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Keywords = Kepler manifold

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15 pages, 595 KiB  
Article
Application of Manifold Corrections in Tidal Evolution of Exoplanetary Systems
by Qian-Qian Xiao, Ying Wang, Fu-Yao Liu, Chen Deng and Wei Sun
Symmetry 2023, 15(1), 253; https://doi.org/10.3390/sym15010253 - 16 Jan 2023
Viewed by 2401
Abstract
The discovery of numerous close-in planets has updated our knowledge of planet formation. The tidal interaction between planets and host stars has a significant impact on the orbital and rotational evolution of the close planets. Tidal evolution usually takes a long time and [...] Read more.
The discovery of numerous close-in planets has updated our knowledge of planet formation. The tidal interaction between planets and host stars has a significant impact on the orbital and rotational evolution of the close planets. Tidal evolution usually takes a long time and requires reliable numerical methods. The manifold correction method, which strictly satisfies the integrals dissipative quasiintegrals of the system, exhibits good numerical accuracy and stability in the quasi-Kepler problem. Different manifold correction methods adopt different integrals or integral invariant relations to correct the numerical solutions. We apply the uncorrected five- and six-order Runge–Kutta–Fehlberg algorithm [RKF5(6)], as well as corrected by the velocity scaling method and Fukushima’s linear transformation method to solve the tidal evolution of exoplanet systems. The results show that Fukushima’s linear transformation method exhibits the best performance in the accuracy of the semimajor axis and eccentricity. In addition, we predict the tidal timescale of several current close exoplanetary systems by using this method. Full article
(This article belongs to the Special Issue Symmetry in Gravity Research)
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15 pages, 301 KiB  
Article
Representing Functions in H2 on the Kepler Manifold via WPOAFD Based on the Rational Approximation of Holomorphic Functions
by Zeyuan Song and Zuoren Sun
Mathematics 2022, 10(15), 2729; https://doi.org/10.3390/math10152729 - 2 Aug 2022
Cited by 1 | Viewed by 1854
Abstract
The central problem of this study is to represent any holomorphic and square integrable function on the Kepler manifold in the series form based on Fourier analysis. Because these function spaces are reproducing kernel Hilbert spaces (RKHS), three different domains on the Kepler [...] Read more.
The central problem of this study is to represent any holomorphic and square integrable function on the Kepler manifold in the series form based on Fourier analysis. Because these function spaces are reproducing kernel Hilbert spaces (RKHS), three different domains on the Kepler manifold are considered and the weak pre-orthogonal adaptive Fourier decomposition (POAFD) is proposed on the domains. First, the weak maximal selection principle is shown to select the coefficient of the series. Furthermore, we prove the convergence theorem to show the accuracy of our method. This study is the extension of work by Wu et al. on POAFD in Bergman space. Full article
(This article belongs to the Special Issue Geometry of Manifolds and Applications)
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