Linear and Regular Celestial Mechanics
Chapters (11)
- ... The most common use of the KS transformation is the numerical integration of perturbed elliptic motion, where many intricacies introduced by the additional degree of freedom can be ignored, although -as recently demonstrated by Roa et al (2016) -they can be quite useful in the assessment of a global integration error. Analytical perturbation methods for KS-transformed problems often follow the way indicated by Kustaanheimo and Stiefel (1965) and developed by Stiefel and Scheifele (1971): variation of arbitrary constants is applied to constant vector amplitudes of the KS co- ordinates and velocities. But those who want to benefit from the wealth of canonical formalism, require a set of action-angle variables of the regularized Kepler problem. ...... The first step in this direction can be found in the monograph by Stiefel and Scheifele (1971), where the symplectic polar coordinates are introduced for each separate de- gree of freedom. However, this approach does not account for degeneracy of the problem and thus is unfit for the averaging-based perturbation techniques. ...... If the motion can be restricted to the linear subspace of H spun by two basis quaternions u and w, such that J (u, w) = 0, the KS transformation reduces to the Levi-Civita transformation (Levi-Civita, 1906). For this reason, a two-dimensional subspace P of quaternions being the linear combinations of u and w, hence such that the form J on any two of them equals 0, was dubbed the 'Levi-Civita plane' by Stiefel and Scheifele (1971). We will use the name 'LC plane', although, strictly speaking, a (hyper-)plane in a space of dimension 4 should be spun by 3 basis quaternions. ...The Kustaanheimo-Stiefel transformation of the Kepler problem with a time-dependent perturbation converts it into a perturbed isotropic oscillator of 4-and-a-half degrees of freedom with additional constraint known as bilinear invariant. Appropriate action-angle variables for the constrained oscillator are required to apply canonical perturbation techniques in the perturbed problem. The Lissajous-Kustaanheimo-Stiefel (LKS) transformation is proposed, leading to the action-angle set which is free from singularities of the LCF variables earlier proposed by Zhao. One of the actions is the bilinear invariant, which allows the reduction back to the 3-and-a-half degrees of freedom. The transformation avoids any reference to the notion of the orbital plane, which allowed to obtain the angles properly defined not only for most of the circular or equatorial orbits, but also for the degenerate, rectilinear ellipses. The Lidov-Kozai problem is analyzed in terms of the LKS variables, which allow a direct study of stability for all equilibria except the circular equatorial and the polar radial orbits.
- ... KS formulation was used by Engels and Junkins[13]and Jezewski[14]for short-term orbit predictions with J 2 effect. The KS uniform regular canonical equations of motion[15]are a particular canonical form where all the ten elements are constant for unperturbed two-body problem and are applicable to elliptic, parabolic and hyperbolic orbital motion. Sharma and James Raj[16]numerically integrated these equations to obtain accurate orbit prediction under the effect of Earth's oblateness with zonal harmonic terms up to J 36. ...... The KS uniformly regular canonical equations of motion for the state vector in terms of the independent variable's' are[15,20] ...... As in[15], for 1 0 x < : ...A new non-singular, analytical theory with respect to the Earth’s zonal harmonic term J2 has been developed for short-periodic motion, by analytically integrating the uniformly regular KS canonical equations of motion using generalized eccentric anomaly ‘E’ as the independent variable. Only one of the eight equations needs to be integrated analytically to generate the state vector, as a result of symmetry in the equations of motion, and the computation for the other equations is by changing the initial conditions. The integrals are much simpler than the earlier obtained one in terms of the independent variable ‘s’. Numerical results indicate that the solution is reasonably accurate for a wide range of orbital parameters during a revolution. The error in computing the most important orbital parameter ‘semi-major axis’ which is the measure of energy is less than five percentage during a revolution. The analytical solution can have number of applications. It can be used for studying the short-term relative motion of two or more space objects. It can also be useful in collision avoidance studies of space objects. It can be used for onboard computation in the navigation and guidance packages, where the modeling of J2 effect becomes necessary.
- ... Let us observe a useful property of the mixed dot product, ¯ u · (vw) = ¯ w · (uv) = ¯ v · (wu), (19) echoing the mixed product rule of the standard vector algebra. Another useful operation is called a quaternion outer product (Morais et al 2014) or a quaternion cross product (Stiefel and Scheifele 1971; Vivarelli 1988; Deprit et al 1994). Conventions vary among the authors; we adopt the one of Deprit et al (1994) u ∧ v = v¯ u − u¯ v 2 = (0, u 0 v − v 0 u + u × v) . ...... Thus we find another argument in favor of the claim that the KS1 transformation attaches a special role to the axis Ox 1 . The existence of some relation between KS variables and rotation was mentioned 'for the record' by Stiefel and Scheifele (1971), who declared the lack of interest in studying it closer. Then Vivarelli (1983) returned to this issue, but her description is based on a statement that since a unit quaternion q 'represents a rotation', so a product qq * also 'represents' some rotation with the axis and angle expressions provided. ...... It differs from the rules of Stiefel and Scheifele (1971), effectively based upon the sign of c · x (with c = e 1 ). An interesting alternative was proposed by Saha (2009), who used the inversion rule implying a pure vector (0, v) form. ...Kustaanheimo-Stiefel (KS) transformation depends on the choice of some preferred direction in the Cartesian 3D space. This choice, seldom explicitly mentioned, amounts typically to the direction of the first or the third coordinate axis in celestial mechanics and atomic physics, respectively. The present work develops a canonical KS transformation with an arbitrary preferred direction, indicated by what we call a defining vector. Using a mix of vector and quaternion algebra, we formulate the transformation in a reference frame independent manner. The link between the oscillator and Keplerian first integrals is given. As an example of the present formulation, the Keplerian motion in a rotating frame is re-investigated.
- ... To test the efficiency of DROMO we select the example 2b at page 122 of the book by Stiefel and Scheifele [11]. The problem, which was also considered by Bond [3] and Pel?ez [10] for comparing different regularized methods, is to determine the position of a satellite after 288.12768941 mean solar days (msd) around the Earth under the perturbing action of the Earth oblateness and the Lunar gravitational attraction . ...... The satellite is initially at the perigee at the distance R 0 = 6800 km. Details on the implementation of the two perturbations are available in the book [11]. We compare in terms of accuracy the following methods: the element formulation derived from the Kustaanheimo-Stiefel regularization scheme by Stiefel and Scheifele [11] , the L-C extension developed by Fukushima [7], DROMO [10] and Cowell's method. ...... Details on the implementation of the two perturbations are available in the book [11]. We compare in terms of accuracy the following methods: the element formulation derived from the Kustaanheimo-Stiefel regularization scheme by Stiefel and Scheifele [11] , the L-C extension developed by Fukushima [7], DROMO [10] and Cowell's method. Apart from the last, the other propagators adopt an independent variable which is not the physical time, but a fictitious time. ...The special perturbation method DROMO developed by Peláez in 2006 for the perturbed two-body problem is employed to propagate the relative motion in spacecraft formation flying, and the performance of the new method, named DROMO-FF, is analyzed. DROMO is a very fast and accurate regularized method which involves a set of seven integrals of the pure Keplerian motion whose physical meaning is described in the paper. We propose to propagate the absolute motion of n spacecraft simultaneously by using DROMO with the introduction of new dependent variables, necessary for the synchronization, and to determine the relative dynamics by differentiating the absolute states. After investigating the influence on the performance due to the numerical integration of the new variables, we show that DROMO-FF is significantly more accurate than Cowell's method for the same computing time, or equivalently, faster for the same accuracy. A second approach to propagate relative motion wherein linearization is performed with respect to the formation baricenter is presented and compared to DROMO-FF. It is shown that for closed formations round-off does not affect the accuracy of DROMO-FF.
- ... These formulations of dynamics have shown excellent numerical performances due to several characteristics: the solution is well-behaved also close to the primary body; the indepen- dent variable is an angle-like quantity (also called fictitious time) which produces an analytical step-size regulation in physical time; the instability of Keplerian motion 1 is miti- gated or suppressed; the equations of motion are linear in 1 Consider at a certain epoch a particle following a Keplerian ellipse and another particle having a slightly different position and velocity. The two orbital energies will differ by some amount, and since the orbital period depends on the energy, for arbitrary small variations of the position and velocity after some time the the Keplerian case (Stiefel & Scheifele 1971;Deprit et al. 1994). Linearization of Equation 1 is also possible with- out eliminating the singularity at collision. ...... 3). We employ two regularized methods: the Kustaanheimo- Stiefel regularization (Stiefel & Scheifele 1971), and a set of non-singular orbital elements Bà u et al. 2015), which stems from the Dromo formulation originally developed by Peláez et al. (2007) for tether bead models. The Dromo elements are strictly related to Burdet's focal variables. ...... The K-S regularization has seen extensive use in dynamical astronomy, and it is described in detail in the text by Stiefel & Scheifele (1971). It is based on the classical Sundman transformation, i.e. f = α = 1 in Equation 1, and on a map- ping from u ∈ R 4 to r ∈ R 3 given by ...We present an efficient strategy for the numerical propagation of small Solar System objects undergoing close encounters with massive bodies. The trajectory is split in severalphases, each of them being the solution of a perturbed two-body problem. Formulations regularized with respect to different primaries are employed in two subsequent phases. In particular, we consider the Kustaanheimo-Stiefel regularization and a novel set of non-singular orbital elements pertaining to the Dromo family. In order to test the proposed strategy, we perform ensemble propagations in the Earth-Sun CR3BP using a variable step-size and order multi-step integrator and an improved version of Everhart’s radau solver of 15th order. By combining the trajectory splitting with regularized equations of motion in short-term propagations (one year), we gain up to six orders of magnitude in accuracy with respect to the classical Cowell’s method for the same computational cost. Moreover, in the propagation of asteroid (99942) Apophis through its 2029 Earth encounter, the position error stays within 100 meters after 100 years. In general, as to improve the performance of regularized formulations, the trajectory must be split between 1.2 and 3 Hill radii from the Earth. We also devise a robust iterative algorithm to stop the integration of regularized equations of motion at a prescribed physical time. The results rigorously hold in the CR3BP, and similar considerations may apply when considering more complex models. The methods and algorithms are implemented in the Naples Fortran 2003 code, which is available online as a GitHub repository.
- ... Next, in Section 4, after reduction to the submanifold of null-twistors and applying respective symplectic diffeomorphisms, see Proposition 4.1 and Proposition 4.2, we show that these Hamiltonian systems are equivalent to a 3-dimensional perturbed Kepler systems written in the "fictitious time" representation, [6,14]. In such a way we obtain a family of 3-dimensional perturbed Kepler systems which are integrated by quadratures. ...... For an exhaustive discussion of the time regularization problem for instance see [6,14]. One obtains the solution ( y(t), x(t)) of the Hamilton equations ...In the framework of the Poisson geometry of twistor space we consider a family of perturbed 3-dimensional Kepler systems. We show that Hamilton equations of this systems are integrated by quadratures. Their solutions for some subcases are given explicitly in terms of Jacobi elliptic functions.
- ... where ρ is equal to the negative of one-half of the Keplerian energy. The readers are referred to Reference [9] for a detailed derivation of the above equation. The independent variable in the above equation is the fictitious time s (derivatives with respect to s are shown using primes), which is introduced to regularize the two-body solution in the regions of close encounters to the central gravitational body. ...... In case of parabolic transfers, the value of z is obviously 0. For the other two cases, a relation between z and the eccentric anomaly or hyperbolic anomaly is sought in the sequel, which can then be used to compute the bounds on z. Stiefel and Scheifele have shown that the parametric trajectory in U 4 is an ellipse for an elliptic trajectory in Cartesian space (see Reference [9], Chapter III). Similarly, it can also be shown that the parametric trajectory in U 4 is a hyperbola for a hyperbolic trajectory in Cartesian space. ...Lambert's problem is widely used in preliminary design and optimization of in-terplanetary as well as planetocentric missions. For preliminary design, it is often necessary to obtain feasible trajectories that satisfy the mission constraints. These solutions can be used for low-fidelity trade studies and as initial guesses for high-fidelity numerical optimization. The classic Lambert's problem only allows for position and transfer time constraints. In this work, various two-body orbital boundary value problems with constraints on velocities, flight path angle, ∆v, final radius, transfer angle, etc. are studied and their exact solutions in universal form are derived via the KS-transformation. All of the solutions are regular and completely analytic if the energy of the transfer orbit is known a priori, otherwise they reduce to solving a single transcendental equation with well-defined bounds on the roots. The formulation presented in the paper results in a single constraint in each case for solving a class of boundary value problems commonly encountered in trajectory design problems.
- ... evaluate P S in the space frame; it depends on the problem at hand 23: The advantages of using an integration method in associattion with a particular formulation may depend on the orbital scenario to which the formulation is applied (see chap. 6 of Roa, 2016), and the discussion of the more efficient integration method for the ideal frame formulation is not approached here, where both differential systems have been integrated numerically with the reliable and widespread DOP853 free code described in (Hairer et al., 2008). Following tradition, the tests were based on the numerical examples in (Stiefel and Scheifele, 1971 , p. 118 and ff.). Namely, the forces model considers the noncentralities of the Geopotential limited to the contribution of the second zonal harmonic, as well as the moon attraction in the simplifying assumption that the moon moves in a circular orbit about the earth. ...An implementation of the ideal frame formulation of perturbed Keplerian motion is presented which only requires the integration of a differential system of dimension 7, contrary to the 8 variables traditionally integrated with this approach. The new formulation is based on the integration of a scaled version of the Eulerian set of redundant parameters, and slightly improves runtime performance with respect to the 8 dimensional case while retaining comparable accuracy.
- ... En este caso utilizaremos las soluciones dadas por Stiefel y Scheifele [4], en su problema dédé orbita inclinada de alta excentricidad (e = 0.95) tras 50 revoluciones con perturbaciones lunares y debidas al achatamiento terrestre. Las condiciones iniciales de dicho problema en coordenadas geocéntricas inerciales son las siguientes: ...El propósito de este proyecto es determinar la eficacia de los métodos de perturbaciones especiales, principalmente del método de Encke. El objetivo es determinar la viabilidad de dichos métodos con órbitas hiperbólicas, de alta excentricidad, y de cometas y satélites reales. Se utilizan rutinas en lenguaje C para la propagación de órbitas perturbadas, con motores de integración Runge-Kutta de alto orden. Se demuestra que el método de Encke es muy eficiente rectificando de forma automática, y que es viable para el cálculo de órbitas de cometas y de satélites artificiales.
- ... Both Levi-Civita and KS regularizations are local transformations in the sense that their application allows one to regularize collisions with only one of the two primaries. A suitable extension of such transformations allows for obtaining a simultaneous regularization with both primaries, thus obtaining a global V.K. Srivastava et al. transformation known as Birkhoff's method (Stiefel and Scheifele 1971;Szebehely 1967). ...In this paper, a time and space-coordinate transformation, commonly known as Kustaanheimo-Stiefel (KS)-transformation is applied to reduce the order of singularities arising due to the motion of infinitesimal body in the vicinity of smaller primary in the three body system. In this work, the Sun-Earth system is considered assuming the Sun as a radiating body and the Earth as an oblate spheroid. The study covers motion around collinear Lagrangian L1 and L2 points. Numerical computations are performed for both regularized and non-regularized equations of motion and results are compared for non-periodic as well as periodic motion. In the transformed space, time is also computed as a function of solar radiation pressure (q ) and oblateness of the Earth (A2 ). The two parameters (q, A2) have a significant impact on time in the transformed space. It is found that KS-regularization reduces the order of pole from five to three at the point of singularity of the governing equations of motion.
- ... The classical Newtonian equations, being unstable, are not very ideal choice for numerical integration as well as for development of analytical theories. Regularization of the classical Newtonian equations by Kustaanheimo and Stiefel[11] and presented in details in Stiefel and Scheifele [15], drew significant attention when J2 perturbed Lambert-type solutions in terms of KS (Kustaanheimo-Stiefel) transformation was derived in closed-form by Andrus [2] and Engels & Junkins [6]. Sharma [13] obtained a singularity-free analytical solution for short-term orbit computation in terms KS elements with J2, in series expanded up to fourth power in eccentricity. ...Sharma’s singularity-free analytical theory for the short-term orbital motion of satellites in terms of KS elements in closed form in eccentricity with Earth’s zonal harmonic term J2, is improved by using King-Hele’s expression for the radial distance ‘r’ which includes the effect of J2, and is suitable for low eccentricity orbits. Numerical experimentation with four test cases with perigee altitude of 200 km and eccentricity varying from 0.01 to 0.3 for different inclinations is carried out. It is found that the orbital elements computed with the analytical expressions in a single step during half a revolution match very well with numerically integrated values and show significant improvement over the earlier theory. The solution can be effectively used for computation of mean elements for near-Earth orbits, where the short-term orbit perturbations due to J2 play most important role. The theory will be very useful in computing the state vectors during the coast phase of rocket trajectories and flight algorithms for on-board implementation.
- A generalized Hamiltonian formalism is established which is invariant not only under canonical transformations but under arbitrary transformations. Moreover the dependent variables, coordinates and momenta, as well as the independent variable are allowed to be transformed. This is to say that instead of the physical time t another independent variable s is used, such that t becomes a dependent variable or, more precisely, an additional coordinate. The formalism under consideration permits also to include nonconservative forces. In case of Keplerian motion we propose to use the eccentric anomaly as the independent variable. By virtue of our generalized point of view a Lyapunov-stable differential system is obtained, such that all coordinates, including the time t, are computed by stable procedures. This stabilization is performed by control terms. As a new result a stabilizing control term also for the time integration is established, such that no longer any kind of time element is needed. This holds true for the usual coordinates as well as for the KS-coordinates.
- This paper deals with remarks on the linearization methods of nonlinear mathematical models of the phenomena. We can classify the linear methods into two distinct types, the «exact methods» and the «approximate methods». The advantages and disadvantages of these methods are clarified by appropriately selected examples.
- The Störmer–Verlet formula is a popular numerical integration method which has played an important role in the numerical simulation of differential equations.
- In last few years, many ERKN methods have been investigated for solving multi-frequency multidimensional second-order ordinary differential equations, and the numerical efficiency has been checked strongly in scientific computation. But in the constructions of (especially high-order) new ERKN methods, lots of time and effort are costed in presenting the practical order conditions firstly and then in adding some reasonable assumptions to get the coefficient functions finally. In this paper, a feasible and effective technique is given which makes the construction of ERKN methods finished in a few seconds or a few minutes, even for high-order integrators. Moreover, this technique does not need any more information and knowledge except the classical RKN method. And this paper also gives the theoretical explanation to guarantee that the ERKN method obtained from this technique has the same order and the same properties as the underlying RKN method.
- The Physics of spinor Lorentz group significantly differs from the one based on the orthogonal Lorentz group L+-↑↓, and only experiments may decide on this problem. In this context, the study of the fermion parity problem based on investigating possible single-valued representations of spinor coverings of the extended Lorentz group shows that P-parity and T-parity do not exist as separate concepts. Instead of this, only some unified concept of (PT)-parity can be determined in group-theoretical terms. The extension procedure which describes a space with spinor structure is performed by relying on cylindrical parabolic coordinates. This is done by expanding the region G(t, u, v, z) → G(t, u, v, z), so that instead of the half-plane (u, v > 0) the entire plane (u, v) is used, while considering new identification rules for the boundary points. In the Cartesian picture, this procedure corresponds to taking the two-sheet surface (x', y') ⊕ (x', y') instead of the one-sheet surface (x, y). The solutions of the Klein-Fock-Gordon equation classify into four types: Ψ++, Ψ--, Ψ+-, Ψ-+. The first two ones, Ψ++ and Ψ--, provide single-valued functions of the vector space points, whereas the last two, Ψ+-, Ψ-+, have discontinuities in the frame of vector spaces, and therefore they are be discarded in this model. However, all the four types of functions are continuous ones, while regarded in the spinor space. It is established that all the solutions Ψ++, Ψ--, Ψ+- and Ψ-+, are orthogonal to each other, provided that integration is done over the extended region of integration which covers the spinor space. Similar results are obtained for the Dirac equation. The solutions of the type (--), (++) are single-valued in the space with vector structure, whereas the solutions of the types (-+), (+-) are not single-valued in the space with vector structure, so the solutions of types (-+) and (+-) must be discarded. However, they are valid solutions in the space with spinor structure.
- A new singularity-free analytical theory for orbit propagation of Earth satellites with the effect of lunar perturbation in terms of KS regular elements is developed. The elegance of the theory is the existence of the symmetry between the equations of motion which allows us to solve only two out of nine equations to compute state vector and time. The analytical solution, when compared with numerically integrated values, exhibited high level of accuracy. In comparison with classical series expansion method and method of averaging in terms of Lagrange planetary equations, the present solution with KS regular element equations provides better accuracy. This theory can be useful for both short-term and long-term orbit propagation.
- We have seen the complexity of the problem when more than two gravitating masses are involved. We have seen two methods of determining the orbits, Cowell’s and Encke’s methods . Now, let us look at the basic mathematical description of the perturbation problem.
- Two fully regular and universal solutions to the problem of spacecraft relative motion are derived from the Sperling–Burdet (SB) and the Kustaanheimo–Stiefel (KS) regularizations. There are no singularities in the resulting solutions, and their form is not affected by the type of reference orbit (circular, elliptic, parabolic, or hyperbolic). In addition, the solutions to the problem are given in compact tensorial expressions and directly referred to the initial state vector of the leader spacecraft. The SB and KS formulations introduce a fictitious time by means of the Sundman transformation. Because of using an alternative independent variable, the solutions are built based on the theory of asynchronous relative motion. This technique simplifies the required derivations. Closed-form expressions of the partial derivatives of orbital motion with respect to the initial state are provided explicitly. Numerical experiments show that the performance of a given representation of the dynamics depends strongly on the time transformation, whereas it is virtually independent from the choice of variables to parameterize orbital motion. In the circular and elliptic cases, the linear solutions coincide exactly with the results obtained with the Clohessy–Wiltshire and Yamanaka–Ankersen state-transition matrices. Examples of relative orbits about parabolic and hyperbolic reference orbits are also presented. Finally, the theory of asynchronous relative motion provides a simple mechanism to introduce nonlinearities in the solution, improving its accuracy.
- Consideration is given to the solution by numerical integration of systems of differential equations that are derived from a Hamiltonian function in the extended phase space plus additional forces not included in the Hamiltonian (that is, nearly-Hamiltonian systems). An extended phase space Hamiltonian which vanishes initially will vanish on any solution of the system differential equations. Furthermore, it vanishes in spite of the additional forces, and defines a surface in the extended phase space upon which the solution is constrained. Direct numerical comparisons are made between (1) nearly-Hamiltonian systems having vanishing Hamiltonians and (2) those having nonvanishing Hamiltonians. It is seen that for some problems, numerical solutions are more stable when computed from systems of the type (1). The problems considered are the harmonic oscillator with the van der Pol perturbation and perturbed Keplerian motion.
- The multi-impulse and multi-criteria optimal trajectories of the spacecraft transfer between orbits of various types are considered. A practical goal is the inspection or service of a number of space objects moving along these orbits at the initial data set. It is required to define a set of the admissible solutions taking into account priorities and possible restrictions on time or on expenses of energy, and to choose the best of possible routes for successively visiting every object in the desired group.
- A uniform treatment of the two-body problem leads to a differential time transformation to introduce the arc length along the orbit as the independent variable. The transformation is integrated in terms of the classical anomalies.
- Scheifele(1970) applied Delaunay-Sirmlar (DS) elliptic Keplerian elements (with the true anomaly as the independent variable) to the J 2 Problem in Artificial Satellite Theory, making an element of the true anomaly. (1981) views Scheifele’s TR-mapping as an extension of Hill’s transformation from a 6-dimensional phase space to an enlarged, 8-dimensional one. To adapt this approach to elliptic-type two-body problems with a time-varying Keplerian parameter μ(t), Fioría (1997, §3, §4) treated a Gylden system (Deprit 1983) and derived “Delaunay-Similar” variables via a TR-like transformation. Now we extend our treatment to perturbed Gylden systems, and modify the TR-map to deal with any kind of two-body orbit. We work out our generalization and the resulting variables within a unified pattern whatever the type of motion, in terms of universal functions (Stiefel & Scheifele 1971, §11; Battin 1987, §4.5, §4.6) and auxiliary angle-like parameters.
- To take advantage of the linear and regular formulation and treatment of Celestial Mechanics problems (Kustaanheimo & Stiefel 1965; Stiefel & Scheifele 1971; Deprit, Elipe & Ferrer 1994), Sharaf & Saad (1997) have given an analytical expansion of the Earth’s gravitational zonal potential in terms of Kustaanheimo-Stiefel (KS) regular elements (Stiefel & Scheifele 1971, §19), with special emphasis on its application to elliptic-type two-body orbits and, consequently, using a generalized (elliptic) eccentric anomaly as the independent variable. Motivated by these and other considerations based on the definition and use of KS elements, and following a treatment similar to that of Stiefel & Scheifele (1971, §19), we develop element equations corresponding to a DEF-formulation of the satellite problem under the effect of the zonal potential.
- Within the framework of linear and regular celestial mechanics, we revise a recent method of Belen’kii (1981). We generalize some of his results, giving a new regularizing function. We make an application to the zonal earth satellite, considering the hamiltonian function through the harmonic J4. After the angular variable u has been removed, we introduce a new time and we reduce the problem to a linear equation.
- The equations governing motion of the satellite under the effect of oblateness of Earth and atmospheric drag have been simulated, for a fixed initial position and three different initial velocities, till satellite collapses on Earth. Simulation of motion of artificial Earth satellite subject to the combined effects of oblate Earth and atmospheric drag is presented. The atmospheric model considered here takes in to account of exponential variation of the density with initial distance of Satellite from Earth's surface, scale height and radial distance. The minimum and maximum values of orbital elements and their variation over a time for different initial velocities have been reported.
- Quaternionic elements in orbital mechanics are usually related to the Kustaanheimo-Stiefel transformation or to the definition of the orbital plane. The new set of regular elements presented in this paper stems from the form of the equations of motion of a rotating solid, which model the evolution of a quaternion defining the orientation of a bodyfixed frame and the change in the angular velocity of such frame. By replacing the body-fixed frame with a special orbital frame and accounting for the radial motion separately, an equivalent solution to orbital motion can be constructed. The variation of parameters technique furnishes a new set of elements that is independent from the orbital plane. A second-order Sundman transformation introduces a fictitious time that replaces the physical time as the independent variable. This technique improves the numerical performance of the method and simplifies the derivation. The use of a time element yields an even smoother evolution of the orbital elements under perturbations. Once the Lagrange and Poisson brackets are obtained, the most general nonosculating version of the set of elements is presented. Regarding the performance, numerical experiments show that the method is comparable to other formulations involving similar stabilization and regularization techniques. © Copyright 2017 by Javier Roa. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
- This chapter presents a class of trigonometric collocation methods based on Lagrange basis polynomials for solving multi-frequency and multidimensional oscillatory systems \(q^{\prime \prime }(t)+Mq(t)=f\big (q(t)\big )\). The properties of the collocation methods are investigated in detail. It is shown that the convergence condition of these methods is independent of \(\left\| M\right\| \), which is crucial for solving multi-frequency oscillatory systems.
- Orbital mechanics refers to the study of the translational motion of bodies in mutual gravitational attraction, and the resulting spatial paths of the centers of mass of the respective bodies are called the orbits. The relative motion of two spherical bodies in mutual gravitational attraction is the fundamental problem of translational space dynamics, called the two-body problem (or the Keplerian motion), and possesses an analytical solution.
- Solar gravity propelled highly elliptical resident Earth orbits has been utilized to improve lunar mission performance. A regularized orbit propagator has been used to perform linear search for initial conditions that produce energy saving pre-trans lunar injection exo-atmospheric highly elliptical orbits. Additional propellant mass margin or smaller piggyback payloads to the Moon/high altitude orbits may be enabled by such transfers.
- About 5% of the catalogued resident space objects (RSO) around Earth are in highly elliptical orbits (HEO). The forecast of the distribution and lifetime of uncontrolled RSO is essential for carrying out stable and safe future space operations. With low perigee and high apogee, large uncontrolled RSO in HEO pass through high spatial density regions and are a potential source for increasing debris population. When RSO survive the re-entry disintegration process at the end of lifetime, the resulting debris fragments may endanger terrestrial life. Re-entry time estimation of decaying HEO objects is a key area of debris mitigation and remediation. The dynamics of the motion in HEO is governed by the influence of luni-solar gravity, atmospheric drag and oblateness of the Earth. The combination of atmospheric drag, varying with upper atmospheric density due to solar activity, and luni-solar gravity alter the perigee altitude leading to lifetime variation from several weeks to decades. The orbital evolution and lifetime estimation of HEO objects are sensitive to initial conditions which should be optimized for obtaining accurate re-entry time. Two initial parameters with significant uncertainties are the ballistic coefficient ({\it B}) and eccentricity ({\it e}). {\it B}, varies with RSO's attitude during atmospheric passage, depends on the drag coefficient and RSO’s area to mass ratio. {\it e} is uncertain since semi-major axis is easily computable from the orbital period. The usage of complex perturbation models in unstable nonlinear Newtonian equations of motion abets computational difficulties in obtaining accurate state vector. Using regularization method due to Kustaanheimo and Stiefel (KS), linear differential equations of a harmonic oscillator with constant frequency is obtained and extended to perturbed motion. A regularized numerical orbit propagator (KSROP), with constant step-size, in terms of KS regular elements is developed. The positions of the Sun and the Moon are computed using Plataforma Solar de Almería (PSA) algorithm and a series expansion algorithm, respectively. An oblate atmospheric model using density scale height varying with altitude for drag and zonal harmonic terms up to {\it J}$ _{6}$ for oblateness effects are considered. In the present study, re-entry prediction of HEO RSOs is treated as an optimal estimation problem. Two-line element sets (TLE) at different epochs of few RSOs are used with SGP4/SDP4 theory to obtain initial osculating orbital elements. Optimal estimates of {\it B} and {\it e} are found using response surface methodology (RSM) with genetic algorithm (GA). KSROP is used to propagate the optimal initial orbital parameters and is able to predict re-entry times with low relative error (<5%) when compared with actual data.
- An improved orbit theory in terms of Kustaanheimo-Stiefel regular elements with J2, J3, and J4 perturbations is developed. Singularity causing terms occurring in the perturbing potential are replaced with their trigonometric equivalents. The existence of symmetrical structure between the eight equations of motion expedites the computation speed to obtain physical time and state vector by solving only two of nine equations and updating three instead of eight elements during integration. Comparison with previous theory, numerical integration and DRI method revealed good accuracy, which makes the present analytical solution suitable for many applications including on-board implementation.
- In this paper, the initial value problem of space dynamics in universal Stumpff anomaly ψ is set up and developed in analytical and computational approach. For the analytical expansions, the linear independence of the functions Uj(ψ;ς); j=0,1,2,3 are proved. The differential and recurrence equations satisfied by them and their relations with the elementary functions are given. The universal Kepler equation and its validations for different conic orbits are established together with the Lagrangian coefficients. Efficient representations of these functions are developed in terms of the continued fractions. For the computational developments we consider the following items: Top-down algorithm for continued fraction evaluation.One-point iteration formulae.Determination of the coefficients of Kepler’s equation.Derivatives of Kepler’s equation of any integer order.Determination of the initial guess for the solution of the universal Kepler equation. Top-down algorithm for continued fraction evaluation. One-point iteration formulae. Determination of the coefficients of Kepler’s equation. Derivatives of Kepler’s equation of any integer order. Determination of the initial guess for the solution of the universal Kepler equation. Finally we give summary on the computational design for the initial value problem of space dynamics in universal Stumpff anomaly. This design based on the solution of the universal Kepler’s equation by an iterative schemes of quadratic up to any desired order ℓ.
- We consider a restricted three body problem on surfaces of constant curvature. As in the classical Newtonian case the collision singularities occur when the position particle with infinitesimal mass coincides with the position of one of the primaries. We prove that the singularities due to collision can be locally (each one separately) and globally (both as the same time) regularized through the construction of Levi-Civita and Birkhoff type transformations respectively. As an application we study some general properties of the Hill’s regions and we present some ejection–collision orbits for the symmetrical problem.
- Equations of motion for the perturbed circular restricted three-body problem have been regularized in canonical variables in a moving coordinate system. Two different L-matrices of the fourth order are used in the regularization. Conditions for generalized symplecticity of the constructed transform have been checked. In the unperturbed case, the regular equations have a polynomial structure. The regular equations have been numerically integrated using the Runge–Kutta–Fehlberg method. The results of numerical experiments are given for the Earth–Moon system parameters taking into account the perturbation of the Sun for different L-matrices.
- Quotient spaces of Keplerian orbits are important instruments for the modelling of orbit samples of celestial bodies on a large time span. We suppose that variations of the orbital eccentricities, inclinations and semi-major axes remain sufficiently small, while arbitrary perturbations are allowed for the arguments of pericentres or longitudes of the nodes, or both. The distance between orbits or their images in quotient spaces serves as a numerical criterion for such problems of Celestial Mechanics as search for common origin of meteoroid streams, comets, and asteroids, asteroid families identification, and others. In this paper, we consider quotient sets of the non-rectilinear Keplerian orbits space \(\mathbb H\). Their elements are identified irrespective of the values of pericentre arguments or node longitudes. We prove that distance functions on the quotient sets, introduced in Kholshevnikov et al. (Mon Not R Astron Soc 462:2275–2283, 2016), satisfy metric space axioms and discuss theoretical and practical importance of this result. Isometric embeddings of the quotient spaces into \(\mathbb R^n\), and a space of compact subsets of \(\mathbb H\) with Hausdorff metric are constructed. The Euclidean representations of the orbits spaces find its applications in a problem of orbit averaging and computational algorithms specific to Euclidean space. We also explore completions of \(\mathbb H\) and its quotient spaces with respect to corresponding metrics and establish a relation between elements of the extended spaces and rectilinear trajectories. Distance between an orbit and subsets of elliptic and hyperbolic orbits is calculated. This quantity provides an upper bound for the metric value in a problem of close orbits identification. Finally the invariance of the equivalence relations in \(\mathbb H\) under coordinates change is discussed.
- It will be convenient to use the term Kepler orbit for any curve x = x(t) in 3-space which arises as a solution to the Newtonian two body problem. Hamilton showed that the velocity vector v = dx/dt, associated with any nondegenerate Kepler orbit, moves along a circle. Following Györgyi, Moser, Osipov and Belbruno, this velocity circle can be interpreted as follows. If we fix the total energy E, then the manifold ME consisting of all vectors v with v · v > 2 E possesses a natural Riemannian metric of constant curvature — 2 E, whose geodesies are precisely the circles associated in this way with Kepler orbits. In other words, ME can be identified with one of the three classical geometries, that is with spherical, Euclidean or Lobachevsky space, so that each “straight line” in this geometry corresponds to a unique Kepler orbit.
- The regularizing techniques known as Kustaanheimo-Stiefel (KS) transformation have investigated. It has proved that it is very useful in n-body simulations, where it helps to handle close encounters. This paper shows how the basic transformation is a starting point for a family of polynomial coupled function. This interpretation becomes simply on writing KS transformations in quaternions form, which also helps to derive concise expressions for regularized equations of motion. Even if the KS reg-ularization method is more easy to use, it is interesting to encapsulate the KS transformation in a family of methods, which all conserve the KS transformations' properties. Further, an interesting point of view is considering, the orbital shapes of the restricted three-body problem (also regularized restricted three-body problem) for different initial conditions has compared with flower pattern.
- The $\mathcal{KS}$ map is revisited in terms of an $S^1$-action in $T^*\mathbb{H}_0$ with the bilinear function as the associated momentum map. Indeed, the $\mathcal{KS}$ transformation maps the $S^1$-fibers related to the mentioned action to single points. By means of this perspective a second twin-bilinear function is obtained with an analogous $S^1$-action. We also show that the connection between the 4-D isotropic harmonic oscillator and the spatial Kepler systems can be done in a straightforward way after regularization and through the extension to 4 degrees of freedom of the Euler angles, when the bilinear relation is imposed. This connection incorporates both bilinear functions among the variables. We will show that an alternative regularization separates the oscillator expressed in Projective Euler variables. This setting takes advantage of the two bilinear functions and another integral of the system including them among a new set of variables that allows to connect the 4-D isotropic harmonic oscillator and the planar Kepler system. In addition, our approach makes transparent that only when we refer to rectilinear solutions, both bilinear relations defining the $\mathcal{KS}$ transformations are needed.
- There is numerical evidence to support the conclusion that appropriate "matching" of the formulation of the equations of motion to the numerical integration method can lead to significant improvement in the accuracy and computational efficiency of the orbit generation process. This paper investigates this possibility from the point of view of matching a "Gaussian" variation of parameter (VOP) formulation with Adams type numerical integrators. The performance of the new orbit generators is then compared to the popular classical Cowell/Gauss-Jackson formulation/integrator pair. Numerical results indicate that the VOP orbit generator can yield significant efficiency advantages. For example, after 28 days of integration of a synchronous orbit, a comparable accuracy (.5m) was attained with a VOP formulation at one-half the number of force model evaluations needed using the Cowell formulation. It is suspected that the reasons one can do better with the VOP formulation include more favorable truncation error/stability region balancing. For example, it has been reported that lower order integration processes coupled with VOP are "better than" higher order processes. This supports the view that the VOP formulation results in a smaller stability region with the advantage of lower truncation error due to the more slowly varying parameters. An analytical approach to the study of this type of matching of the equation formulation to the numerical integration technique is given.
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