Rotational evolution of planetary systems under the action of gravitational and magnetic perturbations

Dynamics of planets around other stars that demonstrate a variety of possible characteristics is of interest from the point of view of realization of new scenarios of evolution which have not been realized in the Solar System. We consider the rotational evolution of exoplanets under the action of gravitational perturbations and magnetic disturbances using the methods of quality analysis and theory of bifurcation of multiparametric differential equations that describe evolution of non-resonant rotation of a dynamically symmetric planet magnetized along its symmetry axis. We analyze 64 phase portraits describing the evolution of angular momentum vector L for all possible values of planet parameters. The values of parameters are determined for the case when the direct rotation of a planet is changed for its retrograde rotation.     

Translational–Rotational Motion of a Viscoelastic Sphere in Gravitational Field of an Attracting Center and a Satellite

We study the translational–rotational motion of a planet modeled by a viscoelastic sphere in the gravitational fields of an immovable attracting center and a satellite modeled as material points. The satellite and the planet move with respect to their common center of mass that, in turn, moves with respect to the attracting center. The exact system of equations of motion of the considered mechanical system is deduced from the D'Alembert–Lagrange variational principle. The method of separation of motions is applied to the obtained system of equations and an approximate system of ordinary differential equations is deduced which describes the translational–rotational motion of the planet and its satellite, taking into account the perturbations caused by elasticity and dissipation. An analysis of the deformed state of the viscoelastic planet under the action of gravitational forces and forces of inertia is carried out. It is demonstrated that in the steady-state motion, when energy dissipation vanishes, the planet's center of mass and the satellite move along circular orbits with respect to the attracting center, being located on a single line with it. The viscoelastic planet in its steady-state motion is immovable in the orbital frame of reference. It is demonstrated that this steady-state motion is unstable.     

Evolution of Motion of a Binary Planet

A model of a binary planet, consisting of a material point of small mass and a deformable viscoelastic sphere, is suggested. The center of mass of the binary planet moves in the gravitational field of a central body in the plane, which contains planets forming the binary planet. A deformable spherical planet rotates around the axis orthogonal to the plane of planetary motion. Planet deformations are described by the linear theory of viscoelasticity. It is shown that with an appropriate approximation of the gravitational potential, there is a class of quasicircular orbits, when the eccentricities of an orbit of the center of mass of a binary planet and an orbit, describing mutual planet motion, are equal to zero. The further evolution of motion is investigated in this class of orbits with the use of the canonical Poincare–Andoyer variables. Corresponding averaged equations are found, and phase pictures are constructed; the stability of stationary solutions is investigated on the basis of equations in variations. For the Solar system planets with their satellites, forming binary planets, the application of the presented model allows us to conclude that satellites sooner or later will fall on the corresponding planets.     

Evolution of motion of two viscoelastic planets in the field of forces of their mutual attraction

Translational-rotational motion of two viscoelastic planets in a gravitational force field is studied. The planets are modeled by homogeneous isotropic viscoelastic bodies. In their natural undeformed state each of the planets represents a sphere. We investigate a specific case when the planet’s centers of mass move in a fixed plane, the axis of rotation for each planet being directed along the normal to this plane. An equation describing the evolution of a slow angular variable (perihelion longitude) is derived. The observed displacement of the perihelion of Mercury is compared with the results obtained in the considered model problem about motion of two viscoelastic planets. Quite important is the fact that the planet of smaller mass (Mercury) moves not in a central Newtonian field of forces, but rather in the gravitational field of a rotating viscoelastic planet (Sun).     

Evolution of almost circular orbits of satellites under the action of noncentral gravitational field of the Earth and lunisolar perturbations

Based on the results of paper [1] by G.V. Mozhaev, joint perturbations produced by nonsphericity of the Earth and by attraction of the Moon and the Sun are investigated using the method of averaging. Arbitrary number of spherical harmonics was taken into account in the force function of the Earth’s gravitational filed, and only the principal term was retained in the perturbing function of the Sun. In the perturbing function of the Moon two parallactic terms were considered in addition to the dominant term. The flight altitude was chosen in such a way that perturbations produced by the Sun and Moon would have the second order of smallness relative to the polar oblateness of the Earth. As a result, the formulas for calculation of satellite coordinates are derived that give a high precision on long time intervals.     

基于受摄轨道模型的小卫星轨道摄动分析研究

小卫星在实际运行中受到多种摄动力的作用,这会对其轨道造成不同程度的影响,因此在小卫星的轨道设计与控制中,摄动是必须考虑的重要因素。本文以500km高的小卫星太阳同步轨道为研究对象,运用轨道摄动的基本理论,估计了地球非球形引力、大气阻力、太阳光压及第三体引力的量级并进行比较;建立小卫星轨道摄动分析模型,并在此模型的基础上,利用计算机仿真技术,对小卫星轨道摄动问题展开仿真研究,验证了几种主要摄动力的量级估计的结果,分析比较了几种主要摄动力对小卫星运行轨道的影响程度及规律,本文结果对小卫星轨道设计与控制具有很好的参考价值。     

Electrostatically inflated gossamer space structure voltage requirements due to orbital perturbations

This paper explores the concept of using electrostatic forces for deployment of gossamer space structures. The Electrostatically Inflated Membrane Structure (EIMS) uses two conducting membranes that are interconnected through membrane ribs. An absolute electrostatic charge is applied to the structure through active charge emission. This causes repulsion between layers of lightweight membranes that inflates the EIMS system and tensions the membranes. Assuming positive tensions, the EIMS system is modeled as a rigid system. Typical orbital perturbations are considered such as solar radiation pressure, differential gravity, and atmospheric drag which may compress the structure leading to shape destabilization. Restricting the analysis in this paper to flat membranes, the minimum potentials required to exactly compensate for the worst case scenario of differential solar radiation pressure at geostationary altitudes are estimated to be on the order of hundreds of volts. In low Earth orbit, voltage magnitudes of several kilovolts are required to reach an inflation pressure to offset the normal compressive drag pressure.     

Motion of a Planet with a Complex Structure

The motion of a planet consisting of a mantle and a core (solid bodies) connected by a viscoelastic layer and interacting with each other and an external point mass by the law of gravitation is considered. The mutual motions of the core and mantle are investigated assuming that the centers of mass of the planet and external point mass moves along undisturbed Keplerian orbits around the common center of mass of the system. The planetary core and mantle have an axial symmetry and different principal moments of inertia, which leads to a displacement of the center of mantle relative to the center of core and to their mutual rotations. The results obtained on the basis of averaged equations are illustrated by the example of the Earth–Moon system.     

Imaging of extrasolar advanced terrestrial planets

The gravitational lens effect of the Sun would allow, by using a detector at one of its foci, to obtain a “telescope” with gigantic amplification and resolution powers opening extraordinary perspectives for the detailed study of extrasolar planets, particularly technologically advanced ones. But, astronautical challenges are raised by the necessity to align precisely and put in an efficient tracking and scanning mode the detector, necessarily modest in size compared to the dimensions of the planet images and ranges of orbital and rotational motions. In the frame of the FOCAL space mission submitted to ESA, we present the dynamical geometry of the images for two typical cases of observational wavelengths: 10 centimeters (radio) and 10 micrometers (infrared), for a solar-type stellar system 10 parsecs away. Plasma thrusters could provide interesting solutions for the control of the detector for tracking and scanning the focal images.     

一种基于图像的星体表面撞击坑自动提取方法

以星体表面的图像数据为基础,根据撞击坑的向阳面呈亮色调而背阳面呈暗色调的原理,提出了一种基于图像的星体表面撞击坑自动提取方法。该方法先用自适应双阈值分割法对星体表面的光学图像进行分割处理,获取图像中明暗区域的形状和位置信息;再用统计学原理分析明暗区域组成结构、像平面上光照方向,结合相关的约束条件来匹配同一撞击坑的明暗区域,同时拟合出撞击坑的外边缘并确定其半径、位置等信息。实验结果表明,该方法能够从常见的星体表面光学图像中快速、可靠地提取出撞击坑的中心位置和半径大小,具有较宽的普适应性和一定的应用价值。     

Motion of three viscoelastic planets in the field of their mutual attraction forces

Translational-rotational motion of three planets modeled by viscoelastic balls in the gravitational field of mutual attraction is studied in this paper. The system of equations of motion for the mechanical system under consideration is deduced from the d’Alembert-Lagrange variational principle. Using the method of separation of motions, an approximate system of ordinary differential equations, describing the translational-rotational motion of the planets, is obtained with taking into account perturbations caused by elasticity and dissipation. The found steady-state motion of the system is an analog to triangular libration points in the classical three-body problem.     

Quasioptimal control with feedback for multiorbit low-thrust transfer between noncoplanar elliptic and circular orbits

The problem of synthesizing stable feedback control is considered based on solving the problem of time minimization for a multiorbit transfer between noncoplanar elliptic and circular orbits in a Newtonian gravitational field. The problem is solved using asymptotic properties and symmetries of optimal control in the unperturbed problem. Stability of the obtained control against external perturbations, deviations of initial conditions, and errors in thrust vector realizations is demonstrated. The obtained quasioptimal control with feedback can be used as an onboard algorithm of spacecraft control and when performing design and ballistic analysis.     

Investigation of the Time of Ballistic Existence for Elliptic Orbits Evolving under the Influence of Gravitational Perturbations of External Bodies

A comparative analysis of the time of ballistic existence for different elliptic orbits evolving under the influence of gravitational perturbations of external bodies is performed. For this purpose, the solution of a satellite variant of the restricted circular double-averaged three-body problem obtained by M.L. Lidov [1] is used. The use of a geometrical interpretation of Lidov's solutions has allowed us to create a visualized tool for a choice of evolving satellite orbits with a long time of ballistic existence.     

卫星编队飞行相对轨道的摄动研究综述

卫星在编队飞行过程中总受到各种摄动力的作用，由于轨道摄动直接影响卫星的寿命，研究卫星编队飞行相对轨道的摄动问题具有重要的意义。本文综述了国内外有关编队飞行卫星相对轨道摄动问题的研究方法以及进展，最后针对目前相对轨道摄动的研究现状，指出了进一步的研究方向。     

地球轨道卫星电推进变轨控制方法

针对地球同步轨道（GEO）卫星,采用电推进系统完成转移轨道变轨。采用基于Lyapunov函数的反馈控制方法确定时间最短变轨策略。首先在开普勒模型下研究变轨过程,然后在开普勒模型的基础上考虑地球J2项摄动和地球阴影,最后在全引力模型下研究变轨过程,即在开普勒模型的基础上考虑地球非球形引力摄动、日月第三体引力摄动、太阳光压摄动和地球阴影。仿真结果显示在变轨过程中摄动项不可忽略,除地球J2项摄动外还应该考虑日月第三体引力摄动和太阳光压摄动。对比上述三组仿真结果,发现考虑摄动后轨道转移时间的增加比燃料消耗的增加更为明显。数值仿真结果表明本文研究对未来的全电推进任务具有良好的通用性和应用参考价值。     

On the features of long-term evolution of a high-apogee orbit of the SPECTR-R spacecraft

The practical tasks related to qualitative investigation of long-term evolution of high-apogee orbits of artificial Earth satellites (AES), for which the main perturbing factors are gravitational perturbations from the Moon and the Sun, are considered. Attention is given to the problem of the ballistic lifetime of similar orbits, and the issues associated with possibilities of the correction of orbits for ensuring the required duration of their ballistic lifetime are considered. The orbit of the SPECTR-R spacecraft launched in July of 2011 is considered as an example.     

The construction of gravitational field models on the basis of satellite measurements of gravitational potential derivatives

In [1] expressions were constructed for the derivatives of all the orders of a planet’s gravitational potential with respect to the rectangular coordinates related to the gravity center of a planet. These expressions are series of spherical functions. The coefficients of the series of first-order derivatives depend on two Stokes constants, whereas the coefficients of next-order derivatives are linear combinations of the coefficients of preceding-order derivatives. In the present paper the derived expressions for the first and second potential derivatives are transformed into the form that is most convenient for solving the inverse problem, i.e., evaluating Stokes constants from satellite measurements of these derivatives. Each term of the new series for a derivative depends on a sum of two Stokes constants multiplied by linear combinations of several spherical functions. The new form of the expansions for the potential derivatives makes it possible to calculate Stokes constants by simultaneously applying satellite data either for all three first-order potential derivatives, or for all six second-order derivatives. The constructed series may be applied for modeling the Earth’s gravitational field from the satellite data obtained in the international CHAMP, GRACE, and GOCE missions.     

Optimal control of reorientation of a spacecraft using free trajectory method

The problem of optimal control over spatial reorientation of a spacecraft is considered. The functional having a sense of propellant consumption is minimized. The analytical solution to the formulated problem is presented. It is shown that the optimal solution can be found in the class of two-impulse control at which the spacecraft’s turn is performed along a free motion trajectory. In order to improve the accuracy of spacecraft guidance into a specified angular position, methods of control are suggested that realize the method of free trajectories. The synthesized controls are invariant with respect to both external perturbations and parametric errors. The results of mathematical modeling are presented that demonstrate high efficiency of developed control algorithms. Propellant consumption for realizing a programmed turn is numerically estimated taking into account considerable gravitational and aerodynamic moments acting upon the spacecraft.     

Long-term evolution of Galileo operational orbits by canonical perturbation theory

Galileo operational orbits are slightly affected by the 3 to 5 tesseral resonance, an effect that can be much more important in the case of disposal orbits. Proceeding by canonical perturbation theory we show that the part of the long-term Hamiltonian corresponding to the non-centralities of the Earth's gravitational potential can be replaced by an intermediary that shows the pendulum dynamics of the 3 to 5 tesseral resonance problem. Inclusion of lunisolar perturbations requires a semi-analytical integration, which is compared with the corresponding results from the well-established Draper Semi-analytical Satellite Theory.