Solving Kepler's equation CORDIC-like

  title={Solving Kepler's equation CORDIC-like},
  author={Mathias Zechmeister},
  journal={arXiv: Instrumentation and Methods for Astrophysics},
  • M. Zechmeister
  • Published 21 August 2018
  • Physics, Computer Science
  • arXiv: Instrumentation and Methods for Astrophysics
Context. Many algorithms to solve Kepler's equations require the evaluation of trigonometric or root functions. Aims. We present an algorithm to compute the eccentric anomaly and even its cosine and sine terms without usage of other transcendental functions at run-time. With slight modifications it is applicable for the hyperbolic case, too. Methods. Based on the idea of CORDIC, it requires only additions and multiplications and a short table. The table is independent of eccentricity and can… 

Figures from this paper

Symbolic iteration method based on computer algebra analysis for Kepler’s equation
The results show that the accuracy is almost one order of magnitude higher than that of those methods (double precision) and the simple codes make the method well-suited for a wide range of algebraic programming languages and computer hardware (GPU and so on).
Fast Switch and Spline Function Inversion Algorithm with Multistep Optimization and k-Vector Search for Solving Kepler’s Equation in Celestial Mechanics
Obtaining the inverse of a nonlinear monotonic function f(x) over a given interval is a common problem in pure and applied mathematics, the most famous example being Kepler’s description of orbital
The Mass of the White Dwarf Companion in the Self-lensing Binary KOI-3278: Einstein versus Newton
KOI-3278 is a self-lensing stellar binary consisting of a white dwarf secondary orbiting a Sun-like primary star. Kruse & Agol noticed small periodic brightenings every 88.18 days in the
The HARPS search for southern extra-solar planets – XLV. Two Neptune mass planets orbiting HD 13808: a study of stellar activity modelling’s impact on planet detection
We present a comprehensive analysis of 10 yr of HARPS radial velocities (RVs) of the K2V dwarf star HD 13808, which has previously been reported to host two unconfirmed planet candidates. We use
Bivariate Infinite Series Solution of Kepler’s Equations
A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This
Comment on ‘An efficient code to solve the Kepler equation: elliptic case’
In a recent MNRAS article, Raposo-Pulido and Pelaez (RPP) designed a scheme for obtaining very close seeds for solving the elliptic Kepler equation with the classical and modified Newton–Raphson
True Masses of the Long-period Companions to HD 92987 and HD 221420 from Hipparcos–Gaia Astrometry
The extensive time span of modern radial velocity surveys has made the discovery of long-period substellar companions more common in recent years; however, measuring the true masses of these objects
Solving Kepler’s equation with CORDIC double iterations
The new shift-and-add algorithm brings Kepler's equation close to hardware and allows to solve it with cheap and simple hardware components.


A method solving kepler's equation without transcendental function evaluations
We developed two approximations of the Newton-Raphson method. The one is a sort of discretization, namely to search an approximate solution on pre-specified grid points. The other is a Taylor series
Kepler Equation solver
Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation, and requires only four transcendental function evaluations.
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.
Dynamic discretization method for solving Kepler’s equation
This paper defines Kepler’s equation for the elliptical case and describes existing solution methods, and presents the dynamic discretization method and shows the results of a comparative analysis, demonstrating that, for the conditions of the tests, dynamicDiscretization performs the best.
An efficient mathematically correct scale free CORDIC
An overview of the CORDIC algorithm for the computation of the circular functions is presented, essentially the scaling free version, and a substential improvement to the commonly used one is given.
The CORDIC Trigonometric Computing Technique
The trigonometric algorithms used in this computer and the instrumentation of these algorithms are discussed in this paper.
Symplectic maps for the N-body problem.
The present study generalizes the mapping method of Wisdom (1982) to encompass all gravitational n-body problems with a dominant central mass. The rationale for the generalized mapping method is
Celestial Mechanics and Dynamical Astronomy Equations for the orbital elements : Hidden symmetry
We revisit the Lagrange and Delaunay systems of equations for the orbital elements, and point out a previously neglected aspect of these equations: in both cases the orbit resides on a certain