Document Type : Research Paper
Author
Applied Science Department , University of Technology, Baghdad, Iraq
Abstract
In the present study, a modification for iterative methods of three order is presented by means of mean anomaly to calculate the values of E and e (true anomaly and eccentricity) respectively, for a planet in an elliptical orbit around the sun. We find that the improved methods converge to the true anomaly value E, solution with less iteration. The efficiency of the modified third order algorithms are examined on many cases of the values of M and e. It is observed that methods are more efficient than third order methods presented by Weerakoon [1], Homerier [2] and Ababneh [3] and classical Newton's method.
Keywords
Main Subjects
[1] Weerakoon S.,and Fernado G. I., 2000,"A variant of Newton's method with Accelerated Third-Order Convergence", Applied Mathematics Letters, 17(8) 87-93.
[2] Homeier H. H. H.,2005, "On Newton-Type Methods with Cubic Convergence", Journal of Computational and Applied Mathematics, 176(2), 425-432.
[3] Ababenh O. Y.,2012, "New Newton's Method with Third-order Convergence For Solving Nonlinear Equations", World Academy of Science, Engineering and Technology, 61(198), 1071-1073.
[4] Wissink, A. M., and Lyrintzis, A. S.,1996, "Efficient Iterative Methods Applied to the solution of Transonic Flows", Journal of Computational Physics, 123(2), 379-393. Available on: IVSL.org.
[5] Li Z. H., and Zhang, H. X., 2004,"Study on Gas Kinetic Unified Algorithm for Flows from Rarefied Transition to Continuum", Journal of Computational Physics, 193(2), 708-738. Available from: IVSL.org.
[6] Arovas D., 2012,"Lecture Notes on Classical Mechanics, (A Work in Progress)", Department of Physics, University of California, San Diego, UCSD book.
[7] Bate R. R., and Mueller D. D.,1971, "Fundamentals of Astrodynamics", Dover Publications, Inc., New York, January .
[8] Boubaker K., and Mohmoud B. ,2010," Kepler’s celestial two-body equation: a second attempt to establish a continuous and integrable solution via the BPES: Boubaker Polynomials Expansion Scheme", Astrophysics and Space Science., 327(1),77–81. Available on: IVSL.org.
[9] Boyd J. P., 2009,"Chebyshev expansion on intervals with branch points with application to the root of Kepler’s equation: A Chebyshev–Hermite–Pad´e method", Journal of Computational and Applied Mathematics, 223(2), 693-702.
[10] Carinena, J. F., Lopez, C., del Olmo, M. A., and Santander, M., 1991,"Conformal Geometry of the Kepler Orbit Space", Celestial Mechanics and Dynamical Astronomy, 52(4), 307-343.
[11] Eisinberg A., Fedele G.,and Ferrise A.,2010,"On an integral Representation of a class of Kapteyn (Fourier_Bessel) Series: Kepler's Equation, Radiation Problems and Meissel's Expansion", Applied Mathematics Letters, 23(5),1331-1335. Available from: IVSL.org.
[12] Toshio F.,1996 "A Fast Procedure Solving Kepler's Equation for Elliptic Case", The Astronomical Journal, 112(6),2858-2861. Available on: IVSL.org.
[13] Toshio F., 1997,"A Method Solving without Transcendental Function Evaluations" Celestial Mechanics and Dynamical Astronomy, 66(3), 309-319. Available on: IVSL.org.
[14] Yves N.,2009, "Computing the distance from a point to a helix and solving Kepler’s equation", Nuclear Instruments and Methods in Physics Research A, 598(3),788-794. Available on: IVSL.org.
[15] Rongfu T., and Dongyun Y.,2008,"TAIC Algorithm for the Visibility of the Elliptical Orbits’ Satellites", Changsha, Hunan Province, 410073, P. R. China, 1-4244-1212-9/07, IEEE International,781-785. Available on: IVSL.org.
[16] Chun Ch., and Kim Y. II.,2010, "Several New Third-Order Iterative Methods for Solving Nonlinear Equations", ACTA Applicandae Mathematicae, 103(3), , 1053-1063. Available on: IVSL.org.
[17] Colwell P.,1991, "Kepler's Equation and Newton's Method", Celestial Mechanics and Dynamical Astronomy, 52(2),203-204. Available on: IVSL.org.
[18] Palacios M.,2002, "Kepler equation and accelerated Newton method", Journal of Computational and Applied Mathematics, 138(2), 335–346. Available on: IVSL.org.
[19] Rasheed M. S.,2010, "Approximate Solutions of Barker Equation in Parabolic Orbits", Engineering & Technology Journal, 28(3) ,492-499.
[20] Rasheed M. S., 2010, "An Improved Algorithm For The Solution of Kepler‘s Equation For An Elliptical Orbit", Engineering & Technology Journal, 28 (7), 1316-1320.
[21] Rasheed M. S.,2012, "Acceleration of Predictor Corrector Halley Method in Astrophysics Application", International Journal of Emerging Technologies in Computational and Applied Sciences, 1(2), 91-94.
[22] Rasheed M. S., 2012,Fast Procedure for Solving Two-Body Problem in Celestial Mechanic, International Journal of Engineering, Business and Enterprise Applications, 1(2), 60-63.
[23] Rasheed M. S., 2013,"Solve the Position to Time Equation for an Object Travelling on a Parabolic Orbit in Celestial Mechanics", Diyala Journal for Pure Sciences, 9(4), 31-38.
[24] Rasheed M. S., 2013,"Comparison of Starting Values for Implicit Iterative Solutions to Hyperbolic Orbits Equation", International Journal of Software and Web Sciences (IJSWS), 1(2), 65-71.
[25] Rasheed M. S.,2014, "On Solving Hyperbolic Trajectory Using New Predictor-Corrector Quadrature Algorithms", Baghdad Science Journal, 11(1), 186-192.
[26] Aisha A., and Ebaid A.,2017, "Accurate analytical periodic solution of the elliptical Kepler equation using the Adomian decomposition method", Acta Astronautica, 140, 27-33.
[27] Aisha,A. 2017,"The Homotopy Perturbation Method for Accurate Orbits of the Planets in the Solar System: The Elliptical Kepler Equation", Zeitschrift für Naturforschung A, 72.10, 933-940.
[28] Calvo M., Elip.A., Montijano J. I., and Rández L,2019, "A monotonic starter for solving the hyperbolic Kepler equation by Newton’s method", Celest Mech Dyn Astr, 131(18), 1-18.
[29] Badolati E., and S. Ciccone, 2015,"On the history of some explicit formulae for solving Kepler's equation", Astronomische Nachrichten, 336.3, 316-320.
[30] Daniele T., and Olivieri D. N.,2020, "Fast switch and spline scheme for accurate inversion of nonlinear functions: The new first choice solution to Kepler’s equation", Applied Mathematics and Computation, 364, , 124677.
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