# 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 t