Dynamical stability of nascent neutron stars

This thesis presents a study of the dynamical stability of nascent neutron stars resulting from the accretion induced collapse of rapidly rotating white dwarfs.

Chapter 2 and part of Chapter 3 study the equilibrium models for these neutron stars. They are constructed by assuming that the neutron stars have the same masses, angular momenta, and specific angular momentum distributions as the pre-collapse white dwarfs. If the pre-collapse white dwarf is rapidly rotating, the collapsed object will contain a high density central core of size about 20 km, surrounded by a massive accretion torus extending to hundreds of kilometers from the rotation axis. The ratio of the rotational kinetic energy to gravitational binding energy, β, of these neutron stars is all found to be less than 0.27.

Chapter 3 studies the dynamical stability of these neutron stars by numerically evolving the linearized hydrodynamical equations. A dynamical bar-mode instability is observed when the β of the star is greater than the critical value β_d ≈ 0.25. It is expected that the unstable mode will persist until a substantial amount of angular momentum is carried away by gravitational radiation. The detectability of these sources is studied and it is estimated that LIGO II is unlikely to detect them unless the event rate is greater than 10^-6/year/galaxy.

All the calculations on the structure and stability of the neutron stars in Chapters 2 and 3 are carried out using Newtonian hydrodynamics and gravity. Chapter 4 studies the relativistic effects on the structure of these neutron stars. New techniques are developed and used to construct neutron star models to the first post-Newtonian (1PN) order. The structures of the 1PN models are qualitatively similar to the corresponding Newtonian models, but the values of β are somewhat smaller. The maximum β for these 1PN neutron stars is found to be 0.24, which is 8% smaller than the Newtonian result (0.26). However, relativistic effects will also change the critical value β_d. A detailed post-Newtonian stability analysis has yet to be carried out to study the relativistic effects on the dynamical stability of these neutron stars.

A probabilistic treatment of uncertainty in nonlinear dynamical systems

In this work, computationally efficient approximate methods are developed for analyzing uncertain dynamical systems. Uncertainties in both the excitation and the modeling are considered and examples are presented illustrating the accuracy of the proposed approximations.

For nonlinear systems under uncertain excitation, methods are developed to approximate the stationary probability density function and statistical quantities of interest. The methods are based on approximating solutions to the Fokker-Planck equation for the system and differ from traditional methods in which approximate solutions to stochastic differential equations are found. The new methods require little computational effort and examples are presented for which the accuracy of the proposed approximations compare favorably to results obtained by existing methods. The most significant improvements are made in approximating quantities related to the extreme values of the response, such as expected outcrossing rates, which are crucial for evaluating the reliability of the system.

Laplace's method of asymptotic approximation is applied to approximate the probability integrals which arise when analyzing systems with modeling uncertainty. The asymptotic approximation reduces the problem of evaluating a multidimensional integral to solving a minimization problem and the results become asymptotically exact as the uncertainty in the modeling goes to zero. The method is found to provide good approximations for the moments and outcrossing rates for systems with uncertain parameters under stochastic excitation, even when there is a large amount of uncertainty in the parameters. The method is also applied to classical reliability integrals, providing approximations in both the transformed (independently, normally distributed) variables and the original variables. In the transformed variables, the asymptotic approximation yields a very simple formula for approximating the value of SORM integrals. In many cases, it may be computationally expensive to transform the variables, and an approximation is also developed in the original variables. Examples are presented illustrating the accuracy of the approximations and results are compared with existing approximations.

Scheme for unconventional geometric quantum computation in cavity QED

In this paper, we present a scheme for implementing the unconventional geometric two-qubit phase gate with nonzero dynamical phase based on two-channel Raman interaction of two atoms in a cavity. We show that the dynamical phase and the total phase for a cyclic evolution are proportional to the geometric phase in the same cyclic evolution; hence they possess the same geometric features as does the geometric phase. In our scheme, the atomic excited state is adiabatically eliminated, and the operation of the proposed logic gate involves only the metastable states of the atoms; thus the effect of the atomic spontaneous emission can be neglected. The influence of the cavity decay on our scheme is examined. It is found that the relations regarding the dynamical phase, the total phase, and the geometric phase in the ideal situation are still valid in the case of weak cavity decay. Feasibility and the effect of the phase fluctuations of the driving laser fields are also discussed.

Polarization evolution of a few-cycle ultrashort laser pulse propagating in a degenerate three-level medium

The propagation of an arbitrary polarized few-cycle ultrashort laser pulse in a degenerate three-level medium is investigated by using an iterative predictor-corrector finite-difference time-domain method. It is found that the polarization evolution of the ultrashort laser pulse is dependent not only on the initial atomic coherence of the medium but also on the polarization condition of the incident laser pulse. When the initial effective area is equal to 2 pi, complete linear-to-circular and circular-to-linear polarization conversion of few-cycle ultrashort laser pulses can be achieved due to the quantum interference effects between the two different transition paths.

Understanding planet formation through high precision photometry and spectroscopy

From studies of protoplanetary disks to extrasolar planets and planetary debris, we aim to understand the full evolution of a planetary system. Observational constraints from ground- and space-based instrumentation allows us to measure the properties of objects near and far and are central to developing this understanding. We present here three observational campaigns that, when combined with theoretical models, reveal characteristics of different stages and remnants of planet formation. The Kuiper Belt provides evidence of chemical and dynamical activity that reveals clues to its primordial environment and subsequent evolution. Large samples of this population can only be assembled at optical wavelengths, with thermal measurements at infrared and sub-mm wavelengths currently available for only the largest and closest bodies. We measure the size and shape of one particular object precisely here, in hopes of better understanding its unique dynamical history and layered composition.

Molecular organic chemistry is one of the most fundamental and widespread facets of the universe, and plays a key role in planet formation. A host of carbon-containing molecules vibrationally emit in the near-infrared when excited by warm gas, T~1000 K. The NIRSPEC instrument at the W.M. Keck Observatory is uniquely configured to study large ranges of this wavelength region at high spectral resolution. Using this facility we present studies of warm CO gas in protoplanetary disks, with a new code for precise excitation modeling. A parameterized suite of models demonstrates the abilities of the code and matches observational constraints such as line strength and shape. We use the models to probe various disk parameters as well, which are easily extensible to others with known disk emission spectra such as water, carbon dioxide, acetylene, and hydrogen cyanide.

Lastly, the existence of molecules in extrasolar planets can also be studied with NIRSPEC and reveals a great deal about the evolution of the protoplanetary gas. The species we observe in protoplanetary disks are also often present in exoplanet atmospheres, and are abundant in Earth's atmosphere as well. Thus, a sophisticated telluric removal code is necessary to analyze these high dynamic range, high-resolution spectra. We present observations of a hot Jupiter, revealing water in its atmosphere and demonstrating a new technique for exoplanet mass determination and atmospheric characterization. We will also be applying this atmospheric removal code to the aforementioned disk observations, to improve our data analysis and probe less abundant species. Guiding models using observations is the only way to develop an accurate understanding of the timescales and processes involved. The futures of the modeling and of the observations are bright, and the end goal of realizing a unified model of planet formation will require both theory and data, from a diverse collection of sources.

Stimulated Raman adiabatic passage from atomic to molecular Bose-Einstein condensates: Feedback laser-detuning control and suppression of dynamical instability

We present a feedback control scheme that designs time-dependent laser-detuning frequency to suppress possible dynamical instability in coupled free-quasibound-bound atom-molecule condensate systems. The proposed adaptive frequency chirp with feedback is shown to be highly robust and very efficient in the passage from an atomic to a stable molecular Bose-Einstein condensate.

Valley evolution by meandering rivers

Fluvial systems form landscapes and sedimentary deposits with a rich hierarchy of structures that extend from grain- to valley scale. Large-scale pattern formation in fluvial systems is commonly attributed to forcing by external factors, including climate change, tectonic uplift, and sea-level change. Yet over geologic timescales, rivers may also develop large-scale erosional and depositional patterns that do not bear on environmental history. This dissertation uses a combination of numerical modeling and topographic analysis to identify and quantify patterns in river valleys that form as a consequence of river meandering alone, under constant external forcing. Chapter 2 identifies a numerical artifact in existing, grid-based models that represent the co-evolution of river channel migration and bank strength over geologic timescales. A new, vector-based technique for bank-material tracking is shown to improve predictions for the evolution of meander belts, floodplains, sedimentary deposits formed by aggrading channels, and bedrock river valleys, particularly when spatial contrasts in bank strength are strong. Chapters 3 and 4 apply this numerical technique to establishing valley topography formed by a vertically incising, meandering river subject to constant external forcing—which should serve as the null hypothesis for valley evolution. In Chapter 3, this scenario is shown to explain a variety of common bedrock river valley types and smaller-scale features within them—including entrenched channels, long-wavelength, arcuate scars in valley walls, and bedrock-cored river terraces. Chapter 4 describes the age and geometric statistics of river terraces formed by meandering with constant external forcing, and compares them to terraces in natural river valleys. The frequency of intrinsic terrace formation by meandering is shown to reflect a characteristic relief-generation timescale, and terrace length is identified as a key criterion for distinguishing these terraces from terraces formed by externally forced pulses of vertical incision. In a separate study, Chapter 5 utilizes image and topographic data from the Mars Reconnaissance Orbiter to quantitatively identify spatial structures in the polar layered deposits of Mars, and identifies sequences of beds, consistently 1-2 meters thick, that have accumulated hundreds of kilometers apart in the north polar layered deposits.

Recent research in the African Great Lakes: fisheries, biodiversity and cichlid evolution

The East African Great Lakes are now well known for (1) their fisheries, of vital importance for their rapidly rising riparian human populations, and (2) as biodiversity hotspots with spectacular endemic faunas, of which the flocks of cichlid fishes unique to each of the three largest lakes, Tanganyika, Malawi and Victoria, offer unique opportunities to investigate how new species evolve and coexist. Since the early 1990s research involving over a hundred scientists, financed by many international bodies, has produced numerous reports and publications in widely scattered journals. This article summarizes their main discoveries and examines the status of, and prospects for, the fisheries, as well as current ideas on how their rich endemic fish faunas have evolved. It first considers fisheries projects in each of the three lakes: the deep rift valley lakes Tanganyika and Malawi and the huge Victoria, all of which share their waters between several East African countries. Secondly it considers the biodiversity surveys of each lake, based on underwater (SCUBA) observations of fish ecology and behaviour which have revealed threats to their fish faunas, and considers what conservation measures are needed. Thirdly, using the lakes as laboratories, what have the international investigations (including DNA techniques and follow-up aquarium experiments) now revealed about the origins and relationships of their cichlid species flocks and mechanisms of evolution?

Study of dynamical behaviour and fermionization of a bosonic gas in funnel potential

This paper presents a funnel external potential model to investigate dynamic properties of ultracold Bose gas. By using variational method, we obtain the ground-state energy and density properties of ultracold Bose atoms. The results show that the ultracold Bose gas confined in a funnel potential experiences the transition from three-dimensional regime to quasi-one-dimensional regime in a small aspect ratio, and undergoes fermionization process as the aspect ratio increases.

Multipole moments in general relativity and dynamical perturbations of black-hole magnetospheres

This thesis consists of two parts. In Part I, we develop a multipole moment formalism in general relativity and use it to analyze the motion and precession of compact bodies. More specifically, the generic, vacuum, dynamical gravitational field of the exterior universe in the vicinity of a freely moving body is expanded in positive powers of the distance r away from the body's spatial origin (i.e., in the distance r from its timelike-geodesic world line). The expansion coefficients, called "external multipole moments,'' are defined covariantly in terms of the Riemann curvature tensor and its spatial derivatives evaluated on the body's central world line. In a carefully chosen class of de Donder coordinates, the expansion of the external field involves only integral powers of r ; no logarithmic terms occur. The expansion is used to derive higher-order corrections to previously known laws of motion and precession for black holes and other bodies. The resulting laws of motion and precession are expressed in terms of couplings of the time derivatives of the body's quadrupole and octopole moments to the external moments, i.e., to the external curvature and its gradient.

In part II, we study the interaction of magnetohydrodynamic (MHD) waves in a black-hole magnetosphere with the "dragging of inertial frames" effect of the hole's rotation - i.e., with the hole's "gravitomagnetic field." More specifically: we first rewrite the laws of perfect general relativistic magnetohydrodynamics (GRMHD) in 3+1 language in a general spacetime, in terms of quantities (magnetic field, flow velocity, ...) that would be measured by the ''fiducial observers” whose world lines are orthogonal to (arbitrarily chosen) hypersurfaces of constant time. We then specialize to a stationary spacetime and MHD flow with one arbitrary spatial symmetry (e.g., the stationary magnetosphere of a Kerr black hole); and for this spacetime we reduce the GRMHD equations to a set of algebraic equations. The general features of the resulting stationary, symmetric GRMHD magnetospheric solutions are discussed, including the Blandford-Znajek effect in which the gravitomagnetic field interacts with the magnetosphere to produce an outflowing jet. Then in a specific model spacetime with two spatial symmetries, which captures the key features of the Kerr geometry, we derive the GRMHD equations which govern weak, linealized perturbations of a stationary magnetosphere with outflowing jet. These perturbation equations are then Fourier analyzed in time t and in the symmetry coordinate x, and subsequently solved numerically. The numerical solutions describe the interaction of MHD waves with the gravitomagnetic field. It is found that, among other features, when an oscillatory external force is applied to the region of the magnetosphere where plasma (e⁺e^-) is being created, the magnetosphere responds especially strongly at a particular, resonant, driving frequency. The resonant frequency is that for which the perturbations appear to be stationary (time independent) in the common rest frame of the freshly created plasma and the rotating magnetic field lines. The magnetosphere of a rotating black hole, when buffeted by nonaxisymmetric magnetic fields anchored in a surrounding accretion disk, might exhibit an analogous resonance. If so then the hole's outflowing jet might be modulated at resonant frequencies ω=(m/2) Ω_H where m is an integer and Ω_H is the hole's angular velocity.

Manganese: Minerals, Microbes, and the Evolution of Oxygenic Photosynthesis

Oxygenic photosynthesis fundamentally transformed our planet by releasing molecular oxygen and altering major biogeochemical cycles, and this exceptional metabolism relies on a redox-active cubane cluster of four manganese atoms. Not only is manganese essential for producing oxygen, but manganese is also only oxidized by oxygen and oxygen-derived species. Thus the history of manganese oxidation provides a valuable perspective on our planet’s environmental past, the ancient availability of oxygen, and the evolution of oxygenic photosynthesis. Broadly, the general trends of the geologic record of manganese deposition is a chronicle of ancient manganese oxidation: manganese is introduced into the fluid Earth as Mn(II) and it will remain only a trace component in sedimentary rocks until it is oxidized, forming Mn(III,IV) insoluble precipitates that are concentrated in the rock record. Because these manganese oxides are highly favorable electron acceptors, they often undergo reduction in sediments through anaerobic respiration and abiotic reaction pathways.

The following dissertation presents five chapters investigating manganese cycling both by examining ancient examples of manganese enrichments in the geologic record and exploring the mineralogical products of various pathways of manganese oxide reduction that may occur in sediments. The first chapter explores the mineralogical record of manganese and reports abundant manganese reduction recorded in six representative manganese-enriched sedimentary sequences. This is followed by a second chapter that further analyzes the earliest significant manganese deposit 2.4 billon years ago, and determines that it predated the origin of oxygenic photosynthesis and thus is supporting evidence for manganese-oxidizing photosynthesis as an evolutionary precursor prior to oxygenic photosynthesis. The lack of oxygen during this early manganese deposition was partially established using oxygen-sensitive detrital grains, and so a third chapter delves into what these grains mean for oxygen constraints using a mathematical model. The fourth chapter returns to processes affecting manganese post-deposition, and explores the relationships between manganese mineral products and (bio)geochemical reduction processes to understand how various manganese minerals can reveal ancient environmental conditions and biological metabolisms. Finally, a fifth chapter considers whether manganese can be mobilized and enriched in sedimentary rocks and determines that manganese was concentrated secondarily in a 2.5 billion-year-old example from South Africa. Overall, this thesis demonstrates how microbial processes, namely photosynthesis and metal oxide-reducing metabolisms, are linked to and recorded in the rich complexity of the manganese mineralogical record.