Gallopade

https://digitalcommons.library.umaine.edu/mmb-ps/2844/thumbnail.jp

Symplectic integration of space debris motion considering several Earth's shadowing models

In this work, we present a symplectic integration scheme to numerically compute space debris motion. Such an integrator is particularly suitable to obtain reliable trajectories of objects lying on high orbits, especially geostationary ones. Indeed, it has already been demonstrated that such objects could stay there for hundreds of years. Our model takes into account the Earth's gravitational potential, luni-solar and planetary gravitational perturbations and direct solar radiation pressure. Based on the analysis of the energy conservation and on a comparison with a high order non-symplectic integrator, we show that our algorithm allows us to use large time steps and keep accurate results. We also propose an innovative method to model Earth's shadow crossings by means of a smooth shadow function. In the particular framework of symplectic integration, such a function needs to be included analytically in the equations of motion in order to prevent numerical drifts of the energy. For the sake of completeness, both cylindrical shadows and penumbra transitions models are considered. We show that both models are not equivalent and that big discrepancies actually appear between associated orbits, especially for high area-to-mass ratios

Chemical composition of zircons from the Cornubian Batholith of SW England and comparison with zircons from other European Variscan rare-metal granites

This is the author accepted manuscript. The final version is available from the Mineralogical Society via the DOI in this record.Zircon from 14 representative granite samples of the late-Variscan Cornubian Batholith in SW England was analyzed for W, P, As, Nb, Ta, Si, Ti, Zr, Hf, Th, U, Y, La, Ce, Pr, Nd, Sm, Gd, Dy, Er, Yb, Al, Sc, Bi, Mn, Fe, Ca, Pb, Cu, S, and F using EPMA. Zircons from the biotite and tourmaline granites are poor in minor and trace elements, usually containing 1.0–1.5 wt% HfO2, <0.5 wt% UO2 and P2O5, <0.25 wt% Y2O3, <0.2 wt% Sc203 and Bi2O3, and <0.1 wt% ThO2. Zircon from topaz granites from the St. Austell Pluton, Meldon Aplite and Megiliggar Rocks are slightly enriched in Hf (up to 4 wt% HfO2), U (1– 3.5 wt% UO2), and Sc (0.5–1 wt% Sc2O3). Scarce metamictized zircon grains are somewhat enriched in Al, Ca, Fe, and Mn. The decrease of the zircon Zr/Hf ratio, a reliable magma fractionation index, from 110-60 in the biotite granites to 30-10 in the most evolved topaz granites (Meldon Aplite and Megiliggar Rocks), supports a comagmatic origin of the biotite and topaz granites via long fractionation of common peraluminous crustal magma. In comparison with other European rare-metal provinces, the overall contents of trace elements in Cornubian zircons are low and the Zr/Hf- and U/Th-ratios show lower degrees of fractionation of the parental melt.This contribution was supported by the Czech Science Foundation, project No. GA14-13600S and RVO 67985831. Bernard Bingen and one anonymous member of the Editorial Board are thanked for careful review and inspiring comments

Stress field and spin axis relaxation for inelastic triaxial ellipsoids

A compact formula for the stress tensor inside a self-gravitating, triaxial ellipsoid in an arbitrary rotation state is given. It contains no singularity in the incompressible medium limit. The stress tensor and the quality factor model are used to derive a solution for the energy dissipation resulting in the damping (short axis mode) or excitation (long axis) of wobbling. In the limit of an ellipsoid of revolution, we compare our solution with earlier ones and show that, with appropriate corrections, the differences in damping times estimates are much smaller than it has been claimed. This version implements corrections of misprints found in the MNRAS published text.Comment: 14 pages, 6 figures, published in Monthly Notices RAS (containing misprints

Note on the generalized Hansen and Laplace coefficients

Recently, Breiter et al (2004) reported the computation of Hansen coefficients $X_k^{\gamma,m}$ for non integer values of $\gamma$. In fact, the Hansen coefficients are closely related to the Laplace $b_{s}^{(m)}$, and generalized Laplace coefficients $b_{s,r}^{(m)}$ (Laskar and Robutel, 1995) that do not require $s,r$ to be integers. In particular, the coefficients X_0^{\g,m} have very simple expressions in terms of the usual Laplace coefficients b_{\g+2}^{(m)}, and all their properties derive easily from the known properties of the Laplace coefficients.Comment: 9/11/200

Yarkovsky-O’Keefe-Radzievskii-Paddack effect on tumbling objects

A semi-analytical model of the Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effect on an asteroid spin in a non-principal axis rotation state is developed. The model describes the spin-state evolution in Deprit–Elipe variables, first-order averaged with respect to rotation and Keplerian orbital motion. Assuming zero conductivity, the YORP torque is represented by spherical harmonic series with vectorial coefficients, allowing us to use any degree and order of approximation. Within the quadrupole approximation of the illumination function we find the same first integrals involving rotational momentum, obliquity and dynamical inertia that were obtained by Cicaló & Scheeres. The integrals do not exist when higher degree terms of the illumination function are included, and then the asymptotic states known from Vokrouhlický et al. appear. This resolves an apparent contradiction between earlier results. Averaged equations of motion admit stable and unstable limit cycle solutions that were not previously detected. Non-averaged numerical integration by the Taylor series method for an exemplary shape of 3103 Eger is in good agreement with the semi-analytical theory

NIMASTEP: a software to modelize, study and analyze the dynamics of various small objects orbiting specific bodies

NIMASTEP is a dedicated numerical software developed by us, which allows one to integrate the osculating motion (using cartesian coordinates) in a Newtonian approach of an object considered as a point-mass orbiting a homogeneous central body that rotates with a constant rate around its axis of smallest inertia. The code can be applied to objects such as particles, artificial or natural satellites or space debris. The central body can be either any terrestrial planet of the solar system, any dwarf-planet, or even an asteroid. In addition, very many perturbations can be taken into account, such as the combined third-body attraction of the Sun, the Moon, or the planets, the direct solar radiation pressure (with the central body shadow), the non-homogeneous gravitational field caused by the non-sphericity of the central body, and even some thrust forces. The simulations were performed using different integration algorithms. Two additional tools were integrated in the software package; the indicator of chaos MEGNO and the frequency analysis NAFF. NIMASTEP is designed in a flexible modular style and allows one to (de)select very many options without compromising the performance. It also allows one to easily add other possibilities of use. The code has been validated through several tests such as comparisons with numerical integrations made with other softwares or with semi-analytical and analytical studies. The various possibilities of NIMASTEP are described and explained and some tests of astrophysical interest are presented. At present, the code is proprietary but it will be released for use by the community in the near future. Information for contacting its authors and (in the near future) for obtaining the software are available on the web site http://www.fundp.ac.be/en/research/projects/page_view/10278201/Comment: Astronomy & Astrophysics - Received: 25 November 2011 / Accepted: 27 February 2012 -- 14 pages, 4 figure

Orbital similarity functions - application to asteroid pairs

The paper expands the idea of Vokrouhlický and Nesvorný who used a modified Zappalà et al. metric with osculating elements in search for pairs of asteroids suspected of having a common origin. Using six different orbital similarity functions, we find that five of them display a similar excess of close pairs in the catalogue of osculating elements. The excess is even higher when mean orbital elements are used. Similarly, when the mean elements are applied, there is a better agreement between the closest pairs found in the same catalogue using different metrics. The common subset of 62 pairs from five lists of 100 closest pairs according to different distance functions is provided. Investigating an artificial sample of asteroid orbital pairs with a known initial orbital velocity difference we find that the Drummond metric best preserves orbital proximity over long time intervals

Where do long-period comets come from? 26 comets from the non-gravitational Oort spike

The apparent source region (or regions) of long-period comets as well as the definition of the dynamically new comet are still open questions.The aim of this investigation is to look for the apparent source of selected long period comets and to refine the definition of dynamically new comets. We show that incorporation of the non-gravitational forces into the orbit determination process significantly changes the situation. We determined precise non-gravitational orbits of all investigated comets and next followed numerically their past and future motion during one orbital period. Applying ingenious Sitarski's method of creating swarms of virtual comets compatible with observations, we were able to derive the uncertainties of original and future orbital elements, as well as the uncertainties of the previous and next perihelion distances. We concluded that the past and future evolution of cometary orbits under the Galactic tide perturbations is the only way to find which comets are really dynamically new. We also have shown that a significant percentage of long-period comets can visit the zone of visibility during at least two or three consecutive perihelion passages.Comment: Accepted for publication in MNRA