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This report is a compilation of the dedassifid chapters of reports LA.-IO2O and LA-1021, which 11.a.sbeen published becWL’3eof the demnd for the hfcmmatimo with the exception d? CaM@mr 3, which was revised in MN& it reports early work in the field of blast phencmma. Ini$smw’$has the authors had h?ft the b$ Ala.mos kb(mktm’y Wkll this z%po~ was compiled, -they have not had ‘the Oppxrtl.mity to review their [email protected] and make mx.m%xtti.cms to reflect later tiitii~e

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CONTENTS

Chapter I INTRODUCTION, by Hans A. Bethe

Chapter 2 THE

Areas of Discussion Comparison of Nuclear and Ordinary Explosion The Sequence of Events in a Blast Wave Produced by a Nuclear Explosion Radiation Reflection of Blast Wave, Altitude Effect, etc. Damage Measurements of Blast

POINT SOURCE SOLUTION, by John von Neumann Introduction Analytical Solution of the Problem Evaluation and Interpretation of the Results

Chapter 3 THERMAL RADIATION PHENOMENA, by John L. Magee and Joseph O. Hirschfelder 3.1 Radiation Hydrodynamics: the Radiation Flux’ 3.1.1 The Hydrodynamical Equations Imluding

. Radiation 3.1.2 The Radiation Flux 3.1.3 Steady State Smearing Out of a Shock Front by Radiation 3.2 The Strong Blast Without Radiation 3.3 The Opacity of Air

Page 11 11 11

Chapter 5

.

.. -

Chapter 6

CONTENTS (Continued)

ASYMPTOTIC THEORY FOR SMALL BLAST PRESSURE, by Hans A. Bethe and Klaus Fuchs

Introduction Acoustic Theory General Theory Second Approximation The Motion of the Shock Front Results for Very Large Distances The Energy The Propagation of the Shock at Intermediate Distances The Negative Phase. Development of the Back Shock The Case of Two Pressure Pulses Up with Each Other The Continuation of the IBM Run

Catching

THE IBM SOLUTION OF THE BLAST WAVE

PROBLEM, by Klaus Fuchs

Introduction The Initial Conditions of the IBM Run 6.2,1 The Isothermal Sphere 6.2.2 Initial Pressure and Density Distribution The Total Energy The IBM Run Results Comparison with TNT Explosion. Efficiency of Nuclear Explosion

6,7 Scaling Laws^202

CONTENTS (Continued)

Chapter 7 THE EQUATION OF STATE OF AIR, by Klaus Fuchs and Rudolph E. Peierls 7.1 Equation of State of Air 2 x 106 Degrees 7.1.1 Outline of Method

Between 2 x 104 and

7.1.2 The Partition Function of the Ions 7.1.3 The Partition Function of Free Electrons 7.1.4 Degree of Ionization 7.1.5 The Free Energy, Thermal Energy$ Entropy, and Pressure 7.1.6 The Hugoniot Curve

Chapter I

Chapter 3

7.2 The Equation of State of Air Below 25,000% 7.2.1 The Range from 12,000 to 25,000”K 7.2.2 The Range from 300 to 12,000”K 7.3 The Equation of State of Air at Low Pressures 7.3.1 Introduction 7.3.2 The Composition of Air 7.3.3 The Thermodynamkal Quantities 7.4 Collected Results 7.5 Approximate Form of Adiabatic for IBM Run of Nuclear Explosion 7.6 Approximate Form Used for IBM Run of HE Explosion

ILLUSTRATIONS

INTRODUCTION

1.1 Path of Acoustic Waves Starting from the Center ~’0~~

THERMAL RADIATION PHENOMENA 3.1 Itosseland Mean Free Path 3.2 Rosseland Mean Free Path

ILLUSTRATIONS (Continued)

6.13 Pressure vs Distance at a Fixed Time 6.14 Deviation of the Positive Phase of a Pulse at a Fixed Distance 6.15 The Positive Impulse and the Energy in the Blast 6.16 Efficiency of Nuclear Bomb as Cornpared to an Equivalent Charge of TNT

Chapter 7 THE^ EQUATION OF STATE^ OF^ AIR

7.1 Degree of Ionization 7.2 Hugoniot Curve; Density vs Pressure 7.3 Hugoniot Curve; Temperature as a Function of Shock Pressure

TABLES

Chapter 2 THE^ POINT^ SOURCE SOLUTION

2.1 x/xshock 2*2 x/xshock

2.3 ~/~shock 2.4 U/Ushock

2.5 P/PShock 2.6 ei/6i shock = T/Tshock 2,7 E#c shock 2.8 Some Other Useful Quantities

Chapter 3 THERMAL RADIATION PHENOMENA 3.1 Data for Opacity Calculation 3.2 Values of Number of Electrons per Cubic Centimeter in Air 3.3 Some Properties of the Blast Wave During Radiation Expansion Period 3.4 Some Properties of the Blast Wave During Radiation Expansion Period

TABLES (Continued)

3.5 Effect of Shielding

L Chapter 4 APPROXIMATION^ FOR SMALL^ y -^1

Chapter 7

THE

7.i

Variation of Ratio of Internal Pressure to Shock Pressure with Gamma

EQUATION OF STATE OF AIR Ionization Energies Energy Levels of Nigrogen and Oxygen Ions The Integrals I(Q) and J(a) The Integrals LV The Degrees of Ionization and Number of Free Electrons Pressure, Energy, and Entropy The Adiabatic The Shock Wave Curve Partition Functions of Molecules and Atoms Low-Density Equation-of-State Data Pressure vs Temperature for Various Compressions Properties of Air Along the Hugoniot Curve Thermodynamic Properties of Air at a Pressure of 1 bar Properties of Air Along the Adiabatic Terminating in the Hugoniot Curve Thermodynamic Properties of Air Properties of Air Along the Adiabatic Rate of Change of Shock Pressure with Entropy Smoothed E@ation of State of Air Values of V., VA, a, and ~ as Functions

of ps/pQ D “ 300

——.——

Chapter 1

INTRODUCTION

by Hans A. Bethe

1.1 Areas of Discussion

Ih this report the general phenomena connected with a blast wave in air will be discussed. The particular features of the blast wave produced by a nuclear explosion will be emphasized, but many of the developments in this volume will apply generally to blast waves produced by any type of ex- plosion. In this introductory chapter we shall try to give a general idea of the various phenomena occurring in a blast wave in air, of their interrelation and their time sequence. In the following chapters the details of the theory will be given, including curves showing the pressure distribution as a func- tion of time and position. We have not included any detailed discussion of the effects of sn atomic bomb other than the blast effect. (^) Only a short discussion of other effects is given in Chapter 3. For further discussion, especially on flash burn and radioactivity, reports on the experience in Japan should be consulted.

1,2 Comparison of Nuclear and Ordinary Explosion

The main difference between a nuclear explosion and^ that^ of an^ ordi- nary explosive (denoted here by HE) is that in the first case the energy is

.. developed^ in^ a very^ much^ smaller^ space.^ The^ temperatures^ reached^ are consequently very much higher, namely of the order of fifty million degrees inside the active material, as compared with about five thousand degrees in- .. (^) side the HE. This has several consequences in making the two types of ex- plosion differ.

TIM most important of these is that a nuclear explosion can be con- sidered more nearly as a point source of energy than an HE explosion, The solution for the blast wave originating from a point source is particularly simple (see Section 1.3 and Chapter 2), so that the blast wave from a nuclear explosion is simpler to treat than that from an HE explosion. lh the case of an HE explosion it takes a considem.ble time until the energy is trans- ferred from the HE to the surrounding air, owing to the great difference in density between these two media. During all this period the pressure in the air shock wave is Iess than it would be for a point source explosion liberat- ing the same energy. This phenomenon was already pointed out by G. I. Taylor in his first treatment of the shock wave produced by a point source explosion. He found that the experimental pressures at small distsnces were lower, and decreased more slowly with increasing distance, than his theory indicated. The effect discussed in the last paragraph is reversed when the over- pressure in the blast wave has decreased to the order of an atmosphere or less. This is due to the irreversible dissipation of energy (increase of en- tropy) at the shock front. This dissipation is greater the higher the shock pressure. As we have seen, in the initisl phase, at a given radius and at a given energy release, the shock pressure is lower in the HE explosion than in the nuclear explosion. Therefore, less energy is dissipated in the blast ,., caused by HE and, therefore, more energy is left in the blast by the time the blast pressure has become small. In other words, the energy released by HE is transformed into blast more efficiently than that released by a

nuclear explosion. The latter, on the other hand, should be just the effi- ciency calculated theoretically for a point source explosion. A further point of difference, again connected with the concentration of the nuclear explosion, is the appearance of large amounts of electromagnetic radiation in a nuclear explosion. Air temperatures of the order of a million degrees can be expected. (^) In the case of an HE explosion, even with a bare charge, the highest temperature reached in the air is about three thousand degrees. At a million degrees, radiation is very powerful and will transport energy through the air very easily, while at three thousand radiation plays only a minor role. It is, therefore, not surprising that a much larger fraction of the energy is emitted in the form of visible, or ultra-violet, radiation if we have a nu- clear explosion. (^) In the Trinity test it was measured that about 15 to 20 per cent of the total energy was emitted in the form of radiation which could penetrate the air to a distance of several miles. This radiation has appre- ciable heating effects which manifested themselves in the melting of the sand near the Trinity explosion Japan the radiation caused

to a distance of about three hundred meters. III intense heat over a large area and caused many

HE. However, the amount of solid material is much smaller in comparison with the energy release than it is in an ordinary HE bomb. Therefore, the transmission of energy to the air will be completed at relatively high pres- sures and from then on the blast wave will have in effect the same shape as for a point source explosion. Calculations have been made shtig the influence of the heavy mate~ rial in the bomb upon the structure of the shock wave in the initial stages. It has been shown that the shock pressure is first ~omewhat lower, and later somewhat higher, than it would be for a point source. This effect makes it rather difficult to obtain the energy release from measurements of the shock pressure at very early stages. These calculations show also how the solution for the shock wave approaches the point source solution when the mass of air in the shock wave exceeds the mass of the bomb. This proves the essential stability of the point source solution of Taylor and von Neumann. Our considerations so far have not taken into account the existence of radiation. However, we have mentioned earlier that the air may be heated t~ as much as a million degrees or more by an atomic bomb. At such high temperatures the effect of radiation is very great; in particular, radiation can transport energy from one piece of material to another very effectively. It is not yet clear to what extent this transport of radiation will actually o determine the transport of energy in the air surrounding a nuclear bomb. In

Section 1.4 and in Chapter 3, we shall describe the phenomena which will occur if the radiation is effective.

Fortunately, it can be shown that, whether or not the propagation of energy by radiation is important, after a certain time the properties of the air shock wave become very similar to the point source solution. This point source solution will, therefore, determine the further development of the shock wave as its radius increases and its pressure decreases. Methods were evolved to study the further development of the point source solution at lower pressures than are covered by the Taylor-von Neumann theory. The first correction which has to be applied to that theory is due to the fact that the equation of state air cannot be described by adiabatic of the form

with constant y

P = Apy

and A another

constant. The assumption of constant ~ is essential in the point source solution of Taylor and von Neumann. It is possible, however, to take into account variations of y as long as y itself does not deviate much from unity. In this case most of the material behind the shock wave is concentrated immediately behind the wave front and the

,.

problem of finding the pressure distribution can, therefore, be reduced es- sentially to a one-dimensional problem which can be solved. This method is developed in Chapter 4. New phenomena will occur when the pressure behind the shock front ceases to be very large compared to atmospheric pressure. (^) In this case the limiting form of the Hugoniot equations is no longer valid. The density behind the shock is no longer constant and similarity no longer holds for the solution. In principle, it is possible to continue calculations into this region, using the method developed in Chapter 4. However, this is quite cumbersome and it is not clear at what shock pressure the approximation will break down. It was, therefore, decided to make a complete numerical calculation for these lower pressures. A similar numerical calculation had previously been done by Penney. The present calculation was facilitated by several circumstances: (1) It was possible to use IBM computers for the integration, which reduced tedious computing work to a minimum and increased the speed of operations very greatly. (2) The calculation was done with an isothermal sphere at the center such as is expected due to the effect of radiation (see Section 1.4); there- fore, from the beginning, temperature, pressure, and material velocity were smooth functions of the distance from the origin, and none of these quantities reached unduly high values near the center. (^) In this way erratic fluctuations in the early values of physical quantities were avoided. (3) The use of mechanical devices for the integration made it possible to use fine intervals in the coordinate and in the time. (4) New data on the equation of state for air were available (see Chapter 7). The numerical calculation of the blast wave was begun at a pressure of 80 atm and continued to a pressure of 1.025 atm in the shock. Details and re suits are reported in Chapter 6. The results show clearly the for- mation of a negative phase (pressure below atmospheric) in the shock wave. The shape of the pressure wave is very similar to the observed shape. As the overpressure in the shock wave decreases to a small fraction of 1 atm, it is possible to use more approximate methods to describe the further development of the shock wave. This theory is a correction of or- dinary acoustic theory and is described in Chapter 5. From this theory it is possible to calculate the asymptotic behavior of a shock wave at very large distances. As in all calculations on shock waves it is assumed that the shock front is infinitely sharp, i.e., that the viscosity of the air is very small. When the pressure decreases to very small values, viscosity and turbulence effeots will occur which will change the shape and size of the pressure pulse.

Another difficulty is due to the very fact that radiation transport may be important in the air and may lead to an immediate loss of energy by those parts of the air which have been heated by the shock coming from the atomic bomb. Fortunately it makes very little difference to the later stages a (^) of the blast wave whether or not there is a radiative stage, lf there is,, initially, a radiative stage, the phenomena are as follows: The radiation will spread very quickly and will heat surrounding air. The heated air is practically transparent to radiation of high energy, at least as long as the air temperature is sufficiently high to remove the K-electrons from the air atoms. With normal density the temperature required for re- moval of half the K-electrons is about a million degrees; therefore, this figure was mentioned above as the critical temperature. The ionization is a very sensitive fuilction of the temperature so that an increase in tempera- ture by 20 per cent will make the ionization nearly complete. The radiation will therefore continue to spread rapidly into new mate- ri&l, and this process will stop only when the air temperature decreases to 1,000,000 degrees. (^) In other words, the radiative stage will continue until the radiation has spread over an air volume of such size that it can just be heated by the available energy of the explosion to a temperature of 1,000, degrees. (^) This volume is clearly proportional to the energy released. For

. (^) an energy release of 10,000 tons of TNT equivalent and with air initially at atmospheric pressure and O°C, the radius of this hot sphere will be about .. 10 meters. ... (^) At such a small distance of only 10 meters, the air shock and the radiation will still be affected by the details of the transmission of the shock from the bomb into air. Delay in transmission will reduce the im- portance of radiation in transporting energy. Further spread of the radiation would lead to a decrease of the tem- perature below the critical value of about one million degrees and, therefore, to considerably higher opacity of the air. The^ further^ spread^ of the^ radia- tion becomes very much slower and will, in general, be slower than the motion of the shock wave. From now on the shock front will move ahead of the radiation. We shall have a comparatively large sphere which has been swept by the shock wave, and inside of this a smaller sphere which has been affected by the radiation. The rate of spreading of the radiation decreases quite rapidly with temperature. It is a good approximation to assume that from a certain . (^) point on, the radiation does not spread at all. In this case we have a

1. A ‘ton^ of TNTT’ by definition^ releases^ an^ amount^ of energy^ equal^ to 4.185 x 1016 ergs (a million kilocalories).

sphere in which radiation has effectively equalized the temperature (iso- thermal sphere). This sphere contains a fixed amount of air, but its vol- ume continues to increase as the shock wave propagates. It is sgain a good approximation to assume that the temperature, the density, and the pressure are uniform in this sphere. In Chapter 4 a theory of the blast wave, including such an isothermal sphere, is developed. The isothermal sphere has also been taken into account in the numerical calculations at lower pressure (Chapter 6); actually these calculations are greatly facilitated by the existence of the isothermal sphere. In reality, of course, the radiation will continue to spread into new material all the time. To follow this process it is necessary to know the opacity of air as a function of temperature and density. Calculations of the opacity are reported in Chapter 3. (^) Our present knowledge of the opacity is still very incomplete because the calculations are extremely complicated. However, we have attempted to obtain at least an approximate theory of the propagation of radiation, which is also reported in Chapter 3. TM theory is in qualitative agreement with our previous assumption that the spread of the radiation sphere is slow compared with that of the shock wave after the temperature falls below about one million degrees. The phenomena so far described determine the distribution of tempera- ture and density inside the shock wave. From the practical point of view, e however, the most important radiation problem is the emission of radiation which can penetrate to large distances. In order to penetrate through ordi- nary cold air, i.e., through the air outside the shock wave, radiation has to

be of long wave length. Nearly all radiation up to an energy hv = 7 ev is transmitted by air without absorption. Between 7 ev and the ionization po- tential of about 15 ev there are many absorption bands which absorb very effectively. There are some gaps of low absorption so that the effective limit of transmission of air lies between 7 and 15 ev, but closer to the lower limit. All radiation of higher energy than 15 ev is absorbed by air suffi- ciently effectively to prevent transmission over distances of more than a few centimeters. Therefore, even though the temperature of the hot air near the atomic explosion is extremely high, corresponding to a spectrum predominantly in the X-ray region, the radiation going out to a distance will still be confined to the visible and the near ultraviolet. This feature of the absorption by cold air reduces enormously the total amount of radiation which can be emitted by the hot air. It all types of.^. radiation could be transmitted, the energy radiated would be proportional to the fourth power of the temperature, and the energy loss by radiation would. soon exhaust the energy the high opacity of cold

supplied to the air by air for high frequency

the explosion. radiation is a

In this sense (^). necessary