Decreasing Computing Time with Symplectic Correctors in Adaptive Timestepping Routines

It has previously been shown that varying the numerical timestep during a symplectic orbital integration leads to a random walk in energy and angular momentum, destroying the phase space-conserving property of symplectic integrators. Here we show that when altering the timestep symplectic correctors can be used to reduce this error to a negligible level. Furthermore, these correctors can also be employed to avoid a large error introduction when changing the Hamiltonian's partitioning. We have constructed a numerical integrator using this technique that is nearly as accurate as widely used fixed-step routines. In addition, our algorithm is drastically faster for integrations of highly eccentricitic, large semimajor axis orbits, such as those found in the Oort Cloud.Comment: Accepted to AJ, 29 pages, 8 figure

Variational Integrators for Almost-Integrable Systems

We construct several variational integrators--integrators based on a discrete variational principle--for systems with Lagrangians of the form L = L_A + epsilon L_B, with epsilon << 1, where L_A describes an integrable system. These integrators exploit that epsilon << 1 to increase their accuracy by constructing discrete Lagrangians based on the assumption that the integrator trajectory is close to that of the integrable system. Several of the integrators we present are equivalent to well-known symplectic integrators for the equivalent perturbed Hamiltonian systems, but their construction and error analysis is significantly simpler in the variational framework. One novel method we present, involving a weighted time-averaging of the perturbing terms, removes all errors from the integration at O(epsilon). This last method is implicit, and involves evaluating a potentially expensive time-integral, but for some systems and some error tolerances it can significantly outperform traditional simulation methods.Comment: 14 pages, 4 figures. Version 2: added informative example; as accepted by Celestial Mechanics and Dynamical Astronom

Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motions

We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the phase error can also be eliminated two orders higher than that of the integrator. The use of fourth order forward time step integrators can result in sixth order accuracy for the phase error and eighth accuracy in the periodic energy. We study the 1-D harmonic oscillator and the 2-D Kepler problem in great details, and compare the effectiveness of some recent fourth order algorithms.Comment: Submitted to Phys. Rev. E, 29 Page

Asteroids in the Inner Solar System I - Existence

Ensembles of in-plane and inclined orbits in the vicinity of the Lagrange points of the terrestrial planets are integrated for up to 100 million years. The integrations incorporate the gravitational effects of Sun and the eight planets (Pluto is neglected). Mercury is the least likely planet, as it is unable to retain tadpole orbits over 100 million year timescales. Both Venus and the Earth are much more promising, as they possess rich families of stable tadpole and horseshoe orbits. Our survey of Trojans in the orbital plane of Venus is undertaken for 25 million years. Some 40% of the survivors are on tadpole orbits. For the Earth, the integrations are pursued for 50 million years. The stable zones in the orbital plane are larger for the Earth than for Venus, but fewer of the survivors are tadpoles. Both Venus and the Earth also have regions in which inclined test particles can endure near the Lagrange points. For Venus, only test particles close to the orbital plane are stable. For the Earth, there are two bands of stability, one at low inclinations (i < 16 degrees) and one at moderate inclinations (between 24 degrees and 34 degrees). The inclined test particles that evade close encounters are primarily moving on tadpole orbits. Our survey of in-plane test particles near the Martian Lagrange points shows no survivors after 60 million years. Low inclination test particles do not persist, as their inclinations are quickly increased until the effects of a secular resonance with Jupiter cause de-stabilisation. Numerical integrations of inclined test particles for timespans of 25 million years show stable zones for inclinations between 14 and 40 degrees.Comment: 20 pages, 21 figures, Monthly Notices (in press

The role of chaotic resonances in the solar system

Our understanding of the Solar System has been revolutionized over the past decade by the finding that the orbits of the planets are inherently chaotic. In extreme cases, chaotic motions can change the relative positions of the planets around stars, and even eject a planet from a system. Moreover, the spin axis of a planet-Earth's spin axis regulates our seasons-may evolve chaotically, with adverse effects on the climates of otherwise biologically interesting planets. Some of the recently discovered extrasolar planetary systems contain multiple planets, and it is likely that some of these are chaotic as well.Comment: 28 pages, 9 figure

The effects of more realistic forms of lead heterogeneity in soil on uptake, biomass and root response of two brassica species

The spatial heterogeneity of soil constituents is known to have significant impacts on plant growth and plant uptake of nutrients and contaminants, yet studies have rarely used patterns of heterogeneity based on those found in the field. Heterogeneity refers to how lumpy materials are distributed in the soil, whilst homogeneity is the uniformity in the distribution of such materials. We identified patterns of lead contamination at historically polluted field sites and conducted pot trials using field–based parameters to determine the pattern of distribution of lead within the pots. We examined plant Pb uptake and growth in simulated low, medium and high heterogeneity environments as well as a control homogeneous treatment. We found a significant effect of Pb spatial heterogeneity on uptake and biomass of two Brassica species (Brassica napus and Brassica juncea), both candidate species for phytoremediation projects. Biomass was 4 to 5 fold lower in the high heterogeneity treatment and total plant Pb uptake as Pb mass in (µg) was 40 to 80% lower, compared to the homogeneous treatment. Plant lead concentration (mg/kg) increased by a factor of 2 with increasing heterogeneity. Peak uptake was observed in low and medium heterogeneity treatments of B. napus and B. juncea respectively. We also explored roots behaviour in the high heterogeneity treatment and found variation in root mass by 20 to 80% between concentric patches with significant (P < 0.05) differences between patches and species. High proportion of roots (40 to 50%) were proliferated in patches of lower Pb concentration. The tap root was a greater proportion of root in B. napus, which was absent in B. juncea. Results suggest that root morphology of this plant species might be a factor influencing the placement of roots in concentric patches and consequently the overall root response to Pb spatial heterogeneity. This is an indication that the root response could be realistic of that experienced by plants in field conditions. Generally result showed that spatial heterogeneity of Pb has a significant effect on plant growth and biomass. This study also demonstrated that the presence and extent of in situ heterogeneity of Pb in soil plays an important role in Pb uptake by plants. This work has implications for improving the phytoremediation of Pb contaminated land, phytomining, the reliability of risk assessment/models of human exposure to Pb and the quality of trace mineral content of agricultural produce

Asteroids in the Inner Solar System II - Observable Properties

This paper presents synthetic observations of long-lived, coorbiting asteroids of Mercury, Venus, the Earth and Mars. Our sample is constructed by taking the limiting semimajor axes, differential longitudes and inclinations for long-lived stability provided by simulations. The intervals are randomly populated with values to create initial conditions. These orbits are re-simulated to check that they are stable and then re-sampled every 2.5 years for 1 million years. The Mercurian sample contains only horseshoe orbits, the Martian sample only tadpoles. For both Venus and the Earth, the greatest concentration of objects on the sky occurs close to the classical Lagrange points at heliocentric ecliptic longitudes of 60 and 300 degrees. The distributions are broad especially if horseshoes are present in the sample. The full-width half maximum (FWHM) in heliocentric longitude for Venus is 325 degrees and for the Earth is 328 degrees. The mean and most common velocity of these coorbiting satellites coincides with the mean motion of the parent planet, but again the spread is wide with a FWHM for Venus of 27.8 arcsec/hr and for the Earth of 21.0 arcsec/hr. For Mars, the greatest concentration on the sky occurs at heliocentric ecliptic latitudes of 12 degrees. The peak of the velocity distribution occurs at 65 arcsec/hr, significantly less than the Martian mean motion, while its FWHM is 32.3 arcsec/hr. The case of Mercury is the hardest of all, as the greatest concentration occurs at heliocentric longitudes close to the Sun.Comment: 16 pages, 11 figures, Monthly Notices (in press). Higher quality figures available at http://www-thphys.physics.ox.ac.uk/users/WynEvans/home.htm

Pseudo-High-Order Symplectic Integrators

Symplectic N-body integrators are widely used to study problems in celestial mechanics. The most popular algorithms are of 2nd and 4th order, requiring 2 and 6 substeps per timestep, respectively. The number of substeps increases rapidly with order in timestep, rendering higher-order methods impractical. However, symplectic integrators are often applied to systems in which perturbations between bodies are a small factor of the force due to a dominant central mass. In this case, it is possible to create optimized symplectic algorithms that require fewer substeps per timestep. This is achieved by only considering error terms of order epsilon, and neglecting those of order epsilon^2, epsilon^3 etc. Here we devise symplectic algorithms with 4 and 6 substeps per step which effectively behave as 4th and 6th-order integrators when epsilon is small. These algorithms are more efficient than the usual 2nd and 4th-order methods when applied to planetary systems.Comment: 14 pages, 5 figures. Accepted for publication in the Astronomical Journa

The impact of a pilot continuing professional development module on hospital pharmacists’ preparedness to provide contemporary advice on the clinical use of vancomycin

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.BACKGROUND: Revised international clinical guidelines for the antibiotic vancomycin have changed the advice pharmacists need to provide to medical and nursing colleagues. OBJECTIVES: (1) To determine the self-reported confidence of hospital pharmacists to provide contemporary advice on vancomycin and (2) to evaluate hospital pharmacists' knowledge to provide contemporary advice on vancomycin following a pilot continuing professional development (CPD) module. METHODS: The study was a prospective two-phase design in an Australian teaching hospital. Phase one: a survey of pharmacist self-reported confidence to eight questions on providing contemporary advice on vancomycin. Responses were recorded using a Likert scales. Phase two: The provision of a pilot online CPD module on vancomycin containing knowledge-based assessment based on a clinical vignette. Likert scales recorded self-reported confidence were reported as median and interquartile range (IQR). Knowledge assessment was reported using descriptive statistics. The main outcome measure were the self-reported confidence, and knowledge of pharmacists regarding provision of contemporary advice on clinical vancomycin use. RESULTS: Response rates for surveys; confidence n = 35 (72.9 %) and knowledge n = 31 (58.5 %). Phase one: confidence was highest regarding vancomycin dosing and monitoring with 71.4-81.6 % of respondents agreeing or strongly agreeing that they were confident in these domains. Respondents agreeing or strongly agreeing were least confident regarding intravenous administration and infusion related reactions, 57.1 and 45.7 % respectively. Respondents who provided advice on vancomycin >10 times in the prior 12 months reported significantly higher confidence in; therapeutic range 1 (IQR 1-2) versus 2 (IQR 1-3) p = 0.02; amending dosage based on therapeutic drug monitoring results 2 (IQR 1-3) versus 3 (IQR 2-3) p = 75 % of pharmacists. CONCLUSION: Pharmacists' self-reported confidence to managing vancomycin was variable but generally high. Knowledge scores were consistently high after pharmacists completed a pilot CPD module on vancomycin. These data provides impetus for a randomised controlled study across multiple sites to determine the extent to which pharmacist knowledge on vancomycin can be attributed to completion of an online CPD

Influence of the coorbital resonance on the rotation of the Trojan satellites of Saturn

The Cassini spacecraft collects high resolution images of the saturnian satellites and reveals the surface of these new worlds. The shape and rotation of the satellites can be determined from the Cassini Imaging Science Subsystem data, employing limb coordinates and stereogrammetric control points. This is the case for Epimetheus (Tiscareno et al. 2009) that opens elaboration of new rotational models (Tiscareno et al. 2009; Noyelles 2010; Robutel et al. 2011). Especially, Epimetheus is characterized by its horseshoe shape orbit and the presence of the swap is essential to introduce explicitly into rotational models. During its journey in the saturnian system, Cassini spacecraft accumulates the observational data of the other satellites and it will be possible to determine the rotational parameters of several of them. To prepare these future observations, we built rotational models of the coorbital (also called Trojan) satellites Telesto, Calypso, Helene, and Polydeuces, in addition to Janus and Epimetheus. Indeed, Telesto and Calypso orbit around the L_4 and L_5 Lagrange points of Saturn-Tethys while Helene and Polydeuces are coorbital of Dione. The goal of this study is to understand how the departure from the Keplerian motion induced by the perturbations of the coorbital body, influences the rotation of these satellites. To this aim, we introduce explicitly the perturbation in the rotational equations by using the formalism developed by Erdi (1977) to represent the coorbital motions, and so we describe the rotational motion of the coorbitals, Janus and Epimetheus included, in compact form