Current State of Research on the Ionosphere

II. Ionospheric Microphysics

Ya. L. Alpert

Contents

No.	Section	Page
1	Introduction	1
2	Formation of ionized layers	2
3	Basic elementary processes in the ionosphere	6
4	Theory of recombination and difficulties in explaining experimental data	10
5	The problem of the $F_2$ layer and the difficulties with the theory of absorption of ultraviolet in it	18
6	Effective number of collisions $\nu_{\mathrm{eff}}$	22
7	Temperature, density, and composition of the ionosphere	25

1. Introduction

In the preceding article^1 the principal experimental results of radio investigations of the ionosphere were summarized, and certain aspects of the theory of the propagation of radio waves in it were also considered. Taken as a whole, all the material presented was descriptive in character and was based on a macrophysical picture of phenomena in the ionosphere.

In the work mentioned, however, the basic microprocesses in the ionized plasma were not discussed at all; the causes of the formation of ionized layers were not examined; no data were given on the temperature, composition, and density of the ionosphere, etc.

Here we shall consider the state of this range of questions, which as a whole constitutes, one may say, the microphysics of the ionosphere.

It should be pointed out that at the present time the state of affairs in this field is, in many respects, rather uncertain. We encounter, on the one hand, a number of difficulties in the theoretical interpretation of repeatedly verified and sufficiently reliable experimental data. On the other hand, there is still a small amount of experimental data of such a type on the basis of which

one could attempt to construct a more or less complete theory of the microprocesses of the ionosphere. Thus, we are dealing with a developing field in which further substantial changes may still be expected.

2. FORMATION OF IONIZED LAYERS

Let us consider the absorption of ultraviolet radiation from the Sun in a homogeneous atmosphere whose density decreases as the height \(z\) increases. Having initially in mind only a single gas, we may assume that the radiation is monochromatic and that, in the frequency region of interest to us, determined by the ionization potential of the given gas, the solar constant is equal to

\[ S_\infty\ \frac{\mathrm{erg}}{\mathrm{cm}^2\,\mathrm{sec}} \tag{2.1} \]

(at \(z=\infty\), i.e., beyond the limits of the atmosphere).

Considering a plane layer of the atmosphere, one may write that, when radiation passes at an angle \(\chi\) to the vertical \(z\) through the layer \((z,z-dz)\), the loss of energy of the given frequency is equal to

\[ dS=-\sigma S\,(n-n_e)\,dz\,\sec\chi, \tag{2.2} \]

and the number of newly formed electrons in one cubic centimeter of gas at the given height is equal to

\[ J=\frac{1}{\varepsilon_i}\frac{dS}{dz}\cos\chi\, \frac{\text{electrons}}{\mathrm{cm}^3\,\mathrm{sec}}. \tag{2.3} \]

In expressions (2.2) and (2.3), \(\varepsilon_i\) is the amount of energy required for a single act of ionization, and \(\sigma\) is the coefficient of photoabsorption in the frequency region of interest to us, i.e., the photoabsorption cross section or the photoionization cross section, if the greater part of the energy in this frequency region goes into ionization. \((n-n_e)\) is the number of neutral particles in \(1\ \mathrm{cm}^3\) of gas, equal to the difference between the density of neutral particles \(n\) before ionization and the density of electrons \(n_e\), or of pairs of ionized gas particles. Strictly speaking, instead of \((n-n_e)\) one must write, for an electrically neutral atmosphere \((n_e+n^- = n^+)\), the expression

\[ n-n_e-n^- = n-n^+, \]

where \(n^-\) and \(n^+\) denote, respectively, the densities of negative and positive ions in the medium. Taking into account, however, that in most cases of interest, apparently, \(n^- \ll n_e\), one may neglect the number of negative ions. In a number of cases one may also neglect in (2.2) the quantity \(n_e\), as is the case, for example, in the region of the layer \(E\), where undoubtedly \(n_e \ll n\). However, for regions of low density \(n\), for example at sufficiently great heights, where the energy is large and the gas must be almost completely ionized, such an assumption is invalid.

From (2.2) and (2.3) it follows directly:

\[ S=S_{\infty}e^{-\sigma \sec\chi\,(N-N_e)}, \tag{2.4} \]

\[ J=\sigma (n-n_e)\frac{S_{\infty}}{\varepsilon_i}e^{-\sigma \sec\chi\,(N-N_e)}, \tag{2.5} \]

where

\[ N=\int_z^{\infty} n\,dz \quad\text{and}\quad N_e=\int_z^{\infty} n_e\,dz \]

are, respectively, the total numbers of neutral particles and electrons in a vertical column of cross section \(1\ \mathrm{cm}^2\) above the level \(z\).

It is physically quite clear that the quantity \(J\) must have one maximum, determined from the condition

\[ \frac{\partial J}{\partial z}=0, \tag{2.6} \]

since the energy decreases with decreasing height \(z\), while the particle density, on the contrary, decreases with increasing \(z\).

If, for example, we assume that the density decreases with height according to an exponential law (Boltzmann distribution in a force field),

\[ n=n_0 e^{-\frac{z}{H}}, \tag{2.7} \]

where \(H=\dfrac{RT}{mg}\) (\(R\) is the gas constant, \(T\) the absolute temperature, \(m\) the molecular weight of the gas, \(g\) the acceleration of gravity), and assume that \(n_e \ll n\), then from (2.4) and (2.6) follow the well-known relations connecting the values of the various parameters for the maximum (index \(m'\)) of \(J\) of the so-called simple layer\(^{2,3}\):

\[ \left. \begin{aligned} z_{m'}&=H\ln(H\sigma n_0\sec\chi),\\ n_{m'}&=\frac{1}{H\sigma\sec\chi},\\ J_{m'}&=\frac{S_{\infty}}{\varepsilon_i eH}\cos\chi =\frac{S_m}{\varepsilon_i H}\cos\chi. \end{aligned} \right\} \tag{2.8} \]

Of interest also is the total number of newly formed electrons in a column of cross section \(1\ \mathrm{cm}^2\), throughout the entire thickness of the atmosphere from \(z_0=0\) to \(z=\infty\), and for the region of the atmosphere from \(z_0=0\) to the height \(z=z_m\) corresponding to the maximum of \(n_e\) (index \(m\)). Integrating (2.3), we obtain:

\[ \left. \begin{aligned} Q_{\infty} &=\int_0^{\infty} J\,dz =\frac{S_{\infty}}{\varepsilon_i}\cos\chi \left(1-\frac{S_0}{S_{\infty}}\right),\\ Q_H &=\int_{z_0}^{z_m} J\,dz =\frac{S_m}{\varepsilon_i}\cos\chi \left(1-\frac{S_0}{S_m}\right)\\ &=\frac{S_m}{\varepsilon_i}\cos\chi \left(1-e^{-\sigma\sec\chi\,(N^{*}-N_{eH}^{*})}\right). \end{aligned} \right\} \tag{2.9} \]

where \(S_0\) and \(S_m\) are the values of the energy at the base and at the maximum of the layer, and

\[ (N^* - N_e^*)_{\mathrm{н}} = \int_{z_0}^{z_m} n\,dz - \int_{z_0}^{z_m} n_e\,dz . \]

Here \(N^*\) and \(N_e^*\) are, respectively, the total numbers of neutral particles and electrons in a column of cross-section \(1\ \mathrm{cm}^2\). The index “н” means that the quantity refers to the entire region lying below the level \(z_m\).

From the preceding consideration, as we have seen, it follows quite clearly that the number of newly formed electrons in \(1\ \mathrm{cm}^3\) of the atmosphere varies with height and, at a certain height determined from (2.6) or (2.8), reaches its maximum value. This, however, is not yet a sufficient condition for the appearance of an ionized layer with a maximum degree of ionization \(n_e\) located at some height \(z_m\). For this it is necessary that the transfer and disappearance of charges in the layer (recombination, attachment, etc.—see in more detail below) should not compensate the maximum of \(n_e\) owing, say, to an increase in the rate of these processes in the region of this maximum. The actual existence of ionized layers indicates that such compensation in fact does not occur. It should be borne in mind, however, that the maximum of \(n_e\) need not coincide at all with the maximum of \(J\).

From (2.6), (2.4), and (2.5) it follows that for the maximum of \(J\) the relation

\[ \frac{\partial n}{\partial z} - \frac{\partial n_e}{\partial z} = -\sigma \sec \chi\, (n - n_e)^2, \tag{2.10} \]

holds, and this determines the value \(z = z_{m'}\), at which \(J = J_{m'}\).

To determine the value \(z = z_m\), at which \(\dfrac{\partial n_e}{\partial z}=0\), it is necessary to know \(n_e(z)\), for which a solution of the equation is required that takes into account both the production and the disappearance of charges in the atmosphere (see below).

The consideration carried out above referred to a single gas, for which the value \(J\) was obtained. Taking into account that the atmosphere consists of a mixture of gases, it would be necessary to rewrite relations (2.2) and (2.3) in the form

\[ dS = \sum dS_i = -\sum \sigma_i S_i \bigl(n^{(i)} - n_e^{(i)}\bigr), \tag{2.2a} \]

\[ J = \sum dJ_i = -\sum \frac{1}{\varepsilon_i^{(i)}} \frac{dS_i}{dz}\cos\chi, \tag{2.3a} \]

in which, for each gas, the corresponding partial densities \(n_e^{(i)}\), \(n^{(i)}\), the energy values \(S_i\) of those frequency regions that ionize these gases, and the corresponding values \(\varepsilon_i^{(i)}\), \(\sigma_i\) are written out. It is quite clear that, by virtue of the additivity of all terms of the equation

(2.2a) a separate consideration for each of the gases and a subsequent summation of the electron production for each of them is quite legitimate. In equations (2.2) and (2.3), however, generally speaking, we use the frequency-averaged values \(\varepsilon_i\), \(\sigma\), and \(S\).

Thus, the electron density of the ionosphere observed experimentally may be of a mixed nature, i.e., arise as a result of the ionization of different gases. However, the fact that several ionization maxima exist indicates that in each layer the ionization of a quite definite principal gas is more pronounced; and, for known conditions and a mixture of gases, selective absorption of the corresponding portion of the frequency range of the Sun’s radiation by this gas gives a maximum of \(n_e\), the position of which depends both on \(n(z)\) and on the value of \(S\) of the part of the spectrum ionizing this gas.

Fig. 1.

Above, the derivation of \(J\) and \(Q\) was considered for a plane layer. This assumption is legitimate for large values of the Sun’s altitude and quite illegitimate for the period of sunrise of the Sun at the Earth’s surface \((\chi = -90^\circ)\) or in the ionosphere \((\chi > 90^\circ)\), which is immediately evident from the expressions written above. In a number of cases, however, it is of interest to compute \(J\) for \(\chi > 90^\circ\), for which it is necessary to carry out the corresponding calculations taking into account the sphericity of the Earth.

In this case the calculation reduces to computing

\[ J = \frac{1}{\varepsilon_i}\frac{\partial S}{dl}, \tag{2.11} \]

where the total differential of the energy is determined by the relation

\[ dS = -\sigma S\,(n-n_e)\,dl \tag{2.12} \]

(see Fig. 1).

To determine from (2.11) and (2.12) the dependence of interest to us of the corresponding expressions on \(\chi\), it is necessary to specify the functions \(n(z)\) and \(n_e(z)\). For an exponential distribution of \(n\) with height (2.7) and \(n_e \ll n\), it is obtained\(^4\) that in the expressions for \(S\) and \(J\), \(\sec \chi\) is replaced by the function \(f\!\left(\dfrac{R+z}{H}, \chi\right)\), where \(R\) is the radius of the Earth. From the analysis of

of the function \(f\left(\dfrac{R+z}{H}, \chi\right)\), it follows, however, that for \(\dfrac{R+z}{H} > 200\) (this corresponds to the majority of practically interesting cases) the function

\[ f\left(\frac{R+z}{H}, \chi\right) \sim f(\chi) \sim \sec \chi \]

for \(\chi \leq (75 \div 80)^\circ\).

Table I gives the values of this function for various values of \(\dfrac{R+z}{H}\).

Table I

Numerical values of the function \(f\left(\dfrac{R+z}{H}, \chi\right)\), characterizing the diurnal variation of the intensity of solar radiation\(^4\)

\(\chi\)	\(30^\circ\)	\(45^\circ\)	\(60^\circ\)	\(75^\circ\)	\(80^\circ\)	\(83^\circ\)	\(85^\circ\)	\(87^\circ\)	\(90^\circ\)	\(93^\circ\)	\(95^\circ\)	\(97^\circ\)	\(100^\circ\)
\(\sec \chi\)	1,15	1,41	2,00	3,86	5,76	8,21	11,48	19,11	\(\infty\)
\(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)	\(f\left(\dfrac{R+z}{H}, \chi\right)\) for different \(\dfrac{R+z}{H}\)
50	1,15	1,39	1,91	3,23	4,10	5,05	5,82	6,81	8,93	12,3	15,7	20,8	32,9
100	1,15	1,40	1,95	3,51	4,61	5,91	7,07	8,68	12,6	20,2	29,7	49,9	107
200	1,15	1,41	1,98	3,64	5,01	6,66	8,28	10,7	17,7	35,9	67,6	150	714
300	1,15	1,41	1,98	3,71	5,28	7,02	8,92	11,9	21,7	53,6	126	398	4108
400	1,15	1,41	1,98	3,74	5,38	—	9,33	12,8	25,1	73,9	220	978	21690
500	1,15	1,41	1,99	3,76	5,44	—	9,62	13,5	28,0	97,9	366	2315	\(1,1 \cdot 10^5\)
600	1,15	1,41	1,99	3,78	5,49	—	9,84	14,0	30,7	126	591	—	\(5,5 \cdot 10^5\)
650	1,15	1,41	1,99	3,79	5,51	—	9,93	14,2	32,0	141	747	—	\(1,2 \cdot 10^6\)
700	1,15	1,41	1,99	3,79	5,53	—	10,0	14,4	33,2	159	940	—	\(2,7 \cdot 10^6\)
800	1,15	1,41	1,99	3,80	5,55	—	10,1	14,7	35,5	197	1476	—	\(1,3 \cdot 10^7\)

3. BASIC ELEMENTARY PROCESSES IN THE IONOSPHERE

The height distribution and the magnitude of the degree of ionization \(n_e\) of each layer, it goes without saying, depend on all the processes occurring in it and leading both to the appearance (production) of free charges (electrons) and to their disappearance. Therefore, before formulating the equation for the equilibrium state of ionization of a layer, we shall analyze the principal ones of these processes and give, for each of them, the known values of the corresponding coefficients.

a) Photoionization

Apparently, the principal ionizing agent of the atmosphere is the ultraviolet radiation of the Sun. A process of the type occurs

\[ \Gamma + h\nu \to \Gamma^{+} + e, \tag{3.1} \]

where \(\Gamma\) is a particle (molecule, atom, ion) of any gas entering into the composition of the atmosphere, i.e. \(\mathrm{O}_2,\ \mathrm{O}_1,\ \mathrm{N}_2,\ \mathrm{N}_1\), etc. (\(+\) denotes the positive charge of the ion). The production of electrons in photoionization is described by the term (2.5), equal to

\[ \sigma(n \to n_e)\frac{S}{\varepsilon_i}. \tag{3.2} \]

The quantity \(\varepsilon_i\) has been measured with sufficient accuracy for the majority of gases entering into the composition of the atmosphere, and is determined from the values of the ionization potentials of these gases.

Table II gives the corresponding values for the principal atmospheric gases. This table also gives the values of \(H\) (see 2.7), needed for calculating \(S\) in the case of an exponential distribution of the density \(n\).

Table II

Values of the ionization potentials of the principal atmospheric gases

(in electron-volts) and of the quantities \(H=\dfrac{RT}{mg}\) (in km) for \(T=273^\circ\) K

\[ (1\ \mathrm{eV}=1.59\cdot10^{-12}\ \mathrm{erg}) \]

Gas	\(\mathrm{O}_2\)	\(\mathrm{O}_1\)	\(\mathrm{N}_2\)	\(\mathrm{N}_1\)
First level . . .	12.5 (987 Å)	13.5 (914 Å)	15.8 (781 Å)	14.5 (851 Å)
Second level . . .	16.1	16.9	18.7	—
Third level . . .	16.9	18.5	—	—
\(H\) . . . . . . .	7.23	14.46	8.26	16.52

As for the quantity \(\sigma\), in the literature there are known only data from quantum calculations\(^{5,6}\) for atomic oxygen \(\mathrm{O}_1\), for which the following values of the photoionization cross section \(\sigma_e\) are given:

\[ \left. \begin{aligned} \sigma_e &= 0.45\cdot10^{-17}\ \mathrm{cm}^2, && \text{for } 13.55\,\mathrm{eV} \leq h\nu \leq 16.86\,\mathrm{eV},\\ \sigma_e &= 1.1\cdot10^{-17}\ \mathrm{cm}^2, && \text{for } 16.86\,\mathrm{eV} \leq h\nu \leq 18.54\,\mathrm{eV},\\ \sigma_e &= 1.6\cdot10^{-17}\ \mathrm{cm}^2, && \text{for } 18.54\,\mathrm{eV} \leq h\nu \leq 25\,\mathrm{eV}. \end{aligned} \right\} \tag{3.3} \]

The authors assume\(^{8}\), moreover, that for \(\mathrm{N}_2\) one may take \(\sigma_e \sim 10^{-17}\ \mathrm{cm}^2\) for the first level and \(\sigma_e \sim 10^{-16}\ \mathrm{cm}^2\) for the second level. For \(\mathrm{O}_2\) they give the value \(\sigma_e \sim 10^{-20}\ \mathrm{cm}^2\).

b) Recombination of electrons

Various types of electron recombination are possible. The process inverse to photoionization (3.1) is photorecombination

\[ \Gamma^{+}+e\to \Gamma+h\nu . \tag{3.4} \]

It is described by a term of the form

\[ -\alpha_e n_e n^{+}=-\alpha_e n_e^{2}, \tag{3.5} \]

where \(\alpha_e\) is the photorecombination coefficient, and \(n_e\sim n^{+}\).

The calculation of \(\alpha_e\) has also been carried out for O\(_1\). In the work cited above the following values of the total photorecombination coefficient are given (see Table III):

Table III

Values of the photorecombination coefficient for atomic oxygen O\(_1\)

\(T^\circ\)	250	500	1000	2000	4000	8000
\(\alpha_e\) in \(10^{-12}\ \mathrm{cm^3/sec}\)	3.7	2.4	1.5	0.9	0.53	0.33

The authors indicate that O\(_2\) and N\(_2\) should have the same values of \(\alpha_e\).

For the process

\[ \begin{aligned} \Gamma^{+}+e&\to \Gamma',\\ \Gamma'&\to \Gamma+h\nu, \end{aligned} \tag{3.6} \]

i.e., for recombination accompanied by an intermediate excited state of the atom or molecule, the value \(\alpha_e\sim 10^{-12}\) is given. (The prime denotes the excited state.)

Recombination accompanied by dissociation has not been calculated at all, namely a process of the form

\[ \Gamma_2^{+}+e\to \Gamma'+\Gamma'', \tag{3.7} \]

leading to excited states of two atoms.

c) Attachment of electrons

The photoattachment reaction

\[ \Gamma+e\to \Gamma^{-}+h\nu \tag{3.8} \]

is described by a term of the form

\[ \beta_n e n, \tag{3.9} \]

for which the following values of \(\beta\) are given\(^7\) for \(\mathrm{O}_1\), at various values of the binding energy (Table IV):

Table IV

Values of the photodetachment coefficients for \(\mathrm{O}_1\)

\(T^\circ\)	250	500	1000	2000
\(\beta\)	\(7.5\cdot 10^{-16}\)—\(6\cdot 10^{-14}\)	\(7.7\cdot 10^{-16}\)—\(3\cdot 10^{-14}\)	\(8.0\cdot 10^{-16}\)—\(1.5\cdot 10^{-14}\)	\(8.3\cdot 10^{-16}\)—\(8\cdot 10^{-15}\)

The authors take as the most probable value of the photodetachment coefficient \(\beta \simeq 1.1\cdot 10^{-15}\), and also suggest that \(\beta\) does not depend on temperature\(^8\).

The values of \(\beta\) for process (3.8) for other gases are completely unknown (\(\mathrm{O}_2\) and \(\mathrm{N}_1\); investigations show\(^9\) that the ion \(\mathrm{N}_2^-\) is unstable).

An electron-attachment reaction of the type is possible

\[ \Gamma + e \to \Gamma', \qquad \Gamma' \to \Gamma^- + h\nu . \tag{3.9a} \]

However, it follows from the literature\(^ {10}\) that this reaction should not play a large role in the ionosphere.

Likewise, in work\(^8\) it is stated that in the ionosphere one may disregard the attachment reaction with subsequent dissociation

\[ \Gamma_2 + e \to \Gamma + \Gamma^- . \tag{3.10} \]

d) Recombination of ions

The situation is most difficult with the data for the ion-recombination coefficient. There are essentially no theoretical data, owing to the great complexity of the calculations. Here one may consider reactions with the participation of three bodies of the following kinds:

\[ \Gamma_1^- + \Gamma^+ + M \to \Gamma_1 + \Gamma + M, \tag{3.11} \]

\[ \Gamma_2^- + \Gamma^+ + M \to \Gamma_2 + \Gamma + M \tag{3.12} \]

(where \(M\) is a molecule or atom of another kind), or reactions with the participation of two bodies:

\[ \Gamma_1^- + \Gamma^+ \to \Gamma_1' + \Gamma'', \tag{3.13} \]

\[ \Gamma_2^- + \Gamma^+ \to \Gamma_2' + \Gamma'' . \tag{3.14} \]

This process is described by the term:

\[ -\alpha_i n^- n^+ . \tag{3.15} \]

For reactions (3.11), (3.12) some indications can be found in works\(^{11,12,13}\). It is noted, for example\(^{9}\), that \(\alpha_i\), apparently, should not be greater than \(10^{-7} \div 10^{-8}\ \text{cm}^3/\text{sec}\).

d) Detachment of electrons from ions

The reverse detachment of an electron from an ion is possible, i.e. the process inverse to attachment:

\[ \Gamma^- + h\nu \to \Gamma + e, \tag{3.16} \]

or

\[ \Gamma^- + h\nu \to \Gamma', \qquad \Gamma' \to \Gamma + e. \tag{3.17} \]

These reactions are described by the term

\[ I n^-, \tag{3.18} \]

where \(I\) is the coefficient of photodetachment of the electron. The quantity \(I\), according to calculations\(^{7}\) for \(O_1\), varies within the limits

\[ I \simeq 0.27 \div 0.85\ \text{sec}^{-1}. \tag{3.19} \]

However, in the literature\(^{8}\) the most probable value recommended is

\[ I \simeq 0.35\ \text{sec}^{-1}. \tag{3.20} \]

Reactions of electron detachment of the form are also possible

\[ \Gamma_1^- + \Gamma_1 \to \Gamma_2 + e, \qquad \Gamma_2^- + \Gamma_1 \to \Gamma_3 + e, \tag{3.21} \]

or, for example,

\[ \Gamma^- + M' \to \Gamma + M + e, \tag{3.22} \]

which are described by the term

\[ \gamma n^- n. \tag{3.23} \]

From experimental works\(^{14,15}\), and also from\(^{8}\), it follows that one may assume

\[ \gamma \sim 10^{-16} \div 10^{-17}\ \text{cm}^3/\text{sec}. \tag{3.24} \]

4. THEORY OF RECOMBINATION

AND THE DIFFICULTY OF EXPLAINING THE EXPERIMENTAL DATA

In the preceding section the principal elementary reactions of production and disappearance of electrons and ions in an ionized layer were considered. Collecting the expressions written above (3.2), (3.5), (3.9), (3.15), (3.18), and (3.23), one can now write the equations for the equilibrium state of ionization \(n_e\) of the layer. However, the change in \(n_e\) is not limited to the processes indicated above. Owing to temperature changes in the layer (expansion and compression), the change of \(J(z)\) with the diurnal course of \(f(\chi)\) (displacement of the maximum of the layer), diffusion\(^{16,17}\) and oth—

these possible motions in the layer, there occurs in it a transport of charges, affecting the overall balance of its ionization.

Temperature changes make it necessary to include in the equation terms of the form

\[ -\frac{n_e}{T}\frac{dT}{dt}, \qquad -\frac{n_e}{V}\frac{dV}{dt}, \tag{4.1} \]

where \(V\) is a [[unclear: adjective before “volume”]] volume; there are no definite data concerning these terms, and it can only be shown that in many cases they are small in comparison with the other terms. If we assume, for example, that in the layer over 10 hours \(T\) changes from \(T=1000^\circ\) to \(T=2000^\circ\), and \(V\) changes by a factor of 4–5 (larger changes of \(T\) and \(V\) are difficult to assume on the basis of the available data for the \(F_2\) layer even in summer), then for \(n_e \sim 10^6\) we obtain:

\[ \frac{n_e}{T}\frac{dT}{dt} \sim 25\ \frac{\text{electrons}}{\text{cm}^3\,\text{sec}}, \qquad \frac{n_e}{V}\frac{dV}{dt} \sim 100\ \frac{\text{electrons}}{\text{cm}^3\,\text{sec}}. \]

The principal term of the ionization equation, i.e. the production of electrons \(J\), is in many cases equal to \(10^3\) electrons per \(\text{cm}^3\text{sec}\). However, from this same estimate it follows that in a number of cases these terms may have substantial significance (see, for example, in § 5 the data for summer), and neglecting them may lead to errors in the calculations. In constructing the general equation for \(n_e\), these terms are usually omitted.

It is also expedient to process the experimental data with the aid of the ionization equation integrated over the entire layer [see (4.10)], i.e. to take into account the total ionization in the layer, which in general should substantially reduce the influence of electron transport on the overall ionization balance.

Indeed, allowance for transport leads to the necessity of including in the ionization equation (see below 4.5), along with (4.1), also a term of the form

\[ \frac{\partial (n_e v)}{\partial z}, \tag{4.1a} \]

where \(v\) is the velocity of motion of the electrons along the \(z\)-axis. However, it is not difficult to see that integrating (4.1a) from \(z_0=0\) to \(z=\infty\), for which \(n_e \sim 0\) and \(\dfrac{\partial n_e}{\partial z} \sim 0\), gives the value of this integral equal to zero. Since in reality we consider data only for the lower half of the layer, then, by excluding the transport terms in equation (4.10), strictly speaking, we do not take into account in it that quantity of charges which passes from its lower half into the upper half through the maximum of the layer, i.e. we discard the term

\[ v_m \cdot n_e, \tag{4.1b} \]

where \(v_m\) and \(n_e\) are, respectively, the values of the transport velocity and the electron density at the maximum of the layer. It is possible to estimate the magnitude

(4.16) on the basis of the available measurement data. Proceeding from the fact that, on the basis of experimental data (see below § 5), the principal term of equation (4.10) is in some cases equal to

\[ Q_{\mathrm n}\sim 5\cdot 10^9 \div 5\cdot 10^8\ \text{electrons}/\text{cm}^2\,\text{sec}, \]

and

\[ n_e \simeq 2\cdot 10^6\ \text{electrons}/\text{cm}^3\,\text{sec}, \]

we obtain from (4.16):

\[ v \ll (25 \div 5)\ \text{m/sec},\quad \text{i.e.}\quad v \ll (90 \div 18)\ \text{km/hour}. \tag{4.1в} \]

At the present time there are not yet sufficient grounds either to deny or to insist that in the layer \(F_2\) there occurs a regular displacement in one direction of the main mass of electrons through the maximum of the layer with the velocities indicated in (4.1в). At the same time, it is seen from (4.1в) that in a number of cases the rejection of (4.1а) may also prove to be unlawful.

Let us suppose that the layer is electrically neutral, i.e. that

\[ n_e + n^- = n^+. \tag{4.2} \]

The equations of ionization change (collecting all the terms of § 3) are written in the form:

\[ \frac{dn_e}{dt}=J-In^-+\gamma n^- n-\beta n_e n-\alpha_e n_e n^+, \tag{4.3} \]

\[ \frac{dn^-}{dt}=\beta n_e n-In^- - \gamma n^- n-\alpha_i n^- n^+. \tag{4.4} \]

The joint solution of (4.3) and (4.4) gives an equation with an effective recombination coefficient \(\alpha\), which is also used in processing experimental data for \(n_e\) (usually this is done for the region of the maximum of the layer), namely:

\[ \frac{dn_e}{dt}=\frac{J}{1+\lambda}-\alpha n_e^2, \tag{4.5} \]

where

\[ \lambda=\frac{n^-}{n_e}=\frac{\alpha_e n_e^2+\beta n_e n-J+\dfrac{dn_e}{dt}}{\gamma n_e n+In_e-\alpha_e n_e^2} \tag{4.6} \]

and

\[ \alpha=\alpha_e+\lambda \alpha_i+\frac{1}{n_e}\frac{d\ln(1+\lambda)}{dt}. \tag{4.7} \]

Formula (4.6) is more conveniently rewritten in the form

\[ \lambda=\frac{\alpha_e n_e^2+\beta n_e n-\alpha n_e^2}{\gamma n_e n+In_e-\alpha_e n_e^2+J_0}, \tag{4.8} \]

taking

\[ \frac{J}{1+\lambda}=J_0, \tag{4.9} \]

the value of which is also determined in the numerical treatment of (4.5) from the results of measurements.

If (4.5) is integrated over the lower half of the layer (from \(z_0=0\) to \(z=z_m\)), bearing in mind that only for this half do we have experimental results at our disposal, then from (4.5) it follows that

\[ \frac{dN_e^*}{dt}=\frac{Q_{\mathrm{H}}}{1+\lambda}-\bar{\alpha}\int_{z_0}^{z_m} n_e^2\,dz, \tag{4.10} \]

where \(Q_{\mathrm{H}}\) was calculated above [see (2.9)], and \(\bar{\alpha}\) is the mean value of \(\alpha\) with respect to \(z\); in integrating (4.5) it has been assumed that \(\lambda=\mathrm{const}\).

For the further consideration and analysis of formulas (4.8) and (4.5) we give in Table V some measurement data for \(\alpha\), \(J_0\), and \(n_e\) for various ionospheric layers, as well as the coefficients of various processes given above, obtained from quantum calculations, which may serve as the initial data for the corresponding estimates.

Table V

Layer	\(E\) in summer	\(F_1\) in summer	\(F_2\) in winter
\(n_e\) (electrons/\(\mathrm{cm}^3\))	\(1.5\cdot 10^5\)	\(3\cdot 10^5\)	\(2.5\cdot 10^6\)
\(\alpha\) (\(\mathrm{cm}^3/\mathrm{sec}\))	\(\sim 10^{-8}\)	\(\sim 3\cdot 10^{-9}\)	\(\sim 1.5\cdot 10^{-10}\)
\(J_0\) \(\left(\dfrac{\text{electrons}}{\mathrm{cm}^3\,\mathrm{sec}}\right)\)	200	300	800
\(\alpha_e\sim 10^{-12},\quad \beta\sim 10^{-15},\quad I\sim 0.35,\quad \gamma\sim 5\cdot 10^{-17}\)	\(\alpha_e\sim 10^{-12},\quad \beta\sim 10^{-15},\quad I\sim 0.35,\quad \gamma\sim 5\cdot 10^{-17}\)	\(\alpha_e\sim 10^{-12},\quad \beta\sim 10^{-15},\quad I\sim 0.35,\quad \gamma\sim 5\cdot 10^{-17}\)	\(\alpha_e\sim 10^{-12},\quad \beta\sim 10^{-15},\quad I\sim 0.35,\quad \gamma\sim 5\cdot 10^{-17}\)

The table gives average values referring to a time of day close to noon, for the period of maximum solar activity, according to data for middle latitudes of the Northern Hemisphere.

It should be pointed out that the values of \(J_0\) given for \(\chi=0\) attain, respectively for \(E\), \(F_1\), and \(F_2\), quantities of the order: \(300\)—\(400\), \(500\)—\(600\), and \(2000\)—\(3000\) electrons/\(\mathrm{cm}^3\,\mathrm{sec}\).

From the data of Table V it is seen that for all layers one may take

\[ \alpha_e n_e^2 \ll \alpha n_e^2,\qquad \alpha_e n_e^2 \ll I n_e,\qquad J_0 \ll I n_e . \tag{4.11} \]

Moreover, from these data it follows that even for the \(E\) layer, in which \(n\simeq(2\div 5)\,10^{12}\) (and certainly for the higher layers \(F_1\) and \(F_2\), in which \(n\) is smaller),

\[ \gamma n_e n \ll I n_e . \tag{4.12} \]

Taking (4.11) and (4.12) into account, instead of (4.8) one may write

\[ \lambda \simeq \frac{\beta n_e n-\alpha n_e^2}{I n_e} =\frac{\beta n-\alpha n_e}{I}. \tag{4.13} \]

It is already clear from the general form of formula (4.13) that it is hardly correct. Indeed, since \(\lambda\) is proportional to \(n\), \(\alpha\) [see (4.7)] is likewise proportional to the density. In view of the fact that \(J_0 \sim n\) [see (3.2)], for \(n_e \ll n\) we obtain that

\[ n_e \sim \sqrt{\frac{\overline{J}_0}{\alpha}} \tag{4.14} \]

has no maximum (strictly speaking, these arguments are valid only for the quasi-stationary state of the layer, when \(\dfrac{dn_e}{dt} \sim 0\)). Quite apart from the fact that in experiment a layer with a clearly expressed maximum is always observed, from various measurements (see also below) one may conclude that \(\alpha\) does not depend on the density of the atmosphere.

Substituting in (4.13) the data for the \(E\)-layer and taking \(n \sim 5 \cdot 10^{12}\), we obtain:

\[ \lambda \simeq 10^{-3}. \tag{4.15} \]

If, further, we assume that \(\lambda\) changes from 0 to \(\lambda = 10^{-3}\) in the course of even a minute, then

\[ \frac{1}{n_e}\frac{d\ln(1+\lambda)}{dt} \sim 10^{-10} \]

and from (4.7) it follows:

\[ \alpha_i \sim 10^{-5}, \tag{4.16} \]

which is clearly impossible, since even for air at atmospheric pressure

\[ \alpha_i \sim 2 \cdot 10^{-6}. \tag{4.17} \]

Still more contradictory data are obtained for the layers \(F_1\) and \(F_2\), since in order that the condition \(\lambda > 0\) be satisfied it is necessary that \(\beta n > \alpha n_e\), and from this it follows that at the maximum of \(F_1\) (\(z \sim 250\) km) the quantity \(n\) must be greater than \(9 \cdot 10^{11}\), and at the maximum of \(F_2\) (\(z \sim 300\) km) \(n > 4 \cdot 10^{11}\). This also is hardly possible, if one takes into account that for the layer \(E\) (\(z \simeq 120\) km) the value \(n \simeq (2 \div 5)\cdot 10^{12}\) is at present quite well justified. From formula (4.13) it is also clear that the value of \(\lambda\) for \(F_1\) and \(F_2\) is still smaller than for \(E\) [see (4.15)].

Thus there arises a need to revise the values of the coefficients given in Table V.

If we assume, as some authors do\(^{8}\), that \(\gamma\) is considerably greater than the value indicated in the table, namely that

\[ \gamma \sim 2 \cdot 10^{-13}, \tag{4.18} \]

then from (4.8), while retaining only condition (4.11), it follows that \(\alpha\) depends little on the density \(n\), and

\[ \lambda \sim 5 \cdot 10^{-3}, \qquad \alpha_i \sim 2 \cdot 10^{-6}. \tag{4.19} \]

This somewhat eases the difficulty for the layer \(E\), but does not at all remove it for \(F_1\) and \(F_2\).

From the preceding it is evident that, when comparing the results of experiments with calculations, we encounter substantial contradictions, indicating that the coefficients given above for the various processes possibly do not correspond to those which actually play a role in the different layers of the ionosphere. The main difficulty, as is clear from the calculations, consists in the fact that the effective recombination coefficient \(\alpha\) is considerably greater than the known values of the photorecombination coefficient \(\alpha_e\).

We saw above that the recombination of electrons occurs not only as a result of process (3.4), but also as a result, for example, of the processes (3.6) or (3.7) indicated in § 3, whose recombination coefficients are unknown. It might turn out that they have values equal to the experimentally measured values of \(\alpha\), which would remove the difficulties noted above. In work \(^{8a}\), for example, the suggestion is made that in the layer \(E\) the reaction

\[ \mathrm{O}_2^{+}+e\to \mathrm{O}'+\mathrm{O}'' \tag{4.20} \]

may play the principal role.

For the layers \(F_1\) and \(F_2\), reactions with the participation of molecules or other heavy particles are also assumed, of the form:

\[ \left. \begin{aligned} \mathrm{O}^{+}+\mathrm{XY}&\to \mathrm{XY}^{+}+\mathrm{O}\\ \mathrm{XY}^{+}+e&\to \mathrm{X}'+\mathrm{Y}' \end{aligned} \right\}, \tag{4.21} \]

or

\[ N_2^{+}+e\to N'+N'', \tag{4.22} \]

\[ \left. \begin{aligned} N_2^{+}+\mathrm{XY}&\to \mathrm{XY}^{+}+N_2\\ \mathrm{XY}^{+}+e&\to \mathrm{X}'+\mathrm{Y}' \end{aligned} \right\}, \tag{4.23} \]

where \(\mathrm{XY}\) may be an NO or \(\mathrm{O}_2\) molecule.

In this connection the authors assume that in \(F_2\) the coefficient \(\alpha\) will depend on density. In the cited work, a hypothesis is also advanced which, in the authors’ opinion, cannot simply be rejected without verification: namely, that in recombination processes in the ionized layers, heavy particles \(D\) of volcanic or meteoric origin may participate, forming reactions of the type

\[ \left. \begin{aligned} D+\Gamma^{+}&\to D^{+}+\Gamma\\ D^{+}+e&\to D \end{aligned} \right\}, \tag{4.24} \]

or

\[ \left. \begin{aligned} O+e&\to D^{-}\\ D^{-}+O_1&\to D+O^{-} \end{aligned} \right\}. \tag{4.25} \]

The difficulty is aggravated, in any case at the present time, still

and also by the fact that \(\alpha\) decreases on passing to higher layers; at the same time, as we have seen, the assumption that \(\alpha\) is proportional to the density of neutral particles \(n\), by itself, creates additional ambiguities.

Until calculations have been carried out for different gases and processes (see § 3), and until a sufficient number of

Table VI

Values of the effective recombination coefficient \(\alpha\), obtained from the processing of daily characteristics of the maximum \(n_e\) of various layers and from measurements during solar eclipses

Layer	Observation time	\(\alpha,\ \dfrac{\mathrm{cm}^3}{\mathrm{sec}}\)	Note
\(E\)	By day	\(1\cdot10^{-8},\ 2\cdot10^{-8},\) \(1.2\cdot10^{-8},\ 3\cdot10^{-9}\)	England, Huancayo, England, Calcutta
\(E\)	At night	\(2\cdot10^{-9},\ 4\cdot10^{-9},\) \(1.1\cdot10^{-9}\)	Huancayo, England, Calcutta
\(E\)	During a solar eclipse	\(1.5\cdot10^{-8},\ 5\cdot10^{-9},\) \(5\cdot10^{-9},\ 1.2\cdot10^{-8}\)	South Africa, Tromsø, Central Brazil, Africa
\(F_1\)	By day	\(1.5\cdot10^{-9}\div 0.5\cdot10^{-9}\)	Moscow
\(F_1\)	During a solar eclipse	\(8\cdot10^{-9},\ 6\cdot10^{-9},\) \(2\cdot10^{-9}\)	Tromsø, South Africa, Central Brazil
\(F_2\)	By day	\(8.5\cdot10^{-11},\ 10^{-10},\) \(3.6\cdot10^{-11},\) \(2\cdot10^{-10}\div 0.5\cdot10^{-10}\)	England, Washington, Calcutta, Moscow
\(F_2\)	At night	\(4\cdot10^{-10},\ 10^{-10},\) \(5\cdot10^{-10},\) \(3\cdot10^{-11}\div 2\cdot10^{-10}\)	England, Washington, Calcutta, Moscow
\(F_2\)	During a solar eclipse	\(4\cdot10^{-11},\ 6\cdot10^{-11},\) \(10^{-10}\)	South Africa, Africa, Central Brazil

results of laboratory experiments and more detailed experimental investigations of the ionosphere, it is, of course, difficult to indicate the cause of this substantial discrepancy between experiment and theory. In this connection it is interesting to note that, at various times, laboratory measurements of the recombination coefficient at low pressures have been made. The behavior of the recombination coefficient, as a function of temperature and pressure, corresponded to a collision reaction involving two bodies. For argon \(^{18}\), cesium and mercury \(^{19}\), and more recently again for argon \(^{20}\), values of

\[ \alpha \sim (2 \div 5)10^{-10}, \]

i.e., several hundred times greater \(^{8,20}\) than the expected theoretical values \(\alpha_e\), were obtained.

Thus, experimental measurements of the effective recombination coefficient, carried out by different methods, are very important. At the same time, as far as is known to the author, a sufficiently systematic and detailed treatment of ionospheric data has not yet been carried out with the aim of determining \(\alpha\) and revealing the dependence of \(\alpha\) on other, more or less regular phenomena of the ionosphere. We shall indicate here the following results of calculations of \(\alpha\) that have appeared in the literature (Table VI).

In recent times \(^{22}\), calculations of the quantities \(\bar{\alpha}\) and \(\alpha\) for winter have also been made, in treating the results of measurements with the aid of (4.10) and (4.5) for the layer \(F\). The values of \(\alpha\) referred to heights of the order of 310–320 km, and \(\bar{\alpha}\) to the mean height in the region 200–320 km. The distribution of \(n_e\) with height indicates in these measurements that \(\bar{\alpha}\) may be referred to \(z\) of the order of 260–270 km. From Fig. 2, in which the averaged values of \(\alpha\) and \(\bar{\alpha}\) are compared, it is seen that they do not differ in any substantial way from one another (there is no dependence on height). Moreover, in these data a tendency of the diurnal variation of \(\alpha\) is revealed. The latter is apparently connected with the diurnal variation of the temperature of the layer.

Fig. 2. Diurnal variation of the recombination coefficients \(\alpha\) (for the maximum of layer \(F\)) and \(\bar{\alpha}\) (mean for the entire lower half of layer \(F\)).

For the same heights, \(z \sim 260\text{–}270\) km, calculations of \(\alpha\) have been carried out for the maximum of the layer \(F_1\). Values

\[ \alpha \sim (0.4 \div 1)10^{-9}, \]

were obtained, i.e. larger than \(\bar{\alpha}\) for \(F\). This is an important result, since it also testifies to the fact that \(\alpha\) depends little on height, and that the values of \(\alpha\) are different in different layers of the ionosphere, apparently because different gases, or mixtures of gases, play an active role in them.

Ya. L. Al’pert

5. THE PROBLEM OF THE \(F_2\) LAYER AND DIFFICULTIES WITH THE THEORY OF PHOTON ABSORPTION IN THE ULTRAVIOLET

Considerable difficulties also arise in explaining the well-known experimental data on the \(F_2\) layer.

The principal anomalies of the \(F_2\) layer, as was already noted in the first part of this article \(^{1}\), consist in the discrepancy between the summer diurnal variation of the degree of ionization \(n_e\) of the layer and the diurnal variation of the intensity of solar radiation (the function \(f(\chi)\) or \(\sec \chi\), see § 2), in the complex geographical distribution of \(n_e\), i.e. also in the absence of agreement with \(f(\chi)\), in the presence of a “longitude” effect, and in the fact that the summer midday values of \(n_e\) are smaller than the corresponding values for winter (see, for example, Fig. 3). Recently the suggestion has been put forward \(^{21}\) that the main cause of the abnormal seasonal, diurnal, and geographical changes in the ionization of \(F_2\) is the tidal phenomena occurring in it, caused by the Sun. In his work the author develops a “dynamic theory of the ionosphere,” which consists in the fact that in the \(F_2\) layer, because of tides, horizontal winds arise, causing the motion of electrons along the lines of the Earth’s magnetic field. These motions lead to vertical displacements of electrons, which indicates the necessity of taking into account the term written above,

\[ \frac{\partial(n_e v)}{\partial z}, \]

in equation (4.5). On the basis of an analysis of experimental data the author shows that, in a number of cases, oscillations of \(n_e\) and of the heights of the beginning and maximum of the \(F_2\) layer are indeed observed with a semidiurnal period, indicating the presence of tides in it. The cited work gives a theoretical calculation of the motion of charges caused by these tides; however, it contains no reliable data on the possible velocities of displacement of electrons in the layer.

Fig. 3. Typical diurnal variation of the electron concentration in the maximum of the \(F_2\) layer for winter and summer.

The author, thus, chiefly on the basis of qualitative considerations, believes that the geographical effect is explained by the asymmetry of the terrestrial magnetic field with respect to the geographical poles. Let us note that the existence of a correspondence between the geographical distribution of \(n_e\) and geomagnetic characteristics (inclination, latitude) has already been pointed out more than once in the literature (see, for example, references \(^{13,17}\) in \(^{1}\)). Further, the author sees confirmation of the point of view developed by him in the semi-diurnal oscillations of the \(F_2\) layer. On the basis of these considerations he explains the high values of ionization \(n_e\) at low latitudes and the greater heights of the ionization maximum at the equator, as well as the maxima of \(n_e\) observed in some places after midday and during the night hours. It should be noted that the author’s indication that the reason for the absence of these phenomena in \(E\) and \(F_1\) is the increase of electron production in these layers compared with \(F_2\), which leads to

\[ \frac{d(n_e v)}{dz} \ll J_0, \]

is puzzling, since it contradicts experimental data. (See, for example, Table V.)

Fig. 4. Typical diurnal variation of the total number of electrons in the lower half of the \(F_2\) layer and of the function \(1/f(\chi)\) for winter and summer.

However, a very important and, perhaps, even fundamental question that is not explained in this work is the abnormally low midday values of \(n_e\) in summer compared with winter (see Fig. 3). The author correctly notes that the total ionization of the layer does not change as a result of these phenomena; however, apparently not having the corresponding data, he does not consider the behavior of \(N_e\) in the light of the theory he has constructed.

Recently, independently of this work, an analysis of data for the \(F_2\) layer has been carried out with the aim of elucidating the causes of the anomalous seasonal and diurnal variation of its changes in the daytime hours in summer \(^{22}\). To exclude or, in any case, reduce the influence of transport, the processing of the diurnal characteristics of \(F_2\) was carried out both by means of equation (4.10) and by means of (4.5), for which purpose also were

calculated, on the basis of experimentally obtained altitude characteristics of the ionosphere, the true distribution of ionization with height (see \(^{35}\), and also \(^{36}\)) and the diurnal variation of the total ionization \(N_e^*\). Fig. 4 shows the diurnal variation of \(N_e^*\) for summer (curve 1) and winter (curve 2), and \(1/f(\chi)\) for summer (curve a) and winter (curve b). Fig. 4 corresponds to the same day for which the data are shown in Fig. 3.

This processing showed that the total ionization \(N_e^*\), as we see, retained an irregular diurnal (for summer) and seasonal variation. Thus, at noon in summer:

\[ N_e^* \sim 10^{13}\ \frac{\text{electrons}}{\text{cm}^2} \quad \text{and} \quad n_e \sim 10^6\ \frac{\text{electrons}}{\text{cm}^3}, \tag{5.1} \]

and in winter:

\[ N_e^* \sim 2.5 \cdot 10^{13}\ \frac{\text{electrons}}{\text{cm}^2} \quad \text{and} \quad n_e \sim 2.8 \cdot 10^6\ \frac{\text{electrons}}{\text{cm}^3}. \tag{5.2} \]

For winter (1947) it was found that in the lower half of the layer there is formed (for \(\chi=0\))

\[ \frac{S_m}{\varepsilon_i} \sim 2 \cdot 10^{10}\ \frac{\text{electrons}}{\text{cm}^2\text{ sec}}, \quad J_m \sim 2 \cdot 10^3\ \frac{\text{electrons}}{\text{cm}^2\text{ sec}}, \tag{5.3} \]

which gives, for the “solar constant” in the region \(1000\text{--}800\ \text{\AA}\) and below,

\[ S_\infty \sim (1 \div 1.5)\ \frac{\text{erg}}{\text{cm}^2\text{ sec}}. \tag{5.4} \]

For summer the number of newly formed electrons in a column of cross section \(1\ \text{cm}^2\), reduced to \(\chi=0\), is equal to

\[ Q_n \sim 5 \cdot 10^8\ \frac{\text{electrons}}{\text{cm}^2\text{ sec}}, \quad J_m \sim 50\ \frac{\text{electrons}}{\text{cm}^3\text{ sec}}. \tag{5.5} \]

Thus, from (5.3) and (5.5) it follows that in summer, when the solar energy entering the layer increases, the total ionization in the layer falls, and the amount of energy going into ionization decreases.

Moreover, from these data it follows that during the day in summer, beginning with sunrise, \(Q_n\) decreases by a factor of 10 or more over several hours. Both of these circumstances indicate, in accordance with the supposition made in \(^{22}\), that equation (4.10), solved for summer, is substantially affected [see (2.9)] by the factor

\[ \left[1 - e^{-\sigma \sec \chi\,(N^* - N_e^*)}\right], \tag{5.6} \]

which becomes considerably less than unity, owing to the fact that

\[ e^{-\sigma \sec \chi\,(N^* - N_e^*)} \to 1. \]

Analysis of this question led to the supposition (see \(^{22}\)) that in summer, owing to the rise of the layer by 80–100 km upward, it passes into a region of lower atmospheric density, where the ionization substantially approaches saturation.

However, in order for this assumption to be satisfied, it is necessary that the magnitude of the photoabsorption cross section in the region of the Sun’s ultraviolet radiation that causes ionization be equal to

\[ \sigma \simeq 10^{-14}\ \text{cm}^2 . \tag{5.7} \]

From (5.7) and (3.3) it is seen that this value of \(\sigma\) exceeds the value of the photoionization cross section \(\sigma_e\), obtained from quantum calculations for \(O_1\), by \(10^3\)—\(10^4\) times. It should be recalled that a discrepancy of the same order was obtained between the effective recombination coefficient and the photorecombination coefficient (§ 4), characterizing the process inverse to photoionization.

What is involved here, and where the root of the possible error lies, is at present still difficult to say.

The second question that remains unclear and arises independently of the magnitude of \(\sigma\) is where in the atmosphere the energy arriving from the Sun in summer in the frequency range responsible for ionization in the layer \(F_2\) is absorbed.* It would be very important to know:

a) Are there any processes possible in the upper layers of the atmosphere that absorb the energy of ultraviolet radiation in the frequency range responsible for ionization?

b) Can this absorption appear in summer and be absent in winter?

c) Are substantially different values of the photoabsorption (photoionization) cross section possible in different layers of the ionosphere? One may attempt to resolve this question by jointly solving equations (4.5) and (4.10), using experimental data for different layers, from which one can determine

\[ \frac{S_m}{\varepsilon_i} \quad \text{and} \quad \sigma (n-n_e)\frac{S_m}{\varepsilon_i}. \tag{5.8} \]

It should be pointed out that, since the values of the density \(n\) in the region of the \(E\) layer are at present known with sufficient reliability on the basis not only of radio data, one could try to determine \(\sigma\) from (5.8). These calculations, which require a careful analysis of a large amount of material, have not yet been carried out.

Let us note here that in \(^{8a}\) a discrepancy is observed between \(\sigma\) theoretical and that determined from measurement data for the maximum of the \(E\) layer. Since in the \(E\) layer the density \(n \sim (2 \div 5)10^{12}\) and \(H \sim 10^6\ \text{cm}\), from (2.8) we have for the maximum of \(E\) \(\sigma \sim (2 \div 5)10^{-19}\), whereas theoretically for \(O_2\) the value obtained is \(\sigma \sim 10^{-20}\) (see § 3,a). Thus, from these data it follows that if the initial value of the photoionization cross section is taken to be \(\sigma \sim 10^{-20}\), then \(n\) must be—

* Calculations show that the ionization of the \(F_1\) layer and of the region intermediate between \(E\) and \(F_1\) absorbs in summer no more than 10–20% of this energy.

of the order of \(10^4\), which is at variance with other data. In connection with this, the authors put forward the supposition that in the layer \(E\) there exists absorption of ultraviolet radiation not connected with ionization.

The examination carried out above of the difficulties arising in the layer \(F_2\), and of their possible explanation, indicates that at present there are still not sufficient grounds for rejecting or defending one or another point of view and for singling out the principal cause of these phenomena. From these data, however, it is clear that we are faced with a very complex and important problem.

6. THE EFFECTIVE NUMBER OF COLLISIONS \(\nu_{\mathrm{eff}}\)

Another parameter, besides \(\alpha\), characterizing microprocesses in the ionosphere and determined from radio measurements, is the number of elastic collisions \(\nu\). In radio investigations it appears as \(\nu_{\mathrm{eff}}\) and determines the absorption of radio waves in the ionosphere; measurements of this absorption make it possible to compute \(\nu_{\mathrm{eff}}\).

Since the principal agent influencing the propagation of radio waves in the ionosphere is the electrons, the main role here must be played by collisions of electrons with neutral particles or ions.

Before proceeding further, it is expedient to clarify the role that may be played by collisions of ions or, respectively, of electrons with one another. Collisions of ions with one another are, in order of magnitude, determined by the same formula as collisions of electrons with ions (see below), with the replacement in it of the electron velocity \(v_e\) by the ion velocity \(v_i\). In view of the fact that \(v_i \ll v_e\) and the concentration of negative ions is apparently always less than the concentration of electrons (and positive ions), this type of collision can practically always be neglected. Taking account of the influence of collisions of electrons with one another is connected with great difficulties; however, for various reasons it may be assumed that this type of collision too may be neglected.

The numbers of collisions of electrons with ions \(\bigl(\nu^{(i)}\bigr)\) or with neutral particles \((n)\), i.e. their scattering by these particles, can be calculated by the method indicated in \({}^{23}\). The problem is solved by means of the kinetic equation \({}^{24}\), neglecting the velocities of the heavy particles in comparison with the velocity of motion of the electrons.

It follows from the calculations, under the assumption \(\nu^2 \ll \omega^2\) (\(\omega\) is the angular frequency of the wave), that the number of collisions with neutral particles is equal to

\[ \nu_{\mathrm{eff}}^{(n)} = \frac{4}{5}\,\pi a^2 n \overline{v}, \tag{6.1} \]

where \(\overline{v}\) is the arithmetic mean value of the electron velocity:

\[ \overline{v} = \frac{2}{\sqrt{\pi}}\sqrt{\frac{2kT}{m}} = 6.2 \cdot 10^5 \sqrt{T}\ \frac{\mathrm{cm}}{\mathrm{sec}}, \tag{6.2} \]

and \(\pi a^2\), according to experimental data for air (for \(O_2\) and \(N_2\)), is equal to

\[ \pi a^2 \simeq 7.2\cdot 10^{-16}\ \text{cm}^2 . \tag{6.3} \]

From (6.2) and (6.3) we obtain:

\[ \nu_{\mathrm{eff}}^{(n)} \simeq 6\cdot 10^{-10}\cdot n\sqrt{T} \tag{6.4} \]

For the number of collisions of electrons with ions, it follows from these calculations (also for \(\nu^2 \ll \omega^2\) and \(\ln \dfrac{kT}{e^2 n^{1/3}} \gg 1\))

\[ \nu_{\mathrm{eff}}^{(ni)} \simeq \frac{\pi e^4}{(kT)^2}\, n^i \bar{v}\ln \frac{kT}{e^2 \sqrt[3]{n^i}}, \tag{6.5} \]

or, using (6.2) and the values of the constants, we have:

\[ \nu_{\mathrm{eff}}^{(ni)} \simeq \frac{5.6}{T^{3/2}}\, n^i \ln\left(670\,\frac{T}{\sqrt[3]{n^i}}\right). \tag{6.6} \]

It follows from (6.4) and (6.6) that the numbers of collisions with ions and with neutral particles have substantially different dependences on temperature, and that the effective cross section for ions is significantly larger than the cross section for neutral particles.

Table VII gives the values of \(\nu_{\mathrm{eff}}^{(n)}\) and \(\nu_{\mathrm{eff}}^{(ni)}\), calculated by formulas (6.4) and (6.6).

Table VII

Numbers of collisions of electrons with ions \(\nu_{\mathrm{eff}}^{(ni)}\) and with neutral particles \(\nu_{\mathrm{eff}}^{(n)}\), as functions of \(T\), \(n\), and \(n^i\)

\(n, n^i\)	\(T=300^\circ\mathrm{K}\) \(\nu_{\mathrm{eff}}^{(ni)}\)	\(T=300^\circ\mathrm{K}\) \(\nu_{\mathrm{eff}}^{(n)}\)	\(T=500^\circ\mathrm{K}\) \(\nu_{\mathrm{eff}}^{(ni)}\)	\(T=500^\circ\mathrm{K}\) \(\nu_{\mathrm{eff}}^{(n)}\)	\(T=1000^\circ\mathrm{K}\) \(\nu_{\mathrm{eff}}^{(ni)}\)	\(T=1000^\circ\mathrm{K}\) \(\nu_{\mathrm{eff}}^{(n)}\)	\(T=2000^\circ\mathrm{K}\) \(\nu_{\mathrm{eff}}^{(ni)}\)	\(T=2000^\circ\mathrm{K}\) \(\nu_{\mathrm{eff}}^{(n)}\)
\(10^3\)	\(10\)	—	\(5\)	—	\(2\)	—	—	—
\(10^6\)	\(7.8\cdot 10^3\)	—	\(4\cdot 10^3\)	—	\(1.5\cdot 10^3\)	—	\(6\cdot 10^2\)	—
\(10^9\)	\(5.4\cdot 10^6\)	\(10\)	\(2.8\cdot 10^6\)	\(13\)	\(1.1\cdot 10^6\)	\(19\)	\(4.5\cdot 10^5\)	\(27\)
\(5\cdot 10^{11}\)	—	\(5\cdot 10^3\)	—	\(6.5\cdot 10^3\)	—	\(9.5\cdot 10^3\)	—	\(13.5\cdot 10^3\)
\(10^{12}\)	\(3.0\cdot 10^9\)	\(10^4\)	\(1.7\cdot 10^9\)	\(1.3\cdot 10^4\)	\(7.3\cdot 10^8\)	\(1.9\cdot 10^4\)	\(3.0\cdot 10^8\)	\(2.7\cdot 10^4\)
\(10^{15}\)	—	\(10^7\)	\(1.6\cdot 10^{11}\)	\(1.3\cdot 10^7\)	\(3.2\cdot 10^{11}\)	\(1.9\cdot 10^7\)	\(1.5\cdot 10^{11}\)	\(2.7\cdot 10^7\)

The value of the number of collisions determined from experiments is equal to

\[ \nu_{\mathrm{eff}}=\nu_{\mathrm{eff}}^{(n)}+\nu_{\mathrm{eff}}^{(ni)}, \tag{6.7} \]

which substantially complicates the analysis of experimental data and the possibility of drawing one conclusion or another. However, with sufficiently complete and detailed measurements of \(\nu_{\mathrm{eff}}\) and study of its time variation, as well as its dependence on altitude, many important data can be obtained. Let us point, for example, to the following. It follows from Table VII that, with increasing temperature, \(\nu_{\mathrm{eff}}^{(n)}\) decreases rather rapidly, whereas \(\nu_{\mathrm{eff}}^{(n)}\), on the contrary, increases, and appreciably more slowly. Thus, when \(\nu_{\mathrm{eff}}^{(m)}\) decreases by a factor of 13, \(\nu_{\mathrm{eff}}^{(n)}\) increases only by a factor of 2.7. Thus, with detailed measurements of the time variation of \(\nu_{\mathrm{eff}}\) and comparison of it with the temperature

Table VIII

Values of \(\nu_{\mathrm{eff}}\) for various layers of the atmosphere

Layer	Time of observation	\(\nu_{\mathrm{eff}}\), collisions/sec	Remark
\(D\)	—	\(1.5\cdot 10^{7}\)	—
\(D\)	—	\(3.2\cdot 10^{6}\)	—
\(D\)	—	\(1.0\cdot 10^{6}\)	—
\(E\)	—	\(4\cdot 10^{4}\)
\(E\)	—	\(3\cdot 10^{5}\)	In the upper part of the layer; England
\(E\)	—	\(2.7\cdot 10^{4}\)	In the lower part of the layer; England
\(E\)	—	\(2.7\cdot 10^{5}\)	In the upper part of the layer; India
\(E\)	—	\(1\cdot 10^{4}\)	In the lower part of the layer; India
\(E\)	—	\(2\cdot 10^{5}\)	According to calculations in the lower part of the layer; England
\(E\)	—	\(5\cdot 10^{5}\)	According to studies of nonlinear effects; at an altitude of 85 km
\(F_{1}\)	—	\(10^{4}\)	According to various estimates
\(F_{2}\)	after noon	\(1.6\cdot 10^{3}\)	England. October
\(F_{2}\)	same	\(1.6\cdot 10^{3}\)	England. October
\(F_{2}\)	noon	\(3.5\cdot 10^{3}\)	London. January–May
\(F_{2}\)	”	\(3\cdot 10^{3}\)	London. September
\(F_{2}\)	”	\(1.6\cdot 10^{3}\)	England. September
\(F_{2}\)	after noon	\(1.2\cdot 10^{4}\)	India. March

with the course of the atmosphere (obtained from other data, say, from measurements of \(\alpha\)) it would apparently be possible, in some cases, to determine which type of collision is predominant, and perhaps even fundamental, in one or another layer; this at once opens the possibility for very important conclusions. Thus, for example, if it were established that in the \(F_2\) layer, with increasing temperature, the value of \(\nu_{\mathrm{eff}}\) decreases in accordance with the calculations, i.e. that the predominant type of collisions is collisions with ions, then it would be possible to say that in this layer \(n \sim 10^6 \div 10^7\), since for \(F_2\) \(\nu_{\mathrm{eff}} \simeq 10^3\) (see Table VIII). The inverse temperature dependence would indicate that here \(n \simeq 10^{10} \div 10^{11}\).

From the example given it is thus clear that the carrying out of sufficiently complete and careful experiments in this direction opens up a number of possibilities for studying the structure and processes of the ionosphere.

Up to the present time, as far as the author knows, no regular and complete experiments on measurements of \(\nu_{\mathrm{eff}}\) have been made, and there are only separate measurements, as also for \(\alpha\) (see § 4). We give in Table VIII the values of \(\nu_{\mathrm{eff}}\) known from the literature data\(^{25--32}\).

Recently it has also been pointed out\(^{33}\) that from experimental studies of nonlinear effects in the ionosphere it might be possible in some cases to form an idea of the role of collisions of electrons with ions or neutral particles.

In conclusion to this paragraph we note the following circumstance. In the formulas for \(\nu_{\mathrm{eff}}\) given above, the electron temperature is meant, whose value may not coincide—especially in the rarefied atmosphere—with the gas temperature. A complete calculation of this problem has not yet been carried out up to the present time, since for its performance it is necessary to know the distribution of the intensity of the Sun’s radiation in the ultraviolet part of its spectrum. From calculations made in\(^{34}\) it follows that, with a high degree of accuracy, it may be assumed for the \(F\)-layer that

\[ T \simeq T_{\mathrm{r}} + 50 \varepsilon, \tag{6.8} \]

where \(T_{\mathrm{r}}\) is the gas temperature, and \(\varepsilon\) is the mean energy of the electron when it is torn from the atom, expressed in electron-volts. The quantity \(\varepsilon\) may reach several electron-volts.

7. TEMPERATURE, DENSITY, AND COMPOSITION OF THE IONOSPHERE

Radio investigations of the ionosphere are one of the methods for obtaining data on the temperature, density, composition, and processes occurring in the atmosphere, especially at great heights. However, since for the detailed processing and analysis of the results of radio measurements it is already necessary to know all these quantities, extraction

from them the necessary information on the structure of the atmosphere is very difficult.

We shall give here a brief account of the data known in the literature on the structure of the upper atmosphere, partly obtained also from radio measurements (see also \(^{37,38}\)).

Temperature

At present there is a large quantity of data from various measurements, providing sufficiently reliable information on the temperature of the atmosphere up to altitudes of \(\sim 110\)—\(120\) km. The distribution of temperature with height has been analyzed in rather great detail from the point of view of radio measurements in \(^{23}\). On the basis of radio measurements \(^{39}\) it was shown that at an altitude of 70 km (the \(D\)-layer) the temperature of the atmosphere is \(\sim 200^\circ\), which agreed with the results of other measurements. For the altitude \(z \sim 100\)—\(110\) km (the \(E\)-layer) \(^{41}\) gave \(T \sim 350\)—\(400^\circ\). However, the basic information on temperature has been obtained by other methods (see, for example, \(^{42,43,44}\)).

Data on the temperature of still higher layers of the atmosphere are more meager. Various works \(^{8,38,45,46}\) contain the corresponding results of temperature measurements up to altitudes exceeding 300 km. It follows from them that at an altitude of 300 km and above the temperature may reach \(1000^\circ\) and more.

Fig. 5. Temperature of the high layers of the atmosphere in winter and summer, according to \(^{22}\).

The principal method of determining temperature on the basis of radio measurements is the computation from these data of the height of the homogeneous atmosphere \(H\) [see (2.7)] for the measured heights of reflection of radio pulses at different frequencies and the height of the layer maximum [see (2.8)]. This gives very meager results. Further development of methods for measuring the recombination coefficient \(\alpha\) (§ 4) and the collision number \(\nu\) (§ 6) will, it seems to us, make it possible to expand considerably the possibilities of radio methods for determining the temperature of the upper layers of the atmosphere.

In the above-cited work \(^{22}\) an attempt was made to compute, on the basis of the assumption put forward there concerning the cause of the anomalies \(F_2\), a possible distribution of atmospheric temperature with height for winter and summer in the daytime.

From these calculations it follows that at an altitude of the order of \((140 \div 160)\) km one may assume a third temperature minimum, reaching \(T \simeq 200^\circ\). At greater altitudes there is a rapid rise in temperature, which may reach \(1000^\circ\) (in winter) and \(2000^\circ\) (in summer). In this case the third temperature minimum may be still deeper (\(T < 200^\circ\)) in summer (see Fig. 5). The latter circumstance, however, is not obligatory if the region of dissociation of atmospheric gases in summer is located 20–30 km higher than in winter. It is interesting in this connection to point out that other authors also independently arrived at the assumption of the existence of a temperature minimum at an altitude of 140–160 km[^47], in the mathematical theory of resonant oscillations in the atmosphere developed by them, which explains, in particular, the tidal phenomena observed in the \(E\) layer[^48].

It goes without saying that the data cited on the temperature at great altitudes are still insufficiently reliable, and only on the basis of future measurements will it be possible to verify their correctness. It may be expedient to note that the tidal phenomena observed, according to the latest reports, in the \(F_2\) layer[^21] may lead to a substantial revision of ideas about the possible temperature at great altitudes (above 300–400 km).

Density

Data on the density of the atmosphere up to 120 km are also contained in works[^42–^44]. For greater altitudes these data have been calculated in works[^8,^45,^46] and others. Determination of the density of neutral particles on the basis of radio measurements is carried out only for the maxima of the layers \(E\), \(F_1\), and \(F_2\) [see formula (2.8)]. In view of the absence of sufficiently reliable values of \(H\) (2.7) and \(\sigma\) (see § 5), the corresponding density calculations cannot yet be considered reliable, although in some cases they agree fairly well with the results of other calculations. From the totality of the data available in the literature, one may indicate the following distribution of the density of neutral particles with height (see Table IX).

Table IX

Distribution of the density of neutral particles with height

\(Z\), km	0	20	40	60	80	100	120	220	300
\(n\), cm\(^{-3}\)	\(2.6 \cdot 10^{19}\)	\(1.5 \cdot 10^{18}\)	\(8.1 \cdot 10^{16}\)	\(7.6 \cdot 10^{15}\)	\(5.2 \cdot 10^{14}\)	\(2.6 \cdot 10^{13}\)	\(2.5 \cdot 10^{12}\)	\(\sim 10^{11}\)	\(2 \cdot 10^{10}\)

In [^22] an attempt was also made to calculate the density of the atmosphere at altitudes above 120 km. From these calculations it follows that there is a seasonal variation of density, consisting in the fact that in summer in the region

250–300 km the density of the atmosphere may be lower than in winter, while above 300 km, on the contrary, in summer the density of the atmosphere is greater than in winter. At an altitude of 220 km, according to these calculations, \(n\) has a value of \(\sim 10^7\), i.e. \(10^4\) times less than that indicated in Table IX. From these same data it follows that during the day above 400 km the atmosphere is essentially completely ionized. Naturally, all these calculations are only tentative, just as are the results of the calculations of \(T\) given above.

Composition

On the basis of the existing data, the principal constituent gases at an altitude of 100–120 km (i.e. in layer \(E\)) are molecular nitrogen \(N_2\) and molecular oxygen \(O_2\). It is assumed that at an altitude of 100–120 km the dissociation of oxygen begins, so that at altitudes of 150 km and above atomic oxygen \(O_1\) and molecular nitrogen \(N_2\) predominate; and recently there have been indications that at still greater altitudes atomic nitrogen \(N_1\) is also present.^49 Most investigators believe that layer \(E\) is the result of ionization of \(O_2\), while layer \(F\) consists of ionized \(N_2\) and \(O_1\). When layer \(F\) is divided into two, \(F_1\) and \(F_2\), one of them (\(F_1\)) consists mainly of \(N_2\), and the other (\(F_2\)) of \(O_1\). Thus, the role of atomic oxygen \(O_1\) in the upper atmosphere is very great (see, for example, ^50,^51), and there are grounds for thinking that at still greater altitudes the role of atomic nitrogen \(N_1\) also increases.

As with temperature and density, information on the gaseous composition at great altitudes (200–300 km and more) is still very unreliable. In ^22 it is assumed that the region of most intense dissociation of atomic oxygen may lie not at 100–120 km, but at altitudes of the order of 180–250 km, i.e. that complete dissociation occurs mainly higher than is usually supposed. In this case, a displacement of this region of dissociation from summer to winter is possible; it is lower in winter than in summer by 20–30 km.

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