Search for and study of nearly periodic orbits in the plane problem of three equal-mass bodies

We analyze nearly periodic solutions in the plane problem of three equal-mass bodies by numerically simulating the dynamics of triple systems. We identify families of orbits in which all three points are on one straight line (syzygy) at the initial time. In this case, at fixed total energy of a triple system, the set of initial conditions is a bounded region in four-dimensional parameter space. We scan this region and identify sets of trajectories in which the coordinates and velocities of all bodies are close to their initial values at certain times (which are approximately multiples of the period). We classify the nearly periodic orbits by the structure of trajectory loops over one period. We have found the families of orbits generated by von Schubart’s stable periodic orbit revealed in the rectilinear three-body problem. We have also found families of hierarchical, nearly periodic trajectories with prograde and retrograde motions. In the orbits with prograde motions, the trajectory loops of two close bodies form looplike structures. The trajectories with retrograde motions are characterized by leafed structures. Orbits with central and axial symmetries are identified among the families found.     

A Hard X-ray Two-Ribbon Flare Observed with Yohkoh/HXT

The Yohkoh hard X-ray telescope (HXT) observed hard X-rays from the impulsive phase of a long-duration event (LDE) occurring on 14 July 2000. The Yohkoh soft X-ray telescope (SXT) and other instruments observed a large arcade, with width and length ∼30 000 km and ∼120 000 km, respectively. In hard X-rays, for the first time, a two-ribbon structure was clearly observed in the energy range above 30 keV. This result suggests that electrons are in fact accelerated in the whole system of this arcade, not merely in a particular dominant loop. We analyzed the motions of bright kernels in the two hard X-ray ribbons in detail. Assuming these bright kernels to be footpoints of newly reconnected loops, we infer from their motions that the loops reconnecting early are highly sheared, while the loops reconnecting later are less sheared. We have also analyzed the hard X-ray spectra of the two ribbons independently. At the outer edge of a ribbon, the spectrum tends to be harder than that in the inner edge. This suggests that higher-energy electrons precipitate at the footpoints of outer loops and lower ones do at those of inner loops. We discuss what kind of model can support this tendency.     

A numerical investigation of the one-dimensional newtonian three-body problem

Numerical orbit integrations have been conducted to characterize the types of trajectories in the one-dimensional Newtonian three-body problem with equal masses and negative energy. Essentially three different types of motions were found to exist. They may be classified according to the duration of the bound three-body state. There are zero-lifetime predictable trajectories, finite lifetime apparently chaotic orbits, and infinite lifetime quasi-periodic motions. The quasi-periodic orbits are confined to the neighbourhood of Schubart's stable periodic orbit. For all other trajectories the final state is of the type binary + single particle in both directions of time. The boundaries of the different orbit-type regions seem to be sharp. We present statistical results for the binding energies and for the duration of the bound three-body state. Properties of individual orbits are also summarized in the form of various graphical maps in a two-dimensional grid of parameters defining the orbit. Supported by the Academy of Finland.     

The extended structure of the Phoenix dwarf galaxy

The surface brightness profile in the V band of the Phoenix dwarf galaxy shows two stellar components: an inner one, which contains all the young stars of the galaxy, and an outer one predominantly populated by red stars. Deep color-magnitude diagrams (CMDs), based on Hubble Space Telescope (HST) observations and reaching the oldest turn-offs, are used to analyze the inner and outer stellar components. Results show that, together with an old stellar population, the outer field contains also an intermediate-age population. These results are compatible with a scenario in which star forming regions are shrinking with time (the shrinking scenario). It seems more difficult to support a halo-disk scenario, which would require extended structures populated only by really old stars. This revised version was published online in August 2006 with corrections to the Cover Date.     

The influence of winds on the time-dependent behaviour of self-gravitating accretion discs

We study the effects of winds on the time evolution of isothermal, self-gravitating accretion discs by adopting a radius-dependent mass-loss rate because of the existence of the wind. Our similarity and semi-analytical solution describes time evolution of the system in the slow accretion limit. The disc structure is distinct in the inner and outer parts, irrespective of the existence of the wind. We show that the existence of wind will lead to a reduction of the surface density in the inner and outer parts of the disc in comparison to a no-wind solution. Also, the radial velocity significantly increases in the outer part of the disc, however, the accretion rate decreases due to the reduced surface density in comparison to the no-wind solution. In the inner part of the disc, mass loss due to the wind is negligible according to our solution. But the radial size of this no-wind inner region becomes smaller for stronger winds.     

Relativistic positioning: four-dimensional numerical approach in Minkowski space-time

We simulate the satellite constellations of two Global Navigation Satellite Systems: Galileo (EU) and GPS (USA). Satellite motions are described in the Schwarzschild space-time produced by an idealized spherically symmetric non rotating Earth. The trajectories are then circumferences centered at the same point as Earth. Photon motions are described in Minkowski space-time, where there is a well known relation, (Coll et al. in Class. Quantum Gravit. 27:065013, 2010a), between the emission and inertial coordinates of any event. Here, this relation is implemented in a numerical code, which is tested and applied. The first application is a detailed numerical four-dimensional analysis of the so-called emission coordinate region and co-region. In a second application, a GPS (Galileo) satellite is considered as the receiver and its emission coordinates are given by four Galileo (GPS) satellites. The bifurcation problem (double localization) in the positioning of the receiver satellite is then pointed out and discussed in detail.     

Sundman surfaces and Hill stability in the three-body problem

Using the famous Sundman inequality, we have constructed for the first time the surfaces for the general three-body problem that we suggest calling Sundman surfaces. These surfaces are a generalization of the widely known Hill surfaces in the restricted circular three-body problem. The Sundman surfaces are constructed in a rectangular coordinate system that uses the mutual distances between the bodies as the Cartesian rectangular coordinates. The singular points of the family of these surfaces have been determined. The possible and impossible regions of motion of the bodies have been constructed in the space of mutual distances. We have shown the existence of Hill stable motions and established sufficient criteria for Hill stability of motions. Some of the astronomical applications are considered.     

The mid-term and long-term solar quasi-periodic cycles and the possible relationship with planetary motions

This work investigates the solar quasi-periodic cycles with multi-timescales and the possible relationships with planetary motions. The solar cycles are derived from long-term observations of the relative sunspot number and microwave emission at frequency of 2.80 GHz. A series of solar quasi-periodic cycles with multi-timescales are registered. These cycles can be classified into three classes: (1) the strong PLC (PLC is defined as the solar cycle with a period very close to the ones of some planetary motions, named as planetary-like cycle) which is related strongly with planetary motions, including nine periodic modes with relatively short period (P<12 yr), and related to the motions of the inner planets and of Jupiter; (2) the weak PLC, which is related weakly to planetary motions, including two periodic modes with relatively long period (P>12 yr), and possibly related to the motions of outer planets; (3) the non-PLC, for which so far there has been found no clear evidence to show the relationship with any planetary motions. Among the planets, Jupiter plays a key role in most periodic modes due to its sidereal motion or spring tidal motions associated with other planets. Among planetary motions, the spring tidal motion of the inner planets and of Jupiter dominates the formation of most PLCs. The relationships between multi-timescale solar periodic modes and the planetary motions will help us to understand the essential nature and prediction of solar activities.     

A photometric study of the double ring structure of NGC 4736

In this study we present photometric results for the galaxy NGC 4736: infrared and visible profiles. After a careful correction for the extinction within the galaxy based on measured neutral gas surface densities, we interpret the profiles in the individual bands and in colour indices, in terms of the radial distribution of stellar populations. We pick out the behaviour of the two rings, an inner ring some 40–50 arc sec from the nucleus, and an outer ring some 300 arc sec away. We show how the photometry allows us to make tentative physical inferences about the nature of these two structures, showing that the inner ring is connected with an outflow of gas observed via itsHii regions, and is probably the result of an axisymmetric starburst, while the outer ring is a site of star formation which appears to be further from the centre than the typical resonant structures associated with a density wave.     

Two-dimensional observations of the velocity field in and around sunspots

Doppler spectroheliograms of sunspots and their surroundings have been obtained with a spatial resolution approaching one second of arc and a time resolution of 20 s per frame. Observations of 5 sunspots, located 18°, 45°, 56°, 60° and 72° from the disk center respectively, showed considerable long-lived fine structure and, in particular, indicated the following:

1. The Evershed outflow terminated in spoke-like structures that constitute the ragged outer boundary of the penumbra. Some of these spokes extended more than 8000 km beyond the average outer boundary.
2. Although there was considerable long-lived fine structure of both Doppler polarities in the extra-penumbral photosphere, the spatially-averaged horizontal flow was outward for roughly 10000 km beyond the outer boundary of the penumbra. This extra-penumbral velocity field was distinct from the Evershed flow, and, in particular, did not represent its extension beyond the end of the penumbral spokes.

Although these results are based on observations of relatively few sunspots, they do suggest that if magnetic fields are carried outward from sunspots by material motions, then these motions are more like the supergranulation than an extension of the Evershed velocity.     

HD 12661行星系统构形的形成与稳定性

最近的多普勒观测表明恒星HD 12661周围存在两颗中等偏心率轨道上运行的行星,内行星的最小质量为2．3木星质量,轨道周期为263．6天;外行星的最小质量为1．57木星质量,轨道周期为1444．5天．该系统的稳定性要求两颗行星处在平运动轨道共振．用数值方法研究了该系统形成初期在恒星气体盘作用下的轨道迁移与稳定性,计算了行星在迁移中被平运动共振俘获的概率．发现这两颗行星目前很可能正处在11:2平运动共振边缘,且运动是混沌的,从而澄清了关于系统目前构形的不同说法,并且很可能在系统形成后行星迁移到目前构形时,气体盘几乎消失了．     

The formation and stability of the configuration of the planetary system HD 126611

Recent Doppler velocity measurements have revealed the existence of two planets orbiting the star HD 12661 on medium-eccentricity orbits. The inner planet has a period of 263.6 d and a mass of 2.3 Jupiter masses, and the outer planet has a period of 1444.5d and a mass of 1.57 Jupiter masses. The stability of this system requires the two planets to be in a state of mean-motion orbit resonances. By numerical method we have studied the orbit migration and stability of the system in its early ages under the action of the proto-stellar disk, and calculated the probabilities of the planets being captured into the mean -motion resonances during their migrations. It is found that at present the two planets are probably situated at the edge of the 11:2 mean-motion resonance and are in chaotic motions. This result may be helpful to clarify the arguments on the present configuration. Besides, it is indicated that very probably, after the formation of the system, the gaseous disk has almost disappeared before the planets migrated to the present configuration.     

Are polar rings indeed polar?

We have considered polar ring galaxy candidates, the images of which can be found in the SDSS survey. The sample of 78 galaxies includes the most reliable candidates from the SPRC and PRC catalogs, some of which already have kinematic confirmations. We analyze the distributions of studied objects by the angle between the polar ring and the central disk, and by the optical diameter of the outer ring structures. In the vast majority of cases, the outer structures lie in the plane close to polar (within 10°–20°) which indicates the stability of the corresponding orbits in the gravitational potential of the halo. Moderately inclined outer structures are observed only in about 6% of objects which probably indicates their short lifetime. In such an unstable configuration, the polar ring would often cross the disk of the galaxy, being smaller than it in the diameter. We show that the inner polar structures and outer large-scale polar rings form a single family in the distribution of diameters normalized to the optical size of the galaxy. At the same time, this distribution is bimodal, as the number of objects with d _ring = (0.4–0.7) d _disk is small. Such a shape of size distribution is most likely due to the fact that the stability of polar orbits in the inner regions of galaxies is maintained by the bulge or the bar, while in the outer regions it is provided by the spheroidal (or triaxial) halo.     

Radio galaxies with a 'double-double morphology'– I. Analysis of the radio properties and evidence for interrupted activity in active galactic nuclei

We present four Mpc-sized radio galaxies which consist of a pair of double-lobed radio sources, aligned along the same axis, and with a coinciding radio core. We call these peculiar radio sources 'double-double' radio galaxies (DDRGs) and propose a general definition of such sources: a 'double-double' radio galaxy consists of a pair of double radio sources with a common centre. Furthermore, the two lobes of the inner radio source must have a clearly extended, edge-brightened radio morphology. Adopting this definition, we find several other candidate DDRGs in the literature. We find that in all sources the smaller (inner) pair of radio lobes is less luminous than the larger (outer) pair, and that the ratio of 1.4-GHz flux density of these two pairs appears to be anticorrelated with the projected linear size of the inner source. Also, the outer radio structures are large, exceeding 700 kpc. We discuss possible formation scenarios of the DDRGs, and we conclude that an interruption of the jet-forming central activity is the most likely mechanism. For one of our sources (B 1834+620) we have been able observationally to constrain the length of time of the interruption to a few Myr. We discuss several scenarios for the cause of the interruption, and suggest multiple encounters between interacting galaxies as a possibility. Finally, we discuss whether such interruptions help the formation of extremely large radio sources.     

Morphology and origin of palimpsests on Ganymede based on Galileo observations

Palimpsests are large, circular, low-relief impact scars on Ganymede and Callisto. These structures were poorly understood based on Voyager-era analysis, but high-resolution Galileo images allow more detailed inspection. We analyze images of four Ganymedean palimpsests targeted by Galileo: Memphis and Buto Faculae, Epigeus, and Zakar. Ganymedean craters and Europan ring structures are used as tools to help better understand palimpsests, based on morphologic similarities. From analysis of Galileo images, palimpsests consist of four surface units: central plains, an unoriented massif facies, a concentric massif facies, and outer deposits. Using as a tie point the location in these structures where secondary craters begin to appear, outer deposits of palimpsests are analogous to the outer ejecta facies of craters; the concentric massif facies of palimpsests are analogous to the pedestal facies of craters; and the unoriented massif facies and central plains are analogous to crater interiors. These analogies are supported by the presence of buried preexisting structure beneath the outer two and absence of buried structure beneath the inner two units. Our observations indicate that palimpsest deposits represent fluidized impact ejecta, rather than cryovolcanic deposits or ancient crater interiors.     

Studies of multiple stellar systems – III. Modulation of orbital elements in the triple-lined system HD 109648

The triple-lined spectroscopic triple system HD 109648 has one of the shortest periods known for the outer orbit in a late-type triple, 120.5 d, and the ratio between the periods of the outer and the inner orbits is small, 22:1. With such extreme values, this system should show orbital element variations over a time-scale of about a decade. We have monitored the radial velocities of HD 109648 with the CfA Digital Speedometers for 8 yr, and have found evidence for modulation of some orbital elements. While we see no definite evidence for modulation of the inner binary eccentricity, we clearly observe variations in the inner and the outer longitudes of periastron, and in the radial velocity amplitudes of the three components. The observational results, combined with numerical simulations, allow us to put constraints on the orientation of the orbits.     

Possible relations between the rotational axis of the Earth's inner core and the magnetic dipole axis

The geomagnetic field is maintained by amagnetohydrodynamic dynamo process within the liquid outer core. The distribution of the associated electric currents is modified if the outer core is bounded by electrically conducting material. Then, eddy currents and the related magnetic fields are generated within these regions. In particular, the relative rigid rotation of the inner core produces a secondary magnetic field, which is superimposed on the dynamo field. The angle between the dipole axis of the total field and the rotational axis of the inner core is an important quantity needed for the theory of polar motion of the Earth. This angle is investigated for a broad spectrum of angular velocities of the inner core. To simplify the mathematical procedure, we model the dynamo field using an axisymmetric field generated by a system of electric currents within the outer core. The conductivity of the mantle is neglected. We find that the position of the dipole axis depends on the angular velocity of the inner core as well as on the distribution of the current system within the outer core. Coincidence of both axes can be reached if the angular velocity is high enough and if the current system is concentrated within a thin sheet near the outer core-inner core boundary.     

Trajectory exploration within asynchronous binary asteroid systems using refined Lagrangian coherent structures

Ground observations have found that asynchronous systems constitute most of the population of the near-Earth binary asteroids. This paper concerns the trajectory of a particle in the asynchronous system which is systematically described using periodic ellipsoidal and spherical body models. Due to the non-autonomous characteristics of the asynchronous system, Lagrangian coherent structures (LCS) are employed to identify the various dynamical behaviors. To enhance the accuracy of LCS, a robust LCS finding algorithm is developed incorporating hierarchical grid refinement, one-dimensional search and variational theory verification. In this way, the intricate dynamical transport boundaries are detected efficiently. These boundaries indicate that a total of 15 types of trajectories exist near asynchronous binary asteroids. According to their Kepler energy variations, these trajectories can be grouped into four basic categories, i.e., transitory, escape, impact and flyby trajectories. Furthermore, the influence of the ellipsoid’s spin period on the dynamical behavior is discussed in the context of the change of dynamical regions. We found that the transitory and impact motions occur easily in the synchronous-like binary systems, in which the rotation period of the ellipsoid is nearly equal to that of the mutual orbit. Meanwhile, the results confirm a positive correlation between the spinning rate of the ellipsoid and the probability of the escape and flyby trajectories. The LCS also reveal a marked increase in trajectory diversity after a larger initial energy is selected.     

Restriction of motions in wide pairs in the Galactic field

The motions of the components of wide binary stars in the solar neighborhood in the regular Galactic gravitational field on time scales ～10¹⁰ yr have been studied numerically. The regions of restricted motions of the components in wide pairs have been found depending on the initial conditions: the magnitude of the relative velocity of the components, their mutual distance, and the inclination of the relative velocity vector to the Galactic plane. The size of the main part of the region of restricted motions is approximately equal to the tidal radius. Profound changes in the eccentricity of the binary orbit occur at inclinations close to 90°, which can lead to close approaches of the stars with a pericenter distance less than 1 AU. In the case of retrograde motions (the binary rotates in a direction opposite to the Galactic rotation), there is a region of restricted motions extending at least to 10 pc. Examples of the trajectories of relative motion of the stars and the change in osculating orbital elements are given for systems with restricted motions.