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Gravitational wave science in the high school classroomBenjamin Farr, GionMatthias Schelbert, and Laura TrouilleCitation:Am. J. Phys. 80, 898 (2012); doi: 10.1119/1.4738365View online: http://dx.doi.org/10.1119/1.4738365View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v80/i10Published by theAmerican Association of Physics TeachersAdditional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/Journal Information: http://ajp.aapt.org/about/about_the_journalTop downloads: http://ajp.aapt.org/most_downloadedInformation for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html

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opposite to that of the perpendicular axis [see Figs. 1(a)1(d)]. The goal of gravitational wave detectors is to measurethese minuscule vibrations, with the hope of learning moreabout their sources.

B. Gravitational wave detectors

Currently, the most sensitive operational gravitationalwave detectors are based on the Michelson interferometer,which uses the interference properties of light to makeincredibly precise measurements of distances.5 As shown in

Fig. 2, these detectors split a coherent light beam from a sin-gle laser into two beams. These two beams travel along dif-ferent paths before recombining and entering thephotodetector. More specifically, the detector is set up in anL formation, with a mirror suspended at the end of eacharm. A laser emits a beam of light that is in phase, meaningthe peaks and troughs of each light wave are aligned. Thisoriginal beam of light is split by the beamsplitter. One beamis reflected off of one mirror while the other beam isreflected off of the other mirror. Once the two beams returnto the beamsplitter they recombine, with some light goingback toward the laser while the rest passes to the photodetec-tor. The beam splitter has a reflective coating on one side ofit, which means that of the two possible paths the light could

take to the photodetector, one has reflected from the glassside of the coating, and the other from the vacuum side.

These two different cases of reflection will result in a 180

phase difference. If the distances traveled by each half of thebeam are equal (i.e., the arms are equal in length), the twolight beams will be exactly out of phase and cancel eachother out, resulting in all light traveling back toward the laserand none reaching the photodetector. If instead one arm isslightly shorter than the other, the light no longer exactlycancels out, and a nonzero light intensity is measured by thephotodetector. The intensity of this light measured at the de-tector depends very sensitively on the phase difference of thetwo halves of the beam. Thus by monitoring the fluctuations

in the intensity of exiting light, the difference in arm lengthscan be determined with incredible accuracy.

This is the underlying principle for the detection of gravi-tational waves. The three largest detectors in the world basedon this design make up the LIGOVirgo Collaboration(LVC). Figure 3 shows the LVC network, consisting of twodetectors in the United States (located in Hanford, WA andLivingston, LA) with arm lengths of 4 km that make up theLIGO,1 as well as Virgo,2 a 3 km detector in Italy.

C. Sources

According to the theory of general relativity, any mass

that is accelerating in a way that is not perfectly sphericallyor cylindrically symmetric will produce gravitational waves.Though this excludes some processes like the sphericallysymmetric pulsations of stars, it does include countless otherevents, ranging from the energetic collision of stars andblack holes to the less spectacular toss of a ball.

Consider a system of two bodies, each about as massive asthe Sun, orbiting about one another. As the bodies orbit,gravitational waves are emitted with a period that is closelyrelated to that of the orbit. These waves carry energy awayfrom the system. As the orbit loses energy, the separationbetween the two objects must shrink, thereby decreasing theorbital period. Furthermore, as the bodies get closer together,the second time derivative of the quadrupole moment varies

more rapidly, resulting in an increase in the gravitationalwave amplitude. This increase in frequency and amplitudecontinues until the orbital radius decreases to the point ofmerger, where the two objects physically combine to form asingle body. Until the time of merger the system is said to bein its inspiral phase, which is modeled fairly accurately bymaking small corrections to the non-general relativisticequations of motion. An example of a gravitational waveproduced during the inspiral phase of such a system is shownin Fig. 4(a). As discussed in Sec. III B, when presenting thisin a high school classroom setting, the explanation for whythe amplitude increases is more qualitative. The focus is onthe students appreciating the connection between the more

Fig. 1. Time evolution of a ring of particles influenced by a passing gravitational wave, where P is the period of the wave. Each consecutive panel is a snapshottaken P/4 later in time. In each figure, both the current state of the ring (black) and the previous state of the ring (gray) are shown.

Fig. 2. Schematic diagram of the Michelson interferometer, showing the

path of the light beam as it is split and then recombined before entering the

photodetector. The dark line on the beam splitter indicates the reflective

coating, and beams reflected from different sides of the coating have a 180

phase difference. This setup is very similar to that used in gravitational

wave detector technology.

899 Am. J. Phys., Vol. 80, No. 10, October 2012 Farr, Schelbert, and Trouille 899



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extreme curvature of spacetime as the bodies get closer to-gether and the increase in amplitude of the signal.

If we were to take the same system, but compress eachobjects mass into a smaller radius, the inspiral phase wouldbe prolonged. In this case, the orbital radius is able to reacheven smaller values before these denser objects merge,thereby increasing the amplitude and frequency reached bythe gravitational wave before merger. Consequently, only bi-nary systems containing the densest objects in the universe,namely neutron stars and black holes, are capable of produc-

ing gravitational waves at amplitudes and frequencies detect-able by current detectors.The fact that we have yet to detect a gravitational wave,

despite being surrounded by sources, is due primarily to thestiffness of spacetime. The stiffness of spacetime refers tothe incredible amounts of energy in gravitational waves thatare required to distort spacetime to a degree we can measurewith our detectors. A second major factor is the relativelylow amount of energy emitted in gravitational waves by sys-tems in the first place. As an example of the latter, theamount of energy per second (power) radiated in gravita-tional waves by the orbit of Jupiter around the Sun is5200 W.6 Even though this is the most energetic source ofgravitational waves in our solar system, the energy radiated

in all directions each year by the orbit of Jupiter would onlybe enough to power a single average household in the UnitedStates.7 To detect gravitational waves, we must look formuch more energetic sources, outside of our solar system.The stiffness of spacetime is apparent if we consider a gravi-tational wave just barely detectable with current detectors,which periodically changes the difference in the distancesalong the arms of the detector by at most 1019 m with a fre-quency of 100 Hz. Such a signal has a flux (power per unit

area) of about 105 W=m2. This is approximately the sameflux in visible light measured 300 m away from a standard60 W light bulb. Thus even though these fluxes are equiva-lent, in the case of gravitational waves, the signal is barelydetectable with state of the art detectors, whereas with elec-tromagnetic radiation, the signal is easily detected by thehuman eye.

D. Data analysis

The LVC detectors shown in Fig. 3 are measuring changesin the difference of the distances along the arms that areorders of magnitude smaller than the diameter of a proton(approximately 1015 m), reaching sensitivities of 1018 to1019 m.5 In addition to gravitational waves, such minutefluctuations in arm length can also be caused by many unin-teresting sources, including seismic vibrations, local high-way traffic, etc. With so many noise sources causing signalsat comparable levels to those we are trying to detect,advanced data analysis techniques are necessary. Many ofthese techniques rely on having a reasonably accurate theo-retical model for the system emitting the gravitational waves,which provide the hypothesized signal that is then looked forin the data. As described in Sec. II C, the LVC network is

particularly sensitive to the mergers of black holes and neu-tron stars. The main search algorithm for merger signals inthe LVC uses the technique of matched filtering,8 in whichwe first construct a bank of possible signals, and then searchthe data for instances of a statistically significant match to asignal in the bank. This method is very efficient at detectingpossible signals in large amounts of data but does a poor jobof determining source properties of individual signals, suchas the masses of the merging objects and where in the sky

Fig. 3. The LIGOVirgo gravitational wave detector network, consisting of (a) LIGO Hanford (Credit: LIGO Laboratory), (b) LIGO Livingston (Credit: LIGOLaboratory), and (c) Virgo (Credit: EGO). LIGO Hanford (Hanford, WA) and LIGO Livingston (Livingston, LA) both have 4 km long arms, while Virgo (Cas-

cina, Italy) has 3 km long arms.

Fig. 4. The two most common responses by students when asked to hypothesize what the gravitational wave from a binary merger should look like. (a) A qual-

itatively accurate model, similar to the majority of student responses. (b) A model with constant amplitude, the most common misconception among students.




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the source is located. To accurately estimate these parame-ters, other algorithms are required that are designed to ana-lyze individual signals found by the main search algorithm.These codes are based on Bayes theorem,9 and extract themaximum amount of information possible from the meas-ured signal in the data, assuming the models used in thesearch accurately represent the signal in the data.1012

III. GRAVITATIONAL WAVE SCIENCE

IN THE HIGH SCHOOL CLASSROOM

Over the course of the 20102011 academic year, theauthors worked to incorporate concepts and ideas from thefield of gravitational wave science into a high school astron-omy curriculum. Curriculum changes and lessons wereimplemented across 8 classes of students, with each classhaving about 25 students. Each individual class had a mix ofjuniors and seniors, as well as non-honors and honors stu-dents. Throughout the year, new lesson topics were addedpertaining to gravitational wave astronomy. Existing lessonswere also modified and tied into gravitational wave science(e.g., waves and interference), creating a common theme forthe years lessons.

In Sec. III A, we describe how we incorporated gravita-tional wave science into the existing unit on waves. In Sec.III B, we present how we used gravitational waves as anintroduction to the basics of general relativity. We alsodescribe the demonstrations, guided inquiry, and manipula-tion of computational models that we designed and used inthe classroom to support student learning of gravitationalwave science. In Sec. III C, we discuss how we taught signalprocessing using Fourier techniques in the context of gravita-tional wave science. Finally, in Sec. III D, we provide a listof other tools for educators interested in teaching gravita-tional wave science, and in Sec. III E, we discuss futurework.

A. Waves and interference

The physics of waves is relevant to both the gravitationalwaves themselves, and the design of the interferometricobservatories built to detect them. Many properties of elec-tromagnetic waves, such as amplitude, frequency, and polar-ization, are also relevant to gravitational waves. Since theMichelson interferometer is designed to utilize the propertiesof wave interference, a tabletop interferometer is an idealdemonstration to supplement discussion of these topics (seeFig. 2). In this modified lesson on wave physics, we firstreviewed transverse waves and their properties in a mini-lecture, material that the students had read the night before.After the review, students broke into small groups to work

on an activity,13

making use of an online applet14

thatallowed students to manipulate overlapping waves andobserve their resulting superposition. After completing theactivity, students were shown a tabletop Michelson interfer-ometer and were introduced to how constructive and destruc-tive interference pertain to its design. By applying veryslight pressure to a mirror of the device, students developedan appreciation for the extreme sensitivity of the instrument.Small changes in the interference pattern on the projectionscreen were observed even when students lightly tapped onthe table holding the device. On this particular apparatus,there was a knob that would move the mirror by very smallamounts. This knob was rotated back and forth as an exag-

gerated example of the effects of a gravitational wave on theinterference pattern, while emphasizing to the students thatduring a gravitational wave event, it is the distance betweenthe mirrors and splitter that is expanding and contractingcausing such an effect, not the physical movement of themirrors. This mixture of hands-on demonstrations and com-putational models provided the students with a variety ofways to develop an understanding of wave physics.

Though we did not do so in this particular demonstration,

future lessons could be improved by embedding a photo-diode in the screen, which would measure fluctuations in theinterference pattern. By feeding to a speaker the varyingvoltage output by the photodiode, the changes in the interfer-ence pattern can be heard. In particular, this would makehigh frequency periodic changes to the pattern much easierto observe. With this configuration, we can potentially showthat these instruments are even capable of picking up vibra-tions caused by speech, acting much like a microphone.

B. General relativity and gravitational waves

To help students begin to grasp the concept of gravity inthe framework of general relativity, a large fabric sheet was

used as a 3D representation (two spatial dimensions plustime) of our 4D spacetime. Spheres of various masses wereplaced on the sheet to demonstrate how the presence of masscurves spacetime. Rolling marbles near these massiveobjects then showed how this curvature acts in a way analo-gous to gravity. Before demonstrating each different scenarioto the students, we asked for their predictionswhat didthey think would happen? In the first scenario, the sheet wasempty and we gave a marble an initial velocity so as to sendit in a straight path across the sheet. In the second scenario, amassive ball was placed in the middle of the sheet and themarble was given the same initial velocity as in the first sce-nario. Since the physics of objects on the fabric sheet is fairlyintuitive to the students, they were able to predict that the

marbles motion would be deflected by the curvature causedby the massive ball on the sheet. The students were also ableto predict that the marbles trajectory would depend on itsspeed and distance of closest approach to the massive ball.To demonstrate circular motion of the marble around themassive ball (simulating, for example, the motion of planetsaround our sun), we found that it worked best to place themarble quite close to the massive ball and give it a small ini-tial transverse velocity. The final demonstration was the caseof a binary orbit, with two massive balls orbiting oneanother. During this demonstration, students were asked tolook at the behavior of the sheet far away from the binarysystem, so that they would observe the distant vibrationscaused by the binary orbit. These vibrations, the students

were told, were analogous to gravitational waves.Following this discussion, students worked in small

groups on an activity designed to guide them through the sci-entific reasoning required to determine the basic characteris-tics of a gravitational wave from a binary system. Thisactivity, which has been made publicly available,13 beginsby having the students analyze a simple sinusoidal wave,measuring its amplitude, period, and frequency. They werethen asked to draw examples of waves whose amplitudes andfrequencies changed with time, to get them thinking aboutthe possibility of waves without a fixed amplitude and fre-quency. Students were then reminded that gravitationalwaves carry energy, and thus a system emitting gravitational




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waves must be losing energy. This situation is analogous towhat they had just seen in the fabric sheet demonstration,where instead of losing energy through gravitational wavesthe orbit lost energy due to friction. This loss of energyresulted in the orbital radius shrinking and the orbital veloc-ity increasing, just as it does during binary evolution.

The last part of the activity had students hypothesize whatthey believed a gravitational wave from an inspiraling binarysystem should look like, then compare that to a more rigor-

ous computational model.

15

The applet used to interfacewith the computational model (provided for free by WolframDemonstrations16) displays the gravitational wave for agiven set of parameters describing the binary system. Sliderbars then allow students to manipulate various parameters inthe model (e.g., inclination, component masses, etc.), toinvestigate how these parameters can affect the modeledwaveform.

This combination of visual demonstrations, guided in-quiry, comparison and manipulation of computational mod-els, and group discussion throughout is essential for havingthe students develop an accurate understanding of gravita-tional waves and overcome common misconceptions. Wefound that through the demonstration and the guided inquiry

worksheet, over 80% of students were able to predict andexplain why the frequency of the gravitational wave willincrease with time as the binary system inspirals. At thisstage, however, only about half of the students were able todeduce that the amplitude of the wave will also increase withtime, producing qualitatively accurate models like the oneshown in Fig. 4(a). The most common misconception wasthat the amplitude of the wave would remain constant withtime, as illustrated in Fig. 4(b). Through manipulating thecomputational model, students with the misconception rec-ognized that the amplitude of the wave does in fact increasewith time. However, even after this stage, many could notexplain the physical cause for this amplitude increase. Onlythrough a final full class discussion, in which students with

more accurate models explained in their own words whythe amplitude increases during the inspiral phase, did theremainder of the students understand this aspect of the sys-tem. More specifically, the students explained to their peersthat the curvature of spacetime becomes more extreme asthe massive objects move closer together (as seen in thefabric demonstration). As a result, we expect the strengthof the gravitational wave (i.e., its amplitude) to also increaseas the black holes approach one another (as seen in the

computational model). The more accurate model thus showsboth an increase in frequency and an increase in amplitudeof the gravitational wave with time during the inspiral phase.This final discussion as well as the small group discussionswas very lively. The students were clearly engaged andexcited to explore and gain understanding of this cutting-edge science. Additional lessons can later be used to showhow such models are crucial to gravitational wave detectionin noisy data.17

C. Signal processing

Signal processing is a skill common to many fields in obser-vational science but is often not taught, or even mentioned,until students are in college. By teaching about gravitationalwaves elsewhere in the curriculum, we were able to providean authentic context in which to learn about signal processing.Not only that, but we were also able to appeal to many stu-dents interest in music and music editing while doing so.

With such high levels of noise, signal processing and dataanalysis for gravitational wave detectors is a challengingtask. One crucial tool in the search forany periodic signal intime-domain data is Fourier analysis.18 Rather than scanning

the data for indicators of the detector arm length changingperiodically as a function of time (something made impossi-ble by the noise causing such similar periodic changes), Fou-rier analysis allows one to transform the data to look atpower as a function of frequency instead. When viewingdata in the frequency domain, the presence of a low-levelperiodic signal in random noise is easily seen, even with sig-nal amplitudes much smaller than that of the noise. In thecontext of gravitational waves, a given waveform is gener-ated according to a computational model. This model is thensubtracted from the data, and the remaining data is comparedto the noise measured in earlier data that did not contain ameasurable signal. How well these data with the model sig-nal subtracted match the previously measured noise gives a

quantitative way to assess how well the model matches thesignal.

Students used AUDACITY,19 an open-source audio editingprogram, to generate specific tone and noise spectra. Theythen added the tone and the noise spectra together. Figure5(a) shows the time-domain track of the combined data, con-taining white noise and a low-amplitude 100 Hz sinusoidalwave. By using AUDACITY, students were able to see that avisual inspection of the time-domain data does not allow

Fig. 5. Students used the open-source audio editing program AUDACITY to explore the use of Fourier analysis in signal processing. (a) A composite track con-

taining white noise and a 100 Hz sine wave, generated by the student. (b) The result of a Fourier analysis of the combined signal and noise, showing the pres-

ence of a strong signal at 100 Hz.




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them to separate the signal from the noise nor can they hearthe signal when playing the audio sample. The students thentook the Fourier transform of the composite track, as shownin Fig. 5(b). The spike corresponds to the frequency of theinput tones low-amplitude sine wave. This demonstrated to

them how using Fourier analysis can aid in the search for aquiet (low amplitude) coherent signal buried in randomnoise. This is a problem very similar to that encountered ingravitational wave data analysis.

Finally, students were shown a simulated noise spectrumof LIGO, seen in Fig. 6(a), and the frequency-domain repre-sentation of a gravitational wave from a binary inspiral,shown in Fig. 6(b). Students were also able to listen to theaudio representation of these data and could distinctivelyhear the difference between white noise and LIGOs noise,as well as listen to the waves they had learned about in a pre-vious activity (see Sec. III B).

D. Other tools for gravitational wave education

As part of its education and public outreach efforts, theLVC has created many other tools to assist with educatingstudents about gravitational waves. The LIGO Science Edu-cation Center20 located on the Livingston, LA detector sitehosts field trips and professional development workshops, aswell as a Research Experience for Teachers program.21 Thisprogram is a 6-week paid internship for K-12 teachers,designed to provide teachers with the opportunity to work ina scientific research environment on topics related to LIGO.The Einsteins Messengers website22 provides many curricu-lar resources including connections to state and nationalstandards, teacher and student study guides, and classroomactivities related to LIGO and gravitational waves. Finally,

there are many web applets and games designed to introducestudents to concepts in gravitational wave physics and detec-tor technology. The gwoptics.org website23 hosts outreachmaterial, including several games developed by the gravita-tional wave group in Birmingham, UK. Black Hole Hunter24

is a game intended to introduce people to gravitational wavedata analysis, having them search for the sound of gravita-tional waves in simulated noisy data.

E. Future work

Now that basic gravitational wave science has been inte-grated into the astronomy curriculum, more advanced topics

can be introduced in coming years. For example, the additionof independent study projects would provide another way toengage student in hands-on activities. One possible projectwould be to have a group of students design and build theirown interferometric detector. The components necessary to

construct a tabletop interferometer are relatively cheap,requiring only a laser, beamsplitter, photodiode, tabletop,and two mirrors. An optical table would be ideal to house theinterferometer, though that could be quite expensive toobtain. Cheaper tables could be constructed from wood orother materials, and table construction itself could even bepart of the project. Once the detector is built, students couldexperiment with ways of making it more sensitive and reduc-ing the effects of environmental noise.

The material presented here was integrated into the astron-omy curriculum throughout the year. Time should now bespent to condense this into a single self-contained unit thatcan be more easily inserted into a K-12 or introductory col-lege physics curriculum.

One main area still to be added to the unit is parameter esti-mation. After the detection of the first gravitational wave, theera of gravitational wave astronomy will begin. In order toextract all the information available from a gravitational wavesignal, one must use a technique specifically designed to doso. Some work has been done to bring parameter estimationconcepts into the classroom;25 however, lessons focusing onthe currently used Bayesian methods have yet to be devel-oped. The Bayesian analysis techniques employed by theLIGOVirgo collaboration are designed to extract the physicalcharacteristics of the source of a measured gravitationalwave.1012 These methods utilize concepts from computa-tional science, statistics, applied math, and other disciplines.By including this in the curriculum, students would be

exposed to topics across the STEM fields that they would nor-mally never see in the high school setting, further demonstrat-ing the importance of interdisciplinarity in science.

IV. SUMMARY

Gravitational wave astronomers are on the verge of directlydetecting gravitational waves for the first time. Within thenext decade, gravitational wave science will change the fieldof astronomy by opening a new window to the universe. Asgravitational wave science assumes a more prominent role inthe astronomical community, it will be important for people tohave at least a basic understanding of what gravitational

Fig. 6. Since LIGO is most sensitive to frequencies around 100 Hz, noise and signals relevant to LIGO can be converted to sound data that is audible to the

human ear. Using sound editing software, students were able to listen to and plot the spectra of (a) simulated LIGO noise and (b) a gravitational wave signal

from a binary merger.




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waves are. This article has provided several examples of howto introduce students to waves, interference, gravity, generalrelativity, and basic data analysis techniques through theirapplications to gravitational wave science. Through theseinteractive demonstrations and computationally based activ-ities, students are able to learn about gravitational waves andthe technology behind their detection, without losing focus ontopics already in the curriculum.

ACKNOWLEDGMENTS

This work was supported by the NSF GK-12 grant, awardDGE-0948017, and the NSF LIGO grant, award PHY-0969820. The authors would like to thank the referees fortheir comments and suggestions.

a)Electronic mail: [email protected]

b)Electronic mail: [email protected]

c)Electronic mail: [email protected]

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