近年來，由於系外行星系統的大量發現，人們對了解行星系統的起源和形成歷
史的興趣激增。在這篇博士論文中，我們探討了行星系統之間以及各種行星類
型之間的各種相關性，以便深入了解它們。我們採用角動量逆差（AMD）模型
作為理解行星系統的關鍵框架。AMD 模型不僅建立了堅實的物理基礎，還提供
了有用的分析表達式來闡明行星系統的特徵。我們根據AMD 模型確定了多行
星系統中相鄰行星的週期比和質量比之間的相關性。後續工作研究系統中不同
行星類型之間的條件機率。我們使用這些結果來限制這些行星類型的形成理論。
這些研究結果對行星系統的形成歷史以及這些系統中不同行星類型的存在所造
成的影響進行了完整的描述。

The quest to understand the origin and formation histories of planetary systems has witnessed a surge in interest, fueled by the numerous discoveries of extra-solar planetary systems in recent years. In this doctoral thesis, we explore the various correlations between planetary systems and between various planetary types to gain insights about them. We employ the Angular-Momentum-Deficit (AMD) model as a pivotal framework for understanding generic planetary systems. The AMD model not only establishes a robust physical foundation but also provides valuable analytical expressions to elucidate the characteristics of planetary systems. We have identified a correlation between the period-ratios and mass-ratios of adjacent planets in multiple planetary systems in accordance with the AMD model. The follow-up work looks into the conditional probabilities between different planetary types in a system. We use these results to constrain the formation scenario for these planetary types. These results yield a complete and descriptive study of the formation histories of generic planetary systems and the effects/changes caused by the presence of different planet types in these systems.

Contents ii
List of Tables iii
List of Figures vii
1 Introduction 1
2 The Period-Ratio-Mass-Ratio Correlation 10
2.1 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The Correlation Between Adjacent Planet Pairs . . . . . . . . . . . 16
2.3 The Correlation Between Non-Adjacent Planet Pairs . . . . . . . . 17
2.4 The Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 The Scaling And Spacing Of Planetary Systems 23
3.1 The Scaling and Spacing Rules from the AMD Model . . . . . . . . 26
3.2 The Data and Method . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 A Gas-Free Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 A Gas-Poor Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 The Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Correlations Between Various Planetary Types 44
4.1 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Conditional Probabilities of Super-Earths and Cold Jupiters . . . . 54
4.3 Conditional Probabilities of Warm Jupiters and Super-Earths . . . 58
4.4 Conditional Probabilities of Warm Jupiters and Cold Jupiters . . . 60
4.5 Planet Multiplicity versus Metallicity . . . . . . . . . . . . . . . . . 61
4.6 The Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 Concluding Remarks 67
The Angular Momentum Deficit Model . . . . . . . . . . . . . . . . . . . 77
A Planetary Hamiltonian . . . . . . . . . . . . . . . . . . . . . 77
B Angular Momentum Deficit (AMD) . . . . . . . . . . . . . . 78
C AMD-stability . . . . . . . . . . . . . . . . . . . . . . . . . 79

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