan et al.
2013) and general properties (Gopalswamy et al. 2012). Rouillard et al. (2016)
recently studied the link between an expanding coronal shock and the energetic
particles measured near Earth during the GLE of 17 May 2012. The analysis showed
that the GLE event occurred inside a clear magnetic cloud (see e.g. Malandraki et
al. 2002). Using a new technique developed to triangulate the three-dimensional
<figure><img src="image_17.png" /><figcaption>**Fig. 1.10** Source plasma temperature T very recently reported by Reames (2016) is plotted vs. the Ne/O/0.157 (30 MeV/nucleon) value for the large Fe-poor (red) and Fe-rich (blue) events during solar cycle 23. The *dashed line* is the least-square fitting result for all the collected events as listed in Table 2 of Tan et al. (2017) (Reproduced from Tan et al. 2017)</figcaption></figure>

(3D) expansion of the shock forming in the corona it was found that the highest Mach number ($M_{FM}$) values appear near the coronal neutral line within a few minutes of the CME onset. This neutral line is usually associated with the source of the heliospheric current sheet (HCS) and plasma sheet. It was shown that the release time of GeV particles occurs when the coronal shock becomes super-critical ($M_{FM} > 3$).

### 1.1.4 Multi-Spacecraft Observations of SEP Events

In this section the differences in the SEP event characteristics as observed from different vantage points in the heliosphere are discussed. Figure 1.11 shows as an example the 1 March 1979 SEP event observed by three different s/c. Helios 1 encounters the event near central meridian and observes a peak in the 3–6 MeV proton intensity near the shock passage time. The intensities at the other s/c, after reaching a peak, begin to track closely those seen at Helios 1 after they enter the so-called ‘reservoir’ region (see also McKibben 1972; Roelof et al. 1992) in which the intensities and energy spectra are nearly identical. These results indicate that only a small number of particles can leak out of the reservoir. Observations have provided strong evidence for the location of magnetic ‘barriers’ in space beyond 1 AU and
<figure><img src="image_18.png" /><figcaption>**Fig. 1.11** The top left panel shows the intensity time-profiles for protons in the 1 March 1979 event at 3 s/c. 'S' denotes the time of shock passage at each s/c. The top right panel shows energy spectra in the 'reservoir' region behind the shock at time 'R'. The lower panel shows the s/c trajectories through a sketch of the CME (Reproduced from Reames 2013, permission for reuse from publisher Springer for both print and electronic publication)</figcaption></figure>

their role in determining the decay phase of SEP events and the establishment and maintenance of particle reservoirs in the heliosphere (Roelof et al. 1992, 2012a, b; Sarris and Malandraki 2003; Tan et al. 2012). Reames et al. (1996) also considered that the decay phase of the SEP events consists of particles propagating between the converging magnetic field near the Sun and a moving shell of strong scattering formed downstream of the distant traveling shocks. After formation, the reservoir slowly dissipates as a result of the nominal diffusion, convection, adiabatic cooling, and drift mechanisms that govern the propagation of SEPs.

The Ulysses European Space Agency (ESA)/National Aeronautics and Space Administration (NASA) mission provided unprecedented observations of the 3D heliosphere inside ~5 AU. Comparison of simultaneous SEP observations near the ecliptic plane with the Ulysses observations at high latitudes showed that most events that produce large high-energy (>20 MeV) proton and near-relativistic
electron flux increases near Earth also produce flux increases at high latitudes, regardless of the longitudinal, latitudinal and radial separation between the s/c, although with somewhat lower maximum intensities and slower rise at Ulysses (McKibben et al. 2003; Lario and Pick 2008; Malandraki et al. 2009). Particle anisotropies during SEP events at high latitudes are typically directed outward from the Sun and aligned with the local magnetic field (McKibben et al. 2003; Malandraki et al. 2009).

The observed field-aligned anisotropies, with components perpendicular to the local magnetic field that are essentially zero, indicate that there is no net flow of particles across the local magnetic field. The Ulysses observations revealed the 3D nature of the reservoir effects in the heliosphere. Dalla et al. (2003) concluded that the presence of a shock is not necessary for creating the near-equality observed at Ulysses and near Earth decay phases, but that these observations are better explained by diffusion across the interplanetary magnetic field (IMF).

More recently, combined observations by the twin STEREO s/c as well as near-Earth observatories revealed the wide longitudinal spreads of large gradual SEP events in the heliosphere and even strongly questioned the constraint of a narrow spread for ³He-rich events (Wiedenbeck et al. 2013). A combination of physical processes appears to cause the large longitudinal spread of high-energy particles. Dresing et al. (2014) concluded that both an extended source region at the Sun and perpendicular transport in the IP medium are involved for most of the wide-spread events under study. The studies (Rouillard et al. 2011, 2012) found that the delayed SEP release times at STEREO and L1 are consistent with either the time required for the CME shock to reach field lines connected to the s/c or with the time required (~30–40 min) for the CME to perturb the corona over a wide range of longitudes. Observations by Gómez-Herrero et al. (2015) indicated that higher SEP fluxes, harder SEP spectra and direct injections of SEPs onto well-connected IMF lines are associated with lateral expansions of CME-driven shocks in the low corona, and may therefore be responsible for the rapid longitudinal spread as observed at vastly distributed s/c in many SEP events. Other factors that may also play a role in distributing SEP events longitudinally include the large-scale IMF configuration inside interplanetary CMEs (ICMEs) (e.g. Kahler and Vourlidas 2013) and the relative strength of the CME shock, which depends on the local Alfvén speed, rather than the actual speed of the CME (e.g. Gopalswamy et al. 2014).

### 1.1.5 Particle Acceleration

Recently, important progress has occurred, both from the theoretical and the observational perspective in the research of small-scale magnetic islands in the solar wind and their role in particle acceleration. Khabarova et al. (2015) presented observations that show the occurrence of small-scale magnetic islands and related plasma energization in the vicinity of the HCS. They found evidence that magnetic islands experience dynamical merging in the solar wind and that increases of
energetic particle fluxes in the keV–MeV range are found to coincide with the presence of magnetic islands confined by strong current sheets (CSs). Moreover, the interaction of ICMEs with the HCS can lead to significant particle acceleration due to plasma confinement. Their observations confirmed the rippled structure of the HCS and since such a structure confines plasma, it makes possible the strong energization of particles trapped inside small-scale magnetic islands. They concluded that although initial particle acceleration due to magnetic reconnection at the HCS may be insufficient to obtain high energies, the presence of magnetic islands inside the ripples of the HCS or between two CSs with a strong guide field offers the possibility of re-accelerating particles in the ways discussed theoretically by Zank et al. (2014) and le Roux et al. (2015a).

Khabarova et al. (2016) further explored the role of the heliospheric magnetic field configurations and conditions that favor the generation and confinement of small-scale magnetic islands associated with the so-called atypical energetic particle events (AEPEs) in the solar wind. Some AEPEs have been found not to align with standard particle acceleration mechanisms, such as flare-related or simple diffusive shock acceleration processes related to ICMEs and corotating interaction regions (CIRs). They provided more observations fully supporting the idea and the theory of particle energization by small-scale-flux-rope dynamics previously developed by Zank et al. (2014, 2015a, b) and le Roux et al. (2015a, b). If the particles are pre-accelerated to keV energies via classical mechanisms, they may be additionally accelerated up to 1–1.5 MeV inside magnetically confined cavities of various origins. Khabarova et al. (2016) showed that particle acceleration inside magnetic cavities may explain puzzling AEPEs occurring far beyond IP shocks, within ICMEs, before approaching CIRs, as well as between CIRs. SEP transport processes are described in detail in Chap. 4 of this volume (see also Desai and Giacalone (2016) which includes a review on this topic).

### 1.1.6 Key Open Questions and Future Missions

Solar Orbiter (SolO) is a unique ESA/NASA joint mission conceived to unveil the Sun-heliosphere connection (Mueller et al. 2013), expected to be launched in 2019. The orbital configuration includes a close perihelion, high inclination intervals allowing the observation of the solar polar regions and quasi-co-rotation periods. One of the top-level science questions is “How do solar eruptions produce energetic particle radiation that fills the heliosphere?” which can be broken down into three inter-related key topics: What are the seed populations for energetic particles? How and where are energetic particles accelerated at the Sun? How are energetic particles released from their sources and distributed in space and time?

The Energetic Particle Detector (EPD) instrument suite onboard SolO (Principal Investigator: Prof. J. Rodríguez-Pacheco, Spain) will measure energetic electrons, protons and ions, operating at partly overlapping energy ranges covering from a few keV to 450 MeV/nucleon. The EPD sensors will measure the composition, spectra
and anisotropies of energetic particles with sufficient temporal, spectral, angular and mass resolution to achieve the mission goals (Gómez-Herrero et al. 2016). Energetic particles escaping from the acceleration sites propagate through the turbulent IMF. Previous observations by the Helios s/c have shown that SEP events near the Sun are much less affected by IP transport effects compared to 1 AU observations. As the s/c moves further away from the Sun, scattering and diffusion processes become more important and multiple injections closely spaced in time cannot be resolved (Wibberenz and Cane 2006). Thus, SolO observations close to the perihelion will be crucial to unveil SEP injection, acceleration, release and transport processes, in view of the ongoing debate about the SEP acceleration sites, disentangling the acceleration at CME-driven shocks and at reconnection sites in solar flares (e.g. Malandraki et al. 2006).

NASA's Solar Probe Plus mission, recently re-named to Parker Solar Probe to honour pioneering physicist Prof. Eugene Parker, will fly within nine solar radii of the Sun's surface and is scheduled to be launched in July 2018. The two Energetic Particle Instruments (EPI) of the Integrated Science Investigation of the Sun (ISIS) (Principal Investigator: Prof. D. J. McComas, USA) will measure lower (EPI-Lo) and higher (EPI-Hi) energy particles. EPI-Lo will measure ions and ion composition from ~20 keV/nucleon–15 MeV total energy and electrons from ~25–1000 keV. EPI-Hi measures ions from ~1–200 MeV/nucleon and electrons from ~0.5–6 MeV. The unique ISIS observations will allow the exploration of the mechanisms of energetic particles dynamics, including their (1) Origin: defining the seed populations and physical conditions necessary for energetic particle acceleration; (2) Acceleration: determining the role of shocks, reconnection, waves and turbulence in accelerating energetic particles; (3) Transport: revealing how energetic particles propagate from the corona out into the heliosphere (McComas et al. 2016).

It is evident that the next decade is expected to revolutionize our understanding of SEP acceleration and transport, by means of state-of-the-art sensors on board these two upcoming missions providing unique and unprecedented measurements for the exploration of the solar corona and inner heliosphere. Synergies between the two missions are of particular relevance, since both missions have overlapping timelines and the Parker Solar Probe perihelion, reaching up to ~9 solar radii, will permit simultaneous in-situ observations at the SEP acceleration region close to the Sun and at larger radial distances, with continuous remote sensing coverage provided by SolO and near-Earth s/c. Radial alignments between the two s/c will enable the observations of plasma 'entities' from the same solar source region at progressive radial distances as well as the study of energetic particle radial gradients. Furthermore, other useful configurations for the optimization of the science return are alignments along the same IMF line allowing the observation of SEPs originating at the same acceleration site by two or more s/c located at different radial distances. SEP event observations by multiple s/c located at widely separated points in the heliosphere, both in longitude and in latitude, will be valuable for the investigation of the spatial distribution of SEPs and the unraveling of the physical
mechanisms responsible for producing wide-spread SEP events (see e.g. Sect. 1.1.4)
(Gómez-Herrero et al. 2016).

**1.2 Applications**

**1.2.1 Why Study SEP Events?**

It has become apparent during the last decades that SEP events pose important challenges for modern society. Due to their unpredictability, specifically for those that reach relativistic velocities (high energies) and peak values in very short time scales, they are of concern. SEPs ranging from protons to heavy ions up to iron have been found to have impacts on space systems (s/c, instruments, electronic components, solar arrays, ...), avionics and living organisms (e.g. Feynman and Gabriel 2000; Jiggens et al. 2014). It has even been suggested that systems with very high safety and reliability requirements (e.g. in the nuclear power industry) may need to take account of superstorm ground level radiation on microelectronic devices within the system.

In the case of nuclear power a Carrington event may not be a sufficient case since relevant
timescales for risk assessment may be as long as 10,000 years.

(Paul Cannon (Cannon et al. 2013))

In the following some of the most important and common SEP induced effects
are presented, as well as mitigation strategies currently being relied on.

**1.2.2 SEP Effects on Technology**

Developments in technology such as miniaturization has no doubt benefited space
industry, but at the same time technical equipment has increasingly become more
vulnerable to the space environment. On a lesser scale the well-known “snow”
effect, resulting from the increase in high energy protons during intense SEP events
is sometimes seen on coronagraphic images, as shown in Fig. 1.12, obscuring the
image of the CME itself. However, in some instances SEP induced effects may be of
such a nature that they can result in long-term damage. Missions that target the inner
solar system are especially vulnerable to high-energy charged particles (DiGregorio
2008).

Table 1.1 presents a summary of SEP induced effects observed onboard s/c and aircraft. It is clearly shown how both the energy and species of the particles being considered is an important factor for evaluating their potential effect. Particle flux intensities at lower energies are important for effects such as solar cell degradation, whereas nuclear interactions are associated with particle flux intensities at higher
<figure><img src="image_19.png" /><figcaption>**Table 1.1** Particle effects on technology observed as a function of the particle energy range</figcaption></figure>

<table><thead><tr><td>Energy range</td><td>Effects</td></tr></thead><tbody><tr><td>Protons &lt;10 MeV</td><td>Material and solar cell effects over time as a result of cumulated dose (e.g. solar cell degradation)</td></tr><tr><td>Protons &gt;10 MeV</td><td>Nuclear interactions (e.g., sensor background noise, ionization, displacement damage)</td></tr><tr><td>Protons &gt;50 MeV and Ions &gt;10 MeV nucleon<sup>-1</sup></td><td>Nuclear interactions (e.g., single event effects in electronics onboard satellites, as well as aircraft)</td></tr></tbody></table>

energies. Single-event effects (SEEs) are classified as either non-destructive or destructive:

* Single Event Upsets: Occurs in logical circuits and is defined as a bit switching from an initial logical state to an opposite logical state.

* Single Event Latchup: Results in a high operating current, above device specifications, and must be cleared by a power reset.

* Single Event Gate Rupture: Occurs in powerful transistors and is manifested by an increase in gate leakage current

* Single Event Burnout: A condition that can cause device destruction due to a high current state in a power transistor.

For s/c mission planning and operations SEP events are considered. In regard to launch operations the SEP environment is also a decisive factor whether to give the go ahead to launch or not for several reasons. Launch vehicles and s/c reaching sufficiently high geomagnetic latitudes could for example see an increase in SEE rates at times of significant SEP events. On the other hand, optical instruments are also vulnerable to SEPs and induced sensor interference can disrupt the operation of star trackers and put critical s/c manoeuvres at risk.
Under normal space weather conditions Earth's magnetosphere acts as a shield and protects us from charged particles and magnetic clouds. Nevertheless at times SEPs may have sufficient energies to “break” through this shield and enter the ionosphere; SEPs have easier access to the polar regions near Earth's magnetic poles than at the equator due to the “open” magnetic field lines. The cutoff latitude is a function of a particle's momentum per unit charge and is referred to as its rigidity (see Chap. 5). Variations in SEP access to latitudes can occur on time scales of an hour or less in response to changes in the solar wind dynamic pressure and IMF (Kress et al. 2010). For this reason high inclination LEO satellites can at times be vulnerable to SEPs, as well as the International Space Station that has an orbital inclination of 51.64°.

SEP events can also effect signal propagation between Earth and satellites. Polar cap absorption (PCA) events result from intense ionisation of the D-layer of the polar ionosphere by strong (>10 MeV) SEP events. Due mainly to protons with energy in the range of 1–100 MeV (corresponds to an altitude between 30 and 80 km) the increased ionisation absorbs radio waves in the HF and VHF bands, resulting in problems for communications (degraded radio propagation through the polar regions) and navigation position errors with the importance being a function of the individual SEP event.

Despite the relative steepness of SEP energy spectra, the small percentage of protons accelerating up to high energies (>500 MeV) still pose considerable problems. These high-energy SEP events such as the September 1989 SEP event (Fig. 1.3) are often associated with GLE events and can result in secondary radiation caused by particles interacting with s/c shielding and other material. This results in the production of particles such as lower energetic protons, neutrons, and pions that in some cases may be more of an obstacle for the s/c designer than the primary SEPs themselves. While the former can induce SEEs, secondary particle background can have more profound effects on sensitive space-borne instrumentation.

Technology onboard commercial airline operations can also be affected by SEP events including avionics (electronic systems), communications and GPS navigation systems (Jones et al. 2005). Specifically ultra-long-haul “over-the-pole” routes and high-latitude flights are susceptible to these SEP induced effects.

### 1.2.3 SEPs and Human Health Effects

In addition to being a threat to technology, SEP events are also an important risk to human health. Since the Apollo missions to Earth's Moon in the 1970s human space exploration has mainly been focused on low-Earth orbit altitudes (e.g. Space Shuttle, International Space Station) and suborbital flights. Outside Earth's magnetosphere SEP events have for the most part been a concern for robotic flight missions up until now. During the last decades the vision for space exploration has changed as space agencies and private companies are contemplating sending humans to Mars and asteroids, and as the population on Earth increases colonizing such targets and pursuing deep space exploration will only become more and more attractive. The
downside is that human interplanetary exploration will expose astronauts not only to
the galactic cosmic ray background but at times also to increased levels of radiation
during SEP events and this may indeed be the most important obstacle to overcome.

The field of radiation biology concerns how the radiation environment of space
affects cells. Radiation effects on astronauts are sub-classed into two categories:

1. Deterministic (early) Effects: Due to exposure to a large dose of radiation for a limited time (ranges from hair loss, nausea, acute sickness, death)

2. Stochastic (late) Effects: Due to random radiation-induced changes at the deoxyribonucleic acid (DNA) molecule level (cancer).

As already mentioned in Sect. 1.1.1 protons with high energies (>30 MeV) are a health risk for astronauts. For this reason protons with energies >10 MeV are continuously monitored and taken into account when planning extra-vehicular activities.

In those instances SEP events reach aviation altitudes they become also a concern
for human health as the radiation dose received can increase. This specifically
applies to high-latitude flights (>50°N) and polar routes (>78°N). For commercial
aviation this can be a risk for frequent flyers and particularly for aircrew. Effective
pilot training programs as well as monitoring, measuring, and regulatory measures
in regard to radiation exposure risks for both human tissue and avionics are
recognized by a broad community (Tobiska et al. 2015).

**1.2.4 Mitigating the Effects of SEPs**

The SEP radiation environment is assessed when designing s/c, for s/c mission
planning and operations, and when human spaceflight is involved. How does one
best go about protecting assets in space and on the ground from the effects of
SEP events? For this purpose post-event analysis “hazard assessment” is performed
after an anomaly occurs (is recorded). Furthermore, mitigation procedures are put in
place before launch (e.g. s/c shielding, redundancy onboard) and during operations
SEP forecasting takes place.

**1.2.4.1 Hazard Assessment**

Analyzing s/c anomalies (hazard assessment) is one way to infer whether an
observed anomaly was due to technical or human error, or whether it was a
direct consequence of space environment conditions. In Fig. 1.13 it is clearly seen
that there was an increase in the number of SEUs in the Ramdisk onboard the
LEO Algerian Alsat-1 satellite during three SEP events (29/10/2003: 790 SEUs,
20/01/2005: 774 SEUs, 13/12/2006: 303 SEUs). Figure 1.14 illustrates that the SEU
rate is directly a function of the SEP energy spectrum; the flatter the spectrum the
higher the number of SEUs (Bentoutou and Bensikaddour 2015).
<figure><img src="image_20.png" /><figcaption>**Fig. 1.13** The daily SEU rate observed on the Alsat-1 Ramdisk during the previous solar cycle (Reproduced from Bentoutou and Bensikaddour 2015, permission for reuse from publisher Elsevier for both print and electronic publication)</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Fig. 1.14** SEU rate (purple bars) with proton energy spectrum composition (>30 MeV, >50 MeV, >100 MeV integral flux) when significant SEP events were observed (28–30/10/2003, 17–18/01/2005, 20–21/01/2005, 7–8/12/2006, 13/12/2006) (Reproduced from Bentoutou and Bensikaddour 2015, permission for reuse from publisher Elsevier for both print and electronic publication)</figcaption></figure>
Information obtained from hazard assessment can provide useful input for both engineering (mitigation) and scientific approaches (forecasting), and establish a one-to-one correction between space environment conditions and technical failures. However, s/c operators are sometimes reluctant in providing anomaly data due to confidentiality issues specifically when the anomaly is an important one (e.g. loss of s/c); this makes it sometimes difficult to assess whether a failure was due to the space environment or not. To complement hazard assessment one therefore relies on efficient mitigation strategies such as s/c shielding and forecasting techniques.

### 1.2.4.2 Mitigation Procedures

The classical engineering approach is based on passive shielding that can protect the crew and hardware (exterior and interior) of the s/c and understanding how the space environment interacts with the shielding. For this reason SEP energy spectra are used as input in engineering tools when computing induced effects such as the dose encountered on technology and humans during SEP events. The shape of the spectrum is important as worst-case scenarios are application dependent, meaning that the flux intensity at lower energies are important for material and solar cell effects. At the other end of the energy spectrum, flux intensities at higher energies are more important for nuclear interactions (e.g., background noise, single event upsets).

Aluminium is generally the material used when building most s/c shielding structures and this type of shielding protects against SEP proton events for the most part (Wilson et al. 1997). However, the higher the particle energy, for example in the case of extreme events, the thicker the shielding material necessary to stop the primary particles. This not only implies the possibility of secondary radiation but also higher costs. For these reasons SEP event forecasting is also relied upon to mitigate against SEP events. Currently, short-term SEP event forecasting systems are based on:

* theoretical understanding (e.g. physical models),

* remote sensing of phenomena such as solar flares, CMEs and active regions

* space-based in-situ observations at L1 (shock arrival, energetic storm particles) and GEO

* historical data

* ground-based observations (e.g. radio, neutron monitors).

and can roughly be divided in two categories:

(a) Physics-based numerical models (e.g. Earth-Moon-Mars Radiation Environment Module (EMMREM) (Schwadron 2010), Predictions of radiation from REleASE, EMMREM and Data Incorporating CRaTER, COSTEP and other SEP measurements (PREDICCS) (Schwadron 2012), Solar Energetic Particle MODel (SEPMOD) (Luhmann et al. 2010), SOLar Particle ENgineering Code (SOLPENCO) (Aran et al. 2006), and SOLPENCO2 (provides SEP modelling
away from 1 AU to the SEP statistical model of the SEPEM project (Crosby et al. 2015)))

(b) Empirical models (e.g. University of Malaga Solar Energetic Particle (UMASEP) system (Núñez 2011), Relativistic Electron Alert System for Exploration (REleASE) (Posner 2007), Proton Prediction System (PPS) (Kahler et al. 2007), PROTONS system (Balch 2008), GLE Alert Plus (Kuwabara et al. 2006; Souvatzoglou et al. 2014) and Laurenza's approach (Laurenza et al. 2009))

In some cases forecasting systems rely on methods from both categories such as the SEPForecast tool built under the EU FP7 COMESEP project (263252) (Crosby et al. 2012), http://www.comesep.eu/alert/.

The EU H2020 HESPERIA project (637324) developed two novel real-time SEP forecasting tools based on the UMASEP and REleASE proven concepts:

* The HESPERIA UMASEP-500 tool makes real-time predictions of the occurrence of GLE events, from the analysis of SXR and differential proton flux measured by the GOES satellite network.

* The HESPERIA REleASE tool generates expected proton flux alerts at two energy ranges (15.8–39.8 MeV and 28.2–50.1 MeV) making use of relativistic electrons ($v > 0.9$ c) provided by the Electron Proton Helium Instrument (EPHIN) on SOHO and near-relativistic ($v < 0.8$ c) electron measurements from the Electron Proton Alpha Monitor (EPAM) aboard the Advanced Composition Explorer (ACE).

Both of these new tools are operational through the project's website (https://www.hesperia.astro.noa.gr/) and described in detail in Chap. 7 of this volume.

**Acknowledgements** Olga E. Malandraki has been partly supported by the International Space Science Institute (ISSI) in the framework of International Team 504 entitled “Current Sheets, Turbulence, Structures and Particle Acceleration in the Heliosphere”.

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<figure><img src="image_22.png" /></figure>
# Chapter 2
## Eruptive Activity Related to Solar Energetic Particle Events

Karl-Ludwig Klein

**Abstract** Solar energetic particle events are associated with solar activity, especially flares and coronal mass ejections (CMEs). In this chapter a basic introduction is presented to the nature of flares and CMEs. Since both are manifestations of evolving magnetic fields in the solar corona, the chapter starts with a qualitative description of the magnetic structuring and electrodynamic coupling of the solar atmosphere. Flares and the radiative manifestations of energetic particles, i.e. bremsstrahlung, gyrosynchrotron and collective plasma emission of electrons, and nuclear gamma-ray emission are briefly presented. Observational evidence on the particle acceleration region in flares is given, as well as a very elementary qualitative overview of acceleration processes. Then CMEs, their origin and their association with shock waves are discussed, and related particle acceleration processes are briefly described.

### 2.1 Introduction

Solar energetic particle events are associated with transient solar activity, especially with flares and coronal mass ejections (CMEs). The understanding of how and when the sun ejects enhanced fluxes of protons, ions and electrons, sometimes up to relativistic energies, needs insight into these basic eruptive processes. In this chapter an elementary introduction is presented. Particle acceleration requires transient electric fields. They are produced in relation with magnetic reconnection and turbulence, and in large-scale shock waves driven by CMEs. Since flares and CMEs often occur together, it is not easy to identify which of the candidate acceleration processes is at work. They may all act together, but provide particles of different energies.

K.-L. Klein (✉)

LESIA-Observatoire de Paris, CNRS, 92190 Meudon, France

PSL Research University, Universités P & M. Curie, Paris-Diderot, Meudon, France

e-mail: ludwig.klein@obspm.fr
The chapter starts with a brief overview of the magnetic structuring of the outer solar atmosphere (Sect. 2.2). Flares and accelerated particle signatures related to them in the solar atmosphere are introduced in Sect. 2.3, together with the radiative processes that make these particles observable at gamma-ray, hard X-ray and radio wavelengths. CMEs, shock waves and the related particle acceleration processes are addressed in Sect. 2.4. Because of the introductory nature of this chapter, references are rather to review papers than to the original literature.

## 2.2 The Scene

The solar corona is a hot plasma with an average ion temperature $T \approx 1.5 \cdot 10^6$ K. The mean energy of the particles in this plasma, to the extent that it can be described by a Maxwellian distribution, is $kT \approx 160$ eV, where *k* is Boltzmann's constant.

A remarkable and significant feature revealed by eclipse observations (Fig. 2.1a) is the non-spherical shape of the corona. In an eclipse image one sees especially light from the photosphere, which is Thomson-scattered by free electrons in the corona. The morphology of the corona hence shows the electron density integrated along the line of sight. The image demonstrates that gravity is not the only force that comes into play. The electron concentrations shown by the bright localized structures are confined by magnetic fields. This is also shown by the EUV multi-wavelength image from the *Solar Dynamics Observatory* in Fig. 2.1b: this emission is a mixture of bremsstrahlung and spectral lines emitted mostly at MK-temperatures. The bright structures (active regions) visualize coronal magnetic field lines connected to the underlying atmosphere and the solar interior. They overlie regions with strong magnetic fields in the photosphere, often including sunspots.

<figure><img src="image_23.png" /><figcaption>**Fig. 2.1** The solar corona during a total eclipse in visible light (a; courtesy C. Viladrich) and as seen in EUV (b; courtesy of NASA/SDO and the AIA, EVE, and HMI science teams)</figcaption></figure>
The localized magnetic fields in active regions are the emerged parts of a global solar magnetic field. The plume-like structures pointing to the lower left and upper right in the eclipse image show clearly that such a global magnetic field exists. In and below the photosphere this field is subject to the motions of the plasma, such as the convective motions revealed by granulation and super granulation in the photosphere. These motions shuffle field lines around in the photosphere, concentrate them into magnetic flux tubes with a strong magnetic field that is surrounded by regions with no or weak magnetic field. As the kinetic and dynamic pressure of the plasma decrease with increasing altitude, the magnetic field fills the entire space in the corona, and dominates the dynamics of the low corona. The confinement of the plasma there creates the structures shown in Fig. 2.1.

The plasma motions in the photosphere and below inject energy into the magnetic field, which is transported by field-aligned electric currents to the overlying corona, and may be temporarily stored there (Forbes 2010). As a result of this electrodynamic coupling with the photosphere and the convection zone, the corona undergoes dynamical evolution on different scales of time and energy, ranging from coronal heating to large eruptive events, the manifestations of which are flares and CMEs. They often occur together. In the following these combined events are referred to as eruptive events. When a flare is not accompanied by a CME, it is often called a confined flare.

## 2.3 Solar Flares: Energy Release and Radiative Signatures of Charged Particle Acceleration

From the observational viewpoint a solar flare is defined as a temporary brightening across the electromagnetic spectrum. Radio and X-ray signatures observed during a major flare are shown in Fig. 2.2. No spatial resolution is involved. The shape of the light curve depends strongly on the nature of the emitting particle population: the relatively slow and smooth time evolution in soft X-rays (top panel) comes from the heating to $T > 10^7$ K of a coronal volume located in an active region. The emission evolves relatively slowly, on minute time scales, due to the thermal inertia of the coronal plasma. Hα emission (not shown here) comes from a cooler plasma volume in the chromosphere and displays also a smooth overall evolution. Hard X-rays (second panel from top) and microwaves, on the other hand, are mostly emitted by non-thermal electron populations, that is electrons accelerated to energies of tens of keV or sometimes several MeV. These are much higher than average energies in the pre-eruptive corona (100–200 eV) or even in the hot flare plasma (a few keV). Emission from non-thermal electrons is usually spiky, in particular during the impulsive flare phase of the flare, which is the rise phase of the soft X-ray time profile. The spikiness most probably reveals the fragmentation of the acceleration process.
<figure><img src="image_24.png" /><figcaption>**Fig. 2.2** Dynamic spectrum and single-frequency records of a complex hard X-ray and radio burst on 2003 Nov 3 (Dauphin et al. 2006). *From top to bottom*: (1) Time histories in two soft X-ray channels (GOES), (2) in one hard X-ray channel (RHESSI), (3) at six individual radio frequencies (Trieste at 610 MHz, Nançay Radioheliograph at the other frequencies), (4) dynamic spectrum from 2000 to 500 MHz (Zurich) and 400 to 40 MHz (Potsdam). Shading from *black* (background) to *white* (bright emission). *Horizontal white bands* are frequency ranges where no observation is possible because of terrestrial emitters. Credit: Dauphin et al., A&A 455, 339, 2006, reproduced with permission ©ESO</figcaption></figure>
Soft X-rays are routinely observed since 1975 by the *Geostationary Operational Environmental Satellites* (GOES) operated by NOAA (USA). Because of their general availability they replaced Hα observations from ground as the standard indicator of solar flares. The reference for the importance of a solar flare is the peak soft X-ray flux in the 0.1–0.8 nm channel of GOES (also referred to as the 1–8 Angstrom channel): bursts with peak flux $n \cdot 10^{-m}$ are referred to as class X ($m = -4$), M ($m = -5$), C ($m = -6$) etc. for classes B and A, followed by a multiplier $n$ of the order of magnitude. A flare of class X3.5 has a peak flux in the 0.1–0.8 nm band of $3.5 \cdot 10^{-4} \text{ W m}^{-2}$.

### 2.3.1 Emission Processes

Emission in different spectral ranges is produced by different mechanisms in different regions of the atmosphere: microwaves (1 GHz—some tens of GHz) are gyrosynchrotron radiation, while hard X-rays are bremsstrahlung of energetic electrons with ambient protons and ions. In the event shown in Fig. 2.2 a first hard X-ray burst (second panel from top, photon energies 100–150 keV) occurs during the impulsive phase of the flare. After its decay minor fluctuations persist. They are followed by a new rise near 09:57 UT. The single-frequency records at radio frequencies between 610 and 164 MHz show some similarities with the hard X-rays, especially the initial impulsive phase emission and (down to 236 MHz) the new rise near 09:57 UT. But the brightest emission at these frequencies has no obvious counterpart in the hard X-ray time profile. This emission comes from higher in the solar atmosphere than the X-rays. In fact the radio spectrum is rather complex, as shown by the dynamic spectrogram in the bottom panel. Broadband features like the late rise near 09:57 UT are clearly visible, but are a relatively faint background emission. The broadband nature points to the gyrosynchrotron mechanism as the basic process. The brightest emissions are very strongly structured in frequency and time. They reveal a variety of different radiation processes, which involve micro instabilities of the plasma that are far from being completely understood. These emissions are collectively referred to as “plasma emission”.

#### 2.3.1.1 Bremsstrahlung

Free electrons travelling through a background of ions are deflected by the Coulomb force. This change of momentum and energy is balanced by the emission of a photon. The long range of the Coulomb force implies that the electron's motion is dominated by multiple interactions at long distance within the Debye sphere. Each individual interaction creates only a small deflection, corresponding to the emission of a photon with small momentum and energy, at radio wavelengths. Close encounters are rare, but each one creates an individual strong deflection, associated with the emission of a high-energy (X-ray or gamma-ray) photon.
Since bremsstrahlung is a collisional process, the volume emissivity depends on the product of the electron number density and the ion number density, or rather on the sum of this product over protons and all ion species in the plasma. If the impacting electrons are non-thermal, with density $n_e^*$, the emissivity is proportional to the product with the ambient ion density, $n_e^*n_i$. In a thermal plasma, since $n_e \sim n_i$, the emissivity is $\sim n_e^2$. Thermal bremsstrahlung produces the radio emission of the quiet solar atmosphere and also a weak microwave emission in bursts. However, much higher intensities can be reached at radio frequency by other processes. Bremsstrahlung of non-thermal electrons produces the hard X-ray emission of solar flares (Tandberg-Hanssen and Emslie 1988). Because of its dependence on the ambient ion density, the emission comes from dense layers of the solar atmosphere, mostly the chromosphere.

### 2.3.1.2 Gyrosynchrotron Radiation

The smooth broadband emission of moderate flux density in Fig. 2.2 is ascribed to gyrosynchrotron radiation by mildly relativistic electrons (energies from about 100 keV to a few MeV). This mechanism is the generally adopted interpretation of the microwave spectrum of solar flares, implying magnetic fields of some hundreds of gauss in the low corona, and may on occasion extend to much lower frequencies.

The radiation is produced by electrons gyrating in magnetic fields. Thermal electrons in the corona, which have rather low energy, may emit at low harmonics of the electron cyclotron frequency

$$v_c = \frac{eB}{2\pi m_e}, \qquad (2.1)$$

where e is the electric charge, $m_e$ the mass of the electron, B the magnetic field intensity. In magnetic fields of order 1000 G = 0.1 T the cyclotron frequency is in the GHz range. As long as the energy of the emitting electron is low, an observer in the plane of the cyclotron motion will see an electric field that varies nearly sinusoidally in the course of time, corresponding to a frequency spectrum that shows a signal at the cyclotron frequency and its low harmonics. For a relativistic electron with speed v the fundamental frequency of the emission is the electron cyclotron frequency divided by the Lorentz factor $\gamma = 1/\sqrt{1 - (v/c)^2}$. Its radiation is strongly beamed in the direction of motion. The observer looking at a single gyrating electron will perceive a flash each time the velocity vector of the electron is along the line of sight, and will record a succession of sharp pulses. The frequency spectrum contains numerous lines at harmonics of the relativistic cyclotron frequency. In practice these lines are broadened, and the spectrum is a continuum. For a highly relativistic electron ($\gamma \gg 1$; synchrotron radiation) with pitch angle $\alpha$ the emitted
spectrum extends from the cyclotron frequency up to the critical frequency

$$v_c = \frac{3}{2} \gamma^2 v_{ce} \sin \alpha . \qquad (2.2)$$

The emission is referred to as gyrosynchrotron in the case of mildly relativistic electrons as observed in solar flares.

### 2.3.1.3 Plasma Emission from Electron Beams

The radio spectrum of Fig. 2.2 shows a wealth of structure at decimetre-metre waves (~1000–40 MHz), which contrast with the smooth evolution of the gyrosynchrotron spectrum. For instance, a cluster of narrow-band features is seen between 200 and 400 MHz in the time interval 09:50–09:53. A structured band of emission drifts from about 500 MHz (09:51–09:53) down to 200 MHz (09:53–09:56). The clear spectral structure with narrow bandwidth as compared to the central frequency (Δv/v ≪ 1) and the high flux density are typical of collective or coherent emission processes, where a perturbation of the plasma makes the entire electron population radiate at the characteristic frequencies of the plasma. In the solar corona the plasma frequency is the relevant frequency, because it is higher than the cyclotron frequency.

Most often plasma emission arises from some micro-instability due to deviations of the electron distribution function from isotropy. We outline this mechanism for a type III burst from a magnetic field-aligned electron beam: when a beam of field-aligned fast electrons is superposed on a Maxwellian, a positive slope arises over some velocity range in the distribution function measured along the magnetic field. The electrons in the beam interact with plasma waves with phase speeds in the range where the distribution function has a positive slope. These are Langmuir waves. If there is more energy in electrons slightly above the phase velocity than in electrons that are slower, waves are excited at the expense of the kinetic energy of the electron beam. Electrons are removed from the beam and transferred to lower speeds so that the beam distribution is flattened to a plateau. The instability ceases when the distribution function has no longer a positive slope. Actual measurements of such distribution functions in space are shown in Ergun et al (1998). A recent review is Sinclair Reid and Ratcliffe (2014).

The Langmuir waves cannot escape from the corona, but can transfer their energy to electromagnetic waves, for instance through coupling with other waves: low-frequency waves like ion sound waves or high-frequency waves, especially other Langmuir waves. The transfer of energy between waves can occur provided the three waves satisfy parametric conditions, which can be formulated like the conservation of energy (hv) and momentum in quantum mechanics. Energy conservation implies that the frequency of the resulting wave must be the sum of the frequencies of the interacting waves. The frequency of Langmuir waves $v_L$ is close to the electron plasma frequency. The frequency $v_{EM}$ of an electromagnetic wave generated through
the coupling process with an ion-sound wave (frequency $v_{lf}$) is $v_{EM} = v_L + v_{lf} \simeq v_L \simeq v_{pe}$. Coupling of two Langmuir waves yields $v_{EM} = v_L + v_{L'} \simeq 2v_{pe}$. The electromagnetic wave is hence generated near the electron plasma frequency or its harmonic. Depending on which process prevails, the resulting electromagnetic emission is referred to as “fundamental” or “harmonic”. Both may occur together, or one of the two modes may prevail.

The dependence of the plasma frequency on the ambient electron density creates the typical frequency behaviour of type III bursts: as the electron beam proceeds to increasing altitude, hence to lower ambient electron density, the frequency of the emission decreases. The result is a short burst that drifts from high to low frequencies. In a hydrostatic atmosphere the frequency drift rate is directly related to the speed of the exciter along the density gradient. Relatively fast drifts in type III bursts are related to a fast exciter, namely an electron beam. The slowly drifting type II bursts are ascribed to a slower exciter, namely a shock wave. An example is the burst between 09:53 and 09:55 in the 200–400 MHz range of Fig. 2.2.

### 2.3.1.4 Gamma-Rays from Accelerated Protons and Ions

So far only electron-related radiative diagnostics were addressed. This is a real bias, since observational signatures of non-thermal protons and ions are more difficult to obtain in the solar atmosphere. These particles emit at gamma-ray wavelengths.

Prominent nuclear lines are produced when accelerated protons or ions with energies in the range 1–100 MeV/nucleon bombard the low solar atmosphere. The nuclei of the ambient medium are excited to high energies, and subsequently relax by emitting the gamma-ray photons. If the impacting high-energy particle is a proton and the target a heavy ion, the target does not move significantly, and the emitted gamma-ray line is narrow. If an accelerated ion hits a target proton or helium, it continues its motion and emits a Doppler-shifted line, which is considerably broadened by the angular distribution of the ions.

The most widely observed nuclear line is the neutron capture line near $h\nu = 2.2$ MeV. When protons with energies exceeding about 30 MeV interact with other nuclei, neutrons are released, initially with high energy. When thermalised after some tens of seconds, they can be captured by ambient protons to form a deuterium nucleus. Its binding energy is released via emission of a photon at 2.223 MeV. At still higher energies, above 300 MeV/nucleon, nuclear interactions of protons and helium nuclei with ambient protons create pions, which decay rapidly. Pion-decay positrons eventually annihilate with electrons. Neutral pions decay into photons and create a specific emission feature at $h\nu > 60$ MeV, which in solar gamma-ray spectra shows up as a high-energy bump on top of the decaying spectrum of electron bremsstrahlung. A recent review of solar gamma-ray emission is given by Vilmer et al. (2011). Pion decay gamma-ray emission from relativistic protons will be addressed in more detail in Chap. 8.
### 2.3.2 Where Are Electrons Accelerated in Solar Flares?

Images of flares in hard X-rays show frequently configurations that look like magnetic loops with a particle acceleration region near or above the top. In the RHESSI image of the 2005 Jan 20 flare (Fig. 2.3a) the red contours outline a thermal X-ray source at temperatures above $10^7$ K, which traces the upper part of a coronal loop. The blue contours are the sources of hard X-ray emission from non-thermal electrons precipitated into the chromospheric footpoints. The two elongated grey bands onto which the footpoints project are flare ribbons seen in UV. They outline regions of the chromosphere heated by energy deposition during the flare. This source morphology is generally interpreted as a signature of energy release near or above the loop top, which heats the plasma in the coronal loop and accelerates electrons. They escape from the primary acceleration site as magnetic field-aligned beams. Because of their high energy they interact very little with the low-density coronal plasma and precipitate into the dense chromosphere at the base of the loop. They lose their energy instantaneously through collisions with this dense environment, while simultaneously emitting a small amount as hard X-rays. The RHESSI image is a snapshot: during the impulsive phase of the event the hard X-ray sources occur in an irregular temporal succession at neighbouring places on the UV ribbons. A standard cartoon scenario as in Fig. 2.3b locates the energy release above the loop top, probably related to magnetic reconnection. The upward field lines may be part of a plasmoid that is ejected upward or they may be open to the high corona. The energy release may equally well be related to magnetic reconnection

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /><figcaption>**Fig. 2.3 (a)** Contours of hard X-ray emission in two spectral bands (RHESSI superposed on a negative TRACE image of chromospheric flare ribbons in UV (Krucker et al. 2008). Credit: Krucker et al., Astrophys. J. 678, L63 (2008). ©AAS. Reproduced with permission. **(b)** Cartoon scenario of particle populations and related electromagnetic emissions during a flare. From Klein and Dalla (2017)</figcaption></figure>
with another closed magnetic structure. A more detailed discussion of hard X-ray source morphology and its interpretation can be found in Holman et al. (2011).

Radio emission of electron beams, such as type III bursts, is another key observation to identify the electron acceleration. In some flares Aschwanden and coworkers were able to identify radio emissions from bidirectional electron beams, with downward-directed beams at high frequency (high ambient electron density) and upward directed beams at low frequency. The authors concluded that these beams came from a common acceleration region where the electron plasma frequency had the value corresponding to the frequency from where the oppositely drifting radio bursts emanate. They derived an ambient density of about (1–10) · 10⁹ cm⁻³. From the timing of peaks at different hard X-ray energies they concluded that the acceleration region is placed at a typical altitude of about 1.5 times the half-length of the magnetic loop above the photosphere (see Sects. 3.3 and 3.6 of Aschwanden 2002). This corresponds to cartoon scenarios such as Fig. 2.3b.

It is not clear if protons and ions are accelerated in the same regions as electrons. A close connection between relativistic electrons (above 300 keV) and protons above 30 MeV is suggested by the observed correlation of the peak fluxes of their respective bremsstrahlung and nuclear line emissions (Shih et al. 2009). But differences become visible in the detailed time evolution (Kiener et al. 2006), and the source locations are in general not identical (see review in Vilmer et al. 2011). While models exist to explain different acceleration regions of these particle species, for instance in terms of resonances with different types of waves, the observations provide a number of challenges which show that our understanding is far from complete.

### 2.3.3 A Qualitative View of Acceleration Processes

An electric field is needed to accelerate charged particles. This is simply because only the electric component of the Lorentz force **F** = *q*(**E** + **v**×**B**) is able to change the energy, $\frac{dW}{dt} = \mathbf{v} \cdot \mathbf{F} = q\mathbf{v} \cdot \mathbf{E}$. Since the solar corona is a highly conducting medium in most parts (comparable to copper), no static electric field can be maintained along the ambient magnetic field. Peculiar configurations where transient magnetic-field-aligned electric fields can exist are current sheets and shock waves. We cannot measure electric fields in the corona, only infer them from the plasma motions.

In the cartoon scenario of Fig. 2.3b, where a reconnection region is depicted by oppositely directed magnetic fields, the most elementary electric field is the motional **E** = −**V** × **B** induced by the inflow of plasma into the reconnecting current sheet. A test particle exposed to this field will **E** × **B**-drift into the current sheet. In the region where the magnetic field is near zero, the particle decouples from the magnetic field. Protons propagate along the induced electric field, while electrons propagate opposite to it. Both particle species are hence accelerated. The stationary situation shows the principle, but is unlikely to be encountered in the solar corona. Current sheets are expected to fragment into magnetic islands. They
correspond to parallel electric currents that attract each other, so that the magnetic islands formed during the fragmentation coalesce. Charged particles are trapped in a highly dynamical medium between coalescing magnetic islands and gain more energy (Cargill et al. 2012).

Reconnection jets are another ingredient of magnetic reconnection that can lead to particle acceleration. They evacuate the plasma from the reconnection region. The jets may generate shock waves when impinging on the underlying magnetic structures, or waves (turbulence) in the ambient plasma. Different particle species interact with different types of waves. This may explain preferential acceleration of some particle species, as observed in those SEP events where the ³He abundance is enhanced by several orders of magnitude with respect to the quiet solar corona. Acceleration processes at shock waves and in turbulent plasmas are discussed in more detail in Chap. 3.

## 2.4 What Is a Coronal Mass Ejection?

The observational definition of a coronal mass ejection (CME) is an extended outward travelling feature in white-light coronographic images. This means that the visible manifestation is the outward motion of plasma. Phenomenologically many CMEs have a three-part structure in coronographic images as shown in Fig. 2.4: an outer bright region, which is understood to be mainly composed of plasma swept up from the ambient corona by the outward propagating piston, a dark cavity, which is low-density material in the ejected magnetic structure, and a bright core consisting of filament material. This basic structure of the CME is created by the magnetic field. A CME is hence the ejection of a large-scale coronal magnetic structure together with the confined plasma.

<figure><img src="image_27.png" /><figcaption>**Fig. 2.4** Coronographic image of a CME in white light, from Riley et al. (2008). Credit: Riley et al., Astrophys. J. 672, 1221 (2008). ©AAS. Reproduced with permission</figcaption></figure>
The time-height diagram constructed from tracking the front of the white-light feature through the field of view of the coronograph shows CMEs that travel at constant speed in the field of view, while others accelerate or decelerate. The fastest part is in general the outward-moving apex of the CME, but in some events fast lateral expansion is also observed. Outward speeds observed by SoHO/LASCO since 1996 range from a poorly defined lower limit of some tens of km s⁻¹ to a maximum of 3400 km s⁻¹ (Gopalswamy 2009). The CME leaves the Sun and travels through the heliosphere. The physical process behind this phenomenology is a large-scale instability of coronal magnetic structures. A recent review on CMEs is Chen (2011).

## 2.4.1 CME Magnetic Structure and Eruption

The magnetic structure outlined by the dark CME cavity (Fig. 2.4) looks like a closed two-dimensional magnetic field. More detailed studies suggest it is the projection of a three-dimensional magnetic flux rope, where the magnetic field lines are helices wound around the confined plasma. Such a flux rope is sketched in Fig. 2.5a. The blue-green loop-like structure is the plasma in the flux rope, the blue and black field lines indicate the helicoidal magnetic field in and around the flux rope. The Lorentz force on this configuration is directed upward, since the magnetic field lines are more densely packed below the flux rope than above. The upward Lorentz force, sometimes called “hoop force”, is balanced in equilibrium by the downward-directed Lorentz force exerted by the surrounding coronal magnetic field, whose field lines are plotted in orange. An excess upward force can be generated for instance by the torsion of one foot of the flux rope and its magnetic field, due to the plasma motions in the photosphere. When this happens, the excess magnetic pressure below the flux rope is enhanced—the flux rope is lifted by the Lorentz force (Fig. 2.5b), ambient coronal plasma and the embedded magnetic field are convected from both sides towards the region where it was located before, and oppositely directed magnetic fields can reconnect. This is illustrated in Fig. 2.5b, c for two field lines, with the reconnection happening in a limited region schematically indicated by the yellow symbol of an explosion. New magnetic field is then added to the flux rope (the upper part of the field line drawn in red colour), and new magnetic loops form in the low corona. These reconnected loops appear as arcades in EUV images. Their formation may continue over several hours.

A 2D projection of this situation is depicted in Fig. 2.5d, together with the consequences of the magnetic reconnection: charged particles accelerated in trans-ient electric fields around the reconnection region, and electromagnetic emissions excited directly or indirectly by these particles in different regions of the erupt-ing configuration. Hard X-rays and gamma-rays are generated respectively by electrons and ions in the dense low atmosphere. Radio emission is generated by energetic electrons in different regions, including the dilute plasma in higher atmospheric layers. Typical radio signatures of electrons accelerated during flux
<figure><img src="image_28.png" /><figcaption>**Fig. 2.5** Cartoon scenario of the magnetic field configuration around a magnetic flux rope in the solar corona (a), and of its evolution during the liftoff of a coronal mass ejection (CME; b, c). The white and grey-shaded areas indicate opposite magnetic polarities in the photosphere, separated by the *grey line*, where the vertical photospheric magnetic field is zero. Figure (d) shows a two-dimensional projection of (c)</figcaption></figure>

rope eruptions are broadband continua like the one created by gyrosynchrotron emission in the bottom panel of Fig. 2.2, or similar features created by plasma emission from trapped electrons. Since the electrons are simultaneously present in a range of ambient electron densities, plasma emission occurs at a corresponding
range of frequencies, i.e. over a broad spectral band. These emissions are called
type IV bursts.

2.4.2 Shock Waves and Particle Acceleration at CMEs

A characteristic signature of the corona observed in EUV in the aftermath of a
CME liftoff is the arcade of loops, which is thought to form as the magnetically
stressed corona reconnects. The progress of the reconnection process is depicted in
the cartoons of Fig. 2.5, where a loop formed in the course of reconnection is plotted
in red. Both reconnection and turbulence in the aftermath of the CME are able to
accelerate particles in a similar way as during the impulsive phase of solar flares.
Because of the outward movement of the CME, the relevant processes occur higher
up than during the impulsive flare phase, and less particles get access to the low
solar atmosphere. All processes involving collisions with the ambient plasma, such
as electron bremsstrahlung or nuclear line radiation, are less efficient. The main
evidence on particle acceleration in the aftermath of a CME is a long-lasting type
IV burst at frequencies below about 1 GHz.

Fast CMEs are likely to exceed the Alfvén speed and the fast magnetosonic speed of the coronal plasma, and to drive shock waves. Such shock waves are observed in situ in the heliosphere, and can be inferred in the corona from type II bursts at radio frequencies, for instance between 09:51 and 09:55 in the 500–100 MHz range in Fig. 2.2. Shock waves have also been inferred from UV spectroscopy in the corona (Mancuso 2011). The associated shocks usually have moderate Mach numbers: model-dependent interpretations of the radio spectra of type II bursts give $M_A = 1.2–2.9$ (Vršnak et al. 2002). But modelling of stereoscopic white-light observations (Rouillard et al. 2016) shows that higher Mach numbers can be found in localized regions of some CMEs.

Particle acceleration at coronal shock waves is discussed in Chap. 3. Two broad
categories of processes are commonly distinguished: shock drift acceleration occurs
predominantly in quasi-perpendicular shock regions, where the normal on the
shock front and the upstream magnetic field include a large angle. In the shock-
drift acceleration process particles gain energy from the convective electric field
**E** = -**V** × **B** in the shock frame, where **V** is the inflow speed of the plasma, **B**
the upstream magnetic field in the rest frame of the shock. In the upstream region,
electrons and ions **E** × **B**-drift towards the shock front. Because the magnetic field
is compressed by the oblique shock, the particles undergo a gradient drift along
the shock front. This drift is directed along the electric field for positively charged
particles, and opposite to the electric field for negatively charged particles. Hence
electrons, protons and ions gain energy. Depending on the energy and pitch angle
before the first encounter with the shock, the energy gained from the drift along
the convection electric field may be such that the particle is again injected into the
upstream medium and may escape. Besides a gradient drift, a particle also undergoes
a curvature drift while its guiding centre travels along the magnetic field, which is
curved in the shock transition. In a planar fast shock, the angle between a field line and the shock normal is larger downstream than upstream, and the curvature drift is opposite to the gradient drift. The curvature drift hence leads to energy losses. But the drift speed decreases as the shock becomes more oblique, so that in quasi-perpendicular shocks the gradient drift, and hence the energy gain, dominates.

In work on shock-acceleration of SEPs the acceleration process most often invoked is diffusive shock acceleration. When ions are reflected at shock waves and stream into the upstream region, they acquire a beam-like distribution and are therefore likely to generate waves, parallel-propagating Alfvén waves as well as obliquely propagating fast magnetosonic waves. When these waves grow to sufficient amplitudes, they can scatter subsequent ions back to the shock. Since the shock propagates faster than these waves, ions find themselves confined between approaching scattering centers downstream and upstream, and gain energy by bouncing back and forth through the shock front, until they eventually escape. In order to interact with the waves, the particles must be able to escape into the upstream medium after the initial reflection. This means that they must stream away from the shock front at a minimum speed $V/\cos\theta_{Bn}$, where $V$ is the speed of the shock as above, and $\theta_{Bn}$ the angle between the shock normal and the upstream magnetic field vector. Since this speed is the smaller, the smaller $\theta_{Bn}$, diffusive shock acceleration is expected to work best at quasi-parallel shocks.

## 2.5 Summary and Conclusion

Charged particles may be accelerated during flares and CMEs to energies largely above the mean energy in the corona. The energy transferred to the particles is drawn from plasma flows in the photosphere and below, transported to the corona along magnetic field lines, and stored in the coronal field. The release involves magnetic reconnection or the loss of equilibrium of large-scale magnetic structures, leading to CMEs.

Particle acceleration processes related to magnetic reconnection, turbulence and shock waves are all supported by observations of electromagnetic emissions and SEPs. The evolution of the coronal magnetic field is usually described by magnetohydrodynamics, and the accelerated particles as test particles. High fluxes of non-thermal particles may, however, develop sufficient pressure and energy to back-react on the magnetic field configuration, invalidating the MHD and test particle hypotheses. This is especially expected in large solar events, where the energetic particles contain a substantial amount of the energy released during the flare or the CME (Emslie et al. 2005; Mewaldt et al. 2005). We are therefore far from a complete understanding of how particle acceleration proceeds.

In the present author's opinion it is not possible to identify an acceleration process that is *a priori* more plausible than others. In astrophysical settings like supernovae, shock waves produced in the course of a gravitational instability are widely considered as a privileged process to explain energetic particle populations.
Shock waves in the solar corona are always the result of plasma processes that are themselves conducive to particle acceleration. The preference for shock waves as particle accelerators is not justified under these conditions. Constraints from observations remain essential to understand how particle acceleration is related to the different manifestations of eruptive solar activity, and which physical processes are at work.

**Acknowledgements** The author is grateful to the HESPERIA consortium for many interesting discussions, and to J. Kiener for his comments on the manuscript.

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<figure><img src="image_29.png" /></figure>
Chapter 3

Particle Acceleration Mechanisms

Rami Vainio and Alexandr Afanasiev

**Abstract** This chapter provides a short tutorial review on particle acceleration in dynamic electromagnetic fields under scenarios relevant to the problem of particle acceleration in the solar corona and solar wind during solar eruptions. It concentrates on fundamental aspects of the acceleration process and refrains from presenting detailed modeling of the specific conditions in solar eruptive plasmas. All particle acceleration mechanisms (in the solar corona) are related to electric fields that can persist in the highly conductive plasma: either electrostatic (or potential) or inductive related to temporally variable magnetic fields through Faraday’s law. Mechanisms involving both kinds of fields are included in the tutorial.

**3.1 Introduction**

Solar energetic particles (SEPs) are accelerated in flares and coronal mass ejections (CMEs) (see Chap. 2 and references therein). In this chapter, we will give a tutorial review of particle acceleration mechanisms in solar eruptions. The aim is not to give a comprehensive review of the original literature but rather give the reader an idea of what are the main ideas that could explain the acceleration of ions and electrons to the relativistic energies, we observe to be produced in the SEP events. For recent reviews on SEPs we direct the reader to Desai and Giacalone (2016) and Reames (2017).

SEPs are accelerated in the electromagnetic fields related to the dynamics of the
solar corona and solar wind during solar eruptions. In quasi-static fields of the quiet-
time solar corona, strong electric fields would not exist and particle acceleration
beyond quasi-thermal energies (up to ~keV) would seldom occur. However, solar
eruptions are manifestations of very non-thermal conditions favorable for particle
acceleration.

Magnetic reconnection is by definition a process, which requires electric fields
to be present in the system. As discussed in Chap. 2 the plasma advected towards

<footnote>R. Vainio (✉) • A. Afanasiev
Department of Physics and Astronomy, University of Turku, Turku, Finland
e-mail: rami.vainio@utu.fi; alexandr.afanasiev@utu.fi</footnote>
a current sheet with a frozen-in magnetic field will be associated with a convective electric field, which can accelerate particles. The situation is most likely a dynamic one, where induced electric fields with closed field lines will be generated by temporally varying magnetic fields. As discussed in Sect. 3.2.1, this can lead to very efficient conditions for particle acceleration. Reconnection outflows from the corona towards the Sun may generate collapsing loop systems, which can also act as particle accelerators. In addition to direct acceleration by large scale fields, reconnection jets in flares can also be turbulent regions. These can give rise to stochastic acceleration in the plasma (see Sect. 3.2.5). Even shock acceleration (Sect. 3.2.3) could in principle occur in reconnection regions, since the super-Alfvénic reconnection jets can be terminated by dense plasma or strong magnetic fields impeding their flow. Flares are commonly thought of as being the source of (at least) the impulsive SEP events (Reames 2017). These events show peculiar abundance ratios of ions difficult to explain in terms of non-selective acceleration processes, as shock acceleration. Presently the abundances are probably best explained by stochastic acceleration (Petrosian 2012).

Especially the gradual SEP events are undoubtedly related to CMEs. Mostly, but not exclusively, because of the morphology of the time-intensity profiles of the MeV and deka-MeV protons, showing clear organization based on the magnetic connectivity of the observer to the CME, the bow shock driven by the fast magnetized eruption has been identified as the prime candidate for accelerating the ions in SEP gradual events (Desai and Giacalone 2016; Reames 2017). There is a large amount of evidence that CME-driven shocks accelerate protons up to some tens of MeVs, but the case becomes more open at energies approaching and exceeding 100 MeV. Theoretical modeling (see, e.g., Chap. 9), however, suggests that CME-driven shocks are able to accelerate ions also to relativistic energies. For electrons, strong correlation between the observed flux properties of deka-MeV protons and MeV electrons (see, e.g., Chap. 7) could point towards shock acceleration as well, but the theoretical foundations of electron acceleration by shocks are not on such solid basis as for ions, mainly because electrons resonate with plasma fluctuations at much lower scales than protons (see Sect. 3.2.2). The main shock acceleration mechanisms are, however, able to accelerate electrons as well. In addition to shock acceleration, the CME downstream region hosts regions of strong plasma compression and turbulence. Such regions may accelerate particles via the compressional acceleration mechanism (Sect. 3.2.4) and stochastically (Sect. 3.2.5), respectively. Note also, that coronal magnetic restructuring (outside the flaring active region) behind the CME may also lead to particle acceleration via many of the same mechanisms.

## 3.2 Acceleration Mechanisms

Particle acceleration in solar plasmas requires electric fields and the ability of particles to propagate along them, since the energy gain rate produced by the
Lorentz force, $q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$, is $\dot{\mathbf{W}} = q\mathbf{v} \cdot \mathbf{E}$. Here $q$, $\mathbf{v}$, and $\mathbf{W}$ are the charge, the velocity and the kinetic energy of the particle and $\mathbf{E}$ and $\mathbf{B}$ are the electric and the magnetic field. Electric fields can be derived from the scalar and vector potential of the electromagnetic field:

$$ \mathbf{E} = -\nabla\phi - \frac{\partial\mathbf{A}}{\partial t}, \qquad (3.1) $$

and the magnetic field from the vector potential $\mathbf{B} = \nabla \times \mathbf{A}$. Particles can move in the direction of the electric field in several possible ways. In the following, we will discuss the different possibilities briefly.

### 3.2.1 Large-Scale Electric Field Acceleration

We will first discuss electric fields in large scales, meaning that the field changes over scales that are large compared to the gyroradii of the accelerated particles. In the simplest case, the electric field has a component parallel to the magnetic field, along which ions and electrons can move without being affected by the magnetic field. Large-scale, static parallel electric fields are, however, not easy to set up in dilute coronal plasmas, since in the near absence of collisions, electrons can usually move very quickly along the magnetic field and build up charge separation to counter any accelerating parallel electric field. In the presence of magnetic field inhomogeneities, however, gradient and curvature drifts can move ions and electrons in the direction perpendicular to the field. This may lead to particles gaining energy, if there is an electric field along the direction of the drift motion. This is the basis of several shock-acceleration mechanisms to be discussed in Sect. 3.2.3.

Magnetic reconnection is perhaps the most obvious plasma process to set up large-scale electric fields. Some models consider static electric fields accelerating particles near magnetic nulls, where particles can propagate along the electric field unimpeded (Litvinenko 1996). This, however, is not the only way to set up a large scale electric field that particles can utilize effectively to get accelerated.

Choosing to work in the Coulomb gauge ($\nabla \cdot \mathbf{A} = 0$), the electric fields due to charge separation are all described by the scalar potential and, thus,

$$ \nabla^2 \phi = -\frac{\rho_q}{\epsilon_0}. \qquad (3.2) $$

Therefore, the part of the electric field related to the vector potential contains inductive fields related to the time derivative of the magnetic field via Faraday's law. These electric fields are then described by fields lines that are either closed loops or extend to the boundaries of the system. Electric fields induced by the temporally variable magnetic field (or vector potential) can sometimes lead to very efficient acceleration of particles. Let us illustrate this with an example.
Consider

$$ \mathbf{B} = B_0 \left( \frac{y\mathbf{e}_x - x\mathbf{e}_y}{r_0} + \gamma_0 t\mathbf{e}_z \right) = \nabla \times \mathbf{A} \qquad (3.3) $$

$$ \mathbf{A} = B_0 \left( -\frac{y\mathbf{e}_x - x\mathbf{e}_y}{2} \gamma_0 t + \frac{x^2 + y^2}{2r_0} \mathbf{e}_z \right), \quad \nabla \cdot \mathbf{A} = 0 \qquad (3.4) $$

$$ \mu_0 \mathbf{J} = \nabla \times \mathbf{B} = -\frac{2B_0}{r_0} \mathbf{e}_z \qquad (3.5) $$

This magnetic configuration is represented by a helical magnetic field, where the axial field increases linearly with time and the poloidal field (driven by a constant axial current) is constant (see Fig. 3.1).

The axial magnetic field is generated by a poloidal external current flying at some radial distance, e.g., $r = r_0$, increasing linearly with time. We neglect the displacement current so the time evolution of the axial field (and external current) has to be slow in the sense that $r_0\gamma_0 \ll c$. With these assumptions, the induced electric field is

$$ \mathbf{E} + \nabla\phi = -\frac{\partial \mathbf{A}}{\partial t} = \gamma_0 B_0 \frac{y\mathbf{i} - x\mathbf{j}}{2} = \frac{1}{2} r_0 \gamma_0 \mathbf{B}_p. \qquad (3.6) $$

Clearly, the induced electric field is in the direction of the poloidal magnetic field and, therefore, particles propagating along the field lines will be accelerated along the field, ions in one direction and electrons in the other. The axial field, however, will make sure that electrons and ions will also propagate along the axial direction,

<figure><img src="image_30.png" /><figcaption>**Fig. 3.1** Electromagnetic fields in the simple model. The magnetic field (**left**) is a helix, where the axial field (**B**<sub>x</sub>) is linearly increasing with time and the poloidal field (**B**<sub>p</sub>) is driven by a homogeneous axial current. The poloidal electric field **E**<sub>p</sub> (**right**) is induced by the time-dependent axial magnetic field and it is in the same direction as the poloidal magnetic field. Charge separation caused by the particles accelerated by **E**<sub>p</sub> along the magnetic field lines will produce an oppositely directed axial electric field. This would eventually turn the electric field perpendicular to the magnetic field if charge separation is let to build up in the plasma</figcaption></figure>
in the opposite directions. Thus, charge separation will be set up. It produces a potential field in the axial and radial direction, which combined to the induced field could produce a total electric field perpendicular to the magnetic field. The condition would be

$$ 
\begin{aligned}
\mathbf{E} \cdot \mathbf{B} &= \frac{r^2 \gamma_0}{2r_0} B_0^2 - B_0 \gamma_0 t \frac{\partial \phi}{\partial z} \sim 0 \\
\Rightarrow \phi \sim \frac{r^2 z B_0}{2r_0 t} \Rightarrow -\nabla \phi \sim -\frac{r z B_0}{r_0 t} \mathbf{e}_r - \frac{r^2 B_0}{2r_0 t} \mathbf{e}_z
\end{aligned}
\quad (3.7) $$

However, it is clear that this field can be approached only after a finite time (as it diverges at $t \to 0$). Thus, during a limited amount of time, the induced electric field can accelerate particles very efficiently.

We notice also that particles could be accelerated by the induced electric field even without the poloidal magnetic field in the system. If only the axial magnetic field is present, particles will perform gyro-motion around it. The contour integral of the Lorentz force along one circular Larmor orbit,

$$ \Delta W_{\perp} = \oint_{J_{\perp}} q \mathbf{E} \cdot d\mathbf{l}, \quad (3.8) $$

is in the left-handed [right-handed] sense around the magnetic field $\mathbf{B} = \mathbf{B}(t) \mathbf{e}_z$ for $q > 0$ [$q < 0$]. Taking the integral in the right-handed sense (denoted below by the minus sign after L) gives

$$ 
\begin{aligned}
\Delta W_{\perp} &= -|q| \oint_{J_{\perp}} \mathbf{E} \cdot d\mathbf{l} = -|q| \int_{A_{\perp}} (\nabla \times \mathbf{E}) \cdot (\mathbf{e}_z \, dS) \\
&= |q| \int_{A_{\perp}} \frac{\partial \mathbf{B}}{\partial t} \cdot (\mathbf{e}_z \, dS) = |q| \pi r_{\perp}^2 \frac{\partial B}{\partial t}
\end{aligned}
\quad (3.9) $$

Since the Larmor radius is $r_L = v_{\perp} \tau_L / 2\pi$, where $\tau_L = 2\pi \gamma m / |q| B$ is the particle gyrotime, and $\gamma$ and $m$ its Lorentz factor and mass, respectively, we have

$$ 
\begin{aligned}
v_{\perp} \Delta p_{\perp} &= \Delta W_{\perp} = |q| \pi \frac{v_{\perp} \tau_L \gamma m v_{\perp}}{2\pi |q| B} \frac{\partial B}{\partial t} \\
\Rightarrow \frac{\dot{p}_{\perp}}{p_{\perp}} &= \frac{1}{p_{\perp}} \frac{\Delta p_{\perp}}{\tau_L} = \frac{1}{2B} \frac{\partial B}{\partial t} = \frac{1}{2} \gamma_0,
\end{aligned}
\quad (3.10) $$

which describes the betatron acceleration process. Clearly, $p_{\perp} = p_{\perp 0} e^{\frac{1}{2}\gamma_0 t}$ gives the perpendicular particle momentum in terms of its initial value $p_{\perp 0}$ at $t = 0$.
### 3.2.2 Resonant Wave Acceleration

As discussed above, generating electric fields with quasi-static components along the magnetic field is not easy. Instead of large scale fields, the fields can also be at the gyromotion scale, and such fields can be carried by various plasma waves. At first sight, if the electric field is fluctuating, it is quasi-periodically pointing in opposite directions and the time integral of the acceleration would seem to vanish in most cases. In this case, however, it is possible for the particle to be in resonance with the wave's electric field: if the period of particle motion agrees with the period of the wave, the phase of the wave at the location of the particle can be constant, leading to a constant accelerating electric field felt by the particle.

This process can be illustrated with a simple linearized model. Consider the electric field of a circularly polarized wave propagating along the mean magnetic field taken to be constant and along the z axis. Thus,

$$ \mathbf{E}_1 = E_1[\mathbf{e}_x \cos(kz - \omega t) - \mathbf{e}_y \sin(kz - \omega t)], \quad (3.11) $$

and the phase speed of the wave is $V_\phi = \omega/k$. The electric and magnetic fields of the wave are related via Faraday's law, i.e.,

$$ \begin{aligned} \frac{\partial \mathbf{B}_1}{\partial t} &= -\nabla \times \mathbf{E}_1 = - \begin{vmatrix} \mathbf{e}_x & \mathbf{e}_y & \mathbf{e}_z \\ \partial_x & \partial_y & \partial_z \\ E_{1x} & E_{1y} & 0 \end{vmatrix} = \mathbf{e}_x \frac{\partial E_{1y}}{\partial z} - \mathbf{e}_y \frac{\partial E_{1x}}{\partial z} \\ &= kE_1[-\mathbf{e}_x \sin(kz - \omega t) - \mathbf{e}_y \cos(kz - \omega t)] \end{aligned} \quad (3.12) $$

$$ \Rightarrow \mathbf{B}_1 = \frac{k}{\omega} E_1 [-\mathbf{e}_x \cos(kz - \omega t) + \mathbf{e}_y \sin(kz - \omega t)] \quad (3.13) $$

$$ \Rightarrow \mathbf{E}_1 = -\frac{\omega}{k} \mathbf{B}_1 = -V_\phi \mathbf{B}_1, \quad (3.14) $$

i.e., the magnetic field is circularly polarized, as well. In case the phase speed is less than the speed of light, we can always make a boost along the z axis to a coordinate system where the wave frequency is zero. In this frame the wave electric field vanishes and the charged particles interacting with the wave conserve their energy. In the laboratory frame the change in the particle energy is, thus, obtained through

$$ \Delta W = \Gamma_{\phi} V_{\phi} p' \Delta \mu', \quad (3.15) $$

where $\Gamma_\phi = (1 - V_\phi^2/c^2)^{-1/2}$, $\mu'$ and $p'$ are the pitch-angle cosine and the (constant) momentum magnitude of the particle in the wave frame, and $\Delta X$ represents the
change in the quantity $X$. The equations of motion in the wave frame (omitting the primes) can be written as

$$ \dot{\mu} = \frac{B_1}{B_0} \sqrt{1 - \mu^2} \Omega_0 \cos(\varphi + kz) \quad (3.16) $$

$$ \dot{\varphi} = -\Omega_0 + \frac{B_1}{B_0} \frac{\mu}{\sqrt{1 - \mu^2}} \Omega_0 \sin(\varphi + kz) \quad (3.17) $$

$$ \dot{z} = v\mu, \quad (3.18) $$

where $\varphi$ is the phase angle measured around the $z$ axis and $\Omega_0 = qB_0/\gamma m$ is the particle's relativistic (signed) gyro-frequency. Now, assuming that the magnetic amplitude of the wave is small compared to the mean magnetic field, $B_1 \ll B_0$, we can approximate the phase angle evolution by $\varphi = \varphi_0 - \Omega_0 t$. Substituting that to the equation for $\mu$ shows a resonance (a constant right-hand side) at $kz - \Omega_0 t = 0$

$$ \Rightarrow kv\mu = \Omega_0, \quad (3.19) $$

which is the cyclotron resonance condition between charged particles and electromagnetic waves propagating parallel to the mean magnetic field, written in the rest-frame of the wave. Transforming back to the laboratory frame gives the resonance condition as

$$ \omega - kv\mu = -\Omega_0, \quad (3.20) $$

where negative (positive) frequencies denote left-handed (right-handed) polarization. The circular polarization of the wave has to match the handedness of the particle motion in the guiding-centre rest-frame (where $v\mu = 0$), so ions (electrons) resonate with left-hand (right-hand) polarized waves in that frame. The wave-particle resonance is sketched for protons and cold-plasma waves in Fig. 3.2. We can see that low-energy protons (with $|v\mu| \lesssim 2V_A$) can only resonate with counter-propagating left-handed waves (Vainio 2000).

Approximating the left-handed dispersion curve with a straight line at $-\Omega_0 < \omega < 0$ simplifies the picture so that the phase speed of the resonant wave is constant. Thus, the relevant wave-frame, where the proton would conserve its energy, is propagating at speed $\pm V_A$ along the background field and protons with positive [negative] parallel speed resonate with waves with $\omega/k_\parallel = -V_A [\omega/k_\parallel = +V_A]$. Thus, their position in $(v_\parallel, v_\perp)$ velocity plane is constrained on semicircles centered at $v_\parallel = \pm V_A$. Starting from low speeds, $v_0 \ll V_A$, wave-frame pitch-angle scattering leads to an increase of perpendicular velocity of protons, as depicted in Fig. 3.3. If the extent of the initial distribution in the parallel direction is $v_0$, then the extent of the final distribution in the perpendicular direction would be
<figure><img src="image_31.png" /><figcaption>**Fig. 3.2** Cyclotron resonance of protons with parallel propagating cold-plasma waves. The resonance condition, Eq. (3.19), is depicted with the black line for a proton with positive parallel speed $v\mu$. The green and the red curves give the dispersion relations of parallel-propagating cold-plasma waves with negative frequencies denoting the left-handed Alfvén-ion-cyclotron waves and positive frequencies the right-handed fast-MHD-whistler waves. The red (green) line gives the wave with positive (negative) phase speed $\omega/k_{\parallel}$. The red (green) dashed lines depict the low-frequency approximation to the dispersion relation, i.e., the Alfvén waves with positive (negative) phase speed, $\omega/k_{\parallel} = \pm V_A$. The wave-particle resonances are found where the black line crosses the dispersion curves</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>**Fig. 3.3** The resonant-wave acceleration process. Particles are scattered off electromagnetic waves propagating in the medium at phase speeds $\pm V_{\phi}$, here approximated with $\pm V_A$. Scattering is elastic in the wave frame, which leads to particles conserving their kinetic energies in that frame. If waves have left-handed polarization, low-energy ions resonate with them when propagating in the opposite direction (in the plasma frame) than the wave. Particles initially in the grey region inside the blue semi-circle will be scattered by the waves to fill the intersection of the green and red sectors, centered at $v_{\parallel} = -V_A$ and $v_{\parallel} = +V_A$, respectively</figcaption></figure>

$\sqrt{(V_A + v_0)^2 - V_A^2} = \sqrt{2V_A v_0 + v_0^2}$ and the ratio of final perpendicular and initial parallel energies is

$$ \frac{W_{\perp, \max}}{W_{\parallel, 0}} = \frac{v_0^2 + 2V_A v_0}{v_0^2} = 1 + \frac{2V_A}{v_0}, \qquad (3.21) $$
so obviously the mechanism alone cannot accelerate ions to very high energies. This mechanism is, however, the basis of the cyclotron heating models of the solar corona (Isenberg 2001).

The resonant wave-acceleration process in solar flares is thought to be responsible for the preferential acceleration of minor ions. The dispersion relations of the waves in a multi-species plasma are not as simple as in Fig. 3.2, but contain additional resonances ($|k_\parallel| \to \infty$) with thermal He⁴ ions at cyclotron frequency half of protons, $\Omega_\alpha = \frac{1}{2}\Omega_p$. Ions with cyclotron frequencies differing from that of protons and alpha particles are much more efficiently accelerated than protons and alphas in such plasmas. We will come back to this in Sect. 3.2.5.

### 3.2.3 Shock Acceleration

Shocks can accelerate particles in various ways. The three most commonly studied mechanisms are shock drift acceleration (SDA), shock surfing acceleration and diffusive shock acceleration (DSA).

SDA occurs, when a particle interacts once with a quasi-perpendicular shock front (Fig. 3.4, panels a and b) (Sarris and Van Allen 1974). The particle drifts (due to the motional electric field, $\mathbf{E} = -\mathbf{u}_1 \times \mathbf{B}_1$) with the upstream bulk speed $u_{1x}$ toward the shock wave. When the ion [electron] hits the shock front, it feels the stronger downstream magnetic field, meaning that its Larmor radius is smaller than in the upstream field, and that its guiding center shifts parallel [anti-parallel] to the electric field. Thus, the particle is accelerated. During their interaction with the shock the particles, at least in an averaged sense, conserve their first adiabatic invariant, $p_\perp^2/B$. For a perpendicular shock, we then get

$$ \frac{p_2^2}{B_2} = \frac{p_1^2}{B_1} \Rightarrow p_2 = p_1 \sqrt{\frac{B_2}{B_1}}, \qquad (3.22) $$

<figure><img src="image_33.png" /><figcaption>**Fig. 3.4** Shock drift acceleration and shock surfing. An energetic charged particle is convected to a quasi-perpendicular shock from upstream by the electric-field drift. In the shock front, due to magnetic field gradient in the front, (a) ions drift parallel and (b) electrons drift anti-parallel to the electric field and, thus, gain energy. Panel (c) depicts an ion surfing on the shock due to multiple reflections by the cross-shock potential electric field. (Panels (a) and (b) from Koskinen 2011)</figcaption></figure>
so the particle momentum in the downstream region, $p_2$, is approximately $\sqrt{B_2/B_1}$ times the particle momentum in the upstream region, $p_1$. The total gain in energy is, therefore, not very significant, but may explain the so-called shock spike events observed during nearly perpendicular interplanetary shocks crossing the spacecraft.

SDA operates in oblique shocks as well, but slightly modified. It is most advantageous to transform to the so-called de Hoffmann-Teller frame, where the flow and the magnetic field are parallel throughout the shock and the motional electric field vanishes. This transformation from the upstream plasma frame is along the magnetic field at speed $u_1 = u_{1x}/\cos \Theta_{Bn}$ away from the shock in the upstream region. Here $\Theta_{Bn}$ is the shock normal angle. Now, particles incident on the shock from the upstream side with high-enough pitch-angle, i.e., with $1-\mu^2 > B_1/B_2$, cannot enter the downstream side of the shock while conserving their adiabatic invariant and, thus, are reflected back to the upstream region. This reflection process, when viewed in the upstream plasma frame, give the reflected particle a parallel momentum addition of the same order as transmission in the quasi-perpendicular shock.

Shock surfing (Fig. 3.4, panel c) is related to SDA in the sense that the accelerating electric field is the same, the motional electric field. However, the drift motion along the field is related to the upstream-directed cross-shock potential electric field, caused by charge separation at the shock front (Shapiro and Üçer 2003). The cross-shock potential electric field can specularly reflect ions incident on the shock and enable their acceleration in the direction of the motional electric field. The specular reflections can occur multiple times at a grazing angle. This leads to ion trajectories surfing on the shock front in the direction of the electric field. The maximum energy gained in the process depends on the thickness of the shock and can be estimated to reach the MeV range at interplanetary shocks, if the shock thickness is as low as an electron skin depth (Koskinen 2011). Note that electrons will not be accelerated in this simple type of model, which has a monotonic potential (and, thus, unidirectional field) inside the shock front.

One encounter with the shock does not typically lead to a very substantial gain of energy. If, however, particles can interact with the shock many times, acceleration becomes more efficient. Shock surfing is not the only way this can happen. Particles can interact with magnetic irregularities in the plasma flow, and this can change the particle's propagation direction relative to the shock front enabling several encounters with the shock. Since particle transport under such conditions is described by diffusion relative to the local plasma flow, this acceleration mechanism is called diffusive shock acceleration (DSA) (Drury 1983; Lee 1983).

DSA can be best understood by considering shock waves propagating parallel to the magnetic field. There, as the particle crosses the shock front, its velocity vector does not change, because the magnetic field is not compressed. When the particle is moving relative to the plasma under the influence of frozen in magnetic turbulence, providing scattering centers,<fnref n="1" /> it conserves its energy in the local plasma frame while

<footnote n="1">¹For simplicity, the turbulence in the vicinity of the shock is often assumed to be magnetostatic, but the DSA theory can be formulated assuming that the scattering centers are propagating Alfvén waves. This simply modifies the velocities of the scattering centers.</footnote>
<figure><img src="image_34.png" /><figcaption>**Fig. 3.5.** Diffusive shock acceleration. An energetic charged particle scatters off magnetic irregularities frozen in to the local plasma flow. The numbered points depict successive crossings of the shock front, where the speed of the scattering centers changes. Because points with odd numbers must have $v_\parallel > 0$ and points with even numbers must have $v_\parallel < 0$, the shock crossings lead to a systematic gain of energy $W = \frac{1}{2}mv^2$. (From: Koskinen 2011)</figcaption></figure>

simultaneously scattering in pitch angle. Upstream [downstream] particles are, thus, staying on semicircles in velocity space, centered at $(v_\parallel, v_\perp) = (u_1[2], 0)$. Due to pitch-angle scattering, energetic ($v' > u$) particles can propagate in either direction relative to the shock. When the flow at the shock is compressed (i.e., $u_2 < u_1$), particles crossing the shock many times gain speed systematically as shown in Fig. 3.5.

When particle speeds are much larger than the fluid speeds, $v \gg u$, particle distributions become almost isotropic as a result of the scattering process. This enables one to calculate the energy spectrum of accelerated particles resulting from DSA. After $n$ shock crossings, the mean particle momentum is

$$ \langle p_n \rangle = p_0 \exp \left\{ \frac{4}{3} \sum_{j=1}^{n} \frac{u_1 - u_2}{v_j} \right\}, \qquad (3.23) $$

where $p_0 \gg mu_1$ is the injection momentum. The probability of a particle of performing at least $n$ crossings of the shock is

$$ P_n = \exp \left\{ -4 \sum_{j=1}^{n} \frac{u_2}{v_j} \right\} = \left( \frac{\langle p_n \rangle}{p_0} \right)^{-3u_2/(u_1-u_2)}. \qquad (3.24) $$

By combing these, the integral momentum spectrum can be given as

$$ N(p > \langle p_n \rangle) = N_0 \left( \frac{\langle p_n \rangle}{p_0} \right)^{-3/(r-1)}, \qquad (3.25) $$
where $N_0$ is the total number of particles injected to the acceleration process and $r = u_1/u_2$ is the compression ratio of the shock. Thus, shock-accelerated particles have a power-law differential momentum spectrum

$$ \frac{dN}{dp} = \frac{3N_0}{r-1} \left(\frac{p_0}{p}\right)^{(r+2)/(r-1)} \qquad (3.26) $$

with spectral index $\sigma = d \ln N / d \ln p = (r+2)/(r-1)$ solely determined by the compression ratio of the shock. While the calculation above was presented for parallel shocks, the final result applies for oblique shocks as well.

The spectral index is actually determined by the shock's compression ratio only if $M_A \gg 1$. If the Mach number of the shock is of the order unity, the magnetic scattering centers in the flow (which are actually magnetohydrodynamic waves or turbulence and not static) are no longer static magnetic fluctuations. Instead, they have non-negligible phase speeds $V_\phi \sim V_A$ relative to the flow. Recall that the scattering is elastic in the frame of the propagating magnetic structure, where the electric field of the fluctuation vanishes. Taking these considerations into account when determining the scattering-center compression ratio of the shock (Vainio and Schlickeiser 1999), one gets

$$ \sigma = \frac{r_{sc} + 2}{r_{sc} - 1}; \quad \text{with} \quad r_{sc} = \frac{u_{1x} + V_{\phi 1}}{u_{2x} + V_{\phi 2}} \underset{M \to \infty}{\stackrel{M \to \infty}{\Rightarrow}} \frac{u_{1x}}{u_{2x}} = r. \qquad (3.27) $$

In slow-mode shocks, the scattering centers (Alfvén waves) always have larger phase speeds than fluid speeds. Thus, the scattering centers do not converge in slow shocks under many circumstances. In such cases, DSA is not operating at the shock.

Upstream of fast-mode shocks, particles can generate their own scattering centers by streaming instabilities of the Alfvén waves (Lee 1983; Afanasiev et al. 2015). This can be figured out most easily by looking at particle scattering in the upstream plasma frame. Particles conserve their energies in the frame co-moving with the MHD waves, because in that frame, the wave has no electric field (since $\delta B$ has no time dependence there, as discussed above). Thus, particle energy in the plasma frame is

$$ W = W' \pm V_{A1} p'_{\parallel}, \qquad (3.28) $$

where the signs denote waves propagating forward (+$V_{A1}$) and backward ($-V_{A1}$) relative to the flow. Particles entering the upstream region from downstream have $p'_{\parallel}/W' < -(u_1 \pm V_{A1}) < 0$. Scatterings make the particles isotropic in the wave frame, so as a result of scatterings in the upstream region, $p'_{\parallel}$ increases. Thus, particle energy in the plasma frame increases (decreases) in scatterings off forward (backward) MHD waves in the upstream region. Since the total energy of particles and waves has to remain constant in the plasma frame, this means that the energy density of backward waves increases and forward waves decreases.
Finally, one should note that the power-law spectrum does not extend to infinite energies, but experiences a cut-off at some high energy determined by the age and the size of the system. Obviously, if there is limited time $\tau$ available to accelerate the particles, they cannot be accelerated beyond energies determined by $\dot{p} \sim p/\tau$, where $\dot{p}$ is the momentum gain rate related to the scattering rates and flow velocities in the system. Likewise, when the particle's diffusion length, $\kappa/u_1$, becomes of the order of the system size, the particle can not be accelerated any further, as it will not be confined to the vicinity of the shock anymore but may escape from the system. Here, $\kappa = \frac{1}{3}\lambda v$ is the spatial diffusion coefficient and $\lambda$ is the scattering mean free path, which cannot be smaller than the Larmor radius of the particle. In an inhomogeneous magnetic field, such as the coronal field, particles also need to be confined by turbulence near the shock strong enough to avoid the escape by adiabatic focusing (Vainio et al. 2014). This leads to yet another condition that $\kappa/L < u_1$, where $L = -B/(\partial B/\partial s)$ is the focusing length, i.e., the scale height of the magnetic field intensity. This represents the relevant system size in the coronal medium.

### 3.2.4 Compressional Acceleration and Collapsing Magnetic Traps

DSA operates because of the convergence of the flow of scattering centers at the shock. The acceleration rate of an isotropic population of particles in a converging flow is given by $\dot{p} = -\frac{1}{3}p\nabla \cdot \mathbf{u}$, where $\mathbf{u}$ is the scattering center velocity. Compressional acceleration can, therefore, work also in presence of compressions ($\nabla \cdot \mathbf{u} < 0$) of non-shock type. If the diffusion length of the particles, $\kappa/u$, is much longer than the gradient scale of the flow, $L \sim u/|\nabla \cdot \mathbf{u}|$, then the compression acts on the particles as a shock, i.e., the resulting spectrum is practically the same as in the case of DSA (Jokipii and Giacalone 2007). In the opposite case, $\kappa \ll u^2/|\nabla \cdot \mathbf{u}|$, the compression will accelerate the particle distribution adiabatically. Because their diffusion length is very small, particles are primarily advected through the compression and all particles regardless of their initial momentum will gain the same factor in momentum.

Let us illustrate this with a simple example. Assume that scattering centers are frozen-in in the plasma flow. Thus, $\nabla \cdot \mathbf{u}$ is given by the hydrodynamic conservation of mass as

$$ \nabla \cdot \mathbf{u} = -\frac{1}{\rho} \frac{d\rho}{dt}, \qquad (3.29) $$

where $\rho$ is the density of the plasma parcel being advected at velocity $\mathbf{u}$. Thus,

$$ \frac{\dot{p}}{p} = -\frac{1}{3} \nabla \cdot \mathbf{u} = \frac{\dot{p}}{3\rho} \Rightarrow \frac{p^3}{\rho} = \text{constant}, \qquad (3.30) $$
which is consistent with the adiabatic equation of state for a monoatomic gas, i.e., $T\rho^{-2/3} = \text{constant}$ at non-relativistic energies.

Similar to compressional acceleration, particles can be accelerated adiabatically if they are confined to a collapsing magnetic trap (see Borissov et al. 2016 and references therein). A simple example is a shrinking magnetic bottle, consisting of two magnetic mirrors in the ends, e.g., a contracting coronal loop. Particles mirroring in the ends of the trap would get accelerated, as in the rest frame of the center of the trap (the loop apex in the simple example), the mirrors in the two ends of the trap would be approaching each other. The acceleration rate is obtained from the conservation of the second adiabatic invariant, $\oint p_{\parallel}ds_{\parallel} = 2|p_{\parallel}|s = \text{constant}$, where the integral is along the magnetic field lines from one end of the trap to the other and back, and $s$ is the length of the trap along the field. Decreasing $s$ has to be compensated by increasing $|p_{\parallel}|$ and the parallel momentum will increase at rate

$$ \frac{d|p_{\parallel}|}{dt} = -|p_{\parallel}|\frac{\dot{s}}{s} \qquad (3.31) $$

As a magnetic bottle cannot trap particles residing in the loss-cone of the weaker magnetic mirror, this mechanism would appear at first sight to be limited to rather modest gains of momentum. However, if the magnetic field inside the trap is simultaneously increasing the betatron effect will increase the perpendicular momentum and help the particles stay trapped. The details of the magnetic configuration and its evolution will determine the acceleration efficiency of a collapsing magnetic trap.

### 3.2.5 Stochastic Acceleration

DSA, compressional acceleration, and collapsing traps are all examples of so-called first-order Fermi acceleration, where particles gain momentum systematically and proportionally to the speed of a moving magnetic structure. Stochastic acceleration, or second-order Fermi acceleration, is a process, where particles gain or lose energy (with a positive net energy gain) by interacting simultaneously with plasma disturbances with different phase speeds in the laboratory frame (Miller 1998). For example, if a high-energy proton is propagating in a medium with counter-propagating Alfvén waves, it can simultaneously scatter off waves propagating in both directions (Fig. 3.6, left panel). As the scatterings conserve the energy of the proton in each wave frame, the scatterings off one wave produce energy change along different characteristics in the $(v_{\parallel}, v_{\perp})$ plane than the scatterings off the other mode. This leads to random walk of the particle in the $(v_{\parallel}, v_{\perp})$ plane, which can be described by momentum diffusion. In this process the net momentum gain rate is proportional to the second power of the wave speed. (Hence, second-order Fermi acceleration.)
<figure><img src="image_35.png" /><figcaption>**Fig. 3.6** The left panel shows the resonance plot for a multi-species plasma with protons and alpha particles as the major ion species. The wave modes with positive (negative) phase speeds are plotted with green (red) curves. The grey shaded regions are those where resonances with thermal ions will damp the wave power efficiently. Cyclotron waves have two branches with resonances with alpha particles and protons, respectively. The lines depict the resonance conditions for energetic protons ($|v\mu| \gg V_A$) and thermal minor ions. The right panel shows the possible velocity-space trajectories of protons in a plasma with counter propagating Alfvén waves, only. The red (green) semicircles are trajectories of particles interacting with Alfvén waves with positive (negative) phase speeds. The red (green) region depicts the velocities at which resonances with and Alfvén waves with positive (negative) phase speeds are not possible. Outside these regions, protons can always resonate with waves propagating in both directions leading to random walk in velocity space, i.e., momentum diffusion</figcaption></figure>

The situation is illustrated in Fig. 3.6, right panel, where energetic protons can interact with two Alfvén waves propagating in opposite directions, when the proton velocity is situated in the region $|v_\| \gtrsim 2V_A$. Scatterings off waves with positive (negative) phase speed will result in motion along the red (green) semi-circles showing that the particle energy in the plasma frame can change in a random fashion, i.e., increase or decrease. As this diffusive process spreads the distribution of protons in energy, the net effect will be acceleration. This is an example of stochastic acceleration, but other types of waves as well as randomly moving coherent structures interacting with the particles can lead to a similar situation of particles diffusing in momentum when interacting with the structures.

Figure 3.6, left panel also shows the resonant interaction of low-energy minor ions ($^3$He and iron) with higher-frequency left-handed wave modes. The figure shows that while thermal protons and alpha particles would typically resonate with heavily damped wave modes (and their resonant wave acceleration should, therefore, be somewhat inefficient), minor ions not only resonate with much less damped waves but also simultaneously with several waves propagating at different phase speeds. Especially $^3$He can be very efficiently accelerated stochastically in this process, starting already from thermal energies. Stochastic acceleration
process, therefore, has the power to explain the peculiar abundances (strong increase in ³He and heavy ion abundances over protons and alphas) observed in impulsive events (cf. Chap. 2). Note that stochastic acceleration may also occur in the turbulent sheath regions of coronal shocks and help to explain the observed double-power-law spectral form often observed in large gradual SEP events (Afanasiev et al. 2014).

## 3.3 Concluding Remarks

The basic acceleration mechanisms at play in erupting coronal plasmas accelerating particles to the highest energies have been described. By making several simplifications the aim has been to convey the principles of the most important SEP acceleration mechanisms, fostering efforts to create realistic and comprehensive models of solar eruptions also from the particle acceleration aspect. The models should involve realistic descriptions of both the macroscopic and microscopic fields in the plasma, as electric fields of practically all scales from the kinetic to the global may contribute to the acceleration of particles in solar eruptions.

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**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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<figure><img src="image_36.png" /></figure>
**Chapter 4**

**Charged Particle Transport
in the Interplanetary Medium**

Angels Aran, Neus Agueda, Alexandr Afanasiev, and Blai Sanahuja

**Abstract** The scenario and fundamentals of the physics of charged particle interplanetary transport are briefly introduced. Relevant characteristics of solar energetic particle (SEP) events and of the interplanetary magnetic field are described. Next, the motion of a charged particle and the main assumptions leading to the description of the focused and diffusive particle transport equations utilised in the next chapters are discussed. Finally, two different models are applied to interpret SEP events.

**4.1 Introduction**

**4.1.1 Energetic Particles in the Solar System**

Major solar eruptive phenomena, solar flares and coronal mass ejections (CMEs), are usually accompanied by outbursts of charged particles that have been accelerated up to several hundred MeV/nucleon, in some instances up to a few GeV/nucleon. These solar energetic particle (SEP) events are mostly composed of ionised H with ~10% He and <1% heavier elements. The acceleration, injection, and propagation of SEPs from their source to an observer in interplanetary space have been investigated over the last decades by a combination of in situ (space) and remote-sensing observations. SEP acceleration processes are described in Chap. 3. Here the focus is on the transport processes of SEPs in interplanetary space.

SEP events are usually classified into two types: impulsive events and gradual events. Impulsive events last for hours, are rich in electrons, ³He and heavy ions, have relatively high charge states, and are produced by solar flares. Gradual events can last for days, are electron poor, have relatively low charge states, and are

<footnote>A. Aran (✉) • N. Agueda • B. Sanahuja
Dep. de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, c. Martí i Franquès, 1, E-08028 Barcelona, Spain
e-mail: angels.aran@fqa.ub.edu; agueda@fqa.ub.edu; blai.sanahuja@ub.edu</footnote>

<footnote>A. Afanasiev
Department of Physics and Astronomy, University of Turku, Turku FI-20014, Finland
e-mail: alexandr.afanasiev@utu.fi</footnote>
associated with coronal and interplanetary shocks driven by CMEs that move rapidly through the solar wind plasma. Gradual SEP events are more intense (i.e., much higher intensities and higher fluences) than impulsive events. Hybrid SEP events have also been observed; they show mixed characteristics which partially correspond to impulsive and gradual events, suggesting that solar particle events can have distinct components (i.e., flare-accelerated particles and shock-accelerated particles).

Impulsive events are generally limited to within a 30° longitude band around the footpoint of the nominal field line magnetically connected to their parent active region. On the other hand, gradual events are able to produce much wider longitudinal distributions due to the extended nature of the propagating interplanetary shock. In this way, observers located at different places in the heliosphere can be magnetically connected to different parts of the front of such a shock by interplanetary magnetic field (IMF) lines. In gradual events, in addition to the various transport effects at play, the shape of the particle intensity temporal profiles depends also on both the dynamic evolution of the shock strength and the relative location of the observer with respect to the traveling shock. The point along the shock front at which successive magnetic field lines connect with the observer is termed the cobpoint (connecting with the observer point, see Heras et al. 1995); and the gradual SEP event intensity-time profiles recorded in interplanetary space can be interpreted in terms of the cobpoint evolution, after deconvolving the transport effects. The SEP propagation in interplanetary space is controlled by the large-scale structure of the magnetic field and the turbulent magnetic field fluctuations superposed on the mean magnetic field.

## 4.1.2 The Interplanetary Magnetic Field

At a certain distance from the Sun, the solar wind flow speed is much higher than the local sound and Alfvén speeds. Near the Earth’s orbit (1 AU), typical values for the sound and aflvénic speeds are ~60 km s⁻¹ and ~40 km s⁻¹, respectively, whereas for the solar wind speed is ~400 km s⁻¹. This implies that the plasma dynamic pressure is much higher than both the magnetic and thermal pressures (see more details in e.g., Hundhausen 1995). The solar wind carries the magnetic field from the Sun to interplanetary space, with the magnetic field *frozen-in* in the nearly radially expanding solar wind flow, given the very high conductivity of the solar wind plasma (for a deduction of the *frozen-in* condition see e.g., Bittencourt 2004). Field lines can then be seen as stream lines of the fluid flow. However, the situation becomes complicated because of the solar rotation which has an average period of 27.3 days. One can interpret the radially outflowing plasma streams as parcels of plasma emitted from the same source region. These parcels carry the magnetic field with them, and because they are tethered to the rotating Sun, the IMF lines that trail behind them are spirals. By assuming a constant solar wind speed, *u*, an IMF line
depicts an Archimedean spiral (known as the Parker spiral after the model proposed by Parker 1958).

The equation of the Archimedean spiral can be derived from the displacement in radial and angular directions. Assuming as initial conditions of a plasma parcel on the Sun, a source longitude $\phi_0$, and a source radius $r_0$, at time $t$, the parcel is found in the equatorial plane at position, $r$

$$r = r_0 + u(\phi - \phi_0)/\Omega \quad (4.1)$$

where $\Omega$ is the solar rotation speed. The angle between the radial direction and the magnetic field **B**, $\psi$, is given by $\tan \psi = r/a$ where $a = u/\Omega$. Figure 4.1 shows a sketch of an IMF line. Assuming a solar wind speed of 400 km s$^{-1}$, $\psi \sim 45^{\circ}$ at 1.0 AU. The spiral configuration represents a smooth average of the large-scale IMF. Variations in solar wind velocity and processes acting in the solar wind and corona, including reconnection, create a spread in directions around the spiral angle. In fact, individual vectors can point to any angle superposed on this average field because of small-scale random fluctuations and, in addition, individual field lines can meander relative to the average direction.

The path length, $z(r)$, along the spiral can be estimated from $dz = \sec \psi dr$:

$$z(r) = \frac{a}{2} \left[ \frac{r}{a} \sqrt{1 + \frac{r^2}{a^2}} + \ln \left( \frac{r}{a} + \sqrt{1 + \frac{r^2}{a^2}} \right) \right] \quad (4.2)$$

<figure><img src="image_37.png" /><figcaption>**Fig. 4.1** Sketch of an Archimedean spiral interplanetary magnetic field line crossing the Earth. $\psi$ is the angle between the radial direction and the magnetic field **B**</figcaption></figure>
close to the Sun (i.e., for $r/a \ll 1$), $z \sim r$, and well beyond 1 AU (i.e., in the limit for $r/a \gg 1$), $z \sim r^2/2a$.

By assuming that the frozen-in condition holds in interplanetary space and that IMF lines are Archimedean spirals, the strength of the IMF is given by:

$$B(r) = B_0(r_0/r)^2 \sqrt{1 + (r/a)^2} \quad (4.3)$$

where $r_0$ is the heliocentric radial distance at which the field is completely frozen into the solar wind, being $r_0 > 2R_\odot$ and $B_0 = B(r_0)$. Thus, close to the Sun, $B(r)$ decreases as $r^{-2}$ while well beyond 1 AU it decreases as $r^{-1}$. Figure 4.2 shows the dependence of the magnetic field line length and the magnetic field strength with the radial distance from the Sun.

Depending on the magnetic polarity of the photospheric footpoint of the field lines, the magnetic field spirals outward (positive) or inward (negative) from the Sun. The global interplanetary positive-negative magnetic domains are separated by a huge electric current system, the heliospheric current sheet (HCS). The HCS is tilted out of the solar equatorial plane a few tens of degrees. Changes in the coronal magnetic field due to solar activity are carried outward to space, and manifest as spatial and temporal variations (e.g., magnetic sectors). This overall picture of the IMF is known as the *ballerina skirt model*. As the HCS rotates along with the Sun, the peaks and troughs of the skirt pass through the Earth's magnetosphere, interacting with it. A detailed description of the global heliospheric magnetic field can be found in Smith (2008).

<figure><img src="image_38.png" /><figcaption>Fig. 4.2 Dependence of the magnetic field line length and the magnetic field strength with the radial distance from the Sun. The dotted line indicates the length of a radial line for comparison</figcaption></figure>
### 4.1.3 Motion of Charged Particles. First Adiabatic Invariant

Space and time variable magnetic fields, with a large variety of characteristic lengths and times, play a key role in the description of the transport of SEPs in interplanetary space. The solar wind is a collisionless plasma highly conductive in which SEPs propagate basically tracking the IMF. It is also assumed here that SEPs are not able to modify the external, non-uniform and time-varying, magnetic fields. This section shortly reviews the most relevant aspects of electrodynamics that will later be used in Sect. 4.2 when introducing the transport of SEPs in interplanetary space. An extended description can be found elsewhere (e.g., Bittencourt 2004; Kallenrode 2004).

Let's consider a charged particle of rest mass $m$, charge $q$, moving with velocity $v$, in a given electric field, **E**, and in a magnetic field, **B**. Neglecting collisions and any other external force (e.g., a gravitation force), the motion of the particle is governed by the Lorentz force

$$ \frac{dp}{dt} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \qquad (4.4) $$

where $v = p/(ym)$ and $\gamma = (1 + p^2/(mc^2))^{1/2}$ is the Lorentz factor of the particle. Then, the Larmor (or gyration) radius of the particle is given by

$$ r_L = \frac{v_{\perp}}{\omega_c} = \frac{p_{\perp}}{|q|B} \qquad (4.5) $$

Where $v_{\perp}$ and $p_{\perp}$ are, respectively, the components of the velocity and momentum perpendicular to the magnetic field. The gyro frequency of the motion is $\omega_c = |q|B/(ym)$.

To give some numbers, at 1 AU, for a typical IMF strength value ($B = 5$ nT), the Larmor radius of a proton with a kinetic energy of 700 MeV is $r_L \sim 9 \times 10^5$ km (~2.3 times the mean distance between the Earth and the Moon).

Each particle describes an helicoidal motion that can be decoupled into motions parallel ($v_{\|}$) and perpendicular ($v_{\perp}$) to the magnetic field. The parallel component describes the motion of the centre of gyration of the particle along **B** while $v_{\perp}$ consists of a gyration motion (characterised by $\omega_c$) around **B** plus a drift velocity component, $v_F$, also perpendicular to **B**. Then the particle motion is described by the gyration and the motion of the guiding centre, which is the addition of the motion parallel to **B** and a drift.

A charged particle moving under the presence of time-varying or non-uniform fields is affected by a number of velocity drifts: plasma drift (e.g., associated with an electric field), the gravitational drift, field line curvature and gradient drifts (associated with the magnetic field), or the polarisation drift (when there is a time-varying electric field). The drift velocity term, $v_F$, represents the drift velocities caused by the aforementioned fields (see detailed descriptions of the various particle drifts in e.g., Bittencourt 2004; Kallenrode 2004).
The pitch angle, $\alpha$, gives the relative size of the perpendicular and parallel components of the particle velocity, it is given by

$$ \alpha = \tan^{-1}(v_{\perp}/v_{\|}) \qquad (4.6) $$

In the description of SEP transport, the cosine of the pitch angle, $\mu = \cos \alpha$, is frequently used.

The magnetic moment, $\mathbf{m}$, of a charged particle moving in a magnetic field is a measure of the magnetic flux traversing the circular section defined by the particle's Larmor radius. The kinetic energy can be written as the sum of its parallel and perpendicular components to the magnetic field, $W = W_{\|} + W_{\perp}$. It can be shown (e.g., Bittencourt 2004) that

$$ |\mathbf{m}| = \frac{v_{\perp}p_{\perp}}{2B} = \frac{W_{\perp}}{B} = \frac{W \sin^2 \alpha}{B} \qquad (4.7) $$

Note that in a static uniform magnetic field, $v_{\|}$ is constant, so the particle moves at constant velocity along $\mathbf{B}$, and $W$ and $W_{\|}$ are constant. Hence, it follows that, $W_{\perp}$ and $v_{\perp}$ are also constants of the motion.

The first adiabatic invariant can be derived from the equation of motion. It states that the magnetic moment of a particle, $|\mathbf{m}|$, is constant when moving in a slowly varying magnetic field (there are also other conditions for specific scenarios; e.g., no wave-particle interactions or that $\omega_c$ does not go through zero).

A particle moving into a converging magnetic field increases its $W_{\perp}$ while its $W_{\|}$ decreases, to keep $|\mathbf{m}|$ and $W$ constant; hence, its Larmor radius decreases and its pitch angle increases. The opposite is true if the magnetic field strength decreases (e.g., diverging IMF).

In the absence of parallel electric fields ($W$ constant) the pitch angle of a particle at two locations, with respective magnetic field strength $B_1$ and $B_2$, must satisfy:

$$ \frac{\sin^2 \alpha_2}{\sin^2 \alpha_1} = \frac{B_2}{B_1} \qquad (4.8) $$

If the magnetic field becomes strong enough, then $v_{\|} = 0$, the direction of the particle is reversed. The particle's speed increases in the direction of decreasing $\mathbf{B}$ by the parallel component of the gradient force of the field, thus, the particle is reflected from the region of converging field lines. The parallel component of the average Lorentz force is called the mirror force because it leads to mirroring trajectories for a particle in a converging field. This effect is relevant for SEP propagation in interplanetary space. For example, it can explain that, for certain SEP events, solar- or near Sun- accelerated particles are later observed travelling back to the Sun (e.g., Tan et al. 2009).

If a particle is in a region of space between two high magnetic field regions, then the particle may be reflected at one side, travel towards the second, and also reflect there. Thus the particle motion is confined to a certain region of space, bouncing
back and forth between two regions of large field strength. Examples of such a magnetic trapping scenario are the bidirectional SEP events sometimes observed in the downstream region of interplanetary shocks.

The same force that causes mirroring in a converging magnetic field, causes particle focusing in a diverging (decreasing) magnetic field. In this latter case, the particle will describe orbits with increasingly larger $r_L$ (i.e., the pitch angle tends to approach zero). This focusing process is particularly important in the context of SEP transport given that energetic particles travel from Sun to Earth along the divergent interplanetary magnetic field. If no particle-diffusion process is considered, an isotropic energetic particle population released at the Sun will appear to come in a very narrow cone pitch angle (~1° wide) when observed at 1 AU.

## 4.2 Particle Transport

The solar wind is a collisionless plasma, hence SEPs mainly experience the effect of the electromagnetic fields. In the interplanetary space, one can assume that the average electric field vanishes due to the large conductivity of the solar wind plasma. The interplanetary magnetic field is turbulent (waves and fluctuations may be treated as perturbations added to the large-scale magnetic field configuration) and as such, it scatters particles in pitch-angle. Given the different scales involved, the transport of particles is often described by the evolution of the particle distribution, i.e., in an ensemble-averaged manner. The main objective of this section is to provide the key elements to contextualise the interplanetary models used later in this book.

### 4.2.1 Particle Transport Equations

The particle distribution function, $f$, is defined so that the number of particles $dN$ in a phase space volume ($\mathbf{r} + d^3\mathbf{r}, \mathbf{p} + d^3\mathbf{p}$) at a time $t$ is given by

$$dN = f(\mathbf{r}, \mathbf{p}, t)d^3\mathbf{r}d^3\mathbf{p} \quad (4.9)$$

The phase-space volume element as well as the number of particles are Lorentz invariants, and therefore, the phase space density, $f(\mathbf{r}, \mathbf{p}, t)$, is an invariant. The Boltzmann equation is the fundamental equation of motion in the phase space. For a particle of charge $q$ and mass $m$ under an external electromagnetic force, it reads

$$\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla f + q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot \frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t}\right)_s \quad (4.10)$$

where the first term of the right hand side does not represent collisions but may describe the scattering of particles by electromagnetic waves and random magnetic
field fluctuations. Given the random nature of the scattering processes, Eq. (4.10) is a Fokker-Planck equation. A thorough derivation of the transport equations in different scenarios can be found in e.g., Zank (2014). Another approach to study SEP transport is to start from the Vlasov equation that describes self-consistently the non-linear coupling between particles and fluctuating wave fields (e.g., Kallenrode 2004). It is widely used in plasma physics and has the same form as Eq. (4.10) but without the right hand side term. A discussion on the relation of the Vlasov and the Boltzmann equations can be found in e.g., Bittencourt (2004).

## 4.2.2 Focused Transport

For describing the transport of SEPs in the interplanetary space, the frame of the focused transport is the most adequate. In the focused transport model, energetic particles are considered to undergo pitch-angle scattering due to small scale irregularities in the IMF, and focusing and mirroring due to the large-scale weakening of the IMF at increasing distance from the Sun. Focused transport for SEP particles is mainly a competition between the focusing effect and the pitch-angle scattering processes. The standard equation derived by Roelof (1969) from Eq. (4.10) is:

$$ \frac{\partial f}{\partial t} + v\mu \frac{\partial f}{\partial z} + \frac{1 - \mu^2}{2L} v \frac{\partial f}{\partial \mu} = \frac{\partial}{\partial \mu} \left( D_{\mu\mu} \frac{\partial f}{\partial \mu} \right) \quad (4.11) $$

where f is a function of the spatial coordinate, z, measured along the IMF line, the particle momentum, p, the pitch-angle cosine, μ, and time t. v is the velocity of the particles, $D_{\mu\mu}$ is the pitch-angle diffusion coefficient describing the stochastic forces, and L is the focusing length which involves spatial variations of the guiding field (assumed to be static), given by $L(z) = -B(z)/(\partial B/\partial z)$. The second term of the left hand side of Eq. (4.11) describes the streaming of the particles along the IMF; the third term, the focusing of particles and the term in the right hand side, accounts for the scattering in pitch-angle.

In order to describe SEP events, a particle source term, Q, is added in the right hand side of Eq. (4.11) to account for the injection of particles by either a fixed source (like flares) or a mobile source (i.e., coronal/interplanetary shock waves).

By following the quasi-linear theory (QLT), the interaction between particles and waves is treated here to the first order only. The irregularities of the IMF are considered to be sufficiently small so that several gyrations of a particle are needed to modify significantly its pitch-angle. Also, it is assumed that the magnetic field irregularities can be described as waves of axially-symmetric transverse components, with wave vectors parallel to the average field (known as the ‘slab model’). The combination of QLT with the slab model is known as the ‘standard model’ of particle scattering. The magnetic field fluctuations can be described by a power-density spectrum, $P(k) \propto k^{-q}$ where $\bar{q}$ is the spectral slope. Under these
assumptions $D_{\mu\mu}$ can be expressed analytically as Jokipii (1971):

$$D_{\mu\mu} = \frac{\nu_0}{2} |\mu|^{\bar{q}-1} (1 - \mu^2), \qquad (4.12)$$

where $\nu_0 = 6v/[2\lambda_\parallel(4-\bar{q})(2-\bar{q})]$ and provides the relation between the diffusion coefficient and the particle's mean free path parallel to the IMF, $\lambda_\parallel$. Values of $\bar{q}$ have been determined from observations in the range $1.3 \le \bar{q} \le 1.9$ with an average value of $\bar{q} = 1.63$ (Kunow et al. 1991). More details of the standard model and other equivalent descriptions for $D_{\mu\mu}$ can be found in e.g., Agueda and Vainio (2013).

Once the form of $D_{\mu\mu}$ is fixed, the other main parameter of the transport models is $\lambda_\parallel$. For protons, there is a dependence of the mean free path on the magnetic rigidity of the particles, $R = pc/q$, where c is the speed of light. For the standard model, $\lambda_\parallel \propto R^{2-\bar{q}}$, as long as $\bar{q} < 2$ (Hasselmann and Wibberenz 1970).

As is generally done, in the models by Pomoell et al. (2015) used in Chap. 9 and by Agueda et al. (2008) used in Chap. 10, the IMF is modelled as an Archimedean spiral. In this case, the radial mean free path of the particles, $\lambda_r$ is related to $\lambda_\parallel$ by $\lambda_r = \lambda_\parallel \cos^2 \psi$.

In the description above, transport of particles perpendicular to the average magnetic field is neglected, because in the inner heliosphere (1) particles move much smaller distances per time unit in the perpendicular direction than in the parallel direction e.g., Bieber et al. (1995) and (2) in the case of a mobile particle source, the continuous injection (during days) of shock-accelerated particles has a stronger contribution in shaping the SEP intensity profiles than cross-field diffusion does.

### 4.2.3 Diffusive Transport

When the spatial scales are larger than the particle's mean free path and the scattering time is small compared to the time scales of the phenomena under study, the standard diffusion equation can describe the variation of the particle distribution function. In this assumption, the spatial diffusion tensor, $\kappa$, reflects the effects of the fluctuations of the magnetic field (Jokipii 1971). In 1965, Parker (1965) was the first to describe the evolution in space of the cosmic-ray distribution using the following Fokker-Planck equation:

$$\frac{\partial f}{\partial t} + \mathbf{u} \cdot \nabla f - \frac{1}{3} p (\nabla \cdot \mathbf{u}) \frac{\partial f}{\partial p} = \nabla \cdot (\kappa \cdot \nabla f), \qquad (4.13)$$

This equation describes the effects of spatial diffusion, advection due to the movement of the scattering centres with the solar wind, and energy changes of the particles. Equation (4.13) is only valid when the pitch-angle distribution of the particles is nearly isotropic. This is the assumption made in Chap. 9 to describe the transport of (quasi)relativistic protons from shocks low in the corona towards
the Sun. However, Parker's equation is not applicable when the anisotropy of the distribution of particles is large (Jokipii 1987), as generally happens in the case of SEP events associated with interplanetary shocks e.g., Heras et al. (1995).

## 4.3 Application: Description of Solar Energetic Particle Events

The main application of interplanetary transport models is for the description of SEP events. First, different methods to solve the transport equations and approaches to perform data fitting are presented. Next, two interplanetary transport models, the Shock-and-Particle (SaP) model by Pomoell et al. (2015), used in Chap. 9, and the inversion model by Agueda et al. (2008), used in Chap. 10, are applied to briefly describe the observations needed for the modelling of SEP events and the derivation of the main transport parameters.

### 4.3.1 Numerical Techniques

Different numerical techniques can be applied to solve transport equations. Particularly, the interplanetary transport models used in Chap. 9 (i.e., Lario et al. 1998) and in Chap. 10 (i.e., Agueda et al. 2008) employ finite-difference and Monte Carlo methods, respectively. Both have advantages and drawbacks, and therefore there are models in the literature that utilise either methods or even a combination of the two. The main advantage of finite-difference methods (e.g., Ruffolo 1995; Lario et al. 1998; Dröge 2000) is that they are computationally fast, whereas the main advantage of Monte Carlo methods (e.g., Agueda et al. 2008; Afanasiev and Vainio 2013) is that tracking of individual particles is allowed by them.

Also, numerical models may follow two methods for data fitting: forward and inversion modelling. Forward models, like SaP (Pomoell et al. 2015), are inductive and based on the prediction of measurements with a given set of model parameters. On the other hand, inversion models, like SEP inversion (Agueda et al. 2008, 2012), are deductive and make use of the measurements to infer the actual values of the model parameters, and hence, no a priori assumption about the particle injection profile, $Q$, is needed.

### 4.3.2 Observations

SEP events are intensity enhancements above a background level detected typically for an extended particle energy range. The particle intensities obtained for a particular energy window are called differential intensities, $J$, or $dJ/dE$. Differential
intensities, with units of [(MeV sr s cm²)⁻¹], are related to the particle distribution function by $J = p²f$, where p is the particle's momentum.

The top panel of Fig. 4.3 shows a gradual SEP event starting on 2000 April 4, with 0.115–101 MeV proton differential intensity enhancements measured by the ACE and SOHO spacecraft. The source of this particle event was a travelling CME-driven shock that originated near to the solar west limb (e.g., Pomoell et al. 2015). This panel exemplifies the energy dependence of the intensity-time profiles observed during a single SEP event. Whereas at high energies (>30 MeV) proton intensities show a rapid onset, a maximum peak intensity, followed by a slow decay with

<figure><img src="image_39.png" /></figure>

**Fig. 4.3** April 4, 2000 SEP event. Top panel: 0.115–101 MeV proton differential intensities measured by 23 energy channels (colour coded) of different detectors (ACE/EPAM (Gold et al. 1998) and SOHO/ERNE (Torsti et al. 1995)). The second panel shows the solar wind speed measured by ACE/SWEPAM (McComas 1998) and the three bottom panels show the magnetic field strength, latitude and longitude (in RTN coordinates) recorded by ACE/MAG (Smith et al. 1998). The solid vertical line marks the time of the shock passage by the ACE spacecraft and the arrow marks the onset time of the associated CME
intensities being already at background level prior to the shock passage (the vertical solid line in Fig. 4.3), low energy ($<$2 MeV) intensities keep increasing with a marked peak at the shock. The smooth transition of the shape of flux profiles from high to low proton energies suggests that the efficiency of the shock at accelerating particles gradually diminishes with energy as it propagates away from the Sun.

Information of the solar wind and IMF is needed in order to perform the modelling and to know how the assumptions made in the models comply with the actual conditions. The bottom panels of Fig. 4.3 show smooth profiles for the solar wind speed and magnetic field strength in the pre-shock region, and variations in the IMF direction that do not modify significantly the shape of the intensity-time profiles; hence, the SaP model can be applied to describe this SEP event. First, the shock propagation is modelled to obtain the position of the particle source, and next the simulation of the transport of particles up to the observer’s position (in this case, located at L1) is performed. For this, particles are assumed to be injected by the shock at the points in the shock front connected with the spacecraft through a Parker IMF line (i.e., at the cobpoints; Heras et al. 1995). In the SaP model, Eq. (4.11) is solved with the inclusion of the solar wind effects on the low-energy protons (Ruffolo 1995; Lario et al. 1998).

From the particle transport models, the evolution of the injection rate, $Q$ and the proton mean free path, $\lambda_\parallel$ can be obtained by fitting the observed time profiles of particle (omnidirectional) intensities<fnref n="1" /> and of the first order anisotropies (when available, e.g., Lario et al. 1998) or by fitting directly directional intensities (e.g., Agueda et al. 2008). For the April 2000 event, the omnidirectional intensity-time profiles are simultaneously fitted for eighteen energy channels and the first order anisotropies for $E < 2$ MeV, to better constrain the values of the parameters used (see details in e.g., Pomoell et al. 2015).

The resulting values of $\lambda_\parallel$ at 9.1 MeV are: 1.30 AU for $t < 11.0$ h, 0.65 AU for $11.0 \le t < 15.8$ h and 0.33 AU for $t \ge 15.8$ h, assuming $\bar{q} = 1.6$ for its rigidity dependence. Figure 4.4 shows the derived evolution of the source function, $Q$, that continuously changes from low to high energies. For $E > 36.4$ MeV, $Q$ decreases rapidly (two orders of magnitude in 10 h). After this time, the shock, located already at $80 R_\odot$ is no longer efficiently injecting >40 MeV protons. For lower energies, $Q$ continuously decreases, this decrease being slow for lower energies. The use of a particle transport model (in this case the SaP model) yields the quantitative description of how the shock is gradually losing efficiency at injecting particles as it moves away from the Sun and as the magnetic connection with the observer varies.

<footnote n="1">¹Hereafter ‘intensity’ is used for ‘differential intensity’.</footnote>
<figure><img src="image_40.png" /><figcaption>**Fig. 4.4** Evolution of the injection rate of shock-accelerated particles, *Q*, derived from the modelling of the SEP event with the SaP model. Each curve, *colour coded* as indicated in the inset, corresponds to the injection rate profile derived for each of the eighteen energy channels modelled and computed from the first cobpoint up to the shock arrival at the ACE spacecraft</figcaption></figure>

### 4.3.3 Inferring Transport Conditions

An important aspect to consider when inferring the transport conditions is the level of freedom in performing the fitting of observed intensities. If $Q$ and $\lambda_r$ values are derived employing only the omnidirectional intensity profile, for a given energy channel, the problem is ill-posed. Additional information is needed and can be extracted from first order anisotropies or from directional intensities. To illustrate this point the inversion model by Agueda et al. (2008, 2012) is utilised. This model assumes a fixed source of near-relativistic electrons placed at $2R_\odot$ from the Sun. The middle column of Fig. 4.5 shows an omnidirectional intensity-time profile (red curve) that is fitted with the model (black curves). In the left column are shown four different electron injection histories, $Q(t)$, and values of $\lambda_r$ resulting from the fitting. The injection profile and $\lambda_r$ shown in the first row do not fit the data, as it is clearly seen in the middle panel; however, the remainder three combinations shown in the next rows do fit the omnidirectional intensities perfectly; thus, indicating that