Chirp mass

In astrophysics the chirp mass of a compact binary system determines the leading-order orbital evolution of the system as a result of energy loss from emitting gravitational waves. Because the gravitational wave frequency is determined by orbital frequency, the chirp mass also determines the frequency evolution of the gravitational wave signal emitted during a binary's inspiral phase. In gravitational wave data analysis it is easier to measure the chirp mass than the two component masses alone.

Definition from component masses

A two-body system with component masses ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ has a chirp mass of

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

The chirp mass may also be expressed in terms of the total mass of the system ${\displaystyle M=m_{1}+m_{2}}$ and other common mass parameters:

• the reduced mass ${\displaystyle \mu ={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}}$:

  ${\displaystyle {\mathcal {M}}=\mu ^{3/5}M^{2/5},}$

• the mass ratio ${\displaystyle q=m_{1}/m_{2}}$:

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

• the symmetric mass ratio ${\displaystyle \eta ={\frac {m_{1}m_{2}}{(m_{1}+m_{2})^{2}}}={\frac {\mu }{M}}={\frac {q}{(1+q)^{2}}}=\left({\frac {m_{\mathrm {g} eo}}{M}}\right)^{2}}$:

  ${\displaystyle {\mathcal {M}}=\eta ^{3/5}M.}$

  The symmetric mass ratio reaches its maximum value ${\displaystyle \eta ={\frac {1}{4}}}$ when ${\displaystyle m_{1}=m_{2}}$, and thus ${\displaystyle {\mathcal {M}}=(1/4)^{3/5}M\approx 0.435\,M.}$

• the geometric mean of the component masses ${\displaystyle m_{geo}={\sqrt {m_{1}m_{2}}}}$:

  ${\displaystyle {\mathcal {M}}=m_{\mathrm {g} eo}\left({\frac {m_{\mathrm {g} eo}}{M}}\right)^{1/5},}$

  If the two component masses are roughly similar, then the latter factor is close to ${\displaystyle (1/2)^{1/5}=0.871,}$ so ${\displaystyle {\mathcal {M}}\approx 0.871\,m_{\mathrm {g} eo}}$. This multiplier decreases for unequal component masses but quite slowly. E.g. for a 3:1 mass ratio it becomes ${\displaystyle {\mathcal {M}}=0.846\,m_{\mathrm {g} eo}}$, while for a 10:1 mass ratio it is ${\displaystyle {\mathcal {M}}=0.779\,m_{\mathrm {g} eo}.}$

Orbital evolution

In general relativity, the phase evolution of a binary orbit can be computed using a post-Newtonian expansion, a perturbative expansion in powers of the orbital velocity ${\displaystyle v/c}$. The first order gravitational wave frequency, ${\displaystyle f}$, evolution is described by the differential equation

${\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}={\frac {96}{5}}\pi ^{8/3}\left({\frac {G{\mathcal {M}}}{c^{3}}}\right)^{5/3}f^{11/3}}$,[1]

where ${\displaystyle c}$ and ${\displaystyle G}$ are the speed of light and Newton's gravitational constant, respectively.

If one is able to measure both the frequency ${\displaystyle f}$ and frequency derivative ${\displaystyle {\dot {f}}}$ of a gravitational wave signal, the chirp mass can be determined.[4][5][note 1]

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

(1)

To disentangle the individual component masses in the system one must additionally measure higher order terms in the post-Newtonian expansion.[1]

See also

• Reduced mass
• Two-body problem in general relativity

Note

1. 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. Cutler, Curt; Flanagan, Éanna E. (15 March 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:gr-qc/9402014. Bibcode:1994PhRvD..49.2658C. doi:10.1103/PhysRevD.49.2658.
2. L. Blanchet; T. Damour; B. R. Iyer; C. M. Will; A. G. Wiseman (1 May 1995). "Gravitational-Radiation Damping of Compact Binary Systems to Second Post-Newtonian order". Phys. Rev. Lett. 74 (18): 3515–3518. arXiv:gr-qc/9501027. Bibcode:1995PhRvL..74.3515B. doi:10.1103/PhysRevLett.74.3515. PMID 10058225.
3. Blanchet, Luc; Iyerddag, Bala R.; Will, Clifford M.; Wiseman, Alan G. (April 1996). "Gravitational waveforms from inspiralling compact binaries to second-post-Newtonian order". Classical and Quantum Gravity. 13 (4): 575–584. arXiv:gr-qc/9602024. Bibcode:1996CQGra..13..575B. doi:10.1088/0264-9381/13/4/002.
4. Abbott, B. P.; et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2016). "Properties of the Binary Black Hole Merger GW150914". Physical Review Letters. 116 (24): 241102. arXiv:1602.03840. Bibcode:2016PhRvL.116x1102A. doi:10.1103/PhysRevLett.116.241102. PMID 27367378.
5. Abbott, B. P.; et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2018). "Properties of the binary neutron star merger GW170817". arXiv:1805.11579. Bibcode:2018arXiv180511579T. Cite journal requires |journal= (help)
6. Tiwari, Vaibhav; Klimenko, Sergei; Necula, Valentin; Mitselmakher, Guenakh (January 2016). "Reconstruction of chirp mass in searches for gravitational wave transients". Classical and Quantum Gravity. 33 (1): 01LT01. arXiv:1510.02426. Bibcode:2016CQGra..33aLT01T. doi:10.1088/0264-9381/33/1/01LT01.