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.     

Dynamic discretization method for solving Kepler’s equation

Kepler’s equation needs to be solved many times for a variety of problems in Celestial Mechanics. Therefore, computing the solution to Kepler’s equation in an efficient manner is of great importance to that community. There are some historical and many modern methods that address this problem. Of the methods known to the authors, Fukushima’s discretization technique performs the best. By taking more of a system approach and combining the use of discretization with the standard computer science technique known as dynamic programming, we were able to achieve even better performance than Fukushima. We begin by defining Kepler’s equation for the elliptical case and describe existing solution methods. We then present our dynamic discretization method and show the results of a comparative analysis. This analysis will demonstrate that, for the conditions of our tests, dynamic discretization performs the best.     

Fast Procedure Solving Universal Kepler's Equation

We developed a procedure to solve a modification of the standard form of the universal Kepler’s equation, which is expressed as a nondimensional equation with respect to a nondimensional variable. After reducing the domain of the variable and the argument by using the symmetry and the periodicity of the equation, the method first separates the case where the solution is so small that it is given an inverted series. Second, it separates the cases where the elliptic, parabolic, or hyperbolic standard forms of Kepler’s equation are suitable. Here the separation is done by judging whether detouring these nonuniversal equations will cause a 1-bit loss of information to their nonuniversal solutions or not. Then the nonuniversal equations are solved by the author’s procedures to solve the elliptic Kepler’s equation (Fukushima, 1997a), Barker’s equation (Fukushima, 1998), and the hyperbolic Kepler’s equation (Fukushima, 1997b), respectively. And their nonuniversal solutions are transformed back to the solution of the universal equation. For the rest of the case, we obtain an approximate solution by solving roughly the approximated cubic equation as we did in solving Barker’s equation. Then the correction to the approximate solution is obtained by Halley’s method precisely. There the special function appeared in the universal equation is rewritten into a combination of similar special functions of small arguments, so that they are efficiently evaluated by their Taylor series. Numerical measurements showed that, in the case of Intel Pentium II processor, the new method is 10–25 times as fast as Shepperd’s method (Shepperd, 1985) and 7–13 times as fast as the standard Newton method. This revised version was published online in July 2006 with corrections to the Cover Date.     

Solving Kepler’s Equation using Bézier curves

This paper presents a non-iterative approach to solve Kepler’s Equation, M = E − e sin E, based on non-rational cubic and rational quadratic Bézier curves. Optimal control point coordinates are first shown to be linear with respect to orbit eccentricity for any eccentric anomaly range. This property yields the development of a piecewise (e.g., 3, 4) solving technique providing accuracies better than 10⁻¹³ degree for orbit eccentricity e ≤ 0.99. The proposed method does not require large pre-computed discretization data, but instead solves a cubic/quadratic algebraic equation and uses a single final Halley iteration in only a few lines of code. The method still provides accuracies better than 10⁻⁵ degree for the near parabolic worst case (e = 0.9999) with very small mean anomalies (M < 0.0517 deg). The complexity of the proposed algorithm is constant, independent of the parameters e and M. This makes the method suitable for extensive orbit propagations. Presented at the 7th Dynamics and Control of Systems and Structures in Space Conference, July 18–22, 2006, Greenwich, England.     

Sequential solution to Kepler’s equation

Seven sequential starter values for solving Kepler’s equation are proposed for fast orbit propagation. The proposed methods have constant complexity (not iterative), do not require pre-computed data, and can be implemented in just a few lines of code. The resulting sequential orbit propagation techniques can be done at different levels of accuracy and speed, depending essentially on the value of orbit eccentricity. Accuracy and algorithmic complexity are evaluated for all the proposed approaches and compared with several existing single-point techniques to solve Kepler’s equation. The new methods obtain improved accuracy at lower computational cost as compared to the best existing methods.     

Gamma-ray emission expected from Kepler’s SNR

Nonlinear kinetic theory of cosmic ray (CR) acceleration in supernova remnants (SNRs) is used to investigate the properties of Kepler’s SNR and, in particular, to predict the γ-eay spectrum expected from this SNR. Observations of the nonthermal radio and X-ray emission spectra as well as theoretical constraints for the total supernova (SN) explosion energy E _sn are used to constrain the astronomical and particle acceleration parameters of the system. Under the assumption that Kepler’s SN is a type Ia SN we determine for any given explosion energy E _sn and source distance d the mass density of the ambient interstellar medium (ISM) from a fit to the observed SNR size and expansion speed. This makes it possible to make predictions for the expected γ-eay flux. Exploring the expected distance range we find that for a typical explosion energy E _sn=10⁵¹ erg the expected energy flux of TeV γ-rays varies from 2×10⁻¹¹ to 10⁻¹³ erg/(cm² s) when the distance changes from d=3.4 kpc to 7 kpc. In all cases the γ-eay emission is dominated by π ⁰-decay γ-rays due to nuclear CRs. Therefore Kepler’s SNR represents a very promising target for instruments like H.E.S.S., CANGAROO and GLAST. A non-detection of γ-rays would mean that the actual source distance is larger than 7 kpc.     

Bimodal brightness oscillations in models of young binary systems

We consider a model for the cyclic activity of young binary stars that accrete matter from the remnants of a protostellar cloud. If the orbit of such a binary system is inclined at a small angle to the line of sight, then the streams of matter and the density waves excited in the circumbinary disk can screen the primary component of the binary from the observer. To study these phenomena by the SPH (smoothed particle hydrodynamics) method, we have computed grids of hydrodynamic models for binary systems based on which we have constructed the light curves as a function of the orbital phase. The main emphasis is on investigating the properties of the brightness oscillations. Therefore, the model parameters were varied within the following ranges: the component mass ratio q = M ₂: M ₁ = 0.2–0.5 and the eccentricity = 0–0.7. The parameter that defined the binary viscosity was also varied. We adopted optical grain characteristics typical of circumstellar dust. Our computations have shown that bimodal oscillations are excited in binaries with eccentric orbits, provided that the binary components do not differ too much in mass. In this case, the ratios of the periods and amplitudes of the bimodal oscillations and their shape depend strongly on the inclination of the binary plane and its orientation relative to the observer. Our analysis shows that the computed light curves can be used in interpreting the cyclic activity of UX Ori stars.     

The Lambert <Emphasis Type="Italic">W</Emphasis> function and solutions to Kepler’s equation

This paper presents a method for the truncation of infinite Fourier–Bessel representations for functions requiring a solution to Kepler’s equation. Use is made of the Lambert W function to solve for the desired index that bounds the remainder terms of the series, within the prescribed tolerance. The enforcement of a maximum on the number of Bessel functions is also useful in truncating the Bessel functions themselves, resulting in an analytical representation of the solution to a desired tolerance, without the use of infinite series.     

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.     

The Restricted Hill Four-Body Problem with Applications to the Earth–Moon–Sun System

Starting from the four-body problem a generalization of both the restricted three-body problem and the Hill three-body problem is derived. The model is time periodic and contains two parameters: the mass ratio ν of the restricted three-body problem and the period parameter m of the Hill Variation orbit. In the proper coordinate frames the restricted three-body problem is recovered as m → 0 and the classical Hill three-body problem is recovered as ν → 0. This model also predicts motions described by earlier researchers using specific models of the Earth–Moon–Sun system. An application of the current model to the motion of a spacecraft in the Sun perturbed Earth–Moon system is made using Hill's Variation orbit for the motion of the Earth–Moon system. The model is general enough to apply to the motion of an infinitesimal mass under the influence of any two primaries which orbit a larger mass. Using the model, numerical investigations of the structure of motions around the geometric position of the triangular Lagrange points are performed. Values of the parameter ν range in the neighborhood of the Earth–Moon value as the parameter m increases from 0 to 0.195 at which point the Hill Variation orbit becomes unstable. Two families of planar periodic orbits are studied in detail as the parameters m and ν vary. These families contain stable and unstable members in the plane and all have the out-of-plane stability. The stable and unstable manifolds of the unstable periodic orbits are computed and found to be trapped in a geometric area of phase space over long periods of time for ranges of the parameter values including the Earth–Moon–Sun system. This model is derived from the general four-body problem by rigorous application of the Hill and restricted approximations. The validity of the Hill approximation is discussed in light of the actual geometry of the Earth–Moon–Sun system. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stability criterion for a light binary attracted by a heavy body

Dynamical behaviour of a small binary with equal components, each of mass m, is considered under attraction of a heavy body of mass M. Differential equations of the general three-body problem are integrated numerically using the code by S. J. Aarseth (Aarseth, Zare 1974) for mass ratios m/M within 10⁻¹¹–10⁻⁴ range. The direct and retrograde orbits of light bodies about each other are considered which lie either in the plane of moving their center of mass or in the plane perpendicular to it. It is shown numerically that the critical separation between the binary components which leads to disruption of binary is proportional to (m/M)^1/3. The criterion can be used for studying (in the first approximation) the motion of double stars and binary asteroids or computing the parameters of magnetic monopol and antimonopol pairs.     

Precise determination of the orbital elements of binary stars using Differential Evolution

This paper presents an ‘adaptive probability of crossover’ technique, as a variation of the differential evolution algorithm (ACDE), for optimal parameter estimation in the general curve-fitting problem. The technique is applied to the determination of orbital elements of a spectroscopic binary system (eta Bootis). In the ACDE, Varying the crossover probability rate (Cr) provides faster convergence than keeping it constant. The Cr is determined for each trial parameter vector (‘individual’) as a function of fit goodness. The adaptation automatically updates control parameter to an appropriate value, without requiring prior knowledge of the relationship between particular parameter settings and a given problem optimization characteristics. The presented analysis of eta Bootis derives best-fitting Keplerian and phasing curves. Error estimation of the optimal parameters is also included. Comparison of the results with previously published values suggests that the ACDE technique has a useful applicability to astrophysical data analysis.     

Brightness oscillations in models of young binary systems with low-mass secondary components

We consider a model for the cyclic brightness variations of a young star with a low-mass companion that accretes matter from the remnants of a protostellar cloud. At small inclinations of the binary orbit to the line of sight, the streams of matter and the density waves excited in the circumbinary disk can screen the primary component of the binary from the observer. To study these phenomena, we have computed grids of hydrodynamic models for binary systems by the SPH method based on which we have calculated the phase light curves for the different orientations of the orbit. The model parameters were varied within the following ranges: the component mass ratio q = 0.01–0.1 and the eccentricity e = 0–0.5. We adopted optical grain characteristics typical of circumstellar dust. Our computations have shown that the brightness oscillations with orbital phase can have a complex structure. The amplitudes and shapes of the light curves depend strongly on the inclination of the binary orbit and its orientation relative to the observer and on the accretion rate. The results of our computations are used to analyze the cyclic activity of UX Ori stars.     

Modulation of circumstellar extinction in a young binary system with a low-mass companion in a noncoplanar orbit

We consider a model of cyclic brightness variations in a young star with a low-mass (q = M ₂/M ₁ ≤ 0.1) companion that accretes matter from the remnants of a protostellar cloud (circumbinary disk). We assume that the orbit of the companion is circular and that its plane does not coincide with the disk plane. We have computed grids of hydrodynamic models for such a binary by the SPH method based on which we have investigated the circumstellar extinction variations produced by the streams of matter and density waves excited in the circumbinary disk by the orbital motion of the companion. We show that, depending on the inclination and orientation of the binary’s line of nodes relative to the observer, the brightness of the primary component can undergo various (in shape and depth) oscillations with a period equal to the orbital one. In contrast to the models with coplanar circular orbits, the accretion rate onto the components of a binary with a noncoplanar orbit depends on the orbital phase. The results of our computations can be used to study the cyclic activity of UX Ori stars and young eclipsing binaries with anomalously long eclipses.     

Classification of Motions for Generalization of the two Centers Problem on a Sphere

The generalization of a test particle motion in a central field of two immovable point-like centers to the case of a constant curvature space, on a three-dimensional sphere, is investigated in the paper. The bifurcation set in the plane of integrals of motion is constructed and the classification of the domains of possible motion is carried out on a two-dimensional sphere. The regularization of the Kepler’s problem on a two-dimensional sphere is carried out. This revised version was published online in July 2006 with corrections to the Cover Date.     

Initial orbit determination using multiple observations

A novel approach for initial orbit determination based on multiple angles-only observations is presented. The proposed technique is iterative and uses Lagrangian coefficients, f and g. The proposed method does not show singularity for the coplanar cases. In addition, the method is capable of handling multiple observations, providing higher accuracy, whereas the level of the algorithm complexity and processor running time remain almost invariant. The technique presented is compared with the Double r-iteration and Gauss’ methods using data corrupted by noise to simulate true measurements. Results show that the proposed method is a valid alternative to the classical methods of orbit determination.     

Dusty disk wind in young binary systems viewed pole-on

A model of a young binary system with companions of unequal mass whose orbital plane is inclined at a large angle to the line of sight is examined. The system components are assumed to accrete matter from the remains of the protostellar cloud. It is found that eclipsing of the primary component by the disk wind of the secondary can be observed when the plane of the orbit is inclined at a large angle to the line of sight or even when the binary system is observed pole-on. __________ Translated from Astrofizika, Vol. 51, No. 2, pp. 267–275 (May 2008).     

Bragg diffraction and the iron crust of cold neutron stars

If cooled-down neutron stars have a thin atomic crystalline–iron crust, they must diffract X-rays of appropriate wavelength. If the diffracted beam is to be visible from Earth (an extremely rare but possible situation), the illuminating source must be very intense and near the reflecting star. An example is a binary system composed of two neutron stars in close orbit, one of them inert, the other an X-ray pulsar. (Perhaps an “anomalous” X-ray pulsar or magnetar, not powered by gas absorption from the companion or surrounding space, would be the cleanest example.) The observable to be searched for is a secondary peak added (quasi-) periodically to the main X-ray pulse. The distinguishing feature of this secondary peak is that it appears at wavelengths related by simple integer numbers, λ,λ/2,λ/3,…,λ/n because of Bragg’s diffraction law.     

Dark matter around binary galaxies: formation of collimated flows

We study here the dynamics of an extended shell of relatively low-mass (almost zero-mass) particles around massive binary systems by computer simulations in the framework of approximately restricted three-body problem with a set of several initial conditions concerning the massesM ₁ andM ₂ of the binary components surrounded byN test particles in uniform random distribution on a spherical envelope of radiusR expanding with a velocityV. We apply this model to binary galaxy systems with a halo of baryonic dark matter, e.g., massive black holes, globular clusters, and giant molecular clouds. It is shown that, initially, the shell expands homologously with decreasing velocity and then, falls back into the system forming zones of compressed matter. At some moment there could be a collapse or these particles onto the heavier component of the binary. Further in time, a number of particles escape from the system. We consider a number of different models with different initial parameters. For models with smallerR andV, about one-half of the particles escape from the system; while for larger values the shell disrupts as a whole. The escaping particles form a collimated flow in the plane of the orbit of the binary. The position of the flow and the directions of motions depend on the position of the heavier component of the binary at the moment of the closest approach of the particles and on the ratioM ₁/M ₂.     

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.