EExplosion analysis from images: Trinity and Beirut
Jorge S. D´ıaz
Physics Department, Indiana University, Bloomington, IN 47405, U.S.A.
Images of an explosion can be used to study some of its physical properties. After reviewing andclarifying the key aspects of the method originally developed to study the first nuclear explosion, theanalysis of the data is discussed in connection to undergraduate laboratory experiences. Followingthe exposition of the procedure for the Trinity explosion, the method is applied to the Beirutexplosion of August 2020 by using the frames of many videos posted online and producing remarkablyaccurate results. The estimate for the explosion yield of the Beirut blast is found to be 4 . +0 . − . TJor 1 . +0 . − . kt of TNT equivalent. A basic modeling of the pressure wave indicates the temporarysupersonic speed of the blast. I. INTRODUCTION
The mathematical description of an explosion size asa function of all the relevant physical quantities has be-come a classic in undergraduate classrooms over the yearsas a way of introducing dimensional analysis. The storygoes that, only using dimensional analysis, physicist Ge-offrey I. Taylor was able to determine the yield of thefirst nuclear explosion (Trinity test). This over-simplifiedaccount is usually accompanied by many other inaccura-cies that have been preserved and exaggerated leadingto a more dramatic narrative. Some versions incorrectlyportrait Taylor as an independent researcher that wasnot involved in the Manhattan Project; whereas otherseven suggest that Taylor revealed to the public a secretnumber from declassified information. These misleadingversions only add an entertainment aspect to an alreadyscientifically interesting story that can be used in class-rooms to show our students that the techniques theylearn in their first courses can have real-world applica-tions and even crucial consequences. Taylor was not anindependent researcher, his first report was the result ofa request from the UK Ministry of Home Security in 1941that shared highly classified information about the poten-tial development of a fission-powered weapon, as narratedby himself [1]. During the Manhattan Project he was, to-gether with Niels Bohr, one of the highly distinguishedconsultants that were made available under British aus-pices as part of the British mission to Los Alamos [2]and one of the selected group of scientists invited to theTrinity test [3]. Furthermore, his now popular work de-termining the yield of the Trinity explosion from declas-sified images was published in 1950 [1, 4], only after histwo technical reports on the blast formation were declas-sified by the U.S. Atomic Energy Commission. At thistime the yield of the Trinity test as well as the two bombsdropped over Japan were already of public knowledge.Nonetheless, most of the inaccuracies appear in theevaluation of a dimensionless factor that cannot be ac-counted by dimensional analysis. Many of the myths be-hind this story have been examined in detail by Deakin[5], including comparisons between the work of Taylor[1, 4] with lesser known developments in parallel thattook place in the U.S. by John von Neumann [6] and in the Soviet Union by Leonid Sedov [7].This article is organized as follows. The general pre-sentation of the evolution of an explosion is discussed inSec. II; the method used by Taylor is presented in Sec.III, and the application of the method to the data fromthe Trinity explosion is shown in Sec. IV. The Beirut ex-plosion and the application of the method to this eventis described in Sec. V, including the selection of imagesand the estimate of the energy yield. Finally, a generalanalysis of the kinematics of the blast is presented in Sec.VI. All the relevant images and data sets are included inthe Appendix.
II. DESCRIPTION OF THE BLAST SIZE
The popular presentation of Taylor’s work follows thedescription of a spherically symmetric explosion charac-terized by its radius R in terms of the energy of the explo-sion E , the time since the detonation t , and the densityof the medium ρ . The assumption is that these quantitiesare related by power laws R = S ( γ ) E a ρ b t c , (1)where a , b , and c are dimensionless constants. The di-mensionless function S ( γ ) has to be determined fromthe thermodynamical evolution of the explosion and itdepends on the adiabatic index of the medium γ . Allpopular accounts of Taylor’s story completely ignore thisobservation and simply assume that the dimensionlessquantity is a constant. Most versions then assume thatthis constant is approximately 1; others go further anddescribe the tale of Taylor experimenting with small-scaleexplosions to determine the constant . This tale probablyarose from an addendum that Taylor included at the endof his first paper at the time of declassification (1949),in which he briefly compares his theoretical descriptionwith newly available data of pressure measurements fromthe conventional explosion of RDX and TNT [1].At this point dimensional analysis is introduced. Theradius has units of length [ R ] = L , whereas the dimen-sions of the quantities on right-hand side are [ S ( γ )] = 1,[ E ] = M L T − , [ ρ ] = M L − , and [ t ] = T . The con-sistency of Eq. (1) implies a system of linear equations a r X i v : . [ phy s i c s . e d - ph ] S e p relating the three exponents as follows:0 = a + b, a − b, (2)0 = − a + c, whose solution a = − b = 1 / , c = 2 / R = S ( γ ) (cid:18) Et ρ (cid:19) / . (3)This is the first equation in Taylor’s first paper [1]. Itshould be emphasized that more than simple applicationof dimensional analysis, the relationship in Eq. (3) wasformaly obtained via a scale-invariance argument thatreduced a system of partial differential equations into or-dinary differential equations [8]. Another critical miscon-ception of Taylor’s work is solving the last equation forthe energy in the form E = ρR S ( γ ) t , (4)because this apparently shows that a single measurementof the fireball size R at time t after the detonation sufficeto determine the energy. This assumes that the func-tional relationship between all the quantities involved iscorrect; unfortunately, a single measurement cannot pro-vide any information about the validity of this assump-tion. Several measurements in the form of pairs ( t, R ) areneeded to first verify the validity of Eq. (3), and then theenergy can be determined. This is exactly what Taylordid in his second paper [4]. III. TAYLOR’S METHOD
In 1941, G. I. Taylor developed a theoretical descrip-tion of the formation of a blast by a hypothetical nuclearexplosion [1]. His report remained classified until 1949.The day after the second anniversary of the Trinity test,the U.S. Atomic Energy Commission declassified a tech-nical report including 25 images of the Trinity explosionindicating timestamps and a length scale [9]. This ledTaylor to write his now-famous second paper, in whichhe makes use of the 25 available pairs ( t, R ) to assess histheoretical formulation. In early laboratory experiencesstudents learn that the analysis of power-law and expo-nential relations can be done with ease by taking advan-tage of the properties of logarithms, that allow convertingexponents into simple multiplicative factors. This is howstudents can analyze the terminal velocity of falling ob-jects, such as coffee filters, in Mechanics class and thetime constant in RC circuits in Electromagnetism class:the determination of exponential and other factors is re-duced to simply estimate the slope and intercept of aline from their experimental data in log-log space. Tay-lor followed this method to reduce the complexity of the relationship between R and t in Eq. (3), which can bewritten as 52 log R = log t + 12 log (cid:18) S ( γ ) Eρ (cid:19) . (5)This expression indicates that if instead of the pairs ( t, R )we use the pairs ( x, y ) = (log t, / R ), then the plotwill be a straight line of the form y ( x ) = mx + n , with m = 1. Here Taylor makes two clear predictions:1. the pairs (log t, / R ) will follow a line withslope 1;2. the intercept n can be used to determine the energyusing n = 12 log (cid:18) S ( γ ) Eρ (cid:19) . (6)Notice that prediction 2 requires all pairs to be along thesame line so that there is a unique value for the intercept n . The high temperatures involved would lead to changesin the value of γ due to the increase of C V via absorptionof energy in the form of molecular vibrations of the gasesin the air as well as the absorption of intense radiation inthe outer layers of the blast. This means that the datacould satisfy prediction 1 but not necessarily prediction2 due the functional dependence of the factor S ( γ ) onthe fluctuating adiabatic index. The validity of the twopredictions implies from Eq. (6) that the energy yield ofthe explosion can be written E = 10 n ρS ( γ ) . (7) IV. TRINITY EXPLOSION
From the technical report declassified by the U.S.Atomic Energy Commission [9] Taylor constructed a ta-ble with the ( t, R ) and (log t, / R ) pairs [4].The result is reproduced in Fig. 1, which shows a re-markable agreement of the data with the theoretical de-scription in Eq. (5). A simple linear fit results in the pa-rameters m = 1 . ± .
02 and n = 6 . ± .
05, with all thedata following a single line confirming the two predictionsof Taylor’s modeling. This last observation suggests thatall the effects leading to variations in the value of the adi-abatic index interplay producing a constant value for theeffective γ [4]. Considering an atmospheric explosion andthe diatomic nature of nitrogen and oxygen that composemost of the air, the adiabatic index is γ = 7 /
5. Taylorcomputed the numerical value of S ( γ ) − for different sit-uations. For γ = 7 / ≈ . S (1 . − = 0 . K in the second paper) [4]. Notice that thisimplies S (1 .
4) = 1 . − − − − t/ s)34567 l og ( R / m ) Fireball radius R at time t after explosionTrinity datalog t + 6 . FIG. 1. Reproduction of Taylor’s logarithmic plot showingthat the data from the Trinity explosion follows the predic-tions of Eq. (5). The dashed line corresponds to the linearfit. Notice that Taylor used CGS units [4]. in close agreement with the formal analysis. Using thedensity of air ρ = 1 .
23 kg/m and the intercept fromthe data produces the estimate for the yield of the firstnuclear explosion to be E = 79 . ± . .
18 TJ [10], the data in Fig. (1) leads toa yield of 19 . ± . n = 6 . ± . E = 66 . ± . . ± . V. BEIRUT EXPLOSION
On August 4, 2020 a devastating explosion occurredat the port of Lebanon’s capital Beirut. Official prelimi-nary reports indicate that a fire affecting a fireworks stor-age facility would have extended within the warehouse towhere a large amount of ammonium nitrate was stored.The main explosion was preceded by an early deflagra-tion of the firework products generating a thick gray col-umn of smoke. This led to many video recordings of
FIG. 2. Frames of the first 167 milliseconds in Video 1 show-ing the expanding fireball. The time interval between framesis 33 msec. and the filming location is indicated in Fig. 3 the events that followed from multiple angles around thecity. Many videos show the explosion of the ammoniumnitrate as an expanding fireball surrounded by a short-lived Wilson cloud that engulfed the buildings near theport before disappearing and making visible an ascendingred-brown mushroom cloud characteristic of ammoniumnitrate explosions, followed by the corresponding blastof the pressure wave that caused great damage. Gov-ernment officials reported that the blast was caused bythe explosion of 2750 metric tons of ammonium nitratestored in one of the port warehouses [11].Footage of the explosion circulated rapidly via social-media channels [12]. Armed with Taylor’s method we cantry to estimate the yield of the Beirut explosion. A frame-by-frame analysis allows a clear identification of the first200 milliseconds of the expanding fireball. Figure 2 showsthe first few frames of one of the videos with a clear viewto the explosion site [12].In order to obtain reliable measurements of the pairs( t, R ), an object of known length nearby the explodingwarehouse is needed as a reference to determine the cor-rect conversion factor between pixels in the image anda length scale. The explosion took place at Warehouse12 located right next to the 150-meters-long grain silosvisible in most videos. For each video, a good conver-sion factor can be obtained by identifying the locationfrom where the footage was obtained, which allows de-termining the angle between the line of sight and thegrain silos. This angle is then used to find the projectionof the building visible from the camera point of view inmeters, which can be related to the corresponding view inpixels. The next step is measuring the size of the fireballin each frame and use the conversion factor to obtain itsradius in meters. Video selection was based on the fol-lowing criteria: 1) enough of the surroundings are visiblefor a clear identification of the filming location; 2) thereis a clear view of the grain silos to determine the conver-sion factor; and 3) at least four points of the evolution ofthe fireball are visible. Out of the many videos publiclyavailable, only four met the criteria [12–15]. Their film-ing locations with respect to the site of the explosion areshown in Fig. 3.The frame extraction for each video was implementedusing
OpenCV , a computer vision library in Python thatalso allows determining the timestamps for each framewith millisecond precision [17]. The determination of the
Warehouse 121 23 4
FIG. 3. Filming locations of the four selected videos. Detailsare presented in Table II. Satellite image of Beirut on July31, 2020.[16] size of the explosion for each frame was achieved by mea-suring the size of the fireball in pixels and then convertingto meters using the corresponding factor for each video.A fully automatic measurement of the fireball is challeng-ing due to the thick dark smoke that appears in the firstmilliseconds covering significant fractions of the fireball.This process was completed using a hybrid manual/semi-automatic method. The manual procedure accounts formost of the uncertainties. By repeating measurementsseveral times an approximate error of a few meters wasfound. Another source of uncertainty is the measurementof the angle between the line of sight and the grain si-los used to determine the conversion factor; nevertheless,this was found to be less than 2% of the error from thefireball measurement. The total uncertainty was roundedup to a moderate 10 m. in the analysis. For the timemeasurements the uncertainty is obtained from the timebetween frames for each video. Subsequently, a carefultreatment in
OpenCV using each color channel for pro-cessing the images allowed the determination of the sin-gle connected component containing the fireball, whosecenter of mass was used to determine the center of thefireball in the image confirming the validity of the manualmeasurements. This hybrid method was used to super-impose the measured fireball on each frame of the videosin Figs. 10–13. A total of 26 pairs ( t, R ) were obtainedand are presented in Table I. Following Taylor’s method,the data from the four selected videos is represented inFig. 4.Remarkably, all the data is consistent with a single line − . − . − . − . . t/ s)4 . . . . . . . . l og ( R / m ) k t k t k t . k t . k t Fireball radius R at time t after explosionlog t + 6 . FIG. 4. Logarithmic plot showing the data from the four se-lected videos following the predictions of Eq. (5). The labelscorrespond to the filming location in Fig. 3. The solid linecorresponds to the linear fit for the mean value of the dis-tribution of the intercept (indistinguishable from the dashedline 1 kt TNT) and the dashed lines show the lines of 0.2, 0.5,2, and 5 kt of TNT equivalent for reference. with a slope consistent with one within the uncertaintyof the measurements, as described by Eq. (5). The for-mulation presented in Sec. III assumes the energy tobe released instantaneously from a single point, whereasthe ammonium nitrate in the warehouse was far from apoint source and the chemical nature of the energy releasemakes it slower than the case of nuclear detonation. Nev-ertheless, Fig. 4 shows that for the first 200 millisecondsEq. (5) describes the evolution of the fireball extremelywell. A Bayesian approach for fitting Taylor’s model tothe data can be followed by using emcee , a Python im-plementation of the affine-invariant ensemble sampler forMarkov Chain Monte Carlo (MCMC) [18, 19]. The re-sulting projections of the posterior probability distribu-tions of the model parameters are shown in Fig. 5 [20].The projections of the posterior probability distribu-tions of the slope m and the intercept n show an excellentagreement with the unitary slope predicted by Taylor’smodel within the uncertainty of the data. The slope isfound to have the value m = 0 . +0 . − . , whereas the in-tercept is given by n = 6 . +0 . − . ; the uncertainties arebased on the 16th, 50th, and 84th percentiles of the sam-ples in the marginalized distribution for each parameter.If instead of two parameters we fix the slope to one andonly fit a model with a single parameter describing theintercept we find n = 6 . +0 . − . . Using Eq. (7), we findthe estimate for the energy of the Beirut explosion to be m = 0 . +0 . − . .
75 0 .
90 1 .
05 1 . m . . . . n .
00 6 .
15 6 .
30 6 . n n = 6 . +0 . − . FIG. 5. One and two dimensional projections of the posteriorprobability distributions of the model parameters. The true-value lines correspond to the slope m = 1 and the mean valueof the distribution of the intercept n . E = 4 . +0 . − . TJ, corresponding to a yield of 1 . +0 . − . kt of TNT equivalent.Some unofficial records have reported the yield of theBeirut explosion in the range 1–2 kt. The high end mightarise from incorrectly using the parameters for the mix-ture of ammonium nitrate and fuel oil (ANFO), whichhas about twice as much explosive heat as the fertilizer-grade ammonium nitrate stored at Warehouse 12 of theport of Beirut. Additionally, many of these estimatesmake use of the relative effectiveness factor. Neverthe-less, the rendition of explosion yield in the form of TNTequivalent can be challenging because of the variety ofexplosion characteristics produced by the different en-ergy release rates of different explosive materials. Noticethat the yield reported in the previous paragraph was ob-tained in joules and expressed as kilotons of TNT usingthe SI unit convention rather than using any TNT equiv-alent factor for ammonium nitrate [10]. A more reliableassessment of the yield found using Taylor’s method isby comparison to proper chemical properties of ammo-nium nitrate. A useful quantity is the heat of explosion,which has a value 346 cal/g for ammonium nitrate [21].It is important to note that heat of explosion can alsolead to some inaccuracies because it does not accountfor the expansion of the gases produced. However, thisis the method recommended by the U.S. Department ofDefense [22]. Using the value of 2750 metric tons of am-monium nitrate according to official report [11], the heatof explosion leads to 3.98 TJ of energy released, equiva- Warehouse 121 23 456 78 910
FIG. 6. Filming locations of the nine selected videos [31].Details are presented in Table II. Satellite image of Beirut onJuly 31, 2020 [16]. lent to 0.95 kt of TNT. If we take this as the true valueof the yield of the explosion we conclude that Taylor’smethod produces a remarkable estimate of the energyreleased with the mean of the distribution within 5% ofthe true value.Furthermore, the result obtained in this work for theyield and its uncertainty is fully contained within theintervals found by other methods including seismic mea-surements (0.5 – 1.1 kt) [23] and audio-visual analysis ofthe blast (0.50 – 1.12 kt) [24].
VI. KINEMATICS OF THE PRESSURE WAVE
One feature that captured a lot a attention and thatis clearly visible in all videos is the Wilson cloud thatexpanded and rapidly disappeared after the explosion. Ifthe air around the explosion has a high content of water-droplet aerosols then the pass of the pressure wave canbecome visible. The sudden drop in pressure behind thepressure wave briefly prompts the relative humidity to su-persaturation, which dramatically enlarges the size of thedroplets producing a visible cloud [25]. Most media out-lets incorrectly called this a “mushroom cloud” becauseof the similarity of historical footage of nuclear tests inthe ocean. The column of dark smoke that followed theexplosion did produce an actual mushroom cloud; never-theless, there was a confusion between the smoke mush-room cloud and the semi-spherical Wilson cloud.Another feature that was clearly noticeable in most t (s)050010001500 x ( m )
123 45 67 89 10
Pressure wave arrival at different locations
FIG. 7. Data for the pressure wave arrival at each of the tenlocations in Fig. 6. Error bars have been multiplied by afactor 5 × to make the uncertainties visible. videos is the arrival of the pressure wave to the filminglocation in the form of a violent and loud blast. Follow-ing the spirit of the previous section and the plenty ofrecordings available, a frame-by-frame scrutiny and theidentification of the location from where each video wasfilmed can be used to analyze the kinematics of the pres-sure wave. The video selection was based on the followingcriteria: 1) enough of the surroundings are visible for aclear identification of the filming location; 2) there is aclear view of the moment of the explosion; and 3) themoment of the arrival of the pressure wave can be iden-tified. In addition to the four videos used in Sec. V,five other videos met the criteria [12–15, 26–31]. Theirfilming locations with respect to the site of the explosionare shown in Fig. 6 and other details including distanceand time between the explosion and the blast arrival aresummarized in Table II. The arrival time of the pressurewave at the different locations is shown in Fig. 7. Thedata reveals that the speed of the wave decreases as itpropagates, as expected. Notice that there is an extradelay of almost 400 ms for the wave to reach location 6(Clap Bar) compared to location 8 (Waterfront) despitethe very similar distance. From the map of locations Fig.6 we can also note that the video filmed from the ClapBar is the only one with a blocked line of sight to theexplosion due to the grain silos. This observation canin fact explain the delay: the pressure wave reaches theWaterfront directly, whereas it must first reach the silosand diffract around the tall building before continuing itspath toward the Clap Bar.The grain silos are first significant obstacle that breaksthe symmetry of the pressure wave; therefore, any real-istic description of the blast propagation requires carefultreatment. Despite the lack of symmetry, we can try abasic description of the blast kinematics. The time de-pendence of the velocity field can be parametrized usingthe Dewey-Friedlander blast profile [32, 33] v ( t ) = v (cid:18) − tτ (cid:19) e − αt , (8)from which we can determine the time dependence of the v = 562 . +17 . − .
510 540 570 600 630 v . . . . . α .
08 0 .
10 0 .
12 0 .
14 0 . α α = 0 . +0 . − . FIG. 8. One and two dimensional projections of the posteriorprobability distributions of the model parameters. The true-value lines correspond to the mean value of each distribution. position of the blast by direct integration x ( t ) = (cid:90) v ( t ) dt = v α (cid:18) tτ + 1 ατ − (cid:19) e − αt . (9)The boundary condition x (0) = 0 implies the relation τ = 1 /α and the last expression reduces to the followingtwo-parameter model x ( t ) = v t e − αt . (10)The fit of this model to the data in Fig. 7 is shown inFig. 8 and resulted in the parameters v = 562 +17 − m/sand α = 0 . +0 . − . s − . The data together with the fitresults are shown for the position of the blast and itsspeed in Fig. 9. The more striking observation from thisbasic treatment is the fact that the blast moves at super-sonic speeds during the first two seconds, correspondingto the region within 890 m. from the site of the explosion.This behavior would be a clear indication of a detonation(instead of a deflagration) of the stored ammonium ni-trate. After 2 seconds the shock wave rapidly becomes asubsonic pressure wave and keeps slowing down. Interest-ingly, the region in which the propagation was in the formof a shock wave is in perfect agreement with the regionof maximal structural damage as captured by NASA’sAdvanced Rapid Imaging and Analysis, that uses radardata to map structural changes [34]. x ( t )( m )
123 45 67 89 10 t (s)0200400600 v ( t )( m / s )
123 45 67 89 10speed of sound
FIG. 9. Data for the pressure wave arrival at each of the tenlocations in Fig. 6. The gray curves correspond to samplesfrom the posterior distributions of the two model parameters.Error bars have been multiplied by a factor 5 × to make theuncertainties visible. VII. CONCLUSION
Images can be powerful representations of events andin the case of explosions can contain rich informationthat can be studied. The same methods for experimen-tal analysis that students learn in their early years inphysics laboratory experiences can serve to validate themodeling of a blast formation from a detonation. Tay-lor’s method for studying the blast from the first nuclearexplosion at the Trinity test confirms that a complex rela-tionship between the parameters describing the evolutionof the explosion can be reduced to a simple linear fit lead-ing to very accurate predictions. The same method hasbeen applied to the tragic ammonium nitrate explosion inBeirut. The availability of plenty of footage of the explo-sion from different angles allows producing a new datasetthat can be used for analyzing the evolution of the fireballas well as determining the yield of the explosion. Despitethe characteristics of this chemical explosion, a remark-able agreement is found between the model and the data.Fitting the model to the data allows determining an es-timate for the energy yield of the Beirut explosion to be4 . +0 . − . TJ or 1 . +0 . − . kt of TNT equivalent.A complementary set of recordings also allow studyingthe arrival times of the pressure wave at different loca-tions. A basic modeling of the blast kinematics shows atemporary shock wave that loses speed to become a stilldestructive subsonic pressure wave around one kilometerfrom the explosion site.Both the energy yield estimate and the shock wavereach appear consistent with the preliminary informationof the amount of ammonium nitrate stored and satelliteimages of the structural damage, respectively. Further-more, the result obtained for the yield is consistent with FIG. 10. Frames from Video 1 (VGQ8+Q9); superimposed isthe fireball size at each stage. The values of the radius andtime are presented in Table I.
FIG. 11. Frames from Video 2 (VGVF+JP); superimposed isthe fireball size at each stage. The values of the radius andtime are presented in Table I. seismic measurements and audio-visual analysis of theblast.
Appendix: Beirut recordings and tables
This appendix includes all the frames for each of thefour selected recordings used in the analysis of the Beirutexplosion. Each frame also includes the size of the fireballdetermined using the hybrid method described in Sec. V.
FIG. 12. Frames from Video 3 (VGWC+FV); superimposedis the fireball size at each stage. The values of the radius andtime are presented in Table I.
FIG. 13. Frames from Video 4 (VGWC+H6); superimposedis the fireball size at each stage. The values of the radius andtime are presented in Table I.TABLE I. Data extracted from the four selected videos forthe fireball evolution and energy estimate in Sec. V. R (px) R (m) t (s) Label 5 / R log t
111 90.36 0.034 1 4.89 -1.47143 116.41 0.067 1 5.16 -1.17163 132.69 0.100 1 5.31 -1.00184 149.78 0.134 1 5.44 -0.87202 164.03 0.167 1 5.54 -0.7880 77.83 0.033 2 4.73 -1.48118 114.31 0.067 2 5.15 -1.17140 136.20 0.100 2 5.34 -1.00150 145.93 0.133 2 5.41 -0.88168 162.95 0.167 2 5.53 -0.78180 175.11 0.200 2 5.61 -0.70185 179.97 0.233 2 5.64 -0.6375 67.39 0.017 3 4.57 -1.7799 88.51 0.035 3 4.87 -1.46112 100.64 0.052 3 5.01 -1.28132 118.16 0.070 3 5.18 -1.15136 122.20 0.087 3 5.22 -1.06147 132.09 0.105 3 5.30 -0.98158 141.52 0.122 3 5.38 -0.91166 149.16 0.139 3 5.43 -0.86171 153.65 0.157 3 5.47 -0.80175 157.25 0.174 3 5.49 -0.76113 75.79 0.033 4 4.70 -1.48176 118.04 0.066 4 5.18 -1.18202 135.14 0.100 4 5.33 -1.00223 149.22 0.133 4 5.43 -0.88 TABLE II. Details of all the videos used in this work. The firstfour were used for the fireball evolution and energy estimate inSec. V; their location is given by their Plus Code in GoogleMaps [35–39]. All videos listed were used in Sec. VI: thelabels are shown in Fig. 6; the distance is measured fromthe site of the explosion; and the time is measured from themoment of the explosion to the arrival of the pressure wave.Label Location Distance (m) Time (s) Ref.1 VGQ8+Q9 1382 3.770 [12]2 VGVF+JP 944 2.133 [13]3 VGWC+H6 550 1.100 [14]4 VGWC+FV 684 1.434 [15]5 Car on 51M 538 1.066 [26]6 Clap Bar 1153 3.169 [28]7 Helou Ave. 600 1.240 [27]8 Waterfront 1144 2.798 [29]9 Chafaka-Helou 493 1.000 [30]10 Chafaka-Armenia 644 1.366 [30]
ACKNOWLEDGMENTS
The author thanks the people who recorded the tragicevents in Beirut and made their audiovisual materialavailable online. The author also thanks S. Rigby andO. Ram for their constructive feedback. This work wassupported in part by the Indiana University Center forSpacetime Symmetries. [1] G. I. Taylor, “The formation of a blast wave by a veryintense explosion: I. Theoretical discussion,” Proc. R.Soc. Lond. A201, 159–174 (1950).[2] D. C. Fakley, “The British Mission,” Los Alamos Science,Winter/Spring, 186-189 (1983).[3] K. T. Bainbridge, “Trinity,” Los Alamos Scientific Lab-oratory, LA-6300-H (1976).[4] G. I. Taylor, “The formation of a blast wave by a very in-tense explosion: II. The atomic explosion of 1945,” Proc.R. Soc. Lond. A201, 175–186 (1950).[5] M. A. B. Deakin, “G.I. Taylor and the Trinity test,” In-ternational Journal of Mathematical Education in Sci-ence and Technology, 42:8, 1069-1079 (2011).[6] J. von Neumann, “The point source solution,” in
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