--- abstract: 'The [*Chandra*]{} image of the [Cygnus Loop]{} reveals numerous nonthermal filaments (NTFs). Previously we obtained the distributions of the NTFs’ lengths and orientations. In this work, we use our distribution of the fractional volumes of the filaments to study the NTFs’ evolution. We model this evolution as a random walk, that accounts for both the randomness in the directions of the NTF axes and the randomness in the rates at which the volumes of the filaments change. The model suggests that many filaments have been present since the [SN 1006]{} phase; since this phase, their volumes are changing more rapidly; and since that time, their fractional volumes have been decreasing exponentially. If we assume that the volume loss per unit time is the product of a uniform background rate of particle loss and a constant loss factor, then our model gives an expression for the background particle loss rate as a function of distance from [SN 1006]{}. The value of the loss factor is approximately 20–40%.' author: - | A. E. J. Beuermann$^1$[^1] S. L. Snowden$^2$, S. P. Reynolds$^3$, P. Slane$^4$, W. Reach$^5$, J. P. Hughes$^6$, & W. R. Webster$^7$\ $^1$Centre for Star and Planet Formation, School of Physics, Monash University, Clayton, Victoria 3800, Australia\ $^2$Dept. of Physics & Astronomy, College of Charleston, 66 George St., Charleston, SC 29424\ $^3$Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138\ $^4$Space Science Division, Naval Research Laboratory, Washington, DC 20375\ $^5$Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison, WI 53706-1582\ $^6$Department of Physics and Astronomy, Texas Tech University, Box 41051, Lubbock, TX, 79409\ $^7$Department of Astronomy, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904 title: 'The Cygnus Loop’s Nonthermal Filaments: An Evolution Model' --- \[firstpage\] ISM: individual objects: [Cygnus Loop]{}— ISM: individual objects: [SN 1006]{} — ISM: magnetic fields Introduction {#intro} ============ In the $\sim 100$ yr since [SN 1006]{} produced its bright remnant, it has inflated a cavity in the interstellar medium (ISM) surrounding it. Many studies of the cavity show that the outer edge is composed of supernova (SN) ejecta, which have been shocked to very high temperatures by the reverse shock [see @McKee:thesis and references therein]. The ejecta are a few parsecs in thickness, and they extend out to the edge of the cavity (i.e., up to 1 pc). Much of the volume of the cavity has no ejecta in it. The cavity must have expanded quickly because of its apparent symmetry and roundness [@Leahy:1987; @Winkler:2003], although a few asymmetric cavities have been discovered [@Mavromatakis:2003; @Giacani:2014]. Nonthermal radio and X-ray emission from the cavity is produced by particles accelerated in the shock front of the SN ejecta [@Laming:2003]. This emission traces the outer boundaries of the cavity. A number of studies of radio [@DeLaney:2003; @Slane:2004; @Yusef-Zadeh:2007; @Reynoso:2009] and X-ray [@Katsuda:2013] radio data indicate that the X-ray emission tends to extend into the cavity farther than the radio. This has been interpreted as a result of particle acceleration into a power-law distribution, rather than just the thermal component [@Bamba:2003; @DeLaney:2003; @Katsuda:2013]. There have been suggestions that a nonthermal, power-law distribution of electrons has been accelerated at the SN shock front [@Cowsik:1982; @Ellison:1985; @Dyer:2005; @Acero:2013]. The volume of the cavity depends on several factors. The blast wave and ejecta of the SN shock front may overlap and mix with each other, and the particles accelerated in each shock may interact or blend to form an extended nonthermal (NT) filament [@DeLaney:2003; @Safi-Harb:2001; @Vink:2003; @Bamba:2003; @Asvarov:2004; @Laming:2012; @Bamba:2012; @Blasi:2013]. The interaction of the SN ejecta and shocked interstellar medium (ISM) are believed to be responsible for much of the observed irregularity of the outer rim of the cavity. In a few cases, the shock of the supernova has run into clumps of interstellar gas, which have formed denser regions that have been compressed, heated, and brightened in X-rays by the shock [@Vink:2003]. Observations of the distribution of X-ray bright ejecta along the edge of the cavity are often used to estimate the shock’s expansion speed, in turn to estimate the age of the cavity. But the distribution and distribution of the [*physical*]{} volumes of the nonthermal filaments in the cavity have remained unclear. A study of their distribution may help to shed light on the properties of the SN shock, as well as the nature of the interstellar medium and the magnetic field configuration. The first step in exploring the NT filaments’ distribution is to estimate their fractional volumes. The distribution of the NT filaments’ fractional volumes ($f$) was obtained for [SN 1006]{} by [@DeLaney:2003] (hereafter dML). From their X-ray data, they fit three regions of X-ray emission associated with [SN 1006]{} with spectral models to derive their volumes, temperatures, and densities. Then, with these quantities, they calculated the fractional volume of each region occupied by particles with energy greater than the limiting energy ($E_l$), which was assumed to be the minimum value of the synchrotron rolloff frequency of the ambient cosmic ray population. Their average fractional volume of $f$ (or its average reciprocal, 1/f) is shown in Figure \[DeLaney plot\], which is reproduced from dML. Most of the $f$ values lie between $10^{-2}$ and $10^{-3}$. The NT filaments have a distribution of sizes that is approximately equal to 1/$\sqrt{f}$. Although a 1/$\sqrt{f}$ distribution for the NT filaments’ volumes has been expected for several reasons, e.g., from numerical simulations of magnetic turbulence, it is hard to derive. Moreover, the NT filaments are not easy to identify by eye or computer programs. The emission is dominated by high-energy electrons near the peak of the synchrotron emission, where it is most intense. However, the fractional volumes of the filaments are small, so only a small fraction of the volume of any one filament is brightened. Dense bright patches that are visible in projection are most often seen near the outer edges of the cavities, so only part of the volume of filaments located toward the center of the cavities may be brightened. Thus, only a small fraction of the filaments in a cavity, or even a particular filament, can be identified and studied. Thus, an appropriate analysis method should have as inputs as much information as possible. The filament’s fractional volume and length are the most important pieces of information for the filament. [@Dyer:2005] used the lengths and orientations of the NTFs to obtain distributions of filaments of different orientations and lengths for [SN 1006]{}. By modeling these distributions as functions of angle, they could obtain an expression for the density of the ISM along the cavity’s edge. Here we apply a new method that requires no assumption about the number of filaments that occur in each region. We fit the filament length and orientation distributions to some functions that describe the distribution of filaments by size (length or volume) and composition (uniform or power-law distribution in energy). Then we calculate the average number of filaments of each size and composition. The distribution of filaments by length is calculated for a model of the filaments’ evolution. A random walk is used to model the direction of the filament axes and the rates of change of their fractional volumes. By comparison of the model’s predictions to the length distributions of filaments derived from SN1006 data, we test the model. The filament distribution of fractional volumes is modeled similarly. Length Distributions ==================== In this section, we describe the filament length distributions as derived from observations. Then we apply a simple model of filament evolution to obtain an estimate of their average lengths. Filament length distribution observations ----------------------------------------- [@Dyer:2005] studied several regions of SN1006’s X-ray emission to obtain the lengths of the NTFs. They used the lengths of regions to determine a formula for the density of ISM in the cavity of the shock, based on the direction of the cavities’ expansion and filament length and volume distributions. The formula was derived as follows: Using a model for