Obtaining the Parametric Equation of the Curve of the Sun’s Apparent Movement by Using Quaternions

Year 2022, Volume: 5 Issue: 2, 42 - 50, 30.06.2022

Deniz Güçler , Nejat Ekmekçi , Yusuf Yaylı , Mustafa Helvacı

https://doi.org/10.32323/ujma.1091832

Cited By: 1

Abstract

The purpose of this article is to express the daily and yearly apparent movement of the Sun in the same curve by using quaternions as a rotation operator. To achieve this, the daily and yearly apparent movement of the Sun, the algebraical structure of quaternions and how quaternions work as rotation operators has been examined. For each of the apparent movements of the Sun, a quaternion that will work as a rotation operator has been determined. Afterwards, these two rotation operators have been applied to the vector that is found between point (0,0,0) and the accepted starting point of the apparent movement of the Sun. As a result, a curve on a sphere is obtained. The importance of this study is to emphasize the use of quaternions in other areas of study and to provide the science of astronomy a new outlook with regards to expressing the apparent movement of the Sun.

Keywords

quaternions, apparent movement of the Sun, rotational movement

References

• [1] Kuipers, J.B, Quaternions and Rotation Secuences, Princeton. New Jersey. 1998.
• [2] Altman, S.L, Quaternions and Double Groups, Oxford Science Publications. 1986.
• [3] Housner, G.W– Hudson, D. E, Applied Mechanics Dynamics, D.Van Nostrand Company, Inc. 1959.
• [4] Kuipers, J.B, “Object Tracking and orentation determination means, system and process,” U.S. Patent 3,868,565, Feburary 25, 1975.
• [5] Kuipers, J.B, “Tracking and determing orientation of object using coordinate transformation means, sistem and prosess,” U.S. Patent 3,983,474, September 26, 1976
• [6] Kuipers, J.B, “Methods and Apparatus for determining remote object orientation and position,” U.S. Patent 4,742,356, May, 1988
• [7] Pence – Dennis, “Spacecraf Attitute, Rotations and Quaternions,” UMAP Unit 652, The UMAP Journal, vol V, no. 2, 1984
• [8] Kızılırmak, A, Küresel Gökbilimi, Bornova – İzmir. 1977.
• [9] Pannecoch, A, A History of Astronomy. London. 1961
• [10] Palmer, C.I. – Leigh, C.W, Plane and Spherical Trigonometry, London. 1934.
• [11] Payne – Gaposckin, C, Introduction to Astronomy, Prentice – Hall, Inc. Englwood Cliffs, N.J 1961.
• [12] Smart, W.M, Foundations of Astronomy, London. 1962.
• [13] Smart, W.M, Text – Book on Spherical Astronomy, Cambridge. 1962.
• [14] Todhunter, M.A. – Leathem, J.G, Spherical Trigonometry, London. 1960.
• [15] Voronston, V. – Rabbitt, P.M, Astronomical Problems, London. 1969.
• [16] Woolard, E.W. – Clemence, G.M, Spherical Astronomy, New York. 1966.
• [17] Kells, L.M. – Kern, W.F. – Bland, J.R, Plane and Spherical Trigonometry, London. 1940.
• [19] Motz, L. – Duveen, A, Essentials of Astronomy, London. 1966.
• [20] Halliday, D - Resnick, R – Walker, J, Foundamentals of Physics, Wiley 10th edition. 2013
• [21] Fisher, R. C. – Zıebur, A. D, Calculus and Analytic Geometry, Prentice Hall. 1965
• [22] Smart, W.M, Celestial Mechanics, Literary Licensing, 2013
• [23] Kummer, Martin, “Reduction in Rotating Kepler Problem and Related Topics,” Contemporary Mathematics, Vol.198, P:155-180, 1996 [24] Cushman, Richard Et Al., “Direction of Hamiltonian Dynamics and Celestial Mechanics,” Contemporary Mathematics, Vol.98, P:229-240, 1996