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A CRITICAL REVIEW OF DATA ON THE TEMPERATURE OF THE IONOSPHERE
N. C. Gerson*)
CONTENTS
§ 1. Introduction . . . . . . . . . . . . . . . . . . . . . . 564
§ 2. Basic equations of the ionosphere . . . . . . . . . . . 566
§ 3. Model of the atmosphere . . . . . . . . . . . . . . . . 569
§ 4. Temperature and number of collisions . . . . . . . . . 572
§ 5. Barometric formula and number of collisions . . . . . 580
§ 6. Temperature and scale height . . . . . . . . . . . . . 583
§ 7. Temperature and electron concentration . . . . . . . . 586
§ 8. Temperature and recombination coefficient . . . . . . 595
§ 9. Temperature and a function describing the inflow of electrons . . . 604
§ 10. Thermal equilibrium in the mesosphere . . . . . . . . 612
§ 11. Electron temperature . . . . . . . . . . . . . . . . . 614
§ 12. Temperature of the ionosphere . . . . . . . . . . . . 615
§ 13. Heating by electric currents . . . . . . . . . . . . . 616
§ 14. Direction of further investigations . . . . . . . . . 617
NOTATION
| \(d\) — radius of action of attractive forces \((\mathrm{cm})\) | \(n_e\) — number of electrons \((\mathrm{cm}^{-3})\) |
| \(e\) — electron charge | \(q\) — number of newly formed electrons \((\mathrm{cm}^{-3}\,\mathrm{sec}^{-1})\) |
| \(f\) — frequency of electromagnetic waves \((\mathrm{cps})\) | \(t\) — time \((\mathrm{sec})\) |
| \(f_c\) — critical frequency | \(u, v\) — velocity of ions and molecules \((\mathrm{cm}/\mathrm{sec})\) |
| \(g\) — acceleration due to gravity \((\mathrm{cm}/\mathrm{sec}^{2})\) | \(z\) — height above the earth’s surface \((\mathrm{cm})\) |
| \(h\) — Planck’s constant \((\mathrm{erg}\cdot\mathrm{sec})\) | \(z'\) — effective height \((\mathrm{cm})\) |
| \(k\) — Boltzmann’s constant \((\mathrm{erg}/\mathrm{degree})\) | \(H=\dfrac{kT}{gmU}\) — height of a homogeneous atmosphere, scale height \((\mathrm{cm})\) |
| \(m\) — mass of a molecule or ion \((\mathrm{g})\) | \(J\) — intensity of radiation |
| \(m_e\) — mass of an electron \((\mathrm{g})\) | |
| \(n\) — number of ions \((\mathrm{cm}^{-3})\) |
*) N. C. Gerson, Reports on Progress in Physics 14, 316—365 (1951). Reviews on the ionosphere and, in particular, some analysis of the questions considered here, see in the articles by Ya. L. Al’pert, published in the journal UFN 143, 144, and also in the book by I. A. Khvostikov 150, in the articles by T. G. Megrelishvili 151, 152 and N. M. Shtaude 153. (Transl. note.)
$M$ — mean molecular weight of the atmosphere
$N$ — number of neutral particles $(\mathrm{cm}^{-3})$
$P$ — total barometric pressure $(\mathrm{dyn}/\mathrm{cm}^{2})$
$R$ — radius of the Earth $(\mathrm{cm})$
$\overline{R}$ — gas constant of dry air $(\mathrm{cm}^{2}/\mathrm{sec}^{2})$
$T$ — gas temperature $(^\circ)$
$U$ — atomic mass unit (g)
$T_e$ — electron temperature $(^\circ)$
$W$ — ionization potential (eV)
$Z=\dfrac{(z-z_{\max})}{H}$ — height, expressed in units of the height scale, measured from $z_{\max}$
$\alpha$ — recombination coefficient
$\beta$ — absorption coefficient
$e$ — base of natural logarithms
$\delta$ — declination of the Sun $(^\circ)$
$\theta$ — additional angle to the latitude
$\eta$ — Stefan’s constant
$\sigma$ — collision cross section of molecules $(\mathrm{cm}^{2})$
$\overline{\lambda}$ — mean free path $(\mathrm{cm})$
$\chi$ — zenith distance of the Sun $(^\circ)$
$\mu$ — refractive index
$\rho$ — atmospheric density $(\mathrm{g}/\mathrm{cm}^{3})$
$\nu$ — number of collisions between neutral particles $(\mathrm{sec}^{-1})$
$\nu'$ — number of collisions between electrons and neutral particles $(\mathrm{sec}^{-1})$
$\nu''$ — number of collisions between electrons and ions $(\mathrm{sec}^{-1})$
$\nu_e$ — number of collisions between electrons $(\mathrm{sec}^{-1})$
CONSTANTS
\[ e=4.8025\cdot 10^{-10}\ \text{abs. el.-stat. units} \]
\[ g_0=980.665\ \mathrm{cm}/\mathrm{sec}^{2} \]
\[ h=6.63\cdot 10^{-27}\ \mathrm{erg}\cdot\mathrm{sec} \]
\[ k=1.3805\cdot 10^{-16}\ \mathrm{erg}/\mathrm{mol}\cdot{}^\circ \]
\[ m_e=9.107\cdot 10^{-28}\ \mathrm{g} \]
\[ M_0=28.979 \]
\[ R=6.3676\cdot 10^{8}\ \mathrm{cm} \]
\[ U=1.660\cdot 10^{-24}\ \mathrm{g} \]
\[ e=2.7183 \]
\[ \eta=5.735\cdot 10^{-5}\ \mathrm{erg}/\mathrm{cm}^{2}/\mathrm{sec} \]
\[ \rho_0=1.2929\ \mathrm{g}/\mathrm{cm}^{3} \]
The subscript “0” denotes the mean value at sea level at $45^\circ$ north latitude.
In order of importance, the following methods may be indicated, which are used for calculating temperature from ionospheric research data:
a) temperature and number of collisions,
b) temperature and the height scale of auroras,
c) temperature and electron concentration,
d) temperature and ion influx,
e) temperature and recombination coefficient.
Method e), owing to its imperfection, does not make it possible to obtain accurate temperature values. With the aid of method d), some indications of the diurnal variations of temperature can be obtained.
The table given below contains temperature values in the altitude region $(100 \div 400)$ km, which were obtained by different methods:
Temperature \((T)\) of the upper atmosphere (in degrees)
| Altitude (km) | from the number of collisions | from the scale height of auroras | from electron concentration |
|---|---|---|---|
| 100 | 300 | 219 | — |
| 150 | 825 | 531 | — |
| 200 | 1350 | 1580 | — |
| 250 | 2175 | 2073 | — |
| 300 | 2400 | 2455 | — |
| 330 | — | — | 1530—2680 |
| 350 | 3225 | 2704 | — |
| 400 | 3450 | — | — |
| 500 | — | — | 2200—3910 |
These values should be regarded as probable average annual values of the temperature at middle latitudes.
The electron temperature and the gas temperature are in fact the same in the ionosphere. If free electrons acquire an excess of energy (during ionization), this energy is rapidly transferred to the gas and to other particles.
At altitudes of \(100 \div 400\) km the atmosphere is in thermal equilibrium and, consequently, the velocities of the particles have a Maxwellian distribution. Therefore the kinetic temperature is equivalent to the gas temperature. It is possible that at still greater heights of the terrestrial atmosphere there are likewise no deviations from the Maxwellian distribution of velocities.
The assumption that the atmosphere below 400 km is completely ionized is not consistent with experimental data; the temperature values calculated on this basis do not agree with the observed density values.
Experimental data show that in the altitude region \((100 \div 400)\) km there are no significant diurnal temperature variations. This conclusion is apparently confirmed by the results of preliminary calculations of the rate of heating of the atmosphere caused by the release of excess energy absorbed during ionization. However, it is not known how the atmosphere is heated as a result of as yet undetected absorption processes, or because of large currents arising in the ionosphere, or else as a result of bombardment of the atmosphere by a large number of high-energy electrons.
It is difficult to suppose that the temperature of the terrestrial atmosphere has values larger than those indicated in the table. Indeed, it can be shown qualitatively that the maximum values of the temperature are reached in the ionospheric layers, near or
somewhat above 400 km. Above the region of this temperature maximum the temperature apparently falls, and the limiting value of the atmospheric temperature corresponds to the temperature of interstellar space. The temperature maximum arises near the ionospheric layers because of various processes of absorption and heating occurring in these regions, which still contain a sufficiently large number of particles.
§ 1. INTRODUCTION
Physicists who have occasion to engage in studies of the upper atmosphere are struck not only by the large number of problems as yet unresolved, but also by the rapid growth of activity in this field. In particular, the various aspects connected with the study of temperature have been the subject of a great many investigations, especially because they are intertwined with such questions as the pressure of the upper atmosphere, density, the mean free path of molecules, etc.
There is much literature on investigations of the temperature of the upper atmosphere, the results of which have been summarized in various reviews. However, the absence of a critical analysis of the basic assumptions of the various works—often entirely different, if not opposite—has been a serious obstacle to obtaining a unified dependence of temperature on altitude.
In order to give a complete picture of the present state of the question of the temperature of the upper atmosphere, it is necessary to mention some early works. Some of them ^27,75,134 are now, to a considerable extent, obsolete. A number of research results ^2,52,81 were subsequently corrected. Following these early works, many investigations were carried out by various methods, and their results are, in general, in agreement for altitudes below 100 km. Above this level the results vary substantially from author to author and depend both on the assumptions made and on the method used.
Although direct temperature measurements have been performed, with varying degrees of accuracy, only up to an altitude of 100 km, theoretical investigations cover regions up to altitudes on the order of 400 km. To determine the temperature below 100 km, the following methods were used:
a) direct sounding by means of kites, balloons, aircraft, and rockets;
b) direct studies of the propagation of atmospheric compression waves;
c) study of the absorption of solar radiation in the ozonosphere;
d) mathematical analysis of atmospheric tides and oscillations;
d) studies of twilight sky light;
e) photographing meteors;
zh) spectrographic studies of the glow of the night sky and auroras; and
z) the theory of equilibrium radiation in the upper atmosphere.
The results of these observations were summarized in various works\(^{46в,112}\).
To obtain data on the temperature above 100 km, conclusions drawn from consideration of various parameters by means of methods zh) and z) were used almost exclusively. It should be expected that radio studies of the ionosphere will serve as the basis for obtaining the diurnal, seasonal, and geographical dependence of the temperature and density of the ionospheric layers. The importance of these questions is discussed in various monographs that consider the connection between the ionosphere and the meteorology of the upper atmosphere. Until now, temperature values obtained in various works have been summarized in reviews\(^{54,98}\).
The methods that use, for determining the temperature, the relation between it and one of the ionospheric parameters are the following:
a) temperature and number of collisions;
b) the barometric formula and number of collisions;
v) temperature and height scale;
g) temperature and electron concentration;
d) temperature and recombination coefficient;
e) temperature and influx of ions.
There are two further methods that are not discussed here, namely: zh) temperature equilibrium between incident radiation, neutral and ionized particles, and z) temperature equilibrium between incident radiation, neutral particles, and dissociated particles.
Method a) requires additional knowledge of the density or number of neutral particles at the height under consideration. Since neither of these parameters is usually known from other sources, the application of a) is severely limited. This limitation is partly removed when method b) is used. In this case, instead of barometric pressure, crude radio measurements of the number of collisions are used. In most cases, determination of the height scale requires knowledge of the electron concentration; therefore, until an independent method exists for determining the height scale, method v) includes method g). Their separation is possible when using studies of auroras, namely from analysis of the attenuation of their glow by a method analogous to the method for determining absorption of radiation in the ionosphere. In determining the temperature by means of method g), certain simplifying assumptions are made which undoubtedly somewhat reduce the accuracy of the corresponding calculations.
In method d), the results of laboratory investigations of the relation between the recombination coefficient and temperature are used as applied to the ionosphere. In method e), diurnal temperature changes are considered which, by causing the nocturnal compression of the atmosphere, explain the nocturnal changes in ionization density.
In order to obtain a better dependence of temperature on altitude in the region \((100 \div 400)\) km, this article critically examines the interrelation between the various parameters of the ionosphere and temperature. Since the analysis of the properties of the upper atmosphere depends on the properties of the lower atmosphere, below a model of the atmosphere is given, based on the best data selected from recent investigations (§ 3). For completeness, the basic equations used in the text are considered in § 2.
In § 14, investigations are enumerated which must be carried out in order to resolve the contradictions existing in current theories and to advance our knowledge of the temperature of the atmosphere.
§ 2. BASIC EQUATIONS OF THE IONOSPHERE
Expressions are well known which establish the functional dependence between temperature and various parameters of the ionosphere; the diurnal dependence of the ionization density has been calculated \(^{26*}\). In them it is assumed that the atmosphere absorbs the incident ultraviolet monochromatic radiation of the Sun, that the absorption coefficient has a constant value, and that, upon absorption of a unit quantity of radiation energy, \(n''\) ions are formed. It is also assumed that ionization processes are not subject to the influence of winds and diffusion, and that between ionized particles there occurs only recombination according to the quadratic law. Under these conditions the character of the change in ionization density is described by the equation
\[ \frac{dn}{dt}=q-\alpha n^2, \tag{2,1} \]
if it is assumed that the particles are singly ionized. The number of newly formed ion pairs is determined in the following way.
Under the assumption that the ionizing radiation falls on the atmosphere at an angle \(\chi\), the following equation is obtained for the intensity of the radiation crossing the atmosphere:
\[ dJ=\beta \rho J \sec \chi \, dz. \tag{2,2} \]
Using further the barometric equation \(dP=-\rho gdz\) and the ideal-gas equation \(\rho=\dfrac{mP}{kT}\), one can rewrite (2,2)
*) The theory of ionization of the atmosphere and of the formation of the so-called simple layer was given independently and earlier than in \(^{25}\) by S. N. Kryukov \(^{148}\). (Translator’s note.)
in the form
\[ \frac{dJ}{J}=\beta\rho_i T_i\sec\chi\left[\exp\left\{-\int\frac{mg}{kT}\,dz\right\}\right]\cdot\frac{dz}{T}. \tag{2,3} \]
If the number of ions formed per \(cm^3/sec\) is equal to \(Q=\beta n''J\rho\), then from (2,3) it follows (the index \(i\) refers to the conditions at the level \(z_i\))
\[ Q=\beta n''J_\infty\rho_i\left(\frac{T_i}{T}\right)e^G, \tag{2,4} \]
where
\[ G=-\int\frac{mg}{kT}\,dz+\rho_iT_i\beta\cdot\sec\chi\int\left[\exp-\int\frac{mg}{kT}\,dz\right]\frac{dz}{T} \]
and \(J_\infty\) is the intensity of the radiation at the top of the atmosphere.
If it is assumed that the temperature varies according to the linear law \(T=T_i(1+bz)\), where \(z\) is measured from the level \(z_i\), and \(g\) is constant, then one obtains
\[ Q=\beta n''J_\infty\rho_i\left(\frac{T}{T_i}\right)^{-(1+A)} \exp-\left[\rho_i\beta H_i\sec\chi\left(\frac{T}{T_i}\right)^{-A}\right], \tag{2,5} \]
where \(A=\dfrac{1}{bH_i}\) and \(H_i=\dfrac{kT_i}{mg}\). When \(b=0\), i.e. the atmosphere is isothermal, (2,5) is transformed into the well-known equation of a “simple” layer
\[ q=\beta n''J\rho\exp\left\{-\left(\frac{z}{H}+\beta\rho He^{-\frac{z}{H}}\sec\chi\right)\right\}. \tag{2,6} \]
In (2,6) \(\rho=\rho_i\), \(T=T_i\), \(H=H_i\), and the index \(\infty\) has been omitted from \(J\). These equations are unsuitable for oblique incidence \((\chi<80^\circ)\). The height at which the number of newly formed ions is maximal is, from (2,6), for a given value of the zenith angle \(\chi\):
\[ z_{\max}=H\cdot\ln(\beta\rho H\sec\chi). \tag{2,7} \]
The number of newly formed ions at this height assumes the value
\[ q=\beta n''\cdot J\cdot\rho\exp-\left[\frac{z_{\max}}{H}+1\right]. \tag{2,8} \]
It is clear that \(q\) reaches a maximum when \(\dfrac{z_{\max}}{H}=0\), i.e. when \(\beta\rho H=\cos\chi\). Consequently, the maximum number of newly formed ions is equal to
\[ q'_{\max}=\beta\rho n''\frac{J}{e}=\frac{n''J\cos\chi}{eH}. \tag{2,9} \]
The value \(q_{\max}\) corresponding to the case when \(\chi=0\), i.e. \(\beta\rho H=1\), is equal to
\[ q_{\max}=\frac{n''J}{eH}. \tag{2,10} \]
From (2.6), (2.9), and (2.10) it follows that
\[ q=q_{\max} e^{f(T)}=q'_{\max} e^{F(T)}, \tag{2.11} \]
where
\[ f(T)=1-Z-e^{-Z}\sec\chi,\qquad F(T)=1-Z-e^{-Z}. \tag{2.12} \]
Under equilibrium conditions, when \(\dfrac{dn}{dt}=0\), from (2.1) one obtains the equation
\[ n=\sqrt{\frac{q}{\alpha}}, \tag{2.13} \]
which in the isothermal case gives, for constant \(\alpha\),
\[ n=n_{\max} e^{\frac{f(T)}{2}}=n'_{\max} e^{\frac{1}{2}F(T)}, \tag{2.14} \]
where \(n_{\max}\) denotes the electron density at the level of maximum ionization \((\chi=0)\), and \(n'_{\max}\) is the electron density at this same level when \(\chi\ne 0\).
Assuming a linear dependence of \(T\) on height, we obtain from (2.5)
\[ n=\sqrt{\left(\frac{n'J\cdot \rho_i\beta}{\alpha}\right)} \cdot \left(\frac{T}{T_i}\right)^{(1+A)} \cdot \exp\frac{1}{2}\left\{\rho_i\beta H\sec\chi\left(\frac{T}{T_i}\right)^{-A}\right\}. \tag{2.15} \]
When \(\chi=0\) and the first three terms of the expansion of the exponential function are used, (2.12) and (2.14) take the form
\[ \frac{1}{2}f(T)=-\frac{1}{4}Z^2;\qquad n=n'_{\max}\left(1-\frac{1}{4}Z^2\right). \tag{2.16} \]
From (2.9) and (2.13) the following relation is obtained between two values of the electron concentration \(n_a\) and \(n_b\), corresponding to different conditions of ion formation:
\[ \frac{n_a}{n_b}= \sqrt{\frac{J_aH_b\sigma_b\cos\chi_a}{J_bH_a\sigma_a\cos\chi_b}}. \tag{2.17} \]
For grazing incidence of the radiation, instead of (2.6) one obtains the equation
\[ q=\beta n''J\cdot\rho\exp\left[-\left(\frac{z}{H}+\beta\rho D\right)\right], \tag{2.18} \]
where
\[ D=\int_{z=\infty}^{z=C\sec i-R} e^{\frac{z}{H}} \sqrt{\frac{(R+z)^2}{(R+z)^2-C^2}}\cdot dz \tag{2.19} \]
and \(C\) is a constant quantity depending on the specified radiation flux. Equation (2.18), which should be used when \(\chi>80^\circ\), may be rewritten in the form
\[ q=q_{\max}e^{\psi(T)}, \tag{2.20} \]
where
\[ \psi(T)=1-\frac{(z-z'_{\max})}{H} -\left(\frac{D}{H}\right)\exp\left(\frac{z'_{\max}}{H}\right). \tag{2.21} \]
The last equation is solved by the method of numerical integration.
In what follows, the refractive index of the wave is also used:
\[ \mu^2=1-\frac{ne^2}{\pi mf^2}. \tag{2.22} \]
At the point of reflection of the wave, where \(\mu=0\),
\[ n=\frac{\pi mf^2}{e^2}. \tag{2.23} \]
The number of ions is calculated from the experimental values of \(f\). In this work the expression for \(\mu\) derived by “Sellmeyer” is used. This derivation is in agreement with the theoretical conclusions of Darwin\(^{30}\) and V. L. Ginzburg\(^{48}\), and with experimental results\(^{84}\).
§ 3. MODEL OF THE ATMOSPHERE
Throughout our entire investigation it will be necessary to refer to a “model of the atmosphere.” As a result of careful discussion of a number of studies\(^{13, 32, 46в, 51, 54, 56, 62в, 64, 74, 98в, 99, 132, 137}\), in the present work the atmospheric model shown in Fig. 1 and in Table I has been chosen (see also \(^{92, 108, 109}\)). The temperature values are upper limits of the quantities obtained by various investigators, and are adopted for \(45^\circ\) north latitude, respectively for the months of January and August. It has been suggested\(^{135}\) that above the \(E\) layer the temperature decreases, but, since this result is still far from reliable, it is not taken into account here*).
Fig. 1. Dependence of temperature on altitude in the adopted model of the atmosphere.
Using the equations of hydrostatics and assuming that the dissociation of molecular oxygen or nitrogen occurs linearly—
) Independently, Ya. L. Al’pert\(^{144, 145}\) suggested that a third temperature minimum with \(T \cong 200^\circ\) may exist in the atmosphere, situated somewhat above the \(E\) layer of the ionosphere. The author takes into account neither this result of works\(^{144, 145}\), nor a number of other results directly related to the questions considered in this article. See the translator’s following notes. (Translator’s note.*)
within the given region and that gravitation is inversely proportional to the square of the distance from the center of the Earth, one can calculate the pressure values; they are also given in Table I.
Table I
Atmospheric data up to an altitude of 400 km
| Altitude (km) | Temperature (° abs. K) | Pressure (mbar) | Number of particles (cm\(^{-3}\)) \(\left(N=\dfrac{P}{kT}\right)\) | Density (g/cm\(^3\)) \(\left(\rho=\dfrac{PM}{M_0 kT}\right)\) | Classical value of the reciprocal mean free path \(\left(\lambda^{-1}=\sqrt{2\pi n\sigma^2}\right)\), \(\sigma=3{,}2\times10^{-8}\) cm |
|---|---|---|---|---|---|
| 0 | 288 | \(1{,}01325\cdot10^3\) | \(2{,}549\cdot10^{19}\) | \(1{,}226\cdot10^{-3}\) | \(8{,}625\cdot10^{-6}\) |
| 10,8 | 218 | \(2{,}349\cdot10^2\) | \(7{,}805\cdot10^{18}\) | \(3{,}756\cdot10^{-4}\) | \(2{,}816\cdot10^{-5}\) |
| 32 | 218 | \(8{,}608\cdot10^0\) | \(2{,}860\cdot10^{17}\) | \(1{,}376\cdot10^{-5}\) | \(7{,}685\cdot10^{-4}\) |
| 62 | 330 | \(2{,}041\cdot10^{-1}\) | \(4{,}480\cdot10^{15}\) | \(2{,}156\cdot10^{-7}\) | \(4{,}906\cdot10^{-2}\) |
| 84 | 200 | \(1{,}204\cdot10^{-2}\) | \(4{,}361\cdot10^{14}\) | \(2{,}098\cdot10^{-8}\) | \(5{,}041\cdot10^{-1}\) |
| 94 | 262,5 | \(2{,}834\cdot10^{-3}\) | \(7{,}821\cdot10^{13}\) | \(3{,}763\cdot10^{-9}\) | \(2{,}811\) |
| 100 | 300 | \(1{,}488\cdot10^{-3}\) | \(3{,}593\cdot10^{13}\) | \(1{,}429\cdot10^{-9}\) | \(6{,}118\) |
| 300 August | 2700 | \(1{,}142\cdot10^{-5}\) | \(3{,}064\cdot10^{10}\) | \(1{,}127\cdot10^{-12}\) | \(7{,}174\cdot10^3\) |
| 300 January | 2000 | \(3{,}983\cdot10^{-5}\) | \(1{,}443\cdot10^{10}\) | \(5{,}307\cdot10^{-13}\) | \(1{,}542\cdot10^4\) |
| 400 August | 3900 | \(5{,}692\cdot10^{-6}\) | \(1{,}057\cdot10^{10}\) | \(3{,}619\cdot10^{-13}\) | \(2{,}079\cdot10^4\) |
| 400 January | 2850 | \(1{,}546\cdot10^{-6}\) | \(3{,}929\cdot10^9\) | \(1{,}345\cdot10^{-13}\) | \(5{,}594\cdot10^4\) |
It is assumed that oxygen dissociates in the region of \(94\div100\) km and that above 100 km it is completely in the atomic state\(^{98,104,140}\). Despite the fact that it is now assumed that at great altitudes nitrogen is dissociated, it is still impossible to indicate with certainty any altitude at which atomic nitrogen would predominate. The ultraviolet radiation of the Sun arriving at the altitudes considered here is possibly not intense enough for direct dissociation of nitrogen. However, atomic nitrogen can also be formed in sufficient quantity as a result of predissociation of \(N_2\), photodissociation of \(N_2^+\)\(^{17,60,120}\), and so forth.
In calculating Table I it was assumed that the concentration of nitrogen compounds with oxygen, such as \(N_2O\), \(NO\), etc., is small. The relative value of the concentration of atomic nitrogen, which is obtained in interpreting spectra of aurorae,
calculation with altitude^34. From photographs of auroras it was established^17 that the fraction of dissociated nitrogen increases from a value of 0.07 near 100 km to 0.75 near (600–700) km. There are, however, grounds for supposing that in the auroral zone the percentage of atomic nitrogen is greater than at lower latitudes; therefore the result just mentioned cannot be applied to the atmosphere over the entire globe.
Within the limits of admissible errors, theory and experiment give the same concentrations of atomic nitrogen and show that nitrogen is not yet completely dissociated in this region of altitudes. It may be pointed out, for example, that the interpretation of the spectra of the highest polar auroras indicates the presence of considerable quantities of molecules \(N_2\) and \(N_2^+\)^12,127,128. It has also been stated in the literature^45 that the dissociation of nitrogen begins below 160 km and reaches, at an altitude of 400 km, 80% by day and 15% by night. Judging from present-day data, this assumption is somewhat arbitrary. In the model of the atmosphere adopted here it is postulated that at an altitude of 200 km nitrogen is still not dissociated; then the dissociation increases linearly with altitude and reaches 25% at an altitude of 400 km. In work^88 it is also assumed that complete mixing of the constituent parts of the atmosphere exists up to 300–350 km and that the temperature gradient is 4° per kilometer; this value differs from that adopted in the atmospheric model given above. In the present work it is assumed that complete mixing of the constituent parts of the atmosphere exists up to 400 km. With the assumptions made above, the mean values of the molecular weight of the atmosphere have the values given in Table II, where \(C_2 = 3.5 \cdot 10^{-4}\) cm; \(C_3 = 9.4 \cdot 10^6\) cm; \(C_4 = 8.09 \cdot 10^{-9}\) cm; \(C_5 = 2 \cdot 10^7\) cm.
Table II
Mean values of the molecular weight of the atmosphere at different altitudes
| Molecular weight | Altitude (km) | Percentage of dissociation \(O_2\) | Percentage of dissociation \(N_2\) |
|---|---|---|---|
| \(M = M_0 = 28.98\) | \(0 \div 94\) | 0 | 0 |
| \(M = M_0[1 - C_2(z - C_3)]^{-1}\) | \(94 \div 100\) | 0—100 | 0 |
| \(M = M' = 23.95\) | \(100 \div 200\) | 100 | 0 |
| \(M = M'[1 - C_4(z - C_5)]^{-1}\) | \(200 \div 400\) | 100 | 0—25 |
| \(M = 20.61\) | 400 | 100 | 25 |
Table III gives the values of pressure, mean free path, and particle density at altitudes of 300 and 400 km, calculated for different temperature values. In the calculations it is assumed that, beginning at 100 km (where \(T = 300^\circ\)), the temperature varies linearly with altitude.
Table III
Values of pressure, density, and mean free path at different temperatures at altitudes of 300 and 400 km
| \(T^\circ\) | \(P\) (µbars) | \(N\) (\(\mathrm{cm}^{-3}\)) | \(\bar{\lambda}\) (cm) |
|---|---|---|---|
| \(z = 300\) km | |||
| 300 | \(4{,}238 \cdot 10^{-11}\) | \(1{,}023 \cdot 10^{6}\) | \(2{,}148 \cdot 10^{8}\) |
| 500 | \(2{,}281 \cdot 10^{-9}\) | \(3{,}305 \cdot 10^{7}\) | \(6{,}652 \cdot 10^{6}\) |
| 800 | \(4{,}850 \cdot 10^{-8}\) | \(4{,}392 \cdot 10^{8}\) | \(5{,}005 \cdot 10^{5}\) |
| 1 200 | \(4{,}328 \cdot 10^{-7}\) | \(2{,}613 \cdot 10^{9}\) | \(8{,}413 \cdot 10^{4}\) |
| 2 000 | \(3{,}983 \cdot 10^{-6}\) | \(1{,}443 \cdot 10^{10}\) | \(1{,}524 \cdot 10^{4}\) |
| 2 700 | \(1{,}142 \cdot 10^{-5}\) | \(3{,}064 \cdot 10^{10}\) | \(7{,}174 \cdot 10^{3}\) |
| 3 000 | \(1{,}589 \cdot 10^{-5}\) | \(3{,}837 \cdot 10^{10}\) | \(5{,}729 \cdot 10^{3}\) |
| 4 000 | \(3{,}557 \cdot 10^{-5}\) | \(6{,}442 \cdot 10^{10}\) | \(3{,}412 \cdot 10^{3}\) |
| 5 000 | \(6{,}087 \cdot 10^{-5}\) | \(8{,}819 \cdot 10^{10}\) | \(2{,}493 \cdot 10^{3}\) |
| 10 000 | \(2{,}166 \cdot 10^{-4}\) | \(1{,}569 \cdot 10^{11}\) | \(1{,}401 \cdot 10^{3}\) |
| \(z = 400\) km | |||
| 300 | \(2{,}230 \cdot 10^{-14}\) | \(5{,}385 \cdot 10^{2}\) | \(4{,}082 \cdot 10^{11}\) |
| 600 | \(3{,}645 \cdot 10^{-11}\) | \(4{,}401 \cdot 10^{5}\) | \(4{,}995 \cdot 10^{8}\) |
| 1 050 | \(4{,}105 \cdot 10^{-9}\) | \(2{,}832 \cdot 10^{7}\) | \(7{,}762 \cdot 10^{6}\) |
| 1 650 | \(8{,}676 \cdot 10^{-8}\) | \(3{,}809 \cdot 10^{8}\) | \(5{,}771 \cdot 10^{5}\) |
| 2 850 | \(1{,}546 \cdot 10^{-6}\) | \(3{,}939 \cdot 10^{9}\) | \(5{,}594 \cdot 10^{4}\) |
| 3 900 | \(5{,}692 \cdot 10^{-6}\) | \(1{,}057 \cdot 10^{10}\) | \(2{,}079 \cdot 10^{4}\) |
| 4 350 | \(8{,}501 \cdot 10^{-6}\) | \(1{,}416 \cdot 10^{10}\) | \(1{,}553 \cdot 10^{4}\) |
| 5 850 | \(2{,}230 \cdot 10^{-5}\) | \(2{,}761 \cdot 10^{10}\) | \(7{,}960 \cdot 10^{3}\) |
| 7 350 | \(4{,}194 \cdot 10^{-5}\) | \(4{,}133 \cdot 10^{10}\) | \(5{,}318 \cdot 10^{3}\) |
| 10 000 | \(8{,}592 \cdot 10^{-5}\) | \(6{,}224 \cdot 10^{10}\) | \(3{,}532 \cdot 10^{3}\) |
§ 4. TEMPERATURE AND NUMBER OF COLLISIONS
4.1. Experimental data
Before conducting a theoretical discussion of the question of the number of collisions, it is useful to present the results of the corresponding experiments. The values of the number of collisions obtained by various authors are given in Table IV and in Fig. 2. Several methods were used to measure the number of collisions. For layer \(E\), the phenomenon of cross-modulation, arising as a result of the interaction, was used.
Table IV
Results of measurements of the collision number
| Altitude (km) | Collision number | Author |
|---|---|---|
| 70 ÷ 95 | 1.1·10⁶ ÷ 2·10⁵ | Ratcliffe and Shaw 106 |
| 85 | 1.2·10⁶ | Huxley, Foster, and Newton 63 |
| 100 | 1.0·10⁶ ÷ 3·10⁵ | Chapman and Bartels 26 |
| 100 | 1.9·10⁵ | Bailey and Martyn 8 |
| 220 | 1.2·10⁴ | Toshiwal, Pant, and Baipai 124 |
| 250 ÷ 400 | 5.0·10³ ÷ 1.2·10³ | Wilbig 130 |
| 265 | 3.6·10³ ÷ 1.6·10³ | Farmer and Ratcliffe 40б |
| 265 | 3.6·10³ ÷ 1.6·10³ | Eckersley 37 |
| 265 | 3.6·10³ ÷ 1.6·10³ | Farmer and Ratcliffe 40a |
| 300 | 3·10³ | Ginzburg and Al’pert 49 |
radio waves in the ionosphere. However, in the main, the results used were measurements of the absorption of radio waves in the ionosphere and/or, otherwise, measurements of the reflection coefficient as a function of frequency. Under certain assumptions these quantities are brought into agreement with the number of collisions in the region under investigation.
4.2. Theoretical consideration
Since the upper atmosphere consists of a mixture of neutral particles of different gases, electrons, and heavy ions, the collision number includes collisions of neutral particles with one another, of neutral particles with electrons, of ions with electrons, and of electrons with one another.
Each of these acts is an individual physical process described by a definite relation. Several such relations are considered below.
Fig. 2. Dependence of the collision number on altitude, obtained from the results of ionospheric investigations.
In a gas with a Maxwellian distribution of velocities, the mean values of the velocity and of the mean free path are respectively equal (in the absence of external force fields)
\[ \bar{u}=\sqrt{\frac{8kT}{\pi m}} \tag{4,1} \]
and
\[ \bar{\lambda}=\frac{\sqrt{3\pi}}{4\sigma^2 N}. \tag{4,2} \]
Since the mean number of collisions is defined as \(\nu=\dfrac{\bar{u}}{\bar{\lambda}}\), we have
\[ \nu=8\sigma^2N\sqrt{\frac{2kT}{3m}}. \tag{4,3} \]
Equation (4,3) can be used to determine the temperature only if the total number of atoms and molecules and the mean value of their molecular weight are known. Formula (4,3) may also be written in the form
\[ \nu=8P\sigma^2\sqrt{\frac{2}{3mkT}}, \tag{4,4} \]
where \(P\) is the partial pressure of atoms and molecules.
Equations (4,3) and (4,4) express the number of collisions between neutral particles; the number of collisions of electrons with neutral particles is increased, in comparison with (4,4), by a factor of
\[ \sqrt{\frac{m}{m_e}}. \]
Furthermore, since the mean free path of an electron is \(4\sqrt{2}\) times greater than the mean free path of uncharged particles, the number of collisions \(\nu'\) must be divided by this quantity. Thus, the number of collisions between electrons and neutral particles is equal to (assuming \(T_e=T\)):
\[ \nu'=2\sigma^2N\sqrt{\frac{kT}{3m_e}} =2\sigma^2P\sqrt{\frac{1}{3m_ekT}}. \tag{4,5} \]
If the classical value of \(\sigma\) is used, then usually \(\dfrac{\nu'}{\nu}\simeq 70\).
Quite different formulas have been obtained for collisions between ions. The number of collisions of an electron with a positive ion was found to be equal to1
\[ \nu''_M= \frac{4\pi e^4}{3\sqrt{2\pi m_ek^3}}\cdot \frac{n}{T^{3/2}} \left[\ln\left(T\bar{\lambda}^2\right)-B\right], \tag{4,6} \]
where \(B=24.58\) and \(\bar{\lambda}''\) is the mean free path of an ion. This relation was derived for singly ionized particles. For calculations one may take \(\bar{\lambda}''\sim\dfrac{\bar{\lambda}}{5}\). An analogous expression for the number of collisions between electrons and singly ionized particles was obtained by V. L. Ginzburg[^2]
\[ \nu''_{\Gamma}= \frac{\pi e^4n}{\sqrt{(kT)^3}} \sqrt{\frac{3}{m_e}}\cdot \ln\left(\frac{kT}{e^2\sqrt[3]{n}}\right). \tag{4,7} \]
Below it is shown that, in order of magnitude, (4.6) and (4.7) give identical numerical results. For collisions of electrons with neutral particles, Ginzburg also uses (4.5).
Fig. 3. Fig. 4.
Results of calculations of the number of collisions at altitudes of 300 km (Fig. 3) and 400 km (Fig. 4). The numbers of collisions of electrons with ions were computed for a fully ionized atmosphere according to the formulas of Mayomdar and Ginzburg. The numbers of collisions of electrons with oxygen atoms were calculated for various states of oxygen atoms, as computed by Bates, Yamanouchi, and Mitra. The numbers of collisions of electrons with nitrogen (with atoms and molecules) were calculated for \(\sigma=\mathrm{const}\).
For completeness, we should also give here the formula for the number of collisions of electrons with one another. This quantity, calculated by Drukarev\(^{33}\), is equal to:
\[ \nu_e= \frac{n_e e^4}{\sqrt{m_e(kT_e)^3}}\, \ln \frac{\sqrt{(kT_e)^3}} {e\bar{\lambda}^{1/2}\sqrt{8\pi n_e}}, \tag{4,8} \]
where
\[ \bar{\lambda}'=4\sqrt{2}\,\bar{\lambda}. \]
It should be recalled that the experimentally measured value \(\nu_{\mathrm{eff}}\) is the sum of the various collision frequencies
\[ \nu_{\mathrm{eff}}=\nu' + \nu'' + \nu_e. \tag{4,9} \]
Figures 3 and 4 show the temperature dependence of all components of \(\nu_{\mathrm{eff}}\), respectively at altitudes of 300 and 400 km. For
in calculating the curves the data of Table III were used, and in formulas (4.6) and (4.7) it was assumed that \(n=\dfrac{N}{2}\), i.e., that the atmosphere is completely ionized. In calculations by means of (4.5) the cases considered were: a) \(\sigma=\mathrm{const}\) (nitrogen) and b) \(\sigma=\sigma(T)\) (atomic oxygen). Several values of \(\sigma(T)\) were used (for more detail see § 4.3). From the figures it is seen that the values of \(\nu_e''\) according to Mitra and Ginzburg are close to one another.
By means of (4.8) Drukarev obtained the values \(\nu_e=30\ \mathrm{s}^{-1}\) and \(\nu_e=3\ \mathrm{s}^{-1}\), respectively for the \(F_2\) and \(E\) layers. Drukarev assumed that the mean free path is equal to \(10^{-8}\ \mathrm{cm}\), which is typical for the atmosphere at sea level. From Tables I and III, however, it is seen that for the \(F_2\) layer \(\bar{\lambda}\sim 10^4\ \mathrm{cm}\), and for the \(E\) layer \(\bar{\lambda}\sim 10\ \mathrm{cm}\). If these values are substituted in (4.8), negative values of the logarithm are obtained, so that by Drukarev’s formula \(\nu_e\) can be calculated only for very small values of \(\bar{\lambda}\).
Under the assumptions made, the number of collisions of electrons with ions at \(T=1000^\circ\) is \((10^4 \div 10^5)\) times greater than the number of collisions of electrons with neutral particles. Further, since the observed collision numbers at an altitude of \(300\ \mathrm{km}\) are of the order of \(10^3\ \mathrm{s}^{-1}\), in order to satisfy this value of \(\nu_{\mathrm{eff}}\) it is necessary that an atmosphere consisting entirely of ions have a temperature of the order of \(200^\circ\). If, however, it is assumed that the basic process is collisions of electrons with neutral particles, then temperature values of \(4000^\circ\) are obtained for nitrogen (\(\sigma=\mathrm{const}\)) or \((1300 \div 7000^\circ)\) for atomic oxygen (for various admissible values of the cross section).
With regard to an ionosphere consisting entirely of ions, one thing is obvious: that at an altitude of \(300\ \mathrm{km}\) a temperature of \(200^\circ\) is impossible, since at such a temperature density values are obtained which are less than the observed ionization density (see Table III). Therefore the possibility of complete ionization of the atmosphere below \(400\ \mathrm{km}\) is directly excluded*).
In individual cases it was assumed that the number of collisions of electrons with ions in the ionosphere is several orders of magnitude greater
) The author excludes here a third alternative, namely, that at an altitude of \(300 \div 400\ \mathrm{km}\) complete ionization of the atmosphere could occur if here the density of neutral particles were \(10^6 \div 10^7\), and not \(10^{10}\), as is now accepted by a number of researchers, and also by the author, but has not yet been finally proved. With \(n\sim 10^6 \div 10^7\) and \(T\sim 1000 \div 2000^\circ\), the number of collisions of electrons with ions, calculated from (4.7), amounts to several units \(\times 10^3\) and agrees better with the results of measurements of \(\nu_{\mathrm{eff}}\) than when considering collisions of electrons with neutral particles (see below). Such a hypothesis, based on the processing of results of measurements of the electron density of the \(F_2\) layer, was put forward in works \(^{144,145}\), where a model of the atmosphere was constructed somewhat different from the model proposed in this article. (Translator’s note.*)
number of collisions of electrons with neutral particles. These results are not confirmed here. The conclusions drawn here are in agreement with the results of Kauling\(^{28}\), who determined that if \(\frac{N}{n} > 10^6\), then collisions between charges of opposite sign are almost impossible.
It should be noted here that, since the temperature is proportional to the fourth power of the collision diameter\(^*\), it is very important to determine \(\sigma\) carefully, since small changes in \(\sigma\) lead to large changes in \(T\).
4.3. Collision cross section of atmospheric oxygen
For slow electrons interacting with oxygen or nitrogen molecules, one may use the gas-kinetic value of the effective collision diameter\(^{42,77}\). At electron energies of 1 eV and less, the collision cross section of atomic oxygen depends not only on the electron energy, but also on the force field of the atom. In particular, in order to determine the coefficient of elastic scattering and the effective collision cross section, it is necessary to determine the radial Hartree wave function, calculated by the self-consistent-field method with allowance for exchange; that is, it is necessary to obtain a solution of the Fock equations\(^{58,90}\).
Because the field of the atom is not known exactly, in one of the papers\(^{15}\) a parameter \(p\) is introduced, characterizing the polarizability of the atom, which takes into account the influence of the \(3s\)-electron on the polarization of the shell. Using the parameter \(p\), one can numerically integrate the corresponding Fock equations. Subsequently, by means of this method, collision numbers were obtained\(^{156}\) for various values of \(p\). With the aid of the self-consistent-field method, the collision cross section of atomic oxygen with electrons was calculated\(^{89}\). The collision cross section was also estimated in another paper\(^{141}\), in which the self-consistent field with exchange was used as the average field of the ground state of the oxygen atom. It should be noted that, whereas in \(^{141}\) the ground state \({}^3P\) is considered, in \(^{15}\) a complex consisting of an oxygen atom in the ground state and a free \(s\)-electron is considered (it is assumed that the atom is in the \({}^3P\) or \({}^4P\) state).
In Fig. 5, curves are plotted for the dependence of the collision diameter on temperature, calculated according to the theories indicated above, for the case of interaction of an oxygen atom with slow
\(^*\) The collision diameter, or mean effective diameter, is defined as \(\sqrt{\frac{S}{\pi}}\), where \(S\) is the collision cross section \((\text{cm}^2)\).
Fig. 5. Dependence of the effective cross section for collisions of oxygen atoms with electrons on temperature, calculated by various researchers.
Fig. 6. Dependence of the number of collisions of electrons with oxygen atoms, nitrogen particles, and ions on temperature at an altitude of 300 km (for various values of the temperature gradient and atomic states).
Fig. 7. Dependence of the number of collisions of electrons with oxygen atoms, nitrogen particles, and ions on temperature at an altitude of 400 km.
electrons. Two cases are taken, where \(p=5.7\) (the predicted value) and \(p=18.3\) (the value corresponding almost to resonance). It is seen that the elastic-collision cross section depends strongly on the polarization factor. From Fig. 5 it also follows that the curve obtained in \({}^{89}\) closely coincides with the curve calculated in \({}^{15}\) for \(p=18.3\).
With such a difference and divergence in the values \(\sigma=\sigma(T)\), the choice of its effective value in the atmosphere is associated with confusion. The value \(p=18.3\) corresponds closely to resonance, when the electron adheres to the oxygen atom. From an analysis of the physical processes occurring in the upper atmosphere, it seems improbable that the atomic shell should here have so large a polarization value. Moreover, it is shown below that the collision numbers calculated for \(p=18.3\) are considerably larger than the observed values. On the other hand, the collision numbers obtained for \(p=5.7\) are so small that, for agreement with experiment, exceedingly large temperature values in the high regions of the atmosphere are required. Apparently, the most suitable values were obtained in \({}^{141}\).
4.4. Effective number of collisions
Since \(\nu_e\) may be neglected, and \(\sigma\) is equal to \(\sigma(T)\) for atomic oxygen, (4.9) should be rewritten in the form
\[ \nu_{\mathrm{eff}}=\nu''+\nu'_{N+N_2}+\nu'_0, \tag{4.10} \]
where
\[ \nu'_{N+N_2}=2\sigma^2 N_{N+N_2}\sqrt{\frac{kT}{3m_e}}, \tag{4.11} \]
\[ \nu'_0=2[\sigma'(T)]^2 N_0\sqrt{\frac{kT_0}{3m_e}}, \tag{4.12} \]
and \(N_{N+N_2}\) and \(N_0\) are, respectively, the numbers of nitrogen molecules and atoms and of oxygen atoms.
The value \(\nu''\) was calculated in equation (4.10) for \(n=2.5\cdot 10^6\ \mathrm{cm}^{-3}\) at an altitude of \(300\ \mathrm{km}\) and \(n=10^6\ \mathrm{cm}^{-3}\) at an altitude of \(400\ \mathrm{km}\), and for the corresponding density values.
The results of the calculations are given in Figs. 6 and 7, respectively, for altitudes of 300 and \(400\ \mathrm{km}\). To determine \(\nu''\), formula (4.6) and the value \(\sigma(T)\) calculated in \({}^{141}\) were used. From a comparison of the measured values of the collision numbers, presented in Fig. 2, with the data of Figs. 6 and 7, it follows that at an altitude of \(300\ \mathrm{km}\) the temperature varies within the limits \((2000 \div 3100^\circ)\), while at an altitude of \(400\ \mathrm{km}\), \(T<4100^\circ\).
§ 5. BAROMETRIC FORMULA AND NUMBER OF COLLISIONS*)
5.1. Theoretical Considerations
It has been suggested \(^{1276, 138}\) that the pressure in the upper atmosphere is subject to the influence of Coulomb and radiation forces, as well as to the influence of gravitational forces. Coulomb forces may be the result of the interaction of space charges located in the atmosphere and encompassing large volumes, while radiation forces are due to radiation from the Sun and, possibly, the stars. The first effect may arise under a suitable separation of charges, i.e., in a process that is effective at great heights. However, calculations \(^{1186}\) have clearly shown that such a hypothesis is untenable because of the large energies required for it. Radiation forces likewise can be significant only at large distances from the Earth, and in the range of heights under consideration they may be ignored. Thus, in this section only the influence of gravitational forces on atmospheric pressure is taken into account.
From the barometric equation one obtains
\[ \frac{dP}{P}=-\left(\frac{mg}{kT}\right)\,dz. \tag{5,1} \]
It is assumed that the temperature varies linearly with height, i.e. that
\[ T=T_1[1-a(z-z_1)]. \tag{5,2} \]
Integrating (5,1) and assuming that, in the height interval considered, \(m\) and \(g\) are constant, we obtain
\[ \frac{P}{P_1}=\left(\frac{T_1}{T}\right)^{\frac{mg}{akT}}. \tag{5,3} \]
Substituting (4,5) into (5,3), we have
\[ \frac{\nu}{\nu_1} = \left(\frac{\sigma}{\sigma_1}\right)^2 \left(\frac{T_1}{T}\right)^{\frac{mg}{akT}+\frac{1}{2}}, \tag{5,4} \]
and, eliminating the coefficient \(a\), we obtain
\[ \frac{\nu}{\nu_1} = \left(\frac{\sigma}{\sigma_1}\right)^2 \left(\frac{T_1}{T}\right)^x, \tag{5,5} \]
where
\[ x=\frac{mg(z-z_1)}{k(T-T_1)}+\frac{1}{2}. \tag{5,6} \]
*) In § 4 and in the present section one and the same question is considered, but from different points of view. In § 3 the number of collisions is considered at a fixed height and the temperature is a variable quantity. In this section the dependence of the number of collisions on height is studied for different laws of temperature variation. Because of the importance of the results, it appears advisable to consider both of these questions.
Calculations by formula (5.5) were carried out for neutral atoms and molecules of nitrogen and atomic oxygen; in the latter case several values of \(\sigma(T)\) were adopted. The initial value of the number of collisions \(\nu'_1\) was determined on the basis of the chosen atmospheric model for an altitude of 100 km. For nitrogen \(\sigma\) was taken as constant.
As was to be expected, the results obtained agree with those of the preceding subsection. The slight discrepancy arising between them is due to the difference in the assumptions made.
5.2. Effect of neutral particles and ions
The results of the calculation for nitrogen are given in Fig. 8 for several values of the temperature gradient \(\dfrac{\Delta T}{\Delta Z}\). The value \(\nu'_1\) corresponds to the number of collisions between electrons and nitrogen particles at an altitude of 100 km. Although \(\dfrac{\Delta T}{\Delta Z}=20^\circ/\mathrm{km}\) better satisfies the observational results, this value of \(\dfrac{\Delta T}{\Delta Z}\) leads to a temperature of \(6300^\circ\) at an altitude of 400 km, which is quite impossible in the terrestrial atmosphere.
Fig. 8. Dependence of the number of collisions of electrons with nitrogen particles (atoms and molecules) on altitude for different values of the temperature gradient.
In order finally to find the most probable value of \(\nu'_1\), calculations by (5.5) were carried out for several values of \(\sigma(T)\) proposed for atomic oxygen (Fig. 9). It may be noted that for \(p=5.7\) the values of \(\nu'\) do not agree with the experimental data; even for \(\dfrac{\Delta T}{\Delta Z}=20^\circ/\mathrm{km}\) the calculated values of \(\nu'\) are \(10^3\) times smaller than the values obtained for nitrogen and, as will be shown below, for ions. For \(p=18.3\) the values of \(\nu'_0\) at an altitude of 100 km are 10 times greater than the observed ones; from this it may be concluded that such values of the polarization of the atomic shell are impossible in the upper atmosphere. The value of \(\sigma(T)\) recommended by Yamanouchi \(^{141}\) gives values of \(\nu'_0\) which are correct in order of magnitude.
It remains to consider collisions of electrons with ions. For this purpose equation (4.7) was used for: a) different values
\(\dfrac{\Delta T}{\Delta Z}\) and b) a linear change of the ionic concentration with height from the value \(n = 2 \cdot 10^5\ \text{cm}^{-3}\) at an altitude of 100 km to \(n = 4 \cdot 10^6\ \text{cm}^{-3}\) at an altitude of 400 km. Under these assumptions, the curves shown in Fig. 10 were obtained. First, it is clear that at an altitude of 100 km the number of collisions of electrons with ions is negligibly small compared with the number of collisions with neutral particles. Further,
Fig. 9. Results of calculating the dependence of the number of collisions of electrons with oxygen atoms on altitude for various values of the temperature gradient and atomic states.
Fig. 10. Dependence of the number of collisions of electrons with ions on altitude for various values of the temperature gradient.
it is evident that for \(\dfrac{\Delta T}{\Delta Z} = \text{const}\), \(\nu''\) is almost constant in the region \((150 \div 400)\) km and, possibly, to still greater altitudes. The constancy of \(\nu''\) is masked at greater altitudes by collisions of electrons with neutral particles.
From the various results (Figs. 8–10) one can now draw conclusions about the conditions existing in the atmosphere. The observed numbers of collisions at an altitude of 100 km can be explained if one considers collisions of electrons with nitrogen particles or atomic oxygen having the value \(\sigma(T)\) obtained in \(^{141}\). At an altitude of 300 km the calculated values of the number of collisions agree with the experimental values for the combined re-
collisions of electrons with ions, neutral particles of nitrogen and atomic oxygen (for \(\sigma(T)\) according to \({}^{141}\)); if \(\dfrac{\Delta T}{\Delta Z} \simeq (5 \div 10)^\circ/\text{km}\), then the same results are obtained also for an altitude of \(400\) km\(*\).
Other values of the number of collisions, obtained from calculations, do not give the same agreement with the results of observations as the combination chosen above. The most probable dependence of temperature on altitude, obtained from the discussion carried out in this section, is given in Table V.
Table V
Values of temperature calculated from the barometric formula and of the number of collisions
| 100 | 150 | 200 | 250 | 300 | 350 | 400 | |
|---|---|---|---|---|---|---|---|
| \(z\) (km) | 100 | 150 | 200 | 250 | 300 | 350 | 400 |
| \(T^\circ\) | 300 | 825 | 1350 | 1875 | 2400 | 2925 | 3450 |
| \(T^\circ *\) | 300 | — | — | \(1200^{71,83}\) | — | — | — |
5.3. Comparison with the data of other investigators
In the work whose results are given in Table V, it was assumed\({}^{83}\) that
\[ \frac{\nu'}{\nu_1'}=\left(\frac{T_1}{T}\right)^{\frac{mg}{akT}}, \tag{5,7} \]
that the number of collisions in the \(F\)-layer is equal to \(\dfrac{1}{200}\) of the number of collisions in the \(E\)-layer (this agrees with the data of Table III), and that the mean velocity of electrons in the \(F\)-layer is 10 times greater than that in the \(E\)-layer (this is not quite exact).
§ 6. TEMPERATURE AND SCALE HEIGHT
6.1. Scale height from auroral data
The scale height is equal to
\[ H=\frac{kT}{mg}=\frac{kT(R+z)^2}{mg_0R^2}. \tag{6,1} \]
\(*\) For an unclear reason the author does not emphasize here the fact that the values of \(\nu''\) obtained by him in the \(F\)-layer agree more closely with the experimental values of \(\nu_{\mathrm{eff}}\). This gives grounds for supposing that in the \(F\)-layer the main collision process is collisions of electrons with ions. Further, the author notes neither here nor in the subsequent sections that, since \(\nu''\) decreases with increasing temperature, whereas \(\nu'\), on the contrary, increases, this may help, with appropriate processing of experimental data, to clarify the question of the type of collisions, and also to determine the density and temperature in the \(F\)-layer\({}^{144}\). (Translator’s note.)
Since \(k\) and \(R\) are constants and, in the range of heights considered, the change in \(z\) amounts to no more than 5%, the changes in \(H\) may be directly ascribed to changes in \(m\) or \(T\). This section considers a method for determining \(H\) from luminosity curves of aurorae when using photographs taken with filters\(^{576,\mathrm{b}}\). The basis of the method for determining luminosity from absorption curves of electrons invading the earth’s atmosphere is the proportionality of the luminosity \(L\), arising in the height interval \(dz\), to the value
\[ \frac{dJ}{dz}, \]
where \(J\) is the intensity of the electron flux. This relation is analogous to the relation used in ionospheric theory for determining the density of ionization arising under the action of monochromatic radiation penetrating a homogeneous atmosphere. Thus, for the luminosity of aurorae, a function is obtained which is analogous to equation (2.3).
Fig. 11. Dependence of the height of the homogeneous atmosphere on height, obtained from the results of auroral observations.
Proceeding in this way, \(^{57}\) values of the height scale were obtained, given in Table VI and in Fig. 11.
Table VI
Values of the height scale and the corresponding temperature values obtained from auroral observation data (winter 1943/1944, Tromsø, Norway)
| Maximum height (km) | \(H\) (km) | \(T\) (degrees) | Maximum height (km) | \(H\) (km) | \(T\) (degrees) |
|---|---|---|---|---|---|
| 100 | 8.0 | 219 | 185 | 51 | 1360 |
| 110 | 8.8 | 240 | 200 | (59.5) | (1580) |
| 120 | 10.3 | 280 | 250 | (82.5) | (2073) |
| 130 | 12.3 | 334 | 270 | 92 | 2263 |
| 140 | 15.2 | 411 | 300 | (103) | (2455) |
| 150 | 19.7 | 531 | — | — | — |
| 160 | 28.3 | 761 | 350 | (119.5) | (2704) |
It should be noted that at a height of about 185 km the data obtained have a discontinuity. Below this height the values of \(H\) were computed ...
from diffuse auroras and drapery-type auroras; at an altitude of about 185 km—from ray auroras, and at an altitude of about 270 km—from studies of the glow of regions illuminated by solar rays. Thus, the temperature values obtained above 270 km correspond to daytime conditions. It should be noted that the cited values of the height scale are minimal, since the spiral character of the orbit of the incident ion is not taken into account.
The temperature values obtained from observations of auroras may differ from the values of \(T\) calculated by other methods. One of the reasons for this discrepancy is connected with the fact that intensive study of auroras is carried out mainly in autumn, winter, and spring and at high latitudes, where the temperature regime of the atmosphere differs from that at lower latitudes. In this connection it should be noted that the temperature values obtained from measurements of the number of collisions (carried out at lower latitudes) are in general higher than the values of \(T\) obtained from auroral data. For example, at an altitude of 350 km the corresponding values \(T = 3300\) and \(2700^\circ\) are obtained. The difference may, however, be explained not only by the difference in latitudes, but also by the fact that the spiral character of the ion orbit was not taken into account in the calculations. At an altitude of 100 km these calculations likewise give a lower temperature.
6.2. Values of the height scale obtained by different methods
The first temperature values from the height scale were determined\(^{98}\) using measurement data in the \(E\)-layer\(^{3\mathrm{v}}\), as well as\(^{97}\) from data for the \(F_2\) layer\(^{3\mathrm{v}}\). The values obtained were, respectively, \(H = 11.4\) km for the \(E\) layer and \(H = 20 \div 30\) km for the \(F_2\) layer. Starting from these values of \(H\) and from various concentrations of molecular and atomic oxygen and nitrogen, values \(T = 308 \div 374^\circ\) were calculated in the \(E\) layer and values \(T = 437 \div 625^\circ\) and \(655 \div 936^\circ\) in the \(F_2\) layer. The low values of \(H\) and, correspondingly, \(T\) for the \(F_2\) layer seem unusual.
Subsequently, from ionospheric studies, values were obtained of \(H = 70\) km and 40 km in the \(F_2\) layer, respectively for summer and winter, \(H = 55\) km and \(H = 90\) km for autumn and spring (at Watheroo, Australia\(^{72}\)) and \(H = 65 \pm 5\) km for autumn\(^{103}\). An average value \(H = 170\) km\(^{93\mathrm{б}}\) is also given, as well as values \(H = 30\) and 60 km, respectively, for the \(F_1\) and \(F_2\) layers\(^{93\mathrm{б}}\).
From the data given in\(^{3\mathrm{v}}\), temperature values of 615 and \(1080^\circ\)\(^{126}\) were also obtained for the \(F_1\) and \(F_2\) layers. In these calculations it was assumed that nitrogen is completely dissociated. The author, using the data on nitrogen dissociation adopted in the above-mentioned
models of the atmosphere, computed from these same data for the layers \(F_1\) and \(F_2\) temperature values of 955 and 1670°. The values of the height scale \(H\) cited in the literature are summarized in Table VII.
Table VII
Values of the height scale (in km) obtained from ionospheric data
| Investigator | \(D\) | \(E\) | \(F\) | \(F_1\) | \(F_2\) |
|---|---|---|---|---|---|
| Appleton \(^{3c}\) | — | 11.4 | 40 | — | 70 |
| Appleton and Beynon \(^{4}\) | — | — | 45 | 55 | 45–70 |
| Booker and Seaton \(^{22}\) | — | — | — | — | 50–90 |
| Budden, Ratcliffe, and Wilkes \(^{24}\) | 6 | — | — | — | — |
| Fuchs \(^{44}\) | — | — | — | — | 110*) |
| Gerson \(^{46}\) | — | — | — | — | 67–167 |
| Gerson \(^{46}\) | — | — | — | — | 105–186**) |
| Graik, Kelso, and Miller \(^{53}\) | — | 10 | — | — | — |
| Guha \(^{55}\) | — | 10 | — | — | 46–54*) |
| Jones \(^{67}\) | — | — | 6–32**) | — | — |
| Jones and Jones \(^{69}\) | — | — | 15–56**) | — | — |
| Kellogg \(^{70}\) | — | — | — | 43*) | 170*) |
| Nicole \(^{936}\) | — | — | — | 30 | 60 |
| Pekeris \(^{97}\) | — | — | — | — | 20–30 |
| Pirs \(^{103}\) | — | — | — | — | \(65 \pm 5\) |
| Rydbeck \(^{110}\) | — | — | — | — | 37–58 |
| Rydbeck \(^{110}\) | — | — | — | — | 51–165*) |
| Rydbeck \(^{1106}\) | — | — | 35 | — | — |
) Equatorial values. *) Arctic values.
§ 7. TEMPERATURE AND ELECTRON CONCENTRATION
7.1. Introduction
For a simple layer, from (2,14) one obtains the dependence of the electron concentration on height in the form
\[ n=n_{\max}\cdot e^{\frac{1}{2} f(T)}=n'_{\max}\cdot e^{\frac{1}{2} F(T)}, \tag{7,1} \]
where
\[ n=\frac{\pi m f^2}{e^2}. \tag{7,2} \]
These equations do not take into account the Lorentz polarization correction. In ionospheric investigations one usually measures the dependence of the effective height on frequency; therefore it is necessary to determine the relationships between the corresponding parameters.
7.2. Approximation of a Simple Layer
Without taking account of the Earth’s magnetic field, the effective height is equal to
\[ z' = z_0 + \int_0^s \frac{ds}{\mu}. \tag{7,3} \]
The refractive coefficient for the quadratic approximation of a simple layer (see (2,16))\(^{3,43}\) is equal to
\[ \mu^2 = 1 - \frac{2s}{A}\left(\frac{f_c}{f}\right)^2 + \left(\frac{f_c}{f}\right)\left(\frac{s}{A}\right)^2 . \tag{7,4} \]
In the equations written above, \(z_0\) is the lower boundary of the layer, where \(n=0\), \(A=(z_{\max}-z_0)\), and \(s\) is measured along \(z\) in such a way that \(s=0\) when \(z=z_0\). As a result of the integration one obtains
\[ z' = z_0 + \frac{kT}{mg}\cdot \frac{f}{f_c} \ln\left(\frac{f_c+f}{f_c-f}\right), \tag{7,5} \]
if \(A=2H\).
For a cubic approximation of a simple layer
\[ n = n_{\max}\left[1-\frac{32}{81}z^2-\frac{64}{729}z^3\right]. \tag{7,6} \]
Fuchs \(^{446}\) obtained:
\[ z' = z_0 + \frac{9}{16}\left(\frac{kT}{mg}\right) B \ln\left(\frac{1+B}{1-B}\right), \tag{7,7} \]
where
\[ A=\frac{9H}{4} \quad \text{and} \quad 2B^2 = -1 + \sqrt{1+\frac{8f^2}{f_c^2}} . \tag{7,8} \]
In Fig. 12 curves are plotted representing the parabolic and cubic approximations of a simple layer.
In order to emphasize the difference between them, one may rewrite equations (7,5) and (7,7) in the general form:
\[ T = \frac{mg}{k}\,\frac{z'-z_0}{\Phi(f)}, \tag{7,9} \]
where for the parabolic approximation
\[ \Phi(f)_A = \frac{f}{f_c}\ln\frac{f_c+f}{f_c-f} \tag{7,10} \]
and for the cubic approximation
\[ \Phi(f)_F = \frac{9B}{16}\ln\frac{1+B}{1-B}. \tag{7,11} \]
It should be recalled that in both cases the Earth’s magnetic field is not taken into account. In Fig. 13 the curves of the functions \(\Phi(f)_A\) and \(\Phi(f)_F\) are plotted.
Fig. 12. Distribution of ionization with height according to Chapman and its quadratic (dashed line) and cubic (dots) approximations. Crosses show the results of calculations by Gukh’s formula.
With the aid of (7,9) and (7,10) it was found that in winter in \(F_2\) the value
\[ H=\frac{kT}{mg}=70\ \text{km} \]
at noon. When (7,11) is used, \(H\) will be larger. The accuracy of the determination of \(H\) depends on the radio frequencies used.
For two successive values \(z_i'\) and \(z_{i+1}'\) of the effective height, from (7,9) one obtains:
\[ T=-\frac{mg}{k}\cdot \frac{\Delta z'}{\Delta \Phi(f)}, \tag{7,12} \]
where \(\Delta z'=z_i'-z_{i+1}'\) and
\[ \Delta \Phi(f)=\Phi(f_i)-\Phi(f_{i+1}). \]
Using (7,12), the temperature was calculated\({}^{44}\) in \(F_1\) and \(F_2\) over Huancayo (Peru) for the value
\[ \frac{mg}{k}=1.77\cdot 10^{-4}\ \text{deg/cm}. \]
This value is apparently underestimated and corresponds to almost complete dissociation of nitrogen. Since experiments do not confirm such a high degree of dissociation of nitrogen, the value of \(\frac{mg}{k}\) must be improved by using the atmospheric model given in the present work. It is obtained, for example, for an altitude of \(185\ \text{km}\),
\[ \frac{mg}{k}=2.67\cdot 10^{-4}\ \text{deg/km}. \]
It should be noted that in \({}^{44}\) there are a number of inaccuracies in the calculations. Correction of these calculations and the use of a more correct value of \(\frac{mg}{k}\) lead to substantially different temperature values (see Table VIII).
Fig. 13. Graphs of the functions
\[ \Phi(f)_A=-\frac{f}{f_c}\ln\left(\frac{f_c+f}{f_c-f}\right) \]
and
\[ \Phi(f)_F=-\frac{9B}{16}\ln\left(\frac{1+B}{1-B}\right), \]
where
\[ B=B\left(\frac{f}{f_c}\right). \]
A third relation was also recommended, allowing the temperature to be calculated from the equation of a simple layer. As a
Table VIII
Temperature over Huancayo (Peru) at noon, February 1934
| $f$ Mc/s | Temperature (degrees), Fuchs | Temperature (degrees), recalculated | Temperature (degrees), recalculated | Altitude (km), Fuchs | Altitude (km), recalculated |
|---|---|---|---|---|---|
| $F_1$, 14.II 1934 | — | — | — | — | — |
| 3.53 | — | — | — | — | — |
| 4.00 | 373 | 553 | 316 | — | — |
| 4.21 | 368 | 599 | 333 | — | — |
| 4.37 | 782 | 1184 | 852 | 190 ÷ 220 | 185 |
| 4.52 | — | — | — | — | — |
| $F_1$, 28.II 1934 | — | — | — | — | — |
| 3.50 | — | — | — | — | — |
| 3.90 | 435 | 623 | 355 | — | — |
| 4.11 | 460 | 737 | 410 | — | — |
| 4.23 | 1040 | 1626 | 897 | 190 ÷ 220 | 185 |
| 4.41 | — | — | — | — | — |
| $F_2$, 14.II 1934 | — | — | — | — | — |
| 6.30 | — | — | — | — | — |
| 6.60 | 1670 | 1942 | 1113 | — | — |
| 6.90 | 1510 | 2086 | 1170 | — | — |
| 7.10 | 1960 | 2448 | 1359 | — | — |
| 7.26 | 1960 | 2384 | 1324 | 350 ÷ 420 | 355 |
| 7.75 | — | — | — | — | — |
| $F_2$, 28.II 1934 | — | — | — | — | — |
| 5.45 | — | — | — | — | — |
| 5.80 | 1175 | 1442 | 826 | — | — |
| 6.20 | 980 | 1198 | 672 | — | — |
| 6.39 | 1430 | 1717 | 947 | — | — |
| 6.49 | 1950 | 2403 | 1332 | 350 ÷ 420 | 395 |
| 6.80 | — | — | — | — | — |
For the approximation of the ionization distribution, equation\(^ {55}\) was obtained
\[ -z = 0.6672 + 1.397 \frac{n}{n_{\max}} - 2.084 \sqrt{\frac{n}{n'_{\max}}} + 2 \sqrt{1 - \frac{n}{n'_{\max}}}. \tag{7.13} \]
In the calculations the usual expression for the refractive index of an ionized gas is used, without taking into account the influence of the Earth’s magnetic field:
\[ \mu^2 = 1 - \frac{ne^2}{\pi m f^2}. \tag{7.14} \]
Substitution of (7.13) into (7.3) and integration directly give:
\[ z' = z_{\max} - 2.667H - 2.794H \cdot \left(\frac{f}{f_c}\right)^2 + 3.274H \cdot \left(\frac{f}{f_c}\right) + 2H \left(\frac{f}{f_c}\right)\operatorname{Arcsh}\sqrt{\left(\frac{f_c}{f}\right)^2 - 1}, \tag{7.15} \]
if the formula for the refractive index is rewritten in the form
\[ \mu = 1 - \left(\frac{f_c}{f}\right)^2\left(\frac{n}{n'_{\max}}\right). \tag{7.16} \]
Equation (7.15) can be solved with respect to \(z_{\max}\) and \(H\), using, as in (7.12), the values of \(z'\) at different points of the experimentally recorded height–frequency characteristic.
The temperature can be determined by methods similar to those described above, also using a fourth method. In all cases of interest, the probing electromagnetic wave begins and ends its path through the ionosphere in a region where the refractive index is close to unity; therefore one may determine the difference between the equivalent and true paths of the wave. This difference is equal to\(^ {65}\) (see (7.3))
\[ X = \int \left(\frac{1}{\mu} - 1\right)\, ds. \tag{7.17} \]
Introducing
\[ x^3 = e^{-\frac{1}{2}z} \tag{7.18} \]
and
\[ a = \sqrt{2e}\left(\frac{f_c}{f}\right)^2, \tag{7.19} \]
we obtain, instead of (7.14),
\[ \mu^2 = 1 - ax \cdot e^{-x^2}, \tag{7.20} \]
and equation (7.17) takes the form
\[ X = -2H \int_{x_1}^{\infty} \left\{\frac{1}{\sqrt{1 - axe^{-x^2}}} - 1\right\}\frac{dx}{x}. \tag{7.21} \]
Using, for the determination of \(x_i\), the expression
\[ ax = e^{x^2}, \tag{7.22} \]
we obtain the value of the height of reflection above the earth’s surface:
\[ z = z_{\max} + H \ln\left(\frac{1}{2}\right) - 2H \ln x_1. \tag{7.23} \]
The effective height of reflection is then obtained as
\[ z' = z_{\max} + H \ln \left( \frac{1}{2} \right) - 2H \ln x_1 + \frac{1}{2}HY\left( \frac{f}{f_c} \right). \tag{7.24} \]
The values \(x_1 = x_1\left( \frac{f}{f_c} \right)\) and \(Y = Y\left( \frac{f}{f_c} \right)\) are tabulated in \({}^{65}\).
Equations (7.5), (7.7), (7.15), and (7.24) were obtained without taking into account the lower layers of the ionosphere. If, however, there is such a retarding layer, the following formulas are obtained:
a) instead of (7.5)
\[ z' = z_0 + H_F\left( \frac{f}{f_c} \right)\ln \left( \frac{f_c + f}{f_c - f} \right) - 4H_E + 2H_E\left( \frac{f}{f_E} \right)\ln \left( \frac{f + f_E}{f - f_E} \right), \tag{7.25} \]
b) instead of (7.15)
\[ z' = z_{\max} + H_F\left\{ -2.667 - 2.794\left( \frac{f}{f_c} \right)^2 + 3.274\left( \frac{f}{f_c} \right) + 2\left( \frac{f}{f_c} \right)\operatorname{Arcsh}\sqrt{\left( \frac{f_c}{f} \right)^2 - 1} \right\} + AH_E, \tag{7.26} \]
c) instead of (7.24)
\[ z' = z_{\max} + H_F\left[ \ln \left( \frac{1}{2} \right) - 2\ln x_1 + \frac{1}{2}Y\left( \frac{f}{f_c} \right) \right] + \frac{1}{2}H_EY'\left( \frac{f}{f_E} \right). \tag{7.27} \]
In the last group of formulas \(H_F\) and \(H_E\), \(f_c\) and \(f_E\) are, respectively, the values of the height scale and the critical frequency of the layers \(F\) and \(E\), and
\[ A = \sqrt{\frac{\pi e}{2}}\left( \frac{f_c}{f} \right)^2 + \frac{3e}{8}\left( \frac{f_c}{f} \right)^4 + \frac{5}{24}\sqrt{\frac{\pi e^3}{6}}\left( \frac{f_c}{f} \right)^6 + \cdots, \tag{7.28} \]
where \(e\) is the base of the natural logarithm.
With the aid of (2.16), the following formula may be obtained for calculating the temperature:
\[ T = \frac{mg(z - z_{\max})}{2k\left(1 - \dfrac{n}{n_{\max}}\right)^{1/\gamma}}. \tag{7.29} \]
By these formulas, very small values \(H = 20 \div 30\) km were obtained \({}^{97}\) for the \(F_2\) layer. From the same data, using (7.29) and (7.5), respectively, the values \(H = 18 \div 35\) km and \(H = 28 \div 32\) km were obtained. Thus, in the case considered, the values of \(H\) over England were indeed small.
7.3. Simple Layer
A solution for \(T\) can be obtained from the complete equations of the simple layer (7.1) and (7.2)*):
\[ 4\ln\left(\frac{f}{f_c}\right)=F(T). \tag{7.30} \]
This equation was used to determine the temperature in the \(F_2\) layer over Washington and Huancayo (Peru) on January 14, 1947. For comparison, the temperature values were calculated by several formulas. The true heights of the ionization maximum, obtained by the method given in \({}^{3}\) and \({}^{80}\), are also compared with the values calculated by the formulas given above. The results of the calculations are presented in Table IX.
Table IX
Comparison of different data for the \(F_2\) layer
(local noon, January 14, 1947)
| Station | Formula No. | \(z_{\max}\) (km) | \(H_F\) (km) | \(H_E\) (km) | \(T\) °K |
|---|---|---|---|---|---|
| Washington | (7.5) | 331 | 67 | — | 1530 |
| Washington | (7.30) | — | 71 | — | 1630 |
| Washington | (7.25) | 186 | 68 | \(-550 \div 440\) | — |
| Washington | (7.7) | — | 117 | — | 2680 |
| Washington | (7.15) | 313 | 68 | — | 1600 |
| Washington | (7.26) | 320 | 81 | \(-305 \div 142\) | — |
| Washington | (7.24) | 309 | 73 | — | 1720 |
| Washington | (7.27) | 284 | 83 | \(-580 \div +110\) | — |
| Huancayo | (7.5) | 500 | 105 | — | 2200 |
| Huancayo | (7.30) | — | 132 | — | 2750 |
| Huancayo | (7.6) | — | 186 | — | 3910 |
| Huancayo | (7.15) | 564 | 160 | — | — |
| Huancayo | (7.24) | 568 | 154 | — | — |
*) It is somewhat surprising that, here as well as in other places in this article, the author analyzes in such detail a number of relations that are consequences of calculations with a simple layer, when it is well known that the \(F_2\) layer is in no way described by the equation of a simple layer, as the author himself notes below. (Trans. note.)
The temperature can also be determined by means of equation (2.5)
\[ Q=\beta n''J_{\infty}\cdot \rho_i\left(\frac{T}{T_i}\right)^{-(1+A)} e^{-\left[\rho_i\beta H_i\sec\chi\left(\frac{T}{T_i}\right)^{-A}\right]}, \tag{7.31} \]
which was obtained for a linear temperature dependence; the index \(i\) denotes the quantity referring to the given height. From (7.31) one obtains:
\[ z_{\max}=z_i+\frac{1}{b}\left\{\frac{\cos\chi}{\rho_i\beta H_i}(1+A)^{-A}-1\right\} \tag{7.32} \]
and
\[ n_{\max}=\sqrt{\frac{\beta\rho_i n''J}{\alpha}\frac{\cos\chi}{\rho_i\beta H_i}(1+A)e^{-\frac{1}{2}(1+A)}}, \tag{7.33} \]
where
\[ A=\frac{1}{bH_i}. \]
Using equations (7.31) and (7.33), temperature values at heights of 200 and 300 km were calculated from ionospheric observations in South Africa\({}^{50}\) for the period from November 1945 to February 1946. In these calculations it was assumed that \(M=15\) (i.e.
\[ \frac{mg}{k}=16.2\ \text{deg}/\text{cm} \]
) and \(\rho_i=10^{-11}\ \text{cm}^{-3}\) at a height of 200 km, and the corresponding values of \(b\), \(T_i\), \(\beta\), and \(\chi\) were used. To determine \(b\) and \(T_i\), the calculated values of the absorption coefficient \(\beta\) and the expression \(\dfrac{\beta\rho_i n''J}{\alpha}\) were successively investigated. For \(\beta=3.2\cdot10^{-17}\ \text{cm}^2\) and heights of 200 and 300 km, the following temperature values were obtained:
| Height (km) | Temperature values | Temperature values | Temperature values | Temperature values |
|---|---|---|---|---|
| Height (km) | November | December | January | February |
| 200 | 485 | 408 | 369 | 447 |
| 300 | 1097 | 1376 | 1392 | 1305 |
7.4. Conclusion
The results presented in Table IX indicate how cautiously one must treat the formulas given above. In the case when the retardation effect in the \(E\) layer is not taken into account, it is found that in the tropics the temperature is higher than at
middle latitudes. The scatter of the obtained temperature values is, of course, important under all conditions, since the same experimental data were used in the calculations.
Analyzing the internal contradictions of the calculation results, it should be pointed out that the basic equation (7.1) is in fact inapplicable to the \(F_2\) layer, since this layer can hardly be a “simple” layer, whose equations were obtained under the assumption of an isothermal atmosphere.
Approximations of the basic (incorrect) equation repeat the errors of the basic equation and introduce additional errors connected with the approximations adopted. It should also be recalled that (7.1) is written under the assumption
\[ \frac{dn}{dt}=0, \]
which is usually incorrect for the \(F_2\) layer.
Obtaining exact experimental values of the critical frequency and the effective height also presents certain difficulties. In a number of cases it is difficult to determine the true value of the electron concentration near the maximum of the layer; moreover, near the critical frequency the determination of the effective height is also subject to errors.
Neglect of the Earth’s magnetic field may also have a noticeable effect on the results of the calculations. Preliminary calculations for the \(E\) layer with allowance for the Earth’s magnetic field change the values of \(H\) by a factor of about 1.2; an even larger error may be introduced in determining \(H\) for the \(F_2\) layer. It has been pointed out \(^{69}\) that allowance for tidal oscillations of the atmosphere may also change the value of \(H\), but their influence has not yet been discussed.
When retardation in the \(E\) layer is taken into account (equations (7.25), (7.26), and (7.27)), values of the height scale \(H_E\) are obtained that differ greatly from the expected values of \(H_E\). The completely obviously incorrect values of \(H_E\) may be a consequence not only of the reasons noted above, but also of the fact that the lower layer is not parabolic or a “simple layer”*). In addition, the retardation depends strongly on the electron concentration between the layers; since it is not taken into account, this may completely spoil the usefulness of these equations. Since the results of the calculations are very sensitive to the value of the effective height, even small errors in its determination may strongly affect the values of \(H_E\). The estimates of the values of \(H_E\) given in Table IX raise serious doubts as to the usefulness of these equations.
) The author somewhat weakens the inadequacy of these calculations by saying: “differ greatly” from the expected values of \(H_E\). In fact, absurd values of \(H_E\) are obtained (see Table IX). The suspicion arises that computational errors have crept into the calculations, but the corresponding check is impossible because the initial data are absent.
(Translator’s note.*)
The use of formula (7.31) is subject to the same doubts indicated above. In calculations with (7.31) and (7.33) an additional difficulty arises, connected with the uncertainty of the exact value of \(\beta\). Moreover, the value \(M\) taken in \(^{50}\) is also doubtful, since it corresponds to an almost complete dissociation of nitrogen. Although this method is undoubtedly better, since instead of an isothermal atmosphere it considers a linear temperature gradient, it cannot be applied directly to \(F_2\) because of the peculiarities in the behavior of this layer.
On the basis of the available results it may be concluded \(^{96}\) that the possibility of using the results of measurements of electron concentration to determine the temperature of the \(F_2\) layer is still an open question. The calculation methods give temperatures that are of the right order of magnitude, but crude; these methods undoubtedly require revision.
§ 8. TEMPERATURE AND THE RECOMBINATION COEFFICIENT
8.1. Recombination at Low Pressures
Before a detailed discussion of this question, it is necessary to obtain a clear understanding of the process called recombination. Recombination of ions means the combination of positive and negative (including electron) ions into a neutral system. The recombination coefficient characterizes the rate of formation of neutral systems.
It is clear from this definition that recombination is a process occurring under all possible conditions. In each particular case, essentially different phenomena may occur. The reasons for this lie in the great diversity of the nature of the charge carriers and of the states in which they find themselves after ionization. For example, the mechanism of combination of a free electron with a positive ion differs radically from the case when charge transfer occurs from an ordinary negative ion to a positive ion, or when two oppositely charged ions are neutralized as a result of contact. Although each of these acts is recombination, the mechanism and characteristic constants in these cases differ from one another and require separate consideration.
The general term “recombination” includes special actions of the electron, when preferential, columnar, and voluminar recombination occur; each of these processes represents a separate mechanism and has a corresponding recombination coefficient*). For the ionosphere
) For more details on recombination processes, see \(^{147}\). (Translator’s note.*)
is determined by the generalized recombination coefficient, which describes the joint action of adhesion and recombination, or the joint action of several recombination mechanisms acting simultaneously or successively. In general, if the term “recombination” is used freely, considerable confusion may arise.
Fortunately, in ionospheric work only two mechanisms should be discussed: (a) volume recombination at low pressures and (b) generalized recombination. The first reaction occurs when equal numbers of ions of different signs are chaotically distributed in a volume; such conditions occur in the \(F_2\) layer. The second mechanism apparently takes place in the \(E\) layer.
Volume recombination at low pressures is determined by the coefficient
\[ \alpha=\pi \sqrt{\overline{u^2}+\overline{v^2}}\left(Rd^3+\frac{\sigma^2}{4}\right), \tag{8,1} \]
where
\[ d=\frac{2e^2}{3kT} \tag{8,2} \]
is the radius of action of the attractive forces, \(u\) and \(v\) are respectively the velocities of ions of each type, and \(R\) is the probability of collision at the distance \(d\). \(\sigma\) may be regarded as the mean effective value of the collision cross section, on the assumption of a Maxwellian velocity distribution.
Depending on whether the value of the ion energy at the last collision is equal to, less than, or greater than the energy acquired by them in falling from infinity under the action of the attractive forces, the orbit of the ion forms an ellipse, a parabola, or a hyperbola. If the value of \(d\) is of the order of, or less than, the mean free path, the ions will approach each other with energies equal to the sum of their thermal energy and the energy formed by their mutual potential field. Thus their orbit will not be closed, and there is a high probability that after the encounter the ions will separate. If, however, within the distance \(d\) the ions undergo elastic collisions with neutral molecules, then they have a great chance of losing part of the energy acquired in the field and therefore of being captured into elliptical orbits.
In one of the papers it is postulated \(^{122}\) that a single collision of one of the ions with a third particle within the distance \(d\) is sufficient for the energy to fall to the value required for the formation of a closed orbit. This assumption is also adopted in this paper.
At low pressures the mean free path between collisions is large, and
\[ R \simeq \frac{4d}{3}\left(\frac{1}{\lambda_1}+\frac{1}{\lambda_2}\right), \]
where the subscripts correspond-
tions to ions of different types. Since \(\frac{\sigma}{2} \simeq 10^{-8}\ \mathrm{cm}\) and \(d \simeq 10^{-7}\ \mathrm{cm}\), the term \(Rd^{2}\) predominates at pressures greater than \(10^{-5}\) mb; for lower pressures the term \(\frac{\sigma^{2}}{4}\) predominates. Thus, when the pressure is greater than \(10^{-5}\) mb, (8.1) reduces to the formula\(^{123}\)
\[ \alpha=\pi R d^{2}\sqrt{\bar u^{2}+\bar v^{2}} . \tag{8.3} \]
At lower pressures one obtains\(^{77}\) the formula
\[ \alpha=\pi \frac{\sigma^{2}}{4}\sqrt{\bar u^{2}+\bar v^{2}} . \tag{8.4} \]
Since \(\bar{\lambda}\) varies as \(T^{1/2}\), \(d\)—as \(T^{-1}\), and \(\sqrt{\bar u^{2}+\bar v^{2}}\)—as \(T^{1/2}\), (8.3) and (8.4), respectively, give:
\[ \alpha=AT^{-3} \tag{8.5} \]
and
\[ \alpha=B\sqrt{T}, \tag{8.6} \]
where \(A\) and \(B\) are constants.
The physical meaning hidden in (8.1) is readily apparent. The term \(Rd^{2}\) describes a mechanism in which, after collision with a third body, one ion moves, within a distance \(d\), in a closed orbit around another ion. However, when the pressure decreases, the probability of the process decreases and \(Rd^{2}\) becomes equal to \(\frac{\sigma^{2}}{4}\). Under these conditions, the probability of a simple collision leading to neutralization and the probability of a collision of three particles\(^{122}\) are equal. With a further decrease in pressure, the first term decreases rapidly in comparison with the second. Since in the \(F_{2}\) layer the pressure is less than \(10^{-6}\) mb, attachment becomes negligibly small there, and the conditions described by equation (8.6) are more applicable to it than (8.5). The applicability of equation (8.5) to the \(E\) layer is discussed below.
It should be noted that equations (8.5) and (8.6) may be regarded as special cases of the more general equation
\[ \alpha=CT^{r}. \tag{8.7} \]
It may also be written in the form
\[ \frac{\alpha}{\alpha_{a}}=\left(\frac{T}{T_{a}}\right)^{r}. \tag{8.8} \]
On the basis of the theory of statistical equilibrium\(^{85,142}\) it was found that \(r=-\frac{1}{2}\), when \(T\) is the electron temperature \(T_{e}\). An analogous result was obtained from quantum-mechanical considerations\(^{126}\). On the basis of theoretical calculations\(^{14}\) it was established\(^{46a}\) that at low pressures radiative recombina-
electron recombination with positive ions occurs according to the law \(\sim T^{-3/4}\). Under ionospheric conditions \({}^{15a}\) \(r=-\dfrac{3}{2}\). In earlier theoretical works the value \(r=-\dfrac{5}{2}\) was obtained at low pressures and \(r=-\dfrac{3}{2}\) at high pressures \({}^{131}\). From quantum-mechanical considerations \({}^{19}\), the value \(r=-0.9\) was obtained.
The variety of these values should be interpreted in the light of the recombination mechanisms considered. For radiative recombination of electrons with positive ions, the most probable value is \(r=-\dfrac{1}{2}\), while for volume recombination of the type occurring in the plasma of a discharge tube, apparently, \(r=-\dfrac{3}{2}\). However, judging from the present state of the question, these reactions are apparently applicable to the ionosphere.
Let us assume that two processes coexist in layer \(E\): recombination and attachment. The combined process can be described by means of the generalized effective recombination coefficient \({}^{111}\)
\[ a' = a_e + K a_i, \tag{8,9} \]
where \(a_e\) is the coefficient of recombination of electrons with positive ions, \(a_i\) is the coefficient of recombination of negative and positive ions, \(K=\dfrac{n_-}{n_e}\), and \(n_-\) is the number of negative ions.
Although \(a_e\) and \(a_i\) may be described by equation (8,7), there is no basis for assuming that in every case the values of \(r\) are identical. Under these circumstances it is impossible to recommend formula (8,5) for layer \(E\).
In several extensive works \({}^{116}\), calculations were carried out for various latitudes of the diurnal variation of temperature and of the distribution of temperature with height in the region \(100 \div 300\) km. For these temperature calculations it was necessary first to determine the variation of the recombination coefficient during the day.
8.2. Diurnal variation of the recombination coefficient
For different times of day, the values of the recombination coefficient were obtained in the following manner:
a) at night, for processing the experimental data the equation was used
\[ \frac{dn_a}{dt} = -a_a n_a^2 \quad (q_a=0); \tag{8,10} \]
b) for comparatively short, but not necessarily equal, time intervals on both sides of sunrise and sunset,
\[ \frac{dn_a}{dt}=-\alpha_a n_a^2,\qquad \frac{dn_b}{dt}=q_b-\alpha_a n_a^2, \tag{8,11} \]
where \(q_a=0,\ n_b=n_a,\ \alpha_b=\alpha_a\);
c) for small and equal time intervals on different sides of noon,
\[ \frac{dn_c}{dt}=q_c-\alpha_c n_c^2,\qquad \frac{dn_d}{dt}=q_c-\alpha_c n_d^2, \tag{8,12} \]
where \(q_c=q_d\) and \(\alpha_c=\alpha_d\). In all these cases the equations can be solved for \(z\). It is true that, if \(n_c^2=n_d^2\), equations (8,12) are incompatible.
It should be noted that equations (8,11) and (8,12) can be correctly applied to the \(F_2\) layer only when the level of its maximum ionization does not fluctuate owing to changes in atmospheric conditions and remains at one and the same height during the time period under consideration. The question of the applicability of these equations during periods of ionospheric disturbances remains open. Preliminary studies\(^{116}\) yielded different results in cases where the time interval on both sides of noon amounted to several hours. However, these conclusions cannot yet be regarded as final. With some exceptions, the use of (8,10) and (8,12) for overlapping time intervals*) should be considered inadmissible, because serious contradictions may arise under such a procedure.
In these calculations\(^{116}\) for the \(E\), \(F_1\), and \(F_2\) layers the relation
\[ \frac{\alpha}{\alpha_a}=\left(\frac{T}{230}\right)^{-3}. \tag{8,13} \]
was used.
The value \(T=230^\circ\) was chosen on the basis of investigations of auroral spectra\(^{129}\). The value of \(\alpha_a\), however, was determined by the following
*) The author does not consider yet another possibility that makes it possible to calculate the diurnal variation of \(q\) and \(\alpha\) in those cases where the smoothed diurnal variation of the ionization of the layer is sufficiently smooth. Equations (8,12) can be solved step by step for two adjacent time intervals, each \((60 \div 30)\) minutes long or less, postulating constancy of \(q\) and \(\alpha\) over these intervals of time. In this way, in works \(^{145}\) and \(^{146}\), in particular, the diurnal variation was calculated for the values of \(\alpha\) averaged over the corresponding time intervals. This same method can, moreover, be applied to calculate \(\alpha\) at different levels of a layer, and not only for the region of the layer maximum, as is usually done, by constructing from height–frequency characteristics the true distribution of ionization with height and, correspondingly, the diurnal variation of electron density at different heights. In this way, in works 3 and 4 cited above, some data were obtained on the diurnal and altitude dependence of \(\alpha\). The author does not consider these data, nor does he discuss such a method of investigating \(\alpha\) and \(T\). (Translator’s note.)
thus: a) for layers \(E\) and \(F_1\) the value for the early morning was chosen; b) for layer \(F_2\), the mean nighttime value calculated from (8,10). (In an earlier work\(^{1166}\), when processing data for three stations, the value \(\alpha_a\) was chosen, obtained as the mean daily value of \(\alpha\) in Tromsø (Norway).) On this basis, temperature values for January 1947 were calculated for 18 stations located at different latitudes (from \(70^\circ\) N to \(43^\circ\) S) and longitudes. It was found that during the daytime in layers \(E\) and \(F_1\) the temperature has the values \(T = 71\div 568^\circ\) and \(T = 35\div 1785^\circ\), respectively. In layer \(F_2\), however, the temperature varied over 24 hours from \(T = 84\) to \(T = 1098^\circ\).
8.3. Limitations of the Method
It is necessary to point out several limitations inherent in any method for determining the temperature of the ionosphere from measurements of the recombination coefficient. These limitations arise: a) because of the methods used to obtain the recombination coefficient, and b) because of the uncertainty in the values of \(\alpha\) and \(T\) employed.
When determining \(\alpha\) by means of equations (8,10) and (8,12), it is evident that when the course of the curve \((n,t)\) is constant (in the vicinity of noon), then \(\alpha \to 0\). Indeed, if in the vicinity of noon the electron concentration increases linearly, so that
\[ \frac{dn_a}{dt} = \frac{dn_b}{dt}, \]
then \(\alpha = 0\), since \(q_a = q_b\) and \(n_a \ne n_b\). In this case \(T \to \infty\) if \(r < 0\), and \(T \to 0\) if \(r > 0\), which is absurd. Further, in some cases it follows from the calculations that \(\alpha < 0\). These strange results can be attributed to the incorrectness of the assumptions made in writing equations (8,11) and (8,12). In addition, the legitimacy of applying the first of these equations to obtain \(\alpha\) over long periods on both sides of the moments of darkness has also not been proved. In order for these two equations to be applicable, it is necessary that: a) the change in ion concentration be described by the quadratic law of recombination, and b) there be no disturbances at the level of maximum ionization. It is also clear that these equations do not take into account the influence of significant diurnal temperature variations, since the terms accounting for the influx and outflow of ions have been discarded (see § 9).
If, however, the method of obtaining the recombination coefficient over the course of a day is considered, then in connection with the determination of \(\alpha\) and \(T\) new questions arise. Early laboratory experiments gave the value \(r \cong -\frac{7}{3}^{39}\) and \(r \cong -2.2^{102}\) (both values are close to \(r = -3\)) for pressures significantly greater than ionospheric pressures and, apparently, for the pure recombination process. Extrapolation of this value, even if on the whole it agrees
with theory, for obtaining an effective recombination coefficient is hardly justified.
Finally, the question arises of the correspondence between the values of $\alpha$ and $T$. Since $\alpha=\alpha(T)$, each value of $\alpha$ corresponds to a definite value of the temperature, and in the equations this value of the temperature should be used, not the value of the rotational temperature obtained from auroral observation data. In fact, it has been shown$^{61,95}$ that the temperature obtained from bands of rotational spectra is not necessarily equal to the gas temperature. Studying the temperature calculated from the negative bands of nitrogen, a number of works$^{35,36}$ concluded that the rotational temperature depends not only on the gas temperature but also on the type of collisions. It was, however, indicated$^{134}$ that although the effective value of the rotational temperature depends on the gas temperature, it is not identical with the latter. It turned out that the rotational temperature calculated from the bands of $N_2^+$ is lower than the temperature of the envelope surrounding the discharge tube*). It was also found$^{33}$ that in collisions with electrons the molecules have anomalously low rotational temperatures; such conditions can easily occur during auroras.
Approximate values of $\alpha$ and $T$ for the $F_2$ layer can be calculated by means of (8.4). Taking $\sigma=3.2\cdot10^{-8}\ \text{cm}$, $\bar u^2=\bar v^2=\dfrac{3kT}{m}$ ($m$ is the mass of an oxygen atom or ion), and $T=2000^\circ$, we obtain $\alpha=2\cdot10^{-10}\ \text{cm}^3/\text{sec}$. This value is rather close to the values obtained from ionospheric data$^{111,136}$.
For the $F_2$ layer there are also cited values of $\alpha$ varying within the limits from $1\cdot10^{-12}$ to $1.3\cdot10^{-9}$, and temperature values varying correspondingly (for $r=-3$) from 1100 to 85°$^{116г}$. In these calculations it was assumed that the mean nighttime value of $\alpha$ corresponds to $T=230^\circ$. If one assumes that $r=+\tfrac{1}{2}$ at $\alpha=2\cdot10^{-10}$ and $T=2000^\circ$, then the range of temperature variation will be $(5\cdot10^{-2}\div84{,}500)^\circ$! It is truly evident that the method of calculating the temperature from the recombination coefficient requires more fundamental study.
For the reasons indicated above, the application of this method for establishing a correspondence between the temperature and the character of the $F_2$ layer (see$^{111}$) is subject to serious criticism.
8.4. Seasonal variations of the temperature of the $F_2$ layer
The relation between the temperature and the recombination coefficient was also used to obtain information concerning the seasonal and latitudinal variation of the temperature. Thus,
*) Under some conditions the temperature values calculated from bands of rotational spectra may be considerably lower than the gas temperature. Such conditions may arise in the ionosphere and mesosphere (see$^{118}$).
under the assumption that \(J_a=J_b\), \(m_a=m_b\), and \(g_a=g_b\), from (2,17) one obtains
\[ \frac{n_a}{n_b}= \sqrt{\frac{T_b a_b \cos \chi_a}{T_a a_a \cos \chi_b}} \tag{8,14} \]
and, with the aid of (8,8),
\[ \frac{n_a}{n_b}= \left(\frac{T_b}{T_a}\right)^{\frac{r+1}{2}} \sqrt{\frac{\cos \chi_a}{\cos \chi_b}} . \tag{8,15} \]
If the difference in conditions denoted in these equations by the indices \(a\) and \(b\) signifies opposite periods of solstice at noon at the same height above the earth’s surface, then
\[ \frac{n_a}{n_b}= \left(\frac{T_b}{T_a}\right)^{\frac{r+1}{2}} \cdot \sqrt{\frac{\sin(\theta+\delta_a)}{\sin(\theta+\delta_b)}} , \tag{8,16} \]
where the index \(a\) now refers to December, and the index \(b\) to June. For middle latitudes, for example at \(\theta=45^\circ\) north or \(45^\circ\) south, \(\delta_a=\delta_b=-23.5^\circ\). Assuming further that \(n_a=2n_b\), which is in general confirmed by experimental data, we obtain, for different values of \(r\) in both hemispheres, the behavior of the ratio of the temperature in June to the value of the temperature in December, shown in Fig. 14.
Fig. 14. Ratio of the temperature values in June and December as a function of \(r\). The solid lines correspond to \(45^\circ\) N latitude, and the dashed lines to \(45^\circ\) S latitude.
As might have been expected, at \(r=-1\) the curves have a pole. When \(r<-1\), the temperature in both hemispheres is lower in June than in December. Thus, at \(r=-3\) the ratio of the temperatures is, respectively, for \(45^\circ\) N and \(45^\circ\) S, 0.25 and 0.8. This result is strange for the northern hemisphere. Although in December the Sun is closer to the Earth, it is unlikely that the closeness of the Sun’s position would outweigh the effect of the increase in the angle of incidence of radiation in the Northern Hemisphere. The same strange result is obtained for \(r>-1\), since in this case in both hemispheres the temperature in June is higher than in December. Thus, at \(r=0.5\) their ratio is, respectively, for \(45^\circ\) N and \(45^\circ\) S, 6 and 1.3.
In the past, attempts were also made to determine seasonal temperature changes. With the aid of (8,14), the following relation was obtained\(^6\)
\[ \frac{n_a}{n_b}= \sqrt{\frac{T_b}{T_a}} \cdot \sqrt{\frac{\sin(\theta+\delta_a)}{\sin(\theta-\delta_a)}} . \tag{8,17} \]
This formula, which assumes that the electron temperature is the same in winter and in summer, means that the recombination coefficient is constant during both periods. Since at that time in England, Japan, and the USA it had been found that the electron density in December is greater than in June, a hypothesis of thermal expansion of the layer was put forward,\(^{6}\) and it was suggested that the larger values of temperature in the \(F_2\) layer in summer lead to the entire layer as a whole expanding upward and, because of this, to a decrease in its electron density. In winter the reverse mechanism operates. It was calculated that the mean noon values of temperature in the \(F_2\) layer in mid-summer are four times greater than those in mid-winter. Taking \(300^\circ\) as the mean temperature value for darkness in the winter period, one obtains for the \(F_2\) layer in England a summer temperature value equal to \(1200^\circ\).\(^{6}\)
If \(r=-0.5\), then the ratio \(\dfrac{T_b}{T_a}\) may vary from 10 to 1, and in summer the temperature value reaches \(3000^\circ\).
Since according to the calculations the temperature is higher in summer than in winter, at first glance it seems that the method described above is a simple and correct method for studying the climate of the ionosphere. However, this is not so. It was shown\(^{16,117}\) that the electron concentration of the \(F_2\) layer in December is greater than in June in both hemispheres. This fact completely destroys the hypothesis of thermal expansion of the layer.
8.5. Latitudinal changes of the temperature of the \(F_2\) layer
From (8.14) one can also theoretically calculate the latitudinal changes of temperature. If the subscripts \(c\) and \(d\) denote different values of the additional angle \(\theta\) to the latitude, then one obtains:
\[ \frac{n_c}{n_d}= \left(\frac{T_d}{T_c}\right)^{\frac{r+1}{2}} \sqrt{\frac{\sin(\theta_c+\delta)}{\sin(\theta_d+\delta)}}. \tag{8.18} \]
For calculations with (8.18), one may use values of the electron concentration obtained at different latitudes for a given value of the declination; however, this equation is still difficult to solve because of the uncertainty that exists at present regarding the value of \(r\).
From a comparison of electron densities over Japan at Hiraiso (\(36^\circ\)) and Asahigawa (\(44^\circ\))\(^{78}\), on the basis of simultaneous experiments at both stations in June 1936, it was found that for \(r=1\) the maximum electron concentration of the \(F_2\) layer over the first station was 20% greater than that over the second. From these data it followed that over the more northern station the temperature was 2.2 times higher. Such latitudinal changes of temperature are hardly plausible,
The values of the temperature in the \(F_2\) layer were also compared for the following three points: the ratio was calculated of the measured temperature at Clyde on Baffin Island (Canada), at Churchill, and at Manitouge (Canada) to the temperature at Trinidad. Monthly mean values at 1800 Greenwich time for January and August were processed. The data obtained are contained in Table X.
Table X
Ratio of temperatures in the \(F_2\) layer at different points
| Station | Latitude | January \(r = \frac{1}{2}\) | January \(r = -3\) | August \(r = \frac{1}{2}\) | August \(r = -3\) |
|---|---|---|---|---|---|
| Clyde . . . . . . . | \(76.5^\circ\ \mathrm{N}\) | — | — | 6.20 | 0.25 |
| Churchill . . . . . | \(58.8^\circ\ \mathrm{N}\) | 0.44 | 1.85 | 3.46 | 0.39 |
| Trinidad . . . . . . | \(10.6^\circ\ \mathrm{N}\) | 1.00 | 1.00 | 1.00 | 1.00 |
CONCLUSION
The various results set forth in this section lead to only one conclusion, namely, that the determination of temperature by means of the recombination coefficient, both under laboratory conditions and in the atmosphere, is not at present yet substantiated. Before any hopes can be entertained for the successful use of the methods described, extensive investigations are necessary. The failure of the methods listed above may be explained by the fact that a) the \(F_2\) layer of the ionosphere is not described by the equation of a “simple” layer, b) it is difficult to apply the still not fully understood results of laboratory investigations to the ionosphere, and also, possibly, by the fact that c) the calculations neglect the influence of the Earth’s magnetic field on recombination processes in the upper atmosphere (see also \(^{78}\)).
§ 9. TEMPERATURE AND THE FUNCTION DESCRIBING THE INFLOW OF ELECTRONS
9.1. Compression of the Atmosphere
The application of a function describing the inflow of electrons to determine the temperature in the \(F_2\) layer of the ionosphere is the result of attempts to explain the nocturnal increase of electron concentration.
The theories currently accepted require that the electron concentration of the \(F_2\) layer decrease during the period of darkness to a greater extent than is observed experimentally. The action, however, on these
...at altitudes of some sources of ionization of non-solar origin is rejected*) just as it is assumed that electron attachment processes usually play only a small role here (see \(^{150,846}\) and \(^{139}\)). Therefore it is postulated that the effect of atmospheric compression leads to the observed stability of the nighttime values of electron density. Since the compression effect depends on temperature, its study makes it possible to estimate reasonably the ratio of the daytime and nighttime values of temperature.
Qualitatively, the expansion effect has already been considered in the literature. The difference between the winter and summer values of the electron concentration in the northern hemisphere was explained by expansion of the layer \(^{36,626}\). The behavior of the \(F_2\) layer was attributed to thermal expansion of the layer and to changes in solar radiation \(^{119}\). It was established \(^{37}\) that the rise in the altitudes of auroras in the polar atmosphere illuminated by the Sun is explained by the fact that the ratio of the nighttime and daytime temperature values changes as \(1:1.57\). It was also indicated that during an eclipse the \(F_2\) layer expands substantially. It was noted that, if the hypotheses of thermal expansion of the layer are correct, then the midnight values of temperature are probably low and relatively constant during the year, and that the summer midday values of temperature are considerably greater than the midnight values (see \(^{10}\)). It was calculated \(^{83}\) that the \(F\) layer cools after sunset on the average by approximately 30%, and that from night to night the degree of its cooling changes. In view of an attempt to explain the diurnal changes in ionization density, the possibility was discussed \(^{47}\) of a decrease in temperature in the \(E\) layer of the ionosphere. It was suggested \(^{62}\) that the expansion of the \(F_2\) layer is caused by the heat that forms winds. There is also a report that a decrease in the ionic concentration of the \(F_2\) layer, caused by its thermal expansion, has been observed \(^{108}\).
9.2. Diurnal thermal variations
It is assumed that the dependence of temperature on height in the region \((0 \div 100)\) km does not change over the course of a day, and that above 100 km there are both diurnal and seasonal variations of temperature.
*) Recently a suggestion has been made \(^{149}\) that interstellar matter penetrating the Earth’s atmosphere as a result of the Sun’s attraction additionally ionizes the \(F_2\) layer at night, causing irregular ionization phenomena in it. This hypothesis was proposed in order to explain oscillations in the intensity of the radio emission of discrete sources in the Galaxy. Processing of a series of experimental data showed good agreement between irregular phenomena in the ionosphere and fluctuations in the intensity of the radio emission of discrete sources. If the indicated ionization mechanism, operating only at night in the unilluminated part of the terrestrial globe, is plausible, then, along with irregular ionization phenomena caused by the spontaneous action of this mechanism, the possibility cannot be excluded that the nonuniformity of the particle flux may cause an increase in the general level of ionization density of the layer. (Translator’s note.)
As a first approximation it is assumed that during the period beginning two hours before sunset and continuing until sunrise,
\[ T=T_m e^{-2at}, \tag{9.1} \]
where \(T_m\) is the temperature value two hours before sunset. At all other times
\[ T=T_s e^{2bt}. \tag{9.2} \]
Functions of this type correspond well to conditions at the earth’s surface, and, for lack of more accurate data, it is assumed that they are also valid in the \(F_2\) layer. It is assumed that the minimum temperature \(T_s=300^\circ\) and is the same throughout the year. For the isobar—the surface of equal pressure \(P_m\)—the number of particles during the darkness period is equal to
\[ N=\frac{P_m e^{2at}}{kT_m}. \tag{9.3} \]
Fig. 15. Dependence of the altitude, at which the equal-pressure surface \(P=P(T_m)\), initially located at an altitude of 300 km from \(T\), is situated by sunrise. An exponential decrease of temperature during the night is postulated. The final value of the temperature at sunrise is \(T_m=300^\circ\).
Another important parameter is the height to which, by sunrise, the isobar \(P_m=P(T_m)\), initially located at an altitude of 300 km, descends. The dependence of the smallest value of the height on the maximum value of the temperature \(T_m\) is shown in Fig. 15.
9.3. Derivation of the Equations
It is easy to obtain an expression quantitatively describing the effect of compression\({}^{466}\). The ion influx function is introduced
\[ s(t)=\frac{n}{N}\frac{dN}{dt}, \tag{9.4} \]
which determines the fraction of ions of one sign flowing into a unit volume under the influence of atmospheric compression. If the adhesion of electrons to neutral particles is neglected, then the equation describing the ionization of the layer has the form
\[ \frac{dn}{dt}=q+s(t)-\alpha n^2. \tag{9.5} \]
Since \(\alpha=\alpha(T)\), one may write:
\[ \alpha=AT^r=\alpha_0 e^{-2art}, \tag{9.6} \]
where \(a_0=AT_m^r\). In the period of darkness, when \(q=0\), we obtain:
\[ -\frac{dn}{dt}=-\frac{n}{N}\frac{dN}{dt}-a_0 n^2 e^{-2art}. \tag{9,7} \]
Taking \(r=-\frac{1}{2}\) and assuming that the \(F_2\) layer lies in a region of equal pressures, as a result of integrating (9,7) we obtain:
\[ \frac{1}{n}=\frac{a_0}{a}e^{-at}+Ce^{-2at}, \tag{9,8} \]
where \(C\) is the constant of integration.
On the basis of § 8 it seems expedient to choose for the \(F_2\) layer the value \(r=+\frac{1}{2}\). This value indicates the existence of a mechanism in which an electron passes directly from a negative ion to a positive ion when they approach to within a certain distance. In this reaction the ion acts as a dynamic catalyst, accelerating the process. The neutral oxygen atom, which is formed as a result of the electron being detached from it, captures a new electron at the next collision. Such a mechanism removes the necessity for the existence in the upper atmosphere of a collision process involving three bodies, which is quite unavoidable under low-pressure conditions.
It is assumed that during the daytime the levels of maximum ionization and maximum ion-formation rate both lie in a region of constant pressure. From (9,5) and (9,4) we obtain:
\[ \frac{dn}{dt}=q(t)+\frac{n}{N}\frac{dN}{dt}-AT_s^{1/2}n^2e^{bt}, \tag{9,9} \]
where in (9,9) the value of \(q(t)\) is taken for a spherical Earth, determined by equations (2,18) and (2,20). The quadrature of (9,9) cannot be obtained in analytic form. It is possible, however, to solve this equation approximately by replacing \(\frac{dn}{dt}\) by \(\frac{\Delta n}{\Delta t}\) and taking \(\Delta t=3600\) sec.
9.4. Results of Numerical Calculations
Figures 16 and 17 give the results of measurements and calculation of the nocturnal decrease of ionization in the \(F_2\) layer over Washington. For better illustration of the effect obtained, calculations are also given for extremely large temperature values, which are impossible in planetary atmospheres.
From consideration of the calculated curves it is evident that, for moderate and large changes of temperature in the ionosphere, the effect of electron influx (caused by compression of the atmosphere) is noticeable only
into the second half of the night. In the first half of the night the recombination effect predominates.
It is evident that the nocturnal course of the measured ionization values does not coincide with any of the curves. Some constant recombination mechanism is acting, while the recombination coefficient in January is by no means necessarily equal to the recombination coefficient in August. It may be concluded that, for the selected monthly median values in the \(F_2\) layer, there are no pronounced diurnal changes in temperature.
Fig. 16. Experimental and calculated data on the nocturnal decrease of ionization of the \(F_2\) layer in January 1947 in Washington.
Fig. 17. Experimental and calculated data on the nocturnal decrease of ionization of the \(F_2\) layer in August 1947 in Washington.
It was originally found \(^{46\mathrm{в}}\) that the nocturnal change of ionization during magnetically quiet periods can be explained by a decrease of the night temperature from 2000 to \(300^\circ\). However, on the basis of the more general results described here, such temperature changes in general seem implausible.
In Fig. 18 ionization curves are presented, obtained for the daytime in January 1947 experimentally, as well as the results of calculations using formula (9,9). For the calculations it was
it was assumed that in the daytime \(n \cdot J'' = 3 \cdot 10^9\) \(1/\mathrm{cm}^2\,\mathrm{sec}\), which corresponds to the obtained values of the number of newly formed ions in January of the order of \(450\) \(1/\mathrm{cm}^3\,\mathrm{sec}\) or \(115\) \(1/\mathrm{cm}^3\,\mathrm{sec}\), respectively, for \(T=300\) and \(5000^\circ\).
In the calculations it was taken into account that the height of the isobar—the surface \(P(T_m)\)—for \(T=5000^\circ\) changes during the day. The value \(D\) was obtained as a result of numerical integration of (2,19).
The theoretical curves give the results that were to be expected, namely, they show that when the expansion of the atmosphere is taken into account the resulting influx of ions decreases; also reduced are both the maximum of the electron density and the rate of change of ionization in comparison with the values expected for an isothermal atmosphere. However, in not a single case do the results of the calculation coincide with the experimental results. The lag in the beginning of the growth of the electron concentration, obtained theoretically, is a substantial discrepancy between theory and observations. The general course of the calculated electron concentration also does not coincide with the experimental values. For intermediate values of the temperature \(T_m\), the theoretical curves could have been fitted to the experimental ones. Because of the complexity, these calculations were not carried out; it should be expected that they would lead to analogous conclusions.
Fig. 18. Experimental (solid line) and calculated (dotted line) data of the diurnal variation of the electron density of the \(F_2\) layer in January 1947 in Washington.
The discrepancy between the calculated and measured values of the electron concentration can apparently once again be attributed to the fact that in the \(F_2\) layer there are no significant changes of temperature during the day.
The results given above may be compared with results obtained independently\(^{59}\) from several other assumptions, namely from consideration of the influence of the expansion of the atmosphere on the change in electron concentration. Instead of (9,9), the equation\(^{59}\) was obtained
\[ \frac{dn}{dt}=q'_{\max}-\alpha n^3-\frac{n}{T}\frac{dT}{dt}, \tag{9,10} \]
where \(q'_{\max}\) is determined from (2,9), and
\[ \frac{dN}{N}=-\frac{dT}{T}. \]
In solving (6,10), two cases were discussed:
\[ \text{a) } \alpha=\mathrm{const} \quad \text{and} \quad \text{b) } \alpha=\frac{C}{\sqrt{T}}, \]
where \(C\) is a constant. Equation (9,10) can be rewritten in the form
\[ dT+\alpha T n\cdot dt+T\cdot \frac{dn}{n} = \frac{n''Jmg}{nke}\cos\chi\,dt \tag{9,11} \]
and solved for \(\dfrac{T}{T_a}\), where \(T_a\) is the value of the temperature at \(t=t_a\). In this way the diurnal variation of the ratio \(\dfrac{T}{T_a}\) was obtained, although \(T_a\) is unknown. The right-hand side of (9,11) was integrated numerically. It was assumed that the temperature variations have a 24-hour cycle and that
\[ \int n\,dt=\frac{1}{2}\sum_{i=0}^{i=24}(n_i+n_{i+1})\,\Delta t, \]
where \(\Delta t=3600\) sec. The results are given in the form of the ratio \(\dfrac{T}{T_a}\) for different values of the recombination coefficient.
For the mean values of the data for the \(F_2\) layer over the period from 6 to 10 July 1939, obtained in Germany\({}^{52}\), the following values of the ratio \(T_{\max}\) to \(T_{\min}\) were calculated:
| \(\alpha\ \dfrac{\mathrm{cm}^3}{\mathrm{sec}}\) | \(T_{\max}/T_{\min}\) |
|---|---|
| \(1.5\cdot 10^{-10}\) and more | 27 and more |
| \(1.0\cdot 10^{-10}\) | 10 |
| \(0.8\cdot 10^{-10}\) | 6 |
Calculations also showed that the temperature is maximal between 14 h 00 min and 16 h 30 min local time and minimal between 04 h 00 min and 06 h 30 min. For the case when
\[ \alpha=\frac{C}{\sqrt{T}}, \]
the equation reduces to the form
\[ \frac{dn}{dt}=\psi\left(n,\frac{dn}{dt},T,\chi\right). \tag{9,12} \]
This equation was integrated graphically, and the values of \(\dfrac{T}{T_a}\) were calculated. The value obtained was \(T_{\max}/T_{\min}=4.5\) for \(\alpha(T_{\max})=4.3\cdot 10^{-11}\). For both values of the recombination coefficient, the resulting number of newly formed electrons was equal to \(60\ \text{cm}^{-3}\,\text{sec}^{-1}\).
Even if one assumes that in the \(F_2\) layer \(T_{\min}=300^\circ\), it is difficult to obtain values of \(\dfrac{T_{\max}}{T_{\min}}\) greater than 10. If the recombination coefficient is constant, then the temperature depends very sensitively on changes in the recombination coefficient, so that small changes in \(\alpha\) lead to substantial changes in \(T\). Therefore the errors inherent in determinations of \(\alpha\) from experimental data increase considerably when calculating the temperature ratio.
If the recombination coefficient is considered to be a function of temperature, then the value \(T_{\max}/T_{\min}=4.5\) is obtained, which is more plausible. Let us recall that this result was obtained\(^ {59}\) for a “flat” Earth and \(r=-1/2\); therefore it cannot be compared with the results presented in Fig. 18, calculated for a spherical Earth and \(r=+1/2\).
Several remarks should be made concerning the use of equations (9.9), (9.11), and (9.12) for determining the diurnal variation of the temperature in the \(F_2\) layer. Since this layer is obviously not a “simple” layer either in its structure or in its formation, the formulas of § 2, derived for the plane and spherical cases, are inapplicable to it. (Let us recall that they were obtained for an isothermal atmosphere.) The assumption that the maximum ionization of the layer lies in a region of constant pressure is itself an open question and may cause considerable discrepancies in the calculations. Further, there is no certainty on the question of the relation between the temperature and the recombination coefficient. The values of \(q\) and \(\alpha\) also usually cannot be determined unambiguously from daytime electron-concentration data. Finally, it should be recalled that \(q'_{\max}\) (in equation (9.10)) was obtained for \(\chi<80\).
For these reasons, it seems inadvisable to apply the above equations to determine the probable diurnal variation of the temperature. A more suitable method is undoubtedly the calculation of the nighttime decrease in temperature, since in this case the complicating circumstances associated with the term describing ion formation in the layer (containing \(q\)) disappear.
On the basis of an analysis of equations that take into account the influx of electrons, no changes in the temperature of the \(F_2\) layer during the day have yet been definitively established. Likewise, at night no significant decrease in temperature has been observed. Thus, it may be concluded that there is no appreciable diurnal cycle of temperature change in the \(F_2\) layer.
§ 10. THERMAL EQUILIBRIUM IN THE MESOSPHERE *)
Before discussing the results obtained, it is necessary to define clearly the concept of temperature in a rarefied atmosphere.
It has already been pointed out that at great heights above the Earth’s surface the concept of temperature differs from that in the laboratory.
The classical concept of temperature is determined from the following law of thermodynamics: if each of two collections of particles is in thermal equilibrium with a third body, then they are in thermal equilibrium with respect to each other. Thus, the condition for thermal equilibrium is the equality of a special single-valued function characterizing the thermodynamic state of these collections. This function is called temperature. Any of the collections may be used as a thermometer and give a certain empirical quantity known as the empirical temperature.
When the number of particles in a collection decreases radically, it is then necessary to consider fluctuation phenomena and deviations from the mean. It is easy to determine the ratio \(Q\) of the standard deviation of the energies to the mean deviation of the energy \(^{76}\); it is equal to
\[ Q=\sqrt{\frac{2}{3N}}. \tag{10,1} \]
*) The density of a planet’s atmosphere decreases exponentially with increasing radial distance from its surface (depending on the change of temperature with height). The mean free path \(\lambda\) increases in this process until, at some height, \(\lambda\) for a particle moving radially upward away from the planet becomes greater than \(\lambda\) for a particle moving in the opposite direction. At this height and higher, particles move more slowly upward than downward. Many particles moving upward will change their direction of motion under the influence of the gravitational field even before encountering another particle and will return back to denser regions of the atmosphere. Thus, in the atmosphere there exists a boundary region where: a) collisions between particles can be neglected, b) the mean free path is very large, and c) gas particles move, depending on their energy, along elliptic, parabolic, or hyperbolic orbits. This region is called the exosphere. Its base is called the critical level, and it lies in the middle of the extended region from which collisions expel the particles populating it \(^{67, 68, 87}\). The mesosphere, or middle part of the atmosphere, extends from the top of the ionospheric layers to the critical level. In the ionosphere and mesosphere of the Earth the content of ionized particles is different, and they exist as a permanent constituent of the atmosphere.
Auroras arise in this region.
The critical level is located at an altitude of \(\sim 1000\) km above the Earth, so that the exosphere extends from this height upward; the mesosphere occupies the region approximately from 400 to 1000 km, and the ionosphere—from 70 to 400 km.
This formula shows that the spread of energies within an ensemble with ordinary collisions is insignificant; thus, under conditions of normal pressure and temperature \(Q \sim 10^{-10}\). At an altitude of 300 km, at a temperature \(T = 3000^\circ\), \(Q \sim 10^{-5}\) (see Table III). Although this value of \(Q\) is still very small, nevertheless the probability of deviations from the Maxwellian distribution is considerably greater. It should be recalled that with a Maxwellian velocity distribution it is impossible to observe emission spectra of atoms and molecules. Therefore, the detection of these spectra in the light of the night sky means (assuming that these lines are not caused by incident particles) a deviation from thermal equilibrium. At the same time, while calculation methods give the same temperature values for an equilibrium atmosphere, the temperature values change in a nonequilibrium atmosphere. On the basis of observational data on the distribution of the intensity of the rotational bands of nitrogen in auroras, temperature values of the order of \(225 \div 240^\circ\) were obtained\({}^{127,129}\). Recently these results\({}^{93}\) have been corrected—values of \(250 \div 265^\circ\) were obtained. Values\({}^{57a}\) of \(225^\circ\) for dark zones and \(260 \div 300^\circ\) for illuminated zones were also obtained. According to some data\({}^{129}\), the temperature of auroras is \(220^\circ\) in the dark zone, while in the zone illuminated by the Sun it is \(240^\circ\).
The distribution of the vibrational energy of some emission bands gives anomalously high temperature values. For excited helium and hydrogen, temperature values of 4600 and \(7600^\circ\) were obtained\({}^{101}\). From studies of the intensity of the spectral lines OII it was concluded\({}^{101}\) that the velocity distribution of excited particles corresponds to \(T = 3000 \div 6000^\circ\). For OI, \(T = 7000^\circ\) was obtained. These results are also applicable to auroras, which apparently arise under the action of bombarding particles.
Studies of thermal equilibrium in interstellar space can undoubtedly help clarify the question under consideration, since the conditions existing in these regions are limiting conditions into which planetary atmospheres degenerate. It was pointed out earlier\({}^{38}\) that elastic collisions between interstellar particles lead to a Maxwellian velocity distribution and an equal distribution of kinetic energy among different atoms.
In interstellar gas, free electrons may undergo elastic or inelastic collisions with other electrons, ions, or neutral particles; the latter of these reactions leads to excitation of heavy particles. Collisions of the second kind may also occur with ions or with atoms and molecules in an excited metastable state, free-free transitions in the vicinity of the nucleus, or capture of electrons by heavy particles. In the study of planetary nebulae it was found\({}^{30}\) that
the velocity distribution is very close to the Maxwellian one and that deviations from equilibrium are quite insignificant. Although these authors came to the conclusion that elastic collisions are most probable here, it should be recalled that the mean lifetime of an electron in an ensemble of particles is of the order of 10 years and, consequently, many orders of magnitude greater than in the Earth’s mesosphere. A numerical treatment of these results \(^{118}\) showed that in interstellar space and in denser nebulae the deviation from the Maxwellian distribution is insignificant.
Analogous conclusions, apparently, are applicable to rarefied regions of the Earth’s atmosphere; the number of radiating particles is only a small fraction of all particles. If a similar analysis is carried out not for the exosphere, but for the ionosphere and mesosphere of the Earth, then in them the possibility of the existence of thermal equilibrium is still more probable. Therefore one may conclude, with small error, that the ionosphere, and also some part of the mesosphere, are essentially in thermal equilibrium.
§ 11. ELECTRON TEMPERATURE
For completeness it is necessary to compare the electron temperature with the gas temperature. Usually, in ionization more energy is absorbed than is necessary for ionization. Because of the difference in the masses of the ion and the electron, this excess energy appears in the form of kinetic energy of the liberated electron. At the moment of ionization the electron temperature may be defined as
\[ T_e = 2B(\bar{h f} - W)/3k, \tag{11,1} \]
where \(\bar{f}\) is the mean value of the frequency, \(B = 7730^\circ \mathrm{K}\) \(^{36}\), and \(W\) is the ionization potential. However, in collisions with heavier particles this excess energy is lost, since such small amounts of energy are lost in each collision. It is easy to make quantitative calculations of this fact. The expression calculated by Drukarev \(^{33}\) is used, establishing the relation between the electron temperature and the gas temperature in the \(F_2\) layer:
\[ T_e = T + 3.88 \cdot 10^5(\bar{h f} - W), \tag{11,2} \]
and the relation for the mean fraction of energy lost in each collision of an electron and expressed as a fraction of the electron’s mean energy:
\[ \frac{\Delta E}{E} = \frac{8m \cdot m_e \left(1 - \dfrac{T}{T_e}\right)} {3(m_e - m)^2}. \tag{11,3} \]
Since \(m \gg m_e\), then
\[ \frac{\Delta E}{E}=\frac{8m_e A}{3m(T+A)}, \tag{11,4} \]
where \(A=3.88\cdot 10^5\cdot(hf-W)\).
In order to visualize what the time is equal to during which the temperature \(T_e\) acquired by an electron becomes equal to \(T\), one may choose the following example: if \(hf=25\ \text{eV}\), \(W=15\ \text{eV}\), \(M=20\), \(T=3000^\circ\), then \(\Delta E/E=7.31\cdot 10^{-5}\) per collision. If one uses the experimental value \(\nu'=2\cdot 10^3\ \text{s}^{-1}\), then \(T_e\) assumes the value \(T\) after 25 seconds in the \(F_2\) layer and in a shorter time in the \(E\) layer.
During a period of high solar activity, the value \(T_e \simeq 40\,000^\circ\) was obtained\(^{44}\). A similarly large value \(T_e=30\,000^\circ\) was obtained during auroras. However, from the example considered it is clear that the excess energy of the electrons is rapidly distributed among the heavy particles. It may be concluded that in the lower part of the mesosphere and in the ionosphere the electron temperature is in fact equal to the gas temperature.
§ 12. TEMPERATURE OF THE IONOSPHERE
Many different values have been proposed for the temperature of the ionosphere. It was assumed\(^{88}\) that from 100 to 300 km the temperature varies linearly from 300 to \(1100^\circ\). In another work\(^{79}\), a temperature value \(T=300^\circ\) is recommended for the \(E\) layer and \(1000^\circ\) for the \(F\) layer. It was indicated\(^{91}\) that in the \(E\) layer the temperature varies from 218 to \(323^\circ\) and that it cannot exceed here the value \(100^\circ\text{C}\). It was also assumed that from 82 to 300 km the temperature varies linearly within the limits \(160\div 1200^\circ\)\(^{9}\). It was indicated that at 120 km the temperature is \(400^\circ\), while between 200 and 300 km the temperature is \(1000\div 2000^\circ\)\(^{15a}\). Values \(T=600^\circ\) above 160 km and \(300^\circ\) between 100 and 160 km were indicated\(^{18}\). In considering the question of diffusion in the ionosphere, the values \(T=1000^\circ\) in the \(F_2\) layer and \(350^\circ\) in the \(F_1\) layer were postulated\(^{41}\).
From the present work it is seen that, in order of magnitude, the temperature values recommended in the literature for the \(E\) layer are correct; however, the values of \(T\) for the \(F\) layer are somewhat underestimated.
In polar regions it was assumed that between 100 and 200 km at night \(T=300^\circ\), and in the daytime \(T\simeq 1000^\circ\), and that above 700 km \(T=300^\circ\) around the clock\(^{31}\). These results do not agree with the values of \(T\) obtained in the present work.
Some remarks should be made regarding the character of the increase in temperature of the upper atmosphere. It was proposed\(^{62a}\) that, owing to the absorption of ultraviolet radiation by molecular oxygen, the temperature increases by \(50^\circ\) per hour. It is easy
to show that if in the layer \(F\) 100 particles are ionized per second and each of them has an excess of energy, on the average equal to 10 ev, then the total excess of energy, distributed among \(10^{10}\) particles, will cause a rise in temperature of only \(2.8^\circ\) per hour. Such a rate of temperature rise is very small, so that it may be assumed that the upper layers are heated at the expense of other processes of insolation or, possibly, by agents of non-solar origin. An attempt was made to explain qualitatively the high temperature of the ionospheric layers by postulating large values of the coefficients of absorption of ultraviolet by the principal constituents of the atmosphere\(^{107}\). When data on the values of the absorption coefficients were used, however, it was found that in the daytime the ionosphere is considerably heated and that the temperature has a diurnal variation.
An attempt was made to determine the temperature of the ionosphere by treating it as an absolutely black body\(^{136}\). On this basis, using the Stefan–Boltzmann law
\[ a=\eta T^4 \tag{12,1} \]
and the value of the solar constant \(\dot a=1.35\cdot 10^6\) erg/sec, it was found that \(T=390^\circ\) is the maximum possible value of the temperature of the atmosphere. In order that the ionosphere, being an absolutely black body, could transmit a considerable part of the solar radiation, it would have to have lower values of the maximum temperature. It was shown, however, that the observed values of the ionization density are satisfied at higher values of the temperature. One may therefore conclude that the upper atmosphere is not an absolutely black body and does not obey the Stefan–Boltzmann equation.
It is more probable that the atmosphere obeys some type of modified Kirchhoff radiation law. According to the usual Kirchhoff law for the selective absorption and emission of a gas, each gas absorbs and emits its own frequency. In a mixture of gases, however, selective heating of some component rapidly leads, owing to heat transfer, to heating of the entire mixture of gases as a whole. Thus, if the energy of radiation is absorbed by only one gas, all the gases are heated. Selective emission, on the other hand, arises earlier in those gases that emit lower frequencies, since their equilibrium state is disturbed at lower temperatures.
§ 13. HEATING BY ELECTRIC CURRENTS
Besides the absorption of solar radiation, another mechanism of heating the atmosphere was also considered. The strength of the electric currents required in the theory of variations of the Earth’s magnetic field\(^{115}\) is sufficient to heat the atmosphere appreciably. It was shown\(^{35}\) that the currents postulated by the dynamo theory can
heat the atmosphere. A reconsideration of this question^256 on the basis of improved data showed that, under certain assumptions, the rate of heating of the atmosphere caused by the currents is less than \(0.05^\circ\) per day. Under more favorable assumptions the maximum rate of heating in the auroral zone is presumably \(18^\circ\) per hour.
It was calculated^100 that, for specified electron currents penetrating the earth’s atmosphere, the temperature of the atmosphere increases during a magnetic storm. Under the assumptions chosen, a large increase in temperature proved possible. The suggestion was made^72, ^73 that magnetic storms may be accompanied by anomalous heating of the atmosphere, leading under certain conditions to an expansion of the atmosphere. Recently it has also been suggested that considerable heating of the atmosphere is connected with electric currents^70. It was assumed^5 that during magnetic storms the \(F_2\) layer expands because of heating, and that the temperature of the atmosphere increases during magnetic storms when it is bombarded by slow charged particles^86. In the latter work it is indicated that during aurorae accompanied by magnetic storms the temperature rises sharply, leading to a more uniform distribution of ionization with height. It is also indicated that the uniform illumination of auroral columns is a consequence of a uniform density distribution caused by heating of the atmosphere under the action of invading particles.
A careful examination of the works cited above makes it possible to draw a definite conclusion that this question is still far from being resolved. Further quantitative investigations in this field are necessary.
§ 14. DIRECTION OF FURTHER INVESTIGATIONS
As a result of the detailed analysis carried out above, we shall try to list the most suitable methods of obtaining temperature from ionospheric parameters.
It is quite clear that, in order to obtain definite data, further experimental and theoretical investigations are necessary. The following unresolved questions should be noted, which are directly connected with the temperature of the ionosphere.
a) Comparison of the rotational temperature with the gas temperature
This question arises because the temperature calculated from auroral data is lower than the temperature obtained from other sources. It is known^31 that the relative intensity of the rotational OH bands does not correspond to the large—
to the Boltzmann distribution, if they are excited by discharges occurring in water vapor.
This phenomenon was interpreted as the result of the simultaneous interaction of the dissociation of H$_2$O and the excitation of OH$^{94}$. Quantitative calculations$^{61,131}$ showed that the observed distribution of intensity can be attributed to two groups of molecules having different values of the equilibrium temperature. Moreover, it was established$^{61}$ that, whereas the rotational energies of the CO$_2$ and N$_2$ molecules correspond to the Boltzmann distribution of intensity (and consequently to the gas temperature), the rotational bands of the N$_2^+$ ions give higher temperature values than the gas temperature. It is undoubtedly necessary to continue the investigations indicated above and to redetermine the structure of the auroral bands. There are many well-known experimental results from the study of band spectra that give higher values of the rotational temperature than of the gas temperature. Rotational temperatures lower than the gas temperature are observed much more rarely in the laboratory, since special excitation conditions are required$^{114}$. The results of these investigations will provide valuable information about the excitation conditions of the upper atmosphere.
b) Determination of the cross section of collisions of electrons with oxygen atoms
Additional theoretical calculations and laboratory investigations are needed in order to remove the uncertainty in the question of the collision cross section of oxygen atoms with electrons having energy in the range $0 \div 0.5$ eV. These investigations are especially important, since this type of collision plays a major role in the upper atmosphere.
It is also necessary to determine the collision cross section for different metastable states.
c) Determination of the collision cross section of electrons with various ions
Both theoretical and experimental investigations of the collision cross sections of an electron with the following ions remain to be carried out:
O$^+$, O$^{++}$, O$_2^+$, O$_2^{++}$, N$^+$, N$^{++}$, N$_2^+$, N$_2^{++}$, NO$^+$, NO$_2^+$, Na$^+$.
d) Determination of the recombination coefficient at low pressures
Theoretical and experimental studies of the dependence of the recombination coefficient on temperature at low pressures lag substantially behind. These investigations are necessary
for considering various probable reactions at relatively low energies of the order of 0–0.5 ev.
d) Investigation of flat currents of the ionosphere
Further investigations are needed of the properties of horizontal motions of electrons in all layers of the ionosphere.
e) Diurnal variation of the number of collisions
Careful studies are important of the diurnal variation of the number of collisions in different layers of the ionosphere and at different latitudes.
The above list of studies that it is important to carry out is not exhaustive; however, the performance of these studies will make it possible to advance substantially the question of temperature in the altitude region 100–400 km.
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It should be noted that the author erroneously substitutes into Ginzburg’s formula the value of the root-mean-square velocity, and not of the mean velocity. (Translator’s note.) ↩