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The Homotopy Perturbation Method for Accurate Orbits of the Planets in the Solar System: The Elliptical Kepler Equation

  • Aisha Alshaery EMAIL logo
Published/Copyright: September 13, 2017
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 72 Issue 10

Abstract

Accurate trajectories for the orbits of the planets in our solar system depends on obtaining an accurate solution for the elliptical Kepler equation. This equation is solved in this article using the homotopy perturbation method. Several properties of the periodicity of the obtained approximate solutions are introduced through some lemmas. Numerically, our calculations demonstrated the applicability of the obtained approximate solutions for all the planets in the solar system and also in the whole domain of eccentricity and mean anomaly. In the whole domain of the mean anomaly, 0≤M≤2π, and by using the different approximate solutions, the residuals were less than 4×10−17 for e∈[0, 0.06], 4×10−9 for e∈[0.06, 0.25], 3×10−8 for e∈[0.25, 0.40], 3×10−7 for e∈[0.40, 0.50], and 10−6 for e∈[0.50, 1.0]. Also, the approximate solutions were compared with the Bessel–Fourier series solution in the literature. In addition, the approximate homotopy solutions for the eccentric anomaly are used to show the convergence and periodicity of the approximate radial distances of Mercury and Pluto for three and five periods, respectively, as confirmation for some given lemmas. It has also been shown that the present analysis can be successfully applied to the orbit of Halley’s comet with a significant eccentricity.

Keywords: Homotopy Perturbation Method; Kepler’s Equation; Series Solution

Acknowledgment

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. G-528-363-38. The authors, therefore, acknowledge with thanks DSR for technical and financial support. I appreciate the referee for his close attention and the time spent to read my article in detail. The reviewer’s precise comments were insightful for making this paper more comprehensive.

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Received: 2017-5-28
Accepted: 2017-8-20
Published Online: 2017-9-13
Published in Print: 2017-9-26

©2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Dyons and Certain Symmetries in Maxwell’s Equations
  3. Shear Alfvén Wave with Quantum Exchange-Correlation Effects in Plasmas
  4. Homotopy Perturbation Method for Creeping Flow of Non-Newtonian Power-Law Nanofluid in a Nonuniform Inclined Channel with Peristalsis
  5. Asymptotic Analysis of a Nonlinear Problem on Domain Boundaries in Convection Patterns by Homotopy Renormalization Method
  6. The Exchange-Correlation Field Effect over the Magnetoacoustic-Gravitational Instability in Plasmas
  7. Structural, Spectroscopic, and Energetic Parameters of Diatomic Molecules Having Astrophysical Importance
  8. The Homotopy Perturbation Method for Accurate Orbits of the Planets in the Solar System: The Elliptical Kepler Equation
  9. Electron-Nuclear Dynamics on Amplitude and Frequency Modulation of Molecular High-Order Harmonic Generation from H2+ and its Isotopes
  10. Interaction Solutions for Lump-line Solitons and Lump-kink Waves of the Dimensionally Reduced Generalised KP Equation
  11. Discrete Solitons and Bäcklund Transformation for the Coupled Ablowitz–Ladik Equations
  12. Rapid Communication
  13. Nonclassical t-Dependent Energy Integral of q″+aq′+b(t)q+c(t)qn=0
https://doi.org/10.1515/zna-2017-0181
Keywords for this article
Homotopy Perturbation Method; Kepler’s Equation; Series Solution

Articles in the same Issue

  1. Frontmatter
  2. Dyons and Certain Symmetries in Maxwell’s Equations
  3. Shear Alfvén Wave with Quantum Exchange-Correlation Effects in Plasmas
  4. Homotopy Perturbation Method for Creeping Flow of Non-Newtonian Power-Law Nanofluid in a Nonuniform Inclined Channel with Peristalsis
  5. Asymptotic Analysis of a Nonlinear Problem on Domain Boundaries in Convection Patterns by Homotopy Renormalization Method
  6. The Exchange-Correlation Field Effect over the Magnetoacoustic-Gravitational Instability in Plasmas
  7. Structural, Spectroscopic, and Energetic Parameters of Diatomic Molecules Having Astrophysical Importance
  8. The Homotopy Perturbation Method for Accurate Orbits of the Planets in the Solar System: The Elliptical Kepler Equation
  9. Electron-Nuclear Dynamics on Amplitude and Frequency Modulation of Molecular High-Order Harmonic Generation from H2+ and its Isotopes
  10. Interaction Solutions for Lump-line Solitons and Lump-kink Waves of the Dimensionally Reduced Generalised KP Equation
  11. Discrete Solitons and Bäcklund Transformation for the Coupled Ablowitz–Ladik Equations
  12. Rapid Communication
  13. Nonclassical t-Dependent Energy Integral of q″+aq′+b(t)q+c(t)qn=0
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