What is basic physics worth? François Roby

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François Roby. What is basic physics worth?: Orders of magnitude, energy, and overconfidence in technical refinements. 2019. ￿hal-02004696v2￿

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Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License What is basic physics worth? Orders of magnitude, energy, and overconfidence in technical refinements

François Roby∗

« Être informé de tout et condamné ainsi à ne rien comprendre, tel est le sort des imbéciles. »

Georges Bernanos (18881948), in La France contre les robots

“It doesn’t make any difference how beautiful your guess is, it doesn’t make any difference how smart you are, who made the guess, or what his name is. If it disagrees with experiment, it’s wrong. That’s all there is to it.”

Richard Phillips Feynman (19181988), in a famous 1964 lecture at Cornell University.

Physics is often perceived as a science of tors, the motive and the technical means are complex and precise calculations, making questioned. As a correct argument cannot possible any sorts of technical “miracles” in rely on improper vocabulary, we shall not the midst of which we live. However, the impede ourselves with such demonizing and basis of the discipline does not lie in these shall only show which “conspiracy theories” refinements, be they enabled by fancy math are compatible with physics laws and which ematics or, today, by computer calculations, ones are not, since even the official version but in a small number of laws that should be belongs to them. rigorously applied; it also lies in the physi A striking and well documented feature of cists’ ability to distinguish the secondary 9/11 attacks in New York City is persisting from the essential and therefore to perform fires in the World Trade Center ruins: the justified approximations. last one was extinguished only 100 days af Strangely, some people often talk about ter the event. This simple fact is intriguing “conspiracy theories” in order to denigrate and needs explanation. Airborne or satellite some alternative interpretations of known infrared thermography measurements have events, even when the very existence of a been made, just after the event as well as conspiracy makes no doubt: this is for in weeks and months later, which allow to es stance the case with 9/11 terrorist attacks timate surface temperature and correspond in the USA, for which only the perpetra ing areas, and the cooling characteristic time of the place. ∗ Université de Pau et des Pays de l’Adour/CNRS Cooling of a hot body in a colder environ IPREM UMR 5254 ment occurs thanks to conduction, convec Technopole Hélioparc tion and radiation. In open air, thermal 2 avenue P. Angot dissipative power due to free convection is 64053 Pau Cedex 9 FRANCE [email protected] / [email protected] easily obtained if one knows the heat trans

1 1 Introduction fer coefficient h, the temperature difference released and on the depth of burial of the ∆T and the area S of the corresponding sur explosive, it is therefore possible to gener face. Taking into account only free convec ate effects at ground level, first when the tion, and performing only orders of magni shock wave travels through the media (in tude calculations because of a lack of accu cluding on materials not usually considered rate data, it is possible to get a lower esti as brittle, such as steel, because of the excep mate of the total heat released at Ground tionally steep shape of the wavefront), later Zero. when the cavity “roof” collapses and creates For fundamental reasons (electrons energy a rubble chimney, and finally during heat levels in atoms, nucleons mass) any kind of diffusion which lasts for months. chemical energy production involves a mini It turns out that the physics community, mum amount of mass. Nuclear energy, in having been too easily intimidated by argu volving the same mass but using nuclear ments being not real ones, or being not a bonding energy roughly 106 larger, releases matter of physics (“likelihood” of a hypoth consequently about 106 more energy per unit esis...), has for too long, and with few ex mass, or, for technical applications such as ceptions, tacitly admitted interpretations of nuclear explosives which include a lot of extremely important events that are merely matter not releasing any nuclear energy, still pseudoscience, if not extravagant science 104 times more. fiction. Combining the minimum total heat estimate Would physicists have worked with academic with the physical limits of chemical energy rigour and starting from the most well estab carriers, we can rule out any chemical energy lished foundations of their science, such as as the source of heat released at Ground Zero the first and second laws of thermodynamics, and therefore consider nuclear energy explo and would have they added to a purely sci sives as the only available solution. For obvi entific work some retrospective critical look ous reasons, only deep underground nuclear on their own enthusiasm partly irrational explosions could remain relatively unnoticed during the postWorld War II era, they could as such; therefore only the opportunistic use have shown that not only the explanation of a builtin nuclear demolition feature, de of the destruction of 3 highrise buildings in signed at the same time as the World Trade New York City on September 11, 2001 by Center itself, is a viable explanation. Some underground nuclear explosions, given more literature search about pacific use of nuclear than a decade ago by someone pretending explosives as envisaged in the 1960s (espe to be a former soviet officer and nuclear cially some books like The Constructive Uses weapons expert (Dimitri Khalezov), was not of Nuclear Explosives by Teller et al., 1968) a crazy one, but that it was actually, with a shows that such an idea, if surprising today, few corrections, the only possible one. was not unthinkable in the context of the time. It comes out that any nuclear explosion in a 1 Introduction bedrock produces a shock wave that turns this material into tiny pieces (the smaller It is usually believed that basic physics, such ones being the closer to the “zero point”), as classical mechanics, electromagnetism, then creates a plasmafilled cavity with ex optics or any other field that students learn tremely high pressures ( 1014 Pa) and tem 7 ∼ at undergraduate levels, is a necessary step peratures ( 10 K) which, after cooling, ∼ towards more elaborate physics specialties ends most of the time filled with rock de but can never by itself lead to striking dis bris falling from a “collapse chimney” located coveries at the fringe of scientific knowledge, above the cavity. Depending on the energy since it addresses only wellestablished con

2 cepts that have been used for decades or simple, clear ideas. even, quite often, for centuries. It is true Our purpose is not to denigrate the use that no one will ever be able to “discover” of computer simulations, which have proven that, for instance, Newton’s laws of motion to be effective, fast and often irreplaceable are false, since it is already known that they tools for physics and engineering, but to are indeed false or, to write it more precisely, show that they should not be used in the that their domain of validity is limited and first place when a direct, humanmade ar does not extend as far as, for instance, high gument gives an answer to the problem energy particle physics. although a simplified one and leads to a Although we do not challenge this obvious deeper, yet easier to share among “ordinary fact, we will show in this paper that much humans”, understanding. Since science is more than a mere “necessary step” is to be not only valuable for its technological ap expected from basic physics, especially at a plications but also for its educative value, time when computer simulations, although such a perspective should not be considered, being extremely valuable tools for solving according to us, as an oldfashioned or a complex problems particularly in engineer limitedbudget way of doing physics, but as ing areas sometimes lead to a “black box” the primary and most important one before thinking that obscures simple and powerful any technological refinement is called on for physics concepts. Because computer simu help. And especially when dealing about lations have become “too easy a method” complex problems where risk of error is high: for solving even simple physics problems “safety first”, as sensible sailors or alpinists some scientists endangering further the un would say. derstanding when talking about “computer experiments” instead of “computer simula 2 Orders of magnitude: the tions” it is of great interest to call back “good old methods” of physics, those of the Fermi approach precomputer era when experiments were only genuine ones and basic understanding 2.1 The classical piano tuners was required before performing them, or be problem fore performing tedious analytical calcula tions, which can also sometimes muddle up A story often narrated to students for educa understanding by diverting too much of a tive purposes is how Enrico Fermi, the fa scientist’s effort in solving equations instead mous Italian physicist and 1938 Nobel Prize of concentrating on the underlying concepts. winner, used to ask his students to find an swers although approximate to almost all A practical example will be given through questions, including the ones which have lit a wellknown, yet poorlyunderstood even tle or no link to physics, using simple logic among the scientific community energy and dimensional analysis [1]. The most pop problem, emphasizing the need to limit the ular example of this, the “classical Fermi modelling level of complexity instead of try problem”, was to guess the number of piano ing to make the model as close to reality tuners in Chicago. Independantly of the nu as possible, as it is generally the rule when merical data, such an answer can be found working with computer simulations. If the using the following reasoning: aim is to rigorously understand what is re ally going on at a fundamental level, it is • A piano needs to be tuned from time to important to avoid elaborating hypothesis time, let’s say n times a year. which are nothing but mere speculations, • The operation takes some time to be and this can be done only when working on performed (including travel time), let’s

3 2 Orders of magnitude: the Fermi approach

call it ∆t. • C = 3.5 106 (1940 value, urban area × not taken into account) • A piano tuner works like other workers, a finite amount of time per year, let’s With these numbers the answer is: say why hours a year. 1 1 3.5 106 2 hours N = × = 87.5 pt year 20 2 hours • Not all households have a piano; more 2000 year over, not all households have a piano (1) that is tuned regularly. Let’s call f the Of course such a noninteger number is ab fraction of households that have got one surd and must be rounded up to 90, or even regularly tuned piano (the number of 100 since one should not expect better than households having more than one reg a crude estimate of the real number; but the ularly tuned piano will be considered real point is that it can’t be only one, nor negligible). 4 10 . • There is an average of p persons per household. 2.2 The pinhole camera problem • There are C inhabitants in Chicago. Dimensional analysis is an important part of When all these parameters are known, then the game, although the classical Fermi prob the answer (let us call N the number of pt lem deals more with numerical estimates and piano tuners) can be computed as follows: straightforward thinking than with checking C • There are p households in Chicago and units. Let us take a slightly different exam C f p pianos that need to be tuned regu ple to illustrate this, where actually no nu larly. merical guess has to be performed but where the right answer comes only from dimen • There are nf C tunings that are made p sional analysis. per year in Chicago, and they necessi C A pinhole camera is the most primitive type tate nf p ∆t working hours. of camera [2], consisting only in a small hole • To perform this work a number of C ∆t punctured on one side of a lightproof box. Npt = nf fulltime piano tuners p why It produces, just as an ordinary camera, in is needed. verted real images of the surroundings on Of course any numerical answer will depend the side of the box facing the pinhole let on the quality of the numerous estimates us call this side the image plane. There is an made; however, each of them should be eas optimal size for this hole: if it is too large, ily performed by anyone if only the right or light rays coming from an object point out der of magnitude is sought for. And fur side of the box will be able to strike the im thermore, there is a reasonable chance that age plane within a rather large image spot, errors in different estimates will more or less only because of straight propagation of light compensate. Let’s give a numerical illustra according to geometrical optics, and this will tion: cause image blur. Conversely, if the hole • n = 1 is too small geometrical optics is not valid year any more, diffraction occurs and enlarges the • ∆t = 2 hours image spot also. A rigorous calculation of hours weeks the optimum pinhole diameter can be per • why = 40 week 50 year = formed using wave optics; Josef Petzval has 2000 hours   year
proposed one in the mid19th century and 1 • f = 20 gave the result[3]: • p = 2 d = 2fλ (2) p

4 where λ is the wavelength of the light and events were only accidents, and everyone ac f the distance between the pinhole and the knowledges they were planned in advance image plane equivalent to the focal length by some criminal individuals which is the in a camera with lens. However this re very definition of a conspiracy. It would be sult, excepted the √2 dimensionless factor more correct in this case, therefore, to call that is rather close to unity, can be inferred them “alternative conspiracy theorists”. As very quickly using only dimensional analysis. physics does not deal with human intentions, The answer must depend on the wavelength it cannot address directly the “conspiracy” λ, because diffraction is involved; it must item which is anyway, as we pointed out, also depend obviously on the distance f be irrelevant here. But as it deals with nat tween the pinhole and the image plane for ural laws, it can refute some explanations equally obvious geometrical reasons. These which do not fulfill the necessary require are the only two parameters of the prob ment of being compatible with these laws, lem, and one must get from them an opti just as a crime laboratory is able to rule out mum diameter d which is also a length. To some murder suspects. get a length from two other lengths could Most “conspiracy theories” about September be done mathematically using a sum or a 11, 2001 events including the widely ac difference, but this would have no physical cepted one, since 4 passenger airliners being meaning since λ being much smaller than hijacked by 19 terrorists is certainly the re f, a sum or a difference would practically sult of a highlevel kind of conspiracy deal keep only the largest quantity. Hence, the with complex phenomena like collision be most straightforward mathematical formula tween airplanes and buildings or the catas for getting the optimal diameter is d = √fλ, trophic collapse of skyscrapers. Surpris which is almost the right one and gives at ingly, people who discuss these issues for least the right order of magnitude, since any instance, arguing about what made three dimensionless factor like √2 here can not al skyscrapers collapse in a few seconds elab ter the order of magnitude. orate from the very beginning some complex scenarii without even checking the most ob vious and welltried laws of physics, such 3 A practical example: as conservation of energy. As we will see, disproving some “conspiracy such an approach turns out to be ineffective and to obscure even more an already dif theories” ficult problem. Complex and questionable arguments should always be used to refine The terrorists attacks that occured in sev the conclusions of straightforward and ro eral places of the USA on September 11, bust ones, not the other way round. 2001 have been since the event the subject Some authors [6] have challenged in 2013, of many controversies, for instance regard using a detailed analysis, a series of papers ing the very fact that a plane really strucked by Bažant et al. who pretended to explain the Pentagone or a missile instead [4]. Most rationally the dominant narrative1 regard mass media, and even some scientists [5], ing the World Trade Center skyscrapers col denigrate people who look for explanations lapses, which attributes them to fires weak of wellknown events that depart from the ening their structure. More recently (2016), one provided by officials, calling them “con 1 spiracy theorists”, although in the case of We use here the adjective “dominant” in the sense where it is massively reported by mass media and September 11, 2001 attacks, the widely ac governments, not in the sense that it should be cepted version is undoubtely also of a con considered as more plausible or more correct on spiracy type. Nobody claims these horrific a scientific basis.

5 3 A practical example: disproving some “conspiracy theories” others [7] using only simple mechanics have sue of the newsletter of his association successfully shown this narrative to be in (p.3): “As of 21 days after the attack, compatible with physics laws. the fires were still burning and molten steel was still running.”[9] Note that We shall here try to go one step further and, the melting point of steel, which de without using mechanics at all but rather pends on its chemical composition, is thermodynamics in its simplest form, ex- close to 1700 K. clude from the range of options some ex planations that have been advanced for the • James Glanz wrote in the New York buildings collapses. For this we will rely Times on November 29, 2001, about almost exclusively on the basic concepts of the “strange collapse of 7 World Trade thermodynamics, being taught at an entry Center”, citing Dr. Barnett, a pro level course of physics: namely, the first law fessor of fire protection engineering at which states that energy is conservative and the Worcester Polytechnic Institute: “A that work and heat are two kinds of en combination of an uncontrolled fire and ergy that can transform in each other, and the structural damage might have been the second law which states that heat can able to bring the building down, some only flow spontaneously from warmer bodies engineers said. But that would not ex- to colder ones, according to Clausius state plain steel members in the debris pile ment. And to make the argument easier to that appear to have been partly evap- expose and above all to understand by orated in extraordinarily high tempera- any reader, we will adopt in the following a tures”. [10] Note that the boiling tem kind of “Fermi approach” and ignore unnec perature of iron, the main component essary details to focus on a simple energetics of steel, is 3134 K. problem, for which experimental data need • William Langewiesche, the only jour only to be known at the order of magnitude nalist to have had unrestricted access level. to Ground Zero during the cleanup op eration, states in the book “American 3.1 Aftermath of 9/11 terrorist Ground: Unbuilding the World Trade attacks: Ground Zero persistent Center” the following (pp. 3132): “He high temperatures would go wandering off through the sub- terranean ruins, [...] with apparently It is a widely documented fact that persis only a vague awareness of the danger tent fires occurred at Ground Zero in Man signs around him — the jolt of a col- hattan as an aftermath of the September 11, lapse far below, [...] or, in the early 2001 terrorist attacks. Numerous newspa days, the streams of molten metal that pers as well as broadcasted news bulletins leaked from the hot cores and flowed narrated the extremely long effort made by down broken walls inside the foundation firefighters to secure the place, and it was hole.”[11] reported that “The underground fire burned • during the public hearing for the Na for exactly 100 days and was finally declared tional Commission on Terrorist At “extinguished” on Dec. 19, 2001.”[8]. We tacks Upon the United States, on April will give here only a few examples of testi 1, 2003, Ken Holden (New York De monies related to these very unusually high partment of Design and Construction) and longlasting temperatures: stated: “Underground, it was still so • James M. Williams, then president of hot that molten metal dripped down the the Structural Engineers Association of sides of the wall from Building 6.”[12] Utah, wrote in the October 2001 is • New York firefighters have recalled

6 3.2 Estimating energy released from Ground Zero: the Fermi approach

“heat so intense they encountered rivers nisms (conduction, convection both free of molten steel.”[13] and forced, and radiation) that can only be accurately calculated using numerical com Famous photographs of redglowing steel re putation based on these experimental data. moved from the pile were published, and some interviews of firefighters were made Furthermore, huge amounts of water per where it was claimed that flows of liquid colated through the debris, contributing to metal were flowing underneath the debris, the cooling process by elevating the temper “like in a steel plant”. Note that there was ature of water which was drained away, but also evidence (see for instance [7], Fig. 6) of also for some part by evaporation, as white glowing molten metal pouring out of WTC2 plumes on the site demonstrate: fires are continuously for 7 minutes before its col very unlikely to produce white plumes, es lapse, but we will not address this particular pecially in an oxygenstarved environment feature since our aim is only to understand like underground remnants. It is very dif the origin of persistant high temperatures in ficult to estimate the cooling contribution Ground Zero ruins, and not to document or of this water, since it would require the study what happened before the collapse of knowledge of both the volumes and the the 3 buildings. temperature differences; an article submit To summarize, ample evidence exists that ted to the 23rd American Chemical Soci can rule out fires as the heat source at ety National Meeting (Orlando, FL, April Ground Zero in the weeks or even months 711, 2002) stated [14] that for the first following the attacks, since the extremely 10 days after the attacks, roughly 30 mil lion gallons ( 114.103 m3) water percolated high temperatures encountered there would ≈ violate the second law of thermodynamics. through the debris, based on the pumping Recalling that steel industry has only suc records. From this volume roughly 1 million ceeded at the end of the nineteenth cen gallon fell on the site (the socalled “bath tury to produce temperatures high enough tub” area) because of rain, 3 million gal to process steel in the molten state, it is clear lons were hosed in the firefighting efforts that fires that were persisting for 100 days and consequently 26 million gallons, i.e. the at Ground Zero were the consequence, not main part, came from leaks in the “bathtub”, the cause, of an extremely large heat source which was proven to be seriously damaged. which temperature was, during a long time, However, if we intend only to give a rough much higher than that of common building estimate at the order of magnitude level of fires, be they collapsed or not. this heat, and furthermore, if we are satisfied with only a lower limit of this energy, then the work becomes much easier and we can 3.2 Estimating energy released from use a kind of “Fermi approach” to get the Ground Zero: the Fermi result. As energy is the integral of power approach over time, and as any cooling process de scribed by linear heat transfer equations in Giving a precise estimate of the amount a fixed temperature environment leads to an of heat released at Ground Zero is an al exponential decay of temperature and ther most impossible task, since it would re mal power (heat transfer rate), we only need quire a huge amount of experimental data to estimate the following: (mainly temperature measurements, in a lot of places and repeated over the cooling down • thermal power released at Ground Zero time which lasted for months) and since at some time, let us call it P (t); this heat transfer in such a complex environment thermal power is proportional to the as the debris pile involves several mecha temperature difference ∆T with the en

7 3 A practical example: disproving some “conspiracy theories”

vironment if the cooling process can be damaged “bathtub”, if not negligible, is also described with linear equations; restricted to the first weeks after the attacks and therefore should not be a dominant part • characteristic time of cooling process, of the cooling process, given the extremely let us call it τ. long characteristic time of it as we will see Then the heat Q can be expressed integrat later. Taking the order of magnitude cited ing thermal power over time (here t = 0 rep above ( 105 m3), the heat capacity of liquid ∼ 3 −1 −1 resents September 11, 2001): water (C = 4.18 10 J.kg .K ) and as p × ∞ suming a maximum temperature difference Q = P (t) dt (3) ∆T = 50 K between water “in” and water ˆ 0 “out”, the water could have taken away some 108 4.18 103 50 2 1013 J 20 TJ and assuming exponential decay of tempera × × × ∼ × ∼ ture difference ∆T , hence of thermal power, of heat, which sounds huge but is still much with a characteristic time τ, the calculation less than the total amount of heat we will is straightforward: estimate in the following. Assuming all the water was transformed into vapor, and con ∞ − t sidering its heat of vaporisation of 2.26 Q = P0 e τ dt = τP0 (4) −1 ≈ ˆ0 MJ.kg , this would translate in more than 250 TJ of heat which is much more but most Again, as several cooling processes are at probably overestimated, although the per play with different characteristic times, this sistence of white “fumes” at Ground Zero for calculation should not be considered as an weeks means that a large quantity of water accurate one, but rather as a way to get a did evaporate because of underground heat. lower boundary for the real amount of heat released at Ground Zero let us call it QGZ It can be argued also that ambient tem since all contributions give positive values. perature was not constant at Ground Zero, One can only hope to get the right order of both because of daily oscillations and be magnitude of QGZ if one chooses the domi cause of weather variations, the cooling pro nant cooling process, which for such a prob cess having taken place during months; how lem (hot ground in contact with atmosphere ever, given the very important temperature for months) is known to be usually a free difference between the place and the air (see convection mechanism within the air. How below), this can not lead to a major change ever, radiation might play an important and in the result. Moreover, since September 11 even dominant role at the beginning where is at the end of summer in northern hemi surface temperatures were proved to be ex sphere, if we take as the ambient temper tremely hot, because of the T 4 dependance ature value the one that New York experi of StefanBoltzmann law. We provide in Ap enced at this date (or, in a more relevant pendix A a crude estimate of heat released way, the mean value of the corresponding by radiative transfer and show that its con week), we underestimate the heat release tribution should have been comparable to rate for the cooler times of autumn and win that of free convection. ter, which is consistent with our approach of giving a lower estimate of Q . Conduction in the ground is difficult to esti real mate but given the poor thermal conductiv ity of it is relatively minor, and anyway gives 3.3 Heat transfer by free convection: a positive contribution which we can neglect the basics if we are satisfied with a lower estimate. Forced convection because of water sprayed We recall here what can be found in any heat by the firefighters or leaked through the transfer textbook; see for instance [15]. Free

8 3.3 Heat transfer by free convection: the basics

or natural convection is a phenomenon degrees of freedom exist and this translates that occurs in a gravitational field because directly into the following results: the volumic mass of a fluid varies with tem 5 perature, inducing buoyancy forces. Heat CV = R (7) transferred by free convection is solely de 2 7 termined by these forces hence by volumic Cp = CV + R = R (8) mass differences and by the fluid heat ca 2 pacity and viscosity. Let us recall here, just to put these results into an historical perspective, that the high 3.3.1 Preliminary remarks about ideal temperature limit of molar heat capacity for gases solids was known to be roughly a constant as soon as 1819, thanks to French physicists In the case of gases, and especially the sim Pierre Louis Dulong and Alexis Thérèse Pe ple case of ideal gases, the variation of volu tit, who expressed it in the socalled Dulong mic mass with temperature does not depend Petit law (molar Cp = 3R), and even though on chemical peculiarities of the fluid but re the ideal gas constant had not been defined lies solely on the ideal gas law: yet. Viscosity of ideal gases can be inferred from pV = nRT (5) the kinetic theory of gases and was shown experimentally to be independent of density where p is the pressure, V the volume, T by James Clerk Maxwell in a famous 1866 the absolute temperature, n the number article [16]. It can be theoretically expressed of moles of the gas, and R = kB A −1 −1 N ≈ as follows (see for instance [17]): 8.314 J.K .mol the ideal gas constant, with k Boltzmann’s constant and Avo 1 B NA η = Nmvl¯ gadro’s number. 3 Using the molar mass M, the volumic mass where N is the number of molecules per unit of the gas is given by: volume, m their mass, v¯ their mean velocity and l = 1 their mean free path, where σ is M Mp σN ρ = = (6) the crosssectional area of the gas molecules. V RT As a consequence, the product Nl does not and therefore varies proportionally with depend on N and the dynamic viscosity only pressure and as the inverse of absolute tem depends on the mass of the molecules and perature. the absolute temperature via the mean ve locity v¯ = 3kB T . Furthermore, the heat capacity of ideal gases m is also independent of the chemical nature These preliminaryq remarks show that heat of the gas, since it is a direct consequence transfer by free convection in the air at stan of the equipartition theorem; any introduc dard pressure is governed by universal, well tory course in thermodynamics demonstrate known physics laws that do not depend on that molar heat capacities at constant vol peculiarities of the problem. Therefore, heat ume, CV , and constant pressure, Cp, depend transfer coefficients used in the following, directly on the number of degrees of freedom although depending on precise geometry of of the molecules. For instance, for diatomic the heated surfaces if one needs to know molecules at room temperature, where ro their precise values, will also obey universal tational degrees of freedom must be taken laws which provide easily numerical values, into account but not vibrational ones, five at least at the order of magnitude level.

9 3 A practical example: disproving some “conspiracy theories”

3.3.2 Similarity considerations Let us then define the ratio of buoyancy forces to viscous forces acting on a fluid, On a more experimental perspective, con through the dimensionless Grashof number vection heat transfer either free or forced Gr: will be expressed through a convection heat 3 buoyancy forces gβ (T T∞) L transfer coefficient, usually written h, by the Gr = s − ≡ viscous forces ν2 following equation: (11) where: q˙ = h (T T∞) (9) S − • g is gravitational acceleration (gravity where q˙ is the convective heat flux (in of Earth); −2 W.m ), and TS and T∞ the surface and bulk fluid temperature, respectively. Conse • β is the volumetric thermal expansion quently, h will be expressed in W.m−2.K−1. coefficient of the fluid, defined as β = 1 ∂ρ or as β = 1 ∂V , in K−1. If the underlying physical mechanism of free − ρ ∂T p V ∂T p   convection is simple, detailed analysis of a Note that in the simple case
of an ideal specific free convection case can be very gas, we have pV = nRT and hence β = 1 nR = 1 complex due to intricacies of fluid dynamics V p T . for various geometries; analytical solutions Now let us define the ratio of viscous diffu are often not available. However, as soon as sion rate to thermal diffusion rate, through we deal only with orders of magnitude as it is the dimensionless Prandtl number Pr: the case in this paper, we only need to know viscous diffusion rate ν how to classify the present problem among a Pr = (12) limited number of typical cases and to apply ≡ thermal diffusion rate α some similarity considerations. We recall be where α is the thermal diffusivity, in m2.s−1. low the basics of these considerations, that Note that the kinematic viscosity ν is also are thoroughly developped, for instance, in called the momentum diffusivity. Chapter 9 of [15]. The convection heat transfer coefficient h, We will only consider free convection flows which is what we’re looking for, will be bounded by a surface, as it is the case of obtained through the dimensionless Nusselt interest for a hot surface in contact with at number, Nu: mosphere. hL Let us first introduce the ratio of inertial to Nu (13) ≡ k viscous forces acting on a fluid, through the dimensionless Reynolds number Re: where k is the thermal conductivity of the fluid, in W.m−1.K−1. In the general case inertial forces VL Re = (10) Nu is a function of Re, Gr and Pr; however, ≡ viscous forces ν forced convection effects may be neglected when Gr/ Re2 1 and in this case, Nu where: ≫ is only a function of Gr and Pr. • V is the maximum velocity of the flow;
• L is a characteristic length of the prob 3.3.3 Special case: upper surface of lem; heated plate • ν is the kinematic viscosity of the fluid, in m2.s−1, or ratio of the dynamic vis Free convection being a gravitydriven phe cosity to the volumic mass ρ of the nomenon, orientation of the surface with re µ fluid: ν = ρ . spect to gravity acceleration vector, as well

10 3.4 Numerical estimate of heat transferred by convection as relative temperature of the surface (hot Rayleigh number Ra; it is therefore neces ter or cooler) with respect to the surround sary to estimate first the value of this num ing fluid, are the main parameters used to ber. Let us recall the expression of the classify the different possible cases. Here Rayleigh number: we recall some empirical laws that are valid 3 gβ (T T∞) L for our case of interest, the upper surface of Ra = s − (17) να heated plate, as well as the symmetrical case of the lower surface of cooled plate. As flow with the same notations as above. The nu conditions are generally not constant over a merical values are the following : surface, only local heat transfer coefficient h • gravity of Earth: g 9.81 m.s−2 or g −2 ≈ ≈ and Nusselt number Nu can in general be 10 m.s which is accurate enough for defined; however, it is always possible to de our purpose. fine average transfer coefficient h and corre • volumetric thermal expansion of the air: sponding Nusselt number by: considering it as an ideal gas and taking 1 ambient temperature T 290 K, we get ¯ −3 −1 ≈ h = h dAs (14) β 3.45 10 K As ˆ ≈ × • surface temperature T : obviously, it where A is the surface area of the zone s s varies during cooling process. Shortly where the average is to be defined. after the attacks, on September 16, Two regimes have been empirically identi 2001, temperatures as high as 1000 K 2 fied [18] depending on the Rayleigh number have been measured using thermal im value, which give the average Nusselt num agery (Airborne Visible/Infrared Imag ber Nu, and hence the average heat transfer ing Spectrometer). However, these tem coefficient h, as a function of the Rayleigh peratures were only “hot spot” tem number Ra which is defined as the product peratures and the visible surface (from of the Grashof and Prandtl numbers: above) of Ground Zero did not exhibit such high values on a large fraction of 1/4 4 7 Nu = 0.54 Ra if 10 . Ra . 10 (15) the area. As we are interested only in an order of magnitude value of the Nu = 0.15 Ra1/3 if 107 . Ra . 1011 (16) temperature difference T T∞, and as s − this difference slowly goes to zero dur We will now determine which expression to ing the cooling process, we will consider use for the peculiar case of heat transfer by T T∞ 100 K since 10 K would s − ∼ free convection at Ground Zero. be obviously too small and 1000 K too large. 3.4 Numerical estimate of heat • characteristic length L: each of the transferred by convection Twin Towers (WTC1 and WTC2) had a roughly square horizontal section of 2 As shown above, the average heat trans (64 m) 4.103 m2 and WTC7 a ≈ fer coefficient h is determined through the slightly smaller but comparable foot average Nusselt number Nu, which itself print. According to thermal imagery is expressed differently depending on the hot zones were slightly larger than the

2 respective footprints of the buildings. The Rayleigh number is the product of the Grashof and Prandtl numbers : Therefore, we will take a characteristic 3 length order of magnitude L 100 m. gβ (Ts − T∞) L ∼ Ra = GrPr = να • kinematic viscosity of the air: at T = −5 2 −1 . 290 K ν 1.5 10 m .s and at T = ≈ ×

11 3 A practical example: disproving some “conspiracy theories”

390 K has a larger value ν 3 10−5 3.4.1 Estimating initial heat transfer rate 2 −1 ≈ × m .s . We can therefore take ν 2 −5 2 −1 ∼ × 10 m .s as an order of magnitude Let us now estimate the initial value P0 value. of heat transfer rate, or thermal power, of the whole Ground Zero site short after • thermal diffusivity of the air: at T = −5 2 −1 the attacks, considering only free convection 290 K α 2 10 m .s . ≈ × mechanism. As said before, heat transfer in The above values give us: volves several mechanisms, namely convec tion (either free or forced), conduction and 9.81 3.45 10−3 100 106 Ra = × × × × radiation. Each of these mechanisms gives 2 10−5 2 10−5 × × × a positive contribution to the global heat 8.5 1015 ≈ × transfer rate, which means that we can only 1016 (18) underestimate the amount of heat released ∼ if we consider, as will be done here, only one This very high Rayleigh number value is of them: free convection. This is precisely well above the range given for calculating our aim: for the sake of our demonstration the Nusselt number in equations 15 and we do not need a correct estimate of the to 16. However some authors [19] have inves tal amount of heat released at Ground Zero tigated thermal transport up to extremely but only a lower estimate of it. It is impor 17 high Rayleigh numbers (Ra 10 ) and find tant keeping this in mind for the following ∼ no significant departure from the 1/3 expo discussion. nent given in equation 16. Therefore, we will use this expression to estimate a numerical In order to estimate the initial heat transfer value of the average Nusselt number: rate, we need to estimate the initial temper ature difference between the hot surface and Nu 0.15 Ra1/3 0.15 1016/3 3.2 104 the ambient air, as well as the correspond ≈ ≈ × ≈ × ing heat exchange area. Of course, tempera And we finally compute an estimate of the ture was not uniform across the hot zones at average heat transfer coefficient h using the Ground Zero, and a precise estimate of heat thermal conductivity of the air at 390 K, k transfer should take into account such local −1 −1 ∼ 0.03 W.m .K , and still L 100 m: variations; however, to get an order of mag ∼ nitude of the initial thermal power we only k h Nu need, since heat transfer laws by free convec ≈ L tion are linear, to replace the real case by an 0.03 3.2 104 equivalent one which is drastically simplified ≈ × 100 − − and reduces to a zone with a given uniform 10 W.m 2.K 1 (19) ∼ temperature and a given area. Actually, h does not depend on L: since Nu Let us begin with the most imprecise guess: scales as Ra1/3 and Ra scales as L3, Nu is defining the initial equivalent uniform tem proportional to L. perature. To do this, we will rely on actual temperature measurements of socalled “hot It could be argued that such a long theoret spots”. Note that the temperature value in ical development was not necessary to get itself is not important per se, but rather it a mere estimate of an amount of heat re is the product of a temperature by a cor leased, since engineers or architects are used responding surface area which must be cor to empirical numerical values for heat trans rectly determined, since it is this product fer coefficients. However, due to the unusual which defines the heat transfer rate. scale of the hot surface area, we considered safer to do so. According to several publicly released docu

12 3.4 Numerical estimate of heat transferred by convection ments [20, 21], three “hot zones” could be observed through thermal imagery (either airborne or satellite), corresponding respec tively to WTC1, WTC2 and WTC7 debris. Surprisingly according to mainstream expla nation of the collapses, and taking into ac count the very different levels of damage that suffered the three buildings (no plane hitting WTC7, and only minor fires com pared to WTC1 and WTC2), all three col lapse piles from WTC 1, WTC 2, and WTC 7 emitted infrared radiation with similar in tensity as shown by a compilation of docu ments by A. Dreger [22], WTC1 and WTC2 hot zones having a slightly greater extent due to the larger spread of the debris during collapse. It is also remarkable that no equiv- alent high temperatures / persisting Figure 1: Some hot spots observed on fires phenomenon was encountered at September 16, 2001 with AVIRIS the Pentagon site, although a similar (Airborne Visible/InfraRed Imag attack with a similar passenger plane ing Spectrometer). was supposed to occur. The Pentagon Source: United States Geological building was obviously very different from Survey[23]. the WTC1, WTC2 and WTC7 buildings, but as the long lasting fires at Ground Zero were supposed to have been triggered by a large amount of jet fuel igniting office furni ture, one should have expected not so large a difference between the WTC fires and the rapidly extinguished Pentagon fires. According to USGS [23], determination of hot spots temperatures could be accurately performed using spectral analysis of the emitted radiation, as examplified in Fig. 2. On September 16, 2001, i.e. 5 days after the terrorist attacks, some surface temperatures as high as 1000 K could be measured, all the spots labeled by letters in Fig. 1 being at temperatures above 700 K. Figure 2: Example of IR radiation spec Note that according to Planck’s law, a black tral analysis for surface tempera body at 1000 K already emits a noticeable ture determination, as explained part of its spectrum in the visible region (it in [23]. The area at 1000 K (on is glowing red), although the maximum radi September 16, 2001) covers 0.56 2 ance, given by Wien’s displacement law, oc m according to authors and is lo curs in the infrared zone at λ 2.9 m. cated in the WTC7 debris. max ≈ A report from the Multidisciplinary Cen

13 3 A practical example: disproving some “conspiracy theories” ter for Earthquake Engineering Research (MCEER), “Emergency response in the wake of the world trade center attack: The remote sensing perspective”[21], gives several other thermal images which clearly show the same three hot zones, of roughly the same mag nitude: WTC1, WTC2 and WTC7 debris. We can also note that this same paper dis plays (p. 17) the very first infrared image (1.58 1.75 m wavelength) of Ground Zero, taken at 11:55 am on September 11, 2001 by a multispectral sensor of French satellite SPOT 4. This was roughly 3 hours after the WTC1 (08:46:40) and WTC2 (09:03:00) Figure 3: Infrared SPOT image, acquired attacks and before the WTC7 collapse. Al three hours after the World though this image has a poor spatial reso Trade Center Attack on the 11th lution of 20 m, it clearly shows only 2 “hot September 2001. From [21]. Note spots” at WTC1 and WTC2 locations, but that although fires are already rag no other one at WTC7 location. Therefore, ing in WTC7 which are supposed although a note below the SPOT 4 image to be the cause of its future col (see Fig. 3) in [21] reads “Hotspots asso- lapse according to NIST no “hot ciated with fires raging at Ground Zero ap- spot” appears there, but only at pear in red”, we suggest that the hot areas WTC1 and WTC2 locations (in displayed in red are the cause of the fires red). rather that the fires themselves, which at that time extended well beyond the WTC1 prints” of the three collapsed buildings ap and WTC2 footprints, and in particular in pear surprisingly similar. the WTC7 building still standing. Further more, the black (false) color of the plume ris According to U.S. National Institute of Stan ing from Ground Zero is consistent with wa dards and Technology (NIST), concerning ter as a majority component of the plumes the Twin Towers (WTC1 & WTC2) “The which appears white in visible light as wa footprint of each tower was a square, about ter is opaque to infrared radiation. We will 210 ft. on a side” (64 m)[24]; the footprint argue on this later in this paper. area for each tower was consequently about S = 642 4.1 103 m2. Still ac On October 7th, 2001, a thermal image tower fp ≈ × cording to NIST[25], WTC7 footprint was of Ground Zero (see Fig. 4) acquired by a trapezoid with a 140 ft. (42.7 m) width, EarthData using a Raytheon airborne sen and 2 parallel sides of 329 ft. (100.3 m) sor ([21], Fig. 3.6, p. 22) still shows a very and 247 ft. (75.3 m); the footprint area of characteristic thermal pattern of roughly WTC7 was consequently about S 7 = the same magnitude for all the three build WTC fp 42.7 100.3+75.3 3.75 103 m2. The sum of ings (WTC1, WTC2 and WTC7) which suf 2 ≈ × the footprints areas for the 3 buildings that fered catastrophic failure on September 11, collapsed on September 11, 2001 was then 2001. Unfortunately no temperature scale about: has been publicly released, but it should be emphasized again that although being ar S = 2S + S 7 chitecturally very different and having suf total fp tower fp WTC fp = 11.95 103 fered very different damages (no plane hit × ting WTC7), the resulting “thermal foot 12 103 m2 (20) ≈ ×

14 3.4 Numerical estimate of heat transferred by convection

Figure 4: Thermal image of Ground Zero ac Figure 5: Superposition of thermal image quired by EarthData on the 7th obtained on October 7th, 2001 by October 2001, using a Raytheon EarthData shown in Fig. 4 and of airborne sensor. From [21], p. 22. a WTC map, with colours altered and scale added.

As can be seen by superposition of thermal images and WTC site plans3, “thermal foot lease rate is S 2 104 m2. hz ≈ × prints” were, in the first weeks, larger than As said before, this area in itself has little the buildings footprints themselves. For in importance if it is associated with the wrong stance, if one uses the EarthData thermal temperature difference ∆T ; that is, taking a image given in Fig. 4, the image of Fig. 5, larger value with a lower ∆T can give the where colours have been altered for the sake same heat release rate. of clarity and a scale added, can be obtained. As cooling down of the 3 buildings debris Any precise determination of the hot zones took place over months, we do not need ei total area is difficult, but we want to recall ther to know a precise value of ambient tem here that it is anyway not our goal, since peratures; it began with rather mild ones in we only deal with orders of magnitude for September (T 290 K on September 11) our demonstration. Since the image shown ≈ but went down to much lower values dur in Fig. 5 was obtained nearly one month af ing the following winter. We take the rather ter the attacks, and since hot zones extend conservative4 value of T = 300 K (27°C, or significantly further than the buildings foot a 80°F) for the following. As said before, sev prints total area which is about 12 103 m2, × eral hot spots were measured at 1000 K on we consider that a good estimation of the September 16, 2001, 5 days after the attacks, hot zones total area for the initial heat re which can be considered as almost an ini 3 For instance the one that can be found on tial value; that makes a ∆T0 = 700 K differ Wikipedia, itself based on NIST report draw ings: 4In the sense where it is rather high, and therefore https://en.wikipedia.org/wiki/World_ will minimize the total amount of heat released Trade_Center_(1973-2001) in our calculations.

15 3 A practical example: disproving some “conspiracy theories” ence with ambient temperature. Of course, there was by no means a uniform tempera ture on the whole “hot zone” defined previ ously, and we should not take this value as a good estimate for our simplified calcula tion, even in the order of magnitude perspec tive. On the other hand, since the average − − heat transfer coefficient h 10 W.m 2.K 1 ≈ we estimated before was based on theoreti cal results for horizontal plane surfaces, and since the hot wreckage zones were obviously not horizontal plane surfaces but exhibited Figure 6: WTC – Thermal Imagery, Febru a much larger contact surface with air, it is ary 12, 2002. New York State, Of clear that we underestimate h in our model fice for Technology (©2001) and and should therefore not be too conservative EarthData International. From in our mean temperature difference estima [20]. tion. Somewhat arbitrarily, but keeping in mind that it should be enough for an order of magnitude estimation of initial heat release ambient air, which is a correct assumption rate, we choose to take half of the maximum if heat transfer laws are linear and ambi value given above for an equivalent hot zones ent temperature a constant, we could take temperature difference: this 100 days value (a little more than 3 ∆T0,hz 350 K (21) months) as a first estimation of the charac ≈ teristic cooling time. However, as rather hot Hence we get the initial heat transfer rate, temperatures are needed to cause a fire, it or thermal power associated with the three might be considered as an underestimated buildings hot zones: value. Furthermore, among the publicly available thermal images of Ground Zero P0 Shz h ∆T ≈ 4 some of them[20], like the one shown in Fig. 2 10 10 350 6, still show a clearly noticeable hot zone on ≈ × 7 × × 7 10 W WTC1 location as far as February 12, 2002, ≈ × 70 MW (22) that is 5 months after the terrorist attacks. ≈ Although no temperature scale is provided, Again, this should not be considered as an the simple fact that some thermal signal sig accurate value but just as an indication that nificantly emerges from background noise on we should expect the actual initial thermal the image proves that Ground Zero is still power to have been in the 100 MW range for cooling down and that the system is not at the whole site, rather than in the 10 MW or thermal equilibrium.This is also a proof 1 GW ranges. that fires were not the source for heat but the consequence of it, since as stated 3.4.2 Estimating characteristic cooling above, the last underground fires were ex time tinguished on December 19, 2001, almost 2 months earlier. As said before, “The underground fire burned for exactly 100 days and was finally declared Consequently, and again somehow arbitrar “extinguished” on Dec. 19, 2001.”[8]. As ily, but consistently with our order of mag suming an exponential decay for the tem nitude approach, we chose an intermediate perature difference between hot surface and value of 4 months as characteristic time, or

16 3.5 The physical limits of energy carriers to express it in seconds: like coal or oil and nuclear fuel used in nu clear reactors. Nuclear power plants need τ 4 30 24 3600 ≈ × × × to be “refilled” with nuclear fuel only rarely 107 s (23) (usually every 3 years, and only partly re ≈ filled), whereas most coal power plants must be designed with a railway track carrying 3.4.3 Estimating amount of heat released millions of tons of coal annually. Let us re call below the physical origin of this differ Now, accordingly to equ. 4, we just have to ence. multiply the initial heat transfer rate given in equ. 22 by the characteristic time given in equ. 23 to find an estimate of the total 3.5.1 Chemical energy carriers amount of heat released by free convection : Every undergraduate student in physical or Qfc P0 τ chemical science knows that a chemical re ≈ 7 7 7 10 10 action is no more than a reorganization of ≈ × × 7 1014 J (24) electrons within bonds which tie atoms to ≈ × gether; he or she knows also that energy lev Let us recall that the total amount of heat els of electrons involved in chemical bonds was released through several mechanisms: considering only the strongest, covalent ones free convection in the air, forced convection do not exceed the electronvolt range. For (water), conduction and radiation. We chose instance, the HH bond energy is 436 kJ/mol to focus on free convection as it is proba which translates into 7.24 10−19J per bond × bly the dominant mechanism in the prob or 4.52 eV. The N N triple bond, one of the ≡ lem, but each mechanism gives a positive strongest ones, has a 9.79 eV dissociation en contribution, so we can only underestimate ergy. the total value if we take into account only one mechanism. As above, this should be The energy involved in a chemical bond is only considered as an indication that the always associated with some mass, but elec actual value ot total heat released at trons are not by far the main mass carriers Ground Zero is in the 1015 J, or peta- in atoms, since proton mass mp and neu tron mass mn are much larger than electron joule, range: −31 mass me: me 9.1 10 kg, mp mn −27 ≈ × ≈ ≈ Q 1015 J (25) 1.67 10 kg. We can therefore estimate GZ × ∼ the energy per unit mass of any chemical This is, indeed, a really huge value which ori energy carrier if we can estimate how many gin has to be questioned. As material sup protons and neutrons are associated with ev ports for energy are well known, we will re ery electron involved in a covalent bond. call below some basic knowledge about the Since stable chemical compounds are always possible origins of energy, and especially the electrically neutral, and since electric charge limits of any kind of chemical energy. in atoms is carried by electrons (for the neg ative charge) and by protons (for the posi 3.5 The physical limits of energy tive charge), it can be inferred that for every carriers electron involved in a covalent bond one can at least associate one proton in the corre It is well known, even to undergraduate stu sponding atom. Since not all electrons are dents, that there is a huge difference in mass necessarily shared in chemical bonds, espe for a given amount of energy released be cially in heavy or moderately heavy atoms, tween conventional chemical energy sources some protons can also be present that will

17 3 A practical example: disproving some “conspiracy theories”

−1 instance HHVCH 55.5 MJ.kg ), hydro 4 ≈ gen being the one and only molecule where each atom consists only in one proton and one electron, which is the most favourable case. Actually this upper limit calculation as sumes that there are no neutrons associated with protons (otherwise we would have to di vide the result by 2 at least) and that every electron is involved in a covalent bond, two conditions that are fulfilled only by hydro gen gas. If we exclude this very special case, we should rather take the following limit:

chemical excl. H2 E − > 5 107 J.kg 1 (27) m ×  max It should be emphasized also that such a mass constraint is entirely independent of the chemical nature of the compound (com Figure 7: Plot of atomic isotopes (Z: number bustible6, explosive7, or even food8) and of of protons, N: number of neutrons) technological refinements, that can act on colored by half life. From [26] energy release rate (i.e. power) but not on energy per unit mass. not be associated with any electron shared in A lot of scientific and technological excite a bond. Moreover, most atoms include neu ment has occured in the last decades in the trons in their nuclei, in a proportion which is field of socalled “nanotechnologies”; how generally of slightly more than one neutron ever, since the very principle of such tech for one proton (see Fig. 7). nologies is to finely divide matter down to Therefore, considering only an order of mag the molecular level, only the surface/volume nitude calculation, one can estimate the ratio of the chemical compounds can be in E creased and consequently the speed of a upper value m max of any chemical en ergy per unit mass by dividing one electron
Heating Values (HHVs) and Lower Heating Val volt by the mass of a nucleon: ues (LHVs) for combustibles. The former equals the thermodynamic heat of combustion (or en E chemical 1 eV thalpy change) whereas the latter does not take > m m into account energy released by water condensa max n tion. For our purpose this difference does not   −19 1.6 10 J actually matter. × 6 ≈ 1.7 10−27 kg The wellknown “ton oil equivalent” or toe equals × 42 GJ, hence the energy per unit mass of oil is 8 −1 − 10 J.kg 4.2 × 107 J.kg 1. ≈ − 7 100 MJ.kg 1 (26) It is of common use to express energy released by ≈ an explosion in “TNT equivalent”; a ton of TNT equals by convention 4.184 GJ, hence the energy It should be emphasized that this value is 6 −1 per unit mass of TNT is 4.184 × 10 J.kg . slightly smaller than hydrogen Higher Heat 8Food industry indicates on every packaging the 5 −1 ing Value (HHVH 142 MJ.kg ) but amount of energy for a given mass (in Europe, 2 ≈ larger than all HHVs of any other fuel (for usually for 100 g); it is therefore very easy to check that food energy content is also in the − 5For practical reasons engineers define Higher 107J.kg 1 range.

18 3.5 The physical limits of energy carriers

exothermic redox reaction and that the ox idizing agent is oxygen contained in the at mosphere, which is considered as free and unlimited. For instance, the combustion of hydrogen can be written as: 1 H2 + O2 H2O 2 → which means that for every molecule of hy drogen, weighing 2 grams per mole, one atom of oxygen, weighing 16 grams per mole, is needed. Therefore the mass of all reac tants is 2+16 = 18 grams per mole, whereas the mass of hydrogen molecule is only 2 grams per mole, i.e. 9 times less. Then if we take the ratio of energy to mass using the total mass instead of the mass of hydrogen alone, we must divide the previously given value by 9 and we get the modified higher ∗ −1 heating value (HHVH2 ) 16 MJ.kg . Figure 8: Food industry provides energy ≈ content on the packaging for The same calculation for methane (80 grams instead of 16) will give its products. Here, “Servietten ∗ −1 (HHVCH4 ) 11 MJ.kg whereas Knödel” from Germany, with a ≈ −1 7 −1 (HHVCH ) 55.5 MJ.kg , and for 0.78 10 J.kg energy content. 4 ≈ × pure carbon (44 grams instead of ∗ −1 12), (HHVC) 9 MJ.kg whereas ≈−1 chemical or physical process, which de (HHVC) 33 MJ.kg . ≈ pends heavily on this ratio. For instance, This is the reason why, when a chemical the battery industry has made a significant compound can release energy by itself with leap in energy content per unit mass when out needing an additional reactant such as replacing leadacid batteries by Liion bat oxygen, its energy content per unit mass is teries, thanks to the small mass of lithium much lower than that of usual fuels. This compared to lead; but it has made much is the case for explosives: one of the most more dramatic improvements in power per famous ones, trinitrotoluene (TNT), has an unit mass using finely divided electrodes, energy content of roughly 4.2 106 J.kg−1, × and can produce now on an industrial scale that is one tenth that of oil. power batteries delivering as much as several kilowatts per kg9, which are very useful for In conclusion, although we computed in electric or hybrid cars, or even motorcycles Equ. 26 a maximum order of magnitude used in drag racing competitions. of chemical energy per unit mass, which is in the 108 J.kg−1 range for hydrogen, this Furthermore, we must keep in mind that in is a very special case of little relevance if most cases, when expressing chemical energy we deal with heat coming from underground per unit mass this mass does not include all fires, since it is rather obvious that such a the reactants, since energy comes from an heat can not originate from the combustion of pure, ideal fuels like H2 (or CH4, or even 9For instance A123 Systems announces “over 4000 W/kg” for its AHP14 Lithiumion pris oil...) but only from complex, solid mate matic cell: http://www.a123systems.com/ rials that burn only partially. We will con prismatic-cell-ahp14.htm sequently retain a more practical, effective

19 3 A practical example: disproving some “conspiracy theories” order of magnitude for chemical energy per unit mass that could be released at Ground Zero by any material:

GZ E − 107 J.kg 1 (28) m ≈  max

3.5.2 Nuclear energy carriers

Just as chemical energy comes from modifi cation of bonds between atoms, nuclear en Figure 9: Average binding energy per nu ergy comes from modification of bonds be cleon in MeV against number of tween nuclei; but as any undergraduate stu nucleons in nucleus, for relatively dent knows, the energy levels of these bonds abundant isotopes. From [29] are much higher, on the order of a million times greater, and this is the reason why atoms are stable. Binding energy of a nu The energy released is ∆E = 198 MeV, cleus can also be expressed, according to for 235 nucleons (236 if we take into Einstein’s equation E = mc2, as a mass dif account the incoming neutron), which ference or “mass defect” between the mass of gives roughly 0.84 MeV per nucleon. a nucleus and the sum of the masses of the • fusion reaction with deuterium: nucleons of which it is composed [27]. 2 2H 3H + 1H (30) It is useful to express the binding energy per → nucleon as a function of the number of nu The energy released is ∆E = 4.54 MeV, cleons in the nucleus; the obtained curve ex for 4 nucleons, which gives 1, 13 MeV hibits a plateau in the vicinity of iron (56Fe) per nucleon. 62 as shown on Fig. 9. Ni has actually the So in both cases, fission or fission, energy re largest binding energy per nucleon (see for 56 leased per unit mass is on the order of mag instance [28]) but Fe has the least aver nitude of 1 MeV divided by the mass of a nu age mass per nucleon, having a smaller neu 6 62 cleon, which is 10 times greater than what tron/protons ratio than Ni. It follows that can be achieved with chemical energy: some energy can be released when heavy nu clei split into parts (nuclear fission), or when E nuclear 1.6 10−13 J > × light nuclei combine together (nuclear fu m 1.7 10−27 kg  max sion). Both solutions have been extensively 14× −1 10 J.kg studied since World War II, for military and ≈ − 100 TJ.kg 1 (31) civilian purposes, and lead to an energy gain ≈ on the order of 1 MeV per nucleon, as can However, in contrast to the case of chemical be seen on the curve on Fig. 9. energy where complete combustion of a fuel Let us give two examples than can be found will give energy per unit mass on the order in many textbooks: of what has been determined in Equ. 26 or a 235 little less, this theoretical limit is of little rel • fission reaction, starting from U: evance for practical applications of nuclear energy, be they nuclear reactors or nuclear 235U + 1n 90Kr + 143Ba + 3 1n → 90 143 bombs. First because not all the “nuclear Zr + Nd fuel” will react in the process, and second → − +3 1n + 8 e (29) because a lot of material which does not take

20 3.5 The physical limits of energy carriers part at the reaction is necessary to build a Note that the bomb contained a 6.2 working nuclear device. Hence the practi kg plutonium mass, of which approx cal energy/mass ratio will be much lower, as imately 1 kg underwent nuclear fis we will see on some examples below, but the sion. If we consider only this one difference with chemical energy/mass ratio kilogram mass, we end with 8.8 1013 14 −1 × ≈ will still remain huge. 10 J.kg , which is the value given in Equ. 31. For instance, the first practical thermonu clear device to be detonated by the USA, Even though these values were more than during the Castle Bravo test on March 1, 100 times lower than the one achieved dur 1954 on the Bikini Atoll, had a mass of ing Castle Bravo nuclear test, they never roughly 10.7 103 kg and a “yield”, or quan could have been reached with any conven × tity of energy released, of 15 megatons of tional chemical explosives, TNT being, at − TNT10, which is approximately 63 1021 J. 4.2 106 J.kg 1, more than 3000 times heav × × That gives an energy/mass ratio of: ier for the same energy released. 15 As an intermediate value, we can also com E 63 10 12 −1 × 5.9 10 J.kg pute the E ratio for a 100kt nuclear ex m ≈ 10.7 103 ≈ × m  CB × (32) plosive, weighing 5 metric tons, given as an This is rather far from the value given in example in Teller et al. book “The Construc- Equ. 31, but several orders of magnitude tive Uses of Nuclear Explosives” [30], p. 129: 8 −1 higher than the 1.42 10 J.kg value for 12 × E 100 4.18 10 the “best” chemical energy source known, × × m Teller etal. ≈ 5000 hydrogen.   − 8.4 1010 J.kg 1 And even much older devices, like the his ≈ × − 1011 J.kg 1 (35) torical fission bombs that exploded over Hi ∼ roshima and Nagasaki on August 6 and 9, respectively, in 1945, outperformed conven This is the reason why nuclear weapons tional, chemical bombs by several orders of are such a strategic asset, since a single magnitude also. Let us recall their charac plane can carry a bomb powerful enough to teristics: cause massive devastation, and since minia turized versions, socalled “tactical nuclear • for “Little Boy”, the 235U bomb that was weapons”, can cause enormous damage com dropped on Hiroshima, the mass was pared to their chemical counterparts if used 4400 kg and the yield 15 kT of TNT, on battlefields, or even for terrorist actions or 63 1012 J, which gives: × as they are easy to conceal inside a small 12 vehicle or even carried by men. The small E 63 10 − × 1.4 1010 J.kg 1 m ≈ 4.4 103 ≈ × est nuclear weapons ever reported had in  LB × (33) deed a smaller yield than that of the largest conventional (chemical) ones: for instance, 239 • for “Fat Man”, the Pu bomb that the USA manufactured the Davy Crockett was dropped on Nagasaki, the mass was Weapon System that had a yield between 4700 kg and the yield 21 kT of TNT, or 10 and 20 tons of TNT (42 to 84 GJ), 88 1012 J, which gives: × whereas the Aviation Thermobaric Bomb 12 of Increased Power (ATBIP) developped re E 88 10 − × 1.9 1010 J.kg 1cently (2007) by Russia has a claimed yield m ≈ 4.7 103 ≈ ×  FM of 44 tons of TNT. But of course, the masses × (34) of the devices are not comparable, with 23 10For an expected yield of “only” 5 megatons. kg for the Davy Crockett and 7 100 kg for

21 3 A practical example: disproving some “conspiracy theories” the ATBIP. Note that the Davy Crockett is, With this simple calculation, we can at E 3 109 J.kg−1only, comparatively therefore rule out any chemical origin m ≈ × “heavy” for a nuclear bomb. of the heat released at Ground Zero during the months following September 11, 2001 terrorist attacks: that would have re 3.5.3 Conclusion regarding Ground Zero quired a significant fraction of the build heat source ings masses to be combustible, which is ab surd. As we already mentioned in subsection Taking back the result obtained in Equ. 25 3.1, heat was not a consequence of fires but for heat released at Ground Zero, we recall the cause of them, because the second law that this energy was in the petajoule range, 15 of thermodynamics precludes heat to flow or 10 J. Combining this result with Equ. spontaneously from the lower temperatures 28 and 35 (for a more realistic case than 31), to the higher ones, which consequently pre we end up with the following mass require vents buildings fires to melt (or worse vapor ments depending on the nature of energy ize) steel. Here, we have demonstrated that source: the first law of thermodynamics also leads • for a chemical energy source, to the same conclusion.

1015 It is therefore impossible that the cause m chemical ∼ 107 for underground fires was some pyrotechnic 108 kg (36) compound like thermite or “nanothermite”, ∼ as some authors have suggested (see for in which can be also expressed as 100 000 stance [33]). We do not claim that such com metric tons. pounds were not used at all in the whole pro cess; we only claim that they cannot explain • for a nuclear energy source comparable the amount of heat that was released after with a nuclear explosive such as the one the attacks. cited by Teller et al., Furthermore, it is also impossible that WTC 1015 mnuclear 11 destruction was done using miniaturized nu ∼ 10 clear bombs planted inside the buildings, as 104 kg (37) ∼ some have suggested, first because of the lack of characteristic effects of aerial nu or 10 metric tons. clear explosions, and second because the The masses of WTC1, WTC2 and WTC7, energy released during the cooling process most of which was structural steel and light of Ground Zero corresponds to energies re concrete used in the floors, and therefore not leased by big nuclear weapons, not small combustible, were in the 108 kg range: for ones or a large number of them would have instance NIST claims that each of the Twin been needed. As there were three distinct Towers used 100 000 metric tons of struc events (WTC2, WTC1 and WTC7 collapses, tural steel (see [31], p. 55). Some authors in chronological order), we claim that 3 nu (see [32]) conclude after a detailed analy clear bombs of respectable size were deto sis of the materials involved in the build nated deep underground. We discuss in ap ing that the inservice mass of WTC1 was pendix B how this surprising conclusion can about 2.9 108 kg (290 000 metric tons), be more easily understood in the technical × a figure consistent with the mass per floor context of the 1960s’, since it is quite obvi unit area of similar and contemporaneous ous that burying three big nuclear devices buildings like John Hancock Center (1969) deep in the ground under three skyscrapers or Sears Tower (1973). could not be a “classical” terrorist operation

22 but only the opportunistic use of a builtin a dramatic effect, and as it is also quite ob feature. vious that there was no aerial11 nuclear ex plosion in New York on September 11, 2001, As it is usual to express the “size” of a nu we need now to check if underground nu clear bomb in TNT equivalent, let us trans clear blasts can explain satisfactorily what late our figure into this nonstandard unity. was observed there. Let us begin with a brief A kiloton of TNT equals, by convention, 12 introduction to the effects of underground 4.184 10 J. We estimated (Equ. 25) that × nuclear explosions. the total amount of heat released at Ground Zero was on the order of Q 1015 J. This GZ ∼ translates into: 4 Nuclear explosions as an 1015 Q 240 kilotons of TNT engineering tool GZ ∼ 4.18 1012 ∼ × (38) Assuming because of the similar “thermal 4.1 Basic knowledge about footprints” of the three collapses as can be underground nuclear explosions seen in Fig. 4 an equal “size” for all three bombs, this translates to 80 kilotons per Underground nuclear explosions have been bomb; however, although it is by far the ma extensively studied since November 29, 1951 jor part, not all the energy of a deep under when the USA performed the first un ground nuclear explosion converts into heat. derground nuclear weapon test within the framework of the Jangle program at the As we deal only with order of magnitude cal Nevada Test Site (see for instance [34], p. culations, we propose then, using this con 8). Such experiments were initially done ventional unity and keeping only a power for military purposes but were also eventu of ten expression, that 3 deep under- ally conducted for civilian ones (see for in ground nuclear explosions occurred on stance [35]), mainly civil engineering and en September 11, 2001 under the World ergy production. In the USSR underground Trade Center site, each of them at nuclear testing began a decade later in the least of 50 kt and more probably on Kazakh Socialist Soviet Republic known to the order of 100 kt of TNT. This is com day as Kazakhstan, and in November 7, parable to the Sedan nuclear test already 1961, France performed its first underground mentioned above, which yield was 104 kt of nuclear test at the Reggane site in the Sa TNT. However, as the device was disposed hara desert, Algeria. in the desert alluvium in the case of Sedan (at a depth of 194 m), it is clear that the Numerous studies have been made since effect of the explosion was very different in about underground nuclear explosions, and New York, since every skyscraper needs to although the most detailed ones are proba be anchored in a lithified rock for obvious bly kept secret for obvious military reasons, stability reasons. enough documents are publicly available to give us a pretty good idea of the overall pic Note that this simple energetic argument ture, which is all what we need to address does not prove in itself that the origin of our investigation. Most of the papers are of the heat at Ground Zero was the explosion the technical report type, emphasizing ex of underground nuclear bombs; it just proves perimental results like cavity size, seismic that the only known type of energy able to 11 do that was nuclear, not chemical. But as it In aerial explosion we include also explosions inside a building, even at underground levels: is quite obvious that only an explosion (and among of the numerous effects of such explosions not a progressive release of energy that oc are the “fireball” and, of course, a tremendous curs with a nuclear reactor) could have such acoustic shock wave.

23 4 Nuclear explosions as an engineering tool signature etc., but do not give a good un nomena produced by nuclear devices consis derstanding of the underlying physics, which tent with our energy release estimate. departs significantly from the one encoun tered with “ordinary” chemical explosions. 12 4.1.1 Some specificities of nuclear However, it appears that at least one book explosions physics stands out that can replace this very spe cific experimental field in the general frame As seen earlier (see subsection 3.5.2), nu wok of physics: The Constructive Uses of clear energy is, per unit mass, roughly 106 Nuclear Explosives, by Edward Teller, Wil larger14 than chemical energy, which also im son K. Talley, Gary H. Higgins and Ger plies that it is contained in a much smaller ald W. Johnson [30]. Some textbooks on volume for the same energy. Moreover, in nuclear explosions are also nowadays freely a nuclear explosion, energy is released in a available, like the classical “The Effects of typical timescale τ 10−6 s, whereas Nuclear Weapons” edited and compiled by nuc ∼ for chemical explosives it is a much longer Samuel Glasstone and Philip J. Dolan [36]. timescale τ 10−3 s. These two simple It is worth noting that the book written by ch ∼ facts account for the extremely high tem Teller et al., published in 1968, was intended peratures (megakelvin range) and pressures to promote as its title says nonmilitary (billions of atmospheres, or 1014 Pa) en use of nuclear explosions, mainly for large ∼ countered in the first stages of nuclear ex and energyconsuming civil engineering. plosions, around the place where the de It is nowadays almost forgotten that such vice was triggered. The temperatures ob uses can exist, and were actually extensively tained are high enough to strip most elec investigated at the beginning of the “nuclear trons from their orbitals and turn everything era”, especially in the 1960’s, both by the into a plasma, which for aerial nuclear explo USA and by former USSR13; most people sions results in the wellknown “fireball”. can only cite today, as civilian uses of nu Although such values are far above most of clear energy, electricity production be it in experimental physics knowledge, they lead big stationary power stations or in smaller to some theoretical simplifications that are mobile ones used in nuclearpowered ships worth pointing out, for two reasons: they [37] or submarines. We address briefly this are a good illustration of the simplicity and technoscientific collective amnesia in Ap power of physics, and they provide us a safe pendix B, where we recall some of the most way to make numerical estimates in a pres striking peaceful engineering projects based sure and temperature range where no probe on nuclear explosives. can resist. As Teller et al. point out in [30], In the following subsections, we will first give nuclear explosions physics bears some simi some overall picture of the kind of physics larities with astrophysics, since in the vicin involved in nuclear underground explosions, ity of the shot point the states of matter mainly relying on [30], and then deduce from are the same, and chemical differences are it, and from comparisons with experimental no longer relevant since everything, includ data, some numerical estimates of the phe ing rocks, is turned into a plasma.

12 A first, somewhat counterintuitive result is We have not made an extensive bibliographical research on this subject. that vaporized rock in the vicinity of the ex 13During 1960s and 1970s, both USA and USSR plosion can be treated as an ideal gas. Stu conducted peaceful nuclear explosions programs, dents are generally told to be cautious with named Operation Plowshare in the USA and Nuclear Explosions for the National Economy 14Or, as seen above, for a technically feasible device (Мирные ядерные взрывы в СССР) in the ∼ 104 times larger, which is much less but still USSR. enough to make a huge difference.

24 4.1 Basic knowledge about underground nuclear explosions ideal gas laws, especially when dealing with magnitude for the initial pressure level, in very high pressures. How then is it possi pascals (translating into 2 109 atm). ∼ × ble that tremendous pressures generated by Now this total energy density can be divided a nuclear explosion do not make this piece in 2 contributions, namely a material and a of advice valid? To understand that, one radiative one: has to remember that an ideal gas is merely a collection of independent particles which E = Emat + Erad (41) energy is purely of thermal nature, i.e., a The material part is the translational kinetic collection of particles which interaction en energy density of particles at temperature ergy can be neglected compared to their ki T in a 3dimension space, that is, in accor netic energy.15 This is precisely the case dance with equipartition of energy: for the collection of electrons and ions form ing a plasma just after16 a nuclear explo 3 Emat = nkBT (42) sion, which temperature and pressure can be 2 derived quite easily following the arguments where n is the number of particles per unit given by Teller et al., that we shall summa volume and kB Boltzmann constant. rize below. The radiative part is the energy density of a photon gas which can be expressed as: Derivation of initial temperature 5 4 E 8π kB 4 The energy released (“yield” of the de rad = 3 T (43) 15 (hc) vice, in common language) is known, as well −34 as the volume of the bomb, within which where h 6.626 10 J.s is Planck con ≈ × 8 −1 one can assume this energy is initially dis stant and c 2.998 10 m.s the speed ≈ × tributed. Teller et al. give as an example, of light in vacuum; or more simply, using 5 4 for a 100kt nuclear explosive, a cylindrical 2π kB StefanBoltzmann constant σ = 15 h3c2 −8 −2 −4 ≈ canister 1 m in diameter and 3 m long, which 5.67 10 W.m .K : volume is: × 4σ 4 Erad = T (44) 2 3 c V = πR h 2.36 m (39) ≈ Whereas both σ and c in Equ. 44 are Since the energy released is E = 100 4.18 well known constants, one has to precise the 12 14 × × 10 4.18 10 J, this accounts for an ≈ × value of n in Equ. 42 which depends on the initial volumic energy density: particular case studied. As stated before, we expect temperatures high enough to strip 4.18 1014 E × electrons from their orbitals and produce a ≈ 2, 36 − plasma consisting only in electrons and nu 1.77 1014 J.m 3 (40) 17 ≈ × clei . Hence, the particles we must count in this ideal gas are the electrons and the nu Note that an energy divided by a volume is clei18. Now, to determine how many parti homogeneous to a pressure, and this simple cles per unit volume are present in the initial result can already give us the right order of 17We use here the word “nuclei” instead of “ions” to 15Actually, the very concept of an “ideal gas” is emphasize that most electrons are free; however, much more general than an idealization or real ionization can still be incomplete. gases, since some applications of it can be found 18It is worth pointing out here that in the ideal gas in solids, for instance in polymer science [38] model, every particle has the same mean kinetic where it is a key to understanding elasticity of energy regardless of its mass. Therefore, elec rubber or gels. trons and nuclei, although having extremely dif 16In the few microseconds following the chain reac ferent masses, give the same contribution to ki tion triggering. netic energy and must be considered equally.

25 4 Nuclear explosions as an engineering tool volume of the canister we can use its volumic is close to the electrons density: mass, and consider the fact that mass comes almost exclusively from nucleons. According n ne ≈ ρ to the figures given by Teller et al. we have ≈ 2mn a mean volumic mass: − 6.34 1029 m 3 (48) 5000 ≈ × ρ ≈ 2.36 In short, we only count the electrons in the − 2.12 103 kg.m 3 (45) ideal gas particles but we only consider nu ≈ × cleons mass. As pointed out by Teller et The mass of a nucleon be it a neutron or a −27 al. in their book (p. 131), this is equiva proton is mn 1.67 10 kg, therefore lent to having a molecular ideal gas with an ≈ × − the number of nucleons per unit volume is: effective molar mass M 2 g.mol 1, like eff ≈ ′ ρ n deuterium. ≈ m n We now have all numerical data to solve 2.12 103 × Equ. 41 for temperature; let us write it ex ≈ 1.67 10−27 × − plicitly: 1.27 1030 m 3 (46) ≈ × 3 4σ E = nk T + T 4 (49) Now we do not look for the number of nu 2 B c cleons but for the total number of particles, or numerically, in SI units: which are nuclei and electrons. As stated − previously in subsection 3.5.1, there are ap E 1.31 107 T + 7.57 10 16 T 4 (50) proximately as many neutrons as protons in ≈ × × Although extremely tedious to solve analyt a nucleus (slightly more neutrons) except ob 20 viously for hydrogen, and there are exactly ically , this equation can easily be solved numerically for temperature; the above val as many protons as electrons for the sake of − ues will give, for E 1.77 1014 J.m 3 as electrical neutrality. From this simple infor ≈ × 21 mation we can infer that the number of elec we found in Equ. 40, a temperature: trons per unit volume is roughly half that of 7 T 1.22 10 K (51) nucleons: init ≈ × n′ ne (47) ≈ 2 Derivation of initial pressure But for an order of magnitude calculation, We have already got thanks to Equ. 40 a given that most chemical elements in the nu crude estimate of the initial pressure, which clear explosive canister will not be of very 20Some online equation solvers (e. g. low atomic number, we can even consider numberempire.com) will give the four so that the number of nuclei not nucleons! lutions in an instant, only one of which is is negligible compared to the number of elec physically acceptable. But its mathematical trons19, so that the desired particle density expression is itself particularly cumbersome and impractical. 19For instance, the chemical element sulfur, in its 21Note that with the same numerical values and ap alpha form, has a volumic mass at room temper proximations, Teller et al. give a slightly dif 3 −3 7 ature and ambient pressure of 2.07×10 kg.m , ferent temperature Tinit ≈ 1.37 × 10 K, and a roughly equal to the volumic mass we just cal slightly different numerical equation E ≈ 1.32 × − culated for the nuclear explosive canister. Sulfur 107 T +7.65 × 10 16 T 4. However, solving nu has an atomic number of 16, therefore counts 16 merically this equation for the same total energy − protons and 16 electrons. Neglecting to count density E ≈ 1.77 × 1014 J.m 3 (100 kt in a cylin the nuclei in a plasma obtained when all elec drical canister 3 m long and 1 m in diameter) 7 trons are separated from their nuclei will then still gives Tinit ≈ 1.22 × 10 K. Anyway, it does give a 1/16 ≈ 0.06 relative error, which is more not change the important result, that is an initial than acceptable for our purpose. temperature in the 10 MK range.

26 4.1 Basic knowledge about underground nuclear explosions is in the 100 TPa range (109 atm). However, A longitudinal wave travels in a medium of since photons and matter behave differently volumic mass ρ with a speed: for generating pressure it is better to give the correct expression as Teller et al. do: 1 ∂p v = = (55) √ρχ ∂ρ 2 1 S s S p = Emat + Erad (52) 3 3 where the subscript S indicates an isen The reason is that pressure exerted on a cav tropic transformation. Shock waves are in ity walls comes from the momentum change trinsically irreversible and thus not entropy upon collision of the particles it contains, conservative, however the above equation and kinetic energy for particles with mass m can explain why any compression wave of 1 2 1 high enough amplitude should evolve to is expressed as Eke = 2 mv = 2 pv whereas it reads for massless photons of velocity c: wards a shock wave, i.e. an almost dis continuous pressure variation that travels Eke = pc. Consequently, energy density will E 3 E through the material. be = 2 p for material particles and = 3p for photons. In other words, for a given en In the crest of a wave23, pressure is higher ergy density, photons exert one half the pres and so is temperature, since the transforma sure a material gas would exert. tion is adiabatic. As a result, the crest goes Let us give Equ. 52 in numerical form: faster than the rest of the wave, which ends in a sawtoothshaped signal instead of the − p 8.75 106 T + 2.52 10 16 T 4 (53) initial sinusoïdal one. This can easily be un ≈ × × derstood in the particular case of ideal gases Reporting Tinit value found above gives an where the speed of sound can be expressed 22 initial pressure: γRT as v = M , where temperature T appears 14 explicitly, but although different the physics pinit 1.13 10 Pa (54) q ≈ × remains qualitatively the same for most ma which comes mainly from material pressure: terials.

14 12 On the other hand, a sawtoothshaped sig p 1.07 10 Pa, p 6 10 Pa mat ≈ × rad ≈ × nal can be decomposed in Fourier analysis as a sum of frequencies which are multiples of This tremendous pressure ( 109 atm) gen ∼ the initial characteristic frequency of the si erates a shock wave which travels at super nusoidal signal: transformation of the wave sonic speed and destroys every material in into a series of discontinuities as it travels tegrity around the explosion point, long be creates frequencies that were absent from fore heat can diffuse; we will discuss this as the signal in the early stages of propagation. pect more precisely below. Now, higher frequencies tend to dissipate en ergy faster than lower ones, and this is the Shock wave effects on materials reason why shock waves do not appear for We will not in the following present a de moderate signal amplitudes: the tempera tailed description of shock waves, which can ture rise in the crests is too low to induce a be found in many textbooks, but only recall sufficient velocity increase before dissipation some basic facts about shockwaves in gen flattens the signal. eral, and what kind of effects they can have However this is entirely different, of course, in the specific case of an underground nu for the pressure levels encountered in nu clear explosion generating initial pressures clear explosions, even at a significant dis as high as the ones derived above. 23We take here, for instance, a sinusoidal wave but 22 14 Teller et al. find, here, pinit ≈ 1.15 × 10 Pa not a shock wave for the moment.

27 4 Nuclear explosions as an engineering tool

We can see from Fig. 10 that at 300 m from zero point the pressure level is still higher than 1 kb, or 100 MPa; at 100 m, it is around 1 GPa. To give some examples, ultimate tensile strength of granite is not higher than 25 MPa24 and that of a classi cal structural steel like ASTM A36, about 550 MPa ([40],[41]) which is a pressure at tained at about 150 m from zero point of a 100kt nuclear explosion in isotropic propa gation conditions. It might look inconsistent to give tensile strength of materials while we are dealing with a compression wave (which has also some tension counterpart but which figures are not the ones given here); actually this is not the case, and here is why. We are interested in explosions which occur underground but not far from the surface: Figure 10: Peak stress (maximal pressure) in deep enough for the explosion to be con kilobars (1 kb = 100 MPa) as tained, but shallow enough to cause great a function of distance from “zero damage to superstructures like a highrise point”, for a 100kt underground building because of the shock wave, and nuclear explosion, according to eventually make it collapse. It is well known Teller et al. ([30], p. 132) that every discontinuity in propagation con ditions of a wave generates a reflected wave at the surface where the discontinuity oc curs. Coming from underground, a shock tance from the “zero point”. Teller et al. pro wave that reaches the surface meets a con vide us a graph of peak stress, or maximum dition of zero pressure which generates a re pressure level, for a 100kt underground nu flected wave in phase opposition, i.e., a ten- clear explosion, both experimental and the sion (or rarefaction) wave. oretical. The loglog scale with a slope close to 2 is a direct consequence of the conser Superposition of incident and reflected − vation of energy: as the shock wave spreads waves gives, as the incident wave continues over a sphere of growing radius r, the en to progress, a large negative pressure about ergy per unit volume in the shock front zone, the same value of the peak stress mentioned 25 which is also a pressure, must scale as r−2 above. The result is spallation, as soon as in order for the energy integral to remain utimate tensile strength of the material ly constant over time. ing at the surface is smaller than the peak

24Some other sources like [39] give 4 to 5 MPa; gran Of course, some energy is dissipated since ite is not a metrology standard but encompasses the shock wave irreversibly alters the a variety of rocks. medium it travels through, but the ability of 25This is, by the way, the same mechanism, al the rock to dissipate energy is very limited though with much lower pressure levels, that make a lot of music instruments work like flutes compared to the energy content of the shock for instance and produce a harmonic sound wave, that is the reason why it can travel spectrum as a result of multiple reflections on rather far in a quasiconservative manner. both sides of a 1dimensional waveguide.

28 4.1 Basic knowledge about underground nuclear explosions

Derivation of cavity radius We shall now derive this cavity size as a function of explosion energy and depth of burial, with some degree of empirical adjust ment, using experimental evidence from real underground nuclear tests to determine scal ing laws prefactors.

Final cavity radius will obviously depend on explosive energy, since converting rock into a highpressure and hightemperature plasma Figure 11: Spallation mechanism occurs costs some energy. As the mass, or volume, near a free surface (zero pressure of rock which is turned into a plasma is pro condition) when the shock wave portional to released energy, we expect the reaches it. From Teller et al. cavity radius to scale as the cubic root of the ([30]), p. 68. σmax is the peak energy: 1/3 stress and σc the ultimate tensile Rc E (56) strength of the material. ∼ But as stated earlier, the final cavity radius achieved in less than one second not only stress, which is schematically illustrated in comes from transforming rock into plasma Fig. 11. Depending on the explosive yield but also from this extremely high pressure and on the depth of burial, spallation can plasma expanding and permanently deform occur even on materials which are tradition ing the surrounding rock; we are not talk nally considered as tensileresistent, such as ing here about the fracturation done by the construction steel. shock wave but about huge hydrostatic pres Not only does spallation occur near the sur sure being able to compress the rock. The face because of the reflected wave, but closer “final” cavity radius, which may be actu to the zero point the peak stress is large ally transitory since cavity roof may collapse enough to brake the rock into fine parts even later, is achieved when plasma pressure equi because of the incident compression wave librates lithostatic pressure, which depends the finest parts being obviously closer to the on depth of burial h through the relation: center of the explosion. Granite or other hard rocks will therefore turn into a sand plith = ρgh (57) like, brittle material, or into discrete blocks which size will increase with the distance where ρ is the rock volumic mass and g grav from point zero. ity acceleration. Later, when heat has had time to diffuse During the expansion phase, as stated earlier through the ground, part of the rock sur plasma can be treated as an ideal gas. Fur rounding the explosion cavity will melt and, thermore, the speed of this expansion allow finally, fall at the bottom of this cavity. us to consider this ideal gas undergoes an Teller et al. give, in the case of a 100kt ex adiabatic expansion, and even an isentropic plosive, a molten layer of about 50 cm thick one since almost no dissipative process can for a final cavity radius26 of about 45 m. take place at this timescale. The Laplace law

26 for an isentropic transformation of an ideal The authors make a distinction between an initial gas between states labeled 1 and 2 reads: cavity radius and a final one. However, the final radius is achieved in no more than a few hundred γ γ milliseconds from time zero. p1V1 = p2V2 (58)

29 4 Nuclear explosions as an engineering tool which can also be written here, since V = kilotons of TNT, ρ in g.cm−3 (equivalent to 4 3 3 −3 3 πR : 10 kg.m ) and h in meters. Translating 3γ 3γ this for every variable expressed in SI units p1R1 = p2R2 (59) (but E in terajoules to avoid large num 1/3γ R2 p1 bers), and taking the mean of the 2 values for = (60) ⇔ R1 p2 C (in granite) given by the authors, we can   also write the following empirical formula: Now we can take for p1 the plasma pres sure at the beginning of its expansion and E1/3 Rc 206 1 4 (62) for p2 its final pressure which is the litho ≈ (ρh) / static pressure plith = ρgh. It follows that −1/3γ the final radius R2 will scale as (ρgh) where E is to be expressed in terajoules, ρ in −1/3γ −3 or, to keep only the variables, as (ρh) . kg.m , and h in meters for Rc to be given in meters. For instance, a 80kt (or 335 TJ) Although values of the heat capacity ratio ≈ explosive we estimated in subsection 3.5.3, γ are well known for “ordinary” ideal gases buried at 100 m depth, would produce a cav and depend solely on the number of degrees 2 ity of final radius Rc 63 m. of freedom f of the molecules (γ = 1+ f , ≈ 5 Note however that for a nuclear explosive which translates in γ = 3 for monoatomic 7 buried at relatively shallow depth, the litho gases and γ = 5 for diatomic ones at not too high temperatures27), it appears that static pressure can vary significantly from the situation is a little bit more complex here the bottom to the top of the cavity, which due to the “bimodal” nature of our ideal gas: will account for a nonspherical, but elon a material part (nuclei and electrons) and a gated shape of the cavity. The overburden massless part (photons). According to Teller pressure being less important near the sur et al. (p. 79), the correct value for a pho face, expansion of the cavity can be more 4 important in its upper part. We will not ton gas is γ = 3 and, “purely by chance”, it is also approximately correct for the mate develop further than this purely qualitative rial part if the temperature is high enough comment. to produce large ionization (but not much greater than 10 000 K, where complete ion 5 4.1.2 Empirical description of ization would make γ approach the 3 limit). phenomena As a result, we can write a scaling law for the cavity radius where a constant C remains to A number of technical reports on under be determined experimentally: ground nuclear tests are freely available on the internet, sometimes previously confiden 1/3 E tial but now unclassified, such as “Some Ba- Rc C 1 4 (61) ≈ (ρh) / sic Principles of Scaling Explosion-Produced Damage to Deep Unlined Openings in Teller et al. give (p. 137) a table of data Rocks” by G. B. Clark [42], “Underground obtained from 15 underground nuclear ex Nuclear Explosion Effects in Granite Rock plosions; let us cite here only the numerical Fracturing” by S. Derlich (CEA France) C values obtained in granite: C = 57.70 and [39], “The containment of Nuclear Under- C = 60.48, for Hardhat and Shoal tests, re ground Explosions” by the U. S. Office of spectively. Technology Assessment [43], or “Visual In- However, these numbers must be read for spection for CTBT Verification” by Ward Rc expressed in meters, energy expressed in Hawkins and Ken Wohletz [44]. Some more 27Otherwise vibrational degree of freedom comes recent scientific papers can also be found into play. easily (see for instance [45]).

30 4.1 Basic knowledge about underground nuclear explosions

All these documents give the same overall The shock wave will continue to expand and picture of underground effects of nuclear ex decrease in strength eventually becoming the plosions, but with numerical values that can "head" (or leading) wave of a train of seis- vary significantly because of the different ge mic waves. ological nature of the test sites and of the During the third stage, the cavity will cool inherently fuzzy definitions of the different and the molten rock material will collect and damaged zones considered. solidify at the bottom of the cavity. Let us first read some qualitative descrip Finally, the gas pressure in the cavity tion of the phenomena that occur just after a decreases to the point when it can no deep underground nuclear explosion (i.e. an longer support the overburden. Then, in explosion that does not produce ejections of a matter of seconds to hours, the roof falls solid matter in the atmosphere), as they are in and this is followed by progressive collapse exposed in “The Effects of Nuclear Weapons” 28 of the overlying rocks. A tall cylinder, com- [36] on pages 6162. The following lines are monly referred to as a "chimney," filled with excerpts from subsections 2.101, 2.102 and broken rock or rubble is thus formed. If the 2.103. Emphasis is added to separate the top of the chimney does not reach the ground four phases. surface, an empty space, roughly equivalent “The phenomena of deep underground deto- to the cavity voIume, will remain at the top nations can be described best in terms of four of the chimney. However, if the collapse of phases having markedly different time scales. the chimney material should reach the sur- face, the ground will sink into to the First, the explosion energy is released empty space thereby forming a subsi- in less than one microsecond. As a dence crater. The collapse of the roof and result, the pressure in the hot gas bubble the formation of the chimney represented the formed will rise to several million atmo- fourth (and last) phase of the underground spheres and the temperature will reach about explosion.” a million degrees within a few microseconds. Although [36] gives a schematic picture of In the second (hydrodynamic) stage, the resulting zones underground (Fig. 2.103 which generally is of a few tens of a p. 62), we prefer to show here (Fig. 12) an second duration, the high pressure of the other illustration found in a paper originat hot gases initiates a strong shock wave which ing from French Commissariat à l’Énergie breaks away and expands in all directions Atomique (CEA) [39] which we think to be with a velocity equal to or greater than the more precise. It introduces several radii (Rc, speed of sound in the rock medium. During R and R ) measured from the shot point the hydrodynamic phase, the hot gases con- b f that will be discussed later. An even more tinue to expand, although more slowly than realistic picture, describing a real case of initially, and form a cavity of substantial deep underground explosion in former USSR size. At the end of this phase the cavity will (“borehole 102” experiment, Balapan test have attained its maximum diameter and its site), can be found in [34], p. 12, and we walls will be lined with molten rock. The give it in Fig. 13 where a spall zone is clearly shock wave will have reached a distance of identified near the surface (see 4.1.1 for ex some hundreds of feet 29 ahead of the cavity planations). and it will have crushed or fractured much of the rock in the region it has traversed. Cavity radius data 28 A similar description can also be found in [43], Chapter 3, p. 32, in [34], p. 35, or in [30], Chap We report in table 1 some empirical scal ter 4, p.133. ing laws provided by the CEA paper [39], 29100 ft. ≈ 30,5 m with W the yield in kilotons and the radii

31 4 Nuclear explosions as an engineering tool

1/3 Rc = 7.3 W cavity radius 1/3 Rb = 10 W crushed zone radius 1/3 Rf = 26 W fractured zone radius 1/3 Rr = 35 W stressed zone radius

Table 1: Different radii as defined in Fig. 12 as a function of bomb yield ex pressed in kilotons TNT equivalent.

1/3 Rc = 4.5 E cavity radius 1/3 Rb = 6.2 E crushed zone radius 1/3 Rf = 16 E fractured zone radius 1/3 Rr = 22 E stressed zone radius

Table 2: Same as table 1 but with bomb en ergy expressed in terajoules.

expressed in meters, for tests conducted in Figure 12: Vertical cross section through a granitic batholith in the Sahara desert near chimney resulting from a deep the Hoggar mountains, with about 1000 m underground nuclear explosion. of overburden pressure. From [39]. As 1 kiloton TNT equivalent equals 4.184 TJ we can also reformulate these laws in table 2 using the energy E released by the bomb in terajoules. Note that all these values are approximate and depend heavily on the boundaries def initions. According to [39], “crushed zone is encountered from 7.3 to 10 W 1/3.” If we compare the cavity radius scaling found here with that given by Teller et al., tak ing into account h = 1000 m and ρ = − 2.63 103 kg.m 3, we find that Equ. 62 gives × 1 3 R 5.1 E / , quite close to the CEA value. c ≈ Other authors like Fokin [46] or Rogers [47] include in their scaling laws the dependence on depth of burial and rock volumic mass we derived above. Rogers gives the following relation, identical to that proposed by Teller et al., for the final cavity radius (in meters): Figure 13: Same kind of diagram as Fig. 12 W 1/3 but for a real nuclear test in for R = C c 1/4 (63) mer USSR (Balapan test site). (ρh) From [34], p. 12, Fig. 4. where C 59 for granite (from 57 to 61), ≈ when W is in kilotons, h in meters and ρ in

32 4.1 Basic knowledge about underground nuclear explosions g.cm−3. Authors Cavity radius (m) On the Russian side, Fokin gives three rela CEA [39] 34 tions, some with detailed parameters like the Teller et al.[30] 38 initial value of the adiabatic exponent, the Rogers [47] 38 adiabatic exponent for the equilibrium part Fokin [46] 36 of the detonation products and the volumic mass of the explosives; and finally gives the Table 3: Some values of final cavity radius following approximated relation: for a 100kt bomb in granite at 1000 m depth (taking ρ = 2.63 1/3 3 −3 × E0 10 kg.m ), according to different Rc = 0.2842 (64) P authors.  h  where E0 is the energy expressed in kgf.m and Ph is the counterpressure at the depth other factors play only a minimal role, some of explosion defined as: authors may choose to use simplified laws where only energy determines the cavity ra Ph = σcomp + ρh 30 dius. with σcomp the maximum strength of the rock under compression expressed in Les us calculate how numerical results com kgf.m−2 and ρ the volumic mass31 of the pare for these different expressions, in the rock in kg.m−3. Translating Equ. 64 in case of a 100kt (418 TJ) explosion at a 1000 SI units (pascal for stress, joule for energy) m depth in granite, and taking for the Fokin we get the same formula since the numerical relation σcomp = 200 Mpa which is the value value of E0 does not change. given by the CEA study. We summarize in Ph Table 3 the final radius cavity obtained from Note that for shallow depths of burial ρh ≪ table 1 and expressions 61, 63 and 64. σcomp, so that the radius can be consid ered independent of h. In order to achieve Note that in this depth range the relation ρh < 0.01 σcomp we must restrict for gran given by Fokin gives almost no dependance 6 ite, taking σcomp = 200 Mpa, to h > 740 m. on h since ρh = 2.63 10 Pa σcomp = 8 × ≪ 2 10 Pa. As we can see, although rela Note also that formulas 63 and 64 seem to × be incompatible since the former does not tions given by different authors may differ take into account the compressive strength formally and even look theoretically incom of the rock, whereas the latter does. More patible, as a practical tool they manage to over, Equ. 63 scales as h−1/4 whereas Equ. give similar results, provided they are used 64 is practically independent of h in the shal in the case they are designed for. −1/3 low depth approximation but scales as h The above example addresses the case of in the opposite case. And expressions given a deeply buried explosive; however we are in 1 do not even take into account depth of more interested in explosions which, al burial whereas 63 and 64 do, because they though still contained, will produce some are restricted to some narrow depth of burial striking effects at the surface. range. Even this is not satisfying from a theoretical point of view, it should be kept On Fig. 14 we give an overall picture of in mind that all these expressions are ap different explosion effects depending on the proximate laws based on experimental val depth of burial of the device, as shown in ues collected after nuclear tests, and that if [36] p. 234. These three cases do not include very deep underground explosions where no 30Compressive strength for granite can vary be tween about 100 and 300 MPa [40]. permanent deformation of the ground sur 31Volumic mass for granite is about 2.7 × face to the vertical of the device, around the − 103 kg.m 3. socalled “ground zero” or “surfaced ground

33 4 Nuclear explosions as an engineering tool

dislocations movements that dissipate en ergy. But it can be also regarded as a transi tion depending on deformation speed, just as the timetemperature superposition princi ple [48] suggests for “soft materials” physics: decreasing temperature is equivalent to in creasing deformation speed.

4.2 Proposed mechanism

We have so far demonstrated that the only known source of energy that can account for the huge amount of heat ( 1 PJ) re ∼ leased from Ground Zero remnants during Figure 14: Different possible effects of un the months following September 11, 2001 derground nuclear explosions de terrorist attacks was nuclear, and have given pending on depth of burial. From some basic review of known effects of un [36], part of Fig. 6.06, p. 234. Ra derground nuclear explosions. In the follow and Da are the apparent radius ing, we will see how a nuclear demolition and depth, respectively. DOB = device could have been rationally designed Depth Of Burial. for skyscrapers and implemented for some of them, without any terrorist intentions but based solely on efficiency and cost criteria. zero” point, occurs. Any discussion about the opportunistic use In case f of Fig. 14, the initial cavity cre of such devices for or during terrorists ated by the explosion finally creates a rub attacks, and about the identity of the peo ble chimney between the shot point and the ple deciding it or their motives, is outside surface, once the pressure inside the cavity the field of physics and will be left up to the has decreased enough and no longer sup readers discretion. ports fractured rock above the cavity. Let us emphasize again that fracturation of the rock occurs just after the explosion before 4.2.1 General mechanism the expansion of the cavity due to the enor mous pressure generated by the shock wave. As stated above, an underground nuclear explosion creates a tremendous shock wave Moreover, because of the almost discon that is able to shatter every material and tinuous, extremely abrupt change in pres turn it into tiny pieces or larger chunks de sure generated by the shock wave much pending on the distance from point zero, more than for chemical explosions it can up to some limit where the wave will have be inferred that there will be very little only elastic and nondestructive behaviour. difference in behaviour of materials that This is the main feature that can ex- will be in the shockwave path: even duc plain why it can be used to collapse tile materials like steel can in these par buildings, even steel-framed ones like ticular conditions appear as brittle. The skyscrapers, provided that the nuclear ex ductilebrittle transition in metals is gen plosive is placed at the right depth under erally considered as a function of temper 32 neath the building in order to create a ature , since low temperatures slow down subsidence crater at ground level that 32in the same way as glass transition in polymers triggers the collapse. An example of

34 4.2 Proposed mechanism

are taken. The aim of an engineer for design ing such a demolition device must therefore be to maximize the desired effect (collaps ing the building) and at the same time to minimize the undesirable effect (radioactive fallout). A compromise has to be found be tween a zero point situated very deep under ground, in order to contain radioactive ele ments as much as possible, and a shallower one, to maximize shock wave shattering. Figure 15: Numerous examples of subsi dence craters created by under If the purpose is to destroy a highrise build ground nuclear explosions can ing, using an underground explosive will nec be found for instance at Nevada essarily shatter more effectively the lower test site in the USA. Although part than the upper part of the building, this one is in tuf, hard rocks since the shock wave maximum pressure de such as granite can also pro creases approximately as the inverse square duce a similar, counterintuitive of distance from point zero, as seen above phenomenon. (subsection 4.1.1 and Fig. 10). However, to some extent, the structure of the building itself might act as a “waveguide” and pro this rather counterintuitive (a depression duce a peak stress at the top higher than it at ground surface resulting from extremely would be for the same traveling distance in high pressures underground) but very well a homogeneous and isotropic material; fur known phenomenon is given on Fig. 15 and thermore, as explained above, reflection of various videos of it are available on the In the wave at the top produces a tension wave ternet, see for instance [49] on AtomCentral which negative pressure is more effective at YouTube channel. breaking materials. Release of a large quantity of heat is only Apart from the shock wave effect itself, a a necessary sideeffect, as well as release of very important feature of a nuclear under some radioactive elements. Because of en ground explosion is, in most cases, the cre- ergy conservation, heat release cannot be ation of a rubble cylindrical chimney as a avoided it is directly proportional to explo consequence of the cavity “roof” collapse. As sive energy or “yield” but is only delayed a consequence, there is no need for the initial due to high thermal resistance of surround cavity to reach the basement of the building ing rock; however radioactive contamina to make it collapse: it is enough to make the tion, although unavoidable, can be restricted upper part of the rubble chimney reach the to some safe levels according to Teller et basement. In this case, the building having 33 al. , provided some precautionary measures been at least in its lower part shattered

33 by the shock wave but kept apparently un More specifically, according to them “The fusion explosions [...] can be handled in such a way as changed in a kind of “metastable” state, it to eliminate most of the ensuing residual radioac- will finally collapse when its basement no tivity.” (p. 3). Note also that although the Cha longer finds mechanical support or a much gan nuclear test performed at the Semipalatinsk weaker one because of the rubble collaps test site on January 15, 1965 used a fission de vice, the resulting lake was declared safe by the ing into the cavity. According to Teller et al. soviet authorities and a small movie [50] even (Fig. 4.8 p. 138), the ratio of the chimney 34 showed some swimmers dipping into it shortly height to the cavity radius is around 4.35, after, wearing only a small swimsuit and using only a simple snorkel. 34Defined as the distance between zero point and

35 4 Nuclear explosions as an engineering tool an experimental mean value obtained from effect that can trigger building final several shots in granite. If we consider, in collapse. However, it can be argued the case of the World Trade Center, an esti that the distance between the cavity mated energy of 80 kt TNT equivalent35 or top (before chimney formation) and the 1 1015 J per explosive as we mentioned buiding basement, about 40 m, could ≈ 3 × earlier in subsection 3.5.3, we can make sev be insufficient to satisfactorily contain eral hypothesis for their depth of burial and radioactive elements that can migrate see which one fulfills the best an engineers from the cavity to the surface through team specifications for building controlled the rubble. In other words, from the demolition. radioactive pollution criterion it should be safer to chose the maximum depth 1. Let us first suppose the explosive is that still can trigger building collapse, buried 100 m below the ground level. that is, a depth for which only the top of The radius of the cavity will be, ac the rubble chimney reaches the building cording to Equ. 62 and taking ρ 3 −3 ≈ basement. 2.7 10 kg.m for the rock volumic × mass: 2. Let us then try this case, and choose a depth of burial of 200 m below ground 1 3 1/3 3 10 surface. Now the cavity radius is Rc 206 1 4 ≈ (2.7 103 100) / slightly reduced: × ×
63 m (65) 1 3 1/3 ≈ 3 10 Rc 206 1 4 As said before, in this case the di ≈ (2.7 103 200) / × ×
mension of the cavity itself is on the 53 m (66) same order of magnitude as the depth of ≈ Now the chimney height is about h burial, so that lithostatic pressure can ≈ not be considered the same at the top 230 m, which is enough to make sure and at the bottom of the cavity and the the basement will fall into a depression cavity itself will not grow to a spherical to trigger the building collapse, but the shape, even before collapse of the frac distance between the initial cavity top tured rock leading to the rubble chim and the building basement is a more se ney. This calculation should therefore cure 150 m. But another problem now be considered as a very crude one and emerges: the distance between the “zero the “radius” of about 60 m only taken point” and the top of the building might as a rough guide. be too large in order to weaken its struc ture sufficiently. In the case of the Twin In this case, the theoretical chimney Towers, distance varies from 200 m in height being h 63 4.35 275 m, ≈ × ≈ the basement to more than 600 m at it is clear that ground surface and the the top. As can be inferred from Fig. building itself stops chimney growth 10 taken out Teller et al., peak stress so that the building basement falls into for a 100kt explosive36 at 600 m dis a local depression, which is a desired 36We choose here a 80kt explosive for our calcu the chimney top. lations although, as said above, the exact value 35Let us recall here that we estimated this energy might well be larger and even larger than 100 only through heat released by free convection at kt. But anyway, it should be kept in mind that Ground Zero, and that it should therefore be initial temperature and pressure values derived underestimated since free convection is not the from Equ. 50 and 53 come from an energy vo- only heat transfer mechanism involved, and since lumic density estimate, and not from an energy explosive energy is not entirely released as heat. estimate. If the 80kt has the same energy per However, we choose here to use this “conserva unit volume as the 100kt one, calculations re tive” value. main unchanged.

36 4.2 Proposed mechanism

tance in a homogeneous and isotropic medium will be in the 350 bar (or 35 MPa) range, and as seen in paragraph 4.1.1 this value is larger than ultimate tensile strength of granite (maximum of 25 MPa) but smaller than that of com mon structural steel (550 MPa). At 200 m distance, it will be roughly 300 MPa. Although real strain on structural ma terials would require much more precise analysis, this order of magnitude calcu lation already tells building destruction is no more certain, or at least that the upper part of the building is likely to remain unshattered. We face here a typical optimisation prob lem where two effects a desired one and an undesirable one pull in opposite directions and a compromise has to be chosen which depends on the weighting of the two; engi neers must solve such problems all the time for cost, safety or regulations reasons. At this point it is impossible to give any precise estimate of the optimum depth of burial of a nuclear demolition device, since we do not know its precise energy content (or “yield”) nor which minimum depth can be considered or was considered, in the 1960’s and in the USA as “safe” for radioactive pollution is sues37. Order of magnitude arguments have their virtues we hope to have demonstrated Figure 16: Possible position of a 80kt nu this but cannot replace more precise and te clear explosive (at center of dious calculations when fine tuning is sought sphere), initial cavity (sphere) after. and approximate boundary of We therefore conclude that a nuclear explo rubble chimney (cylinder) for nu sive, on the order of 100 kt TNT equivalent clear demolition of WTC1 or in energy and buried at a depth on the order WTC2, drawn to scale with the of 100 m and this should really be taken as building and its basement. Here mere powers of ten estimates can destroy a a depth of 120 m under ground highrise building using both the shock wave surface has been chosen. Col created by the explosion and the subsidence lapse chimney may have not ex crater produced by the collapse chimney cre actly the same diameter as cavity, ated underground, once the cavity pressure which would in this case adopt has decreased to a level where broken rock a vertically elongated shape (not can no longer sustain overburden. As an il displayed) because of lithostatic pressure differences from top to 37The radioactive issue will be addressed more pre bottom. cisely in subsection ??.

37 4 Nuclear explosions as an engineering tool lustration, we draw in Fig. 16 the approxi am and 10:28 am EDT38, respectively) af mate position of a nuclear explosive relative ter having allegedly been struck by airlin to WTC1 or WTC2 buildings, as well as ap ers, whereas WTC7 collapsed only in the af proximate dimensions of initial cavity and ternoon of the same day (at 5:20 pm EDT) rubble chimney. Note that the cavity hence without having been hit by any plane, but the chimney diameter is larger than the also because the last collapse appeared to building footprint (a square of 64 m width, many building demolition experts as an ex which has a 90m diagonal). ample of a properly made controlled demoli tion, whereas the first two were rather catas Finally, we would like to emphasize again trophic events from this perspective even if the very large time scale difference between we do not take into account the tragic fact the first effects of the explosion and the that many people died in the process. last mentioned by the different authors (col lapse of the “roof” and chimney formation), Such a difference needs to be explained, which are actually not the last ones since at least qualitatively, since we claim that the enormous amount of energy generated the demolition method employed was the converts mainly into heat and heat diffusion same. The first obvious difference between through the surrounding medium will take the Twin Towers and WTC7 resides in their place on a much larger time scale, even if the height: according to NIST in document NC chimney top is close to the ground surface. STAR 11 ([52]), p.7, WTC1 was 1368 ft 39 This characteristic cooling time will be sim (417 m) high , not including the antenna, ilar to the ones encountered for thick lava and WTC2 about the same (1362 ft or 415 flows after volcanic eruptions, i.e., several m), both consisting of square structures with months, which corresponds to the timescale a side dimension of 207 ft (63 m), whereas we estimated in Equ. 23. In a 1972 paper the overall dimensions of WTC7 ([52], p. 13) [51], the French Commissariat à l’Énergie were approximately 330 ft (100.6 m) long, Atomique investigated the temperature dis 140 ft (42.7 m) wide, and 610 ft (186 m) tribution in the rock 178 and 221 days af high. ter a fully contained nuclear test in gran The Twin Towers were then more than twice ite. It turned out that the maximum tem as tall as WTC7, which makes a huge differ peratures measured, in the vicinity of the ence if a demolition process uses a powerful shot point, were respectively about 600°C underground explosive designed to shatter (for 178 days 1.5 107s) and 500°C (for ≈ 7× the building structure before making it col 221 days 1.9 10 s). This is entirely con ≈ × lapse thanks to an underground depression. sistent with the very long characteristic time It appears from infrared imaging that ther of τ 107 s (Equ. 23) we have chosen for ≈ mal energy released at the three building lo estimating the total amount of heat released cations was roughly the same, at least on at Ground Zero. an order of magnitude level, therefore shock wave pressure levels must have been also similar. Since they decrease as a function 4.2.2 Differences between WTC1, of height in the building structure, it turns WTC2 and WTC7 out that the much smaller WTC7 could have its structure much more effectively shattered than those of WTC1 and WTC2, which cor There have been obvious differences in responds to observation: both WTC1 and WTC1 and WTC2 collapses, on one side, WTC2 collapsed with an upper part falling and WTC7 collapse, on the other side. Not only because the Twin Towers collapsed in 38Eastern Daylight Time, or UTC4. the morning of September 11, 2001 (at 9:59 39above the Concourse level

38 Figure 17: Fig. 51, p. 53 of [53], shows Con Edison power substa tion footprint, which differs from that of WTC7.

Figure 18: Aerial picture of WTC7 rub apparently as a single entity. ble pile during cleanup opera Furthermore, the structure of WTC7 was tion, cropped from [55]. Some very different from that of the Twin Towers. almost undamaged facade walls Still according to NIST in document NC parts are clearly visible, cover STAR 1A ([25]), p. 5, in WTC7 “the layout ing some much more shattered of the columns did not align with the building debris. foundation and the Con Edison columns40. Therefore, a set of column transfers were consequence, exterior columns up to the 7th constructed within the volume bounded by floor experienced no or little shattering from the 5th and 7th floor slabs”. In short, the this shock wave, and exterior walls up to building had a larger footprint than its foun this height could not be destroyed. This dations, and up to the 7th floor exterior is consistent with what was observed as nu columns had no structural role but could merous photographs of the WTC7 ruins can rather be seen as “hanging” from the upper prove; see for instance Fig. 18 from Na part rather than sustaining it. The Federal tional Oceanic and Atmospheric Adminis Emergency Management Agency gives in a tration [55]. 2002 document [53] the foundation plan of WTC7 that we reproduce in Fig. 17. We conclude about WTC7 that both its much lower height and its very specific struc According to [53] again (p. 54), “An inte- ture can easily explain the observed dif rior braced core extended from the founda- ferences between its collapse and those of tion to the 7th floor. The horizontal shear WTC1 and WTC2, assuming the same kind was transferred into the core at the 5th and of underground nuclear explosive was used the 7th floors.”. So, if a shock wave was for its demolition. transmitted to the building from a deep un derground located explosive, it was necessar ily through this core only and not through 5 Open questions the entire footprint of the building. As a

40WTC7 was built on top of an existing Con Edison We have so far demonstrated that the en electric power substation. See for instance [54]. ergy source responsible for the huge amount

39 5 Open questions of heat released at Ground Zero after the September 11, 2001 terrorists attacks was necessarily nuclear, and that basic knowl edge about underground nuclear explosions could give us a sensible demolition mech anism for the WTC1, WTC2 and WTC7 buildings, consistent with the very common idea in the 1960s that nuclear explosives could be used not only for military purposes, but also as a peaceful and powerful engineer ing tool. In the following, we will make no demon stration but rather suggest our colleagues to investigate secondary issues that may proba bly be related to the use of underground nu clear explosives. However, having made no Figure 19: On this aerial photograph of detailed investigations by ourselves on the World Trade Center, WTC6 following questions, some of them may have building exhibits very large and little or even no relevance to the main sub deep holes (cropped from [55]). ject of our paper; we welcome in any case all efforts to clarify these items.

5.1 Accidental destruction of WTC6 September 11, 2001 in New York City were surprisingly small compared to their stand The WTC1, WTC2 and WTC7 collapses ing size. We have not investigated this were spectacular events by themselves, but question, but if the observation is exact, it some neighbouring buildings also suffered could be explained precisely by the subsi some extremely unusual damages, especially dence crater that is formed after an under WTC6 which was located between WTC1 ground nuclear explosion occurs. and WTC7. Numerous aerial pictures show very large, round holes on top of the build ing, but that appear to go very deep inside Similarly, it could be interesting to study up to ground level or even deeper. Although LIDAR pictures taken at Ground Zero af they are generally attributed to very large ter the attacks in order to detect possi parts of WTC1 falling on top of WTC6, we ble depressions originating in underground suggest that another explanation could be nuclear explosions. Such pictures have investigated: namely, that some unwanted widely circulated, one of them even being underground collapse mechanism could have the cover picture of a book: “American partially destroyed WTC6 in just the same Ground Unbuilding the World Trade Cen way it destroyed WTC1, the foundations los ter” by William Langewiesche, already cited ing support as a sideeffect of WTC1 cavity here [11]. LIDAR being commonly used in formation. geodesy and seismology (among other uses) to produce accurate measurements of alti 5.2 Surprisingly low debris pile tude, it should be possible to know if ground level around WTC1, WTC2 and WTC7 ex Many observers noted that the debris piles hibited some anomalies that could be at for the 3 buildings which collapsed on tributed to collapse chimney formation.

40 5.4 Seismic signals of characteristic type

trolled demolition methods had been used. These evidences for severe damage, if not to tal destruction of some of the underground structures of the World Trade Center imply that some destructive device was necessarily located underground, since a building col lapse per itself can not produce such an ef fect.

5.4 Seismic signals of characteristic Figure 20: One of numerous LIDAR pictures type that were made of Ground Zero. Orange levels, around WTC1 and As it is the case for any kind of power WTC2 ruins and in the WTC6 ful underground explosion, detonating a nu “big hole” on the left, are nega clear underground explosive produces a seis tive elevations (0 to 40 feet) rel mic event, the magnitude of which depends ative to street level. (EarthData, on the energy or “yield” of the explosive. September 17, 2001. Photo by As seismic signals can travel very large dis NYC Office of Emergency Man tances, and be detected thousands of kilome agement/Getty Images). ters away from the explosion location when the explosive event is large enough, seismic signals have been used for decades for de 5.3 “Bathtub” partially destroyed tecting unclaimed underground nuclear tests since the Treaty on the NonProliferation If a building is to be destroyed thanks to a of Nuclear Weapons entered into force in shock wave coming from deep underground, 1970. Naturally occuring seismic events dif one must expect underground structures to fer markedly from artificial ones, be they be severely damaged, whereas if destruction caused by chemical explosives in the case is achieved only thanks to aerial explosions of mining activity for instance or nuclear as it is the case with conventional controlled ones. demolition no aftermath is to be expected Some authors have tried to give a relation underground. Precisely, in the case of the beween the explosive “yield” (energy) and World Trade Center we have some evidence the seismic magnitude which results from that the slurry wall around the complex, the 41 the explosion. For instance Teller et al. [30] socalled “bathtub” , was severely damaged propose in The constructive Uses of Nuclear [56] at least near WTC2 southwest facade, Explosives (p. 300): although it was not directly in the footprint of any of the three collapsed buildings. Fur M = 3.64 + log10 Y (67) thermore, there is also some evidence [57] that the basement levels of the Twin Tow where Y is the yield in kilotons and M the ers were totally destroyed, which would be magnitude from a WoodAnderson torsional impossible if, as it is often said, collapse seismograph. Applying this relation to a 80 had been triggered by thermal weakening of kt nuclear device would give M 5.5. Other ≈ the structures, but also if conventional con authors like Argo et al. [58] give the follow ing rule: 41which was built in order to prevent water from Hudson River, permeating through the soil, to “A 1 kiloton (kt) explosion close- flood underground levels. coupled in hard rock is roughly

41 5 Open questions

equivalent to a seismic magnitude any detectable seismic signals. The only ef of mb 4.0: a 1 kt explosion decou- fect a physicist would expect is a vibration pled (that is fired in a large cavity) motion of the buildings, which are basically is equivalent to around mb 2.0.” cantilever beams, at their natural frequency. It appears that formula 67 gives M = 3.64 for a 1kt nuclear device, which is consistent 5.5 White fumes caused by water with above rule. However Argo et al. let a evaporation/condensation much wider range open, since a difference of 2 units in magnitude, which is a logarithmic As it has been said before, fires lasted for scale, is indeed a very large one. more than three months at Ground Zero, which in itself deserves an explanation. But There has been a comprehensive work on what appeared to emerge from Ground Zero seismic waves recorded on September 11, into the atmosphere were not mainly fire 2001 by Kim et al. [59]. However, this paper smokes, which are usually rather black es gives a maximum seismic magnitude (M ) of L pecially in the case of oxygenstarved fires, 2.3 for WTC1 collapse, of 2.1 for WTC2 col but white fumes which looked very much like lapse and does not address the case of WTC7 clouds of water condensation. This is ex collapse, although some minor but signifi actly what is to be expected from a zone cant signals (M = 0.9 and M = 0.7, in L L where coexist a huge heat source and large chronological order) were recorded for what amounts of water in the soil, coming from authors say to be airplane impacts. These Hudson River, especially once the “bathtub” 2.1 and 2.3 magnitudes, even in the high de was partially destroyed. In order for fumes coupling hypothesis, is clearly inconsistent to be white, small droplets or solid parts that with the underground detonation of a 80kt constitute it must be transparent, and there nuclear device, or even a 50kt one. fore it excludes carbonrich particles which We consider so far this paper as inconsis are generally found in fires aerosols, which tent with our findings, and would appreciate are black. comments and further work by seismology specialists, especially regarding WTC7 col Geysers springing from the street lapse records. Despite some efforts, we did Even more conclusive, some large gey not manage to get several independent sci sers springing from the ground level have entific papers addressing seismic waves gen been recorded on video during the collapse erated by September 11, 2001 events in New of WTC2 ([61] at 9:51). This observation, York City, and as science needs always to be which is strange if one accepts the idea reproducible, we consider this situation as of a “natural” collapse or even that of a unsatisfactory. conventional controlled demolition, becomes Moreover, we consider that Kim et al. paper much less mysterious if one knows that an raises some questions for which we have no extremely large heat source resides under answer; for instance, even with modest mag ground, and that water is present both be nitudes of ML = 0.9 and ML = 0.7, the so cause of soil humidity and of New York City called “plane impacts” seem to defy the laws underground water supply system. It is of physics, since it seems extremely unlikely therefore not surprising that some accidental that even a big airliner hitting horizontally contact between extremely hot temperature 42 at full speed a skyscraper can give rise to material and water can produce a powerful vapor eruption. 42Some observers, especially professional pilots, have noticed [60] that full speed was not attain We believe this aspect should equally be investi able at sea level in any case for these planes. gated, but it is not the subject of our work.

42 5.7 Extremely rapidly corroding steel remains

5.6 Extremely large amount of dust, that was not mainly concrete dust but also steel dust

A striking fact about the collapses that oc curred at the World Trade Center was the extremely large amount of dust generated, which particles were particularly small and consequently deposited onto the streets only very slowly. It was also a rather dark dust, as numerous photographs have pictured it, and a short “night” was observed at street Figure 21: Some orange fumes observed at level in NYC just after WTC1 and WTC2 Ground Zero, from [63] collapses, even they did occur in the morn ing of a sunny day. Apart from foundations, concrete was a mi nor component of the three skyscrapers; Zero. It is of course not surprising that un lightweight concrete was used in WTC1 and protected steel rusts over time, especially in WTC2 as a 10 cm thick slab (see NIST re a moisty and hot environment which was the port [24] p. 10) at each floor, mainly for case here. However, the pace of the process acoustic reasons. was described as surprising, and may be ex plained by a side effect of a nuclear explo Although producing steel dust usually in sion, namely gamma rays production. It is volves abrasion rather than explosions, we known for long that gamma irradiation in suggest that the extremely sharp wavefront an air/water environment can produce nitric generated by a nuclear explosion, compared acid because of radiolysis of nitrogen gas and to a chemical one, could produce on steel water; see for instance Etoh et al. [62]. some effects that are usually encountered on fragile materials like concrete. A fre quency analysis (Fourier transform) of a nu clear shockwave will contain much higher There were huge amounts of water present at frequencies that that of a chemical explosion Ground Zero after the attacks, both because shockwave. Given that submitting materi of firefighters action and because of a partial als to higher frequencies is generally equiv destruction of the “bathtub”, or slurry wall, alent to lowering temperature (something which is in itself a significant event as ex well known in the field of polymer physics as plained above. If a large amount of gamma timetemperature superposition principle), radiation was produced because of a nuclear we suggest that the possibility for a nuclear explosion, a subsequent production of nitric underground explosion to produce fine parti acid was unavoidable. As nitric acid is a cles of steel because of the shockwave should known factor of rapid corrosion for struc be investigated. tural steels, we suggest that fast corrosion could be a side effect of nuclear reactions that occurred on the site. Furthermore, ni 5.7 Extremely rapidly corroding steel tric acid fumes are known to be orange, and remains this was precisely the colour of some fumes that were observed at Ground Zero in the Number of observers have reported an ex first days after the attacks. Colour itself is tremely fast corrosion process occuring at no proof of nitric acid presence, of course, so the surface of steel remnants at Ground we consider this idea as an open question.

43 5 Open questions

5.8 Tritiated water and removal with the water flow. Our modeling suggests that such a Traces of tritium, in the form of tritiated scenario would require a minimum water, (HTO) have been found at Ground of 120 equipped weapons destroyed Zero [14] although this particular chemical and a quantitative capturing of element is usually considered as a signature tritium, which is too high, since of the occurence of a nuclear reaction, and in many weapons were found with any case the result of human activity. Ccos only minor damage and tritium mic rays do produce tritium when interact sights are shielded in a metal. ing with atmospheric gases, but this effect Therefore, such a mechanism is not enough to explain what was found at alone is not sufficient to account Ground Zero. for the measured HTO concen- The possible sources for tritiated water was, trations. This indicates that the according to authors, tritium radiolumines weapons/watches are consistent cent devices (emergency signs) that were with the missing source, which supposed to be present in the 2 planes that would have complemented the presumably hit the Twin Towers (WTC1 airplane source.” and WTC2), watches belonging to the air According to us, this does not sound like planes passengers and possibly, weapons be an objective, scientific explanation of a phe longing to federal and lawenforcement agen nomenon but rather like an attempt to make cies, accidentally destroyed at the WTC. data stick to a preconceived theory. We However, their conclusion, that we report would suggest to investigate the possibility below, seems not really convincing: of a nuclear explosion source, and not to “The modeling implies that the limit this investigation to tritium but to all contribution from the aircraft alone fission or fusion byproducts. would yield the HTO deposition fraction of 2.5%. This value is too 5.9 Cancer epidemy among first high by a comparison with other in- responders cidents involving fire and tritium. Therefore, the source term from Sadly, it appears that a very large number the airplanes alone is too small of first responders and workers that were to explain the measured concentra- breathing Ground Zero atmosphere during tions, and another missing source the weeks following the attacks developped is needed. cancers. According to an August 2018 New There is evidence that weapons York Post article [64], “nearly 10 000 people belonging to federal and law- have gotten cancer from toxic 9/11 dust”. enforcement agencies were present and destroyed at the WTC. Such As we have no medical competence, we will weapons contain tritium sights not elaborate on this but suggest that a nu by design. The exact activity of clear origin of the diseases should also be in tritium from the weapons was not vestigated, as there are many radioinduced determined. The data and model- cancers. ing are consistent with the tritium However, it should be noted also that an un source from the weapon sights derground nuclear explosion does not pro (plus possibly tritium watches) in duce an extremely large amount of radioac the debris, from which tritium was tive fallout, especially for contained explo slowly released in the lingering sions. As we already pointed out in a foot fires, followed by an oxidation note on subsection 4.2.1, Teller et al. wrote

44 5.10 Visible cavity when cleaning Ground Zero in their 1968 book [30] (p. 3), suggesting to use fusion rather than fission devices to minimize health problems:

“The fusion explosions, however, can be handled in such a way as to eliminate most of the ensuing residual activity.”

5.10 Visible cavity when cleaning Ground Zero

Although the idea of a nuclear destruction of the World Trade Center has not, to our Figure 22: Location of 2 large depressions knowledge, been reported in the scientific lit found at Ground Zero, according erature so far, it is not new in itself and to Moss and Merguerian [66](Fig. has been claimed for at least ten years by 8 p. 8). Black line and arrows number of individuals all over the Internet. indicate the location of a cross Some consider that miniaturized nuclear ex section not shown here. plosives were planted in the buildings, which is false as we have shown, and others do claim that the explosives were planted deep this could a nuclear demolition device be de underground, like we do. However they very signed in a “normal” i.e. noncriminal often consider, among the proofs that lead way, since radioactive contamination had to to such a conclusion, the fact that strange be confined as much as possible; although cavities were found at Ground Zero during Teller et al. [30] do not specifically address the cleaning process and before new build the case of nuclear explosives used for high ing constructions, and they claim these cav rise building demolition, it can be logically ities were indeed the initial cavities left by induced from their book that in the mind of the underground explosives, which contra some engineers at this time the 1960’s it dicts reports made by newspapers like the was not a taboo and could even be seen as New York Times [65], or more specialized an economical and efficient way to achieve academic papers [66]. such a complex work. We claim here that there is no chance these Furthermore, we would like to point out the cavities could be the ones left by the un obvious fact that if the plunge pools and derground nuclear explosives, and that on potholes found at Ground Zero were rem the contrary, in order to admit the idea that nants of previously achieved demolitions, three buildings were demolished using un they would necessarily be located under the derground nuclear explosives at the World three buildings that collapsed on September Trade Center, one must understand that it 11, 2001. This was not the case, as can be was necessary for the cavities to be pro found in Moss and Merguerian paper [66]; duced deep underground and never to open they were found beneath the Tower 4 site, onto the ground surface or even close to it. i.e. a place where was previoulsly standing Such cavities had necessarily to be filled with a small 9story office building, WTC4, and rubble after the collapse of the cavity roof where it was necessary to dig deeper in order and subsequent formation of a rubble chim to secure the foundations of a new and much ney, as exposed earlier. Only because of higher (297 m) building, directly onto the

45 6 Concluding remarks bedrock. There is consequently no mistery A similar danger comes from higher math in the fact that these large depressions re ematics methods when they are used not mained unnoticed during the first construc because they are really necessary, but as tion of the World Trade Center. There is a secondbest solution when no simple ex also no need to suspect some geologists to planation has been found, or worse, when hide the truth and to be part of a plot the authors wants to hide some question against American citizens. able aspects of their demonstration. If it is true that structural engineers need some refined mathematics methods or powerful 6 Concluding remarks computer simulations to get a precise solu tion of some complex problems, it is also true that even the most complicated calculations 6.1 The slippery slope of higher rely on simple physics laws, like Newton’s mathematics and computer laws of motion, not the other way round. simulations refinements And as a consequence must reflect the cause, any obviously invalid conclusion drawn from We have shown in this paper that physics, cumbersome calculations like violation of even at its basic level, can be an extremely energy conservation must be rejected in the powerful tool for solving complex problems, first place, without any need to “check the provided it is used bearing in mind that no equations”. As our Ph. D. advisor put it, computer simulation, no fancy mathemat “Never make a calculation if you don’t know ics gan give a human mind a better under in advance what it gives.”. Although it can sanding than a rigorous and simple appli sound a bit rough, it is basically correct. cation of its fundamental laws expressed in natural, human language such as, in our 6.2 Coming back to “simple” physics case, mainly first and second laws of ther modynamics, and the fundamental origins of as our glorious ancesters did chemical and nuclear energies. Recently, the much celebrated physicist We live in a time where mathematical tools Stephen Hawking passed away and it has and computing power especially the latter been recalled that his life and scientific activ can be both extremely complex and power ity, much longer than what was anticipated ful, which opens new research fields or tech because of his illness, had been only possi nical applications that were some decades ble thanks to a huge level of technical assis ago only to dream about. This is not to be tance and computerized achievements, such regretted, just the opposite, but scientists as a speech machine he could control with must now face the idea that some of their tiny cheek muscle contractions. This is cer activity has turned into something different tainly true, but only part of the story. As from rational thinking, with high risks of French anthropologist of science and tech producing a new kind of “magic thinking” or nology Hélène Mialet showed [67], Hawking obscurantism. The fact that virtual reality managed to live and to be scientifically pro is less and less discernable from real world ductive such a long time thanks to an ex should be a real concern since it tends to ob tremely important human support he was scure understanding and make people sci by no way “alone” and also because he man entists or not believe some events are real aged to formulate problems in a rather sim whereas they can be mere fabrications that ple, visual form; PenroseCarter diagrams violate very simple and thoroughly checked were an especially useful tool for him, since laws of physics, and that were planned by he was unable to use his hands to write down illintentioned individuals to fool others. even with the help of a computer com

46 plicated equations and to solve them. So consider it false but to think investigating even the “pure mind” Stephen Hawking had the question was not worthy of their skills to find shortcuts in demonstrations, and to rather to be seen accepting discussion with practice physics in a rather direct and in “weird” people. This is a wrong way to do tuitive way rather than by solving pages of science, since it introduces some psychologi equations. cal criteria does this subject really deserve my expertise? Will I look ridiculous if I ac cept to work with these people? where only 6.3 Power of physics, which is not rigorous thinking and approved knowledge psychology nor politics, comes should prevail. For a physicist, conservation from its “coldness” and rigor of energy is not optional.

Richard P. Feynman was famous not only Science is defined by its method, not by its for his scientific contributions and excellent subject. The unavoidable specialization of teaching skills, but also for his ability to be scientific research has crumbled the scientific “politically incorrect” as we would say to community into numerous peoples that use day. Being a physicist, he considered physics different tools and can hardly speak a com as the queen of sciences, which was maybe mon language, but the very basis of their a personal view, but he was right when he work human language, which conditions pointed out that some demonstrations in logic is the same. We hope this work other sciences do not fulfill the logical rigor will help scientists all around the world, and criteria that a physics demonstration is sup especially physicists, remind that the most posed to have. He was therefore considered valuable part of their work resides in a rig sometimes as contemptuous with respect to orous use of fundamental knowledge, and other fields like psychology, which he denied not in fancy calculations or computer simu the label of “true science”. lations that remain obscure for most people and maybe even for themselves. But he could probably give us a lesson to day, as a whole scientific community, with a few exceptions a notable one is a paper published in Europhysics News by Jones et 7 Acknowledgments al. in 2016 [7] has done almost nothing to question the dominant narrative around the It would be particularly unfair to end this terrorist attacks of September 11, 2001, even article without telling who is at the origin of though it was obviously false from numerous our investigation. As we already said, num scientific aspects. How was this possible? It ber of people, who do not pretend to be sci is in our opinion a very interesting question, entists for most of them, have proposed for but not a physics one, ant this is why we will years a nuclear demolition explanation for let others try to answer it. the World Trade Center disaster. However, We will only give a hint which is by no way to our knowledge, only one person originated a proof: professional scientists love to solve the “deep underground nuclear demolition” problems and hate to stumble over a demon scheme: a man named Dimitri Khalezov, stration. They also do not want to be seen as who pretends to be a former soviet officer obscurantists, and it is true that among the specialized in nuclear weapons. He has also written a huge “report” of more than 1000 people that question the dominant narrative 43 we can find a large number that have only pages he pretends to be a “witness”, not a faint idea of what science is. Therefore, 43Its title is “9/11thology: The “third” truth about as irrational as the “official story” was, they 9/11 or Defending the US Government, which preferred to consider it true or, at best, to has only the first two...”.

47 7 Acknowledgments a “conspiracy theorist” that he has made do acknowledge that his attitude is at the available for download on a website [68]. same time openminded, benevolent and We have read this report (July 2013 ver brave. Were only a small fraction of profes sional physicists in the world be able of such sion), with the exception of its appendices. 44 Although it contains some errors (regard a “Mut zur Wahrheit ” attitude, would the ing steel thickness in WTC1 and WTC2 9/11 mystery be entirely solved and would columns, for instance, or the fact that it con physics as a science gain enormous sympa siders the “WTC potholes” as nuclear cavi thy and respect. ties), we consider it gives a rather correct ex Finally, we would like to thank deeply our planation of the way 3 skyscrapers were de “old” friend François Sebesi who, although molished on September 11, 2001. On other not a scientist according to academic crite aspects however, this report might be con ria, has produced a great deal of scientific sidered as utterly unscientific and we do not discourse in order to draw his fellows in claim to agree with all its content. cluding ourselves out of a big collective il But interestingly, our first appreciation of lusion. this work was that it looked entirely unbe lievable, especially because Dimitri Khale zov gives the yield of the nuclear devices: 150 kt each. It was precisely this feeling that made us performing a backoftheenvelope calculation based on energy conservation, and to find out that, at least at the order of magnitude level, he was right if one con siders the first law of thermodynamics to be valid. This was for us a lesson: never call someone “stupid” before checking what he (or she) says. Some other people, like a German citizen named Heinz Pommer, who has also a back ground in nuclear energy (as a graduate physicist), have resumed and developped the idea of underground nuclear devices as an at tempt to fully explain the strange phenom ena that occurred at the World Trade Center on September 11, 2001. Heinz Pommer has made a great deal of effort making his work public for a large audience, and maintains currently two websites, one in German and in English [69], and one in German, Russian and English [70]. We consider that these websites contain a lot of useful information, but also some speculative arguments that do not fulfill the criteria of scientific research, which should proceed by elimination rather than by accumulation. However, we are extremely grateful to Heinz Pommer for stimulating discussions and we 44“Courage to truth”

48 Appendix A: estimating energy released by radiative transfer

We argue in this paper that free convection is the main contributor to heat transfer in the cooling process of socalled “Ground Zero” after the terrorist attacks of September 11, 2001. Actually, since our aim is to find a lower estimate of the total amount of heat released, and since every mechanism adds a positive contribution to the total process because of the second law of thermodynamics which forces heat to flow spontaneously from a hot zone towards a colder one, knowing which mechanism is dominant is of secondary importance, and taking the wrong one only leads to underestimate the total amount of heat. However, it is worth considering separately radiative transfer, since the nonlinearity of StefanBoltzmann law (which scales as T 4) can lead to an important contribution, at least in the first part of the decay when the surface temperature is very high. For the same reason, it is not possible to take as a model an equivalent area of ground with uniform temperature like we did for free convection, but we must estimate the heat transfer rate by integrating power over some surface temperature distribution. In the following, we will assume that temperature at a given time t is distributed according to a gaussian law. Using a polar coordinate system, and taking for the origin the maximum temperature point, we use for the temperature distribution the following:

−αr2 T (r, t)= T∞ +∆Te (68) where T∞ is the ambient temperature, far from the hot zones, ∆T (which is a function of 1 ∞ time) is the maximum temperature difference with T at r = 0, and α = 2σ2 , σ being the standard deviation which is here a length measuring the spatial extension of the hot zones. We will avoid in the following to use the σ notation for standard deviation since the same letter is also the usual notation for StefanBoltzmann constant. As can be found in any standard textbook, the net radiative transfer rate between a surface of area S at temperature TS with emissivity ǫ, considered as a “grey body”, and a surrounding medium with a farfield temperature limit T∞ reads:

4 4 P = ǫσS T T∞ (69) rad S − 2 5 4
π kB −8 −2 −4 where σ = 3 2 5.67 10 W m K is the StefanBoltzmann constant. 15h c ≈ × It follows that the net radiative transfer rate for a hot surface with a radial temperature distribution T (r, t) will be expressed as the following integral over the surface, in polar coordinates:

2π +∞ 4 4 Prad (t) = ǫσ dθ T (r, t) T∞ rdr ˆ0 ˆ0 − +∞   4 4 = 2πǫσ T (r, t) T∞ rdr (70) ˆ0 −   4 Let us calculate T (r, t) according to Equ. 68:

4 4 −αr2 T (r, t) = T∞ +∆Te 2 3 4 h 4 3 −iαr2 2 −αr2 −αr2 −αr2 = T∞ + 4T∞∆Te + 6T∞ ∆Te + 4T∞ ∆Te + ∆Te      

49 Appendix A: estimating energy released by radiative transfer

Therefore the integrand in Equ. 70 reduces to: 2 3 4 3 −αr2 2 −αr2 −αr2 −αr2 I (r) = 4T∞ ∆Te + 6T∞ ∆Te + 4T∞ ∆Te + ∆Te r   2  2  2   2  3 −αr 2 2 −2αr 3 −3αr 4 −4αr = 4T∞ ∆Te + 6T∞ ∆T e + 4T∞ ∆T e +∆T e r (71) h i and the net radiative transfer rate can be written as the sum of 4 integrals:

Prad (t) = 2πǫσ [I1 + I2 + I3 + I4] where: ∞ 3 −αr2 I1 = 4T∞ ∆Te rdr ˆ0 ∞ 2 2 −2αr2 I2 = 6T∞ ∆T e rdr ˆ0 ∞ 3 −3αr2 I3 = 4T∞ ∆T e rdr ˆ0 ∞ 4 −4αr2 I4 = ∆T e rdr ˆ0

Let us now calculate each∞ of these sums, which is quite straightforward since ∞ 2 2 e−βr rdr = 1 e−βr = 1 : 0 2β 0 2β ´ − h i ∞ 3 −αr2 I1 = 4T∞ ∆T e rdr ˆ0 2 3 = T∞ ∆T α

∞ 2 2 −2αr2 I2 = 6T∞ ∆T e rdr ˆ0 3 2 2 = T∞ ∆T 2α

∞ 3 −3αr2 I3 = 4T∞ ∆T e rdr ˆ0 2 3 = T∞ ∆T 3α

∞ 4 −4αr2 I4 = ∆T e rdr ˆ0 1 = ∆T 4 8α Finally we get for the net radiative transfer rate45:

Prad (t) = 2πǫσ [I1 + I2 + I3 + I4] 2 3 3 2 2 2 3 1 4 = 2πǫσ T∞ ∆T + T∞ ∆T + T∞ ∆T + ∆T (72) α 2α 3α 8α   45Note that although we did not write it explicitly to keep equations readable, ∆T is a function of time whereas T∞ is assumed to be a constant.

50 Now to calculate the total energy transferred by radiation during the cooling process, we need to assume some particular form for ∆T as a function of time. When only linear equations are involved for heat transfer we get an exponential decay; in the case of radiative transfer we have nonlinear equations because StefanBoltzmann law scales as T 4. At the beginning of the cooling process, for high temperature differences, the relative contribution of radiative transfer will therefore be larger than at the end; the temperature decay will consequently be faster in the early times than for a purely exponential decay. However, let us try a selfconsistent calculation and suppose first that radiative transfer is a relatively small contributor compared to other mechanisms (free convection, forced convec tion, conduction): assuming this we can say that ∆T will be approximatively exponentially decaying with time: − t ∆T ∆T0 e τ (73) ≈ where ∆T0 is the maximum temperature difference at t = 0 and τ the characteristic time of the exponential. We can now calculate the energy released by radiative transfer: ∞ Erad = Prad (t) dt ˆ0 = Erad,1 + Erad,2 + Erad,3 + Erad,4 where: ∞ 4πǫσ 3 − t 4πǫσ 3 Erad,1 = T∞ ∆T0 e τ dt = T∞ ∆T0 τ α ˆ0 α ∞ 3πǫσ 2 2 − 2t 3πǫσ 2 2 Erad,2 = T∞ ∆T0 e τ dt = T∞ ∆T0 τ α ˆ0 2α ∞ 4πǫσ 3 − 3t 4πǫσ 3 Erad,3 = T∞ ∆T0 e τ dt = T∞ ∆T0 τ 3α ˆ0 9α ∞ πǫσ 4 − 4t πǫσ 4 Erad,4 = ∆T0 e τ dt = ∆T0 τ 8α ˆ0 32α Putting everything together we finally get:

πǫστ 3 3 2 2 4 3 1 4 E = 4T∞ ∆T0 + T∞ ∆T + T∞ ∆T + ∆T (74) rad α 2 0 9 0 32 0   This energy is to be compared with the heat released by free convection for the same surface temperature distribution. As stated before in Equ. 9, the convective heat flux q˙ coming from a surface can be expressed through a convection heat transfer coefficient h in the following way: q˙ = h (T T∞) S − where TS and T∞ are the surface and bulk fluid temperature, respectively. For our gaussian temperature distribution the total heat flux, or free convection heating power (let us call it Pfc) will be:

2π +∞ −αr2 Pfc = h dθ ∆Te rdr ˆ0 ˆ0 +∞ − 2 = 2πh ∆T e αr rdr ˆ0

51 Appendix A: estimating energy released by radiative transfer

∞ 1 − 2 = 2πh ∆T e αr −2α  0 πh ∆T = (75) α And finally, integrating over time we get the total amount of heat Qfc released by free convection during the cooling process: +∞ Qfc = Pfc dt ˆ0 +∞ πh − t = ∆T0 e τ dt α ˆ0 πhτ = ∆T0 (76) α We can now compare the two values from Equ. 74 and 76. Let us call RE the ratio between the energy released by radiative transfer and the heat released by free convection:

Erad RE = Qfc 3 3 2 2 4 3 1 4 ǫσ 4T∞ ∆T0 + T∞ ∆T0 + T∞ ∆T0 + ∆T0 = 2 9 32 (77) h  ∆T0  To estimate the relative importance of radiative transfer compared to free convection we must now choose some numerical values for ǫ, h, T∞ and T0. As proposed earlier (Equ. 46 −2 −1 19) we retain h 10 W.m .K and T∞ 300 K. Emissivity ǫ of a surface can vary ≈ ≈ 47 from very low values for highly polished metals (ǫ = 0.02 for polished silver at 300 K according to [15]) to values close to 1 for rough, nonconducting surfaces (ǫ 0.9 for 48 ≈ concrete and several building materials at 300 K according to the same source). Note however that emissivity can vary with temperature and is lower (as low as 0.4 for T 1000 ≈ K) for some refractory materials at high temperatures according to several sources. We choose therefore ǫ 0.8 although we could as well put ǫ 1 for this rough estimate ≈ ≈ calculation. The maximum temperature difference (both in time and in space) ∆T0 must be different from the 350 K value taken in Equ. 21 since this one was an equivalent value for a uniform temperature hot zone, a simplification which was possible because of the linearity of convective heat transfer equations. But as we showed from several sources in subsection 3.4, 5 days after the terrorist attacks some temperatures as high as 1000 K could be measured by IR radiation spectral analysis. We consequently take ∆T0 1000 300 = 700 K. ≈ − Let us now perform a numerical calculation: 3 2 2 3 4 − 3 4 1 0.8 5.67 10 8 4 (300) 700 + 2 (300) (700) + 9 300 (700) + 32 (700) R × × × × × × E ≈ 10 h 700 i 9 9 9 9 − 75.6 10 + 66.15 10 + 45.73 10 + 7.50 10 4.54 10 9 × × × × ≈ × 700 9 − 195 10 4.54 10 9 × ≈ × 700 1.3 (78) ≈ 46We used then the slightly different notation h for an average heat transfer coefficient. 47hemispherical emissivity 48hemispherical emissivity

52 Figure 23: Variation of RP as a function of ∆T . The larger the temperature difference, the more important becomes radiative transfer compared to free convection.

Given all the uncertainties of the problem49, all we can conclude is that radiative and free convection contributions to the whole cooling process are about the same magnitude, which is contradictory to the hypothesis we have made earlier for imposing an exponential decay to temperature difference in Equ. 73. However, as stated earlier, our purpose is to get an order of magnitude estimate of heat released at Ground Zero, using as simple calculations as possible. All mechanisms giving a positive contribution, we can only underestimate the sum if we choose to estimate the contribution of a mechanism which is not the dominant one. Furthermore, because free convection heat transfer equations are linear, calculation is much less sensitive to data errors than for radiative transfer and it may be safer to consider free convection mechanism instead of radiative transfer to get a total heat estimate, even if it is not the dominant mechanism, especially because we only need a lower boundary for this estimate.

Note that it is also possible to express a power ratio RP instead of an energy ratio, giv ing the relative importance of radiative transfer compared to free convection for a given temperature difference at any time of the cooling process, irrespective of the temporal evolution of the temperature difference. Dividing Equ. 72 by Equ. 75 we get:

2 ǫσ 3 3 2 2 2 1 3 R = 2T∞ + T∞ ∆T + T∞ ∆T + ∆T (79) P h 2 3 8   which we can plot as a function of ∆T to make this result more explicit. Using the same values as above for ǫ, h and T∞ we get the graph in Fig. 23.

49And especially because, although “hot spots” at T ≈ 1000 K have really been measured 5 days after the attacks, we can not ascertain that such high temperatures were distributed across the surface in a way that makes a gaussian distribution realistic.

53 Appendix A: estimating energy released by radiative transfer

Appendix B: The 1960s as the golden era of nuclear energy - a little reminder

We would like here to recall how common and “fashionable” was the use of nuclear en ergy in the 1960s that is, when the World Trade Center was erected and even for the use of nuclear explosives. This appendix is not strictly speaking of scientific nature, but rather is an attempt to prove that the so lution we found for our problem50 was not as “unthinkable” for engineers without any criminal intents as it may seem today, and that it looks even rather “rational” once put into historical context. Although we claim that our proof is a real one in the physics sense, and therefore does not need any other argument to be accepted, we think it is im portant for such a sensitive subject to make things as clear and precise as possible. Ex cept in Orwell’s dystopia, ignorance is never a strength. As said before in subsection 4.1, some large programs about nonmilitary use of nuclear explosives have been conducted both in the USA and in the USSR during the 1960s and the 1970s and even up to the 1980s in the case of USSR. These programs were absolutely not secret, but just the oppo site widely presented in mainstream media Figure 24: In 1958, French popular science as great achievements of science and tech magazine Science & Vie presents nology that could allow to carry out large several civil engineering tech projects, especially within the field of civil niques using underground nu engineering, at a fraction of the cost needed clear explosives. Depending on with conventional methods. We will below the depth of burial, different ef give two examples of such mainstream pub fects can be achieved. lications in French magazines, Sciences & Avenir 51 (in 1965) and Science & Vie52 (in 1958), two wellknown popular science mag azines. Between October 29, 1956 and November 7,

50A solution that, let us recall it here, is the only possible one given the physical limitations of en ergy carriers. 51which translates into Sciences and Future 52which translates into Science and Life

54 1956 the socalled Suez Crisis gave number of engineers and scientists a good opportu nity to begin promoting nuclear explosives as peaceful tools for large civil engineering projects. This invasion of Egyptian Sinai by Israel, followed by the United Kingdom and France, just after Egyptian President Gamal Abdel Nasser nationalized the Suez Canal, ended in a withdrawal of the three in vaders after some political pressure of both USA and USSR. The event showed that the availability of such a strategic route as the Suez Canal especially as a conduit for the shipment of oil could be questioned, since Nasser responded to the attack by sinking all 40 ships present in the canal, which re mained completely blocked until early 1957. As a consequence, it was suggested that a second canal could be made not depending on Egypt’s will, using Israel’s territory. But to carry out such a project at reasonable cost and time, only nuclear explosives appeared to be a viable solution. If the project itself was abandoned, the idea remained and gave birth to similar projects. In October 1958, Science & Vie published an article [71] where the project of a nuclearmade harbour in Alaska was pre Figure 25: In a 1965 paper, French popu sented: it was Project Chariot (also aban lar science magazine Sciences & doned later). But the paper not only men Avenir presents an example of tioned this project: it also gave numerous nuclear explosives use: “300 nu- other examples of what would be possible clear explosions to open a second thanks to nuclear explosives in a near future. Panama Canal”. The title pic Among them: closing Bering Strait in order ture is the Sedan crater, created to warm the climate of some northern terri after the Sedan nuclear test in tories and make agriculture possible53, dou 1962, which was part of the Op- bling the Canal des Deux Mers in France54 eration Plowshare program. by a second one for large ships, or creat ing a Saharian inland sea... nothing seemed impossible to nuclear explosives, which were not only seen as modern tools, but also as cheap ones. For instance, it was written that “If the port being prepared to dig in Alaska will be accessible to vessels of 90 m draft, it is for the sake of economy, and because it would cost too much to dig its basins only at

53Global warming was not, at that time, an issue. 54between Mediterranean Sea and Atlantic Ocean

55 Appendix A: estimating energy released by radiative transfer

15 m depth.”55 In 1965 popular science magazine Science & Avenir also advocated the use of nuclear explosives for civil engineering in an article [72] about opening a second Panama Canal. The nuclear techniques were explained and although such explosions had obviously not to be confined underground, the issue of radioactive fallout was said to be control lable, provided thermonuclear explosives are used: “It is obvious that only thermonuclear and not fission explosives should be used and that, in any case, it would be necessary to evacuate the population from relatively large areas.”56. Anyway, the amount of energy re quired strongly pushed towards thermonu clear explosives: depending on the option chosen, a total between 170 and 270 Mt TNT equivalent should have been necessary, using explosives between 100 kt and 10 Mt each. We will not argue further on this point and only advise readers to research the publica tions of this period in order to understand how much perception of nuclear energy has changed since, even in the case of nuclear explosives. We recall also that as we men tioned on a footnote in section 4.2, although it was made with a fission explosive, the so viet nuclear test Chagan in 1965, which re sulted in an artificial lake, was declared safe by the authorities and that a short movie showed swimmers dipping into it shortly af ter, wearing only a small swimsuit.[50]

55“Si le port qu’on s’apprête à creuser dans l’Alaska sera accessible aux navires de 90 m de tirant d’eau, c’est par raison d’économie, et parce qu’il coûterait trop cher de creuser ses bassins seule- ment à 15 m de profondeur.” 56“Il est évident qu’il ne faudrait employer que des explosifs thermonucléaires et non pas à fission et que, de toute façon, il serait nécessaire de procéder à l’évacuation de la population de zones assez étendues.”

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