A QUEST FOR AN ELUSIVE COMPANION STAR

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1 Draft version July 31, 2012 Preprint typeset using L A TEX style emulateapj v. 12/16/11 A QUEST FOR AN ELUSIVE COMPANION STAR Shengkai Alwin Mao University of California, Berkeley, CA Draft version July 31, 2012 ABSTRACT The mechanism behind Thermonuclear (Type Ia) Supernovae is believed to involve a white dwarf and a companion star, which could be a normal star (single-degenerate model) or a white dwarf (double-degenerate model). Finding the companion star would show that the companion star was not destroyed in the collision, implying a single-degenerate progenitor. Due to the sheer size of the supernovae and the number of candidate stars contained therein, it is necessary to determine the star s rough location before making an exhaustive search. It is thought that the companion star can be found at the explosion center, but assumptions that the explosion center would be at the remnant s geometric center have proved fruitless. However, these searches did not factor in the possibility that a nonuniform interstellar medium (ISM) density could cause an asymmetric expansion, which would cause the apparent geometric center to shift from the true explosion site. By running numerical hydrodynamics simulations modeling the effects of ISM density gradients with differing steepness and shape, a collection of remnants can be formed, and a more reliable search location can be determined. To that end, it has been found that Type Ia supernovae simulated by exponential density models are significantly offset by jump discontinuity and hyperbolic tangent ambient density profiles. However, young supernovae simulated with an exponential density model in linear ambient density profiles are not significantly offset until they reach the Sedov expansion phase. 1. INTRODUCTION Supernovae, best known for their distinctly high luminosities, are categorized into several groups. One of these classifications is known as Type Ia, defined by the presence of strong SiII spectral lines but a distinct lack of hydrogen and helium spectral lines. Type Ia supernovae are also unique in that they are believed to be thermonuclear explosions of white dwarfs in close binaries which have managed to gain enough mass from a companion star to exceed the Chandrasekhar limit. However, some of the greatest significance of Type Ia supernovae can be found in their importance to cosmology and to understanding the chemical evolution of galaxies. The convenient uniformity of their brightness allows them to be used as standard candles (not to mention that the magnitude of said brightness makes them very visible and observable). Then, using Type Ia supernovae as standard candles, distances, redshifts, and velocities can all be measured. This is done by assuming the absolute brightness of a Type Ia supernova based on the Phillips relation, observing the apparent magnitude, and using these luminosities to determine the distance (simply put, light sources appear dimmer linearly with distance). Such measurements have brought to light the acceleration of the expansion of the Universe. Even our knowledge of the cosmological parameters themselves depends upon the accuracy of these measurements. In addition, Type Ia supernovae are responsible for the majority of iron production in the universe. To better understand Type Ia supernovae is to better understand the chemical evolution of galaxies.(wang & Han 2012) However, the accuracy and reliability of the results and the understanding garnered from Type Ia supernova studies are arguably questionable. The nature of the progenitor and the companion star is not fully understood, which affects the reliability of necessary assumptions such as those of uniform luminosity. In order to better understand the Universe and the galaxies within, it is critical to improve our understanding of Type Ia supernovae, and more specifically, their progenitors. (Wang & Han 2012) Currently, the two most prominent proposed models are the single-degenerate model and the doubledegenerate model. In the single-degenerate model, the white dwarf accretes mass from a non-degenerate companion star. By accreting hydrogen and helium, the white dwarf can gradually increase its mass to the Chandrasekhar mass, at which point a supernova is triggered. Alternatively, according to the double-degenerate model, the companion star is instead a white dwarf. The two white dwarfs approach each other as they radiate gravitational waves. If the two white dwarfs combine, and if they have a combined mass greater than the Chandrasekhar mass, a supernova results. (Wang & Han 2012) Both models have advantages and disadvantages. For example, the single-degenerate model requires the accretion of hydrogen, which has been shown to be difficult to drive (Badenes et al. 2007). On the other hand, the gradual accretion to a Chandrasekhar mass involved in the single-degenerate model also effectively explains why Type Ia supernovae would be fairly uniform (Wang & Han 2012). The double-degenerate model is successful in explaining a lack of hydrogen and helium in Type Ia supernovae spectra, since only carbon-oxygen white dwarfs are involved. However, this model presents the troubling possibility of neutron star formation instead of supernova ignition, as well as predicting a wide range of supernova masses. Since arguments and debates have proven unable to identify the correct model, a different approach must be taken (Schaefer & Pagnotta 2012). A distinguish-

2 2 Mao Fig. 1. This image of SNR from Schaefer & Pagnotta (2012) reveals the scope of the problem: the vast number of stars that must be checked in a supernova remnant. The circle is a projection of where the progenitor could be, with a 99.73% chance of being within the circle, ignoring the asymmetries in the remnant and any effects due to the interstellar medium. ing property between the single-degenerate and doubledegenerate models is that the companion star in a singledegenerate supernova should survive, while the combination of two white dwarfs in a double-degenerate supernova will not leave anything (Wang & Han 2012). Thus, finding the companion star in a supernova remnant would provide absolute proof of the single-degenerate scenario. Unfortunately, this task is in and of itself prone to difficulty and controversy. In order to find the companion star, it must be identified among the other stars in the supernova remnant. This is possible because of a few peculiarities that are imparted to a hypothetical companion star by the exploding white dwarf. The supernova affects the motion of the nearby companion star, resulting in a uniquely high velocity. The supernova also deposits iron and nickel onto the surface of the companion star, providing it with unusual spectral lines. Indeed, it is possible to distinguish the progenitor star from other stars contained within a remnant. However, supernova remnants are enormous, and efforts to find the companion star are too resource-intensive due to the sheer number of candidate stars contained within a remnant. Thus, searches must be narrow, oftentimes limited to small areas near the remnant s apparent geometric center. Such an approach has not been very successful in finding companion stars. The only claimed discovery of a companion star in a Type Ia supernova remnant was the supposed progenitor of Tycho s supernova of 1572 (Ruiz-Lapuente et al. 2004). However, the validity of this result was highly controversial, not absolute (Schaefer & Pagnotta 2012). Hence, a progenitor of a Type Ia supernova is yet to be found. It is known that density irregularities in the interstellar medium surrounding a supernova remnant can cause the remnant to become non-spherical (Hnatyk & Petruk 1999). As the remnant expands into a nonuniform inter- Fig. 2. An illustration of the principle of apparent center shift due to a density gradient. stellar medium, the interactions between remnant and ISM form asymmetric remnants and shift the apparent geometric center of the remnant away from the true supernova origin (Dohm-Palmer & Jones 1996). To illustrate, a supernova will expand at a slower rate into a high density interstellar medium (ISM) than into a low density ISM. If a supernova encounters a higher density ISM on one side, that edge of the supernova will have a lesser distance from the true center due to slower expansion. With a little thought, it becomes clear that, since the supernova edge on the low density side is farther away from the true center, a density difference will shift the apparent center towards the low density area, away from the true center. Since past searches did not account for these effects, it is possible that past surveys did not consider distances far away enough from the apparent geometric center. If a supernova has expanded into a density gradient such that portions of the explosion have traveled significantly more than others, the apparent geometric center would be significantly offset. Only by factoring in these effects can a more definitive result be attained. By conducting a search which considers these effects and by checking a larger area which is far more likely to contain the true supernova center, the result will be far more likely to be valid, accepted, and applicable. Since density gradients surrounding a supernova remnant should cause brightness variations (Petruk 1999), observational data of a supernova remnant can be used to describe the density gradient of its surrounding interstellar medium. By determining a relation between interstellar medium density gradients and apparent center shifts for a supernova remnant, a better search location can be formulated in order to find or debunk a possible companion star and determine the nature of its progenitor.

3 A Quest For An Elusive Companion Star 3 2. COMPUTATIONAL METHODS 2.1. Sedov Blast Wave Simulations In order to simulate supernovae, I made use of VH-1, a PPMLR hydrodynamics simulation code (utilizing a piecewise parabolic method and formulated as a Lagrangian (fluid-frame) calculation followed by a remap of the conserved quantities onto the original grid ) maintained and provided by Dr. John Blondin (Chevalier et al. 1992). I began with simulating a 1-Dimensional Sedov Blast Wave (assuming spherical symmetry) in order to achieve proper starting conditions for the pressure and density at the explosion center. I moved to a 2-Dimensional Sedov Blast Wave (in R-Z coordinates with cylindrical symmetry) in order to reveal asymmetries and irregularities in the explosion. The coordinate geometry chosen made it simpler to analyze the output data. Arrays of data containing pressure and density values for each zone could easily be read from output files. Since the grid was essentially a 2-Dimensional Cartesian grid, it was convenient for array element coordinates to correspond to zone coordinates (i.e. the bottom right zone in a 400x200 grid simulation would have zone coordinates (200,200). Thus, its pressure and density values could be found in the array in element (400,200). The x-coordinate shifts by 200 since the simulation center has array coordinates (200,0) and simulation coordinates (0,0)). The coordinate geometry chosen also avoided problems with choosing coordinates reliant on angles, such as varying zone widths which could become too small for the simulation code to process. The explosion center was placed on the midpoint of the axis of symmetry (Z-axis). The axis of symmetry was set to have reflecting boundary conditions. The other boundaries had inflow/outflow zero-gradient boundary conditions, allowing material to flow freely past the boundary. I added an external density gradient which increased linearly parallel to the axis of symmetry, according to the relation ρ(z) = slope Z + constant, (1) and began finding irregularities. However, the time allotted to the simulation to evolve those irregularities was limited because the explosion would become larger than the simulation grid size. To remedy this, the VH-1 code was altered in order to expand the grid automatically if the supernova explosion grew to a certain fraction of the grid size. This was accomplished through a routine which checked certain grid points for a higher pressure characteristic of the supernova edge. When the supernova edge reached a certain distance from the center, the grid expansion was triggered. This grid expansion increased all of the distances and sizes of each grid zone by a scale factor kept small enough for the grid zones to retain their original values for pressure, density, and velocity after the expansion. This would happen during every computational cycle wherein the supernova edge had grown to a certain fraction of the grid size, thus ensuring that the supernova would not interact with any boundary other than the axis of symmetry. Unfortunately, it could not scale the densities to maintain the linear relation between density and distance. This was fixed by recalculating the den- Fig. 3. Example of a Linear external density profile with ρ(z) = 2 Z + 7 sity values at the boundaries perpendicular to the axis of symmetry during each cycle in order to match it to the initial formula for density as a function of position. As a note, although these limitations to simulation time had been removed, a linear density gradient makes one limit unavoidable. Since grid expansion allows distances to increase with time, the position of the left boundary becomes more and more negative. Eventually, as prescribed by Equation 1, densities of negative or 0 values are assigned. This ends the simulation. This is currently the limiting factor for simulation time for cases using a linear external density profile. Several simulations were run with various linear external density gradients, altering the slope and the density value at Z = 0 between test cases Exponential Ejecta Simulations After finding that significant offsets could be found in remnants formed with a linear external density gradient and a 2-Dimensional Sedov Blast Wave, I adapted the VH-1 code to simulate an exponential ejecta density profile as described in Dwarkadas & Chevalier (1998) to more accurately and realistically portray a Type Ia supernova. The simulation was still run in two dimensions with cylindrical (R-Z) coordinates with the same boundary conditions. The code used the initial simulation time to determine the initial sizes of the simulation grid. It also determined the initial pressure, velocity, and density for the grid zones within a reasonably large defined radius from the center. Outer zones were assigned a small pressure and a linear external density gradient. The density values of the boundary between the outer zones and inner zones was adjusted to ensure a smooth density transition between the external medium and the initialized explosion. The same grid expansion code from the Sedov blast simulations was implemented. The values of the innermost zones were recalculated each simulation cycle based on the same analytic solution used to initialize the grid, in order to avoid computational errors. However, an unavoidable error was the growth of ear-like features on the axis of symmetry which would artificially increase the distance of the supernova edge from the true center on both sides. This error was minimized through a dissipation of the zones where the ear-like features formed and were present. The linear external density gradients

4 4 Mao Fig. 4. This image of a 2-D VH-1 simulation of a Sedov blast wave clearly shows a linear external density gradient (ρ(z) = 10Z+ 12) which increases along the axis of symmetry. It also reveals a pileup of material on the denser side of the ISM. Fig. 5. This image of the same 2-D VH-1 simulation of a Sedov blast wave shows the corresponding increased pressure on the denser side of the interstellar medium. Furthermore, the supernova is visibly off-center, and slightly asymmetric, while remaining elliptical. tested with the Sedov Blast were retested with these new initial conditions and model. In order to form a varied collection of remnants, it was necessary to test various forms of external density profiles. After implementing many different versions of a linear external density profile, a jump discontinuity external density profile was tested using the function ρ for Z < 0 ρ(z) = ρmin for Z > 0. (2) Fig. 6. The evolution of a 2-D VH-1 simulation initialized in a 400 x 200 grid with an exponential ejecta density profile and a linear external density profile. In the bottom right image, the linear density profile and the contrast in density from left to right become visible. max The simulation was initialized as before with a 2-D grid in R-Z coordinates assuming cylindrical symmetry and a reflecting axis of symmetry. The same exponential ejecta density profile was placed into the grid. The primary difference was the initialization of the external density profile in the outer zones, which was set to a smaller constant value (ρmin ) for Z < 0 and a larger constant value (ρmax ) for Z > 0 (hence, a jump discontinuity at Z = 0). This was repeated for several different density ratios ( ρρmax ). min Although it was more difficult to ensure a smooth transition between the innermost zones initialized with the exponential ejecta density profile and the outermost zones with the ambient density, the difference was contained within an order of magnitude. The most extreme case set ρmax = 100 and ρmin = 5. Both values were within an order of magnitude of the outer-inner-zone Fig. 7. Example of a Jump Discontinuity external density profile with ρmin = 11 and ρmax = 88

5 A Quest For An Elusive Companion Star 5 Fig. 9. Example of a Hyperbolic Tangent external density proz file ρ(z) = tanh( 0.3parsecs ) 700 Fig. 8. A depiction of the evolution of a 2-D VH-1 simulation initialized in a 400 x 200 grid with an exponential ejecta density profile and a jump discontinuity external density profile, which becomes visible in the third row. The image is mirrored over the Z-axis (axis of symmetry) for aesthetic purposes. boundary density value of 50. Afterwards, I attempted to reproduce the hyperbolic tangent external density profile used in Dohm-Palmer & Jones (1996) using the exponential ejecta density profile again as initial conditions rather than their model. The hyperbolic tangent function used was 1 1 Z ρ(z) = (ρ0 + ρ1 ) + (ρ0 ρ1 ) tanh( ), (3) 2 2 H where ρ0 is a maximum density, ρ1 is a minimum density, Z is position along the Z-axis (axis of symmetry), and H determines the transition width. Based on their values for the energy, mass, and ambient density of the explosion, it was possible to determine how distance and time scaled between my dimensionless simulation units and real units through equations prescribed in Dwarkadas & Chevalier (1998). This meant that upon testing their hyperbolic tangent external den- sity profile, I could run other simulations with other variations of the hyperbolic tangent external density profile, including a few which approximate observed supernovae. Furthermore, the transition width of the hyperbolic tangent external density profile and minimum and maximum densities could be changed to form a varied collection of remnants with this ISM density profile. Finally, simulations were run in order to approximate the conditions found in SNR , the remnant searched by Schaefer & Pagnotta (2012). Using the distance scaling found from the equations in Dwarkadas & Chevalier (1998) and an approximate value for the diameter of SNR , the corresponding diameter in the simulation s dimensionless units was calculated. Different ratios ( ρρ01 ) between the minimum and maximum densities were tested, as were varying transition widths (H) of various fractions of the supernova remnant s approximate diameter (6.8 pc). The values tested were H = 0.3 pc, 1.7 pc, 3.4 pc, 5.0 pc, and 6.8 pc. Upon determining a satisfactory value of H, a ratio ρρ10 of 4 and 6 were tested. These ratios approximated a possible density ratio between the density of the ISM on one side of the remnant and the density of the ISM on the other side. In this manner, SNR was roughly simulated. Data was taken from the simulation time frame at which point the simulation radius matched the observed radius of SNR in order to compare simulations to observed data. In this manner, by determining the asymmetry and apparent center shift for time frames in the simulation, an apparent center shift could be estimated for SNR Analysis Methods As a numerical representation of asymmetry and apparent center shift, I used axial ratio and fractional offset, respectively. To do this, for each time frame, the position of the edge of the supernova was calculated using a Python script. First, it read in the simulation output files. Then, the diameter (D) of the supernova was calculated from the two positions (let R0 and Rπ denote the distance of these positions from the true center) of the supernova edge lying on the axis of symmetry (several zones were found and averaged to provide more accurate values). The perpendicular radius R of the supernova

6 6 Mao Fig. 11. This is a plot of axial ratio versus time (x-axis) of a 2-D VH-1 Sedov blast wave simulation set in a linear external density gradient which reveals that over time (from 1000 to 1140) the axial ratio increases steadily. Fig. 10. These are a series of images depicting the evolution of a 2-D VH-1 simulation initialized in a 400 x 200 grid with an exponential ejecta density profile and the hyperbolic tangent external density profile, which becomes visible in the second row and appears similar to a jump discontinuity at the large distance scales of the fourth row. The image is mirrored over the Z-axis (axis of symmetry) for aesthetic purposes. Fig. 12. The plot of fractional offset (y-axis) against time (xaxis) for the same simulation shows that the fractional offset also grows over time. was calculated from the position (R) of the supernova at a range of zones around the center (Z=0). This easily produced an axial ratio between width and height. axial ratio = D R0 + Rπ = 2 R 2 R (4) The offset was calculated from the difference between the average position of the supernova edge along the axis of symmetry and the center of the supernova along the axis of symmetry. fractional offset = R0 Rπ R0 + Rπ (5) Fig. 13. This is the result of plotting the fractional offset against the corresponding axial ratio for that time slice. It reveals that they are somewhat linearly related (ignoring outliers and error) and that for a small axial ratio change, there is large fractional offset. Thus, for each time interval, an axial ratio and a fractional offset was calculated. The axial ratios were then plotted against the fractional offsets. Observing that the graph of axial ratio versus fractional offset was fairly linear, ignoring noise, I applied a method of least squares in order to fit a line to the graph. This method would also return the slope of the fitted line in order to quantify the relation between density, asymmetry, and offset in a numerical table. Fig. 14. This is the result of taking the fractional offset and axial ratio data, removing unrealistic axial ratio values significantly less than 1.0, and computing a least squares fit with the remaining data.

7 A Quest For An Elusive Companion Star 7 Fig. 15. These are fractional offset versus axial ratio plots of 3 cases where ρ(z) = slope Z + constant, constant = 7, and slope = 4, 3, and 2. Many of the earlier points are clustered around a fractional offset of 0 and an axial ratio of 1, showing little change in asymmetry or offset until later Sedov phases. The cyan marker delineates the axial ratio and fractional offset of SNR RESULTS In the linear external density cases, early time periods of the supernova dominated by the exponential ejecta density profile showed little to no change in fractional offset and axial ratio. It is only the later stages dominated by Sedov expansion where the supernova remnant experiences significant changes in offset and axial ratio. The slope of plots of the fractional offset versus the axial ratio of select time frames in the simulation is surprisingly similar between several linear external density cases. Furthermore, for a small change in aspect ratio, there is a relatively large change in fractional offset. The remnant remained fairly elliptical over time. In the jump discontinuity external density cases, early time periods of the supernova dominated by the exponential ejecta density profile showed significant changes in fractional offsets. The remnant also ceased being elliptical, quickly becoming asymmetric. As might be expected, cases wherein the ratio between the two constant densities were close to 1 appeared more similar to a uniform external density profile. Cases wherein the ratio between the two constant densities was high were more likely to reach higher offsets and axial ratios. Later time periods of the supernova dominated by Sedov expansion were relatively stagnant. The presence of a density discontinuity in the center of the grid had an insignificant effect on offset and axial ratio when the supernova had reached a sizeable radius. Thus, the jump discontinuity external density cases led to rapid change during early time periods and little to no change during later time periods. In the hyperbolic tangent external density cases, including the attempted reproduction of a simulation found in Dohm-Palmer & Jones (1996), the earliest time periods of the supernova showed little to no change in fractional offset and axial ratio. After sufficient time had passed for the supernova to sweep up surrounding material, the remnant experienced very high offset changes for very low axial ratio changes (plots of fractional off- Fig. 16. These are fractional offset versus axial ratio plots of 2 cases. Line 1 corresponds to a simulation run with an external density profile of ρ(z) = tanh( ). Line 2 1.7pc Z corresponds to a simulation run with an external density profile of ρ(z) = tanh( Z ). The Magenta points correspond to approximate values of axial ratio and fractional offset for SNR pc The 4 Cyan points on the green Line 1 correspond to the 4 timeframes depicted in Figure 10. It is clear that the early stages have static fractional offsets and axial ratios, followed by a period of rapid growth, and a period of slower fractional offset gain and rapid axial ratio change. The second Cyan point from the bottom on green Line 1 and the only Cyan point on red Line 2 correspond to the time frame in the simulation where the radius of the supernova is approximately equal to that of SNR sets versus axial ratios had a nearly vertical slope). Once the supernova expanded past the transition phase of the hyperbolic tangent external density profile, the remnant behaved similarly to a late-stage remnant in a jump discontinuity external density profile, exhibiting heavily reduced offset growth. However, the axial ratio continued to change rapidly as the remnant lost its elliptical shape. The slope of fractional offset versus axial ratio approached 0 over time. By examination of an image found in Schaefer & Pagnotta (2012) of SNR , the axial ratio was found to be The fractional offset necessary to place a companion star outside of the 99.73% error circle searched in Schaefer & Pagnotta (2012) was found to be The diameter of the remnant was taken to be 6.8 parsecs, corresponding to a distance of 11.6 in simulation dimensionless units. These observed values of axial ratio and fractional offset for SNR were used in comparisons to the calculated axial ratio and fractional offsets for simulations. The values of the diameter was used for certain simulations using a hyperbolic tangent external density profile in order to compare the simulation to the observed data at a similar time and growth stage. In linear cases with slopes as low as 2.5, by the time the remnant reached an axial ratio of , the calculated fractional offset was greater than 0.09, placing the companion star outside of the search area for that level of asymmetry. The other linear cases with lower densities had not evolved long enough to reach an axial ratio of 1.07 but were well on their way to reaching 0.09 if allowed to continue, which reveals that even small

8 8 Mao asymmetries can indicate high offsets. In simulations run with a jump discontinuity external density profile, density ratios ( ρmax ρ min ) between 2 and 8 reached 0.09 fractional offset before reaching an axial ratio of Simulations run with smaller ratios would not experience significant axial ratio and fractional offset changes. Simulations run with a ratio of 20 experienced larger axial ratio changes than fractional offset changes and reached 0.09 fractional offset after reaching an axial ratio of A hyperbolic tangent external density simulation was run using the same parameters as SNR3 in Dohm- Palmer & Jones (1996) but a different model for the supernova (exponential ejecta density profile), using Equation 3. As prescribed in Dohm-Palmer & Jones (1996), ρ 0 = 1200 m p cm 3, ρ 1 = 200 m p cm 3, and H = 0.3 parsecs. Their simulation ended at the age of 1200 years, which corresponded to an age of in simulation units, based on the scaling formulas described in Dwarkadas & Chevalier (1998). At this end time, the axial ratio and fractional offset of their remnant was approximately 0.93 and 0.18, respectively. At this time, the remnant in our simulation had an axial ratio of and a fractional offset of When our simulation reached their level of asymmetry (axial ratio of 0.93), our fractional offset was only Our fractional offset only reached 0.18 when our remnant had an axial ratio of 0.9, much more asymmetric than their results. Hence, our simulations did not predict as high an offset for a given age or level of asymmetry as the simulations run in Dohm-Palmer & Jones (1996). This discrepancy may be attributed to differing initial conditions (the exponential ejecta density profile) or differing computational methods. Other simulations run with the hyperbolic tangent external density profile often resulted in plots of fractional offset versus axial ratio which would allow them to reach 0.09 fractional offset before reaching 1.07 axial ratio. However, by the time they reached the size of SNR , only simulations run with a transition width H = 1.7pc would reach 0.09 fractional offset, doing so before reaching 1.07 axial ratio. The simulations using a transition width of 1.7 pc were run with densities of 0.1 on the low density side and 0.4 and 0.6 on the high density side. 4. CONCLUSIONS Supernovae initialized in a linear external density gradient would only experience significant apparent center shifts during late-stage Sedov expansion. Supernovae placed in a jump discontinuity external density gradient would only experience significant apparent center shifts during early times, and changing slowly during later times. Remnants in a hyperbolic tangent external density gradient would be offset quickly during early stages and slowly during later stages. They would also experience greater axial ratio change in the later stages. The majority of the density cases tested were able to experience significant geometric center shifts without becoming extremely asymmetric. Although the jump discontinuity cases would quickly cease to be elliptical, linear ISM density gradient remnants remained elliptical, and hyperbolic tangent ISM density gradient remnants would remain elliptical for some time. Since a jump discontinuity density profile is fairly unrealistic, a small asymmetry in an elliptical supernova remnant could easily belie a large offset, and the density gradient should be considered when deciding upon a search radius. A direct comparison with SNR by analyzing simulation remnants with an approximate density gradient at the time at which the simulation remnant reached the size of SNR revealed that a roughly realistic density gradient could offset the apparent geometric center from the true center considerably by that time. The offset alone could easily have been greater than the radius of the search area used in Schaefer & Pagnotta (2012). Since a hypothetical companion star could easily be outside of the search area, the absence of a companion star reported therein may not be valid in arguing for a double-degenerate progenitor for SNR Hence, the single-degenerate progenitor model is still a plausible possibility for SNR In order for these results to be useful, knowledge of the actual density gradients is necessary. These can be determined from observational data through brightness contrasts throughout a supernova remnant, as discusssed in Hnatyk & Petruk (1999), since a non-uniform density will cause certain areas of a supernova remnant to be brighter. The density gradients used in our simulations could be compared to actual observational data to determine the accuracy of our results by examining the validity of our density assumptions. Furthermore, by analyzing simulation remnants as though they are viewed in the plane of the sky, the calculated offset is potentially overestimated. At an angle (θ), the offset would be visibly scaled down by a factor of sin(θ). Since a typical angle is 60, it would be reasonable to say that calculated offsets should be scaled down by sin(60 ), or approximately Furthermore, asymmetries are overestimated, since viewing the simulated supernovae in the direction of the axis of symmetry would obviously result in a completely symmetrical image. Future work could correct for these assumptions. Finally, there are still many density irregularities left untested, and future work could examine these new cases. I am grateful towards the National Science Foundation (NSF) and the North Carolina State University (NCSU) for providing the funding and resources for this research to be conducted under the Undergraduate Research in Computational Astrophysics (URCA) program and the Physics REU program at NCSU. I would also like to acknowledge Dr. John Blondin for providing me with crucial access to VH-1, not only by giving me the program itself, but also through mentorship and guidance as I learned how to utilize it properly. Last but not least, I would like to acknowledge Dr. Stephen Reynolds for his instrumental part in my understanding of the project and his ability to transform a large problem into several smaller manageable pieces. REFERENCES Badenes, C., Hughes, J. P., & Langer, N. 2007, ApJ, 662, 472 Chevalier, R. A., Blondin, J. 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