Sections · Gravitational waves

Vol. Science · Astronomy · The chirp

Gravitational waves
and the chirp

Two masses in a tightening orbit radiate gravitational waves, mostly at twice the orbital frequency. The leading-order sweep depends on one combination of the masses, the chirp mass (m₁ m₂)^3/5 / (m₁ + m₂)^1/5. Time to coalescence is the integral of the frequency derivative printed in the GW150914 discovery paper. The face-on strain is the Newtonian quadrupole amplitude at the distance you set. c = 299,792,458 m/s exactly. A mass in solar masses uses the IAU 2015 Resolution B3 nominal GM☉ = 1.3271244×10²⁰ m³ s⁻². Distance uses the parsec, (648,000/π) astronomical units, IAU 2015 Resolution B2, and the astronomical unit is 149,597,870,700 m. The GW150914 button types that paper’s source-frame central values, 36 and 29 solar masses, at 410 Mpc, starting at 35 Hz. The formula returns 28.10 M☉. The ISCO line is a rough end-of-inspiral marker. The radius itself is the black holes desk. The same c, used as a clock, is the light-time desk.

Educational calculator. Leading-order Newtonian quadrupole. No spin, no higher-order post-Newtonian terms, no tides, and no detector antenna pattern. The masses are not multiplied by (1+z). The coalescence in the integral is a mathematical blow-up, not a merger. Not a waveform model and not for data analysis.

Interactive · A compact binary

Strain above · frequency below · the dashed line is the ISCO marker

Strain
Frequency
ISCO

Published events, typed as the numbers below

Mass 1

Mass 2

Distance

Start f

Sonification · not a detector recording

Opens on GW150914. The tone’s frequency is the gravitational-wave frequency times the pitch factor. Loudness is normalized so the tone is audible. It is not the physical strain, and it is not a LIGO recording.

GW150914 · source-frame central values

28.10 M☉

0.190 s to coalescence · 35 Hz

Masses36 and 29 M☉
Total65 M☉
Chirp mass28.10 M☉
To coalescence0.190 s
To the ISCO marker0.157 s
ISCO frequency67.65 Hz
Strain at start8.054×10⁻²²
Strain at ISCO1.25×10⁻²¹

GW150914, the discovery abstract’s source-frame central values, 36 and 29 solar masses, at 410 Mpc, starting at 35 Hz. The chirp mass is 28.10 M☉. Time to the Newtonian coalescence from 35 Hz is 0.190 s. The face-on strain at that frequency is 8.054×10⁻²². The Schwarzschild test-particle ISCO for the total mass is 67.65 Hz, 0.157 s from this start on the same integral. That marker is not a merger time. Leading-order quadrupole. Not a waveform model.

Presets · each row is this page’s formulas, not a stored posterior
Event	Masses	Distance	Start	Chirp	To coalescence	ISCO	Strain
GW150914	36 and 29 M☉	410 Mpc	35 Hz	28.10 M☉	0.190 s	67.65 Hz	8.054×10⁻²²
GW170817	1.37 and 1.37 M☉	40 Mpc	24 Hz	1.193 M☉	100.65 s	1,605 Hz	3.316×10⁻²³

The buttons need JavaScript. The arithmetic is still Mc = (m1 m2)^(3/5) / (m1 + m2)^(1/5), τ = (5/256) (π f)^(−8/3) (G Mc / c³)^(−5/3), and a face-on strain (4/D) (G Mc / c²)^(5/3) (π f / c)^(2/3). GW150914 at 36 and 29 solar masses is a chirp mass of 28.10, 0.190 s from 35 Hz, ISCO marker 67.65 Hz. The equal split of 2.74 solar masses is a chirp mass of 1.193 and 100.65 s from 24 Hz. Not a waveform model.

The buttons and the papers

28.10 from 36 and 29. The posterior is 28.0.

The GW150914 button types 36 and 29 solar masses. Their Newtonian chirp mass is 28.10 M☉. Abbott et al. 2016, the properties paper, Table I, overall column, gives a source-frame chirp mass of 28.0 solar masses, plus 2.0 and minus 1.7. The same paper’s detector-frame component medians are 38.9 and 31.6. Typed into this formula they give 30.49 M☉. The paper’s detector-frame chirp median is 30.4, plus 2.1 and minus 1.9. Marginal medians are not a joint sample, and this page does not multiply by (1+z). The discovery paper’s rough estimate from the observed frequency and its derivative is about 30 solar masses in the detector frame. From 35 Hz to 150 Hz the Newtonian integral on the button’s masses is 0.186 s. The discovery paper says the signal increases from 35 to 150 Hz in about 0.2 s, and the sweep continues to 250 Hz. The face-on strain of the button at 150 Hz is 2.125×10⁻²¹. The discovery paper’s peak strain is 1.0×10⁻²¹. That peak is the strain in the detectors. It is not this face-on formula. The GW170817 button’s Newtonian chirp mass is 1.193 M☉. The paper’s source-frame chirp mass is 1.188, plus 0.004 and minus 0.002. From 24 Hz the Newtonian time to coalescence is 100.65 s. The paper’s time in band, calculated from 24 Hz, is about 100 s.

What the sweep is

One combination. The chirp mass is the one combination of the two masses that sets the leading-order sweep. Individual masses matter at higher order, and this page does not carry those terms. A heavier chirp mass reaches coalescence sooner from the same frequency.

An integral, not a merger. Time to coalescence is the integral of the discovery paper’s frequency derivative, from the frequency you set up to the Newtonian blow-up. The blow-up is not a merger. Stop reading the curve at the ISCO marker if you want the rough end of the inspiral.

Face-on, and only that. The strain is the face-on Newtonian amplitude at the distance you set. It grows as frequency to the two-thirds, and it falls as one over distance. An edge-on binary and a detector’s antenna pattern are smaller. Nothing here is a matched-filter template.

A marker, not the ringdown. The marker is a test particle’s innermost stable circular orbit around a non-spinning hole of the total mass, the same 6GM/c² the black-hole desk draws, read as a gravitational-wave frequency. GW150914 was observed well above that frequency. Spin and comparable masses move the real last orbit. For a neutron-star pair the stars meet before this black-hole orbit.

What the buttons are

GW150914 is 36 and 29, not the posterior. The discovery abstract prints source-frame masses 36+5−4 and 29+4−4 solar masses, a luminosity distance 410+160−180 Mpc, and a sweep from 35 Hz to 250 Hz. The button types 36, 29, 410, and 35. The uncertainties stay in the citation. The properties paper’s source-frame chirp median is 28.0. This page’s 28.10 is the Newtonian combination of those two central values.

GW170817 is an equal split of 2.74. The discovery paper’s best-measured mass is the chirp mass, 1.188+0.004−0.002 solar masses in the source frame. It does not publish one component pair. Under the low-spin prior the components lie between 1.17 and 1.60 solar masses and the total is 2.74+0.04−0.01. The button splits that 2.74 into 1.37 and 1.37. The paper’s figure marks an equal-mass line at 1.36. That line is not the button. The broader high-spin ranges run from 0.86 to 2.26. Distance is 40+8−14 Mpc. The in-band time is calculated from 24 Hz. The same paper puts the onset of strong tidal effects near 600 Hz. This curve has no tides.

Two bands, both cited. Advanced LIGO’s band in Martynov et al. 2016 is about 10 Hz to 10 kHz. The LISA Consortium describes LIGO, Virgo, and KAGRA as optimized for the tens to hundreds of hertz, and LISA for millihertz waves. The LISA Definition Study Report gives LISA’s survey band as 0.1 mHz to 1 Hz. The same ISCO marker at a total mass of one million solar masses is 4.397 mHz. That line is the formula, not a detected event.

Literacy · the words on the readout

Chirp mass: Mc = (m1 m2)^(3/5) / (m1 + m2)^(1/5). At this order it is the mass that sets how fast the frequency rises. The discovery paper writes it that way, and also as the combination of f and f-dot that the data measure.
Time to coalescence: τ = (5/256) (π f)^(−8/3) (G Mc / c³)^(−5/3). Integrate df/f-dot from the frequency you set up to infinity. Infinity here is the Newtonian blow-up. f(t) = f0 (1 − t/τ)^(−3/8).
Face-on strain: h = (4/D) (G Mc / c²)^(5/3) (π f / c)^(2/3). The same number is 4 (G μ / c² D) (π G M f / c³)^(2/3), with μ the reduced mass and M the total. Face-on means the coefficient of both h+ and h×. Other inclinations are smaller. D is the distance you type.
ISCO marker: A test particle’s innermost stable circular orbit at 6GM/c². The gravitational-wave frequency is c³ / (π 6^(3/2) G M), twice the orbital frequency. The trace is allowed to continue past the dashed line so you can see that the formula has not stopped. The approximation already has.
Detector frame: The frequency evolution of a wave that has traveled a cosmological distance measures Mc(1+z), not the source-frame chirp mass. This page does not apply that factor. Type the detector-frame masses, 38.9 and 31.6 for GW150914, if you want the formula next to the observed sweep.
Sonification: Play turns the trace into a tone. The tone’s frequency is the gravitational-wave frequency times 1, 4, or 8. The loudness is normalized. A short fade keeps the start and the end from clicking. It is not a whitened LIGO strain, and it is not a sound the merger made in air. GW150914’s sweep, 35 Hz and up, is already in the audible band, which is why 1× is the default.

Sources · the constants and the events

Cited, not invented

NIST — speed of light in vacuum and BIPM — the metre — c = 299,792,458 m/s exactly.
IAU 2015 Resolution B3 (Prša et al. 2016) — nominal solar mass parameter 1 (GM)^N_☉ = 1.3271244 × 10²⁰ m³ s⁻². A conversion constant. This page multiplies it by the solar-mass number. The same resolution’s archive copy is the IAU 2015 English PDF, which also carries Resolution B2.
IAU 2012 Resolution B2 — the astronomical unit is 149,597,870,700 m exactly.
IAU 2015 Resolution B2 — the parsec is exactly (648,000/π) astronomical units. A megaparsec is 10⁶ of those.
Abbott et al. 2016, Phys. Rev. Lett. 116, 061102 — GW150914 discovery. Source-frame masses 36+5−4 and 29+4−4 solar masses. Final mass 62+4−4. Radiated energy 3.0+0.5−0.5 solar masses. Luminosity distance 410+160−180 Mpc, redshift 0.09+0.03−0.04. The signal sweeps from 35 to 250 Hz, with a peak strain of 1.0×10⁻²¹. Over about 0.2 s it rises from 35 to 150 Hz. The chirp-mass equation in the text is the f and f-dot relation this page integrates. Their rough detector-frame estimate from that relation is about 30 solar masses. Open copy: arXiv:1602.03837.
Abbott et al. 2016, Phys. Rev. Lett. 116, 241102 — GW150914 properties, Table I, overall column. Source-frame chirp mass 28.0+2.0−1.7. Detector-frame chirp mass 30.4+2.1−1.9. Detector-frame component medians 38.9 and 31.6. Source-frame total mass 65.0. Cited for comparison. The button does not type the detector-frame pair. Open copy: arXiv:1602.03840.
Abbott et al. 2017, Phys. Rev. Lett. 119, 161101 — GW170817. Source-frame chirp mass 1.188+0.004−0.002. Detector-frame chirp mass 1.1977+0.0008−0.0003. Low-spin total mass 2.74+0.04−0.01, components in 1.17–1.60. High-spin components in 0.86–2.26. Luminosity distance 40+8−14 Mpc. About 100 s in band, calculated from 24 Hz. Strong tidal effects near 600 Hz. The button is half of 2.74, twice. Open copy: arXiv:1710.05832.
Martynov et al. 2016, Phys. Rev. D 93, 112004 — Advanced LIGO is designed for gravitational waves from about 10 Hz to 10 kHz. That is the LIGO sentence in the bands paragraph. It is not a Virgo number.
LISA Consortium — how LISA differs from LIGO, Virgo, and KAGRA — ground-based detectors are optimized for the tens to hundreds of hertz. LISA is optimized for millihertz waves.
LISA Definition Study Report, arXiv:2402.07571 — LISA will survey the gravitational-wave sky from 0.1 mHz to 1 Hz.
Flanagan and Hughes 2005, New J. Phys. 7, 204 — a review of the quadrupole waveform. This page uses the leading-order face-on amplitude and checks the chirp-mass form against the reduced-mass form. It does not import a higher-order template.

Classroom chirp. c is the SI constant. GM☉ is the IAU 2015 nominal 1.3271244×10²⁰ m³ s⁻². The parsec is IAU 2015 Resolution B2. Event buttons type published central values, or an equal split of a published total, and the formulas do the rest. Leading-order quadrupole. Not a waveform model and not for data analysis. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

The radius

Events	GW150914 types 36 and 29 solar masses, 410 Mpc, and 35 Hz. GW170817 types 1.37 and 1.37, half of the published low-spin total 2.74, at 40 Mpc and 24 Hz.
Masses	Each from 0.5 to 200 solar masses, on a logarithmic slider. The number accepts the same range.
Distance	0.1 to 10,000 Mpc, logarithmic. Used as the D in the face-on strain.
Start f	1 to 2,000 Hz, logarithmic. The gravitational-wave frequency where the clock starts. Left and Right on the chart step it.
Play	A sonification of the trace on the chart. Pitch 1×, 4×, or 8× multiplies the gravitational-wave frequency. Loudness is normalized. Stop ends it. Changing a mass, the distance, or the frequency stops it too.

Chirp mass	The Newtonian combination of the two masses. The headline.
To coalescence	The integral from the frequency you set up to the Newtonian blow-up.
ISCO	The test-particle Schwarzschild gravitational-wave frequency, and the time on the same integral to reach it. If you start above it, the card says so.
Strain	Face-on amplitude at the start, and at the marker when the start is still below it.
Chart	Strain against time, then frequency against time. A long chirp shows only its last 4 seconds, where the rise is visible. The card still prints the full time.

Buttons	Enter or Space selects. The pressed button is aria-pressed. Targets are at least 44 pixels on a side.
Phone	Below 900 pixels the long section row becomes a single Sections control. Open it for the same links. The site menu is the Menu button in the top bar.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Gravitational waves
and the chirp

Cited, not invented

Black holes

Light time

Time dilation

Redshift

Solar map

Gravitational-wave desk

Controls

Readouts

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Gravitational wavesand the chirp

Cited, not invented

Black holes

Light time

Time dilation

Redshift

Solar map

Gravitational waves
and the chirp