Quaternion methods and models of regular celestial mechanics and astrodynamics

This paper is a review,which focuses on our work,while including an analysis of many works of other researchers in the field of quaternionic regularization.The regular quaternion models of celestial mechanics and astrodynamics in the Kustaanheimo-Stiefel(KS)variables and Euler(Rodrigues-Hamilton)parameters are analyzed.These models are derived by the quaternion methods of mechanics and are based on the differential equations of the perturbed spatial two-body problem and the perturbed spatial central motion of a point particle.This paper also covers some applications of these models.Stiefel and Scheifele are known to have doubted that quaternions and quaternion matrices can be used efficiently to regularize the equations of celestial mechanics.However,the author of this paper and other researchers refuted this point of view and showed that the quaternion approach actually leads to efficient solutions for regularizing the equations of celestial mechanics and astrodynamics.This paper presents convenient geometric and kinematic interpretations of the KS transformation and the KS bilinear relation proposed by the present author.More general(compared with the KS equations)quaternion regular equations of the perturbed spatial two-body problem in the KS variables are presented.These equations are derived with the assumption that the KS bilinear relation was not satisfied.The main stages of the quaternion theory of regularizing the vector differential equation of the perturbed central motion of a point particle are presented,together with regular equations in the KS variables and Euler parameters,derived by the aforementioned theory.We also present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters,developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame.This paper also gives new results using quaternionic methods in the perturbed spatial restricted three-body problem.

Research career of an astronomer who has studied celestial mechanics

Celestial mechanics has been a classical field of astronomy. Only a few astronomers were in this field and not so many papers on this subject had been published during the first half of the 20th century. However, as the beauty of classical dynamics and celestial mechanics attracted me very much, I decided to take celestial mechanics as my research subject and entered university, where a very famous professor of celestial mechanics was a member of the faculty. Then as artificial satellites were launched starting from October 1958, new topics were investigated in the field of celestial mechanics. Moreover, planetary rings, asteroids with moderate values of eccentricity, inclination and so on have become new fields of celestial mechanics. In fact I have tried to solve such problems in an analytical way. Finally, to understand what gravitation is I joined the TAMA300 gravitational wave detector group.

A Rendezvous Mission to the Second Earth Trojan Asteroid 2020 XL_5 with Low-Thrust Multi-Gravity Assist Techniques

As the second of Earth's Trojan asteroids, 2020 XL_(5) is worthy of rendezvous and even sample return missions in many aspects. In this paper, a rendezvous mission to Earth's second Trojan asteroid 2020 XL_(5) is proposed.However, due to its high inclination and large eccentricity, direct impulsive transfer requires large amounts of fuel consumption. To address this challenge, we explore the benefits of electric propulsion and multi-gravity assist techniques for interplanetary missions. These two techniques are integrated in this mission design. The design of a low-thrust gravity-assist(LTGA) trajectory in multi-body dynamics is thoroughly investigated,which is a complex process. A comprehensive framework including three steps is presented here for optimization of LTGA trajectories in multi-body dynamics. The rendezvous mission to 2020 XL_(5) is designed with this three-step approach. The most effective transfer sequence among the outcomes involves Earth–Venus–Earth–Venus-2020 XL_(5). Numerical results indicate that the combination of electric propulsion and multi-gravity assists can greatly reduce the fuel consumption, with fuel consumption of 9.03%, making it a highly favorable choice for this rendezvous mission.

Time Dilation Cosmology 2

This paper is a further elaboration of the author’s Time Dilation Cosmology (TDC) holographic model that ties gravitation and celestial mechanics and kinematics directly to time dilation, resolving all the major conundrums in astrophysics, and ties astrophysics directly to quantum physics. It begins with a brief summary of the TDC model and contains the new derivation for the time dilation version of the formula for summing relativistic velocities, Einstein’s gravitational constant and the time dilation versions for the Lorentz factor and the Euclidean norm of the 3d velocity vector, the two of which can then be used in the Four-velocity formula. It is demonstrated how orbital curvature is manifested as the resultant of two time dilation-manifested velocities. It also explains why an interferometer cannot distinguish free fall from zero gravity and further elaborates on the author’s previous explanations of how spiral galaxies are formed, and contains mathematical proof that Black Holes are actually Magnetospheric Eternally Collapsing Objects (MECOs) that are massless spacetime vortices.

Micro-arcsecond Celestial Reference Frames:definition and realization——Impact of the recent IAU Resolutions

The adoption of the International Celestial Reference System （ICRS）, based on Very Long Baseline Interferometry （VLBI） observations of extragalactic radiosources by the International Astronomical Union （IAU） since 1998 January 1, opened a new era for astronomy. The ICRS and the corresponding frame, the International Celestial Reference Frame （ICRF）, replaced the Fundamental Catalog （FK5） based on positions and proper motions of bright stars, with the Hipparcos cat- alog being adopted as the primary realization of the ICRS in optical wavelengths. According to its definition, the ICRS is such that the barycentric directions of distant extragalactic objects show no global rotation with respect to these objects; this pro- vides a quasi-inertial reference for measuring the positions and angular motions of the celestial objects. Other resolutions on reference systems were passed by the IAU in 2000 and 2006 and endorsed by the International Union of Geodesy and Geophysics （IUGG） in 2003 and 2007, respectively. These especially concern the definition and re- alization of the astronomical reference systems in the framework of general relativity and transformations between them. First, the IAU 2000 resolutions refined the con- cepts and definition of the astronomical reference systems and parameters for Earth＇s rotation, and adopted the IAU 2000 precession-nutation. Then, the IAU 2006 resolutions adopted a new precession model that is consistent with dynamical theories; they also addressed definition, terminology or orientation issues relative to reference systems and time scales that needed to be specified after the adoption of the IAU 2000 resolutions. An additional IUGG 2007 resolution defined the International Terrestrial Reference System （ITRS） so that it strictly complies with the IAU recommendations. Finally, the IAU 2009 resolutions adopted a new system of astronomical constants and an improved realization of the ICRF. These fundamental changes have led to significant improvements in the fields of astrometry, celestial mechanics, geodynam- ics, geodesy, etc. Of special interest are the improvements in the model for variations in Earth＇s rotation, which, in turn, can provide better knowledge of the dynamics of the Earth＇s interior. These have also contributed to a significant improvement in the accuracy of the ephemerides of the solar system bodies as determined from modern measurements, with a large number of scientific applications. This paper recalls the main aspects of the recent IAU resolutions on reference systems as well as their con- sequences on the concepts, definitions, nomenclature and models that are suitable for the definition, realization and transformation of reference frames at a microarcsecond level.

Notes on the Motion of Celestial Bodies

A novel method for the computation of the motion of multi-body systems is proposed against the traditional one, based on the dynamic exchange of attraction forces or using complex field equations, that hardly face two-body problems. The Newton gravitational model is interpreted as the emission of neutrino/gravitons from celestial bodies that combine to yield a cumulative flux that interacts with single bodies through a momentum balance. The neutrino was first found by Fermi to justify the energy conservation in β decay and, using his model;we found that the emission of neutrino from matter is almost constant independently from the nuclides involved. This flux can be correlated to Gauss constant G, allowing the rebuilding of Newton law on the basis of nuclear data, the neutrino weight and the speed of light. Similarly to nature, we can therefore separate in the calculations the neutrino flux, that represents the gravitational field, is dependent on masses and is not bound to the number of bodies involved, from the motion of each body that, given the field, is independent of the mass of bodies themselves. The conflict between exchanges of forces is avoided, the mathematics is simplified, the computational time is reduced to seconds and the stability of result is guaranteed. The example of computation of the solar system including the Sun and eight planets over a period of one to one hundred years is reported, together with the evolution of the shape of the orbits.

Effect of Orbital Characteristic of Inclined Third-body on Motion of Secondary-body for a Hierarchical Triple Systems

The influence of a third-body's orbital elements on the second-body's motion in a hierarchical triple system is a crucial problem in astrophysics.Most prolonged evaluation studies have focused on a distant zero-inclined thirdbody.This study presents a new perspective on second-body motion equations that addresses a perturbing-body in an elliptic orbit derived with consideration of the axial-tilt(obliquity)of the primary.The proposed model is compared by the dual-averaged method and the N-body problem algorithm.After validation,a generalized threebody model is derived to investigate the effects of the third-body's orbital elements on secondary-body motion behavior.The proposed model considers short-time oscillations that affect secular evaluation and applies to exoplanets with all the primary and third body eccentricities,inclinations,and mass ratios.It is shown that the obliquity of the primary(or third-body's inclination)must be considered for precise long-term assessment,even in highly-hierarchical systems.

Studying the Equilibrium Points of the Modified Circular Restricted Three-body Problem: The Case of Sun–Haumea System

We intend to study a modified version of the planar Circular Restricted Three-Body Problem(CRTBP) by incorporating several perturbing parameters. We consider the bigger primary as an oblate spheroid and emitting radiation while the small primary has an elongated body. We also consider the perturbation from a disk-like structure encompassing this three-body system. First, we develop a mathematical model of this modified CRTBP.We have found there exist five equilibrium points in this modified CRTBP model, where three of them are collinear and the other two are non-collinear. Second, we apply our modified CRTBP model to the Sun–Haumea system by considering several values of each perturbing parameter. Through our numerical investigation, we have discovered that the incorporation of perturbing parameters has resulted in a shift in the equilibrium point positions of the Sun–Haumea system compared to their positions in the classical CRTBP. The stability of equilibrium points is investigated. We have shown that the collinear equilibrium points are unstable and the stability of non-collinear equilibrium points depends on the mass parameter μ of the system. Unlike the classical case, non-collinear equilibrium points have both a maximum and minimum limit of μ for achieving stability. We remark that the stability range of μ in non-collinear equilibrium points depends on the perturbing parameters. In the context of the Sun–Haumea system, we have found that the non-collinear equilibrium points are stable.

Time Dilation Cosmology

This model ties gravitation and celestial mechanics and kinematics directly to time dilation. It is a new theory of cosmology and the evolution of galaxies. Space and time are not two separate things, but two aspects of a single thing, “spacetime”. Whatever affects space, affects time, and vice-versa. If time speeds up, space must contract to maintain the speed of light, c, and when space thickens into a mass, it is harder to evolve forward, and time appears to slow. If spatial events are spinning as time passes, then the forward direction of time is spinning. This is Einstein’s curvature in the forward direction of time. Herein, the basis is outlined for time dilation cosmology in a spacetime/quantum continuum, including the time dilation-based derivation of the mass of the Cosmic Microwave Background Radiation (CMBR), and time dilation formulas are derived for stellar system orbital, and galactic rotation, velocities, the force in time in Newtons, the Hamiltonian, the Hubble shift, the empirical gravitational constant, G, and other formulas, showing their direct relationship to the difference in the rate of time between the far distant observer’s invariant 1 s/s rate of time and the slower rate of time at the coordinate point, proving the universe is not composed of separate bodies moving through space, but is an evolving 3-dimensional holographic continuum containing varying densities evolving forward in the forward direction of time, the 4th dimension, at apparently different rates of time, the velocities merely being compensation for those slower rates of time in a continuum evolving forward overall at c, which is why light propagates at c, even from a moving source. As per General Relativity, if there is no rate of time difference between coordinate points, there is no gravitational attraction between those points, and no gravitationally induced velocity. This model resolves all the major conundrums in astrophysics, eliminating Dark Energy and Dark Matter, and ties astrophysics directly to quantum physics.

Modification of the Lagrange-Jacobi Equation and Its Application

The Lagrange-Jacobi equation is one of the significant tools for the qualitative analysis of the n-body problem. In this paper, we present the modified Lagrange-Jacobi equation by introducing a new formal parameter of n-body problem and propose its application to the dynamical study of clusters of galaxies which are large-scale structures of Universe. We put forward and study a new dynamical problem which is related to the stage of relaxation of observed stationary clusters of galaxies which are considered as a non-equilibrium systems of point masses. We also received the analytical form of the potential energy of such galaxy clusters. One of the applications of this analytical form is the analytical relation between the time of setting up the virial equilibrium in relaxing clusters of galaxies and the cosmological epoch T.

CHES: A Space-borne Astrometric Mission for the Detection of Habitable Planets of the Nearby Solar-type Stars

The Closeby Habitable Exoplanet Survey(CHES) mission is proposed to discover habitable-zone Earth-like planets of nearby solar-type stars(~10 pc away from our solar system) via microarcsecond relative astrometry.The major scientific objectives of CHES are:to search for Earth Twins or terrestrial planets in habitable zones orbiting100 FGK nearby stars;further to conduct a comprehensive survey and extensively characterize nearby planetary systems.The primary payload is a high-quality,low-distortion,high-stability telescope.The optical subsystem is a coaxial three-mirror anastigmat(TMA) with a 1.2 m-aperture,0°.44 × 0°.44 field of view and 500 nm-900 nm working wave band.The camera focal plane is composed of a mosaic of 81 scientific CMOS detectors each with4 k × 4 k pixels.The heterodyne laser interferometric calibration technology is employed to ensure microarcsecond level(1 μas) relative astrometry precision to meet the requirements for detection of Earth-like planets.The CHES satellite operates at the Sun-Earth L2 point and observes all the target stars for 5 yr.CHES will offer the first direct measurements of true masses and inclinations of Earth Twins and super-Earths orbiting our neighbor stars based on microarcsecond astrometry from space.This will definitely enhance our understanding of the formation of diverse nearby planetary systems and the emergence of other worlds for solar-type stars,and finally provide insights to the evolution of our own solar system.

Experiment on diffuse reflection laser ranging to space debris and data analysis

Space debris poses a serious threat to human space activities and needs to be measured and cataloged. As a new technology for space target surveillance, the measurement accuracy of diffuse reflection laser ranging （DRLR） is much higher than that of microwave radar and optoelectronic measurement. Based on the laser ranging data of space debris from the DRLR system at Shanghai Astronomical Observatory acquired in March-April, 2013, the characteristics and precision of the laser ranging data are analyzed and their applications in orbit determination of space debris are discussed, which is implemented for the first time in China. The experiment indicates that the precision of laser ranging data can reach 39 cm-228 cm. When the data are sufficient enough （four arcs measured over three days）, the orbital accuracy of space debris can be up to 50 m.

Analysis of the wide area differential correction for BeiDou global satellite navigation system

The regional BeiDou Satellite System, or BDS2, broadcasts a differential correction as Equivalent Satellite Clock Correction to correct both orbit and satellite clock errors. For the global BDS, or BDS3, satellite orbit and clock corrections conforming with RTCA standards will be broadcast to authorized users. The hybrid constellation and regional monitoring network pose challenges for the high precision separation of orbit and satellite clock corrections. Three correction models of kinematic,dynamic and Two-way Satellite Time Frequency Transfer(TWSTFT)-based dynamic were studied to estimate the satellite orbit and clock corrections. The correction accuracy of the three models is compared and analyzed based on the BDS observation data. Results show that the accuracies(root mean square, RMS) of dual-frequency real-time positioning for the three models are about 1.76 m, 1.78 m and 2.08 m respectively, which are comparable with the performance of WAAS and EGNOS. With dynamic corrections, the precision of Precise Point Positioning(PPP) experiments may reach about 23 cm after convergence.

A particle-linkage model for non-axisymmetric elongated asteroids

This paper investigates a simplified model for describing the gravitational fields of non- axisymmetric elongated asteroids. The connection between the simplified model and the target aster-oid is built by considering the positions of equilibrium points. To improve the performance of po-sition matching for the equilibrium points associated with these non-axisymmetric asteroids, a non-axisymmetric triple-particle-linkage model is proposed based on two existing axisymmetric particle- linkage models. The unknown parameters of the simplified model are determined by minimizing the matching error using the nonlinear optimization method. The proposed simplified model is applied for three realistic elongated asteroids, 243 Ida, 433 Eros and （8567） 1996 HW1. The simulation re-sults verify that the current particle-linkage model has better matching accuracy than the two existing particle-linkage models. The comparison, between the simplified model and the polyhedral model, on the topological cases of the equilibrium points and the distribution of gravitational potential further validate the rationality and accuracy of the simplified model.

Optimized transfer trajectories in the earth-moon system

Investigations of low energy transfer trajectories are important for both celestial mechanics and astronautics. Methodologies using the theories from dynamical systems are developed in recent years. This paper investigates the dynamics of the earth-moon system. Low energy transfer trajectories are solved numerically by employing a hybrid strategy： first, a genetic hide and seek method performs a search in large domain to confine the global minimum f（η） （objective function） region; then, a deterministic Nelder-Mead method is utilized to refine the minimum quickly. Some transfer trajectories of the spacecraft in the earth-moon system are successfully simulated which verify the desired efficiency and robustness of the method of this paper.

Orbit Determination Using Satellite-to-Satellite Tracking Data

Satellite-to-Satellite tricking (SST) data can be used to determine the orbits of spacecraft in two ways. One is combined orbit determination, which combines SST data with ground-based tracking data and exploits the enhanced tracking geometry. The other is the autonomous orbit determination, which uses only SST. The latter only fits some particular circumstances since it suffers the rank defect problem in other circumstances. The proof of this statement is presented. The nature of the problem is also investigated in order to find an effective solution. Several. methods of solution are discussed. The feasibility of the methods is demonstrated by their application to a simulation.

Capturing near earth objects

Recently, Near Earth Objects （NEOs） have been attracting great attention, and thousands of NEOs have been found to date. This paper examines the NEOs＇ orbital dynamics using the framework of an accurate solar system model and a Sun- Earth-NEO three-body system when the NEOs are close to Earth to search for NEOs with low-energy orbits. It is possible for such an NEO to be temporarily captured by Earth; its orbit would thereby be changed and it would become an Earth-orbiting object after a small increase in its velocity. From the point of view of the Sun-Earth- NEO restricted three-body system, it is possible for an NEO whose Jacobian constant is slightly lower than C1 but higher than C3 to be temporarily captured by Earth. When such an NEO approaches Earth, it is possible to change its orbital energy to nearly the zero velocity surface of the three-body system at point L1 and make the NEO become a small satellite of the Earth. Some such NEOs were found; the best example only required a 410 m s^-1 increase in velocity.

Feasible region and stability analysis for hovering around elongated asteroids with low thrust

This paper investigates properties of low-thrust hovering, including the feasible region and stability, in terms of the dynamical parameters for elongated asteroids. An approximate rotating mass dipole model, by which the description of the rotational gravitational field is reduced to two independent parameters, is employed to construct normalized dynamical equations. The boundaries of the feasible region are determined by contours representing the magnitude of the active control. The effects of a rotating gravitational field and maximal magnitude of the low thrust on the feasible hovering regions are analyzed with numerical results. The stability conditions are derived according to the forms of the eigenvalues of the linearized equation near the hovering position. The stable regions are then determined by a grid search and the effects of the relevant parameters are analyzed in a parametric way. The results show that a close hovering can be easier to realize near the middle of the asteroid than near the two ends in the sense of both required control magnitude and stability.

Application of two special orbits in the orbit determination of lunar satellites

Using inter-satellite range data, the combined autonomous orbit determina- tion problem of a lunar satellite and a probe on some special orbits is studied in this paper. The problem is firstly studied in the circular restricted three-body problem, and then generalized to the real force model of the Earth-Moon system. Two kinds of spe- cial orbits are discussed： collinear libration point orbits and distant retrograde orbits. Studies show that the orbit determination accuracy in both cases can reach that of the observations. Some important properties of the system are carefully studied. These findings should be useful in the future engineering implementation of this conceptual study.

On the Station-Keeping and Control of the World Space Observatory/Ultraviolet

Collinear libration points play an important role in deep space exploration because of their special positions and dynamical characteristics. Since motion around them is unstable, we need to control the spacecraft if we wish to keep them around such a libration point for a long time. Here we propose a continuous low-thrust control strategy, illustrated with numerical simulations combined with the orbit design and control of the World Space Observatory/UltraViolet （WSO/UV）.