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# Introduction

## Background and Motivation

Cosmic rays (CRs) are extremely energetic particles that propagate throughout the universe, consisting of protons, electrons, heavy nuclei, etc. They can be accelerated to energies beyond $$10^{20}~{\rm eV}$$ with possible sources including relativistic jets and powerful astrophysical shocks. Much effort has been made starting from the early twentieth century to present day in understanding their origin, properties, and role in various environments in the universe, from the interstellar medium (ISM), to galaxies, and to galaxy clusters. The study of CRs has aided in advances in a variety of fields, including particle physics, cosmology, high-energy astrophysics, and plasma astrophysics. Open questions still remain to this day, such as the origin of the ultra-high-energy CRs and the nature of particle acceleration. Continual advances, however, in numerical methods, high-performance computing architectures, observational technology, and laboratory-astrophysics techniques show promise for further unraveling the mysteries behind CRs.

The two main deviations from a pure power-law are the knee, where the energy spectrum steepens slightly near $$10^{15}~{\rm eV}$$, and at the ankle, where the spectrum slightly flattens near $$10^{18}~{\rm eV}$$. These deviations in the power-law may indicate a transition of different populations of CRs, where it is theorized that CRs below the knee originate in the Galaxy and CRs above the ankle are extra-galatic in origin (Aharonian 2004, Beatty and Westerhoff 2009).

One of the widely accepted sources of CRs is supernova remnants (SNRs) through diffusive shock acceleration (DSA), where particles are accelerated from thermal energies to non-thermal energies at the shock front. DSA has been mostly studied in the context of CR transport in collisionless shocks, where injected particles into the shock excite flow instabilities as DSA progresses. A complete understanding of how the mechanism manifests in supernovae has yet to be achieved. Surveys by the Chandra X-Ray Observatory and Fermi Gamma-Ray Telescope have provided a wealth of data for validating theoretical models and studying supernova processes at thermal and non-thermal energies.

Supernovae are stellar explosions that generate strong shocks. This work does not focus on the cause of such explosions, but rather on the conditions after the explosion that may be suitable for CR production. Supernovae explosions eventually become supernova remnants (SNRs). Not only are supernovae of interest to the astrophysics community, they are also of interest to the high energy-density physics (HEDP) community. Technological advancements have made it possible to control and study astrophysical processes in the laboratory.

## Supernova-remnant theory

Supernovae can occur either by the thermonuclear disruption of a white dwarf or by the core-collapse of a massive star. The first case is referred to as a type Ia supernova, where it is assumed that the explosion occurs when a massive white dwarf accretes matter from a companion and the mass exceeds the Chandrasekhar mass limit. Here, the explosion mechanism is reasonable well understood. The second case is referred to as a type II supernova, where the process is reasonably well understood with the exception of the late stages of the explosion mechanism. Several models have been proposed for this last stage. For example, the proper motion of a neutron star can be investigated because it depends on the kick a neutron star gains during the explosion when they are formed. Therefore, the proper motion strongly depends on the final explosion mechanism. The focus of this work will be the remnants of such explosions. SNRs are generally grouped by their morphology into shell-type SNRs, plerions, composite, and thermal-composite SNRs. A shell-like structure is created when the shock wave propagates through the ISM and heats the plasma. Hence, an SNR that shows such a structure is named a shell-type SNR and this group includes most known remnants. A plerion is a highly energetic wind of relativistic electrons and positrons, accelerated to ultra-relativistic energies by a rapidly rotating neutron star (pulsar), also referred to as a pulsar-wind nebulae (Kargaltsev et al. 2013). Remnants with a pulsar-wind nebula surrounded by a shell are designated as composite SNRs, but only a few of this kind have been detected so far. The thermal-composite SNRs exhibit a shell-like structure in the radio band and thermal x-ray emission from the center of the remnant.

### Evolution

Based on the energetics, supernovae were first proposed as CR-acceleration sites (Baade and Zwicky 1934). The observed CR energy density can be explained if a portion of the kinetic energy of SNe can be transferred into the ambient medium to accelerate particles. Each supernova injects kinetic energy on the order of $$\sim 10^{51}~{\rm erg}$$ into the ambient medium. The first phase of SNR evolution is the free-expansion phase, where the shock front propagates into the ISM. The ISM material is separated from the eject by a contact discontinuity. Behind the contact discontinuity, a reverse shock develops in the ejected material. Ahead of the contact discontinuity, the ISM material is compressed to form a thin shell. The phase lasts for the first $${\sim} 1000~{\rm yr}$$ of the SNR’s existence. During this phase, a fast, collision-less shock at the edge of the expanding SNR exhibits a featureless x-ray spectrum, characterized by a power-law with a photon index $$\Gamma\sim 2.5$$. This power-law emission is attributed to synchrotron radiation (Reynolds and Chevalier 1981) from relativistic electrons interacting with ambient magnetic-field lines swept up by the expanding shock. The emitting electrons at the shock front are accelerated to relativistic speeds by the first-order Fermi process (Fermi 1949, Blandford and Ostriker 1978), where the electrons repeated cross back and forth across the moving shock front and gain a substantial amount of energy per shock crossing.

The interior portion of the remnant exhibits a hot thermal spectrum dominated by stellar ejecta located behind the forward shock. This material has high elemental abundances produced both in late nuclear-burning stages and explosive nucleo-synthesis. This material has been reheated by the reverse shock, which is still propagating outward in this phase, separated from the forward shock by a contact discontinuity.

When the mass of the accumulated ISM material on the expanding shell is approximately equal to the mass of the initial explosion, the SNR enters the energy-conserving (Sedov) phase. The energy lost to radiative cooling is negligible up to this phase. As the remnant ages, however, the accumulated losses become significant. The third phase, the momentum-conserving phase, enters around this time. The final phase is the fade-away phase where the shock structure begins to break up. When the interior of the SNR has cooled down sufficiently, the expansion speed falls below the sound speed and the shock wave turns into an ordinary sound wave. The outer shell breaks up into clumps and gradually merges with the ambient medium.

SNRs emit at radio, infrared (IR), optical, ultraviolet (UV), x-ray, and gamma-ray wavelengths. The radiative processes that yield this multi-wavelength emission can be categorized into 2 groups. The first group includes bremsstrahlung, radiative recombination, and collisional excitation. The second group includes synchrotron emission, inverse Compton (IC) scattering, nuclear-decay lines from unstable isotopes, and warm dust emission. The details of each of the non-thermal processes are explained thoroughly in the following sections. The shock wave of a supernova is observable as hot, x-ray emitting plasma, which is optically thin in a first-order approximation. Furthermore, the plasma in young SNR is mostly not in equilibrium. The observed thermal x-ray spectra of SNR consists of absorbed thermal continuum emission by bremsstrahlung, recombination, two-photon processes, and line-emission from collision-excitation of ions (Vink 2012). In order to show how to extract important properties of a SNR, there processes are summarized below (Mewe 1999 and Kaastra et al. 2008):

• Bremsstrahlung or free-free emission is caused by the collision of an ion with a free electron where the emissivity is $\epsilon_{\rm ff} = \epsilon_0 g_{\rm ff}(T_{\rm e}) T_{\rm e}^{-1/2} \exp \left(-\frac{h\nu}{kT_{\rm e}}\right) \ ,$ where the gaunt factor $$g_{\rm ff} \approx 1$$, $$T_{\rm e}$$ is the electron temperature, and $$\epsilon_0$$ is a normalization constant referred to as the emission measure, which depends on the electron and ion number densities and the ion charge.

• Recombination or free-bound emission occurs when a free electron is captured by an ion and a photon is emitted (the reverse process of photo-ionization).

• Two-photon emission is caused by excitation of the meta-stable $$2{\rm s}$$ level and emission of two photons when decaying into the $$1{\rm s}$$ ground state. A normal decay from $$2{\rm s}$$ to $$1{\rm s}$$ is forbidden by quantum-mechanical selection rules. The total energy of the two photons is equal to the difference in energy between the two states and a symmetric spectrum around this energy difference is measurable. This process can only happen when the density is low enough such that no collisional excitation in a higher state occur before the decay (Mewe et al. 1986).

• Line emission is caused when an excited ion in state $$j$$ decays back into a lower state $$i$$ and emits a photon with an energy $$E = E_j - E_i$$. In SNRs, the dominant process is collisional excitation, in most cases, electron-ion collisions.

All these emission processes have to be modeled when fitting a spectrum of a SNR. As mentioned before, SNRs usually are not in collisional equilibrium. In young SNRs, after the shock wave changes its temperature rapidly, the plasma does not have enough time to reach ionization balance again.

### Magnetic-field amplification

The ISM is turbulent.

## Cosmic-ray theory

### Acceleration mechanisms

Fermi acceleration refers to the mechanism by which particles are accelerated from thermal velocities to relativistic speeds. Such a mechanism is thought to occur in collision-less shocks, where the mean-free-path of collisions is larger than the shock width. Particle acceleration at collision-less shocks is often referred to as shock acceleration. The magnetic field around the shock domain is expected to be turbulent, such that scattered particle motions are co-moving with the shock. Each time the particle crosses the shock front, it gains energy. Shock acceleration thus produces a power-law CR flux naturally, which can explain the power-law behavior of the measured CR spectrum up to the knee.

### Second-order Fermi acceleration

Second-order Fermi acceleration occurs within the magnetized ISM. A charged particle moving through the ISM is reflected by randomly moving magnetic mirrors. These can be clouds of magnetized plasma. Depending on the direction in which the magnetic mirror moves relative to the incoming particle, i.e., if it is approaching or receding, the particle will either gain or lose kinetic energy, respectively.

### First-order Fermi acceleration

First-order Fermi acceleration occurs in the vicinity of a shock front. In contrast to the second-order process, the motion of a shock front is not random, but rather directional. It was therefore argued by theorists that the acceleration by shock waves would be more efficient, since one would expect far more head-on collisions.

### Energy losses

Once CRs are accelerated, they can propagate along magnetic-field lines. CR protons interact with the interstellar medium (ISM) via pion production ${\rm p}_{\rm CR}^+ + {\rm p}_{\rm ISM}^+ \to {\rm p}^+ + {\rm p}^+ + \pi^0 \ ,$ where the pions rapidly decay into gamma rays ($$\pi^0 \to \gamma + \gamma$$) with energies $$> 100~{\rm MeV}$$. CR electrons interact with both the ISM and interstellar radiation field (ISRF) via Bremsstrahlung and inverse Compton scattering, respectively

\begin{aligned} {\rm e}_{\rm CR}^- + {\rm p}_{\rm ISM}^+ &\to {\rm e}^- + {\rm p}^+ + \gamma \ , \\ {\rm e}_{\rm CR}^- + \gamma_{\rm ISRF} &\to {\rm e}^- + \gamma \ .\end{aligned}

CR electrons also produce synchrotron emission as they propagate along magnetic-field lines. The energy-loss rates for these processes are as described by Schlickeiser 2002:

\begin{aligned} -\left. \frac{{\rm d} E}{{\rm d} t} \right|_{\rm brem} &\propto n_{\rm ISM} E_{\rm e} \ , \\ -\left. \frac{{\rm d} E}{{\rm d} t} \right|_{\rm IC} &\propto U_{\rm rad} E_{\rm e}^2 \ , \\ -\left. \frac{{\rm d} E}{{\rm d} t} \right|_{\rm synch} &\propto U_{\rm mag} E_{\rm e}^2 \ . \end{aligned}

Thus, CR electrons at $${\rm GeV}$$ energies lose the majority of their energy to bremsstrahlung. At higher energies, inverse Compton and synchrotron processes are comparable in significance, with the dominant one determined by the energy density in the magnetic and radiation fields.

Accelerated electrons streaming through the compressed-amplified magnetic field near a SNR shock emit non-thermal (synchrotron) radiation, most prominently in the radio band. Hard x-ray non-thermal synchrotron tails as distinguished from soft x-ray from thermal bremsstrahlung can be observed if the maximum electron energy reaches several hundred TeV. The energy-loss rate of electrons through synchrotron is (Rybicki and Lightman 1979) $\dot{E}_{\rm synch} = - \frac{4}{3} c \sigma_{\rm T} \gamma_{\rm e}^2 \beta_{\rm e}^2 \left(\frac{B^2}{8\pi}\right) \ ,$ where $$\sigma_{\rm T}$$ is the Thomson cross section, $$\gamma_{\rm e}$$ is the electron Lorentz factor and $$B$$ is the magnitude of the magnetic field.

### Inverse-Compton scattering

Relativistic electrons up-scatter low-energy photons in the background radiation fields to gamma-ray energies through the IC mechanism. The seed radiation field is dominated by the CMB photons, along with the less well-determined local IR/optical backgrounds. For electron energies below $$\sim 10~{\rm TeV}$$, for which the seed photon energy in the electron rest-frame is always much less than the electron rest energy, the process can be described accurately in the classical regime at the Thomson limit.

### The fluid approximation for cosmic rays

Capturing the fine details of CR transport can be computationally prohibitive on macroscopic scales, i.e., $$\gg 1~{\rm m}$$. On such scales, CRs are often treated as a relativistic fluid.

## Supernova-remnant simulations

Modeling supernova-explosions is a difficult endeavor. There have been several groups that try to simulate the onset of the explosion () with several physical processes: hydrodynamics, magnetic fields, neutrino transport, general relativity, etc. The physics spans several orders of length and time scales. We focus on the expansion of the blast wave into the surrounding medium.

## Cosmic-ray simulations

Generally, simulations of CRs either consist of kinetic particle-in-cell (PIC) simulations at small scales $$({\sim} 1~{\rm cm})$$ or hydrodynamic simulations on galactic scales. We review the advances made in both regimes in this section.

Astrophysical particle acceleration is a long-standing problem in plasma astrophysics. DSA has been the main acceleration model (Reynolds 2008 and references therein); however, a major criticism of this model is that the maximum energy that CRs achieve via this mechanism is about 1 magnitude less than where the knee in the spectrum is (Lagage and Cesarsky 1983). Nonlinear effects, magnetic field amplification, etc., may also play a significant role in accelerating particles to relativistic energies (Aharonian 2004, Zweibel 2013).

Direct simulation of DSA requires expensive kinetic techniques and is still an active area of research. Hybrid particle-in-cell (PIC) simulations of CRs self-consistently coupled to magnetized fluids have emphasized their role in exciting flow instabilities. While it is computationally prohibitive to perform a PIC simulation on galactic scales, appropriate CR-MHD coupling can be achieved via a moment-equation approach. Results suggest that while the nonlinear back-reaction due to CRs modifies the shock structure, in realistic shocks with Mach numbers $${\sim} 10$$, the shocks and CR spectra achieve steady state. Previous work, such as Skillman et al. 2008, has treated shock acceleration by using an algorithm that detects shocks, estimates the upstream Mach numbers and adds a local source term to the CR energy density. We do not consider DSA in this work, but future investigation will involve using astronomical observations to constrain sub-grid acceleration models.

## Supernova-remnant observations

There are a multitude of SNR observations. For this work, I will review a few that are of particular interest in terms of characterizing transient behavior. From studies of the Milky Way, Fermi has provided compelling evidence that SNRs are CR accelerators (Ackermannn et al. 2013). By extension, this suggests that star-forming regions are sites of CR acceleration.

## Cosmic-ray observations

In terms of observations, CRs can either be observed indirectly through their radiation signatures or directly when they pe