Detailed demonstrations of the relations between coordinate systems in the celestial sphere
DOI:
https://doi.org/10.5902/2179460X65707Keywords:
Positional Astronomy, Spherical Astronomy, Mathematical Astronomy, Spherical Geometry, Spherical TrigonometryAbstract
Spherical Geometry is the basis for what we know as Positional Astronomy. This is one of the oldest approaches to Astronomy as a science, being used by ancient Greeks and possibly by other people before that. Concerning a more formal current mathematical description, there are several types of coordinate systems with respect to the celestial sphere. Each system differs in the choice of its referential plan. Several connections between elements of different types of coordinates are well known once it is a well-developed field. However, the reader will not find in the literature complete mathematical proofs for such formulas. Thus, this article fills that gap. We present in this text several formulas relating the coordinates Zenital Distance, Hour Angle, Azimuth, Declination and Geographic Latitude and their mathematical proofs in detail, explaining all possible steps. Our main contribution is in the form of the presentation and deduction of classical results from Positional Astronomy.
Downloads
References
ARCARI, I. Um texto de geometria hiperbólica. Master’s thesis (Mathematics) – Universidade Estadual de Campinas, UNICAMP, Campinas, 2008.
BARBOSA, J. L. M. Geometria hiperbólica. 3ª impressão. Rio de Janeiro: IMPA, 2002, p. 167.
CARITÁ, L. A. Lei dos cossenos e senos esféricos: demonstrações matemáticas para ferramentas astronômicas. Revista Univap, [s.l.], v. 24, n. 46, 10-18, 2018.
CHODOROVÁ, M.; JUKLOVÁ, L.; SMETANOVÁ, D. Navigation course in mathematical examples – utilization of the spherical geometry. NAŠE MORE : znanstveni časopis za more i pomorstvo, [s.l.], v. 64, n. 3, 67-70, 2017.
MCNALLY, D. Positional astronomy. New York: John Wiley & Sons, 1974, p. 375.
MURRAY, D. A. Spherical Trigonometry, for Colleges and Secondary Schools. California: University of California Libraries, 1900, p. 136.
ROZENFELD, B. A. A history of non-Euclidean geometry: evolution of the concept of a geometric space, v. 12. Germany: Springer Science and Business Media, 2012, p. 471.
SANTOS, C. P. F. d. Fundamentos de geometria e astronomia esférica. Master’s thesis (Mathematics), Universidade Estadual Paulista “Júlio de Mesquisa Filho”, UNESP, Rio Claro, 2020.
SANTOS, C. P. F.; CARITÁ, L. A.; GADOTTI, M. C. Contribuições das navegações portuguesas para a geometria da superfície terrestre. HIPÁTIA - Revista Brasileira de História, Educação e Matemática, [s.l.], v. 6, n. 1, 40-50, 2021.
SMART, W. M. Textbook on spherical astronomy. New York: Cambridge University Press, 1977, p. 448.
VAN BRUMMELEN, G. Heavenly mathematics: The forgotten art of spherical trigonometry. New Jersey: Princeton University Press, 2017, p. 216.
WHITTLESEY, M. A. Spherical geometry and its applications. Florida: Chapman and Hall/CRC Press, 2019, p. 347.
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Ciência e Natura
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
To access the DECLARATION AND TRANSFER OF COPYRIGHT AUTHOR’S DECLARATION AND COPYRIGHT LICENSE click here.
Ethical Guidelines for Journal Publication
The Ciência e Natura journal is committed to ensuring ethics in publication and quality of articles.
Conformance to standards of ethical behavior is therefore expected of all parties involved: Authors, Editors, Reviewers, and the Publisher.
In particular,
Authors: Authors should present an objective discussion of the significance of research work as well as sufficient detail and references to permit others to replicate the experiments. Fraudulent or knowingly inaccurate statements constitute unethical behavior and are unacceptable. Review Articles should also be objective, comprehensive, and accurate accounts of the state of the art. The Authors should ensure that their work is entirely original works, and if the work and/or words of others have been used, this has been appropriately acknowledged. Plagiarism in all its forms constitutes unethical publishing behavior and is unacceptable. Submitting the same manuscript to more than one journal concurrently constitutes unethical publishing behavior and is unacceptable. Authors should not submit articles describing essentially the same research to more than one journal. The corresponding Author should ensure that there is a full consensus of all Co-authors in approving the final version of the paper and its submission for publication.
Editors: Editors should evaluate manuscripts exclusively on the basis of their academic merit. An Editor must not use unpublished information in the editor's own research without the express written consent of the Author. Editors should take reasonable responsive measures when ethical complaints have been presented concerning a submitted manuscript or published paper.
Reviewers: Any manuscripts received for review must be treated as confidential documents. Privileged information or ideas obtained through peer review must be kept confidential and not used for personal advantage. Reviewers should be conducted objectively, and observations should be formulated clearly with supporting arguments, so that Authors can use them for improving the paper. Any selected Reviewer who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the Editor and excuse himself from the review process. Reviewers should not consider manuscripts in which they have conflicts of interest resulting from competitive, collaborative, or other relationships or connections with any of the authors, companies, or institutions connected to the papers.
