An efficient code to solve the Kepler equation

Context. This paper introduces a new approach for solving the Kepler equation for hyperbolic orbits. We provide here the Hyperbolic Kepler Equation–Space Dynamics Group (HKE–SDG), a code to solve the equation. Methods. Instead of looking for new algorithms, in this paper we have tried to substantially improve well-known classic schemes based on the excellent properties of the Newton–Raphson iterative methods. The key point is the seed from which the iteration of the Newton–Raphson methods begin. If this initial seed is close to the solution sought, the Newton–Raphson methods exhibit an excellent behavior. For each one of the resulting intervals of the discretized domain of the hyperbolic anomaly a fifth degree interpolating polynomial is introduced, with the exception of the last one where an asymptotic expansion is defined. This way the accuracy of initial seed is optimized. The polynomials have six coefficients which are obtained by imposing six conditions at both ends of the corresponding interval: the polynomial and the real function to be approximated have equal values at each of the two ends of the interval and identical relations are imposed for the two first derivatives. A different approach is used in the singular corner of the Kepler equation – |M| < 0.15 and 1 < e < 1.25 – where an asymptotic expansion is developed. Results. In all simulations carried out to check the algorithm, the seed generated leads to reach machine error accuracy with a maximum of three iterations (∼99.8% of cases with one or two iterations) when using different Newton–Raphson methods in double and quadruple precision. The final algorithm is very reliable and slightly faster in double precision (∼0.3 s). The numerical results confirm the use of only one asymptotic expansion in the whole domain of the singular corner as well as the reliability and stability of the HKE–SDG. In double and quadruple precision it provides the most precise solution compared with other methods.

The Homotopy Perturbation Method for Accurate Orbits of the Planets in the Solar System: The Elliptical Kepler Equation

Accurate trajectories for the orbits of the planets in our solar system depends on obtaining an accurate solution for the elliptical Kepler equation. This equation is solved in this article using the homotopy perturbation method. Several properties of the periodicity of the obtained approximate solutions are introduced through some lemmas. Numerically, our calculations demonstrated the applicability of the obtained approximate solutions for all the planets in the solar system and also in the whole domain of eccentricity and mean anomaly. In the whole domain of the mean anomaly, 0≤M≤2π, and by using the different approximate solutions, the residuals were less than 4×10−17 for e∈[0, 0.06], 4×10−9 for e∈[0.06, 0.25], 3×10−8 for e∈[0.25, 0.40], 3×10−7 for e∈[0.40, 0.50], and 10−6 for e∈[0.50, 1.0]. Also, the approximate solutions were compared with the Bessel–Fourier series solution in the literature. In addition, the approximate homotopy solutions for the eccentric anomaly are used to show the convergence and periodicity of the approximate radial distances of Mercury and Pluto for three and five periods, respectively, as confirmation for some given lemmas. It has also been shown that the present analysis can be successfully applied to the orbit of Halley’s comet with a significant eccentricity.

Jacobi–Maupertuis metric and Kepler equation

This paper studies the application of the Jacobi–Eisenhart lift, Jacobi metric and Maupertuis transformation to the Kepler system. We start by reviewing fundamentals and the Jacobi metric. Then we study various ways to apply the lift to Kepler-related systems: first as conformal description and Bohlin transformation of Hooke’s oscillator, second in contact geometry and third in Houri’s transformation [T. Houri, Liouville integrability of Hamiltonian systems and spacetime symmetry (2016), www.geocities.jp/football_physician/publication.html ], coupled with Milnor’s construction [J. Milnor, On the geometry of the Kepler problem, Am. Math. Mon. 90 (1983) 353–365] with eccentric anomaly.