ON THE DEPENDENCE OF THE LINEAR ELECTRON DENSITY IN A METEOR TRAIL ON THE VELOCITY AND MASS OF THE METEOR BODY

E. I. FIALKO

(Presented by the scientific seminar of the Radio Engineering Faculty)

Preliminary Remarks. Statement of the Problem

Knowledge of the dependence of the linear electron density in a meteor trail is necessary in considering a number of problems. As is known [1],

\[ \alpha=\frac{9}{4}\alpha_{\max}\cdot\frac{p}{p_m}\left(1-\frac{1}{3}\cdot\frac{p}{p_m}\right)^2, \tag{1} \]

where \(\alpha\) and \(p\) are, respectively, the linear electron density in the meteor trail and the atmospheric pressure at the point under consideration; \(p_m\) is the pressure corresponding to the point with the maximum linear electron density \(\alpha_{\max}\), equal to

\[ \alpha_{\max}=\frac{4}{9}\beta\,\frac{m\cos z}{\mu H}, \tag{2} \]

where \(\beta\) is the probability that, as a result of the evaporation of one atom of the meteor body, one free electron will be released; \(H\) is the height of the homogeneous atmosphere at the point under consideration; \(m\) is the initial mass of the meteor body; \(z\) is the zenith distance of the meteor radiant; \(\mu\) is the mass of an atom of the meteor body. As follows from (1) and (2), there is no explicit dependence of \(\alpha\) on the meteor velocity \(v\). However, \(\alpha\) depends on \(v\), since \(\alpha_{\max}\) and \(p_m\) depend on \(v\):

\[ p_m=\frac{2lg}{\Lambda A}\cdot\frac{m^{1/3}\cos z}{v^2}, \tag{3} \]

where \(l\), \(\Lambda\), \(A\) are constants characterizing the physical and geometric properties of the meteor body [1]; \(g\) is the acceleration of gravity.

Kaiser assumes that \(\alpha_{\max}\) may depend on velocity in connection with the dependence of the ionization probability \(\beta\) on velocity [1]; at the same time

the dependence of $H$ on $v$ is not taken into account. Neglect of the dependence of $H$ on $v$ in revealing the dependence of $\alpha$ on $v$ is not justified, since Kaiser [1] and Evans [2] arrive at the conclusion that the meteor velocity $v$ has only a weak influence on $\beta$.

Most of the detected meteors are weak meteors, giving reflected signals with a power close to that of the threshold signal. Therefore, naturally, for a considerable (if not the greater) part of the detected meteors, the section that normally reflects radio waves will be located in the region of the characteristic height $h_m$ (i.e., the height at which the most intense evaporation of the meteoroid body occurs and $\alpha=\alpha_{\max}$).

Consequently, it is of special interest to determine the dependence $\alpha(v)$ in the region $\alpha \simeq \alpha_{\max}$.

Large meteors can be detected in the case when the height of the normally reflecting section considerably exceeds the characteristic height. Naturally, in this region the dependence $\alpha(v)$ must be different from that in the region $\alpha \simeq \alpha_{\max}$.

Fig. 1. Change in the linear electron density along the trail (illustration). $l$ is measured along the trail axis. $L_0$, $L_1$, $L_2$ are points of possible location of the radar. $R$ is the slant range. $h$ is height. $\alpha$ is the linear electron density in the meteor trail.

It is also possible to detect a meteor in the case $h<h_m$ (Fig. 1). Thus it is necessary to determine the dependence of $\alpha$ on $v$ when the normally reflecting section is located in different parts of the meteor trail and, first of all, in the region $h \simeq h_m$.

Let us first consider the dependence of $\alpha_{\max}$ on $v$.

Dependence of the maximum linear electron density on the meteor velocity $\alpha_{\max}(v)$

We write (2) in the form

\[ \alpha_{\max}=B\cdot \frac{\beta}{H}, \tag{4} \]

where \(B=\dfrac{4}{9}\cdot\dfrac{m}{\mu}\cos\chi\) is a coefficient independent of the velocity \(v\).

We shall use the commonly applied approximation

\[ \beta=a\cdot v^x . \tag{5} \]

By analogy with (5), let us represent \(H\) in the form

\[ H=b\cdot v^y, \tag{6} \]

where \(x, y, a, b\) are coefficients independent of \(v\). Substituting (5) and (6) into (4), we obtain

\[ \alpha_{\max}=B_1\cdot v^n, \tag{7} \]

where \(n=x-y,\quad B_1=B\dfrac{a}{b}\).

As is known, the dependence of the ionization probability on velocity \(\beta(v)\) cannot at present be regarded as established. The values of the coefficient \(x\) given in some works [2, 3] differ and lie within the range \(0\div 5.6\).

Thus, the value of \(x\) still remains to be refined.

To determine the value of the exponent \(y\), we shall use the dependences of the characteristic height on velocity, \(h_m(v)\), and \(H(h_m)\), presented as graphs [1, 2]. From \(h_m(v)\) and \(H(h_m)\) we find the dependence of the homogeneous-atmosphere height on meteor velocity, \(H(v)\), which in the range \(v=20–60\ \text{km/sec}\) is approximated by the parabola (6) with coefficients \(y\simeq 0.5\) and \(b\simeq 1\). Thus,

\[ \alpha_{\max}\sim v^n, \tag{8} \]

\[ n\simeq x-0.5\quad (\text{and for } x\simeq 2,\ n\simeq 1.5). \]

Dependence \(\alpha(v)\) in the region \(p\ll p_m\)

In the case where the portion of the meteor trail that normally reflects radio waves is located at a height \(h>h_m\), and the pressure at height \(h\) is considerably less than the pressure at the characteristic height \((p\ll p_m)\), formula (1) is simplified and takes the form

\[ \alpha \simeq \frac{9}{4}\cdot \alpha_{\max}\cdot \frac{p}{p_m}. \tag{9} \]

Substituting into (9) the expressions for \(\alpha_{\max}\) and \(p_m\), (3) and (7), taking into account expressions (5) and (6), we obtain

\[ \alpha \simeq D\cdot v^{n+2}, \tag{10} \]

where

\[ D=\frac{a}{b}\cdot \frac{\Lambda A}{2\mu g}\cdot m^{2/3}\cdot p . \]

If we compare two meteors with different velocities but identical trajectories (more precisely, with the same position of the trail axis relative to the station) with the position of the locator unchanged, then \(p=\mathrm{const}\) (Fig. 2). Thus, in the region \(p \ll p_m\),

\[ \alpha \sim v^{x-y+2}. \tag{11} \]

From a comparison of expressions (8) and (11) it follows that in the region \(h \gg h_m\) (for \(p \ll p_m\)) the influence of the meteor velocity on the value of the linear electron density is considerably sharper than in the region \(h \simeq h_m\) (\(p \simeq p_m\)). This is also clear from simple physical considerations. In the region \(h \simeq h_m\), an increase in the velocity leads to an increase in \(\alpha\) mainly due to an increase in \(\alpha_{\max}\). In the region \(p \ll p_m\), the increase in \(\alpha\) with increasing \(v\) occurs not only as a result of the increase in \(\alpha_{\max}\), but also as a result of the shift of the curve \(\dfrac{\alpha}{\alpha_{\max}}\) toward greater heights (Fig. 2).

Fig. 2. Influence of the velocity of a meteoric body on the change in the linear electron density along the trail (illustration). 1 — \(\alpha(l)\) for \(v=v_1\); 2 — \(\alpha(l)\) for \(v=v_2\) (\(v_2>v_1\)); I — region \(h \simeq h_m\) (\(p \simeq p_m\) and \(\alpha \simeq \alpha_{\max}\)); II — region \(h > h_m\) (\(p \ll p_m\)); III — region \(h < h_m\).

It is also clear from simple considerations that at some height \(h_1\) the change in \(\alpha\) is practically absent (when \(v\) changes within small limits); at heights \(h<h_1\), an increase in velocity leads to a decrease in the linear electron density.

Let us find the range of heights in which an increase in \(v\) leads to a decrease in \(\alpha\) (and conversely), and also consider in more detail the character of the change \(\alpha(v)\) at different heights.

Change of \(\alpha(v)\) at different heights

To clarify the dependence \(\alpha(v)\) at different heights, let us find the relation between the increment of the linear electron density \(\Delta \alpha\) and the increment of velocity \(\Delta v\). We note that we are comparing two meteors with different, but constant, velocities, and are not considering the case of a single meteor with variable velocity.

In the general case, as is evident from (1),

\[ \alpha=\alpha\left(\alpha_{\max}; \frac{p}{p_m}\right), \tag{12} \]

where \(\alpha_{\max}=\alpha_{\max}(v)\) and \(\dfrac{p}{p_m}=\dfrac{p}{p_m}(v)\).

The derivative of \(\alpha\) with respect to \(v\) is

\[ \frac{d\alpha}{dv} = \frac{\partial \alpha}{\partial \alpha_{\max}}\cdot \frac{d\alpha_{\max}}{dv} + \frac{\partial \alpha}{\partial\left(\dfrac{p}{p_m}\right)} \cdot \frac{d\left(\dfrac{p}{p_m}\right)}{dv}. \tag{13} \]

Using (1), (7), (3) and carrying out simple transformations, we find:

\[ \frac{\partial \alpha}{\partial \alpha_{\max}} = \frac{\alpha}{\alpha_{\max}}; \tag{14} \]

\[ \frac{\partial \alpha}{\partial\left(\dfrac{p}{p_m}\right)} = \alpha\cdot\frac{p_m}{p}\cdot \frac{1-\dfrac{p}{p_m}} {1-\dfrac{1}{3}\cdot\dfrac{p}{p_m}}; \tag{15} \]

\[ \frac{d\alpha_{\max}}{dv} = n\,\frac{\alpha_{\max}}{v}; \tag{16} \]

\[ \frac{d\left(\dfrac{p}{p_m}\right)}{dv} = 2\cdot\frac{1}{v}\cdot\frac{p}{p_m}. \tag{17} \]

Substituting (14)—(17) into (13), after the corresponding transformations we obtain

\[ \frac{d\alpha}{dv} = n\cdot\frac{\alpha}{v} \left( 1+\frac{2}{n} \frac{1-\dfrac{p}{p_m}} {1-\dfrac{1}{3}\dfrac{p}{p_m}} \right). \tag{18} \]

Passing from differentials to increments, we find the relation between the relative changes in velocity and in the linear electron density

\[ \frac{\Delta \alpha}{\alpha} = \frac{\Delta v}{v} \left( n+2 \frac{1-\dfrac{p}{p_m}} {1-\dfrac{1}{3}\cdot\dfrac{p}{p_m}} \right). \tag{19} \]

Thus, \(\dfrac{\Delta \alpha}{\alpha}\) is a function of \(\Delta v\), \(v\), \(n\), and \(h\). Let us note that \(p_m\) depends on \(h_m\), and \(h_m\) on \(v\); \(p\) depends on the altitude \(h\) at which the portion of the trail under consideration is located.

It is quite obvious that
\[ \left|\frac{\Delta \alpha}{\alpha}\right|\sim \left|\frac{\Delta v}{v}\right|. \]
For given \(\Delta v\) and \(v\), the quantity \(\dfrac{\Delta \alpha}{\alpha}\) depends substantially on the value \(n=x-y\).

Let us consider the dependence of \(\dfrac{\Delta \alpha}{\alpha}\) on \(h\).

For \(h>h_m\) and \(p\ll p_m\),

\[ \frac{\Delta \alpha}{\alpha}\simeq \frac{\Delta v}{v}(n+2). \tag{20} \]

In the region \(h\simeq h_m\) \((p\simeq p_m)\),

\[ \frac{\Delta \alpha}{\alpha}\simeq \frac{\Delta v}{v}\cdot n . \tag{21} \]

As is seen from (20) and (21), for \(h>h_m\) the sign of \(\Delta \alpha\) coincides with the sign of \(\Delta v\).

As follows from (18) and (19), the sign of \(\Delta \alpha\) coincides with the sign of \(\Delta v\) when

\[ n+2\frac{1-\dfrac{p}{p_m}}{1-\dfrac{1}{3}\cdot\dfrac{p}{p_m}}>0, \]

i.e., when

\[ \frac{p}{p_m}<\frac{1+\dfrac{n}{2}}{1+\dfrac{n}{6}}. \tag{22} \]

When

\[ \frac{p}{p_m}=\frac{1+\dfrac{n}{2}}{1+\dfrac{n}{6}}, \qquad \frac{\Delta \alpha}{\alpha}=0. \tag{23} \]

Condition (23) is satisfied at some height \(h_1\), depending on \(v\) and \(n\) (since \(p_m\) depends on \(v\)).

For

\[ \frac{p}{p_m}>\frac{1+n/2}{1+n/6} \]

the sign of \(\Delta \alpha\) is opposite to the sign of \(\Delta v\). As follows from (1), \(p_{\max}=3p_m\), i.e., the minimum height \(h_{\min}\) to which the meteor trail descends corresponds to a pressure exceeding by a factor of 3 the pressure at the characteristic height.

Thus, in the region \(h>h_1\) \(\left[\text{i.e., } p<\dfrac{1+\dfrac{n}{2}}{1+\dfrac{n}{6}}\,p_m\right]\), \(\Delta \alpha\sim \Delta v\);

in the region \(h_{\min}<h<h_1\) \(\left[\text{i.e., } p>\dfrac{1+\dfrac{n}{2}}{1+\dfrac{n}{6}}\cdot \Delta \alpha\sim -\Delta v\, p_m\right]\); when \(h\simeq h_1\)

\[ \left(\text{i.e. } \rho=\frac{1+\dfrac{n}{2}}{1+\dfrac{n}{6}}\,\rho_m\right)\Delta\alpha \simeq 0 \]
for any sign of \(\Delta v\) (Fig. 3).

Fig. 3. Relative change in the linear density of electrons at different heights for given values of \(\dfrac{\Delta v}{v}\) and \(n\) (illustration). \(h_{\max}\) and \(h_{\min}\) are the heights at which the formation of the meteor trail respectively begins and ends.

On the Dependence of the Linear Density of Electrons in an Ionized Trail on the Mass of the Meteoroid

In solving a number of problems in meteor radar, one has to consider normal reflection of radio waves from meteor trails formed by meteoroids “piercing” a small area lying in the plane of the echo (i.e., oriented normally with respect to the direction toward the radar; Fig. 4).

In this case the question arises: how does the linear density of electrons \(\alpha\) change in the normally reflecting portion of the ionized trail if, at the same meteoroid velocity \(v\) and the same height of the reflecting portion \(h\) (above ground level), a meteoroid of a different mass appears?

The normally reflecting portion may be located at different heights \(h\), depending on the orientation of the meteor trail relative to the radar. It may turn out that \(h\) exceeds the characteristic height \(h_m\); cases \(h \simeq h_m\) and \(h < h_m\) may also occur (see Fig. 1).

Fig. 4. On the dependence \(q(v,m)\). \(L\) is the location point of the radar; \(P\) is the cross section of the radiation pattern by the plane of the echo, i.e., the plane passing through point \(L\), perpendicular to the direction toward the radar; \(\Delta S\) is an elementary area in the plane of the echo, at a distance \(R\) from the radar, through which meteoroids pass (1 and 2).

Will the transition in all these cases from a meteoroid with mass $m_1$ to a body with mass $m_2$ ($m_2>m_1$) lead to an increase in $a$ (at the point of normal reflection)?

This question arises because a larger mass of the meteoroid corresponds to more intense evaporation at the characteristic height; however, the characteristic height is then somewhat reduced.

This question also arises because the transition from velocity $v_1$ to $v_2$ ($v_2>v_1$), at the same meteoroid mass, leads to an increase in $a$ at $h>h_1$ and to a decrease in $a$ at $h<h_1$ ($h_{\min}<h_1<h_m$) (see above).

In order to determine how $a$ changes with changing $m$ (i.e., when comparing meteors that differ only in mass), let us find the dependence of $\Delta a$ on $\Delta m$. For this purpose we differentiate $a$ with respect to $m$:

\[ \frac{d a}{dm} = \frac{\partial a}{\partial a_{\max}}\cdot \frac{d a_{\max}}{dm} + \frac{\partial a}{\partial \left(\dfrac{p}{p_m}\right)} \cdot \frac{d\left(\dfrac{p}{p_m}\right)}{dm}. \tag{23} \]

As follows from (1) and (3),

\[ \frac{d a_{\max}}{dm}=\frac{a_{\max}}{m}, \tag{24} \]

\[ \frac{d\left(\dfrac{p}{p_m}\right)}{dm} = -\frac{1}{3}\cdot \frac{p}{p_m}\cdot \frac{1}{m}. \tag{25} \]

Substituting (24) and (25), as well as (14) and (15), into (23), we obtain

\[ \frac{d a}{dm} = \frac{a}{m} \left( 1-\frac{1}{3}\cdot \frac{1-\dfrac{p}{p_m}} {1-\dfrac{1}{3}\dfrac{p}{p_m}} \right), \tag{26} \]

whence

\[ \frac{\Delta a}{a} = \frac{\Delta m}{m}\cdot B, \tag{27} \]

where

\[ B = 1-\frac{1}{3}\cdot \frac{1-\dfrac{p}{p_m}} {1-\dfrac{1}{3}\cdot\dfrac{p}{p_m}}. \tag{28} \]

As follows from (27), the relative change in electron density

\[ \frac{\Delta a}{a} \]

is proportional to the relative change in mass. The coefficient $B$ is always positive. Indeed, as is seen from (28), for ...

\[ p \ll p_m B \simeq \frac{2}{3}, \quad \text{for } p \simeq p_m \quad B \simeq 1, \quad \text{for } p \simeq p_{\max} \simeq 3p_m \ \text{(see 28)} \quad B \to \infty . \]

The dependence

\[ B=\frac{\Delta a/a}{\Delta m/m} \]

as a function of \(\frac{p}{m}\) and \(h\) is illustrated in Fig. 5.

Fig. 5. On the dependence \(a(m)\).

\[ B=\frac{\frac{\Delta a}{a}}{\frac{\Delta m}{m}} \]

\(p\) is the pressure at height \(h\); \(p_m\) is the pressure at the characteristic height \(h_m\); \(h_{\max}\) and \(h_{\min}\) are the heights at which intense evaporation of the meteoric body respectively begins and ends.

Thus, at any heights \(h\) within the limits of the heights at which intense evaporation of the meteoric body appears and ceases, larger masses of meteoric bodies correspond to larger electron densities.

Conclusions

1. If the section that normally reflects radio waves is located in the region of the most intense ionization, i.e., at \(h \simeq h_m\) \((x \simeq x_{\max})\), then

\[ \alpha \sim v^n . \]

In the case of distant meteors, i.e., for \(h > h_m\) and \(p \ll p_m\),

\[ \alpha \sim v^{n+2}. \]

To determine the quantitative dependence of the linear electron density \(\alpha\) on the meteor velocity \(v\), it is necessary to refine the numerical value of the exponent \(n\), which is connected with refining the dependence of the ionization probability \(\beta\) on \(v\).

1. Larger masses of meteoric bodies, under identical conditions (identical \(v\), \(\chi\), etc.), correspond to larger linear electron densities at any point of the meteor trail.

References

1. T. R. Kaiser, Phil. Mag., 1953, 2, No. 8, 495.
2. S. Evans, M. N. R. A. S., 1954, 114, No. 1, 63.
3. G. S. Hawkins, Astroph. journ., 1956, 124, No. 1, 311.