Planetary Stability Zones in Hierarchical Triple Star Systems

A symplectic integrator algorithm suitable for hierarchical triple systems is formulated and tested. The positions of the stars are followed in hierarchical Jacobi coordinates, whilst the planets are referenced purely to their primary. The algorithm is fast, accurate and easily generalised to incorporate collisions. There are five distinct cases -- circumtriple orbits, circumbinary orbits and circumstellar orbits around each of the stars in the hierarchical triple -- which require a different formulation of the symplectic integration algorithm. As an application, a survey of the stability zones for planets in hierarchical triples is presented, with the case of a single planet orbiting the inner binary considered in detail. Fits to the inner and outer edges of the stability zone are computed. Considering the hierarchical triple as two decoupled binary systems, the earlier work of Holman & Wiegert on binaries is shown to be applicable to triples, except in the cases of high eccentricities and close or massive stars. Application to triple stars with good data in the multiple star catalogue suggests that more than 50 per cent are unable to support circumbinary planets, as the stable zone is almost non-existent.Comment: 16 pages, MNRAS, in pres

A New Superintegrable Hamiltonian

We identify a new superintegrable Hamiltonian in 3 degrees of freedom, obtained as a reduction of pure Keplerian motion in 6 dimensions. The new Hamiltonian is a generalization of the Keplerian one, and has the familiar 1/r potential with three barrier terms preventing the particle crossing the principal planes. In 3 degrees of freedom, there are 5 functionally independent integrals of motion, and all bound, classical trajectories are closed and strictly periodic. The generalisation of the Laplace-Runge-Lenz vector is identified and shown to provide functionally independent isolating integrals. They are quartic in the momenta and do not arise from separability of the Hamilton-Jacobi equation. A formulation of the system in action-angle variables is presented.Comment: 11 pages, 4 figures, submitted to The Journal of Mathematical Physic

HD 98800: A most unusual debris disc

The dynamics of planetesimals in the circumbinary debris disc of the quadruple star system HD 98800 are investigated. Evolving a spherical shell of test particles from a million years ago to the present day indicates that both coplanar and retrograde warped discs could exist, as well as a high inclination halo of material. Significant gaps are seen in the discs, as well as unexpected regions of stability due to the retrograde nature of the stellar orbits. Despite a viewing angle almost perpendicular to the direction of the warp of the planetesimal disc it is still intersected by the line of sight for eccentricities of the outer orbit of 0.5 or less.Comment: MNRAS, in pres

High Inclination Planets in Multistellar Systems

The Kozai mechanism often destabilises high inclination orbits. It couples changes in the eccentricity and inclination, and drives high inclination, circular orbits to low inclination, eccentric orbits. In a recent study of the dynamics of planetesimals in the quadruple star system HD98800 (Verrier & Evans 2008), there were significant numbers of stable particles in circumbinary polar orbits about the inner binary pair which are apparently able to evade the Kozai instability. Here, we isolate this feature and investigate the dynamics through numerical and analytical models. The results show that the Kozai mechanism of the outer star is disrupted by a nodal libration induced by the inner binary pair on a shorter timescale. By empirically modelling the period of the libration, a criteria for determining the high inclination stability limits in general triple systems is derived. The nodal libration feature is interesting and, although effecting inclination and node only, shows many parallels to the Kozai mechanism. This raises the possibility that high inclination planets and asteroids may be able to survive in multistellar systems.Comment: MNRAS, submitte

Crack Detectability in Vertical Axis Cooling Pumps During Operation

The problem which is faced in this paper is the analysis of the effects of a transverse propagating crack on the vibrational behaviour of a vertical axis cooling pump. The crack is assumed to develop in a section between the impeller and a seal, which prevents the hot water to flow upwards along the rotor shaft. The pressurized seal is fed with an injection of cold water. Crack initiation may be due to a thermal striping phenomenon. Afterwards, crack growth could be driven by a combination of thermal and mechanical loads, causing alternate cyclic stress in the shaft. Cracking instances of this type have been reported worldwide in several machines of similar design. In this paper, the fact is emphasized that the crack behavior is likely to be influenced by the thermal field and by the water pressure in the cracked area. A dynamical lineshaft model, integrated by an original representation of the crack, has been developed to investigate the possible vibratory symptoms related to a crack propagation. The vibrations are generally measured in correspondence of a rigid coupling which connects the motor shaft to the pump shaft, in position which is rather far away from crack. 1x rev., 2x rev and 3x rev. vibration components, which are generally displayed by the machine condition monitoring system and are the most significative symptoms of the presence of a transverse crack in a rotating shaft, are calculated

Superintegrability of the Caged Anisotropic Oscillator

We study "the Caged Anisotropic Harmonic Oscillator", which is a new example of a superintegrable, or accidentally degenerate Hamiltonian. The potential is that of the harmonic oscillator with rational frequency ratio (l:m:n), but additionally with barrier terms describing repulsive forces from the principal planes. This confines the classical motion to a sector bounded by the principal planes, or cage. In 3 degrees, there are five isolating integrals of motion, ensuring that all bound trajectories are closed and strictly periodic. Three of the integrals are quadratic in the momenta, the remaining two are polynomials of order 2(l+m-1) and 2(l+n-1). In the quantum problem, the eigenstates are multiply degenerate, exhibiting multiple copies of the fundamental pattern of the symmetry group SU(3).Comment: Submitted to the Journal of Mathematical Physic

Level I fieldwork today: A study of contexts and perceptions.

The last comprehensive examination of the Level I fieldwork experience was performed 15 years ago (Shalik, 1990) and addressed the different types of settings in which fieldwork occurred; amounts and types of supervision; structure and scheduling of the Level I experiences; and the effects of supervising Level I students on productivity. Although every occupational therapy and occupational therapy assistant student encounters a number of Level I fieldwork opportunities, little is available describing the process and contexts of the Level I fieldwork experience today. This study, which examines 1,002 student reports on Level I fieldwork experiences, finds that Level I fieldwork today occurs in a wide variety of physical disability, pediatric, mental health, and emerging practice settings. Findings also indicate that, whereas most fieldwork educators are occupational therapy practitioners, more fieldwork educators are non-occupational therapists than in the past. Furthermore, although students reported opportunities to practice observation and communication across all settings, practice of other clinical skills was specific to type of settings, and opportunities to practice were limited. Student perceptions about opportunities for experiencing occupation-based practice, observation of theory in practice, and how students value different types of fieldwork experiences are addressed. In addition, this study explores the expansion of Level I fieldwork into emerging practice arenas and how students perceive those experiences

Effects of hydroxyapatite and PDGF concentrations on osteoblast growth in a nanohydroxyapatite-polylactic acid composite for guided tissue regeneration

The technique of guided tissue regeneration (GTR) has evolved over recent years in an attempt to achieve periodontal tissue regeneration by the use of a barrier membrane. However, there are significant limitations in the currently available membranes and overall outcomes may be limited. A degradable composite material was investigated as a potential GTR membrane material. Polylactic acid (PLA) and nanohydroxyapatite (nHA) composite was analysed, its bioactive potential and suitability as a carrier system for growth factors were assessed. The effect of nHA concentrations and the addition of platelet derived growth factor (PDGF) on osteoblast proliferation and differentiation was investigated. The bioactivity was dependent on the nHA concentration in the films, with more apatite deposited on films containing higher nHA content. Osteoblasts proliferated well on samples containing low nHA content and differentiated on films with higher nHA content. The composite films were able to deliver PDGF and cell proliferation increased on samples that were pre absorbed with the growth factor. nHA–PLA composite films are able to deliver active PDGF. In addition the bioactivity and cell differentiation was higher on films containing more nHA. The use of a nHA–PLA composite material containing a high concentration of nHA may be a useful material for GTR membrane as it will not only act as a barrier, but may also be able to enhance bone regeneration by delivery of biologically active molecules