On Selecting a Mode for Photographic Recording of Meteor Radio Echoes
Abstract Generated abstract
This paper analyzes how to choose the film transport speed for photographic recording of meteor radio echoes when the observations are used to estimate the exponent characterizing the meteoroid mass distribution. It derives approximate relations connecting the error in this exponent to errors in measuring radio echo duration, then relates the duration error to the recorded streak length, spot size, film width, screen diameter, and film speed. For typical recording parameters, the study estimates that a film speed of about 18 to 36 mm per minute gives echo duration errors of roughly 10 to 20 percent and permits determination of the mass distribution exponent with an accuracy of about 5 to 10 percent.
Full Text
On the Choice of a Photoregistration Mode for Meteor Radio Echoes
E. I. Fialko
(Presented by the scientific seminar of the Radio Engineering Faculty)
Introduction
In designing a radar intended for meteor observations, primary attention is usually devoted to the solution of radio-engineering problems; the questions of photoregistration of meteor radio echoes, however, are not subjected to careful analysis.
This may lead to a considerable reduction in the accuracy of measuring certain parameters of the meteor and of the atmosphere.
One of the essential characteristics of a photoregistering installation is the speed at which the photographic film is drawn, \(v_{\text{fl}}\).
The accuracy of measurements, the magnitude of the threshold signal (i.e., the sensitivity of the apparatus), film consumption, the promptness of processing, etc., depend on \(v_{\text{fl}}\).
The choice of the optimal film-drawing speed, ensuring a prescribed measurement accuracy while taking economic factors into account, may prove to be a difficult problem.
In the present work a special case is considered—the choice of \(v_{\text{fl}}\) in studying the mass distribution of meteoroid bodies.
As is known, one of the very effective methods for investigating the mass distribution of meteoroid bodies is based on constructing the distribution of radio-echo durations.
If the differential distribution of meteor masses \(m\) is represented in the form \(p(m)\sim m^{-s}\) (where \(s\) is a constant) [1], then the integral distribution of stable radio echoes will have the form
\[ N \sim \tau^{-\frac{3}{4}(s-1)}, \tag{1} \]
where \(N\) is the number of reflections with duration \(>\tau\) [2].
Thus, the coefficient \(s\) proves to be equal to [2]
\[ s = 1+\frac{4}{3}\, \frac{\lg \dfrac{N_2}{N_1}} {\lg \dfrac{\tau_1}{\tau_2}}, \tag{2} \]
where \(N_1\) (and \(N_2\)) is the number of reflections with durations \(>\tau_1\) (and \(>\tau_2\)), recorded during the observation time.
The accuracy of measuring \(s\), in particular, depends on the error allowed in counting \(\tau\), which depends on \(v_{n.l.}\).
Error in measuring the exponent \(s\)
As is not difficult to show, starting from (1), the relative error \(\dfrac{\Delta s}{s}\) will be equal to
\[ \frac{\Delta s}{s} \approx \left(1-\frac{1}{s}\right) \left\{ \frac{\dfrac{\Delta N_2}{N_2}-\dfrac{\Delta N_1}{N_1}} {\lg \dfrac{N_2}{N_1}} + \frac{\dfrac{\Delta \tau_2}{\tau_2}-\dfrac{\Delta \tau_1}{\tau_1}} {\lg \dfrac{\tau_1}{\tau_2}} \right\}, \tag{3} \]
where \(\Delta N_2\) and \(\Delta N_1\) are the errors with which \(N_2\) and \(N_1\) are counted, and \(\Delta \tau_1\) and \(\Delta \tau_2\) are the errors with which the durations \(\tau_1\) and \(\tau_2\) are counted.
Taking (1) into account, formula (3) can be represented in the form
\[ \frac{\Delta s}{s} \approx \left(1-\frac{1}{s}\right) \left\{ \frac{\dfrac{\Delta N_2}{N_2}-\dfrac{\Delta N_1}{N_1}} {\dfrac{3}{4}(s-1)\lg \dfrac{\tau_1}{\tau_2}} + \frac{\dfrac{\Delta \tau_2}{\tau_2}-\dfrac{\Delta \tau_1}{\tau_1}} {\lg \dfrac{\tau_1}{\tau_2}} \right\}. \tag{4} \]
Without carrying out a detailed analysis of (4), we shall confine ourselves to considering the special case in which the relative error in measuring the duration is much greater than the relative error in counting the number.
Taking also into account that, for \(\Delta \tau_1 \simeq \Delta \tau_2\) and \(\tau_2 > \tau_1\), one may assume
\[ \left| \frac{\Delta \tau_2}{\tau_2} - \frac{\Delta \tau_1}{\tau_1} \right| \simeq \frac{|\Delta \tau|}{\tau_1}, \]
we obtain (omitting the subscripts at \(\tau\))
\[ \frac{\Delta s}{s} \approx \left(1-\frac{1}{s}\right) \frac{1}{\left|\lg \dfrac{\tau_1}{\tau_2}\right|} \cdot \frac{|\Delta \tau|}{\tau}. \tag{5} \]
As follows from (5), in order to reduce the error in measuring \(s\), one should increase the accuracy of measuring the duration of the echo signal and increase the interval \(\tau_1 : \tau_2\) within which the dependence \(N(\tau)\) is approximated. However, the lower limit \(\tau_{\min}\) is restricted by the region of stable echoes; the upper limit \(\tau\) is also limited, since the peculiarities of long-duration reflections are not taken into account by dependence (1).
The distribution \(N(\tau)\) often has a break in the region \(\tau \simeq 10\) sec (at wavelength \(\lambda = 10\) m) [3].
As a result, the ratio \(\dfrac{\tau_{\max}}{\tau_{\min}}\) is limited and is of the order of 10.
Remark. Here we do not consider questions of the accuracy of measuring \(s\) that are associated with the fluctuating character of long-duration echoes.
As follows from (5), the accuracy of determining \(s\) (with other conditions being equal) increases with increasing \(s\). Taking \(s \simeq \bar{s} \simeq 2\) [3] and \(\tau_{\max}/\tau_{\min} \simeq 10\), we find approximately
\[ \frac{|\Delta s|}{s} \simeq 0.5\,\frac{\Delta \tau}{\tau}. \tag{6} \]
Thus, the accuracy of determining the index \(s\) depends mainly on the error in measuring the duration of the radio echo \(\tau\).
Error in measuring the duration of the echo signal
There are several methods of recording echo signals and, correspondingly, several methods of measuring their duration.
With amplitude marking, the duration of a reflection is usually measured at a certain level relative to the greatest value of the echo-signal amplitude [4, etc.]. With brightness indication and amplitude limiting of the signal recorded on a continuously (slowly) moving film, the duration \(\tau\) is measured by a photometric method [5, etc.].
In the case of brightness marking and a sufficient film transport speed, the duration of the echo signal can be measured by the length of the “dash” [6, 7, etc.].
The formulas obtained above for estimating the error \(\Delta s/s\) are valid for any of these methods.
Here we shall dwell on the choice of the film speed only for the third method of photographic recording.
If the length of the streak corresponding to the echo signal is equal to \(l\) (Figs. 1 and 2), and the diameter of the spot on the film is equal to \(\delta\), then the duration of the echo signal \(\tau\) will be
\[ \tau = \frac{l-\delta}{v_{\mathrm{pl}}}. \tag{7} \]
However, it is quite obvious that under real conditions the diameter of the spot cannot be measured exactly, and therefore the error in measuring \(\tau\) will be
\[ |\Delta \tau| \simeq \frac{1}{n}\cdot \frac{\delta}{v_{\mathrm{pl}}}, \tag{8} \]
where \(n\) is a coefficient of the order of several units.
If a screen diameter \(D\) is projected onto a film of width \(L\), then for a light spot (on the screen) with diameter \(d\), we obtain
\[ \delta = d\cdot \frac{L}{D}. \tag{9} \]
Substituting (9) into (8), we obtain
\[ |\Delta \tau| \simeq \frac{1}{n}\cdot \frac{d}{D}\cdot \frac{L}{v_{\mathrm{pl}}}. \tag{10} \]
The relative error in measuring the duration will be equal to
\[ \frac{|\Delta \tau|}{\tau} \simeq \frac{a}{v_{\text{f}}}, \tag{11} \]
where
\[ a=\frac{dL}{nD\tau}. \tag{12} \]
Thus, in order to increase the accuracy of measuring \(\tau\) (in the given range of \(\tau\)), one should increase the film speed, improve the focusing of the beam on the indicator screen (which reduces \(\frac{d}{D}\)), measure the size of the spot more carefully (which increases \(n\)), and also use narrow film.
Fig. 1. Schematic representation of photographic recording of a meteor radio echo. \(\Phi\)—photographic film; \(R\)—axis of slant range; \(t\)—time axis; \(M\)—recording of the echo signal (extent \(l\) mm).
An excessive increase in \(v_{\text{f}}\) leads to increased consumption of film and complicates frame processing. In this connection it is expedient to choose the minimum film-transport speed at which the required accuracy of measuring \(s\) is achieved.
Choice of \(v_{\text{f}}\)
As follows from (6) and (11), in order to measure the index \(s\) with a prescribed accuracy, it is necessary to transport the film at a speed
\[ v_{\text{f}} \geq \frac{0.5 \cdot a \cdot s}{|\Delta s|}\ \frac{\text{mm}}{\text{sec}}. \tag{13} \]
For
\[ \frac{d}{D}\simeq \frac{1}{200},\quad L=25\ \text{mm},\quad n\simeq 2,\quad \text{and}\quad \tau\simeq 1\ \text{sec}, \]
the coefficient \(a\), as follows from (12), will be equal to
\[ a\simeq 0.06\ \frac{\text{mm}}{\text{sec}}, \]
and condition (13) will take the form
\[ v_{\text{pl.}} > 0.03\,\frac{s}{|\Delta S|}\,\frac{\text{mm}}{\text{sec}} = 1.8\,\frac{s}{|\Delta S|}\,\frac{\text{mm}}{\text{min}} . \tag{14} \]
To ensure a measurement accuracy of \(\dfrac{\Delta s}{s} \simeq 5 : 10\%\), it is necessary to have \(v_{\text{pl.}}\), as follows from (14),
\[ v_{\text{pl.}} \simeq 18 : 36\ \text{mm/min}. \]
Fig. 2. On the determination of the error in measuring the duration of a meteor radio echo. \(l\)—length of the “dash” in the recording of the meteor radio echo; \(\delta\)—size (diameter) of the spot image on the film; \(l'\)—size corresponding to the true duration of the reflection.
In this case the reflection durations \(\tau\) will be measured, as is evident from (11) and (12), with an error
\[ \frac{|\Delta \tau|}{\tau} = \frac{0.06 \cdot 60\ \text{mm/min}}{18 \div 36\ \text{mm/min}} \simeq 10 : 20\%, \]
which is in agreement with relation (6).
Conclusions
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The accuracy of measuring the coefficient \(s\) (which characterizes the distribution of meteor bodies by mass) and the accuracy of measuring the durations \(\tau\) of meteor radio echoes depend substantially on the film-pull speed \(v_{\text{pl.}}\).
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For \(v_{\text{pl.}} \simeq 18 : 36\ \text{mm/min}\), the error in measuring the echo durations is \(\simeq 10 : 20\%\), and the index \(s\) can be measured with an accuracy of
\[ \frac{\Delta S}{s} \simeq 5 : 10\%. \]
References
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T. R. Kaiser, “Theory of the meteor height distribution obtained from radio-echo observations,” M. N. R. A. S., 1954, v. 114, No. 1, pp. 39–62.
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E. I. Fialko, “Distribution of Stable Meteor Radio Echoes by Duration,” Izvestiya TPI, 1960, vol. 105, p. 14.
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E. I. Fialko, “Some Results of an Investigation of the Mass Distribution of Meteor Bodies,” Astr. Tsirk., 1958, No. 195, p. 22.
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J. S. Greenhow, E. L. Neufeld, “The diffusion of ionized meteor trails in the upper atmosphere,” Journ. Atm. Terr. Phys., 1955, v. 6, No. 2–3, p. 133.
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K. V. Kostylev. Apparatus of the Engelhardt Astronomical Observatory for radar observations of meteors with automatic recording. Astronomical Journal, 1958, 35, issue 4, p. 643.
-
K. R. Bowden, J. G. Davies. “The time distribution of meteors,” J. Atm. Terr. Phys., 1957, v. 11, No. 1, pp. 62–66.
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E. I. Fialko, F. I. Peregudov, E. K. Nemirova. Preliminary results of radar observations of meteors at $\lambda = 10$ meters, Bulletin of the Commission on Comets and Meteors of the Astronomical Council of the Academy of Sciences of the USSR, No. 2, pp. 39–43, 1958.