Solving Kepler’s equation via Smale’s α-theory

• Published in: Celestial mechanics and dynamical astronomy Vol. 119; no. 1; pp. 27 - 44
• Main Authors: Avendano, Martín, Martín-Molina, Verónica, Ortigas-Galindo, Jorge
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.05.2014
• Subjects: Aerospace Technology and Astronautics; Astrophysics and Astroparticles; Classical Mechanics; Dynamical Systems and Ergodic Theory; Geophysics/Geodesy; Original Article; Physics; Physics and Astronomy; Smale’s; Newton’s method; Optimal starter; Kepler’s equation; theory
• ISSN: 0923-2958; 1572-9478
• DOI: 10.1007/s10569-014-9545-8

We obtain an approximate solution E ~ = E ~ ( e , M ) of Kepler’s equation E - e sin ( E ) = M for any e ∈ [ 0 , 1 ) and M ∈ [ 0 , π ] . Our solution is guaranteed, via Smale’s α -theory, to converge to the actual solution E through Newton’s method at quadratic speed, i.e. the n -th iteration produces a value E n such that | E n - E | ≤ ( 1 2 ) 2 n - 1 | E ~ - E | . The formula provided for E ~ is a piecewise rational function with conditions defined by polynomial inequalities, except for a small region near e = 1 and M = 0 , where a single cubic root is used. We also show that the root operation is unavoidable, by proving that no approximate solution can be computed in the entire region [ 0 , 1 ) × [ 0 , π ] if only rational functions are allowed in each branch.