The proximity of the Galactic center has allowed for the detailed study of the environs of a supermassive black hole (SMBH). While the region is inhabited predominantly by old, late-type giants, there exists a population of ∼200 young (6 ± 2 Myr), massive (10-100 msun) stars within the central parsec. Their presence here is puzzling since the standard mode of star formation cannot proceed in the face of the strong tidal forces from the SMBH. Given their youth, the dynamics of these stars can be used to understand their origin and lend insight into star formation processes in the hostile environment surrounding a supermassive black hole. In this thesis, I use high resolution infrared imaging from the W. M. Keck telescopes in order to determine precise orbital parameter estimates of the young stars and understand the Galactic center's most recent epoch of star formation.

First, we present a new optical distortion model for the Keck/NIRC2 narrow camera that is based upon on-sky measurements of a globular cluster. With an improved distortion model, we show that a stable astrometric reference frame for the GC can be established with Sgr A* at rest to within 0.09 mas/yr (∼3.4 km/s at a distance of 8 kpc), thereby improving the stability of the reference frame. Accurate proper motions of the central stellar cluster are presented and the stars are shown to have significant net rotation parallel to Galactic rotation. These stars can be used as astrometric standards for defining a reference frame without requiring the assumption of no net motion of the central stellar cluster, as has been done in earlier proper motion studies.

Second, we use high-precision astrometry, measured in the newly constructed reference frame, and radial velocities of ∼115 young stars at projected radii between R = 0.8" - 13.3" in order to estimate their orbital properties. This constitutes the largest sample of stars used for this type of study to date. The median proper motion uncertainty for the stars within R=6", for which we have up to a 16-year baseline of measurements, is 0.03 mas/yr (∼1.2 km/s). Acceleration uncertainties are typically 10 microarcsec/yr/yr (∼0.4 km/s/yr), which has allowed for the detection of six significant accelerations in the plane of the sky outside the central arcsecond. Such measurements provide direct calculations of the line of sight distance and therefore precise orbital parameter estimates. We detect the clockwise stellar disk reported in previous studies, but find that the fraction of young stars within the disk is much smaller than once thought. We do not find evidence for the previously-claimed counter-rotating disk. The clockwise disk has an inclination of ∼130° and an angle to the ascending node of ∼96°, with an opening angle of 15.2°. The orientation of the disk plane does not change with radius, contrary to recent claims of a highly-warped disk. We identify a bias in the orbital solutions of disk stars near the line of nodes that stems from a previously adopted line-of-sight distance prior and show that this bias leads to an apparently-warped disk. The candidate disk members have orbital eccentricities of e ∼ 0.3. This can be explained by dynamical relaxation in an initially circular disk with a moderately top-heavy mass function (Gamma ∼ 1.6), consistent with the latest estimates of the young star population's IMF. This cannot, however, account for the high inclinations of the out-of-disk population, which makes up at least half of the central parsec's young stars. Thus, if all of the young stars formed in a single disk, an additional dynamical mechanism must be invoked to explain their orbits.

Stellar orbits around the supermassive black hole (SMBH) at Galactic center (GC) provide unique insight into the physics and astrophysics of SMBHs and how they interact with their host galaxies. In this dissertation, I use stellar orbits at the GC to directly measure the distribution of extended mass around the Milky Way’s central SMBH. Dynamical theories predict that there should be a steep increase in mass density, or a cusp, towards the center of relaxed stellar populations around supermassive black holes. While the theory of stellar cusp formation is well motivated by physics and therefore widely adopted, it has never been observationally verified in the direct vicinity (within 0.5 parsecs) of a supermassive black hole. Observational verification is important because the assumption that cusp models are correct underlies a wide range of calculations that impact understandings of astrophysical processes such as galaxy evolution, the growth of and dynamics around SMBHs, and the origins and rates of gravitational wave emission. The GC provides a unique laboratory to test cusp theories as it hosts stellar populations old enough to be dynamically relaxed, and is the only galactic nucleus close enough to measure the stellar density distribution of the innermost region around an SMBH. This work uses multi-star General Relativistic orbit fits to probe the extended mass within the central 0.01 pc around the SMBH at the GC. Assuming a canonical Bahcall-Wolf cusp with a power-law index of 1.5, measurements of orbital precession of the 16-year-period star S0-2 suggest ∼18,000 � 12,000 � 6,000 M⊙ of extended mass within the central 0.01 pc of the Galaxy. This measurement from S0-2 alone is consistent with predictions from dynamical theories and allows for the possibility of a dark cusp, though still allows for the possibility of a missing cusp (no extended mass) at the 2σ-level. One of the key advancements of this work is the ability to move beyond fits to S0-2 alone and simultaneously constrain multiple stars with a relativistic orbit model. Added constraints from additional stars make it possible to leave the power-law index of the density profile as a free parameter. Multi-star relativistic orbit fits with the power-law index left as a free parameter suggest evidence for a non-zero extended mass within the central 0.01 pc of the GC that follows a core-like density profile rather than a cusp-like density profile. The inferred value of extended mass is ∼60,000 � 20,000 � 6,000M⊙, following a density profile with a 1σ upper limit of γ=0.46 (2σ<0.76, 3σ<1.66)—consistent with the core-like profile inferred from observations of late-type stars in the GC. Updated values for the mass of and distance to the SMBH at the GC, which are fundamental parameters for modeling the central SMBH, are also presented: MBH = (4.18 � 0.03 � 0.01) � 10^6 M⊙ and R0= (8.16 � 0.04 � 0.01) kpc.

This thesis presents Adaptive Optics (AO) imaging and spectroscopic measurements of infrared excess sources in the Galactic Center to determine if they are similar to G2, the first example of a spatially-resolved object interacting with the supermassive black hole at the center of the Galaxy, Sgr A$^{*}$. Our goal is to understand the debated nature of these infrared excess sources and to understand their relationship with the supermassive black hole. Our objects have been monitored with AO from he past decade (2004 - 2015) and are within the inner 1\farcs75 of the supermassive black hole.

We initially focus on one source, G1, which gets comparably close to the supermassive black hole compared to G2 ($a_{min}\sim200-300$ AU) and lies on a very eccentric orbit ($e_{G1} = 0.99$). While G2 has been tracked before and during periapse passage ($T_{0} \sim$ 2014.2), G1 has been followed since soon after emerging from periapse ($T_{0} \sim$ 2001.3). Our observations of G1 double the previously reported observational time baseline, which improves its orbital parameter determinations. G1's orbital trajectory appears to be in the same plane as that of G2, but with a significantly different argument of periapse ($\Delta\omega$ = 21$\pm$4 degrees). This suggests that G1 is an independent object and not part of a gas stream containing G2 as has been proposed. Furthermore, we show for the first time that: (1) G1 is extended in the early epochs (those closest to periapse) along the direction of orbital motion and (2) G1 becomes significantly smaller over time, changing from 450 AU in 2004 to less than 170 AU in 2009 and thereafter. Based on these observations, G1 appears to be the second example of an object tidally interacting with a supermassive black hole. G1's continued survival 14 years after periapse, along with its compactness in epochs further from the time of periapse, suggests that this source is stellar in nature and may be consistent with a black-hole driven stellar binary merger induced by the Kozai mechanism.

We then observationally characterize several other infrared excess sources to see whether or not they share similar properties. We find another source, G3, has nearly identical observation properties to G1 and G2 but lies further away from Sgr A${*}$. G3 does not lie on the same orbital plane to G1 or G2 at all, suggesting that these objects do not necessarily all come from the same region. G3 does not lie on a highly eccentric orbit at all ($e\sim0.4$) and its periapse passage distance is significantly further away than G1 or G2 ($a_{min}\sim5000$ AU). Several other sources also exhibit Br-$\gamma$ emission, but also do not lie on the same orbital plane as G1 or G2.

G3 and other infrared excess sources supports the hypothesis that there is a population of objects that share similar observational qualities to G1 and G2 -- high infrared luminosities, cold dust temperatures, and Br-$\gamma$ emission line features. These new objects might therefore exist as a new population of sources in the Galactic center, and might not necessarily have the same physical manifestation.

The proximity of the Milky Way Galactic center (GC) allows a unique perspective into how supermassive black holes (SMBHs) affect their surrounding stellar population. Several uncertainties still remain about the stellar population at the GC, notably the frequency of binary star systems at the GC and the distribution of dark stellar remnants surrounding the SMBH. In this thesis, we used over 13 years of Keck Observatory adaptive optics (AO) photometric flux measurements of the stellar population within approximately a half parsec of the GC SMBH in an effort to address the uncertainties about the GC stellar binary population and to probe the density of unseen stars or compact objects near the SMBH.

First, we present an ≈ 11.5 year adaptive optics (AO) study of stellar variability and search for eclipsing binaries in the central ∼ 0.4 pc (∼ 10′′) of the Milky Way nuclear star cluster. In the study, we measured the photometry of 563 stars using the Keck II NIRC2 imager (K′-band, λ0 = 2.124 μm). We achieved a photometric uncertainty floor of ∆mK′ ∼ 0.03 (≈ 3%), comparable to the highest precision achieved in other AO studies. Approximately half of our sample (50 � 2%) shows variability. 52 � 5% of known early-type young stars and 43 � 4% of known late-type giants are variable. These variability fractions are higher than those of other young, massive star populations or late-type giants in globular clusters, and can be largely explained by two factors. First, our experiment time baseline is sensitive to long-term intrinsic stellar variability. Second, the proper motion of stars behind spatial inhomogeneities in the foreground extinction screen can lead to variability. We recovered the two known Galactic center eclipsing binary systems: IRS 16SW and S4-258 (E60). We constrained the Galactic center eclipsing binary fraction of known early-type stars to be at least 2.4 � 1.7%. We found no evidence of an eclipsing binary among the young S-stars nor among the young stellar disk members. These results are consistent with the local OB eclipsing binary fraction. We additionally identified a new periodic variable at the GC, S2-36, with a 39.43 day photometric period.

Second, we present the discovery of the first known eclipsing, ellipsoidal binary star system (S2-36) at the GC, with a binary orbital period of ≈ 78.8 days at a projected distance of ≈ 2 arcsec from Sgr A*. As the first dynamically soft stellar binary discovered in the region, S2-36 offers powerful constraints on the dark cusp density within ∼ 0.1 pc of the SMBH. We used new and existing near-infrared adaptive optics Keck Observatory imaging data collected over 13 years to measure the photometric and astrometric properties of S2-36. S2-36’s proper motion suggests a distance < ≈ 0.2 pc from the SMBH. A trended multi-band periodicity search of the photometric measurements in the K′- (λ0 = 2.124 μm) and H-band −4 (λ0 = 1.633 μm) reveals a periodic signal with false-alarm probability of less than 10 . These results, combined with our flux and color measurements, suggest S2-36 is an ellipsoidal binary at the GC. Best-fit astrophysical binary models to the observed light curve consist of two red giant stars where the photometric variability originates from ellipsoidal variation and eclipses. However, with photometry measurements alone, our model fits have degeneracies in age and metallicity of the component stars (age ≈ 3 to 13.5 Gyr, [Fe/H] ≈ −1.5 to +0.5). The presence of an old, dynamically soft binary at the GC over its entire lifetime implies few encounters with sources that would disrupt the binary. Under the assumption that the binary has spent its lifetime in the GC environment, the binary’s survival places an upper limit of ∼ 104 stellar-mass black holes (M ∼ 10M⊙) within ≈ 0.1 pc of the SMBH, and a local two-body relaxation time < ≈ 5 � 10^8 yr. Our observations are consistent with X-ray estimates of compact objects in the GC and provide a powerful new constraint on the presence of the dark cusp at the GC.

Finally, we present new methodology and initial results for a photometric study to constrain the stellar binary fraction of the GC. With 873 stars, a larger stellar sample than previously ever used for a photometric binary search of GC stars, we detected a total of five likely periodic signals with our updated methods; two of the likely stars (S4-308 and S3-438) have not been previously identified as periodic variables in any previous study. We then provide an overview of the calculation of our experiment’s sensitivity to photometric binaries and detail a Bayesian framework to combine the sensitivity and our experiment’s detections to constrain the intrinsic GC stellar binary fraction. The methods outlined can be employed on future photometric studies of the GC stellar population to constrain its binary fraction, particularly for short-period binaries.

This thesis performs unprecedented investigations for spectroscopic binaries at the Galactic center and utilizes the development and extended observing baseline of integral field spectroscopy behind adaptive optics. The second chapter of this thesis describes the first direct search for binarity of its kind for the star S0-2, the most well-studied star of the S-star cluster. Almost two decades of spectroscopic with adaptive optics has led to the capability to look for a binary signal in S0-2's spectroscopic data. We introduce a methodology for investigating periodicity and placing limits on binary parameters with radial velocity data. We do not detect any signs of binarity for S0-2 and place limits on a hypothetical companion mass for S0-2 to be 1.6 solar masses, which is below current detection limits. We also investigate and find that a feasible spectroscopic binary system would not bias the detection of relativistic redshift, a post-Newtonian spectroscopic measurement.

The third chapter of this thesis explains the improvements in spectral analysis that lead to greater sensitivity to spectroscopic binary systems. We use a new method for measuring radial velocity measurements of stars at the Galactic center. This new method utilizes spectral fitting software and a new infrared spectral grid that finally covers effective temperatures of 30,000 K, putting the young, massive S-stars at the Galactic center within reach. We improve the precision of our radial velocities by a factor of 1.7. We also test our methodology on radial velocity standard stars and find that the new method eliminates the slight bias of our previous method. These characteristics of this method make us more sensitive to detecting the radial velocity variations of spectroscopic binary stars.

The fourth chapter presents a systematic search for spectroscopic binaries at the Galactic Center. With over two decades of integral field spectroscopy data, advanced tools for fitting mid-infrared stellar spectra, and 1000 radial velocity data points, we conduct a spectroscopic binary search of 29 stars at the Galactic center. After subtracting a star's motion around the supermassive black hole, we search for a periodic signal using a Lomb-Scargle analysis and fitting the residual radial velocity curve to a binary system radial velocity curve. We find no significant periodic detections in our sample, suggesting there are no binaries among the S-stars. We also place limits on the hypothetical companion masses of these S-stars. We also place a limit on the intrinsic binary fraction of these stars at 42%, which disagrees with the binary fraction for massive field stars. These results favor S-star formation mechanisms that result in a low binary fraction.

The final chapter presents new integral field spectroscopic observations of stars in the nuclear star cluster. We present data taken over 6 years and add the spectral types of 169 stars: 47 of which are Wolf-Rayet and O/B stars, and the remaining 122 are K and M giants. This addition to the rich Galactic center integral field spectroscopy data set provides opportunities to further study stellar demographics and dynamics of the nuclear star cluster.

As the closest galactic nucleus, the Galactic center provides a unique opportunity for learning about the supermassive black hole at the center of the Milky Way, Sgr A*, and its environment. High-angular-resolution observations in the near-infrared further this advantage. This thesis presents an investigation of the long-term near-infrared accretion properties of Sgr A*, and characterizes the star formation history in this extreme environment. This work utilizes the unique data gathered by the UCLA Galactic Center Group over 25 years to answer fundamental questions about the supermassive black hole and the formation of stars around it.

First, we investigate the near-infrared variability of Sgr A* with the longest time baseline yet considered. The recently improved speckle holography technique has led to the capability to detect Sgr A* in the early years of the group's monitoring program. We carry out a new analysis of speckle imaging data (1995 - 2005) obtained from the Keck observatory, enabling new detections of Sgr A* in the near-infrared over a decade that was previously inaccessible at these wavelengths. We report that the near-infrared brightness of Sgr A* and its variability are consistent over 22 years, which addresses variability timescales that are 10 times longer than earlier published studies. The power spectral density of Sgr A* shows a plateau between $\sim$80 days and 7 years, further confirming that it is uncorrelated in time beyond the previously-proposed, single power-law break of $\sim$245 minutes. We also investigate and note that the closest approach of the extended and dusty object G1, experiencing tidal disruption, had no apparent effect on the near-infrared emission from the accretion flow onto Sgr A*.

Second, we report the first star-formation history study of the Milky Way's nuclear star cluster (NSC) which includes observational constraints from a large sample of stellar metallicity measurements from Gemini and VLT. Along with stellar photometry and spectroscopically derived stellar temperatures, a Bayesian inference approach is developed to derive the NSC's star formation history. By including metallicity measurements, the low-temperature red giants that were previously difficult to constraint are now accounted for, and the best fit favors a two-component model. The dominant ($\sim$93\% of the mass), metal-rich component has an age of 5$^{+3}_{-2}$ Gyr, which is likely $\sim$3 Gyr younger than earlier studies with fixed (solar) metallicity; this younger age challenges the co-evolution models of the supermassive black hole, the NSC and the bulge. The minor component ($\sim$7\%) is metal-poor and its age is uncertain though likely less than that of the dominant component. We present updated estimates of the number of compact objects at the Galactic center and their rates of mergers in order to inform interpretations of gravitational-wave detections. Of particular note, when metallicity measurements are included, the predictions result in 2 - 4 times fewer neutron stars compared to earlier predictions that assume solar metallicity, introducing a possible new path to understand the so-called ``missing pulsar problem" at the Galactic center.