ERRORS IN MEASURING METEOR VELOCITIES BY THE SPACED-RECEPTION METHOD

E. K. Nemirova

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

In installations operating by the spaced-reception method, one transmitter and three receivers are usually used; one of the receivers is located together with the transmitter, while the other two are placed at a distance on the order of several kilometers. In such systems it is possible to measure meteor velocities even in the case where the meteor does not give a clear diffraction pattern.

By measuring the range at three receiving points, one can find the positions on the trail of three reflecting points \(O_1\), \(O_2\), and \(O_3\). In this case

\[ v = \frac{O_1 O_2}{\tau_1} = \frac{O_2 O_3}{\tau_2}, \]

where \(\tau_1\) (\(\tau_2\)) is the time taken by the meteor to traverse the distance \(O_1 O_2\) (\(O_2 O_3\)). Thus, velocity measurement is reduced to measuring the flight time \(\tau\) between two reflecting points. The time \(\tau\) can be measured by fixing some point on the diffraction pattern, for example, the moment of the amplitude maximum, which is usually clearly expressed.

In some systems using a remote receiving point, knowledge of the flight time is necessary for determining the position of the trail in space (the direction toward the radiant).

Installations for measuring velocities and determining meteor radiants by the spaced-reception method are used mainly for measurements on weak meteors, i.e., meteors of the nonpersistent type. As is known, resonance phenomena are possible in trails of this type [1]. Resonance is accompanied by a shift (delay) of the diffraction pattern.

The magnitude of the delay is determined by the reflection conditions (the electron density \(a\), the diffusion coefficient \(D\), the magnitude of the initial radius \(r_n\), the range \(R\), the wavelength \(\lambda\), etc.). In the case where the reflecting points are located in different parts of the trail, these conditions may be different; in particular, there may be considerable differences in the values of the electron densities \(a\) [2]. Owing to the different heights of the reflecting points, the diffusion coefficients and initial radii may be unequal. This, in turn, will lead to inequality of the delay times and, consequently, to an error in measuring the flight time, i.e., the velocity.

In estimating this error, we shall restrict ourselves to the plane problem (Fig. 1; the trail lies in one plane with the baseline \(a\)). When the baseline \(a\) is several kilometers,

\[ O_1O_2 \simeq \frac{a\sin\chi}{2}. \]

Fig. 1. Reflection geometry for spaced reception (plane case).

As calculations by the method set forth in [3] have shown, the magnitude of the lag \(\Delta t\) depends on the ratio of the length of the resonating section of the trail \(l_{\mathrm{rez}}\) to \(\sqrt{R\lambda}\). For \(r_n \simeq 0\) this dependence is well approximated by the expression

\[ \frac{\Delta t}{T_1} \simeq 0.78\left(1-e^{-l_{\mathrm{rez}}/\sqrt{R\lambda}}\right). \]

For \(r_n \ne 0\),

\[ \frac{\Delta t}{T_1} \simeq 0.78\left[\left(1-e^{-l_{\mathrm{rez}}/\sqrt{R\lambda}}\right)-r_n^{1.08}\cdot 10^{-2}\right], \]

where \(r_n\) is in cm.

\(T_1\) is the magnitude of the first period of the ideal diffraction pattern,

\[ T_1 \simeq 0.62\sqrt{R\lambda}/v. \]

The velocity is

\[ v=\frac{O_1O_2}{\tau}, \]

and the relative error of measuring \(v\) is

\[ \delta v=-\frac{\delta\tau}{1+\delta\tau}, \]

where

\[ \delta\tau=\frac{\Delta\tau}{\tau} \qquad\text{and}\qquad \Delta\tau=\Delta t_2-\Delta t_1. \]

Taking into account that

\[ l_{\mathrm{rez}}=B\frac{n\lambda^2}{\pi^2D}\ [3], \]

where \(B\simeq 0.65\cdot 10^{-13}\) cm,

\[ \delta\tau \simeq 0.98\,\frac{\sqrt{R\lambda}}{a\sin\chi} \left[ e^{-B\frac{n_1\lambda^2}{\pi^2D_1}\cdot \frac{v}{\sqrt{R\lambda}}} \left( 1- e^{-B_1\frac{\lambda^2 v}{\pi^2\sqrt{R\lambda}} \left(\frac{n_2}{D_2}-\frac{n_1}{D_1}\right)} \right) - \left(r_{n_1}^{1.08}-r_{n_2}^{1.08}\right)\cdot 10^{-2*} \right]. \]

The difference in electron density, diffusion coefficients, and initial radii at the reflecting points is determined by the difference in the heights of these points,

\[ \Delta h=\frac{a\sin\chi}{2}\cdot\cos\chi, \]

as well as by the height and the position on the trail of one of them relative to the point of maximum ionization.

\[ \text{* The formula is applicable in the case when } n_1 \text{ and } n_2 \ll 2.4\cdot 10^{12} \text{ electrons per cm.} \]

The error \(\delta\tau\) reaches its maximum value in the case when the meteor trail near one of the reflecting points is in a state of resonance, while near the other point resonance is impossible. Such conditions may arise because of a considerable difference in electron densities, when at one of the reflecting points the trail is stable, and at the other unstable. Such an anomalous difference in electron densities (\(\sim 10\) times) with a small difference in heights (\(\sim 2.5\ \text{km}\)) was often observed in the experiments of Greenhow and Neufeld [2].

The magnitude of the maximum error is

\[ \delta\tau_{\max} \simeq \frac{\sqrt{R\lambda}}{a\sin\chi} \left[(1-e^{-l_{\mathrm{rez}}/\sqrt{R\lambda}})-r_n^{1.08\cdot 10^{-2}}\right]; \]

here \(l_{\mathrm{rez}}\) and \(r_n\) correspond to the point with the smaller electron density. \(\delta\tau_{\max}\) is close to its greatest value at \(r_n \simeq 0\) and

\[ l_{\mathrm{rez}}/\sqrt{R\lambda} \simeq 3 \quad (\text{Fig. 2}). \tag{1} \]

Fig. 2. Dependence of \(\Delta t/T_1\) on \(l_{\mathrm{rez}}/\sqrt{R\lambda}\)

Physically, these conditions mean that the point with the smaller electron density lies at a small height, significantly less than the characteristic one, while \(a\) is sufficiently large (remaining \(\ll 2.4\cdot 10^{13}\ \text{cm}^{-1}\)). Since \(l_{\mathrm{rez}}\sim \lambda^2\), conditions (1) should be encountered more often in observations at longer wavelengths.

From Fig. 2, \(\Delta t_{\max}/T_1 \simeq 0.75\); hence

\[ \delta\tau_{\max\max} \simeq 0.93\cdot \frac{\sqrt{R\lambda}}{a\sin\chi}. \]

The magnitude of the greatest value of the error \(\delta\tau_{\max}\) for different baselines is shown in Fig. 3. As can be seen from the figure, the error for small baselines (\(\sim 2\div 4\ \text{km}\)) reaches several tens of percent. However, as the baseline is decreased, the probability of a sharp difference in electron densities decreases. For trails in which the electron density changes with height in accordance with the ionization curve [2], the error depends on the difference in the heights of the reflecting points and differs in each of the following three cases:

First case — both points lie near the characteristic height \(h_m\).

Second case — the heights of both points are greater than \(h_m\).

Third case — the heights are less than \(h_m\).

The dependence of \(\delta\tau\) on the base \(a\) for a number of cases is shown in Fig. 4. The error was calculated for \(R=200\ \text{km}\), \(\lambda=10\ \text{m}\), \(\chi=45^\circ\), \(v=35\ \text{km/sec}\) \((h_m=92\ \text{km})\).

The most significant cause of the error is the change in electron density along the trail. Therefore, in the first case, when both points lie near the characteristic height (\(\alpha\) changes little in this case), the error is insignificant.

When \(h_1\) and \(h_2>h_m\), the error \(\delta\tau\) amounts to several percent (\(\leq 10\%\)) and is almost independent of the magnitude of the base and the height, increasing somewhat with increasing \(\alpha\).

Fig. 3. Largest value of the error \(\delta\tau_{\max}\) as a function of the magnitude of the base \(a\) for \(R=200\ \text{km}\), \(\lambda=10\ \text{m}\), and \(\chi=45^\circ\).

Fig. 4. Error \(\delta\tau\) as a function of the base for several particular cases:
1 — \(h_1=90\ \text{km}\), \(D_1=2.2\ \text{m}^2/\text{sec}\), \(\alpha_1=3\cdot10^{11}\ \text{cm}^{-1}\);
2 — \(h_1=88\ \text{km}\), \(D_1=1.58\ \text{m}^2/\text{sec}\), \(\alpha_1=3\cdot10^{11}\ \text{cm}^{-1}\);
3 — \(h_1=100\ \text{km}\), \(D_1=10\ \text{m}^2/\text{sec}\), \(\alpha_1=3\cdot10^{11}\ \text{cm}^{-1}\);
4 — \(h_1=102\ \text{km}\), \(D_1=13.3\ \text{m}^2/\text{sec}\), \(\alpha_1=10^{11}\ \text{cm}^{-1}\).

When \(h_1\) and \(h_2<h_m\), the error increases with increasing base; the magnitude of the error is the greater, the lower the reflecting points are located. This is explained by the sharper change of \(\alpha\) with height at the beginning of the trail.

For meteors with high velocities, the length of the resonant section is greater (other conditions being equal), and therefore \(\delta\tau\) should be greater.

The error decreases sharply as the wavelength decreases, since in this case the magnitude of the delay \(\Delta t\) decreases sharply.

When working in the long-wavelength part of the meteor range (\(\lambda\sim 8\div10\ \text{m}\)), the error is significant; it can be eliminated, when observing a stream, by ensuring parallel polarization, and, when observing the sporadic background, by rejecting reflections with perpendicular polarization.

REFERENCES

1. T. R. Kaizer, R. L. Closs, Philos. Mag. 43, 1, 1952.
2. J. S. Greenhow, E. L. Neufeld, Mon. Not. RAS, 117, No. 4, 1957.
3. Nemirova E. K. On the role of resonance effects in measuring meteor velocities, Astron. Zhurnal, vol. XXXVI, issue 3, 1959.