# Chirp mass

The chirp mass of a compact binary star system with component masses ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ is given by ${\displaystyle {\mathcal {M}}={\frac {(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}}}}$.[1][2] In general relativity, the chirp mass determines the leading-order amplitude and frequency evolution of the gravitational-wave signal emitted by the binary during its inspiral. To lowest order in a post-Newtonian expansion, the evolution of the waveform’s phase depends only on the chirp mass:

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

(1)

where ${\displaystyle c}$, ${\displaystyle G}$, ${\displaystyle f}$ and ${\displaystyle {\dot {f}}}$ are the speed of light, Newton's gravitational constant, the observed gravitational wave frequency (twice the orbital frequency) and the first time derivative of ${\displaystyle f}$, respectively.[3][4] Accordingly, in gravitational-wave astronomy, the chirp mass can be accurately measured by detectors from frequency and gravitational strain of the gravitational wave.[5]

Rewrite equation (1) to obtain the frequency evolution of gravitational waves from a coalescing binary:[6]

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

(2)

Integrating equation (2) with respect to time gives:[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)

where C is the constant of integration. Furthermore, on identifying ${\displaystyle x\equiv t}$ and ${\displaystyle y\equiv {\frac {3}{8}}f^{-8/3}}$, the chirp mass can be calculated from the slope of the line fitted through the data points (x, y).

## References

1. ^ L. Blanchet; T. Damour; B. R. Iyer; C. M. Will; A. G. Wiseman (1995). "Gravitational-Radiation Damping of Compact Binary Systems to Second Post-Newtonian order". Phys. Rev. Lett. 74 (3515): 3515–3518. arXiv:. Bibcode:1995PhRvL..74.3515B. doi:10.1103/PhysRevLett.74.3515.
2. ^ L. Blanchet; B. R. Iyer; C. M. Will; A. G. Wiseman (1996). "Gravitational waveforms from inspiralling compact binaries to second-post-Newtonian order". Classical Quantum Gravity. 13 (575): 575–584. arXiv:. Bibcode:1996CQGra..13..575B. doi:10.1088/0264-9381/13/4/002.
3. ^ B. P. Abbott (LIGO Scientific Collaboration and Virgo Collaboration) et al. (2016). "Observation of Gravitational Waves from a Binary Black Hole Merger". Physical Review Letters. 116 (6): 061102. arXiv:. Bibcode:2016PhRvL.116f1102A. doi:10.1103/PhysRevLett.116.061102. PMID 26918975.
4. ^ Curt Cutler and Éanna E. Flanagan (1994). "Gravitational waves from merging compact binaries: How accurately can one extract the binary's parameters from the inspiral waveform?". Physical Review D. 49 (6): 2658–2697. arXiv:. Bibcode:1994PhRvD..49.2658C. doi:10.1103/PhysRevD.49.2658.
5. ^ Jim Wheeler (2013), Lecture Notes: Gravitational waves (PDF), Department of Physics - Utah State University, retrieved 14 February 2016
6. ^ a b V Tiwari, S Klimenko, V Necula and G Mitselmakher (2016). "Reconstruction of chirp mass in searches for gravitational wave transients". Classical and Quantum Gravity. 33 (1). Bibcode:2016CQGra..33aLT01T. doi:10.1088/0264-9381/33/1/01LT01.