A Solution of Kepler’s Equation
Author(s)
John N. Tokis*
ABSTRACT
The present study deals with a traditional physical problem: the solution of the Kepler’s equation for all conics (ellipse, hyperbola or parabola). Solution of the universal Kepler’s equation in closed form is obtained with the help of the two-dimensional Laplace technique, expressing the universal functions as a function of the universal anomaly and the time. Combining these new expressions of the universal functions and their identities, we establish one biquadratic equation for universal anomaly (χ) for all conics; solving this new equation, we have a new exact solution of the present problem for the universal anomaly as a function of the time. The verifying of the universal Kepler’s equation and the traditional forms of Kepler’s equation from this new solution are discussed. The plots of the elliptic, hyperbolic or parabolic Keplerian orbits are also given, using this new solution.
The present study deals with a traditional physical problem: the solution of the Kepler’s equation for all conics (ellipse, hyperbola or parabola). Solution of the universal Kepler’s equation in closed form is obtained with the help of the two-dimensional Laplace technique, expressing the universal functions as a function of the universal anomaly and the time. Combining these new expressions of the universal functions and their identities, we establish one biquadratic equation for universal anomaly (χ) for all conics; solving this new equation, we have a new exact solution of the present problem for the universal anomaly as a function of the time. The verifying of the universal Kepler’s equation and the traditional forms of Kepler’s equation from this new solution are discussed. The plots of the elliptic, hyperbolic or parabolic Keplerian orbits are also given, using this new solution.
KEYWORDS
Keplerian Motion, Universal Kepler’s Equation, Universal Anomaly, Two-Dimensional Laplace Transforms
Keplerian Motion, Universal Kepler’s Equation, Universal Anomaly, Two-Dimensional Laplace Transforms
Cite this paper
Tokis, J. (2014) A Solution of Kepler’s Equation. International Journal of Astronomy and Astrophysics, 4, 683-698. doi: 10.4236/ijaa.2014.44062.
Tokis, J. (2014) A Solution of Kepler’s Equation. International Journal of Astronomy and Astrophysics, 4, 683-698. doi: 10.4236/ijaa.2014.44062.
References
[1] Dandy, J.M.A. (2003) Fundamentals of Celestial Mechanics. 2nd Edition, Willmann-Bell, Virginia. [2] Fukushima, T. (1999) Fast Procedure Solving Universal Kepler’s Equation. Celestial Mechanics and Dynamical Astronomy, 75, 201-226.
http://dx.doi.org/10.1023/A:1008368820433 [3] Battin, R.H. (1999) An Introduction to the Mathematics and Methods Astrodynamics. Revised Edition, AIAA Education Series, New York.
http://dx.doi.org/10.2514/4.861543 [4] Voelkel, J.R. (2001) The Composition of Kepler’s Astronomia Nova. Princeton University Press, New York.
http://press.princeton.edu/titles/7187.html [5] Bruce, S. (1987) Kepler’s Physical Astronomy. Springer-Verlag, New York. [6] Tisserand, F. (1894) Mecanique Celeste. Vol. 3. Gauthier-Villars, Paris. [7] Siewert, C.E. and Burniston, E.E. (1972) An Exact Analytical Solution of Kepler’s Equation. Celestial Mechanics, 6, 294-304.
http://link.springer.com/article/10.1007/BF01231473#page-1 [8] Tokis, J.N. (1973) Effects of Tidal Dissipative Processes on Stellar Rotation. PhD Thesis, Victoria University of Manchester, Manchester. [9] Fernandes, S.daS. (2003) Universal Closed-form of Lagrangian Multipliers for Coast-Arcs of Optimum Space Trajectories. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 25, 1-9.
http://dx.doi.org/10.1590/S1678-58782003000400004 [10] Condurache, D. and Martinusi, V. (2006) Vectorial Regularization and Temporal Means in Keplerian Motion. Journal of Nonlinear Mathematical Physics (World Scientific), 13, 420-440. http://dx.doi.org/10.2991/jnmp.2006.13.3.7 [11] Pathan, A. and Collyer, T. (2006) A Solution to a Cubic—Barker’s Equation for Parabolic Trajectories. Mathematical Gazette, 90, 398-403. [12] Boubaker, K. (2010) Analytical Initial-Guess-Free Solution to Kepler’s Transcendental Equation Using Boubaker Polynomials Expansion Scheme BPES. Apeiron, 17, 1-12.
http://redshift.vif.com/JournalFiles/V17NO1PDF/V17N1BOU.pdf [13] Colwell, P. (1993) Solving Kepler’s Equation over Three Centuries. Willmann-Bell, Richmond.
http://www.willbell.com/math/mc12.htm [14] Floria, L. (1996) A Proof of Universality of Arc Length as Time Parameter in Kepler Problem. Extracta Mathematicae, 11, 315-324. http://dmle.cindoc.csic.es/pdf/EXTRACTAMATHEMATICAE_1996_11_02_09.pdf [15] Fukushima, T. (1998) A Fast Procedure Solving Gauss Form of Kepler’s Equation. Celestial Mechanics and Dynamical Astronomy, 70, 115-130. http://link.springer.com/article/10.1023%2FA%3A1026479306748#page-1 [16] Sharaf, M.A. and Sharaf, A.A. (1998) Closest Approach in Universal Variables. Celestial Mechanics and Dynamical Astronomy, 69, 331-346. http://link.springer.com/article/10.1023%2FA%3A1008223105130#page-1 [17] Sharaf, M.A. and Sharaf, A.A. (2003) Homotopy Continuation Method of Arbitrary Order of Convergence for the Two-Body Universal Initial Value Problem. Celestial Mechanics and Dynamical Astronomy, 86, 351-362.
http://link.springer.com/article/10.1023/A:1024544523868#page-1 [18] Jia, L. (2013) Approximate Kepler’s Elliptic Orbits with the Relativistic Effects. International Journal of Astronomy and Astrophysics, 3, 29-33.
http://dx.doi.org/10.4236/ijaa.2013.31004 [19] Aghili, A. and Salkhordeh-Moghaddam, B. (2008) Laplace Transform Pairs of N-Dimensions and Second Order Linear Partial Differential Equations with Constant Coefficients. Annales Mathematicae et Informaticae, 35, 3-10.
http://www.emis.de/journals/AMI/2008/ami2008-aghili-salkhordeh.pdf [20] Valko, P.P. and Abate, J. (2005) Numerical Inversion of 2-D Laplace Transforms Applied to Fractional Diffusion Equation. Applied Numerical Mathematics, 53, 73-88. http://dx.doi.org/10.1016/j.apnum.2004.10.002 [21] Ditkin, V.A. and Prudnicov, A.P. (1962) Operational Calculus in Two Variables and Its Application. Pergamum Press, New York.
[1] Dandy, J.M.A. (2003) Fundamentals of Celestial Mechanics. 2nd Edition, Willmann-Bell, Virginia. [2] Fukushima, T. (1999) Fast Procedure Solving Universal Kepler’s Equation. Celestial Mechanics and Dynamical Astronomy, 75, 201-226.
http://dx.doi.org/10.1023/A:1008368820433 [3] Battin, R.H. (1999) An Introduction to the Mathematics and Methods Astrodynamics. Revised Edition, AIAA Education Series, New York.
http://dx.doi.org/10.2514/4.861543 [4] Voelkel, J.R. (2001) The Composition of Kepler’s Astronomia Nova. Princeton University Press, New York.
http://press.princeton.edu/titles/7187.html [5] Bruce, S. (1987) Kepler’s Physical Astronomy. Springer-Verlag, New York. [6] Tisserand, F. (1894) Mecanique Celeste. Vol. 3. Gauthier-Villars, Paris. [7] Siewert, C.E. and Burniston, E.E. (1972) An Exact Analytical Solution of Kepler’s Equation. Celestial Mechanics, 6, 294-304.
http://link.springer.com/article/10.1007/BF01231473#page-1 [8] Tokis, J.N. (1973) Effects of Tidal Dissipative Processes on Stellar Rotation. PhD Thesis, Victoria University of Manchester, Manchester. [9] Fernandes, S.daS. (2003) Universal Closed-form of Lagrangian Multipliers for Coast-Arcs of Optimum Space Trajectories. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 25, 1-9.
http://dx.doi.org/10.1590/S1678-58782003000400004 [10] Condurache, D. and Martinusi, V. (2006) Vectorial Regularization and Temporal Means in Keplerian Motion. Journal of Nonlinear Mathematical Physics (World Scientific), 13, 420-440. http://dx.doi.org/10.2991/jnmp.2006.13.3.7 [11] Pathan, A. and Collyer, T. (2006) A Solution to a Cubic—Barker’s Equation for Parabolic Trajectories. Mathematical Gazette, 90, 398-403. [12] Boubaker, K. (2010) Analytical Initial-Guess-Free Solution to Kepler’s Transcendental Equation Using Boubaker Polynomials Expansion Scheme BPES. Apeiron, 17, 1-12.
http://redshift.vif.com/JournalFiles/V17NO1PDF/V17N1BOU.pdf [13] Colwell, P. (1993) Solving Kepler’s Equation over Three Centuries. Willmann-Bell, Richmond.
http://www.willbell.com/math/mc12.htm [14] Floria, L. (1996) A Proof of Universality of Arc Length as Time Parameter in Kepler Problem. Extracta Mathematicae, 11, 315-324. http://dmle.cindoc.csic.es/pdf/EXTRACTAMATHEMATICAE_1996_11_02_09.pdf [15] Fukushima, T. (1998) A Fast Procedure Solving Gauss Form of Kepler’s Equation. Celestial Mechanics and Dynamical Astronomy, 70, 115-130. http://link.springer.com/article/10.1023%2FA%3A1026479306748#page-1 [16] Sharaf, M.A. and Sharaf, A.A. (1998) Closest Approach in Universal Variables. Celestial Mechanics and Dynamical Astronomy, 69, 331-346. http://link.springer.com/article/10.1023%2FA%3A1008223105130#page-1 [17] Sharaf, M.A. and Sharaf, A.A. (2003) Homotopy Continuation Method of Arbitrary Order of Convergence for the Two-Body Universal Initial Value Problem. Celestial Mechanics and Dynamical Astronomy, 86, 351-362.
http://link.springer.com/article/10.1023/A:1024544523868#page-1 [18] Jia, L. (2013) Approximate Kepler’s Elliptic Orbits with the Relativistic Effects. International Journal of Astronomy and Astrophysics, 3, 29-33.
http://dx.doi.org/10.4236/ijaa.2013.31004 [19] Aghili, A. and Salkhordeh-Moghaddam, B. (2008) Laplace Transform Pairs of N-Dimensions and Second Order Linear Partial Differential Equations with Constant Coefficients. Annales Mathematicae et Informaticae, 35, 3-10.
http://www.emis.de/journals/AMI/2008/ami2008-aghili-salkhordeh.pdf [20] Valko, P.P. and Abate, J. (2005) Numerical Inversion of 2-D Laplace Transforms Applied to Fractional Diffusion Equation. Applied Numerical Mathematics, 53, 73-88. http://dx.doi.org/10.1016/j.apnum.2004.10.002 [21] Ditkin, V.A. and Prudnicov, A.P. (1962) Operational Calculus in Two Variables and Its Application. Pergamum Press, New York.