= Stipulated chirps and detector projections
To construct an original example rather than imitate <GW150914>, set $G=6.67430\times10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}$, $c=299\,792\,458\,\mathrm{m\,s^{-1}}$, and adopt the rounded <solar mass> $M_\odot=1.98847\times10^{30}\,\mathrm{kg}$. Compare $10+10$ and $20+20$ <solar mass> binaries starting at $20\,\mathrm{Hz}$. Stop each at the stipulated <orbital compactness> threshold
$$
x=\frac{GM}{ac^2}=\left(\frac{\pi GMf}{c^3}\right)^{2/3}=0.10,\qquad
f_{\rm cut}=\frac{c^3}{\pi GM}(0.10)^{3/2}.
$$
This is a conservative pedagogical stopping rule, not a claimed error bound: the relative speed already reaches $\sqrt{0.10}\,c$. Higher-order corrections accumulate <wave phase> well before merger, and precision inference requires better models. The two cutoffs are approximately $102.2$ and $51.1\,\mathrm{Hz}$ respectively; doubling the <chirp mass> multiplies $\dot f$ at fixed $f$ by $2^{5/3}$.
For a nearby stipulated source at distance $D$, neglect <redshift> and define inclination $\iota$ as the angle between the orbital <angular momentum> and the line of sight. The leading <quadrupole formula> yields
$$
A(f)=\frac{4(G\mathcal M)^{5/3}(\pi f)^{2/3}}{c^4D},\qquad
h_+=A(f)\frac{1+\cos^2\iota}{2}\cos\Phi,\qquad
h_\times=A(f)\cos\iota\sin\Phi.
$$
The numerator of $A$ has units $\mathrm{m^5\,s^{-4}}$, matching $c^4D$, so both polarizations are dimensionless. Its $f^{2/3}$ growth and the rising <frequency> produce the tightening, strengthening oscillations. For face-on viewing the polarization components have equal magnitudes in <phase quadrature>; for edge-on viewing the cross component vanishes, agreeing with <circular-binary quadrupole radiation>.
\Image[/newtonian-binary-chirps-detector-projections.png]
{title=Stipulated <Newtonian binary chirps> and two ideal detector projections}
{description=Calculated examples only: no measured <LIGO> strain or noise. Upper panels compare equal-mass binaries up to $x=0.10$. Lower panels use $20+20$ <solar masses>, $D=500\,\mathrm{Mpc}$ and $\iota=60$ degrees. Ideal overhead detectors have $(F_+,F_\times)=(1,0)$ and $(1/2,\sqrt3/2)$; the second record has a stipulated $7\,\mathrm{ms}$ delay, then is shifted back for comparison. These geometries and delay are not fitted Hanford–Livingston responses.}
Shifting an arrival time aligns the chirp evolution, but different polarization projections need not overlay after only a sign flip. The last panel divides the ideal projected records by their known envelope factors to expose this phase difference. A physical source model must jointly explain those differences, not erase them by arbitrary rescaling. Independent integration of the separation equation, the radiated <power> and $d\Phi/dt=2\pi f$ checks the analytic chirp against orbital <energy> balance over the entire stipulated interval.
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