在本論文中，主要探討雙黑洞動態系統在星系重力場的影響下，速度比與質量比之間的關係。在天文學的研究裡，普遍認為活躍星系核中心是超大質量黑洞。Wang et al. [11] 在雙活躍星系核的研究中，發現雙活躍星系核速度比與質量比之間有著顯著的相關性。因此，在考慮星系重力場的影響下，我們提供不同初始條件的雙黑洞模型，對速度比與質量比的關係進行研究。

In this thesis, we primarily study the relationship between velocity ratio and mass ratio in the binary black holes with galactic potential. In the field of astronomy, it is widely accepted that the center of active galactic nuclei hosts supermassive black hole. In the study of dual active galactic nuclei (AGNs), Wang et al. [11] discovered a significant correlation between the velocity ratio and mass ratio of
dual AGNs. Therefore, considering the influence of galactic potential, we explore the relationship between velocity ratio and mass ratio by choosing different initial conditions in binary black holes models.

摘要 . . . . . . . . . . . . . . . . . . i

Abstract . . . . . . . . . . . . . . . ii

Chapter 1 Introduction . . . . . . . . . . . . . . . . 1

Chapter 2 Analytical Formulas . . . . . . . . . . . . .3
2.1 Circular Orbit . . . . . . . . . . . . 3
2.2 Elliptical Orbit . . . . . . . . . . . 7

Chapter 3 The Orbit Behaviors of Supermassive Binary Black Hole in Galaxical Potential . . . . . . . . . . . . . . . . . .14
3.1 Equations of Motion . . . . . . . . . . . . . 16
3.2 c = 0 Case with a circular orbit . . . . . . 19
3.3 c ̸= 0 Case with elliptical orbits . . . . . 22
3.3.1 Equal mass case . . . . . . . . . . . 22
3.3.2 Unequal masses cases . . . . . . . . .24
3.3.3 Different total mass of galaxy . . . .31

Chapter 4 Conclusion . . . . . . . . . . . . . . . . . .33

A Kepler’s Laws of Planetary Motion . . . . 34
A.1 Kepler’s 2nd Law . . . .34
A.2 Kepler’s 1st Law . . . .36
A.3 Kepler’s 3rd Law . . . .39

B Three Dimensional Transformation . . . . .41

Reference . . . . . . . . . . . . . . . . . . . . . . . .43

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