Axial tilt angles of active regions

Separate Mount Wilson plage and sunspot group data sets are analyzed in this review to illustrate several interesting aspects of active region axial tilt angles. (1) The distribution of tilt angles differs between plages and sunspot groups in the sense that plages have slightly higher tilt angles, on average, than do spot groups. (2) The distributions of average plage total magnetic flux, or sunspot group area, with tilt angle show a consistent effect: those groups with tilt angles nearest the average values are larger (or have a greater total flux) on average than those farther from the average values. Moreover, the average tilt angles on which these size or flux distributions are centered differ for the two types of objects, and represent closely the actual different average tilt angles for these two features. (3) The polarity separation distances of plages and sunspot groups show a clear relationship to average tilt angles. In the case of each feature, smaller polarity separations are correlated with smaller tilt angles. (4) The dynamics of regions also show a clear relationship with region tilt angles. The spot groups with tilt angles nearest the average value (or perhaps 0-deg tilt angle) have on average a faster rotation rate than those groups with extreme tilt angles.All of these tilt-angle characteristics may be assumed to be related to the physical forces that affect the magnetic flux loop that forms the region. These aspects are discussed in this brief review within the context of our current view of the formation of active region magnetic flux at the solar surface.Dedicated to Cornelis de JagerOperated by the Association of Universities for Research in Astronomy, Inc., under Cooperative Agreement with the National Science Foundation.     

On the relationship between instability and Lyapunov times for the three-body problem

In this study we consider the relationship between the survival time and the Lyapunov time for three-body systems. It is shown that the Sitnikov problem exhibits a two-part power-law relationship as demonstrated by Mikkola & Tanikawa for the general three-body problem. Using an approximate Poincaré map on an appropriate surface of section, we delineate escape regions in a domain of initial conditions and use these regions to analytically obtain a new functional relationship between the Lyapunov time and the survival time for the three-body problem. The marginal probability distributions of the Lyapunov and survival times are discussed and we show that the probability density function of Lyapunov times for the Sitnikov problem is similar to that for the general three-body problem.     

The use of angles and angular rates

Earlier work on the angles and angular rate initial orbit determination problem has been extended to allow the incorporation of arbitrary amounts and mixtures of angles and angular rate data. The statistical estimation technique used is that of Maximum Likelihood. Numerical tests on six widely different satellite orbits indicate that the orbital elements can generally be computed to 1% from data acquired during a single apparition.This work was sponsored by the Department of the Air Force.     

Stability of Co-Orbital Motion in Exoplanetary Systems

The stability of co-orbital motions is investigated in such exoplanetary systems, where the only known giant planet either moves fully in the habitable zone, or leaves it for some part of its orbit. If the regions around the triangular Lagrangian points are stable, they are possible places for smaller Trojan-like planets. We have determined the nonlinear stability regions around the Lagrangian point L₄ of nine exoplanetary systems in the model of the elliptic restricted three-body problem by using the method of the relative Lyapunov indicators. According to our results, all systems could possess small Trojan-like planets. Several features of the stability regions are also discussed. Finally, the size of the stability region around L₄ in the elliptic restricted three-body problem is determined as a function of the mass parameter and eccentricity.     

Spectra of stretching numbers and helicity angles in dynamical systems

We define a stretching number (or Lyapunov characteristic number for one period) (or stretching number) a = In % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaada% Wcaaqaaiabe67a4jaadshacqGHRaWkcaaIXaaabaGaeqOVdGNaamiD% aaaaaiaawEa7caGLiWoaaaa!3F1E!\[\left| {\frac{{\xi t + 1}}{{\xi t}}} \right|\]as the logarithm of the ratio of deviations from a given orbit at times t and t + 1. Similarly we define a helicity angle as the angle between the deviation % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdGNaam% iDaaaa!3793!\[\xi t\]and a fixed direction. The distributions of the stretching numbers and helicity angles (spectra) are invariant with respect to initial conditions in a connected chaotic domain. We study such spectra in conservative and dissipative mappings of 2 degrees of freedom and in conservative mappings of 3-degrees of freedom. In 2-D conservative systems we found that the lines of constant stretching number have a fractal form.     

An attempt to test Ambartsumian's idea of the origin of the galaxy . II. location of galaxies within clusters and galaxy position angles

The location of galaxies in 377 rich Abell galaxy clusters is discussed. We compared the distributions of galaxies in the sample containing all galaxies with the sample compiled from the 20 brightest objects. Counts in circular sectors with angle width equal to 30î show isotropy. Only in the case of BM I clusters and the coordinate system related to the cluster major axis did we find anisotropic distributions. We investigated also the distributions of galaxy position angles within clusters exhibiting isotropy. The structure position angles for both samples have been studied as well. They appeared to be different. The difference in location of galaxies in the case of cD clusters, as well as the more elliptical shape and different position angles of samples containing bright galaxies is observed.     

Damped Lyα systems at high redshift and models of protogalactic discs

We employ observationally determined intrinsic velocity widths and column densities of damped Lyman alpha (Lyα) systems at high redshift to investigate the distribution of baryons in protogalaxies within the context of a standard cold dark matter (CDM) model. We proceed under the assumption that damped Lyα systems represent a population of cold, rotationally supported, protogalactic discs, and that the abundance of dark matter haloes is well approximated by a CDM model with critical density and vanishing cosmological constant. Using conditional cross-sections to observe a damped system with a given velocity width and column density, we compare observationally inferred velocity width and column density distributions to the corresponding theoretically determined distributions for a variety of disc parameters and CDM normalizations. In general, we find that the observations cannot be reproduced by the models for most disc parameters and CDM normalizations. Whereas the column density distribution favours small discs with large neutral gas fraction, the velocity width distribution favours large and thick discs with small neutral gas fraction. The possible resolutions of this problem in the context of this CDM model may be (1) an increased contribution of rapidly rotating discs within massive dark matter haloes to damped Lyα absorption, or (2) the abandoning of simple disc models within this CDM model for damped Lyα systems at high redshift. Here the first possibility may be achieved by supposing that damped Lyα system formation occurs only in haloes with fairly large circular velocities, and the second possibility may result from a large contribution of mergers and double discs to damped Lyα absorption at high redshift.     

The use of angles and angular rates

MIT's Lincoln Laboratory has developed a computer driven, rapidly slewing (4° s^–1), electro-optical (3 resolution) telescope. This enables the rapid measurement of angles and instantaneous angular rates for artificial satellites. The simultaneous acquisition of angles and angular rates constitutes a new initial orbit problem which has been solved. Three different methods of solution are presented including an exact, analytical one. Numerical tests on six widely different satellite orbits indicate that the topocentric distance can be determined to better than 1% (and usually as well as 0.1%) for most satellites after a 5–10 min observation interval.The views and conclusion contained in this document are those of the contractor and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the United States Government.This work was sponsored by the Department of the Air Force.     

Effects of stellar wind,dynamical friction,and star mergers on the dynamical evolution of multiple stars

The dynamical evolution of small stellar groups composed of N=6 components was numerically simulated within the framework of a gravitational N-body problem. The effects of stellar mass loss in the form of stellar wind, dynamical friction against the interstellar medium, and star mergers on the dynamical evolution of the groups were investigated. A comparison with a purely gravitational N-body problem was made. The state distributions at the time of 300 initial system crossing times were analyzed. The parameters of the forming binary and stable triple systems as well as the escaping single and binary stars were studied. The star-merger and dynamical-friction effects are more pronounced in close systems, while the stellar wind effects are more pronounced in wide systems. Star-mergers and stellar wind slow down the dynamical evolution. These factors cause the mean and median semimajor axes of the final binaries as well as the semimajor axes of the internal and external binaries in stable triple systems to increase. Star mergers and dynamical friction in close systems decrease the fraction of binary systems with highly eccentric orbits and the mean component mass ratios for the final binaries and the internal and external binaries in stable triple systems. Star mergers and dynamical friction in close systems increase the fraction of stable triple systems with prograde motions. Dynamical friction in close systems can both increase and decrease the mean velocities of the escaping single stars, depending on the density of the interstellar medium and the mean velocity of the stars in the system.     

Galactic models in the light of collision dynamics

A simple and flexible model of a spiral galaxy is developed in the light of the study of the dynamics of colliding galaxies. As such interactions are strongly characterised by the binding energy distributions of the systems involved; hence, the main criterion used in constructing the model is its compatibility with the observed shapes and intensity distribution of light of spirals. The model basically consists of thick exponential disk component and a spherical polytropic bulge component. The relevant quantities pertaining to the model, significant for dynamical studies, are computed and analysed.The study of the dynamics of colliding galaxies require galactic models with and without halos, depending upon the problem under study and the effects to be accessed. Consequently, we have preferred to tackle the problem of galactic models with massive halos separately in the next paper.     

Close triple approaches and escape in the three-body problem

We have studied a total of 5000 close triple approaches resulting in escape, for equal-mass systems with zero initial velocities. Escape is shown to take place in the majority of the cases after a fly-by close triple approach when the escaper passes near the centre of mass along an almost straight-line orbit. A number of configurational and kinematical parameters are introduced in order to characterize the triple approach. The distributions of these parameters are investigated. A comparison with 831 examples in the vicinity of the so-called Pythagorean problem is carried out. We find that the general features of close triple approaches which result in escape are the same for both types of systems.     

Statistical distribution of gravitational-lensing excursion angles: winding ways to us from the deep Universe

We investigate statistical distributions of differences in gravitational-lensing deflections between two light rays, the so-called lensing excursion angles. A probability distribution function of the lensing excursion angles, which plays a key role in estimates of lensing effects on angular clustering of objects (such as galaxies, quasi-stellar objects and also the cosmic microwave background temperature map), is known to consist of two components: a Gaussian core and an exponential tail. We use numerical gravitational-lensing experiments in a ΛCDM cosmology for quantifying these two components. We especially focus on the physical processes responsible for generating these two components. We develop a simple empirical model for the exponential tail which allows us to explore its origin. We find that the tail is generated by the coherent lensing scatter by massive haloes with   M > 10¹⁴  h ⁻¹ M_⊙  at   z < 1  and that its exponential shape arises due to the exponential cut-off of the halo mass function at that mass range. On scales larger than 1 arcmin, the tail does not have a practical influence on the lensing effects on the angular clustering. Our model predicts that the coherent scatter may have non-negligible effects on angular clustering at subarcminute scales.     

On the filling factor of emitting material in the upper atmosphere of ε Eri (K2 V)

The emission measure distribution in the upper transition region and corona of ε Eri is derived from observed emission-line fluxes. Theoretical emission measure distributions are calculated assuming that the radiation losses are balanced by the net conductive flux. We discuss how the area factor of the emitting regions as a function of temperature can be derived from a comparison between these emission measure distributions. It is found that the filling factor varies from ∼0.2 in the mid-transition region to ∼1.0 in the inner corona. The sensitivity of these results to the adopted ion fractions, the iron abundance and other parameters is discussed. The area factors found are qualitatively similar to the observed structure of the solar atmosphere, and can be used to constrain two-component models of the chromosphere. Given further observations, the method could be applied to investigate the trends in filling factors with indicators of stellar activity.     

A Statistical Study of the Mass Distribution of Neutron Stars

By reviewing the methods of mass measurements of neutron stars in four different kinds of systems, i.e., the high-mass X-ray binaries (HMXBs), low-mass X-ray binaries (LMXBs), double neutron star systems (DNSs) and neutron star-white dwarf (NS-WD) binary systems, we have collected the orbital parameters of 40 systems. By using the boot-strap method and the Monte-Carlo method, we have rebuilt the likelihood probability curves of the measured masses of 46 neutron stars. The statistical analysis of the simulation results shows that the masses of neutron stars in the X-ray neutron star systems and those in the radio pulsar systems exhibit different distributions. Besides, the Bayes statistics of these four different kind systems yields the most-probable probability density distributions of these four kind systems to be (1.340 ± 0.230)M8, (1, 505 ± 0.125)M8,(1.335 ± 0.055)M8 and (1.495 ± 0.225)M8, respectively. It is noteworthy that the masses of neutron stars in the HMXB and DNS systems are smaller than those in the other two kind systems by approximately 0.16M8. This result is consistent with the theoretical model of the pulsar to be accelerated to the millisecond order of magnitude via accretion of approximately 0.2M8. If the HMXBs and LMXBs are respectively taken to be the precursors of the BNS and NS-WD systems, then the influence of the accretion effect on the masses of neutron stars in the HMXB systems should be exceedingly small. Their mass distributions should be very close to the initial one during the formation of neutron stars. As for the LMXB and NS-WD systems, they should have already under- gone the process of suffcient accretion, hence there arises rather large deviation from the initial mass distribution.     

The formation and evolution of star cluster systems in different environments

The age, mass, and size distributions of star clusters in nearby star-forming galaxies provide important clues to the formation and evolution of cluster systems. In particular, the similarities and differences between these cluster distributions in very different environments can help to disentangle formation and disruption processes. We present the age and mass distributions for clusters younger than ≈1 Gyr in the Magellanic Clouds, which are typical, star-forming irregular galaxies, and compare the results with the more “extreme” environment found in the merging Antennae galaxies. In addition, we describe some new results on the interpretation of ancient globular cluster systems, and present an emerging picture for the life cycle of star clusters.     

Stochasticity of Planetary Orbits in Double Star Systems

Cosmogonical theories as well as recent observations allow us to expect the existence of planets around many stars other than the Sun. On an other hand, double and multiple star systems are established to be more numerous than single stars (such as the Sun), at least in the solar neighborhood. We are then faced to the following dynamical problem: assuming that planets can form in a binary early environment (I do not deal here with), does long-term stability for planetary orbits exist in double star systems.Although preliminary studies were rather pessimistic about the possibility of existence of stable planetary orbits in double or multiple star systems, modern computation have shown that many such stable orbits do exist (but possible chaotic behavior), either around the binary as a whole (P-type) or around one component of the binary (S-type), this latter being explored here.The dynamical model is the elliptic plane restricted three-body problem; the phase space of initial conditions is systematically explored, and limits for stability have been established. Stable S-type planetary orbits are found up to distance of their "sun" of the order of half the periastron distance of the binary; moreover, among these stable orbits, nearly-circular ones exist up to distance of their "sun" of the order of one quarter the periastron distance of the binary; finally, among the nearly-circular stable orbits, several stay inside the "habitable zone", at least for two nearby binaries which components are nearly of solar type.Nevertheless, we know that chaos may destroy this stability after a long time (sometimes several millions years). It is therefore important to compute indicators of chaos for these stable planetary orbits to investigate their actual very long-term stability. Here we give an example of such a computation for more than a billion years.     

Kinematics and evolution of local features of the large-scale magnetic field – II. Relation to Active Regions

Relationships have been studied between the background magnetic field and the distribution of active regions over the solar surface and time. A series of magnetic-field synoptic maps covering a 20-year period has been cross-correlated with spatio-temporal distributions of three types of active formations (sunspots, calcium plages, and solar flares) used as indicators of the active regions. To make the data analysis more effective, we expanded both the magnetic-field and the active-region distributions in terms of Fourier series in longitude, and then cross-correlated the latitude-dependent Fourier harmonics. Cross-correlation functions calculated from the lower-order Fourier harmonics exhibit prolonged maxima of the amplitude. For the first-order harmonic, the maxima can be tracked throughout a long time interval of at least 13 Carrington rotations, but the time of cross-correlation decreases down to 2 rotations, as the harmonic order increases up to 8. The maxima of the cross-correlation functions indicate moreover a poleward directed drift of the magnetic features that occurred with a velocity of 10–15 m s^–1. The cross-correlation functions calculated separately by using the three types of active formations as indicators of the active regions are similar to each other, although they differ in some details of minor significance. The results of the data analysis make it possible to conclude that the cross-correlation between the magnetic-field and the active-region distributions displays long-term evolution of the magnetic features emerged in the photosphere in the form of the active regions, and that the evolution occurs in accordance with Leighton's (1964) concept known at present as the flux transport model. In order to verify this conclusion, we applied the cross-correlation technique to analyze a magnetic field distribution simulated by means of the flux transport equation by using an ensemble of local-scale magnetic bipoles as a source of magnetic flux. Results of the simulated magnetic field analysis exhibit a substantial qualitative agreement with those obtained by examining the observational data.     

Equivalent ionospheric current systems representing IMF sector effects on the polar geomagnetic field

Equivalent ionospheric current systems representing IMF sector effects on the geomagnetic field in high latitudes are examined for each of the twelve calendar months by spherical harmonic analyses of geomagnetic hourly data at 13 northern polar stations for seven years. The main feature of obtained equivalent current systems includes circular currents at about 80° invariant latitude mostly in the daytime in summer and reversed circular currents at about 70° invariant latitude mainly at night in winter. Field-aligned current distributions responsible for equivalent currents, as well as vector distributions of electric fields and ionospheric currents, are approximated numerically from current functions of equivalent current systems by taking assumed distributions of the ionospheric conductivity. Two sets of upward and downward field-aligned current pairs in the auroral region, and also a field-aligned current region near the pole show seasonal variations. Also, ionospheric electric-field propagation along geomagnetic field lines from the summer hemisphere to the winter hemisphere with auroral Hall-conductivity effects may provide an explanation for the winter reversal of sector effects.     

New adaptive time step symplectic integrator:an application to the elliptic restricted three-body problem

The time-transformed leapfrog scheme of Mikkola Aarseth was specifically designed for a second-order differential equation with two individually separable forms of positions and velocities.It can have good numerical accuracy for extremely close two-body encounters in gravitating few-body systems with large mass ratios,but the non-time-transformed one does not work well.Following this idea,we develop a new explicit symplectic integrator with an adaptive time step that can be applied to a time-dependent Hamiltonian.Our method relies on a time step function having two distinct but equivalent forms and on the inclusion of two pairs of new canonical conjugate variables in the extended phase space.In addition,the Hamiltonian must be modified to be a new time-transformed Hamiltonian with three integrable parts.When this method is applied to the elliptic restricted three-body problem,its numerical precision is explicitly higher by several orders of magnitude than the nonadaptive one's,and its numerical stability is also better.In particular,it can eliminate the overestimation of Lyapunov exponents and suppress the spurious rapid growth of fast Lyapunov indicators for high-eccentricity orbits of a massless third body.The present technique will be useful for conservative systems including N-body problems in the Jacobian coordinates in the the field of solar system dynamics,and nonconservative systems such as a time-dependent barred galaxy model in a rotating coordinate system.