Axially symmetric solution in teleparallel equivalent of general relativity, energy, momentum and angular momentum

We apply the energy-momentum tensor which is coordinate independent, of the gravitational field established in the Hamiltonian structure of the teleparallel equivalent of general relativity (TEGR), to an axially symmetric tetrad field to calculate energy, momentum and angular momentum. Also the definition of the gravitational energy is used to investigate the energy within the external event horizon of this tetrad.     

Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite

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.     

Hot accretion flow with ordered magnetic field, outflow, and saturated conduction

The importance of thermal conduction on hot accretion flow is confirmed by observations of hot gas that surrounds Sgr A^? and a few other nearby galactic nuclei. On the other hand, the existence of outflow in accretion flows is confirmed by observations and magnetohydrodynamic (MHD) simulations. In this research, we study the influence of both thermal conduction and outflow on hot accretion flows with ordered magnetic field. Since the inner regions of hot accretion flows are, in many cases, collisionless with an electron mean free path due to Coulomb collision larger than the radius, we use a saturated form of thermal conduction, as is appropriate for weakly collisional systems. We also consider the influence of outflow on accretion flow as a sink for mass, and the radial and the angular momentum, and energy taken away from or deposited into the inflow by outflow. The magnetic field is assumed to have a toroidal component and a vertical component as well as a stochastic component. We use a radially self-similar method to solve the integrated equations that govern the behavior of such accretion flows. The solutions show that with an ordered magnetic field, both the surface density and the sound speed decrease, while the radial and angular velocities increase. We found that a hot accretion flow with thermal conduction rotates more quickly and accretes more slowly than that without thermal conduction. Moreover, thermal conduction reduces the influences of the ordered magnetic field on the angular velocities and the sound speed. The study of this model with the magnitude of outflow parameters implies that the gas temperature decreases due to mass, angular momentum, and energy loss. This property of outflow decreases for high thermal conduction.     

Thin-Shell Magnetohydrodynamic Equations for the Solar Tachocline

We present a new system of equations designed to study global-scale dynamics in the stably-stratified portion of the solar tachocline. This system is derived from the 3D equations of magnetohydrodynamics in a rotating spherical shell under the assumption that the shell is thin and stably-stratified (subadiabatic). The resulting thin-shell model can be regarded as a magnetic generalization of the hydrostatic primitive equations often used in meteorology. It is simpler in form than the more general anelastic or Boussinesq equations, making it more amenable to analysis and interpretation and more computationally efficient. However, the thin-shell system is still three-dimensional and as such represents an important extension to previous 2D and shallow-water approaches. In this paper we derive the governing equations for our thin-shell model and discuss its underlying assumptions, its context relative to other models, and its application to the solar tachocline. We also demonstrate that the dissipationless thin-shell system conserves energy, angular momentum and magnetic helicity.     

Solar wind energy transfer regions inside the dayside magnetopause: Accelerated heavy ions as tracers for MHD-processes in the dayside boundary layer

Plasma and magnetic field data from PROGNOZ-7 have revealed that solar wind (magnetosheath) plasma elements may penetrate the dayside magnetopause surface and form high density regions with enhanced cross-field flow in the boundary layer.The injected magnetosheath plasma is observed to have an excess drift velocity as compared to the local boundary layer plasma, comprising both “cold” plasma of terrestrial origin and a hot ring current component. A differential drift between two plasma components can be understood in terms of a momentum transfer process driven by an injected magnetosheath plasma population. The braking action of the injected plasma may be described as a dynamo process where particle kinetic energy is transferred into electromagnetic energy (electric field). The generated electric field will force the local plasma to $\text{ε×}\text{B}\text{-drift}$, and the dynamo region therefore also constitutes an accelerator region for the local plasma. Whenever energy is dissipated from the energy transfer process (a net current is flowing through a load), there will also be a difference between the induced electric field and the $\text{v}\text{×}\text{B}$ term of the generator plasma. Thus, the local plasma will drift more slowly than the injected generator plasma.We will present observations showing that a relation between the momentum transferred, the injected plasma and the momentum taken up by the local plasma exists. For instance, if the local plasma density is sufficiently high, the differential drift velocity of the injected and local plasma will be small. A large fraction of the excess momentum is then transferred to the local plasma. Conversely, a low local plasma density results in a high velocity difference and a low fraction of local momentum transfer.In our study cases the “cold” plasma component was frequently found to dominate the local magnetospheric plasma density in the boundary layer. Accordingly, this component may have the largest influence on the local momentum transfer process. We will demonstrate that this also seems to be the case. Moreover we show that the accelerated “cold” plasma component may be used as a tracer element reflecting both the momentum and energy transfer and the penetration process in the dayside boundary layer.The high He⁺ percentage of the accelerated “cold” plasma indicates a plasmaspheric origin. Considering the quite high densities of energetic He⁺ found in the boundary layer, the overall low abundance of He⁺ (as compared to e.g. O⁺) found in the plasma sheet and outer ring current evidently reduces the importance of the dayside boundary layer as a plasma source in the large scale magnetospheric circulation system.     

The origin of the density distribution of disc galaxies: a new problem for the standard model of disc formation

We present new models for the formation of disc galaxies that improve upon previous models by following the detailed accretion and cooling of the baryonic mass, and by using realistic distributions of specific angular momentum. Under the assumption of detailed angular momentum conservation, the discs that form have density distributions that are more centrally concentrated than an exponential. We examine the influence of star formation, bulge formation, and feedback on the outcome of the surface brightness distributions of the stars. Low angular momentum haloes yield disc galaxies with a significant bulge component and with a stellar disc that is close to exponential, in good agreement with observations. High angular momentum haloes, on the other hand, produce stellar discs that are much more concentrated than an exponential, in clear conflict with observations. At large radii, the models reveal distinct truncation radii in both the stars and the cold gas. The stellar truncation radii result from our implementation of star formation threshold densities, and are in excellent agreement with observations. The truncation radii in the density distribution of the cold gas reflect the maximum specific angular momentum of the gas that has cooled. We find that these truncation radii occur at H  i surface densities of roughly 1 M_⊙ pc⁻², in conflict with observations. We examine various modifications to our models, including feedback, viscosity, and dark matter haloes with constant-density cores, but show that the models consistently fail to produce bulge less discs with exponential surface brightness profiles. This signals a new problem for the standard model of disc formation: if the baryonic component of the protogalaxies out of which disc galaxies form has the same angular momentum distribution as the dark matter, discs are too compact.     

Mean zonal and meridional accelerations and mean heating induced by solar tides for equinox and solstice conditions

Evaluations are presented of the momentum and energy flux divergences of the diurnal and semidiurnal tidal fields calculated by Forbes (1982a, b) from 0 to 400 km altitude. Results are presented in the form of meridional cross-sections from 0 to 78°N or S latitude with a 6° latitude interval. Comparisons are made with evaluations of the momentum flux divergences of the diurnal tide by Miyahara (1981, 1983) and good agreement is obtained in the lower thermosphere (below about 130 km) but a large disparity arises in the upper thermosphere. In the lower thermosphere momentum flux divergences of the semidiurnal tide are comparable with those of the diurnal tide and should be included in general circulation calculations of the 90–120 km region.     

Energy Distribution of a Gödel-Type Space–Time

We calculate the energy and momentum distributions associated with a Gödel-type space–time, using the well-known energy–momentum complexes of Landau–Lifshitz and Møller. We show that the definitions of Landau–Lifshitz and Møller do not furnish a consistent result.     

N-body integrators with individual time steps from Hierarchical splitting

We review the implementation of individual particle time-stepping for N-body dynamics. We present a class of integrators derived from second order Hamiltonian splitting. In contrast to the usual implementation of individual time-stepping, these integrators are momentum conserving and show excellent energy conservation in conjunction with a symmetrized time step criterion. We use an explicit but approximate formula for the time symmetrization that is compatible with the use of individual time steps. No iterative scheme is necessary. We implement these ideas in the HUAYNO¹ code and present tests of the integrators and show that the presented integration schemes shows good energy conservation, with little or no systematic drift, while conserving momentum and angular momentum to machine precision for long term integrations.     

Possible conditions for formation of satellite from protoearth by ejection of material

Conditions for the formation of a large gaseous protoearth and the formation of a satellite are considered. The energy required is assumed to come from the gravitational contraction to form the Sun, the gravitational contraction to form the protoearth, core overturn and planetesimal impacts on the protoearth. The model is compatible with geochemical, tidal and angular momentum data.     

Mass-losing accretion discs around supermassive black holes

We study the effects of outflow/wind on the gravitational stability of accretion discs around supermassive black holes using a set of analytical steady-state solutions. Mass-loss rate by the outflow from the disc is assumed to be a power-law of the radial distance and the amount of the energy and the angular momentum which are carried away by the wind are parameterized phenomenologically. We show that the mass of the first clumps at the self-gravitating radius linearly decreases with the total mass-loss rate of the outflow. Except for the case of small viscosity and high accretion rate, generally, the self-gravitating radius increases as the amount of mass-loss by the outflow increases. Our solutions show that as more angular momentum is lost by the outflow, then reduction to the mass of the first clumps is more significant.     

Angular momentum and kinetic energy of rotating deformable bodies

The general equations of angular momentum and kinetic energy of a rotating deformable (or not rigid) body are discussed for a fixed and a rotating coordinate system. A new system of equations is developed for a deformable body of arbitrary form using the Lagrangian (vector) cisplacement up to the first order terms. The equations are, then, illustrated for a self-gravitating ceformable body perturbed by tides.     

The distribution function of dark matter in massive haloes

We study the distribution function (DF) of dark matter particles in haloes of mass range  10¹⁴–10¹⁵ M_⊙  . In the numerical part of this work we measure the DF for a sample of relaxed haloes formed in the simulation of a standard Λ cold dark matter (ΛCDM) model. The DF is expressed as a function of energy E and the absolute value of the angular momentum L , a form suitable for comparison with theoretical models. By proper scaling we obtain the results that do not depend on the virial mass of the haloes. We demonstrate that the DF can be separated into energy and angular momentum components and propose a phenomenological model of the DF in the form     . This formulation involves three parameters describing the anisotropy profile in terms of its asymptotic values (β₀ and  β_∞  ) and the scale of transition between them ( L ₀). The energy part   f _E ( E )  is obtained via inversion of the integral for spatial density. We provide a straightforward numerical scheme for this procedure as well as a simple analytical approximation for a typical halo formed in the simulation. The DF model is extensively compared with the simulations: using the model parameters obtained from fitting the anisotropy profile, we recover the DF from the simulation as well as the profiles of the dispersion and kurtosis of radial and tangential velocities. Finally, we show that our DF model reproduces the power-law behaviour of phase-space density   Q =ρ( r )/σ³( r )  .     

A massive thick disc around a massive black hole and its runaway instability

We investigate the runaway instability of configurations consisting of a massive dense but non-self-gravitating thick disc/torus surrounding a massive black hole (MBH). We limit our model parameters to values that result in a self-consistent thick disc around an MBH. We identify, analytically, the index of the angular momentum distribution that will form a thick disc as an initial equilibrium state, and obtain the mass ratio of the disc to the central black hole for which the disc is dominated by the radiation pressure. In our theoretical framework we find that a self-consistent thick disc with constant angular momentum leads to a runaway instability on a dynamical time-scale. However, even a slight increase of the specific angular momentum outwards has a strong stabilizing effect on the accretion process. Finally, we discuss our results and present possible applications to high-energy emission.     

Energy and stability in the Full Two Body Problem

The conditions for relative equilibria and their stability in the Full Two Body Problem are derived for an ellipsoid–sphere system. Under constant angular momentum it is found that at most two solutions exist for the long-axis solutions with the closer solution being unstable while the other one is stable. As the non-equilibrium problem is more common in nature, we look at periodic orbits in the F2BP close to the relative equilibrium conditions. Families of periodic orbits can be computed where the minimum energy state of one family is the relative equilibrium state. We give results on the relative equilibria, periodic orbits and dynamics that may allow transition from the unstable configuration to a stable one via energy dissipation.