### A. The Discovery

Sometime, maybe 1.5 billion years ago, two large collapsed stars began a long, slow mating dance. Wedded by gravity, they patiently circled their center of mass, inching together imperceptibly (perhaps a few meters each year, like the Hulse-Taylor binary), all the while quickening their speed. As they approached one another – over millions of years – they shed vast amounts of energy via gravitational radiation, and seconds before the end of their dance, their gyration increased feverishly as their relative speed approached half that of light. Then, in a split second, they embraced to form a single black hole, emitting a final burst of gravitational radiation that *briefly outshone the light emitted by the entire visible universe*! 1.3 billion years later, on September 15, 2014, this blast of energy reached planet Earth, where it was detected by two gargantuan, souped-up Michelson interferometers, each with arms 4 km long, collectively called the Laser Interferometer Gravitational Wave Observatory (LIGO). (See Fig. 2.)

The passing gravitational wave differentially changed the length of each detector’s arms by about one-thousandth the diameter of an atomic nucleus, inducing an unmistakable signal 24 times greater than background noise. The age of gravitational-wave observational astronomy had begun.

The LIGO Collaboration first reported its discovery in the February 12, 2016 issue of *Physical Review Letters*. (See Ref. 1.) Their main conclusions, summarized in Section II of that paper, are described clearly and simply, and best of all, they can be understood by students having an introductory-level comprehension of classical mechanics. (Because of its groundbreaking significance, the *PRL* paper is available open access — no subscription or institutional affiliation is necessary to download it.) Two other recently published papers will surely be helpful to educators wishing to discuss the LIGO discovery in their classrooms. The first (Ref. 2), by Lior Burko, describes an introductory lab exercise using data taken from the LIGO Open Science Center ( https://losc.ligo.org/events/GW150914/ ). It is clearly written and discusses the LIGO results in more detail than the present post. The second (Ref. 3), by Louis Rubbo *et al.*, illustrates how to extract astrophysical information from LIGO-like (simulated) data, and a separate link offers a complete worksheet and teacher’s guide suitable for use by first-year physics or astronomy students (http://cgwp.gravity.psu.edu/outreach/activities/handson_activity/downloads/handson_teachers_guide.pdf ).

- When the two bodies merged, they shed about 3 solar masses of energy in less than 0.1 s. Show that the average power emitted during this fleeting time interval was greater than the total electromagnetic radiation emitted by the entire visible universe. (The luminosity of the Milky Way galaxy is roughly , where is the luminosity of the Sun. There are about galaxies in the observable universe.)

### B. A Quick Review of Gravitational Wave Physics

Our goal is to introduce LIGO’s exciting discovery into the introductory mechanics curriculum, in a way that supports and complements the fundamental topics of the course. For the intended first-year audience, the discussion must rest primarily on Newtonian physics, and concepts from general relativity must be strictly limited: no tensors allowed! In the previous post, we introduced an expression (Eqn. 1) for the gravitational luminosity of two equal-mass stars in circular orbit, and later generalized (Eqn. 3) that expression to treat binary systems with *un*equal masses in *elliptical* orbit:

(7)

where is the system’s reduced mass, and is a correction associated with the orbit eccentricity ε*. *In the following discussion, we’ll use Eqn. 7 rather than Eqn. 1 to conform to the notation commonly found in the literature. Eqn. 7 – plus a lot of tedious algebra – is all we need to extract physical information from the LIGO signal. That signal was produced by two massive in-spiraling bodies in the split second before they merged to form a single entity, and by that time the energy drain due to gravitational radiation had circularized their orbits (, ). As usual, the total orbital energy of the binary system is half the potential energy (see *PPE*, section 10.6): . Setting , and proceeding just as we did in the previous post to derive Eqn. 4, we find

.

(8)

For spectroscopic binaries (*PPE* section 8.11), where we can measure a radial velocity curve for each star, and can be determined using Kepler’s 3rd law and momentum conservation. In the LIGO case, we have no radial velocity curve, but the detected signal allows us to measure the orbital period *T* and its rate of change . This will be sufficient to draw conclusions about the mass and nature of the system. In particular, we can understand why the two merging bodies were most likely black holes.

### C. Extracting Physical Information from the LIGO Signal

Like a radial velocity curve, the period (or frequency) of the LIGO signal is directly related to the orbital period (or frequency) of the binary system. In the LIGO case, however, the period of the detected gravitational wave is* half* that of the orbital period (), so the wave frequency is *twice* that of the orbit: , where we have added the subscripts “gw” and “orb” to distinguish between wave and orbit. These relationships are easy to understand in the case of two identical bodies (Figure 3).

When the two bodies execute half an orbit, they exchange their original positions, and the gravitational effects seen far away are the same as at the start of the orbit. Therefore, the radiated wave is periodic with half the period of the orbit.

We cannot determine the orbital radius *a* from the detected signal, so let’s use Kepler’s 3rd law to eliminate *a* from Eqn. 8:

and, after MUCH messy algebra (which might be left as a student exercise), we obtain

(9)

where is called the *chirp mass. *Finally, using

and , we obtain

.

Solving for the chirp mass,

(10)

where we have now dropped the subscript “gw” to adopt the notation used in the literature.

Figure 4 is taken from Ref. 1. It shows a calculated waveform derived from the actual detected signals shown in Fig. 1 at the beginning of this post. According to Ref. 1, the signal increases in frequency from 35 to 150 Hz in the time interval 0.250 to 0.425 s immediately before the two bodies coalesce. Let’s use these numbers to calculate the chirp mass.

2. Let (in SI units), and rewrite Eqn. 10 as

.

Next, integrate both sides over the time interval ,

.

Finally, plug in the LIGO numbers to show .

3. Use Eqn. 10 to show that the total mass of the system must be greater than about . (Hint: let and derive expressions for and *M *in terms of and . Then minimize *M*.)

4. Just before the two bodies merged, their orbital frequency was about 75 Hz. Assuming , estimate the separation *a* between the bodies at this time. (Ans: )

Your answers to questions 3 and 4 should show that the two bodies must have been highly compact objects, either black holes or neutron stars. (Compare your answer to question 4 to the radius of the Sun.) Since a neutron star has a mass of about , they could not both have been neutron stars. Could one of them have been a neutron star?

5. Assuming one of the merging bodies was a neutron star, what was the mass of the other? (Hint: let , , and derive an expression for in terms of and . Solve for , noting that .) (Ans: , so .)

The effective radius of a black hole of mass *m* is given by its Schwarzchild radius . General relativity is needed to understand this properly, but a crude qualitative explanation follows from Newtonian reasoning: is the distance from a point source *m* at which the escape speed equals *c*. (Nothing, including light, can escape from a body whose radius is less than , so the body is “black.”)

6. Calculate for the mass found in Question 5. Assuming the bodies merge when their separation is equal to , calculate the orbital period immediately before merging. Then find the frequency of the gravitational wave emitted at that time. (Don’t forget the factor of 2.) (Ans: ; ; .)

7. So why did the LIGO team rule out a black hole – neutron star merger?

References:

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

2. Lior Burko, “Gravitational Wave Detection in the Introductory Lab,” arXiv:1602.04666 [physics.ed-ph]

3. Louis J. Rubbo *et al*., “Hands-on Gravitational Wave Astronomy: Extracting astrophysical information from simulated signals,” *Am. J. Phys*. **75**, 597 (2007) and accompanying teacher’s guide: http://cgwp.gravity.psu.edu/outreach/activities/handson_activity/downloads/handson_teachers_guide.pdf