凌星時間變化被認為是一項能夠偵測系外行星與研究多行星系統軌道參數的工具。我們決定研究的主要目標系統是克卜勒感興趣天體：KOI 168.03-01、KOI 834.01-05 和 KOI 1270.01-02。為了進一步瞭解克卜勒太空望遠鏡觀測到的行星系統，我們使用解析公式和數值模型來模擬其觀測到的凌星時間變化。透過馬可夫鏈蒙地卡羅抽樣方式，解析公式將被用於擬合凌星時間變化數據，並獲得我們感興趣的行星系統可能的軌道性質，這使我們能夠從凌星時間變化訊號中得到行星質量。接著，我們利用凌星時間變化的數值工具瞭解行星系統的其他物理特性。透過這兩個過程，我們能夠更加瞭解如何利用凌星時間變化研究行星系統的物理參數。

The transit timing variation has known to be a fundamental tool for exoplanet detection and understanding the orbital parameters within multi-planetary systems. The main target systems we decided to study are Kepler Objects of Interest: KOI 168.03-01, KOI 834.01-05 and KOI 1270.01-02. In order to further understand the nature of the systems observed by the Kepler Space Telescope, we try to model their observed transit timing variations by both analytic formulae and numerical model. Analytic formulae will be employed to fit the transit timing variation data through Markov chain Monte Carlo sampling and get the probable orbital properties of interested planetary systems, thus this approach allowed us to extract nominal planet masses from the transit timing variation signals. Subsequently, we utilize the numerical tool for transit timing variation to reveal more other physical properties of the planetary systems. Through the dual process, we can enrich our understanding of how transit timing variation discovers the physical parameters of planetary systems.

Contents ii
List of Tables . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . iv

1 Introduction . . . . . . . . . . . . . . . . . . 1
2 The Data . . . . . . . . . . . . . . . . . . . . . . 4
3 Methodology . . . . . . . . . . . . . . . . . 7

3.1 Analytic Formulae for TTV signals . . . . . . . . . . . . . . . . . . 8
3.2 Applications of Analytic Formulae . . . . . . . . . . . . . . . . . . 12
3.3 Realizations of Zfree . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 A Numerical Tool for TTV : TTVFast . . . . . . . . . . . . . . . . 16

4 Results . . . . . . . . . . . . . . . . . . 22
4.1 Target Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Analytic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2.1 Preliminary Results of Fitting . . . . . . . . . . . . . . . . . 24
4.2.2 Realizations of Zfree . . . . . . . . . . . . . . . . . . . . . . 28
i
4.2.3 The Discussion of Zfree Realizations . . . . . . . . . . . . . 30
4.3 TTVFast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.1 Fitting Results of TTVFast . . . . . . . . . . . . . . . . . . 31
4.3.2 Parameters Adapted in TTVFast . . . . . . . . . . . . . . . 32
4.3.3 Parameters between Numerical and Analytic Results . . . . 35

5 Conclusions and Future Works . . . . . . . . . . . . . . . . . 45

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