有關天鵝座X-1及蟹狀星雲於X射線/伽瑪射線波段的輻射機制，可以透過量測其偏極狀態來進一步了解。本篇論文中，我們使用搭載於立方衛星上的康普頓偏光儀 (Compol) 來觀測上述兩個目標的偏極性。並使用 MEGAlib 進行模擬分析。與先前的探討不同的是，此處我們組合四個探測器單元作為新的模型，相對於過去增加約四倍的有效面積。我們將模型放置於低地軌道並以正軸方向觀測天鵝座 X-1 一千萬秒，其可量測最小偏極值 (MDP) 在 160-250 keV 波段中可低至 12.4 %、250-400keV 可低至 12.5 % 及 400-2000 keV 可到 27.6 %。此最小偏極值結果比過去利用 INTEGRAL/IBIS 或 INTEGRAL/SPI 研究的偏極值都還要低，代表 Compol 足夠靈敏而得以量測在軟伽瑪射線波段的偏極資訊。
我們也對蟹狀星雲做同樣的觀測，並得到比觀測天鵝座 X-1 還低的MDP。在160-250 keV 範圍量到 5.8 %、250-400 keV 到 6.4 % 而 400-2000 keV 到 11.3 %。這個最小偏極值也低於其他X射線/伽瑪射線儀器所測量到的偏極值，因此Compol可以提供在項位積分(包含脈衝星及其脈衝星星雲)下的蟹狀星雲在軟伽瑪射線的偏極資訊。此外，我們對蟹狀星雲進行相位分辨並測量不同相位的最小偏極值。透過估計不同相位的光通量可以計算出在離脈相位的最小偏極值在 160-250 keV 可低至 8.35 %、250-400 keV 至 11.56 % 而 400-2000 keV 至 24.16 %。並且，此結果可以驗證由離脈相位的偏極值及偏極角度是否會隨著相位改變 (由AstroSat CZTI測量到的結果)。而在第一峰值及第二峰值相位，我們估算的最小偏極值在兩個較低的波段中，已經到達 PoGO+ 及 POLAR 兩儀器所量測到的偏極值上限。

Measuring the polarization state of Cygnus X-1 and Crab in the X-ray/gamma-ray band is of great importance in understanding their emission mechanism. In this thesis, we propose to use a small Compton polarimeter (Compol) on board a Cubesat to observe the gamma-ray polarization of Cygnus X-1 and Crab. Compared to previous results in Yang et al. 2020 and Hsiang (2021), we increase our detector units by a factor of 4, gaining a larger effective area. Therefore, we estimate the improvement of the polarization measurement with a four times larger detector effective area by running simulations with MEGAlib package. Results show that a 10-Ms, on-axis observation in low-Earth orbit (LEO), the minimum detectable polarization (MDP) for Cygnus X-1 can reach to about 12.4 % in 160-250 keV, 12.5 % in 250-400 keV and about 27.6 % in 400-2000 keV. The MDPs we derived are lower than the polarization degree measured in previous studies (e.g. INTEGRAL/IBIS and INTEGRAL/SPI), which means this instrument is sensitive enough to yield useful polarization information in the gamma-ray regime.
We performed the same procedure to study the MDP of the phase-integrated Crab, and found that we can reach a lower MDP than that of Cygnus X-1. The MDP in 160-250 keV can reach 5.8 %, 6.4 % in 250-400 keV, and 11.3 % in 400-2000 keV. These results are also lower than the polarization degrees reported by those measured by other X-ray/gamma-ray instruments, therefore we are able to measure the polarization degree of the phase-integrated Crab using Compol. Moreover, we evaluate the polarization information on the phase-resolved Crab. Estimations on the flux in different phases show that the MDP in off-pulse phase can reach to about 8.35 % in 160-250 keV, 11.56 % in 250-400 keV, and 24.16 % in 400-2000 keV. We can further investigate whether there exist a variation in polarization angle and polarization degree reported by AstroSat CZTI in the off-pulse phase as well. For P1 phase and P2 phase, the estimated MDP in lower energy bands have reached the upper limit that recent studies from PoGO+ and POLAR measured.

1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Compton Polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Instrument Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Simulation and Data Analysis . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Minimum Detectable Polarization of Cygnus X-1 with Compol 12
2.1 Introduction of Cygnus X-1 . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Input Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Data Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 MDP of Cygnus X-1 using Compol . . . . . . . . . . . . . . . . . . . 18

3 Minimum Detectable Polarization of Crab with Compol 20
3.1 Introduction of the Crab . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Input Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Data Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 MDP of the Crab using Compol . . . . . . . . . . . . . . . . . . . . . 23

4 Discussion 25
4.1 Comparison on different model designs . . . . . . . . . . . . . . . . . 25
4.2 The phase-resolved polarization measurement of the Crab . . . . . . . . 31
4.3 Polarization variation of the Crab in off-pulse phase . . . . . . . . . . . 36

5 Summary 40

A Detailed Information in deriving the MDP 42
A.1 MDP information of 4 detector-unit model described in Section 1.3 for CygnusX-1................................ 43
A.2 MDP information of 4 detector-unit model described in Section 1.3 for theCrab.................................. 47
A.3 MDP Information of 1 detector-unit model described in Section 4.1 for CygnusX-1................................ 51
A.4 MDP Information of 1 detector-unit model described in Section 4.1 for theCrab.................................. 63

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