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Gravitational deflection of starlight around the Sun during the 1919 total solar eclipse provided measurements that confirmed Einstein’s general theory of relativity. We have used the Hubble Space Telescope to measure the analogous process of astrometric microlensing caused by a nearby star, the white dwarf Stein 2051 B. As Stein 2051 B passed closely in front of a background star, the background star’s position was deflected. Measurement of this deflection at multiple epochs allowed us to determine the mass of Stein 2051 B—the sixth-nearest white dwarf to the Sun—as 0.675 ± 0.051 solar masses. This mass determination provides confirmation of the physics of degenerate matter and lends support to white dwarf evolutionary theory.

The last twenty-five years have seen our understanding of the formation and abundance of planets revolutionised, thanks to the first detections of debris discs, and, a decade later, of the first extrasolar planets. Hardly a week now goes by without a planet discovery, and the range of methods used to search for planets has expanded to include techniques that are efficient at detecting different types of planets. By combining the discoveries of the various methods, we therefore have the opportunity to build a picture of planet populations across the Galaxy. In this thesis, I am presenting work done as a basis towards such an effort: first I present work carried out to improve modelling methods for gravitational microlensing events. Since the first microlensing observing campaigns, the amount of data of anomalous events has been increasing ever faster, meaning that the time required to model all observed anomalous events is putting a strain on available human and computational resources. I present work to develop a method to fit anomalous microlensing events automatically and show that it is possible to conduct a thorough and unbiased search of the parameter space, illustrating this by analysing an event from the 2007 observing season. I then discuss the possible models found with this method for this event, and their implication (Kains et al. 2009), and find that this algorithm locates good-fit models in regions of parameters that would have been very unlikely to be found using standard modelling methods. Results indicate that it is necessary to use a full Bayesian approach, in order to include prior information on the parameters. I discuss the analytical priors calculated by Cassan et al. (2009) and suggest a possible form of an automatic fitting algorithm by incorporating these priors in the algorithm used by Kains et al. (2009). Another topic with which this thesis is concerned is the evolution of debris discs around solar-type stars. Late-type stars are expected to be the most numerous host stars of planets detected with the microlensing technique. Understanding how their debris discs evolve equates to understanding the earliest stages of planet formation around these stars, allowing us to truly put our Solar System in perspective. Using the analytical model of Wyatt et al. (2007a), I modelled the evolution of infrared excess flux at 24 and 70 microns using published data of debris discs around solar-type (spectral types F, G and K) stars from the Spitzer Space Telescope. By comparing the results of this study to an analogous study carried out by for A stars by Wyatt et al. (2007b), I find that although best-fit parameters are significantly different for solar-type stars, this may be due to the varying number of inefficient emitters around stars of different spectral types. I suggest that although effective properties are different by an order of magnitude or more, intrinsic properties, while still different, are so by a much smaller factor. These differences may be due to the longer timescales over which solar-type stars evolve, which allow for the formation of larger and stronger planetesimals.

In November 2019, the nearby single, isolated DQ-type white dwarf LAWD 37 (WD 1142-645) aligned closely with a distant background source and caused an astrometric microlensing event. Leveraging astrometry from \\Gaia{} and followup data from the \\textit{Hubble Space Telescope} we measure the astrometric deflection of the background source and obtain a gravitational mass for LAWD~37. The main challenge of this analysis is in extracting the lensing signal of the faint background source whilst it is buried in the wings of LAWD~37's point spread function. Removal of LAWD 37's point spread function induces a significant amount of correlated noise which we find can mimic the astrometric lensing signal. We find a deflection model including correlated noise caused by the removal of LAWD~37's point spread function best explains the data and yields a mass for LAWD 37 of \\(0.56\\pm0.08 M_{\\odot}\\). This mass is in agreement with the theoretical mass-radius relationship and cooling tracks expected for CO core white dwarfs. Furthermore, the mass is consistent with no or trace amounts of hydrogen that is expected for objects with helium-rich atmospheres like LAWD 37. We conclude that further astrometric followup data on the source is likely to improve the inference on LAWD 37's mass at the \\(\\approx3\\) percent level and definitively rule out purely correlated noise explanations of the data. This work provides the first semi-empirical test of the white dwarf mass-radius relationship using a single, isolated white dwarf and supports current model atmospheres of DQ white dwarfs and white dwarf evolutionary theory.

We present a general framework for matching the point-spread function (PSF), photometric scaling, and sky background between two images, a subject which is commonly referred to as difference image analysis (DIA). We introduce the new concept of a spatially varying photometric scale factor which will be important for DIA applied to wide-field imaging data in order to adapt to transparency and airmass variations across the field-of-view. Furthermore, we demonstrate how to separately control the degree of spatial variation of each kernel basis function, the photometric scale factor, and the differential sky background. We discuss the common choices for kernel basis functions within our framework, and we introduce the mixed-resolution delta basis functions to address the problem of the size of the least-squares problem to be solved when using delta basis functions. We validate and demonstrate our algorithm on simulated and real data. We also describe a number of useful optimisations that may be capitalised on during the construction of the least-squares matrix and which have not been reported previously. We pay special attention to presenting a clear notation for the DIA equations which are set out in a way that will hopefully encourage developers to tackle the implementation of DIA software.

We report on the first results from a large-scale observing campaign aiming to use astrometric microlensing to detect and place limits on the mass of single objects, including stellar remnants. We used the Hubble Space Telescope to monitor stars near the Galactic Center for 3 years, and we measured the brightness and positions of \\(\\sim\\)2 million stars at each observing epoch. In addition to this, we monitored the same pointings using the VIMOS imager on the Very Large Telescope. The stars we monitored include several bright microlensing events observed from the ground by the OGLE collaboration. In this paper, we present the analysis of our photometric and astrometric measurements for 6 of these events, and derive mass constraints for the lens in each of these. Although these constraints are limited by the photometric precision of ground-based data, and our ability to determine the lens distance, we were able to constrain the size of the Einstein ring radius thanks to our precise astrometric measurements, the first routine measurements of this type from a large-scale observing program. This demonstrates the power of astrometric microlensing as a tool to constrain the masses of stars, stellar remnants, and, in the future, of extrasolar planets, using precise ground- and space-based observations.

Gravitational deflection of starlight around the Sun during the 1919 total solar eclipse provided measurements that confirmed Einstein's general theory of relativity. We have used the Hubble Space Telescope to measure the analogous process of astrometric microlensing caused by a nearby star, the white dwarf Stein 2051 B. As Stein 2051 B passed closely in front of a background star, the background star's position was deflected. Measurement of this deflection at multiple epochs allowed us to determine the mass of Stein 2051 B -- the sixth nearest white dwarf to the Sun -- as 0.675 +/- 0.051 solar masses. This mass determination provides confirmation of the physics of degenerate matter and lends support to white dwarf evolutionary theory.