# An efficient code to solve the Kepler equation. Elliptic case

@article{RaposoPulido2017AnEC, title={An efficient code to solve the Kepler equation. Elliptic case}, author={Virginia Raposo-Pulido and Jes{\'u}s Pel{\'a}ez}, journal={Monthly Notices of the Royal Astronomical Society}, year={2017}, volume={467}, pages={1702-1713} }

A new approach for solving Kepler equation for elliptical orbits is developed in this paper. This new approach takes advantage of the very good behaviour of the modified Newton?Raphson method when the initial seed is close to the looked for solution. To determine a good initial seed the eccentric anomaly domain [0, ?] is discretized in several intervals and for each one of these intervals a fifth degree interpolating polynomial is introduced. The six coefficients of the polynomial are obtained…

## Figures and Tables from this paper

## 10 Citations

An efficient code to solve the Kepler equation

- Mathematics, PhysicsAstronomy & Astrophysics
- 2018

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…

An Efficient Iterative Method for Solving the Elliptical Kepler’s Equation

- Physics, Mathematics
- 2021

In this paper, a numerical technique, based on Banach’s fixed point theorem, is proposed to obtain an approximate solution of the elliptical Kepler equation which is often used to describe the motion…

Solving Kepler's equation via nonlinear sequence transformations

- Mathematics
- 2021

Since more than three centuries Kepler’s equation continues to represents an important benchmark for testing new computational techniques. In the present paper, the classical Kapteyn series solution…

The solution of the generalized Kepler's equation

- Physics
- 2018

In the context of general perturbation theories, the main problem of the artificial satellite analyses the motion of an orbiter around an Earth-like planet, only perturbed by its equatorial bulge or…

Fast Switch and Spline Function Inversion Algorithm with Multistep Optimization and k-Vector Search for Solving Kepler’s Equation in Celestial Mechanics

- Physics
- 2020

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…

orvara: An Efficient Code to Fit Orbits Using Radial Velocity, Absolute, and/or Relative Astrometry

- Physics, GeologyThe Astronomical Journal
- 2021

We present an open-source Python package, Orbits from Radial Velocity, Absolute, and/or Relative Astrometry (orvara), to fit Keplerian orbits to any combination of radial velocity, relative…

Dynamical Mass of the Young Substellar Companion HD 984 B

- Physics, GeologyThe Astronomical Journal
- 2022

Model-independent masses of substellar companions are critical tools to validate models of planet and brown dwarf cooling, test their input physics, and determine the formation and evolution of these…

Comment on ‘An efficient code to solve the Kepler equation: elliptic case’

- PhysicsMonthly Notices of the Royal Astronomical Society
- 2021

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…

'exoplanet': Gradient-based probabilistic inference for exoplanet data & other astronomical time series

- PhysicsJ. Open Source Softw.
- 2021

This work presents a meta-anatomy ofoplanets and Habitable Worlds, a probabilistic model that ranks Planet Earths according to their ability to accommodate greenhouse gases, and its implications for climate change.

Bivariate Infinite Series Solution of Kepler’s Equations

- MathematicsMathematics
- 2021

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…

## References

SHOWING 1-10 OF 14 REFERENCES

A method solving kepler's equation without transcendental function evaluations

- Physics
- 1996

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…

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

- Physics
- 1986

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…

An analytical solution for Kepler's problem

- Physics
- 2009

In this paper, we present a framework which provides an analytical (i.e. infinitely differentiable) transformation between spatial coordinates and orbital elements for the solution of the…

Kepler Equation solver

- Computer Science
- 1995

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.

The hyperbolic Kepler equation (and the elliptic equation revisited)

- Physics, Mathematics
- 1988

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…

Long-term changes in the semimajor axes of the outer planets

- Physics, GeologyNature
- 1986

One of the oldest problems of celestial mechanics is that of the long-term behaviour of the semimajor axes a of the planetary orbits. Analytical theories1,2 predict periodic variations in a, some of…

Solving Kepler's equation with high efficiency and accuracy

- Physics
- 1991

We present a method for solving Kepler's equation for elliptical orbits that represents a gain in efficiency and accuracy compared with those currently in use. The gain is obtained through a starter…

Chaotic behaviour in the newton iterative function associated with kepler's equation

- Physics
- 1999

The chaotic behaviour observed when Newton's method is used to solve Kepler's equation is analysed using methods borrowed from chaos theory. The result of the analysis is compared with previous…

An introduction to the mathematics and methods of astrodynamics

- Physics, Mathematics
- 1987

Part 1 Hypergeometric Functions and Elliptic Integrals: Some Basic Topics In Analytical Dynamics The Problem of Two Bodies Two-Body Orbits and the Initial-Value Problem Solving Kepler's Equation…

Ideal frames for perturbed Keplerian motions

- Physics
- 1976

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…