Geosynchronous Transfer Orbit Research Articles

Earth shadow time/accelerometer/GNSS-based geosynchronous transfer orbit determination method for electric propulsion satellite

Autonomous orbital maneuvering of an electric propulsion satellite in geosynchronous transfer orbit (GTO) requires more accurate autonomous orbit determination (OD). Implementing the traditional autonomous OD method for the electric propulsion satellites may result in the accumulation of measurement errors and dynamic errors from orbit extrapolation, which significantly restrict the OD accuracy. Fortunately, an electric propulsion satellite in GTO can sense the transit time in or out of the Earth shadow according to the output power change of the solar array. Therefore, an Earth shadow time/accelerometer/GNSS-based geosynchronous transfer OD method for electric propulsion satellites is proposed in this paper. First, a system state model with dynamics is established for the entire orbit transfer process. Second, measurement models that use the measurement information from the accelerometer, GNSS, and the transit time into or out of the Earth shadow are established. Third, a sequential filtering estimation approach is designed to solve the problem of asynchronous data fusion for autonomous OD. Comparison simulations demonstrate the effectiveness and superiority of the proposed method.

Probabilistic assessment of disposal orbit lifetime for CubeSat rideshares on resonant decaying geosynchronous transfer orbits

Recent years have seen an increase in CubeSat missions on rideshares to geosynchronous orbit. The typical practice for these missions is to deploy the CubeSat on a geosynchronous transfer orbit (GTO) which results in a perigee altitude that is low enough that atmospheric drag will cause the apogee altitude to decay and eventual reentry. However, for GTOs, demonstrating compliance with limits on orbital lifetime in orbital debris mitigation guidelines is not straightforward due to a solar resonance phenomenon that exists which can cause high variability of the orbital lifetime of the satellite. This paper presents a procedure for determining the likelihood that orbital lifetime of a rideshare CubeSat on a resonant GTO will have an orbital lifetime below a 25-year and 5-year limit. The procedure uses a Monte Carlo analysis in which uncertain parameters are randomly varied, including the launch time/initial RAAN and the drag coefficient. The Aerospace precision propagation tool TRACE is used with a high-fidelity force model to enable precision integration through the atmosphere at perigee. It is shown in the study that there is a systematic variation in likelihood of staying below a 25- and 5-year orbital lifetime limit and that drag enhancement devices may be needed to meet the 5-year limit for CubeSat rideshares. The study also presents findings on a solar radiation pressure induced resonance that was observed for high area-to-mass ratios which suggests that there can be a diminishing return when increasing the area of a drag enhancement device to quicken deorbit.

A multi-satellite survey scheme for addressing open questions on the Earth’s outer radiation belt dynamics

The Earth’s outer radiation belt is highly dynamic, containing relativistic electron fluxes that can increase by several orders of magnitude during magnetospheric disturbances. This greatly increases the likelihood of spacecraft malfunction or failure and significantly influences the solar-terrestrial system’s energy and mass coupling, highlighting the importance of fully understanding the mechanisms governing these dynamics from both theoretical and practical perspectives. Although many theories have been proposed, further research is essential to quantify the specific contributions of different dynamic mechanisms for improving space weather forecasting. To address this, observations of the outer radiation belt with high spatial–temporal resolution to distinguish the spatial and temporal variations are essential. We introduce a 10-CubeSat constellation survey scheme in the geosynchronous transfer orbit (GTO) to achieve this required observation. Three baseline instruments are proposed to be employed: the high energy electron detector (HEED), the search coil wave detector (SCWD), and the magnetometer (MAG). Two groups of physical processes will be investigated: wave-particle interactions involving charged particles interacting with whistler-mode waves, electromagnetic ion cyclotron (EMIC) waves, and ultra-low frequency (ULF) waves; and radial transport encompassing shock-induced injections, substorm injections, storm convection and magnetopause shadowing. The performance parameters of instruments and platform of the constellation are presented. Additionally, aligned with the concept of constellation survey, we outline the COSPAR-coordinated space program, COnstellation of Radiation BElt Survey (CORBES), which will provide a crucial scientific contribution in the absence of the Van Allen Probes. The program’s excellent observational capability enables a comprehensive understanding of the underlying physical mechanisms governing the outer radiation belt dynamics and improved space weather forecasting.

Comparison of Proton and Gamma Irradiation on Single-Photon Avalanche Diodes

In this paper, the effects of proton and gamma irradiation on reach-through single-photon avalanche diodes (SPADs) are investigated. The I–V characteristics, gain and spectral response of SPAD devices under proton and gamma irradiation were measured at different proton energies and irradiation bias conditions. Comparison experiments of proton and gamma irradiation were performed in the radiation environment of geosynchronous transfer orbit (GTO) with two different radiation shielding designs at the same total ionizing dose (TID). The results show that after 30 MeV and 60 MeV proton irradiation, the leakage current and gain increase, while the spectral response decreases slightly. The leakage current degradation is more severe under the “ON”-bias condition compared to the “OFF”-bias condition, and it is more sensitive to the displacement radiation damage caused by protons compared to gamma rays under the same TID. Further analysis reveals that the non-elastic and elastic cross-section of protons in silicon is 1.05 × 105 times greater than that of gamma rays. This results in SPAD devices being more sensitive to displacement radiation damage than ionizing radiation damage. Under the designed shielding conditions, the leakage current, gain and spectral response parameters of SPADs do not show significant performance degradation in the orbit.

Design and Instrument Characterization of the High‐Energy Resolution Relativistic Electron Telescope (HERT) Using Geant4 Simulation

AbstractEarth's outer radiation belt is filled with relativistic (∼100s of keV–3 MeV) and ultrarelativistic (∼>3 MeV) electrons. The acceleration of electrons to these high energies are believed to be caused by inward radial transport and local acceleration, but the relative importance of the two mechanisms to electrons with different energies are still under considerate debate. In the post Van Allen Probes era, measurements of radiation belt populations must rely on small space missions such as CubeSats and SmallSats. The Miniaturized High‐Energy‐Resolution relativistic electron Telescope (HERT) is a compact (≤3U) telescope designed for a CubeSat mission in geosynchronous transfer orbit (GTO). HERT's main objective is to provide high‐energy‐resolution measurements of outer belt electrons in an energy range of ∼1–7 MeV to help differentiate the contribution of inward radial transport and local acceleration. Geant4 simulations were conducted to characterize the instrument responses. A novel method of using a spherical cap particle source in Geant4 simulation was developed for more efficient instrument characterization. Combined with Bow tie analysis, it is demonstrated that HERT will have an energy resolution of 5% for ∼1.5–3 MeV electrons and 12% for ∼3–7 MeV with current electronics design. In addition, using the AE9 model, we showed the instrument would have statistically sufficient count rates in the outer belt while not saturating the electronics. With a compact configuration and higher energy resolution in comparison to previous instruments, HERT will significantly contribute to the quantitative understanding of the radiation belt electron dynamics.

Study on geosynchronous satellite launch vehicle propellants and combustion mechanism of each stage

GSLV is a “geosynchronous satellite launch vehicle” which aims at launching space objects into GTO- Geosynchronous transfer orbit. This paper provides a brief idea of the chemistry of propellants used in all three stages and strap-on motors of geosynchronous satellite launch vehicles and their combustion mechanism. It includes all series of GSLVs launched till now and a description of criteria for choosing fuel. This paper also includes a brief description on hydrazine which is a basic part of so many propellant combinations. Furthermore, it includes basic reasons which result in unsuccessful ignition especially at cryogenic. It also emphasizes on reasons for modification in fuels over time and advancement in efficiency obtained due to respective modified fuels, further, giving an overview of future ideas in the development of propellants.

Low-Energy Transfer Design of Heliocentric Formation Using Lunar Swingby on the Example of LISA

Space-based gravitational wave (GW) detection at low frequencies is of great scientific significance and has received extensive attention in recent years. This work designs and optimizes the low-energy transfer of the heliocentric formation of GW detectors, which starts from a geosynchronous transfer orbit and targets an Earth-like orbit. Based on the example of the Laser Interferometer Space Antenna (LISA), the transfer is first designed in two-body dynamical models and then refined in simplified high-fidelity dynamical models that only consider the major orbital perturbations evaluated here. The main contributions of this work are to present an adaptive model continuation technique and to exploit the lunar swingby technique to reduce the problem-solving difficulty and velocity increment of orbital transfer, respectively. The adaptive model continuation technique fully reveals the effect of perturbations and rapidly iterates the solutions to the simplified models. The simulation results show that the lunar swingby does reduce the energy needed to escape the Earth’s sphere of influence. It is found that the gravitation of the Earth–Moon system has a significant contribution to reducing the velocity increment. The solution of low-energy transfer in the simplified models is that the duration is 360.6615 days and the total velocity increment is 0.8468 km/s.

Assessment of in-house algorithms on re-entry time prediction of uncontrolled space objects

Accurate prediction of re-entry time and location of an uncontrolled re-entering space object is crucial but challenging. This paper presents in-house re-entry prediction algorithms: RSMGA, STKOptim, STKLTOptim, and ABPro. Case studies on re-entry prediction of Starlink-26 (COSPAR ID 2019-029F or NORAD ID 44240) and CZ-5B (COSPAR ID 2021-035B or NORAD ID 48275) during the IADC re-entry test campaigns 2021/1 and 2021/2, respectively are presented. Simultaneous application of these re-entry prediction algorithms facilitates decision-making on automation of switching between these algorithms during various phases (short term or long-term) of the re-entry campaign. The effectiveness and applicability of these algorithms are studied based on percentage error. The study focuses on the overall and time-based performance of the four methods. Six more re-entered objects are selected (three Geosynchronous transfer orbit (GTO) objects and 3 Low-Earth orbit (LEO) objects), and their re-entry time prediction has been carried out using the four methods for different time instances before re-entry. Prediction estimates are compared among the methods for all the time instances in terms of percentage error. Also, the study helps in fixing the suitability of the re-entry time prediction algorithm for objects from GTO and LEO.

Ridesharing Options from Geosynchronous Transfer Orbits to the Sun–Earth Region

Ridesharing options for secondary payloads to be delivered to regions beyond geostationary altitude are increasingly available with propulsive evolved expendable launch vehicle secondary payload adapter rings. The preliminary mission design for secondary payloads must consider the variability of the dropoff orbit orientation: a factor that is a function of the primary mission. In this analysis, assume that the dropoff orbit is an Earth-centered geosynchronous transfer orbit (GTO). Then, propellant-efficient transfers to periodic and quasi-periodic orbits near the Sun–Earth Lagrange point are constructed from a range of GTO orientations. Dynamical structures (such as hyperbolic invariant manifolds) associated with orbits near Sun–Earth Lagrange point are leveraged to identify and summarize various types of transfer opportunities.

Swarm Optimization of Lunar Transfers from Earth Orbit with Operational Constraints

Due to the frequent launches of primary satellites into geosynchronous transfer orbits, lunar transfer from an initial geosynchronous transfer orbit represents an attractive option for secondary payloads with lunar mission objectives. The development of advanced launch vehicle upper stages with low boil off rates and restartable engines further enhances the potential of geosynchronous transfer orbits to serve as starting points for lunar transfers. Lunar trajectory design from arbitrarily oriented Earth orbits involves complex tradeoffs that can be successfully addressed with particle swarm optimization. An important mission design consideration is radiation dose, since coasting in a geosynchronous transfer orbit involves repeated passes through the Van Allen radiation belts. In this investigation, a mixture density neural network is trained on the state-of-the-art radiation environment models to provide a surrogate model capable of making probabilistic estimates of radiation dosage. The network is combined with the particle swarm method to optimize the coasting and transfer arcs of a lunar trajectory. The end-to-end design of a trajectory from geosynchronous transfer orbit into a specified lunar orbit is demonstrated by modeling the optimized trajectory in a high-fidelity orbit propagator, including B-plane targeting to overfly a desired lunar landing site.

Multi-Fidelity Modeling and Adaptive Co-Kriging-Based Optimization for All-Electric Geostationary Orbit Satellite Systems

Abstract All-electric geostationary orbit (GEO) satellite systems design is a challenging multidisciplinary design optimization (MDO) problem, which is computation-intensive due to the employment of expensive simulations. In this paper, the all-electric GEO satellite MDO problem with multi-fidelity models is investigated. The MDO problem involving six inter-coupled disciplines is formulated to minimize the total mass of the satellite system subject to a number of engineering constraints. To reduce the computational cost of the multidisciplinary analysis (MDA) process, multi-fidelity transfer dynamics models and finite element analysis (FEA) models are developed for the geosynchronous transfer orbit (GTO) and structure disciplines, respectively. To effectively solve the all-electric GEO satellite MDO problem using multi-fidelity models, an adaptive Co-Kriging-based optimization framework is proposed. In this framework, the samples from a high-fidelity MDA process are integrated with those from a low-fidelity MDA process to create a Co-Kriging metamodel with a moderate computational cost for optimization. Besides, for refining the Co-Kriging metamodels, a multi-objective adaptive infill sampling approach is developed to produce the infill sample points in terms of the expected improvement (EI) and the probability of feasibility (PF) functions. Optimization results show that the proposed optimization framework can significantly reduce the total mass of satellite system with a limited computational budget, which demonstrates the effectiveness and practicality of the multi-fidelity modeling and adaptive Co-Kriging-based optimization framework for all-electric GEO satellite systems design.

Mission Design and Performance Analysis of a Smallsat Aerocapture Flight Test

An Earth smallsat mission architecture is developed to demonstrate the feasibility of aerocapture. The mission concept proposes to transfer from a geosynchronous transfer orbit rideshare trajectory to a low Earth orbit. Single-event drag modulation is used as a straightforward means of achieving the control required during the maneuver. Low- and high-fidelity guidance algorithm choices are considered. Numeric trajectory simulations and Monte Carlo uncertainty analyses are performed to show the robustness of the system to day-of-flight environments and uncertainties. Related investigations are described for Mars mission concepts to show the relevance of the proposed flight test to potential future applications.

Breakup prediction under uncertainty: Application to upper stage controlled reentries from GTO orbit

More and more human-made space objects re-enter the atmosphere, and yet the risk for human populations remains often unknown because predicting their reentry trajectories is formidably complex. While falling back on Earth, the space object absorbs large amounts of thermal energy that affects its structural integrity. It undergoes strong aerodynamic forces and heating that lead to one or several breakups. Breakup events have a critical influence on the rest of the trajectory but are extremely challenging to predict and subject to uncertainties. In this work, we present an original model for robustly predicting the breakup of a reentering space object. This model is composed of a set of individual solvers that are coupled together such as each solver resolves a specific aspect of this multiphysics problem. This paper deals with two levels of uncertainties. The first level is the stochastic modeling of the breakup while the second level is the statistical characterization of the model input uncertainties. The framework provides robust estimates of the quantities of interest and quantitative sensitivity analysis. The objective is twofold: first to compute a robust estimate of the breakup distribution and secondly to identify the main uncertainties in the quantities of interest. Due to the significant computational cost, we use an efficient framework particularly suited to multiple solver predictions for the uncertainty quantification analysis. Then, we illustrate the breakup model for the controlled reentry of an upper stage deorbited from a Geosynchronous Transfer Orbit (GTO), which is a classical Ariane mission.

Development of Cryogenic Engine for GSLV MkIII: Technological Challenges

Cryogenic engine capable of delivering 200 kN thrust is being developed for the first time in the country by ISRO for powering the upper stage of GSLV Mk-III, the next generation launch vehicle of ISRO capable of launching four tonne class satellites to Geo-synchronous Transfer Orbit(GTO). Development of this engine started a decade ago when various sub-systems development and testing were taken up. Starting with injector element development, the design, realization and testing of the major sub-systems viz the gas generator, turbopumps, start-up system and thrust chamber have been successfully done in a phased manner before conducting a series of developmental tests in the integrated engine mode. Apart from the major sub-systems, many critical components like the igniter, control components etc were independently developed and qualified. During the development program many challenges were faced in almost all areas of propulsion engineering. Systems engineering of the engine was another key challenge in the realization. This paper gives an outlook on various technological challenges faced in the key areas related to the engine development, insight to the solutions and measures taken to overcome the challenges.

Improving Space Object Catalog Maintenance Through Advances in Solar Radiation Pressure Modeling

This paper investigates the weaknesses of using the cannonball model to represent the solar radiation pressure force on an object in an orbit determination process, and it presents a number of alternative models that greatly improve the orbit determination performance. These weaknesses are rooted in the fact that the cannonball model is not a good representation of the true solar radiation pressure force acting on an arbitrary object. Using an erroneous force model results in poor estimates, inaccurate trajectory propagation, unrealistic covariances, and the inability to fit long and/or dense arcs of data. The alternative models presented are derived from a Fourier series representation of the solar radiation pressure force. The simplest instantiation of this model requires only two more parameters to be estimated, however, this results in orders of magnitude improvements in tracking accuracy. This improvement is illustrated through numerical examples of a discarded upper stage in a geosynchronous transfer orbit, and more drastically for a piece of high-area-to-mass ratio debris in a near-geosynchronous orbit. Implementation of improved solar radiation pressure models in this manner will alleviate track correlation, object identification, and sensor tasking issues that plague current catalog maintenance due to the inaccurate standard dynamic model.

ORDEM 3.0 and MASTER-2009 modeled debris population comparison

The latest versions of the two premier orbital debris engineering models, NASA׳s ORDEM 3.0 and ESA׳s MASTER-2009, have been publically released. Both models have gone through significant advancements since inception, and now represent the state-of-the-art in orbital debris knowledge of their respective agencies. The purpose of these models is to provide satellite designers/operators and debris researchers with reliable, and timely, estimates of the artificial debris environment in -Earth orbit, from about 200 km through about 40,000 km. The small debris environment within the “critical size range” of 1cm to 1mm is of particular interest to both human and robotic spacecraft programs. These objects are much more numerous than larger trackable debris and are still large enough to cause significant, if not catastrophic, damage to spacecraft upon impact. They are also small enough to elude routine detection by existing observation systems. Without reliable detection the modeling of these populations has always coupled theoretical origins with supporting observational data in different degrees.This paper offers the first cooperative comparison of the latest versions of the two models. No attempt to review model internal populations, or compare subsystems, or supporting data sets is made at this time. The models are simply run in the “Spacecraft Mode” for four test cases representing four orbital regimes, ISS (International Space Station orbit), SSO (sun synchronous orbit), GTO (geosynchronous transfer orbit), and GEO (geosynchronous orbit). Specific satellites are chosen within those categories for the analysis. Debris cumulative fluxes at three debris sizes (1m, 10cm, and the critical size range of 1cm to 1mm) are compared. At 1m there is a very good match between ORDEM 3.0 and MASTER-2009 fluxes in the non-GEO orbits, less so in the GEO orbit. The 10cm fluxes do not match as well, indicating possibly different treatment of uncatalogued objects. Within the critical size range it is noted that ORDEM 3.0 flux overtakes that of MASTER-2009 as debris size decreases in all test cases.In the analysis of separate model populations, the disparate usage of debris source populations (MASTER-2009) versus debris material density populations (ORDEM 3.0) is presently unresolved. MASTER-2009 populations of SRM (solid rocket motor) slag and explosion and collision fragments dominate with lower populations of ejecta debris. ORDEM 3.0 populations are dominated by medium and high density material with minor low density material populations. Undoubtedly, more in-depth collaborations will follow to hone in on population differences and sources.

Weak stability boundary transfer to the Moon from GTO as a piggyback payload on Ariane 5

In cooperation with the German non-profit amateur satellite organisation (AMSAT-DL), the German Aerospace Center developed the idea of using AMSAT’s Earth satellite P3-D as a baseline for a Moon mission. For cost-effectiveness, P3-D was launched as an auxiliary payload on Ariane 5 into a geosynchronous transfer orbit (GTO) and used its on-board propulsion system to achieve a Molniya orbit. The present study describes how a similar satellite can reach a 100 × 100 km lunar orbit with the same launch strategy. A delta-v saving transfer scenario using the weak stability boundary transfer was found to be feasible taking a P3-D-like satellite bus into account. It contains phasing orbits as a solution for non-dedicated launch dates and deals with the constraints of Ariane’s GTO. This approach opens up the opportunity to accomplish a low-cost mission to the Moon with public and scientific value.

The Solar Dynamics Observatory (SDO)

The Solar Dynamics Observatory (SDO) was launched on 11 February 2010 at 15:23 UT from Kennedy Space Center aboard an Atlas V 401 (AV-021) launch vehicle. A series of apogee-motor firings lifted SDO from an initial geosynchronous transfer orbit into a circular geosynchronous orbit inclined by 28° about the longitude of the SDO-dedicated ground station in New Mexico. SDO began returning science data on 1 May 2010. SDO is the first space-weather mission in NASA’s Living With a Star (LWS) Program. SDO’s main goal is to understand, driving toward a predictive capability, those solar variations that influence life on Earth and humanity’s technological systems. The SDO science investigations will determine how the Sun’s magnetic field is generated and structured, how this stored magnetic energy is released into the heliosphere and geospace as the solar wind, energetic particles, and variations in the solar irradiance. Insights gained from SDO investigations will also lead to an increased understanding of the role that solar variability plays in changes in Earth’s atmospheric chemistry and climate. The SDO mission includes three scientific investigations (the Atmospheric Imaging Assembly (AIA), Extreme Ultraviolet Variability Experiment (EVE), and Helioseismic and Magnetic Imager (HMI)), a spacecraft bus, and a dedicated ground station to handle the telemetry. The Goddard Space Flight Center built and will operate the spacecraft during its planned five-year mission life; this includes: commanding the spacecraft, receiving the science data, and forwarding that data to the science teams. The science investigations teams at Stanford University, Lockheed Martin Solar Astrophysics Laboratory (LMSAL), and University of Colorado Laboratory for Atmospheric and Space Physics (LASP) will process, analyze, distribute, and archive the science data. We will describe the building of SDO and the science that it will provide to NASA.

Magnetic field dipolarization in the deep inner magnetosphere and its role in development of O+‐rich ring current

We studied magnetic field dipolarization and associated ion acceleration in the deep inner magnetosphere, using magnetic field data obtained by the magnetometer on board the Mission Demonstration Satellite 1 (MDS‐1) and the energetic neutral atom (ENA) flux data obtained by the high‐energy neutral analyzer imager on board the Imager for Magnetopause‐to‐Aurora Global Exploration satellite. Because the MDS‐1 satellite has a geosynchronous transfer orbit, we could survey magnetic field variations at L = 3.0–6.5. Analyzing data in the period from February to July 2002, we found that (1) dipolarization can be detected over a wide range of L (i.e., L = 3.5–6.5, which is far inside the geosynchronous altitude); (2) when the MDS‐1 satellite was located close to auroral breakup longitude, the occurrence probability of dipolarization was about 50% just inside the geosynchronous altitude and about 16% at L = 3.5–5.0, suggesting that dipolarization in the deep inner magnetosphere is not unusual; (3) magnetic storms were developing whenever dipolarization was found at L = 3.5–5.0; (4) dipolarization was accompanied by magnetic field fluctuations having a characteristic timescale of 3–5 s, which is comparable to the local gyroperiod of O+ ions; and (5) after dipolarization, the oxygen ENA flux in the nightside ring current region was predominantly enhanced by a factor of 2–5 and stayed at an enhanced level for more than 1 h, while clear enhancement was scarcely seen in the hydrogen ENA flux. From these results, we conjectured a scenario for the generation of an O+‐rich ring current, in which preexisting thermal O+ ions in the outer plasmasphere (i.e., an oxygen torus known from satellite observations) experience local and nonadiabatic acceleration by magnetic field fluctuations that accompany dipolarization in the deep inner magnetosphere (L = 3.5–5.0).

Design of Solar Sail Trajectories with Applications to Lunar South Pole Coverage

Potential orbits for continuous surveillance of the lunar south pole with just one spacecraft and a solar sail are investigated. Displaced periodic orbits are first computed in the Earth–moon restricted three-body problem, using Hermite–Simpson and seventh-degree Gauss–Lobatto collocation schemes. The schemes are easily adapted to include path constraints favorable for lunar south pole coverage. The methods are robust, generating a control history and a nearby solution with little information required for an initial guess. Five solutions of interest are identifiedand,usingcollocation,transitionedtothefullephemerismodel(includingtheactualsun-to-spacecraft line andlunarlibrations).Oftheoptionsinvestigated,orbitsneartheEarth–moonL2 pointyieldthebestcoverageresults. Propellant-free transfers from a geosynchronous transfer orbit to the coverage orbits are also computed. A steering law is discussed and refined by the collocation methods. The study indicates that solar sails remain an option for constant lunar south pole coverage. Nomenclature a = altitude from the lunar south pole a = sail acceleration vector DF = Jacobian matrix E = two-body energy with respect to Earth F = full constraint vector gi = path constraint vector H = two-body angular momentum with respect to Earth hk = general node states and/or control constraint � = characteristic acceleration l = sun-to-spacecraft unit vector n = number of nodes R, V = position and velocity vectors, Earth mean equator of J2000 inertial frame r, v = position and velocity vectors, Earth–moon rotating frame Ti = segment time t = time U = potential function u = control variable vector X = total design variable vector x = state variable vector � = sail pitch angle � i = defect vector � = sail clock angle � = elevation angle from the lunar south pole i = control-magnitude constraint !s = angular rate of sun, Earth–moon rotating frame