High-Precision Measurements of the Solar Diameter and Oblateness by the Solar Disk Sextant (SDS) Experiment

We reduce and analyze, in a uniform way, all of the data obtained by the Solar Disk Sextant (SDS) experiment, concerning high-precision measurements of the solar radius and oblateness, in the bandwidth 590 {–} 670 nm, made onboard stratospheric balloons during a series of flights carried out in 1992, 1994, 1995, and 1996. The measured radius value appears anti-correlated with the level of solar activity, ranging from about 959.5 to 959.7 arcsec. Its variation from year to year is outside the error range, which is mostly due to a systematic diurnal behavior, particularly evident in the 1996 flight. The oblateness shows an analogous temporal behavior, ranging from about (4.3 to 10.3) × 10⁻⁶. We regret that Prof. Caccin died on June 19, 2004.     

Extension of the rotating dipole model with oblateness of both primaries

A rotating mass dipole can be used to understand the dynamical behaviors around elongated asteroids as well as binary asteroids. In this paper an improved dipole model with oblateness in both primaries is investigated. The dynamical equations of a particle around the improved model are first derived by introducing the oblateness coefficients. The characteristic equations of equilibrium points are obtained, resulting in the emergence of new equilibria in the equatorial plane and the plane xoz depending on the shape of the spheroid. Numerical simulations are performed to illustrate the distribution of these equilibrium points. Significant influence from the oblateness of the primaries on the topological structure is also analyzed via zero-velocity curves.     

Oblateness effect of Saturn on periodic orbits in the Saturn-Titan restricted three-body problem

We explore the effect of oblateness of Saturn (more massive primary) on the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system in the framework of planar circular restricted three-body problem. First order interior and exterior mean motion resonances are located. The effect of oblateness is studied on the location, nature and size of periodic and quasi-periodic orbits, using the numerical technique of Poincare surface of sections. Some of the periodic orbits change to quasi-periodic orbits due to the effect of oblateness and vice-versa. The stability of the orbits around Saturn, Titan and both varies with the inclusion of oblateness. The centers of the periodic orbits around Titan move towards Saturn, whereas those around Saturn move towards Titan. For the orbit around Titan at C=2.9992, x=0.959494, the apocenter becomes pericenter. By incorporating oblateness effect, the orbit around Titan at C=2.99345, x=0.924938 is captured by Saturn, remains in various trajectories around Saturn, and as time progresses it spirals away around both the primaries.     

Application of binary pulsars to axisymmetric bodies in the Elliptic R3BP

Binary systems hosting astrophysical compact objects such as white dwarfs and/or neutron stars provide excellent test beds for studying the impact of the oblateness of the main bodies in the restricted three-body problem (R3BP). The case is investigated when the primary bodies are non-luminous, non-spherical (oblate) bodies and the third body of infinitesimal mass is also an oblate spheroid. The existence of extra solar planets orbiting these systems constitutes a three-body problem which makes them excellent models for this axisymmetric ER3BP. The positions of the equilibrium points are affected by the oblateness parameters of the three-bodies; this is shown for double neutron star binaries. The triangular points are stable for 0<μ<μ _c ; where μ is the mass ratio (μ≤1/2) and μ _c is the critical mass value influenced by the eccentricity, semi major axis and oblateness factors. The size of the region of stability increases with decreasing values of the oblateness. The oblateness of the system’s bodies does not affect the nature of the stability of the collinear points since they remain unstable. Due to the almost equal masses of the primaries, our study shows that even the triangular points of these systems are unstable.     

The evolution of prominences and their relationship to active centers

The radiation field, emergent from an inhomogeneous atmosphere, may differ significantly from that calculated using a mean model for such an atmosphere. In the solar case, horizontal anisotropy of the granulation pattern leads to azimuthal dependence of the emergent intensity, and this appears as a latitude-dependent limb flux which may mimic oblateness. We examine this latitude-dependence for several two and three-dimensional models of the inhomogeneous solar atmosphere, with varying degrees of anisotropy in the granulation pattern. Elongation along an east-west axis of about 7% would yield a signal somewhat imperfectly mimicking an excess oblateness of 4 × 10^–5. Using the Babcock-Leighton model of the general solar magnetic field we show that some stretching of granules, of this order of magnitude, should be expected. However, it may vary with the solar activity cycle, and in any case the result is very sensitive to the parameters adopted. Even if study of granulation observations should exclude elongations as high as 7%, smaller essentially undetectable elongations may exist. We find that 1 % elongation can account for 25–50 % of a signal corresponding to excess oblateness 4 × 10^–5. We conclude that anisotropy of the granulation pattern may influence oblateness determinations; when this is considered together with other effects, much of the claimed oblateness may be eliminated.     

Combined effects of perturbations,radiation, and oblateness on the nonlinear stability of triangular points in the restricted three-body problem

This paper investigates the nonlinear stability of the triangular equilibrium points under the influence of small perturbations in the Coriolis and centrifugal forces together with the effect of oblateness and radiation pressures of the primaries. It is found that the triangular points are stable for all mass ratios in the range of linear stability except for three mass ratios depending upon above perturbations, oblateness coefficients and mass reduction factors.     

Exoplanet's Figure and Its Interior

Along with the development of the observing technology, the observation and study on the exoplanets’ oblateness and apsidal precession have achieved significant progress. The oblateness of an exoplanet is determined by its interior density profile and rotation period. Between its Love number k₂ and core size exists obviously a negative correlation. So oblateness and k₂ can well constrain its interior structure. Starting from the Lane-Emden equation, the planet models based on different polytropic indices are built. Then the flattening factors are obtained by solving the Wavre's integro-differential equation. The result shows that the smaller the polytropic index, the faster the rotation, and the larger the oblateness. We have selected 469 exoplanets, which have simultaneously the observed or estimated values of radius, mass, and orbit period from the NASA (National Aeronautics and Space Administration) Exoplanet Archive, and calculated their flattening factors under the two assumptions: tidal locking and fixed rotation period of 10.55 hours. The result shows that the flattening factors are too small to be detected under the tidal locking assumption, and that 28% of exoplanets have the flattening factors larger than 0.1 under the fixed rotation period of 10.55 hours. The Love numbers under the different polytropic models are solved by the Zharkov's approach, and the relation between k₂ and core size is discussed.     

Trajectories of satellites under the combined influences of earth oblateness and air drag

Trajectories of satellites under the influences of earth oblateness and air drag are derived by the asymptotic method in nonlinear mechanics. Based on the assumptions: (1) the dominant oblateness factor of the earth is the second harmonic (J ₂), (2) a non-rotating, spherically symmetric atmosphere and an exponential distribution of atmospheric density, (3) original elliptical orbits being of small eccentricity, closed-form solutions for the improved first order approximation are obtained. After finding the osculating orbital elements of the resulting trajectories, we expose the behavior of osculating orbits at various inclinations.     

The influence of triaxiality and oblateness on the triangular points of double pulsars in the ER3BP

We consider the motion of a test particle around a triaxial primary and an oblate companion orbiting each other in elliptic orbits about their common barycenter in the neighborhood of triangular libration points. The positions and stability of these points are influenced by the triaxiality and oblateness of the primary and secondary, and by the semi-major axis and eccentricity of the orbits. The triangular points are stable for 0<μ<μ _c ; where μ is the mass ratio (μ≤1/2) and μ _c is the critical mass value influenced by the eccentricity, oblateness, semi major axis and triaxiality factors. The size of the region of stability increases with decreasing values of triaxiality and oblateness. An application of the results obtain to double neutron star binaries results show that the positions and stability of the triangular points of PSR J1518+4904, PSR B1534+12, PSR B1914+16 and PSR B2127+11c are affected by the parameters in the systems’ dynamics.     

The Sitnikov family and the associated families of 3D periodic orbits in the photogravitational RTBP with oblateness

We consider the photogravitational restricted three-body problem with oblateness and study the Sitnikov motions. The family of straight line oscillations exists only in the case where the primaries are of equal masses as in the classical Sitnikov problem and have the same oblateness coefficients and radiation factors. A perturbation method based on Floquet theory is applied in order to study the stability of the motion and critical orbits are determined numerically at which families of three-dimensional periodic orbits of the same or double period bifurcate. Many of these families are computed.     

Effect of oblateness and radiation pressure on angular frequencies at collinear points

In the three-dimensional restricted three-body problem, by considering the more massive primary as an oblate spheroid with its equatorial plane coincident with the plane of motion as well as source of radiation, it is found that the collinear point L ₁ comes nearer to the primaries with the increase in oblateness and radiation pressure, while L ₂ and L ₃ move away from the more massive primary with the increase in oblateness and come nearer to it with the increase in radiation pressure. It is noted that the angular frequency s ₁ at L ₁ increases with oblateness as well as with radiation pressure. s ₂ increases with oblateness and decreases with radiation pressure and s ₃ decreases with oblateness and increases with radiation pressure. A study on the norms of the characteristic roots λ and s at L ₁, L ₂ and L ₃ is carried out. It is established that for certain oblateness and radiation pressure parameters there is a one-to-one commensurability at the collinear points L ₂, L ₃ between the planar angular frequencies (s _2,3) and the corresponding angular frequency (s _z ) in the z-direction, and that at L ₁ no such commensurability exists. At L ₂ and L ₃, the value of oblateness parameter providing the commensurability decreases with the increase in the radiation pressure. However, the commensurable angular frequencies and eccentricity of the periodic orbits decrease at L ₂ and increase at L ₃, with the increase in the radiation pressure.     

考虑地球扁率摄动影响的初轨计算方法

在二体问题意义下的短弧定轨,Laplace型方法是最主要最典型的一种初轨计算方法。若测角资料达到10^-4-10^-5精度(相当于2″—20″之间),那么要使定轨精度达到与其相应的程度,地球非球形引力位中的扁率项摄动应该考虑,在此前提下,同样可以采用相应的Laplace型定轨方法。即给出这种严格包含扁率摄动的初轨计算方法的原理和具体计算过程以及计算实例,除采用多资料定轨方法外,这种方法也是提高初轨计算精度的一种途径,它同样可用于多资料的情况,这种方法对于大扁率主天体(即中心天体)的卫星定轨将更有实用价值。     

Einstein-Maxwell field equations in isotropic coordinates: an application to neutron star and quark star

The photogravitational restricted three bodies within the framework of the post-Newtonian approximation is carried out. The mass of the primaries are assumed changed under the effect of continuous radiation process and oblateness effects of the two primaries. New perturbed locations of the triangular points are computed. In order to introduce a semi-analytical view, A Mathematica program is constructed so as to draw the locations of triangular points versus the whole range of the mass ratio μ taking into account the photo-gravitational effects, the relativistic corrections and/or oblateness effects. All the obtained figures are analyzed.     

Non-linear stability zones around triangular equilibria in the plane circular restricted three-body problem with oblateness

Non-linear stability zones of the triangular Lagrangian points are computed numerically in the case of oblate larger primary in the plane circular restricted three-body problem. It is found that oblateness has a noticeable effect and this is identified to be related to the resonant cases and the associated curves in the mass parameter versus oblateness coefficientA ₁ parameter space.     

Collinear equilibria and their characteristic exponents in the restricted three-body problem when the primaries are oblate spheroids

Celestial Mechanics and Dynamical Astronomy - In this paper location of the collinear libration points is investigated numerically, by taking the oblateness of the primaries into consideration, for...     

Combined effect of oblateness,radiation and a circular cluster of material points on the stability of triangular liberation points in the R3BP

This paper studies the motion of an infinitesimal mass in the framework of the restricted three-body problem (R3BP) under the assumption that the primaries of the system are radiating-oblate spheroids, enclosed by a circular cluster of material points. It examines the effects of radiation and oblateness up to J ₄ of the primaries and the potential created by the circular cluster, on the linear stability of the liberation locations of the infinitesimal mass. The liberation points are found to be stable for 0<μ<μ _c and unstable for $\mu_{c}\le\mu\le\frac{1}{2}$ , where μ _c is the critical mass value depending on terms which involve parameters that characterize the oblateness, radiation forces and the circular cluster of material points. The oblateness up to J ₄ of the primaries and the gravitational potential from the circular cluster of material points have stabilizing propensities, while the radiation of the primaries and the oblateness up to J ₂ of the primaries have destabilizing tendencies. The combined effect of these perturbations on the stability of the triangular liberation points is that, it has stabilizing propensity.     

Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory

This paper deals with the stability analysis of the triangular equilibrium points for the generalized problem of the photogravitational restricted three body where both the primaries are radiating. The problem is generalized in the sense that the eccentricity of the orbits and the oblateness due to both the primaries and infinitesimal are considered. The stability in the case of linear resonance are analyzed based on the Floquet’s theory for finding the characteristic exponent for a system containing periodic coefficients. It was found that the critical value of μ for the stability boundary for parametric excitation is dependent on the oblateness of the primaries as well as infinitesimal.     

The equilibrium state in the Magnetic-Binary problem when the more massive primary is an oblate spheroid

The author's aim is to study in this paper the stationary state of the planar Magnetic-Binary problem by taking into consideration the oblateness of the more massive primary, and then to investigate the stability of motion about the equilibrium points by means of the characteristic polynomial of the linearised variational equations.     

Second-order state transition for relative motion near perturbed,elliptic orbits

This paper develops a tensor and its inverse, for the analytical propagation of the position and velocity of a satellite, with respect to another, in an eccentric orbit. The tensor is useful for relative motion analysis where the separation distance between the two satellites is large. The use of nonsingular elements in the formulation ensures uniform validity even when the reference orbit is circular. Furthermore, when coupled with state transition matrices from existing works that account for perturbations due to Earth oblateness effects, its use can very accurately propagate relative states when oblateness effects and second-order nonlinearities from the differential gravitational field are of the same order of magnitude. The effectiveness of the tensor is illustrated with various examples.