Stability of motion in the Sitnikov 3-body problem

We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m ₁ = m ₂ = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 ≤ m ₃ < 10⁻³, placed initially on the z-axis. We begin by finding for the restricted problem (with m ₃ = 0) an apparently infinite sequence of stability intervals on the z-axis, whose width grows and tends to a fixed non-zero value, as we move away from z = 0. We then estimate the extent of “islands” of bounded motion in x, y, z space about these intervals and show that it also increases as |z| grows. Turning to the so-called extended Sitnikov problem, where the third particle moves only along the z-axis, we find that, as m ₃ increases, the domain of allowed motion grows significantly and chaotic regions in phase space appear through a series of saddle-node bifurcations. Finally, we concentrate on the general 3-body problem and demonstrate that, for very small masses, m ₃ ≈ 10⁻⁶, the “islands” of bounded motion about the z-axis stability intervals are larger than the ones for m ₃ = 0. Furthermore, as m ₃ increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away “disperse” at larger m ₃ values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries, as observed in the m ₃ = 0 case.     

Platinum-Group Mineral Characterization in Concentrates from High-Grade PGE Al-rich Chromitites of Korydallos Area in the Pindos Ophiolite Complex (NW Greece)

The Pindos ophiolite complex, located in the north-western part of continental Greece, hosts various podiform chromite deposits generally characterized by low platinum-group element (PGE) grades. However, a few locally enriched in PPGE + Au (up to 29.3 ppm) chromitites of refractory type are also present, mainly in the area of Korydallos (south-eastern Pindos). The present data reveal that this enrichment is strongly dependant on chromian spinel chemistry and base metal sulfide and/or base metal alloy (BMS and BMA, respectively) content in chromitites. Consequently, we used super-panning to recover PGM from the Al-rich chromitites of the Korydallos area. The concentrate of the composite chromitite sample contained 159 PGM grains, including, in decreasing order of abundance, the following major PGM phases: Pd-Cu alloys (commonly non-stoichiometric, although a few Pd-Cu alloys respond to the chemical formula PdCu₄), Pd-bearing tetra-auricupride [(Au,Pd)Cu], nielsenite (PdCu₃), sperrylite (PtAs₂), skaergaardite (PdCu), Pd-bearing auricupride [(Au,Pd)Cu₃], Pt and Pd oxides, Pt-Fe-Ni alloys, hollingworthite (RhAsS) and Pt-Cu alloys. Isomertieite (Pd₁₁Sb₂As₂), zvyagintsevite (Pd₃Pb), native Au, keithconnite (Pd₂₀Te₇), naldrettite (Pd₂Sb) and Rh-bearing bismuthotelluride (RhBiTe, probably the Rh analogue of michenerite) constitute minor phases. The bulk of PGE-mineralization is dominated by PGM grains that range in size from 5 to 10 µm. The vast majority of the recovered PPGM are associated with secondary BMS and BMA, thus confirming that a sulphur-bearing melt played a very important role in scavenging the PGE + Au content of the silicate magma from which chromian spinel had already started to crystallize. The implemented technique has led to the recovery of more, as well as noble, PGM grains than the in situ mineralogical examination of single chromitite samples. Although, the majority of the PGM occur as free particles and in situ textural information is lost, single grain textural evidence is observed. In summary, this research provides information on the particles, grain size and associations of PGM, which are critical with respect to the petrogenesis and mineral processing.     

A multiphysics and multiscale software environment for modeling astrophysical systems

We present MUSE, a software framework for combining existing computational tools for different astrophysical domains into a single multiphysics, multiscale application. MUSE facilitates the coupling of existing codes written in different languages by providing inter-language tools and by specifying an interface between each module and the framework that represents a balance between generality and computational efficiency. This approach allows scientists to use combinations of codes to solve highly coupled problems without the need to write new codes for other domains or significantly alter their existing codes. MUSE currently incorporates the domains of stellar dynamics, stellar evolution and stellar hydrodynamics for studying generalized stellar systems. We have now reached a “Noah’s Ark” milestone, with (at least) two available numerical solvers for each domain. MUSE can treat multiscale and multiphysics systems in which the time- and size-scales are well separated, like simulating the evolution of planetary systems, small stellar associations, dense stellar clusters, galaxies and galactic nuclei. In this paper we describe three examples calculated using MUSE: the merger of two galaxies, the merger of two evolving stars, and a hybrid N-body simulation. In addition, we demonstrate an implementation of MUSE on a distributed computer which may also include special-purpose hardware, such as GRAPEs or GPUs, to accelerate computations. The current MUSE code base is publicly available as open source at http://muse.li.     

Complex statistics in Hamiltonian barred galaxy models

We use probability density functions (pdfs) of sums of orbit coordinates, over time intervals of the order of one Hubble time, to distinguish weakly from strongly chaotic orbits in a barred galaxy model. We find that, in the weakly chaotic case, quasi-stationary states arise, whose pdfs are well approximated by q-Gaussian functions (with 1 <?q < 3), while strong chaos is identified by pdfs which quickly tend to Gaussians (q =?1). Typical examples of weakly chaotic orbits are those that ??stick?? to islands of ordered motion. Their presence in rotating galaxy models has been investigated thoroughly in recent years due to their ability to support galaxy structures for relatively long time scales. In this paper, we demonstrate, on specific orbits of 2 and 3 degree of freedom barred galaxy models, that the proposed statistical approach can distinguish weakly from strongly chaotic motion accurately and efficiently, especially in cases where Lyapunov exponents and other local dynamic indicators appear to be inconclusive.     

Mineralogy, composition and PGM of chromitites from Pefki, Pindos ophiolite complex (NW Greece): evidence for progressively elevated fAs conditions in the upper mantle sequence

The Pindos ophiolite complex, located in the northwestern part of continental Greece, hosts various chromite deposits of both metallurgical (high-Cr) and refractory (high-Al) type. The Pefki chromitites are banded and sub-concordant to the surrounding serpentinized dunites. The Cr# [Cr/(Cr?+?Al)] of magnesiochromite varies between 0.75 and 0.79. The total PGE grade ranges from 105.9 up to 300.0?ppb. IPGE are higher than PPGE, typical of mantle hosted ophiolitic chromitites. The PGM assemblage in chromitites comprises anduoite, ruarsite, laurite, irarsite, sperrylite, hollingworthite, Os-Ru-Ir alloys including osmium and rutheniridosmine, Ru-bearing oxides, braggite, paolovite, platarsite, cooperite, vysotskite, and palladodymite. Iridarsenite and omeiite were also observed as exsolutions in other PGM. Rare electrum and native Ag are recovered in concentrates. This PGM assemblage is of great petrogenetic importance because it is significantly different from that commonly observed in podiform mantle-hosted and banded crustal-hosted ophiolitic chromitites. PGE chalcogenides of As and S are primary, and possibly crystallized directly from a progressively enriched in As boninitic melt before or during magnesiochromite precipitation. The presence of Ru-bearing oxides implies simultaneous desulfurization and dearsenication processes. Chemically zoned laurite and composite paolovite-electrum intergrowths are indicative of the relatively high mobility of certain PGE at low temperatures under locally oxidizing conditions. The PGM assemblage and chemistry, in conjunction with geological and petrologic data of the studied chromitites, indicate that it is characteristic of chromitites found within or close to the petrologic Moho. Furthermore, the strikingly different PGM assemblages between the high-Cr chromitites within the Pindos massif is suggestive of non-homogeneous group of ores.     

Compositional variation in Hg-bearing sphalerite from the polymetallic Eskay Creek deposit, British Columbia, Canada

The Eskay Creek, British Columbia, Canada, is a polymetallic, gold- and silver-rich, volcanic-hosted, massive sulfide deposit. The ore in the deposit is divided into subzones distinguished by mineralogy, texture, grade and metallurgical characteristics. This study presents the results of a mineralogical examination of three composite field samples, with emphasis on the chemistry of sphalerite. Sphalerite is associated with variable amounts of Hg-tetrahedrite and cinnabar, and an array of sulfides, sulphosalts and non-opaque minerals. Electron micro probe analyses of sphalerite in the three composite samples reveal wide variations in compositions. The Hg content in sphalerite in the three samples varies between 0.08 and 16.35 wt%, whereas the Fe content ranges from 0.33 to 2.29 wt%. The chemical formula of the sphalerite shows the compositional range (Zn_0.89–0.98Hg_0.01–0.09.Fe_0.005–0.02)S. Sphalerite exhibits an almost perfect substitution of Hg and Zn, as shown by the negative covariance between them. Sphalerite with the highest Hg contents tends to have the lowest Fe concentrations. The highest Hg contents in sphalerite are recorded in the samples with the highest bulk Hg concentrations and with the highest cinnabar contents.

The compositional variations of sphalerite are important because they can be used in mapping ore forming fluids and indicate possible temporal variations. Second, determination of the compositional variation of the sphalerite in the mine has metallurgical implications because the mineral is an important Zn source. The mineralogical data indicate that non-physical processes (e.g. pyrometallurgy) must be used to separate Hg from Zn concentrates, with direct environmental implications, that is, release of metals, such as Hg, into the environment during mining and processing.     

The stability of vertical motion in the N-body circular Sitnikov problem

We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. In particular, we consider ν = N ? 1 equal mass primary bodies which rotate on a circle, while the Nth body (of negligible mass) moves perpendicularly to the plane of the primaries. Thus, we extend previous work on the 4-body Sitnikov problem to the N-body case, with N = 5, 9, 15, 25 and beyond. We find, for all cases we have considered with N ≥ 4, that the Sitnikov family has only one stability interval (on the z-axis), unlike the N = 3 case where there is an infinity of such intervals. We also show that for N = 5, 9, 15, 25 there are, respectively, 14, 16, 18, 20 critical Sitnikov periodic orbits from which 3D families (no longer rectilinear) bifurcate. We have also studied the physically interesting question of the extent of bounded dynamics away from the z-axis, taking initial conditions on x, y planes, at constant z(0) = z ₀ values, where z ₀ lies within the interval of stable rectilinear motions. We performed a similar study of the dynamics near some members of 3D families of periodic solutions and found, on suitably chosen Poincaré surfaces of section, “islands” of ordered motion, while away from them most orbits become chaotic and eventually escape to infinity. Finally, we solve the equations of motion of a small mass in the presence of a uniform rotating ring. Studying the stability of the vertical orbits in that case, we again discover a single stability interval, which, as N grows, tends to coincide with the stability interval of the N-body problem, when the values of the density and radius of the ring equal those of the corresponding system of N ? 1 primary masses.     

Periodic orbits and bifurcations in the Sitnikov four-body problem

We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż₀ varies within a finite interval (while z ₀ tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity.