Gravitational Wave Strain Amplitude Calculator

Calculate gravitational wave strain amplitude for compact binaries or rotating neutron stars. Uses the quadrupole approximation.

🌊 Gravitational Wave Strain Amplitude Calculator
Component Mass m₁ (M☉)
M☉
Component Mass m₂ (M☉)
M☉
GW Frequency fGW (Hz)
Hz
Luminosity Distance
NS Mass (M☉)
M☉
NS Radius (km)
km
Equatorial Ellipticity ε
(dimensionless)
Rotation Frequency (Hz)
Hz
Distance
Strain Amplitude h
—
Chirp Mass Mc
—
GW Luminosity
—
rs of m₁
—
rs of m₂
—
Strain Amplitude h
—
Moment of Inertia I
—
GW Frequency
—
GW Luminosity
—

🌊 What is a Gravitational Wave Strain Amplitude Calculator?

Gravitational wave strain amplitude h is the dimensionless quantity that characterizes how strongly a gravitational wave distorts spacetime as it passes through a detector. Concretely, h = Delta-L / L, where Delta-L is the change in arm length and L is the original arm length. For GW150914, LIGO's 4 km arms moved by about 4 x 10^-18 m, less than one-thousandth the diameter of a proton. This calculator implements the two leading GW source models in the quadrupole approximation: compact binary inspiral (BBH, BNS, or BH-NS) and continuous emission from a rotating neutron star with non-zero equatorial ellipticity.

For compact binaries, the strain is h = (4/r)(GMc/c^2)(pi*G*Mc*f_GW/c^3)^(2/3), where Mc is the chirp mass -- the particular combination (m1*m2)^(3/5)/(m1+m2)^(1/5) that dominates the inspiral waveform. Chirp mass can be measured to sub-percent accuracy from the rate of frequency evolution alone, making it the most precisely determined parameter in GW astronomy. GW150914 (Mc ~ 28.3 M_sun, 410 Mpc) and GW170817 (Mc ~ 1.19 M_sun, 40 Mpc) are the landmark events that opened the era of multi-messenger GW astrophysics.

For rotating neutron stars, a non-axisymmetric mass distribution (ellipticity epsilon) generates continuous monochromatic gravitational waves at exactly twice the rotation frequency. The strain is h = 4*G*I*epsilon*(2*pi*f_rot)^2 / (c^4*r), where I is the moment of inertia. Even epsilon = 10^-6 (a mountain only ~0.1 mm high on a 10 km star) gives strain near LIGO's detection floor for pulsars within a few kpc. Current searches have placed astrophysically interesting upper limits on epsilon for dozens of known pulsars.

The GW luminosity -- the power radiated as gravitational waves -- reaches staggering levels during compact binary mergers. GW150914 briefly radiated ~3.6 x 10^49 W, more than 10 times the total electromagnetic luminosity of all stars in the observable universe. Even the quieter neutron star binary GW170817 emitted ~10^46 W during its final inspiral seconds. These numbers illustrate why LIGO's precision interferometers, capable of measuring motions smaller than an atomic nucleus, represent one of the most demanding engineering achievements in physics.

📐 Formula

Compact Binary:   h  =  (4/r) × (GMc/c²) × (πGMcfGW/c³)2/3
h = dimensionless strain amplitude
Mc = chirp mass = (m1m2)3/5/(m1+m2)1/5
fGW = gravitational wave frequency (Hz) = 2 × orbital frequency
r = luminosity distance to source
G = 6.674 × 10−11 m³ kg−1 s−2,   c = 2.998 × 108 m/s
Rotating NS:   h  =  4GIε(2πfrot)² / (c4r)
I = moment of inertia ~ 0.4 MR² for uniform sphere (kg m²)
ε = equatorial ellipticity = (Ixx − Iyy) / Izz
frot = rotation frequency (Hz); GW frequency = 2frot
Example: GW150914 at f = 35 Hz, 410 Mpc: Mc = 28.3 M⊙, h ≈ 1.1 × 10−21

📖 How to Use This Calculator

Steps

1
Select mode - Choose Compact Binary for BBH, BNS, or BH-NS mergers, or Rotating NS for continuous gravitational waves from an asymmetric pulsar.
2
Enter masses and frequency - For binary mode, type the two component masses in solar masses and the GW frequency in Hz. For NS mode, enter mass (M_sun), radius (km), ellipticity (dimensionless), and rotation frequency (Hz).
3
Enter distance - Type the source distance and choose units. Typical values: galactic pulsars in kpc, extragalactic GW events in Mpc or Gpc.
4
Click Calculate - Get the dimensionless strain h, chirp mass or moment of inertia, GW luminosity, and Schwarzschild radii for binary sources.

💡 Example Calculations

Example 1 - GW150914 Binary Black Hole Merger

GW150914: m₁ = 36 M☉, m₂ = 29 M☉, f = 35 Hz, r = 410 Mpc

1
Chirp mass: Mc = (36 × 29)3/5 / (36+29)1/5 = 28.27 M⊙
2
Strain: h = (4/r)(GMc/c²)(piGMcf/c³)2/3 with r = 410 Mpc = 1.265 × 1025 m
3
GW luminosity at 35 Hz: L ~ 8.1 × 1048 W (peak near merger is ~3.6 × 1049 W at ~150 Hz)
h ≈ 1.12 × 10−21 | Mc = 28.27 M☉ | LGW ≈ 8.1 × 1048 W
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Example 2 - GW170817 Binary Neutron Star Merger

GW170817: m₁ = 1.36 M☉, m₂ = 1.17 M☉, f = 100 Hz, r = 40 Mpc

1
Chirp mass: Mc = (1.36 × 1.17)3/5 / (1.36+1.17)1/5 = 1.098 M⊙ (the measured 1.186 M⊙ uses the precise pulsar-timing masses)
2
Much lighter Mc but closer distance (40 Mpc vs 410 Mpc) gives comparable strain at higher frequency
3
GW170817 was accompanied by a short gamma-ray burst and kilonova, marking the first multi-messenger GW detection
h ≈ 2.3 × 10−22 | Mc = 1.098 M☉ | r = 40 Mpc
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Example 3 - Crab Pulsar Continuous GW (upper limit)

Crab Pulsar: M = 1.4 M☉, R = 10 km, ε = 10⁻⁴, f_rot = 30 Hz, r = 2 kpc

1
Moment of inertia: I = 0.4 × 1.4 × 1.989 × 1030 × (104)2 = 1.11 × 1038 kg m²
2
Strain: h = 4 × G × I × epsilon × (2pi × 30)2 / (c4 × 2 kpc) = ~3.3 × 10−25
3
This epsilon = 10^-4 is an upper limit; LIGO data limit true Crab ellipticity to below ~10^-5
h ≈ 3.3 × 10−25 | fGW = 60 Hz | just below Advanced LIGO sensitivity
Try this example →

❓ Frequently Asked Questions

What is gravitational wave strain amplitude h?+
The strain amplitude h is the fractional change in length Delta-L/L induced by a gravitational wave passing through a detector. It is dimensionless. For GW150914, h ~ 10^-21 caused LIGO's 4 km arms to oscillate by ~4 x 10^-18 m, far smaller than a proton. LIGO measures this by comparing the travel time of laser light in two perpendicular arms using Fabry-Perot cavities and power recycling to store ~100 kW of light.
What is the formula for GW strain from a compact binary?+
For a compact binary in circular orbit, the leading quadrupole strain is h = (4/r)(GMc/c^2)(pi*G*Mc*f_GW/c^3)^(2/3). Here Mc = (m1*m2)^(3/5)/(m1+m2)^(1/5) is the chirp mass, f_GW is the GW frequency (twice orbital frequency), and r is the luminosity distance. This formula applies during the inspiral phase when orbital velocities are well below c.
What is chirp mass and why is it important?+
Chirp mass Mc = (m1*m2)^(3/5)/(m1+m2)^(1/5) controls both the strain amplitude and the chirp rate df_GW/dt = (96/5)*pi^(8/3)*(G*Mc/c^3)^(5/3)*f^(11/3). It can be measured to sub-percent accuracy from the observed frequency evolution even when individual masses are poorly determined. GW150914 had Mc = 28.3 M_sun; GW170817 had Mc = 1.188 M_sun.
How does distance affect gravitational wave strain?+
Strain h falls as 1/r with luminosity distance, since h is a wave amplitude, not a power flux. Doubling the distance halves the strain. For detectability, this means LIGO's survey volume grows as r^3 with sensitivity: a factor-10 improvement in strain sensitivity expands the detectable volume by 1,000, increasing the accessible merger rate by the same factor.
What is the formula for GW strain from a rotating neutron star?+
A non-axisymmetric NS with ellipticity epsilon emits continuous GW at f_GW = 2*f_rot with strain h = 4*G*I*epsilon*(2*pi*f_rot)^2 / (c^4*r). Using I ~ 10^38 kg m^2 (canonical 1.4 M_sun, 10 km NS), epsilon = 10^-6, f_rot = 100 Hz, r = 1 kpc gives h ~ 10^-27, below current LIGO sensitivity but within reach of next-generation detectors.
What is GW luminosity and how large is it?+
GW luminosity for a binary is L = (32/5)*(c^5/G)*(G*Mc*pi*f/c^3)^(10/3). At peak, GW150914 radiated about 3.6 x 10^49 W, briefly outshining all stars in the observable universe. Even the quieter GW170817 emitted roughly 10^46 W near merger. For rotating NS with epsilon = 10^-6, luminosity is typically 10^20 to 10^30 W, minuscule but potentially detectable over long integrations.
Why is GW frequency twice the orbital or rotation frequency?+
GW emission is quadrupole radiation. A binary system has two equal masses separated by a diameter; as they orbit, the mass quadrupole repeats its orientation twice per orbital revolution, so GW emission peaks at 2*f_orb. Likewise, a non-axisymmetric spinning star presents the same quadrupole face twice per rotation, giving f_GW = 2*f_rot.
What LIGO sensitivity is needed to detect these signals?+
Advanced LIGO (O3) achieved noise floor ~3 x 10^-24 per root-Hz near 100 Hz. A transient with h ~ 10^-21 achieves SNR above 10 in ~1 second. For continuous signals, coherent integration over months is needed; sensitivity improves as 1/sqrt(observation time), so 1 year of data reaches h_min ~ 10^-26 in the best frequency bands, beginning to probe astrophysically plausible NS ellipticities.
What values of NS ellipticity are physically realistic?+
Theory suggests NS crusts can support mountains with epsilon ~ 10^-7 to 10^-5 depending on the breaking strain of nuclear pasta phases. Strong magnetic fields (magnetars, B above 10^15 G) can sustain epsilon ~ 10^-4 to 10^-3. LIGO's targeted searches have placed upper limits below 10^-7 for many known ms pulsars, beginning to constrain NS equation-of-state models.
What are Einstein Telescope and Cosmic Explorer?+
These proposed third-generation detectors plan to improve strain sensitivity by a factor of 10 over Advanced LIGO. Einstein Telescope (EU) would use 10 km triangular underground arms at cryogenic temperatures. Cosmic Explorer (US) proposes 40 km L-shaped surface arms. Both would detect BBH mergers to redshift z ~ 20, covering virtually all BH mergers in the observable universe.
How is strain measured from GW150914?+
GW150914 was detected on 14 September 2015 by LIGO Hanford and Livingston with a time delay of 6.9 ms. The matched-filter template analysis recovered masses 36 and 29 M_sun at 410 Mpc, with peak strain h ~ 10^-21 at about 150 Hz. The signal chirped from ~35 Hz to ~150 Hz in 0.2 seconds, sweeping through LIGO's most sensitive band and providing the definitive first direct detection of gravitational waves.
Tags: Gravitational Wave Strain Calculator Gw Strain Amplitude Chirp Mass Calculator LIGO Strain Compact Binary Gravitational Waves Neutron Star Ellipticity Calculator Formula Free Online

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What is gravitational wave strain amplitude h?

The strain amplitude h is the fractional change in length Delta-L/L induced by a gravitational wave as it passes through a detector. It is dimensionless: h = 10^-21 means two mirrors 4 km apart oscillate by about 4 x 10^-18 m. LIGO measures strain by laser interferometry with sensitivity near 10^-23 per root-Hz in its most sensitive frequency band.

What is the formula for GW strain from a compact binary?

For a compact binary (BBH, BNS, or BH-NS) in circular orbit the leading-order quadrupole formula gives h = (4/r)(GMc/c^2)(pi*G*Mc*f_GW/c^3)^(2/3), where Mc is the chirp mass, f_GW is the gravitational wave frequency (twice the orbital frequency), and r is the luminosity distance to the source.

What is chirp mass and why is it important?

Chirp mass Mc = (m1 m2)^(3/5)/(m1+m2)^(1/5) is the combination of component masses that controls both the strain amplitude and the rate of frequency evolution (chirp rate df/dt = (96/5) pi^(8/3) (G Mc/c^3)^(5/3) f^(11/3)). It can be measured extremely precisely from the observed chirp even when individual masses are uncertain. GW150914 had Mc = 28.3 M_sun.

How does distance affect gravitational wave strain?

Strain h falls as 1/r with luminosity distance, just like electromagnetic flux falls as 1/r^2 in power (because h is an amplitude, not a flux). Doubling the distance halves the strain. This means LIGO's detectable volume grows as r^3 with sensitivity, and a factor-of-10 strain improvement in the detector opens a factor-of-1,000 larger survey volume.

What is the formula for GW strain from a rotating neutron star?

A neutron star with non-zero equatorial ellipticity epsilon = (Ixx - Iyy)/Izz emits continuous gravitational waves at twice its rotation frequency. The strain is h = 4 G I epsilon (2 pi f_rot)^2 / (c^4 r), where I is the moment of inertia (canonical value ~10^38 kg m^2 for a 1.4 M_sun, 10 km NS). Even epsilon = 10^-6 (a tiny mountain) gives detectable strain from nearby pulsars.

What is GW luminosity and how large is it?

For compact binaries, GW luminosity L = (32/5) c^5/G * (G Mc pi f / c^3)^(10/3). At peak, GW150914 radiated about 3.6 x 10^49 W, briefly outshining all stars in the observable universe combined. Even the quieter GW170817 (BNS) emitted roughly 10^46 W at 100 Hz. For rotating neutron stars the luminosity is much lower, typically 10^20 to 10^30 W.

Why is GW frequency twice the orbital or rotation frequency?

Gravitational waves are quadrupole radiation. A binary system has two arms of the mass quadrupole rotating at the orbital frequency f_orb; because the quadrupole completes a full cycle of symmetry twice per orbit, GW emission occurs at 2 f_orb. Similarly, a non-axisymmetric rotating star presents the same quadrupole orientation twice per rotation, giving f_GW = 2 f_rot.

What is the LIGO sensitivity needed to detect these signals?

LIGO Advanced (O3 run) achieved a design sensitivity noise floor of about 3 x 10^-24 per root-Hz near 100 Hz. A signal with h ~ 10^-21 stands well above the noise with SNR above 10 in a 1-second integration. At 410 Mpc (GW150914 distance), only extremely massive mergers are detectable; third-generation detectors like Einstein Telescope aim for sensitivity of 10^-25 per root-Hz.

What values of neutron star ellipticity are physically realistic?

Theoretical models suggest the NS crust can support mountains with epsilon up to roughly 10^-7 to 10^-5 depending on the breaking strain of nuclear pasta. Strong internal magnetic fields (magnetars) can distort the star to epsilon ~ 10^-4 to 10^-3. Current LIGO continuous-wave searches have set upper limits of epsilon below 10^-7 for many known millisecond pulsars.

What are the Einstein Telescope and Cosmic Explorer?

Einstein Telescope (ET) and Cosmic Explorer (CE) are proposed third-generation ground-based GW detectors planned for the 2030s. ET aims for 10 km triangular underground arms and would improve strain sensitivity by a factor of 10 over Advanced LIGO, detecting BBH mergers to redshift z ~ 20. CE proposes 40 km L-shaped surface arms with even greater reach.

How is strain measured from the example event GW150914?

GW150914 was a BBH merger of 36 and 29 solar masses at 410 Mpc. The observed peak strain was h ~ 1.0 x 10^-21 at about 150 Hz. This calculator returns h ~ 1.12 x 10^-21 at 35 Hz (an earlier inspiral point); the peak near merger is higher because the frequency and chirp mass are higher. Use the GW150914 preset to explore the parameter space.

📌 Quick Tips

💡Strain h is dimensionless and represents the fractional change in distance: if h = 10^-21, two test masses 4 km apart move by about 4 x 10^-18 m, less than one-thousandth the diameter of a proton.
💡LIGO's noise floor is about 10^-23 per root-Hz near 100 Hz, so events with h above roughly 10^-21 can be detected. GW150914 peaked at h ~ 10^-21 at 150 Hz.
💡Chirp mass Mc is the combination of component masses that dominates the early inspiral waveform. It can be measured very accurately from the rate of frequency evolution even when the mass ratio is poorly constrained.
💡For a rotating neutron star, the GW frequency is exactly twice the rotation frequency. A 30 Hz rotation gives 60 Hz GW emission, squarely in the LIGO band.
💡Ellipticity epsilon ~ 10^-6 to 10^-4 is theoretically plausible for neutron stars with mountains supported by the solid crust or by strong magnetic fields. Current LIGO upper limits rule out epsilon above about 10^-7 for nearby known pulsars.