If A is an nxn complex matrix and λ is an eigenvalue of A with geometric multiplicity k, then λ is in at least k of the Geršgorin discs Di of A. Let k, r, t be positive integers with k ≤ r ≤ t. Then there is a txt complex matrix A and an eigenvalue λ of A such that λ has geometric multiplicity k and algebraic multiplicity t, and λ is in precisely r Geršgorin Discs of A. Some examples and related results are also provided.

If A is an nxn complex matrix and λ is an eigenvalue of A with geometric multiplicity k, then λ is in at least k of the Geršgorin discs Di of A. Let k, r, t be positive integers with k ≤ r ≤ t. Then there is a txt complex matrix A and an eigenvalue λ of A such that λ has geometric multiplicity k and algebraic multiplicity t, and λ is in precisely r Geršgorin Discs of A. Some examples and related results are also provided.

In this thesis I have studied how discs around young stars evolve and disperse. In particular, I build models which combine viscous evolution with photoevaporation, as previous work suggests that photoevaporation can reproduce the observed disc evolution and dispersal time-scales. The main question this thesis attempts to address is: Can photoevaporation provide a dominant dispersal mechanism for the observed population of young stars? Photoevaporation arises from the heating that high energy (UV and X-ray) photons provide to the surface layers of a disc. Before I started this work, only photoevaporation from a pure EUV radiation field was described within a hydrodynamic framework. Therefore, I start by building a hydrodynamic solution to the pure X-ray photoevaporation problem, and then extend this solution to the entire high energy spectrum. This hydrodynamic model leads me to conclude that it is the X-ray radiation field that sets the mass-loss rates. These mass-loss rates scale linearly with X-ray luminosity, are independent of the underlying disc structure and explicitly independent of stellar mass. I build a radiation-hydrodynamic algorithm, based on previous work, to describe the process of X-ray heating in discs. I then use this algorithm to span the full range of observed parameter space, to fully solve the X-ray photoevaporation problem. I further extend the algorithm to roughly approximate the heating an FUV radiation field would have on the photoevaporative flow, as well as separately testing the effect an EUV radiation field will have. These numerical tests are in agreement with the hydrodynamic model derived. Specifically, it is the X-rays that are driving the photoevaporative flow from the inner disc. Armed with an accurate description of the photoevaporative mass-loss rates from young stars, I consider the evolution of a population of disc-bearing, young (

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No description available.

The work performed here studies particle dynamics in local two-dimensional simulations of self-gravitating accretion discs with a simple cooling law. It is well known that the structure which arises in the gaseous component of the disc due to a gravitational instability can have a significant effect on the evolution of dust particles. Previous results using global simulations indicate that spiral density waves are highly efficient at collecting dust particles, creating significant local over-densities which may be able to undergo gravitational collapse. This thesis expand on these findings, using a range of cooling times to mimic the conditions at a large range of radii within the disc. The PENCIL Code is used to solve the 2D local shearing sheet equations for gas on a fixed grid together with the equations of motion for solids coupled to the gas solely through aerodynamic drag force. The work contained here shows that spiral density waves can create significant enhancements in the surface density of solids, equivalent to 1-10cm sized particles in a disc following the profiles of Clarke (2009) around a solar mass star, causing it to reach concentrations several orders of magnitude larger than the particles mean surface density. These findings suggest that the density waves that arise due to gravitational instabilities in the early stages of star formation provide excellent sites for the formation of large, planetesimal-sized objects. These results are expanded on, with subsequent results introducing the effects of the particles self-gravity showing these concentrations of particles can gravitationally collapse, forming bound structures in the solid component of the disc.

by Li Chong, Victor Kun Wa. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 99-101). / Abstracts in English and Chinese. / ABSTRACT --- p.i / 摘要 --- p.iii / ACKNOWLEDGEMENT --- p.v / TABLE OF CONTENTS --- p.vii / LIST OF FIGURES --- p.x / LIST OF TABLES --- p.xv / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- CD Manufacturing Technology --- p.1 / Chapter 1.1.1 --- Electroforming --- p.2 / Chapter 1.1.2 --- Computer Numerical Control (CNC) --- p.2 / Chapter 1.1.3 --- Photolithography --- p.4 / Chapter 1.1.4 --- Laser --- p.5 / Chapter 1.2 --- Research Objective --- p.6 / Chapter 1.3 --- Thesis Outline --- p.6 / Chapter 2 --- Conversion Software (AutoGEN) --- p.9 / Chapter 2.1 --- Computer-Aided Design --- p.9 / Chapter 2.2 --- AutoCAD Programming --- p.10 / Chapter 2.3 --- AutoCAD Development System (ADS) --- p.11 / Chapter 2.4 --- AutoCAD Runtime Extension (ARX) --- p.12 / Chapter 2.5 --- AutoLISP Programming --- p.12 / Chapter 2.5.1 --- Advantages of AutoLISP --- p.13 / Chapter 2.6 --- Caltech Intermediate Format (CIF) --- p.14 / Chapter 2.6.1 --- Structure of CIF Format --- p.15 / Chapter 2.7 --- Architecture of Conversion Software --- p.15 / Chapter 2.7.1 --- Stage 1 - AutoGEN (DLTM) Module --- p.16 / Chapter 2.7.2 --- Stage 2 - AutoGEN (DCRM) Module --- p.17 / Chapter 2.8 --- DLTM Input Screen --- p.17 / Chapter 2.9 --- DCRM Data Screen --- p.18 / Chapter 2.10 --- Conversion from 2D to 3D --- p.18 / Chapter 2.11 --- AutoGEN - Geometric Primitive --- p.19 / Chapter 2.12 --- AutoGEN - Geometric Transformation --- p.19 / Chapter 2.13 --- Conversion of Simplified and Complex Drawings --- p.22 / Chapter 3 --- Manufacturing Process --- p.24 / Chapter 3.1 --- Stamper Manufacturing --- p.25 / Chapter 3.2 --- CD Manufacturing --- p.30 / Chapter 3.3 --- Internal Stress of Deposit in Electroforming --- p.34 / Chapter 4 --- CNC Approach --- p.37 / Chapter 4.1 --- Computer-Aided Manufacturing --- p.37 / Chapter 4.2 --- CNC Machining --- p.37 / Chapter 4.2.1 --- Experiment --- p.39 / Chapter 4.3 --- Advantages of CNC Approach --- p.42 / Chapter 4.4 --- Limitations of CNC Approach --- p.42 / Chapter 4.5 --- CNC and Effects of Heat Generated --- p.43 / Chapter 5 --- Photolithography Approach --- p.45 / Chapter 5.1 --- Experiment --- p.47 / Chapter 5.2 --- Channel Analysis --- p.49 / Chapter 6 --- Laser Approach --- p.53 / Chapter 6.1 --- Dual Beam Laser Machine --- p.53 / Chapter 6.2 --- Creation of Pits and Lands --- p.54 / Chapter 6.2.1 --- Experiment --- p.54 / Chapter 6.3 --- Creation of Continuous Channel --- p.56 / Chapter 6.4 --- Procedure of Channel Creation (NA set at a fixed constant) --- p.57 / Chapter 6.4.1 --- Experiment 1 --- p.59 / Chapter 6.4.2 --- Experiment 2 --- p.60 / Chapter 6.4.3 --- Experiment 3 --- p.61 / Chapter 6.5 --- Procedure of Channel Creation (ILV set at a fixed constant) --- p.62 / Chapter 6.5.1 --- Experiment 1 --- p.63 / Chapter 6.5.2 --- Experiment 2 --- p.64 / Chapter 6.5.3 --- Experiment 3 --- p.66 / Chapter 7 --- Photolithography Approach (Enhancement) --- p.68 / Chapter 7.1 --- Creation of High-Aspect-Ratio Channel --- p.68 / Chapter 7.1.1 --- Experiment 1 --- p.76 / Chapter 7.1.2 --- Experiment 2 --- p.80 / Chapter 8 --- Conclusion and Future Proposal --- p.83 / Chapter 8.1 --- Conclusion --- p.83 / Chapter 8.2 --- Future Proposal --- p.86 / APPENDIX --- p.89 / Chapter A.1 --- Additional Information on CNC Approach --- p.88 / Chapter A.2 --- Channel Dimension of Design Mask --- p.89 / Chapter A.3 --- Additional Information on Photolithography Approach --- p.94 / Chapter A.4 --- Additional Information on Laser Approach --- p.95 / BIBLIOGRAPHY --- p.98

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

A rich variety of discs are found orbiting massive bodies in the universe. These could be accretion discs composed of gas around stellar mass compact objects fueling micro-quasar activity; protoplanetary discs, mainly composed of dust and gas, are the progenitors for planet formation; accretion discs composed of stars and gas around super-massive black holes at the centers of galaxies fueling the active galactic nuclei activity; discs in spiral galaxies; and many more. Structural and kinematic properties of these discs in several astrophysical systems are correlated to the global properties; for example, over a sample of thousands of galaxies, a correlation has been found between lopsidedness, black hole growth, and the presence of young stellar populations in the centers of galaxies. Galaxy formation and evolution of the central BH are some of the contexts in which such correlations become important. Studying the dynamics of these discs helps to explain their structural properties and is thus of paramount importance. In most astrophysical discs(a notable exception being the stellar discs in spiral galaxies),the dynamics are usually dominated by the gravity of the central object, and is thus nearly Keplerian. However, there is a small contribution to the total force experienced by the disc due to the disc material. Discs mentioned above differ from each other due to different underlying force that dominates the non-Keplerian dynamics of these discs. Two important numbers which are useful in describing physical properties of any disc structure in astrophysics are: (1) Mach number M , and(2) Toomre Q parameter. If thermal pressure gradient and/or random motion dominate the non-Keplerian forces, then M « Q, and in the case when the self-gravity of the disc is more important then Particles constituting the disc orbit under Keplerian potential due to central object, plus the small contribution from the non-Keplerian potential due to disc self-gravity, or the thermal pressure gradient. For a Keplerian potential, the radial and azimuthal frequencies are in 1 : 1 ratio w.r.t. each other and hence there is no precession in the orbits. In case of nearly Keplerian potential(when non-Keplerian contributions are small), the orbits precess at a rate proportional to the non-Keplerian forces. It is this non-zero but small precession that allows the existence of modes whose frequencies are proportional to the precession rate. These modes are referred to as slow modes (Tremaine 2001). Such modes are likely to be the only large-scale or long-wavelength modes. The damping they suffer due to viscosity, collisions, Landau damping, or other dissipative processes is also relatively less. Hence, these modes can dominate the overall appearance of discs. In this thesis we intend to study slow modes for nearly Keplerian discs. Slow modes innear-Kepleriandiscscantobethereasonforvariousnon-axisymmetricfeatures observed in many systems: 1 Galactic discs: Of the few galaxies for which the observations of galactic nuclei exist, two galaxies: NGC4486B(an elliptical galaxy) andM31(spiral galaxy), show an unusual double-peak distribution of stars at their centers. In order to explain such distributions, Tremaine in 1995 proposed an eccentric disc model for M31; this model was then further explored by many authors. In addition, lopsidedness is observed in many galaxies on larger scales, and such asymmetries need to be explained via robust modeling of galactic discs. 2 Debris disc: Many of the observed discs show non-axisymmetric structures, such as lopsided distribution in brightness of scattered light, warp, and clumps in the disc around β Pictoris; spiral structure inHD141569A,etc. Most of these features have been attributed to the presence of planets, and in some cases planets have also been detected. However, Jalali & Tremaine(2012)proposed that most of these structures can be formed also due to slow (m =1 or 2) modes. 3 Accretion Discs around stellar mass binaries have also been found to be asymmetric. One plausible reason for this asymmetry can be m =1slowmodes in these systems. Slow modes are studied in detail in this thesis. The main approaches that we have used, and the major conclusions from this work are as follows: Slow pressure modes in thin accretion disc Earlier work on slow modes assumed that the self-gravity of the disc dominates the pressure gradient in the discs. However, this assumption is not valid for thin and hot accretion discs around stellar mass compact objects. We begin our study of slow modes with the analysis of modes in thin accretion discs around stellar mass compact objects. First, the WKB analysis is used to prove the existence of these modes. Next, we formulate the eigenvalue equation for the slow modes, which turns out to be in the Sturm-Liouville form; thus all the eigenvalues are real. Real eigenvalues imply that the disc is stable to these perturbations. We also discuss the possible excitation mechanisms for these modes; for instance, excitation due to the stream of matter from the secondary star that feeds the accretion disc, or through the action of viscous forces. Slow modes in self-gravitating, zero-pressure fluid disc We next generalize the study of slow m = 1 modes for a single self-gravitating disc of Tremaine(2001) to a system of two self-gravitating counter–rotating, zero-pressure fluid discs, where the disc particles interact via softened-gravity. Counter– rotating streams of matter are susceptible to various instabilities. In particular, Touma(2002)found unstable modes in counter–rotating ,nearly Keplerian systems. These modes were calculated analytically for a two-ring system, and numerically for discs modeled assuming a multiple–ring system. Motivated by this, the corresponding problem for continuous discs was studied by Sridhar & Saini(2010),who proposed a simple model, with dynamics that could be studied largely analytically in the local WKB approximation. Their work, however, had certain limitations; they could construct eigenmodes only for η =0&12, where η is the mass fraction in the retrograde disc. They could only calculate eigenvalues but not the eigen functions. To overcome the above mentioned limitations, we formulate and analyze the full eigenvalue problem to understand the systematic behaviour of such systems. Our general conclusions are as follows 1 The system is stable for m = 1 perturbations in the case of no-counter rotation. 2. For other values of mass fraction , the eigenvalues are generally complex, and the discs are unstable. For η =12,theeigenvalues are imaginary, giving purely growing modes. 2 The pattern speed appears to be non-negative for all values of , with the growth(or damping) rate being larger for larger values of pattern speed. 3 Perturbed surface density profile is generally lopsided, with an overall rotation of the patterns as they evolve in time, with the pattern speed given by the real part of the eigenvalue. Local WKB analysis for Keplerian stellar disc We next urn to stellar discs, whose dynamics is richer than softened gravity discs. Jalali & Tremaine(2012)derived the dispersion relation for short wavelength slow modes for a single disc with Schwarzschild distribution function. In contrast to the softened gravity discs(which have slow modes only for m = 1), stellar discs permit slow modes for m 1. The dispersion relation derived by Jalali & Tremaine makes it evident that all m 1 slow modes are neutrally stable. We study slow modes for the case of two counter–rotating discs, each described by Schwarzschild distribution function, and derive the dispersion relation for slow m 1 modes in the local WKB limit and study the nature of the instabilities. One of the important applications of the dispersion relation derived in this chapter is the stability analysis of the modes. For fluid discs, it is well known that the stability of m = 0 modes guarantees the stability of higher m modes; and the stability criterion for such discs is the well known Toomre stability criterion. However, this is not the case for collisionless discs. Even if the discs are stable for axisymmetric modes, they can still be unstable for non-axisymmetric modes. The stability of axisymmetric modes is governed by the Toomre stability criterion The non-axisymmetric perturbations were found to be unstable if the mass in the retrograde component of the disc is non-zero. We next solve the dispersion relation using the Bohr-Sommerfeld quantization condition to obtain the eigen-spectrum for a given unperturbed surface density profile and velocity dispersion. We could obtain only the eigenvalues for no counter– rotation η = 0, where η is the mass fraction in the retrograde disc and equal counter–rotation(η =12). All the eigenvalues obtained were real for no counter– rotation, and purely growing/damping for equal counter–rotation. The eigenvalue trends that we get favour detection of high ω and low m modes observationally. We also make a detailed comparison between the eigenvalues for m = 1 modes that we obtain with those obtained after solving the integral eigenvalue problem for the softened gravity discs for no counter–rotation and equal counter–rotation. The match between the eigenvalues are quite good, confirming the assertion that softened gravity discs can be reasonable surrogates for collisionless disc for m =1 modes. Non-local WKB theory for eigenmodes One major limitation of the above method is that eigenfunctions cannot be obtained as directly as in quantum mechanics because the dispersion relation is transcendental in radial wave number . We overcome this difficulty by dropping the assumption of locality of the relationship between perturbed self-gravity and surface density. Using the standard WKB analysis and epicyclic theory, together with the logarithmic-spiral decomposition of surface density and gravitational potential, we formulate an integral equation for determining both WKB eigenvalues and eigenfunctions. The application of integral equation derived is not only restricted to Keplerian disc; it could be used to study eigenmodes in galactic discs where the motion of stars is not dominated by the potential due to a central black hole (however we have not pursued the potential application in this thesis). We first verify that the integral equation derived reduces to the well known WKB dispersion relation under the local approximation. We next specialize to slow modes in Keplerian discs. The following are some of the general conclusions of this work 1 We find that the integral equation for slow modes reduces to a symmetric eigenvalue problem, implying that the eigenvalues are all real, and hence the disc is stable. 2 All the non-singular eigenmodes we obtain are prograde, which implies that the density waves generated will have the same sense of rotation as the disc, albeit with a speed which is compared to the the rotation speed of the disc. 3 Eigenvalue ω decreases as we go from m =1 to 2. In addition, for a given , the number of nodes for m =1 are larger than those for m =2. 4 The fastest pattern speed is a decreasing function of the heat in the disc. Asymmetric features in various types of discs could be due to the presence of slow m =1 or 2 modes. In the case of debris discs, these asymmetric features could also be due to the presence of planets. Features due to the presence of slow modes or due to planets can be distinguished from each other if the observations are made for a long enough time. The double peak nucleus observed in galaxies like M31 and NGC4486B differ from each other: stellar distribution in NGC4486B is symmetric w.r.t. its photocenter in contrast to a lopsided distribution seen in M31. It is more likely that the double peak nucleus in NGC4486B is due to m = 2 mode, rather than m = 1 mode as is the case for M31. NGC4486B being an elliptical galaxy, it is possible that the excitation probability for m =2 mode is higher.

Disc material in the corotation region librates with respect to low-mass planets on horseshoe trajectories. The exchange of angular momentum associated with this libration gives rise to the non-linear corotation torque (the horseshoe drag). For the first project described herein, we ran a suite of high-resolution 2D hydrodynamic simulations of low-mass (5 Earth mass) planets, at eccentricities 0 e < 0.3, embedded in both viscous protoplanetary discs with entropy relaxation and inviscid discs without. The attenuation of the corotation torque was obtained from these simulations and found to be well-fitted by an exponential decay with a characteristic ‘e-folding eccentricity’ that scales linearly with disc scale height. These results were tested with different disc scale heights between 0.03 and 0.1 and with a 10 Earth mass planet. In the second project in this thesis we sought to extend on these results by examining the case of an embedded 5 Earth mass planet in three dimensional discs. We found that our scaling relation held in this new case, confirming that it is possible to use 2D simulations with a softening parameter to capture the behaviour of the corotation torque. We investigated the time-averaged horseshoe width as a function of altitude and found that the corotation region extends from the midplane to around three scale heights, changing most near the midplane for eccentric planets. The final project looked at 3D radiative discs, under the influence of stellar irradiation, with more massive embedded planets capable of triggering gap formation. We use the pluto code to simulate a Jupiter mass planet at 5 AU in a protoplanetary disc. We describe our progress in understanding the process of gap formation in a case study of this class of hitherto undescribed disc.