The influence of mutual perturbations on the eccentricity excitation by jet acceleration in extrasolar planetary systems

We study the influence of mutual planetary perturbations on the process of eccentricity excitation by jet acceleration suggested by Namouni (Astron. J. 130, 280–294). Modeling the jet’s action by a constant-direction acceleration, we solve the linear secular equations of the combined planetary perturbations and the jet acceleration of the host star for a two-planet system. The effects of the acceleration’s strength, relative mass ratio and the relative distance of the two planets are investigated. The model is applied to the extrasolar planetary systems of HD 108874, 47 Uma, and HD 12661.     

Canonical Modelling of Relative Spacecraft Motion Via Epicyclic Orbital Elements

This paper presents a Hamiltonian approach to modelling spacecraft motion relative to a circular reference orbit based on a derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations within the context of the Clohessy–Wiltshire solution. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton–Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic elements. The influence of higher order terms and perturbations, such as Earth’s oblateness, are incorporated into the analysis by a variation of parameters procedure. As an example, closed-form solutions for J₂-invariant orbits are obtained.     

Distances to galaxies from the brightest stars in the Universe

Blue Supergiants (BSGs) are the brightest stars in the universe at visual light with absolute magnitudes up to M _V =−10 mag. They are ideal stellar objects for the determination of extragalactic distances, in particular, because the perennial uncertainties troubling most of the other stellar distance indicators, interstellar extinction and metallicity, do not affect them. The quantitative spectral analysis of low resolution spectra of individual BSGs provides accurate stellar parameters and chemical composition, which are then used to determine accurate reddening and extinction from photometry for each individual object. Accurate distances can be determined from stellar gravities and effective temperatures using the “Flux Weighted Gravity–Luminosity Relationship (FGLR)”.     

Orbit Determination of Binary Asteroids

In addition to the detection of an asteroid moon or a binary asteroid, the knowledge of the satellite’s true orbit is of high importance to derive fundamental physical parameters of the binary system such as its mass and to shed light on its possible formation history and dynamical evolution (prograde/retrograde orbit, large/small eccentricity or inclination, etc.). A new methodology for preliminary orbit determination of binary asteroids – and visual binaries in general – is proposed. It is based on Thiele–Innes method combined with a ‘trial and error’ Monte-Carlo technique. This method provides the full set of solutions (bundle of orbits, with the 7 orbital elements) even for a reduced number of observations. The mass is a direct by-product of this orbit determination, from which one can next infer the bulk-density and porosity. In addition to the bundle of orbits, the method provides the marginal probability densities of the foreseen parameters. Such error analysis – since it avoids linear approximation – can be of importance for the prediction of the satellite’s position in the plane-of-sky during future stellar occultations or subsequent observations, but also for the analysis of the orbit’s secular evolution. After briefly describing the method, we present the algorithm and its application to some practical cases, with particular emphasis on asteroids binaries and applications on orbital evolution.     

Injection of Oort Cloud comets: the fundamental role of stellar perturbations

We present Monte Carlo simulations of the dynamical evolution of the Oort cloud over the age of the Solar System, using an initial sample of one million test comets without any cloning. Our model includes perturbations due to the Galactic tide (radial and vertical) and passing stars. We present the first detailed analysis of the injection mechanism into observable orbits by comparing the complete model with separate models for tidal and stellar perturbations alone. We find that a fundamental role for injecting comets from the region outside the loss cone (perihelion distance q > 15 AU) into observable orbits (q < 5 AU) is played by stellar perturbations. These act in synergy with the tide such that the total injection rate is significantly larger than the sum of the two separate rates. This synergy is as important during comet showers as during quiescent periods and concerns comets with both small and large semi-major axes. We propose different dynamical mechanisms to explain the synergies in the inner and outer parts of the Oort Cloud. We find that the filling of the observable part of the loss cone under normal conditions in the present-day Solar System rises from <1% for a < 20 000 AU to about 100% for a ? 100 000 AU.     

The evidence of an early stellar encounter in Edgeworth-Kuiper belt

We investigate the influence of a stellar fly-by encounter on the Edgeworth-Kuiper belt objects through numerical orbital calculations, in order to explain both mass depletion and high orbital inclinations of the classical Edgeworth-Kuiper belt (CEKB) objects, which have semimajor axis of 42-48 AU and perihelia beyond 35 AU. The observationally inferred total mass of the CEKB is ∼1/10 Earth masses, which is only ∼0.02 of that extrapolated from the minimum-mass solar nebula model. The CEKB consists of bimodal population: “hot population” with inclinations i?0.2-0.6 radians and “cold population” with i?0.1. The observationally suggested difference in size and color of objects between the two populations may imply different origins of the two populations. We find that both the depletion of solid materials in the CEKB and the formation of the hot population are accounted for by a single close stellar encounter with pericenter distance of 80-100 AU and inclination relative to the initial protoplanetary disk ?50°-70°. Such a stellar encounter highly pumps up eccentricities of most objects in the CEKB and then their perihelia migrate within 35 AU. These objects would be removed by Neptune's perturbations after Neptune is formed at or migrates to the current position (30 AU). Less than 10% of the original objects remain in stable orbits with small eccentricities and perihelion distances larger than 35 AU, in the CEKB, which is consistent with the observation. We find that i of the remaining objects are as large as that of the observed hot population. The only problem is how to stop Neptune's migration at ∼30 AU, which is addressed in a separate paper. The depletion by the stellar encounter extends deeply into ∼30-35 AU, which provides the basis of the formation model for the cold population through Neptune's outward migration by Levison and Morbidelli (2003, Nature, 426, 419-421). The combination of our model with Levison and Morbidelli's model could consistently explain the mass depletion, truncation at 50 AU, bimodal distribution in i, and differences in size and color between the hot and the cold populations in the CEKB.     

Spatial Structure of Regular and Chaotic Orbits in A Self-Consistent Triaxial Stellar System

We created a triaxial stellar system through the cold dissipationless collapse of 100,000 particles whose evolution was followed with a multipolar code. Once an equilibrium system had been obtained, the multipolar expansion was freezed and smoothed in order to get a stationary smooth potential. The resulting model was self-consistent and the orbits and Lyapunov exponents could then be computed for a randomly selected sample of 3472 of the bodies that make up the system. More than half of the orbits (52.7 % ) turned out to be chaotic. Regular orbits were then classified using the frequency analysis automatic code of Carpintero and Aguilar (1998, MNRAS 298(1), 1–21). We present plots of the distributions of the different kinds of orbits projected on the symmetry planes of the system. We distinguish chaotic orbits with only one non-zero Lyapunov exponent from those with two non-zero exponents and show that their spatial distributions differ, that of the former being more similar to the one of the regular orbits. Most of the regular orbits are boxes and boxlets, but the minor axis tubes play an important role filling in the wasp waists of the boxes and helping to give a lentil shape to the system. We see no problem in building stable triaxial models with substantial amounts of chaotic orbits; the difficulties found by other authors may be due not to a physical cause but to a limitation of Schwarzschild’s method.     

On quasi-periodic motions around the collinear libration points in the real Earth–Moon system

Due to various perturbations, the collinear libration points of the real Earth–Moon system are not equilibrium points anymore. Under the assumption that the Moon’s motion is quasi-periodic, special quasi-periodic orbits called dynamical substitutes exist. These dynamical substitutes replace the geometrical collinear libration points as time-varying equilibrium points. In the paper, the dynamical substitutes of the three collinear libration points in the real Earth–Moon system are computed. For the points L ₁ and L ₂, linearized motions around the dynamical substitutes are described, and the variational equations of the dynamical substitutes are reduced to a form with a near constant coefficient matrix. Then higher order analytical formulae of the central manifolds are constructed. Using these analytical solutions as initial seeds, Lissajous orbits and halo orbits are computed with numerical algorithms.     

On The Origin of The High-Perihelion Scattered Disk: The Role of The Kozai Mechanism And Mean Motion Resonances

We study the transfer process from the scattered disk (SD) to the high-perihelion scattered disk (HPSD) (defined as the population with perihelion distances q > 40 AU and semimajor axes a>50 AU) by means of two different models. One model (Model 1) assumes that SD objects (SDOs) were formed closer to the Sun and driven outwards by resonant coupling with the accreting Neptune during the stage of outward migration (Gomes 2003b, Earth, Moon, Planets 92, 29–42.). The other model (Model 2) considers the observed population of SDOs plus clones that try to compensate for observational discovery bias (Fernández et al. 2004, Icarus , in press). We find that the Kozai mechanism (coupling between the argument of perihelion, eccentricity, and inclination), associated with a mean motion resonance (MMR), is the main responsible for raising both the perihelion distance and the inclination of SDOs. The highest perihelion distance for a body of our samples was found to be q = 69.2 AU. This shows that bodies can be temporarily detached from the planetary region by dynamical interactions with the planets. This phenomenon is temporary since the same coupling of Kozai with a MMR will at some point bring the bodies back to states of lower-q values. However, the dynamical time scale in high-q states may be very long, up to several Gyr. For Model 1, about 10% of the bodies driven away by Neptune get trapped into the HPSD when the resonant coupling Kozai-MMR is disrupted by Neptune’s migration. Therefore, Model 1 also supplies a fossil HPSD, whose bodies remain in non-resonant orbits and thus stable for the age of the solar system, in addition to the HPSD formed by temporary captures of SDOs after the giant planets reached their current orbits. We find that about 12 – 15% of the surviving bodies of our samples are incorporated into the HPSD after about 4 – 5 Gyr, and that a large fraction of the captures occur for up to the 1:8 MMR (a ⋍ 120 AU), although we record captures up to the 1:24 MMR (a ≃ 260 AU). Because of the Kozai mechanism, HPSD objects have on average inclinations about 25°–50°, which are higher than those of the classical Edgeworth–Kuiper (EK) belt or the SD. Our results suggest that Sedna belongs to a dynamically distinct population from the HPSD, possibly being a member of the inner core of the Oort cloud. As regards to 2000 CR₁₀₅ , it is marginally within the region occupied by HPSD objects in the parametric planes (q,a) and (a,i), so it is not ruled out that it might be a member of the HPSD, though it might as well belong to the inner core.     

Continuity and stability of families of figure eight orbits with finite angular momentum

Numerical solutions are presented for a family of three dimensional periodic orbits with three equal masses which connects the classical circular orbit of Lagrange with the figure eight orbit discovered by C. Moore [Moore, C.: Phys. Rev. Lett. 70, 3675–3679 (1993); Chenciner, A., Montgomery, R.: Ann. Math. 152, 881–901 (2000)]. Each member of this family is an orbit with finite angular momentum that is periodic in a frame which rotates with frequency Ω around the horizontal symmetry axis of the figure eight orbit. Numerical solutions for figure eight shaped orbits with finite angular momentum were first reported in [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and mathematical proofs for the existence of such orbits were given in [Marchal, C.: Celest. Mech. Dyn. Astron. 78, 279–298 (2001)], and more recently in [Chenciner, A. et al.: Nonlinearity 18, 1407–1424 (2005)] where also some numerical solutions have been presented. Numerical evidence is given here that the family of such orbits is a continuous function of the rotation frequency Ω which varies between Ω = 0, for the planar figure eight orbit with intrinsic frequency ω, and Ω = ω for the circular Lagrange orbit. Similar numerical solutions are also found for n > 3 equal masses, where n is an odd integer, and an illustration is given for n = 21. Finite angular momentum orbits were also obtained numerically for rotations along the two other symmetry axis of the figure eight orbit [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and some new results are given here. A preliminary non-linear stability analysis of these orbits is given numerically, and some examples are given of nearby stable orbits which bifurcate from these families.     

Spacecraft trajectories to the L<Subscript>3</Subscript> point of the Sun–Earth three-body problem

Of the three collinear libration points of the Sun–Earth Circular Restricted Three-Body Problem (CR3BP), L₃ is that located opposite to the Earth with respect to the Sun and approximately at the same heliocentric distance. Whereas several space missions have been launched to the other two collinear equilibrium points, i.e., L₁ and L₂, taking advantage of their dynamical and geometrical characteristics, the region around L₃ is so far unexploited. This is essentially due to the severe communication limitations caused by the distant and permanent opposition to the Earth, and by the gravitational perturbations mainly induced by Jupiter and the close passages of Venus, whose effects are more important than those due to the Earth. However, the adoption of a suitable periodic orbit around L₃ to ensure the necessary communication links with the Earth, or the connection with one or more relay satellites located at L₄ or L₅, and the simultaneous design of an appropriate station keeping-strategy, would make it possible to perform valuable fundamental physics and astrophysics investigations from this location. Such an opportunity leads to the need of studying the ways to transfer a spacecraft (s/c) from the Earth’s vicinity to L₃. In this contribution, we investigate several trajectory design methods to accomplish such a transfer, i.e., various types of two-burn impulsive trajectories in a Sun-s/c two-body model, a patched conics strategy exploiting the gravity assist of the nearby planets, an approach based on traveling on invariant manifolds of periodic orbits in the Sun–Earth CR3BP, and finally a low-thrust transfer. We examine advantages and drawbacks, and we estimate the propellant budget and time of flight requirements of each.     

Physical parameters of multiple systems like the trapezium of early spectral types, derived fromuvbyβ photometry. II

The effective temperatures T_eff, gravitational force g at the stellar surface, and Fe/H have been calculated for 50 components of 15 Trapezia of early spectral types from electrophotometric observations in the six-color uvbyβ photometric system. The results of the calculation showed that the log T_eff of Trapezia components lie in the range 4.03–4.54, while their log g lie in the range 2.58–4.02. The atmospheric parameters T_eff and g were used to determine the stars’ masses and ages based on Hejlesen’s calculated evolutionary model sequences. All the Trapezia components have several times more mass than the sun and almost half of them have over 10 times that mass. Over 70% of the components are less than 40.10⁶ years old. Translated from Astrofizika, Vol. 41. No. 2. pp. 235–244, April-June, 1998.     

A Monte Carlo investigation of Jovian perturbations on short-period comet orbits

We present statistical distributions of Jovian perturbations on short-period comet orbits resulting from accurate numerical integrations. Our sample of 60, 000 cometary orbits with low inclinations and random orientations is characterized by perihelia between 0 and 7 AU and aphelia between 4 and 13 AU. The perturbations considered are those experienced because of Jupiter's gravitation per orbital revolution by the comets. Regularization and accurate step-length control in the numerical integration gives statistical results appreciably different from those computed by Rickman and Vaghi (1978). Their use of a crude method of integration led to erroneous results for close encounters. Strong asymmetries of the $\text{δ(}\text{1}\text{a}\text{)}$ distributions, in particular for the extreme tails, are observed for perihelion- or aphelion-tangent orbits. These orbits are also shown to experience the strongest energy perturbations on the average. Some results concerning the perturbations of Tisserand parameters are indicated. The perturbation distributions for the angular elements are described and discussed. The role of the minimum distance from Jupiter as an indicator of perturbations is investigated.     

Long-term evolution of orbits about a precessing oblate planet. 2. The case of variable precession

We continue the study undertaken in Efroimsky [Celest. Mech. Dyn. Astron. 91, 75–108 (2005a)] where we explored the influence of spin-axis variations of an oblate planet on satellite orbits. Near-equatorial satellites had long been believed to keep up with the oblate primary’s equator in the cause of its spin-axis variations. As demonstrated by Efroimsky and Goldreich [Astron. Astrophys. 415, 1187–1199 (2004)], this opinion had stemmed from an inexact interpretation of a correct result by Goldreich [Astron. J. 70, 5–9 (1965)]. Although Goldreich [Astron. J. 70, 5–9 (1965)] mentioned that his result (preservation of the initial inclination, up to small oscillations about the moving equatorial plane) was obtained for non-osculating inclination, his admonition had been persistently ignored for forty years. It was explained in Efroimsky and Goldreich [Astron. Astrophys. 415, 1187–1199 (2004)] that the equator precession influences the osculating inclination of a satellite orbit already in the first order over the perturbation caused by a transition from an inertial to an equatorial coordinate system. It was later shown in Efroimsky [Celest. Mech. Dyn. Astron. 91, 75–108 (2005a)] that the secular part of the inclination is affected only in the second order. This fact, anticipated by Goldreich [Astron. J. 70, 5–9 (1965)], remains valid for a constant rate of the precession. It turns out that non-uniform variations of the planetary spin state generate changes in the osculating elements, that are linear in , where is the planetary equator’s total precession rate that includes the equinoctial precession, nutation, the Chandler wobble, and the polar wander. We work out a formalism which will help us to determine if these factors cause a drift of a satellite orbit away from the evolving planetary equator.By “precession,” in its most general sense, we mean any change of the direction of the spin axis of the planet—from its long-term variations down to nutations down to the Chandler wobble and polar wander.     

Statistical analysis of the motion of globular clusters in the galaxy

The laws of motion of globular stellar clusters, capable of answering many interesting questions about their evolution and the overall structure of the Galaxy, are investigated. Solving the problem of the orbits of these objects requires knowledge of the initial conditions of their motion and assigning an expression for the gravitational potential of the Galaxy’s regular field. The situation here is complicated, in general by the fact that, despite some progress in determining the proper motions of globular clusters, only their radial velocities can be estimated from observations. One must therefore apply statistical methods of analysis of the possible shapes of the Galactic orbits of these formations. In the present work, on the basis of new, relatively rich observational material, we have carried out statistical investigations of the possible shapes of orbits for about 100 globular clusters and studied the influence of a change in the Galaxy’s gravitational potential on the results of the investigations. Translated from Astrofizika, Vol. 41, No. 1, pp. 101–112, January-March, 1998.     

Constructive analytical solution of the evolution hill problem

A new analytical solution of the system of differential equations describing secular perturbations and long-period solar perturbations of mean orbits of outer satellites of giant planets was obtained. As distinct from other solutions, the solution constructed using von Zeipel’s method approximately takes into account, in the secular part of the perturbing function, the totality of fourth order with respect to the small parameter m of the ratio of the mean motions of the primary planet and the satellite. This enables us to describe more accurately the evolution of satellite orbits with large apocentric distances, which in the course of evolution may exceed the halved radius of the Hill sphere of the planet with respect to the Sun. Among these are the orbits of the two outermost Neptunian satellites N10 (Psamathe) and N13 (Neso). For these satellites, the parameter m amounts to 0.152 and 0.165, respectively. Different from a purely analytical solution, the proposed solution requires preliminary calculations for each satellite. More precisely, in doing so, we need to construct some simple functions to approximate more complex ones. This is why we use the phrase “constructive analytical.” To illustrate the solution, we compare it with the results of the numerical integration of the strict motion equations of the satellites N10 and N13 over time intervals 5–15 thousand years.     

Deprit’s reduction of the nodes revisited

We revisit a set of symplectic variables introduced by Andre Deprit (Celest Mech 30, 181–195, 1983), which allows for a complete symplectic reduction in rotation invariant Hamiltonian systems, generalizing to arbitrary dimension Jacobi’s reduction of the nodes. In particular, we introduce an action-angle version of Deprit’s variables, connected to the Delaunay variables, and give a new hierarchical proof of the symplectic character of Deprit’s variables.     

The Changes of the Orbital Elements and Estimation of the Initial Velocities of Stream Meteoroids Ejected from Comets and Asteroids

The values of the initial velocity of the meteoroids ejected from the parent bodies are small and as a result, the most of the young meteoroid streams have similar orbits to their parent bodies. Assuming that the members of the observed meteor stream evolved under the influence of gravitational perturbations mostly, Pittich [1991, Proceedings of the Conference on Dynamic of Small Bodies of the Solar System, Polish-Slovak Conference, Warsaw, October 25–28, 1988, pp. 55-61], Williams [1996, Earth, Moon, Planets 72, 321–326; 2001, Proceedings of the Meteoroids 2001 conference, Kiruna, Sweden, August 6–10, 2001, pp. 33–42] estimated the ejection velocities of the stream meteoroids. Equation relating the ejection velocity Δυ and the change Δa of the semi-major axis, Williams (2001), was applied with two slightly different variations. In the first one (M1) as Δa the difference between the mean orbit of the stream and the orbit of the parent body was substituted, in the second one (M2), as Δa the dispersion of semi-major axes around the mean orbit of the stream was used. The results obtained by these two methods are not free from discrepancies, partly explained by the particular orbital structure of the stream. Kresak [1992, Contrib. Astron. Obs. Skalnate Pleso 22, 123–130] strongly criticized the attempts to determine the initial velocities of the stream using the statistics of the meteor orbits. He argued that this is essentially impossible, because the dispersion of the initial velocities are masked by much larger measuring errors and by the accumulated effects of planetary perturbations. In our paper, we study the reliability of M1 and M2 methods. We made a numerical experiment consisting of formation of several meteor streams and their dynamical evolution over 5000 years. We ejected meteoroids particles from the comets: 1P/Halley, 2P/Encke, 55P/Tempel-Tuttle, 109P/Swift-Tuttle and from minor planets (3200) Phaethon and 2002 SY₅₀. During the integration, the ejection velocities were estimated using both M1 and M2 methods. The results show that the velocities obtained by M1 method are unstable: too high or too low, when compared with the known ejection velocities at the time of the stream formation. On the other hand, the velocities obtained using M2 method are too small, mostly. In principle, M2 estimates the dispersion of the distribution of the ejection velocities around the mean value, not the mean value itself. Applying more accurate Equation relating Δυ and Δa we decreased the bias of the results, but not their variation observed during the evolution of the streams and the parent bodies. We have found that the variability of the estimated ejection velocities was caused mainly by the gravitational changes of the semi-major axis and eccentricity of the parent body. In brief, we have found that the reliability of the results obtained by M1 or M2 method are low, and have to be used with great care.     

Slow modes in stellar systems with nearly harmonic potentials: I. Spoke approximation,radial orbit instability

Using a consistent perturbation theory for collisionless disk-like and spherical star clusters, we construct a theory of slow modes for systems having an extended central region with a nearly harmonic potential due to the presence of a fairly homogeneous (on the scales of the stellar system) heavy, dynamically passive halo. In such systems, the stellar orbits are slowly precessing, centrally symmetric ellipses (2: 1 orbits). Depending on the density distribution in the system and the degree of halo inhomogeneity, the orbit precession can be both prograde and retrograde, in contrast to systems with 1: 1 elliptical orbits where the precession is unequivocally retrograde. In the first paper, we show that in the case where at least some of the orbits have a prograde precession and the stellar distribution function is a decreasing function of angular momentum, an instability that turns into the well-known radial orbit instability in the limit of low angular momenta can develop in the system. We also explore the question of whether the so-called spoke approximation, a simplified version of the slow mode approximation, is applicable for investigating the instability of stellar systems with highly elongated orbits. Highly elongated orbits in clusters with nonsingular gravitational potentials are known to be also slowly precessing 2: 1 ellipses. This explains the attempts to use the spoke approximation in finding the spectrum of slow modes with frequencies of the order of the orbit precession rate. We show that, in contrast to the previously accepted view, the dependence of the precession rate on angular momentum can differ significantly from a linear one even in a narrow range of variation of the distribution function in angular momentum. Nevertheless, using a proper precession curve in the spoke approximation allows us to partially “rehabilitate” the spoke approach, i.e., to correctly determine the instability growth rate, at least in the principal (O(α _T^−1/2) order of the perturbation theory in dimensionless small parameter α _T, which characterizes the width of the distribution function in angular momentum near radial orbits.