A general algorithm for the solution of Kepler's equation for elliptic orbits

Ng, Edward W.

doi:10.1007/bf01371365

Cited by 23 publications

(7 citation statements)

References 8 publications

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“…The region near the (1, 0) corner where a cubic root is needed can be reduced as much as desired but cannot be completely avoided, as the following two results show. Other authors have found similar obstructions in handling values of the eccentricity near 1 ([7]; [10]; [11]). Finally, Theorem 1.4 and Remark 5.1 show that the classical starters S 1 , .…”

Section: Introductionsupporting

confidence: 57%

“…6M e 2 otherwise is an approximate zero of f e,M for all e ∈ [0, 1) and M ∈ [0, π]. This way of constructing an approximate solution by a piecewise function can be compared to Ng's approach (see Figure 2 of [10]). However, our function is computationally simpler because Ng's formula outside the corner uses rational functions involving many terms and near the corner uses S 10 , which requires at least a cubic and a square root for its computation.…”

Section: Introductionmentioning

confidence: 99%

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Solving Kepler’s equation via Smale’s $$\alpha $$ α -theory

Avendaño

Martín-Molina

Ortigas-Galindo

2014

Celest Mech Dyn Astr

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We obtain an approximate solutionẼ =Ẽ(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 En such that |En −E| ≤ ( 1 2 ) 2 n −1 |Ẽ −E|. The formula provided forẼ 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. √ 1 − e 2 and T can be obtained from a using the third law). For a derivation of these formulas, and a detailed introduction to Kepler's equation, see [1].By a symmetry argument, the equation can be easily reduced to the case M ∈ [0, π]. The existence and uniqueness of solution E ∈ [0, π] follows from the fact that the function f e,M : [0, π] → [0, π] given by f e,M (E) = E − e sin(E) − M is strictly increasing.Several solutions to the problem have been proposed since it was stated 400 years ago. Some authors have tried non-iterative methods to solve the equation up to a fixed predetermined accuracy ([6]; [8]). However, we want to calculate the solution with arbitrary precision, hence our interest in iterative techniques.Kepler himself proposed to use a fixed-point iteration to solve the equation ([3, Ch. 1]), i.e. guess E 0 , an approximation of the exact solution E, and then iterate E n+1 = M + e sin(E n ). This sequence converges to E, sinceThe problem with this approach is that the convergence is slow for values of e near 1. For the orbit of Mercury, which has e ≈ 0.2, about 5 iterations are needed to reduce the error by a factor of 10 −3 , while for values of eccentricity e > 0.5 the fixed-point iteration is even slower than a bisection method.Although the fixed-point iteration does not provide an efficient solution to Kepler's equation, it exhibits the structure of most of the current methods to solve it: first, guess an approximationẼ of the solution (called starter ), and then use some iterative technique to produce a sequence quickly converging to the actual solution (see [4], [5], [9], [13]). For the second part, Newton's method seems to be the most used iteration, mainly due to its conceptual simplicity, generality and fast convergence. The guessing part, however, requires some specific understanding on the equation and has been the subject of many recent papers ([2]; [7]; [10]; [11]; [12]; [15]).Starters have been compared (and optimized) using different criteria, such as the number of iterations needed to reach certain precision, the distance to the actual solution, the number of floating point operations needed for its computation, etc. For this purpose, we adopt a criterion which is very specific to Newton's method and guarantees that the iterations reduce the error at quadratic sp...

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Section: Introductionsupporting

confidence: 57%

Section: Introductionmentioning

confidence: 99%

Solving Kepler’s equation via Smale’s $$\alpha $$ α -theory

Avendaño

Martín-Molina

Ortigas-Galindo

2014

Celest Mech Dyn Astr

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“…While there are many articles discussing solutions for the elliptic case (see Avendano et al 2014;Colwell 1993;Danby and Burkardt 1983;Montenbruck and Pfleger 1994;Ng 1979;Odell and Gooding 1986;Taff and Brennan 1989, among others), the hyperbolic case has received less attention. Prussing (1977) and Serafin (1986) gave upper bounds for the actual solution of the hyperbolic Kepler's equation, which can be used as starters for Newton's method since f g,L ≥ 0.…”

Section: Introductionmentioning

confidence: 98%

Approximate solutions of the hyperbolic Kepler equation

Avendaño¹,

Martín-Molina²,

Ortigas-Galindo³

2015

Celest Mech Dyn Astr

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We provide an approximate zero S(g, L) for the hyperbolic Kepler's equation S − g arcsinh(S) − L = 0 for g ∈ (0, 1) and L ∈ [0, ∞). We prove, by using Smale's α-theory, that Newton's method starting at our approximate zero produces a sequence that converges to the actual solution S(g, L) at quadratic speed, i.e. if S n is the value obtained after n iterations, then |S n − S| ≤ 0.5 2 n −1 | S − S|. The approximate zero S(g, L) is a piecewisedefined function involving several linear expressions and one with cubic and square roots. In bounded regions of (0, 1) × [0, ∞) that exclude a small neighborhood of g = 1, L = 0, we also provide a method to construct simpler starters involving only constants.

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“…This paper focuses on the situation of nearly-circular motions ( ). We simplify the problem by limiting and within the range of 2 , as shown in Eq. ( 2 ).…”

Section: Introductionmentioning

confidence: 99%

“…However, limited by historical scientific conditions, these traditional methods usually contain numerical iterative calculations, such as the standard Newton–Raphson method 2 , 4 and the Halley's method 2 , which have to reset the initial values and iterate from the beginning every time the eccentricity e changes 22 . Meanwhile, some traditional numerical methods 4 contain tremendously complex functions, which costs too much time to compute.…”

Section: Introductionmentioning

confidence: 99%

Symbolic iteration method based on computer algebra analysis for Kepler’s equation

Zhang

Bian

2022

Sci Rep

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The Kepler’s equation of elliptic orbits is one of the most significant fundamental physical equations in Satellite Geodesy. This paper demonstrates symbolic iteration method based on computer algebra analysis (SICAA) to solve the Kepler’s equation. The paper presents general symbolic formulas to compute the eccentric anomaly (E) without complex numerical iterative computation at run-time. This approach couples the Taylor series expansion with higher-order trigonometric function reductions during the symbolic iterative progress. Meanwhile, the relationship between our method and the traditional infinite series expansion solution is analyzed in this paper, obtaining a new truncation method of the series expansion solution for the Kepler’s equation. We performed substantial tests on a modest laptop computer. Solutions for 1,002,001 pairs of (e, M) has been conducted. Compared with numerical iterative methods, 99.93% of all absolute errors δE of eccentric anomaly (E) obtained by our method is lower than machine precision $$\epsilon$$ ϵ over the entire interval. The results show that the accuracy is almost one order of magnitude higher than that of those methods (double precision). Besides, the simple codes make our method well-suited for a wide range of algebraic programming languages and computer hardware (GPU and so on).

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A general algorithm for the solution of Kepler's equation for elliptic orbits

Cited by 23 publications

References 8 publications

Solving Kepler’s equation via Smale’s $$\alpha $$ α -theory

Solving Kepler’s equation via Smale’s $$\alpha $$ α -theory

Approximate solutions of the hyperbolic Kepler equation

Symbolic iteration method based on computer algebra analysis for Kepler’s equation

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