Large-Amplitude Shock Waves in Gases

Ya. B. Zel’dovich and Yu. P. Raizer

Introduction

The phenomena occurring in shock waves are of great interest from many points of view. We encounter shock waves in such practically important processes as explosions and the motion of bodies at supersonic speed.

Fundamental interest is connected with the distinctive nature of compression in a shock wave, which occurs very rapidly and, being accompanied by a sharp increase in the entropy of the gas, is irreversible. During compression, high temperatures are reached, considerably higher than in adiabatic compression to the same pressure. Still higher temperatures arise in interactions of shock waves—their reflection from a solid wall and collisions of waves.

This makes shock waves one of the most important laboratory methods for obtaining high temperatures for the study of the thermodynamic properties of gases and a whole series of phenomena occurring at high temperatures: dissociation, chemical reactions, ionization, and emission of light.

Of very great importance is the possibility that opens up of studying not only thermodynamically equilibrium states, which occur behind the front of a shock wave, but also the kinetics of the processes that take place as the gas passes through the thin layer of the wave front and that determine the internal structure of the latter.

Ever since the time of Mach, who in 1889 first experimentally recorded, with the aid of shadow photography, the density jump at the front of a shock wave1, researchers have been attracted by the problem of studying the internal structure of the front. An especially large number of theoretical and experimental works in this direction, both in our country and abroad, appeared after the war, in connection with the rapidly developing experimental technique that makes it possible to obtain shock waves of very large amplitude, and with the increased interest in phenomena occurring at high temperatures.

Another, purely hydrodynamic, direction, which has enormous applied significance, is also developing with extraordinary vigor.

In the Soviet Union, a very large amount of work has been done on the hydrodynamic theory of the propagation of shock waves, especially in explosions: this includes the works of S. P. D’yakov, N. E. Kochin, M. A. Lavrent’ev, L. D. Landau, D. E. Okhotsimskii, M. A. Sadovskii, L. I. Sedov, K. P. Stanyukovich, F. I. Frankl’, S. A. Khristianovich, and others. The works of G. N. Abramovich, G. M. Bam-Zelikovich,

S. Z. Belenkii, L. A. Vulis, A. A. Grib, V. A. Prokof'ev, L. I. Shchelkin, and others.

In the present article we leave entirely aside the technically important questions of supersonic flow around wings with shock waves, etc.

The subject of the article is the consideration of various physical processes occurring in the front of a shock wave, and their influence on the internal structure of the front. Especially great attention will be devoted to the processes of emission and absorption of light and to radiative heat exchange in the front, which chiefly determine the structure and thickness of the front of a shock wave of large amplitude, as well as its brightness.

Theoretical works concerning the question of the structure of the front of shock waves are, to a considerable extent, represented in the list of references; as regards experimental works on shock waves, the number of which is very large, the list in no way claims completeness.

§ 1. BRIEF REVIEW OF THE EXPERIMENT

There are several principal methods for the laboratory production of strong shock waves.

Investigations are at present widely carried out with the aid of so-called shock tubes \(^{2-5}\). A shock tube is a long cylinder, of the order of several meters, usually of rectangular cross section, divided by a diaphragm into two parts. Into one of them—the compression chamber—the working gas is pumped under a pressure of several atmospheres, the upper limit of which is determined by the strength of the diaphragm. Into the second—the rarefaction chamber—the gas under investigation is placed, often strongly rarefied. A shock wave arises in it upon the sudden rupture of the diaphragm, when the compressed gas rushes into the rarefaction chamber (Fig. 1). The fragments of the diaphragm are thereby pressed against the walls of the tube and have almost no effect on the flow.

Fig. 1.

The predecessor of the shock tube was an apparatus used by Ya. K. Gershanik, Ya. B. Zel'dovich, and A. N. Rozlovskii \(^{6}\) for the purpose of studying chemical reactions, in which atmospheric air flowed into a small tube containing the mixture under investigation, which was at low pressure, upon rapid removal of a stopper.

An idea of the dependence of the parameters of a shock wave on the initial pressures and the properties of both gases can be obtained from the solution of the problem of the decay of an arbitrary discontinuity \(^{7}\), which was first solved in general form by N. E. Kochin.

It turns out that the upper limit of the temperature in a shock wave, which is reached when the degree of rarefaction in the gas under investigation is very high, is proportional to the temperature of the working gas and to the ratio of the molecular weights of the investigated and working gases. The highest temperatures, of the order of \(20\,000^\circ\), are obtained when hydrogen is used as the working gas and a heavy monatomic gas, for example krypton or xenon, is used as the investigated gas. Various methods are often employed which increase the effectiveness of the tube in the sense of attaining high temperatures: the use, as the working gas, of a combustible mixture, ignit—

…present at the moment the diaphragm ruptures, reflection of a shock wave from a solid wall placed at the end of the tube, and the collision of two shock waves traveling toward one another.

Another widespread method of obtaining strong shock waves is the use of explosions. A detonation wave, propagating through an explosive substance and emerging at the boundary with a gas, creates in the latter a strong shock wave. Calculations of the motion of the expanding detonation products were made by L. D. Landau and K. P. Stanyukovich8. From the experimental point of view, it is more convenient to place a metal plate9 between the gas and the explosive; this plate is accelerated by the detonation products and pushes the shock wave ahead of it, although in this case the shock wave is somewhat weakened in comparison with the case of direct expansion of the detonation products into the gas.

Shock waves are also obtained when bodies move in a gas at supersonic velocity10–12. Very high temperatures have been achieved in this way. Thus, O. I. Leipunsky and Ya. B. Zel’dovich10, 35 were the first to observe high temperatures attained during the motion of a bullet in mercury vapor.

Methods that make use of the cumulative effect of shock waves are promising for attaining high temperatures. A. F. Belyaev13 carried out experiments on the collision of shock waves from many symmetrically arranged explosive charges and thus obtained a pressure considerably exceeding the pressure in each of the waves. Perry and Kantrovitz14 achieved high temperatures owing to the cumulative effect in a cylindrical converging wave.

A. M. Gurevich15, K. S. Vul’fson and I. Sh. Libin16, Fowler et al.17 obtained strong shock waves in studying spark discharges in gases. A theory of the development of the spark-discharge channel on the basis of shock-hydrodynamic processes was developed by S. I. Drabkina18. Koski et al.19 observed a shock wave produced by the electric explosion of metal wires. Shock waves of very large amplitude, with record temperatures of several hundred thousand degrees, were obtained in a powerful pulsed discharge in a highly rarefied gas by L. A. Artsimovich and co-workers20 in studying the possibilities of creating a controlled thermonuclear reaction. Finally, atomic explosions serve as a source of powerful shock waves in air with temperatures of tens and hundreds of thousands of degrees. Some information about them is given in an American book21.

Perhaps the most characteristic phenomenon indicating the attainment of high temperatures, of the order of ten thousand degrees and higher, is the glow of the gas in a shock wave.

The glow was discovered and investigated by many authors7, 17, 22–28. Murour27 had already noted that the glow observed in an explosion is not a chemiluminescent reaction of decomposition of the explosive substance and is not the thermal glow of the explosion products—the surrounding air through which the shock wave propagates is what glows. O. I. Leipunsky and Ya. B. Zel’dovich observed the glow in the work cited above10. Hollmer et al.22 found a glow in a wave reflected from the end wall in a shock tube filled with xenon; moreover, the number of lines in the line spectrum increased as the wave amplitude increased. At very high temperatures a continuous emission spectrum appeared. Petschek et al.23 investigated the glow of argon at a temperature of \(18\,000^\circ\), which also contained lines against the background of a continuous spectrum. The glow in argon when through it…

of a strong shock wave is used as a powerful pulsed light source ^29.

The first quantitative data on the intensity of the glow were obtained by I. Sh. Modelem ^30, who measured the brightness of the surface of the front of shock waves. As a result of the measurements he succeeded in finding the absolute value of the coefficient of light absorption in air at two temperature values of the order of 10,000°. I. Sh. Model’s measurement of the brightness of the front of large-amplitude shock waves in various gases revealed interesting regularities in the dependence of the brightness on the wave amplitude, on which we shall dwell in detail in § 7.

In investigations of shock waves, two kinds of questions are usually posed. Some of them concern the equilibrium final state of the gas behind the wave front. This is the study of the thermodynamic properties of gases at high temperatures, or the determination of the constants on which these properties depend. Measurement of any two parameters of a shock wave, with the aid of the three known relations connecting the quantities before and behind the wave front with the velocity of the front (see § 3), makes it possible to determine all the remaining parameters and to construct the shock adiabat. By comparing the experimental adiabat with the theoretically calculated one, one can draw a conclusion about the magnitude of the constants entering into the calculation, say the dissociation energies ^8,30. For example, the velocity of the wave front is measured by recording the instants at which the front passes through specified points, and the pressure (with a piezoelectric sensor) in the shock tube; or the velocity of the front and the velocity of the gas behind the front are measured (from the velocity of a plate in explosion experiments). On the other hand, knowledge of the thermodynamic properties of the gas makes it possible to determine all the parameters of the wave from the measurement of only one of them; this is used as a practical method for determining the amplitude of the wave from the velocity of its propagation in explosions or from the velocity of a body moving at supersonic speed.

The second problem in investigations is the experimental determination of the thickness of the shock-wave front; knowledge of it makes it possible to judge the kinetics of the processes occurring in the front. For this purpose chiefly optical methods are used. Three photographic methods are widely employed, in which the wave is photographed from the side with the aid of an external pulsed light source placed on the other side of the transparent windows in the side surface of the shock tube: shadow photography, the schlieren method, and the interference method ^31.

The shadow and schlieren methods are based on the change in the refractive properties of the substance in the thin layer inside the wave front, where the density of the gas, and consequently also the refractive index, changes sharply along the coordinate in the direction in which the wave moves. The regions where the density and refractive index are constant, i.e., before and behind the wave front, give uniform illumination of the photographic film; the nonuniformity corresponds only to the image of the front layer itself. The interference method is also based on the change in the refractive index in the wave front.

An interesting method was proposed by Hornig, who, together with coworkers, carried out a number of measurements ^32–34. The thickness of the shock-wave front is measured from the reflecting power of the surface of the front for light incident on it at an angle. This proves possible owing to the circumstance that the thickness of the compression jump is of the order of the wavelength of visible light. The method requires advanced experimental technique, since the reflection coefficient is extremely small: of the order of \(10^{-5}\)—\(10^{-6}\).

Shock waves are widely used for studying the kinetics of chemical reactions, since they make it possible to produce an extremely rapid

heating the reacting mixture to very high and, moreover, easily regulated temperatures. The ignition of a combustible mixture by a bullet was studied by Ya. B. Zel’dovich and I. Ya. Shlyapintokh[^36]. In the work mentioned above by Ya. K. Gershanik, Ya. B. Zel’dovich, and A. N. Rozlovsky[^6], the wave was also used to study chemical reactions. Carrington and Davidson[^37] measured the kinetics of the dissociation of \(N_2O_4\) into \(NO_2\) in a shock tube from the change in the transparency of the gas behind the wave front (\(NO_2\) strongly absorbs visible light).

§ 2. THE SHOCK ADIABAT UNDER CONDITIONS OF DISSOCIATION AND IONIZATION

Let us consider a plane shock wave (in which the gas moves normally to the surface of the front) in a coordinate system attached to the discontinuity.

As is known, the hydrodynamic quantities of the final state of a gas that has undergone a compression jump and has reached thermodynamic equilibrium (denoted by the subscript “1”) are related to the quantities of the initial state (subscript “0”) by the laws of conservation of mass, momentum, and energy:

\[ \left. \begin{aligned} \rho_1 u_1 &= \rho_0 D,\\ p_1+\rho_1 u_1^2 &= p_0+\rho_0 D^2,\\ \varepsilon_1+\frac{p_1}{\rho_1}+\frac{u_1^2}{2} &= \varepsilon_0+\frac{p_0}{\rho_0}+\frac{D^2}{2}. \end{aligned} \right\} \tag{2.1} \]

Here \(p,\rho,u,\varepsilon\) are the pressure, density, velocity, and specific internal energy. The velocity of the gas flowing into the discontinuity (\(u_0\)) will henceforth always be denoted by \(D\). It is equal in absolute value to the velocity of propagation of the shock-wave front through the initial gas. Since the internal energy of a thermodynamically equilibrium state depends only on any two quantities, for example density and pressure, equations (2.1) make it possible to find all quantities behind the front from the initial characteristics of the gas and one arbitrary parameter determining the amplitude of the shock wave, say the front velocity \(D\). The corresponding formulas for an ideal gas with constant heat capacity are derived in all textbooks of hydrodynamics.

In the limiting case of strong waves (\(p_1 \gg p_0\)), the Hugoniot adiabat \(p_1=f(\rho_1,p_0,\rho_0)\) leads to a compression value equal to

\[ \frac{\rho_1}{\rho_0}=\frac{\gamma+1}{\gamma-1}, \tag{2.2} \]

where \(\gamma=c_p/c_v\) is Poisson’s adiabatic exponent. For a monatomic gas \(\gamma=5/3\) and \(\rho_1/\rho_0=4\); for a diatomic gas with unexcited vibrations \(\gamma=7/5\) and \(\rho_1/\rho_0=6\).

With increasing amplitude of the shock wave and temperature behind the front, vibrational degrees of freedom are excited in the molecules. When the temperature becomes of the order of several thousand degrees, the vibrations are fully excited; in a diatomic gas \(\gamma=9/7\) and the compression \(\rho_1/\rho_0=8\). Calculations of the shock adiabat in the intermediate region, where the vibrations are not fully excited and the heat capacity is variable, have been carried out by a number of authors, for example for air[^38],[^42]. With a further increase in amplitude, dissociation of molecules begins, and then, at temperatures of the order of ten thousand degrees and higher, ionization.

It would seem at first glance that dissociation of molecules and transformation of a polyatomic gas into a monatomic one should lead to a shock compression equal to 4. In reality, however, dissociation and ionization only increase the density jump, since they require an expenditure of energy and therefore increase the heat capacity of the gas. The fact that an increase in heat capacity leads to an increase in compression is already evident from formula (2.2), corresponding to the case of constant heat capacity. This tendency also occurs in the general case of variable heat capacity.

The specific internal energy is made up of the energy of translational motion of the particles (atoms, molecules, electrons), which is related to the pressure by the formula

\[ \varepsilon_{\text{trans}}=\frac{3}{2}\frac{p}{\rho}, \tag{2.3} \]

the internal energy of molecules and atoms (the energy of electronic excitation, rotational and vibrational energy in molecules) \(\varepsilon_{\text{in}}\), and the energy expended on dissociation and ionization \((\varepsilon_{\text{dis}}\) and \(\varepsilon_{\text{ion}})\), i.e.

\[ \varepsilon=\frac{3}{2}\frac{p}{\rho}+\varepsilon_{\text{in}}+\varepsilon_{\text{dis}}+\varepsilon_{\text{ion}} =\frac{3}{2}\frac{p}{\rho}+Q . \tag{2.4} \]

Neglecting the initial pressure and the internal energy of the gas, we find from equations (2.1) the compression behind the front of a strong shock wave in the form

\[ \frac{\rho_1}{\rho_0}=\frac{4}{1-\dfrac{3Q}{\varepsilon_{\text{trans}}}} . \tag{2.5} \]

It follows from this that the compression is the greater, the greater the contribution to the total internal energy made by the expenditures on dissociation and ionization.

Since the statistical weight of the free state of particles is always considerably larger than that of the bound state, dissociation and ionization usually begin at such temperatures that the kinetic energy of the translational motion of particles \(\frac{3}{2}kT\) is much (by a factor of 7) smaller than the binding energy. Therefore the internal energy expended on breaking up molecules and detaching electrons from atoms, at sufficiently high temperatures, proves to be greater than the translational energy of the particles, and the compression reaches 10–12.

Hydrogen, already at temperatures of the order of \(50\,000^\circ\) and at densities corresponding to the normal initial state, is practically completely ionized. In the case of a low initial density, complete ionization occurs even earlier. Under conditions of complete ionization, the relative contribution of the ionization energy to the internal energy decreases as the amplitude of the wave increases, and the compression tends to the value corresponding to a monatomic gas, \(\rho_1/\rho_0=4\).

In heavy gases, whose atoms contain many electrons, the region of increased compression is strongly extended, since after the outer electrons are detached, the detachment of the next ones begins—second, third, etc., ionization. The compression tends to the value \(\rho_1/\rho_0=4\) only after complete ionization of the atoms has occurred. Thus, in air at normal density in the initial state, temperatures of the order of several million degrees are required for this. The magnitude of the compression in the ionization region does not remain constant: as the amplitude of the wave increases, the relative contribution of the ionization energy after

the passage of the maximum in the period of the first ionization gradually decreases, since the contribution of translational energy increases owing to the increase in the number of particles. This continues until all electrons have been stripped from some closed shell.

Between the ionization potentials of the last electron of this shell and the first electron of the next there is always a large gap. For example, in nitrogen (the \(L\)- and \(K\)-shells) this is \(97\) eV and \(550\) eV; in oxygen, \(137\) eV and \(735\) eV. Therefore, in air there exists a rather wide interval of wave amplitudes, approximately with temperatures from \(\simeq 500\,000^\circ\) to \(\simeq 700\,000^\circ\), when the stripping of all electrons from the \(L\)-shells of nitrogen and oxygen atoms has been completed, while stripping from the \(K\)-shells has not yet begun; almost all atoms have become helium-like ions. When stripping of the \(K\)-electrons begins, it requires very large energy expenditures, the relative contribution of the ionization energy increases together with the compression, which thus passes through a second maximum.

The pressure behind the front of a strong wave, as follows from the first two equations, is little sensitive to the magnitude of the compression and, consequently, to all other quantities, and for a compression of order 10 is, to an accuracy of \(\simeq 10\%\), proportional to the square of the wave velocity:

\[ p_1 = \rho_0 D^2 \left( 1 - \frac{\rho_0}{\rho_1} \right). \tag{2.6} \]

Specific enthalpy

\[ w_1 = \varepsilon_1 + \frac{p_1}{\rho_1} = \frac{5}{2}\,\varepsilon_{\mathrm{trans}} + Q \tag{2.7} \]

is, with still greater accuracy, proportional to the square of the velocity,

\[ w_1 = \frac{D^2}{2}\left( 1 - \frac{\rho_0^2}{\rho_1^2} \right), \tag{2.8} \]

whereas the temperature increases with increasing \(D\) much more slowly. In the region of the first ionization this occurs because of the relative increase in the energy expenditure for ionization, i.e. the quantity \(Q/\varepsilon_{\mathrm{trans}} \sim Q/T\); subsequently, when the share of ionization energy in the internal energy decreases in comparison with the translational energy, the slowed growth of temperature is explained by the increase in the number of particles \(\beta(T)\) per initial molecule, i.e. by the increase in the translational part of the heat capacity:

\[ \begin{aligned} \varepsilon_{\mathrm{trans}} &= \frac{3}{2}\,\beta(T)RT,\\ p &= \beta(T)R\rho T. \end{aligned} \tag{2.9} \]

The considerations set forth above are illustrated by a table, which contains the results of calculating the shock adiabat of air up to the temperature \(T_1 = 500\,000^\circ\). The values for low temperatures \(T_1\) from \(273\) to \(2260^\circ\), in the region of vibrational excitation, are taken from the book by Ya. B. Zel’dovich\({}^{42}\); in the dissociation region \(4000\)—\(14\,000^\circ\) the calculations were made by Davis\({}^{74}\). In the wide temperature interval from \(20\,000\) to \(500\,000^\circ\) C the shock adiabat was calculated by V. V. Selivanov\({}^{39}\).

Calculations for hydrogen and argon in the region of the first ionization were made by V. A. Prokof’ev\({}^{40}\) and in work\({}^{2}\) for comparison with the results of experiments in a shock tube; for xenon, in work\({}^{41}\). It should

It should be noted that in the region of comparatively low temperatures, of the order of 20,000°, the energy of electronic excitation plays an insignificant role. It is important only at high temperatures and was taken into account by V. V. Selivanov in his calculations for air. The calculated adiabats in argon and xenon agree well with experimental data.

Table

Parameters of the shock-wave front in air

As for air, in the region of dissociation, i.e., temperatures of the order of 2000–10,000°, the shock adiabat depends strongly on which of the two disputed values of the dissociation energy of nitrogen—7.38 ev or 9.74 ev—is adopted in the calculation. Christian et al.^9 showed that the experimental data (the authors measured the velocities of the front and of the air behind the front, or rather the velocity equal to the latter of the plate accelerated by the explosion, which played the role of a piston in creating the shock wave) are closer to the adiabat corresponding to the value 9.74 ev. The front velocities and temperatures measured by I. Sh. Model^30 (by the optical method) also speak in favor of this value.

§ 3. VISCOUS DISCONTINUITY OF COMPRESSION

As is known, the equations of hydrodynamics of an ideal fluid admit the existence of discontinuous solutions that describe shock waves. The hydrodynamic quantities on both sides of the discontinuity are related to one another and to the velocity of propagation of the discontinuity by the laws of conservation of mass, momentum, and energy (2.1), just as the equations of hydrodynamics are an expression of these laws in the region of continuous flow.

In a certain sense paradoxical is the fact that the equations of adiabatic motion of a fluid admit the existence of such surfaces on which the entropy, as follows from equations (2.1), undergoes a jump. The irreversibility of shock compression indicates that dissipative processes participate in it; because of them there occurs the abrupt braking of the fluid incident on the discontinuity surface (in the coordinate system connected with the latter) and the irreversible conversion of a significant part of the kinetic energy of the hydrodynamic flow into heat.

Dissipation occurs only in a very thin layer. Outside this layer, which within the framework of the hydrodynamics of an ideal fluid is replaced by a mathematical surface, dissipative processes play practically no role and the motion of the fluid is adiabatic.

If one is interested in the mechanism of shock compression, the internal structure and the thickness of the transition layer (called the shock-wave front), which separates the thermodynamically equilibrium regions of the initial and final states, one must turn to a theory that includes a description of dissipative and nonequilibrium proces-

processes. As a first step in this direction it is natural to consider a compression discontinuity within the framework of the hydrodynamics of a real fluid, taking viscosity and thermal conductivity into account.

For this purpose one must solve the problem of a one-dimensional steady flow with boundary conditions expressing the vanishing of the gradients at \(\pm\infty\) and the tendency of the hydrodynamic quantities at \(\pm\infty\) to the values corresponding to the final and initial states:

\[ \left. \begin{aligned} &\frac{d}{dx}\rho u = 0,\\ &\rho u\frac{du}{dx}+\frac{dp}{dx}-\frac{4}{3}\frac{d}{dx}\mu\frac{du}{dx}=0,\\ &\rho uT\frac{d\Sigma}{dx}=\frac{4}{3}\mu\left(\frac{du}{dx}\right)^2-\frac{dS}{dx}. \end{aligned} \right\} \tag{3.1} \]

Here \(\Sigma\) is the specific entropy, \(\mu\) is the coefficient of viscosity\(^*\), and \(S\) is the non-hydrodynamic flux of energy, which in the case of ordinary thermal conductivity is equal to

\[ S=-\chi\frac{dT}{dx}, \tag{3.2} \]

where \(\chi\) is the coefficient of thermal conductivity.

Taking into account the equation of state

\[ p=R\rho T \tag{3.3} \]

and the second law of thermodynamics

\[ Td\Sigma=d\varepsilon+p\,d\left(\frac{1}{\rho}\right)=dw-\frac{1}{\rho}\,dp, \tag{3.4} \]

it is easy to obtain the first integrals of equations (3.1)

\[ \left. \begin{aligned} &\rho u=\rho_0D,\\ &p+\rho u^2-\frac{4}{3}\mu\frac{du}{dx}=p_0+\rho_0D^2,\\ &w+\frac{u^2}{2}+\frac{1}{\rho_0D}\left(S-\frac{4}{3}\mu u\frac{du}{dx}\right)=w_0+\frac{D^2}{2}. \end{aligned} \right\} \tag{3.5} \]

The constants of integration are expressed here in terms of the parameters of the initial state of the gas at \(x=-\infty\), denoted by the subscript “0.” If equation (3.5) is referred to the final state at \(+\infty\), we arrive at the shock relations (2.1). Thus, the jump of entropy as a result of shock compression is entirely independent of the mechanism of dissipation and of the magnitudes of the coefficients of viscosity and thermal conductivity. As the solution will show, only the thickness of the shock-wave front depends on their magnitudes, measured by the molecular mean free path \(\lambda\), to which the coefficients \(\mu\) and \(\chi\) are proportional. In the limit \(\lambda\to0\), the hydrodynamics of a real fluid in the region of continuous flow becomes the hydrodynamics of an ideal fluid. As for the shock-wave front, in the limit \(\lambda\to0\) it becomes a mathematical surface of discontinuity, and the gradients of all hydrodynamic quantities in it tend to \(\infty\) as \(1/\lambda\).

Usually in gases molecular viscosity and thermal conductivity play approximately the same role, since the transport coefficients (kinematic viscosity and thermal diffusivity) are close to one another. However, the roles of these dissipative processes in the formation of a compression discontinuity

\(^*\) The first and second coefficients of viscosity are assumed equal.

...are far from equivalent. In order to be convinced of this, it is enough to consider the problem of one-dimensional steady flow, assuming that viscosity is absent \((\mu=0)\); see, for example, \(^{42}\).

It turns out that without viscosity, with heat conduction alone, a continuous solution can be constructed only in the case of sufficiently weak waves, when the shock compression does not exceed two, which corresponds to a wave amplitude \(p_1/p_0=1.5\) for a diatomic gas with \(\gamma=7/5\). In stronger waves a discontinuity inevitably arises, which shows that in the mechanism of smearing of the density jump the essential role is played precisely by viscosity, and not by thermal conductivity. This was understood already by Rayleigh \(^{43}\), who considered the other limiting case, when only viscosity is present and there is no thermal conductivity \((\chi=0)\), and obtained a continuous solution of equations (3.5) satisfying the boundary conditions. Indeed, the density jump arises as a result of the abrupt braking of the gas impinging on the discontinuity and of the irreversible transformation of the kinetic energy of hydrodynamic motion into heat under the action of viscous forces.

Below, when considering real physical processes (radiative and electronic heat conduction in waves of large amplitude), we shall return once more to the case where thermal conductivity is much greater than viscosity (in the sense that the thermal diffusivity coefficient is much greater than the kinematic viscosity), clarify the role of thermal conductivity, and show the necessity of the existence of a discontinuity.

Here, however, in studying the structure of the density jump itself, we shall proceed from the usual molecular viscosity and thermal conductivity.

In the general case the system of equations (3.5) cannot be solved analytically; the problem can be carried through to the end only in the particular case of a definite relation between the coefficients of viscosity and thermal conductivity, namely when the Prandtl number

\[ \operatorname{Pr}=\frac{\mu c_p}{\chi}=\frac{3}{4}, \]

and the viscosity, thermal conductivity, and heat capacity are constant (Becker \(^{44}\), Morduchow and Libby \(^{45}\)). In this case the expression in parentheses in the third of equations (3.5) becomes the complete differential of the expression \(w+\frac{u^2}{2}=c_pT+\frac{u^2}{2}\), and the equation is easily integrated, giving, by virtue of the boundedness conditions on the solutions at \(+\infty\) and \(-\infty\), the Bernoulli integral

\[ w+\frac{u^2}{2}=w_0+\frac{D^2}{2}. \tag{3.6} \]

All the principal regularities of the structure of a density jump are visible in this particular example; therefore we shall dwell on it in somewhat greater detail. Equation (3.6), together with the first two of equations (3.5), reduces the problem to a quadrature.

The constant of integration is additive with respect to \(x\) and, consequently, may be chosen arbitrarily. It is convenient to do this so that the point of inflection lies at the origin of coordinates.

We give the final solution

\[ \frac{1-\eta}{(\eta-\eta_1)^{\eta_1}} = \frac{1-\sqrt{\eta_1}}{(\sqrt{\eta_1}-\eta_1)^{\eta_1}} \,e^{\beta(1-\eta_1)M\frac{x}{\lambda_0}}, \tag{3.7} \]

\[ \eta=\rho_0/\rho,\qquad \eta_1=\rho_0/\rho_1. \]

Here \(M\) is the Mach number, equal to the ratio of the velocity of the wave front \(D\) to the speed of sound in the initial state; \(\beta\) is a numerical constant of order unity (for air with \(\gamma=1.4\), \(\beta=1.36\)); \(\lambda_0\) is the mean free path of the molecules in the initial state of the gas, which entered the theory through the proportional viscosity coefficient. The velocity, density, and pressure vary monotonically in the wave, having, as was to be expected, a point of inflection (Fig. 2).

The entropy passes through a maximum at this point. This does not contradict the second law of thermodynamics and does not mean, contrary to the opinion of the authors\(^{45}\), that the reason for the decrease in the entropy of a gas particle during its approach to the final state lies, perhaps, in an imperfection of the theory, namely in the neglect of certain nonequilibrium processes.

The entropy of the gas as a whole, which constitutes an isolated system, naturally increases upon passage through the shock discontinuity. An individual layer of gas passing through the wave, however, is no longer isolated: its entropy first increases when heat is supplied to it by thermal conduction and by the work of viscous forces, and then decreases, since the outflow of heat by thermal conduction toward the gas layers following it exceeds the influx due to viscosity.

Fig. 2.

The thickness of the shock-wave front, as is seen from formula (3.7), has the scale of the mean free path and decreases with increasing wave amplitude, i.e., with increasing Mach number,

\[ \frac{\Delta x}{\lambda_0}\sim \frac{1}{M}. \]

For example, in a diatomic gas with \(\gamma=1.4\), at Mach number \(M=2\) (\(p_1/p_0=4.5\)), the effective thickness of the wave front is approximately equal to four mean free paths\(^*\).

After the war many works appeared in which qualitative investigations of equations (3.5) are carried out for arbitrary Prandtl number; the influence of the value of \(\mathrm{Pr}\), of the temperature dependences of the transport coefficients, and of the heat capacity is clarified; and various approximate solutions are found for these more general cases\(^{38,45-54}\). All these investigations introduce nothing fundamentally new in comparison with the particular case considered above and are of interest only for weak waves.

In the case of not too weak waves, when the front thickness becomes of the order of several mean free paths, the hydrodynamic theory of the compression jump in general loses its meaning. The condition for the applicability of the hydrodynamic treatment of transport phenomena is the smallness of the gradients of the hydrodynamic quantities, namely that the distances over which these quantities undergo changes comparable with the quantities themselves be much greater than the mean free path. In strong waves this condition, as we have seen, is not fulfilled. An attempt to refine the hydrodynamic approximation by taking into account second derivatives in the expressions for the transport terms (the so-called Burnett approximation),

\(^*\) The effective thickness according to Prandtl is defined by the formula

\[ \Delta x=(D-u_1)\bigg/\left(\frac{du}{dx}\right)_{\max}. \]

undertaken by Tsoller\(^{55}\), somewhat refines the results for weak waves and, in essence, merely indicates the limit of applicability of hydrodynamic theory.

This limit is very low: already at a wave amplitude \(p_1/p_0 = 1.5\) the wave thickness according to Tsoller is \(\dfrac{\Delta x}{\lambda_0} \approx 17\), while at \(p_1/p_0 = 4\) it is equal to 6.1.

It is physically clear that the thickness of the compression jump in strong waves is of the order of the mean free path. It cannot become smaller than the mean free path, since the gas molecules impinging on the discontinuity must undergo several collisions in order to scatter their momentum, converting the kinetic energy of directed motion into the kinetic energy of chaotic motion, i.e., into heat. At the same time the thickness cannot amount to many mean free paths, since a molecule of the gas impinging on the discontinuity transfers, on the average, half of its momentum in each collision, and only a few collisions with chaotically moving molecules are already sufficient for it to lose its directed momentum.

The problem of the structure and thickness of the compression jump in strong waves must be considered on the basis of the molecular-kinetic theory of gases. Of interest is the attempt made in this direction by Mott-Smith\(^{65}\), who sought an approximate solution of the Boltzmann kinetic equation in the form of a superposition of two Maxwellian distributions with temperatures and macroscopic velocities corresponding to the initial and final states of the gas. The relative weight of both functions changes through the wave from 0 to 1. The thickness of the wave front, as its amplitude is increased without bound, tends in this case to a finite limit. For air under normal initial conditions the wave thickness turned out to be equal to two mean free paths at Mach number \(M = 4\).

Experimental determination of the thickness of weak waves by reflection of light from them in the work of Hornig\(^{32-34}\) gives, for monatomic gases, the best agreement with Tsoller’s calculations (for \(M = 1.1;\ 1.5;\ 2.5\), \(\dfrac{\Delta x}{\lambda_0} = 30,\ 19,\ 13\), respectively).

§ 4. BROADENING OF THE SHOCK-WAVE FRONT DUE TO DELAYED EXCITATION OF PART OF THE HEAT CAPACITY

The internal energy of a gas is distributed among different degrees of freedom: it consists of the kinetic energy of translational motion of the particles, the rotational and vibrational energy of the molecules, chemical energy when a reversible chemical reaction can occur in the gas (for example, dissociation of molecules), ionization energy, and electronic excitation energy. The excitation of each of these degrees of freedom*) requires finite relaxation times \(\tau\), which, generally speaking, differ greatly from one another. Since by the front of a shock wave we mean that layer in which the gas passes from the initial to the final, thermodynamically equilibrium state, the thickness of the front is determined by the longest of the processes and is of the order \(\Delta x \approx D \tau_{\max}\).

Most rapidly, obviously, the translational degrees of freedom are excited. The thickness of the viscous compression jump in sufficiently strong waves is, as we have seen, several mean free paths, for as a result of only a few collisions the kinetic energy of directed motion—

*) We allow an imprecision of terminology, assigning chemical energy and ionization energy also to the degrees of freedom of gases.

of molecules is converted into the kinetic energy of their chaotic motion. The time for establishment of the Maxwellian distribution, i.e., the temperature, is also of the order of the time of several collisions: the collision time for normal densities and temperatures is of the order \(\tau_{\text{trans}}\sim 10^{-10}\) sec. Measurement of the relaxation time of the rotational part of the heat capacity from ultrasonic dispersion shows that rotation in most molecules is excited, as classical theory requires, very rapidly, as a result of a small number of collisions: at temperatures close to normal, in nitrogen and oxygen 3 collisions are sufficient, in methane—10, in carbon dioxide—16. The exceptions are the lightest gases, which have a small moment of inertia and, consequently, a large rotational quantum, so that quantum effects become substantial; in hydrogen the relaxation time is 300 times greater than the time of one collision, and in deuterium—150 times greater \(^{57}\).

The time for excitation of vibrations in molecules at not too high temperatures is several orders of magnitude greater because of quantum effects: the vibrational quanta are greater than, or of the order of, \(kT\) (at normal temperature \(\tau_{\text{vib}}\sim 10^{-6}-10^{-5}\) sec). It decreases with increasing temperature, as L. D. Landau \(^{58}\) showed, according to the exponential law

\[ \tau_{\text{vib}}\sim \frac{T^{1/6}}{p}\, e^{\frac{\mathrm{const}}{\sqrt{T}}}. \tag{4.1} \]

At temperatures of the order of several thousand degrees, vibrations are excited very rapidly; the attainment of thermodynamic equilibrium in a diatomic (polyatomic) gas is mainly delayed by the reversible chemical reaction, dissociation of molecules, and at still higher temperatures, of the order of tens of thousands of degrees, by ionization.

The rates of all processes usually increase rapidly with rising temperature; therefore those processes which, at some amplitude of the wave, were slow and, in the main, determined the thickness of the front, become rapid in a wave of larger amplitude, and are replaced by new processes: for example, at \(T \approx 20\,000^\circ\) dissociation proceeds very rapidly and the thickness is determined by the first ionization (the second ionization is still very small and makes a negligible contribution to the heat capacity). At \(T \approx 50\,000^\circ\) the first ionization is replaced by the second, and so on. Only with complete ionization should the thickness of the shock-wave front monotonically decrease with increasing amplitude and tend to a value of the order of several mean free paths in the limit, when ionization occurs practically at every impact. However, at such high temperatures an essential role is played by other processes leading to a broadening of the front, which will be discussed below.

Thus, for a shock wave of any amplitude, we can divide the processes of excitation of the various parts of the heat capacity into fast and slow ones.

The question of the structure of the shock-wave front in a gas with delayed excitation of part of the heat capacity was first considered by Ya. B. Zel’dovich \(^{42,59}\), using the example of a reversible chemical reaction.

The structure of the front can still be described by the one-dimensional stationary equations of hydrodynamics (3.1). The viscosity and thermal conductivity of the gas play a role only in the region of large gradients, i.e., in the zone of excitation of the fast processes; in the zone of the slow processes, which is greatly extended, the role of transport phenomena is insignificant. But then, separating out the region of the sharp jump and denoting the hydrodynamic quantities beyond it by primes, we can write the relation

initial quantities with these intermediate ones in the form of the conservation equations (2.1), and for the stretched zone of the wave the integrals of equations (3.1), again in the form of equations (2.1), but now valid at any point \(x\),

\[ \left. \begin{aligned} \rho u &= \rho' u' = \rho_0 D,\\ p+\rho u^2 &= p' + \rho' u'^2 = p_0+\rho_0D^2,\\ \varepsilon+\frac{p}{\rho}+\frac{u^2}{2} &= \varepsilon' + \frac{p'}{\rho'}+\frac{u'^2}{2} = \varepsilon_0+\frac{p_0}{\rho_0}+\frac{D^2}{2}. \end{aligned} \right\} \tag{4.2} \]

We shall not be concerned with the zone of rapid compression, treating it as infinitely thin and placing it at the origin of coordinates \((x=0)\).

Let us plot on the \(pv\) diagram (\(v=1/\rho\) is the specific volume) the Hugoniot adiabat \(AC\) (see Fig. 3), corresponding to the attainment of complete thermodynamic equilibrium, i.e. the final state, and the adiabat \(AB\), corresponding to the excitation only of the rapid part of the heat capacity. The point in the \(pv\) plane describing the successive states of a particle in a sufficiently strong wave*) evidently jumps from the initial state

Fig. 3.      Fig. 4.

\(A(p_0v_0)\) to the intermediate state \(B(p'v')\), and then moves to the final state \(C(p_1v_1)\) along the straight line

\[ p=p_0+\rho_0D^2\left(1-\frac{v}{v_0}\right). \tag{4.3} \]

In this process the compression and the pressure increase as the final state is approached, while the temperature falls (see formulas (2.5)—(2.7)), as is shown in Fig. 4, where the profiles of density, temperature, and pressure in the wave front are depicted schematically.

It is evident from formula (4.3) that the pressure increases only slightly, since \(v'/v_0 \leqslant 1/4\); the density, however, may increase, and the temperature may fall, very substantially—by a factor of 2–3—the more so the greater the contribution to the equilibrium internal energy made by the lagging part of the heat capacity. The appreciable change in the density profile in the wave front in the case of a lag of part of the heat capacity makes it possible to use optical methods for investigating the phenomenon.

\[ \text{*) We do not consider weak waves, whose velocity is less than the speed of sound corresponding to the frozen slow heat capacity, when the straight line does not intersect the adiabat } AB \text{ (see } {}^{59}\text{).} \]

Study of the density profile by light reflection in waves with \(M<2\) showed (Hornig et al.\(^{22-34}\)) that hydrogen is compressed initially as a monatomic gas: to compress it to the final density corresponding to \(\gamma=7/5\), no fewer than 150 collisions are required. In oxygen and nitrogen the rotational degrees of freedom are excited rapidly, and the density at once increases to the value corresponding to \(\gamma=7/5\).

The broadening of the wave front due to delayed excitation of vibrations in polyatomic gases was investigated by the interferometric method by Griffiths et al.\(^{56}\)*). Measurements of the density distribution in the front of waves of different amplitudes made it possible to estimate the relaxation time for vibrations \(\tau_{\mathrm{vib}}\) as a function of temperature. The experimental points lay rather well on the theoretical straight line \(\ln \tau=a+bT^{-1/2}\) and were in agreement with measurements from sound dispersion. Thus, in \(\mathrm{CO}_2\) the relaxation time, reduced to normal pressure, proved to be \(\tau_{\mathrm{vib}}\simeq 5\cdot 10^{-6}\) sec at \(T\simeq 300^\circ\), and \(\tau_{\mathrm{vib}}\simeq 0.7\cdot 10^{-6}\) sec at \(T=1000^\circ\).

In recent years papers have appeared\(^{2,22-24}\) in which shock waves in monatomic gases, chiefly in argon, were studied. High temperatures were obtained, at which ionization plays an essential role and luminescence of the heated gas is observed. In this case the hydrodynamic quantities in the front undergo a sharp jump over a distance of the order of the mean free path to the values

\[ \rho'=4\rho_0,\qquad p'=\frac{3}{4}\rho_0D^2,\qquad T'=\frac{3}{16}\frac{D^2}{R}, \]

after which there follows a very extended region in which ionization equilibrium is reached and the degree of ionization \(\alpha\) increases from 0 to the equilibrium value \(\alpha_1\) corresponding to the final state. In this region the temperature falls from \(T'\) to \(T_1\); moreover, in a wave of sufficiently large amplitude, behind whose front the degree of ionization is appreciable, this drop is quite considerable. For example (according to the calculation of Resler et al.\(^{2}\)), at an initial gas pressure \(p_0=50\) mm Hg and Mach number \(M=18\),

\[ \alpha_1=0.185,\qquad T'=31\,500^\circ,\qquad T_1=15\,700^\circ . \]

In the experiments of Petschek\(^{23}\), in waves of comparatively small amplitude the region of gas behind the shock-wave front, at a high degree of purification, when the luminescence of dust and metal vapors was excluded, did not shine; only a layer of thickness of the order of 100 mean free paths, situated immediately behind the wave, shone. Evidently the front itself was luminous, i.e. the nonequilibrium region, where the temperature was significantly higher than \(T_1\) and was sufficient for luminescence. In stronger waves the region behind the front also shone.

To calculate the density and temperature profiles in the wave front with allowance for ionization, one should write the equation of ionization kinetics:

\[ \frac{d\alpha}{dt}=D\frac{d\alpha}{dx}=f(\alpha,T,\rho), \]

\[ x=0,\qquad T=T',\qquad \rho=\rho',\qquad \alpha=0, \]

which reduces to quadrature if, for example, \(T\) and \(\rho\) are expressed in the current—

*) It is extremely surprising that the authors, referring to later theoretical works, make no mention at all of the work of Ya. B. Zel’dovich\(^{42,59}\), in which the broadening of the wave front due to delayed excitation of vibrational degrees of freedom was first considered.

... point \(x\) behind the density discontinuity through \(\alpha\) with the aid of equations (4.2). The degree of ionization \(\alpha\) enters into the specific internal energy:

\[ \varepsilon=\frac{3}{2}RT+\alpha q \]

(\(q\) is the energy required to ionize one gram of gas). Calculations of this type for argon in the range of shock-wave amplitudes for which only the first ionization is significant were carried out by Bond\({}^{60}\).

Of the three principal mechanisms of ionization—collisions of atoms with one another, collisions of electrons with atoms, and photoionization—the last, at not too low a density, plays an insignificant role*). Immediately behind the density discontinuity, where there are as yet no electrons, ionization occurs through collisions of atoms with one another. The effective cross section of such a process is very small; therefore ionization proceeds slowly and the temperature behind the discontinuity decreases rather weakly.

However, even at a small concentration of electrons, ionization by electron impact, because of its large effective cross section, becomes stronger than ionization in collisions of atoms. From this moment on, \(\alpha\) increases exponentially with time, or with distance from the density discontinuity, and rapidly reaches its final value \(\alpha_1\). At the same time the temperature rapidly approaches the final value \(T_1\), following from the shock relations**).

The width of the shock-wave front decreases as its amplitude increases, since at higher temperatures the rate of ionization increases. Thus, in waves in argon with an initial pressure of \(59\) mm Hg, the front width, according to Bond’s calculations, is approximately

\[ \Delta x=2.5\cdot 10^{-3}\ \text{cm}\quad \text{at } D=6\ \text{km/sec},\quad T_1\simeq 18\,000^\circ \]

and

\[ \Delta x\simeq 2\cdot 10^{-2}\ \text{cm}\quad \text{at } D=5\ \text{km/sec},\quad T_1\simeq 15\,000^\circ . \]

§ 5. STRUCTURE OF THE SHOCK-WAVE FRONT WITH RADIATION TAKEN INTO ACCOUNT

At high temperatures in gases, emission and absorption of light play an essential role.

At temperatures of the order of several thousand degrees, the band spectra of molecules are excited; at higher temperatures—the line spectra of atoms; with the onset of ionization the spectrum becomes continuous. Emission of light occurs when free electrons are captured into various orbits in ions, and also when electrons are decelerated in the field of ions; absorption occurs as the result of the reverse processes.

When we speak of the thermodynamic equilibrium of a gas behind the front of a shock wave, it must also be borne in mind that, in the final state, the radiation is likewise in equilibrium with the matter. However, at the temperatures of interest to us the energy density of equilibrium radiation

\[ U_p=\frac{4\sigma T^4}{c} \]

(\(\sigma\) is the Stefan–Boltzmann constant, \(c\) is the speed of light) is extremely small in comparison with the internal energy of the substance and so—

* Light quanta are the principal ionizing agent in strongly rarefied stellar atmospheres.

** See note 1 in the proof corrections at the end of the article.

has absolutely no effect on the shock relations (2.1). Likewise, the radiation pressure, equal to \(\dfrac{U_p}{3}\), is much smaller than the pressure of the material.

The ratio of the energy fluxes carried by radiation and by matter is of a different order; it is approximately \(c/D\), i.e., \(10^3\)—\(10^4\) times greater than the ratio of the corresponding energy densities:

\[ \frac{S}{D\rho_\varepsilon} = \frac{\sigma T^4}{D\rho_\varepsilon} = \frac{U_p}{\rho_\varepsilon}\cdot\frac{c}{4D}. \]

The two fluxes become comparable with one another in air at a temperature of the order of \(300\,000^\circ\), while the density of radiant energy is still very small.

It would seem that the removal of energy by radiation from a shock wave of large amplitude should play an important role, and that in the third of the shock relations (2.1) one should include the radiation energy flux \(\sigma T_1^4\), carried away from the wave front, which at high amplitudes would substantially affect the final state of the gas. In a powerful explosion, when a shock wave propagates from the center of the explosion, whose radius at constant heat capacity, according to L. I. Sedov’s solution\({}^{61}\), is \(R \sim t^{2/5}\), while the temperature behind the front is \(T_1 \sim t^{-6/5}\), the removal of energy by radiation would be proportional to

\[ \int \sigma T_1^4 R^2\,dt \sim \int \frac{dt}{t^4}. \]

This expression diverges as \(t \to 0\)\(*\), i.e., it leads to an instantaneous flashing of the volume of gas heated to a very high temperature.

In reality, nothing of the kind occurs. The point is that gases are transparent only for comparatively small quanta. Molecular gases, as a rule, already absorb ultraviolet radiation (for example, air is transparent only in the visible part of the spectrum with \(\lambda > 3000\ \text{\AA}\), \(h\nu < 4\ \text{eV}\)). Atoms (as well as molecules) very strongly absorb quanta larger than the ionization potential, i.e., of the order of \(10\ \text{eV}\) and higher, which cause the photoelectric effect. At high temperature the quanta carrying energy away from the shock-wave front to “infinity” lie in the Rayleigh–Jeans region and constitute a small fraction of the energy of the whole spectrum. For example, at \(T = 50\,000^\circ\), the part of the spectrum with \(h\nu < 4\ \text{eV}\) contains only \(3\%\) of the energy.

The removal of energy by radiation has the greatest influence on the final state behind the wave front, evidently, at that largest amplitude for which the principal energy of the spectrum still falls within the transparency region of the cold gas. Thus, for example, in air at \(T_1 \simeq 10\,000^\circ\) the absorbed quanta lie immediately beyond the maximum of the Planck spectrum, which is of the order of \(3\ \text{eV}\), and the additional compression behind the front due to the removal of energy amounts to only \(0.1\%\).

Thus, the energy \(S_\infty\) carried away by radiation to infinity has practically no effect on the final state of a gas undergoing shock compression, and the shock relations (2.7) remain valid also for

\(*\) Taking into account the actual dependence of heat capacity on temperature changes the degree of divergence only slightly, but does not remove it.

waves of large amplitude. Radiation emerging from the surface of a shock discontinuity, behind which there lies a region of high temperature, is almost completely absorbed in a thin layer of gas ahead of the discontinuity, heating it. Radiative heat exchange, occurring in the shock wave and consisting in the radiative cooling of gas layers that have already undergone the density jump and the heating of those layers that have not yet undergone the jump, has a substantial influence on the structure of the wave front, if by the latter, as before, one understands the transitional nonequilibrium layer separating the initial and final thermodynamically equilibrium states.

Radiative heat exchange takes place over distances measured by the mean free path for absorption of light. Usually the effective cross sections for absorption of light are several orders of magnitude smaller than the effective cross sections for collisions of atoms or ions; therefore the thickness of the front of a shock wave, when radiation is taken into account, proves to be several orders of magnitude greater than the thickness of the viscous density jump. For this reason atomic or ionic and electronic viscosity and thermal conductivity play a very small role in the zone of radiative heat exchange, and they may be neglected in the hydrodynamic equations (3.5) describing the internal structure of the wave front, which now take the form:

\[ \left. \begin{aligned} \rho u &= \rho_0 D,\\ p+\rho u^2 &= \rho_0 D^2,\\ \varepsilon+\frac{p}{\rho}+\frac{u^2}{2}+\frac{S}{\rho_0 D} &= \frac{D^2}{2}, \end{aligned} \right\} \tag{5.1} \]

\[ S=0 \quad \text{for } x=\pm\infty . \tag{5.2} \]

Here \(S\) is the radiation energy flux; the initial pressure and internal energy have been omitted, since the shock wave is assumed to be strong. Within the framework of equations (5.1), the viscous density jump will correspond to a discontinuity of the hydrodynamic quantities, just as in the consideration of the structure of a wave with delayed excitation of part of the heat capacity.

The structure of the shock-wave front with radiation taken into account was considered by V. A. Prokof’ev\(^{40}\) for the examples of hydrogen and argon in the region of first ionization. To determine the radiation flux he wrote down the second-order differential equation well known from astrophysics, which, in comparison with the exact radiation-transfer equation, takes into account the angular distribution of the radiation only approximately. Such a simplification is quite permissible, since it introduces only an insignificant quantitative error, without distorting the qualitative features of the phenomenon.

However, proceeding from the correct equations of hydrodynamics and radiation, V. A. Prokof’ev attempted to construct continuous distributions of the hydrodynamic quantities in the wave. In the question of the existence of a discontinuity lies the fundamental disagreement of V. A. Prokof’ev with the views of the authors, according to which radiative heat exchange by itself cannot lead to shock compression and by no means replaces the viscous density jump, which always exists*). The further exposition, concerning the structure and brightness of the fronts of shock waves, will be based on the authors’ works\(^{62-64}\).

In the limiting case of sufficiently weak waves, when the role of radiation is negligibly small, the profiles of temperature, density, and pressure in the shock

*) For a criticism of V. A. Prokof’ev’s work, see work\(^{62}\).

in the wave, if one disregards the delayed excitation of part of the heat capacity, have the form of a step (see Fig. 5, a). As the amplitude of the wave increases, the radiation flux \(\sigma T_1^4\), emerging from the surface of the discontinuity, increases; being absorbed in the cold gas ahead of the discontinuity, it heats it. The compression jump now propagates not through the cold gas, but through the heated gas; therefore behind the jump the temperature \(T_+\) is higher than \(T_1\). In other words, ahead of the jump the gas particle is heated by radiation, while behind the jump it is cooled, i.e. radiative heat exchange consists in the transfer of energy from the region behind the discontinuity to the region ahead of the discontinuity. Heating of the gas by radiation ahead of the discontinuity leads to a slight compression of it and to an increase in pressure. In the compression jump the gas is compressed to a density somewhat less than the final one. Its compression to the final state occurs in the zone of radiative cooling; in this process the pressure increases somewhat. The profiles of temperature, density, and pressure in the front of a not-too-strong wave are shown in Fig. 5, b.*)

Fig. 5.

As was already indicated above, radiative heat exchange occurs over distances of the order of the mean free path for absorption of light. In air, for example, the mean free paths are of the order of \(10^{-2}\)—\(10^{-1}\) cm in a very wide interval of quantum energies (from 10 to 100 eV) and temperatures (up to hundreds of thousands of degrees). Therefore the front thickness of a not-too-strong shock wave is also of this order.

Very convenient for considering the structure of the front of a shock wave with radiation, along with the \(pv\)-, are also the \(Tv\)- and \(Sv\)-diagrams.**) The state of a gas particle in the \(pv\)-diagram is represented by a straight line, which follows from the first two equations

Fig. 6.

Fig. 7.

(5.1) (Fig. 6), in which there is no viscous transfer of momentum. The curves \(T(v)\) and \(S(v)\) can also be obtained from equations (5.1) (Fig. 7).

*) The case of very strong waves will be considered somewhat below.
**) This was noted by V. A. Prokof’ev.

The heating zone corresponds to continuous portions of the curves \(p(v)\), \(T(v)\), \(S(v)\)—\(AB\). In the compression shock the compression occurs along the adiabat, the Hugoniot, connecting the intermediate states \(B\) and \(C\) on both sides of the discontinuity: the state of the particle changes discontinuously from \(B\) to \(C\), while the radiation flux \(S_0\) in the compression shock remains continuous because of the stationarity of the phenomenon and the boundedness of the radiation sources. The zone of cooling by radiation corresponds to the continuous portions of the curves \(CD\).

The temperature of heating ahead of the discontinuity is proportional to the radiation flux \(S_0 \approx \sigma T_1^4\) emerging from the surface of the discontinuity, and therefore increases very rapidly with increasing amplitude of the wave or temperature behind the front \(T_1\). For example, in air, for \(T_1 = 50\,000^\circ\), \(T_- = 4000^\circ\), while for \(T_1 = 150\,000^\circ\), \(T_- = 60\,000^\circ\).

At a certain temperature behind the front \(T_1 = T_{\mathrm{cr}}\), the heating temperature \(T_-\) reaches the value \(T_1\). This temperature, approximately equal to \(300\,000^\circ\) for air, may be called critical, since it separates two essentially different cases of front structure (the subcritical case \(T_1 < T_{\mathrm{cr}}\) was considered above).

At the critical amplitude the hydrodynamic and radiation energy fluxes become comparable at the point of discontinuity. In a supercritical wave \(T_1 > T_{\mathrm{cr}}\), the radiation energy flux, limited, according to the equations of hydrodynamics, by a quantity of the order of the hydrodynamic flux \(D\rho_0 \varepsilon(T_1)\) and proportional to \(T_1^{3/2}\) for constant heat capacity, when \(\varepsilon \sim T\), becomes smaller than the Stefan–Boltzmann flux \(\sigma T_1^4\). This occurs as a result of compensation of one-sided fluxes in the opposite direction, each of which is of order \(\sigma T_1^4\), since, in contrast to the subcritical case, the temperature in the heating zone is now high, comparable with \(T_1\), and in this zone radiation is not only absorbed but also generated. The radiation density at every point of the heating zone is now close to the equilibrium density corresponding to the temperature of the point, while the temperature changes little over a distance of the order of the mean free path of the radiation, i.e., there exists the so-called local thermodynamic equilibrium, and the transfer of radiation has the character of radiative heat conduction. The temperature ahead of the discontinuity \(T_-\) can never exceed the temperature behind the front \(T_1\), since then the radiation density ahead of the discontinuity would be higher than the density behind the front and the flux behind the discontinuity would be directed in the opposite direction, which is impossible (the flux in the wave does not change sign).

A rigorous proof of the assertion \(T_- \leq T_1\), which at the same time is a proof of the necessity of the discontinuity, is given in work \(^{62}\) on the basis of a qualitative investigation of the equations of hydrodynamics and radiation.

Fig. 8.

In the approximation of radiative heat conduction the radiation energy flux \(S\) is proportional to the temperature gradient and, consequently, the temperature, unlike the pressure and density, is continuous in the wave: an isothermal shock occurs (see the book by L. D. Landau and E. M. Lifshitz \(^{65}\)). The temperature profile in the wave front has the form shown in Fig. 8. However, as is seen from the \(Tv\) and \(Sv\) diagrams, the continuity of temperature is associated with a discontinuity of the flux in the compression shock. The state before the discontinuity is represented by the point \(B'\), at which \(T_- = T_1\), and beyond the discontinuity there immediately sets in a final—

state \(D\) with \(T = T_1\). This is a consequence of the approximation of radiant thermal conductivity, which leaves out of consideration effects occurring at distances smaller than the mean free path of the radiation.

On a more exact examination, in which allowance is made for the possibility of a deviation of the radiation density from its equilibrium value, an isothermal discontinuity proves impossible, since in reality the flux is continuous, whereas the temperature has a discontinuity. In this case, as follows from the \(Tv\) and \(Sv\) diagrams, on which the state before the discontinuity is still represented by the point \(B'\) \((T_- \simeq T_1)\), behind the compression jump there is a temperature peak (point \(C'\)). The actual temperature profile in the wave is shown in Fig. 9. The approximation of radiant thermal conductivity simply cuts off the temperature peak (dashed line in Fig. 9).

As was to be expected, the thickness of the temperature peak proves to be smaller than the mean free path of the radiation. Indeed, the emissive power of the gas cooling behind the discontinuity, proportional to \(\sigma T_1^4\), is very large in the supercritical case, and therefore a cooling of a layer of gas small in comparison with the mean free path is sufficient to give the required radiation flux from the surface of the discontinuity, which is much less than \(\sigma T_1^4\). It can be shown \(^{63}\) that in the supercritical case the optical thickness of the temperature peak \(\Delta \tau\) is of the order

\[ \left(\frac{T_{\mathrm{cr}}}{T_1}\right)^{2.5}, \]

and the optical thickness of the heating zone is, in order of magnitude, the reciprocal \(\left(T_1/T_{\mathrm{cr}}\right)^{2.5*}\), so that the geometrical thickness of a shock wave with a temperature, say, \(T_1 = 10^6\) degrees in air proves to be quite considerable, of the order of several centimeters.

Fig. 9.

§ 6. BRIGHTNESS OF THE FRONT OF LARGE-AMPLITUDE SHOCK WAVES

The optical method is one of the most important methods for measuring high temperatures. The usual procedure consists in comparing the degrees of blackening produced in a photographic plate by light coming from the object under study and from a standard source. For greater accuracy the photography is usually carried out in a narrow wavelength interval, since the object under study and the standard source, having different temperatures, have different emission spectra, and, in addition, the sensitivity of photographic materials depends on the wavelength, which creates difficulty in converting the degree of blackening into temperature. The optical method can also be used to measure the brightness of the surface of the front of a shock wave. If there is confidence that the surface of the wave front radiates as an absolutely black body, then from the brightness one can directly determine the temperature behind the front, i.e. the amplitude of the shock wave, which is important not only for experimental investigations but is also of great practical interest. The temperature calculated from the brightness of the shock-wave front under the assumption that the latter radiates as an absolutely black body is usually called effective. Experience shows that, in a certain interval of amplitudes, the effective temperature does indeed coincide with the true temperature behind the front, which can be found with the aid of the shock adiabat by measuring some other parameter

\[ \rule{3cm}{0.4pt} \]

*) At constant heat capacity.

shock wave, for example, its velocity or the velocity of the gas behind the front, which indicates that the wave front radiates as an absolutely black body.

However, at sufficiently large amplitudes the brightness of the front passes through a maximum, and then decreases and, as it were, reaches saturation: an increase in amplitude does not lead to an increase in brightness and effective temperature. The saturation effect was observed by Glaser \(^{66}\) in argon, and by K. S. Vul'fson, I. Sh. Libin, and F. A. Charnyi \(^{67}\) in a pulsed discharge in inert gases. I. Sh. Model' \(^{30}\) found that the effective temperatures of shock waves in air, equal to \(\simeq 8000^\circ\) and \(\simeq 11000^\circ\), are very close to the true ones. In krypton and xenon the brightness temperatures proved to be considerably lower than those calculated from the shock adiabat. In these laboratory experiments the brightness maximum predicted by the theory was not observed, since I. Sh. Model' made measurements only at one wave amplitude.

The effects of maximum and saturation of the brightness of the front are readily explained on the basis of the ideas, set forth in the preceding paragraph, concerning the structure of the front of strong shock waves. Most gases, in particular all monatomic gases, are transparent to visible light at ordinary temperatures. They absorb a continuous spectrum of quanta with energies of the order of \(2\text{--}3\ \mathrm{eV}\), which lie in the visible region, only at high temperatures, mainly by means of the photoeffect from highly excited electronic levels of atoms (and molecules). Quanta with energy \(h\nu\) can produce the photoeffect only in those atoms in which the electrons are on levels higher than \(I-h\nu\), where \(I\) is the ionization potential of the atom. Therefore the coefficient of continuous absorption of light is proportional to the Boltzmann factor

\[ \varkappa_{\nu}\sim \exp\left[-\frac{I-h\nu}{kT}\right] \tag{6.1} \]

and depends extremely strongly on the temperature when \(kT \ll I-h\nu\). For example, in air of normal density at a temperature of \(11000^\circ\), \(\varkappa_{\nu}\simeq 0.09\ \mathrm{cm}^{-1}\) (the mean free path \(l_{\nu}=\dfrac{1}{\varkappa_{\nu}}\simeq 11\ \mathrm{cm}\)), while at \(20000^\circ\), \(\varkappa_{\nu}\simeq 60\ \mathrm{cm}^{-1}\), \(l_{\nu}\simeq 0.017\ \mathrm{cm}\) (for red light).

The surface of discontinuity in a shock wave of not too large amplitude, for which the temperature peak behind the discontinuity is small, radiates as an absolutely black body of temperature close to the temperature behind the front, \(T_1\). Quanta greater than the ionization potential \(I\) are absorbed ahead of the discontinuity by the photoeffect from ground levels, heating the gas in a region whose width is of the order of \(10^{-1}\text{--}10^{-2}\ \mathrm{cm}\) at normal density. As long as the temperature in the heating zone is low, visible light passes through the heating zone without hindrance, and the effective radiation temperature coincides with the true one. But when the temperature in the heating zone becomes so high that the mean free path of visible light at this temperature proves comparable with the width of the heated region, the latter absorbs the visible quanta emitted by the discontinuity surface, i.e. it screens the discontinuity.

Moreover, owing to the extremely sharp increase of the absorption of visible light with temperature, the screening in the heating zone increases significantly faster than the radiation intensity of the discontinuity as the amplitude grows, so that the brightness and the effective temperature then rapidly fall after passing through a maximum. The temperature behind the wave front,

at which shielding begins, and to which the maximum value of the effective temperature is close, varies from one gas to another mainly because of the strong dependence on the ionization potential. In air this temperature, according to the calculation of \(^{64}\), is approximately \(90\,000^\circ\). As the amplitude of the wave increases, soon after shielding begins the latter grows so much that the visible light emitted by the high-temperature surface of discontinuity is completely absorbed before the discontinuity. This, however, does not mean that the brightness of the front falls to zero. The point is that visible light is also generated in the heating zone itself. Therefore the brightness falls not to zero, but to the value corresponding to the radiation of the heating zone.

A detailed examination of the temperature profile in the wave, of the spectrum of the radiation producing the heating, and of the absorption coefficients (see \(^{64}\)) shows that the effective temperature, after passing through a maximum, very rapidly approaches a limiting value, definite for each gas (weakly dependent on the initial density), which practically does not change with unlimited growth of the wave amplitude. For air this limiting brightness of the front corresponds to an effective temperature close to \(18\,000^\circ\).

The dependence of the effective temperature of the shock-wave front in air on the true temperature behind the front is illustrated by Fig. 10. The straight line \(T_{\mathrm{eff}} = T_1\) has not been extended to zero, since at comparatively low temperatures behind the shock-wave front, below \(\simeq 6000^\circ\), the front ceases to radiate as an absolutely black body, and the equality \(T_{\mathrm{eff}} = T_1\) is again violated. The brightness and effective temperature of the shock-wave front now depend substantially on the specific conditions: the dimensions of the region of air heated by the shock wave, and the time during which it exists in the heated state. Very peculiar optical phenomena in shock waves with temperatures behind the front of several thousand degrees are observed in a strong (atomic) explosion. These phenomena are described in an American book \(^{21}\).

Fig. 10.

A strong blast wave propagating through air glows brightly, forming the so-called fireball, whose surface coincides with the surface of the shock-wave front. The effective temperature of the fireball falls with time in accordance with the decrease in the amplitude of the shock wave.

However, when the temperature behind the wave front decreases to \(\simeq 2000^\circ\), the glow of the front abruptly ceases, and the boundary of the fireball, having detached from the shock-wave front, lags behind the latter. In this process the fireball, as it were, begins to flare up again: the brightness of its surface, after passing through a minimum, increases. This occurs at a time of the order of \(10^{-2}\) sec from the beginning of the explosion. The brightness of the fireball increases to a maximum and then slowly (over a time of the order of a second) decreases; at the same time the invisible shock wave has already moved far ahead of the boundary of the luminous region.

In Figs. 11 and 12 the dependence of the effective temperature of the fireball on time and the trajectory of the fireball are shown schematically.

and of the shock wave for an explosion with the aid of \(\sim 10^{21}\) ergs (the curves are taken from the book \(^{21}\)). The question of the glow of air at temperatures of several thousand degrees and the phenomena described were considered theoretically by Yu. P. Raizer \(^{68,69*}\).

It turns out that the principal mechanism of absorption and emission of visible light in air at such temperatures is the molecular absorption of nitrogen dioxide formed in the heated air. At temperatures of the order of several thousand degrees, an intense reaction of nitrogen combustion takes place in air, as a result of which nitrogen oxide NO is formed, its concentration reaching several percent. Approximately one percent of the oxide is rapidly oxidized to dioxide, whose concentration, consequently, is of the order of \(10^{-4}\). However, despite such a low concentration, in a system of large dimensions nitrogen dioxide strongly absorbs and emits visible light \(^{**}\). The system of molecular bands is so complex and varied that it forms a practically continuous spectrum. The mean free path of light in air proves to be of the order of a meter. All other conceivable mechanisms of absorption of visible light in air at temperatures \(\sim 5000\text{--}2000^\circ\), as estimates show, play an insignificant role \(^{***}\).

Fig. 11.

Fig. 12.

The absorption of light by highly excited atoms and molecules of nitrogen and oxygen considered above, which depends extremely sharply on temperature (see formula (6.1)), is characterized at \(T \sim 5000^\circ\) by a mean free path measured in kilometers \(^{****}\).

Thanks to the formation of nitrogen dioxide, the shock wave glows down to a temperature behind the front \(\approx 2000^\circ\); however, the effective temperature of the front \(T_{\mathrm{eff}}\) is higher than the true \(T_1\), since the absorption of light behind the front is insufficiently large and a layer of the order of several meters behind

\(*\) See note 2 in the proof correction at the end of the article.

\(**\) Cold nitrogen dioxide is a dark-brown, opaque gas.

\(***)\) The molecular spectra of \(\mathrm{O_3}\), \(\mathrm{N_2}\), and NO lie, as is known, in the ultraviolet region. A certain shift of the spectra toward the red at high temperature does not create sufficient absorption in the visible region. The absorption by negative oxygen ions is likewise small. These absorption mechanisms play some role only at temperatures above \(5000^\circ\).

\(****\) Nitric oxide molecules, having a significantly lower ionization potential \(I = 9.4\) eV, absorb more strongly than nitrogen and oxygen, but likewise only at \(T \sim 5000^\circ\). Absorption by nitric oxide plays a substantial role at \(T \sim 8000\text{--}6000^\circ\).

by the wave front partly transmits high-temperature radiation coming from deeper layers with a high temperature of the order of \(10\,000^\circ\). (As is known \(^{61}\), the temperature in an explosive wave increases rapidly from the periphery toward the center.) The glow ceases at \(T_1\) below \(\simeq 2000^\circ\), since at such low temperatures in the layers of air captured by the shock wave the formation of nitrogen oxides practically stops.

The point is that the rate of the nitrogen-combustion reaction, which was studied in detail theoretically and experimentally by Ya. B. Zel’dovich, P. Ya. Sadovnikov, and D. A. Frank-Kamenetskii \(^{70}\), falls extremely sharply with decreasing temperature, and if the characteristic time for the formation of an appreciable concentration of oxide at \(T \simeq 3000\text{--}4000^\circ\) is \(\sim 10^{-4}\text{--}10^{-6}\) sec, then at \(T \simeq 2000^\circ\) it is already of the order of a second, and at \(T \simeq 1700^\circ\) of the order of minutes, i.e., considerably greater than the time during which an explosion in air lasts. The reason for the “flaring up” of the fireball after the moment of minimum brightness is as follows.

After the temperature of the shock wave has reached \(2000^\circ\), and nitrogen oxide and dioxide practically no longer form in the layers of air newly captured by the wave, the total amount of nitrogen dioxide already formed changes comparatively slowly*).

Owing to the hydrodynamic expansion of the air in the explosive wave, this almost constant amount of dioxide is distributed over a spherical layer of ever larger and larger radius. The radial optical thickness of the dioxide layer continuously decreases, and this layer becomes increasingly transparent to radiation going outward from deeper layers with higher temperatures.

The effective temperature of the fireball, however, does not rise above \(\simeq 10\,000^\circ\) (the temperature at maximum brightness), since layers of air with this temperature absorb visible light very strongly and are completely opaque to still higher-temperature radiation generated in the central region.

§ 7. ELECTRON THERMAL CONDUCTIVITY AND THE SEPARATION OF ELECTRON AND ION TEMPERATURES IN SHOCK WAVES OF LARGE AMPLITUDE

Above, in studying the structure of the front of strong shock waves, we everywhere operated with the macroscopic concept of the internal energy of a gas and with temperature, assuming that the substance is in thermodynamic equilibrium at every point of the wave and that the redistribution of energy among the degrees of freedom cannot introduce substantial changes into the general character of the structure of the front.

On closer examination, taking into account the finite time of energy exchange between the ionic and electronic gases in shock waves of large amplitude, as well as in shock waves propagating through plasma, interesting effects are found, associated with the large difference between the masses of electrons and ions. These effects were considered in work \(^{62}\) and independently by V. D. Shafranov \(^{71}\). Thermodynamic equilibria in the electronic and ionic gases are established very rapidly: it is known that the relaxation time for establishing a Maxwellian velocity distribution is of the order of the collision time. The exchange, however—

*) Gradually all the oxide in the air is oxidized to dioxide; however, this process lasts minutes and occurs only after the explosion. The observed brown tint of the explosion cloud is due to this dioxide (the amount of which is of the order of \(\sim 100\) tons) \(^{21}\).

energy between electrons and ions proceeds considerably more slowly owing to the enormous difference between the masses of electrons and ions. Therefore the electron and ion gases may be assigned their own temperatures \(T_e\) and \(T_i\), which, generally speaking, differ from one another. An especially sharp difference between the two temperatures arises in the compression shock.

The slightest separation of the electron and ion gases leads to the formation of strong electric fields that impede the separation. Therefore each element of the gas is electrically neutral, and the compressions of the ion and electron gases in the compression shock are identical.

In a coordinate system attached to the wave front, in the compression shock an irreversible conversion of a considerable part of the kinetic energy of the hydrodynamic motion into heat takes place. By virtue of the condition of electrical neutrality, the electron gas is decelerated in the shock in exactly the same way as the ion gas; however, because of the small mass of the electrons, the kinetic energy of the macroscopic motion of the electron gas is extremely small, and therefore the increase in the entropy of the electrons in the compression shock is also negligibly small. If the jump in the temperature of the ion gas in the compression shock is of the order

\[ k\Delta T_i \sim m_i D^2, \]

then the jump in the temperature of the electron gas is

\[ k\Delta T_e \sim m_e D^2, \]

that is, \(\dfrac{m_i}{m_e}\) times smaller. After the particles pass through the compression shock, both temperatures gradually equalize and in the final thermodynamically equilibrium state reach the value \(T_1\), which follows from the shock relations for a wave of the given velocity.

The ionic temperature immediately behind the compression shock now proves to be higher than the \(T_+\) which is obtained under the assumption of an infinite rate of energy exchange between electrons and ions, when \(T_e = T_i\). In fact, the energy of the hydrodynamic motion in the compression shock is now spent only on increasing the temperature of the ions, and not of the electrons, i.e. the heat capacity of the gas behind the compression shock is smaller than usual.

Fig. 13.                Fig. 14.

If one abstracts from the effects of nonhydrodynamic energy transfer, i.e. from radiative and electronic thermal conductivity, then instead of the stepwise profile of the mean gas temperature (see Fig. 13, a) we would obtain the curves of the ionic and electronic temperatures shown in Fig. 13, b.

Connected with the smallness of the electron mass is yet another effect—the large electronic thermal conductivity, which considerably exceeds the ionic one.

Electron thermal conductivity leads to heating of the electron gas ahead of the discontinuity. If one disregards energy transfer by radiation, the profiles of both temperatures would have the form shown in Fig. 14; ahead of the jump the ion temperature lags behind the electron temperature; in the opposite direction behind the discontinuity, the electron temperature is continuous. A detailed calculation of this picture is given in the work of V. D. Shafranov^71. Such a case may occur under conditions of a gas of extremely low density (and with small system dimensions), when the radiation does not have time to come into equilibrium with the matter behind the wave front and its density is considerably less than the equilibrium one, so that the transfer of energy by radiation is negligible. In gases of normal density, for example in air, as estimates show^64, radiant energy transfer exceeds electron thermal conductivity.

Proof correction notes.

1. In the recently published article by Petschek and Byron^72, the process of establishing ionization equilibrium behind the front of a shock wave in argon is considered, and the results of the corresponding experiments are presented.

The rate of ionization by electron impact is limited by the rate of energy exchange between the ion and electron gases. At the beginning of ionization, immediately behind the discontinuity, impurities play a large role, even with a high degree of purification of the argon.

1. Quite recently a theoretical paper by Kivel, Mayer, and Bethe^73 was published, in which the radiation of light formed in heated air by nitrogen oxide is considered.

The calculation of the intensities of different frequencies of the molecular spectrum of the radiation pertains to the temperature interval \(5000\)—\(10000^\circ\), in which radiation by nitrogen oxide is substantial.

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