What is the Expression for the Waveform of an In-Spiralling Compact Binary?

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SUMMARY

The waveform expression for an in-spiralling compact binary is given by the formula: h(t; θ) = (1/r) Q(φ) (π M F)^(2/3) cos Φ(t). Here, φ represents the angles of the binary's position and orientation, while the chirp mass is defined as ℳ = μ^(3/5) M^(2/5). The frequency function F(t) and phase Φ(t) are integral to the waveform's behavior. The discussion highlights the need for further exploration of the function Q and its derivation, referencing Kip Thorne's work for deeper insights.

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  • Understanding of gravitational waveforms
  • Familiarity with chirp mass calculations
  • Knowledge of phase and frequency functions in wave mechanics
  • Basic concepts of binary systems in astrophysics
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  • Research the derivation of gravitational waveforms in "300 Years of Gravitation"
  • Examine Kip Thorne's book for detailed explanations of Q and its implications
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  • Explore advanced topics in gravitational wave physics and their mathematical formulations
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Astrophysicists, gravitational wave researchers, and students studying compact binary systems will benefit from this discussion.

ergospherical
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This paper gives the following expression for the waveform from an in-spiralling compact binary:\begin{align*}
h(t;\boldsymbol{\theta}) = \frac{1}{r} Q(\boldsymbol{\phi}) \mathcal{M}(\pi \mathcal{M} F)^{2/3} \cos \Phi(t)
\end{align*}where
  • ##\boldsymbol{\phi} = (\theta, \varphi, \psi, \iota)## is a set of angles describing position & orientation of binary
  • ##\mathcal{M} \equiv \mu^{3/5} M^{2/5}## is the chirp mass
  • ##F(t)## is the wave frequency & ##\Phi(t) \equiv 2\pi \int F(t) dt## is the phase
I've been trying to find a derivation of this guy for quite a while, with no luck. The references lead to the book "300 years of Gravitation", which I'd have to wait until tomorrow to have a look at.

Also, what's the function ##Q## explicitly?
 
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(Disclaimer: I am not an expert in this area.) That paper cites an earlier paper (also found on arxiv here), where equation 15 is comparable to OP. The earlier paper points to Kip Thorne's book for the derivation, but also apparently goes into detail about the definition of Q in section IV (edit: see equation 66).
 
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