Artificial frozen orbits around Mercury

Orbits around Mercury are influenced by the strong elliptic third-body perturbation, especially for high eccentricity orbits, the periapsis altitude changes dramatically. Frozen orbits whose mean eccentricity and argument of perigee remain constants are obviously a good choice for space missions, but the forming conditions are too harsh to meet practical needs. To deal with this problem, a continuous control method that combines analytical theory and parameter optimization is proposed to build an artificial frozen orbit. The artificial frozen orbits are investigated on the basis of double averaged Hamiltonian, of which the second and third zonal harmonics and the perturbation of elliptic third-body gravity are considered. In this paper, coefficients of perturbations which satisfy the conditions of frozen orbits are involved as control parameters, and the relevant artificial perturbations are compensated by the control strategy. So probes around Mercury can be kept on frozen orbit under the influence of continuous control force. Then complex method of optimization is used to search for the energy optimized artificial frozen orbits. The choosing of optimal parameters, the objective function setting and other issues are also discussed in the study. Evolution of optimal control parameters are given in large ranges of semi-major axis and eccentricity, through the variation of these curves, the fuel efficiency is discussed. The result shows that the control method proposed in this paper can effectively maintain the eccentricity and argument of perigee frozen.     

Artificial halo orbits for low-thrust propulsion spacecraft

We consider periodic halo orbits about artificial equilibrium points (AEP) near to the Lagrange points L ₁ and L ₂ 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 L ₁ and beyond L ₂ 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 Poincaré and differential corrector method respectively. Both the period and minimum amplitude of halo orbits about artificial equilibrium points inside L ₁ decreases with an increase in low-thrust acceleration. The halo orbits about artificial equilibrium points beyond L ₂ 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.     

A survey of orbits of co-orbitals of Mars

Many asteroids with a semimajor axis close to that of Mars have been discovered in the last several years. Potentially some of these could be in 1:1 resonance with Mars, much as are the classic Trojan asteroids with Jupiter, and its lesser-known horseshoe companions with Earth. In the 1990s, two Trojan companions of Mars, 5261 Eureka and 1998 VF₃₁, were discovered, librating about the L₅ Lagrange point, 60° behind Mars in its orbit. Although several other potential Mars Trojans have been identified, our orbital calculations show only one other known asteroid, 1999 UJ₇, to be a Trojan, associated with the L₄ Lagrange point, 60° ahead of Mars in its orbit. We further find that asteroid 36017 (1999 ND₄₃) is a horseshoe librator, alternating with periods of Trojan motion. This asteroid makes repeated close approaches to Earth and has a chaotic orbit whose behavior can be confidently predicted for less than 3000 years. We identify two objects, 2001 HW₁₅ and 2000 TG₂, within the resonant region capable of undergoing what we designate “circulation transition”, in which objects can pass between circulation outside the orbit of Mars and circulation inside it, or vice versa. The eccentricity of the orbit of Mars appears to play an important role in circulation transition and in horseshoe motion. Based on the orbits and on spectroscopic data, the Trojan asteroids of Mars may be primordial bodies, while some co-orbital bodies may be in a temporary state of motion.     

Lunar frozen orbits revisited

Lunar frozen orbits, characterized by constant orbital elements on average, have been previously found using various dynamical models, incorporating the gravitational field of the Moon and the third-body perturbation exerted by the Earth. The resulting mean orbital elements must be converted to osculating elements to initialize the orbiter position and velocity in the lunar frame. Thus far, however, there has not been an explicit transformation from mean to osculating elements, which includes the zonal harmonic \(J_2\), the sectorial harmonic \(C_{22}\), and the Earth third-body effect. In the current paper, we derive the dynamics of a lunar orbiter under the mentioned perturbations, which are shown to be dominant for the evolution of circumlunar orbits, and use von Zeipel’s method to obtain a transformation between mean and osculating elements. Whereas the dynamics of the mean elements do not include \(C_{22}\), and hence does not affect the equilibria leading to frozen orbits, \(C_{22}\) is present in the mean-to-osculating transformation, hence affecting the initialization of the physical circumlunar orbit. Simulations show that by using the newly-derived transformation, frozen orbits exhibit better behavior in terms of long-term stability about the mean values of eccentricity and argument of periapsis, especially for high orbits.     

Analytical method for the effects of the asteroid belt on planetary orbits

Analytic expressions are derived for the perturbation of planetary orbits due to a thick constant density asteroid belt. The derivations include extensions and adaptations of Plakhov's analytic expressions for the perturbations in five of the orbital elements for closed orbits around Saturn's rings. The equations of Plakhov are modified to include the effect of ring thickness and additional equations are derived for the perturbations in the sixth orbital element, the mean anomaly. The gravitational potential and orbital perturbations are derived for the asteroid belt with and without thickness, and for a hoop approximation to the belt. The procedures are also applicable to Saturn's rings and the newly discovered rings of Uranus.The effects of the asteroid belt thickness on the gravitational potential coefficients and the orbital motions are demonstrated. Comparisons between the Mars orbital perturbations obtained using the analytic expressions and those obtained using numerical integration are discussed. The effects of the asteroid belt on the Earth based ranging to Mars are also demonstrated.     

Analytical investigations of quasi-circular frozen orbits in the Martian gravity field

Frozen orbits are always important foci of orbit design because of their valuable characteristics that their eccentricity and argument of pericentre remain constant on average. This study investigates quasi-circular frozen orbits and examines their basic nature analytically using two different methods. First, an analytical method based on Lagrangian formulations is applied to obtain constraint conditions for Martian frozen orbits. Second, Lie transforms are employed to locate these orbits accurately, and draw the contours of the Hamiltonian to show evolutions of the equilibria. Both methods are verified by numerical integrations in an 80 × 80 Mars gravity field. The simulations demonstrate that these two analytical methods can provide accurate enough results. By comparison, the two methods are found well consistent with each other, and both discover four families of Martian frozen orbits: three families with small eccentricities and one family near the critical inclination. The results also show some valuable conclusions: for the majority of Martian frozen orbits, argument of pericentre is kept at 270° because J ₃ has the same sign as J ₂; while for a minority of ones with low altitude and low inclination, argument of pericentre can be kept at 90° because of the effect of the higher degree odd zonals; for the critical inclination cases, argument of pericentre can also be kept at 90°. It is worthwhile to note that there exist some special frozen orbits with extremely small eccentricity, which could provide much convenience for reconnaissance. Finally, the stability of Martian frozen orbits is estimated based on the trace of the monodromy matrix. The analytical investigations can provide good initial conditions for numerical correction methods in the more complex models.     

Model orbits of asteroid 3040 Kozai

The orbital evolutions of the asteroid 3040 Kozai and model asteroids with similar orbits have been investigated. Their osculating orbits for an epoch 1991 December 10 were numerically integrated forward within the interval of 20,000 years, using a dynamical model of the solar system consisting of all inner planets, Jupiter, and Saturn.The orbit of the asteroid Kozai is stable. Its motion is affected only by long-period perturbations of planets. With change of the argument of perihelion of the asteroid Kozai, the evolution of the model asteroid orbits changes essentially, too. The model orbits with the argument of perihelion changed by the order of 10% show that asteroids with such orbital parameters may approach the Earth orbit, while asteroids with larger changes may even cross it, at least after 10,000 years. Long-term orbital evolution of asteroids with these orbital parameters is very sensitive on their angular elements.     

Multi-SunSynchronous Orbits in the Solar System

Performances of a planetary observation system are strongly related to the choice of the orbit used. Trajectories with characteristics of periodicity are very useful for the assessment of time-varying phenomena and thus Periodic SunSynchronous and Periodic Multi-SunSynchronous Orbits are particularly suitable to this end. In this paper, the research into these kinds of orbits, previously proposed for the Earth and Mars, has been extended to planets of the Solar System and to their principal moons. In general, these trajectories are typically obtained under the hypothesis that the J₂ harmonic is predominant with respect to the other orbital perturbations, since this allows an analytical solution. However, the hypothesis of J₂ predominant is not always verified in the Solar System and so analytical techniques must be replaced by numerical simulations. Interesting results have been obtained for the planets Mars and Jupiter and for the moons Europa, Callisto and Titan, where periodic trajectories with reduced revisit times and low altitudes have been found. These solutions allow the observation of time-varying phenomena with high spatial and temporal resolution.     

Sensitivity study of high eccentricity orbits for Mars gravity recovery

By linear perturbation theory, a sensitivity study is presented to calculate the contribution of the Mars gravity field to the orbital perturbations in velocity for spacecrafts in both low eccentricity Mars orbits and high eccentricity orbits(HEOs). In order to improve the solution of some low degree/order gravity coefficients, a method of choosing an appropriate semimajor axis is often used to calculate an expected orbital resonance, which will significantly amplify the magnitude of the position and velocity perturbations produced by certain gravity coefficients. We can then assess to what degree/order gravity coefficients can be recovered from the tracking data of the spacecraft. However, this existing method can only be applied to a low eccentricity orbit, and is not valid for an HEO. A new approach to choosing an appropriate semimajor axis is proposed here to analyze an orbital resonance. This approach can be applied to both low eccentricity orbits and HEOs. This small adjustment in the semimajor axis can improve the precision of gravity field coefficients and does not affect other scientific objectives.     

Frozen orbits at high eccentricity and inclination: application to Mercury orbiter

We hereby study the stability of a massless probe orbiting around an oblate central body (planet or planetary satellite) perturbed by a third body, assumed to lay in the equatorial plane (Sun or Jupiter for example) using a Hamiltonian formalism. We are able to determine, in the parameters space, the location of the frozen orbits, namely orbits whose orbital elements remain constant on average, to characterize their stability/unstability and to compute the periods of the equilibria. The proposed theory is general enough, to be applied to a wide range of probes around planet or natural planetary satellites. The BepiColombo mission is used to motivate our analysis and to provide specific numerical data to check our analytical results. Finally, we also bring to the light that the coefficient J ₂ is able to protect against the increasing of the eccentricity due to the Kozai-Lidov effect and the coefficient J ₃ determines a shift of the equilibria.     

Earth–Mars transfers with ballistic escape and low-thrust capture

In this paper novel Earth–Mars transfers are presented. These transfers exploit the natural dynamics of n-body models as well as the high specific impulse typical of low-thrust systems. The Moon-perturbed version of the Sun–Earth problem is introduced to design ballistic escape orbits performing lunar gravity assists. The ballistic capture is designed in the Sun–Mars system where special attainable sets are defined and used to handle the low-thrust control. The complete trajectory is optimized in the full n-body problem which takes into account planets’ orbital inclinations and eccentricities. Accurate, efficient solutions with reasonable flight times are presented and compared with known results.     

Five new and three improved mutual orbits of transneptunian binaries

We present three improved and five new mutual orbits of transneptunian binary systems (58534) Logos-Zoe, (66652) Borasisi-Pabu, (88611) Teharonhiawako-Sawiskera, (123509) 2000 WK₁₈₃, (149780) Altjira, 2001 QY₂₉₇, 2003 QW₁₁₁, and 2003 QY₉₀ based on Hubble Space Telescope and Keck II laser guide star adaptive optics observations. Combining the five new orbit solutions with 17 previously known orbits yields a sample of 22 mutual orbits for which the period P, semimajor axis a, and eccentricity e have been determined. These orbits have mutual periods ranging from 5 to over 800 days, semimajor axes ranging from 1600 to 37,000 km, eccentricities ranging from 0 to 0.8, and system masses ranging from 2 × 10¹⁷ to 2 × 10²² kg. Based on the relative brightnesses of primaries and secondaries, most of these systems consist of near equal-sized pairs, although a few of the most massive systems are more lopsided. The observed distribution of orbital properties suggests that the most loosely-bound transneptunian binary systems are only found on dynamically cold heliocentric orbits. Of the 22 known binary mutual orbits, orientation ambiguities are now resolved for 9, of which 7 are prograde and 2 are retrograde, consistent with a random distribution of orbital orientations, but not with models predicting a strong preference for retrograde orbits. To the extent that other perturbations are not dominant, the binary systems undergo Kozai oscillations of their eccentricities and inclinations with periods of the order of tens of thousands to millions of years, some with strikingly high amplitudes.     

g Dependent particle concentration due to sedimentation

Sedimentation of particles in a fluid has long been used to characterize particle size distribution. Stokes’ law is used to determine an unknown distribution of spherical particle sizes by measuring the time required for the particles to settle a known distance in a fluid of known viscosity and density. In this paper, we study the effects of gravity on sedimentation by examining the resulting particle concentration distributed in an equilibrium profile of concentration C _m,n above the bottom of a container. This is for an experiment on the surface of the Earth and therefore the acceleration of gravity had been corrected for the oblateness of the Earth and its rotation. Next, at the orbital altitude of the spacecraft in orbit around Earth the acceleration due to the central field is corrected for the oblateness of the Earth. Our results show that for experiments taking place in circular or elliptical orbits of various inclinations around the Earth the concentration ratio C _m,n /C _m,ave , the inclination seems to be the most ineffective in affecting the concentration among all the orbital elements. For orbital experiment that use particles of diameter d _p =0.001 μm the concentration ratios for circular and slightly elliptical orbits in the range e=0–0.1 exhibit a 0.009 % difference. The concentration ratio increases with the increase of eccentricity, which increases more for particles of larger diameters. Finally, for particles of the same diameter concentration ratios between Earth and Mars surface experiments are related in the following way .     

Frozen orbits for satellites close to an Earth-like planet

We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ ₂ throughJ ₉. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J ₂ only). In particular, we examine the manner in which the odd zonalJ ₃ breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J ₄+J ₂ ² =0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.     

Effect of 3rd-degree gravity harmonics and Earth perturbations on lunar artificial satellite orbits

In a previous work we studied the effects of (I) the J ₂ and C ₂₂ terms of the lunar potential and (II) the rotation of the primary on the critical inclination orbits of artificial satellites. Here, we show that, when 3rd-degree gravity harmonics are taken into account, the long-term orbital behavior and stability are strongly affected, especially for a non-rotating central body, where chaotic or collision orbits dominate the phase space. In the rotating case these phenomena are strongly weakened and the motion is mostly regular. When the averaged effect of the Earth’s perturbation is added, chaotic regions appear again for some inclination ranges. These are more important for higher values of semi-major axes. We compute the main families of periodic orbits, which are shown to emanate from the ‘frozen eccentricity’ and ‘critical inclination’ solutions of the axisymmetric problem (‘J ₂ + J ₃’). Although the geometrical properties of the orbits are not preserved, we find that the variations in e, I and g can be quite small, so that they can be of practical importance to mission planning.     

Elliptical multi-sun-synchronous orbits for Mars exploration

The multi-sun-synchronous orbits allow cycles of observation of the same area in which solar illumination repetitively changes according to the value of the orbit elements and returns to the initial condition after a temporal interval multiple of the repetition of observation. This paper generalizes the concept of multi-sun-synchronous orbits, whose classical sun-synchronous orbits represent particular solutions, taking into consideration the elliptical case. The feasibility of using this typology of orbits, referred to as elliptical periodic multi-sun-synchronous orbits, has been investigated for the exploration of Mars and particular solutions have been selected. Such solutions considerably reduce the manoeuvre of velocity variation at the end of the interplanetary transfer with respect to the case of a target circular orbit around Mars. They are based on the use of quasi-critical inclinations in order to minimize the apsidal line motion and thus reduce orbit maintenance costs. Moreover, in the case of high eccentricities, the argument of pericentre may be set in order to obtain, around the apocentre, a condition of quasi-synchronism with the planet (the footprint of the probe on the surface presents a small shift with respect to a fixed point on the Martian surface). The low altitude of pericentre allows observation of the planet at a higher spatial resolution, while the orbit arc around the apocentre may be used to observe Mars with a wide spatial coverage in quasi-stationary conditions. This latter characteristic is useful for analysing atmospheric and meteorological phenomena and it allows for most of the orbital period a link between a rover on the surface of Mars and a probe orbiting around the planet.     

Integrable approximation of J 2-perturbed relative orbits

Most existing satellite relative motion theories utilize mean elements, and therefore cannot be used for calculating long-term bounded perturbed relative orbits. The goal of the current paper is to find an integrable approximation for the relative motion problem under the J ₂ perturbation, which is adequate for long-term prediction of bounded relative orbits with arbitrary inclinations. To that end, a radial intermediary Hamiltonian is utilized. The intermediary Hamiltonian retains the original structure of the full J ₂ Hamiltonian, excluding the latitude dependence. This formalism provides integrability via separation, a fact that is utilized for finding periodic relative orbits in a local-vertical local-horizontal frame and determine an initialization scheme that yields long-term boundedness of the relative distance. Numerical experiments show that the intermediary-based computation of orbits provides long-term bounded orbits in the full J ₂ problem for various inclinations. In addition, a test case is shown in which the radial intermediary-based initial conditions of the chief and deputy satellites yield bounded relative distance in a high-precision orbit propagator.     

Three-dimensional time optimal double angular momentum reversal trajectory using solar sails

A new concept of three dimensional non-Keplerian trajectories with double angular momentum reversal is investigated with high performance solar sails. The main discussion of this paper is about such 3D solar inverse orbits with inner constraints. The problem is addressed in a time optimal control framework solved by an indirect method. Two typical solar inverse orbits have been achieved and presented in a 3D non-dimensional dynamic model in the Heliocentric Inertial Frame. Starting from the Earth orbit ecliptic plane, a sailcraft in the inverse orbit exhibits a butterfly shape trajectory. As such, the new orbits are symmetrical with respect to a plane which contains the Sun-perihelion line. The relation of the sail attitude angles between the two symmetrical parts of the orbits are used to reduce the simulation effort. The quasi-heliostationary property at its aphelia is demonstrated with variation of the orbital radius. Evolutions of the orbital velocity and optimal sail orientations are also outlined and discussed to benefit future design work. As is suited for space observation guaranteed by its butterfly shape, the inverse orbits are thoroughly studied in terms of the concerned parameters. The discussion of the parametric influence is ranked in order as perihelion distance r _E , required maximum position z _max, perihelion position z _f and the sail lightness number β. Suitable ranges of each parameter are adopted to illustrate the orbital variation trend. Through numerical simulations the features of such inverse orbits are further emphasized to provide an initial reference for future researchers.     

The Characteristics and Related Problems of the Orbits Around the Earth-Moon Libration Points

Due to the specific dynamics, the probes located at the halo orbits or Lissajous orbits around the Earth-Moon collinear libration point L₁ or L₂ are always studied in the synodic system to understand their trajectories. In fact, they are also orbiting the Earth in a distant Keplerian ellipse. Because of their intrinsic orbital instability, in the orbit prediction the initial errors propagate more prominently than those of the normal orbiting satellites, this requires special attention in the orbit design, maneuver, and control. Despite of all this, they are similar to the normal orbiting satellites in orbit determination and hardly require other special attentions. In this paper, the quantitative results of error propagation under the unstable dynamics, together with the theoretical analysis are presented. The results of precise orbit determination and short-arc orbit predictions are also shown, and compared with the results from the Beijing Aerospace Control Center.     

Approximate analytic method for high-apogee twelve-hour orbits of artificial Earth’s satellites

We propose an approach to the study of the evolution of high-apogee twelve-hour orbits of artificial Earth’s satellites. We describe parameters of the motion model used for the artificial Earth’s satellite such that the principal gravitational perturbations of the Moon and Sun, nonsphericity of the Earth, and perturbations from the light pressure force are approximately taken into account. To solve the system of averaged equations describing the evolution of the orbit parameters of an artificial satellite, we use both numeric and analytic methods. To select initial parameters of the twelve-hour orbit, we assume that the path of the satellite along the surface of the Earth is stable. Results obtained by the analytic method and by the numerical integration of the evolving system are compared. For intervals of several years, we obtain estimates of oscillation periods and amplitudes for orbital elements. To verify the results and estimate the precision of the method, we use the numerical integration of rigorous (not averaged) equations of motion of the artificial satellite: they take into account forces acting on the satellite substantially more completely and precisely. The described method can be applied not only to the investigation of orbit evolutions of artificial satellites of the Earth; it can be applied to the investigation of the orbit evolution for other planets of the Solar system provided that the corresponding research problem will arise in the future and the considered special class of resonance orbits of satellites will be used for that purpose.