What it does
Two black holes or neutron stars orbiting each other lose energy to gravitational waves, spiral closer and orbit faster. The waves’ frequency and strength rise together into a chirp that ends when the bodies merge. At leading order, the whole chirp depends on one combination of the masses, the chirp mass 𝓜 = (m₁m₂)^(3/5)/(m₁ + m₂)^(1/5).
This calculator gives, for your binary:
- the chirp mass, total mass and symmetric mass ratio;
- the frequency at the innermost stable circular orbit (ISCO), where the inspiral ends and the merger begins;
- the time and number of cycles from your start frequency (for example a detector’s lower limit) to the ISCO;
- the chirp rate df/dt, the orbit size and orbital period at the start;
- the strain h for a face-on binary, and the characteristic strain, which is what is compared with detector noise;
- charts of the frequency sweep, a sample of the waveform over the final cycles, and strain against frequency.
How to use it
- Type the two masses in solar masses.
- Type the luminosity distance in megaparsecs, and the redshift for distant sources (masses then look (1 + z) times heavier to the detector).
- Type the start frequency in hertz: 10 to 20 Hz suits ground-based detectors; space detectors work at millihertz.
- Read the results and charts, and download the sweep as CSV or a report.
Worked example
A made-up pair of black holes of 30 and 25 solar masses at 400 Mpc has a chirp mass of 23.82 M☉ and a total of 55 M☉, so its inspiral ends at the ISCO near 79.95 Hz. From 20 Hz the page shows 1.09 s and 32.1 cycles from 20 Hz to 79.9 Hz. At 20 Hz the frequency rises at 6.74 Hz per second, the two bodies are about 1,227 km apart, and the face-on strain is 4.32 × 10⁻²², rising to 1.09 × 10⁻²¹ at the ISCO.
Change the masses to two 1.4 M☉ neutron stars at 40 Mpc: the inspiral from 20 Hz now lasts about 2.6 minutes and over 5,000 cycles, ending near 1.57 kHz (in reality the stars touch and merge a little before that).
Tips
- Equal-mass and unequal binaries with the same chirp mass chirp the same way at this order: the mass ratio shows up only at higher order.
- Doubling the distance halves the strain but leaves the timing unchanged.
- For millihertz sources (white dwarf pairs, massive black holes), lower the start frequency to 0.001 Hz or below.
FAQ
How accurate is this? It is the leading-order formula. Post-Newtonian terms, spin and the merger itself change times and cycle counts by roughly 10 to 20 % near the end. It is for study and planning, not for analysing detector data.
Why is the strain so small? A strain of 10⁻²¹ stretches a 4 km arm by about 4 × 10⁻¹⁸ m, a thousandth of a proton’s width.
Is the source frequency the same as the detected one? No: with redshift z, frequencies are lower and durations longer by (1 + z) at the detector. The tool works in the detector frame.
Where the numbers come from
- Leading-order (quadrupole) inspiral: chirp mass, frequency evolution, time to coalescence, number of cycles, phase and strain amplitude (Peters and Mathews, 1963); standard results as in Maggiore, M., Gravitational Waves, Volume 1 (2008), chapter 4.
- ISCO frequency of a Schwarzschild black hole of the total mass, and the characteristic strain h√(2f²/ḟ): standard approximations.
- Checked by fibub against SciPy (solve_ivp of df/dt and quad for the cycle count) and NumPy; GM☉/c³ and GM☉/c² are the IAU nominal values, rounded, and agree with CODATA G.
- Example masses and distance are made up. No detector sensitivity curves are included.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.