Exact analytic solutions for the rotation of an axially symmetric rigid body subjected to a constant torque

New exact analytic solutions are introduced for the rotational motion of a rigid body having two equal principal moments of inertia and subjected to an external torque which is constant in magnitude. In particular, the solutions are obtained for the following cases: (1) Torque parallel to the symmetry axis and arbitrary initial angular velocity; (2) Torque perpendicular to the symmetry axis and such that the torque is rotating at a constant rate about the symmetry axis, and arbitrary initial angular velocity; (3) Torque and initial angular velocity perpendicular to the symmetry axis, with the torque being fixed with the body. In addition to the solutions for these three forced cases, an original solution is introduced for the case of torque-free motion, which is simpler than the classical solution as regards its derivation and uses the rotation matrix in order to describe the body orientation. This paper builds upon the recently discovered exact solution for the motion of a rigid body with a spherical ellipsoid of inertia. In particular, by following Hestenes’ theory, the rotational motion of an axially symmetric rigid body is seen at any instant in time as the combination of the motion of a “virtual” spherical body with respect to the inertial frame and the motion of the axially symmetric body with respect to this “virtual” body. The kinematic solutions are presented in terms of the rotation matrix. The newly found exact analytic solutions are valid for any motion time length and rotation amplitude. The present paper adds further elements to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.     

Stellar models under homoaxial rotation: Eulerian equations and simulation of non-uniformly rotating stars

The Eulerian equations are set up for a model subject to homoaxial rotation and suitable for simulation of a non-uniformly rotating star. These equations are formulated in a non-inertial frame of reference, rotating uniformly (i.e., rigidly) with respect to the inertial common frame.     

Dynamical and kinematical near homoaxial rotations suitable for representation of nonuniformly rotating stars with extended atmospheres

In this paper, a nontrivial velocity tensor is suitably defined to represent in the common frame the so-called classes of dynamical and kinematical near homoaxial rotations of a deformable finite material continuum. These classes have simple and interesting physical interpretation, especially for treating of nonuniform rotation and its applications to astrophysics. Some important subclasses and derived classes (in the sense of related superclasses) are also discussed.Two coordinate systems are further introduced, one of which rotates uniformly while the other rotates nonuniformly, the latter defined by means of a generalized orthogonal coordinate transformation. Suitable conditions are then given, asserting that the above systems are preferred in the sense of preserving the motion of the continuum in its inertial class.Finally, the required concepts of homotropy and distributivity are defined and the method, by which the established mathematical framework will be subsequently used in applications, is discussed.     

Hydrodynamic equations of differentially rotating stellar systems and the velocity ellipsoid

Anisotropic hydrodynamic equations for differentially rotating collisionless stellar systems are derived. These equations can describe the evolution of the systems in a time span longer than their rotation periods.As a by-product of derivation of hydrodynamic equations, the well-known relation that the ratio of the principal axes of the velocity ellipse in a differentially rotating stellar disk is [B/(B-A)]^1/2 is re-found if the system is in a purely circular rotation, whereA andB are the Oort's constants. In addition, we find a systematic mean motion superposed on a purely circular differential rotation makes the directions of axes of the velocity ellipse deviate from the radial and the transverse direction. The observed deviation of directions of axes in our neighbourhood in the Galaxy can be explained if in the mean motion superposed on a purely circular differential rotatin the gas of stars near us is compressed in the radial direction or rarefied in the transverse directions, with irregularities of the order of 5 km/sec in amplitude of velocity and 1 kpc in size. These magnitudes of irregularities agree with those actually observed or with those anticipated from other theoretical considerations.     

Effect of 3rd-degree gravity harmonics and Earth perturbations on lunar artificial satellite orbits

In a previous work we studied the effects of (I) the J ₂ and C ₂₂ terms of the lunar potential and (II) the rotation of the primary on the critical inclination orbits of artificial satellites. Here, we show that, when 3rd-degree gravity harmonics are taken into account, the long-term orbital behavior and stability are strongly affected, especially for a non-rotating central body, where chaotic or collision orbits dominate the phase space. In the rotating case these phenomena are strongly weakened and the motion is mostly regular. When the averaged effect of the Earth’s perturbation is added, chaotic regions appear again for some inclination ranges. These are more important for higher values of semi-major axes. We compute the main families of periodic orbits, which are shown to emanate from the ‘frozen eccentricity’ and ‘critical inclination’ solutions of the axisymmetric problem (‘J ₂ + J ₃’). Although the geometrical properties of the orbits are not preserved, we find that the variations in e, I and g can be quite small, so that they can be of practical importance to mission planning.     

Kinematics and Evolution of Local Features of the Large-Scale Magnetic Field – I. Kinematical Characteristics

Kinematics of local magnetic features (LMFs) have been investigated by analyzing a 22-year series of synoptic maps of the radial magnetic field of the Sun (the term ‘local’ refers hereinafter to magnetic features with an effective size of the order of an arc min). We applied the cross-correlation technique to analyse separately each of the harmonics obtained by using a one-dimensional Fourier transform of the magnetic field in longitude. Such an approach allowed us to trace the motion of the LMFs for a time interval as long as 12 Carrington rotations. The analysis also has shown that the effective size of the magnetic tracers grows significantly with increasing age, which indicates that the local-scale magnetic features undergo diffusion-like expansion and weakening, in agreement with Leighton's model of magnetic field evolution. The LMFs emerging at latitudes between 10° and 55° reveal a poleward directed motion with a maximum velocity of about 14 m s^-1 near the latitude of 37°. The profile of the meridional velocity agrees approximately with that derived by Komm, Howard, and Harvey (1993) for small-scale, short-lived magnetic features. We have found that the LMFs rotate differentially at latitudes of up to 55°, and do not exhibit the ‘quasi-rigid’ rotation that is assumed to be characteristic of long-lived magnetic features. This disagrees with the results obtained by Stenflo (1989) and by Latushko (1994), who applied direct cross-correlation analysis of the synoptic maps. Such a discrepancy may be treated as being a consequence of inhomogeneity of the large-scale solar magnetic field that consists of several components with different kinematic characteristics.     

The equations of relative motion in the orbital reference frame

The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill–Clohessy–Wiltshire equations. Circular motion is not, however, a solution when the Earth’s flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the \(J_2\) effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the \(J_2\) perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a \(J_2\)-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill–Clohessy–Wiltshire equations for circular reference motion, or the de Vries/Tschauner–Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the \(J_2\) perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Hill–Clohessy–Wiltshire, or the Tschauner–Hempel equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a \(J_2\)-perturbed reference orbit is also introduced.     

Chaos in Barred Galaxy Models

We investigate the regular and chaotic motion in a model potential found using the recent developments of the Inverse Problem of Dynamics. The potential describes the motion in the central parts of a barred galaxy. In the absence of rotation chaotic motion is observed when the perturbation strength is near the escape perturbation for a fixed value of the energy. In the rotating cases one observes that the area of chaotic motion on the surface of section decreases as the angular velocity Ω increases and finally all orbits become regular. The character of motion is also checked by computing the Liapunov characteristic exponents in all cases. This revised version was published online in July 2006 with corrections to the Cover Date.     

‘Similar’ coordinate systems and the Roche geometry: application

A new equivalence relation, named relation of ‘similarity’ is defined and applied in the restricted three-body problem. Using this relation, a new class of trajectories (named ‘similar’ trajectories) are obtained; they have the theoretical role to give us new details in the restricted three-body problem. The ‘similar’ coordinate systems allow us in addition to obtain a unitary and an elegant demonstration of some analytical relations in the Roche geometry. As an example, some analytical relations published by Seidov (in Astrophys. J. 603:283, 2004) are demonstrated.     

Wisdom system: Dynamics in the adiabatic approximation

A simple approximate model of the asteroid dynamics near the 3:1 mean–motion resonance with Jupiter can be described by a Hamiltonian system with two degrees of freedom. The phase variables of this system evolve at different rates and can be subdivided into the ‘fast’ and ‘slow’ ones. Using the averaging technique, wisdom obtained the evolutionary equations which allow to study the long-term behavior of the slow variables. The dynamic system described by the averaged equations will be called the ‘Wisdom system’ below. The investigation of the, wisdom system properties allows us to present detailed classification of the slow variables’ evolution paths. The validity of the averaged equations is closely connected with the conservation of the approximate integral (adiabatic invariant) possessed by the original system. Qualitative changes in the behavior of the fast variables cause the violations of the adiabatic invariance. As a result the adiabatic chaos phenomenon takes place. Our analysis reveals numerous stable periodic trajectories in the region of the adiabatic chaos.     

Semi-analytic solution of a two-body problem with drag

An approximate semi-analytic solution of a two-body problem with drag is presented. The solution describesnon-lifting orbital motion in a central, inverse-square gravitational field. Drag deceleration is a non-linear function of velocity relative to a rotating atmosphere due to dynamic pressure and velocity-dependent drag coefficient. Neglected are aerodynamic lift, gravitational perturbations of the inverse-square field, and kinematic accelerations due to coordinate frame rotation at earth angular rate. With these simplifications, it is shown that (i) orbital motion occurs in an earth-fixed invariable plane defined by the radius and relative velocity vectors, and (ii) the simplified equations of motion are autonomous and independent of central angle measured in the invariable plane. Consequently, reduction of the differential equations from sixth to second-order is possible. Solutions for the radial and circumferential components of relative velocity are reduced to quadratures with respect to radial distance. Since the independent variable is radial distance, the solutions are singular at zero radial velocity (e. g., for circular orbits). General atmospheric density and drag coefficient models may be used to evaluate the velocity quadratures. The central angle and time variables are recovered from two additional quadratures involving the velocity quadratures. Theoretical results are compared with numerical simulation results.Presently affiliated with AVCO Systems Division, Wilmington, MA 01887, U.S.A.     

Shift in the bifurcation points of rotating magnetized newtonian polytropes caused by a magnetic field

Criteria are formulated for determining the critical points and bifurcation points of rotating, magnetized, newtonian polytropes, which coincide in the absence of a magnetic field. The magnitude of the shift in these points is estimated in terms of the flatness parameter e and rotation speed ε. The dependence of the total energy of a polytrope near the bifurcation and critical points is calculated as a function of the asymmetry parameters X for the distribution of mass relative to the axis of rotation and of the rotation speed ε for ε ≪ 1. The stability of the rotating polytrope with respect to the parameter X is analyzed. __________ Translated from Astrofizika, Vol. 51, No. 2, pp. 321–327 (May 2008).     

Regularization of the circular restricted three-body problem using ‘similar’ coordinate systems

The regularization of a new problem, namely the three-body problem, using ‘similar’ coordinate system is proposed. For this purpose we use the relation of ‘similarity’, which has been introduced as an equivalence relation in a previous paper (see Roman in Astrophys. Space Sci. doi:, 2011). First we write the Hamiltonian function, the equations of motion in canonical form, and then using a generating function, we obtain the transformed equations of motion. After the coordinates transformations, we introduce the fictitious time, to regularize the equations of motion. Explicit formulas are given for the regularization in the coordinate systems centered in the more massive and the less massive star of the binary system. The ‘similar’ polar angle’s definition is introduced, in order to analyze the regularization’s geometrical transformation. The effect of Levi-Civita’s transformation is described in a geometrical manner. Using the resulted regularized equations, we analyze and compare these canonical equations numerically, for the Earth-Moon binary system.     

Zero velocity hypersurfaces for the general three-dimensional three-body problem

The equation of zero velocity surfaces for the general three-body problem can be derived from Sundman's inequality. The geometry of those surfaces was studied by Bozis in the planar case and by Marchal and Saari in the three-dimensional case. More recently, Saari, using a geometrical approach, has found an inequality stronger than Sundman's. Using Bozis' algebraic method, and a rotating frame which does not take into account entirely the rotation of the three-body system, we also find an inequality stronger than Sundman's. The comparison with Saari's inequality is more difficult. However, our result can be expressed in four-dimensional space and the regions where motion is allowed can be seen (numerically) to lie inside those obtained by the use of Sundman's inequality.Agrégé de Faculté.     

Ohm’s law for plasma in general relativity and Cowling’s theorem

The general-relativistic Ohm’s law for a two-component plasma which includes the gravitomagnetic force terms even in the case of quasi-neutrality has been derived. The equations that describe the electromagnetic processes in a plasma surrounding a neutron star are obtained by using the general relativistic form of Maxwell equations in a geometry of slow rotating gravitational object. In addition to the general-relativistic effect first discussed by Khanna and Camenzind (Astron. Astrophys. 307:665, 1996) we predict a mechanism of the generation of azimuthal current under the general relativistic effect of dragging of inertial frames on radial current in a plasma around neutron star. The azimuthal current being proportional to the angular velocity ω of the dragging of inertial frames can give valuable contribution on the evolution of the stellar magnetic field if ω exceeds 2.7×10¹⁷(n/σ) s⁻¹ (n is the number density of the charged particles, σ is the conductivity of plasma). Thus in general relativity a rotating neutron star, embedded in plasma, can in principle generate axial-symmetric magnetic fields even in axisymmetry. However, classical Cowling’s antidynamo theorem, according to which a stationary axial-symmetric magnetic field can not be sustained against ohmic diffusion, has to be hold in the general-relativistic case for the typical plasma being responsible for the rotating neutron star.     

Prediction Test for the Two Extremely Strong Solar Storms in October 2003

In late October and early November 2003, a series of space weather hazard events erupted in solar-terrestrial space. Aiming at two intense storm (shock) events on 28 and 29 October, this paper presents a Two-Step method, which combines synoptic analysis of space weather–`observing’ and quantitative prediction – ‘palpating’, and uses it to test predictions. In the first step, ‘observing’, on the basis of observations of the source surface magnetic field, interplanetary scintillation (IPS) and ACE spacecraft, we find that the propagation of the shock waves is asymmetric and northward relative to the normal direction of their solar sources due to the large-scale configuration of the coronal magnetic fields, and the Earth is located near the direction of the fastest speed and greatest energy of the shocks. Being two fast ejection shock events, the fast explosion of extremely high temperature and strong magnetic field, and background solar wind velocity as high as 600 and 1000 km s⁻¹, are also helpful to their rapid propagation. According to the synoptic analysis, the shock travel times can be estimated as 21 and 20 h, which are close to the observational results of 19.97 and 19.63 h, respectively. In the second step, ‘palpating’, we adopt a new membership function of the fast shock events for the ISF method. The predicted results here show that for the onset time of the geomagnetic disturbance, the relative errors between the observational and the predicted results are 1.8 and 6.7%, which are consistent with the estimated results of the first step; and for the magnetic disturbance magnitude, the relative errors between the observational and the predicted results are 4.1 and 3.1%, respectively. Furthermore, the comparison among the predicted results of our Two-Step method with those of five other prevailing methods shows that the Two-Step method is advantageous in predicting such strong shock event. It can predict not only shock arrival time, but also the magnitude of magnetic disturbance. The results of the present paper tell us that understanding the physical features of shock propagation thoroughly is of great importance in improving the prediction efficiency.     

Jerk in Planetary Systems and Rotational Dynamics,Nonlocal Motion Relative to Earth and Nonlocal Fluid Dynamics in Rotating Earth Frame

Some results following from the implications of nonlocal-in-time kinetic energy approach introduced recently by Suykens in the framework of rotational dynamics and motion in a non-inertial frame are discussed. Their roles in treating aspects concerning the nonlocal motion relative to Earth, the free-fall problem, the Foucault pendulum and the motion of a massive body in a rotating tube are analyzed. Governing nonlocal equations of fluid dynamics in particular the nonlocal-in-time Navier–Stokes equations are constructed under the influence of Earth rotation. Their properties are analyzed and a number of features were revealed and discussed accordingly.     

Production of pulses and interpulses in pulsars

Pulsars accelerate the charged particles moving along their magnetic field lines due to their rapidly spinning motion. Particles gain maximum energy from pulsars within the light cylinder when they are moving along the field lines perpendicular to the rotation velocity. In pulsars with non-aligned rotation and magnetic axes, the production of two intense and sharp pulses (main pulse and interpulse) separated by 180° longitude occur at the two regions near the light cylinder where the rotation velocity is perpendicular to the magnetic field. Since the radiating particles move radially along the relativistically compressed magnetic field lines, the observer in the stationary frame receives beamed and transversely compressed radiation pulse. Near the light cylinder position angle varies smoothly during pulsar rotation in a way as Radhakrishnan and Cook (1969) expect its variation near the magnetic pole, as the field lines experience relativistic compression in the direction of rotation. The motion of two charge species along the field lines produce orthogonal modes at each pulse longitude.     

Exact solution to the relative orbital motion in eccentric orbits

This paper studies the relative orbital motion between arbitrary Keplerian trajectories. A closed-form vectorial solution to the nonlinear initial value problem that models this type of motion with respect to a noninertial reference frame is offered. Without imposing any particular conditions on the leader or the deputy satellites trajectories, exact expressions for the relative law of motion and relative velocity are obtained in a closed form. This solution allows the parameterization of the relative motion manifold and offers new methods to study its geometrical and topological properties. The result presented in this paper opens the way to obtain new classes of approximate solutions to the equations of relative motion with time, an eccentric or true anomaly as independent variables. Published in Russian in Solar System Research, 2009, Vol. 43, No. 1, pp. 44–55. The text was submitted by the autors in English.