Stability of the Triangular Points Under Combined Effects of Radiation and Oblateness in the Restricted Three-Body Problem

This paper deals with the existence of triangular points and their linear stability when the primaries are oblate spheroid and sources of radiation considering the effect of oblateness up to 10^?6 of main terms in the restricted three-body problem; we see that the locations of the triangular points are affected by the oblateness of the primaries and solar radiation pressure. It is further seen that these points are stable for 0 ≤ μ ≤μ _c ; and unstable for μ _c  ≤ μ ≤1/2; where μ _c is the critical mass value depending on terms which involve parameters that characterize the oblateness and radiation repulsive forces such that $ \mu_{c} \in (0,1/2) $ ; in addition to this an algorithm has been constructed to calculate the critical mass value.     

Long term evolution of quasi-circular Trojan orbits

Trojan asteroids undergo very large perturbations because of their resonance with Jupiter. Fortunately the secular evolution of quasi circular orbits remains simple—if we neglect the small short period perturbations. That study is done in the approximation of the three dimensional circular restricted three-body problem, with a small mass ratio μ—that is about 0.001 in the Sun Jupiter case. The Trojan asteroids can be defined as celestial bodies that have a “mean longitude”, M + ω + Ω, always different from that of Jupiter. In the vicinity of any circular Trojan orbit exists a set of “quasi-circular orbits” with the following properties: (A) Orbits of that set remain in that set with an eccentricity that remains of the order of the mass ratio μ. (B) The relative variations of the semi-major axis and the inclination remain of the order of ${\sqrt{\mu}}$ . (C) There exist corresponding “quasi integrals” the main terms of which have long-term relative variations of the order of μ only. For instance the product c(1 – cos i) where c is the modulus of the angular momentum and i the inclination. (D) The large perturbations affect essentially the difference “mean longitude of the Trojan asteroid minus mean longitude of Jupiter”. That difference can have very large perturbations that are characteristics of the “horseshoes orbit”. For small inclinations it is well known that this difference has two stable points near ±60° (Lagange equilibrium points L₄ and L₅) and an unstable point at 180° (L₃). The stable longitude differences are function of the inclination and reach 180° for an inclination of 145°41′. Beyond that inclination only one equilibrium remains: a stable difference at 180°.     

The changing rotational excitation of C3 in comet 9P/Tempel 1 during Deep Impact

The 4050 Å band of C₃ was observed with Keck/HIRES echelle spectrometer during the Deep Impact encounter. We perform a 2-dimensional analysis of the exposures in order to study the spatial, spectral, and temporal changes in the emission spectrum of C₃. The rotational population distribution changes after impact, beginning with an excitation temperature of ～45 K at impact and increasing for 2 hr up to a maximum of 61±5 K. From 2 to 4 hours after impact, the excitation temperature decreases to the pre-impact value. We measured the quiescent production rate of C₃ before the encounter to be 1.0×10²³ s^?1, while 2 hours after impact we recorded a peak production rate of 1.7×10²³ s^?1. Whereas the excitation temperature returned to the pre-impact value during the observations, the production rate remained elevated, decreasing slowly, until the end of the 4 hr observations. These results are interpreted in terms of changing gas densities in the coma and short-term changes in the primary chemical production mechanism for C₃.     

The distance to the Galactic center determined by OB stars

The OB stars are concentrated near the Galactic plane and should permit a determination of the distance to the Galactic center. van Leeuwen’s new reduction of the Hipparcos catalog provides, after 824 Gould belt stars have been excluded, 6288 OB stars out to 1 kpc and Westin’s compilation an additional 112 stars between 1 kpc and 3 kpc. The reduction model involves 14 unknowns: the Oort A and B constants, the distance to the Galactic center R ₀, 2 second-order partial derivatives, the 3 components of solar motion, a K term, a first order partial derivative for motion perpendicular to the Galactic plane, a second-order partial for acceleration perpendicular to the plane, two terms for a possible expansion of the OB stars, and a C constant. The model is nonlinear, and the unknowns are calculated by use the simplex algorithm for nonlinear adjustment applied to 14313 equations of condition, 12694 in proper motion and 1619 in radial velocity. Various solutions were tried: an L₁ solution, a least squares solution with modest (2.7 %) trim of the data, and two robust least squares solutions (biweight and Welsch weighting) with more extreme trimming. The Welsch solution seems to give the best results and calculates a distance to the Galactic center 6.72±0.39 kpc. Statistical tests show that the data are homogeneous, that the reduction model seems adequate and conforms with the assumptions used in its derivation, and that the post-fit residuals are random. Inclusion of more terms, such as streaming motion induced by Galactic density waves, degrades the solution.     

Velocities and Temperatures of an Ellerman Bomb and Its Associated Features

We investigated the velocity and temperature characteristics of an Ellerman bomb (EB) and its associated features based on observations made with the Fast Imaging Solar Spectrograph (FISS) and a broadband TiO filter of the 1.6 meter New Solar Telescope at Big Bear Solar Observatory. In the TiO images of the photospheric level, we found a granular cell expanding in two opposite directions near the site of the EB. When one end of this granule reached the EB site, the transverse speed of the tip of the expanding granule rapidly decreased and the EB brightened. The wings of the Hα profile of the EB indicated that the EB was blueshifted up to 7 km?s^?1. About 260 s after the EB brightening, a surge was seen in absorption and varied from a blueshift of 20 km?s^?1 to a redshift of 40 km?s^?1 seen in the Hα and Ca ii 8542 Å lines. From the Doppler absorption width of the two lines determined by applying the cloud model, we estimated the mean temperature of the surge material to be about 29000 K and the mean speed of nonthermal motion to be about 11 km?s^?1. We discuss the physical implications of our results in terms of magnetic reconnection and processes related to it.     

The stability of vertical motion in the N-body circular Sitnikov problem

We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. In particular, we consider ν = N ? 1 equal mass primary bodies which rotate on a circle, while the Nth body (of negligible mass) moves perpendicularly to the plane of the primaries. Thus, we extend previous work on the 4-body Sitnikov problem to the N-body case, with N = 5, 9, 15, 25 and beyond. We find, for all cases we have considered with N ≥ 4, that the Sitnikov family has only one stability interval (on the z-axis), unlike the N = 3 case where there is an infinity of such intervals. We also show that for N = 5, 9, 15, 25 there are, respectively, 14, 16, 18, 20 critical Sitnikov periodic orbits from which 3D families (no longer rectilinear) bifurcate. We have also studied the physically interesting question of the extent of bounded dynamics away from the z-axis, taking initial conditions on x, y planes, at constant z(0) = z ₀ values, where z ₀ lies within the interval of stable rectilinear motions. We performed a similar study of the dynamics near some members of 3D families of periodic solutions and found, on suitably chosen Poincaré surfaces of section, “islands” of ordered motion, while away from them most orbits become chaotic and eventually escape to infinity. Finally, we solve the equations of motion of a small mass in the presence of a uniform rotating ring. Studying the stability of the vertical orbits in that case, we again discover a single stability interval, which, as N grows, tends to coincide with the stability interval of the N-body problem, when the values of the density and radius of the ring equal those of the corresponding system of N ? 1 primary masses.     

Drifting ELF/VLF Emissions Observed at Low Latitude Ground Station During Geomagnetic Storm

We have reported for the first time total seven strong events of drifting ELF/VLF discrete emissions observed on 28th–29th April, 1990 in the pre-midnight sector at Varanasi (Geomag. lat. 14°55′N, long. 154°E, L = 1.07). The events exhibit a regular increasing as well as decreasing frequency drifts and are mainly discrete periodic emissions of riser, faller and hook types observed during a geomagnetic storm period, with minimum Dst-index ?98 nT and K _p -index ≥ 5. The frequency drift in ELF/VLF emissions at low latitudes seems to be a rare phenomenon. The repetition period and the frequency drift rate have been evaluated for all the recorded events. The frequency drifts have been interpreted in terms of a combined effect of L-shell drift of interacting energetic electrons and the change in convection electric fields during the storm developments. The computed maximum spectral power density $ \left\langle {B_{f}^{2} } \right\rangle_{\max } $ of the wave varies between 1.8 × 10^?21 and 4.08 × 10^?22 Gauss²/Hz, whereas frequency drift rates are in agreement with the observed values.     

Variation of Emission Line Width in Mid- and High-Latitude Corona

Spectroscopic studies of the solar corona, using the high spatial and spectral resolution 25-cm coronagraph at the Norikura Solar Observatory for equatorial off-limb observations, indicated that the variation of radiance and line width with height is different for different temperature lines. The line width of the forbidden red emission line [Fe x] 6374 Å was found to increase with height, and that of the green emission line [Fe xiv] 5303 Å decreased with height. This had been interpreted in terms of the interaction between different temperature plasmas but needed to be confirmed. Further observations were made on several days during 2004, in two emission lines simultaneously covering the mid-latitude and polar regions to investigate the existence of the observed variation in other parts of the solar corona. In this study, we have analysed several raster scans that cover mid- and high-latitude regions of the off-limb corona in all four bright emission lines [Fe x] 6374 Å, [Fe xi] 7892 Å, [Fe xiii] 10747 Å, and [Fe xiv] 5303 Å. We find that the FWHM of the red line increases with height and that of the green line decreases with height, similar to the observations in the equatorial regions. The line widths are higher in the polar regions for all of the observed emission lines except the green line. Higher values of FWHM in polar regions may imply higher non-thermal velocities, which could be further linked to a non-thermal source powering the solar-wind acceleration, but the reason for the behaviour of the green emission line remains to be explored.     

The Rotational Periods of Three C-Type Asteroids 431, 521 and 524

The determination for asteroids’ spin parameters is very important for the physical study of asteroids and their evolution. Sometimes, the low amplitude of light curves and kinds of systematic errors in photometric data prevent the determination of the asteroids’ spin period. To solve such a problem, we introduced the de-correlation methods developed in searching for exoplanetary transit signal into the asteroid’s data reduction in this paper. By applying the principle of Collier Cameron (MNRAS 373:799–810, 2006) and Tamuz et al. (MNRAS 356:1466–1470, 2005)’s, we simulated the systematic effects in photometric data of asteroid, and removed those simulated errors from photometric data. Therefore the S/N of intrinsic signals of three selected asteroids are enhanced significantly. As results, we derived the new spin periods of 18.821 ± 0.011 h, 28.202 ± +0.071 h for (431) and (521) respectively, and refined the spin period of (524) as 14.172 ± 0.016 h.     

Non-linear stability of L 4 in the restricted problem when the primaries are finite straight segments under resonances

The non-linear stability of L ₄ in the restricted three-body problem when both primaries are finite straight segments in the presence of third and fourth order resonances has been investigated. Markeev’s theorem (Markeev in Libration Points in Celestial Mechanics and Astrodynamics, 1978) is used to examine the non-linear stability for the resonance cases 2:1 and 3:1. It is found that the non-linear stability of L ₄ depends on the lengths of the segments in both resonance cases. It is also found that the range of stability increases when compared with the classical restricted problem. The results have been applied in the following asteroids systems: (i) 216 Kleopatra–951 Gaspara, (ii) 9 Metis–433 Eros, (iii) 22 Kalliope–243 Ida.     

Large retrograde Centaurs: visitors from the Oort cloud?

Among all the asteroid dynamical groups, Centaurs have the highest fraction of objects moving in retrograde orbits. The distribution in absolute magnitude, H, of known retrograde Centaurs with semi-major axes in the range 6–34 AU exhibits a remarkable trend: 10 % have H<10 mag, the rest have H>12 mag. The largest objects, namely (342842) 2008 YB₃, 2011 MM₄ and 2013 LU₂₈, move in almost polar, very eccentric paths; their nodal points are currently located near perihelion and aphelion. In the group of retrograde Centaurs, they are obvious outliers both in terms of dynamics and size. Here, we show that these objects are also trapped in retrograde resonances that make them unstable. Asteroid 2013 LU₂₈, the largest, is a candidate transient co-orbital to Uranus and it may be a recent visitor from the trans-Neptunian region. Asteroids 342842 and 2011 MM₄ are temporarily submitted to various high-order retrograde resonances with the Jovian planets but 342842 may be ejected towards the trans-Neptunian region within the next few hundred kyr. Asteroid 2011 MM₄ is far more stable. Our analysis shows that the large retrograde Centaurs form an heterogeneous group that may include objects from various sources. Asteroid 2011 MM₄ could be a visitor from the Oort cloud but an origin in a relatively stable closer reservoir cannot be ruled out. Minor bodies like 2011 MM₄ may represent the remnants of the primordial planetesimals and signal the size threshold for catastrophic collisions in the early Solar System.     

Modeling and Measuring the Flux Reconnected and Ejected by the Two-Ribbon Flare/CME Event on 7 November 2004

Observations of the large two-ribbon flare on 7 November 2004 made using SOHO and TRACE data are interpreted in terms of a three-dimensional magnetic field model. Photospheric flux evolution indicates that ?1.4×10⁴³ Mx² of magnetic helicity was injected into the active region during the 40-hour buildup prior to the flare. The magnetic model places a lower bound of 8×10³¹ ergs on the energy stored by this motion. It predicts that 5×10²¹ Mx of flux would need to be reconnected during the flare to release the stored energy. This total reconnection compares favorably with the flux swept up by the flare ribbons, which we measure using high-time-cadence TRACE images in 1?600 Å. Reconnection in the model must occur in a specific sequence that would produce a twisted flux rope containing significantly less flux and helicity (10²¹ Mx and ?3×10⁴² Mx², respectively) than the active region as a whole. The predicted flux compares favorably with values inferred from the magnetic cloud observed by Wind. This combined analysis yields the first quantitative picture of the flux processed through a two-ribbon flare and coronal mass ejection.     

Spectral Characteristics of the He <Emphasis Type="SmallCaps">i</Emphasis> D<Subscript>3</Subscript> Line in a Quiescent Prominence Observed by THEMIS

We analyze the observations of a quiescent prominence acquired by the Téléscope Heliographique pour l’Étude du Magnetisme et des Instabilités Solaires (THEMIS) in the He?i 5876 Å (He?i D₃) multiplet aiming to measure the spectral characteristics of the He?i D₃ profiles and to find for them an adequate fitting model. The component characteristics of the He?i D₃ Stokes I profiles are measured by the fitting system by approximating them with a double Gaussian. This model yields an He?i D₃ component peak intensity ratio of \(5.5\pm0.4\), which differs from the value of 8 expected in the optically thin limit. Most of the measured Doppler velocities lie in the interval ±?5 km?s^?1, with a standard deviation of ±?1.7 km?s^?1 around the peak value of 0.4 km?s^?1. The wide distribution of the full-width at half maximum has two maxima at 0.25 Å and 0.30 Å for the He?i D₃ blue component and two maxima at 0.22 Å and 0.31 Å for the red component. The width ratio of the components is \(1.04\pm0.18\). We show that the double-Gaussian model systematically underestimates the blue wing intensities. To solve this problem, we invoke a two-temperature multi-Gaussian model, consisting of two double-Gaussians, which provides a better representation of He?i D₃ that is free of the wing intensity deficit. This model suggests temperatures of 11.5 kK and 91 kK, respectively, for the cool and the hot component of the target prominence. The cool and hot components of a typical He?i D₃ profile have component peak intensity ratios of 6.6 and 8, implying a prominence geometrical width of 17 Mm and an optical thickness of 0.3 for the cool component, while the optical thickness of the hot component is negligible. These prominence parameters seem to be realistic, suggesting the physical adequacy of the multi-Gaussian model with important implications for interpreting He?i D₃ spectropolarimetry by current inversion codes.     

The solar modulation of electrons in the heliosphere

The propagation and modulation of electrons in the heliosphere play an important part in improving our understanding and assessment of the modulation processes. A full three-dimensional numerical model is used to study the modulation of galactic electrons, from Earth into the inner heliosheath, over an energy range from 10 MeV to 30 GeV. The modeling is compared with observations of 6–14 MeV electrons from Voyager 1 and observations at Earth from the PAMELA mission. Computed spectra are shown at different spatial positions. Based on comparison with Voyager 1 observations, a new local interstellar electron spectrum is calculated. We find that it consists of two power-laws: In terms of kinetic energy E, the results give E ^?1.5 below ～500 MeV and E ^?3.15 at higher energies. Radial intensity profiles are computed also for 12 MeV electrons, including a Jovian source, and compared to the 6–14 MeV observations from Voyager 1. Since the Jovian and galactic electrons can be separated in the model, we calculate the intensity of galactic electrons below 100 MeV at Earth. The highest possible differential flux of galactic electrons at Earth with E=12 MeV is found to have a value of 2.5×10^?1 electrons m^?2?s^?1?sr^?1?MeV^?1 which is significantly lower (a factor of 3) than the Jovian electron flux at Earth. The model can also reproduce the extraordinary increase of electrons by a factor of 60 at 12 MeV in the inner heliosheath. A lower limit for the local interstellar spectrum at 12 MeV is estimated to have a value of (90±10) electrons m^?2?s^?1?sr^?1?MeV^?1.     

Equilibrium Convection on a Tidally Heated and Stressed Icy Shell of Europa for a Composite Water Ice Rheology

Water ice I rheology is a key factor for understanding the thermal and mechanical state of the outer shell of the icy satellites. Ice flow involves several deformation mechanisms (both Newtonian and non-Newtonian), which contribute to different extents depending on the temperature, grain size, and applied stress. In this work I analyze tidally heated and stressed equilibrium convection in the ice shell of Europa by considering a composite viscosity law which includes diffusion creep, basal slip, grain boundary sliding and dislocation creep, and. The calculations take into account the effect of tidal stresses on ice flow and use grain sizes between 0.1 and 100 mm. An Arrhenius-type relation (useful for parameterized convective models) is found then by fitting the calculated viscosity between 170 and 273 K to an exponential regression, which can be expressed in terms of pre-exponential constant and effective activation energy. I obtain convective heat flows between ~40 and ~60 mW m^?2, values lower than those usually deduced (~100 mW m^?2) from geological indicators of lithospheric thermal state, probably indicating heterogeneous tidal heating. On the other hand, for grain sizes larger than ~0.3 mm the thicknesses of the ice shell and convective sublayer are ~20–30 km and ~5–20 km respectively, values in good agreement with the available information for Europa. So, some fundamental geophysical characteristics of the ice shell of Europa could be arising from the properties of the composite water ice rheology.     

Mars interplanetary trajectory design via Lagrangian points

With the increase in complexities of interplanetary missions, the main focus has shifted to reducing the total delta-V for the entire mission and hence increasing the payload capacity of the spacecraft. This paper develops a trajectory to Mars using the Lagrangian points of the Sun-Earth system and the Sun-Mars system. The whole trajectory can be broadly divided into three stages: (1) Trajectory from a near-Earth circular parking orbit to a halo orbit around Sun-Earth Lagrangian point L₂. (2) Trajectory from Sun-Earth L₂ halo orbit to Sun-Mars L₁ halo orbit. (3) Sun-Mars L₁ halo orbit to a circular orbit around Mars. The stable and unstable manifolds of the halo orbits are used for halo orbit insertion. The intermediate transfer arcs are designed using two-body Lambert’s problem. The total delta-V for the whole trajectory is computed and found to be lesser than that for the conventional trajectories. For a 480 km Earth parking orbit, the total delta-V is found to be 4.6203 km/s. Another advantage in the present approach is that delta-V does not depend upon the synodic period of Earth with respect to Mars.     

On the double averaged three-body problem

The Hamiltonian of three point masses is averaged over fast variablel and ll (mean anomalies) The problem is non-planar and it is assumed that two of the bodies form a close pair (stellar three-body problem). Only terms up to the order of (a/á)⁴ are taken into account in the Hamiltonian, wherea andá are the corresponding semi-major axes. Employing the method of elimination of the nodes, the problem may be reduced to one degree of freedom. Assuming in addition that the angular momentum of the close binary is much smaller than the angular momentum of the motion of the binary around a third body, we were able to solve the equation for the eccentricity changes in terms of the Jacobian elliptic functions.     

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

We consider the planar restricted three-body problem and the collinear equilibrium point L ₃, as an example of a center × saddle equilibrium point in a Hamiltonian with two degrees of freedom. We explore numerically the existence of symmetric and non-symmetric homoclinic orbits to L ₃, when varying the mass parameter μ. Concerning the symmetric homoclinic orbits (SHO), we study the multi-round, m-round, SHO for m ≥ 2. More precisely, given a transversal value of μ for which there is a 1-round SHO, say μ ₁, we show that for any m ≥ 2, there are countable sets of values of μ, tending to μ ₁, corresponding to m-round SHO. Some comments on related analytical results are also made.     

A study of line widths and kinetic parameters of ions in the solar corona

Solar extreme-ultraviolet (EUV) lines emitted by highly charged ions have been extensively studied to discuss the issue of coronal heating and solar wind acceleration. Based on observations of the polar corona by the SUMER/SOHO spectrometer, this paper investigates the relation between the line widths and kinetic parameters of ions. It is shown that there exists a strongly linear correlation between two variables (σ/λ)² and M ^?1, where σ, λ and M are the half-width of the observed line profile at \(1/\sqrt{e}\) , the wavelength and the ion mass, respectively. The Pearson product-moment correlation coefficients exceed 0.9. This finding tends to suggest that the ions from a given height of polar corona have a common temperature and a common non-thermal velocity in terms of existing equation. The temperature and non-thermal velocity are obtained by linear least-square fit. The temperature is around 2.8 MK at heights of 57″ and 102″. The non-thermal velocity is typical 21.6 km?s^?1 at height of 57″ and 25.2 km?s^?1 at height of 102″.