Lagrangian Point Research Articles

The two roles of the SPH Kernel and dissipation on accretion disc modelling in semi-detached Low-Mass Close Binaries

This study addresses two aspects of the SPH technique. The first aspect regards the 3D (2D) numerical integration. The second aspect deals with the necessary damping for handling inviscid flow discontinuities. Accretion discs in Low-Mass Close Binaries (LMCBs) are a context pushing the limit of SPH codes since turbulence, shocks, and shear flows coexist. Thus, an LMCB with high mass-transfer from the inner Lagrangian point is considered. This choice is made for an in-depth understanding of the numerical and physical aspects related to pressure forces computed through SPH Kernel spatial gradients in high-speed collisional flows. In this regard, gas compressibility also plays a role. Firstly, we pay attention to the algebraic term 4πr2 coming from the cubic differential d3r and affecting any 3D SPH integration. This is made through a comparison of physically inviscid SPH structures referring to the same LMCB and mass-transfer conditions. Then, a reformulation of the non-viscous damping is also considered by adopting a Lagrangian-free physical formulation. Inviscid accretion discs should develop a steady toroidal structure because radial transport mechanisms should be excluded by missing any radial shear flow damping. Therefore, thinner discs, also including a steady toroidal ring, identify that SPH inviscid modelling free of any incorrect pressure gradient excess among thick disc structures. This task is simplified since initial conditions deliberately favour thick discs. The physically viscous hydrodynamics of the disc is also addressed by adopting a Prandtl-like turbulent kinematic viscosity coefficient in the Navier–Stokes equations, distinguishing the roles of the bulk and shear viscosities.

On Libration Points in the Asteroid–Research Probe System

This paper is dedicated to a study of relative motion in the field of homogeneous spheroid bodies. It is assumed that an asteroid’s satellite does not affect its translational and rotational motion by inertia. The latter motion has the form of rotation in the mode of regular precession. The case in which the asteroid is an ellipsoid of revolution is considered. A system of conditions that the libration points must satisfy is constructed; it is shown that its solution is two types of libration points, which differ in their location with respect to the constant vector of the angular momentum and the axis of symmetry of the ellipsoid. Thereafter, the stability of libration points is investigated numerically and analytically in the first approximation.

On Resonant Values of Parameters in the Problem on the Stability of Lagrangian Solutions in the Near-Circular Restricted Three-Body Problem

The restricted problem of three bodies (material points) moving under the action of gravitational attraction according to Newton’s law is considered. The orbits of the main attracting bodies are considered to be ellipses with a small eccentricity, and a passively gravitating body can perform arbitrary spatial motion near the triangular libration point. For the Hamiltonian function corresponding to such a motion, the structure of the normal form is pointed in the cases of third-order resonances. In the planar restricted three-body problem, the equations up to the second degree of eccentricity for resonance curves for all resonances up to the sixth order inclusive are obtained.

Direct Lyapunov Method in the Problem of Ensuring the Stability of a Compact Minimum Set of a Dynamic System and the Formation of a Halo-Orbit Near the Lagrange Point L2

The efficiency of using the direct Lyapunov method for ensuring the stability of motion in compact invariant sets of finite-dimensional dynamical systems is shown. In the linear model of the restricted three-body problem, the possibility of ensuring the asymptotic stability of the periodic motion of a spacecraft (SC) in the vicinity of the collinear Lagrange point L2 using light pressure forces without the consumption of the working fluid is considered. The required area of control surfaces is estimated depending on the mass of the SC.

Space weather investigation Frontier (SWIFT)

The Space Weather Investigation Frontier (SWIFT) mission will aim at making major discoveries on the three-dimensional structure and dynamics of heliospheric structures that drive space weather. The focus will be on Interplanetary Coronal Mass Ejections (ICMEs) that originate from massive expulsions of plasma and magnetic flux from the solar corona. They cause the largest geomagnetic storms and solar energetic particle events, threatening to endanger life and disrupt technology on Earth and in space. A big current problem, both regarding fundamental solar-terrestrial physics and space weather, is that we do not yet understand spatial characteristics and temporal evolution of ICMEs and that the existing remote-sensing and in-situ observatories are not suited for resolving multi-layered and evolutionary structures in these massive storm drivers. Here, we propose a groundbreaking mission concept study using solar sail technology that, for the first time, will make continuous, in-situ multi-point observations along the Sun-Earth line beyond the Lagrange point L1 (sub-L1). This unique position, in combination with L1 assets, will allow distinguishing between local and global processes, spatial characteristics, temporal evolution, and particle energization mechanisms related to ICMEs. In addition, measurements of the magnetic field in earthbound ICMEs and their sub-structures from the SWIFT location will double the current forecasting lead-times from L1. This concept also paves the way for missions with increasingly longer forecasting lead-times, addressing NASA and NOAA’s space weather goals, as set forth by the Decadal Survey. The objective of this communication is to inform the community of the ongoing effort, including plans to further develop the mission concept, supported by the Heliophysics Flight Opportunities Studies (HFOS) program under NASA’s Research Opportunities in Space and Earth Sciences (ROSES).

A theory of mass transfer in binary stars

ABSTRACT Calculation of the mass transfer (MT) rate $\dot{M}_\text{d}$ of a Roche lobe overflowing star is a fundamental task in binary star evolution theory. Most of the existing MT prescriptions are based on a common set of assumptions that combine optically thick and optically thin regimes with different flow geometries. In this work, we develop a new model of MT based on the assumption that the Roche potential sets up a nozzle converging on the inner Lagrangian point and that the gas flows mostly along the axis connecting both stars. We derive a set of 1D hydrodynamic equations governing the gas flow with $\dot{M}_\text{d}$ determined as the eigenvalue of the system. The inner boundary condition directly relates our model to the structure of the donor obtained from 1D stellar evolution codes. We obtain an algebraic solution for the polytropic equation-of-state (EOS). This gives $\dot{M}_\text{d}$ within a factor of 0.9 to 1.0 of existing optically thick prescriptions and reduces to the existing optically thin prescription for isothermal gas. For a realistic EOS, we find that $\dot{M}_\text{d}$ differs by up to a factor of 4 from existing models. We illustrate the effects of our new MT model on a $30\, M_\odot$ low-metallicity star undergoing intensive thermal time-scale MT and find that it is more likely to become unstable to L2 overflow and common-envelope evolution than expected according to MT prescriptions. Our model provides a framework to include additional physics such as radiation or magnetic fields.

Multi-orbit lunar GNSS constellation design with distant retrograde orbit and Halo orbit combination

The Moon is the closest natural satellite to mankind, with valuable resources on it, and is an important base station for mankind to enter deep space. How to establish a reasonable lunar Global Navigation Satellite System (GNSS) to provide real-time positioning, navigation, and timing (PNT) services for Moon exploration and development has become a hot topic for many international scholars. Based on the special spatial configuration characteristics of Libration point orbits (LPOs), the coverage capability of Halo orbits and Distant Retrograde Orbit (DRO) in LPOs is discussed and analyzed in detail. It is concluded that the Halo orbit with a period of 8 days has a better coverage effect on the lunar polar regions and the DRO has a more stable coverage effect on the lunar equatorial regions, and the multi-orbital lunar GNSS constellation with the optimized combination of DRO and Halo orbits is proposed by combining the advantages of both. This multi-orbital constellation can make up for the fact that a single type of orbit requires a larger number of satellites to fully cover the Moon, using a smaller number of satellites for the purpose of providing PNT services to the entire lunar surface. We designed simulation experiments to test whether the multi-orbital constellations meet the full lunar surface positioning requirements, and compare the coverage, positioning, and occultation effects of the four constellation designs that pass the test, and finally obtain a set of well-performing lunar GNSS constellations. The results indicate that the multi-orbital lunar GNSS constellation combining DRO and Halo orbits can cover 100% of the Moon surface, provides there are more than 4 visible satellites at any time on the Moon surface, which meets the navigation and positioning requirements, and the Position Dilution of Precision (PDOP) value is stable within 2.0, which can meet the demand for higher precision Moon surface navigation and positioning.

Jupiter, the great celestial organizer: On the origin of orbits of some asteroid families in mean motion resonances with Jupiter

The mechanism that forms the different resonant families of the asteroid belt between the orbits of Jupiter and Mars is not completely understood. To study the resonant capture mechanism and the origin of orbits of some asteroid families in mean motion resonances with Jupiter, furthermore to give a possible explanation of this problem, we performed in situ numerical simulations of the 3-dimensional semi-analytical model of the Sun-Jupiter-particle system. We examined the numbers and orbits of particles captured along the Jupiter's present orbit (including the L3, L4 and L5 Lagrange points) for 7000 Jupiter years. In our simulations we did not distinguish between planetesimals, interplanetary dust and asteroids, because we investigated the restricted three-body problem taking into account only gravitation. We found that the asteroid families in 4:3, 1:1, 4:5, 3:4, 2:3, 5:8, 4:7 and 5:9 resonances may have formed from the asteroids captured, launched and kept on their orbits not only by the L4 and L5 points, but also by the unstable L3 point and certain areas along the Jupiter's path. Furthermore, we found possible new asteroid families in 4:5 and 3:4 resonances, no any member of which have yet been identified. Thus, Jupiter forms a shield which protects the inner Solar System from asteroids moving with speeds not significantly differing from that of Jupiter.

Generation of Multiple-Revolution Many-Impulse Optimal Spacecraft Maneuvers

Impulsive trajectories provide time- and -reachability insights. Two novel homotopy-based methods are proposed for generating optimal many-impulse, multirevolution maneuvers. The first method is based on the continuation over the specific impulse value, which is shown to enhance convergence performance of the resulting two-point boundary-value problems. The second method is based on the formulation of impulsive trajectories using a linear acceleration term. The two methods are used in a hybrid framework. The utility of the proposed methods is demonstrated on four problems: Two interplanetary trajectories 1) from Earth to Mars, 2) from Earth to asteroid Dionysus, 3) a planet-centric transfer maneuver from a geostationary transfer orbit (GTO) to geostationary orbit (GEO) consisting of 21 revolutions, and 4) a 50-revolution transfer from the considered GTO to a circular orbit at the first Lagrange point (L1) of the Earth–moon system. The last two problems leading to 18 and 50 impulses are tackled to give an optimal near-impulsive solution. The impulsive solution with 18 impulses is shown to satisfy Lawden’s conditions.

A mixed three‐field total Lagrangian material point method for phase‐field fracture modeling of nearly incompressible rubber‐like solids

AbstractThe rubber‐like solids, as commonly observed materials in nature and engineering areas, often show high extensibility and weak compressibility, which lead to difficulties for modeling the fracture behavior of these materials. In this paper, we propose a mixed three‐field total Lagrangian material point method (TLMPM) to deal with the fracture of nearly incompressible rubber‐like solids. In this method, the phase‐field method combining with perturbed Lagrangian approach is developed to describe the fracture of nearly incompressible materials. Based on the Lagrangian equation, the governing equations of the displacement, pressure and phase fields are derived for the system considering dynamic effect. Then, a mixed three‐field TLMPM discrete formulation with a remapping strategy is then built to solve the problem involving large deformation. Besides, a staggered solving scheme with explicit time integration is developed for the three‐field problem. The accuracy and capabilities of the proposed method for dealing with the fracture of incompressible materials are demonstrated by modeling several quasi‐static and dynamic numerical problems and comparing the results with other numerical methods and experiments. Moreover, the applicability of the method is further proved by an out‐of‐plane tearing case.

An assessment of the total Lagrangian material point method: Comparison to conventional MPM, higher order basis, and treatment of near-incompressibility

The total Lagrangian material point method (TLMPM) is a recent modification to the conventional material point method (MPM). It poses the weak form in the undeformed configuration, and has shown great potential in solid mechanics applications because it seamlessly eliminates the so-called cell-crossing instability and overcomes numerical fracture. Despite its promise, it has not been widely adopted by the MPM community yet, and only a handful of research groups have employed it. At the same time, due to TLMPM’s recent appearance, there are open research questions, such as the subject of near-incompressibility that has not been thoroughly studied within its context. In this paper we attempt to strengthen TLMPM’s position in the field by providing an independent assessment of the method in a set of benchmark problems. Furthermore, we evaluate the need of using higher order spline shape functions, which have been shown to be crucial for the success of the conventional MPM. Finally, we present a simple projection technique for addressing solid mechanics problems that fall within the near-incompressible regime. The numerical results show that TLMPM is indeed a strong alternative to conventional MPM, whereas our near-incompressibility approach produces solutions that are free of volumetric locking and pressure oscillations, while introducing very minor modifications to existing codes.

A Tether System at the L1, L2 Collinear Libration Points of the Mars–Phobos System: Analytical Solutions

This paper is dedicated to identifying stable equilibrium positions of the tether systems attached to the L1 or L2 libration points of the Mars–Phobos system. The orbiting spacecraft deploying the tether is at the L1 or L2 libration point and is held at one of these unstable points by the low thrust of its engines. In this paper, the analysis is performed assuming that the tether length is constant. The equation of motion for the system in the polar reference frame is obtained. The stable equilibrium positions are found and the dependence of the tether angular oscillation period on the tether length is determined. An analytical solution in the vicinity of the stable equilibrium positions for small angles of deflection of the tether from the local vertical is obtained in Jacobi elliptic functions. The comparison of the numerical and analytical solutions for small angles of deflection is performed. The results show that the dependencies of the oscillation period on the length of the tether are fundamentally different for L1 and L2 points. Analytical expressions for the tether tension are derived, and the influence of system parameters on this force is investigated for static and dynamic cases.

Study the Non-Linear Stability of Non-Collinear Libration Point in the Restricted Three-Body Configuration When the Shapes of the Primaries are Taken as Heterogeneous and Finite-Straight Segment

Study the Non-Linear Stability of Non-Collinear Libration Point in the Restricted Three-Body Configuration When the Shapes of the Primaries are Taken as Heterogeneous and Finite-Straight Segment

Direct Lyapunov Method in the Problem of Ensuring the Stability of a Compact Minimum Set of a Dynamic System and the Formation of a Halo-Orbit Near the Lagrange Point L2

Direct Lyapunov Method in the Problem of Ensuring the Stability of a Compact Minimum Set of a Dynamic System and the Formation of a Halo-Orbit Near the Lagrange Point L2

The Origin of Extremely Nonradial Solar Wind Outflows

The origin of nonradial solar wind flows and their effect on space weather are poorly understood. Here we present a detailed investigation of 12 nonradial solar wind events during solar cycles 23–24, covering the period 1995–2017. In all these events the azimuthal flow angles of the solar wind exceed 6° as measured at the L1 Lagrangian point of the Sun–Earth system, for periods of 24 hr. In addition, all the events were selected during periods when coronal mass ejections (CMEs) and/or corotating interaction regions (CIRs) were absent. For most of the events, the near-Earth solar wind density was <5 cm−3 for periods exceeding 24 hr, similar to the well-known “solar wind disappearance events” wherein near-Earth solar wind densities dropped by two orders of magnitude for periods exceeding 24 hr. The solar source regions determined for all the cases were found to be associated with active region–coronal hole (AR–CH) pairs located around the central meridian. Further, the dynamical evolution of the source regions, studied using both the Extreme-ultraviolet Imaging Telescope and the Michelson Doppler Imager, showed a clear reduction in the CH area accompanied by the emergence of new magnetic flux regions. This dynamic evolution in the AR–CH source regions eventually disturbed the stable CH configurations, thereby giving rise to the extremely nonradial solar wind outflows. We discuss, based on our results, a possible causative mechanism for the origin of these highly nonradial flows that were not associated with either CMEs or CIRs.

The Comet Interceptor Mission—Making a case for Solar Electric Propulsion

The Comet Interceptor plans to be the first mission to visit a Dynamically New Comet — by definition, a comet visiting the inner Solar System for the first time. Dynamically New Comets are detected very shortly before their closest passage to the Sun; the mission and spacecraft will have to be designed, and may be even launched, without a known target. The mission is scheduled for launch in 2028 and will stay parked in the Sun–Earth Lagrange Point 2 until the start of the transfer to the target. We analyzed a Solar Electric Propulsion (SEP) transfer for The Comet Interceptor Mission. Specifically, the possible encounter locations were parametrically defined and mapped according to the Δv required to reach them with SEP; a transfer to a Comet Interceptor backup target was also studied. We compared the results obtained with SEP with the available performance data for the Chemical Propulsion (CP) baseline system, establishing the superiority of the SEP option in maximizing both payload mass and reachable locations.

Seven Sisters: a mission to study fundamental plasma physical processes in the solar wind and a pathfinder to advance space weather prediction

This paper summarizes the Seven Sisters solar wind mission concept and the outstanding science questions motivating the mission science objectives. The Seven Sisters mission includes seven individual spacecraft designed to uncover fundamental physical processes in the solar wind and provides up to ≈ 2 days of advanced space weather warnings for 550 Earth days during the mission. The mission will collect critical measurements of the thermal and suprathermal plasma and magnetic fields, utilizing, for the first time, Venus–Sun Lagrange points. The multi-spacecraft configuration makes it possible to distinguish between spatial and temporal changes, define gradients, and quantify cross-scale transport in solar wind structures. Seven Sisters will determine the 3-D structure of the solar wind and its transient phenomena and their evolution in the inner heliosphere. Data from the Seven Sisters mission will allow the identification of physical processes and the quantification of the relative contribution of different mechanisms responsible for suprathermal particle energization in the solar wind.

Analysis of Resonant Periodic Orbits in the Framework of the Perturbed Restricted Three Bodies Problem

In this work, the perturbed equations of motion of the infinitesimal body are constructed in the framework of the circular restricted three-body problem when the main two bodies are oblate and radiating. Under the perturbations effects of the oblateness and the radiation pressure the positions of collinear Lagrange points are evaluated, the interior and exterior first-order resonant periodic orbits are also studied. In addition, the initial positions of the periodic orbits and the size of loops have been estimated under these effects. Thus, the characteristics of periodic orbits have been studied under the combine effects of two, three and four perturbations for all the possible combinations of the perturbed parameters. The different order of resonant periodic orbits have been also analysed under the effects of Jacobi constant, mass factor, order of resonance and number of loops.

Embrace through the universe: Sound design with cosmic muons and the parameters of solar wind

Although humans are destined to live in a world of chaos, there is also hidden beauty within it. Analogically, this implies that there can be a creative way of living through the exploration of the unknown. Considering humans are part of the dynamism of the whole universe that consists of matter and energy, what would it be like if they could feel being embraced through the universe? Having continuous dialogues with the organizer throughout the planning phase, an artistic performance with the sounds of the universe is presented at the cultural event Connected Ink 2021 in Tokyo. This performance is entitled Life, the Universe and Everything in homage to The Hitchhiker's Guide to the Galaxy by Douglas Adams. In this paper, the background, concepts, physics, sound design, and the process and result of the project are discussed in collaboration with artists, sound designers, and experimental physicists. Regarding the sound design, it had been developed, in part, based on what the authors have shown through the preliminary sound the installation: Particle Post – Letters from the Universe in 2019, through which the signals from cosmic muons that are captured by a scintillation detector are poetically designed. Transforming imperceivable elements of the universe into an appreciable form in a poetic manner is one of the crucial essences of the authors' research and practice. With this, we believe that we could approach the wonders of the universe aesthetically. Hence, imperceivable elements such as cosmic muons and the parameters of solar wind (density, speed, and temperature in particular) captured from the satellite DSCOVR orbiting the L1 Lagrange point, are brought into the audio system in almost real time. Five iron fuurin, Japanese wind bells, ring triggered by cosmic muons that constantly arrive on Earth, while the stylized sounds of solar wind surround the large space at the venue. Consequently, these two different kinds of ‘wind’ sounds create harmony within the space, hoping to create a similar harmony through the bodies and minds of the audience as well. This project also signifies the value of genuine dialogue and communication with the purpose of deepening an interdisciplinary collaboration and opening up further creation.

Stability analysis in the R3BP under the effect of heterogeneous spheroid

In this paper, we investigate motion of infinitesimal mass m in the restricted three-body problem using heterogeneous spheroid with three layers. We consider the less massive primary m2 as a heterogeneous spheroid with three layers having different densities, and more massive primary m1 is considered as a point mass. We also derived the potential of proposed model and equations of motion of infinitesimal mass m. It is found that there exist five libration points Li(i=1,2,3,4,5) in the proposed model, three are collinear and two are non-collinear. The locations of all libration points have been investigated for the mass parameter μ(0<μ<0.585688k1) analytically and numerically. Furthermore, the collinear libration points Li(i=1,2,3) are found unstable while non-collinear libration points L4,5 are conditionally stable for the mass parameter μ∈(0,μc),μc=0.585688k1.