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.     

Optimization of passive orbit with the use of gravity maneuver

A high-precision method of calculating gravitational interactions is applied in order to determine optimal trajectories. A number of problems, necessary for determination of optimal parameters at a launch of a spacecraft and during its flyby near celestial bodies, are considered. The spacecraft trajectory was determined by numerical integration of the equations of passive motion of the spacecraft and of the equations of motion for planets, the Sun, and the Moon. The optimal trajectory of the spacecraft approaching the Sun is determined by fitting its initial conditions.     

Construction of theories of motion, ephemerides, and databases for natural satellites of planets

A knowledge base for natural satellites of planets is created. On the basis of observations, new numerical models of motion are constructed for all 96 outer satellites of Jupiter, Saturn, Uranus, and Neptune. A special database is compiled from all available observational data for natural satellites of planets, as well as a bibliographical database and information system of physical and orbital parameters of planets and satellites. The tools for calculations of ephemerides of all natural satellites (except for the Moon) of planets are developed. They represent the final result of studies and actually incorporate the entire knowledge about dynamics of the satellites of planets. Also developed are special ephemerides in order to observe singular phenomena in the apparent motion of the satellites of planets. A computer environment and the Internet allowed us to concentrate all above-listed options in a single toolkit easily available to any user in the world.     

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).     

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.     

Tides and the evolution of planetary habitability

Tides raised on a planet by the gravity of its host star can reduce the planet's orbital semi-major axis and eccentricity. This effect is only relevant for planets orbiting very close to their host stars. The habitable zones of low-mass stars are also close in, and tides can alter the orbits of planets in these locations. We calculate the tidal evolution of hypothetical terrestrial planets around low-mass stars and show that tides can evolve planets past the inner edge of the habitable zone, sometimes in less than 1 billion years. This migration requires large eccentricities (>0.5) and low-mass stars ( less or similar to 0.35 M(circle)). Such migration may have important implications for the evolution of the atmosphere, internal heating, and the Gaia hypothesis. Similarly, a planet that is detected interior to the habitable zone could have been habitable in the past. We consider the past habitability of the recently discovered, approximately 5 M(circle) planet, Gliese 581 c. We find that it could have been habitable for reasonable choices of orbital and physical properties as recently as 2 Gyr ago. However, when constraints derived from the additional companions are included, most parameter choices that indicate past habitability require the two inner planets of the system to have crossed their mutual 3:1 mean motion resonance. As this crossing would likely have resulted in resonance capture, which is not observed, we conclude that Gl 581 c was probably never habitable.     

Trojans in habitable zones

With the aid of numerical experiments we examined the dynamical stability of fictitious terrestrial planets in 1:1 mean motion resonance with Jovian-like planets of extrasolar planetary systems. In our stability study of the so-called "Trojan" planets in the habitable zone, we used the restricted three-body problem with different mass ratios of the primary bodies. The application of the three-body problem showed that even massive Trojan planets can be stable in the 1:1 mean motion resonance. From the 117 extrasolar planetary systems only 11 systems were found with one giant planet in the habitable zone. Out of this sample set we chose four planetary systems--HD17051, HD27442, HD28185, and HD108874--for further investigation. To study the orbital behavior of the stable zone in the different systems, we used direct numerical computations (Lie Integration Method) that allowed us to determine the escape times and the maximum eccentricity of the fictitious "Trojan planets."     

含板类柔性附件的充液航天器动力学研究

对含有板类柔性附件和曲壁轴对称充液储腔的复杂航天器系统进行动力学建模和耦合机理研究。首先,采用Kirchhoff-Love薄板理论对航天器的板类柔性附件进行研究,通过D’Alembert原理得到柔性附件的振动方程,运用模态假设法将混合方程转换为常微分方程。其次,通过推导充液航天器储腔内任意点的运动,得到储腔液体的牵连速度势函数,采用Gauss超几何级数得到液体相对速度势函数的解析形式,通过Hamilton变分原理推导液体晃动的运动方程,以及液体速度势函数模态系数的控制方程。最后采用准坐标Lagrange方程得到耦合航天器系统的状态方程,通过数值仿真校验系统动力学模型的有效性。研究结果表明,刚性平台、液体、柔性附件的相互耦合效应使得航天器系统存在复杂动力学行为,在复杂航天器系统动力学建模过程中需要充分考虑液体晃动和柔性附件振动的影响,柔性附件的安装位置对于耦合航天器系统的动力学行为也有着重要影响。     

Rotational motion of <Emphasis Type="Italic">Foton M-4</Emphasis>

The actual controlled rotational motion of the Foton M-4 satellite is reconstructed for the mode of single-axis solar orientation. The reconstruction was carried out using data of onboard measurements of vectors of angular velocity and the strength of the Earth’s magnetic field. The reconstruction method is based on the reconstruction of the kinematic equations of the rotational motion of a solid body. According to the method, measurement data of both types collected at a certain time interval are processed together. Measurements of the angular velocity are interpolated by piecewise-linear functions, which are substituted in kinematic differential equations for a quaternion that defines the transition from the satellite instrument coordinate system to the inertial coordinate system. The obtained equations represent the kinematic model of the satellite rotational motion. A solution of these equations that approximates the actual motion is derived from the condition of the best (in the sense of the least squares method) match between the measurement data of the strength vector of the Earth’s magnetic field and its calculated values. The described method makes it possible to reconstruct the actual rotational satellite motion using one solution of kinematic equations over time intervals longer than 10 h. The found reconstructions have been used to calculate the residual microaccelerations.     

Estimating the accuracy of the technique of reconstructing the rotational motion of a satellite based on the measurements of its angular velocity and the magnetic field of the Earth

The paper has studied the accuracy of the technique that allows the rotational motion of the Earth artificial satellites (AES) to be reconstructed based on the data of onboard measurements of angular velocity vectors and the strength of the Earth magnetic field (EMF). The technique is based on kinematic equations of the rotational motion of a rigid body. Both types of measurement data collected over some time interval have been processed jointly. The angular velocity measurements have been approximated using convenient formulas, which are substituted into the kinematic differential equations for the quaternion that specifies the transition from the body-fixed coordinate system of a satellite to the inertial coordinate system. Thus obtained equations represent a kinematic model of the rotational motion of a satellite. The solution of these equations, which approximate real motion, has been found by the least-square method from the condition of best fitting between the data of measurements of the EMF strength vector and its calculated values. The accuracy of the technique has been estimated by processing the data obtained from the board of the service module of the International Space Station (ISS). The reconstruction of station motion using the aforementioned technique has been compared with the telemetry data on the actual motion of the station. The technique has allowed us to reconstruct the station motion in the orbital orientation mode with a maximum error less than 0.6° and the turns with a maximal error of less than 1.2°.     

旋转充液系统全飞行过程非线性动力学仿真

本文考虑一类刚认耦合的旋转充液系统的飞行动力学问题，在依据力学变分原理给出基本方程，导出全飞行过程运动状态方程的基础上，建立了相应的仿真模型。可能通过对Navier-Stokes方程求解和液体流动的惯性波振动特性，给出液体的反作用力矩。经过仿真实验，分析角运动的时间序列，给出了比较详尽的飞行不稳定机制分析结果。     

GPS/运动方程组合导航

提出了一种不增加硬件设备而仅用软件（运动方程）－－全球卫星定位系统／运动方程（GPS／EM）组合导航方案，分析了其适用范围和特点。根据传统运动方程建模中的各种误差源，了包含误差源在内的飞行器扰动方程，并得出其解析解。建立了GPS／EM组合系统状态方程和观测方程，并进行仿真，验证了其可行性。结果表明，GPS／EM组合系统导航精度高于GPS，而且较单独GPS增加了信息冗余度。     

Stochastic Correlation Models for the Motion of the Pole of the Deformable Earth

In this paper we develop an unconventional approach to investigations of possible stochastic phenomena in the rotational motions of celestial bodies. The approach is based on special equations of the rotational motion, which represent a three-dimensional nonlinear system of differential equations with random parameters.     

On the possibility of Earth-type habitable planets in the 55 Cancri system

We discuss the possibility of Earth-type planets in the planetary system of 55 Cancri, a nearby G8 V star, which is host to two, possibly three, giant planets. We argue that Earth-type planets around 55 Cancri are in principle possible. Several conditions are necessary. First, Earth-type planets must have formed despite the existence of the close-in giant planet(s). In addition, they must be orbitally stable in the region of habitability considering that the stellar habitable zone is relatively close to the star compared to the Sun because of 55 Cancri's low luminosity and may therefore be affected by the close-in giant planet(s). We estimate the likelihood of Earth-type planets around 55 Cancri based on the integrated system approach previously considered, which provides a way of assessing the long-term possibility of photosynthetic biomass production under geodynamic conditions.     

The use of refined model of aerodynamic moment in the problem of reconstruction of the <Emphasis Type="Italic">Foton</Emphasis> satellite rotational motion

A new mathematical model of the uncontrolled rotational motion of the Foton satellite is presented. The model is based on the Euler dynamic equations of rigid body motion and takes into account the action upon the satellite of four external mechanical moments: gravitational, restoring aerodynamic, moment with constant components in the satellite-fixed coordinate system, and moment arising due to interaction of the Earth’s magnetic field with the satellite’s proper magnetic moment. To calculate the aerodynamic moment a special geometrical model of the outer satellite shell is used. Detailed form of the formulas giving above-mentioned moments in the equations of satellite motion is agreed with the form of the considered motion. Model testing is performed by determining with its help the rotational motion of the Foton M-2 satellite (it was in orbit from May 31, 2005 to June 16, 2005) using the data of the onboard measurements of the Earth’s magnetic field strength. The use of the new model has led to a relatively small improvement in the accuracy of the motion determination, but allowed us to obtain physically real estimates of some parameters.     

Determination of the Attitude Motion of a Satellite Based on a Kinematical Model of Motion and Using Data of Measurements of Its Angular Velocity and the Earth’s Magnetic Field Strength

An integral statistical procedure of determination of the attitude motion of a satellite using the data of onboard measurements of angular velocity vectors and the strength of the Earth’s magnetic field (EMF) is suggested. The procedure uses only the equations of kinematics of a solid body and is applicable to determining both controlled and uncontrollable motions of a satellite at any external mechanical moments acting upon it. When applying this procedure, the data of measurements of both types, accumulated during a certain interval of time, are processed jointly. The data of measuring the angular velocity are smoothed by discrete Fourier series, and these series are substituted into kinematical Poisson equations for elements of the matrix of transition from a satellite-fixed coordinate system to the orbital coordinate system. The equations thus obtained represent a kinematical model of the satellite motion. The solution to these equations (which approximate the actual motion of a satellite) is found from the condition of the best (in the sense of the least squares method) fit of the data of measuring the EMF strength vector to its calculated values. The results of testing the suggested procedure using the data of measurements of the angular velocity vectors onboard the Foton-12 satellite and measurements of EMF strengths are presented.__________Translated from Kosmicheskie Issledovaniya, Vol. 43, No. 4, 2005, pp. 295–305.Original Russian Text Copyright © 2005 by Abrashkin, Volkov, Voronov, Egorov, Kazakova, Pankratov, Sazonov, Semkin.     

Pervasive orbital eccentricities dictate the habitability of extrasolar earths

The long-term habitability of Earth-like planets requires low orbital eccentricities. A secular perturbation from a distant stellar companion is a very important mechanism in exciting planetary eccentricities, as many of the extrasolar planetary systems are associated with stellar companions. Although the orbital evolution of an Earth-like planet in a stellar binary system is well understood, the effect of a binary perturbation on a more realistic system containing additional gas-giant planets has been very little studied. Here, we provide analytic criteria confirmed by a large ensemble of numerical integrations that identify the initial orbital parameters leading to eccentric orbits. We show that an extrasolar earth is likely to experience a broad range of orbital evolution dictated by the location of a gas-giant planet, which necessitates more focused studies on the effect of eccentricity on the potential for life.     

随动推力作用下柔性旋转飞行器稳定性分析

针对柔性旋转飞行器动力学问题,考虑了飞行器转速的作用,建立了计入陀螺力矩及随动推力影响的运动方程并分析了系统的动力稳定性问题。将柔性旋转飞行器简化为带有附加质量的非均匀转子结构,考虑陀螺力矩及随动推力的影响,采用剪切变形对轴向位移有影响的Timoshenko梁模型,基于有限元方法建立了运动方程,分析转速、剪切刚度及附加质量等因素对系统稳定性的影响,发现了转速作用下旋转飞行器刚体和弹性体模态耦合的新现象。结果表明：剪切刚度较小时,剪切变形对临界推力有较大影响;附加质量的大小及位置对临界推力和不稳定域有重要的影响;转速一般会诱发非均匀转子系统的弯曲模态与刚体模态合并为一个耦合模态,致使系统产生动态失稳。     

Optimal control of a spinning double-pyramid Earth-pointing tethered formation

The dynamics and control of a tethered satellite formation for Earth-pointing observation missions is considered. For most practical applications in Earth orbit, a tether formation must be spinning in order to maintain tension in the tethers. It is possible to obtain periodic spinning solutions for a triangular formation whose initial conditions are close to the orbit normal. However, these solutions contain significant deviations of the satellites on a sphere relative to the desired Earth-pointing configuration. To maintain a plane of satellites spinning normal to the orbit plane, it is necessary to utilize “anchors”. Such a configuration resembles a double-pyramid. In this paper, control of a double-pyramid tethered formation is studied. The equations of motion are derived in a floating orbital coordinate system for the general case of an elliptic reference orbit. The motion of the satellites is derived assuming inelastic tethers that can vary in length in a controlled manner. Cartesian coordinates in a rotating reference frame attached to the desired spin frame provide a simple means of expressing the equations of motion, together with a set of constraint equations for the tether tensions. Periodic optimal control theory is applied to the system to determine sets of controlled periodic trajectories by varying the lengths of all interconnecting tethers (nine in total), as well as retrieval and simple reconfiguration trajectories. A modal analysis of the system is also performed using a lumped mass representation of the tethers.     

Motion of deformable planets and stability of their stationary rotations

A mechanical system consisting from N deformable spheres interacting according to the law of gravity is considered as a model of planetary system. Deformations of the viscoelastic spheres are described according to the model of the theory of elasticity of small deformations, the Kelvin-Voigt model of viscous forces, and occur under the action of gravitational fields and fields of centrifugal forces. Approximate equations describing motions of the centers of mass of the spheres and their rotations relative to the centers of mass are constructed by the method of separation of motions on the basis of solving quasistatic problems of the theory of viscoelasticity with allowance made for smallness of sphere deformations. Using the first integral of conservation of the angular momentum of the system relative to its center of mass, the expression for the changed potential energy is obtained with the use of the Routh method. An investigation of stationary rotations is carried out, and it is shown that all of them are unstable, if the number of planets is more than two.