Numerical solutions of differential equations

Various numerical methods for solving differential equations were analyzed and refined in an effort to develop a method which was adaptable to a large class of problems. The prime capabilities of the method included accuracy, numerical stability, and economic use of computer time. In multistep processes the corrector was changed at each step

Predictor-corrector process in which the selection of an optimum corrector is made each step Quarterly technical report, oct. - Dec. 1965

Numerical solution of differential equations by predictor-corrector process which selects optimum corrector at each ste

The timing and location of dust formation in the remnant of SN 1987A

The discovery with the {\it Herschel Space Observatory} of bright far infrared and submm emission from the ejecta of the core collapse supernova SN\,1987A has been interpreted as indicating the presence of some 0.4--0.7\,M$_\odot$ of dust. We have constructed radiative transfer models of the ejecta to fit optical to far-infrared observations from the literature at epochs between 615 days and 24 years after the explosion, to determine when and where this unexpectedly large amount of dust formed. We find that the observations by day 1153 are consistent with the presence of 3$\times$10$^{-3}$M$_\odot$ of dust. Although this is a larger amount than has previously been considered possible at this epoch, it is still very small compared to the amount present in the remnant after 24 years, and significantly higher dust masses at the earlier epochs are firmly ruled out by the observations, indicating that the majority of the dust must have formed at very late times. By 8515-9200 days after the explosion, 0.6--0.8\,M$_\odot$ of dust is present, and dust grains with radii greater than 2\,$\mu$m are required to obtain a fit to the observed SED. This suggests that the dust mass increase at late times was caused by accretion onto and coagulation of the dust grains formed at earlier epochs. These findings provide further confirmation that core collapse supernovae can create large quantities of dust, and indicate that the reason for small dust masses being estimated in many cases is that the vast majority of the dust forms long after most supernovae have been detectable at mid-infrared wavelengths.Comment: 13 pages, 16 figures. Accepted for publication in MNRA

Extra symmetry in the field equations in 5D with spatial spherical symmetry

We point out that the field equations in 5D, with spatial spherical symmetry, possess an extra symmetry that leaves them invariant. This symmetry corresponds to certain simultaneous interchange of coordinates and metric coefficients. As a consequence a single solution in 5D can generate very different scenarios in 4D, ranging from static configurations to cosmological situations. A new perspective emanates from our work. Namely, that different astrophysical and cosmological scenarios in 4D might correspond to the same physics in 5D. We present explicit examples that illustrate this point of view.Comment: Typos corrected. Accepted for publication in Classical and Quantum Gravit

Constraints on Kaluza-Klein gravity from Gravity Probe B

Using measurements of geodetic precession from Gravity Probe B, we constrain possible departures from Einstein's General Relativity for a spinning test body in Kaluza-Klein gravity with one additional space dimension. We consider the two known static and spherically symmetric solutions of the 5D field equations (the soliton and canonical metrics) and obtain new limits on the free parameters associated with each. The theory is consistent with observation but must be "close to 4D" in both cases.Comment: 9 pages, 1 figure; General Relativity and Gravitation, in pres

MHD Memes

The celebration of Allan Kaufman's 80th birthday was an occasion to reflect on a career that has stimulated the mutual exchange of ideas (or memes in the terminology of Richard Dawkins) between many researchers. This paper will revisit a meme Allan encountered in his early career in magnetohydrodynamics, the continuation of a magnetohydrodynamic mode through a singularity, and will also mention other problems where Allan's work has had a powerful cross-fertilizing effect in plasma physics and other areas of physics and mathematics.Comment: Submitted for publication in IOP Journal of Physics: Conference Series for publication in "Plasma Theory, Wave Kinetics, and Nonlinear Dynamics", Proceedings of KaufmanFest, 5-7 October 2007, University of California, Berkeley, US

The bang of a white hole in the early universe from a 6D vacuum state: Origin of astrophysical spectrum

Using a previously introduced model in which the expansion of the universe is driven by a single scalar field subject to gravitational attraction induced by a white hole during the expansion (from a 6D vacuum state), we study the origin of squared inflaton fluctuations spectrum on astrophysical scales.Comment: Final version to be published in Eur. Phys. J.

The extinction and dust-to-gas structure of the planetary nebula NGC 7009 observed with MUSE

The large field and wavelength range of MUSE is well suited to mapping Galactic planetary nebulae (PN). The bright PN NGC 7009 was observed with MUSE on the VLT during the Science Verification of the instrument in seeing of 0.6". Emission line maps in hydrogen Balmer and Paschen lines were formed from analysis of the MUSE cubes. The measured electron temperature and density from the MUSE cube were employed to predict the theoretical hydrogen line ratios and map the extinction distribution across the nebula. After correction for the interstellar extinction to NGC 7009, the internal dust-to-gas ratio (A_V/N_H) has been mapped for the first time in a PN. The extinction map of NGC 7009 has considerable structure, broadly corresponding to the morphological features of the nebula. A large-scale feature in the extinction map, consisting of a crest and trough, occurs at the rim of the inner shell. The nature of this feature was investigated and instrumental and physical causes considered; no convincing mechanisms were identified to produce this feature, other than mass loss variations in the earlier asymptotic giant branch phase. The dust-to-gas ratio A_V/N_H increases from 0.7 times the interstellar value to >5 times from the centre towards the periphery of the ionized nebula. The integrated A_V/N_H is about 2 times the mean ISM value. It is demonstrated that extinction mapping with MUSE provides a powerful tool for studying the distribution of PN internal dust and the dust-to-gas ratio. (Abridged.)Comment: 10 pages, 7 figures. Accepted by A&

Stabilization of test particles in Induced Matter Kaluza-Klein theory

The stability conditions for the motion of classical test particles in an $n$-dimensional Induced Matter Kaluza-Klein theory is studied. We show that stabilization requires a variance of the strong energy condition for the induced matter to hold and that it is related to the hierarchy problem. Stabilization of test particles in a FRW universe is also discussed.Comment: 15 pages, 1 figure, to appear in Class. Quantum Gra

Induced Matter and Particle Motion in Non-Compact Kaluza-Klein Gravity

We examine generalizations of the five-dimensional canonical metric by including a dependence of the extra coordinate in the four-dimensional metric. We discuss a more appropriate way to interpret the four-dimensional energy-momentum tensor induced from the five-dimensional space-time and show it can lead to quite different physical situations depending on the interpretation chosen. Furthermore, we show that the assumption of five-dimensional null trajectories in Kaluza-Klein gravity can correspond to either four-dimensional massive or null trajectories when the path parameterization is chosen properly. Retaining the extra-coordinate dependence in the metric, we show the possibility of a cosmological variation in the rest masses of particles and a consequent departure from four-dimensional geodesic motion by a geometric force. In the examples given, we show that at late times it is possible for particles traveling along 5D null geodesics to be in a frame consistent with the induced matter scenario.Comment: 29 pages, accepted to GR