Gravitational Radiation 2: The Chirp Heard Round the World

ligo-gravitational-waves-detection
Figure 1: Gravitational wave GW150914 “chirp” detected in Hanford WA and Livingston LA. The bright arc in each panel plots the signal’s frequency vs. time, while the brightness registers its intensity. The two wiggly lines are the strain measured at each detector. Image courtesy Caltech/MIT/LIGO Laboratory

 

 

 

 

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.

ligo-livingston-aerial
Figure 2: LIGO Livingston, courtesy Caltech/MIT/LIGO Laboratory

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 ).

  1.   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 \large 10^{10}L_{Sun}, where \large L_{Sun}=3.9\times 10^{26} \textup{W} is the luminosity of the Sun.  There are about \large 10^{11} 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 unequal masses in elliptical orbit:

\large L=\frac{32}{5}\frac{G}{c^{^{5}}}\mu ^{2}a^{4}\omega_{orb}^{6}f(\varepsilon )

(7)

where \large \mu =m_{1}m_{2}/(m_{1}+m_{2}) is the system’s reduced mass, and \large f(\varepsilon ) 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 (\large \varepsilon =0\large f(\varepsilon )=1).  As usual,  the total orbital energy of the binary system is half the potential energy (see PPE, section 10.6): \large E=-Gm_{1}m_{2}/2a.  Setting \large dE/dt=-L, and proceeding just as we did in the previous post to derive Eqn. 4, we find

\large \frac{1}{T}\frac{dT}{dt}=-\frac{96}{5}\frac{G^{3}}{a^{4}c^{5}}\frac{\mu ^{2}(m_{1}+m_{2})^{3}}{m_{1}m_{2}} .

(8)

For spectroscopic binaries (PPE section 8.11), where we can measure a radial velocity curve for each star, \large m_{1} and \large m_{2} 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 \large dT/dt.  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 (\large T_{gw}=T_{orb}/2), so the wave frequency is twice that of the orbit: \large f_{gw}=2f_{orb}, 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).

Picture1
Figure 3: The location of two orbiting bodies at the start of an orbit, and half a period later. The gravitational effects are the same at the two times.

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:

\large a^{4}=\left ( \frac{T_{orb}^{2}}{4\pi ^{2}}G(m_{1}+m_{2}) \right )^{4/3}

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

\large \frac{1}{T_{orb}}\frac{dT_{orb}}{dt}=-\frac{96}{5c^{5}}G^{5/3}\mathfrak{M}^{5/3}(2\pi)^{8/3}T_{orb}^{-8/3}

(9)

where \large \mathfrak{M}=\frac{(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}} is called the chirp mass.  Finally, using

\large \frac{1}{T_{orb}}\frac{dT_{orb}}{dt}=-\frac{1}{f_{orb}}\frac{df_{orb}}{dt}

and \large f_{gw}=2/T_{orb}, we obtain

\large \frac{1}{f_{gw}}\frac{df_{gw}}{dt}=\frac{96}{5c^{5}}G^{5/3}\mathfrak{M}^{5/3}\pi ^{8/3}f_{gw}^{8/3} .

Solving for the chirp mass,

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

(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.

Abbott Fig. 2
Figure 4: (top) LIGO signal reconstructed from data shown in Fig. 1; (bottom) Black curve is separation of the two bodies in units of the sum of their Schwarzchild radii. Green curve is their relative speed in units of c

2.     Let\large A=\frac{c^{5}}{G^{5/3}}\, \frac{5}{96}\, \pi^{-8/3}\simeq 5.45\times 10^{56} (in SI units), and rewrite Eqn. 10 as

\large \mathfrak{M}^{5/3}=Af^{-11/3}df/dt .

Next, integrate both sides over the time interval \large \Delta t=t_{2}-t_{1}=0.425-0.250=0.175\: s,

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

Finally, plug in the LIGO numbers to show \large \mathfrak{M}\simeq 30M_{Sun}.

3.     Use Eqn. 10 to show that the total mass of the system \large M=m_{1}+m_{2} must be greater than about \large 70M_{Sun}.  (Hint: let \large m_{2}=\alpha m_{1} and derive expressions for \large \mathfrak{M} and in terms of \large \alpha and \large m_{1}.  Then minimize M.)

4.     Just before the two bodies merged, their orbital frequency was about 75 Hz.  Assuming \large M=70M_{Sun}, estimate the separation a between the bodies at this time.  (Ans: \large a\simeq 350\: \textup{km})

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 \large 1.4M_{Sun}, 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 \large m_{1}=1.4M_{Sun}\large m_{2}=\alpha m_{1}, and derive an expression for \large \mathfrak{M} in terms of \large \alpha and \large m_{1}.  Solve for \large \alpha , noting that \large \alpha \gg 1.)  (Ans: \large \alpha \simeq 2160, so \large m_{2}\simeq 3\times 10^{3}\, M_{Sun} .)

The effective radius of a black hole of mass m is given by its Schwarzchild radius \large R_{s}=2Gm/c^{2}.  General relativity is needed to understand this properly, but a crude qualitative explanation follows from Newtonian reasoning: \large R_{s} 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 \large R_{s}, so the body is “black.”)

6.     Calculate \large R_{s} for the mass \large m_{2} found in Question 5.  Assuming the bodies merge when their separation is equal to \large R_{s}, calculate the orbital period \large T_{orb} immediately before merging.  Then find the frequency of the gravitational wave emitted at that time.  (Don’t forget the factor of 2.)  (Ans:  \large R_{s}\simeq 9\times 10^{3}\: \textup{km}\large T_{orb}\simeq 0.26 \: \textup{s}\large f_{gw}\simeq 7.6 \: \textup{Hz} .)

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

 

 

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