Splendid isolation: local uniqueness of the centered co-circular relative equilibria in the <Emphasis Type="Italic">N</Emphasis>-body problem     

Marshall?HamptonEmail author 《Celestial Mechanics and Dynamical Astronomy》2016,124(2):145-153

We study the neighborhood of the equal mass regular polygon relative equilibria in the N-body probem, and show that this relative equilibirum is isolated among the co-circular configurations (in which each point lies on a common circle) for which the center of mass is located at the center of the common circle. It is also isolated in the sense that a sufficiently small mass cannot be added to the common circle to form a \(N+1\)-body relative equilibrium. These results provide strong evidence for a conjecture that the equal mass regular polygon is the only co-circular relative equilibrium with its center of mass located at the center of the common circle.  相似文献   

The equations of relative motion in the orbital reference frame     

Stefano?CasottoEmail author 《Celestial Mechanics and Dynamical Astronomy》2016,124(3):215-234

The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill–Clohessy–Wiltshire equations. Circular motion is not, however, a solution when the Earth’s flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the \(J_2\) effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the \(J_2\) perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a \(J_2\)-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill–Clohessy–Wiltshire equations for circular reference motion, or the de Vries/Tschauner–Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the \(J_2\) perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Hill–Clohessy–Wiltshire, or the Tschauner–Hempel equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a \(J_2\)-perturbed reference orbit is also introduced.  相似文献   

Uncertainty propagation in orbital mechanics via tensor decomposition     

Yifei?SunEmail author  Mrinal?Kumar 《Celestial Mechanics and Dynamical Astronomy》2016,124(3):269-294

Uncertainty forecasting in orbital mechanics is an essential but difficult task, primarily because the underlying Fokker–Planck equation (FPE) is defined on a relatively high dimensional (6-D) state–space and is driven by the nonlinear perturbed Keplerian dynamics. In addition, an enormously large solution domain is required for numerical solution of this FPE (e.g. encompassing the entire orbit in the \(x-y-z\) subspace), of which the state probability density function (pdf) occupies a tiny fraction at any given time. This coupling of large size, high dimensionality and nonlinearity makes for a formidable computational task, and has caused the FPE for orbital uncertainty propagation to remain an unsolved problem. To the best of the authors’ knowledge, this paper presents the first successful direct solution of the FPE for perturbed Keplerian mechanics. To tackle the dimensionality issue, the time-varying state pdf is approximated in the CANDECOMP/PARAFAC decomposition tensor form where all the six spatial dimensions as well as the time dimension are separated from one other. The pdf approximation for all times is obtained simultaneously via the alternating least squares algorithm. Chebyshev spectral differentiation is employed for discretization on account of its spectral (“super-fast”) convergence rate. To facilitate the tensor decomposition and control the solution domain size, system dynamics is expressed using spherical coordinates in a noninertial reference frame. Numerical results obtained on a regular personal computer are compared with Monte Carlo simulations.  相似文献   

Dynamics of the Jupiter Trojans with Saturn’s perturbation when the two planets are in migration     

Xiyun?HouEmail author  Daniel?J.?Scheeres  L.?Liu 《Celestial Mechanics and Dynamical Astronomy》2016,125(4):451-484

In a previous paper (Hou et al. in Celest Mech Dyn Astron 119:119–142, 2014a), the problem of dynamical symmetry between two Jupiter triangular libration points (TLPs) with Saturn’s perturbation in the present configuration of the two planets was studied. A small short-time scale spatial asymmetry exists but gradually disappears with the time going, so the planar stable regions around the two Jupiter TLPs should be dynamically symmetric from a longtime perspective. In this paper, the symmetry problem is studied when the two planets are in migration. Several mechanisms that can cause asymmetries are discussed. Studies show that three important ones are the large short-time scale spatial asymmetry when Jupiter and Saturn are in resonance, the changing orbits of Jupiter and Saturn in the planet migration process, and the chaotic nature of Trojan orbits during the planet migration process. Their joint effects can cause an observable difference to the two Jupiter Trojan swarms. The thermal Yarkovsky effect is also found to be able to cause dynamical differences to the two TLPs, but generally they are too small to be practically observed.  相似文献   

Integrability of motion around galactic razor-thin disks     

Ronaldo?S.?S.?VieiraEmail author  Javier?Ramos-Caro 《Celestial Mechanics and Dynamical Astronomy》2016,126(4):483-500

We consider the three-dimensional bounded motion of a test particle around razor-thin disk configurations, by focusing on the adiabatic invariance of the vertical action associated with disk-crossing orbits. We find that it leads to an approximate third integral of motion predicting envelopes of the form \(Z(R)\propto [\varSigma (R)]^{-1/3}\), where R is the radial galactocentric coordinate, Z is the z-amplitude (vertical amplitude) of the orbit and \(\varSigma \) represents the surface mass density of the thin disk. This third integral, which was previously formulated for the case of flattened 3D configurations, is tested for a variety of trajectories in different thin-disk models.  相似文献   

Secular tidal changes in lunar orbit and Earth rotation     

James?G.?WilliamsEmail authorView author&#;s OrcID profile  Dale?H.?Boggs 《Celestial Mechanics and Dynamical Astronomy》2016,126(1-3):89-129

Small tidal forces in the Earth–Moon system cause detectable changes in the orbit. Tidal energy dissipation causes secular rates in the lunar mean motion n, semimajor axis a, and eccentricity e. Terrestrial dissipation causes most of the tidal change in n and a, but lunar dissipation decreases eccentricity rate. Terrestrial tidal dissipation also slows the rotation of the Earth and increases obliquity. A tidal acceleration model is used for integration of the lunar orbit. Analysis of lunar laser ranging (LLR) data provides two or three terrestrial and two lunar dissipation parameters. Additional parameters come from geophysical knowledge of terrestrial tides. When those parameters are converted to secular rates for orbit elements, one obtains dn/dt = \(-25.97\pm 0.05 ''/\)cent\(^{2}\), da/dt = 38.30 ± 0.08 mm/year, and di/dt = ?0.5 ± 0.1 \(\upmu \)as/year. Solving for two terrestrial time delays and an extra de/dt from unspecified causes gives \(\sim \) \(3\times 10^{-12}\)/year for the latter; solving for three LLR tidal time delays without the extra de/dt gives a larger phase lag of the N2 tide so that total de/dt = \((1.50 \pm 0.10)\times 10^{-11}\)/year. For total dn/dt, there is \(\le \)1 % difference between geophysical models of average tidal dissipation in oceans and solid Earth and LLR results, and most of that difference comes from diurnal tides. The geophysical model predicts that tidal deceleration of Earth rotation is \(-1316 ''\)/cent\(^{2}\) or 87.5 s/cent\(^{2}\) for UT1-AT, a 2.395 ms/cent increase in the length of day, and an obliquity rate of 9 \(\upmu \)as/year. For evolution during past times of slow recession, the eccentricity rate can be negative.  相似文献   

The Effect of Key Parameters on the Strength of a Dispersive Soil Stabilized with Sustainable Binders     

Filho  Hugo Carlos Scheuermann  Martins  Carlos Guilherme  Menezes  Ricardo José Wink  Dornelles  Lucas Eduardo  Consoli  Nilo Cesar 《Geotechnical and Geological Engineering》2021,39(7):5395-5404

Geotechnical and Geological Engineering - Dispersive soils are susceptible to phenomena of internal and external erosion when in contact with relatively pure water due to its particle’s...  相似文献   

Would different methodologies for assessing carbon leakage exposure lead to different risk levels? A case study of the Brazilian industry     

Luan Santos  André F. P. Lucena  Rafael Garaffa 《Climate Policy》2019,19(9):1102-1116

Energy-intensive industries play an important role in low-carbon development, being particularly exposed to climate policies. Concern over possible carbon leakage in this sector poses a major challenge for designing effective carbon pricing instruments (CPI). Different methodologies for assessing carbon leakage exposure are currently used by different jurisdictions, each of them based on different approaches and indicators. This paper aims to analyse the extent to which the use of different methodologies leads to different results in terms of exposure to the risk of carbon leakage, using the Brazilian industry sector as a case study. Results indicate that carbon leakage exposure is an expected outcome of eventual CPI implementation in Brazilian industry. However, results vary according to the chosen methodology, so the definition of the criteria is paramount for assessing sectoral exposure to the risk of carbon leakage.

Key policy insights

• Despite increasing discussion about the implementation of carbon pricing on the Brazilian industrial sector, the evaluation of carbon leakage risks is still neglected.

• Assessments of the risk of carbon leakage are directly related to the indicators and criteria used by each methodology. Thus, a given subsector may present different levels of exposure to carbon leakage depending on the methodological choice.

• More than a purely technical discussion, the methodological definition of carbon leakage risk is a political discussion – it can be well-conducted, leading to the success of a CPI, or even sabotaged, by implicitly subsidizing energy-intensive industries.

  相似文献   

Impact of different sampling rates on precise point positioning performance using online processing service     

Serdar Erol  Reha Metin Alkan  ?.Murat Ozulu  Veli Il?i 《地球空间信息科学学报》2021,24(2):302-312

In this study, the effect of different sampling rates (i.e. observation recording interval) on the Precise Point Positioning (PPP) solutions in terms of accuracy was investigated. For this purpose, a field test was carried out in ?orum province, Turkey, on 11 September 2019. Within this context, a Geodetic Point (GP) was established and precisely coordinated. A static GNSS measurement was occupied on the GP for about 4-hour time at 0.10 second (s)/10 Hz measurement intervals with the Trimble R10 geodetic grade GNSS receiver. The original observation file was converted to RINEX format and then decimated into the different data sampling rates as 0.2 s, 0.5 s, 1 s, 5 s, 10 s, 30 s, 60 s, and 120 s. All these RINEX observation files were submitted to the Canadian Spatial Reference System-Precise Point Positioning (CSRS-PPP) online processing service the day after the data collection date by choosing both static and kinematic processing options. In this way, PPP-derived static coordinates, and the kinematic coordinates of each measurement epoch were calculated. The PPP-derived coordinates obtained from each decimated sampling intervals were compared to known coordinates of the GP for northing, easting, 2D position, and height components. According to the static and kinematic processing results, high data sampling rates did not change the PPP solutions in terms of accuracy when compared to the results obtained using lower sampling rates. The results of this study imply that it was not necessary to collect GNSS data with high-rate intervals for many surveying projects requiring cm-level accuracy.  相似文献   

Cartographic Design and the Space–Time Cube     

Irma Kveladze  Menno-Jan Kraak  Corné P.J.M. van Elzakker 《The Cartographic journal》2019,56(1):73-90

This paper reports on the results of an empirical evaluation that aimed to define the effectiveness and efficiency of different visual variables in depicting the Space–Time Cube’s (STC) content. Existing STC applications demonstrate that the most used visual variables are size and colour hue. Less is known, however, about their usability metrics. The research sets design criteria for STC contents, such as space–time paths, based on the cartographic design theory. The visual variables colour hue, colour value, colour saturation, size and orientation have been applied in two different use case studies. Besides, to support the three-dimensional visual environment, depth cues such as shading and transparency were considered too. User tests have been executed based on real-world problems with particular attention for the visualization strategy and data complexity. The outcomes revealed the most efficient and effective visual variables to represent data of various complexities in the STC.  相似文献