Gravitational Wave Calculator

Enter the two masses, distance, and starting frequency to see the chirp mass, merger timeline, and signal strength.

Starting values — Gravitational Wave Calculator

Primary mass m₁ (M☉)	36
Secondary mass m₂ (M☉)	29
Luminosity distance (Mpc)	440
Start frequency f₀ (Hz)	20

Change any control above and the results recompute from these.

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A four-kilometre arm moves less than a proton 🖖

LIGO detects a gravitational wave by watching two 4-kilometer laser arms stretch and squeeze by about one part in 10²¹ — a distortion a few hundred times smaller than the width of a proton. That signal only becomes trustworthy because the same waveform has to show up in detectors thousands of kilometers apart, letting a real chirp be told apart from local noise like passing trucks or distant earthquakes. The frequency doesn't stay put either: as the two objects spiral inward, orbital energy bleeds away as gravitational radiation, so the wave sweeps upward in frequency right up to merger — the 'chirp' this tool's curve is named for.

What the chirp mass tells you 🖖

When two compact objects spiral together they don't broadcast their two masses separately. The waveform is set almost entirely by one combination, the chirp mass M_c = (m₁m₂)^3/5/(m₁+m₂)^1/5. That single number controls how fast the frequency sweeps upward, so measuring the rate of the chirp hands you the chirp mass directly — which is why this tool puts it front and centre.

Mergers as standard sirens 🖖

The amplitude of a gravitational wave depends on the source's true luminosity distance in a way physicists can compute from first principles — no calibration against Cepheids or supernovae required. That makes each merger a standard siren: read the chirp for the masses, read the amplitude for the distance. Pair that with a redshift and you get an independent measurement of the Hubble constant — exactly how the 2017 neutron-star merger GW170817 weighed in on how fast the universe expands.

Problems solved in full

36 and 29 solar masses at 440 Mpc 6 steps
36 and 29 solar masses at 440 Mpc. Find the chirp mass, how long the signal lasts, and how far LIGO's mirrors actually move — then find the one number on the panel that does not depend on any of the inputs.
1. Two masses enter the waveform in only one combination, and this is it. The chirp mass is what a detector measures directly — the individual masses have to be teased out later, from the part of the signal this formula cannot see.
  
  $M = \frac{( m _{1} m _{2} ) ^{3/5}}{( m _{1} + m _{2} ) ^{1/5}} = \frac{( 36 \times 29 ) ^{3/5}}{6 5 ^{1/5}} = \frac{104 4 ^{0.6}}{6 5 ^{0.2}} = \frac{64.77}{2.305} = 28.1 M_{⊙}$
2. Time to merger scales as the chirp mass to the −5/3 and the entry frequency to the −8/3. Both exponents are large, so both inputs matter far more than they look.
  
  $τ \approx 2.18 (\frac{1.21}{M})^{5/3} (\frac{100}{f _{0}})^{8/3} = 2.18 (0.04306)^{5/3} (5)^{8/3} = 0.843 s$
3. The innermost stable circular orbit is where the inspiral has to stop, and its frequency goes inversely with total mass: a heavier binary merges at a lower pitch. At 65 solar masses that is 68 Hz — near the bottom of LIGO's band, which is why this event was heard as a thump rather than a chirp.
  
  $f_{ISCO} = \frac{c ^{3}}{6 6 π G M _{tot}} = \frac{2.694 \times 1 0 ^{25}}{14.70 \times 2.711 \times 1 0 ^{22}} = 67.6 Hz$
4. Strain is dimensionless: it is the fractional stretch of space. Multiply by the arm length to get a distance.
  
  $h_{peak} \approx 1.16 \times 1 0^{- 21} ⟹ Δ L = 4000 \times 1.16 \times 1 0^{- 21} = 4.6 \times 1 0^{- 18} m$
5. Put that distance beside something. A proton is 8.4 × 10⁻¹⁶ m across its charge radius, and the mirrors moved by less than a hundredth of that.
  
  $\frac{4.6 \times 1 0 ^{- 18}}{8.41 \times 1 0 ^{- 16}} = \frac{1}{181} of a proton’s radius$
6. Now the row that never changes. Nothing about the binary survives into it — this is a ceiling, not a measurement.
  
  $L_{GW} \sim \frac{c ^{5}}{5 G} = \frac{2.42 \times 1 0 ^{42}}{3.34 \times 1 0 ^{- 10}} = 7.26 \times 1 0^{51} W = 1.90 \times 1 0^{25} L_{⊙}$
Answer

The luminosity, and it is c⁵/5G. Every other row moves when you touch a slider; that one is 1.90 × 10²⁵ L☉ whatever the masses, the distance or the starting frequency, because it is built from the speed of light and the gravitational constant and nothing else. It is the largest power any process in general relativity can radiate, and for a few milliseconds this merger was putting out more of it than every star in the observable universe puts out in light. The other end of the scale is on the same panel: 1.16 × 10⁻²¹ of strain across LIGO's 4 km arms is a length change of 4.6 × 10⁻¹⁸ m, which is a 181st of the radius of a proton. Both numbers describe the same event.

$M = 28.1 M_{⊙}, τ = 843 ms, f_{ISCO} = 67.6 Hz, L_{GW} = \frac{c ^{5}}{5 G}$
Two neutron stars of 1.4 solar masses each at 100 Mpc 5 steps
Two neutron stars of 1.4 solar masses each, at 100 Mpc. Predict every row from the black-hole answer above without recomputing anything, using only the exponents.
1. Recompute only the chirp mass. For equal masses it collapses to 2^−1/5 times the common mass, which is 1.22 for a pair of 1.4s.
  
  $M = \frac{( 1.4 \times 1.4 ) ^{3/5}}{2. 8 ^{1/5}} = \frac{1.9 6 ^{0.6}}{2. 8 ^{0.2}} = \frac{1.4869}{1.2200} = 1.22 M_{⊙}$
2. Everything else is a ratio. The duration exponent is −5/3, so form the mass ratio first and raise it once.
  
  $\frac{τ _{NS}}{τ _{BH}} = (\frac{M _{BH}}{M _{NS}})^{5/3} = (\frac{28.10}{1.219})^{5/3} = 23.0 5^{5/3} = 187$
3. Apply it to the 0.843 s from the previous problem. No integral, no constants, no formula beyond the exponent.
  
  $τ = 0.843 \times 187 = 157 s = 2.6 min$
4. The ISCO frequency is inversely proportional to total mass, so the same trick with a much smaller exponent gives the pitch at the end.
  
  $f_{ISCO} \propto \frac{1}{M _{tot}} : 67.6 \times \frac{65}{2.8} = 1570 Hz$
5. Strain carries two dependencies at once — up with chirp mass to the 5/3, down with distance — and this pair is both lighter and nearer, so the effects partly cancel.
  
  $h \propto \frac{M ^{5/3}}{d _{L}} : 1.16 \times 1 0^{- 21} \times (\frac{1.219}{28.10})^{5/3} \times \frac{440}{100} = 2.2 \times 1 0^{- 22}$
Answer

The whole difference is one power law: 187. The chirp mass drops from 28.10 to 1.22, and because the inspiral time scales as ℳ^−5/3, dividing by 23 in mass multiplies the time by 23^5/3 = 187. That turns 0.84 seconds into 157, which is why a neutron-star merger is the one LIGO can announce while it is still happening — 2.6 minutes is long enough to point telescopes, and GW170817 is why anyone saw the kilonova. The same lever works against you at the other end: f_ISCO goes as 1/M, so the merger finishes at 1570 Hz, well above where the detectors are most sensitive. Light binaries are easy to hear coming and hard to hear arrive.

$M = 1.22 M_{⊙} ⟹ τ \times 187, f_{ISCO} \times 23, h \div 5$

References (3)

The detection the first block describes, and its measured strain: B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), "Observation of Gravitational Waves from a Binary Black Hole Merger." Physical Review Letters 116(6), 061102, 2016.
The neutron-star merger behind the gw170817 preset: B. P. Abbott et al., "GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral." Physical Review Letters 119(16), 161101, 2017.
The chirp-mass and peak-strain expressions the tool evaluates: M. Maggiore, Gravitational Waves, Volume 1: Theory and Experiments, §4.1. Oxford University Press, 2008. ISBN 978-0-19-857074-5.

By the 42Math team · Updated 2026-07-05

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

GW150914 - GW150914: first detection, chirp mass ~28.1 Msun
GW170817 - Neutron stars of 1.17 and 1.36 solar masses give a chirp mass of 1.10, under both. That single number is what the waveform encodes, and it buys 3.1 minutes in band from 20 Hz. Move the start frequency up by ten hertz and that falls to 1.1 minutes — two thirds of the signal lives in the lowest slice of the band, the hardest place to build a detector.
NS-NS Merger - Two textbook 1.4 solar-mass neutron stars: chirp mass 1.22, and 2.6 minutes from 20 Hz to merger. That is shorter than GW170817’s run, and GW170817 is the lighter pair. Mass drives the inspiral, so the heavier binary is the one you get less time to listen to. The 1570.1 Hz below is the ISCO frequency, where a stable circular orbit stops existing and the two stars fall together.
BH-BH Merger - 160 solar masses of black hole, and the ISCO frequency falls to 27.5 Hz — barely above the 20 Hz where the chart begins. The whole event lasts 185.7 ms inside the band: no rising chirp you could hum, one thump. It is still the loudest of the four presets, 1.28 × 10⁻²¹, from 1,000 Mpc, more than twenty times GW170817’s distance.