Abstract
In this study, we used the fact that unit circle for elliptic numbers is an ellipse to model motion of a planet around a star. For that purpose we first have given a standard derivation of elliptic orbits in Newtonian two-body problem. Then we translated the variables found in the coordinates where the origin is at the focus of the ellipse to elliptic number parameters where the origin is at the center of the ellipse. We noted that a similar argument may be used to model hyperbolic orbits in Newtonian gravity with hyperbolic numbers. However it seems that modelling parabolic orbits is not possible within the context of p-complex numbers.
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References
Donahue, W., Gingerich, O.: Johannes Kepler New Astronomy. Cambridge University Press (1992)
Ghez, A.M., Salim, S., Weinberg, N., Lu, J., Do, T., Dunn, J., Matthews, K., Morris, M., Yelda, S., Becklin, E., et al.: Measuring distance and properties of the Milky Way’s central supermassive black hole with stellar orbits. Astrophys. J. 689(2), 1044 (2008)
Goldstein, H., Poole, C., Safko, J.: Classical Mechanics, third international edn. Pearson (2002)
Harkin, A.A., Harkin, J.B.: Geometry of generalized complex numbers. Math. Mag. 77(2), 118–129 (2004)
Yaglom, I.M.: Complex numbers in geometry. Academic Press (1968)
Özen, K.E.: On the trigonometric and \(p\)-trigonometric functions of elliptical complex variables. Commun. Adv. Math. Sci. 3, 143–154 (2020). https://doi.org/10.33434/cams.789085
Acknowledgements
We would like to thank Kahraman Esen Özen for pointing out various references and useful discussions. Figure 1 is drawn with PGF-TikZ.
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Communicated by Rafał Abłamowicz.
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Dündar, F.S. A Use of Elliptic Complex Numbers in Newtonian Gravity. Adv. Appl. Clifford Algebras 32, 20 (2022). https://doi.org/10.1007/s00006-022-01208-0
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DOI: https://doi.org/10.1007/s00006-022-01208-0