The iodine-plutonium-xenon age of the Moon-Earth system revisited

From iodine-plutonium-xenon isotope systematics, we re-evaluate time constraints on the early evolution of the Earth-atmosphere system and, by inference, on the Moon-forming event. Two extinct radioactivites (129I, T1/2 = 15.6 Ma, and 244Pu, T1/2 = 80 Ma) have produced radiogenic 129Xe and fissiogenic 131-136Xe, respectively, within the Earth, which related isotope fingerprints are seen in the compositions of mantle and atmospheric Xe. Recent studies of Archean rocks suggest that xenon atoms have been lost from the Earth's atmosphere and isotopically fractionated during long periods of geological time, until at least the end of the Archean eon. Here we build a model that takes into account these results. Correction for Xe loss permits to compute new closure ages for the Earth's atmosphere that are in agreement with those computed for mantle Xe. The minimum Xe formation interval for the Earth- atmosphere is 40 (-10+20) Ma after start of solar system formation, which may also date the Moon-forming impact.Comment: 27 pages, 3 figures, 2 table

High-order half-integral conservative post-Newtonian coefficients in the redshift factor of black hole binaries

The post-Newtonian approximation is still the most widely used approach to obtaining explicit solutions in general relativity, especially for the relativistic two-body problem with arbitrary mass ratio. Within many of its applications, it is often required to use a regularization procedure. Though frequently misunderstood, the regularization is essential for waveform generation without reference to the internal structure of orbiting bodies. In recent years, direct comparison with the self-force approach, constructed specifically for highly relativistic particles in the extreme mass ratio limit, has enabled preliminary confirmation of the foundations of both computational methods, including their very independent regularization procedures, with high numerical precision. In this paper, we build upon earlier work to carry this comparison still further, by examining next-to-next-to-leading order contributions beyond the half integral 5.5PN conservative effect, which arise from terms to cubic and higher orders in the metric and its multipole moments, thus extending scrutiny of the post-Newtonian methods to one of the highest orders yet achieved. We do this by explicitly constructing tail-of-tail terms at 6.5PN and 7.5PN order, computing the redshift factor for compact binaries in the small mass ratio limit, and comparing directly with numerically and analytically computed terms in the self-force approach, obtained using solutions for metric perturbations in the Schwarzschild space-time, and a combination of exact series representations possibly with more typical PN expansions. While self-force results may be relativistic but with restricted mass ratio, our methods, valid primarily in the weak-field slowly-moving regime, are nevertheless in principle applicable for arbitrary mass ratios.Comment: 33 pages, no figure; minor correction

Half-integral conservative post-Newtonian approximations in the redshift factor of black hole binaries

Recent perturbative self-force computations (Shah, Friedman & Whiting, submitted to Phys. Rev. {\bf D}, arXiv:1312.1952 [gr-qc]), both numerical and analytical, have determined that half-integral post-Newtonian terms arise in the conservative dynamics of black-hole binaries moving on exactly circular orbits. We look at the possible origin of these terms within the post-Newtonian approximation, find that they essentially originate from non-linear "tail-of-tail" integrals and show that, as demonstrated in the previous paper, their first occurrence is at the 5.5PN order. The post-Newtonian method we use is based on a multipolar-post-Minkowskian treatment of the field outside a general matter source, which is re-expanded in the near zone and extended inside the source thanks to a matching argument. Applying the formula obtained for generic sources to compact binaries, we obtain the redshift factor of circular black hole binaries (without spins) at 5.5PN order in the extreme mass ratio limit. Our result fully agrees with the determination of the 5.5PN coefficient by means of perturbative self-force computations reported in the previously cited paper.Comment: 18 pages, no figures, references updated and minor corrections include

Two-equation modeling of turbulent rotating flows

The possibility to take into account the effects of the Coriolis acceleration on turbulence is examined in the framework of two-equation eddy-viscosity models. General results on the physical consistency of such turbulence models are derived from a dynamical-system approach to situations of time-evolving homogeneous turbulence in a rotating frame. Application of this analysis to a (k,epsilon) model fitted with an existing Coriolis correction [J. H. G. Howard, S. V. Patankar, and R. M. Bordynuik, "Flow prediction in rotating ducts using Coriolis-modified turbulence models", ASME Trans. J. Fluids Eng. 102, (1980)] is performed. Full analytical solutions are given for the flow predicted with this model in the situation of homogeneously sheared turbulence subject to rotation. The existence of an unphysical phenomenon of blowup at finite time is demonstrated in some range of the rotation-to-shear ratio. A direct connection is made between the slope of the mean-velocity profile in the plane-channel flow with spanwise rotation, and a particular fixed point of the dynamical system in homogeneously sheared turbulence subject to rotation. The general analysis, and the understanding of typical inaccuracies and misbehavior observed with the existing model, are then used to design a new model which is free from the phenomenon of blowup at finite time and able to account for both of the main influences of rotation on turbulence: the inhibition of the spectral transfer to high wave numbers and the shear/Coriolis instability

Dimensional regularization of the IR divergences in the Fokker action of point-particle binaries at the fourth post-Newtonian order

The Fokker action of point-particle binaries at the fourth post-Newtonian (4PN) approximation of general relativity has been determined previously. However two ambiguity parameters associated with infra-red (IR) divergencies of spatial integrals had to be introduced. These two parameters were fixed by comparison with gravitational self-force (GSF) calculations of the conserved energy and periastron advance for circular orbits in the test-mass limit. In the present paper together with a companion paper, we determine both these ambiguities from first principle, by means of dimensional regularization. Our computation is thus entirely defined within the dimensional regularization scheme, for treating at once the IR and ultra-violet (UV) divergencies. In particular, we obtain crucial contributions coming from the Einstein-Hilbert part of the action and from the non-local tail term in arbitrary dimensions, which resolve the ambiguities.Comment: 25 pages, published versio

Conservative Dynamics of Binary Systems of Compact Objects at the Fourth Post-Newtonian Order

We review our recent derivation of a Fokker action describing the conservative dynamics of a compact binary system at the fourth post-Newtonian (4PN) approximation of general relativity. The two bodies are modeled by point particles, which induces ultraviolet (UV) divergences that are cured by means of dimensional regularization combined with a renormalization of the particle's wordlines. Associated with the propagation of wave tails at infinity is the appearance of a non-local-in-time conservative tail effect at the 4PN order in the Lagrangian. In turn this implies the appearance of infrared (IR) divergent integrals which are also regularized by means of dimensional regularization. We compute the Noetherian conserved energy and periastron advance for circular orbits at 4PN order, paying special attention to the treatment of the non-local terms. One ambiguity parameter remaining in the current formalism is determined by comparing those quantities, expressed as functions of the orbital frequency, with self-force results valid in the small mass ratio limit.Comment: 7 pages; contribution to the proceedings of the 52nd Rencontres de Moriond, "Gravitation

Fokker action of non-spinning compact binaries at the fourth post-Newtonian approximation

The Fokker action governing the motion of compact binary systems without spins is derived in harmonic coordinates at the fourth post-Newtonian approximation (4PN) of general relativity. Dimensional regularization is used for treating the local ultraviolet (UV) divergences associated with point particles, followed by a renormalization of the poles into a redefinition of the trajectories of the point masses. Effects at the 4PN order associated with wave tails propagating at infinity are included consistently at the level of the action. A finite part procedure based on analytic continuation deals with the infrared (IR) divergencies at spatial infinity, which are shown to be fully consistent with the presence of near-zone tails. Our end result at 4PN order is Lorentz invariant and has the correct self-force limit for the energy of circular orbits. However, we find that it differs from the recently published result derived within the ADM Hamiltonian formulation of general relativity [T. Damour, P. Jaranowski, and G. Sch\"afer, Phys. Rev. D 89, 064058 (2014)]. More work is needed to understand this discrepancy.Comment: 47 pages; references added; Sec. VD enhanced; a few more minor improvement

Numerical error evaluation for tip clearance flow calculations in a centrifugal compressor

Since globally mesh independent solution are still beyond available computer resources for industrial cases, a method to quantify locally the numerical error is proposed. The design of experiments method helps selecting mesh parameters that influence the tip clearance solution, so that additional meshes are computed to evaluate the numerical error on the shroud friction coefficient. In the field of CFD applied to turbomachinery, this study results from a partnership between ENSICA, Liebherr-Aerospace Toulouse and Numeca International. This paper focuses on numerical error evaluation for RANS simulations, applied to centrifugal compressor flow field calculations. CFD is now commonly used for centrifugal compressor design optimization, but, as Hutton and Casey develop in [1], there is an urging demand for improved quality and trust in industrial CFD. Indeed, this stresses the need for comprehensive and thorough numerical error evaluation, namely the process of verification, as defined for example by Oberkampf and Trucano in [2]. Unfortunately, 3D turbulent calculations for turbomachinery components are still very demanding in computational resources and, to the knowledge of the author, there is no published result concerning comprehensive verification of the entire flow field in centrifugal compressors. As a first step on the way to achieve that, this paper presents a method aiming at the obtention of a numerical solution that can be regarded as locally mesh-independent. In other words, the objective is to compute the flow field on a grid such that the solution obtained has a specific region where the numerical error is negligible. It has long been recognized that the tip clearance of a centrifugal compressor is of paramount importance for aerodynamic performances, which means that accurately predicting the flow field in this region is crucial for accurate prediction of performances by means of CFD codes. Numerous studies have been published that compare numerical and experimental results in the tip region. However, in these studies, numerical error still remains an issue; for instance Basson and Lakshminarayana [3] show excellent comparisons with experiments, but they attribute the remaining discrepancies to insufficient grid resolution. Indeed, accurate predictions of global effects, such as efficiency, require a fine description of flow details. Therefore, friction at the shroud endwall is the concern of the study, since it is a very sensitive indicator of the quality of the velocity profile’s prediction at the wall