Lagrangian Point Research Articles

The Lunar Space Elevator: A Key Technology for Realising the Greater Earth Lunar Power Station (GEO-LPS)

The Greater Earth Lunar Power Station (GE -LPS) is a multi-purpose concept that aims to solve several critical issues related to lunar development and terrestrial energy production. As the GE -LPS concept and its energy production functions may be scaled to any dimension, larger versions could be positioned in Earth orbit to provide clean solar energy for terrestrial purposes and thereby reduce the mass needed to be launched from Earth to build SPS units by 80% or more. The construction of the GE -LPS from mostly lunar materials requires the establishment of industrial-scale automated mining and manufacturing processes on the Moon. A key technology is a Lunar Space Elevator (LSE) deployed as a transportation system to move SPS components from an anchor point on the surface of the Moon to a docking and assembly station at Earth-Moon Lagrange point 1 (EM-L1). Significantly, a LSE could be built today with existing tether materials such as Dyneema or Zylon which are already commercially available. Additionally, lunar sourced basalt fibre may be sufficient for reinforcing and extending the LSE once it becomes operational. An Earth-pointing LSE could become a valuable cislunar infrastructure asset – “the Suez Canal of cislunar space” – linking the Earth and Moon economies. Note: The ‘O’ symbol is an ancient European symbol for planet Earth: it is used here to mean “Greater Earth”, a region defined by the Earth’s gravitational field, which includes the Moon. Keywords: Solar Power Satellite, Lunar ISRU, Lunar Space Elevator, Energy, Europe

Five-impulse low-energy earth-moon transfer using manifolds

The libration points in the Earth-Moon and Sun-Earth systems are widely regarded as cost-effective gateways into interstellar space. The associated manifold tubes have been utilized by numerous researchers in the development of transfers between distinct systems. In this paper, a five-impulse transfer method, based on the manifold, has been proposed for designing the low-energy transfer trajectory for the Earth-Moon system. The Sun-Earth-Moon-Spacecraft four-body problem is decomposed into two circular restricted three-body problems: Sun-Earth-Spacecraft and Earth-Moon-Spacecraft. Moreover, a time parameter is employed to describe the positional relationship between the Sun-Earth and Earth-Moon systems, thereby facilitating their conversion. A phase angle is utilized to characterize the position of the Poincare section, and multiple orbital elements, such as orbital altitude and inclination, are considered in the constraints. To solve the design of multi-constrained transfer trajectories, sequential differential correction algorithms are proposed. The results demonstrate that this method can be used to create compliant low-energy transfer trajectories for various combinations of time parameters and phase angles. Additionally, excellent velocity increments and time-of-flight metrics can be found within a single Sun-Earth-Moon conjunction cycle. In parallel, the method is validated by reconstructing transfer trajectories in a bicircular restricted four-body problem model (BCR4BP) of the Sun-Earth/Moon system.

Earth-Venus Mission Analysis via Weak Capture and Nonlinear Orbit Control

Exploration of Venus is recently driven by the interest of the scientific community in understanding the evolution of Earth-size planets, and is leading the implementation of missions that can benefit from new design techniques and technology. In this work, we investigate the possibility to implement a microsatellite exploration mission to Venus, taking advantage of (i) weak capture, and (ii) nonlinear orbit control. This research considers the case of a microsatellite, equipped with a high-thrust and a low-thrust propulsion system, and placed in a highly elliptical Earth orbit, not specifically designed for the Earth-Venus mission of interest. In particular, to minimize the propellant mass, phase (i) of the mission was designed to inject the microsatellite into a low-energy capture around Venus, at the end of the interplanetary arc. The low-energy capture is designed in the dynamical framework of the circular restricted 3-body problem associated with the Sun-Venus system. Modeling the problem with the use of the Hamiltonian formalism, capture trajectories can be characterized based on their state while transiting in the equilibrium region about the collinear libration point L1. Low-energy capture orbits are identified that require the minimum velocity change to be established. These results are obtained using the General Mission Analysis Tool, which implements planetary ephemeris. After completing the ballistic capture, phase (ii) of the mission starts, and it is aimed at driving the microsatellite toward the operational orbit about Venus. The transfer maneuver is based on the use of low-thrust propulsion and nonlinear orbit control. Convergence toward the desired operational orbit is investigated and is proven analytically using the Lyapunov stability theory, in conjunction with the LaSalle invariance principle, under certain conditions related to the orbit perturbing accelerations and the low-thrust magnitude. The numerical results prove that the mission profile at hand, combining low-energy capture and low-thrust nonlinear orbit control, represents a viable and effective strategy for microsatellite missions to Venus.

The dehydration carousel of stratospheric water vapor in the Asian summer monsoon anticyclone

Abstract. During the StratoClim Geophysica campaign, air with total water mixing ratios up to 200 ppmv and ozone up to 250 ppbv was observed within the Asian summer monsoon anticyclone up to 1.7 km above the local cold-point tropopause (CPT). To investigate the temporal evolution of enhanced water vapor being transported into the stratosphere, we conduct forward trajectory simulations using both a microphysical and an idealized freeze-drying model. The models are initialized at the measurement locations and the evolution of water vapor and ice is compared with satellite observations of MLS and CALIPSO. Our results show that these extremely high water vapor values observed above the CPT are very likely to undergo significant further freeze-drying due to experiencing extremely cold temperatures while circulating in the anticyclonic “dehydration carousel”. We also use the Lagrangian dry point (LDP) of the merged back-and-forward trajectories to reconstruct the water vapor fields. The results show that the extremely high water vapor mixed with the stratospheric air has a negligible impact on the overall water vapor budget. The LDP mixing ratios are a better proxy for the large-scale water vapor distributions in the stratosphere during this period.

Analysis of plate reinforced by straight and curved stiffeners by using novel plate elements with refined through-the-thickness expansion

In this paper, two-dimensional (2D) structural models are used to analyze thin-walled stiffened structures. A novel approach is presented to model both skins and stiffeners using 2D finite elements (FEs) with enhanced capabilities. The deformation of the stiffeners is accurately described using higher order Lagrange polynomials, and the mechanical continuity between the reinforcements and the skin is automatically ensured in correspondence of the Lagrange points. The 2D FEs are built using the Carrera Unified Formulation (CUF), which allows arbitrary refinement of the stiffener kinematics. In this way, any type of deformation can be described and, if a higher-order model is used, the three-dimensional (3D) stress distribution can be evaluated. The proposed model is tested on a stiffened plate and compared with results obtained using traditional FEA, including 3D and 2D models and a combination of 2D and one-dimensional (1D) models for the stiffeners. The results show and demonstrate the reliability of the proposed method, both in terms of displacements and stresses, when compared to the 3D solution. In addition, the computational cost, evaluated here as the number of degrees of freedom (DOFs), is much lower. Finally, the ability of this model to deal with curved stiffeners is demonstrated with practical examples.

Radiation driven-dust hydrodynamics in late-phase AGB stars

The interplay of stellar luminosity variations and dust hydrodynamics in Asymptotic Giant Branch (AGB) stars and the consequences for dust survival and mass loss remain elusive. In this work, we broadly investigate the role of dust and radiation hydrodynamics in forming dust and gas structures, heterogeneous clumps observable in AGB remnants and planetary nebulae (PNe). Of interest in this study are the spatial perturbations driven by instabilities in the space that the mass travels through. These spatial perturbations in the dust and gas field may be responsible for forming larger clumps, such as cometary knots, seen in the PNe phase. Previous studies have considered similar physics in dust-driven winds at shorter lengths and time scales, using either 1D simulations or 2D simulations with a single mixed particle–gas fluid. Here we present an Eulerian–Lagrangian method for studying this problem at larger length and time scales. Simulations are performed in 2D, solving the Euler equations with source terms resulting from the particle phase, represented by free Lagrangian points. Radiation coupling was implemented for the particle phase, modeling radiation heating and acceleration of the particles and subsequent coupling to the gas phase through non-continuum heat and momentum transfer models. Spatial perturbations of the dust and radiation fields were found to drive the formation of small dust mass clumps that survive to late times, though these remain below the size of those observed in many PNe.

Structure of the accretion flow of IX Velorum as revealed by high-resolution spectroscopy

Context. Several high mass-transfer cataclysmic variables show evidence of outflow from the system, which could play an important role in their evolution. We investigate the system IX Vel, which was proposed to show similar characteristics. Aims. We study the structure of the IX Vel system, particularly the structure of its accretion flow and accretion disc. Methods. We used high-resolution time-resolved spectroscopy to construct radial velocity curves of the components in IX Vel. We computed Doppler maps of the system, which we used to estimate the temperature distribution maps. Results. We have improved the spectroscopic ephemeris of the system and its orbital period Porb = 0.19392793(3) d. We constructed Doppler maps of the system based on hydrogen and helium emission lines and the Bowen blend. The maps show features corresponding to the irradiated face of the secondary star, the outer rim of the accretion disc, and low-velocity components located outside the accretion disc and reaching towards L3. We constructed a temperature distribution map of the system using the Doppler maps of Balmer lines. Apart from the features found in the Doppler maps, the temperature distribution map shows a region of high temperature in the accretion disc connecting the expected position of a bright spot and the inner parts of the disc. Conclusions. We interpret the low-velocity emission found in the Doppler map as emission originating in the accretion disc wind and in an outflow region located in the vicinity of the third Lagrangian point L3. This makes IX Vel a member of the RW Sex class of cataclysmic variables.

Investigating the impact of non-spherical bodies and three-body interactions on equilibrium dynamics in the circular restricted three-body problem

In this paper, we study a modified version of the classical restricted 3-body problem, which introduces an additional mutual interaction force. Moreover, we extend our analysis to include non-spherical shapes, specifically prolate or oblate shapes, for the primary bodies within the system. Our main objective is to investigate how the additional interaction and non-sphericity of the primaries influence the locations and dynamical characteristics of the equilibrium points in the system. To accomplish this, we employ standard numerical methods and techniques. Through a meticulous examination of the system’s parameter space, we have identified a range of 1 to 13 libration points. We have observed that the total number of equilibrium points is directly related to the sign and intensity of the three-body interaction term. Consequently, our findings reveal a substantial difference when compared to scenarios where only three-body interactions or the non-sphericity of the primaries are considered independently.

Technical Overview and Prospect of India’s First Solar Mission - Aditya L1 Spacecraft

The Aditya-L1 mission represents a significant milestone in solar research, aimed at unlocking the mysteries of our closest star, the Sun. This paper provides an overview of the mission's objectives, scientific instruments, and key findings. By placing a spacecraft in a Lagrangian point L1 orbit, Aditya-L1 offers an unprecedented vantage point for continuous solar observations. The primary scientific goals include studying the Sun's dynamic atmosphere, monitoring solar variability, and enhancing our understanding of space weather and its impact on Earth. The paper discusses the advanced instrumentation onboard, such as the Visible Emission Line Coronagraph (VELC), the Solar Ultraviolet Imaging Telescope (SUIT), and the Aditya Solar Wind Particle Experiment (ASPEX), highlighting their contributions to solar science. Furthermore, this paper presents early results and insights obtained from Aditya-L1's observations, shedding light on solar phenomena, solar eruptions, and their influence on Earth's space environment. The Aditya-L1 mission stands as a testament to international collaboration and technological advancements, poised to reshape our understanding of the Sun and its profound effects on our solar system.

PRIME: a probabilistic neural network approach to solar wind propagation from L1

Introduction: For the last several decades, continuous monitoring of the solar wind has been carried out by spacecraft at the first Earth-Sun Lagrange point (L1). Due to computational expense or model limitations, those data often must be propagated to some point closer to the Earth in order to be usable by those studying the interaction between Earth’s magnetosphere and the solar wind. The current most widely used tool to propagate measurements from L1 (roughly 235 RE upstream) to Earth is the planar propagation method, which includes a number of known limitations. Motivated by these limitations, this study introduces a new algorithm called the Probabilistic Regressor for Input to the Magnetosphere Estimation (PRIME).Methods: PRIME is based on a novel probabilistic recurrent neural network architecture, and is capable of incorporating solar wind time history from L1 monitors to generate predictions of near-Earth solar wind as well as estimate uncertainties for those predictions.Results: A statistical validation shows PRIME’s predictions better match MMS magnetic field and plasma measurements just upstream of the bow shock than measurements from Wind propagated to MMS with a minimum variance analysis-based planar propagation technique. PRIME’s continuous rank probability score (CRPS) is 0.214σ on average across all parameters, compared to the minimum variance algorithm’s CRPS of 0.350σ. PRIME’s performance improvement over minimum variance is dramatic in plasma parameters, with an improvement in CRPS from 2.155 cm−3 to 0.850 cm−3 in number density and 16.15 km/s to 9.226 km/s in flow velocity VX GSE.Discussion: Case studies of particularly difficult to predict or extreme conditions are presented to illustrate the benefits and limitations of PRIME. PRIME’s uncertainties are shown to provide reasonably reliable predictions of the probability of particular solar wind conditions occurring.Conclusion: PRIME offers a simple solution to common limitations of solar wind propagation algorithms by generating accurate predictions of the solar wind at Earth with physically meaningful uncertainties attached.

Flow-induced diffusion in a packed lattice of squirmers

Mass transport in suspensions of swimming microorganisms is one of the most important factors for the colonisation and growth of microorganisms. Hydrodynamic interactions among swimming microorganisms play an important role in mass transport, especially in highly concentrated suspensions. To elucidate the influence of highly concentrated cells on mass transport, we numerically simulated mass transport in lattices of squirmers that were fixed in space and oriented in the same direction. The effects of different volume fractions, Péclet numbers ( $Pe$ ) and lattice configurations on mass transport were quantified by tracking Lagrangian material points that move with background flow with Brownian diffusivity. Although the flow field became periodic in space and each streamline basically extended in one direction, the motion of tracer particles became diffusive over long durations due to Brownian motion and cross-flows. Flow-induced diffusion was anisotropic and significantly enhanced over Brownian diffusion in the longitudinal direction. We also investigated mass transport in random configurations of squirmers to reproduce more general conditions. Similar enhanced diffusion was also observed in the random configurations, indicating that the flow-induced diffusion appears regardless of the configurations. The present flow-induced diffusion did not follow $Pe$ dependency of the conventional Taylor dispersion due to the cross-flows. The time and velocity scales were proposed, which enabled us to predict the flow-induced diffusivity from the data of the flow field and Brownian diffusivity without solving the mass conservation equation. The findings reported here improve our understanding of the transport phenomena in packed suspensions of swimming microorganisms.

A Method for Calculating the Trajectory of a Single-Impulse Flight to a Halo Orbit around the L2 Libration Point of the Earth–Moon System

A Method for Calculating the Trajectory of a Single-Impulse Flight to a Halo Orbit around the L2 Libration Point of the Earth–Moon System

Motion Primitive Approach to Spacecraft Trajectory Design in a Multi-body System

The increasing number and variety of spacecraft that are expected to operate within cislunar space and other multi-body gravitational environments throughout the solar system necessitates the continued development of strategies for rapid trajectory design and design space exploration. In the field of robotics, similar needs have been addressed using motion primitives that capture the fundamental building blocks of motion and are used to rapidly construct complex paths. Inspired by this concept, this paper leverages motion primitives to construct a framework for rapid and informed spacecraft trajectory design in a multi-body gravitational system. First, motion primitives of fundamental solutions, e.g., selected periodic orbits and their stable and unstable manifolds, are generated via clustering to form a discrete summary of segments of the phase space. Graphs of motion primitives are then constructed and searched to produce primitive sequences that form candidate initial guesses for transfers of distinct geometries. Continuous transfers are computed from each initial guess using multi-objective constrained optimization and collocation. This approach is demonstrated by constructing an array of geometrically distinct transfers between libration point orbits in the Earth-Moon circular restricted three-body problem with impulsive maneuvers.

LISAmax: improving the low-frequency gravitational-wave sensitivity by two orders of magnitude

Within its Voyage 2050 planning cycle, the European Space Agency is considering long-term large class science mission themes. Gravitational-wave astronomy is among the topics under study. Building on previous work by other authors (Folkner 2011 (NNH11ZDA019L); Ni 2010 Mod. Phys. Lett. A 25 922–35; Sesana et al 2021 Exp. Astron. 51 1333–83), this paper studies a gravitational-wave interferometer concept, dubbed ‘LISAmax’, consisting of three spacecraft, each located close to one of the Sun–Earth libration points L3, L4 and L5, forming a triangular constellation with an arm length of 259 million kilometers (to be compared to LISA’s 2.5 million kilometer arms). We argue that this is the largest triangular formation that can be reached from Earth without a major leap in mission complexity and cost (hence the name). The sensitivity curve of such a detector is at least two orders of magnitude lower in amplitude than that of LISA, at frequencies below 1 mHz. This makes the observatory sensitive to gravitational waves in the µHz range and opens a new window for gravitational-wave astronomy, not covered by any other planned detector concept. We analyze in detail the constellation stability for a 10 year mission in the full numerical model including insertion dispersion, and self-gravity-induced accelerations. We compute the orbit transfers using a European launcher and chemical propulsion. Different orbit options, such as precessing, inclined orbits, the use of flybys for the transfer, and the launch strategy, are discussed. The payload design parameters are assessed, and the expected sensitivity curve is compared with a number of potential gravitational-wave sources. No show stoppers are identified at this point of the analysis.

X-ray Spectroscopic Study of Low-Mass X-ray Binaries: A Review of Recent Progress via the Example of GX 339-4

Low-mass X-ray binaries (LMXB) serve as natural laboratories, where the predictions of general relativity can be tested in the strong field regime. The primary object of such sources can be a neutron star (NS) or a black hole (BH), and this object captures material from the secondary object through the inner Lagrange point via a process called Roche lobe overflow. Because of the angular momentum of the infalling matter, an accretion disk is formed, in which viscous effects transport the angular momentum radially outward. In the high/soft state of these sources, the accretion disk can extend all the way to the innermost stable circular orbit (ISCO); therefore, when the primary object is a BH, its X-ray spectrum contains information about the region very close to the event horizon. This paper aims to review the theoretical and observational works related to the X-ray spectroscopy of such sources via the example of GX 339-4, which is one of the most well-known and well-studied LMXBs.

A Maneuver of Escape from the Vicinity of a Collinear Libration Point Using Light Pressure Forces

The orbital motion of a spacecraft with a solar sail in the vicinity of the Sun–Earth collinear libration point is studied. The study is carried out within the Hill approximation of the circular restricted three-body problem. The collinear libration point is unstable, but instability can be a positive factor in the tasks of maneuvering in interplanetary space. This is because a small change in velocity near an unstable libration point can lead to a significant change in orbital motion. This property can be used in the development of projects to counteract the comet–asteroid hazard. A scheme for constructing the trajectories of interception of a spacecraft with a potentially hazardous object is proposed. One of the stages of this scheme, an optimal escape from the vicinity of the libration point using light pressure forces, is considered. The results of numerical simulations are presented.

Analysis of the dynamics of a spacecraft in the vicinity of an asteroid binary system with equal masses

In this work, we performed a dynamical analysis of a spacecraft around a nearly equal-mass binary near-Earth asteroid with application to the asteroid 2017 YE5, which is also a possible dormant Jupiter-family comet. Thus, we investigated the motion of a particle around this binary system using the circular restricted three-body problem. We calculated the locations of the Lagrangian points of the system and their Jacobi constant. Through numerical simulations, using the Poincaré Surface of Sections, it was possible to find several prograde and retrograde periodic orbits around each binary system's primary, some exhibiting significantly-sized higher-order behavior. We also calculated the stability of these orbits. After finding the periodic orbits, we investigated the influence of solar radiation pressure on these orbits. For this analysis, we considered that the area-to-mass ratio equals 0.01 and 0.1. We also performed a spacecraft lifetime analysis considering the physical and orbital characteristics of the 2017YE5 system and investigated the behavior of a spacecraft in the vicinity of this system. We analyzed direct and retrograde orbits for different values of Jacobi's constant. This study investigated orbits that survive for at least six months, not colliding or escaping the system during that time. We also analyze the initial conditions that cause the spacecraft to collide with M1 or M2, or escape from the system. In this work, we take into account the gravitational forces of the binary asteroid system and the solar radiation pressure (SRP). Finally, we calculated optimal bi-impulsive orbital maneuvers between the collinear Lagrangian points. We found a family of possible orbital transfers considering times of flight between 0.1 and 1 day.

Designing Low-Energy Low-Thrust Flight to the Moon on a Temporary Capture Trajectory

The study considers the problem of calculating the low-energy trajectories of a low-thrust spacecraft to the Moon during the ballistic capture. The transfer is carried out using a transit trajectory in the vicinity of one of the collinear libration points L1 or L2 of the Earth-Moon system. Using a transit trajectory allows us to reduce fuel consumptions for the transfer by applying spacecraft dynamic in the Earth-Moon system. After exit from the orbit of ballistic capture, depending on the goal of mission the required lunar orbit can be formed, or the maneuver can be completed for inserting into the required interplanetary trajectory. A method for solving the problem is proposed, which consists in selecting the suitable transit trajectory to ensure sufficiently long duration of staying a spacecraft in the sphere of influence of the Moon, and in calculating the optimal low-thrust trajectories from initial lunar orbit to the transit trajectory to the Moon. To solve the problem of optimal control and calculate the optimal exit point to the transit trajectory, the Pontryagin’s maximum principle is used in combination with the continuation method by parameter. Numerical examples are given for calculating low-energy trajectories to the Moon during the ballistic capture with the optimization of exit point to the transit trajectory.

A Maneuver of Escape from the Vicinity of a Collinear Libration Point Using Light Pressure Forces

A Maneuver of Escape from the Vicinity of a Collinear Libration Point Using Light Pressure Forces

Optimization of a Low-Thrust Heliocentric Trajectory between the Collinear Libration Points of Different Planets

The aim of this study is to optimize a low-thrust interplanetary trajectory using collinear libration points L1 and L2 as the junction points of the geocentric or planetocentric segments of the trajectory with the heliocentric segment. The problem of optimizing the heliocentric segment of perturbed low-thrust interplanetary transfer is considered in the four-body ephemeris model, which includes the Sun, Earth, target planet, and spacecraft. To optimize the trajectories, an indirect approach is used based on Pontryagin’s maximum principles and the continuation method. The possibility of reducing the characteristic velocity in comparison with the estimates obtained through the method of zero sphere of influence is shown.