On solving Kepler's equation for nearly parabolic orbits

We deal here with the efficient starting points for Kepler's equation in the special case of nearly parabolic orbits. Our approach provides with very simple formulas that allow calculating these points on a scientific vest-pocket calculator. Moreover, srtarting with these points in the Newton's method we can calculate a root of Kepler's equation with an accuracy greater than 0.001 in 0–2 iterations. This accuracy holds for the true anomaly || 135° and |e – 1| 0.01. We explain the reason for this effect also.Dedicated to the memory of Professor G.N. Duboshin (1903–1986).     

A Fast Procedure Solving Gauss' Form of Kepler's Equation

We developed a procedure solving Gauss' form of Kepler's equation, which is suitable for determining position in the nearly parabolic orbits. The procedure is based on the combination of asymptotic solutions, the method of bisection, and the Newton method of succesive correction. It runs 3–4 times faster than the original Gauss' method. This revised version was published online in July 2006 with corrections to the Cover Date.     

Closest Approach in Universal Variables

In this paper, universal formulations of the closest approach problem are established and solved by two methods. The first method uses the technique of one-dimensional unconstraint minimization and needs the solution of the universal Kepler's equation twice, while for the second method, a constraint minimization technique is developed and needs the solution of two nonlinear simultaneous equations. Flexible iterative schemes of quadratic up to any positive integer order are developed for the solution of the universal Kepler's equation. The two methods of the minimization process are applied for the closest approach of Hyakutake and Hale–Bopp comets, while the first method is applied to obtain the minimum angular separation of ADS 9159, ADS 2959 and ADS 11632 visual binaries as typical examples of elliptic, parabolic and hyperbolic orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

'On Solving Kepler's Equation For Nearly Rectilinear Hyperbolic Orbits'

We deal here with efficient starting points for Kepler's equation in a special case of nearly rectilinear hyperbolic orbits, that is these ones with the eccentricities e1. These orbits appear in stellar dynamics when considering encounters of stars. We test efficiency of these starters for the method for successive approximation (MSA) in its two often applied variants, that is the Newton's method with the quadratic convergence (NM) and in the fixed point method (FPM). Moreover, we determine a dynamical domain of Kepler's equation for this motion.This revised version was published online in October 2005 with corrections to the Cover Date.     

The solution of Kepler's equation,II

Starting values for the iterative solution of Kepler's equation are considered for hyperbolic orbits, and for generalized versions of the equation, including the use of universal variables.     

Area-preserving mappings and deterministic chaos for nearly parabolic motions

The present work investigates a mechanism of capturing processes in the restricted three-body problem. The work has been done in a set of variables which is close to Delaunay's elements but which allows for the transition from elliptic to hyperbolic orbits. The small denominator difficulty in the perturbation theory is overcome by embedding the small denominator in an analytic function through a suitable analytic continuation. The results indicate that motions in nearly parabolic orbits can become chaotic even though the model is deterministic. The theoretical results are compared with numerical results, showing an agreement of about one percent. Some possible applications to cometary orbits are also given.     

An exact analytical solution of Kepler's equation

Complex-variable analysis is used to develop an exact solution to Kepler's equation, for both elliptic and hyperbolic orbits. The method is based on basic properties of canonical solutions to appropriately posed Riemann problems, and the final results are expressed in terms of elementary quadratures.     

Simple integrals for solving of Kepler's equation

In this paper we derive integral representations for the solution of Kepler's equations for elliptic and hyperbolic orbits. The integrands consist merely of rational expressions of the integration variable and its exponential.     

An improved algorithm due to laguerre for the solution of Kepler's equation

A root-finding method due to Laguerre (1834–1886) is applied to the solution of the Kepler problem. The speed of convergence of this method is compared with that of Newton's method and several higher-order Newton methods for the problem formulated in both conventional and universal variables and for both elliptic and hyperbolic orbits. In many thousands of trials the Laguerre method never failed to converge to the correct solution, even from exceptionally poor starting approximations. The non-local robustness and speed of convergence of the Laguerre method should make it the preferred method for the solution of Kepler's equation.     

Motion close to the Hopf Bifurcation Of the vertical family of periodic orbits of L4

The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L₄ in the 3D restricted three-body problem (RTBP) for the mass parameter, μ greater than (and close to) μ_R (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic orbits and the 3D nearby tori are also described.     

A general timing condition for consecutive collision orbits in the limiting case µ = 0 of the elliptic restricted problem

Consecutive collision orbits in the limiting case µ = 0 of the elliptic restricted three-body problem are investigated. in particular those in which the infinitesimal mass collides twice with the smaller (massless) primary. A timing condition is presented that allows the extension of previous results to the case of arbitrary relative orientation of the orbits of the infinitesimal mass and the smaller primary. The timing condition is expressed in two general forms - in terms of orbit parameters and eccentric (or hyperbolic) anomalies at the times of collision - for the specific cases of elliptic. parabolic or hyperbolic orbits of the infinitesimal mass. Some families of solutions are presented.     

Unified Symbolic Algorithm of Gauss Method for Near-Parabolic Orbits

In this paper, a unified algorithm of Gauss method for near‐parabolic orbits that is valid for both elliptic and hyperbolic cases is established symbolically and numerically. This revised version was published online in July 2006 with corrections to the Cover Date.     

Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem

We analytically prove the existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. This is an extension of our previous proof of the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar fully symmetric equal mass four-body problem. We then use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar pairwise symmetric 1, m, 1, m four-body problem for m between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability when 0.54 ≤ m ≤ 1, and are linearly unstable when 0 < m ≤ 0.53.     

The hyperbolic Kepler equation (and the elliptic equation revisited)

A procedure is developed that, in two iterations, solves the hyperbolic Kepler's equation in a very efficient manner, and to an accuracy that proves to be always better than 10^–20 (relative truncation error). Earlier work on the elliptic equation has been extended by the development of a new procedure that solves to a maximum relative error of 10^–14.     

Symmetries and Bifurcations in the Sitnikov Problem

We present some qualitative and numerical results of the Sitnikov problem, a special case of the three-body problem, which offers a great variety of motions as the non-integrable systems typically do. We study the symmetries of the problem and we use them as well as the stroboscopic Poincarée map (at the pericenter of the primaries) to calculate the symmetry lines and their dynamics when the parameter changes, obtaining information about the families of periodic orbits and their bifurcations in four revolutions of the primaries. We introduce the semimap to obtain the fundamental lines l ₁. The origin produces new families of periodic orbits, and we show the bifurcation diagrams in a wide interval of the eccentricity (0 0.97). A pattern of bifurcations was found.This revised version was published online in October 2005 with corrections to the Cover Date.     

The manifolds of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem

This paper deals with the Sitnikov family of straight-line motions of the circular restricted three-body problem, viewed as generator of families of three-dimensional periodic orbits. We study the linear stability of the family, determine several new critical orbits at which families of three dimensional periodic orbits of the same or double period bifurcate and present an extensive numerical exploration of the bifurcating families. In the case of the same period bifurcations, 44 families are determined. All these families are computed for equal as well as for nearly equal primaries (μ = 0.5, μ = 0.4995). Some of the bifurcating families are determined for all values of the mass parameter μ for which they exist. Examples of families of three dimensional periodic orbits bifurcating from the Sitnikov family at double period bifurcations are also given. These are the only families of three-dimensional periodic orbits presented in the paper which do not terminate with coplanar orbits and some of them contain stable parts. By contrast, all families bifurcating at single-period bifurcations consist entirely of unstable orbits and terminate with coplanar orbits.     

Chaos in Relativity and Cosmology

Chaos appears in various problems of Relativity and Cosmology. Here we discuss (a) the Mixmaster Universe model, and (b) the motions around two fixed black holes. (a) The Mixmaster equations have a general solution (i.e. a solution depending on 6 arbitrary constants) of Painlevé type, but there is a second general solution which is not Painlevé. Thus the system does not pass the Painlevé test, and cannot be integrable. The Mixmaster model is not ergodic and does not have any periodic orbits. This is due to the fact that the sum of the three variables of the system (α + β + γ) has only one maximum for τ = τ_m and decreases continuously for larger and for smaller τ. The various Kasner periods increase exponentially for large τ. Thus the Lyapunov Characteristic Number (LCN) is zero. The "finite time LCN" is positive for finite τ and tends to zero when τ → ∞. Chaos is introduced mainly near the maximum of (α + β + γ). No appreciable chaos is introduced at the successive Kasner periods, or eras. We conclude that in the Belinskii-Khalatnikov time, τ, the Mixmaster model has the basic characteristics of a chaotic scattering problem. (b) In the case of two fixed black holes M₁ and M₂ the orbits of photons are separated into three types: orbits falling into M₁ (type I), or M₂ (type II), or escaping to infinity (type III). Chaos appears because between any two orbits of different types there are orbits of the third type. This is a typical chaotic scattering problem. The various types of orbits are separated by orbits asymptotic to 3 simple unstable orbits. In the case of particles of nonzero rest mass we have intervals where some periodic orbits are stable. Near such orbits we have order. The transition from order to chaos is made through an infinite sequence of period doubling bifurcations. The bifurcation ratio is the same as in classical conservative systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Meteors in the IAU Meteor Data Center on Hyperbolic Orbits

The hyperbolic meteor orbits among the 4,581 photographic and 62,906 radar meteors of the IAU MDC have been analysed using statistical methods. It was shown that the vast majority of hyperbolic orbits has been caused by the dispersion of determined velocities. The large proportion of hyperbolic orbits among the known meteor showers strongly suggests the hyperbolicity of the meteors is not real. The number of apparent hyperbolic orbits increases inversely proportional to the difference between the mean heliocentric velocity of meteor shower and the parabolic velocity limit. The number of hyperbolic meteors in the investigated catalogues does not, in any case, represent the number of interstellar meteors in observational data. The apparent hyperbolicity of these orbits is caused by a high spread in velocity determination, shifting a part of the data through the parabolic limit.     

Improvement of the position of planet X based on the motion of nearly parabolic comets

Based on the motion of nearly parabolic comets, we have improved the position of planet X in its orbit obtained by Batygin and Brown (2016). By assuming that some of the comets discovered to date could have close encounters with this planet, we have determined the comets with a small minimum orbit intersection distance with the planet. Five comets having hyperbolic orbits before their entry into the inner Solar system have been separated out from the general list. By assuming that at least one of them had a close encounter with the planet, we have determined the planet’s possible position. The planet’s probable ephemeris positions at the present epoch have been obtained by assuming the planet to have prograde and retrograde motions. In the case of a prograde motion, the planet is currently at a distance Δ whose value belongs to the interval Δ ∈ (1110, 1120) AU and has a right ascension α and declination δ within the intervals α ∈ (83?, 90?) and δ ∈ (8?, 10?); the true anomaly υ belongs to the interval υ ∈ (176?, 184?). In the case of a retrograde motion: α ∈ (48?, 58?), δ ∈ (?12?, ?6?), Δ ∈ (790, 910) AU, and υ ∈ (212?, 223?). It should be noted that in the case of a retrograde motion of the planet, its ephemeris position obtained from the motion of comets agrees with the planet’s position obtained byHolman and Payne (2016) from highly accurate Cassini observations and is consistent with the results of Fienga et al. (2016).     

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.