Lagrangian Point Research Articles

Periodic Orbits Around the Triangular Points with Prolate Primaries

ABSTRACT Periodic orbits play a fundamental role in the study and deep understanding of the behavior of dynamical systems. In the current work, we investigated the periodic orbits around the triangular libration points of the restricted three-body problem. The equations of motion of the restricted problem are presented when both primaries are prolate triaxial. Periodic orbits around the triangular points are obtained and then illustrated graphically for some selected initial conditions and for the entire domain of the mass ratio μ, as well. The eccentricities of the periodic orbits are obtained and then represented graphically. It is observed that the periodic orbits about the triangular stationary points are elliptical, and the frequencies of short and long orbits of the periodic motion are influenced by the shape of the primary bodies. Furthermore, we found that the perturbing forces influence the period, the orientation, and the eccentricities of the short and long periodic orbits.

A Gap-sharing Planet Pair Shaping the Crescent in HD 163296: A Disk Sculpted by a Resonant Chain

The Atacama Large Millimeter Array observations of the disk around HD 163296 have resolved a crescent-shape substructure at around 55 au, inside and off-center from a gap in the dust that extends from 38 to 62 au. In this work we propose that both the crescent and the dust rings are caused by a compact pair (period ratio ≃4:3) of sub-Saturn-mass planets inside the gap, with the crescent corresponding to dust trapped at the L 5 Lagrange point of the outer planet. This interpretation also reproduces well the gap in the gas recently measured from the CO observations, which is shallower than what is expected in a model where the gap is carved by a single planet. Building on previous works arguing for outer planets at ≈86 and ≈137 au, we provide a global model of the disk that best reproduces the data and shows that all four planets may fall into a long resonant chain, with the outer three planets in a 1:2:4 Laplace resonance. We show that this configuration is not only an expected outcome from disk–planet interaction in this system, but it can also help constrain the radial and angular position of the planet candidates using three-body resonances.

An immersed boundary-lattice Boltzmann flux solver for simulation of flows around structures with large deformation

In this paper, we present an immersed boundary-lattice Boltzmann flux solver (IB-LBFS) to simulate the interactions of viscous flow with deformable elastic structures, namely, two-dimensional (2D) and three-dimensional (3D) capsules formed by elastic membranes. The IB-LBFS is based on a finite-volume formulation and makes use of hydrodynamic conservation equations with fluxes computed by a kinetic approach; thus, it is more flexible and efficient than the standard immersed boundary-lattice Boltzmann methods. The membrane of the 2D capsule is represented by a set of discrete Lagrangian points, with in-plane and bending forces acting on the membrane obtained by a finite difference method. In contrast, the membrane of a 3D capsule is discretized into flat triangular elements with membrane forces calculated by an energy-based finite-element method. The IB-LBFS is first validated by studying the deformation of a circular capsule in a linear Newtonian and a power-law shear flow. Next, the deformation dynamics of a spherical, an oblate spheroidal, and a biconcave capsule in a simple shear flow are simulated. For an initially spherical capsule, the tank-treading motion of its membrane is reproduced at the steady state; while for oblate spheroidal and biconcave capsules, the swinging and tumbling motions are observed. Furthermore, under certain parameter settings, the transient mode from tumbling to swinging motions is also found, showing a rich and complex dynamic behavior of non-spherical capsules. These results indicate that the IB-LBFS can be employed in future studies concerning the dynamics of a capsule suspension in more realistic flows.

Equilibrium points and their stability in a new generalized planar version of the collinear restricted four-body problem

We introduce a new generalized planar version of the collinear restricted four-body problem. In this version, all three main bodies are located on the horizontal x-axis, while they have unequal masses. The dynamical system contains two free parameters related to the masses and positions of the main bodies. We investigate the equilibrium dynamics of the system by determining how these two free parameters affect the dynamical properties of the equilibria, such as their position and linear stability. By deploying some standard numerical methods combined with rigorous and systematic analysis, we studied the equilibrium dynamics of the system and revealed its properties regarding the nature of the libration points and the connections with the parameters of the system.

End-to-End Optimization of Power-Limited Earth–Moon Trajectories

The aim of this study is to analyze lunar trajectories with the optimal junction point of geocentric and selenocentric segments. The major motivation of this research is to answer two questions: (1) how much of the junction of the trajectory segments at the libration point between the Earth and the Moon is non-optimal? and (2) how much can the trajectory be improved by optimizing the junction point of the two segments? The formulation of the end-to-end optimization problem of power-limited trajectories to the Moon and a description of the method of its solution are given. The proposed method is based on the application of the maximum principle and continuation method. Canonical transformation is used to transform the costate variables between geocentric and selenocentric coordinate systems. For the initial guess, a collinear libration point between the Earth and the Moon is used as a junction point, and the transformation to the optimal junction of these segments is carried out using the continuation method. The developed approach does not require any user-supplied initial guesses. It provides the computation of the optimal transfer duration for trajectories with a given angular distance and facilitates the incorporation of the perturbing accelerations in the mathematical model. Numerical examples of low-thrust trajectories from an elliptical Earth orbit to a circular lunar orbit considering a four-body ephemeris model are given, and a comparison is made between the trajectories with an optimal junction point and the trajectories with a junction of geocentric and selenocentric segments at the libration point.

Attitude Motion of Cylinder in Variable Electrostatic Field Near Libration Point

This paper explores the spatial attitude motion of an electrostatic cylindrical container (E container) in a variable attracting electrostatic field. This field generated by the E container and the orbiting spacecraft (orbiter) located at the L1 libration point of the Mars–Phobos system is used to capture the E container by the orbiter in the Phobos Sample Return mission. Based on the Lagrange formalism and using the Euler angles, the equations of attitude motion of a cylindrical E container are constructed in the framework of the circular restricted three-body problem with the addition of the electrostatic force and torque acting on the E container. For modeling electrostatic force and torque acting on a cylindrical E container, the multisphere method is used, which was developed by Stevenson and Schaub (Multi-Sphere Method for Modeling Electrostatic Forces and Torques,” Advances in Space Research, Vol. 51, No. 1, 2013, pp. 10–20). The influence of gravitational, centrifugal, and electrostatic moments on angular motion for different initial conditions has been studied analytically by analyzing the equations of both attitude motion and numerical simulations. It is shown that the action of the aforementioned torques as the E container approaches the orbiter causes the excitation of high-frequency nutation oscillations with large amplitudes. On the other hand, there are no natural and artificial factors of physical nature that could lead to damping of oscillations of large amplitude, and this fact corresponds to the obtained equations of attitude motion. Also, the electrostatic torque is the greatest contributor to the oscillations with large amplitude. Hence, it is clear that the E container must have a spherical current-conducting shell: in which case, the electrostatic torque turns to zero.

Fully Autonomous Orbit Determination and Synchronization for Satellite Navigation and Communication Systems in Halo Orbits

This paper presents a solution for autonomous orbit determination and time synchronization of spacecraft in Halo orbits around Lagrange points using inter-satellite links. Lagrange points are stable positions in the gravitational field of two large bodies that allow for a sustained presence of a spacecraft in a specific region. However, a challenge in operating at these points is the lack of fixed landmarks for orbit determination. The proposed solution involves using inter-satellite links to perform range and range-rate measurements, allowing for accurate computation of the spacecraft’s orbit parameters without the need for any facilities on Earth. Simulations using a fleet of three satellites in Near Rectilinear Halo Orbits around the Earth–Moon Lagrange point, proposed for the Lunar Gateway stations, were conducted to demonstrate the feasibility of the approach. The results show that inter-satellite links can provide reliable and accurate solutions for orbit determination with a DRMS error lower than one meter (90th percentile) and synchronization errors of around one nanosecond. This solution paves the way for a fully autonomous fleet of spacecraft that can be used for observation, telecommunication, and navigation missions.

Space-Time Properties of Extreme RN Black Holes in Static Triangular Distribution

We studied the space-time properties of the triangular symmetric black hole in the case of extreme RN black hole. Because the neutral test particle is only affected by space-time in the curved space-time, we chose the triangular symmetric black hole as the model with which to study the motion of the test particle in this case. The curvature tensor and curvature scalar were calculated by giving the metric and the Christoffel Symbol, and then the kinematics equation of the test particle was obtained and analyzed by using these quantities. Then we analyzed the relationship between the coordinate distance and the inherent distance, the relationship between the coordinate time and the inherent time, the inherent velocity and the coordinate velocity of light, and then verified the correctness of general relativity. Next, the one-dimensional effective potential and two-dimensional effective potential of the system under different separation distances were analyzed. Finally, we analyzed and explored the innermost stable circular orbit, calculated all the Lagrange points under this model, and expounded some applications of circular orbit in astrophysics.

Naturally bounded relative motion for formation flying near triangular libration points

This paper proposes a numerical searching and ergodic mappings method to seek the naturally bounded relative orbits near the triangular libration points in the Earth-Moon system. An orbital energy saved numerical continuation method along one axis is proposed and the Poincaré map near the triangular libration points is obtained, based on which two characteristic parameters are defined to generate the naturally bounded relative motions, i.e., the crossing interval period and the reference angle. Then, a numerical searching method is employed to generate an ergodic mapping and a neural network is utilized to realize the transformation from the ergodic mappings to orbital states. By pairing the ergodic mappings of different initial orbital states, the naturally bounded relative motion realized can remain for a long time and does not diverge.

Analyzing the engineering feasibility of the direct fusion drive

The Direct Fusion Drive (DFD) and its terrestrial counterpart, the Princeton Field Reversed Configuration (PFRC) reactor, have seen significant developments in the past decade. Various groups conducted detailed research on the required specifications of the engine and associated technology for power delivery to onboard avionics and payloads. Multiple studies have also addressed the thrust generation mechanism using empirical specific power scaling relations and plasma flow simulations. Recent studies have designed spacecraft for missions to Earth’s second Lagrange point, Mars, transneptunian bodies like Pluto, and the neighboring star systems Alpha Centauri A and B. However, significant work is needed to design the engine components in detail using scientific scaling relations and ab inito calculations to develop the physical systems for prototyping and testing. After critically analyzing the reference design of the DFD and the underlying fusion reactor, this paper addresses the technological gaps and suggests avenues to improve specifications toward targets outlined in previous studies while considering costs. Further, the authors present a prototype engine and magnetohydrodynamic power conversion system design to study the engineering hurdles relevant to the practical implementation of the DFD.

Dust as a solar shield

We revisit dust placed near the Earth–Sun L1 Lagrange point as a possible climate-change mitigation measure. Our calculations include variations in grain properties and orbit solutions with lunar and planetary perturbations. To achieve sunlight attenuation of 1.8%, equivalent to about 6 days per year of an obscured Sun, the mass of dust in the scenarios we consider must exceed 1010 kg. The more promising approaches include using high-porosity, fluffy grains to increase the extinction efficiency per unit mass, and launching this material in directed jets from a platform orbiting at L1. A simpler approach is to ballistically eject dust grains from the Moon’s surface on a free trajectory toward L1, providing sun shade for several days or more. Advantages compared to an Earth launch include a ready reservoir of dust on the lunar surface and less kinetic energy required to achieve a sun-shielding orbit.

Collinear and triangular solutions to the coplanar and circular three-body problem in the parametrized post-Newtonian formalism

This paper investigates the coplanar and circular three-body problem in the parametrized post-Newtonian (PPN) formalism, for which we focus on a class of fully conservative theories characterized by the Eddington-Robertson parameters $\beta$ and $\gamma$. It is shown that there can still exist a collinear equilibrium configuration and a triangular one, each of which is a generalization of the post-Newtonian equilibrium configuration in general relativity. The collinear configuration can exist for arbitrary mass ratio, $\beta$, and $\gamma$. On the other hand, the PPN triangular configuration depends on the nonlinearity parameter $\beta$ but not on $\gamma$. For any value of $\beta$, the equilateral configuration is possible, if and only if three finite masses are equal or two test masses orbit around one finite mass. For general mass cases, the PPN triangle is not equilateral as in the post-Newtonian case. It is shown also that the PPN displacements from the Lagrange points in the Newtonian gravity $L_1$, $L_2$ and $L_3$ depend on $\beta$ and $\gamma$, whereas those to $L_4$ and $L_5$ rely only on $\beta$.

Target selection for Near-Earth Asteroids in-orbit sample collection missions

This work presents a mission concept for in-orbit particle collection for sampling and exploration missions towards Near-Earth asteroids. Ejecta is generated via a small kinetic impactor and two possible collection strategies are investigated: collecting the particle along the anti-solar direction, exploiting the dynamical features of the L$_2$ Lagrangian point or collecting them while the spacecraft orbits the asteroid and before they re-impact onto the asteroid surface. Combining the dynamics of the particles in the Circular Restricted Three-Body Problem perturbed by Solar Radiation Pressure with models for the ejecta generation, we identify possible target asteroids as a function of their physical properties, by evaluating the potential for particle collection.

An assessment of different relay network topologies to improve Earth–Mars communications

The future of deep space communications encompasses a challenging situation where the current facilities used to communicate with different spacecraft may become saturated as a result of an increasing number of missions and their complexity. From this forecast, the present study intends to provide a solution to saturation problems through strategically-located upgradable relays for Earth–Mars communications. The foremost goal of this paper is to quantitatively uncover the potential enhancements coming from relay placement in strategic orbits between Earth and Mars. Herein, two relay configurations – a.k.a. network topologies – are analyzed: the Lagrange-relays network topology and a circular, homogeneously-distributed satellite constellation, acknowledged here as pearl constellation. The first uses the Earth–Sun system Lagrange points L3, L4 and L5 as potential locations for the relays, whilst the second defines an optimized orbit between Earth and Mars with 3 or 4 relay satellites.To aid in the analysis, the authors developed an open-sourced piece of software that obtains the link availability as well as the data rate at which two nodes may communicate, taking as a reference the Deep Space Network (DSN) for Earth, and the Mars Reconnaissance Orbiter for Mars. For complex topologies with more than two communicating nodes, the software outputs the end-to-end bit rate and optimal communication route at each time step. Moreover, this product is extensible to analyze and optimize any network topology and could be adapted to be used for contact management and mission planning in the future.The results show that the network-topology proposals are an advantageous option to significantly increase the link availability of Earth–Mars communications. Nevertheless, the Direct-To-Earth (DTE) link always outperforms the multi-hop path due to the limited telecommunication system’s capabilities of both the spacecraft and the relays. As a result of this, the study includes an analysis on the requirements of the relay’s design in order to make the constellation a beneficial and comparable alternative to the DTE link. This way, the proposed network topologies become a suitable option whom to share with the DSN communications workload, providing enhanced bit rates and data volumes as well as higher availability of the communication.

Debris Collision Risk Analysis Following Simulated Cislunar Spacecraft Explosions

This research simulates spacecraft explosions in cislunar space and analyzes the resulting risk to other notional spacecraft. Three case studies are considered, which include simulations of spacecraft explosions during Earth–moon Hohmann transfers, along the manifold of the Earth–moon Lagrange point, and at the stable and Earth–moon Lagrange points. Debris from the explosions is propagated using a four-body dynamical model that incorporates the gravitational forces of the Earth, moon, and sun. Finally, the risk to nearby cislunar spacecraft is quantified using two different models for the vulnerability of spacecraft to debris, which include a logistic vulnerability model that incorporates only debris particle mass and a ballistic limit vulnerability model that incorporates both particle mass and velocity. The results from each model are compared to evaluate their application to cislunar debris scenarios. The results of this study showed potential debris hazards to spacecraft near Earth from a mishap during an Earth–moon transfer and hazard probabilities to nearby cislunar spacecraft on the order of from Lagrange point mishaps. Overall, research into cislunar debris propagation enhances operational planning outside the traditional near-Earth paradigm of spacecraft mission operations and increases understanding of the debris-related consequences of mishaps within this orbital regime.

The dynamics of co-orbital giant exomoons – applications for the Kepler-1625 b and Kepler-1708 b satellite systems

ABSTRACT Exomoons are a missing piece of exoplanetary science. Recently, two promising candidates were proposed, Kepler-1625 b-I and Kepler-1708 b-I. While the latter still lacks a dynamical analysis of its stability, Kepler-1625 b-I has already been the subject of several studies regarding its stability and origin. Moreover, previous works have shown that this satellite system could harbour at least two stable massive moons. Motivated by these results, we explored the stability of co-orbital exomoons using the candidates Kepler-1625 b-I and Kepler-1708 b-I as case studies. To do so, we performed numerical simulations of systems composed of the star, planet, and the co-orbital pair formed by the proposed candidates and another massive body. For the additional satellite, we varied its mass and size from a Mars-like to the case where both satellites have the same physical characteristics. We investigated the co-orbital region around the Lagrangian equilibrium point L4 of the system, setting the orbital separation between the satellites from θmin = 30° to θmax = 90°. Our results show that stability islands are possible in the co-orbital region of Kepler-1708 b-I as a function of the co-orbital companion’s mass and angular separation. Also, we identified that resonances of librational frequencies, especially the 2:1 resonance, can constrain the mass of the co-orbital companion. On the other hand, we found that the proximity between the host planet and the star makes the co-orbital region around Kepler-1625 b-I unstable for a massive companion. Finally, we provide TTV profiles for a planet orbited by co-orbital exomoons.

Formation of asymmetric arms in barred galaxies

ABSTRACTWe establish a dynamical mechanism to explain the origin of the asymmetry between the arms observed in some barred disc galaxies, where one of the two arms emanating from the bar ends is very well defined, while the second one displays a ragged structure, extending between its ridge and the bar. To this purpose, we study the invariant manifolds associated with the Lyapunov periodic orbits around the unstable equilibrium points at the ends of the bar. Matter from the galaxy centre is transported along these manifolds to the periphery, forming this way the spiral arms that emanate from the bar ends. If the mass distribution in the galaxy centre is not homogeneous, because of an asymmetric bar with one side stronger than the other, or because of a non-centred bulge, the dynamics about the two unstable Lagrange points at the ends of the bar will not be symmetric as well. One of their invariant manifolds becomes more extended than the other, enclosing a smaller section and the escaping orbits on it are fewer and dispersed in a wider region. The result is a weaker arm and more ragged than the one at the other end of the bar.

Catching a Milky Way open cluster in its last breath

ABSTRACT Theoretical models have suggested peculiar velocity dispersion profiles of star clusters facing dissolution. They predicted that, besides bound stars that still belong to the star cluster, and unbound ones already stripped off, there is an intermediate population of stars that having acquired the enough energy to escape the cluster are still within the cluster Jacobi radius. Both, potential escapers and unbound stars, show hot kinematics, not observed along tidal tails of star clusters. We report on the first evidence of an open cluster with stars crossing such a transitional scenario, namely: ASCC 92. The open cluster gathers nearly 10 per cent of its initial total mass, and is moving towards Galactic regions affected by higher interstellar absorption. Precisely, the obscured appearance of the cluster could have hampered disentangling its true internal dynamical evolutionary stage, previously.

A novel coupling method for unresolved CFD-DEM modeling

In CFD-DEM (computational fluid dynamics-discrete element method) simulations particles are considered Lagrangian point particles. The details of the flow near the particle surface are therefore not fully resolved. When the particle scale is larger than the resolved flow scale, the coupling between the CFD model and the DEM model is critical. An effective coupling scheme should minimize the risk of artificial influences on the results from choices of numerical parameters in implementations and consider efficiency and robustness. In this work, a novel coupling method is developed. The method includes both the smoothing of the particle data and the sampling of the gas phase quantities. The smoothing employs the diffusion-based method. The gas sampling method can reconstruct the filtered fluid quantities at the particle center. The sampling method is developed based on the diffusion-based method with higher efficiency. The new method avoids mesh searching and it can be easily implemented in parallel computing. The developed method is validated by the simulation of a forced convection experiment for a fixed bed with steel spheres. With the well-posed grid-independent coupling scheme, the simulation results are in good agreement with the experimental measurements. The coupling effects and the computational cost are discussed in detail.

Space probe of the gravitational constant G

The empirical Gravitational constant G is of fundamental interest in metrology, Newtonian physics, and the field equations in General Relativity, particle physics, geophysics, and astrophysics. The precise value of G has yet to be known with the desired level of accuracy. In the 233 years since Cavendish, hundreds of experiments sought more accurate results but produced inconsistent results. Although they have had much more advanced supporting technology, past experiments have mostly stayed the same from the Cavendish approach. This paper shows that a different method is feasible to enhance the accuracy of G. A sphere made of metal with a large uniform mass density having a channel through the center with a bead oscillating back and forth in a simple-harmonic fashion. Monitoring the positions of the bead experimentally and comparing the predictions of the channel model will allow refining the value of G. The study develops a gravitational force function for the bead in a channeled sphere to provide more accuracy in the prediction of the bead's motion. This study determines that the best place to perform future gravitation constant experiments is in space far away from interfering effects. A near-perfect place is likely to be in outer space at the Sun-Earth Lagrange point L1, located about 1.5 million kilometers from the Earth on a line toward the Sun where other probes reside. It is estimated that three to six significant figures may be added to the current value of G.