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Classical Cosmology II. The Einstein Ring

DOI: 10.4236/jhepgc.2024.102036, PP. 574-598

Keywords: Cosmology, Observational Cosmology, Gravitational Lenses, Luminous Arcs

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Abstract:

The Einstein ring is usually explained in the framework of the gravitational lens. Conversely here we apply the framework of the expansion of a superbubble (SB) in order to explain the spherical appearance of the ring. Two classical equations of motion for SBs are derived in the presence of a linear and a trigonometric decrease for density. A relativistic equation of motion with an inverse square dependence for the density is derived. The angular distance, adopting the minimax approximation, is derived for three relativistic cosmologies: the standard, the flat and the wCDM. We derive the relation between redshift and Euclidean distance, which allows fixing the radius of the Einstein ring. The details of the ring are explained by a simple version of the theory of images.

References

[1]  Einstein, A. (1936) Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field. Science, 84, 506-507.
https://doi.org/10.1126/science.84.2188.506
[2]  Einstein, A. (1994) The Collected Papers of Albert Einstein, Volume 3: The Swiss Years: Writings, 1909-1911. Princeton University Press, Princeton.
[3]  Chwolson, O. (1924) ÜBer Eine Mögliche Form Fiktiver Doppelsterne. Astronomische Nachrichten, 221, 329-330.
https://doi.org/10.1002/asna.19242212003
[4]  Lee, C.H. (2017) A Computer Vision Approach to Identify Einstein Rings and Arcs. Publications of the Astronomical Society of Australia, 34, e014.
https://doi.org/10.1017/pasa.2017.7
[5]  Tian, Y. and Ko, C.M. (2017) Mass Discrepancy—Acceleration Relation in Einstein Rings. Monthly Notices of the Royal Astronomical Society, 472, 765-771.
https://doi.org/10.1093/mnras/stx2056
[6]  Birrer, S., Refregier, A. and Amara, A. (2018) Cosmic Shear with Einstein Rings. The Astrophysical Journal Letters, 852, L14.
https://doi.org/10.3847/2041-8213/aaa1de
[7]  Moffat, J., Rahvar, S. and Toth, V. (2018) Applying MOG to Lensing: Einstein Rings, Abell 520 and the Bullet Cluster. Galaxies, 6, Article 43.
https://doi.org/10.3390/galaxies6020043
[8]  Chen, M.C., Broadhurst, T., Lim, J., Molnar, S.M., Diego, J.M., Oguri, M. and Lee, L.L. (2020) Geometric Support for Dark Matter by an Unaligned Einstein Ring in A3827. The Astrophysical Journal, 898, Article No. 81.
https://doi.org/10.3847/1538-4357/ab9ebc
[9]  Hogg, N.B., Fleury, P., Larena, J. and Martinelli, M. (2023) Measuring Line-of-Sight Shear with Einstein Rings: A Proof of Concept. Monthly Notices of the Royal Astronomical Society, 520, 5982-6000.
https://doi.org/10.1093/mnras/stad512
[10]  Valls-Gabaud, D. (2006) The Conceptual Origins of Gravitational Lensing. Albert Einstein Century International Conference, Paris, 18-22 July 2005, 1163.
https://doi.org/10.1063/1.2399715
[11]  Meneghetti, M. (2021) A Brief History of Gravitational Lensing. In: Meneghetti, M., Ed., Introduction to Gravitational Lensing, Springer, Cham, 3-19.
https://doi.org/10.1007/978-3-030-73582-1_1
[12]  Heiles, C. (1979) H I Shells and Supershells. The Astrophysical Journal, 229, 533-537, 539-544.
https://doi.org/10.1086/156986
[13]  Sánchez-Cruces, M., Rosado, M., Rodrguez-González, A. and Reyes-Iturbide, J. (2015) Kinematics of Superbubbles and Supershells in the Irregular Galaxy, NGC 1569. The Astrophysical Journal, 799, Article 231.
https://doi.org/10.1088/0004-637X/799/2/231
[14]  Zaninetti, L. (2012) Evolution of Superbubbles in a Self-Gravitating Disc. Monthly Notices of the Royal Astronomical Society, 425, 2343-2351.
https://doi.org/10.1111/j.1365-2966.2012.21649.x
[15]  Zaninetti, L. (2020) Energy Conservation in the Thin Layer Approximation: I. The Spherical Classic Case for Supernovae Remnants. International Journal of Astronomy and Astrophysics, 10, 71-88.
https://doi.org/10.4236/ijaa.2020.102006
[16]  Freund, J. (2008) Special Relativity for Beginners: A Textbook for Undergraduates. World Scientific Press, Singapore.
https://doi.org/10.1142/6601
[17]  Zaninetti, L. (2020) Energy Conservation in the Thin Layer Approximation: III. The Spherical Relativistic Case for Supernovae. International Journal of Astronomy and Astrophysics, 10, 285-301.
https://doi.org/10.4236/ijaa.2020.104015
[18]  Williams, B.J., Chomiuk, L., Hewitt, J.W., Blondin, J.M., Borkowski, K.J., Ghavamian, P., Petre, R. and Reynolds, S.P. (2016) An X-Ray and Radio Study of the Varying Expansion Velocities in Tycho Supernova Remnant. The Astrophysical Journal Letters, 823, L32.
https://doi.org/10.3847/2041-8205/823/2/L32
[19]  Patnaude, D.J. and Fesen, R.A. (2009) Proper Motions and Brightness Variations of Nonthermal X-Ray Filaments in the Cassiopeia a Supernova Remnant. The Astrophysical Journal, 697, 535-543.
https://doi.org/10.1088/0004-637X/697/1/535
[20]  Chiad, B.T., Ali, L.T. and Hassani, A.S. (2015) Determination of Velocity and Radius of Supernova Remnant after 1000 Yrs of Explosion. International Journal of Astronomy and Astrophysics, 5, 125-132.
https://doi.org/10.4236/ijaa.2015.52016
[21]  Uchida, H., Yamaguchi, H. and Koyama, K. (2013) Asymmetric Ejecta Distribution in SN 1006. The Astrophysical Journal, 771, Article 56.
https://doi.org/10.1088/0004-637X/771/1/56
[22]  Winkler, P.F., Tuttle, J.H., Kirshner, R.P. and Irwin, M.J. (1988) Kinematics of Oxygen-Rich Filaments in Puppis A. International Astronomical Union Colloquium, 101, 65-68.
https://doi.org/10.1017/S0252921100102131
[23]  Aschenbach, B. (2015) Age and Distance of Puppis A Revised—The Supernova Remnant of the ‘Star of Bethlehem’. Proceedings Frascati Workshop 2015 on Multifrequency Behaviour of High Energy Cosmic Sources—XI (MULTIF15), Palermo, 25-30 May 2015, 21.
https://doi.org/10.22323/1.246.0021
[24]  Reynoso, E.M., Cichowolski, S. and Walsh, A.J. (2017) A High-Resolution H I Study towards the Supernova Remnant Puppis A and Its Environments. Monthly Notices of the Royal Astronomical Society, 464, 3029-3039.
https://doi.org/10.1093/mnras/stw2219
[25]  Marcaide, J.M., Mart-Vidal, I., Alberdi, A. and Pérez-Torres, M.A. (2009) A Decade of SN 1993J: Discovery of Radio Wavelength Effects in the Expansion Rate. Astronomy & Astrophysics, 505, 927-945.
https://doi.org/10.1051/0004-6361/200912133
[26]  Jones, D.O., Scolnic, D.M., Riess, A.G., et al. (2018) Measuring Dark Energy Properties with Photometrically Classified Pan-STARRS Supernovae. II. Cosmological Parameters. The Astrophysical Journal, 857, Article 51.
https://doi.org/10.3847/1538-4357/aab6b1
[27]  Scolnic, D.M., Jones, D.O., Rest, A., et al. (2018) The Complete Light-Curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample. The Astrophysical Journal, 859, Article 101.
https://doi.org/10.3847/1538-4357/aab9bb
[28]  Zaninetti, L. (2021) Sparse Formulae for the Distance Modulus in Cosmology. Journal of High Energy Physics, Gravitation and Cosmology, 7, 965-992.
https://doi.org/10.4236/jhepgc.2021.73057
[29]  Peebles, P.J.E. (1993) Principles of Physical Cosmology. Princeton University Press, Princeton.
[30]  Etherington, I.M.H. (1933) On the Definition of Distance in General Relativity. Philosophical Magazine, 15, 761-775.
https://doi.org/10.1080/14786443309462220
[31]  Remez, E. (1934) Sur la détermination des polynômes d’approximation de degré donnée. Communions Societatis Mathematicae Kharkovensis, 10, 41.
[32]  Remez, E. (1957) General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Ki-ev.
[33]  Braatz, J.A., Reid, M.J., Humphreys, E.M.L., Henkel, C., Condon, J.J. and Lo, K.Y. (2010) The Megamaser Cosmology Project. II. The Angular-Diameter Distance to UGC 3789. The Astrophysical Journal, 718, Article 657.
https://doi.org/10.1088/0004-637X/718/2/657
[34]  Kuo, C.Y., Braatz, J.A., Reid, M.J., Lo, K.Y., Condon, J.J., Impellizzeri, C.M.V. and Henkel, C. (2013) The Megamaser Cosmology Project. V. An Angular-Diameter Distance to NGC 6264 at 140 Mpc. The Astrophysical Journal, 767, Article 155.
https://doi.org/10.1088/0004-637X/767/2/155
[35]  Melia, F. and Yennapureddy, M.K. (2018) The Maximum Angular-Diameter Distance in Cosmology. Monthly Notices of the Royal Astronomical Society, 480, 2144-2152.
https://doi.org/10.1093/mnras/sty1962
[36]  Baes, M., Camps, P. and Van De Putte, D. (2017) Analytical Expressions and Numerical Evaluation of the Luminosity Distance in a Flat Cosmology. Monthly Notices of the Royal Astronomical Society, 468, 927-930.
https://doi.org/10.1093/mnras/stx537
[37]  Olver, F.W.J., Lozier, D.W., Boisvert, R.F. and Clark, C.W. (2010) NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge.
[38]  Turner, M.S. and White, M. (1997) CDM Models with a Smooth Component. Physical Review D, 56, R4439.
https://doi.org/10.1103/PhysRevD.56.R4439
[39]  Tripathi, A., Sangwan, A. and Jassal, H.K. (2017) Dark Energy Equation of State Parameter and Its Evolution at Low Redshift. Journal of Cosmology and Astroparticle Physic, 6, 012.
https://doi.org/10.1088/1475-7516/2017/06/012
[40]  Wei, J.J., Ma, Q.B. and Wu, X.F. (2015) Utilizing the Updated Gamma-Ray Bursts and Type Ia Supernovae to Constrain the Cardassian Expansion Model and Dark Energy. Advances in Astronomy, 2015, Article ID: 576093.
https://doi.org/10.1155/2015/576093
[41]  Abramowitz, M. and Stegun, I.A. (1965) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.
https://doi.org/10.1063/1.3047921
[42]  Von Seggern, D. (1992) CRC Standard Curves and Surfaces. CRC Press, New York.
[43]  Thompson, W.J. (1997) Atlas for Computing Mathematical Functions. Wiley-Interscience, New York.
[44]  Gradshteyn, I.S., Ryzhik, I.M., Jeffrey, A. and Zwillinger, D. (2007) Table of Integrals, Series, and Products. Academic Press, New York.
[45]  Zaninetti, L. (2019) The Distance Modulus in Dark Energy and Cardassian Cosmologies via the Hypergeometric Function. International Journal of Astronomy and Astrophysics, 9, 231-246.
https://doi.org/10.4236/ijaa.2019.93017
[46]  Ashmore, L.E. (2022) Data from 14,577 Cosmological Objects and 14 FRBs Confirm the Predictions of New Tired Light (NTL) and Lead to a New Model of the IGM. Journal of Physics: Conference Series, 2197, Article ID: 012003.
https://doi.org/10.1088/1742-6596/2197/1/012003
[47]  Narayan, R. and Bartelmann, M. (1996) Lectures on Gravitational Lensing. arXiv: astro-ph/9606001.
[48]  Bettinelli, M., Simioni, M., Aparicio, A., Hidalgo, S.L., Cassisi, S., Walker, A.R., Piotto, G. and Valdes, F. (2016) The Canarias Einstein Ring: A Newly Discovered Optical Einstein Ring. Monthly Notices of the Royal Astronomical Society: Letters, 461, L67-L71.
https://doi.org/10.1093/mnrasl/slw097
[49]  Eales, S., Dunne, L., Clements, D. and Cooray, A. (2010) The Herschel ATLAS. Publications of the Astronomical Society of the Pacific, 122, 499.
[50]  ALMA Partnership, Vlahakis, C., Hunter, T.R. and Hodge, J.A. (2015) The 2014 ALMA Long Baseline Campaign: Observations of the Strongly Lensed Submillimeter Galaxy HATLAS J090311.6+003906 at . The Astrophysical Journal Letters, 808, L4.
https://doi.org/10.1088/2041-8205/808/1/L4
[51]  Tamura, Y., Oguri, M., Iono, D., Hatsukade, B., Matsuda, Y. and Hayashi, M. (2015) High-Resolution ALMA Observations of SDP.81. I. The Innermost Mass Profile of the Lensing Elliptical Galaxy Probed by 30 Milli-Arcsecond Images. Publications of the Astronomical Society of Japan, 67, 72.
https://doi.org/10.1093/pasj/psv040
[52]  Rybak, M., Vegetti, S., McKean, J.P., Andreani, P. and White, S.D.M. (2015) ALMA Imaging of SDP.81—II. A Pixelated Reconstruction of the CO Emission Lines. Monthly Notices of the Royal Astronomical Society: Letters, 453, L26-L30.
https://doi.org/10.1093/mnrasl/slv092
[53]  Hatsukade, B., Tamura, Y., Iono, D., Matsuda, Y., Hayashi, M. and Oguri, M. (2015) High-Resolution ALMA Observations of SDP.81. II. Molecular Clump Properties of a Lensed Submillimeter Galaxy at . Publications of the Astronomical Society of Japan, 67, 93.
https://doi.org/10.1093/pasj/psv061
[54]  Wong, K.C., Suyu, S.H. and Matsushita, S. (2015) The Innermost Mass Distribution of the Gravitational Lens SDP.81 from ALMA Observations. The Astrophysical Journal, 811, Article 115.
https://doi.org/10.1088/0004-637X/811/2/115
[55]  Hezaveh, Y.D., Dalal, N. and Marrone, D.P. (2016) Detection of Lensing Substructure Using ALMA Observations of the Dusty Galaxy SDP.81. The Astrophysical Journal, 823, Article 37.
https://doi.org/10.3847/0004-637X/823/1/37
[56]  Lee, C.H. (2016) A Closer Look at the Canarias Einstein Ring. Monthly Notices of the Royal Astronomical Society, 462, 3006-3010.
https://doi.org/10.1093/mnras/stw1885
[57]  Rybicki, G. and Lightman, A. (1991) Radiative Processes in Astrophysics. Wiley-Interscience, New York.
[58]  Hjellming, R.M. (1988) Radio Stars in Galactic and Extragalactic Radio Astronomy. Springer-Verlag, Berlin.
https://doi.org/10.1007/978-1-4612-3936-9_9
[59]  Condon, J.J. and Ransom, S.M. (2016) Es-sential Radio Astronomy. Princeton University Press, Princeton.
https://doi.org/10.1515/9781400881161
[60]  Lang, K.R. (1999) Astrophysical Formulae. 3rd Edition, Springer-Verlag, Berlin.

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