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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.08377 |
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| _version_ | 1866913651088162816 |
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| author | Li, Kaye Jiale Wu, Kinwah Younsi, Ziri Li, Tjonnie G. F. |
| author_facet | Li, Kaye Jiale Wu, Kinwah Younsi, Ziri Li, Tjonnie G. F. |
| contents | Relativistic secular perturbation theory has ignited significant interest in uncovering intricate cross-term effects, especially the interplay between 1PN and quadrupole terms. While most existing studies rely on the Lagrangian planetary perturbation method for computing cross terms, a comprehensive Hamiltonian framework for the field has been missing. In this work, we introduce a framework based on von Zeipel transformation, utilizing two sequential canonical transformations to systematically compute cross terms to arbitrary orders. Our results reveal secular cross terms up to quadrupole-squared order, showcasing remarkable consistency with both the Lagrangian method [1] and the effective-field-theory approach [2]. We present leading-order periodic cross terms arising from the interactions between 1PN and quadrupole, and present estimates of higher-order cross terms. It is demonstrated that this method not only accurately predicts the long-term evolution of hierarchical systems but also captures fast oscillations observed in N-body simulations. We identify and validate resonances caused by quadrupole-squared effects, highlighting both consistencies and discrepancies when compared to N-body simulations. These discrepancies underscore the importance of mean-motion resonances, a factor overlooked in current secular perturbation frameworks. Finally, we provide a comprehensive review of the subtleties and limitations inherent to secular perturbation theory, paving the way for future research and advancements in this field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08377 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complete Hamiltonian Framework of Relativistic Hierarchical Triple Systems: Capabilities and Limitations of Secular Perturbation Theory Li, Kaye Jiale Wu, Kinwah Younsi, Ziri Li, Tjonnie G. F. High Energy Astrophysical Phenomena General Relativity and Quantum Cosmology Relativistic secular perturbation theory has ignited significant interest in uncovering intricate cross-term effects, especially the interplay between 1PN and quadrupole terms. While most existing studies rely on the Lagrangian planetary perturbation method for computing cross terms, a comprehensive Hamiltonian framework for the field has been missing. In this work, we introduce a framework based on von Zeipel transformation, utilizing two sequential canonical transformations to systematically compute cross terms to arbitrary orders. Our results reveal secular cross terms up to quadrupole-squared order, showcasing remarkable consistency with both the Lagrangian method [1] and the effective-field-theory approach [2]. We present leading-order periodic cross terms arising from the interactions between 1PN and quadrupole, and present estimates of higher-order cross terms. It is demonstrated that this method not only accurately predicts the long-term evolution of hierarchical systems but also captures fast oscillations observed in N-body simulations. We identify and validate resonances caused by quadrupole-squared effects, highlighting both consistencies and discrepancies when compared to N-body simulations. These discrepancies underscore the importance of mean-motion resonances, a factor overlooked in current secular perturbation frameworks. Finally, we provide a comprehensive review of the subtleties and limitations inherent to secular perturbation theory, paving the way for future research and advancements in this field. |
| title | Complete Hamiltonian Framework of Relativistic Hierarchical Triple Systems: Capabilities and Limitations of Secular Perturbation Theory |
| topic | High Energy Astrophysical Phenomena General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2501.08377 |