Circumstellar (C-S) dust is present around stars throughout their entire life cycles. About 20-50% of sun-like stars on the main sequence (MS) have belts of rocky material like the Solar System's asteroid and Kuiper Belts. Observations of white dwarfs (WDs), the endpoints of stellar evolution for stars 0.8 to 8 times the mass of our sun, reveal that 20-50% of them have accreted metals typically found in rocky bodies like the Earth. Between the MS and WD stages, stars expand, engulfing any nearby companions, and lose mass, destabilizing their planetary system. So, this ``pollution" is unexpected because the inner regions of system near a WD should be barren.

The focus of this thesis is on the observation and characterization of C-S gas and dust around stars of increasing age, with the goal of studying the evolution of the material in planetary systems. Are we seeing unchanged Earth-like bodies survive long enough to pollute a WD? What are the stellar systems like along the way? To address these questions, we explore several case studies.

We present submillimeter observations with the Submillimeter Array (SMA), the Combined Array for Research in Millimeter-wave Astronomy (CARMA), and the Atacama Large Millimeter/submillimeter Array (ALMA) at $\sim$2'' resolution that spatially resolve the C-S dust or debris disks around five nearby ($d\sim$50 pc) young solar analogues. We perform an MCMC analysis to fit for basic structural parameters, including the inner radius and width of the debris ring, the total mass of the disk, and the characteristic dust grain size. We find that the cold outer belts around the solar analogues in our sample generally exhibit properties consistent with scaled-up versions of the Solar System's Kuiper Belt. The composition of the dust is consistent with an astronomical silicate.

Dust around post-MS stars is expected as a result of stellar evolution, but if a star has not yet reached a stage of predicted mass loss, then the dust is unexpected. We analyze a set of post-MS stars with excess IR emission, using the most recent \textit{Gaia} data release (DR2) to investigate the stellar age, and \textit{Herschel} observations at far-IR and submillimeter wavelengths to constrain the thermally emitting dust. We find that all but one of the 20 stars are post-MS. For the stars detected at submillimeter wavelengths, we find that their spectral indices, $\alpha$ are more shallow than the $\alpha$ that would be expected of grains in the interstellar medium (ISM), pointing toward grain growth in these systems. A fraction of the sample presents characteristics (fast rotation, enhanced lithium abundance, and excess IR emission) that point toward an external source causing the dust excess. One explanation for this trifecta of properties is that these stars have recently engulfed a planet.

Due to the relatively fast gravitational settling times of heavy elements in a WD atmosphere, the presence of those heavy elements is linked to the accretion of planetesimals perturbed by one or more outer planets to within the white dwarf's tidal disruption radius. We present an updated high signal-to-noise ratio spectrum, a new Keck HIRES spectrum, an updated white dwarf atmosphere analysis, and a self-consistent model of the C-S gas around white dwarf WD 1124-293. We constrain the abundances of Ca, Mg, Fe, and a number of other elements in both the photosphere of the white dwarf and the circumstellar disk. We find the location of the gas is approximately one hundred white dwarf radii, the C-S and photospheric compositions are consistent, the gas is not isothermal, and the amount of C-S Ca has not changed in two decades. We also demonstrate for the first time how the radiative transfer code Cloudy can be used to model C-S gas viewed in absorption around a polluted white dwarfs.

Modeling the abundances of C-S gas around polluted white dwarfs with Cloudy provides a new method to measure the instantaneous composition of the material sublimating from the polluting planetesimals. Expanding the work with WD 1124-293, we use Cloudy to predict the physical conditions of C-S gas around model WD photospheres with temperatures ranging from 9000 K to 24,000 K. The models can be used to determine the gas temperature with distance from the star, predict optical depths, constrain the hydrogen number density, and possibly constrain the location of the gas. We compare our models to a small sample of WDs with emission features, and show how the equivalent width can be useful as a diagnostic tool in these systems.
We find, unsurprisingly, that it is difficult to determine the composition of dust around stars on and off the MS. However, the modeling of gas around WDs offers an opportunity to determine the composition of planetesimals in evolved stellar systems. We find that the compositions of C-S gas around polluted WDs are consistent with the bulk Earth, but recommend further modeling. The radiative transfer code, Cloudy can and should be used to search for trends among these planetary systems, to provide unprecedented insight to the composition of extrasolar planetesimals that have survived the evolution of their host stars.