Existence and stability of triangular points in the restricted three-body problem with numerical applications

In this paper, we prove that the locations of the triangular points and their linear stability are affected by the oblateness of the more massive primary in the planar circular restricted three-body problem, considering the effect of oblateness for J ₂ and J ₄. After that, we show that the triangular points are stable for 0<μ<μ _c and unstable when , where μ _c is the critical mass parameter which depends on the coefficients of oblateness. On the other hand, we produce some numerical values for the positions of the triangular points, μ and μ _c using planets systems in our solar system which emphasis that the range of stability will decrease; however this range sometimes is not affected by the existence of J ₄ for some planets systems as in Earth–Moon, Saturn–Phoebe and Uranus–Caliban systems.     

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.     

Effects of axis-symmetric body,zonal harmonic and potential from a belt on the stability of triangular libration points in the CR3BP

Within the frame work of the circular restricted three-body problem (CR3BP) we have examined the effect of axis-symmetric of the bigger primary, oblateness up to the zonal harmonic J ₄ of the smaller primary and gravitational potential from a belt (circular cluster of material points) on the linear stability of the triangular libration points. It is found that the positions of triangular libration points and their linear stability are affected by axis-symmetric of the bigger primary, oblateness up to J ₄ of the smaller primary and the potential created by the belt. The axis-symmetric of the bigger primary and the coefficient J ₂ of the smaller primary have destabilizing tendency, while the coefficient J ₄ of the smaller primary and the potential from the belt have stabilizing tendency. The overall effect of these perturbations has destabilizing tendency. This study can be useful in the investigation of motion of a particle near axis-symmetric—oblate bodies surrounded by a belt.     

Relativistic effects and solar oblateness from radar observations of planets and spacecraft

We used more than 250 000 high-precision American and Russian radar observations of the inner planets and spacecraft obtained in the period 1961–2003 to test the relativistic parameters and to estimate the solar oblateness. Our analysis of the observations was based on the EPM ephemerides of the Institute of Applied Astronomy, Russian Academy of Sciences, constructed by the simultaneous numerical integration of the equations of motion for the nine major planets, the Sun, and the Moon in the post-Newtonian approximation. The gravitational noise introduced by asteroids into the orbits of the inner planets was reduced significantly by including 301 large asteroids and the perturbations from the massive ring of small asteroids in the simultaneous integration of the equations of motion. Since the post-Newtonian parameters and the solar oblateness produce various secular and periodic effects in the orbital elements of all planets, these were estimated from the simultaneous solution: the post-Newtonian parameters are β = 1.0000 ± 0.0001 and γ = 0.9999 ± 0.0002, the gravitational quadrupole moment of the Sun is J₂ = (1.9 ± 0.3) × 10^?7, and the variation of the gravitational constant is ?/G = (?2 ± 5) × 10^?14 yr^?1. The results obtained show a remarkable correspondence of the planetary motions and the propagation of light to General Relativity and narrow significantly the range of possible values for alternative theories of gravitation.     

New values of gravitational moments J 2 and J 4 deduced from helioseismology

By applying the theory of slowly rotating stars to the Sun, the solar quadrupole and octopole moments J ₂ and J ₄ were computed using a solar model obtained from CESAM stellar evolution code (Morel, 1997) combined with a recent model of solar differential rotation deduced from helioseismology (Corbard et al., 2002). This model takes into account a near-surface radial gradient of rotation which was inferred and quantified from MDI f-mode observations by Corbard and Thompson (2002). The effect of this observational near-surface gradient on the theoretical values of the surface parameters J ₂, J ₄ is investigated. The results show that the octopole moment J ₄ is much more sensitive than the quadrupole moment J ₂ to the subsurface radial gradient of rotation.     

Long-term orbit computations with KS uniformly regular canonical elements with oblateness

This paper concerns with the study of KS uniformly regular canonical elements with Earth's oblateness. These elements, ten in number, are all constant in the unperturbed motion and even in the perturbed motion, the substitution is straightforward and elementary due to the transformation laws being explicit and closed expression. By utilizing the recursion formulas of Legendre's polynomials, we are able to include any number of Earth's zonal harmonics J _n in the package and also economize the computations. A fixed step-size fourth-order Runge-Kutta-Gill method is employed for numerical integration of the canonical equations.Utilizing 5 test cases covering a large range of semimajor axis and eccentricity, we have carried out computations to study the effects of Earth's zonal harmonics (up to J ₃₆) and integration step-size variation. Bilinear relations and energy equation are used for checking the accuracies of numerical integration. From the application point of view, the package is utilized to study the behaviour of 900 km height near-circular sun-synchronous satellite orbit over a longer duration of 220 days time (nearly 3078 revolutions) and the necessity of including more number of Earth's zonal harmonic terms is noticed. The package is also used to study the effect of higher zonal harmonics on three 900 km height near-circular orbits with inclinations of 60, 63.2, and 65 degrees, by including Earth's zonal harmonics up to J ₂₄. The mean eccentricity (e _m) is found to have long-periods of 459.6, 6925.1 and 1077.6 days, respectively. Sharp changes in the variation of _m near the minima to e_m are noticed. The values of _m are found to be very near to +-90 degrees at the extrema of e_m. The same orbit is employed to study the effect of variation of inclination from 0 to 180 degrees on long-period (T) of eccentricity with J ₂ to J ₂₄ terms. T is found to increase rapidly as we proceed towards the critical inclinations.     

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.     

Effects of zonal harmonics and a circular cluster of material points on the stability of triangular equilibrium points in the R3BP

We have studied a modified version of the classical restricted three-body problem (CR3BP) where both primaries are considered as oblate spheroids and are surrounded by a homogeneous circular planar cluster of material points centered at the mass center of the system. In this dynamical model we have examined the effects of oblateness of both primaries up to zonal harmonic J ₄; together with gravitational potential from the circular cluster of material points on the existence and linear stability of the triangular equilibrium points. It is found that, the triangular points are stable for 0<μ<μ _c and unstable for $\mu_{c} \le \mu \le \frac{1}{2}$ , where μ _c is the critical mass ratio affected by the oblateness up to J ₄ of the primaries and potential from the circular cluster of material points. The coefficient J ₄ has stabilizing tendency, while J ₂ and the potential from the circular cluster of material points have destabilizing tendency. A practical application of this model could be the study of the motion of a dust particle near oblate bodies surrounded by a circular cluster of material points.     

Solar quadrupole moment and purely relativistic gravitation contributions to Mercury's perihelion advance

The perihelion advance of the orbit of Mercury has long been one of the observational cornerstones for testing General Relativity (G.R.).The main goal of this paper is to discuss how, presently, observational and theoretical constraints may challenge Einstein's theory of gravitation characterized by β=γ=1. To achieve this purpose, we will first recall the experimental constraints upon the Eddington-Robertson parameters γ,β and the observational bounds for the perihelion advance of Mercury, Δω_obs. A second point will address the values given, up to now, to the solar quadrupole moment by several authors. Then, we will briefly comment why we use a recent theoretical determination of the solar quadrupole moment, J ₂=(2.0 ± 0.4) 10^-7, which takes into account both surfacic and internal differential rotation, in order to compute the solar contribution to Mercury's perihelion advance. Further on, combining bounds on γ and J ₂ contributions, and taking into account the observational data range for Δω_obs,we will be able to give a range of values for β. Alternatively, taking into account the observed value of Δω_obs, one can deduce a dynamical estimation of J ₂ in the setting of G.R. This point is important as it provides a solar model independent estimation that can be confronted with other determinations of J ₂ based upon solar theory and solar observations (oscillation data, oblateness...). Finally, a glimpse at future satellite experiments will help us to understand how stronger constraints upon the parameter space (γω J ₂) as well as a separation of the two contributions (from the quadrupole moment, J ₂, or purely relativistic, 2α²+2αγ–β) might be expected in the future. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

A noncanonical analytic solution to theJ 2 perturbed two-body problem

The motion of a satellite subject to an inverse-square gravitational force of attraction and a perturbation due to the Earth's oblateness as theJ ₂ term is analyzed, and a uniform, analytic solution correct to first-order inJ ₂, is obtained using a noncanonical approach. The basis for the solution is the transformation and uncoupling of the differential equations for the model. The resulting solution is expressed in terms of elementary functions of the independent variable (the ‘true anomaly’), and is of a compact and simple form. Numerical results are comparable to existing solutions.     

Interplanetary magnetic field structure and solar wind parameters as inferred from solar magnetic field observations and by using a numerical 2-D MHD model

An attempt is made to infer parameters of the solar corona and the solar wind by means of a numerical, self-consistent MHD simulation. Boundary conditions for the magnetic field are given from the observations of the large-scale magnetic field at the Sun. A two-region, planar (the ecliptic plane is assumed) model for the solar wind flow is considered. Region I of transonic flow is assumed to cover the distances from the solar surface up to 10R _S (R _S is the radius of the Sun). Region II of supersonic, super-Alfvénic flow extends between 10R _S and the Earth's orbit. Treatment for region I is that for a mixed initial-boundary value problem. The solution procedure is similar to that discussed by Endler (1971) and Steinolfson, Suess, and Wu (1982): a steady-state solution is sought as a relaxation to the dynamic equilibrium of an initial state. To obtain a solution to the initial value problem in region II with the initial distribution of dependent variables at 10R _S (deduced from the solution for region I), a numerical scheme similar to that used by Pizzo (1978, 1982) is applied. Solar rotation is taken into account for region II; hence, the interaction between fast and slow solar wind streams is self-consistently treated. As a test example for the proposed formulation and numerical technique, a solution for the problem similar to that discussed by Steinolfson, Suess, and Wu (1982) is obtained. To demonstrate the applicability of our scheme to experimental data, solar magnetic field observations at Stanford University for Carrington rotation 1682 are used to prescribe boundary conditions for the magnetic field at the solar surface. The steady-state solution appropriate for the given boundary conditions was obtained for region I and then traced to the Earth's orbit through region II. We compare the calculated and spacecraft-observed solar wind velocity, radial magnetic field, and number density and find that general trends during the solar rotation are reproduced fairly well although the magnitudes of the density in comparison are vastly different.     

The effect of dense cores on the structure and evolution of Jupiter and Saturn

We have calculated evolutionary and static models of Jupiter and Saturn with homogeneous solar composition mantles and dense cores of material consisting of solar abundances of SiO₂, MgO, Fe, and Ni. Evolutionary sequences for Jupiter were calculated with cores of mass 2, 4, 6, and 8% of the Jovian mass. Evolutionary sequences for Saturn were calculated with cores of mass 16, 18, 20, and 22% of total mass. Two envelope mixtures, representative of the solar abundances were used: X (mass fraction of hydrogen) = 0.74, Y (mass fraction of helium) = 0.24 and X = 0.77 and Y = 0.21. For Jupiter, the observations of the temperature at 1 bar pressure (T_1bar), radius and internal luminosity were best fit by evolutionary models with a core mass of ～6.5% and chemical composition of X = 0.77, Y = 0.21. The calculated cooling time for Jupiter is approximately 4.9 × 10⁹ years, which is consistent, within our error bars, with the known age of the solar system. For Saturn, the observations of the radius, internal luminosity and T_1BAR can be best fit by evolutionary models with a core mass of ～21% and chemical composition of X = 0.77, Y = 0.21. The cooling time calculated for Saturn is approximately 2.6 × 10⁹ years, almost a factor 2 less than the present age of the solar system. Static models of Jupiter and Saturn were calculated for the above chemical compositions in order to investigate the sensitivity of the calculated gravitational moments, J₂ and J₄, to the mass of the dense core, T_1BAR and hydrogen/helium ratio. We find for Jupiter that a model having a core mass of approximately 7% gives values of J₂, J₄, and T_1BAR that are within observational limits, for the mixture X = 0.77, Y = 0.21. The static Jupiter models are completely consistent with the evolutionary results. For Saturn, the quantities J₂, J₄, and J₆ determined from the static models with the most probable T_1BAR of 140°K, using modeling procedures which result in consistent models for Jupiter, are considerably below the observed values.     

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.     

Stability of Triangular Equilibrium Points in the Photogravitational Restricted Three-Body Problem with Oblateness and Potential from a Belt

We have examined the effects of oblateness up to J ₄ of the less massive primary and gravitational potential from a circum-binary belt on the linear stability of triangular equilibrium points in the circular restricted three-body problem, when the more massive primary emits electromagnetic radiation impinging on the other bodies of the system. Using analytical and numerical methods, we have found the triangular equilibrium points and examined their linear stability. The triangular equilibrium points move towards the line joining the primaries in the presence of any of these perturbations, except in the presence of oblateness up to J ₄ where the points move away from the line joining the primaries. It is observed that the triangular points are stable for 0 < μ < μ _c and unstable for \(\mu_{\mathrm{c}} \le \mu \le \frac {1}{2},\) where μ _c is the critical mass ratio affected by the oblateness up to J ₄ of the less massive primary, electromagnetic radiation of the more massive primary and potential from the belt, all of which have destabilizing tendencies, except the coefficient J₄ and the potential from the belt. A practical application of this model could be the study of motion of a dust particle near a radiating star and an oblate body surrounded by a belt.     

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.     

Photometric Variability of Neptune, 1972-2000

We present Strömgren b (472-nm) and y (551-nm) photometry of Neptune based on photoelectric measurements obtained at every apparition from 1972 to 2000. Neptune has brightened by 11% in b and 10% in y since 1980 with most of the increase occurring after 1990. By appending b data to published B magnitudes measured at Lowell from 1950 to 1966 and transformed to b, we show that Neptune is now brighter than at any time during the past half century. The nature of the year-to-year variations changed around 1990 when a steady rising trend overshadowed what appeared to be an inverse correlation with cyclic solar activity. By matching observations in b and y with near-infrared J (1.2-μm) and K (2.2-μm) photometry before, during, and after Neptune's 1976 infrared outburst, we show that the pattern of visible albedo variation parallels the infrared variation but with an amplitude 20-50 times smaller. A detailed comparison of photometry with ground-based and Voyager images at visible and red wavelengths during the 1989 Voyager encounter shows that small brightness variations occur when large discrete features rotate across Neptune's disk. This provides a rough association between visible features and photometric effects that we use to infer the state of Neptune's atmosphere for years when only photometry was available. A year-by-year analysis of variance of the photometry suggests that the 1976 and 1986-1989 infrared outbursts were isolated episodes of unusually vigorous atmospheric activity. Detrended magnitudes of Neptune are correlated with solar activity over the entire 29-year interval as well as 22-year subintervals, with solar UV now being favored as a causative mechanism rather than solar modulated galactic cosmic rays.     

Perturbation of a satellite orbiter by the oblateness of the primary planet

The perturbation of an orbiter around a large satellite of a giant planet (Jupiter, Saturn, Uranus or Neptune) produced by the oblateness of the planet is investigated. The perturbing force of theJ ₂-term (general case) and theJ ₄-term (special case of small eccentricity and inclination) is expanded in an appropriate form and the main term and the parallactic term are given explicitly. The variations of the orbital elements are derived using the stroboscopic method. An example shows that the perturbation of the orbit cannot be neglected.     

Dynamic and Thermal Disappearance of Prominences and Their Geoeffectiveness

This paper is a qualitative study of 42 events of solar filament/prominence sudden disappearances (“disparitions brusques”; henceforth DBs) around two solar minima, 1985 – 1986 and 1994. The studied events were classified as 17 thermal and 25 dynamic disappearances. Associated events, i.e. coronal mass ejections (CMEs), type II bursts, evolution of nearby coronal holes, as well as solar wind speed, and geomagnetic disturbances are discussed. We have found that about 50% of the thermal DBs with adjacent (within 15° from the DB) coronal holes were associated with CMEs within a selected time window. All the studied thermal disappearances with adjacent coronal holes or accompanied by dynamic disappearances were associated with weak and medium geomagnetic storms. Also, nearly 64% of dynamic DBs were associated with CMEs. Ten (40%) dynamic disappearances were associated with intense geomagnetic storms, even when no CMEs was reported, six (24%) dynamic disappearances corresponded to extreme storms, and five (20%) corresponded to medium geomagnetic storms. The extreme geomagnetic storms appeared to be related to combined events, involving dynamic disappearances with adjacent coronal holes or including thermal disappearances. Furthermore, the geomagnetic activity (Dst index) increased if the source was close to the central meridian (±30°). The highest interplanetary magnetic field (B), longest duration, lowest southward direction B _z component, and lowest Dst were highly correlated for all studied events. The Sun – Earth transit time computed from the starting time of the sudden disappearance and the time its effect was measured at Earth was about 4.3 days and was mainly well correlated with the solar wind speed measured in situ (daily value).