On a Method for Determining Atmospheric Parameters in the Meteor Zone from Radar Observations of Meteors
Abstract Generated abstract
This paper discusses the determination of atmospheric temperature, pressure, and density in the meteor zone from radar observations of meteors. It reviews Evans’s method and identifies a major difficulty in its reliance on an uncertain dependence of meteor ionization probability on velocity, then proposes an alternative procedure based on Kaiser’s theory of meteor height distributions. The method derives the atmospheric scale height from the observed spread and mean heights of radar meteor echoes in velocity groups, uses it to infer temperature as a function of height, and obtains pressure and density through hydrostatic relations, with the lower boundary pressure supplied independently. The paper notes limitations due to dissociation, ionization, and boundary pressure accuracy, and recommends comparison with Evans’s method and direct measurements.
Full Text
On the Methodology for Determining Atmospheric Parameters in the Meteor Zone from Radar Observations of Meteors
E. I. Fialko
(Presented by the scientific seminar of the Radio Engineering Faculty)
Introduction
During the International Geophysical Year, certain organizations carried out systematic studies with the aim of investigating temperature, pressure, density, and other atmospheric parameters at high altitudes. A significant place was assigned to indirect methods and, in particular, to radar methods of meteor observation.
In this connection, the proposed methods deserve attention and, above all, Evans’s method [1].
Evans’s method, based on the theory of the distribution of meteor heights developed by Kaiser [2], consists of the following.
As the basic dependence of pressure \(p\) on height \(h\), the formula giving the pressure at the point of most intense evaporation of the meteoric body, \(p_{(\max)}\), is adopted:
\[ p_{(\max)} = 2 g Q \cdot \frac{\cos^2 \chi}{v^2}\,(9 \cdot H \cdot z_{\max}), \tag{1} \]
where \(v\) is the meteor velocity;
\(\chi\) is the zenith distance of the meteor radiant;
\(H\) is the height of the homogeneous atmosphere (corresponding to the characteristic height \(h_m\), i.e., the height at which the linear electron density is maximal — \(z_{\max}\) and \(p = p_{(\max)}\));
\(g\) is the acceleration due to gravity;
\[ Q = \frac{l}{\Lambda A}\left(\frac{\mu}{s}\right)^{1/3}, \]
where \(\mu\) is the mean mass of an atom of the meteoric body;
\(s\) is the probability of ionization;
\(\Lambda\) is the heat-transfer coefficient;
\(l\) is the heat of vaporization;
\(A\) is the shape coefficient;
\[ A = \lambda \cdot m_m^{2/3}; \]
\(m_m\) is the mass of the meteoroid body;
\(A\) is the cross-sectional area of the meteor (in the plane normal to the meteor trajectory).
Formula (1) gives the dependence of \(p_{\max}\) on \(h\) in implicit form. \(p_{\max}\) depends on the meteor velocity both explicitly and implicitly, since \(H\) and \(\zeta\) also depend on \(v\).
For each velocity interval of sporadic meteors, Evans found the mean (and then the characteristic) height and the corresponding height of the homogeneous atmosphere. Then, having made a number of assumptions (taking some mean value of \(\chi\), setting \(\alpha_{\max}\) equal to the minimum electron density corresponding to the apparatus parameters at the mean height and mean zenith angle, and finding that, in a first approximation, \(\zeta\) does not depend on velocity), Evans, substituting \(v\) and the corresponding values of \(H\) into (1), and knowing the dependence \(h_m(v)\), found the variation of pressure with height \(p(h)\).
Although the method described, like the dependence \(p(h)\) obtained by Evans, is undoubtedly of interest, it gives rise to a number of objections.
A particularly serious objection is raised by the assertion that the coefficient \(\zeta\), in a first approximation, does not depend on \(v\).
It is usually assumed that
\[ \zeta \sim v^n, \tag{2} \]
where different investigators arrive at completely different estimates of the magnitude of the coefficient \(n\): from 0 to 5.6 [1, 3, etc.].
At present it is unclear what the actual value of \(n\) is, but in all probability it is of the order of unity [8].
In this connection it seems advisable to use Kaiser’s works [2], which make it possible to find \(H\) for various heights of the meteor zone and, without making questionable assumptions, to pass to pressures and densities.
As for Evans’s method, it will undoubtedly have a significant effect after certain dependences and coefficients have been clarified and refined.
Basic relations
As is known [4, 5, 6], the height of the homogeneous atmosphere is equal to
\[ H = \frac{\kappa}{m_B g}\cdot T = \frac{R}{\mu_B \cdot g}\cdot T, \tag{3} \]
where \(T\) is the absolute temperature;
\[
\kappa = 1.37 \cdot 10^{-16}\ \frac{\mathrm{erg}}{\mathrm{grad}}
\]
is Boltzmann’s constant;
\(g\) is the acceleration of gravity;
\(m_B\) and \(\mu_B\) are, respectively, the mean molecular mass and the molecular weight of air.
\[
R = 8.31 \cdot 10^7\ \frac{\mathrm{erg}\cdot \mathrm{grad}}{\mathrm{mol}}
\]
is the gas constant.
The relation between the pressure \(p\), the density \(\rho\), and the height of the homogeneous atmosphere is expressed by the simple dependence
\[ p = g \cdot \rho \cdot H. \tag{4} \]
The pressure at height \(h\) is equal to
\[ p = p_0 e^{-\frac{m_{\mathrm{a}} g}{k}\int_0^h \frac{dh}{T}} = p_0 e^{-\int_0^h \frac{dh}{H}}, \tag{5} \]
where \(p_0\) is the pressure at height \(h=0\);
\(T\) is the temperature at height \(h\).
In view of the fact that meteor observations cover a limited zone \((h_1 < h < h_2)\), we shall write (5) in the form
\[ p = p_{h_1} e^{-\int_{h_1}^{h}\frac{dh}{H}}, \tag{6} \]
where
\[ p_{h_1}=p_0 e^{-\frac{m_{\mathrm{a}}g}{k}\int_0^{h_1}\frac{dh}{T}}. \tag{7} \]
The quantity \(p_{h_1}\), naturally, is not determined from radar observations of meteors.
The value of \(p_{h_1}\) may either be measured by a direct method, or found from photographic observations of meteors, or, finally, taken from standard data [7].
The dependence \(H(h)\) can be constructed from radar observations of meteors.
Kaiser [2] developed a theory of the height distribution of meteors observed by radar means for the case of normal reflection of radio waves from a meteor trail. Kaiser showed that the root-mean-square deviation from the mean height of the reflection points \(\delta h\) depends on the height of the homogeneous atmosphere and on the properties of the law of distribution of meteoric bodies by mass, i.e.,
\[ \delta h = f(H;s), \tag{8} \]
where
\[ F(m_{\mathrm{M}})\sim \frac{1}{m_{\mathrm{M}}^{s}} \]
is the law of distribution of meteoric bodies by masses \(m_{\mathrm{M}}\);
\(s\) is a constant.
Knowing the quantity \(s\) and finding \(\delta h\) from experiment, we determine \(H\).
The mean height \(\bar h\) is a function of the velocity \(v\).
Thus, it is easy to establish the relation \(H(h)\). For this purpose one may observe showers (with different velocities), or carry out observations of sporadic meteors, dividing them into groups with different velocities [1]. Then, for each interval of \(v\), the corresponding \(\bar h\) and \(\delta h\) are determined; from \(\delta h\), for a given value of \(s\), \(H\) is found [2], and in this way the dependence \(H(\bar h)\) is established. Passing from \(\bar h\) to \(h_m\) [2], one can construct \(H(h_m)\), i.e., find the variation of \(H\) as a function of height.
Sequence of operations
1) After the systematization of the radar-observation materials, \(\delta h\) and \(\bar h\) are determined for each velocity interval.
2) From \(\delta h\) and \(\bar h\), for a given value of \(s\), using the graphs given by Kaiser [2], there are found, respectively, the height of the homogeneous atmosphere \(H\) and the characteristic height \(h_m\), and \(H(h_m)\), that is \(H(h)\), is determined.
3) The variation of temperature with altitude is found from
\[ T(h)=H(h)\cdot \frac{m_k g}{k}. \tag{9} \]
4) The pressure in the region of the meteor zone is determined by
\[ p(h)=p_{h_1}\cdot e^{-\int_{h_1}^{h}\frac{dh}{H(h)}}. \tag{10} \]
5) From \(p(h)\) and \(H(h)\), the variation of atmospheric density is found:
\[ \rho(h)=\frac{p(h)}{g\cdot H(h)}. \tag{11} \]
Some remarks
To improve the accuracy of measuring \(H\), \(T\), \(p\), and \(\rho\), the presence of dissociation and ionization processes in the region of the meteor zone should be taken into account (consideration of this question, as well as an analysis of measurement accuracy, lies beyond the scope of the present work).
The accuracy of measuring \(p(h)\) and \(\rho(h)\) (by the method described in Sec. 2) depends to a considerable extent on the accuracy with which the pressure at the lower boundary of the meteor zone, \(p_{h_1}\), has been measured.
Thus, the overall accuracy of the measurements of \(p(h)\) and \(\rho(h)\) is determined not only by the accuracy of the meteor measurements, but also by the accuracy of those measurements as a result of which the value \(p_{h_1}\) was obtained.
Evans’s method is free of this shortcoming, since it makes it possible to measure the pressure in the region of the meteor zone independently of whether the parameters of the lower atmospheric layers are known or unknown.
However, in order to implement Evans’s method it is necessary to know the dependence of the ionization probability \(\beta\) on the meteor velocity, which is still unknown.
Conclusion
It is of interest to compare Evans’s method (as well as other indirect methods) with the method described in Secs. 2–3 of the present work, using experimental material and subsequently comparing it with the results of measurements of atmospheric parameters carried out by direct methods.
LITERATURE
- S. Evans. Scale Heights and Pressures in the upper Atmosphere from Radio-echo Observations of Meteors, Mon. Not. R.A.S., 1954, v. 114, No. 1, pp. 63—73.
- T. R. Kaiser. Theory of the Meteor Height Distribution obtained from Radio-echo Observations, I,II Mon. Not. R.A.S., 1954, v. 114, No. 1, pp. 39—62.
- G. S. Hawkins. Meteor Ionization and its Dependence on Velocity, Astrophys. J., 1956, v. 124, No. 1, pp. 311—313.
- V. L. Ginzburg. Theory of Radio-Wave Propagation in the Ionosphere, Ogiz, Gostekhizdat, 1949.
- S. K. Mitra. The Upper Atmosphere, I.L., 1955.
- B. Yu. Levin. Physical Theory of Meteors and Meteoritic Matter in the Solar System, Publishing House of the Academy of Sciences of the USSR, 1956.
- Ed. by G. P. Kuiper. The Earth as a Planet, the University of Chicago Press, 1954, pp. 500, 506.
- E. I. Fialko. Approximate estimate of the probability of meteor ionization, Astr. Zhurn., 1959, vol. 36, no. 3, 491.