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.     

Time-varying transformations for Hill–Clohessy–Wiltshire solutions in elliptic orbits

The relative motion of chief and deputy satellites in close proximity with orbits of arbitrary eccentricity can be approximated by linearized time-periodic equations of motion. The linear time-invariant Hill–Clohessy–Wiltshire equations are typically derived from these equations by assuming the chief satellite is in a circular orbit. Two Lyapunov–Floquet transformations and an integral-preserving transformation are here presented which relate the linearized time-varying equations of relative motion to the Hill–Clohessy–Wiltshire equations in a one-to-one manner through time-varying coordinate transformations. These transformations allow the Hill–Clohessy–Wiltshire equations to describe the linearized relative motion for elliptic chief satellites.     

Two problems about the motion of low-moon-orbit satellites

A detailed theoretical analysis on the orbital lifetime and frozen orbit of low-moon-orbit satellites (LMOS) is carried out, and their relationships with the orbital inclination, as well as some mutual relationships are presented. Taking account of the main perturbing sources of low-orbit satellites, we carried out numerical simulations under a comprehensive force model, and the results not only confirm the correctness of the theoretical analysis, but also provide some valuable insights on the orbital design of LMOS.     

Motion of Pasiphae’s perijove and node

We investigated the motion of the perijove and ascending node of the 8th satellite of Jupiter, Pasiphae. The main perturbations by the Sun on the satellite permitted to use an intermediate orbit obtained by approximated solutions of differential equations previously transformed by the Von Zeipel method. The orbit is a non-Keplerian ellipse. The secular motion of the ascending node, argument of perijove, and essential periodic perturbations were taken into account. Using our theory we showed that the inclination and eccentricity of Pasiphae can acquire values by which the orbit becomes a librating one; but, within Pasiphae’s observation period, the motion of its perijove is circulating. Taking into account the results of our previous works on Pasiphae motion, we can conclude that the mean motion of the ascending node is similar for different values of the satellite inclination and eccentricity. But the mean motion of the perijove strongly depends on the orbit inclination and eccentricity, according to the Lidov–Kozai mechanism.     

Satellite Relative Motion Using Differential Equinoctial Elements

For precise control, to minimize the fuel consumption, and to maximize the lifetime of satellite formations a precise analytic solution is needed for the relative motion of satellites. Based on the relationship between the relative states and the differential orbital elements, the state transition matrix for the linearized relative motion that includes the effects due to the reference orbit eccentricity and the gravitational perturbations is derived. This method is called the Geometric Method. To avoid any singularities at zero eccentricity and zero inclination, equinoctial variables are used to derive the relative motion state transition matrices for both mean and osculating elements. This approach can be extended easily to include other perturbing forces.     

Approximate analysis for relative motion of satellite formation flying in elliptical orbits

This paper studies the relative motion of satellite formation flying in arbitrary elliptical orbits with no perturbation. The trajectories of the leader and follower satellites are projected onto the celestial sphere. These two projections and celestial equator intersect each other to form a spherical triangle, in which the vertex angles and arc-distances are used to describe the relative motion equations. This method is entitled the reference orbital element approach. Here the dimensionless distance is defined as the ratio of the maximal distance between the leader and follower satellites to the semi-major axis of the leader satellite. In close formations, this dimensionless distance, as well as some vertex angles and arc-distances of this spherical triangle, and the orbital element differences are small quantities. A series of order-of-magnitude analyses about these quantities are conducted. Consequently, the relative motion equations are approximated by expansions truncated to the second order, i.e. square of the dimensionless distance. In order to study the problem of periodicity of relative motion, the semi-major axis of the follower is expanded as Taylor series around that of the leader, by regarding relative position and velocity as small quantities. Using this expansion, it is proved that the periodicity condition derived from Lawden’s equations is equivalent to the condition that the Taylor series of order one is zero. The first-order relative motion equations, simplified from the second-order ones, possess the same forms as the periodic solutions of Lawden’s equations. It is presented that the latter are further first-order approximations to the former; and moreover, compared with the latter more suitable to research spacecraft rendezvous and docking, the former are more suitable to research relative orbit configurations. The first-order relative motion equations are expanded as trigonometric series with eccentric anomaly as the angle variable. Except the terms of order one, the trigonometric series’ amplitudes are geometric series, and corresponding phases are constant both in the radial and in-track directions. When the trajectory of the in-plane relative motion is similar to an ellipse, a method to seek this ellipse is presented. The advantage of this method is shown by an example.     

月球卫星轨道力学综述

月球探测器的运动通常可分为3个阶段，这3个阶段分别对应3种不同类型的轨道：近地停泊轨道、向月飞行的过渡轨道与环月飞行的月球卫星轨道。近地停泊轨道实为一种地球卫星轨道；过渡轨道则涉及不同的过渡方式(大推力或小推力等)；环月飞行的月球卫星轨道则与地球卫星轨道有很多不同之处，它决不是地球卫星轨道的简单克隆。针对这一点，全面阐述月球卫星的轨道力学问题，特别是环月飞行中的一些热点问题，如轨道摄动解的构造、近月点高度的下降及其涉及的卫星轨道寿命、各种特殊卫星(如太阳同步卫星和冻结轨道卫星等)的轨道特征、月球卫星定轨等。     

关于星座小卫星的编队飞行问题

从轨道力学角度来看星座小卫星编队飞行和星星跟踪中的伴飞，遵循着如下动力学机制：(1)在各小卫星绕地球运动过程中轨道摄动变化的主要特征决定了星-星之间的空间构形，(2)当星星之间相互距离较近时，在退化的限制性三体问题(实为限制性二体问题)中，共线秤动点附近的条件周期运动亦可在一定时间内制约星-星之间的空间构形．将具体阐明这两种动力学机制的原理和相应的星星之间的相对构形，并用仿真计算来证实这两种动力学机制的适用范围，为星座小卫星编队飞行和在伴飞运动过程中进行轨控提供理论依据和具体的轨控条件．     

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.     

Orbital evolution of new distant Neptunian satellites and ω-librators in the satellite systems of Saturn and Jupiter

We used data on the recently discovered three outer Neptunian satellites to analyze the long-period evolution of their orbits. We estimated the ranges of eccentricities and inclinations as well as approximate circulation periods of the pericenter arguments and the longitudes of the ascending nodes. The results were mainly obtained by using two different versions of the averaged Hill problem. Plane sections of the phase space of satellite orbital elements are given. We discuss the peculiarity of the evolution of several satellite orbits related to the librational variation of the pericenter argument ω. The ω-librators of Saturn’s system were found to qualitatively differ from the libration orbit in the system of Jupiter.     

A “Spring–mass” model of tethered satellite systems: properties of planar periodic motions

This paper is devoted to the dynamics in a central gravity field of two point masses connected by a massless tether (the so called “spring–mass” model of tethered satellite systems). Only the motions with straight strained tether are studied, while the case of “slack” tether is not considered. It is assumed that the distance between the point masses is substantially smaller than the distance between the system’s center of mass and the field center. This assumption allows us to treat the motion of the center of mass as an unperturbed Keplerian one, so to focus our study on attitude dynamics. A particular attention is given to the family of planar periodic motions in which the center of mass moves on an elliptic orbit, and the point masses never leave the orbital plane. If the eccentricity tends to zero, the corresponding family admits as a limit case the relative equilibrium in which the tether is elongated along the line joining the center of mass with the field center. We study the bifurcations and the stability of these planar periodic motions with respect to in-plane and out-of-plane perturbations. Our results show that the stable motions take place if the eccentricity of the orbit is sufficiently small.     

J2 Invariant Relative Orbits for Spacecraft Formations

An analytic method is presented to establish J ₂ invariant relative orbits. Working with mean orbit elements, the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly are set equal between two neighboring orbits. By having both orbits drift at equal angular rates on the average, they will not separate over time due to the J₂ influence. Two first order conditions are established between the differences in momenta elements (semi-major axis, eccentricity and inclination angle) that guarantee that the drift rates of two neighboring orbits are equal on the average. Differences in the longitude of the ascending node, argument of perigee and initial mean anomaly can be set at will, as long as they are setup in mean element space. For near polar orbits, enforcing both momenta element constraints may result in impractically large relative orbits. It this case it is shown that dropping the equal ascending node rate requirement still avoids considerable relative orbit drift and provides substantial fuel savings.This revised version was published online in October 2005 with corrections to the Cover Date.     

Evaluation of disturbances from solar radiation in orbital elements of geosynchronous satellites based on harmonics

Using specialized codes for the search of periodic and linear components we show that direct solar radiation leads to short-period variations of all the orbital elements of geosynchronous satellites. The variation period of the semimajor axis a, orbit inclination i and the longitude of the ascending node Ω is 1 day. Eccentricity e, the argument of perigee ω and the mean anomaly M vary with a period of 0.5 days. Direct solar radiation also leads to long-period variations in e, ω and M with a period of 1 year. The elements a, i and Ω undergo variations only in the amplitude of diurnal variations with a period of 1 or 0.5 years. Secular variability (linear components) are not detected. To obtain the initial value array of the orbital elements we used the Lagrange equations of perturbed motion in the form of a Gaussian with their subsequent integration via a special method of harmonics: the values of the derived orbital elements, obtained from the Lagrange equations, were presented through the periodic functions that are easy to integrate.     

A method of astronomical autonomous orbit and attitude determinations for satellites

A method of realtime autonomous orbit determination for earth satellites using the extended Kalman filtering is proposed. The observed quantities are: the satellite-sun direction vector measured by a sun sensor, the satellite-earth and satellite-moon direction vectors measured by an ultraviolet sensor, and the geocentric distance measured by a radar altimeter. At the same time the satellite attitude to the earth is also determined. Results of our simulation of the autonomous orbit determination show that the precision of the orbit determinations is better than 200 m. The effects of the sampling period, orbital inclination, orbital eccentricity and orbital altitude on the precision of orbit determination are analyzed and compared, and certain principles helpful for improving the precision of orbit determination are suggested.     

《大气一号》气球卫星轨道倾角变化分析

引起《大气一号》两颗气球卫星（ＤＱ－１Ａ和ＤＱ－１Ｂ）轨道倾角变化的摄动因素主要是太阳光压摄动、大气旋转和日月引力摄动。太阳光压摄动引起气球卫星轨道倾角增大，平均每天变化约０．００１７，大气旋转引起轨道倾角减小，平均每天变化不到０．０００１，但随着高度下降，变化量亦增大，陨落前达０．００２。本文根据卫星轨道摄动理论，给出气球卫星轨道倾角变化的一种定量分析方法，得到的分析结果为：（１）由太阳光压摄动     

About Tidal Evolution of Quasi-Periodic Orbits of Satellites

Tidal interactions between Planet and its satellites are known to be the main phenomena, which are determining the orbital evolution of the satellites. The modern ansatz in the theory of tidal dissipation in Saturn was developed previously by the international team of scientists from various countries in the field of celestial mechanics. Our applying to the theory of tidal dissipation concerns the investigating of the system of ODE-equations (ordinary differential equations) that govern the orbital evolution of the satellites; such an extremely non-linear system of 2 ordinary differential equations describes the mutual internal dynamics for the eccentricity of the orbit along with involving the semi-major axis of the proper satellite into such a monstrous equations. In our derivation, we have presented the elegant analytical solutions to the system above; so, the motivation of our ansatz is to transform the previously presented system of equations to the convenient form, in which the minimum of numerical calculations are required to obtain the final solutions. Preferably, it should be the analytical solutions; we have presented the solution as a set of quasi-periodic cycles via re-inversing of the proper ultra-elliptical integral. It means a quasi-periodic character of the evolution of the eccentricity, of the semi-major axis for the satellite orbit as well as of the quasi-periodic character of the tidal dissipation in the Planet.     

Tidal effects on the LAGEOS–LARES satellites and the LARASE program

The effects of solid and ocean tides have been computed on the right ascension of the ascending node of the two LAGEOS and LARES satellites and on the argument of pericenter of LAGEOS II. Their effects—together with the possible mis-modeling related to systematic errors in the estimate of the tidal coefficients, especially in the case of ocean tides—are quite important to be well established for the key role of the LAGEOS satellites, as well as of the newly LARES, in space geodesy and geophysics as well as in fundamental physics measurements. In the case of the measurement of the Lense–Thirring effect, the mis-modeling of long-period tides may mimic a secular effect on the cited orbital elements, thus producing a degradation in the measurement of the relativistic precession. A suitable combination of the orbital elements of the three satellites can help in avoiding the effects of the long-period tides of degree \(\ell =2\) (as for the Lunar solid tides with periods of 18.6 and 9.3 years) and \(\ell =4\), but other long-period tides, as the ocean \(K_1\) tide, which has the same periodicities of the right ascension of the ascending node \(\varOmega \) of the satellites, may strongly influence the measurement, especially if it is performed over a relatively short time span. These results are particularly important in the case of LARES, since they are new and because of the role that the orbit of LARES, and especially of its ascending node right ascension, will have in a new measurement of the Lense–Thirring effect by the joint analysis of its orbit with that of the two LAGEOS.     

The excitation of planetary orbits by stellar jet variability and polarity reversal

Planets form in active protoplanetary disks that sustain stellar jets. Momentum loss from the jet system may excite the planets’ orbital eccentricity and inclination (Namouni in Astron. J. 130:280, 2005). Evaluating quantitatively the effects of such excitation requires a realistic modeling of the momentum loss profiles associated with stellar jets. In this work, we model linear momentum loss as a time-variable stochastic process that results in a zero mean stellar acceleration. Momentum loss may involve periodic or random polarity reversals. We characterize orbital excitation as a function of the variability timescale and identify a novel excitation resonance between a planet’s orbital period and the jet’s variability timescale where the former equals twice the latter. For constant variability timescales, resonance is efficient for both periodic and random polarity reversals, the latter being stronger than the former. For a time variable variability timescale, resonance crossing is a more efficient excitation mechanism when polarity reversals are periodic. Each polarity reversal type has distinct features that may help constrain the magnetic history of the star through the observation of its planetary companions. For instance, outward planet migration to large distances from parent stars is one of the natural outcomes of periodic polarity reversal excitation if resonance crossing is sufficiently slow. Applying the excitation mechanism to the solar system, we find that the planet-jet variability resonance with periodic polarity reversal momentum loss is a possible origin for the hitherto unexplained inclination of Jupiter’s orbit by 6^° with respect to the Sun’s equator.     

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.     

The Orbital Evolution of Two Sounding Satellites and Analysis of the Accuracy of Orbit Determination

The satellites TC-1 and TC-2 are the two Chinese satellites with great elliptical orbits which are still in orbit around the earth at present. Since the launch the orbits of the two satellites have continuously evolved, which has a certain effect on the orbit determination and prediction precision. The regularities of the orbital evolution of the two sounding satellites are qualitatively and quantitatively analyzed. Under the current tracking mode the corresponding prediction precision of orbit determination is analyzed based on the different stages of the orbital evolution, thereby providing the basis for the adjustment of planning mode by the satellite application departments and the guarantee of normal satellite payload. Finally, the orbital lifetimes of the two satellites are predicted through the trend of the orbital evolution.