Lagrangian Point Research Articles

Adaptive phase-field smoothed total Lagrangian material point method for fracture analysis of soft materials involving large deformation

Abstract The Interstellar Dust Experiment (IDEX) onboard NASA’s Interstellar Mapping and Acceleration Probe (IMAP) is dedicated to measuring the flux, size distribution, and composition of Interstellar (ISD) and Interplanetary (IDP) Dust Particles while stationed at Lagrange point L1 of the Earth-Sun system. IDEX is an impact ionization Time-of-Flight (TOF) mass spectrometer that measures the elemental and/or molecular and selected isotopic composition of impacting dust particles. Due to its high sensitivity and large detection area, IDEX is expected to detect and analyze approximately 200 ISD and 1250 IDP particles over the first two years of the mission.

This research presents unsupervised machine learning and statistical methods to identify and analyze tidal tails in open star clusters using data from the Gaia DR3 catalog. We aim to identify member stars and to detect and analyze tidal tails in five open clusters, BH 164, Alessi 2, NGC 2281, NGC 2354, and M67, of ages between 60 Myr and 4 Gyr. These clusters were selected based on the previous evidence of extended tidal structures. We utilized machine learning algorithms such as Density-based Spatial Clustering of Applications with Noise (DBSCAN) and principal component analysis (PCA), along with statistical methods to analyze the kinematic, photometric, and astrometric properties of stars. Key characteristics of tidal tails, including radial velocity, the color-magnitude diagram, and spatial projections in the tangent plane beyond the cluster's Jacobi radius (r_J), were used to detect them. We used N-body simulations to visualize and compare the observables with real data. Further analysis was done on the detected cluster and tail stars to study their internal dynamics and populations, including the binary fraction. We also applied the residual velocity method to detect rotational patterns in the clusters and their tails. We identified tidal tails in all five clusters, with detected tails extending farther in some clusters and containing significantly more stars than previously reported (tails ranging from 40 to 100 pc, one to four times their r_J, with 100-200 tail stars). The luminosity functions of the tails and their parent clusters were generally consistent, and tails lacked massive stars. In general, the binary fraction was found to be higher in the tidal tails. Significant rotation was detected in M67 and NGC 2281 for the first time.

This study extends the classical circular restricted three-body problem (CR3BP) by introducing a dominant central primary, forming a collinear restricted four-body problem (CR4BP) that better reflects the dynamics of real planetary systems. The model remains dynamically consistent and non-degenerate when the central mass parameter μ0 lies in (½, 1) and the peripheral mass μ satisfies 0 < μ < ½ (1 – μ0). It generalizes to the CR3BP by setting μ0 = 0, recovering classical results. The system exhibits six libration points: four collinear and two symmetric non-collinear points forming an isosceles triangle with the peripheral primaries. Non-collinear points emerge via a saddle-node bifurcation at a critical μ = μc and as μ increases further within the range μc < μ < ½ (1 – μ0), these points move away from the x-axis and gradually align closer to the y-axis, while remaining symmetric with respect to the x-axis. The stability analysis reveals that collinear libration points L1, L3 and L4 are linearly unstable under all conditions while L2 is stable in the interval 0 < μ < μ* where μ* is a critical threshold for L2. The non-collinear points are linearly stable within a defined interval μc < μ < μc1. Finally, these results are applied to the Saturn–Janus–Epimetheus system to illustrate the model’s practical relevance.

The Deep Space Climate Observatory (DSCOVR), launched in 2015, is the first Earth-observing mission to a Sun-Earth first Lagrange point (L1) orbit, about 1.5 million km from Earth on the Sun-Earth line. The goal of the mission is to provide continuous solar wind measurements for accurate space weather forecasting and observe the sunlit side of the Earth for enhancing climate science. The Earth Polychromatic Imaging Camera (EPIC) is one of the two Earth-observing instruments on DSCOVR. It takes images of nearly the entire sunlit side of the Earth in 10 spectral channels at a relatively high temporal resolution to monitor the changing planet. EPIC’s view contains polar regions that are barely visible from geostationary satellite (GEOs), providing observations of the global reflected spectral radiation. Among other capabilities of EPIC, such as observing atmospheric and surface properties, the well calibrated reflected global spectral radiation observed by EPIC and EPIC-based broadband shortwave (SW) radiance and flux can be used to monitor the changing planet of the Earth. However, to assess the long-term change of the Earth in terms of its spectral brightness and reflected SW radiation, the natural variability of global spectral reflectance and SW radiation must be quantitatively determined. This work provides quantitative estimates of the variability of global spectral reflectance and SW radiance and flux on different time scales. The main finds of this work are: (1) the hourly variability of global average reflectance in red and NIR bands is much larger than the variation in UV and blue bands, and the 24-h variability in boreal summer is significantly larger than in winter; (2) the presence of Antarctica and the Arctic is primarily responsible for seasonal variation in spectral reflectance and SW radiance and flux; (3) the global average SW radiance is highly anisotropic, particularly over land, and assumption of Lambertian reflection will overestimate the SW flux by 20%–30%. Furthermore, the responsible physical mechanisms are provided.

The utilization of the natural resources of our Moon and the near-earth asteroids (NEAs) for the benefit of humankind will need industrial plants in space. There are a number of possible locations for the deep-space processing of extracted space-based materials and future industrial activities in cis-lunar space. Prime among these are the Moon itself and the Earth’s five Lagrange points which provide equilibrium between the gravity forces of the Earth and Moon. Especially in Lagrange points L4 and L5, objects remain in stable positions because of the triangle between the object, Earth, and Moon. Building the first space factory in L5, for example, will enable the processing of material and production of goods in zero gravity. Unlike on Earth or the Moon, solar power would be available for 24 h. The industrialisation of cis-lunar space will start with the mining of our Moon. The Lagrange Space Factory (LSF) would start with the processing of lunar material and extract aluminum, iron, titanium, and other materials from lunar regolith. When the metals are extracted from oxides, oxygen is a byproduct. An additional source for material would be the recycling of orbital debris to clean up Earth’s orbit. In the long run, the LSF would also process NEA material, including gold, platinum, and carbon. C-type (carbonaceous) asteroids also contain water ice and organic molecules. The goal would be to produce building material like steel bars and aluminum panels, tubes, and bricks for future space habitats. Oxygen and space-made propellant could also be produced. The isotope helium-3 is abundant on the Moon and can be used for future nuclear fusion in space and on Earth.

Mass transfer is crucial in binary evolution, yet its theoretical treatment has long relied on analytic models whose key assumptions remain debated. We present a direct and systematic evaluation of these assumptions using high-resolution 3D hydrodynamical simulations including the Coriolis force. We simulate streams overflowing from both the inner and outer Lagrangian points, quantify mass transfer rates, and compare them with analytic solutions. We introduce scaling factors, including the overfilling factor, to render the problem dimensionless. The donor-star models are simplified, with either an isentropic initial stratification and adiabatic evolution or an isothermal structure and evolution. However, the scalability of this formulation allows us to extend the results for a mass-transferring system to arbitrarily small overfilling factors for the adiabatic case. We find that the Coriolis force – often neglected in analytic models – strongly impacts the stream morphology: breaking axial symmetry, reducing the stream cross section, and shifting its origin toward the donor’s trailing side. Contrary to common assumptions, the sonic surface is not flat and does not always intersect the Lagrangian point: instead, it is concave and shifted, particularly toward the accretor’s trailing side. Despite these structural asymmetries, mass transfer rates are only mildly suppressed relative to analytic predictions and the deviation is remarkably small – within a factor of two (ten) for the inner (outer) Lagrangian point over seven orders of magnitude in mass ratio. We use our results to extend the widely used mass-transfer rate prescriptions by Ritter (1988, A&amp;A, 202, 93) and Kolb &amp; Ritter (1990, A&amp;A, 236, 385), for both the inner and outer Lagrangian points. These extensions can be readily adopted in stellar evolution codes like MESA, with minimal changes where the original models are already in use.

Abstract For more than six years SRG/eROSITA is the first X-ray telescope in orbit around the second Lagrangian point in the Sun-Earth system. We present an updated study of its instrumental background based on cleaned data and improved simulations, carried out in the framework of the eROSITA Background Working Group. We examine the standard observing mode with minimum ionising particle rejection switched on as well as the non-standard mode with particle rejection switched off, which was set from time to time to monitor the overflow trails and patterns left by energetic particles. We show that, in the first case, Geant4 simulations allow to reproduce quite well the measured residual background; in the second case, the simulations support the analysis and interpretation of the observed trails, though a peculiar feature of a small minority of them, which apparently split from one frame into the next one, is not reflected in the simulations.

Abstract A star’s luminosity increases as it evolves along the Main Sequence (MS), which inevitably results in a higher surface temperature for planets in orbit around the star. Technologically advanced civilizations may tackle this issue by installing artificial structures – starshades – which can reduce the radiation received by the planet. Starshades, if they exist, are potentially detectable with current or near-future technology. We have simulated phase curve signatures in direct imaging of hypothetical starshades in systems targeted by the upcoming Habitable Worlds Observatory (HWO), which will be tasked with searching for Earth-like exoplanets orbiting nearby stars. The starshade is assumed to be a circular, reflecting surface placed at the inner Lagrange point between the star and the planet. Our results show that the phase curve of a starshade has a distinct shape compared to that of a typical planet. The phase curve signature lies above the expected 1σ = 10−11 single-visit precision in contrast ratio of the telescope for 70.8% of the target stars for the expected inner working angle (IWA) of around 60 mas. If the IWA can be reduced to 45 mas, the percentage of stars above the 1σ limit increases to 96.7%. With a sufficiently small IWA, HWO should be able to detect anomalies in light curves caused by starshades or similar highly-reflective surfaces – which could serve as key indicators for technologically advanced civilizations.

This paper investigates the impact of radiation pressure of the primary body on a third body in the vicinity of the out-of-plane libration points of the restricted problem of three bodies under a rotating mass dipole effect. It studies the impact of radiation factor , force ratio and mass-ratio parameter on the location, zero-velocity curves and instability of the out-of- plane libration points in the frame work of the CRTBP . A symmetrical out-of-plane libration points are identified numerically on the and the three system’s parameters, that is and , are found to have impact on the positions of these points. In addition, the force ratio and radiation factor have significant effect on the configuration of the zero-velocity contours. The linear stability of the points for a broad scope of the parameters’ combinations was investigated and it is found that the out-of-plane libration points are generally points of instability.

ABSTRACTA smoothed total Lagrangian material point method (STLMPM) is developed in this study to effectively simulate dynamic problems involving large deformation in nearly incompressible soft materials. In this method, the governing equations are spatially discretized within the framework of the total Lagrangian material point method (TLMPM) and temporally discretized using an explicit time integration scheme. To address the issue of decreased computational accuracy of the material point method (MPM) near physical domain boundaries, a Gaussian kernel function with kernel correction is employed to establish the interpolation formulas between particles and the background grid. Furthermore, to mitigate volumetric locking caused by the nearly incompressible nature of soft materials, the F‐bar method is further developed within the framework of TLMPM. The accuracy and efficiency of the proposed STLMPM are demonstrated by several representative numerical examples, and the simulation results are compared with analytical solutions and other numerical methods.

Abstract The Earth–Moon libration points no longer exhibit the dynamical characteristics of “equilibrium points” due to perturbation effects when applying the ephemeris model. By decoupling the forced motions within the ephemeris model and computing the dynamical substitute trajectories, we can reconstruct a dynamical system that recovers the “equilibrium points” feature. Diverging from the conventional analytical approach rooted in the framework of Newtonian mechanics, this paper presents a novel method for calculating dynamical substitute based on the Hamiltonian mechanics framework. First, the Hamiltonian equations for the ephemeris model are formulated. Subsequently, the problem of decoupling forced motions is reformulated as solving a nonautonomous differential equation through canonical transformations. Then, an iterative method based on frequency analysis is employed for the computation. Eventually, approximate analytical solutions for five libration points over a 360 yr period are provided. Simulation results demonstrate that the computed approximate analytical solutions are in excellent agreement with the numerical integration results derived from the ephemeris model, thereby validating the efficacy of the proposed method. The Hamiltonian dynamical system derived herein enables the analysis of nonlinear central manifold motions via canonical transformations, facilitating the construction of higher-order analytical solutions for libration point orbits. This framework also provides a robust foundation for exploring characterization parameters of libration point orbits within the real Earth–Moon system.

ABSTRACT The gas-drag capture hypothesis for the origin of the Martian moons Phobos and Deimos, fairly consistent with their reflectance spectra like carbonaceous meteorites, faces difficulties in explaining their small orbital inclinations and the dense gas required for capture. Here, we show through numerical and theoretical analysis that gas-drag capture via temporary capture of small bodies can overcome these difficulties. Temporary capture occurs for small bodies entering the Martian Hill sphere at low speeds from the $\mathrm{L}_1$ or $\mathrm{L}_2$ Lagrange points, allowing them to orbit Mars repeatedly even without dissipation. Such a long-range orbit around Mars enables complete capture by weak gas drag under the minimum-mass solar nebula and more dilute gas densities, with minimal dependence on the specific gas flow structure. The orbital inclination with respect to the Mars orbital plane is suppressed to within a few degrees due to the quasi-conservation law of absolute angular momentum around Mars, and the periapsis immediately after complete capture ranges around several tens of Mars radii. The continuous gas drag with torque exerted by the Martian equatorial bulge can transfer captured bodies towards the primordial low-inclined moons’ orbits inferred from tidal evolution models. If the typical planetesimal masses are close to those of the Martian moons, there are numerous opportunities for complete capture through temporary capture even at late accretion stages with depleted nebula. The last population of captured bodies may have undergone the slowest gas drag-induced migration towards Mars and some of them may have survived as the present-day moons after the nebular dissipation.

Libration point orbit (LPO) family classification is vital for improving the initial orbit determination accuracy of LPOs. This paper proposed a long short-term memory (LSTM)-based neural network for LPO family classification using angle-only measurements. The proposed neural network comprises an LSTM module combined with three deep neural network (DNN) modules. The LSTM module can handle the time-series angle measurements with varying lengths, making the proposed method appropriate for severe cases where continuous and fixed-step measurements are unavailable. In addition, the DNN modules are designed for data preprocessing and postprocessing to improve classification accuracy. Two sample forms, one in the inertial frame and the other in the rotating frame, are analyzed. It is found that the sample form in the rotating frame has better training performance in classifying LPOs. The proposed method is successfully applied to classify LPOs in cislunar space. The accuracy and performance of the proposed method are demonstrated via comparisons with traditional machine learning approaches and standard DNNs. Numerical results show that the maximal false rate of the proposed method is less than 2.9%, while the maximal false rate of the standard DNN is 7.3%. In addition, compared with the competitive methods, the proposed model demonstrates stronger robustness to observation noise, as evidenced by its superior classification accuracy under various noise levels.

Analysis of the space environment for the CHES spacecraft at Lagrange point L2 between the sun and earth

A frequency-based hierarchy of dynamic models in cislunar space is introduced in the recent literature, where multiple models between the circular restricted three-body problem (CR3BP) and a higher-fidelity ephemeris model (HFEM) are investigated within a common rotating reference frame. Intermediate models selectively introduce perturbations associated with specific frequencies, enabling a systematic assessment of their influence on dynamic structures. This study conducts an extensive numerical investigation across cislunar space to complement the hierarchy and evaluate the suitability of various intermediate models. A total of six intermediate models, rendering (quasi-)periodically forced systems within the common frame, are analyzed. The numerical analysis examines key CR3BP structures, including Lagrange points and periodic orbits, and compares their counterparts in the intermediate models and the HFEM. Frequency-domain analysis is leveraged to assess capabilities and limitations of each model. The findings supply a criterion for selecting appropriate intermediate models, improving i) the characterization of perturbations inherent in higher-fidelity models and ii) transitions between models within the cislunar domain.

The cislunar space navigation satellite system is essential infrastructure for lunar exploration in the next phase. It relies on high-precision orbit determination to provide the reference of time and space. This paper focuses on constructing a navigation constellation using special orbital locations such as Earth–Moon libration points and distant retrograde orbits (DRO), and it discusses the simplification of planetary perturbation models for their autonomous orbit determination on board. The gravitational perturbations exerted by major solar system bodies on spacecraft are first analyzed. The minimum perturbation required to maintain a precision of 10 m during a 30-day orbit extrapolation is calculated, followed by a simulation analysis. The results indicate that considering only gravitational perturbations from the Moon, Sun, Venus, Saturn, and Jupiter is sufficient to maintain orbital prediction accuracy within 10 m over 30 days. Based on these findings, a method for simplifying the ephemeris is proposed, which employs Hermite interpolation for the positions of the Sun and Moon at fixed time intervals, replacing the traditional Chebyshev polynomial fitting used in the JPL DE ephemeris. Several simplified schemes with varying time intervals and orders are designed. The simulation results of the inter-satellite links show that, with a 6-day orbit arc length, a 1-day lunar interpolation interval, and a 5-day solar interpolation interval, the accuracy loss for cislunar space navigation satellites remains within the meter level, while memory usage is reduced by approximately 60%.

Almost 6000 exoplanets have thus far been confirmed, revolutionizing our understanding of planetary habitability. Yet, despite the identification of Earth-like exoplanets, definitive evidence of extraterrestrial life remains elusive. Studying Earth, the only confirmed habitable and inhabited planet, as a proxy exoplanet provides critical insights for interpreting forthcoming exoplanet direct-imaging data. Observations from the Deep Space Climate Observatory/Earth Polychromatic Imaging Camera (DSCOVR/EPIC), located at the first Sun-Earth Lagrangian point (L1), offer a unique opportunity to analyze Earth’s full-disk, single-point multi-spectrum light curves. Here, we review progress that treat EPIC data as if Earth were an unresolved, distant world. These studies reveal information about planetary rotation, cloud patterns, and surface types. Autocorrelation of the time series recovers the 24 h rotation period, while principal component analysis (PCA) highlights the land-ocean spectral contrast, enabling the reconstruction of a coarse two-dimensional surface map. Modeling studies further quantify the contributions of different planetary surfaces and clouds to Earth’s observable brightness, with low-level clouds playing a dominant role. Additionally, the effects of Earth’s atmosphere, particularly within strong oxygen bands, have been simulated and evaluated. The rich temporal–spectral “light-curve complexity” produced by its heterogeneous surface and dynamic atmosphere has emerged as a practical, observation-based metric of habitability. Comparisons with simulations and other solar system planets demonstrate that Earth’s light curves exhibit the highest complexity, underscoring its unique status as the only known habitable and inhabited exoplanet. These findings provide a valuable observational baseline for future exoplanet studies, refining our ability to recognize life-supporting worlds beyond the Solar System.

Temporary capture about the Moon involving Sun–Earth libration point dynamics

Particle-laden flows are simulated at various scales using numerical techniques that range from particle-resolved direct numerical simulations (pr-DNS) for small-scale systems to Lagrange point–particle methods for laboratory-scale problems, and Euler–Euler approaches for larger-scale applications. Recent research has been particularly focused on the development of both physics-based and data-driven closures to enhance the accuracy of the Lagrangian point–particle approach by leveraging highly resolved data from pr-DNS. In this study, a data-driven methodology is presented for the prediction of hydrodynamic forces acting on spherical particles immersed in an ambient flow field, where neighboring particle information is represented by volume fractions. The volume fractions are computed on an auxiliary grid with cell sizes on the order of the particle diameter. The volume fraction values in the vicinity of each particle are used as input features for the data-driven model to predict the corresponding hydrodynamic forces and moments. The training data were generated by a series of pr-DNS of flow through arrays of randomly distributed, fixed-position particles at various Reynolds numbers and particle volume fractions. The data-driven model is built using fully connected neural networks (FCNN). Improved prediction accuracy of hydrodynamic forces and torques is demonstrated in comparison to FCNN models that rely on direct particle position inputs. In addition, the proposed volume-fraction-based approach exhibits greater flexibility than previously introduced models by accommodating systems with particles of different sizes and shapes.