The first-order,non-singular Keplerian differential state transition matrix

The Keplerian differential state transition matrix (KDSTM) is a fundamental tool in investigations of the sensitivity of orbital evolution to changes in initial conditions, in perturbation analysis, as well as in targeting and rendezvous operations. Several different forms of the KDSTM are available in the literature. They differ in the choice of state space variables, as well as in derivation methods. Here, a new method for constructing the KDSTM is presented, which is based on the well-known theorem on the differentiability of the solution of a system of ordinary differential equations with respect to initial conditions. A peculiarity of the method is that it allows the direct construction of analytical expressions for both the direct and the inverse fundamental matrices needed to form the KDSTM. The KDSTM is first built in the inertial reference frame and then transformed to the orbital, or Hill reference frame. The resulting expressions contain the full set of Keplerian elements and are hence readily extensible to perturbed Keplerian reference motion. The results are compared with some of the best known KDSTM’s available in the literature, with which they are proven to be fully equivalent, despite their sometimes dramatically different appearance.     

The gravity-perturbed Lambert problem: A KS variation of parameters approach

A new formulation is presented for the perturbed Lambert problem. The formulation employs the variation-of-parameters method in the KS transformed state space to determine perturbations of a Keplerian Lambert solution. The approach is universal (in that its validity is not restricted to a particular energy domain). For the case of the second zonal harmonic (oblateness) perturbation, first order perturbations are carried out entirely analytically; non-iterative corrections are determined through solution of a pair of algebraic equations. For more general perturbations, numerical quadratures are required.     

On the Milankovitch orbital elements for perturbed Keplerian motion

We consider sets of natural vectorial orbital elements of the Milankovitch type for perturbed Keplerian motion. These elements are closely related to the two vectorial first integrals of the unperturbed two-body problem; namely, the angular momentum vector and the Laplace–Runge–Lenz vector. After a detailed historical discussion of the origin and development of such elements, nonsingular equations for the time variations of these sets of elements under perturbations are established, both in Lagrangian and Gaussian form. After averaging, a compact, elegant, and symmetrical form of secular Milankovitch-like equations is obtained, which reminds of the structure of canonical systems of equations in Hamiltonian mechanics. As an application of this vectorial formulation, we analyze the motion of an object orbiting about a planet (idealized as a point mass moving in a heliocentric elliptical orbit) and subject to solar radiation pressure acceleration (obeying an inverse-square law). We show that the corresponding secular problem is integrable and we give an explicit closed-form solution.     

Normal forms for the epicyclic approximations of the perturbed Kepler problem

We compute the normal forms for the Hamiltonian leading to the epicyclic approximations of the (perturbed) Kepler problem in the plane. The Hamiltonian setting corresponds to the dynamics in the Hill synodic system where, by means of the tidal expansion of the potential, the equations of motion take the form of perturbed harmonic oscillators in a rotating frame. In the unperturbed, purely Keplerian case, the post-epicyclic solutions produced with the normal form coincide with those obtained from the expansion of the solution of the Kepler equation. In all cases where the perturbed problem can be cast in autonomous form, the solution is easily obtained as a perturbation series. The generalization to the spatial problem and/or the non-autonomous case is straightforward.     

Dynamics of rotation of super-Earths

We numerically investigate the dynamics of rotation of several close-in terrestrial exoplanet candidates. In our model, the rotation of the planet is disturbed by the torque of the central star due to the asymmetric equilibrium figure of the planet. We model the shape of the planet by a Jeans spheroid. We use surfaces of section and spectral analysis to explore numerically the rotation phase space of the systems adopting different sets of parameters and initial conditions close to the main spin–orbit resonant states. One of the parameters, the orbital eccentricity, is critically discussed here within the domain of validity of orbital circularization timescales given by tidal models. We show that, depending on some parameters of the system like the radius and mass of the planet, eccentricity etc., the rotation can be strongly perturbed and a chaotic layer around the synchronous state may occupy a significant region of the phase space. 55 Cnc e is an example.     

Ideal frames for perturbed Keplerian motions

Cartan's exterior calculus is used to refer a perturbed Keplerian motion to an ideal frame by means of either the Eulerian parameters or the Eulerian angles, in which case the equations are given a Hamiltonian form. The results are compared with the corresponding systems in the orbital and nodal frames.     

Dealing with uncertainties in angles-only initial orbit determination

A method to deal with uncertainties in initial orbit determination (IOD) is presented. This is based on the use of Taylor differential algebra (DA) to nonlinearly map uncertainties from the observation space to the state space. When a minimum set of observations is available, DA is used to expand the solution of the IOD problem in Taylor series with respect to measurement errors. When more observations are available, high order inversion tools are exploited to obtain full state pseudo-observations at a common epoch. The mean and covariance of these pseudo-observations are nonlinearly computed by evaluating the expectation of high order Taylor polynomials. Finally, a linear scheme is employed to update the current knowledge of the orbit. Angles-only observations are considered and simplified Keplerian dynamics adopted to ease the explanation. Three test cases of orbit determination of artificial satellites in different orbital regimes are presented to discuss the feature and performances of the proposed methodology.     

The theory of asynchronous relative motion I: time transformations and nonlinear corrections

Using alternative independent variables in lieu of time has important advantages when propagating the partial derivatives of the trajectory. This paper focuses on spacecraft relative motion, but the concepts presented here can be extended to any problem involving the variational equations of orbital motion. A usual approach for modeling the relative dynamics is to evaluate how the reference orbit changes when modifying the initial conditions slightly. But when the time is a mere dependent variable, changes in the initial conditions will result in changes in time as well: a time delay between the reference and the neighbor solution will appear. The theory of asynchronous relative motion shows how the time delay can be corrected to recover the physical sense of the solution and, more importantly, how this correction can be used to improve significantly the accuracy of the linear solutions to relative motion found in the literature. As an example, an improved version of the Clohessy-Wiltshire (CW) solution is presented explicitly. The correcting terms are extremely compact, and the solution proves more accurate than the second and even third order CW equations for long propagations. The application to the elliptic case is also discussed. The theory is not restricted to Keplerian orbits, as it holds under any perturbation. To prove this statement, two examples of realistic trajectories are presented: a pair of spacecraft orbiting the Earth and perturbed by a realistic force model; and two probes describing a quasi-periodic orbit in the Jupiter-Europa system subject to third-body perturbations. The numerical examples show that the new theory yields reductions in the propagation error of several orders of magnitude, both in position and velocity, when compared to the linear approach.     

Orbital evolution dynamics of two satellites in encounter phase using multiple scales expansion

An approximate solution of the encounter problem of two small satellites describing initially elliptical orbits around a massive oblate primary is obtained. The equations of motion of the center of mass of the two masses are developed in the most general form without any restrictions on the orbital elements. The method of multiple scales which seeks a solution whose behavior depends on several time scales is used. To overcome the singularity the equations of motion are transformed to the Struble variables. An analytical second order theory of the evolution dynamics is obtained. A MATHEMATICA program is constructed. The evolution dynamics of the orbital parameters between the perturbed and the unperturbed cases are plotted. The effect of changing eccentricity and changing inclination on the orbital parameters are highlighted.     

Linearization of dynamical systems using integrals of the motion

A method is presented which transforms certain non-linear differential equations of dynamics into linear equations by introducing a new independent variable and by utilizing the integrals of motion. As examples of special interest the linearizations of unperturbed and perturbed Keplerian motions are discussed.     

Matrix perturbation methods using regularized coordinates

Matrix methods for computing perturbations of non-linear perturbed systems, as formulated by Alexeev, involve an expression for the full solution of the first variational equations of the system evaluated about a reference orbit. These cannot be immediately applied to a regularized system of equations where perturbations about Keplerian motion are considered since the solution of the variational equations of regularized Keplerian motion does not in general correspond to the solution of the variational equations of the unregularized equations. But, as Kustaanheimo and Stiefel have pointed out, the regularized equations of Keplerian motion should be excellent for the initiation of a perturbation theory since they are linear in form. This paper describes a method for applying Alexeev's theorem to a regularized system where full advantage is taken of the basic linear form of the unperturbed equations.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

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.     

On the attitude dynamics of perturbed triaxial rigid bodies

Attitude dynamics of perturbed triaxial rigid bodies is a rather involved problem, due to the presence of elliptic functions even in the Euler equations for the free rotation of a triaxial rigid body. With the solution of the Euler–Poinsot problem, that will be taken as the unperturbed part, we expand the perturbation in Fourier series, which coefficients are rational functions of the Jacobian nome. These series converge very fast, and thus, with only few terms a good approximation is obtained. Once the expansion is performed, it is possible to apply to it a Lie-transformation. An application to a tri-axial rigid body moving in a Keplerian orbit is made.     

Coordinates for perturbed keplerian systems with axial symmetry

A coordinate system is defined on the phase space of a perturbed Keplerian system after the mean anomaly has been averaged out, for the purpose of explaining how eliminating the longitude of the ascending node reduces the orbital space to a two-dimensional sphere in case the system admits an axial symmetry. Concomitantly, on the submanifold of direct osculating ellipses, the CDM variables are replaced by functions which form the basis of a Poisson algebra isomorphic to the Lie algebra so(3) of the rotation group SO(3); furthermore, in these variables, the doubly reduced phase flow appears like a rotation of the reduced phase space.     

Element sets for high-order Poincaré mapping of perturbed Keplerian motion

The propagation and Poincaré mapping of perturbed Keplerian motion is a key topic in Celestial Mechanics and Astrodynamics, e.g., to study the stability of orbits or design bounded relative trajectories. The high-order transfer map (HOTM) method enables efficient mapping of perturbed Keplerian orbits using the high-order Taylor expansion of a Poincaré or stroboscopic map. The HOTM is only accurate close to the expansion point and therefore the number of revolutions for which the map is accurate tends to be limited. The proper selection of coordinates is of key importance for improving the performance of the HOTM method. In this paper, we investigate the use of different element sets for expressing the high-order map in order to find the coordinates that perform best in terms of accuracy. A new set of elements is introduced that enables extremely accurate mapping of the state, even for high eccentricities and higher-order zonal perturbations. Finally, the high-order map is shown to be very useful for the determination and study of fixed points and center manifolds of Poincaré maps.     

Velocity-space maps and transforms of tracking observations for orbital trajectory state analysis

The orbital state of a satellite in a central force field can be uniquely described by its velocity hodograph, a circle, rather than the Keplerian conic. Also, its coordinate-frame rotation about the attracting center is definable, without singularity, by the four-parameter set of Euler parameters. A unified state model of orbital trajectory and attitude dynamics has previously been developed by use of state variables of the orbital velocity hodograph and Euler parameters. The dynamical constraint equations of this orbital state model are especially effective in advanced techniques of state estimation, used for orbit determination and prediction. External observations of orbital vehicles, such as provided by optical and radar sensors of tracking systems, are transformable into corresponding velocity state maps, as presented in this paper. These transformations and the consequent state maps are essential for development of the orbit observation matrix used with the unified state matrix, in recursive estimators such as the Kalman filters. Line-of-sight rays and range spheres (or hemispheres) of observations map conformally into orthogonal spherical surfaces in velocity space, as the result of the point-contact transformations. In bispherical coordinates, the field of observation maps for a ground-based tracking system site is shown to be a reduced (or degenerate) form of the general field of observation maps for a satellite-based tracking site. These orbital state maps and transforms are directly useful in development of observation matrices for candidate observation sets, such as range only, angle only, or range plus range-rate tracking schemes. Also, surface coverage patterns can be generated for proposed new tracking systems, in mission analysis and system synthesis studies.     

Generalized perturbation equations of celestial mechanics

Generalized perturbation equations of celestial mechanics in terms of orbital elements are derived. The most general case is considered: Keplerian motion of two bodies caused by gravitational forces between them is disturbed by disturbing acceleration acting on each of the bodies separately and by changes of masses of these bodies. It is also pointed out why derivation presented in Klaka (1992a) is completely physically correct only for constant masses.     

The dynamical stability of a Kuiper Belt-like region

The dynamics of the Kuiper Belt region between 33 and 63 au is investigated just taking into account the gravitational influence of Neptune. Indeed the aim is to analyse the information which can be drawn from the actual exoplanetary systems, where typically physical and orbital data of just one or two planets are available. Under this perspective we start our investigation using the simplest three-body model (with Sun and Neptune as primaries), adding at a later stage the eccentricity of Neptune and the inclinations of the orbital planes to evaluate their effects on the Kuiper Belt dynamics. Afterwards we remove the assumption that the orbit of Neptune is Keplerian by adding the effect of Uranus through the Lagrange–Laplace solution or through a suitable resonant normal form. Finally, different values of the mass ratios of the primary to the host star are considered in order to perform a preliminary analysis of the behaviour of exoplanetary systems. In all cases, the stability is investigated by means of classical tools borrowed from dynamical system theory, like Poincaré mappings and Lyapunov exponents.     

The spectroscopic orbits of three double-lined eclipsing binaries: I. BG Ind,IM Mon,RS Sgr

We present the spectroscopic orbit solutions of three double-lines eclipsing binaries, BG Ind, IM Mon and RS Sgr. The first precise radial velocities (RVs) of the components were determined using high resolution echelle spectra obtained at Mt. John University Observatory in New Zealand. The RVs of the components of BG Ind and RS Sgr were measured using Gaussian fittings to the selected spectral lines, whereas two-dimensional cross-correlation technique was preferred to determine the RVs of IM Mon since it has relatively short orbital period among the other targets and so blending of the lines is more effective. For all systems, the Keplerian orbital solution was used during the analysis and also circular orbit was adopted because the eccentricities for all targets were found to be negligible. The first precise orbit analysis of these systems gives the mass ratios of the systems as 0.894, 0.606 and 0.325, respectively for BG Ind, IM Mon and RS Sgr. Comparison of the mass ratio values, orbital sizes and minimum masses of the components of the systems indicates that all systems should have different physical, dynamical and probable evolutionary status.