Keplerian Orbit Research Articles

An analytic solution of full-sky spherical geometry for satellite relative motions

Herein, an exact and efficient analytic solution for an unperturbed satellite relative motion was developed using a direct geometrical approach. The derivation of the relative motion geometrically interpreted the projected Keplerian orbits of the satellites on a sphere (Earth and celestial spheres) using the solutions of full-sky spherical triangles. The results were basic and computationally faster than the vector and plane geometry solutions owing to the advantages of the full-sky spherical geometry. Accordingly, the validity of the proposed solution was evaluated by comparing it with other analytic relative motion theories in terms of modeling accuracy and efficiency. The modeling accuracy showed an equivalent performance with Vadali’s nonlinear unit sphere approach, which is essentially equal to the Yan–Alfriend nonlinear theory. Moreover, the efficiency was demonstrated by the lowest computational cost compared with other models. In conclusion, the proposed modeling approach illustrates a compact and efficient closed-form solution for satellite relative motion dynamics.

Low-Thrust Reconfiguration Strategy and Optimization for Formation Flying Using Jordan Normal Form

This article proposes a low-thrust reconfiguration strategy for formation flying based on the Jordan normal form and evaluates its performance based on the required control acceleration and fuel cost. First, the relative dynamics of reconfiguration is disposed by Jordan decomposition to describe the unstable component, i.e., the drift velocity along the track direction, which is not explicit in eigenstructure. Next, the initial states of the Jordan-reduced dynamics are regarded as orbital invariants without control to characterize a formation configuration, similar to the definition of orbital elements that are invariants of the unperturbed Keplerian orbit. Based on the derived differential form of ISJD with control, the reconfiguration trajectory is parameterized by a functional integral, the existence of which is then proved analytically by the proposed polynomial series method. The specified optimal trajectories are yielded by the practical Radau pseudospectral method numerically. Finally, the effects of reconfiguring positions and relative orbital orientation on the required acceleration and fuel cost are investigated using a technique for order preference based on similarity to the ideal solution.

The Black Hole Mass of the z = 2.805 Multiply Imaged Quasar SDSS J2222+2745 from Velocity-resolved Time Lags of the C iv Emission Line

Abstract We present the first results of a 4.5 yr monitoring campaign of the three bright images of multiply imaged z = 2.805 quasar SDSS J2222+2745 using the Gemini North Multi-Object Spectrograph and the Nordic Optical Telescope. We take advantage of gravitational time delays to construct light curves surpassing 6 yr in duration in the observed frame and achieve an average spectroscopic cadence of 10 days during the 8 months of visibility per season. Using multiple secondary calibrators and advanced reduction techniques, we achieve percent-level spectrophotometric precision and carry out an unprecedented reverberation mapping analysis, measuring both integrated and velocity-resolved time lags for the C iv emission line. The full line lags the continuum by τ cen = 36.5 − 3.9 + 2.9 rest-frame days. We combine our measurement with published C iv emission line lags and derive the r BLR − L relationship log 10 ​ ( τ / d a y ) = ( 0.99 ± 0.07 ) + ( 0.48 ± 0.03 ) log 10 ​ [ λ L λ ( 1350 A A ) / 10 44 e r g s − 1 ] with 0.30 ± 0.06 dex intrinsic scatter. The velocity-resolved lags are consistent with circular Keplerian orbits, with τ cen = 86.2 − 5.0 + 4.5 , 25 − 15 + 11 , and 7.5 − 3.5 + 4.2 rest-frame days for the core, blue wing, and red wing, respectively. Using σ line with the mean spectrum and assuming log 10 ( f mean , σ ) = 0.52 ± 0.26 , we derive log 10 ( M BH / M ⊙ ) = 8.63 ± 0.27 . Given the quality of the data, this system represents a unique benchmark for calibration of M BH estimators at high redshift. Future work will present dynamical modeling of the data to constrain the virial factor f and M BH.

The Space of Keplerian Orbits and a Family of Its Quotient Spaces

The distance functions on the set of Keplerian orbits play an important role in solving the problems of searching for the parent bodies of meteoroid streams. A special kind of function is the distances in the quotient spaces of orbits. Three metrics of this type were developed earlier. These metrics allow disregarding the longitude of the ascending node or the argument of pericenter, or both. Here, we introduce one more quotient space, where two orbits are considered to be identical if they differ only in their longitudes of nodes and arguments of pericenters, but have the same sum of these elements (the longitude of pericenter). Function $${{\varrho }_{6}}$$ is defined to calculate the distance between two equivalence classes of orbits. The algorithm for calculation of $${{\varrho }_{6}}$$ from given elements of the orbits is provided along with a reference to the corresponding program written in the C++ programming language. Unfortunately, function $${{\varrho }_{6}}$$ is not a full-fledged metric. We have proven that it satisfies the first two axioms of a metric space; however, it does not satisfy the third axiom: the triangle inequality does not hold, at least in the case of large eccentricities. However, there are two important cases when the triangle axiom is satisfied: one of three orbits is circular and the longitudes of pericenters of all three orbits coincide. Perhaps, the inequality holds for all elliptic orbits, but this is a matter of further research.

General relativistic effects acting on the orbits of Galileo satellites

The first pair of satellites belonging to the European Global Navigation Satellite System (GNSS)—Galileo—has been accidentally launched into highly eccentric, instead of circular, orbits. The final height of these two satellites varies between 17,180 and 26,020 km, making these satellites very suitable for the verification of the effects emerging from general relativity. We employ the post-Newtonian parameterization (PPN) for describing the perturbations acting on Keplerian orbit parameters of artificial Earth satellites caused by the Schwarzschild, Lense–Thirring, and de Sitter general relativity effects. The values emerging from PPN numerical simulations are compared with the approximations based on the Gaussian perturbations for the temporal variations of the Keplerian elements of Galileo satellites in nominal, near-circular orbits, as well as in the highly elliptical orbits. We discuss what kinds of perturbations are detectable using the current accuracy of precise orbit determination of artificial Earth satellites, including the expected secular and periodic variations, as well as the constant offsets of Keplerian parameters. We found that not only secular but also periodic variations of orbit parameters caused by general relativity effects exceed the value of 1 cm within 24 h; thus, they should be fully detectable using the current GNSS precise orbit determination methods. Many of the 1-PPN effects are detectable using the Galileo satellite system, but the Lense–Thirring effect is not.

Charged black hole mergers: orbit circularisation and chirp mass bias

We consider the inspiral of black holes carrying U(1) charge that is not electromagnetic, but corresponds to some dark sector. In the weak-field, low-velocity regime, the components follow Keplerian orbits. We investigate how the orbital parameters evolve for dipole-dominated emission and find that the orbit quickly circularises, though not as efficiently as for a gravitationally dominated emission. We then regard circular orbits, and look for modifications in the gravitational waveform from the components carrying small charges. Taking this into account we populate the waveform with simplified LIGO noise and put it through a matched filtering procedure where the template bank only consists of uncharged templates, focusing on the charges’ effect on the chirp mass estimation. We find a consistent overestimation of the ‘generalised’ chirp mass, and a possible over- and underestimation of the actual chirp mass. Finally, we briefly consider the effect of such charges on hyperbolic encounters, finding again a bias arising from interpreting the generalised chirp mass as the actual chirp mass.

The spin–spin model and the capture into the double synchronous resonance

The aim of this article is to propose a model, that is a planar version of the full two-body problem, and discuss the existence and stability of a relevant periodic solution. Consider two homogeneous ellipsoids orbiting around each other in fixed coplanar Keplerian orbits. Moreover, their respective spin axes are assumed to be perpendicular to the orbital plane, that is also a common equatorial plane. The spin–spin model deals with the coupled rotational dynamics of both ellipsoids. For a non-zero orbital eccentricity, it has the structure of a non-autonomous system of coupled pendula. This model is a natural extension of the classical spin–orbit problem for two extended bodies. In addition, we consider dissipative tidal torques, that can trigger the capture of the system into spin–orbit and spin–spin resonances. In this paper we give some theoretical results for both the conservative model and the dissipative one. The conservative model has a Hamiltonian structure. We use properties of Hamiltonian systems to give some sufficient conditions in the space of parameters of the model, that guarantee existence, uniqueness and linear stability of an odd periodic solution. This solution represents a double synchronous resonance in the conservative regime. Such solution can be continued to the dissipative regime, where it becomes asymptotically stable. We see asymptotic stability as a dynamical mechanism for the capture into the double synchronous resonance. Finally we apply our results to several cases including the Pluto–Charon binary system and the Trojan binary asteroid 617 Patroclus, target of the LUCY mission.

A test of the fermionic dark matter nature of the supermassive compact object at the center of our galaxy

AbstractThe extensive and continuous monitoring of the closest stars to the Galactic center has been producing over decades a large amount of high‐quality data of their positions and velocities. The explanation of these data, especially the S2 star motion, reveals the presence of a compact source, Sagittarius A* (Sgr A*), whose mass must be about 4 × 106 M⊙. This result has been protagonist of the awarded Nobel Prize in Physics 2020 to Reinhard Genzel and Andrea Ghez “for the discovery of a supermassive compact object at the centre of our galaxy.” Traditionally, the nature of Sgr A* has been attributed to a supermassive black hole (SMBH), even though a proof of its existence is absent. Further, recent data on the motion of the G2 cloud show that its post‐peripassage velocity is lower than the expected one from a Keplerian orbit around the hypothesized SMBH. An attempt to overcome this difficulty has used a friction force, produced (arguably) by an accretion flow whose presence is also observationally unconfirmed. We advance an alternative scenario that identifies the nature of the supermassive compact object in a highly concentrated core of dark matter (DM) made of fermions, that is, inos, which we here refer to as darkinos. This DM concentration is predicted by the core‐halo profile obtained from the Ruffini‐Argüelles‐Rueda fermionic DM model, which also fits the Galactic‐halo rotation curves. We show that the sole DM core, for 56 keV darkinos, explains the orbits of S2 and G2. No drag force or other external agents are needed, that is, their motion is purely geodesic. A core made of these darkinos becomes unstable against gravitational collapse into a BH for a mass of ∼108M⊙; hence, they can provide the BH seeds from which grow the observed central SMBHs in active galaxies, for example, M87.

Attitude dynamics of a rigid body in Keplerian motion

This paper studies the attitude dynamics of a rigid body in a Keplerian orbit. We show that the use of Classical Rodrigues Parameters for the attitude motion of the rigid body subject to gravity-gradient torques enables us to characterize the equilibria associated with the rotational motion about its mass center. A parametric study of the stability of equilibria is conducted to show that large oscillations are induced due to the energy exchange between the pitch and roll–yaw motions, specifically near the 2:1 resonant commensurability regions. A visualization tool is developed to study these pitch oscillations and gain insight into the rigid body motion near internal resonance conditions. A measure of coupling between the pitching and roll–yaw motions is developed to quantify the energy exchange utilizing information from the state transition matrix.

Elastic Cable-Connected Satellites System Under Several Influences of General Nature: Equations of Motion in Elliptical Orbit

In the present work, we have applied the analytical approach to obtain the equations of motion of a system of two cable connected artificial satellites under the effect of various perturbative forces of general nature. These perturbative forces are the earth’s shadow, solar radiation pressure, earth’s oblateness and the earth’s magnetic field. Our aim is to investigate the motion of the centre of mass of the two artificial cable connected satellite system in Keplerian elliptical orbit. We establish the equations of motion for the centre of mass of the system and also for the relative motion of the system. We obtain equations of motion in the rotating frame of reference as well as in Nechvile’s co-ordinate system. The cable under consideration is light, flexible, non-conducting and elastic in nature.

Fault-tolerant estimation of satellite orbital states

This paper is devoted to global positioning system (GPS)-based fault-tolerant satellite orbital states estimation which is used for the pseudo-ranging model. GPS is widely used by many researchers to determine the orbit state vectors of satellites, but errors in GPS measurements can cause major errors in satellite location detection. Therefore, there is a need for an architecture that will resist possible measurement errors and GPS malfunctions. This study aims to present the novel approaches to tolerate the errors that may take place in GPS measurements. The satellite localisation has been detected via a pseudo-ranging model between four GPS satellites and the target satellite. The Keplerian equations and orbit elements are used for the satellite dynamic model with orbital J2 perturbation which is due to the flattened shape of the earth is taken into account via the extended Kalman filter (EKF) algorithms. The reconfigurable and the adaptive version of fault-tolerant localisation architecture eliminate the GPS measurement errors during estimating satellite position vectors. The comparison of the prediction architectures of the fault tolerant satellite state vectors originally proposed. At the same time, it is possible to use fault tolerant architectures designed to detect whether GPS satellites are malfunctioning or not.

The space of Keplerian orbits and a family of its quotient spaces

Distance functions on the set of Keplerian orbits play an important role in solving problems of searching for parent bodies of meteoroid streams. A special kind of such functions are distances in the quotient spaces of orbits. Three metrics of this type were developed earlier. These metrics allow to disregard the longitude of ascending node or the argument of pericenter or both. Here we introduce one more quotient space, where two orbits are considered identical if they differ only in their longitudes of nodes and arguments of pericenters, but have the same sum of these elements (the longitude of pericenter). The function q is defined to calculate distance between two equivalence classes of orbits. The algorithm for calculation of ̺6 value is provided along with a reference to the corresponding program, written in C++ language. Unfortunately, ̺6 is not a full-fledged metric. We proved that it satisfies first two axioms of metric space, but not the third one: the triangle inequality does not hold, at least in the case of large eccentricities. However there are two important particular cases when the triangle axiom is satisfied: one of three orbits is circular, longitudes of pericenters of all three orbits coincide. Perhaps the inequality holds for all elliptic orbits, but this is a matter of future research.

Research on the method of determining satellite state vector based on Kepler orbital parameters

to simplify the process of satellite orbit determination, can use limited conditions known to the calculation of satellite orbit dynamic state vector to determine satellite position to implement the standardization of satellite orbit, this paper introduces the satellite state vector at any time using the Kepler orbital parameters method, the calculation method of Kepler orbit parameters are mostly only for satellite without disturbing force movement. This paper is based on the three-point method which is used to calculate the orbit parameters of Kepler satellite in the case of satellite unphotographed.

Enckehh: an integrator for gravitational dynamics with a dominant mass that achieves optimal error behaviour

ABSTRACT We present enckehh, a new, highly accurate code for orbital dynamics of perturbed Keplerian systems such as planetary systems or galactic centre systems. It solves Encke’s equations of motion, which assume perturbed Keplerian orbits. By incorporating numerical techniques, we have made the code follow optimal roundoff error growth. In a 1012 d integration of the outer Solar system, enckehh was 3.5 orders of magnitude more accurate than ias15 in a fixed timestep test. Adaptive steps are recommended for ias15. Through study of efficiency plots, we show that enckehh reaches significantly higher accuracy than the rebound integrators ias15 and whckl for fixed step size.

Anomalous Sun Flyby of 1I/2017 U1 (`Oumuamua)

The findings of Micheli et al. (Nature2018, 559, 223–226) that 1I/2017 U1 (`Oumuamua) showed anomalous orbital accelerations have motivated us to apply an impact model of gravity in search for an explanation. A small deviation from the 1/r potential, where r is the heliocentric distance, is expected for the gravitational interaction of extended bodies as a consequence of this model. This modification of the potential results from an offset of the effective gravitational centre from the geometric centre of a spherically symmetric body. Applied to anomalous Earth flybys, the model accounts for energy gains relative to an exact Kepler orbit and an increased speed of several spacecraft. In addition, the flat rotation profiles of eight disk galaxies could be explained, as well as the anomalous perihelion advances of the inner planets and the asteroid Icarus. The solution in the case of `Oumuamua is also based on the proposal that the offset leads to an approach and flyby trajectory different from a Kepler orbit without postulating cometary activity. As a consequence, an adjustment of the potential and centrifugal orbital energies can be envisaged outside the narrow uncertainty ranges of the published post-perihelion data without a need to re-analyse the original data. The observed anomalous acceleration has been modelled with respect to the orbit solutions JPL 16 and “Pseudo-MPEC” for 1I/`Oumuamua.

Gravitational potential of a planet modeled by a visco-elastic sphere

The paper investigates the gravitational potential of a planet moving in a gravitational field of an attracting center and a satellite. The planet is modeled by a homogeneous isotropic visco-elastic body that in its natural undeformed state has a full sphere shape. The satellite and the attracting center are represented by material points. The planet-satellite system moves relatively to the common center of gravity, which in its turn moves along a Keplerian orbit relatively to a motionless attracting center. Based on the solution of the quasi-static problem of elasticity theory for the current problem, the authors deduced a formula for evaluating the gravitational potential of a planet and also determined the Earth’s gravitational potential, taking into account the Moon and Sun’s tidal effects both in an external point.

A Likely Magnetic Activity Cycle for the Exoplanet Host M Dwarf GJ 3512

Abstract Current radial velocity data from specialized instruments contain a large amount of information that may pass unnoticed if their analysis is not accurate. The joint use of Bayesian inference tools and frequency analysis has been shown as effective in revealing exoplanets but they have been used less frequently to investigate stellar activity. We intend to use radial velocity data of the exoplanet host star GJ 3512 to investigate its magnetic activity. Our study includes the analysis of the photometric data available. The main objectives of our work are to constrain the orbital parameters of the exoplanets in the system, to determine the current level of activity of the star and to derive an activity cycle length for it. An adaptive importance sampling method was used to determine the parameters of the exoplanets orbit. Generalized Lomb–Scargle periodograms were constructed with both radial velocity curve and photometric data. A careful analysis of the harmonic frequencies was conducted in each periodogram. Our fit to multiple Keplerian orbits constrained the orbital parameters of two giant gas planets orbiting the star GJ 3512. The host star showed an increase of its magnetic activity during the last observing campaign. The accurate fit of the radial velocity curve data to the multi-Keplerian orbit permitted to reveal the star rotation in the residuals of the best fit and estimate an activity cycle length of ∼14 yr.

Attitude motion of a space object during its contactless ion beam transportation

The development of systems for transporting space objects by an active spacecraft is an urgent task of modern astronautics. One of the promising areas in this field is contactless ion beam transport systems. The goals of this work are to study a space object attitude motion dynamics during its contactless transportation and the influence of this motion on the magnitude of the force transmitted by the ion beam. A mathematical model of a mechanical system consisting of the space object and the active spacecraft was developed using the Lagrange formalism. A simplified equation of the object's angular motion in Kepler's orbit was obtained. The influence of the object's center of mass position and the ratio of ion beam and gravity gradient torques on the phase portrait view in the case of cylindrical object in a circular orbit was studied. The most favorable areas on phase portrait from the point of view of the magnitude of the force transferred by the ion beam to the object were found. It is shown that in the case of an elliptical orbit chaos exists. Entering a phase trajectory in a chaotic layer can lead to a noticeable decrease in the ion beam force.

An automatic tree search algorithm for the Tisserand graph

The Tisserand graph (TG) is a graphical tool commonly employed in the preliminary design of gravity-assisted trajectories. The TG is a two-dimensional map showing essential orbital information regarding the Keplerian orbits resulting from the close passage by one or more massive bodies, given the magnitude of the hyperbolic excess speed (v∞) and the minimum allowed pericenter height for each passage. Contours of constant v∞ populate the TG. Intersections between contours allow to link consecutive flybys and build sequences of encounters en route to a selected destination. When the number of perturbing bodies is large and many v∞ levels are considered, the identification of all the possible sequences of encounters through visual inspection of the TG becomes a laborious task. Besides, if the sequences are used as input for a numerical code for trajectory design and optimization, an automated examination of the TG is desirable. This contribution describes an automatic technique to explore the TG and find all the encounter paths. The technique is based on a tree search method, and the intersections between contours are computed using the regula-falsi scheme. The method is validated through comparisons with solutions available in the open literature. Examples are given of application to interplanetary mission scenarios, including the coupling with a trajectory optimizer.

Black Holes and the Supermassive Compact Object at the Galactic Center: Multi-arts of Thought and Nature

Black Holes and the Supermassive Compact Object at the Galactic Center: Multi-arts of Thought and Nature