Artificial Equilibrium Points Research Articles

Artificial Equilibrium Points in the Low-Thrust Restricted Three-Body Problem When Both the Primaries Are Oblate Spheroids

This paper studies the existence and stability of the artificial equilibrium points (AEPs) in the low-thrust restricted three-body problem when both the primaries are oblate spheroids. The artificial equilibrium points (AEPs) are generated by canceling the gravitational and centrifugal forces with continuous low-thrust at a non-equilibrium point. Some graphical investigations are shown for the effects of the relative parameters which characterized the locations of the AEPs. Also, the numerical values of AEPs have been calculated. The positions of these AEPs will depend not only also on magnitude and directions of low-thrust acceleration. The linear stability of the AEPs has been investigated. We have determined the stability regions in the xy, xz and yz-planes and studied the effect of oblateness parameters A1(0<A1<1) and A2(0<A2<1) on the motion of the spacecraft. We have found that the stability regions reduce around both the primaries for the increasing values of oblateness of the primaries. Finally, we have plotted the zero velocity curves to determine the possible regions of motion of the spacecraft.

Spacecraft motion around artificial equilibrium points

The main goal of this paper is to describe the motion of a spacecraft around an artificial equilibrium point in the circular restricted three-body problem. The spacecraft is under the gravitational influence of the Sun and the Earth, as primary and secondary bodies, subjected to the force due to the solar radiation pressure and some extra perturbations. Analytical solutions for the equations of motion of the spacecraft are found using several methods and for different extra perturbations. These solutions are strictly valid at the artificial equilibrium point, but they are used as approximations to describe the motion around this artificial equilibrium point. As an application of the method, the perturbation due to the gravitational influence of Jupiter and Venus is added to a spacecraft located at a chosen artificial equilibrium point, near the $$L_3$$ Lagrangian point of the Sun–Earth system. The system is propagated starting from this point using analytical and numerical solutions. Comparisons between analytical–analytical and analytical–numerical solutions for several kinds of perturbations are made to guide the choice of the best analytical solution, with the best accuracy.

Artificial equilibrium points in binary asteroid systems with continuous low-thrust

The positions and dynamical characteristics of artificial equilibrium points (AEPs) in the vicinity of a binary asteroid with continuous low-thrust are studied. The restricted ellipsoid–ellipsoid model of binary system is employed for the binary asteroid system. The positions of AEPs are obtained by this model. It is found that the set of the point \(L_{1}\) or \(L_{2}\) forms a shape of an ellipsoid while the set of the point \(L_{3}\) forms a shape like a “banana”. The effect of the continuous low-thrust on the feasible region of motion is analyzed by zero velocity curves. Because of using the low-thrust, the unreachable region can become reachable. The linearized equations of motion are derived for stability’s analysis. Based on the characteristic equation of the linearized equations, the stability conditions are derived. The stable regions of AEPs are investigated by a parametric analysis. The effect of the mass ratio and ellipsoid parameters on stable region is also discussed. The results show that the influence of the mass ratio on the stable regions is more significant than the parameters of ellipsoid.

Invariant manifolds around artificial equilibrium points for low-thrust propulsion spacecraft

Low-thrust propulsion is incorporated into circular restricted three-body problem to balance the gravitational and centrifugal forces, and then artificial equilibrium points can be generated. The linear dynamics indicates that there are stable and unstable artificial equilibrium points. Around the unstable artificial equilibrium points, there are center and hyperbolic invariant manifolds. In this work, invariant manifolds around artificial equilibrium points are expressed as formal series of amplitudes corresponding to hyperbolic and center dynamics, and high-order series solutions are constructed up to an arbitrary order. By taking advantage of the series expansions constructed, the motions around unstable artificial equilibrium points can be parameterized. In order to check the validity, the practical convergence of series solutions truncated at different orders is considered. Finally, series expansions of invariant manifolds are applied to designing transfer trajectories from the primary to periodic orbits around artificial equilibrium points which are located inside $L_{1}$ and beyond $L_{2}$ points.

On the Artificial Equilibrium Points in the Circular Restricted Problem of 2 + 2 Bodies

The present work studies the stability condition of central control artificial equilibrium points of the planar circular restricted problem of 2 + 2 bodies (PCRP2 + 2B) and also its variant when the shape of larger mass is taken to be an oblate spheroid. We find that the paper will be of great application in choosing an artificial equilibrium point (AEP) in the neighbourhood of numerous planets e.g. Jupiter or the bodies which provide a model of the problem studied. The minimum thrust will save a quantum of energy to be applied to have an arbitrary point as a chosen starter. For solar sailing and magnetic force this minimum thrust will be of great use.

Solar sail equilibria with albedo radiation pressure in the circular restricted three-body problem

Solar Radiation Pressure (SRP) and albedo effects are investigated in the circular restricted three-body problem for a system consisting of the Sun, a reflective minor body and a solar sail. As an approximation of albedo radiation pressure (ARP), the minor body is treated as Lambertian with reflected flux scattered by the bidirectional reflectance distribution function. Incorporating ARP, which is a function of SRP, into the solar sail equations of motion renders additional artificial equilibrium points in a volume between the L1 and L2 points which is defined as the region of influence. Based on the model, characterization of the findings are provided that are theoretically applicable to any body with discernible albedo such as for instance Earth, Mars or an asteroid. Example results are presented for a Sun-Vesta system which show that the inclusion of ARP generates artificial equilibrium points requiring solar sail designs with very low mass-to-area ratio. The equilibrium points are found to be unstable in general but asymptotic stability may be enforced with sail attitude feedback control.

High-order solutions around the stable libration points in the low-thrust restricted three-body system

Non-equilibrium point in the circular restricted three-body system can be changed into an artificial equilibrium point by means of low-thrust acceleration, and an spacecraft located at the artificial equilibrium point remains static relative to the primaries in the synodic coordinate system. In the classical restricted three-body system, the libration point orbit resource is limited, and the existence of artificial equilibrium point could overcome the shortage. The richness of artificial equilibrium points leads to the flexibility of mission design in deep space exploration. Due to the stability of stable artificial equilibrium points, for a spacecraft moving around them, less propellant is required for station keeping. In this paper, we expand the general motion around stable artificial equilibrium points as formal series of long-period, short-period and vertical periodic amplitudes. Then Lindstedt-Poincare method is adopted to construct the series solutions up to an arbitrary order. By taking advantage of the series expansions constructed, the motions around artificial equilibrium points can be parameterized, and these parameters are beneficial to the optimization process of libration point mission design. At last, in order to provide the available range of series expansions constructed, the practical convergence is considered.

On the R4BP when third primary is an oblate spheroid

The present paper deals with the restricted four-body problem, when the third primary placed at the triangular libration point of the restricted three-body problem is an oblate body. The third primary m 3 is not influencing the motion of the dominating primaries m 1 and m 2. We have studied the motion of m 4, moving under the influence of the three primaries m i , i=1,2,3, but the motion of the primaries is not being influenced by infinitesimal mass m 4. The aim of this study is to find the locations of equilibrium points and their stability. We obtain three collinear and five non-collinear equilibrium points. The collinear equilibrium points are unstable for all the mass parameter. The non-collinear points are stable for different mass parameter and oblateness factor. Also, we have considered the autonomous coplanar circular restricted four-body problem with the infinitesimal mass as a low-thrust spacecraft. Artificial equilibrium points are created with the use of continuous low-thrust propulsion. The obtained results show that, in absence of thrust there are unstable equilibrium points close to the third primary. Also, using the constant low-thrust, the artificial equilibrium points can be generated, which move from the natural equilibrium points. Further, it is proved that in certain region closed to the third primary, these points are stable. We have drawn the zero velocity surfaces to determine the possible allowed boundary regions. We observed that for increasing values of oblateness coefficient A, the corresponding possible boundary regions increase where the particle can freely move from one side to another side. Further, for different values of Jacobi constant C, we can find the boundary region where the particle can move in possible allowed partitions. The stability regions of the equilibrium points expanded due to presence of oblateness coefficient and various values of C.

Artificial equilibrium points for a solar balloon in the α Centauri system

This paper discusses the generation, stability, and control of artificial equilibrium points for a solar balloon spacecraft in the α Centauri A and B binary star system. The continuous propulsive acceleration provided by a solar balloon is shown to be able to modify the position of the (classical) Lagrangian equilibrium points of the three-body system on a locus whose geometrical form is known analytically. A linear stability analysis reveals that the new generated equilibrium points are usually unstable, but part of them can be stabilized with a simple feedback control logic.

On the Artificial Equilibrium Points in a Generalized Restricted Problem of Three Bodies

The present article studies the stability conditions of central control artificial equilibrium generalized restricted problem of three bodies. It is generalized in the sense that here we have taken the larger primary body to be in shape of an oblate spheroid. The equilibrium points are sought by the application of the propellant for which it would just balance the gravitational forces. The launching flight of such a satellite is seen to be applicable for having arbitrary space stations for these different missions. Specialty of the result of the investigation lies in the fact that an arbitrary space station can be formed to attain any specified mission.

Novel mission concepts for polar coverage: An overview of recent developments and possible future applications

The paper provides a survey of novel mission concepts for continuous, hemispheric polar observation and direct-link polar telecommunications. It is well known that these services cannot be provided by traditional platforms: geostationary satellites do not cover high-latitude regions, while low- and medium-orbit Sun-synchronous spacecraft only cover a narrow swath of the Earth at each passage. Concepts that are proposed in the literature are described, including the pole-sitter concept (in which a spacecraft is stationary above the pole), spacecraft in artificial equilibrium points in the Sun–Earth system and non-Keplerian polar Molniya orbits. Additionally, novel displaced eight-shaped orbits at Lagrangian points are presented. For many of these concepts, a continuous acceleration is required and propulsion systems include solar electric propulsion, solar sail and a hybridisation of the two. Advantages and drawbacks of each mission concept are assessed, and a comparison in terms of high-latitude coverage and distance, spacecraft mass, payload and lifetime is presented. Finally, the paper will describe a number of potential applications enabled by these concepts, focusing on polar Earth observation and telecommunications.

Artificial equilibrium points for a generalized sail in the elliptic restricted three-body problem

Artificial equilibrium points for a generalized sail in the elliptic restricted three-body problem

Low-thrust propulsion in a coplanar circular restricted four body problem

This paper formulates a circular restricted four body problem (CRFBP), where the three primaries are set in the stable Lagrangian equilateral triangle configuration and the fourth body is massless. The analysis of this autonomous coplanar CRFBP is undertaken, which identifies eight natural equilibria; four of which are close to the smaller body, two stable and two unstable, when considering the primaries to be the Sun and two smaller bodies of the Solar System. Following this, the model incorporates ‘near term’ low-thrust propulsion capabilities to generate surfaces of artificial equilibrium points close to the smaller primary, both in and out of the plane containing the celestial bodies. A stability analysis of these points is carried out and a stable subset of them is identified. Throughout the analysis the Sun-Jupiter-asteroid-spacecraft system is used, for conceivable masses of a hypothetical asteroid set at the libration point L 4. It is shown that eight bounded orbits exist, which can be maintained with a constant thrust less than 1.5 × 10−4 N for a 1000 kg spacecraft. This illustrates that, by exploiting low-thrust technologies, it would be possible to maintain an observation point more than 66% closer to the asteroid than that of a stable natural equilibrium point. The analysis then focusses on a major Jupiter Trojan: the (624) Hektor asteroid. The thrust required to enable close asteroid observation is determined in the simplified CRFBP model. Finally, a numerical simulation of the real Sun-Jupiter-(624) Hektor-spacecraft is undertaken, which tests the validity of the stability analysis of the simplified model.

Low-Thrust-Enabled Highly-Non-Keplerian Orbits in Support of Future Mars Exploration

The technology of high specific impulse propulsion systems with low thrust is improving, opening up numerous possibilities for future missions applying continuous thrust to force a spacecraft out of a natural Keplerian orbit into a displaced non-Keplerian orbit. A systematic analysis is presented as to the applicability of highly non-Keplerian orbits throughout the Solar System. Thereafter, two applications of such orbits in support of future high-value asset exploration of Mars are detailed: a novel concept for an Earth-Mars interplanetary communications relay, on which the paper largely focuses, and a solar storm warning mission. In the former the relay makes use of artificial equilibrium points, allowing a spacecraft to hover above the orbital plane of Mars and thus ensuring communications when the planet is occulted by the Sun with respect to the Earth. The spacecraft’s power requirements and communications band utilized are taken into account to determine the relay architecture. A detailed contingency analysis is considered for recovering the relay after increasing periods of spacecraft propulsion failure, combined with a consideration of how to deploy the relay spacecraft to maximise propellant reserves and mission duration. For such a relay, a combination of solar sail and solar electric propulsion may prove advantageous, but only under specific circumstances of the relay architecture suggested. For highly non-Keplerian orbits the dynamics of the spacecraft is also briefly extended to consider the elliptic restricted three-body problem and the effects of orbit eccentricity.

Artificial equilibrium points for a generalized sail in the circular restricted three-body problem

This paper introduces a new approach to the study of artificial equilibrium points in the circular restricted three-body problem for propulsion systems with continuous and purely radial thrust. The propulsion system is described by means of a general mathematical model that encompasses the behavior of different systems like a solar sail, a magnetic sail and an electric sail. The proposed model is based on the choice of a coefficient related to the propulsion type and a performance parameter that quantifies the system technological complexity. The propulsion system is therefore referred to as generalized sail. The existence of artificial equilibrium points for a generalized sail is investigated. It is shown that three different families of equilibrium points exist, and their characteristic locus is described geometrically by varying the value of the performance parameter. The linear stability of the artificial points is also discussed.

On the stability of artificial equilibrium points in the circular restricted three-body problem

The article analyses the stability properties of minimum-control artificial equilibrium points in the planar circular restricted three-body problem. It is seen that when the masses of the two primaries are of different orders of magnitude, minimum-control equilibrium is obtained when the spacecraft is almost coorbiting with the second primary as long as their mutual distance is not too small. In addition, stability is found when the distance from the second primary exceeds a minimum value which is a simple function of the mass ratio of the two primaries and their separation. Lyapunov stability under non-resonant conditions is demonstrated using Arnold’s theorem. Among the most promising applications of the concept we find solar-sail-stabilized observatories coorbiting with the Earth, Mars, and Venus.

Artificial halo orbits for low-thrust propulsion spacecraft

We consider periodic halo orbits about artificial equilibrium points (AEP) near to the Lagrange points L1 and L2 in the circular restricted three body problem, where the third body is a low-thrust propulsion spacecraft in the Sun–Earth system. Although such halo orbits about artificial equilibrium points can be generated using a solar sail, there are points inside L1 and beyond L2 where a solar sail cannot be placed, so low-thrust, such as solar electric propulsion, is the only option to generate artificial halo orbits around points inaccessible to a solar sail. Analytical and numerical halo orbits for such low-thrust propulsion systems are obtained by using the Lindstedt Poincare and differential corrector method respectively. Both the period and minimum amplitude of halo orbits about artificial equilibrium points inside L1 decreases with an increase in low-thrust acceleration. The halo orbits about artificial equilibrium points beyond L2 in contrast show an increase in period with an increase in low-thrust acceleration. However, the minimum amplitude first increases and then decreases after the thrust acceleration exceeds 0.415 mm/s2. Using a continuation method, we also find stable artificial halo orbits which can be sustained for long integration times and require a reasonably small low-thrust acceleration 0.0593 mm/s2.

Artificial Three-Body Equilibria for Hybrid Low-Thrust Propulsion

This paper proposes a new concept of creating artificial equilibrium points in the circular restricted three body problem, where the third body uses a hybrid of solar sail and solar electric propulsion. The work aims to investigate the use of a hybrid sail for artificial equilibrium points that are technologically di±cult with either of these propulsion systems alone. The hybrid sail has freedom in specifying the sail lightness number, then minimizing the required thrust acceleration from the solar electric propulsion thruster while satisfying the equilibrium condition. The stability analysis of such artificial equilibrium points by a linear method results in a linear time varying (mass) system. The freezing time method then provides unstable and marginally stable regions for hybrid solar sail artificial equilibria. We compare these propulsion systems with a given payload mass and mission life for a polar observation mission. For a near term sail assembly loading we find for the hybrid sail a substantially lower propellant mass compared to solar electric propulsion and lower sail length with respect to a solar sail, and a lower initial spacecraft mass.

Artificial Equilibrium Points in the Low-Thrust Restricted Three-Body Problem

T HE present Note describes the concept of the artificial equilibrium point (AEP) assisted by continuous thrust in the restricted three-body problem (RTBP). In the RTBP, there are five libration points called Lagrange points. Each of the Lagrange points is in equilibrium between the gravitational forces of the two primary bodies and the centrifugal force in the rotating frame. The Lagrange points have been investigated in a number of studies in the field of celestial mechanics. Since the 1950s, space engineers have been interested in these Lagrange points and have investigated the applicability of these points to space missions. Farquhar [1] introduced the concept of telecommunication systems using Lagrange points in the Earth–moon system, and in subsequent studies, he investigated ballistic periodic orbits about equilibrium points not only in the Earth–moon system [2–5], but also in the sun– Earth system [6,7]. Currently, low-thrust propulsion systems such as electric propulsion and the solar sail are being developed not only for controlling satellite orbits, but also asmain engines for interplanetary transfer. These low-thrust propulsion systems are able to provide continuous control acceleration to the spacecraft and thus to increase mission design flexibility. When we attempt to use Lagrange points, the positions are normally restricted to only five points. In terms of mission design, however, Lagrange points are not always the best positions. For example, we must operate the satellite at midnight every day when a spacecraft is placed at L2 points, unless we can use a deep space network. L4 and L5 are stable points, but these points are far from the Earth. Therefore, the transfer time required to reach L4/L5 and to telecommunicate is longer than that for points closer to the Earth. To achieve various mission objectives, positions other than the Lagrange points might be suitable in some cases. These points are in nonequilibrium, but it is possible to keep the spacecraft at these positions by using continuous thrust. Libration points with continuous control acceleration have also been studied (Duseck [8], Simmons et al. [9], McInnes et al. [10,11], and Broschart and Sheeres [12]). In [11], McInnes investigated the magnitude of control acceleration and stability for a two-body problem. These studies reported specific libration points with a certain mass ratio or those with certain low-thrust accelerations. However, for more flexible and generic mission designs, we must analyze arbitral points for the general mass-ratio range with continuous control acceleration by an idealized continuous thrust. In a previous paper [13], we focused on resonant periodic orbits existing on the line connecting two primary bodies with a continuous-low-thrust propulsion system. On the other hand, in the present Note, we investigate the magnitude and direction of the required acceleration creating the AEP in three-dimensional space. In addition, we discuss its stability by linearizing the equations of motion and carrying out a linear stability analysis.

Periodic Orbits with Low-Thrust Propulsion in the Restricted Three-body Problem

Periodic orbits around nonequilibrium points are generated systematically by using continuous low-thrust propulsion in the restricted three-body problem, with a mass ratio varying from 0 to 1/2. A continuous constant acceleration is applied to cancel the gravitational forces of two primary bodies and the centrifugal force at a nonequilibrium point, which is changed into an artificial equilibrium point. The equations of motion are linearized to analytically generate periodic orbits with constant acceleration. Then, periodic orbits around artificial equilibrium points which exist on the line connecting two primary bodies are investigated. The frequencies of these periodic motions are expressed by a parameter that is a function of the mass ratio and the position of the orbits around artificial equilibrium points. By choosing the frequencies of motions that are small-integer resonant, we have found the existence of points at which in-plane and out-of-plane motions are synchronized.