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A study to demonstrate how the dynamics of galaxies may be investigated through the creation of galaxies within a computer model is presented. The numerical technique for simulating galaxies is shown to be both highly efficient and highly robust. Consideration is given to the anatomy of a galaxy, the gravitational N-body problem, numerical approaches to the N-body problem, use of the Poisson equation, and the symplectic integrator.
Different methods are proposed and tested for transforming a non-linear differential system, and more particularly a Hamiltonian one, into a map without integrating the whole orbit as in the well-known Poincare return map technique. We construct piecewise polynomial maps by coarse-graining the phase-space surface of section into parallelograms and using either only values of the Poincare maps at the vertices or also the gradient information at the nearest neighbors to define a polynomial approximation within each cell. The numerical experiments are in good agreement with both the real symplectic and Poincare maps.
The point form is used as a framework for formulating a relativistic quantum mechanics, with the mass operator carrying the interactions of underlying constituents. A symplectic Lie algebra of mass operators is introduced from which a relativistic harmonic oscillator mass operator is formed. Mass splittings within the degenerate harmonic oscillator levels arise from relativistically invariant spin-spin, spin-orbit, and tensor mass operators. Internal flavor (and color) symmetries are introduced which make it possible to formulate a relativistic SU(6) model of baryons (and mesons). Careful attention is paid to the permutation symmetry properties of the hadronic wave functions, which are written as polynomials in Bargmann spaces.
Recently a new class of numerical integration methods - 'mixed variable symplectic integrators' - has been introduced for studying long-term evolution in the conservative gravitational few-body problem. These integrators are an order of magnitude faster than conventional ordinary differential equations (ODE) integration methods. Here we present a simple modification of this method to include small non-gravitational forces. The new scheme provides a similar advantage of computational speed for a larger class of problems in Solar System dynamics.
The generalized Hamming weights of linear codes were first introduced by Wei. These are fundamental parameters related to the minimal overlap structures of the subcodes and very useful in several fields. It was found that the chain condition of a linear code is convenient in studying the generalized Hamming weights of the product codes. In this paper we consider a class of codes defined over some varieties in projective spaces over finite fields, whose generalized Hamming weights can be determined by studying the orbits of subspaces of the projective spaces under the actions of classical groups over finite fields, i.e., the symplectic groups, the unitary groups and orthogonal groups. We give the weight hierarchies and generalized weight spectra of the codes from Hermitian varieties and prove that the codes satisfy the chain condition.
The petrology of a massive olivine-sulphide interplanetary dust particle shows melting of Fe,Ni-sulphide plus complete loss of sulphur and subsequent quenching to a mixture of iron-oxides and Fe,Ni-metal. Oxidation of the fayalite component in olivine produced maghemite discs and cellular intergrowths with olivine and rare andradite-rich garnet. Cellular reactions require no long-range solid-state diffusion and are kinetically favourable during pyrometamorphic oxidation. Local melting of the cellular intergrowths resulted in three dimensional symplectic textures. Dynamic pyrometamorphism of this asteroidal particle occurred at approx. 1100 C during atmospheric entry flash (5-15 s) heating.
The key to studying central regions by means of nobody numerical experiments is to concentrate on the central few parsecs of a galaxy, replacing the remainder of the galaxy by a suitable boundary condition, rather after the manner in which stellar interiors can be studied without a detailed stellar atmosphere by replacing the atmosphere with a boundary condition. Replacements must be carefully designed because the long range gravitational force means that the core region is sensitive to mass outside that region and because particles can exchange between the outer galaxy and the core region. We use periodic boundary conditions, coupled with an iterative procedure to generate initial particle loads in isothermal equilibrium. Angular momentum conservation is ensured for problems including systematic rotation by a circular reflecting boundary and by integrating in a frame that rotates with the mean flow. Mass beyond the boundary contributes to the gravitational potential, but does not participate in the dynamics. A symplectic integration scheme has been developed for rotating coordinate systems. This combination works well, leading to robust configurations. Some preliminary results with this combination show that: (1) Rotating systems are extremely sensitive to non-axisymmetric external potentials, and (2) that a second core, orbiting near the main core (like the M31 second core system), shows extremely rapid orbital decay. The experimental setups will be discussed, along with preliminary results.
We made computer simulations of orbital evolution for intervals of at least 5-10 Myr of N=2000 Jupiter-crossing objects (JCOs) with initial orbits close to those of real comets with period P less than 10 yr, 500 objects with orbits close to that of Comet 10P, and the asteroids initially located at the 3:1 and 5:2 resonances with Jupiter at initial eccentricity e(sub 0)=0.15 and initial inclination i(sub 0)=10(sup 0). The gravitational influence of all planets, except for Mercury and Pluto, was taken into account (without dissipative factors). We calculated the probabilities of collisions of bodies with the terrestrial planets, using orbital elements obtained with a step equal to 500 yr, and then summarized the results for all bodies, obtaining, the total probability Psigma of collisions with a planet and the total time interval Tsigma during which perihelion distance q of bodies was less than a semimajor axis of the planet. The values of p(sub r) =10(exp 6)Psigma/N and T(sub r)=T/1000 yr (where T=Tsigma/N) are presented in a table together with the ratio r of the total time interval when orbits were of Apollo type (at a greater than 1 AU, q less than 1.017 AU, e less than 0.999) to that of Amor type (1.017 less than q less than 1.33 AU), r(sub 2) is the same as r but for Apollo objects with e less than 0.9. For asteroids we present only results obtained by direct integration, as a symplectic method can give large errors for these resonances.
Most stars reside in multiple star systems; however, virtually all models of planetary growth have assumed an isolated single star. Numerical simulations of the collapse of molecular cloud cores to form binary stars suggest that disks will form within such systems. Observations indirectly suggest disk material around one or both components within young binary star systems. If planets form at the right places within such circumstellar disks, they can remain in stable orbits within the binary star systems for eons. We are simulating the late stages of growth of terrestrial planets within binary star systems, using a new, ultrafast, symplectic integrator that we have developed for this purpose. We show that the late stages of terrestrial planet formation can indeed take place in a wide variety of binary systems and we have begun to delineate the range of parameter space for which this statement is true. Results of our initial simulations of planetary growth around each star in the alpha Centauri system and other 'wide' binary systems, as well as around both stars in very close binary systems, will be presented.
Most stars reside in multiple star systems; however, virtually all models of planetary growth have assumed an isolated single star. Numerical simulations of the collapse of molecular cloud cores to form binary stars suggest that disks will form within such systems. Observations indirectly suggest disk material around one or both components within young binary star systems. If planets form at the right places within such circumstellar disks, they can remain in stable orbits within the binary star systems for eons. We are simulating the late stages of growth of terrestrial planets around close binary stars, using a new, ultrafast, symplectic integrator that we have developed for this purpose. The sum of the masses of the two stars is one solar mass, and the initial disk of planetary embryos is the same as that used for simulating the late stages of terrestrial planet growth within our Solar System and in the Alpha Centauri wide binary star system. Giant planets &are included in the simulations, as they are in most simulations of the late stages of terrestrial planet accumulation in our Solar System. When the stars travel on a circular orbit with semimajor axis of up to 0.1 AU about their mutual center of mass, the planetary embryos grow into a system of terrestrial planets that is statistically identical to those formed about single stars, but a larger semimajor axis and/or a significantly eccentric binary orbit can lead to significantly more dynamically hot terrestrial planet systems.
Directly imaging extrasolar terrestrial planets necessarily means contending with the astrophysical noise of exozodiacal dust and the resonant structures created by these planets in exozodiacal clouds. Using a custom tailored hybrid symplectic integrator we have constructed 120 models of resonant structures created by exo-Earths and super-Earths on circular orbits interacting with collisionless steady-state dust clouds around a Sun-like star. Our models include enough particles to overcome the limitations of previous simulations that were often dominated by a handful of long-lived particles, allowing us to quantitatively study the contrast of the resulting ring structures. We found that in the case of a planet on a circular orbit, for a given star and dust source distribution, the morphology and contrast of the resonant structures depend on only two parameters: planet mass and (square root)ap/Beta, where ap is the planet's semi-major axis and Beta is the ratio of radiation pressure force to gravitational force on a grain. We constructed multiple-grain-size models of 25,000 particles each and showed that in a collisionless cloud, a Dohnanyi crushing law yields a resonant ring whose optical depth is dominated by the largest grains in the distribution, not the smallest. We used these models to estimate the mass of the lowest-mass planet that can be detected through observations of a resonant ring for a variety of assumptions about the dust cloud and the planet's orbit. Our simulations suggest that planets with mass as small as a few times Mars' mass may produce detectable signatures in debris disks at ap greater than or approximately equal to 10 AU.
Almahata Sitta (AS) is a polymict breccia mainly composed of variable ureilite lithologies with small amounts of chondritic lithologies [1]. Fe metal is a common accessory phase in ureilites, but our earlier study on Fe metals in one of AS fragments (#44) revealed a unique mineralogy never seen in other ureilites [2,3]. In this abstract we report detailed transmission electron microscopy (TEM) on these metal grains to better understand the thermal history of ureilites. We prepared FIB sections of AS#44 by JEOL JIB-4000 from the PTS that was well characterized by SEM-EBSD in our earlier study [2]. The sections were then observed by STEM (JEOL JEM- 2100F). One of the FIB sections shows a submicron-sized symplectic intergrown texture composed of Fe metal (kamacite), Fe carbide (cohenite), Fe phosphide (schreibersite), and Fe sulfide (troilite). Each phase has an identical SAED pattern in spite of its complex texture, suggesting co-crystallization of all phases. This is probably caused by shock re-melting of pre-existing metal + graphite to form a eutectic-looking texture. The other FIB section is mostly composed of homogeneous Fe metal (93 wt% Fe, 5 wt% Ni, and 2 wt% Si), but BF-STEM images exhibited the presence of elongated lathy grains (approx. 2 microns long) embedded in the interstitial matrix. The SAED patterns from these lath grains could be indexed by alpha-Fe (bcc) while interstitial areas are gamma-Fe (fcc). The elongated alpha-Fe grains show tweed-like structures suggesting martensite transformation. Such a texture can be formed by rapid cooling from high temperature where gamma-Fe was stable. Subsequently alpha-Fe crystallized, but gamma-Fe remained in the interstitial matrix due to quenching from high temperature. This scenario is consistent with very rapid cooling history of ureilites suggested by silicate mineralogy.
Introduction: Analyses of comet 81P/Wild 2 samples re-turned from the Stardust mission have uncovered surprising simi-larities to meteoritic material, including the identification of inner solar system grains [1-3]. The TEM characterization of terminal particle (TP) 4 from Stardust track #147 revealed an assemblage consisting of symplectically intergrown pentlandite and nanocrys-talline maghemite coexisting with high-Ca pyroxene [4]. Mineral-ogically similar cosmic symplectites (COS) containing pentlandite and magnetite in the primitive Acfer 094 meteorite are highly de-pleted in 16O (17O, 18O ~ 180 per mille) [5-7]. This isotopic signature is proposed to record alteration with primordial solar nebula water. Conversely, the normal O isotopic composition of the Stardust COS indicates alteration by a different aqueous reservoir, perhaps on the comet [8]. In this study, we analyzed the Wild 2 COS for S isotopes to further constrain its origin. Experimental: Thin sections of TP4 (12 m) were produced and their mineralogy was thoroughly characterized by TEM. Two of the sections were analyzed for O isotopes by isotopic imaging in the JSC NanoSIMS 50L. The sample in one of the slices was completely consumed. The remaining material in the adjacent slice was analyzed simultaneously for 16O, 32S, 33S, 34S, and 56Fe16O in electron multipliers using a Cs+ primary ion beam. Quasi-simulta-neous arrival (QSA) can have a significant effect on S isotopic ra-tios when using electron multipliers, resulting in undercounting of 32S [9]. Canyon Diablo troilite (CDT) was measured numerous times to deduce a correction factor for QSA and ensure measure-ment reproducibility. Isotopic ratios are reported relative to CDT. Results and Discussion: The Wild 2 COS is enriched in the heavy S isotopes relative to CDT (33S = 6.5 +/- 1.6 per mille; 34S = 5.1 +/- 0.7 per mille; 1). The degree of 33S enrichment indicates mass-inde-pendent fractionation (MIF) with 33S = 3.9 +/- 1.7 per mille. MIF of S has been observed in some chondrules (33S up to 0.11per mille) [10], but this effect has not been identified in sulfides from carbonaceous chondrites [11] or IDPs [12]. S isotopic analysis of Stardust impact craters also did not reveal MIF or anomalies, save for one potential 32S-rich presolar sulfide [13]. Measurement errors on these impact craters were much larger than those in this study, however. MIF of S has been proposed to result from heterogeneities in the solar neb-ula from nucleosynthetic components [14] or photochemical irra-diation of solar nebula gas [10]. Presolar SiC grains are observed to have 32S enrichments [15, 16] contrary to the S isotopic compo-sition of the cometary COS. The S isotopic composition more likely reflects irradiation of nebular gas.