Weak lensing signal in unified dark matter models

This paper is an extension of the work done by Pierens & Nelson in which they investigated the behaviour of a two-planet system embedded in a protoplanetary disc. They put a Jupiter mass gas giant on the internal orbit and a lower mass planet on the external one. We consider here a similar problem taking into account a gas giant with mass in the range 0.5 to  1 M _J  and a Super-Earth (i.e. a planet with mass  ≤10 M _⊕  ) as the outermost planet. By changing disc parameters and planet masses, we have succeeded in getting the convergent migration of the planets which allows for the possibility of their resonant locking. However, in the case in which the gas giant has the mass of Jupiter, before any mean-motion first-order commensurability could be achieved, the Super-Earth is caught in a trap when it is very close to the edge of the gap opened by the giant planet. This confirms the result obtained by Pierens & Nelson in their simulations. Additionally, we have found that, in a very thin disc, an apsidal resonance is observed in the system if the Super-Earth is captured in the trap. Moreover, the eccentricity of the small planet remains low, while the eccentricity of the gas giant increases slightly due to the imbalance between Lindblad and corotational resonances. We have also extended the work of Pierens & Nelson by studying analogous systems in which the gas giant is allowed to take sub-Jupiter masses. In this case, after conducting an extensive survey over all possible parameters, we have succeeded in getting the 1:2 mean-motion resonant configuration only in a disc with low aspect ratio and low surface density. However, the resonance is maintained just for a few thousand orbits. Thus, we conclude that for typical protoplanetary discs the mean-motion commensurabilities are rare if the Super-Earth is located on the external orbit relative to the gas giant.     

On the Motion of Trapped Particles in the Vicinity of Corotation Centers

In the present paper we analyse the motion of a massless particle during the capture process in an exterior mean-motion resonance under the effects of an external dissipative force. In particular, we study the orbital evolution from its initial approach to the commensurability up to the final nesting place in the periodic orbit around the equilibrium solution. This revised version was published online in July 2006 with corrections to the Cover Date.     

Approximate Analytical Theory of Satellite Orbit Prediction Presented in Spherical Coordinates Frame

A comparative review of analytic theories for the motion of Earth satellites in quasi-circular orbits written in the spherical coordinate frame is presented. The theory of motion is developed for satellites in quasi-circular and quasi-equatorial orbits subjected to geopotential, luni-solar and solar radiation pressure force perturbations. The intermediate orbit is Keplerian and the equations of motion are solved by the Lyapunov–Poincaré small parameter method. Both resonant and non-resonant cases are considered. The results can be useful for the development of a complete theory of weakly eccentric orbits.     

Exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral harmonics

A novel approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral and sectorial spherical harmonics is presented in this work. It is shown that the exact solution for the Delaunay normalization can be reduced to quadratures by the application of Deprit’s Lie-transform-based perturbation method. Two different series representations of the quadratures, one in powers of the eccentricity and the other in powers of the ratio of the Earth’s angular velocity to the satellite’s mean motion, are derived. The latter series representation produces expressions for the short-period variations that are similar to those obtained from the conventional method of relegation. Alternatively, the quadratures can be evaluated numerically, resulting in more compact expressions for the short-period variations that are valid for an elliptic orbit with an arbitrary value of the eccentricity. Using the proposed methodology for the Delaunay normalization, generalized expressions for the short-period variations of the equinoctial orbital elements, valid for an arbitrary tesseral or sectorial harmonic, are derived. The result is a compact unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes, which is nonsingular for the resonant orbits, and is closed-form in the eccentricity as well. The accuracy of the proposed theory is validated by comparison with numerical orbit propagations.     

Influence of fast interstellar gas flow on the dynamics of dust grains

The orbital evolution of a dust particle under the action of a fast interstellar gas flow is investigated. The secular time derivatives of Keplerian orbital elements and the radial, transversal, and normal components of the gas flow velocity vector at the pericentre of the particle’s orbit are derived. The secular time derivatives of the semi-major axis, eccentricity, and of the radial, transversal, and normal components of the gas flow velocity vector at the pericentre of the particle’s orbit constitute a system of equations that determines the evolution of the particle’s orbit in space with respect to the gas flow velocity vector. This system of differential equations can be easily solved analytically. From the solution of the system we found the evolution of the Keplerian orbital elements in the special case when the orbital elements are determined with respect to a plane perpendicular to the gas flow velocity vector. Transformation of the Keplerian orbital elements determined for this special case into orbital elements determined with respect to an arbitrary oriented plane is presented. The orbital elements of the dust particle change periodically with a constant oscillation period or remain constant. Planar, perpendicular and stationary solutions are discussed. The applicability of this solution in the Solar System is also investigated. We consider icy particles with radii from 1 to 10 μm. The presented solution is valid for these particles in orbits with semi-major axes from 200 to 3000 AU and eccentricities smaller than 0.8, approximately. The oscillation periods for these orbits range from 10⁵ to 2 × 10⁶ years, approximately.     

Low-eccentricity motion of asteroids near the 2/1 Jovian resonance

(903) Nealley moves on an orbit of low eccentricity with a mean motion that is slightly larger than the 2/1 value of resonance. This orbit and some related fictious orbits are studied by numerical integrations of the four-body problem Sun-Jupiter-Saturn-asteroid over an interval of 110000 yr. The author's experience on related cases of resonance allows a study of the variation of suitably defined orbital parameters. The long-term evolution of the orbits is compared with earlier predictions. Some of the librating orbits are temporarily captured in a secondary resonance that refers to three-dimensional motion and is demonstrated by a special example.     

On the character of evolution of orbits in the vicinity of the 1 : 3 Kirkwood gap

The character of orbital evolution for bodies moving near the if 1 : 3 commensurability with Jupiter was studied by model calculations for the time interval of ～500 years. A comparison of oscillations of the orbital elements a, e, q and q′ is made for ensembles of bodies along three starting orbits in the vicinity of the sharp commensurability with Jupiter. These orbits are eccentric ones of low inclinations having perihelia near the Earth's orbit. Examples of a deceleration of the rate of orbital evolution near the sharp commensurability are revealed. The existence of a group of asteroids connected with the Kirkwood gap, i.e., being in a resonant motion with Jupiter, is suggested. A connection of asteroids 887 Alinda and 1915 Quetzalcoatl with this gap is confirmed.     

The 1/1 resonance in extrasolar planetary systems

We study orbits of planetary systems with two planets, for planar motion, at the 1/1 resonance. This means that the semimajor axes of the two planets are almost equal, but the eccentricities and the position of each planet on its orbit, at a certain epoch, take different values. We consider the general case of different planetary masses and, as a special case, we consider equal planetary masses. We start with the exact resonance, which we define as the 1/1 resonant periodic motion, in a rotating frame, and study the topology of the phase space and the long term evolution of the system in the vicinity of the exact resonance, by rotating the orbit of the outer planet, which implies that the resonance and the eccentricities are not affected, but the symmetry is destroyed. There exist, for each mass ratio of the planets, two families of symmetric periodic orbits, which differ in phase only. One is stable and the other is unstable. In the stable family the planetary orbits are in antialignment and in the unstable family the planetary orbits are in alignment. Along the stable resonant family there is a smooth transition from planetary orbits of the two planets, revolving around the Sun in eccentric orbits, to a close binary of the two planets, whose center of mass revolves around the Sun. Along the unstable family we start with a collinear Euler–Moulton central configuration solution and end to a planetary system where one planet has a circular orbit and the other a Keplerian rectilinear orbit, with unit eccentricity. It is conjectured that due to a migration process it could be possible to start with a 1/1 resonant periodic orbit of the planetary type and end up to a satellite-type orbit, or vice versa, moving along the stable family of periodic orbits.     

On the habitability of the OGLE-2006-BLG-109L planetary system

We investigate the dynamics of putative Earth-mass planets in the habitable zone (HZ) of the extrasolar planetary system OGLE-2006-BLG-109L, a close analogue of the Solar system. Our work is inspired by the work of Malhotra & Minton. Using the linear Laplace–Lagrange theory, they identified a strong secular resonance that may excite large eccentricity of orbits in the HZ. However, due to uncertain or unconstrained orbital parameters, the subsystem of Jupiters may be found in a dynamically active region of the phase space spanned by low-order mean-motion resonances. To generalize this secular model, we construct a semi-analytical averaging method in terms of the restricted problem. The secular orbits of large planets are approximated by numerically averaged osculating elements. They are used to calculate the mean orbits of terrestrial planets by means of a high-order analytic secular theory developed in our previous works. We found regions in the parameter space of the problem in which stable, quasi-circular orbits in the HZ are permitted. The excitation of eccentricity in the HZ strongly depends on the apsidal angle of jovian orbits. For some combinations of that angle, eccentricities and semimajor axes consistent with the observations, a terrestrial planet may survive in low eccentric orbits. We also study the effect of post-Newtonian gravity correction on the innermost secular resonance.     

Aegaeon (Saturn LIII), a G-ring object

Aegaeon (Saturn LIII, S/2008 S1) is a small satellite of Saturn that orbits within a bright arc of material near the inner edge of Saturn’s G-ring. This object was observed in 21 images with Cassini’s Narrow-Angle Camera between June 15 (DOY 166), 2007 and February 20 (DOY 051), 2009. If Aegaeon has similar surface scattering properties as other nearby small saturnian satellites (Pallene, Methone and Anthe), then its diameter is approximately 500 m. Orbit models based on numerical integrations of the full equations of motion show that Aegaeon’s orbital motion is strongly influenced by multiple resonances with Mimas. In particular, like the G-ring arc it inhabits, Aegaeon is trapped in the 7:6 corotation eccentricity resonance with Mimas. Aegaeon, Anthe and Methone therefore form a distinctive class of objects in the Saturn system: small moons in corotation eccentricity resonances with Mimas associated with arcs of debris. Comparisons among these different ring-arc systems reveal that Aegaeon’s orbit is closer to the exact resonance than Anthe’s and Methone’s orbits are. This could indicate that Aegaeon has undergone significant orbital evolution via its interactions with the other objects in its arc, which would be consistent with the evidence that Aegaeon’s mass is much smaller relative to the total mass in its arc than Anthe’s and Methone’s masses are.     

Generalized Lagrangian orbital elements for central force problems

We consider the definitions and resulting equations of motion for the Lagrangian orbital elements associated with conventional osculating orbit theory for central forces. The analysis indicates that the definitions themselves lead to difficulties which are most apparent in the circular limit. An alternate set of defining relations is presented which eliminates the problems associated with osculating elements. The remaining equation of motion based on these new definitions is reduced to quadratures. This solution completely expresses the orbits for central force problems with no restriction on the eccentricity. Both bounded and open orbits are considered. A generalized Laplace-Runge-Lenz vector is developed and a number of example solutions are presented.     

Some properties of the dumbbell satellite attitude dynamics

The dumbbell satellite is a simple structure consisting of two point masses connected by a massless rod. We assume that it moves around the planet whose gravity field is approximated by the field of the attracting center. The distance between the point masses is assumed to be much smaller than the distance between the satellite’s center of mass and the attracting center, so that we can neglect the influence of the attitude dynamics on the motion of the center of mass and treat it as an unperturbed Keplerian one. Our aim is to study the satellite’s attitude dynamics. When the center of mass moves on a circular orbit, one can find a stable relative equilibrium in which the satellite is permanently elongated along the line joining the center of mass with the attracting center (the so called local vertical). In case of elliptic orbits, there are no stable equilibrium positions even for small values of the eccentricity. However, planar periodic motions are determined, where the satellite oscillates around the local vertical in such a way that the point masses do not leave the orbital plane. We prove analytically that these planar periodic motions are unstable with respect to out-of-plane perturbations (a result known from numerical investigations cf. Beletsky and Levin Adv Astronaut Sci 83, 1993). We provide also both analytical and numerical evidences of the existence of stable spatial periodic motions.     

A “Spring–mass” model of tethered satellite systems: properties of planar periodic motions

This paper is devoted to the dynamics in a central gravity field of two point masses connected by a massless tether (the so called “spring–mass” model of tethered satellite systems). Only the motions with straight strained tether are studied, while the case of “slack” tether is not considered. It is assumed that the distance between the point masses is substantially smaller than the distance between the system’s center of mass and the field center. This assumption allows us to treat the motion of the center of mass as an unperturbed Keplerian one, so to focus our study on attitude dynamics. A particular attention is given to the family of planar periodic motions in which the center of mass moves on an elliptic orbit, and the point masses never leave the orbital plane. If the eccentricity tends to zero, the corresponding family admits as a limit case the relative equilibrium in which the tether is elongated along the line joining the center of mass with the field center. We study the bifurcations and the stability of these planar periodic motions with respect to in-plane and out-of-plane perturbations. Our results show that the stable motions take place if the eccentricity of the orbit is sufficiently small.     

Distance in the space of energetically bounded Keplerian orbits

In this paper we derive an explicit, analytic formula for the geodesic distance between two points in the space of bounded Keplerian orbits in a particular topology. The specific topology we use is that of a cone passing through the direct product of two spheres. The two spheres constitute the configuration manifold for the space of bounded orbits of constant energy. We scale these spheres by a factor equal to the semi-major axis of the orbit, forming a linear cone. This five-dimensional manifold inherits a Riemannian metric, which is induced from the Euclidean metric on \mathbbR⁶{\mathbb{R}^6}, the space in which it is embedded. We derive an explicit formula for the geodesic distance between any two points in this space, each point representing a physical, gravitationally bound Keplerian orbit. Finally we derive an expression for the Riemannian metric that we used in terms of classical orbital elements, which may be thought of as local coordinates on our configuration manifold.     

Detection of Earth-mass and super-Earth Trojan planets using transit timing variation method

We have carried out an extensive study of the possibility of the detection of Earth-mass and super-Earth Trojan planets using transit timing variation method with the Kepler space telescope. We have considered a system consisting of a transiting Jovian-type planet in a short period orbit, and determined the induced variations in its transit timing due to an Earth-mass/super-Earth Trojan planet. We mapped a large section of the phase space around the 1:1 mean-motion resonance and identified regions corresponding to several other mean-motion resonances where the orbit of the planet would be stable. We calculated transit timing variations (TTVs) for different values of the mass and orbital elements of the transiting and perturbing bodies as well as the mass of central star, and identified orbital configurations of these objects (ranges of their orbital elements and masses) for which the resulted TTVs would be within the range of the variations of the transit timing of Kepler’s planetary candidates. Results of our study indicate that in general, the amplitudes of the TTVs fall within the detectable range of timing precision obtained from the Kepler’s long-cadence data, and depending on the parameters of the system, their magnitudes may become as large as a few hours. The probability of detection is higher for super-Earth Trojans with slightly eccentric orbits around short-period Jovian-type planets with masses slightly smaller than Jupiter. We present the details of our study and discuss the implications of its results.     

Improved equations for eccentricity generation in hierarchical triple systems

In a series of papers, we developed a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular. However, for certain combinations of the masses and the orbital elements, the secular part of the solution failed. In this paper, we derive a new solution for the secular part of the inner eccentricity, which corrects the previous weakness. The derivation applies to hierarchical triple systems with coplanar and initially circular orbits. The new formula is tested numerically by integrating the full equations of motion for systems with mass ratios from 10⁻³ to 10³. We also present more numerical results for short-term eccentricity evolution, in order to get a better picture of the behaviour of the inner eccentricity.     

Intensity of gravitational wave emitted by an oscillating Keplerian binary

This paper attempts to formulate a way for calculating the intensity of gravitational wave from two point masses in Keplerian circular and elliptic orbits. The intensity is calculated with the assumption that the orbital plane of the binary undergoes small oscillation about the equilibrium x-y plane. This problem is simplification of a physically possible orbit where one of the point masses is spinning whereby the spin-orbit force drives the orbital plane to wobble in a complicated manner. It is shown that the total energy of gravitational wave emitted by the binary in this case is dominated by the parameters which take into account the oscillation of the plane. The results presented are in fact a generalization of the classic results of Landau and Lifshitz.     

Explicit Symplectic Integrator for Rotating Satellites

An explicit symplectic integrator is constructed for the problem of a rotating planetary satellite on a Keplerian orbit. The spin vector is fixed perpendicularly to the orbital plane. The integrator is constructed according to the Wisdom-Holman approach: the Hamiltonian is separated in two parts so that one of them is multiplied by a small parameter. The parameter depends on the satellite’s shape or the eccentricity of its orbit. The leading part of the Hamiltonian for small eccentricity orbits is similar to the simple pendulum and hence integrable; the perturbation does not depend on angular momentum which implies a trivial ‘kick’ solution. In spite of the necessity to evaluate elliptic function at each step, the explicit symplectic integrator proves to be quite efficient. This revised version was published online in July 2006 with corrections to the Cover Date.     

Calculation of the intermediate perturbed orbit from three or more positions of a small body on the celestial sphere

Two new methods are described for finding the orbit of a small celestial body from three or more pairs of angular measurements and the corresponding time points. The methods are based on, first, the approach that has been developed previously by the author to the determination, from a minimum number of observations, of intermediate orbit considering most of the perturbations in the bodies’ motion and, second, Herget’s algorithmic procedure enabling the introduction of additional observations. The errors of orbital parameters calculated by the proposed methods are two orders of magnitude smaller than the corresponding errors of the traditional approach based on the construction of an unperturbed Keplerian orbit. The thus-calculated orbits of the minor planets 1566 Icarus, 2002 EC1, and 2010 TO48 are used to compare the results of Herget’s multiposition procedure and the new methods. The comparison shows that the new methods are highly effective in the study of perturbed motion. They are particularly beneficial if high-precision observational data covering short orbital arcs are available.