Binary inspiral chirp mass

Let M be total mass, mu the reduced mass and nu their dimensionless ratio mu over M. For a Newtonian circular orbit the binding energy is negative and becomes more negative as separation shrinks. The quadrupole luminosity is positive, so orbit-averaged energy balance makes the separation decrease. Kepler's relation then makes the orbital angular frequency increase. Losing energy does not mean that the orbital speed or wave frequency must decrease.

The dominant quadrupole strain oscillates at twice the orbital cyclic frequency. Thus its gravitational-wave frequency is orbital angular frequency divided by pi. Combining energy balance with that frequency relation produces a power law whose mass dependence is captured by chirp mass. The derivation assumes a slowly evolving, approximately circular, isolated binary, with radiation as the dominant secular energy loss and no important spin, tidal or eccentric corrections at this leading order.

Integrating the frequency-sweep power law gives a remaining time proportional to frequency to the minus eight-thirds power. The integration endpoint is a formal infinite-frequency time in the leading model. It approximates an inspiral timescale when the system is still in the controlled weak-orbital-field regime, rather than predicting an exact merger instant through the strong-field plunge. A local combination of frequency and its derivative provides the same leading time without explicitly solving for the mass.

The formula makes useful diagnostics possible: doubling frequency increases its derivative by two to the eleven-thirds power while reducing remaining leading time by two to the eight-thirds power. A fixed sign or dimension error in energy balance would violate this consistency. Higher-order phasing and environmental effects can alter the relation, so applying it to data requires a waveform hypothesis rather than simply observing any signal with an increasing pitch.

Cosmological propagation stretches observed times by one plus redshift and lowers observed frequencies by that factor. The frequency sweep therefore measures a redshifted chirp mass equal to one plus redshift times the source chirp mass in the usual expanding-universe treatment. Converting it to intrinsic masses needs independent redshift information or additional distance and cosmological assumptions. Gravitational-wave phasing alone does not label the source redshift without those inputs.

At a fixed chirp mass, different mass ratios can share the leading phase evolution. Higher-order phase terms, spins and harmonics help distinguish them. The strain amplitude also depends on luminosity distance, inclination and antenna projection, creating additional degeneracies. A louder event is not automatically a more massive source, and a single detector does not separately measure both polarizations. These distinctions make the leading chirp useful as a controlled parameter combination rather than a complete source reconstruction.