Type Ia Supernovae (SNe Ia) mark the explosive terminations of the otherwise quiet, dead cores oflow-mass stars known as White Dwarfs (WDs). A foundational question in the SN Ia field is: What is the complete progenitor system of these events, and what type of binary interaction prompts thermonuclear runaway inside a WD to produce a SN Ia in the Universe? Theorists have proposed a large number of theoretically viable progenitor systems for these energetic transients. To move forward and solidify our understanding of SNe Ia, the field requires a way to test the proposed the- oretical progenitor scenarios and discriminate between them. One particularly promising method of discrimination is to examine stars within Supernova Remnants (SNRs) to identify or rule out the existence of surviving companions to exploded WDs. Different SN Ia progenitor theories predict different surviving companions which makes surviving companion identification a uniquely pow- erful discriminant. Surviving companion investigations are accomplished by backwards modeling the potential surviving companion stars that reside near the site of the explosion to understand each star’s properties, and then comparing the constrained properties of those stars to the predicted properties of surviving companions. This thesis adds to the growing body of evidence that will disambiguate the progenitor system of SNe Ia by systematically studying stars within SNRs Ia and releasing a new stellar radiative transfer code that will newly enable accurate spectral synthesis of surviving companions. In the first surviving companion investigation presented in this thesis, I examine the interior stellar population of the SN 1006 remnant. The goal of this investigation is to test a recently popular and highly promising SN Ia progenitor theory known as the Dynamically Driven Double- Degenerate Double-Detonation (D6) scenario. This theory predicts the existence of a hypervelocity WD within the remnant, an anomalously bright WD moving at greater than 1000 km s−1. I perform high-precision astrometry to extract the proper motion of each star within the remnant and compare the stars to the predicted properties of a theoretical hypervelocity WD. I do not detect any star within the remnant matching the description of a hypervelocity WD predicted by the D6 scenario, which suggests that the scenario is not responsible for the SN 1006 remnant. In the second surviving companion investigation presented in this thesis, I analyze SNR 0509−67.5, a SN Ia remnant in the Large Magellanic Cloud (LMC). In this investigation, I utilize the constraints of the LMC (i.e., known distances and foreground extinctions to the enclosed stars) to model the stellar population interior to the remnant and extract astrophysical properties of each star inside the remnant. I review the literature to gather a list of predicted properties of the surviving companion on timescales similar to the age of the remnant (e.g., the star will be inflated or heated due to interaction with SN ejecta), and then compare those properties to the properties of the modeled stars. All progenitor scenarios I consider predict that a surviving companion would be clearly identifiable against the unrelated stellar population. I do not detect a star within the remnant matching any of the peculiar properties predicted by the individual progenitor scenarios, and rule out the existence of a surviving companion within the remnant in accordance with those progenitor scenarios. The stellar modeling of the surviving companion searches presented here makes use of pre- existing stellar spectral synthesis codes. While existing codes make it possible to identify surviving companions against the unrelated stellar population, they do not allow for detailed investigations of surviving companions themselves. To enable the modeling of peculiar systems that require different physical treatments and approximations to normal stellar atmospheres, I present a new stellar spectral synthesis code called stardis. stardis is open-source, is written in Python, and is intended to be both modular and easily modifiable. I include a review of physics and equations solved by the code, describe the state of the code, and validate its output with a detailed comparison against similar existing codes. The work in this dissertation to develop a modern, approachable, and extensible stellar spectral synthesis code will enable future investigations of detailed surviving companion spectra.
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