The Solar Oscillation Equation and Some of its Particular Solutions

Shankhachur Mukherjee; Koushik Ghosh

doi:10.17485/IJST/v15i16.1523

Article

The Solar Oscillation Equation and Some of its Particular Solutions

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Indian Journal of Science and Technology

DOI: 10.17485/IJST/v15i16.1523

Year: 2022, Volume: 15, Issue: 16, Pages: 760-766

Original Article

The Solar Oscillation Equation and Some of its Particular Solutions

Shankhachur Mukherjee¹, Koushik Ghosh^2*

¹Department of Physics, The University of Burdwan, Golapbag, Burdwan-713104, West Bengal, India
²Department of Mathematics, University Institute of Technology, The University of Burdwan, Golapbag (North), Burdwan 713104, West Bengal, India

*Corresponding Author
Email: [email protected]

Received Date:15 August 2021, Accepted Date:28 December 2021, Published Date:09 May 2022

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

Background: The phenomenon that sun oscillates in characteristic eigen frequencies witnessed on the solar surface has fascinated many solar physicists to go for exact measurements of the internal properties of our nearest star that we are most familiar with. The present work bears the objective to understand the dynamic and nonlinear profile of solar oscillation on the basis of some particular solutions of the fundamental equation of solar oscillation. After the detection of global solar oscillation during 1970’s different physical aspects of solar oscillation started to draw the attention of several astrophysicists both from the perspectives of theoretical investigation as well as observational evidences. Methods: The functional forms of the wave representation term appeared in the wave-equation of our present system has been qualitatively studied by performing linear stability analysis, where the ‘Jacobian’ provides the basis of understanding of the nature of equilibrium points. Analytical means or approaches have been employed to determine the solutions of the said wave equation without using any kind of computational method or simulation at different regions of the number line. Findings: The corresponding equilibrium points of the wave-equations followed from the solar oscillation equation at various chosen functional forms of the complicated wave representation term have been nicely calculated. Sincere analytical attempts have also been taken for the first time to derive the solutions of the solar oscillation equation segmented into several mathematically chosen scenarios by us. Novelty: The analytic solutions of the solar oscillation equation obtained for different particular functional structures explain the existence of nonlinear acoustic modes of frequency in the observed solar oscillations of long period. This result will surely motivate any further quest of chaotic oscillations in both radial and non-radial modes.

Keywords: Solar oscillation; Equilibrium point; Jacobian; Bessel function; Eigenvalues

References

Basu S. Global seismology of the Sun. Living Reviews in Solar Physics. 2016;13(1):1–126. Available from: https://doi.org/10.1007/s41116-016-0003-4
Christensen-Dalsgaard J. Helioseismology. Reviews of Modern Physics. 2002;74(4):1073–1129. Available from: https://dx.doi.org/10.1103/revmodphys.74.1073
Christensen-Dalsgaard J, Däppen W, Ajukov SV, Anderson ER, Antia HM, Basu S, et al. The Current State of Solar Modeling. Science. 1996;272(5266):1286–1292. Available from: https://dx.doi.org/10.1126/science.272.5266.1286
Bazighifan O, Ruggieri M, Santra SS, Scapellato A. Qualitative Properties of Solutions of Second-Order Neutral Differential Equations. Symmetry. 2020;12(9):1520. Available from: https://dx.doi.org/10.3390/sym12091520
Chaplin WJ, Basu S. Perspectives in Global Helioseismology and the Road Ahead. Solar Physics. 2008;251(1-2):53–75. Available from: https://dx.doi.org/10.1007/s11207-008-9136-5
Christensen-Dalsgaard J, Gough DO. Towards a heliological inverse problem. Nature. 1976;259(5539):89–92. Available from: https://dx.doi.org/10.1038/259089a0
Deubner FL. Observations of Low Wavenumber Nonradial Eigenmodes of the Sun. Astronomy & Astrophysics. 1975;44:371–375. Available from: http://adsabs.harvard.edu/pdf/1975A&A....44..371D
Deubner FL, Gough D. Helioseismology: Oscillations as a Diagnostic of the Solar Interior. Annual Review of Astronomy and Astrophysics. 1984;22(1):593–619. Available from: https://dx.doi.org/10.1146/annurev.aa.22.090184.003113
Evans JW, Michard R. Observational Study of Macroscopic Inhomogeneities in the Solar Atmosphere. III. Vertical Oscillatory Motions in the Solar Photosphere. The Astrophysical Journal. 1962;136:493. Available from: http://adsabs.harvard.edu/pdf/1962ApJ...136..493E
Zahn JP, Zinn-Justin J. Course 7. Linear Adiabatic Stellar Pulsation. In: Astrophysical Fluid Dunamics, Les Houches Session XLVII. (pp. 399) Amsterdam. Elsevier. 1993.
Gough DO. What Have We Learned from Helioseismology, What Have We Really Learned, and What Do We Aspire to Learn? Solar Physics. 2013;287(1-2):9–41. Available from: https://doi.org/10.1007/s11207-012-0099-1
Kamke E, Differentialgleichungen. Differentialgleichungen: Lӧsungsmethoden und Lӧsungen. I. Gewӧhnliche Differentialgleichungen. Teubner B.G., 1977; Leipzig. 1977. Available from: https://www.springer.com/gp/book/9783663059264
Kosovichev AG, Guzik JA, Bradley PA. Solar Oscillations. AIP Conference Proceedings. 2010. Available from: https://doi.org/10.1063/1.3246561
Leighton RB, Noyes RW, Simon GW. Velocity Fields in the Solar Atmosphere. I. Preliminary Report. The Astrophysical Journal. 1962;135:474. Available from: https://dx.doi.org/10.1086/147285
Polyanin AD, Zaitsev VF. Handbook of Exact Solutions for Ordinary Differential Equations. Boca Raton. Chapman & Hall/CRC Press. 2003.
Rhodes EJJ, Ulrich RK, Simon GW. Observations of nonradial p-mode oscillations on the sun. The Astrophysical Journal. 1977;218:901. Available from: https://dx.doi.org/10.1086/155745
Thompson MJ. Helioseismology and the Sun's interior. Astronomy and Geophysics. 2004;45(4):4.21–4.25. Available from: https://doi.org/10.1046/j.1468-4004.2003.45421.x
Ulrich RK. The Five-Minute Oscillations on the Solar Surface. The Astrophysical Journal. 1970;162:993. Available from: http://adsabs.harvard.edu/pdf/1970ApJ...162..993U
Arnaut LG, Ibáñez S. Self-sustained oscillations and global climate changes. Scientific Reports. 2020;10(1). Available from: https://dx.doi.org/10.1038/s41598-020-68052-9
Li YH, Tian XJ, Li XQ. Nonlinear Second-Order Model and Equilibrium Point Characteristic Analysis of DC Traction Power Supply System. ISRN Applied Mathematics. 2013;2013:1–8. Available from: https://doi.org/10.1155/2013/981692
He T, Li S, Wu S, Li K. Small-Signal Stability Analysis for Power System Frequency Regulation with Renewable Energy Participation. Mathematical Problems in Engineering. 2021;2021:1–13. Available from: https://dx.doi.org/10.1155/2021/5556062
Gizon L, Fournier D, Albekioni M. Effect of latitudinal differential rotation on solar Rossby waves: Critical layers, eigenfunctions, and momentum fluxes in the equatorial β plane. Astronomy & Astrophysics. 2020;642:A178. Available from: https://dx.doi.org/10.1051/0004-6361/202038525
García RA, Ballot J. Asteroseismology of solar-type stars. Living Reviews in Solar Physics. 2019;16(1):4–5. Available from: https://dx.doi.org/10.1007/s41116-019-0020-1
Christensen-Dalsgaard J, Däppen W, Lebreton Y. Solar oscillation frequencies and the equation of state. Nature. 1988;336(6200):634–638. Available from: https://dx.doi.org/10.1038/336634a0
Zhou Y, Asplund M, Collet R. The Amplitude of Solar p-mode Oscillations from Three-dimensional Convection Simulations. The Astrophysical Journal. 2019;880(1):13. Available from: https://dx.doi.org/10.3847/1538-4357/ab262c
Gough D. Some Glimpses from Helioseismology at the Dynamics of the Deep Solar Interior. Space Science Reviews. 2015;196(1-4):15–47. doi: 10.1007/s11214-015-0159-6
Chae J, Litvinenko YE. Nonlinear Effects in Three-minute Oscillations of the Solar Chromosphere. I. An Analytical Nonlinear Solution and Detection of the Second Harmonic. The Astrophysical Journal. 2017;844(2):129. Available from: https://dx.doi.org/10.3847/1538-4357/aa7be9
Chae J, Cho K, Song D, Litvinenko YE. Nonlinear Effects in Three-minute Oscillations of the Solar Chromosphere. II. Measurement of Nonlinearity Parameters at Different Atmospheric Levels. The Astrophysical Journal. 2018;854(2):127. Available from: https://dx.doi.org/10.3847/1538-4357/aaa8e2
Litvinenko YE, Chae J. Analytical description of nonlinear acoustic waves in the solar chromosphere. Astronomy & Astrophysics. 2017;599:A15. Available from: https://dx.doi.org/10.1051/0004-6361/201629568
Däppen W, Perdang J. Non-linear stellar oscillations. Non-radial mode interactions. Astronomy & Astrophysics. 1985;151:174–188. Available from: http://adsabs.harvard.edu/pdf/1985A%26A...151..174D
Gough D, Scherrer PH. The Solar Interior. In: The Century of Space Science. (pp. 1035-1063) Kluwer Academic Publishers. 2001.
Cesarano C, Bazighifan O. Qualitative Behavior of Solutions of Second Order Differential Equations. Symmetry. 2019;11(6):777. Available from: https://dx.doi.org/10.3390/sym11060777
Park C, Moaaz O, Bazighifan O. Oscillation Results for Higher Order Differential Equations. Axioms. 2020;9(1):14. Available from: https://dx.doi.org/10.3390/axioms9010014

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© 2022 Mukherjee & Ghosh. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Published By Indian Society for Education and Environment (iSee)