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Gravitational Waves: Inspiral, Merger, and Detection

An interactive compact-binary inspiral. The main view is the physical system: two black holes orbit on a Keplerian separation $a = (G M / \omega_{\mathrm{orb}}^2)^{1/3}$ that shrinks as gravitational waves carry away energy, while the leading quadrupole radiation spreads outward as a two-arm spiral that tightens and brightens as the frequency rises, then merges and rings down. The chirp mass $M_c = (m_1 m_2)^{3/5}/(m_1+m_2)^{1/5}$ sets the sweep $df/dt = (96/5) \pi^{8/3}(G M_c/c^3)^{5/3} f^{11/3}$, so $f \sim \tau^{-3/8}$ climbs to merger and the strain amplitude $h = (4/D)(G M_c/c^2)^{5/3}(\pi f/c)^{2/3}$ grows with it (about $10^{-21}$ for a 30+30 solar-mass binary at 400 Mpc, the GW150914 case). Diagnostic strips show the chirp strain $h(t)$, a matched filter that peaks sharply at coalescence and recovers the chirp mass, and a LIGO arm whose 4 km length changes by $h L / 2$, a sub-proton displacement.

Figure 1. Two black holes spiral together on a shrinking orbit and radiate a two-arm gravitational-wave ripple that tightens into the merger and rings down; the chirp strain, a matched filter that recovers the chirp mass, and a LIGO arm changing length by h L / 2 (a sub-proton displacement) are shown as diagnostics. Method: Keplerian inspiral with leading-quadrupole radiation, the chirp-mass frequency sweep, antenna response, and matched filtering; Canvas2D, deterministic.
m1 (Msun)30
m2 (Msun)30
distance (Mpc)400

WHAT TO TRY

  • Raise the component masses m1 and m2: the chirp mass climbs, the inspiral sweeps to higher frequency faster, and the two-arm radiation spiral tightens and brightens into merger. Heavier binaries chirp louder and quicker.
  • Push the distance out in Mpc: the strain h_peak falls as 1/distance and the matched-filter SNR drops with it. Sensitivity volume, not range, is what sets detection rates.
  • Watch the orbital separation a = (G M / omega^2)^(1/3) shrink in real time as the waves carry off energy: the frequency rising is the binary spiraling in, not a free parameter.