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1.
Marcello Romano 《Celestial Mechanics and Dynamical Astronomy》2008,100(3):181-189
The exact analytic solution is introduced for the rotational motion of a rigid body having three equal principal moments of
inertia and subjected to an external torque vector which is constant for an observer fixed with the body, and to arbitrary
initial angular velocity. In the paper a parametrization of the rotation by three complex numbers is used. In particular,
the rows of the rotation matrix are seen as elements of the unit sphere and projected, by stereographic projection, onto points
on the complex plane. In this representation, the kinematic differential equation reduces to an equation of Riccati type,
which is solved through appropriate choices of substitutions, thereby yielding an analytic solution in terms of confluent
hypergeometric functions. The rotation matrix is recovered from the three complex rotation variables by inverse stereographic
map. The results of a numerical experiment confirming the exactness of the analytic solution are reported. The newly found
analytic solution is valid for any motion time length and rotation amplitude. The present paper adds a further element to
the small set of special cases for which an exact solution of the rotational motion of a rigid body exists. 相似文献
2.
P. B. Jones 《Astrophysics and Space Science》1976,45(2):369-381
The time dependences of the inertia tensor and of a dissipative torque caused by the nonleptonic weak interaction have been investigated for a certain class of pulsars with no solid core. Early in the life of the pulsar, the angular velocity vector is predicted to move with respect to fixed body axes in such a way that it becomes perpendicular to the magnetic dipole moment. During this motion, the solid outer shell suffers plastic deformation so that the dipole moment becomes approximately collinear with a principal axis. After 104 or 105 yr, the dissipative torque is negligibly small compared with the electromagnetic torque, the Euler equations are those for a simple rigid body, and alignment of spin and dipole moment occurs. If the dipole moment discussed by Lyneet al. (1975) is interpreted as being equal to the component perpendicular to the spin, its secular decay is a natural property of this model and is not a consequence of field decay through electrical resistivity. 相似文献
3.
The secular effect of YORP torque on the rotational dynamics of an asteroid in non-principal axis rotation is studied. The
general rotational equations of motion are derived and approximated with an illumination function expanded up to second order.
The resulting equations of motion can be averaged over the fast rotation angles to yield secular equations for the angular
momentum, dynamic inertia and obliquity. We study the properties of these secular equations and compare results to previous
research. Finally, an application to several real asteroid shapes is made, in particular we study the predicted rotational
dynamics of the asteroid Toutatis, which is known to be in a non-principal axis state. 相似文献
4.
M. G. Abrahamyan 《Astrophysics》2008,51(2):163-180
A vortical mechanism for generation of astrophysical jets is proposed based on exact solutions of the hydrodynamic equations
with a generalized Rankine vortex. It is shown that the development of a Rankine vortex in the polar layer of a rotating gravitating
body creates longitudinal fluxes of matter that converge toward the vortex trunk, providing an exponential growth in the angular
rotation velocity of the trunk and a pressure drop on its axis. The increased rotational velocity of the vortex trunk and
the on-axis pressure drop cease when the discontinuity in the azimuthal velocity at the surface of the trunk reaches the sound
speed. During this time, ever deeper layers of the gravitating body are brought into the vortical motion, while the longitudinal
velocity of the flow along the vortex trunk builds up, producing jet outflows of mass from its surface. The resulting vortices
are essentially dissipationless.
Dedicated to the 100-th birthday of Academician V. A. Ambartsumyan
__________
Translated from Astrofizika, Vol. 51, No. 2, pp. 201–218 (May 2008). 相似文献
5.
Martin Lara 《Celestial Mechanics and Dynamical Astronomy》2014,118(3):221-234
A simple rearrangement of the torque free motion Hamiltonian shapes it as a perturbation problem for bodies rotating close to the principal axis of maximum inertia, independently of their triaxiality. The complete reduction of the main part of this Hamiltonian via the Hamilton–Jacobi equation provides the action-angle variables that ease the construction of a perturbation solution by Lie transforms. The lowest orders of the transformation equations of the perturbation solution are checked to agree with Kinoshita’s corresponding expansions for the exact solution of the free rigid body problem. For approximately axisymmetric bodies rotating close to the principal axis of maximum inertia, the common case of major solar system bodies, the new approach is advantageous over classical expansions based on a small triaxiality parameter. 相似文献
6.
A. P. Markeev 《Astronomy Letters》2005,31(9):627-633
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. 相似文献
7.
Gravity-gradient perturbations of the attitude motion of a tumbling tri-axial satellite are investigated. The satellite center of mass is considered to be in an elliptical orbit about a spherical planet and to be tumbling at a frequency much greater than orbital rate. In determining the unperturbed (free) motion of the satellite, a canonical form for the solution of the torque-free motion of a rigid body is obtained. By casting the gravity-gradient perturbing torque in terms of a perturbing Hamiltonian, the long-term changes in the rotational motion are derived. In particular, far from resonance, there are no long-period changes in the magnitude of the rotational angular momentum and rotational energy, and the rotational angular momentum vector precesses abound the orbital angular momentum vector.At resonance, a low-order commensurability exists between the polhode frequency and tumbling frequency. Near resonance, there may be small long-period fluctuations in the rotational energy and angular momentum magnitude. Moreover, the precession of the rotational angular momentum vector about the orbital angular momentum vector now contains substantial long-period contributions superimposed on the non-resonant precession rate. By averaging certain long-period elliptic functions, the mean value near resonance for the precession of the rotational angular momentum vector is obtained in terms of initial conditions. 相似文献
8.
We study the dynamical interactions of mass systems in equilibrium under their own gravity that mutually exert and ex‐perience gravitational forces. The method we employ is to model the dynamical evolution of two isolated bars, hosted within the same galactic system, under their mutual gravitational interaction. In this study, we present an analytical treatment of the secular evolution of two bars that oscillate with respect to one another. Two cases of interaction, with and without geometrical deformation, are discussed. In the latter case, the bars are described as modified Jacobi ellipsoids. These triaxial systems are formed by a rotating fluid mass in gravitational equilibrium with its own rotational velocity and the gravitational field of the other bar. The governing equation for the variation of their relative angular separation is then numerically integrated, which also provides the time evolution of the geometrical parameters of the bodies. The case of rigid, non‐deformable, bars produces in some cases an oscillatory motion in the bodies similar to that of a harmonic oscillator. For the other case, a deformable rotating body that can be represented by a modified Jacobi ellipsoid under the influence of an exterior massive body will change its rotational velocity to escape from the attracting body, just as if the gravitational torque exerted by the exterior body were of opposite sign. Instead, the exchange of angular momentum will cause the Jacobian body to modify its geometry by enlarging its long axis, located in the plane of rotation, thus decreasing its axial ratios. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
9.
Yoshio Kubo 《Celestial Mechanics and Dynamical Astronomy》2011,110(2):143-168
Under perturbations from outer bodies, the Earth experiences changes of its angular momentum axis, figure axis and rotational
axis. In the theory of the rigid Earth, in addition to the precession and nutation of the angular momentum axis given by the
Poisson terms, both the figure axis and the rotational axis suffer forced deviation from the angular momentum axis. This deviation
is expressed by the so-called Oppolzer terms describing separation of the averaged figure axis, called CIP (Celestial Intermediate
Pole) or CEP (Celestial Ephemeris Pole), and the mathematically defined rotational axis, from the angular momentum axis. The
CIP is the rotational axis in a frame subject to both precession and nutation, while the mathematical rotational axis is that
in the inertial (non-rotating) frame. We investigate, kinematically, the origin of the separation between these two axes—both
for the rigid Earth and an elastic Earth. In the case of an elastic Earth perturbed by the same outer bodies, there appear
further deviations of the figure and rotational axes from the angular momentum axis. These deviations, though similar to the
Oppolzer terms in the rigid Earth, are produced by quite a different physical mechanism. Analysing this mechanism, we derive
an expression for the Oppolzer-like terms in an elastic Earth. From this expression we demonstrate that, under a certain approximation
(in neglect of the motion of the perturbing outer bodies), the sum of the direct and convective perturbations of the spin
axis coincides with the direct perturbation of the figure axis. This equality, which is approximate, gets violated when the
motion of the outer bodies is taken into account. 相似文献
10.
We study the motion of the free dual-spin gyrostat spacecraft that consists of the platform with a triaxial ellipsoid of inertia and the rotor with a small asymmetry with respect to the axis of rotation. The system with perturbations caused by a small asymmetry of the rotor and the time-varying moments of inertia of the rotor is considered. The dimensionless equations of the system are written in Serret–Andoyer canonical variables. The system’s phase space is described. It is shown that changes in the moments of inertia of the gyrostat leads to the deformation of the phase space. The internal torque control law is proposed that keeps the system at the center point in the phase space. The effectiveness of the control is shown through a numerical simulation. It’s shown that the uncontrolled gyrostat can lose its axis orientation. Proposed internal torque keeps the initial angle between the axis of the gyrostat and the total angular momentum vector. 相似文献
11.
We investigate the Cassini's laws which describe the rotational motion in a 1:1 spin-orbit resonance. When this rotational motion follows the conventional Cassini's laws, the figure axis coincides with the angular momentum axis. In this case we underline the differences between the rotational Hamiltonian for a 'slow rotating' body like the Moon and for a 'fast rotating' body like Phobos. Then, we study a more realistic rotational Hamiltonian where the angle J between the figure axis and the angular momentum axis could be different from zero. This Hamiltonian has not been studied before. We have found a new particular solution for this Hamiltonian which could be seen as an extension of the Cassini's laws. In this new solution the angle J is constant, which is not zero, and the precession of the angular momentum plane is equal to the mean motion of the argument of pericenter of the rotating body. This type of rotational motion is only possible when the orbital eccentricity of the rotating body is not zero. This new law enables describing in particular, the Moon mean rotational motion for which the mean value of the angle J is found to be equal to 103.9±0.7 s of arc. 相似文献
12.
Evolution of Comet Nucleus Rotation 总被引：1，自引：0，他引：1
The secular evolution of comet nucleus rotation states subject to outgassing torques is studied. The dynamical model assumes that the nucleus inertia ellipsoid is axially symmetric. The outgassing torques acting on the surface are modeled using standard cometary activity formulae. The general rotational equations of motion are derived and separately averaged over the fast rotational dynamics terms and the comet orbit. Special cases where the averaging assumptions cannot be applied are evaluated separately. The modification of the comet orbit due to comet outgassing is neglected. Resulting from this analysis is a system of secular differential equations that describes the dynamics of the comet nucleus angular momentum and rotation state. We find that the qualitative secular evolution of the rotation state is controlled by a single parameter that combines parameters related to the comet orbit and parameters related to the nucleus surface geometry and activity. From this solution, we find qualitatively different evolutionary paths for comet nuclei whose entire surface is active, as compared to nuclei with only a single active region. For surface activity models between these extremes, we show that certain evolutionary paths are more likely than others. Additionally, our solution indicates that a comet nucleus' rotational angular momentum will tend to increase over time, potentially contributing to the observed phenomenon of comet nucleus splitting. 相似文献
13.
Jason W. Mitchell David L. Richardson 《Celestial Mechanics and Dynamical Astronomy》2001,81(1-2):13-25
Euler's equations, describing the rotation of an arbitrarily torqued mass asymmetric rigid body, are scaled using linear transformations that lead to a simplified set of first order ordinary differential equations without the explicit appearance of the principal moments of inertia. These scaled differential equations provide trivial access to an analytical solution and two constants of integration for the case of torque-free motion. Two additional representations for the third constant of integration are chosen to complete two new kinetic element sets that describe an osculating solution using the variation of parameters. The elements' physical representations are amplitudes and either angular displacement or initial time constant in the torque-free solution. These new kinetic elements lead to a considerably simplified variation of parameters solution to Euler's equations. The resulting variational equations are quite compact. To investigate error propagation behaviour of these new variational formulations in computer simulations, they are compared to the unmodified equations without kinematic coupling but under the influence of simulated gravity-gradient torques. 相似文献
14.
Alan H. Jupp 《Celestial Mechanics and Dynamical Astronomy》1974,9(1):3-20
It is well known that the equations governing the motion of a freely-rotating rigid body possess an exact analytical solution, involving Jacobi's elliptic functions. Andoyer (1923) and Deprit (1967) have shown that the problem may be very usefully reduced to a one-degree-of-freedom Hamiltonian system. When two of the body's principal moments of inertia are very nearly equal, the Hamiltonian system has the same form as the Ideal Resonance Problem. In earlier publications (Jupp, 1969, 1972, 1973), the author has constructed formal power-series solutions of the latter problem.In this article, the general solution of the Ideal Resonance Problem is employed to formulate a second-order formal series solution of the problem of a freely-rotating rigid body which has two of its principal moments of inertia differing by a small quantity. This solution is firstly expressed in terms of the mean elements, and then in terms of the initial conditions. The latter solution is global in nature being applicable over the whole phase plane. It is demonstrated that the exact solution and the second-order formal series solution, written in terms of the initial conditions, differ by terms of at most third order in the small parameter, over the whole domain of possible motions. This serves as an important check on the general results published in the earlier articles. 相似文献
15.
V. T. Kondurar 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):327-344
The present paper is a direct continuation of the paper (Duboshin, 1973) in which was proved the existence of one kind of Lagrange (triangle) and Euler (rectilinear) solutions of the general problem of the motion of three finite rigid bodies assuming different laws of interaction between the elementary particles of the rigid bodies. In particular, Duboshin found that the general problem of three rigid bodies permits such solutions in which the centres of mass of the bodies always form an equilateral triangle or always remain on one straight line, and each body possesses an axial symmetry and a symmetry with respect to the plane of the centres of mass and rotates uniformly around its axis orthogonal to this plane. The conditions for the existence of such solutions have also been found. The results in Duboshin's paper have greatly interested the author of the present paper. In another paper (Kondurar and Shinkarik, 1972) considering a more special problem, when two of the three bodies are spheres, either homogeneous or possessing a spherically symmetric distribution of the densities or of the material points, and the third is an axially symmetrical body possessing equatorial symmetry, the present author obtained analogous solutions of the ‘float’ type describing the motion of the indicated dynamico-symmetrical body in assuming its passive gravitation. In the present paper new Lagrange solutions of the considered general problems of three rigid bodies of ‘level’ type are found when the axes of geometrical and mechanical symmetry of all three bodies always lie in the triangle plane, and the bodies themselves rotate inertially around the symmetry axis, independently of the parameters of the orbital motion of the centres of mass as in the ‘float’ case. The study of particular solutions of the general problem of the translatory-rotary motion of three rigid bodies, which are a generalization of Lagrange solutions, is in the author's opinion, a novelty of some interest for both theoretical and practical divisions of celestial mechanics. For example, in recent times the problem of the libration points of the Earth-Moon system has acquired new interest and value. A possible application which should be mentioned is that to the orbits of artificial satellites near the triangular libration points to serve as observation stations with the aim of specifying the physical parameters in the Earth-Moon system (e.g., the relation of the Earth's mass to the Moon's mass for investigating the orientation of the satellite, solar radiation, etc.). 相似文献
16.
17.
L. A. Dávalos-Orozco 《Astrophysics and Space Science》1996,243(2):291-313
In this paper the Rayleigh-Taylor instability (RTI) of a two-fluid layer system under the simultaneous action of a general rotation field and a horizontal magnetic field is presented. An approximate and an exact solution of the eigenvalue equation are calculated. These solutions are important not only to understand more deeply the physical problem but also to determine the correct numerical solutions. Numerical calculations are done for an unstable density stratification in the cases of horizontal magnetic field parallel and perpendicular to the horizontal component of the angular velocity. For an adverse density stratification, it is shown that in comparison to previous works, the horizontal magnetic field creates new angular areas (of the angle of propagation of the perturbation) at which the perturbation is stable and propagates as traveling waves. It is also shown that the vertical component of the angular velocity has a destabilizing effect because it works to eliminate the stable angular areas. 相似文献
18.
Andrzej J. Maciejewski Maria Przybylska 《Celestial Mechanics and Dynamical Astronomy》2003,87(4):317-351
In this paper we analyse the integrability of a dynamical system describing the rotational motion of a rigid satellite under the influence of gravitational and magnetic fields. In our investigations we apply an extension of the Ziglin theory developed by Morales-Ruiz and Ramis. We prove that for a symmetric satellite the system does not admit an additional real meromorphic first integral except for one case when the value of the induced magnetic moment along the symmetry axis is related to the principal moments of inertia in a special way. 相似文献
19.
H. Greiner-Mai 《Astronomische Nachrichten》1997,318(1):63-71
The geomagnetic field is maintained by amagnetohydrodynamic dynamo process within the liquid outer core. The distribution of the associated electric currents is modified if the outer core is bounded by electrically conducting material. Then, eddy currents and the related magnetic fields are generated within these regions. In particular, the relative rigid rotation of the inner core produces a secondary magnetic field, which is superimposed on the dynamo field. The angle between the dipole axis of the total field and the rotational axis of the inner core is an important quantity needed for the theory of polar motion of the Earth. This angle is investigated for a broad spectrum of angular velocities of the inner core. To simplify the mathematical procedure, we model the dynamo field using an axisymmetric field generated by a system of electric currents within the outer core. The conductivity of the mantle is neglected. We find that the position of the dipole axis depends on the angular velocity of the inner core as well as on the distribution of the current system within the outer core. Coincidence of both axes can be reached if the angular velocity is high enough and if the current system is concentrated within a thin sheet near the outer core-inner core boundary. 相似文献
20.
Paolo Lanzano 《Astrophysics and Space Science》1969,5(3):300-322
We consider two spheroidal rigid bodies of comparable size constituting the components of an isolated binary system. We assume that (1) the bodies are homogeneous oblate ellipsoids of revolution, and (2) the meridional eccentricities of both components are small parameters.We obtain seven nonlinear differential equations governing simultaneously the relative motion of the two centroids and the rotational motion of each set of body axes. We seek solutions to these equations in the form of infinite series in the two meridional eccentricities.In the zero-order approximation (i. e., when the meridional eccentricities are neglected), the equations of motion separate into two independent subsystems. In this instance, the relative motion of the centroids is taken as a Kepler elliptic orbit of small eccentricity, whereas for each set of body axes we choose a composite motion consisting of a regular precession about an inertial axis and a uniform rotation about a body axis.The first part of the paper deals with the representation of the total potential energy of the binary system as an infinite series of the meridional eccentricities. For this purpose, we had to (1) derive a formula for representing the directional derivative of a solid harmonic as a combination of lower order harmonics, and (2) obtain the general term of a biaxial harmonic as a polynomial in the angular variables.In the second part, we expound a recurrent procedure whereby the approximations of various orders can be determined in terms of lower-order approximations. The rotational motion gives rise to linear differential equations with constant coefficients. In dealing with the translational motion, differential equations of the Hill type are encountered and are solved by means of power series in the orbital eccentricity. We give explicit solutions for the first-order approximation alone and identify critical values of the parameters which cause the motion to become unstable.The generality of the approach is tantamount to studying the evolution and asymptotic stability of the motion.Research performed under NASA Contract NAS5-11123. 相似文献