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Numerical Results of the Global Polytropic Model for the Orbits of 15 Planetary Systems

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Numerical Results of the Global Polytropic Model for the Orbits of 15 Planetary Systems 12th Hel.A.S Conference Thessaloniki, 28 June - 2 July, 2015 CONTRIBUTED POSTER Numerical results of the Global Polytropic Model for the orbits of 15 planetary systems V.S. Geroyannis 1, F.N. Valvi 2, and T.G. Dallas 3 1Department of Physics, University of Patras 2Department of Mathematics, University of Patras 3Department of History, Archaeology and Social Anthropology, University of Thessaly Abstract: In the framework of the Global Polytropic Model, we assume hydrostatic equilibrium for a planetary system, and solve the resulting Lane-Emden differential equation in the complex plane. We thus obtain poly- tropic spherical shells defined by successive roots of the Lane-Emden function θ. These shells provide hosting orbits for the planets of the systems under consideration. In the present poster, we present the results within this framework, for 15 stars hosting planetary systems: 55 Cnc, µ Ara, υ And, HD 40307, HD 10180, Kepler-11, Kepler-20, Kepler-26, Kepler-32, Kepler-33, Kepler-62, Kepler-90, Kepler-102, Kepler-186, Kepler-275. b c d e υ And F n = 2:949 dif ≈ 4:5% b c d e f g h Kepler-90 F n = 2:819 dif ≈ 2:0% b c d e f Kepler-33 G n = 2:592 dif ≈ 6:2% c d b e µ Ara G n = 2:554 dif ≈ 1:3% b c i d e j f g h HD 10180 G n = 3:060 dif ≈ 11:9% b c d e f g Kepler-11 G n = 2:779 dif ≈ 8:7% be c f d Kepler-20 G n = 2:779 dif ≈ 4:0% e b c f d 55 Cnc K n = 3:125 dif ≈ 8:0% b c d e f g HD 40307 K n = 2:531 dif ≈ 6:3% b c d e f Kepler-62 K n = 2:773 dif ≈ 4:1% d b c e Kepler-26 K n = 2:531 dif ≈ 4:6% f e b c d Kepler-32 M n = 2:608 dif ≈ 9:1% b c d e g f Kepler-186 M n = 2:530 dif ≈ 6:9% b c d Kepler-275 ? n = 2:837 dif ≈ 3:5% b c d e f Kepler-102 ? n = 2:605 dif ≈ 4:6% 10−2 10−1 100 101 Semi-major axis (AU) 1 GEROYANNIS, VALVI AND DALLAS: Numerical results of GPM for 15 planetary systems • Bars represent the radii of the host stars. At the end of the bar we display the spectral type of the star; the calculated polytropic index n; the average error for the orbit radii dif computed by the Global Polytropic Model. • Black diamonds represent the semi-major of the planets. • Gray error bars represent the respective measurement errors. • Gray dotted lines represent the maximum values for periastron and aphiastron, as calculated from the measured eccentricity. • Gray diamonds represent additional planets, not yet verified. • Gray vertical lines represent the limits of the corresponding polytropic cell. • Black squares represent the computed semi-major axis of the planets. • Gray squares represent the computed semi-major axis of the unverified planets. It also marks an alternative computed semi-major axis for Kepler-32 b, supposing that there is a third undiscovered planet in that polytropic cell. References [1] Geroyannis, V.S. (1988): A Complex-Plane Strategy for computing rotating polytropic models efficiency and accuracy of the complex first-order perturbation theory. The Astrophysical Journal, 327, 273-283. DOI: 10.1086/166188 [2] Geroyannis, V.S., Valvi, F.N. (2012): A Runge-Kutta-Fehlberg code for the complex plane: Comparing with similar codes by applying to polytropic models. International Journal of Modern Physics C, 23, 5-20. DOI: 10.1142/S0129183112500386 [3] Geroyannis, V.S. (1993): A global polytropic model for the solar system: Planetary distances and masses resulting from the complex Lane-Emden differential equation. Earth Moon Planets, 61, 131-139. DOI: 10.1007/BF00572408 [4] Geroyannis, V.S., Valvi, F.N. (1993): The inverse planetary problem: A numerical treatment. Earth Moon Planets, 63, 15-21. DOI: 10.1007/BF00572135 [5] Geroyannis, V.S., Valvi, F.N. (1994): Application of a global polytropic model to the Jupiters system of satellites: A numerical treatment. Earth Moon Planets, 64, 217-225. DOI: 10.1007/BF00572149 [6] Geroyannis, V.S., Dallas, T.G. (1994): Comments on a global polytropic model for the solar and Jovian systems. Earth Moon Planets, 65, 15-19. DOI: 10.1007/BF00572196 [7] Geroyannis V.S., Valvi F.N, Dallas T.G. (2014): Gravitationally quantized orbits in the Solar system: Com- putations based on the Global Polytropic Model. International Journal of Astronomy and Astrophysics, 4, 464-473. DOI: 10.4236/ijaa.2014.43042 [8] Geroyannis V.S. (2014): Gravitational quantization of satellite orbits in the giant planets. arXiv: 1410.5844 [astro-ph.EP] [9] Geroyannis V.S. (2014): Gravitational quantization of exoplanet orbits in 55 Cnc, υ And, Kepler-11, Kepler- 20, and Kepler-90. arXiv: 1411.5390 [astro-ph.EP] [10] Geroyannis V.S. (2015): Gravitational quantization of exoplanet orbits in HD 40307, µ Ara, Kepler-26, Kepler-62, and Kepler-275: Comparing predicted orbits. arXiv: 1501.04189 [astro-ph.EP] [11] Geroyannis V.S. (2015): Gravitational quantization of exoplanet orbits in HD 10180, Kepler-32, Kepler-33, Kepler-102, and Kepler-186. arXiv: 1506.06344 [astro-ph.EP] 12th Hel.A.S Conference 2.
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