# 啁啾質量

${\displaystyle {\mathcal {M}}={\frac {(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}}}}$

## 定義

${\displaystyle {\mathcal {M}}={\frac {(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}}}}$

${\displaystyle {\mathcal {M}}=\mu ^{3/5}M^{2/5}}$
${\displaystyle {\mathcal {M}}=\left[{\frac {q}{(1+q)^{2}}}\right]^{3/5}M}$
${\displaystyle {\mathcal {M}}=\eta ^{3/5}M}$

## 軌道演化

${\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}={\frac {96}{5}}\pi ^{8/3}\left({\frac {G{\mathcal {M}}}{c^{3}}}\right)^{5/3}f^{11/3}}$

${\displaystyle {\mathcal {M}}={\frac {c^{3}}{G}}\left({\frac {5}{96}}\pi ^{-8/3}f^{-11/3}{\dot {f}}\right)^{3/5}}$

(1)

## 註釋

1. ^ 重寫方程式（1）為[6]

${\displaystyle {\dot {f}}={\frac {96}{5}}\pi ^{8/3}\left({\frac {G{\mathcal {M}}}{c^{3}}}\right)^{5/3}f^{11/3}}$

(2)

將方程式（2）對於時間做積分，則可得到[6]

${\displaystyle {\frac {96}{5}}\pi ^{8/3}\left({\frac {G{\mathcal {M}}}{c^{3}}}\right)^{5/3}t+{\frac {3}{8}}f^{-8/3}+C=0}$

(3)

其中，${\displaystyle C}$是積分常數。

然後，設定${\displaystyle x\equiv t}$${\displaystyle y\equiv {\frac {3}{8}}f^{-8/3}}$，則可對多個數據點(x, y)做直線擬和，從直線的斜率計算出啁啾質量,

## 參考文獻

1. Cutler, Curt; Flanagan, Éanna E. Gravitational waves from merging compact binaries: How accurately can one extract the binary's parameters from the inspiral waveform?. Physical Review D. 1994, 49 (6): 2658–2697. Bibcode:1994PhRvD..49.2658C. arXiv:gr-qc/9402014. doi:10.1103/PhysRevD.49.2658.
2. L. Blanchet; T. Damour; B. R. Iyer; C. M. Will; A. G. Wiseman. Gravitational-Radiation Damping of Compact Binary Systems to Second Post-Newtonian order. Phys. Rev. Lett. (Submitted manuscript). 1995, 74 (3515): 3515–3518. Bibcode:1995PhRvL..74.3515B. PMID 10058225. arXiv:gr-qc/9501027. doi:10.1103/PhysRevLett.74.3515.
3. L. Blanchet; B. R. Iyer; C. M. Will; A. G. Wiseman. Gravitational waveforms from inspiralling compact binaries to second-post-Newtonian order. Classical and Quantum Gravity. 1996, 13 (575): 575–584. Bibcode:1996CQGra..13..575B. arXiv:gr-qc/9602024. doi:10.1088/0264-9381/13/4/002.
4. ^ Abbott, B. P. Properties of the Binary Black Hole Merger GW150914. Physical Review Letters. 2016, 116 (24): 241102. Bibcode:2016PhRvL.116x1102A. PMID 27367378. arXiv:1602.03840. doi:10.1103/PhysRevLett.116.241102.
5. ^ Abbott, B. P. Properties of the binary neutron star merger GW170817. 2018. Bibcode:2018arXiv180511579T. arXiv:1805.11579.
6. Tiwari, Vaibhav; Klimenko, Sergei; Necula, Valentin; Mitselmakher, Guenakh. Reconstruction of chirp mass in searches for gravitational wave transients. Classical and Quantum Gravity. January 2016, 33 (1): 01LT01. Bibcode:2016CQGra..33aLT01T. arXiv:1510.02426. doi:10.1088/0264-9381/33/1/01LT01.