Gravitational Radiation 3: GW151226, Encore from LIGO

Figure 1: A filtered, reconstructed image of the gravitational wave detected by LIGO on December 26, 2015 (UTC).  The signal was generated from the merger of two black holes, of approximately 14.2 and 7.7 solar masses, roughly 1.4 billion light years from  Earth.  Figure taken from Abbott et al, Ref. 3.

LIGO (Laser Interferometer Gravitational-Wave Observatory) consists of two gargantuan Michelson-like interferometers, with arms 4 km long, located 3000 km apart in Livingston LA and Hanford WA.  Its first observing session began on September 12, 2015.  Incredibly, just two days later, it captured the fleeting signal (designated GW150914) produced by the merger of two inspiraling black holes more than 1 billion light years from Earth.  (Ref. 1)  That groundbreaking discovery, announced on February 11, 2016, ended a 60 year race to directly detect gravitational waves, and came 100 years after the phenomenon was first predicted by Albert Einstein.

On Christmas Day (in the USA), LIGO scientists detected a second event, GW151226 (Ref. 3), proving that the earlier discovery was not a fluke, and that black hole mergers are frequent events.  (Ref. 4,5)  Since January, LIGO’s sensitivity has been improved, and it will begin a new observing run in Autumn, 2016.  This time, the two US detectors will be joined by VIRGO, a similar instrument with arms 3 km long, located near Pisa, Italy.  The addition of VIRGO will greatly assist in localizing – by triangulation – the sources of gravitational radiation.  In the near future, when LIGO and VIRGO reach their full design sensitivities, they may be capable of detecting one or more black hole collisions daily!  Indeed, the 21st Century seems destined to become the era of gravitational wave astronomy.

In this brief post, we will use the mathematical treatment outlined in Ref. 2 to analyze the published data given in Ref. 3 for GW151226.  Our educational goal is to add to the collection of homework exercises related to gravitational waves that are suitable for first year physics students.  In this way, we hope to inspire instructors to include this exciting topic in their mechanics syllabi.


Analyzing GW151226

In Ref. 2, the chirp mass of the binary black hole system was defined as \large \mathfrak{M}=\frac{(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}}, where \large m_{1} and \large m_{2} are the masses of the black holes.  This quantity is important because it can be found directly from the time dependence of the gravitational wave’s (GW) frequency (Eqn. 10 of Ref. 2):

\large \mathfrak{M}=\frac{c^{3}}{G}\left ( \frac{5}{96}\: \pi ^{-8/3}f^{-11/3}\: \frac{df}{dt} \right ) .


Following Ref. 2, define \large A=\frac{c^{5}}{G^{5/3}}\: \frac{5}{96}\: \pi ^{-8/3}=5.45\times 10^{56} (SI units), and integrate Eqn. 1 over the time interval \large \Delta t=t_{2}-t_{1} to obtain

\large \mathfrak{M}^{5/3}\Delta t=A\int_{t_{1}}^{t^{_{2}}}f^{-11/3}df=-\frac{3}{8}Af^{-8/3}\mid _{t_{1}}^{t_{2}} .


1.  Referring to Fig. 1 above, at 0.80 s before the two black holes coalesce, the frequency of the GW was 39.2 Hz.  Later, 0.40 s before the merger, the frequency was 50.3 Hz. (Ref. 6)  Calculate the chirp mass and express your answer in terms of \large M_{Sun}.  (Ans: \large 9.7\: M_{Sun}.)

2.  Show that the total mass \large M=m_{1}+m_{2} of the binary system before coalescence was at least \large 22\: M_{Sun}.  (Hint: Let \large m_{2}=\alpha m_{1}.  Express \large M\large \mathfrak{M}, and \large M/\mathfrak{M} in terms of \large m_{1} and \large \alpha.  Show that the ratio is a minimum when \large \alpha=1.)

3.  The GW frequency at the moment of coalescence of the two black holes was 420 Hz.  (See Fig. 1.)  Recall from Ref. 2 that the orbital frequency of the binary system is half the frequency of the GW.   Use the total mass given in Ref. 3, \large M=22\: M_{Sun}, to find the distance between the bodies just before they merged.  (Ans: 120 km)

4.  Assume that one black hole was twice as massive as the other, i.e., \large m_{2}=2m_{1}, which is about what the LIGO team concluded.  If the orbits were circular, what were their radii just before coalescence?  What were their speeds? Ignore relativity.   (Ans: 40 and 80 km; \large v_{1}=0.35\, c)


1.  B. P. Abbott et al, “Observation of Gravitational Waves from a Binary Black Hole Merger,”, Phys. Rev. Lett. 116, 061102 (2016)

2.  this blog:  “Gravitational Radiation 2: The Chirp Heard Round the World”

3.  B. P. Abbott et al, “Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence,”, Phys. Rev. Lett. 116, 241103 (2016)

4.  A. Cho, “LIGO Detects Another Black Hole Crash,” Science 352, 1374, 17 June 2016

5.  D. Castelvecchi, “LIGO Sees a Second Black Hole Crash,” Nature 534, 448, 23 June 2016

6.  We thank Jonah Kanner at the LIGO Open Science Center for providing these numbers, which were calculated using the tutorial on the Center’s webpage:  The numbers obtained from the tutorial differ slightly from those reported in Ref. 3 because the analysis is not identical to the one used for the paper.  (In Ref. 3, \large \mathfrak{M}=8.9\pm 0.3\: M_{Sun}.)  The LIGO Open Science Center is a service of LIGO Laboratory and the LIGO Scientific Collaboration.    LIGO is funded by the U.S. National Science Foundation.


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