Three-dimensional Orbits Research Articles

Potential kick velocity distribution of black hole X-ray binaries and implications for natal kicks

Abstract We use very long baseline interferometry to measure the proper motions of three black hole X-ray binaries (BHXBs). Using these results together with data from the literature and Gaia DR2 to collate the best available constraints on proper motion, parallax, distance, and systemic radial velocity of 16 BHXBs, we determined their three-dimensional Galactocentric orbits. We extended this analysis to estimate the probability distribution for the potential kick velocity (PKV) a BHXB system could have received on formation. Constraining the kicks imparted to BHXBs provides insight into the birth mechanism of black holes (BHs). Kicks also have a significant effect on BH–BH merger rates, merger sites, and binary evolution, and can be responsible for spin–orbit misalignment in BH binary systems. 75 per cent of our systems have potential kicks $\gt 70\, \rm {km\,s^{-1}}$. This suggests that strong kicks and hence spin–orbit misalignment might be common among BHXBs, in agreement with the observed quasi-periodic X-ray variability in their power density spectra. We used a Bayesian hierarchical methodology to analyse the PKV distribution of the BHXB population, and suggest that a unimodal Gaussian model with a mean of 107 $\pm \,\,16\, \rm {km\,s^{-1}}$ is a statistically favourable fit. Such relatively high PKVs would also reduce the number of BHs likely to be retained in globular clusters. We found no significant correlation between the BH mass and PKV, suggesting a lack of correlation between BH mass and the BH birth mechanism. Our python code allows the estimation of the PKV for any system with sufficient observational constraints.

Plane Gravitational Waves in Spatially-Homogeneous Models of type-(3.1) Stäckel Spaces

A classification is performed of the spaces with plane-wave metrics in accordance with the presence of three-dimensional subgroups of the group of space-time motion with three-dimensional spatial orbits. In order to obtain spatially-homogeneous models, exact solutions to Einstein’s vacuum equations are found. To obtain these solutions, the Petrov and Bianchi classifications are performed. The results obtained on the classification of the plane-wave metrics using the assumed or three-dimensional subgroups of the group of motion with three-dimensional spatial orbits can be used for any metric gravitational theories.

Three-dimensional micro-billiard lasers: The square pyramid

Microlasers are of ample interest for advancing quantum chaos studies at the intersection of wave dynamics and geometric optics in resonators. However, the mode structures of three-dimensional microlasers without rotational symmetry remained largely unexplored due to fabrication limitations which have been overcome by now. Previous studies of such cavities revealed lasing modes localized on periodic orbits exclusively confined to a single plane. In this work, we report on the characterization of pyramidal, polymer-based microlasers and demonstrate that the lasing modes are localized on a genuine three-dimensional periodic orbit. The consequences on the laser features are further discussed, in particular stability and polarization issues.

Secular Transport during Disk Dispersal: The Case of Kepler-419

Abstract Due to fortuitous circumstances, the two giant planets around Kepler-419 have well characterized three-dimensional orbits. They are nearly coplanar to each other; the inner one has a large eccentricity (≃0.82); and the apses of the two orbits librate around anti-alignment. Such a state defies available proposals for large eccentricities. We argue that it is instead uniquely produced by a decaying protoplanetary disk. When the disk was massive, its precessional effect on the planets forced the two apses to center around an anti-aligned state. And as the disk is gradually eroded, the pair of planets are adiabatically transported to a new state where most of the eccentricity (or rather, the angular momentum deficit) is transferred to the inner planet, and the two apses are largely anti-aligned. During this transport, any initial mutual inclination may be reduced or enhanced; either may be compatible with the current constraints. So a primordial disk can drive up planet eccentricities both in resonant planet pairs (as has been shown for GJ 876) and in secularly-interacting, non-resonant pairs. The mechanism discussed here may be relevant for forming hot Jupiters and for explaining the observed eccentricities of warm and cold giant planets.

Lossless Positron Injection into a Magnetic Dipole Trap.

The high-efficiency injection of a low-energy positron beam into the confinement volume of a magnetic dipole has been demonstrated experimentally. This was accomplished by tailoring the three-dimensional guiding-center drift orbits of positrons via optimization of electrostatic potentials applied to electrodes at the edge of the trap, thereby producing localized and essentially lossless cross-field particle transport by means of the E×B drift. The experimental findings are reproduced and elucidated by numerical simulations, enabling a comprehensive understanding of the process. These results answer key questions and establish methods for use in upcoming experiments to create an electron-positron plasma in a levitated dipole device.

Neighboring optimal guidance and constrained attitude control applied to three-dimensional lunar ascent and orbit injection

Future human or robotic missions to the Moon will require efficient ascent path and accurate orbit injection maneuvers, because the dynamical conditions at injection affect the subsequent phases of spaceflight. This research is focused on the original combination of two techniques applied to lunar ascent modules, i.e. (i) the recently-introduced variable-time-domain neighboring optimal guidance (VTD-NOG), and (ii) a constrained proportional-derivative (CPD) attitude control algorithm. VTD-NOG belongs to the class of implicit guidance approaches, aimed at finding the corrective control actions capable of maintaining the spacecraft sufficiently close to the reference trajectory. CPD pursues the desired attitude using thrust vector control and side jet system, while constraining the rates of both the thrust deflection angle and the roll control torque. After determining the optimal two-dimensional ascent path, which represents the reference trajectory, VTD-NOG & CPD is applied in the presence of nonnominal flight conditions, namely those due to navigation and actuation errors, incorrect initial position, unpredictable oscillations of the propulsive thrust, and imperfect modeling of the spacecraft mass distribution and variation. These stochastic deviations are simulated in the context of extensive Monte Carlo campaigns, and yield three-dimensional perturbed trajectories. The numerical results obtained in this work unequivocally demonstrate that VTD-NOG & CPD represents an accurate and effective methodology for guidance and control of lunar ascent path and orbit injection.

Global search for periodic orbits in the irregular gravity field of a binary asteroid system

Periodic orbits are important keys to understand the motion of a massless particle in the vicinity of a binary asteroid system. Due to the complex gravity generated by the irregular-shaped asteroids, it is difficult to generate periodic orbits with an analytical method, except with the linearized dynamics in a small region around the libration points. This paper has presented a numerical method to search the three-dimensional periodic orbits in a global space. The grid searching method is used to find nearly periodic orbits, and then the shooting method is used to get the accurate periodic orbits. The method is applied to the binary asteroid (66391) 1999 KW4 to solve the periodic orbits in the vicinity. The periodic orbits are then classified into five categories according to their shapes and locations. The search method can also be used to find periodic orbits in the vicinity of other binary systems.

The Three-dimensional Orbit, Orbital Parallax, and Individual Masses of the Double-lined Spectroscopic Binaries HD 183255, HD 114882, and HD 30712

Abstract Recent speckle observations performed at the Discovery Channel Telescope, the Gemini North Telescope, and the Special Astrophysical Observatory 6 m Telescope have permitted us to calculate the visual orbit of SB2 HD 114882 for the first time and to improve the visual orbits of two other SB2 systems, HD 30712 and HD 183255, using algorithms published by the authors of this research. Recently, new high-quality spectroscopic orbits have been obtained for these binaries by other authors. We determine their 3D orbits, individual masses, and orbital parallaxes, and present them in this paper. The parallaxes are compared with those available from the Gaia mission, and a comparison between the values confirms the precision of the results obtained here.

USING ANDROID-BASED EQUATION PLOTTERS AS SUPPORTING TOOLS FOR TEACHING AND LEARNING ATOMIC ORBITALS

Chemistry undergraduate students are required to master the theory of quantum mechanics as the basic requirement to study the properties of the submicroscopic particle. However, many studies showed the students’ difficulties to bring the quantum mechanical concept to the macroscopic world and vise-versa. The use of computer visualization tool will very useful for teaching and learning the quantum concept, especially electron and its related atomic orbitals. This technological report paper demonstrates some of the Android-based applications i.e FuncXY, TriPlot 3D Graphing, and Graphing calculator to effectively, quickly, and accurately convert wave functions into various hydrogen atomic orbital visualizations. By selecting spherical coordinate and entering wave equation, these smartphone graphical devices visualize three-dimensional atomic orbitals. Users are also easy to rotate, zoom and manipulate the object. Finally, these user-friendly plotters are very recommended to assist chemistry educators in teaching quantum mechanics.

Constraining the three-dimensional orbits of galaxies under ram pressure stripping with convolutional neural networks

Ram pressure stripping (RPS) of gas from disk galaxies has long been considered to play vital roles in galaxy evolution within groups and clusters. For a given density of intracluster medium (ICM) and a given velocity of a disk galaxy, RPS can be controlled by two angles (theta and phi) that define the angular relationship between the direction vector of the galaxy's three-dimensional (3D) motion within its host cluster and the galaxy's spin vector. We here propose a new method in which convolutional neutral networks (CNNs) are used to constrain theta and phi of disk galaxies under RPS. We first train a CNN by using ~10^5 synthesized images of gaseous distributions of the galaxies from numerous RPS models with different theta and phi. We then apply the trained CNN to a new test RPS model to predict theta and phi. The similarity between the correct and predicted theta and $\phi$ is measured by cosine similarity (cos-Theta) with cos-Theta =1 being perfectly accurate prediction. We show that the average cos-Theta among test models is ~0.95, which means that theta and phi can be constrained well by applying the CNN to the spatial distributions of their gas. This result suggests that if the ICM is in hydrostatic equilibrium (thus not moving), the 3D orbit of a disk galaxy within its host cluster can be constrained by the spatial distribution of the gas being stripped by RPS. We discuss the applications of the method to HI surveys such as WALLABY and SKA projects.

Long-term evaluation of orbital dynamics in the Sun-planet system considering axial-tilt

In this paper, the axial-tilt (obliquity) effect of planets on the motion of planets’ orbiter in prolonged space missions has been investigated in the presence of the Sun gravity. The proposed model is based on non-simplified perturbed dynamic equations of planetary orbiter motion. From a new point of view, in this work, the dynamic equations regarding a disturbing body in elliptic inclined three-dimensional orbit are derived. The accuracy of this non-simplified method is validated with dual-averaged method employed on a generalized Earth–Moon system. It is shown that the neglected short-time oscillations in dual-averaged technique can accumulate and propel to remarkable errors in the prolonged evolution. After validation, the effects of the planet’s axial-tilt on eccentricity, inclination and right ascension of the ascending node of the orbiter are investigated. Moreover, a generalized model is provided to study the effects of third-body inclination and eccentricity on orbit characteristics. It is shown that the planet’s axial-tilt is the key to facilitating some significant changes in orbital elements in long-term mission and short-time oscillations must be considered in accurate prolonged evaluation.

Numerical periodic orbits of charged grains around magnetic planets

We apply a numerical searching method to investigate three-dimensional periodic orbits of charged dust particles in planetary magnetospheres. A classic generalized Stormer model of magnetic planets along with the parameters of Saturn is employed. More periodic orbits are found, besides the already known circular periodic orbits in or parallel to the equatorial plane. We divide all these orbits into six categories based on their appearances. By calculating the characteristic multipliers of the orbits, we investigate the stabilities of these periodic orbits.

Computation of three-dimensional periodic orbits in the sun-earth system

In this paper, a third-order analytic approximation is described for computing the three-dimensional periodic halo orbits near the collinear L1 and L2 Lagrangian points for the photo gravitational circular restricted three-body problem in the Sun-Earth system. The constructed third-order approximation is chosen as a starting initial guess for the numerical computation using the differential correction method. The effect of the solar radiation pressure on the location of two collinear Lagrangian points and on the shape of the halo orbits is discussed. It is found that the time period of the halo orbit increases whereas the Jacobi constant decreases around both the collinear points taking into account the solar radiation pressure of the Sun for the fixed out-of-plane amplitude.

Periodic solutions around the collinear equilibrium points in the perturbed restricted three-body problem with triaxial and radiating primaries for binary HD 191408, Kruger 60 and HD 155876 systems

The collinear equilibrium points and periodic motion around them are studied in the framework of the restricted three-body problem where the two primaries are triaxial rigid bodies which emit radiation. Firstly, the positions and stability of the collinear equilibria are studied for the HD 191408, Kruger 60 and HD 155876 binary systems. Then, the planar and three-dimensional periodic motion about these points is considered. Our study includes both semi-analytical and numerical determination of these motions. It is found that all families of planar periodic orbits emanating from these points terminate with asymptotic periodic orbits at the triangular equilibrium points while the corresponding families of three-dimensional periodic orbits terminate with planar periodic orbits. Families of Halo orbits bifurcating from the first vertical critical periodic orbit of the three planar Lyapunov families were also considered.

Non-invasive study of the three-dimensional structure of nanoporous triblock terpolymer membranes.

Nanoporous media are of great importance for drug delivery or filtration. Typically the pore structure of such media is characterized using high-resolution techniques such as electron microscopy or atomic force microscopy. However, these techniques are restricted to the surface of the material and/or are highly invasive. In a proof-of-concept experiment we have employed three-dimensional single-particle orbit tracking for testing the three-dimensional pore structure of a liquid filled nanoporous polystyrene-block-polyisoprene-block-poly(N-isopropylacrylamide) (PS-b-PI-b-PNiPAAm) triblock terpolymer membrane. Using fluorescent tracers with a diameter of about 10% of the relevant void structures, the tracking experiments yielded results that were comparable to those obtained from reference experiments using environmental scanning electron microscopy (eSEM). This testifies that single-particle orbit tracking can serve as a useful non-invasive alternative for characterising the structure of nanoporous materials.

Orbit Determination of High-Earth-Orbit Satellites by Satellite Laser Ranging

This study presents the application of satellite laser ranging (SLR) to orbit determination (OD) of high-Earth-orbit (HEO) satellites. Two HEO satellites are considered: the Quasi-Zenith Satellite-1 (QZS-1), a Japanese elliptical-inclinedgeosynchronous-orbit (EIGSO) satellite, and the Compass-G1, a Chinese geostationary-orbit (GEO) satellite. One week of normal point (NP) data were collected for each satellite to perform the OD based on the batch least-square process. Five SLR tracking stations successfully obtained 374 NPs for QZS-1 in eight days, whereas only two ground tracking stations could track Compass-G1, yielding 68 NPs in ten days. Two types of station bias estimation and a station data weighting strategy were utilized for the OD of QZS-1. The post-fit root-mean-square (RMS) residuals of the two week-long arcs were 11.98 cm and 10.77 cm when estimating the biases once in an arc (MBIAS). These residuals were decreased significantly to 2.40 cm and 3.60 cm by estimating the biases every pass (PBIAS). Then, the resultant OD precision was evaluated by the orbit overlap method, yielding three-dimensional errors of 55.013 m with MBIAS and 1.962 m with PBIAS for the overlap period of six days. For the OD of Compass-G1, no station weighting strategy was applied, and only MBIAS was utilized due to the lack of NPs. The post-fit RMS residuals of OD were 8.81 cm and 12.00 cm with 49 NPs and 47 NPs, respectively, and the corresponding threedimensional orbit overlap error for four days was 160.564 m. These results indicate that the amount of SLR tracking data is critical for obtaining precise OD of HEO satellites using SLR because additional parameters, such as station bias, are available for estimation with sufficient tracking data. Furthermore, the stand-alone SLR-based orbit solution is consistently attainable for HEO satellites if a target satellite is continuously trackable for a specific period.

Boxy Orbital Structures in Rotating Bar Models

Abstract We investigate regular and chaotic two-dimensional (2D) and three-dimensional (3D) orbits of stars in models of a galactic potential consisting of a disk, a halo, and a bar to find the origin of boxy components that are part of the bar or (almost) the bar itself. Our models originate in snapshots of an N-body simulation, which develops a strong bar. We consider three snapshots of the simulation, and, for the orbital study, we treat each snapshot independently, as an autonomous Hamiltonian system. The calculated corotation–to–bar length ratios indicate that in all three cases, the bar rotates slowly, while the orientation of the orbits of the main family of periodic orbits changes along its characteristic. We characterize the orbits as regular, sticky, or chaotic after integrating them for a 10 Gyr period by using the GALI2 index. Boxiness in the equatorial plane is associated either with quasi-periodic orbits in the outer parts of stability islands or with sticky orbits around them, which can be found in a large range of energies. We indicate the location of such orbits in diagrams, which include the characteristic of the main family. They are always found about the transition region from order to chaos. By perturbing such orbits in the vertical direction, we find a class of 3D nonperiodic orbits, which have boxy projections both in their face-on and side-on views.

Periodic orbits around the collinear equilibrium points for binary Sirius, Procyon, Luhman 16, α-Centuari and Luyten 726-8 systems: the spatial case

An investigation of three-dimensional periodic orbits and their stability emanating from the collinear equilibrium points of the restricted three-body problem with oblate and radiating primaries is presented. A simulation is done by using five binary systems: Sirius, Procyon, Luhman 16, α-Centuari and Luyten 726-8. Firstly, based on the topological degree theory, the total number of the collinear equilibrium points for the five binary systems were obtained and then, their positions were determined numerically. The linear stability of these equilibrium points was also examined and found to be unstable in the Lyapunov sense. An analytical approximation of three-dimensional periodic solutions around them was established via the Lindstedt–Poincaré local analysis. Finally, using the analytical solution to obtain starting orbits, the families of three-dimensional periodic orbits emanating from these equilibria have been continued numerically.

Mars spacecraft tracking and analysis of VLBI orbit determination

The Chinese Mars mission will be launched in 2020, which consists of an orbiter and a rover. Mars spacecraft tracking and precise orbit determination are very important for achieving engineering and scientific detection. This paper first analyzes the Mars spacecraft tracking technology, and then introduces VLBI orbit determination results used in the mission of Mars Science Laboratory. Further, for a Mars spacecraft launched in July 2020, we give the angular distance between the spacecraft and the sun from launch to disorbit, the solar plasma delay, and observation elevation, at Shanghai, Beijing, Kunming and Urumqi stations in the arc segments of trajectory correction maneuver, Martian orbit insertion, plane maneuver, and disorbit. Orbit determination and prediction errors in the arc segment of Martian orbit insertion are also presented. Three-dimensional orbit determination error is reduced from 45.7 km to 18.8 km, and the periareon height prediction error (1 sigma) is reduced from 28.2 km to 7.6 km when combined VLBI data with range and Doppler data.

Resolved astrometric orbits of ten O-type binaries

Aims. Our long-term aim is to derive model-independent stellar masses and distances for long period massive binaries by combining apparent astrometric orbit with double-lined radial velocity amplitudes (SB2). Methods. We followed-up ten O+O binaries with AMBER, PIONIER and GRAVITY at the VLTI. Here, we report on 130 astrometric observations over the last seven years. We combined this dataset with distance estimates to compute the total mass of the systems. We also computed preliminary individual component masses for the five systems with available SB2 radial velocities. Results. Nine of the ten binaries have their three-dimensional orbit well constrained. Four of them are known to be colliding wind, non-thermal radio emitters, and thus constitute valuable targets for future high angular resolution radio imaging. Two binaries break the correlation between period and eccentricity tentatively observed in previous studies. This suggests either that massive star formation produces a wide range of systems, or that several binary formation mechanisms are at play. Finally, we found that the use of existing SB2 radial velocity amplitudes can lead to unrealistic masses and distances. Conclusions. If not understood, the biases in radial velocity amplitudes will represent an intrinsic limitation for estimating dynamical masses from SB2+interferometry or SB2+Gaia. Nevertheless, our results can be combined with future Gaia astrometry to measure the dynamical masses and distances of the individual components with an accuracy of 5 to 15%, completely independently of the radial velocities.