Attitude dynamics of a rigid body on a Keplerian orbit: A simplification

An infinitestimal contact transformation is proposed to simplify at first order the Hamiltonian representing the attitude of a triaxial rigid body on a Keplerian orbit around a mass point. The simplified problem reduces to the Euler-Poinsot model, but with moments of inertia depending on time through the longitude in orbit. Should the orbit be circular, the moments of inertia would be constant.     

Sur les mouvements réguliers dans le probléme des deux corps solides

In the present paper the problem of translatory-rotatory motion of two rigid bodies is discussed. Author has shown that this problem admits particular solutions, when each body possesses axial symmetry. In these solutions the centre of mass of one body described the circular orbit around the other body and each body keeps the invariable orientation about this orbit.     

Multiple resonance in one problem of the stability of the motion of a satellite relative to the center of mass

We investigate the stability of the periodic motion of a satellite, a rigid body, relative to the center of mass in a central Newtonian gravitational field in an elliptical orbit. The orbital eccentricity is assumed to be low. In a circular orbit, this periodic motion transforms into the well-known motion called hyperboloidal precession (the symmetry axis of the satellite occupies a fixed position in the plane perpendicular to the radius vector of the center of mass relative to the attractive center and describes a hyperboloidal surface in absolute space, with the satellite rotating around the symmetry axis at a constant angular velocity). We consider the case where the parameters of the problem are close to their values at which a multiple parametric resonance takes place (the frequencies of the small oscillations of the satellite’s symmetry axis are related by several second-order resonance relations). We have found the instability and stability regions in the first (linear) approximation at low eccentricities.     

Orbital dynamics and equilibrium points around an asteroid with gravitational orbit–attitude coupling perturbation

The strongly perturbed dynamical environment near asteroids has been a great challenge for the mission design. Besides the non-spherical gravity, solar radiation pressure, and solar tide, the orbital motion actually suffers from another perturbation caused by the gravitational orbit–attitude coupling of the spacecraft. This gravitational orbit–attitude coupling perturbation (GOACP) has its origin in the fact that the gravity acting on a non-spherical extended body, the real case of the spacecraft, is actually different from that acting on a point mass, the approximation of the spacecraft in the orbital dynamics. We intend to take into account GOACP besides the non-spherical gravity to improve the previous close-proximity orbital dynamics. GOACP depends on the spacecraft attitude, which is assumed to be controlled ideally with respect to the asteroid in this study. Then, we focus on the orbital motion perturbed by the non-spherical gravity and GOACP with the given attitude. This new orbital model can be called the attitude-restricted orbital dynamics, where restricted means that the orbital motion is studied as a restricted problem at a given attitude. In the present paper, equilibrium points of the attitude-restricted orbital dynamics in the second degree and order gravity field of a uniformly rotating asteroid are investigated. Two kinds of equilibria are obtained: on and off the asteroid equatorial principal axis. These equilibria are different from and more diverse than those in the classical orbital dynamics without GOACP. In the case of a large spacecraft, the off-axis equilibrium points can exist at an arbitrary longitude in the equatorial plane. These results are useful for close-proximity operations, such as the asteroid body-fixed hovering.     

A method of intermediate orbit determination based on four positions of a minor body on the celestial sphere

A new method of computing the preliminary orbit of a celestial body based on four pairs of angle measurements has been suggested. The method makes use of preliminary orbit previously constructed by the author based on two position vectors and a corresponding time interval, taking into account the main part of the perturbations in the motion of the body under study. Using the example of constructing the orbit of the minor planet 1383 Limburgia, the results obtained using a four-position procedure of the Gaussian type based on the solution of a two-body problem have been compared with those of the new method. The comparison showed the new method to be highly efficient for perturbed motion studies. It is especially advantageous in the case of high-accuracy observation data on small orbital arcs.     

The perturbing effects on the planetary satellite

The problem triaxial satellite having a plane of dynamical symmetry in the restricted problem of three bodies has been studied. The first integrals are established and the general solution of the problem can be written in quadratures. The results show that the semi-major axis of the satellite orbit and its rotational angular momentum remain unchanged. The singular solution of this problem has been considered and the elements of satellite orbit can be determined.     

Properties of linearized mappings associated with periodic orbits in the three-body problem

In this paper three results on the linearized mapping associated with the plane three body problem near a periodic orbit are established. It is first shown that linear stability of such an orbit is independent of initial position on the orbit and of coordinate system. Second, the relation of Hénon connecting the rates of change of rotation angle and period on an isoenergetic family of periodic orbits is proved, together with a similar relation for families of orbits closing exactly in a rotating coordinate system. Finally, a condition for a critical orbit is given which is applicable to any family of periodic orbits.     

Attitude stability of a spinning symmetric satellite in a planar periodic orbit

This paper contains an analysis of the attitude stability of a spinning axisymmetric satellite whose mass center moves in any known planar periodic orbit of the restricted three-body problem while the spin axis remains normal to the orbit plane. A procedure based on Floquet theory is developed for constructing attitude instability charts, and examples of these are presented for two stable periodic orbits of the Earth-Moon system—one direct and one retrograde. The physical significance of these instability predictions is then explored by means of numerical integration of the full nonlinear equations of motion. Finally, an analysis based on averaging is performed, leading to approximate instability charts and indicating a possible connection between certain orbital-attitude resonance conditions and unstable attitude motions.     

Families of periodic orbits in the planar three-body problem

A periodic orbit of the restricted circular three-body problem, selected arbitrarily, is used to generate a family of periodic motions in the general three-body problem in a rotating frame of reference, by varying the massm ₃ of the third body. This family is continued numerically up to a maximum value of the mass of the originally small body, which corresponds to a mass ratiom ₁:m ₂:m ₃?5:5:3. From that point on the family continues for decreasing massesm ₃ until this mass becomes again equal to zero. It turns out that this final orbit of the family is a periodic orbit of the elliptic restricted three body problem. These results indicate clearly that families of periodic motions of the three-body problem exist for fixed values of the three masses, since this continuation can be applied to all members of a family of periodic orbits of the restricted three-body problem. It is also indicated that the periodic orbits of the circular restricted problem can be linked with the periodic orbits of the elliptic three-body problem through periodic orbits of the general three-body problem.     

Orbital perturbations due to radiation pressure for a spacecraft of complex shape

We analyze the perturbations due to solar radiation pressure on the orbit of a high artificial satellite. The latter is modelled in a simplified way (axisymmetric body plus despun antenna emitting a radio beam), which seems suitable to describe the main effects for existing telecommunication satellites. We use the regularized general perturbation equations, by expressing the force in the moving Gauss' reference frame and by expanding the results in terms of some small parameters, referring both to the orbit (small eccentricity and inclination) and to the spacecraft's attitude. Some interesting results are derived, which assess the relative importance of different physical effects and of different parts of the spacecraft in determining the long-term evolution of the orbital elements.     

Orbit computations with the Eulerian topological regular elements with oblateness

In this paper, the Eulerian set of topological regular elements was utilized to develop an orbit computation package for the initial value problem for any conic motion of artificial satellites in the Earth's gravitational field with axial symmetry. Applications for the two types of short- and long-term predictions are considered. The numerical results proved the high efficiency and flexibility of the package.     

Attitude stability of a gravity-stabilized gyrostat satellite

This paper treats analytically the problem of the stability of the attitude motions of a gravity-stabilized gyrostat satellite that is in a circular orbit around a spherical planet. The vehicle considered consists of a body with no special symmetries that has any number of rotors attached to it. The internal angular momentum vector due to these rotors is parallel to one of the principal axes of the entire satellite; this axis is aligned with (or close to) the normal to the orbit plane. Both the cases in which each rotor is driven by a motor at a constant spin rate relative to the main body of the vehicle and the one in which each rotor is rotating freely, without any friction, are treated. Stability (both infinitesimal and in the sense of Liapunov) of the attitude motions of the vehicle can be quickly predicted by using the results derived here, which are summarized in the form of a continuous, three-dimensional, stability diagram.National Research Council Post-Doctoral Research Associate at NASA-Ames Research Center, Moffett Field, Calif. Parts of this research were included in [12].     

Application of the Gauss' Method to the Stellar Three Body Problem

In this paper, we deal with the stellar three body problem, that is one star is far away from the other two stars. The outer orbit is assumed to be Keplerian. To analyze the effect of the distant star on the orbit of the close stars, we use the Gauss method; this method consist in replacing the gravitational attraction of the third star by the gravitational attraction of an infinitesimal non-homogeneous elliptic ring. We obtain the force vector for the Gauss method in terms of elliptic integrals. Finally we compare the results obtained by this model with the classical third body model. This revised version was published online in August 2006 with corrections to the Cover Date.     

月球探测器过渡轨道的短弧定轨方法

论述的短弧定轨,是指在无先验信息情况下又避开多变元迭代的初轨计算方法,它需要相应的动力学问题有一能反映短弧内达到一定精度的近似分析解．探测器进入月球引力作用范围后接近月球时可以处理成相对月球的受摄二体问题,而在地球附近,则可处理成相对地球的受摄二体问题,但在整个过渡段的力模型只能处理成一个受摄的限制性三体问题．而限制性三体问题无分析解,即使在月球引力作用范围外,对于大推力脉冲式的过渡方式,相对地球的变化椭圆轨道的偏心率很大(超过Laplace极限),在考虑月球引力摄动时亦无法构造摄动分析解．就此问题,考虑在地球非球形引力(只包含J2项)和月球引力共同作用下,构造了探测器飞抵月球过渡轨道段的时间幂级数解,在此基础上给出一种受摄二体问题意义下的初轨计算方法,经数值验证,定轨方法有效,可供地面测控系统参考．     

On stability of orbit of artificial moon moving around a small celestial body perturbed by an external body

We examine the stability of the orbit of an artificial moon of a small celestial body in the presence of an external massive perturbing body in terms of the restricted three-body problem. The orbit of this moon is shown to be dependent on the shape of the small body and central gravitational field of the external body. We study how these factors interact with each other and how they affect the stability of the orbit.     

考虑地球扁率摄动影响的初轨计算方法

在二体问题意义下的短弧定轨,Laplace型方法是最主要最典型的一种初轨计算方法。若测角资料达到10^-4-10^-5精度(相当于2″—20″之间),那么要使定轨精度达到与其相应的程度,地球非球形引力位中的扁率项摄动应该考虑,在此前提下,同样可以采用相应的Laplace型定轨方法。即给出这种严格包含扁率摄动的初轨计算方法的原理和具体计算过程以及计算实例,除采用多资料定轨方法外,这种方法也是提高初轨计算精度的一种途径,它同样可用于多资料的情况,这种方法对于大扁率主天体(即中心天体)的卫星定轨将更有实用价值。     

The limiting case of the double-averaged parabolic restricted three-body problem

The planar case of the parabolic restricted three-body problem is considered. The equations of motion are integrated within the framework of the double-averaged problem taking into account only the first term in the expansion of the perturbing function. It is demonstrated that, at moderate approaches to the central body, the size and the shape of the orbit of the perturbing body are invariable and only the orientation of the orbit changes.     

The motion of artificial satellites in the set of Eulerian redundant parameters

In this paper, the connections between orbit dynamics and rigid body dynamics are established throughout the Eulerian redundant parameters, the perturbation equations for any conic motion of artificial satellites are derived in terms of these parameters. A general recursive and stable computational algorithm is also established for the initial-value problem of the Eulerian parameters for satellites prediction in the Earth's gravitational field with axial symmetry. Applications of the algorithm are considered for the two cases of short and long term predictions. For the short-term prediction, we consider the problem of the final state prediction of some typical ballistic missiles in the geopotential model with zonal harmonic terms up to J ₃₆, while for the long-term prediction, we consider the perturbed J ₂ motion of Explorer 28 over 100 revolutions.     

Poisson equations of rotational motion for a rigid triaxial body with application to a tumbling artificial satellite

Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w ₀)², wheren is the satellite's orbital mean motion andw ₀ its initial rotational angular speed.