Collisional heating of interplanetary gas: Fokker-Planck treatment

Interstellar gas streaming through the solar system undergoes both elastic collisions with solar wind ions and destructive, ionizing processes. The Boltzmann equation is set up, with linear Fokker-Planck terms describing the glancing elastic collisions. Solutions combining the dynamical effects of the central force field and the diffusion in velocity space are derived, appropriate to cool gas.For the He component of the streaming gas, if initially at 100 K, the collisional heating dominates inside 2 a.u. upstream and 5 a.u. downstream. A modified formula is given for the density in the downstream wake, as enhanced by gravitational focussing. Calculations of the helium resonant radiation backscatter require substantial modification.     

Unified analytical solutions to two-body problems with drag

The two-body problem with a generalized Stokes drag is discussed. The drag force is proportional to the product of the velocity vector and the inverse square of the distance. The generalization consists of allowing two different proportionality constants for the radial and the transverse components of the force. Under the 'generalized Robertson transformation', the equation of the orbit takes the form of the Lommel equation and admits solutions in terms of Bessel and Lommel functions. The exact, analytical solutions for this type of drag reveal a paradoxical effect of increasing eccentricity for all trajectories. The Poynting–Robertson drag and Poynting–Plummer–Danby problems are discussed as particular cases of the general solution.     

Stochastic diffusion in inverse square fields: General formulation for interplanetary gas

The Fokker-Planck equation for small stochastic changes to particles in Kepler orbits has to be formulated in terms of the integrals of motion. We generalize the modelling of proton and electron collisional perturbations to gas particles on trajectories through the solar system in order to include both spatial and velocity diffusion. The general solution is obtained in terms of a 4-dimensional normal distribution. Treatment of the singularity in the Fokker-Planck operator reduces the dimensionality by one. In addition to extending earlier results for anisotropic collisional heating in the thermal approximation, the present formulation gives the changes in density due to the mean repulsive force and to perturbations of trajectories (spatial diffusion). The net diffusion is almost everywhere towards the sun and the density increase is significant in the downstream hydrogen wake, particularly where destructive depletion is strong and gravitational focussing weak.     

Protodiscs around hot magnetic rotator stars

We develop equations and obtain solutions for the structure and evolution of a protodisc region that is initially formed with no radial motion and super-Keplerian rotation speed when wind material from a hot rotating star is channelled towards its equatorial plane by a dipole-type magnetic field. Its temperature is around 10⁷ K because of shock heating and the inflow of wind material causes its equatorial density to increase with time. The centrifugal force and thermal pressure increase relative to the magnetic force and material escapes at its outer edge. The protodisc region of a uniformly rotating star has almost uniform rotation and will shrink radially unless some instability intervenes. In a star with angular velocity increasing along its surface towards the equator, the angular velocity of the protodisc region decreases radially outwards and magnetorotational instability (MRI) can occur within a few hours or days. Viscosity resulting from MRI will readjust the angular velocity distribution of the protodisc material and may assist in the formation of a quasi-steady disc. Thus, the centrifugal breakout found in numerical simulations for uniformly rotating stars does not imply that quasi-steady discs with slow outflow cannot form around magnetic rotator stars with solar-type differential rotation.     

Supersonic motions of galaxies in clusters

We study motions of galaxies in galaxy clusters formed in the concordance Λ cold dark matter cosmology. We use high-resolution cosmological simulations that follow the dynamics of dark matter and gas and include various physical processes critical for galaxy formation: gas cooling, heating and star formation. Analysing the motions of galaxies and the properties of intracluster gas in a sample of eight simulated clusters at z = 0, we study the velocity dispersion profiles of the dark matter, gas and galaxies. We measure the mean velocity of galaxy motions and gas sound speed as a function of radius and calculate the average Mach number of galaxy motions. The simulations show that galaxies, on average, move supersonically with the average Mach number of ≈1.4, approximately independent of the cluster-centric radius. The supersonic motions of galaxies may potentially provide an important source of heating for the intracluster gas by driving weak shocks and via dynamical friction, although these heating processes appear to be inefficient in our simulations. We also find that galaxies move slightly faster than the dark matter particles. The magnitude of the velocity bias,   b _v ≈ 1.1  , is, however, smaller than the bias estimated for subhaloes in dissipationless simulations. Interestingly, we find velocity bias in the tangential component of the velocity dispersion, but not in the radial component. Finally, we find significant random bulk motions of gas. The typical gas velocities are of order ≈20–30 per cent of the gas sound speed. These random motions provide about 10 per cent of the total pressure support in our simulated clusters. The non-thermal pressure support, if neglected, will bias measurements of the total mass in the hydrostatic analyses of the X-ray cluster observations.     

On acceleration and motion of ions in corona and solar wind

Assuming a stationary, radial, spherically symmetric solar wind and a radial magnetic field direction in the vicinity of the sun, an equation of motion for ions heavier than protons in the solar wind is derived. The general properties of this equation are discussed and the results of numerical integrations are given. These results are based on the assumption of maxwellian velocity distribution functions for electrons, protons and ions, but the effects of first order deviations from such distributions are also presented and discussed. It is shown that dynamical friction, i.e. momentum transfer from protons to heavier ions accounts for the observed fact that heavier ions - if accelerated at all - normally reach the same velocity as the protons in the solar wind. Because of the non-linear relation between dynamical friction and proton-ion velocity difference a minimum proton flux is required to carry a certain ion species in the solar wind. Formulae comparing the minimum fluxes for different ions are given. It is shown that elements up to and beyond iron will be carried along in the solar wind as long as helium is carried along. Substantial isotopic fractionation is possible, in particular in the case of helium. The effects of ion motion and escape on abundances in the corona and in the outer convective zone of the sun are discussed.     

Velocity dispersion in particulate disks with size distribution

Quasi-equilibrium solutions for the pre-planetary disk are studied in terms of Hämeen-Anttila's theory (1984) of collisional, self-gravitating systems. The distribution of particle sizes is assumed to follow simple power-law distributions, with a power index in the range of 1.5–5.0. The treatment includes mutual impacts with a velocity dependent coefficient of restitution, as well as gravitational encounters with dynamical friction. The mean gravitational field of the disk is also taken into account. The results indicate that the energy(equi)-partition depends mainly on the index of size distribution, but is also affected by the optical thickness of the system, as well as on the vertical thickness as compared to the particle size. The vertical component of the gravitational field is found to be important, especially when the mass of the system is concentrated on the large particles.     

Models of Motion in a Central Force Field with Quadratic Drag

Some general properties for the motion of a particle in a central force field with a general power law drag are derived. Exact and approximate solutions of the equations of motion are found in various cases. Emphasis is placed on inverse square gravitation and drag that varies with the square of the speed and inversely with the distance from the center of attraction. For this model two results stand out. The first is a particular solution in closed form that demonstrates the decay of an initially circular orbit under drag. The second, found from an approximation of the equations of motion when the radial speed is small compared with the tangential speed, demonstrates the decay of an initially elliptic orbit that is not highly eccentric. Formulas for calculation of the time of flight are presented for the two principal results.     

The height distribution of the kinetic temperature and turbulent velocity of solar Hα spicules

The height distribution of the kinetic temperature of solar H spicules is determined using the widths of optically thin hydrogen and metallic lines obtained at the total solar eclipse of 1966: the temperature was found to be 8600 K at the height of 2200 km measured from the radial optical depth of unity at 5000 Å, and to decrease to a minimum of 5000 K ± 180 K at 3200 km, and to increase again to 8200 K at 6000 km.The height distribution of the non-thermal turbulent velocity is also determined and is shown to be consistent with the neutral helium line widths emitted at the kinetic temperature of 5000–8000 K.     

Self-similar solution of hot accretion flows with ordered magnetic field and outflow

Observations and numerical magnetohydrodynamic (MHD) simulations indicate the existence of outflows and ordered large-scale magnetic fields in the inner region of hot accretion flows. In this paper, we present the self-similar solutions for advection-dominated accretion flows (ADAFs) with outflows and ordered magnetic fields. Stimulated by numerical simulations, we assume that the magnetic field has a strong toroidal component and a vertical component in addition to a stochastic component. We obtain the self-similar solutions to the equations describing the magnetized ADAFs, taking into account the dynamical effects of the outflow. We compare the results with the canonical ADAFs and find that the dynamical properties of ADAFs such as radial velocity, angular velocity and temperature can be significantly changed in the presence of ordered magnetic fields and outflows. The stronger the magnetic field is, the lower the temperature of the accretion flow will be and the faster the flow rotates. The relevance to observations is briefly discussed.     

Collisional evolution of Trans-Neptunian populations: Effects of fragmentation physics and estimates of the abundances of gravitational aggregates

The Trans-Neptunian region is yet another example of a collisional system of small bodies in the Solar System. In the last decade the number of TNOs with reliable orbital elements is steadily increasing and even if it is still premature to compare models with observations, we can start to have some idea of the orbital structure and magnitude distribution, so that some loose constraints may be set on the critical parameters that affect collisional evolution. With this aim we have developed a model for the collisional evolution of the Trans-Neptunian region by dividing it into three main different populations, corresponding to the dynamical classification proposed by Gladman et al. [2001.The structure of the Kuiper Belt: size distribution and radial extent. Astrophys. J. 122, 1051] (Resonant region, Classical Belt and Scattered Disk). A multi-zone collisional model is developed, in which each zone can collisionally interact with each other. The model takes into account the known physics of the fragmentation of icy/rocky bodies at the typical relative velocities of TNOs, according to velocity distributions corresponding to each evolving zone. The dependence of the evolution of the considered populations on physically critical collisional parameters is investigated and the corresponding results are presented, including estimates of the abundance of gravitational aggregates in the studied populations.     

Evolution of a Keplerian disk of colliding and fragmenting particles: a kinetic model with application to the Edgeworth-Kuiper belt

We present a kinetic model of a disk of solid particles, orbiting a primary and experiencing inelastic collisions. In distinction to other collisional models that use a 2D (mass-semimajor axis) binning and perform a separate analysis of the velocity (eccentricity, inclination) evolution, we choose mass and orbital elements as independent variables of a phase space. The distribution function in this space contains full information on the combined mass, spatial, and velocity distributions of particles. A general kinetic equation for the distribution function is derived, valid for any set of orbital elements and for any collisional outcome, specified by a single kernel function. The first implementation of the model utilizes a 3D phase space (mass-semimajor axis-eccentricity) and involves averages over the inclination and all angular elements. We assume collisions to be destructive, simulate them with available material- and size-dependent scaling laws, and include collisional damping. A closed set of kinetic equations for a mass-semimajor axis-eccentricity distribution is written and transformation rules to usual mass and spatial distributions of the disk material are obtained. The kinetic “core” of our approach is generic. It is possible to add inclination as an additional phase space variable, to include cratering collisions and agglomeration, dynamical friction and viscous stirring, gravity of large perturbers, drag forces, and other effects into the model. As a specific application, we address the collisional evolution of the classical population in the Edgeworth-Kuiper belt (EKB). We run the model for different initial disk's masses and radial profiles and different impact strengths of objects. Our results for the size distribution, collisional timescales, and mass loss are in agreement with previous studies. In particular, collisional evolution is found to be most substantial in the inner part of the EKB, where the separation size between the survivors over EKB's age and fragments of earlier collisions lies between a few and several tens of km. The size distribution in the EKB is not a single Dohnanyi-type power law, reflecting the size dependence of the critical specific energy in both strength and gravity regimes. The net mass loss rate of an evolved disk is nearly constant and is dominated by disruption of larger objects. Finally, assuming an initially uniform distribution of orbital eccentricities, we show that an evolved disk contains more objects in orbits with intermediate eccentricities than in nearly circular or more eccentric orbits. This property holds for objects of any size and is explained in terms of collisional probabilities. The effect should modulate the eccentricity distribution shaped by dynamical mechanisms, such as resonances and truncation of perihelia by Neptune.     

Radial Velocity and Light Curves Analysis of the Contact Binary V839 Ophiuchi

Complete UBV light curves of the W Ursae Majoris binary V839 Ophobtained in the year 2000 are presented. The available spectroscopic data of V839 Oph is new and we used the first radial velocity data of this system obtained by Rucinski and Lu (1999)for analysis. The radial velocity and light curves analysis was made with the latest version of Wilson programme (1998) and the geometric and physical elements of the system are derived. By searching the simultaneous solutions of the system we have determined the masses and radii of the components: 1.61M _⊙and 1.49402R _⊙ for the primary component; 0.50M _⊙and 0.90147R _⊙ for the secondary component. We estimate deffective temperatures of 6650±18 (K) for the primary and6554±15 (K) for the secondary component. This revised version was published online in July 2006 with corrections to the Cover Date.     

Radially accelerated optimal feedback orbits in central gravity field with linear drag

The solution of a feedback optimal control problem arising in orbital mechanics is addressed in this paper. The dynamics is that of a massless body moving in a central gravitational force field subject also to a drag and a radial modulated force. The drag is linearly proportional to the velocity and inversely proportional to the square of the distance from the center of attraction. The problem is tackled by exploiting the properties of a suitably devised linearizing map that transforms the nonlinear dynamics into an inhomogeneous linear system of differential equations supplemented by a quadratic objective function. The generating function method is then applied to this new system, and the solution is back transformed in the old variables. The proposed technique, in contrast to the classical optimal control problem, allows us to derive analytic closed-loop solutions without solving any two-point boundary value problem. Applications are discussed.     

Radial velocity and light curves analysis of the eclipsing binary NN Vir

The eclipsing binary NN Vir is a short period system showing an EW‐type light curve. Photometric observations of NN Vir were done by Gomez‐Ferrellad & Garcia‐Melendo (1997) at Esteve Duran Observatory. We used photometric data of NN Vir for light curve analysis. The available spectroscopic data of NN Vir is new and we also used the first radial velocity data of this system obtained by Rusinski & Lu (1999) for analysis. The radial velocity and light curves analysis was made with the latest version ofWilson program(1998) and the geometric and physical elements of the system are derived. By searching the simultaneous solutions of the system, we have determined the masses and radii of the components : 1.89(M_⊙) and 1.65(R_⊙) for the primary component; 0.93(M_⊙) and 1.23(R_⊙) for the secondary component. We estimated effective temperatures of 7030(K) for the primary and 6977(K) for the secondary component. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Eruptive prominences and coronal transients

The observed interrelationship between coronal transients and eruptive prominences is used as the basis for a theoretical MHD model of these events. The model begins with an equilibrium configuration consisting of a coronal loop or arcade with a filament lying underneath with its axis oriented perpendicular to the overlying field. The lifting of the filament from the solar surface produces an increase in magnetic pressure under the helmet which drives it outward. This increased pressure is associated with the internal field of the filament as well as the field beneath it. The underlying field could be that which produced the filament eruption or, alternatively, reconnected field lines formed by the inward collapse of the legs of the transient towards the neutral line beneath the rising prominence. We do not attempt to explain the filament eruption which may be due to internal forces in the prominence or, alternatively, from forces imposed from beneath as would be produced by emerging flux. In the latter case, the filament is passive and merely acts as a tracer for the more fundamental underlying process.It is shown that the outward force per unit mass produced by the driving magnetic field and the inward restoring forces in the overlying field due to magnetic tension and gravity all decrease with distance at the same rate - namely, as the inverse square of the distance from the solar center. Hence, the ratio of net outward to inward force is independent of radial distance from the Sun. A stability analysis shows that this situation is one of neutral stability.A mathematical model of this physical process is described in which the MHD equations in simplified form, neglecting gas pressure forces, are solved in time for the velocity, width, density, and magnetic field strength of the transient. The solutions show that the velocity increases sharply close to the Sun but quickly approaches a constant value. The width increases linearly with radial distance. Both of these results are in agreement with observations. An examination of the forces exerted on the legs of the transient shows that their motion should be horizontally inward.On leave from the High Altitude Observatory, National Center for Atmospheric Research, Boulder, Colo., U.S.A.     

19 low mass hypervelocity star candidates from the first data release of the LAMOST survey

Hypervelocity stars are believed to be ejected out from the Galactic center through dynamical interactions between(binary) stars and the central supermassive black hole(s). In this paper, we report 19 low mass F/G/K type hypervelocity star candidates from over one million stars found in the first data release of the LAMOST regular survey. We determine the unbound probability for each candidate using a MonteCarlo simulation by assuming a non-Gaussian proper-motion error distribution, and Gaussian heliocentric distance and radial velocity error distributions. The simulation results show that all the candidates have unbound possibilities over 50% as expected,and one of them may even exceed escape velocity with over 90% probability. In addition, we compare the metallicities of our candidates with the metallicity distribution functions of the Galactic bulge, disk, halo and globular clusters, and conclude that the Galactic bulge or disk is likely the birth place for our candidates.     

Global distribution of helium in the upper atmosphere during solar minimum

Helium in the Earth's thermosphere traces the dynamical systems that redistribute energy and mass. Measurements of the global helium distribution in the thermosphere, using Atmosphere Explorer satellite C. (AE-C), show a gradual seasonal change in the number density of helium for all latitudes. The enhancement in helium over the winter pole (the helium bulge) changes in magnitude slowly as seasons progress. The bulge builds and recedes following the progression of winter North to South and back again. This progression of the winter helium enhancement is presented in this paper using latitudinal profiles of helium number density for each month during the year. The absolute magnitude of the winter helium enhancement in the auroral regions is affected by auroral heating at low altitudes. The reduction in the winter helium bulge at low altitudes shown in AE-C data can be traced to this localized heating. The gradual variation in helium concentration measured at many latitudes for all seasons of the year implies that global thermospheric wind systems change gradually with the seasons.     

Hyper-elliptic orbits

A survey of classes of elliptical orbits outside the usual ones due to the attractive direct power and inverse square force laws reveals some extremely interesting orbits with surprising dynamical characteristics. Particular elliptical orbits of interest in celestial mechanics are discussed.