THE PROBLEM OF MEASURING THE PRESSURE AND DENSITY OF THE UPPER LAYERS OF THE ATMOSPHERE USING AN ARTIFICIAL EARTH SATELLITE

B. S. Danilin, V. V. Mikhnevich, A. I. Repnev,
E. G. Shvidkovsky

Determining the laws governing the distribution of the pressure, density, temperature, and composition of the Earth’s atmosphere with altitude is one of the most important tasks of modern atmospheric physics.

At present, thanks to rocket investigations, the nature of the variation of these parameters with altitude is known approximately up to 160 km. For the higher regions of the atmosphere there exist only very general, and to a considerable extent contradictory, notions. It may be assumed that the use of an artificial Earth satellite as a measuring instrument will make it possible to introduce some clarity into this question as well.

The present article is devoted to an analysis of the physical content of the problem of measuring the pressure and density of the upper layers of the atmosphere when a satellite is used for measurement purposes. In what follows, by an instrument for measuring pressure and density we shall mean a “manometer” of any suitable type; such an instrument may in particular be an ionization manometer or an omegatron.

§ 1. MODELS OF THE UPPER ATMOSPHERE

As has already been said, up to the present time the structure of the atmosphere at altitudes of 200–500 km is almost unknown. Theoretical calculations and ionospheric and spectroscopic observations so far make it possible to establish only the limits of possible mean values of the atmospheric parameters.

As knowledge of the physical processes in the atmosphere has developed, various assumptions have been made in determining the pressure and density of the atmosphere from indirect measurements.

Historically, the first model of the upper atmosphere was proposed by Jeans (1916),¹ on the assumption that there is no mixing of gases in the atmosphere, the molecules do not dissociate, and the temperature is constant and equal to 219 °C.

Gerson (1951),² as a result of analyzing a number of ionospheric and spectroscopic investigations carried out up to 1951, constructed another model of the atmosphere. The temperature values adopted by Gerson are the upper limit of the values found by various investigators and refer to 45° north latitude for the months of January and August. In doing so, Gerson assumed that dissociation of oxygen begins at an altitude of 94 km and reaches 100% at an altitude of 100 km. Nitrogen dissociates beginning at an altitude of 200 km. The atmosphere is completely mixed. The pressure

was determined from the equation of hydrostatics, taking into account the change of the acceleration due to gravity with altitude.

On the basis of rocket-research data and information on the spectra of the night sky, Kalman (1956)^3 calculated, using the barometric formula, a physical model of the atmosphere up to 300 km. In doing so,

Fig. 1.

Visible labels in Fig. 1: height, km; exosphere; northern auroras illuminated by the Sun; dissipation region; particle concentration (number of particles in 1 cm³); ionosphere; upper atmosphere; layer F; \(N_2O\); height reached by V-2 rockets; meteors; layer E; transition region \(O_2 \to O + O\); lower boundary of the region of polar auroras; ozonosphere; troposphere, lower atm.; \(N_2O\); °K.

she proceeded from the following assumptions: oxygen dissociates above 90 km; near 130 km about 30% of the oxygen is in the undissociated state; nitrogen begins to dissociate above 220 km; the concentrations of molecular oxygen and nitrogen decrease exponentially with altitude; in the exosphere region (above 360 km) isothermy is observed; the density of particles at the critical level must not be less than \(10^7\ \text{cm}\); diffusive separation up to an altitude of 300 km is not taken into account. Table 1 gives the atmospheric parameters obtained by Kalman.

Table I

At what altitude the diffusive separation of gases actually occurs is still unknown. Therefore, in their calculations, authors either completely neglect this phenomenon or, on the basis of various more or less justified considerations, prescribe the altitudes starting from which diffusive separation takes place.

Nicolet⁴ believes that this altitude is 160 km. Using the results of rocket measurements up to an altitude of 220 km and assuming that oxygen at altitudes above 160 km is completely dissociated, while nitrogen is dissociated only slightly, Nicolet obtained the distributions of pressure, number of particles per unit volume, and temperature with altitude that are given in Table II.

Table II

Nicolet’s atmospheric model

Mitra⁵, proceeding from a different distribution of temperature and degree of mixing of the atmosphere at different altitudes, obtained a somewhat different distribution of atmospheric parameters. Mitra’s atmospheric model is presented in Fig. 1.

It should be assumed that the last two models are closest to the real distribution of pressure and density in the atmosphere. Naturally, depending on particular assumptions about the variation of temperature and air composition with altitude, the degree of its mixing and dissociation, different values of pressure and density are obtained, differing by several orders of magnitude (see Table III and Fig. 2).

Table III

Concentrations and pressure according to various atmospheric models

Further refinement of these quantities can be carried out primarily by rocket investigations and measurements on an artificial Earth satellite. In investigations of this kind a number of questions arise, connected with the interaction of a rapidly moving body and a rarefied gas. We shall dwell on the consideration of some of these questions.

§ 2. FLUXES OF PARTICLES, MOMENTUM, AND ENERGY FOR THE CASE OF A HOMOGENEOUS GAS

Beginning at an altitude of about \(200\) km, the interaction between a body moving with a speed of the order of several kilometers per second and the atmosphere will proceed according to the laws of free-molecular flow, the analysis of which has been treated in a number of works \(^{6-11}\), based on the following assump-

A) The existence of statistical equilibrium in the medium is postulated, i.e., it is assumed that the velocity distribution in phase space is described by the Maxwell function.

B) It is assumed that \(\frac{l}{L} \gg 1\), where \(l\) is the mean free path of the molecules and \(L\) is the characteristic dimension of the body being flowed around. This condition makes it possible to neglect collisions between the molecules of the impinging stream and the molecules leaving the moving surface; such collisions would cause a violation of the Maxwellian distribution for molecules striking the wall.

Fig. 2.

The feasibility of the second condition at altitudes of \(200\)—\(500\) km, where \(l > 10^3\) cm, is beyond doubt. As for the first assumption, it is connected with the possibility of calculating mean values of functions of the velocity. It is known from statistical physics that the relative fluctuation \(\delta\) of any additive function of the state of a system of \(N\) particles is, in order of magnitude, equal to \(N^{-1/2}\). If \(\delta\) is small, then the deviation from the mean is small, i.e., the number of collisions between molecules is sufficient to ensure statistical equilibrium in the medium. At an altitude of \(400\) km one may expect \(N\) to be of the order of \(10^8\) \(^{5,7}\). Consequently, below \(400\) km \(\delta < 10^{-4}\), and, apparently, statistical equilibrium exists in the atmosphere. Above \(500\) km \(\delta\) is probably greater than \(10^{-3}\) or \(10^{-2}\), and therefore the feasibility of a Maxwellian distribution becomes doubtful.

Let us consider an element \(P\), moving with velocity \(-U\) relative to the Earth. Introduce coordinates \(x, y, z\), connected with the element so that the \(x\)-axis is perpendicular to the plane of the element (Fig. 3). In these coordinates the vector \(\mathbf{U}\) denotes the velocity of the ordered motion of the gas

B. S. Danilin, V. V. Mikhevich, A. I. Repnev, E. G. Shvidkovsky

relative to \(\Pi\). Let \(\theta\) be the angle between the \(y\)-axis, lying in the plane of the area element \(\Pi\), and the vector \(\mathbf U\), lying in the \(xy\)-plane; \(v\) is the most probable velocity of the thermal motion of a particle; \(m\) is its mass; \(T\) is the gas temperature; \(k\) is Boltzmann’s constant; \(N\) is the number of particles per unit volume,

\[ \beta=\frac{U\sin\theta}{v}. \tag{1} \]

Within the framework of the assumptions formulated, using the well-known methods of the kinetic theory of gases, one can obtain the following expressions. For the number of particles incident on a unit area \(\Pi\) per unit time from the region of the half-space \(x\leqslant 0\) (the impinging flux), we shall have:

\[ n_{+}=\frac{Nv}{2\sqrt{\pi}}\chi(\beta), \tag{2} \]

where

\[ \chi(\beta)=e^{-\beta^{2}}+\sqrt{\pi}\,\beta[1+\Phi(\beta)] \tag{3} \]

and

\[ \Phi(\beta)=\frac{2}{\sqrt{\pi}}\int_{0}^{\beta} e^{-s^{2}}\,ds. \tag{4} \]

Fig. 3.

The number of particles incident from the region of the half-space \(x\geqslant 0\) (the catching-up flux) is equal to

\[ n_{-}=\frac{Nv}{2\sqrt{\pi}}\chi'(\beta), \tag{5} \]

where

\[ \chi'(\beta)=e^{-\beta^{2}}-\sqrt{\pi}\,\beta[1-\Phi(\beta)]. \tag{6} \]

Graphs of the functions \(\chi\) and \(\chi'\) are given in Figs. 4 and 5\({}^{10}\).

From the general expression for the momentum-flux density tensor \(p_{ik}\), in what follows we shall be interested only in components of the type \(p_{ir}\), corresponding to the transfer of various components of momentum through the chosen area element \(\Pi\) from the region of the half-space \(x\leqslant 0\) (the impinging flux).

As can be shown, the component of the tensor \(p_{xx}\), representing the flux of the component of momentum directed along the positive \(x\)-axis through a unit surface of the area element \(\Pi\), is expressed as follows:

\[ p_{xx}=\frac{P\beta}{\sqrt{\pi}}\,\varkappa(\beta), \tag{7} \]

where

\[ \varkappa(\beta)=\sqrt{\pi}\left(\beta+\frac{1}{2\beta}\right)[1+\Phi(\beta)]+e^{-\beta^{2}}, \tag{8} \]

and

\[ P=\frac{Nmv^{2}}{2}=NkT \]

is the pressure in the medium at rest.

The component of the tensor \(p_{yx}\), representing the flux of the tangential \(y\)-component of momentum through the same area element, has the form

\[ p_{yx}=\frac{P\beta U\cos\theta}{v}\lambda(\beta), \tag{9} \]

where

\[ \lambda(\beta)=\frac{e^{-\beta^2}}{\sqrt{\pi}\,\beta}+[1+\Phi(\beta)]. \tag{10} \]

\(p_{zr}=0\), since \(U_z=0\), and the thermal motion is chaotic.

Fig. 4.

Fig. 5.

The expressions given for \(p_{xx}\) and \(p_{yx}\) coincide with Tyan’s results\(^8\).

For the components of the momentum-flux tensor from the side of the half-space \(x>0\) (the overtaking flow), \(p'_{xx}\) and \(p'_{yx}\), the following expressions are obtained:

\[ p'_{xx}=\frac{p\beta}{\sqrt{\pi}}\,\chi'(\beta), \tag{11} \]

where

\[ \chi'(\beta)=\sqrt{\pi}\left(\beta+\frac{1}{2\beta}\right)[1-\Phi(\beta)]-e^{-\beta^2}, \tag{12} \]

and

\[ p'_{yx}=\frac{p\beta U\cos\theta}{v}\,\lambda'(\beta). \tag{13} \]

where

\[ \lambda'(\beta)=\frac{e^{-\beta^2}}{\sqrt{\pi}\,\beta}\,[1-\Phi(\beta)]. \tag{14} \]

\[ p'_{zx}=0. \]

These expressions coincide with Zanger’s results. The functions \(\chi\) and \(\chi'\), \(\lambda\) and \(\lambda'\) are shown in Figs. 6 and 7.

Fig. 6.

Finally, the amounts of translational energy of the particles transported through the area from the side \(x<0\) and from the side \(x>0\), denoted by \(E_+\) and \(E_-\), respectively, are expressed by the formulas

\[ E_+=mn_+\left[\frac{U^2}{2}+\frac{v^2}{2}\Psi(\beta)\right] =n_+\left[\frac{mU^2}{2}+\Psi kT\right], \tag{15} \]

where

\[ \Psi(\beta)=1+ \frac{e^{-\beta^2}+\dfrac{3}{2}\sqrt{\pi}\,\beta[1+\Phi(\beta)]} {e^{-\beta^2}+\sqrt{\pi}\,\beta[1+\Phi(\beta)]}, \tag{16} \]

and

\[ E_-=mn_-\left[\frac{U^2}{2}+\frac{v^2}{2}\Psi'(\beta)\right] =n_-\left[\frac{mU^2}{2}+\Psi' kT\right], \tag{17} \]

where

\[ \Psi'(\beta)=1+ \frac{e^{-\beta^2}-\dfrac{3}{2}\sqrt{\pi}\,\beta[1-\Phi(\beta)]} {e^{-\beta^2}-\sqrt{\pi}\,\beta[1-\Phi(\beta)]}. \tag{18} \]

In Figs. 8 and 9 the dependence of \(\Psi\) and \(\Psi'\) on \(\beta\) is presented.

Polyatomic molecules, in addition to the energy of translational motion, will carry the energy of rotations and vibrations. At the temperatures existing in the upper layers of the atmosphere, the rotational degrees of freedom are completely excited, and the mean rotational energy is determined by the theorem of equipartition of energy over the degrees of freedom. The vibrational degrees of freedom may be incompletely excited. Here one cannot neglect quantum effects: although the vibrational frequencies of the molecules \(O_2\) and \(N_2\) lie in the infrared region, nevertheless the temperature may prove insufficiently high for their excitation.

Fig. 7.

Fig. 8.

Owing to the independence of the rotational energy from the relative translational velocity of the molecules, for the amount of rotational energy carried through the area \(P\) we shall have:

\[ E_R=\frac{1}{2}jkTn_{\pm}, \tag{19} \]

where \(j\) is the number of rotational degrees of freedom. We shall not consider the vibrational energy, for the reasons set forth below.

Fig. 9.

§ 3. PRESSURE OF A STREAM OF A HOMOGENEOUS GAS

We shall regard the area \(P\) introduced above as a plate impermeable to particles. The normal pressure produced by particles falling on \(P\) from the half-space \(x \leq 0\) will consist of three terms, corresponding to the change in the momentum flux of absorbed, specularly reflected, and diffusely reflected particles. The change in the momentum flux of absorbed particles is numerically equal to the flux itself; for specularly reflected particles it is equal to twice the momentum flux. Denoting the normal pressure on the area by \(P_+\), we shall have:

\[ P_+=fp_{xx}+2(1-f)p_{xx}+P_r=(2-f)p_{xx}+P_r, \tag{20} \]

where \(f\) is the fraction of diffusely reflected particles and \(P_r\) is their pressure.

From the kinetic theory of gases it is known that

\[ P_r=\frac{\sqrt{\pi}}{2}mv_rfn_{\pm}, \tag{21} \]

where \(v_r\) is the most probable velocity of the diffusely reflected particles. Thus, the total pressure on the plate will be equal to

\[ P_+=\frac{P}{\sqrt{\pi}}\left[(2-f)\beta\chi+\frac{\sqrt{\pi}}{2}f\frac{v_r}{v}\chi\right]. \tag{22} \]

It follows from formula (22) that, for the motion of a satellite with a velocity \(\sim 8\) km/sec \(^{12}\), an average molecular weight of air \(\simeq 20\), and specular reflection of molecules \((f=0)\), the frontal pressure on its surface is \(P_+\simeq 10^2P\), i.e. it exceeds the pressure in the free atmosphere by two orders of magnitude.

The tangential stress due to the momentum flux \(p_{yx}\) on the lateral surface of a moving body under specular reflection will, of course, be equal to zero. In the case of complete absorption of particles \((f=1)\) with subsequent diffuse emission, the tangential stress on the lateral surface due to \(P_{yx}\) will be \(\sim 5P\).

§ 4. THE PHENOMENON OF ACCOMMODATION

The amount of energy of translational motion of molecules brought to the surface \(\Pi\) is determined by formulas (15) and (17). The presence in them of the term \(n_+ \dfrac{mU^2}{2}\) makes it possible, by analogy with gas dynamics, to introduce the stagnation temperature, i.e., the temperature of a stagnated gas whose translational energy has been expended on its heating. The introduction of the stagnation temperature means, in essence, the replacement of the properties of a gas of high velocity by the properties of a gas of equivalently high temperature.

In the process of interaction with the wall, a redistribution of energy among the degrees of freedom may occur. Part of the energy of the translational motion of molecules may pass into their rotational and vibrational energy; some time is required for the establishment of a new equilibrium distribution. It is known that excitation of vibrational degrees of freedom requires many \((10^4—10^5)\) collisions. Consequently, under conditions of a rarefied gas, a long time is required to establish an equilibrium distribution of energy between the translational and vibrational degrees of freedom. We shall exclude this process from consideration, assuming that the accommodation coefficient for the vibrational degrees of freedom of molecules is equal to zero.

Let us recall that the accommodation coefficient is the ratio
\(a=\dfrac{E-E_r}{E-E_w}\), where \(E\) is the energy of the translational, or vibrational, or rotational degrees of freedom carried by the impinging molecules, \(E_r\) is the energy of the corresponding degrees of freedom after reflection, and \(E_w\) is the energy of these degrees of freedom after reflection in the case in which it corresponded to the wall temperature \(T_w\). The accommodation coefficient of one or another degree of freedom indicates the effectiveness of the exchange of energy of the corresponding degrees of freedom of the molecule with the wall in a single collision. For vibrational degrees of freedom the accommodation coefficient is small. Thus, in what follows, only the redistribution of energy between translational and rotational degrees of freedom will be considered.

For brevity, we shall restrict ourselves to the case of interaction of molecules with the surface \(\Pi\) from the side \(x \leq 0\).

Let, as a result of energy redistribution, the fraction of the amount of energy of translational motion equal to \(\eta E_+\) fall to the share of the translational degrees of freedom, and the fraction of this energy equal to \((1-\eta)E_+\) to the share of the rotational degrees of freedom. The flux of energy of translational motion at \(U=0\), according to (15) and (16), is equal to \(2n_+kT\); therefore the stagnation temperature of the translational degrees of freedom \(T_{et}\) is determined by the relation

\[ \eta E_+ = 2n_+kT_{et}. \tag{23} \]

The energy of rotational motion brought by the molecules incident on the surface, according to (19), is equal to \(\dfrac{j}{2}n_+kT\); therefore, to determine the stagnation temperature of the rotational degrees of freedom \(T_{eR}\), one should adopt the expression

\[ (1-\eta)E_+ + \frac{jn_+kT}{2} = \frac{jn_+kT_{eR}}{2}. \tag{24} \]

Generally speaking, the temperatures \(T_{et}\) and \(T_{eR}\) are not equilibrium temperatures with respect to energy exchange between degrees of freedom. If, as a result of collisions with the wall, complete exchange of energy between the translational and rotational degrees of freedom takes place and is established

B. S. Danilin, V. V. Mikhnevich, A. I. Repnev, E. G. Shvidkovskii

an equilibrium energy distribution corresponding to the temperature \(T_e=T_{eR}=T_{et}\), then from the last two equations, with the aid of (15), we obtain the value of \(\eta\) corresponding to equilibrium,

\[ \eta=\frac{2jT+4\left(\dfrac{mU^2}{2k}+\Psi T\right)} {(4+j)\left(\dfrac{mU^2}{2k}+\Psi T\right)} . \tag{25} \]

The equilibrium, in the sense indicated above, braking temperature is determined after substituting the value of \(\eta\) found into (23) and replacing \(T_{et}\) by \(T_e\):

\[ T_e=\frac{jT+2\left(\dfrac{mU^2}{2}+\Psi T\right)}{4+j}. \tag{26} \]

The accommodation coefficient of the translational degrees of freedom is defined by the relation

\[ \alpha=\frac{\eta E_+-E_r}{\eta E_+-E_w}, \tag{27} \]

where \(E_r\) is the energy of the reflected molecules, composed of the energy of molecules that have undergone specular reflection \((1-f)\eta E_+\), and the energy of diffusely reflected (emitted) molecules at the translational temperature \(T_{rt}\) (the energy of the latter is equal to \(2fn_+ kT_{rt}\)); \(E_w=2n_+ kT_w\) is the energy of diffusely emitted molecules at the wall temperature \(T_w\) \(^{13}\). Hence we find

\[ \alpha=\frac{T_{et}-T_{rt}}{T_{et}-T_w}\,f. \tag{28} \]

The accommodation coefficient for rotational energy, as can be shown, is determined in an analogous way:

\[ \alpha_R=\frac{T_{eR}-T_{rR}}{T_{eR}-T_w}\,f, \tag{29} \]

where \(T_{rR}\) is the rotational temperature of diffusely reflected emitted molecules.

For the vibrational degrees of freedom, in accordance with what was said above, we take

\[ \alpha_v=0. \tag{30} \]

Measurements of the accommodation coefficient have shown that it depends on the kind of gas, its temperature, the temperature and nature of the surface, and the presence of impurities \(^{14-17}\). Numerical values of the accommodation coefficient may vary within the limits from 0.1 to 1. The accommodation coefficients of translational and rotational degrees of freedom under ordinary conditions turn out to be close in magnitude \(^{18}\). There are indications \(^{19,20}\) that the accommodation coefficient increases with increasing degree of coverage of the surface by gas molecules. With time, in the process of degassing the satellite surfaces, their accommodation coefficient will decrease. The properties of the surface themselves will change under the action of bombardment by high-velocity molecules and ions, as well as by micrometeorites. In a high-velocity stream of rarefied gas, where, despite the small particle density, their flux to the surface is large, the individual process of accommodation may be interrupted by the impact of a new molecule of high velocity. This applies especially to diffusely scattering surfac-

surfaces for which the residence time of the molecule on the surface is comparatively large. In this case the accommodation coefficient may depend on the velocity and angle of incidence of the molecules on the surface.

§ 5. EQUILIBRIUM PRESSURE IN THE MANOMETER CAVITY

The preceding consideration of the properties of free-molecular flow makes it possible to establish a relation between the pressure and the number of particles per unit volume in the manometer cavity and the corresponding parameters of the surrounding medium.

Let us consider the simplest case: the manometer cavity is connected with the atmosphere through a diaphragm of radius \(r\) (without a connecting tube). Let the satellite move with velocity \(U\) in a medium with parameters \(N, T, P\); let the gas parameters in the cavity be \(N_1, T_1, P_1, v_1\).

The change of pressure in the manometer volume \(W\) during the time \(dt\), caused by the flux of particles into the volume, equal to \(n_{+}S\,dt\), will be

\[ \frac{kT_1}{W}\, n_{+}S\,dt = \frac{Nv}{2\sqrt{\pi}}\,\chi Sdt\,\frac{kT_1}{W}. \tag{31} \]

The change of pressure during the same time, caused by the flux of particles out of the manometer volume, equal to

\[ \frac{N_1v_1}{2\sqrt{\pi}}S\,dt, \]

will be

\[ \frac{N_1v_1}{2\sqrt{\pi}}\,S\,dt\,\frac{kT_1}{W}, \tag{32} \]

where \(S\) is the area of the manometer opening and \(v_1\) is the most probable velocity of the particles in the manometer.

The total change of pressure is

\[ dP_1 = \frac{Nv\chi SkT_1}{2\sqrt{\pi}W}\,dt + kN_1\cdot dT_1 - \frac{N_1v_1SkT_1}{2\sqrt{\pi}W}\,dt. \tag{33} \]

The second term characterizes the change of pressure due to the equalization of the temperature of the particles entering the manometer and of those already present there. The condition of pressure equilibrium \(\dfrac{dP_1}{dt}=0\) leads to the expression

\[ N=N_1\frac{v_1}{v\chi}(1-\xi), \tag{34} \]

where

\[ \xi=\frac{2\sqrt{\pi}W}{Sv_1T_1}\frac{dT_1}{dt}. \tag{35} \]

An estimate of the magnitude of \(\xi\) can be made only very roughly. There are two processes of temperature equalization: 1) by collisions of the particles with the manometer wall, 2) as a result of collisions between particles in the manometer cavity.

The first process is determined completely by accommodation. From equation (28), for \(f=1\) and \(T_{et}=T_e\), assuming \(T_w=\mathrm{const}\) (which corresponds to intensive heat removal from the wall surface) and replacing \(T_{rt}\) by \(T_1\), we have

\[ \frac{\delta T_1}{\delta t} = (1-\alpha)\frac{\delta T_e}{\delta T_1}. \tag{36} \]

Here the symbol \(\delta\) has the meaning of the mean per-particle change in the act of collision, so that \(\delta t\) is the collision time. \(T_e\) is determined by formula (26). Replacing in (35) \(\dfrac{dT_1}{dt}\) by \(\dfrac{\delta T}{\delta t}\), we see that for \(\alpha=1\), \(\xi=0\). In the case

At \(\alpha=0\) specular reflection of the molecular flux takes place, in which \(U^2\) (the square of the velocity of the flux relative to the surface) remains unchanged in the collision process, and then from (27) and (25) we find

\[ \frac{j}{4}\frac{\partial T_1}{\partial t} = \frac{j+2\Psi}{j+4}\frac{\partial T}{\partial t}. \tag{37} \]

Since \(2\leqslant \Psi \leqslant 2.5\), then \(\dfrac{\partial T_1}{\partial t} \simeq \dfrac{\partial T}{\partial t}\). The mean change in the temperature of the flux during its interaction with the wall is \(\simeq 10^3\,^\circ\mathrm{K}\), which leads to the result \(\xi \simeq 10^{-3}/\partial t\).

With multiple specular reflections, collisions between particles in the cavity of the manometer will take place. For this (second) process, \(\partial t\) has the meaning of a relaxation time. Then instead of (37) we shall have the full expression for \(\dfrac{\partial T_1}{\partial t}\). Putting in it \(\partial\,\dfrac{mU^2}{2k} \simeq \dfrac{mU^2}{2k}\) and, consequently, \(T_1 \simeq \dfrac{mU^2}{2k}\), as a result of an order-of-magnitude calculation we obtain \(\xi\partial t \simeq 10^{-2}-10^{-3}\).

Thus, it may be considered that in both mechanisms of temperature equalization

\[ \xi\partial t \simeq 10^{-2}-10^{-3}. \tag{38} \]

In order that the quantity \(\xi\) in (34) may be neglected, it is necessary to have \(\partial t > 10^{-1}-10^{-2}\).

Under real conditions, the relaxation time of the translational degrees of freedom and the time of collision with the wall will probably be large because of the low particle density and the large accommodation coefficient inside the cavity of the manometer. Therefore, in what follows we shall neglect the second term in formula (34).

In this case, the relations connecting the number of particles \(N\) per unit volume of the atmosphere and the atmospheric pressure \(P\) with the number of particles \(N_1\) per unit volume and the pressure \(P_1\) in the manometer have the form:

\[ N=N_1\frac{v_1}{v}\frac{1}{\chi}, \tag{39} \]

\[ P=P_1\frac{Tv_1}{T_1v\chi}. \tag{40} \]

The most interesting is the case of large \(\beta\), for which, in accordance with formulas (3) and (4),

\[ \chi \simeq 2\sqrt{\pi}\,\beta. \tag{41} \]

Then for \(N\) and \(P\) we obtain

\[ N=N_1\sqrt{\frac{kT_1}{2\pi m}}\frac{1}{U\sin\theta}, \tag{42} \]

\[ P=P_1\sqrt{\frac{k}{2\pi m}}\frac{T}{\sqrt{T_1}\,U\sin\theta}. \tag{43} \]

Let us recall that the consideration of the properties of a free-molecular flux was made for the case of a homogeneous gas. Therefore the formulas written refer to partial quantities.

On the basis of the foregoing, one may conclude that the problem of interpreting manometer readings is rather complicated, since for its

solution requires knowledge of such characteristics of the atmosphere as its composition and temperature. Determination of the density appears to be a simpler problem, since knowledge of the temperature of the medium is not required. In the case of calibrating the manometer by the number of particles per unit volume, formula (42) is used. When calibrating the manometer by pressure, the number of particles per unit volume is determined from the formula

\[ N=P_1\sqrt{\frac{1}{2\pi m k}}\,\frac{1}{\sqrt{T_1}\,U\sin\theta}. \tag{44} \]

In all cases, for the conversion it is necessary to know the velocity of the satellite, the angle between the velocity and the normal to the cut of the manometer aperture \((90^\circ-\theta)\), and the temperature of the gas in the manometer.

For \(\theta\simeq 0\), as is evident from (3) and (4), \(\chi\simeq 1\), and for \(N\) and \(P\), respectively, we have (thermal effusion)

\[ N=N_1\sqrt{\frac{T_1}{T}}, \tag{45} \]

\[ P=P_1\sqrt{\frac{T}{T_1}}. \tag{46} \]

However, this case is the least favorable with respect to the influence of errors in the determination of \(\theta\) on the measured \(N\) and \(P\).

If between the manometer cavity and the external atmosphere there is not only a diaphragm but also a pipeline of considerable length, then the pressure inside the manometer must increase. Indeed, if the instrument is oriented with the axis of the tube along the vector \(\mathbf U\) (the incident flow, \(\theta=90^\circ\)), then the reverse flow of particles from the manometer cavity into the atmosphere would be a Knudsen flow, for which the tube presents a more considerable resistance than the diaphragm does \(^{21}\). When the axis of the tube is deviated from the direction of the flow, the process becomes more complicated owing to the emergence of an additional resistance also for the incident flow, which, moreover, depends on the molecular weight of the gases (mass selection).

§ 6. TIME CONSTANT OF THE MANOMETER

Measurements of the pressure or of the number of particles per unit volume of the atmosphere are impossible when the manometer aperture is on the rear side of the satellite. The number of particles flying into the manometer will be negligibly small. This follows from formulas (5) and (6) for large \(\beta\). However, there will exist a flow of particles from the manometer, which entered it at the time when the aperture was on the front side of the satellite.

The number of particles flying out per unit time, according to (32), will be equal to

\[ n_1=\frac{N_1(t)\,\bar v_1 S}{2\sqrt{\pi}}, \tag{47} \]

and the corresponding change in pressure over the time \(dt\) will be determined by the quantity

\[ -\frac{dP_1}{P_1}=\frac{\bar v_1 S}{2\sqrt{\pi}\,W}\,dt. \tag{48} \]

As a result, the pressure in the cavity will decrease according to the law

\[ P_1(t)=P_{10}e^{-at}, \tag{49} \]

where \(P_{10}\) is the pressure in the cavity at the moment when the flow of particles into the manometer ceases, and

\[ a=\frac{v_1 S}{2\sqrt{\pi W}}=\sqrt{\frac{k}{2\pi m}}\,\frac{S\sqrt{T_1}}{W}. \tag{50} \]

The time constant \(t_0=\frac{1}{a}\) will be of the order of \(2\cdot 10^{-3}\) sec for \(S=3.14\ \text{cm}^2\), \(W=100\ \text{cm}^3\), \(T_1=300^\circ\text{K}\), and \(m=2.66\cdot 10^{-23}\ \text{g}\) (atomic oxygen). Thus, the pressure in the cavity will rapidly fall to values not measurable even by a very sensitive manometer. Therefore it is possible that, for approximately half the time of revolution of the satellite about its own axes, the manometer will not operate. During this time intensive degassing of the inner surface of the manometer will take place.

§ 7. SOME QUESTIONS CONNECTED WITH THE MEASUREMENT OF PRESSURE BY MEANS OF ROCKETS AND ARTIFICIAL EARTH SATELLITES

When measuring atmospheric pressure by means of rockets and artificial Earth satellites, a number of specific questions arise. We shall briefly discuss some of them.

a) “Impact” ionization. At a satellite velocity of \(\sim 8\ \text{km/sec}\), the braking temperature amounts to tens of thousands of degrees.

Using Saha’s method \(^{22,23}\), one can calculate the possible ionization at this temperature, assuming the existence of thermal equilibrium. In this case atomic oxygen proves to be ionized up to the third ionization potential. At the same time, a simple comparison of the kinetic energies of particles at a velocity \(\sim 8\ \text{km/sec}\) (5.5 and 10 eV for O and \(N_2\)) and the ionization potentials (13.55 and 15.51 eV, respectively) shows that these energies are insufficient for even single ionization to arise. The explanation of the contradiction should be sought in the nonfulfillment of the condition of thermal equilibrium. The braking temperature in the macroscopic sense will not develop. Energy sufficient for ionization will be possessed only by those particles for which the \(x\)-component of the thermal velocity (see Fig. 3) exceeds the value corresponding to the difference between the ionization energy and the energy of particles at a satellite velocity of \(8\ \text{km/sec}\) (5–8 eV). The number of such particles will be equal to:

\[ N_x=\frac{N}{2}\left[1-\Phi\left(\frac{v_x'}{v}\right)\right]. \tag{51} \]

The thermal velocity \(v_x'\), corresponding to an energy of 5 eV, is equal to \(7.3\ \text{km/sec}\). Even at the most probable particle velocity \(v=1.5\ \text{km/sec}\), the number of such molecules will amount to only about \(10^{-9}\%\). Thus it is quite obvious that “impact” ionization may be disregarded.

b) “Impact” dissociation. At high altitudes the atmosphere consists mainly of atomic oxygen and atomic and molecular nitrogen.

As a result of collisions of nitrogen molecules with the body of the satellite, dissociation of nitrogen is possible. The kinetic energy of an \(N_2\) molecule at a velocity \(\sim 8\ \text{km/sec}\) is 9–10 eV; its dissociation energy is 9.764 eV (Gaydon) or 7.347 eV (Herzberg). Ordinary thermal dissociation is the result of strong excitation of vibrational levels produced by many collisions, which is a comparatively lengthy process. If dissociation of a molecule does not occur when it strikes the po—

whether dissociation as a result of collisions of gas molecules will have time to occur during their stay in the manometer cavity (which will be determined by the speed of rotation of the satellite about its own axes), and, apparently, there is no need to take its effect on the manometer readings into account. The number of molecules capable of dissociation can also be estimated by means of formula (51), choosing the corresponding \(v_{\chi'}\).

c) Gas evolution. For some time the surface of the satellite will give off gases captured from the lower layers of the atmosphere. This process will apparently be quite active because of the low density of the surrounding medium. The mass of the satellite will not be large enough for it to form any appreciable “atmosphere” of its own. Therefore one may assume that, after a certain interval of time has elapsed, the gas evolution will greatly decrease and will cease to affect the manometer readings. Successive readings referring to the same altitude will tend toward some limiting value. The time of degassing of the satellite surface will depend on its hermetic sealing and on the choice of surface material. The outer shell of the satellite must be made of materials having a low vapor pressure.

d) Electric charge of the satellite. The satellite will acquire charge under the action of a number of opposing factors. Collisions with electrons will create a negative charge. Collisions with positive ions, as well as the photoelectric effect under the action of the ultraviolet radiation of the Sun, will impart a positive charge to the body. Micrometeorites, which for the same reasons will be charged, may, upon collision with the satellite, change its potential.

In one of the rocket experiments at an altitude of \(140\) km the potential of the rocket reached \(-17\) eV\(^{24}\). At the same time, a sharp increase of the potential was observed in the region of layer \(E\).

One should expect changes in the potential of the satellite at each passage through layer \(F\).

Depending on the sign and magnitude of the charge of the body, the fluxes of ions and electrons in the manometer will change.

e) Knocking atoms out of the surface of the satellite. The sublimation energy of most metals is \(3\)—\(4.5\) eV. A particle moving with a velocity \(\sim 8\) km/sec has an energy of \(5\)—\(10\) eV. Therefore the possibility arises of intensive ejection of atoms from the surface, which may affect the operation of the manometer. Investigations\(^{25}\) have shown that, up to extremely high energies, electrons cannot knock out atoms, while the tearing out of atoms by ions is not thermal in character.

Two mechanisms of the process of tearing out are possible:

1) chemical, which occurs in the formation of volatile surface chemical compounds;

2) physical, depending on two groups of parameters: a) the gas pressure, the nature of the ions, their energy, the density of their flux, the angle of incidence; b) the nature of the target material, its temperature, and the surface configuration of the bombarded crystalline surfaces.

At ion energies of several tens of eV the tearing out is negligibly small; it becomes appreciable only at energies of the order of hundreds of electron-volts. All this, in its main features, may also be applied to bombardment of the surface by neutral atoms and molecules.

f) Natural ionization of the atmosphere. The presence of considerable ionization of the upper layers of the atmosphere (up to \(10^6\) ions/cm\(^3\)) may introduce errors into the measurement of pressure. At the velocity of motion

for the satellite \(8\ \mathrm{km/sec}\), with an inlet aperture diameter of the manometer of \(2\ \mathrm{cm}\) and ionization of \(10^6\ \mathrm{ions/cm^3}\), the maximum value of the ion current caused by natural ionization may reach a value of the order of \(10^{-7}\ \mathrm{a}\). In order that these ions should not enter the working cavity of the manometer and thereby distort the measurements, it is advisable to install special ion traps at the inlet of the manometer; these can be made in the form of the simplest capacitors.

The dimensions of the capacitor plates and the voltage applied between them must be calculated so that all ionized particles remain inside the ion trap.

In this case the distortion of the measured atmospheric pressure will be small, since the concentration of charged particles amounts to no more than \(1\%\) of the concentration of neutral particles.

k) Photoemission. Photocurrents from the electrodes of the manometer, arising owing to the ultraviolet radiation of the Sun, may distort the results of pressure measurements.

As an approximate calculation shows, at an altitude of \(100\ \mathrm{km}\) the maximum value of the photocurrent from one square centimeter of an irradiated metallic surface does not exceed \(2\cdot 10^{-8}\ \mathrm{a}\). At greater altitudes, although harder components of solar radiation are present, in view of their low intensity one should hardly expect a substantial increase in photocurrents. At the same time, it does not appear possible in the present case to achieve a considerable reduction of photocurrents by using materials possessing a high work function, since all conducting surfaces in the region of hard ultraviolet \((\lambda < 1500\ \text{\AA})\) have a quantum yield of approximately one order of magnitude. In order to reduce photocurrents it is best to arrange the manometers so that they are not exposed to direct solar radiation; and if this is impossible, one must strive to reduce as much as possible the irradiated surfaces of the electrodes.

§ 8. APPARATUS

The problem of measuring atmospheric pressure with the aid of an artificial Earth satellite imposes on the instruments a whole series of specific requirements, among which the foremost are: low inertia, small dimensions and weight, low sensitivity to temperature effects, low power consumption for electrical supply, great vibration strength and insensitivity to considerable overloads, stability of the calibration curve over time, ease of servicing, and reliability of autonomous operation.

In rocket investigations of the atmosphere up to altitudes of \(200\ \mathrm{km}\), membrane, thermal, radioactive, magneto-discharge, and ionization manometers with a hot cathode have been used to measure pressure. The last two manometers were intended for measuring the lowest pressures \(^{26}\).

Ordinary magnetic manometers are suitable for measuring pressure in the range \(10^{-2}\)—\(10^{-5}\ \mathrm{mm\ Hg}\); however, with appropriate selection of the electric and magnetic fields and the geometry of the electrodes, the range of pressures measured by manometers of this type can be extended down to \(1\cdot 10^{-7}\ \mathrm{mm\ Hg}\) \(^{27}\).

With ionization manometers of the usual design, widely used in laboratory investigations, it is practically impossible to measure pressures below \(5\cdot 10^{-8}\ \mathrm{mm\ Hg}\). The reason for this is the so-called “parasitic” collector current, which limits the sensiti-

...the sensitivity of the ionization manometer. This current is produced by photoemission from the collector under the action of the soft X-ray radiation incident upon it, which arises as a result of the braking of the electrons emitted by the cathode in the material of the grid.

In the literature28–31 several special designs of ionization manometers have been described which are capable of measuring very low pressures, down to \(10^{-10}\) mm Hg, by sharply reducing the magnitude of the parasitic collector current.

One possible design variant is shown in Fig. 10. Inside the cylindrical grid \(1\), along its axis, a thin wire is stretched, serving as the ion collector \(2\), and outside the grid, parallel to the collector, a rectangular cathode \(3\) is mounted. Since the collector has an extremely small irradiated surface, photoemission from it is considerably reduced. In addition, in order to reduce the energy of the X-ray quanta arising when electrons are braked on the surface of the grid, the latter is made of a material with a low atomic number.

The ionization manometer requires stabilization of the emission current of the heated cathode. Stabilization is necessary because, during operation of the manometer, the emission current, owing to local poisoning of the cathode, does not remain constant even at an unchanged value of the heating current. The simplest method by which it is possible to stabilize the cathode emission current is to introduce into the manometer an additional grid \(4\), located in the immediate vicinity of the cathode and having, relative to it, a small negative potential (of the order of several volts), the magnitude of which is automatically regulated during operation of the instrument.

Fig. 10.

The calibration curves of manometers depend on the type of gas, and therefore, in the case where the composition of the gas is not known exactly, an instrument capable not only of measuring the pressure and density of the rarefied gas but also of determining its composition is of considerable interest. Such instruments are the radio-frequency mass spectrometer and the omegatron32.

Although the absolute value of the sensitivity of the omegatron is the same as that of the ionization manometer (of the order of 10 per 1 mm Hg), nevertheless, in view of the fact that the emission current of the ion source of the omegatron is approximately 1000 times smaller than that of the tungsten cathode used in the ionization manometer, at a pressure of \(10^{-9}\) mm Hg currents of \(10^{-13}\)–\(10^{-14}\) are obtained in the omegatron, for the measurement of which it is necessary to use special electrometric amplifiers.

The sensitivity of manometers depends on the nature of the gas, since different gases have different ionization potentials and different courses of the ionization-probability curves (the number of ions formed by each electron over a path length of 1 cm).

Table IV gives the values of the relative sensitivity for an ionization manometer when measuring the pressure of various gases, and the ionization potentials of these gases. Since calibration of the manometer is, as a rule, carried out with dry air, its sensitivity in this case is taken as unity.

As can be seen from the table presented, the change in sensitivity even in the case of complete replacement of air by nitrogen, argon, or molecular oxygen does not exceed \(\pm 10\%\).

The calibration curves of magnetic manometers for various gases are arranged in the same sequence as for ionization manometers; however, the sensitivity of a magnetic manometer is not a constant quantity, but changes with pressure. Consequently, it becomes necessary to calibrate each magnetic manometer individually, whereas for an ionization manometer one may restrict oneself to determining its sensitivity in air only.

Table IV

Replacement of molecular oxygen by atomic oxygen can hardly lead to a significant change in the sensitivity of an ionization manometer, since the ionization potential in this case changes by only \(1\) eV. Nevertheless, the presence of atomic oxygen, whose chemical activity is considerably higher than that of molecular oxygen, may cause a change in the work function and emissive capacity of the heated filament of the ionization manometer, owing to the fact that, as the pressure increases, an ever larger amount of oxygen will be adsorbed on the filament. This once again confirms the necessity of continuous monitoring of the emission current.

In addition, as a result of the chemical action of oxygen, the heated filament may be destroyed rather intensively, which limits the service life of the manometer. However, it may be assumed that a heated tungsten filament \(0.1\) mm thick, operating in an oxygen atmosphere at a pressure of \(1 \cdot 10^{-5}\) mm Hg, should not burn out for approximately 100 hours.

As for the magnetic-discharge manometer, the presence of atomic oxygen apparently may lead to the formation of surface oxide films on the cathode plates; this, however, is unlikely to have a substantial effect on changes in the sensitivity of the manometer, since preliminary studies show that the material of the cathode plates has practically no influence on the characteristics of the manometer.

One may attempt to carry out density measurements also with an instrumentless satellite, if it is observed visually \(^{33}\), or with a satellite having only a transmitter serving to determine changes in its orbit by radio methods \(^{34}\).

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