Intermediate Orbit Research Articles

Control strategy for trading off solar power and control input while rendezvous and docking

This paper presents a novel strategy for decoupling the attitude and orbit equations of a CubeSat for rendezvous and docking with an uncontrollable cooperative target. For computing a safe trajectory, the proposed control algorithm takes into account the solar energy received by the CubeSat. The CubeSat is equipped with a thruster for orbit maneuvers, magnetic coils for attitude control, and four fixed single-sided rectangular solar panels. The coupled dynamics arising from the dynamical model of the spacecraft is solved using a novel weighted vector-based approach. This paper presents a linearized nonlinear optimal control problem arising in small spacecrafts while trying to maximize solar energy input in the rendezvous and docking process. An intermediate orbit is defined and used to divide the problem into two different optimal control problems: rendezvous optimization and docking optimization problems. A geometrical approach based on the shape of the chaser and the target is contemplated for collision avoidance while docking. The proposed controller design is modified whenever the sunlight is obstructed by the Earth, and the maneuvering controls are redesigned accordingly for optimal rendezvous and docking. Numerical simulations have been carried out to show the efficacy of the proposed concept for steering the trade-off between solar energy and control input during the rendezvous and docking of CubeSat with a tumbling target.

Low-Thrust Resonance Gravity-Assist Trajectory Design for Lunar Missions

This paper introduces a procedure to optimize a low-thrust gravity-assist trajectory to the Earth–moon L1 periodic orbit utilizing the resonance-orbital structure as a guideline. The Earth–moon circular restricted three-body problem formulation is used to describe the problem. The proposed procedure determines the gravity-assist geometry and then finds the gravity-assist linking based on the multiple-point boundary value problem. The gravity-assist geometry determination step designs the periapsis rotation angle by solving a gradient descent optimization problem, yielding trajectories that break the symmetry of the resonance orbits. The multiple-point boundary-value problem seeks to solve a minimum-fuel problem linking two intermediate resonance-like orbits with rotated periapses. The first step of the optimal control problem establishes and solves a relatively easy two-point boundary problem approximating the original problem. The solution is used as the initial guess for the more complex multiple-point boundary value problem. The low-thrust resonance gravity-assist trajectory is compared to the trajectories designed based on traditional approaches involving low-thrust propulsion, demonstrating its validity and efficiency.

Counteracting contingencies in fast rendezvous with ISS

The existing rendezvous profiles of Russian crew vehicles in some flight phases are not adapted to contingencies which can be caused by an autonomous rendezvous system failure. If it occurs, the crew can complete the rendezvous in manual mode only at a range of less than a kilometer. However, at a long range the manual rendezvous mode is complicated because the on-board algorithm of the motion control system generates free trajectories that do not provide the crew with a convenient manual control. As a rule, in this case re-rendezvous is required with shifting the docking to the next day. In the case of fast rendezvous profiles (which allow to get to the ISS in 3 h) re-docking should be scheduled only in two days because of the crew adaptation to the weightlessness. In order to allow the crew to complete the rendezvous in manual mode without delaying the docking, it is required to create a more simple procedure for calculating and performing transfer burns. From the Gemini, Apollo and ATV spaceflight heritage, the technique for approaching the target from a coelliptical orbit has been known, that provides a uniform motion and convenient control of the crew vehicle in manual mode. The paper proposes the fast rendezvous profile with the insertion of the crew vehicle into an intermediate orbit coelliptical with the target orbit. In this approach, in case of automatic rendezvous failure, the crew, having a complete understanding of the relative motion, is able to perform the transfer to the target in manual mode without increasing the duration of the flight till the docking. The paper presents the results of testing this approach on a simulator, which in the long term will allow to reduce the rendezvous duration of Russian vehicles to a single orbit.

Optimization of the launch profile of geostationary spacecraft with electric propulsion system using Falcon-9 launch vehicle

A combined flight profile is being considered, in which the Falcon 9 launch vehicle launches the spacecraft into elliptical intermediate orbit from the Cape Canaveral launch site. In the process of analyzing the problem, the value of the height of the perigee and the inclination of the intermediate orbit are fixed, and the height of the apogee of the intermediate orbit varies. After separation from the last stage of the launch vehicle, the spacecraft carries out transfer to geostationary orbit using electric propulsion system. At the stage of insertion spacecraft from intermediate orbit to geostationary orbit using electric propulsion system, the problem of minimizing the mass of the propellant, multi-revolutionary transfer is considered. The number of revolution and the height of the apogee of the intermediate orbit vary in order to analyze the effect of these parameters on the duration of the transfer and the delivered mass of the spacecraft into geostationary orbit. The main purpose of this paper is to calculate the optimal values of the apogee height of the intermediate orbit and the optimal number of revolution that ensure the delivery of the maximum mass of the spacecraft to the geostationary orbit in a given time delta(t)*. To solve the optimization problem, the Pontryagin maximum principle is applied. After apply ing the maximum principle, the optimization problem is reduced to solving the boundary value problem, which is solved by the continuation method by parameter. The paper presents the results of the optimization problem of multi-revolutionary transfer and analysis of the energy characteristics of combined flight profile for insertion of spacecraft into geostationary orbit.

Launching Earth radiation monitoring satellites using the combined thrust

The paper considers the problem of launching satellites into the orbits for Earth radiation monitoring using the upper stage previously inserted into a reference circular orbit and equipped with a combined propulsion system, i.e. chemical rocket engines and electric rocket propulsion system (ERPS). The combined launch scheme implies formation of the intermediate orbit using the chemical rocket engines and further successive satellites insertion into the target orbits using the electric rocket propulsion system. Calculation of the satellite group launch trajectory parameters is provided, single accompanying and two more in cluster formation, to the given elliptical orbits. The spacecraft flight is approximated at the chemical rocket engines operation phase by a pulse action using the double-pulse apsidal maneuvers; and the spacecraft flight trajectory in the ERPS phase is calculated under the condition of minimizing the flight time with the Pontryagin maximum principle. The results obtained demonstrate technical feasibility of the task of launching satellites into specified orbits using the combined thrust, as well as advantages of a combined scheme for launching a payload in order to reduce flight duration compared to using the ERPS throughout the entire flight.

A mesh refinement method for low‐thrust orbit transfer problems

AbstractThis paper researches the fuel‐optimal geocentric orbit transfer problem using a low‐thrust, electric propulsion. First, the transfer trajectory is divided into N burn‐coast arcs via introducing (N−1) intermediate orbits between the initial and the target orbits. By restricting the time duration of each burn, the changes in orbital states caused by one time of ignition can be represented by a discrete velocity impulse; therefore, a considerable large N, a great many optimization variables, and a months and revolutions long trajectory will be calculated to solve the optimization problem. By proposing a new transfer coordinate system, the performance index is analytically calculated, and strict constraints derived from the continuous condition between adjacent orbits are removed from the mathematical optimization model. Meanwhile, an improved particle swarm optimization (PSO) method and a mesh refinement method are proposed to address the optimization problem. The mesh refinement algorithm begins with a simple bi‐impulse transfer problem, automatically increases/decreases N, and finally converges to the optimal solution. Numerical simulation shows the efficiency of the method with some classical and representative scenarios.

Simplified optimization model for low-thrust perturbed rendezvous between low-eccentricity orbits

Trajectory optimization of low-thrust perturbed orbit rendezvous is a crucial technology for space missions in low Earth orbits, which is difficult to solve due to its initial value sensitivity, especially when the transfer trajectory has many revolutions. This paper investigated the time-fixed perturbed orbit rendezvous between low-eccentricity orbits and proposed a priori quasi-optimal thrust strategy to simplify the problem into a parametric optimization problem, which significantly reduces the complexity. The optimal trajectory is divided into three stages including transfer to a certain intermediate orbit, thrust-off drifting and transfer from intermediate orbit to the target orbit. In the two transfer stages, the spacecraft is assumed to use a parametric law of thrust. Then, the optimization model can be then obtained using very few unknowns. Finally, a differential evolution algorithm is adopted to solve the simplified optimization model and an analytical correction process is proposed to eliminate the numerical errors. Simulation results and comparisons with previous methods proved this new method’s efficiency and high precision for low-eccentricity orbits. The method can be well applied to premilitary analysis and high-precision trajectory optimization of missions such as in-orbit service and active debris removal in low Earth orbits.

Оптимизация программы выведения полезной нагрузки на заданную траекторию

In this paper, the problem of choosing and optimizing a program for launching the payload (PL) into a specified circular orbit is considered. Both direct insertion into the circular orbit and the insertion using an intermediate elliptical orbit are examined. The mass range of the PL, which can be placed into the circular orbit with specified parameters, is calculated. The PL is launched using a medium-class launch vehicle (LV), which is proposed to be a hypothetical ballistic missile withdrawn from service. The LV movement program is divided into two sections: the escape from the atmosphere and the launch section. Both sections use a control program with the pitch angle as a parameter. The optimal parameters for the program are determined. Direct insertion into the circular orbit is revealed to be possible for a very limited range of PL masses. It is shown that the range can be significantly extended if an intermediate elliptical orbit is used. The study results prove that the use of the selected LV allows placing the PL in the specified circular and elliptical orbits. For the formation of the orbital trajectories of the PL launch, it is possible to use an almost invariable parametric class of LV flight control programs.

Виртуальные гравитационные маневры при проектировании межпланетных перелётов

It is shown that the backward in time extension of the spacecraft interplanetary launch trajectory from the point of the low orbit of the departure planet generates a trajectory structurally coinciding with the virtual gravity assist trajectory near the departure planet. The pericenters of the auxiliary beam of the flying by hyperbolas and the time of their pericenter passage in this case differ slightly from the corresponding parameters of the designed departure orbit when starting from the specified point. Thus, the search for the trajectory of an interplanetary flight can be separated from the need to take into account the boundary conditions of the launch from an intermediate low pre-launch orbit. The pre-launch orbit is refined on the results of the search for the trajectory of the interplanetary flight. A structurally uniform scheme of ballistic design of spacecraft flight paths using multiple gravitational maneuvers based on the consideration of planetary ephemerides is presented.

Optimal Control of Transfer to Vertical Orbits from Lyapunov Orbits Using Low-Thrust Engine

There are many different families of periodic orbits in the Earth-Moon system, such as Lyapunov orbits, halo orbits, vertical orbits, etc. The establishment of a lunar space station requires a spacecraft to be able to transfer among these orbits. Lyapunov orbits have been used by some missions and are well-studied orbits, while periodic vertical orbits can provide large amplitudes of spacecraft motion outside the plane of the Moon’s motion, which makes it possible to avoid shadowing of the orbits and use them as relay satellite in cislunar space. Modern researchers mainly consider the use of high-thrust engines for transfer. With the development of electric propulsion technology, the use of low, but long-acting thrust for deep space exploration has become especially relevant. This is due to the high specific characteristics of the propulsion systems of this type. In this article, an algorithm has been developed for determining the optimal control with a low-thrust engine for a transfer from Lyapunov orbit to a vertical orbit. The minimum time of flight or the minimum costs of the working body are used as criteria for optimality. In the calculation for solving the two-point boundary value problem of the optimal control theory, the parameter continuation algorithm is used, which allows to gradually get the transfer from some simple results to the final transfer trajectory. The results obtained make it possible to assert that the use of intermediate axial orbits allows the use of propulsion systems with lower thrust levels. In this case, the duration of the flight increases slightly with an almost unchanged consumption of the working body. Moreover, the homotopy method makes it possible to reduce the consumption of working body, while the control of the engine throttling becomes discrete. The results of this study and the algorithms proposed in this article can be used to determine the optimal program control and ballistic design of lunar missions.

ГЕОФИЗИЧЕСКИЙ АСПЕКТ КАТАСТРОФИЧЕСКОГО ЗАПУСКА СПУТНИКОВ STARLINK

On February 3, 2022, as part of the SpaceX program, the next Falcon 9 carrier with Starlink satellites was launched. The launch was unsuccessful, which led to the loss of 40 out of 49 satellites. According to SpaceX experts, the cause of this event was an increase in the density of the atmosphere in the intermediate orbit, due to a magnetic storm, followed by deceleration and loss of satellites. Based on the analysis of the geomagnetic situation, the article shows that it is not enough to limit ourselves to consider only a possible change in the density of the atmosphere as the cause of the catastrophe. Most likely, the reason is complex and involves, among other issues, a failure in the operation of the radio-electronic equipment of the satellites due to induced currents and their electrification.

Substantiation of the Western Trajectory of the Tunguska Cosmic Body

Although more than 100 years have passed since the Tunguska event, no hypothesis has yet been proposed that would be consistent with the entire set of available facts and observations. A hypothesis has been put forward that the Tunguska Cosmic Body was an ice comet and its fall was accompanied by the fall of smaller fragments, whereas the TCB and fragments fell from intermediate Earth orbits. This explains all the observed phenomena without exception, including those that occurred long before the catastrophe. In this paper, we present eyewitnesses’ reports and objective materials obtained in the event area and confirming the TCB passage in the direction from West to East. The western trajectory has been traced for 3500 km from the Volga to the Lower Tunguska. We have carried out a detailed analysis of the eyewitnesses’ reports revealing the circumstances of passage of a huge fireball. We provide criticism of trajectories constructed without taking new information into account. We have estimated the energy released during this event and determined the mechanical properties of the comet matter. Science may obtain with new objects to study the direct impact of the comet matter on the Earth’s surface. This may be useful for evaluating the comet-related hazard.

Multi-impulse smooth trajectory design for long-range rendezvous with an orbiting target using multi-objective non-dominated sorting genetic algorithm

This paper develops a methodology to generate Pareto optimal trajectories for long-range rendezvous of a servicing satellite with a moving target, in an on-orbit servicing mission. The methodology employs a multi-impulse shape-based trajectory planning algorithm for in-plane orbit transfer, based on the two-body problem. We first derive the necessary and sufficient conditions that determine the set of smooth impulsive trajectories connecting the servicing satellite to the orbiting target. The Pareto optimal trajectories from this set are then obtained using a constrained multi-objective optimization algorithm developed based on the Non-dominated Sorting Genetic Algorithm-II (NSGA-II). In a mission, an optimal solution from the Pareto frontier set may be selected based on the mission requirements. Transfer time and control effort are the two Pareto cost functions that are considered in the multi-objective optimization. To reduce the risk of collision in populated orbits and to remain in an orbital regime, we include restrictions on orbital elements as part of the constraints. Further, a maximum available impulse is considered as an upper-bound for velocity changes in an impulsive trajectory. The number of impulses along with the location of the first impulse in the parking orbit and the orbital parameters of the intermediate orbits form the set of design variables. The key advantage of the proposed trajectory optimization methodology compared to its counterparts using continuous thrust is the significant reduction of the number of design variables. Finally, we demonstrate the superiority of the developed trajectory planner by comparing its results with those obtained from another multi-objective evolutionary algorithm called the Multi-Objective Genetic Algorithm and an optimal Lambert approach based on single-objective optimization.

The Rise and Fall of the Eclipsing Binary HS Hydrae

Abstract HS Hydrae is a short period eclipsing binary (P orb = 1.57 day) that belongs to a rare group of systems observed to have rapidly changing inclinations. This evolution is due to a third star on an intermediate orbit, and results in significant differences in eclipse depths and timings year to year. Zasche & Paschke revealed that HS Hydrae’s eclipses were rapidly fading from view, predicting they would cease around 2022. Using 25 days of photometric data from Sector 009 of the Transiting Exoplanet Survey Satellite (TESS), we find that the primary eclipses for HS Hydrae were only 0.00173 ± 0.00007 mag in depth in March 2019. This data from TESS likely represents the last eclipses detected from HS Hydrae. We also searched the Digitization of the Harvard Astronomical Plate Collection archive for historic data from the system. With a total baseline of over 125 yr, this unique combination of data sets—from photographic plates to precision space-based photometry—allows us to trace the emergence and decay of eclipses from HS Hydrae, and further constrain its evolution. Recent TESS observations from Sector 035 confirm that eclipses have ceased for HS Hya, and we estimate they will begin again in 2195.

Сравнительный анализ эффективности дросселированных электрических ракетных двигателей постоянной мощности для межорбитальных перелетов на геоста

Осуществление перелетов космических аппаратов в околоземном пространстве с промежуточных эллиптических орбит на геостационарную с использованием электрических ракетных двигателей (ЭРД) малой тяги является одной из самых актуальных задач современной космонавтики. ЭРД, скорость истечения реактивной струи которых на порядок больше, чем у химических РД, лучше для межорбитальных околоземных перелетов с максимальной полезной нагрузкой в случае, когда существенное увеличение продолжительности маневра является допустимым. Дросселирование тяги ракетного двигателя традиционно рассматривается как один из способов понижения массы двигательной установки и необходимого запаса топлива для выполнения заданного маневра. Использование концепции идеально регулируемого двигателя обеспечивает получение верхних оценок массы полезной нагрузки межорбитальных перелетов для выбранного уровня мощности. Учет свойств реальных двигателей приводит к необходимости рассмотрения математических моделей с более жесткими ограничениями на управление. Проведено исследование эффективности трех режимов управления тягой ЭРД при выполнении практически интересных перелетов с высокоэллиптических промежуточных околоземных орбит на геостационарные. Построена математическая модель релейного ЭРД постоянной мощности. Дана формулировка вариационной задачи типа Майера о выполнении заданного динамического маневра космического аппарата для дро-сельированного и нерегулируемого двигателей постоянной мощности. С использованием принципа максимума Понтрягина проведен анализ оптимальных управляющих функций, для которых выписаны конечные соотношения, позволившие записать систему дифференциальных уравнений оптимального движения космического аппарата, оснащенного релейным двигателем. Полученные численные и качественные результаты исследования эффективности различных режимов управления тягой для увеличения полезной нагрузки заданного орбитального маневра подтвердили корректность математических моделей дросселированного и релейного двигателей и, в целом, эффективность использования решений усредненных уравнений оптимального движения космического аппарата для численного решения соответствующих краевых задач в точной. постановке.

Transfer between libration orbits through the outer branches of manifolds for Phobos exploration

The transfer between libration orbits in the Mars–Phobos system is challenging under the strong gravitational perturbation of Mars and Phobos. In this paper, the outer branches of manifolds of libration orbits around Phobos are employed for impulse and continuous thrust transfer between orbits, respectively. The 1:1 resonant orbits are found to be appropriate intermediate orbits for impulse transfer, which can connect the manifolds of various kinds of libration orbits. For the continuous thrust transfer, the convex optimization method is adopted to design the low-thrust trajectory in the three-body problem. The Poincaré section is established to find the best matching points on the outer manifolds. The L1/L2 homoclinic and L1−L2 heteroclinic transfer trajectories are designed and discussed. The simulations show the feasibility of the transfer by outer manifolds with both impulse and continuous thrust. The proposed method can avoid the potential hazard caused by the complicated dynamics near Phobos and provide a reliable transfer method, which can be used in future Phobos exploration missions.

Analytical solutions for extremal orbit transfers utilizing three intermediate thrust arcs in a Newtonian field

Analytical synthesis of extremal transfer trajectories with up to three intermediate thrust arcs between elliptical orbits is presented using Mayer’s variational problem. Solution uniqueness requires three such arcs connected via intermediate transfer orbits. Junction conditions provide state vector continuity at junction points. Fuel efficiency is compared with impulsive trajectories.

Constraints on Sub-Neptune Planet Candidate KOI-972.01 via Joint Variability/Gravity-darkening Analysis

Abstract We analyze Kepler photometry of transiting planet candidate KOI-972.01, accounting for both stellar variability and gravity darkening. KOI-972.01 stands out because of its small radius, less than that of Neptune, and because of its intermediate orbit period at 13.12 days, long enough to avoid significant tidal evolution, and thus it represents an underexplored exoplanet class. The parent star of KOI-972.01 is a rapidly rotating δ-Scuti variable, complicating transit lightcurve interpretation but also offering a potential independent source of stellar parameters. We measure the stellar rotation period (16.2 hr) by identifying the stellar rotation frequency and subsequently place a constraint on the stellar obliquity of no greater than 10, but have difficulty isolating individual oscillation modes in the periodogram owing to time variation of the δ-Scuti oscillations. After subtracting the stellar oscillations, lightcurve fits place the transiting object radius at 3.07 ± 0.09 R ⊕, but the shallow transit prevents useful constraints on the system’s spin–orbit alignment.

An Earth Trojans search mission with asteroid flybys and rendezvous to 2010TK7

In this paper, a mission about Earth’s Trojan asteroids is proposed, which is a rendezvous mission to 2010TK7 with two flybys to Near Earth asteroids (NEAs), and then cruise around L4 point with a survey for potential Earth’s Trojan asteroids around L4. The flyby targets are searched in NEAs first. Flyby sequence 1949 ​MA-2003LC5 is chosen among the search results. The low-thrust transfer trajectories are designed with two parts using indirect method. One part is the transfer trajectory corresponding to the Earth-1949MA-2003LC5-2010TK7 flyby sequence, and the other is the transfer trajectory from 2010TK7 to the cruise orbit. Intermediate orbit is designed to observe asteroid 2010TK7. The cruise orbit is a quasi-periodic orbit restricted in the stable region of Earth’s Trojan asteroids so as to search more Earth’s Trojan asteroids. For a spacecraft with initial mass of 300 ​kg, the total fuel consumption of this mission is 42.184%.

A Highly Eccentric Warm Jupiter Orbiting TIC 237913194

Abstract The orbital parameters of warm Jupiters serve as a record of their formation history, providing constraints on formation scenarios for giant planets on close and intermediate orbits. Here, we report the discovery of TIC 237913194b, detected in full-frame images from Sectors 1 and 2 of the Transiting Exoplanet Survey Satellite (TESS), ground-based photometry (Chilean–Hungarian Automated Telescope, Las Cumbres Observatory Global Telescope), and Fiber-fed Extended Range Optical Spectrograph radial velocity time series. We constrain its mass to = and its radius to = , implying a bulk density similar to Neptune’s. It orbits a G-type star ( = , V = 12.1 mag) with a period of 15.17 days on one of the most eccentric orbits of all known warm giants (e ≈ 0.58). This extreme dynamical state points to a past interaction with an additional, undetected massive companion. A tidal evolution analysis showed a large tidal dissipation timescale, suggesting that the planet is not a progenitor for a hot Jupiter caught during its high-eccentricity migration. TIC 237913194b further represents an attractive opportunity to study the energy deposition and redistribution in the atmosphere of a warm Jupiter with high eccentricity.