Dynamics of the Jupiter Trojans with Saturn’s perturbation in the present configuration of the two planets

The dynamics of the two Jupiter triangular libration points perturbed by Saturn is studied in this paper. Unlike some previous works that studied the same problem via the pure numerical approach, this study is done in a semianalytic way. Using a literal solution, we are able to explain the asymmetry of two orbits around the two libration points with symmetric initial conditions. The literal solution consists of many frequencies. The amplitudes of each frequency are the same for both libration points, but the initial phase angles are different. This difference causes a temporary spatial asymmetry in the motions around the two points, but this asymmetry gradually disappears when the time goes to infinity. The results show that the two Jupiter triangular libration points should have symmetric spatial stable regions in the present status of Jupiter and Saturn. As a test of the literal solution, we study the resonances that have been extensively studied in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006). The resonance structures predicted by our analytic theory agree well with those found in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006) via a numerical approach. Two kinds of chaotic orbits are discussed. They have different behaviors in the frequency map. The first kind of chaotic orbits (inner chaotic orbits) is of small to moderate amplitudes, while the second kind of chaotic orbits (outer chaotic orbits) is of relatively larger amplitudes. Using analytical theory, we qualitatively explain the transition process from the inner chaotic orbits to the outer chaotic orbits with increasing amplitudes. A critical value of the diffusion rate is given to separate them in the frequency map. In a forthcoming paper, we will study the same problem but keep the planets in migration. The time asymmetry, which is unimportant in this paper, may cause an observable difference in the two Jupiter Trojan groups during a very fast planet migration process.     

On the existence of a low-dimensional chaotic attractor in the short-term solar UV time series

A mixture of periodic and chaotic components makes the detection of chaos difficult. The periodic components are sought in the solar UV time series by the Maximum Entropy Method and are removed by a digital notch filter. The filtered output is subjected to investigation for chaotic behavior by three different techniques. (1) Fixed-size method for attractor dimension determination; (2) sensitive initial dependence via prediction error; (3) trajectory direction estimation. All the investigation points to the existence of a chaotic attractor of fractional dimension.     

地极移动的非线性动力机制

用非线性动力学原理和非线性振动方法,结合Chandler摆动的时间序列反演,证实了Chandler摆动的衰减能量是由周年激发的非线性共振所补充的．由高布锡归算的ILS参数,按共振激发模型反演出的Chandler摆动衰减指数和频率品质因数等结果,符合已有的大量基本结论,反演出的非齐次项的强迫频率基本都在周年频率的周围．非线性共振模型还表明,Chandler频率的不稳定性是由于存在频率转换和漂移,摆动振幅的不稳定性是由于存在共振跳跃,非线性的存在还将引出组合共振频率．同时用推广的最小公倍数证实和频差频的存在,提供了非整数的最小公倍数算法,为数论的非线性推广和在天文中的应用提供了工具．     

Identification of bright lenses from the astrometric observations of gravitational microlensing events

When a source star is gravitationally microlensed by a dark lens, the centroid of the source star image is displaced relative to the position of the unlensed source star, with an elliptical trajectory. Recently, routine astrometric follow-up measurements of these source star image centroid shifts by using high-precision interferometers have been proposed to measure the lens proper motion, which can resolve the lens parameter degeneracy in the photometrically determined Einstein time-scale. When an event is caused by a bright lens, on the other hand, the astrometric shift is affected by the light from the lens, but one cannot identify the existence of the bright lens from the observed trajectory because the resulting trajectory of the bright lens event is also an ellipse. As results, lensing parameters determined from the trajectory differ from those of a dark lens event, causing an incorrect identification of the lens population. In this paper, we show that although the shape and size of the astrometric centroid shift trajectory are changed because of the bright lens, the angular speed of centroid shifts around the apparent position of the unlensed source star is not affected by the lens brightness. Therefore, one can identify the existence of a bright lens and determine its brightness by comparing the lens parameters determined from the 'angular speed curve' with those determined from the trajectory of observed centroid shifts. Once the lens brightness is determined, one can correct for the lens proper motion. As the proposed method provides information about both the lens brightness (dark or bright) and the corrected values of the physical parameters of the lens, one can constrain the nature of massive compact halo objects (MACHOs) significantly better.     

Seasonal oscillations in length-of-day

Variations of annual and semiannual oscillations in rotation parameters have been investigated on the basis of length-f-ay(LOD) as well as atmospheric-ngular-omemtum (AAM) series. These oscillations were determined using band-ass filters. In order to show the character of variations of seaasonal oscillations, amplitudes, phases and periods were computed by a least-quares adjustment with the method of modified harmonic analysis at quarterly intervals. In addition, the seasonal imbalances in LOD and AAM budgets were determined and analysed in a similar way. These discrepancies were corrected for tidally excited effects. The non-tmospheric oscillations without the annual tide effect Sa and the semiannual tide effect Ssa have variable amplitudes between 0.02 and 0.10 msec.     

Orbital structure in N-body models of barred-spiral galaxies

We study the orbital structure in a series of self-consistent N -body configurations simulating rotating barred galaxies with spiral and ring structures. We perform frequency analysis in order to measure the angular and the radial frequencies of the orbits at two different time snapshots during the evolution of each N -body system. The analysis is done separately for the regular and the chaotic orbits. We thereby identify the various types of orbits, determine the shape and percentages of the orbits supporting the bar and the ring/spiral structures, and study how the latter quantities change during the secular evolution of each system. Although the frequency maps of the chaotic orbits are scattered, we can still identify concentrations around resonances. We give the distributions of frequencies of the most important populations of orbits. We explore the phase-space structure of each system using projections of the 4D surfaces of section. These are obtained via the numerical integration not only of the orbits of test particles, but also of the real N -body particles. We thus identify which domains of the phase space are preferred and which are avoided by the real particles. The chaotic orbits are found to play a major role in supporting the shape of the outer envelope of the bar as well as the rings and the spiral arms formed outside corotation.     

On the modulational instability of electrostatic ion cyclotron waves

The instability of electrostatic ion cyclotron waves to low frequency density modulations is considered and nonlinear equations are derived which describe its development in terms of a coherent four wave interaction. A dispersion relation for the linear phase of the instability is obtained and threshold conditions for marginal stability determined. It is shown, using data from recent optical observations, that the conditions necessary for the instability to occur in the auroral plasma would probably be satisfied and that modulational frequencies in agreement with the observations are obtained for plausible wave amplitudes. The nonlinear development of the instability is then studied and it is shown that substantial modulation can occur. It is suggested therefore that this instability could lead to the development of a strongly turbulent state.     

On searching for observational manifestations of Alfvén waves in solar faculae

In an effort to detect torsional oscillations, we have studied the periodic half-width variations for several spectral lines in solar faculae. The duration of the series being analyzed was from 40 to 150 min. We have determined the dominant frequencies and amplitudes of the half-width oscillations and considered their phase relations to the intensity and line-of-sight velocity oscillations. Five-minute profile halfwidth oscillations with a peak-to-peak amplitude of ～10 m ?A are recorded with confidence in the upperphotospheric Si I 10 827 ?A line in faculae. The chromospheric He I 10 830 A? and Hα line profiles shows ～40–60 m ?A variations in two frequency bands, 2.5–4 and 1–1.9 mHz. No center-to-limb dependence that, according to the theory, must accompany the torsional oscillations has been revealed in the behavior of the oscillation amplitudes. According to present views, these variations cannot be caused by periodic temperature and magnetic field changes. Our observations do not allow us to explain these variations by the sausage mode action either, which should manifest itself at the double frequency.     

Third Order Effect of Rotation on Stellar Oscillations of a B Star

We aim at investigating the effect of rotation up to the third order in the angular velocity of a star on the p and g modes, based on the formalism developed by Soufi et al. Our ultimate goal is the study of oscillations of β Cephei stars which are often rapidly rotating stars. Our results show that the third-order perturbation formalism presented by Soufi et al. should be corrected for some missing terms and some misprints in the equations. As a first step in our study of β Cephei stars, we quantify by numerical calculations the effect of rotation on the oscillation frequencies of a uniformly rotating zero-age main-sequence star with 12 M<,??>. For an equatorial velocity of 100km s-1, it is found that the second-and third-order corrections for (l, m)=(2, 2), for instance, are of the order of 0.01% of the frequency for radial order n=6 and reaches up to 0.5% for n=14.     

The resonant structure of Jupiter's Trojan asteroids – II. What happens for different configurations of the planetary system

In a previous paper, we have found that the resonance structure of the present Jupiter Trojan swarms could be split up into four different families of resonances. Here, in a first step, we generalize these families in order to describe the resonances occurring in Trojan swarms embedded in a generic planetary system. The location of these families changes under a modification of the fundamental frequencies of the planets and we show how the resonant structure would evolve during a planetary migration. We present a general method, based on the knowledge of the fundamental frequencies of the planets and on those that can be reached by the Trojans, which makes it possible to predict and localize the main events arising in the swarms during migration. In particular, we show how the size and stability of the Trojan swarms are affected by the modification of the frequencies of the planets. Finally, we use this method to study the global dynamics of the Jovian Trojan swarms when Saturn migrates outwards. Besides the two resonances found by Morbidelli et al. which could have led to the capture of the current population just after the crossing of the 2:1 orbital resonance, we also point out several sequences of chaotic events that can influence the Trojan population.     

Delta Scuti stars in the Praesepe cluster observed by the MOST satellite

The Praesepe cluster contains a number of δ Sct and γ Dor pulsators. Asteroseismology of cluster stars is simplified by the common distance, age and stellar abundances. Since asteroseismology requires a large number of known frequencies, the small pulsation amplitudes of these stars require space satellite campaigns. The present study utilizes photometric MOST satellite measurements in order to determine the pulsation frequencies of two evolved (EP Cnc, BT Cnc) and two main‐sequence (BS Cnc, HD 73872) δ Sct stars in the Praesepe cluster. The frequency analysis of the 2008 and 2009 data detected up to 34 frequencies per star with most amplitudes in the submillimag range. In BS Cnc, two modes showed strong amplitude variability between 2008 and 2009. The frequencies ranged from 0.76 to 41.7 cd^–1. After considering the different evolutionary states and mean stellar densities of these four stars, the differences and large ranges in frequency remain (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the frequency and amplitude variations of the δ Scuti star CD−24 7599 (=XX Pyx)

We present 132 h of new time-series photometric observations of the δ Scuti star CD−24 7599 acquired during 86 nights from 1993 to 1996 to study its frequency and amplitude variations. By using all published observations we demonstrate that the three dominating pulsation modes of the star can change their photometric amplitudes within one month at certain times, while the amplitudes can remain constant within the measurement errors at other times. CD−24 7599 also exhibits frequency variations, which do not show any correspondence between the different modes.   The typical time-scale for the amplitude variations is found to be several hundred days, which is of the same order of magnitude as the inverse linear growth rates of a selected model. We find no evidence for periodic amplitude modulation of two of the investigated modes ( f ₂ and f ₃), but f ₁ may exhibit periodic modulation. The latter result could be spurious and requires confirmation. The observed frequency variations may either be continuous or reflect sudden frequency jumps. No evidence for cyclical period changes is obtained.   We exclude precession of the pulsation axis and oblique pulsation for the amplitude variations. Beating of closely spaced frequencies cannot explain the amplitude modulations of two of the modes, while it is possible for the third. Evolutionary effects, binarity, magnetic field changes or avoided crossings cannot be made responsible for the observed period changes. Only resonance between different modes may be able to explain the observations. However, at this stage a quantitative comparison is not possible. More observations, especially data leading to a definite mode identification and further measurements of the temporal behaviour of the amplitudes and frequencies of CD−24 7599, are required.     

Comparative Study of the 2:3 and 3:4 Resonant Motion with Neptune: An Application of Symplectic Mappings and Low Frequency Analysis

The 2:3 and 3:4 exterior mean motion resonances with Neptune are studied by applying symplectic mapping models. The mappings represent efficiently Poincaré maps for the 3D elliptic restricted three body problem in the neighbourhood of the particular resonances. A large number of trajectories is studied showing the coexistence of regular and chaotic orbits. Generally, chaotic motion depletes the small bodies of the effective resonant region in both the 2:3 and 3:4 resonances. Applying a low frequency spectral analysis of trajectories, we determined the phase space regions that correspond to either regular or chaotic motion. It is found that the phase space of the 3:4 resonant motion is more chaotic than the 2:3 one.     

Fourier Analysis of Gapped Time Series: Improved Estimates of Solar and Stellar Oscillation Parameters

Quantitative helioseismology and asteroseismology require very precise measurements of the frequencies, amplitudes, and lifetimes of the global modes of stellar oscillation. The precision of these measurements depends on the total length (T), quality, and completeness of the observations. Except in a few simple cases, the effect of gaps in the data on measurement precision is poorly understood, in particular in Fourier space where the convolution of the observable with the observation window introduces correlations between different frequencies. Here we describe and implement a rather general method to retrieve maximum likelihood estimates of the oscillation parameters, taking into account the proper statistics of the observations. Our fitting method applies in complex Fourier space and exploits the phase information. We consider both solar-like stochastic oscillations and long-lived harmonic oscillations, plus random noise. Using numerical simulations, we demonstrate the existence of cases for which our improved fitting method is less biased and has a greater precision than when the frequency correlations are ignored. This is especially true of low signal-to-noise solar-like oscillations. For example, we discuss a case where the precision of the mode frequency estimate is increased by a factor of five, for a duty cycle of 15%. In the case of long-lived sinusoidal oscillations, a proper treatment of the frequency correlations does not provide any significant improvement; nevertheless, we confirm that the mode frequency can be measured from gapped data with a much better precision than the 1/T Rayleigh resolution.     

The pulsating DA white dwarf star EC 14012−1446: results from four epochs of time-resolved photometry

The pulsating DA white dwarfs are the coolest degenerate stars that undergo self-driven oscillations. Understanding their interior structure will help us to understand the previous evolution of the star. To this end, we report the analysis of more than 200 h of time-resolved CCD photometry of the pulsating DA white dwarf star EC 14012−1446 acquired during four observing epochs in three different years, including a coordinated three-site campaign. A total of 19 independent frequencies in the star's light variations together with 148 combination signals up to fifth order could be detected. We are unable to obtain the period spacing of the normal modes and therefore a mass estimate of the star, but we infer a fairly short rotation period of  0.61 ±0.03 d  , assuming the rotationally split modes are  ℓ= 1  . The pulsation modes of the star undergo amplitude and frequency variations, in the sense that modes with higher radial overtone show more pronounced variability and that amplitude changes are always accompanied by frequency variations. Most of the second-order combination frequencies detected have amplitudes that are a function of their parent mode amplitudes, but we found a few cases of possible resonantly excited modes. We point out the complications in the analysis and interpretation of data sets of pulsating white dwarfs that are affected by combination frequencies of the form   f _A + f _B − f _C   intruding into the frequency range of the independent modes.     

Limb-darkening corrections for interferometric uniform disc stellar angular diameters

Stellar angular diameters determined interferometrically are generally established by fitting the observed visibility data with a curve appropriate for a uniformly illuminated disc. The resulting uniform-disc diameters must be corrected for the effects of limb darkening in order to determine the true angular diameters of the stars. An extensive grid of limb-darkening corrections, based directly on the centre-to-limb intensity variations for Kurucz model stellar atmospheres, has been computed without the intermediate step of a parametrized representation of the centre-to-limb variation. The limitations of this method of correction are discussed.     

A Helioseismic Perspective on the Depth of the Minimum Between Solar Cycles 23 and 24

The minimum in the solar-activity cycle observed between Cycles 23 and 24 is generally regarded as being unusually deep and long. This minimum is being followed by a cycle with one of the smallest amplitudes in recent history. We perform an in-depth analysis of this minimum with helioseismology. We use Global Oscillation Network Group (GONG) data to demonstrate that the frequencies of helioseismic oscillations are a sensitive probe of the Sun’s magnetic field: The frequencies of the helioseismic oscillations were found to be systematically lower in the minimum following Cycle 23 than in the minimum preceding it. This difference is statistically significant and may indicate that the Sun’s global magnetic field was weaker in the minimum following Cycle 23. The size of the shift in oscillation frequencies between the two minima is dependent on the frequency of the oscillation and takes the same functional form as the frequency dependence observed when the frequencies at cycle maximum are compared with the cycle-minimum frequencies. This implies that the same near-surface magnetic perturbation is responsible. Finally, we determine that the difference in the mean magnetic field between the minimum preceding Cycle 23 and that following it is approximately 1 G.     

The characteristics of Pc 3 and Pc 4 geomagnetic micropulsations recorded at Sanae,Antarctica

Digital dynamic spectra of micropulsations recorded at SANAE (L ～ 4) show that Pc 3 pulsations have frequencies which decrease throughout the day. Both the onset frequency and the rate of decrease of frequency depend on the level of magnetic activity during the previous night. The variation of Pc 3 amplitudes and frequencies is explained in terms of the position of the plasmapause and the associated Pc 3 resonance region in the plasmatrough.For Pc 4 pulsations a constant frequency is observed on most days and it is not possible to infer the presence of a Pc 4 resonance region.     

Three new pulsating sdB stars from the Edinburgh–Cape survey

We report the discovery of very rapid pulsations in three sdB stars from the Edinburgh–Cape blue object survey. The short periods, small amplitudes and multi-periodicity clearly establish these stars as members of the EC 14026 class. EC 11583−2708 has pulsation periods near 149, 144 and 114 s, though evidence is presented that the 149-s period is resolved into two periods at 148.87 and 148.55 s by the full photoelectric data set. The amplitudes of the detected variations are in the range 0.002–0.006 mag. The light variation of EC 20338−1925 is dominated by a period near 147 s with a very large amplitude for a variable sdB star (0.025 mag), though four other frequencies are detected with periods near 168, 151, 141 and 135 s and amplitudes in the range 0.002–0.005 mag. The third star, EC 09582−1137, displays a light curve which is virtually a textbook example of frequency beating, being produced by two pulsations of almost equal amplitude (∼0.008 mag) and periods near 136.0 and 151.2 s.     

Baroclinic Wave Transitions in the Martian Atmosphere

Surface pressure data from the Viking Lander mission and from GCM simulations of the martian atmosphere have been analyzed using singular systems analysis. Very regular oscillations are found with frequencies that are distributed bimodally with peaks corresponding to periods of approximately 2–4 days and 5–7 days, respectively. Reconstructions of the amplitudes of the two oscillations are often negatively correlated; i.e., when the amplitude of one oscillation is large, that of the other is small. The GCM simulations show that the negative correlation in the amplitudes of the two oscillations can be explained as a flipping between two different wavenumber modes. In the absence of diurnal forcing in the model, transition from an unrealistically regular high frequency mode to a similarly unrealistic regular low frequency mode occurs at most once during the northern winter season. The diurnal cycle in the model, however, acts in a non-linear sense to stimulate the transitions between the two wavenumbers and thus increases the frequency of mode flipping events. The corresponding simulations bear a closer resemblance to the observations.