Nodal and periastron precession of inclined orbits in the field of a rotating black hole

The inclination of low-eccentricity orbits is shown to significantly affect orbital parameters, in particular, the Keplerian, nodal precession, and periastron rotation frequencies, which are interpreted in terms of observable quantities. For the nodal precession and periastron rotation frequencies of low-eccentricity orbits in a Kerr field, we derive a Taylor expansion in terms of the Kerr parameter at arbitrary orbital inclinations to the black-hole spin axis and at arbitrary radial coordinates. The particle radius, energy, and angular momentum in the marginally stable circular orbits are calculated as functions of the Kerr parameter j and parameter s in the form of Taylor expansions in terms of j to within O[j ⁶]. By analyzing our numerical results, we give compact approximation formulas for the nodal precession frequency of the marginally stable circular orbits at various s in the entire range of the Kerr parameter.     

Determining the pole and the siderial rotation period of a conic artificial Earth satellite from mirror flash moments

We used the mirror-cone model to derive formulas and construct an algorithm for determining the pole position and the siderial period of the rotation of an artificial Earth satellite about its center of inertia. The apex angle of cone and the precession angle are determined as well. An algorithm is also constructed for calculating a model mirror flash series at fixed satellite rotation parameters.     

Polarization of radiation from a strongly magnetized accretion disk: The asymptotic spectral distribution

We calculate the polarization of the radiation from an optically thick accretion disk with a vertical averaged magnetic field. The polarization arises from the scattering of light by free electrons in a magnetized disk plasma. The Faraday rotation of the polarization plane during the propagation of a photon in a medium with a magnetic field is considered as the main effect. We discuss various models of optically thick accretion disks with a vertical averaged magnetic field. Our main goal is to derive simple asymptotic formulas for the polarization of radiation in the case where the Faraday rotation angle Ψ ≫ 1 at the Thomson optical depth τ = 1. The results of our calculations allow the magnetic field strength in the region of the marginally stable orbit near a black hole to be estimated from polarimetric observations, including X-ray observations expected in the future. Since the polarization spectrum of the radiation strongly depends on the accretion disk model, a realistic physical model of the accretion disk can be determined from data on the polarization of its radiation.     

The precession and nutation of deformable bodies

The aim of the present paper will be to derive from the fundamental equations of hydrodynamics the explicit form of the Eulerian equations which govern the motion about the centre of gravity of self-gravitating bodies, consisting of compressible fluid of arbitrary viscosity, in an arbitrary external field of force. If the problem is particularized so that the external field of force represents the attaction of the sun and the moon, this motion would represent the luni-solar precession and nutation of a fluid viscous earth; if, on the other hand, the external field of force were governed by the earth (and the sun), the motion would define the physical librations of the moon regarded as a deformable body. The same equations are, moreover, equally applicable to the phenomena of precession and nutation of rotating fluid components in close binary systems, distorted by mutual tidal action; and the present paper contains the first formulation of the effects of viscosity on such phenomena.Investigation supported in part by the U.S. National Aeronautics and Space Administration under Contract No. NASW-1470.     

Numerical simulation of the Moon's rotation in a rigorous relativistic framework

This paper describes a numerical simulation of the rigid rotation of the Moon in a relativistic framework.Following a resolution passed by the International Astronomical Union(IAU) in 2000,we construct a kinematically non-rotating reference system named the Selenocentric Celestial Reference System(SCRS) and give the time transformation between the Selenocentric Coordinate Time(TCS) and Barycentric Coordinate Time(TCB).The post-Newtonian equations of the Moon's rotation are written in the SCRS,and they are integrated numerically.We calculate the correction to the rotation of the Moon due to total relativistic torque which includes post-Newtonian and gravitomagnetic torques as well as geodetic precession.We find two dominant periods associated with this correction:18.6 yr and 80.1 yr.In addition,the precession of the rotating axes caused by fourth-degree and fifth-degree harmonics of the Moon is also analyzed,and we have found that the main periods of this precession are 27.3 d,2.9 yr,18.6 yr and 80.1 yr.     

The precession and nutation of deformable bodies,III

In preceding papers of this series (Kopal, 1968; 1969) the Eulerian equations have been set up which govern the precession and nutation of self-gravitating fluid globes of arbitrary structures in inertial coordinates (space-axes) as well as with respect to the rotating body axes; with due account being taken of the effects arising from equilibrium as well as dynamical tides.In Section 1 of the present paper, the explicit form of these equations is recapitulated for subsequent solations. Section 2 contains then a detailed discussion of the coplanar case (in which the equation of the rotating configuration and the plane of its orbit coincide with the invariable plane of the system); and small fluctuations in the angular velocity of axial rotation arising from the tidal breathing in eccentric binary systems are investigated.In Section 3, we consider the angular velocity of rotation about theZ-axis to be constant, but allow for finite inclination of the equator to the orbital plane. The differential equations governing such a problem are set up exactly in terms of the time-dependent Eulerian angles and , and their coefficients averaged over a cycle. In Section 4, these equations are linearized by the assumption that the inclinations of the equator and the orbit to the invariable plane of the system are small enough for their squares to be negligible; and the equations of motion reduced to their canonical form.The solution of these equations — giving the periods of precession and nutation of rotating components of close binary systems, as well as the rate of nodal regression which is synchronised with precession — are expressed in terms of the physical properties of the respective system and of its constituent components; while the concluding Section 6 contains a discussion of the results, in which the differences between the precession and nutation of rigid and fluid bodies are pointed out.     

Angular momentum and gyroscope in flat space-time theory of gravitation

The study of a previously proposed theory of gravitation in flat space-time (Petry, 1981a) is continued. A conservation law for the angular momentum is derived. Additional to the usual form, there must be added a term coming from the spin of the gravitational field. The equations of motion and of spin angular momentum for a spinning test particle in a gravitational field are given. An approximation of the equations of the spin angular momentum in the rest frame of the test particle is studied. For a gyroscope in an orbit of a rotating massive body (e.g., the Earth) the precession of the spin axis agrees with the result of Einstein's general theory of relativity.     

Rotating ‘star-in-a-box’ experiments

Using the radiation hydrodynamics code CO⁵BOLD in its ‘star-in-a-box’ setup, we have performed exploratory simulations of global convection in a rotating reference frame. The goal is to study the interaction of convection and rotation by direct numerical simulation. For these first experiments, we chose an idealized configuration (a scaled-down, fast rotating Sun) whose properties resemble those of red supergiants in some respect. We describe the setup and time evolution of these models, and discuss the particular problems we have encountered. Finally, we derive the resulting differential rotation pattern and meridional flow field by temporal and azimuthal averaging of the simulation data. We find anti-solar differential rotation for all cases studied so far. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determining the orientation and spin period of TOPEX/Poseidon satellite by a photometric method

We present the results of photometric observations of the TOPEX/Poseidon satellite performed during 2008–2016. The satellite become space debris after a failure in January, 2006, in a low Earth orbit. In the Laboratory of Space Research of Uzhhorod National University 73 light curves of the spacecraft were obtained. Standardization of photometric light curves is briefly explained. We have calculated the color indices of reflecting surfaces and the spin rate change. The general tendency of the latter is described by an exponential decay function. The satellite spin periods based on 126 light curves (including 53 light curves from the MMT-9 project operating since 2014) were taken into account. In 2016 the period of its own rotation reached its minimum of 10.6 s.A method to derive the direction of the spin axis of an artificial satellite and the angles of the light scattered by its surface has been developed in the Laboratory of Space Research of Uzhhorod National University. We briefly describe the “Orientation” program used for these purposes. The orientation of the TOPEX/Poseidon satellite in mid-2016 is given. The angle of precession β = 45°–50° and period of precession P _pr = 141.5 s have been defined. The reasons for the identified nature of the satellite’s own rotation have been found. They amount to the perturbation caused by a deviation of the Earth gravity field from a central-symmetric shape and the presence of moving parts on the satellite.     

On the local magnetorotational instability of astrophysical jets

We present the local linear stability analysis of rotating jets confined by a toroidal magnetic field. Under the thin flux tube approximation, we derive the equation of motion for slender magnetic flux tubes. In addition to the terms responsible for the conventional instability of the toroidal magnetic field, a term related to the magnetic buoyancy and a term corresponding to the differential rotation become relevant for the stability properties. We find that the rigid rotation stabilizes while the differential rotational destabilizes the jet in a way similar to the Balbus–Hawley instability. Within the frame of our local analysis, we find that if the azimuthal velocity is of the order of or higher than the Alfvén azimuthal speed, the rigidly rotating part of the jet interior can be completely stabilized, while the strong shearing instability operates in the transition layer between the rotating jet interior and the external medium. This can explain the limb-brightening effect observed in several jets. However, it is still possible to find jet equilibria that are stable all across the jet, even in the presence of differential rotation. We discuss observational consequences of these results.     

Secular resonance,solar spin down,and the orbit of Mercury

A mechanism capable of accounting for the large mean eccentricity (0.175) and inclination (7°.2) of Mercury is discussed. Provided the gravitational field of the rapidly rotating primordial Sun had a sufficiently large second degree harmonic (i.e., J₂ ? order 10^?3), subsequent solar spin down would drive the orbit of Mercury through two secular resonances with Venus, one involving the precession of the line of apsides, the other one involving the regression of the nodal line. Resonance passage generates contributions to the eccentricity and inclination that are proportional to the square root of the characteristic solar spin down time. We find that an initial solar rotation l period of $\text{P ? 5}\text{1}\text{2}\text{hr}$ guarantees passage through resonance and that a spin down time of $\text{τ = Ω|dΩ/dt|}{}^{\mathrm{?1}}$ of order 10⁶ years could have produced the observed eccentricity and inclination. Such a primordial rotation rate is comparable to the measured rotations of very young stars and the spin down time appears consistent with the time scale derived for magnetic braking of the Sun's rotation by an intense solar wind during a T-Tauri stage of solar evolution.     

A Test on Two Codes for Extrapolating Solar Linear Force-free Magnetic Fields

We evaluate the performances of two newly-implemented codes for extrapolating the solar linear force-free magnetic fields, by measuring their quantified responses to the lower boundary vector field data on a finite region using analytical models. The codes are based on two boundary integral formulas with different mechanisms in utilizing the transverse boundary field: the first one only utilizes the transverse boundary field to derive the value of the force-free parameter, while the other one explicitly utilizes the whole transverse boundary field, in addition to the vertical field component. Studies on the test cases show that both of the codes could reproduce the analytical model fields with reasonable accuracy within the valid domain, provided a sufficient amount of data is available at the lower boundary. The code utilizing explicitly all three components of the boundary field shows generally better performances and requires a smaller boundary-data area in order to achieve the same degree of accuracy in the metric quantities such as the normalized vector error, vector correlation, vector angular difference, and magnetic energy; however, the accuracy in the divergence-free metric and especially the force-free metric are less ideal.     

Normal modes of synchronous rotation

The dynamics of synchronous rotation and physical librations are revisited in order to establish a conceptually simple and general theoretical framework applicable to a variety of problems. Our motivation comes from disagreements between the results of numerical simulations and those of previous theoretical studies, and also because different theoretical studies disagree on basic features of the dynamics. We approach the problem by decomposing the orientation matrix of the body into perfectly synchronous rotation and deviation from the equilibrium state. The normal modes of the linearized equations are computed in the case of a circular satellite orbit, yielding both the periods and the eigenspaces of three librations. Libration in longitude decouples from the other two, vertical modes. There is a fast vertical mode with a period very close to the average rotational period. It corresponds to tilting the body around a horizontal axis while retaining nearly principal-axis rotation. In the inertial frame, this mode appears as nutation and free precession. The other vertical mode, a slow one, is the free wobble. The effects of the nodal precession of the orbit are investigated from the point of view of Cassini states. We test our theory using numerical simulations of the full equations of the dynamics and discuss the disagreements among our study and previous ones. The numerical simulations also reveal that in the case of eccentric orbits large departures from principal-axis rotation are possible due to a resonance between free precession and wobble. We also revisit the history of the Moon's rotational state and show that it switched from one Cassini state to another when it was at 46.2 Earth radii. This number disagrees with the value 34.2 derived in a previous study.     

Dissipation of energy through tidal deformation

The aim of the present paper will be to derive an equation of dissipation of energy for a rotating body of arbitrary viscosity distorted by tides, which arise from the gravitational field of its companion in a close pair of such bodies.By a transformation of the fundamental equation of energy dissipation in terms of velocity of tidal deformation (Section 2), the dissipation function is constructed for a tidally-distorted body (Section 3). From this equation, the rate of dissipation of tidal energy is formulated for a nearly-spherical rotating body distorted by second harmonic longitudinal tides (Section 4); the coefficients of viscosity (or the bulk modulus) are treated as arbitrary functions of spatial coordinates. Finally (Section 5), expressions for the total energy dissipation within the orbital cycle are given for axial rotation of the distorted body, provided its angular velocity is constant (for example, with the Keplerian angular velocity).Research financed in part by the Division of Scientific Research and Development of Ministry of Sciences and Culture of Greece.     

YORP-Induced Long-Term Evolution of the Spin State of Small Asteroids and Meteoroids: Rubincam's Approximation

The rotation states of small asteroids and meteoroids are determined primarily by their collisions, gravitational torques due to the Sun and planets (in the case of close encounters), and internal dissipative effects (that relax the free-precession energy toward the fundamental state of principal-axis rotation). Rubincam has recently pointed out that thermal reemission on irregular-shaped bodies also results in a torque that may secularly change both the rotation rate and the orientation of the spin axis (the so-called YORP effect). Here we pursue investigation of this effect. Keeping the zero thermal-relaxation approximation of Rubincam and the assumption of the principal-axis rotation, we study the YORP effect both for precisely determined shapes of near-Earth asteroids and also for a large statistical sample of automatically generated shapes by the Gaussian-sphere technique of Muinonen. We find that the asymptotic state of the YORP evolution is characterized by an arbitrary value of the obliquity, with higher but nearly equal likelihood of 0°/180° and 90° states. At the adopted approximation, the most typical feature of this end state of the YORP evolution is secular deceleration of the rotation rate, which means that at some instant collisions will randomize the rotation state. In a minority of cases, the final state of the obliquity evolution leads to a permanent acceleration of the body's rotation, eventually resulting in rotational fission. The YORP-induced slow evolution may also play an important role in driving the rotation state of small asteroids toward the resonances between the forced precession due to the solar torque and perturbations of the orbital node and inclination. We find that for small Themis asteroids these resonances are isolated in the relevant range of frequencies, and the YORP evolving rotation may be either temporarily captured or rapidly jump across these resonances. In contrast, the possible values of the forced precession for small Flora asteroids may be resonant with clustered, nonisolated lines of the orbital perturbation. The individual rotation histories of small Flora asteroids may be thus very complicated and basically unpredictable. We comment on possible astronomical consequences of these results.     

Waves and instabilities in a differentially rotating disc containing a poloidal magnetic field

The theory of waves and instabilities in a differentially rotating disc containing a poloidal magnetic field is developed within the framework of ideal magnetohydrodynamics. A continuous spectrum, for which the eigenfunctions are localized on individual magnetic surfaces, is identified but is found not to contain any instabilities associated with differential rotation. The normal modes of a weakly magnetized thin disc are studied by extending the asymptotic methods used previously to describe the equilibria. Waves propagate radially in the disc according to a dispersion relation which is determined by solving an eigenvalue problem at each radius. The dispersion relation for a hydrodynamic disc is re-examined and the modes are classified according to their behaviour in the limit of large wavenumber. The addition of a magnetic field introduces new, potentially unstable, modes and also breaks up the dispersion diagram by causing avoided crossings. The stability boundary to the magnetorotational instability in the parameter space of polytropic equilibria is located by solving directly for marginally stable equilibria. For a given vertical magnetic field in the disc, bending of the field lines has a stabilizing effect and it is shown that stable equilibria exist which are capable of launching a predominantly centrifugally driven wind.     

Tidal dissipation at arbitrary eccentricity and obliquity

Expressions for tidal dissipation in a body in synchronous rotation at arbitrary orbital eccentricity and obliquity are derived. The rate of tidal dissipation for a synchronously rotating body is compared to that in a body in asymptotic nonsynchronous rotation.     

Non-linear dynamics of the corotation torque

The excitation of spiral waves by an external perturbation in a disc deposits angular momentum in the vicinity of the corotation resonance (the radius where the speed of a rotating pattern matches the local rotation rate). We use matched asymptotic expansions to derive a reduced model that captures non-linear dynamics of the resulting torque and fluid motions. The model is similar to that derived for forced Rossby wave critical layers in geophysical fluid dynamics. Using the model we explore the saturation of the corotation torque, which occurs when the background potential (specific) vorticity is redistributed by the disturbance. We also consider the effects of dissipation. If there is a radial transport of potential vorticity, the corotation torque does not saturate. The main application is to the creation, growth and migration of protoplanets within discs like the primordial solar nebula. The disturbance also nucleates vortices in the vicinity of corotation, which may spark further epochs of planet formation.