A Fast and Accurate Universal Kepler Solver without Stumpff Series

We derive and present a fast and accurate solution of the initial value problem for Keplerian motion in universal variables that does not use the Stumpff series. We find that it performs better than methods based on the Stumpff series.Comment: 10 pages, 7 figures. Accepted by MNRAS, resubmitted because of a typo in the title, added author affiliation

Swimming in Space-Time

Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead to net translation and/or rotation of the body in the manifold. Presuming space-time is a curved manifold as portrayed by general relativity, translation in space can be accomplished simply by cyclic changes in the shape of a body, without any thrust or external forces

Dynamic Elastic Tides

This is an exploration of dynamic tides on elastic bodies. The body is thought of as a dynamical system described by its modes of oscillation. The dynamics of these modes are governed by differential equations that depend on the rheology. The modes are damped by dissipation. Tidal friction occurs as exterior bodies excite the modes and the modes act back on the tide raising body. The whole process is governed by a closed set of differential equations. Standard results from tidal theory are recovered in a two-timescale approximation to the solution of these differential equations

The Role of Programming in the Formulation of Ideas

Classical mechanics is deceptively simple. It is surprisingly easy to get the right answer with fallacious reasoning or without real understanding. To address this problem we use computational techniques to communicate a deeper understanding of Classical Mechanics. Computational algorithms are used to express the methods used in the analysis of dynamical phenomena. Expressing the methods in a computer language forces them to be unambiguous and computationally effective. The task of formulating a method as a computer-executable program and debugging that program is a powerful exercise in the learning process. Also, once formalized procedurally, a mathematical idea becomes a tool that can be used directly to compute results

Precession of the lunar core

Goldreich (Goldreich, P. [1967]. J. Geophys. Res. 72, 3135) showed that a lunar core of low viscosity would not precess with the mantle. We show that this is also the case for much of lunar history. But when the Moon was close to the Earth, the Moon’s core was forced to follow closely the precessing mantle, in that the rotation axis of the core remained nearly aligned with the symmetry axis of the mantle. The transition from locked to unlocked core precession occurred between 26.0 and 29.0 Earth radii, thus it is likely that the lunar core did not follow the mantle during the Cassini transition. Dwyer and Stevenson (Dwyer, C.A., Stevenson, D.J. [2005]. An Early Nutation-Driven Lunar Dynamo. AGU Fall Meeting Abstracts GP42A-06) suggested that the lunar dynamo needs mechanical stirring to power it. The stirring is caused by the lack of locked precession of the lunar core. So, we do not expect a lunar dynamo powered by mechanical stirring when the Moon was closer to the Earth than 26.0–29.0 Earth radii. A lunar dynamo powered by mechanical stirring might have been strongest near the Cassini transition

Functional Differential Geometry

An explanation of the mathematics needed as a foundation for a deep understanding of general relativity or quantum field theory.Physics is naturally expressed in mathematical language. Students new to the subject must simultaneously learn an idiomatic mathematical language and the content that is expressed in that language. It is as if they were asked to read Les Misérables while struggling with French grammar. This book offers an innovative way to learn the differential geometry needed as a foundation for a deep understanding of general relativity or quantum field theory as taught at the college level.The approach taken by the authors (and used in their classes at MIT for many years) differs from the conventional one in several ways, including an emphasis on the development of the covariant derivative and an avoidance of the use of traditional index notation for tensors in favor of a semantically richer language of vector fields and differential forms. But the biggest single difference is the authors' integration of computer programming into their explanations. By programming a computer to interpret a formula, the student soon learns whether or not a formula is correct. Students are led to improve their program, and as a result improve their understanding

1. The Origin of the Kirkwood Gaps: A Mapping for Asteroidal Monitor Near the 3/1 Commensurability. 2. The Resonance Overlap Criterion and the Onset of Stochastic Behavior in the Restricted Three-Body Problem

Paper 1: A mapping of the phase space onto itself with the same low order resonance structure as the 3/1 commensurability in the planar elliptic three-body prob­lem is derived. This mapping is approximately one thousand (1000) times faster than the usual method of numerically integrating the averaged equations of motion (as used by Schubart, Froeschlé and Scholl in their studies of the asteroid belt). This mapping exhibits some very surprising behavior that might provide the key to the origin of the gaps. A test asteroid placed in the gap may evolve for a million years with low eccentricity (&#60; 0.05) and then suddenly jump to large eccentricity (&#62; 0.3) becoming a Mars crosser. The asteroid can then be removed by a close encounter with Mars. To test this hypothesis a dis­tribution of 300 test asteroids in the neighborhood of the 3/1 commensurability was evolved for two million years. When the Mars crossers are removed the dis­tribution of initial conditions displays a gap at the location of the 3/1 Kirkwood gap. While this is the first real demonstration of the formation of a gap, the gap is too narrow. The planar elliptic mapping is then extended to include the incli­nations and the secular perturbations of Jupiter's orbit. The two million year evolution of the 300 test asteroids is repeated using the full mapping. The resulting gap is somewhat larger yet still too small. Finally the possibility that over longer times more asteroids will become Mars crossers is tested by studying the evolution of one test asteroid near the border of the gap for a much longer time. A jump in its eccentricity occurs after 18 million years indicating that indeed it may simply be a matter of time for the full width of the gap to open. Paper 2: The resonance overlap criterion for the onset of stochastic behavior is applied to the planar circular-restricted three-body problem with small mass ratio (µ). Its predictions for µ = 10-3, µ = 10-4 and µ = 10-5 are compared to the transitions in the numerically determined Kolmogorov-Sinai entropy and found to be in remarkably good agreement. In addition, an approximate scaling law for the onset of stochastic behavior is derived.</p

Exchange of ejecta between Telesto and Calypso: Tadpoles, horseshoes, and passing orbits

We have numerically integrated the orbits of ejecta from Telesto and Calypso, the two small Trojan companions of Saturn's major satellite Tethys. Ejecta were launched with speeds comparable to or exceeding their parent's escape velocity, consistent with impacts into regolith surfaces. We find that the fates of ejecta fall into several distinct categories, depending on both the speed and direction of launch. The slowest ejecta follow sub-orbital trajectories and re-impact their source moon in less than one day. Slightly faster debris barely escape their parent's Hill sphere and are confined to tadpole orbits, librating about Tethys' triangular Lagrange points L4 (leading, near Telesto) or L5 (trailing, near Calypso) with nearly the same orbital semi-major axis as Tethys, Telesto, and Calypso. These ejecta too eventually re-impact their source moon, but with a median lifetime of a few dozen years. Those which re-impact within the first ten years or so have lifetimes near integer multiples of 348.6 days (half the tadpole period). Still faster debris with azimuthal velocity components >~ 10 m/s enter horseshoe orbits which enclose both L4 and L5 as well as L3, but which avoid Tethys and its Hill sphere. These ejecta impact either Telesto or Calypso at comparable rates, with median lifetimes of several thousand years. However, they cannot reach Tethys itself; only the fastest ejecta, with azimuthal velocities >~ 40 m/s, achieve "passing orbits" which are able to encounter Tethys. Tethys accretes most of these ejecta within several years, but some 1 % of them are scattered either inward to hit Enceladus or outward to strike Dione, over timescales on the order of a few hundred years

The influence of fasting and post-load glucose levels on maternal and neonatal outcomes in women with hyperglycaemia in pregnancy in Uganda: A prospective observational cohort study.

AIMS: The study aims to evaluate the strength of fasting versus post-load glucose levels in predicting adverse outcomes in women with hyperglycaemia in pregnancy (HIP). METHODS: Women attending antenatal clinics in urban and peri-urban Uganda had oral glucose tolerance test between 24 and 28 weeks of gestation to screen for HIP, and were followed up to collect data on maternal and neonatal outcomes. Univariable and multivariable Poisson regression models were used to estimate the relative risk adverse outcome associated with fasting hyperglycaemia alone post-load hyperglycaemia alone, or elevation of both fasting and post-load glucose levels. RESULTS: We included 3206 participants in the final analysis. HIP was associated with increased risk of Caesarean section, large for gestaional age babies, and neonatal intensive care admission. The risk was highest (2.54-fold compared to normal glycaemic women) when both FBG and post-load glucose levels were elevated. After adjustment for potential confounders, having elevated post-load glucose alone was not associated with increased risk of any of the outcomes, but elevated FBG alone increased the risk of Caesarian section by 1.36-fold. CONCLUSION: Fasting hyperglycemia appears to be more strongly associated with adverse pregnancy outcomes than post-load hyperglycaemia, but the risk is even higher in women with elevation of both fasting and post-load glucose levels

RESOLVING THE PERICENTER

The Wisdom–Holman mapping method and its variations have become a mainstay of research in solar system dynamics. But the method is not without its limitations. Rauch & Holman noted that at large eccentricities sufficiently small steps must be taken to resolve the pericenter. In this paper, I explore in more detail what it means to resolve the pericenter