Smithsonian

Contributions to Astrophysics

VOLUME 5, NUMBER 1

THE METEORIC HEAD ECHO

by ALLAN F. COOK AND GERALD S. HAWKINS

SMITHSONIAN INSTITUTION

Washington, D. C.

1960

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Publications of the Astrophysical Observatory

This series, Smithsonian Contributions to Astrophysics, was inaugurated in 1956 to provide a proper communication for the results of research conducted at the Astrophysical Observatory of the Smithsonian Institution. Its purpose is the “increase and diffusion of knowledge” in the field of astrophysics, with particular emphasis on problems of the sun, the earth, and the solar system. Its pages are open to a limited number of papers by other investigators with whom we have common interests.

Another series is Annals of the Astrophysical Observatory. It was started in 1900 by the Observatory’s first director, Samuel P. Langley, and has been published about every 10 years since that date. These quarto volumes, some of which are still available, record the history of the Observatory’s researches and activities.

Many technical papers and volumes emanating from the Astrophysical Observatory have appeared in the Smithsonian Miscellaneous Collections. Among these are Smithsonian Physical Tables, Smithsonian Meteorological Tables, and World Weather Records.

Additional information concerning these publications may be secured from the Editorial and Publications Division, Smithsonian Institution, Washington, D. C.

FRED L. WHIPPLE, Director,

Astrophysical Observatory,

Smithsonian Institution.

Cambridge, Mass.

II

Contents of Volume 5

No. 1. The Meteoric Head Echo, by Allan F. Cook and Gerald S. Hawkins. Pages 1–7. Published April 8, 1960.

No. 2. The Upper Atmosphere and Satellite Drag, by R. J. Stirton. Pages 9–15. Published March 22, 1960.

No. 3. On the Structure of the Sunspot Zone, by Barbara Bell. Pages 17–28. Published March 28, 1960.

No. 4. Observations of Simulated Meteors, by Richard E. McCrosky. Pages 29–37. Published February 28, 1961.

No. 5. On the Motion of Satellites with Critical Inclination. Pages 39–58. Published May 15, 1961. Libration of an Earth Satellite with Critical Inclination, by Yusuke Hagihara (pages 39–51); Motion of a Particle with Critical Inclination in the Gravitational Field of a Spheroid, by Yoshihide Kozai (pages 53–58).

No. 6. Gaps in the Distribution of Asteroids, by Yusuke Hagihara. Pages 59–67. Published May 10, 1961.

No. 7. Major Flares and Geomagnetic Activity, by Barbara Bell. Pages 69–83. Published May 10, 1961.

No. 8. An Annotated Bibliography on Interplanetary Dust, by Paul W. Hodge, Frances W. Wright, and Dorrit Hoffleit. Pages 85–111. Published May 22, 1961.

No. 9. Rotation of an Earth Satellite in Flight along its Orbit, by Yusuke Hagihara. Pages 113–143. Published September 28, 1961.

No. 10. Sampling Dust from the Stratosphere, by Paul W. Hodge. Pages 145–152. Published August 8, 1961.

No. 11. On the Δ and Φ Operators of Radiative Transfer Theory, by S. S. Kumar. Pages 153–185. Published Feburary 21, 1962.

No. 12. North-South Asymmetry in Solar Spottedness and in Great-Storm Sources. Pages 187–208. Published September 24, 1962. A Long-Term North-South Asymmetry in the Location of Solar Sources of Great Geomagnetic Storms, by Barbara Bell (pages 187–194); On the Unequal Spottedness of the Two Solar Hemispheres, by John G. Wolbach (pages 195–202); On Short-Period Relations Between North-South Asymmetry in Spottedness and in Great-Storm Sources, by Barbara Bell and John G. Wolbach (pages 203–208).

No. 13. Neutral Hydrogen Between Galactic Longitudes 200° and 265°, by R. J. Davis. Pages 209–230. Published September 24, 1962.

No. 14. The Space Density of Atmospheric Dust in the Altitude Range 50,000 to 90,000 Feet, by Paul W. Hodge and Frances W. Wright. Pages 231–238. Published December 4, 1962.

No. 15. Solar Radio Bursts of Spectral Types II and IV: Their Relations to Optical Phenomena and to Geomagnetic Activity, by Barbara Bell. Pages 239–257. Published April 24, 1963.

Contents

Publications of the Astrophysical Observatory

The Meteoric Head Echo

Observational material

Ultraviolet light hypothesis

Solution of the kinetic equation

Application to the observations

Conclusions and discussion

References

Abstract

The Meteoric Head Echo[1]

By ALLAN F. COOK[2] AND GERALD S. HAWKINS[3]

Radar observers have in certain instances detected an echo associated with the head of a meteor. The head echo has been observed at a frequency of 30 mc by McKinley and Millman (1949) and at a frequency of 60 mc by Hey, Parsons, and Stewart (1947). Apart from a general description of the phenomenon very little information has been published on head echoes and no statistical analysis is available that describes their general properties. We may surmise that head echoes are a relatively rare phenomenon and that they are associated with bright meteors. The target that reflects the echo is almost spherical in shape because it usually gives no marked increase in signal strength as it crosses the line of sight. The target cross section is large—many hundreds of meters—whereas the solid meteor itself cannot be more than a few centimeters in diameter. From this we are led to postulate that a coma of ionization surrounding the meteor particles produces the reflection.

An alternative explanation was put forward by Browne and Kaiser (1953), who suggested that the head echo was produced by the discontinuity at the end of the extending cylinder of ionization which the meteoroid was generating. This hypothesis predicts an increase in echo strength of the order of 1000 as the meteoroid passes the minimum range position. Such an increase does not occur, however, and the theory is therefore disproved (McKinley, 1955).

McKinley and Millman (1949) have suggested a hypothesis to explain the coma of ionization: that the meteoroid acts as a source of intense ultraviolet radiation which ionizes an extensive region surrounding the meteoroid. The approximately spherical shape of the coma is maintained by recombination of the ionized particles after passage of the meteoroid. The present paper is an attempt to examine the ultraviolet-light hypothesis on a quantitative basis.

[1] The research in this document was supported Jointly by the U.S. Army, Nary, and Air Force under contract with the Massachusetts Institute of Technology.

[2] Harvard College Observatory, Cambridge, Mass.

[3] Boston University, Boston, Mass., and Harvard College Observatory.

Observational material

McKinley (1955, table 1) has estimated the strength of the head echo for two meteors, both of which occurred during the daytime.

Meteor I was observed from a single station only and was assumed to be a Perseid with a radiant at R.A. 46°, dec +58°. Its maximum visual magnitude is inferred from the duration of the echo by use of the correlation between brightness and duration determined by Millman (1950). McKinley showed that the difference in heights between the beginning and end points was 53 km. Comparison with photographic measurements makes it reasonable to assume that the end height was 70 km and the beginning height was 123 km. On this scale the head echo reached maximum intensity at a height of 77 km. Meteor II was observed simultaneously at three separate stations, so that its trajectory could be accurately determined.

The magnitudes of the meteors were calculated from the following equations (Greenhow, 1952; Hawkins, 1955):

where q is the electron line density in the trail, D is the diffusion coefficient of the atmosphere, ne is the critical electron density for the wavelength of the radar, t is the duration of the backscatter echo from the meteor trail, and M is the visual magnitude of the meteor.

The parameters of the radar equipment used by Millman and McKinley at Ottawa are as follows:

G: Antenna gain over an isotropic radiator, 3.0.

є: Noise level in the receiver, 2×10–14 watts.

P: Transmitter power, 3×10^5 watts.

λ: Radar wavelength, 9.2 meters.

ne: Critical electron density, 1.28×10^13m–3

The echo power p given by a sphere of unit reflection coefficient and radius a at a range R is given by the radar formula:

Using McKinley’s published estimates of echo strength and the parameters given above, we can calculate the radius of the ionized sphere from equation (3), as shown in figure 1.

For Meteor II, observations on echo strength were made from two stations, at Ottawa and at Arnprior. The peaks of the curve for Meteor II, at 47.25 and 48.75 seconds, correspond to the times at which the meteor was passing the minimum range position for the two stations. These peaks are probably due to an elongation of the surface of critical electron density in the direction of the meteor. The discrepancy in the diameters as measured from Ottawa and Arnprior is probably due to an error in the estimated sensitivities of the equipments.

FIGURE 1.—The radius of the ultraviolet coma as a function of time.

Ultraviolet light hypothesis

We will assume that the reflection of radio energy occurs at the surface where the electron density is equal to the critical density ne for the radar. The problem is then to determine the shape of this surface of constant density, or isosurface. The equation of radiative transfer in an elementary volume may be written in terms of distance, ρ, from the center of the meteoric coma as follows:

where Ivdvdω is the intensity of the radiation through the solid angle dω between frequency v and v+dv; σv is the absorption cross section; n is the number of molecules per unit volume; and Sv is the source function (Chandrasekhar, 1950). We assume that the oxygen molecule is the one ionized.

Let us choose a coordinate system at rest with respect to the atmosphere. The kinetic equation for the problem may be written in the form

where ne is the number of electrons (or ions) per unit volume, t is time, h is Planck’s constant, v is the frequency of the radiation, v_0 is the threshold frequency for ionization, dω is an elementary solid angle, α is the volume recombination coefficient of the ions, and the symbol ∮ denotes integration over all directions. The term under the integral sign indicates that within a given solid angle, dω, the intensity of photons (number per cm^2) produces ionizations at a rate n(σvIv/hv) over the range of frequency, dv.

Finally, if n_0 is the initial number of oxygen molecules per unit volume before passage of the meteor, the conservation of particles requires that

where n'0 is the number of oxygen atoms released per unit volume by dissociative recombination.

The following approximations may reasonably be made and should be quite accurate:

The first expression is justified because ne≃ne≪n_0; that is to say, a small fraction of the oxygen molecules is ionized. The source function, Sv, may be neglected because the main recombination process of the ionized molecule involves the dissociation of the molecule which does not produce ultraviolet radiation. These approximations permit the integration of equation (4), to give the expression

where dv dω is the intensity along the direction of integration in the absence of absorption. The average intensity, Jv, is given by the integration

where denotes the average intensity at unit distance from the meteor in the absence of absorption. We have assumed that ρ is large when compared with the radiating coma of the meteor. Thus equation (9) in effect corresponds to the case of a point source. Substitution of this expression and n=n_0 into equation (5) yields the equation,

We will now introduce a mean absorption cross section σ defined as follows:

where is the average photon intensity at unit distance from the meteor in the absence of absorption:

This is often called the “gray” approximation, and is valid when the medium is optically thin.

In addition to substituting approximation (11) into the kinetic equation (10), we shall find it convenient to use a frame of reference that moves with the meteor. The time derivative in this new frame of reference is given by the equation,

where cylindrical coordinates are used; r is the distance from the meteor path, V is the velocity of the meteoroid, and z is the distance along the path, taken positive in front of the meteor. It is reasonable to assume that the steady-state approximation holds, that is to say . The kinetic equation then becomes

Solution of the kinetic equation

A general solution of the kinetic equation (14) is not known. The best that can be done is to treat various limiting cases; in particular, those in which one of the three terms is small compared to the other two.

Case I: Nearly steady state.

Neglecting the term, ∂ne/∂z, we may write the lowest order solution as:

Note that equation (15) involves only the distance from the meteoroid ρ. A solution of the form

is now sought. The following result is obtained:

where a change has been made to polar coordinates ρ and θ,

So long as γ is small, the term with γ as a factor indicates that the surfaces of equal density, ne, are spheres with centers along the z axis. The third term with γ^2 as a factor in these surfaces introduces an elongation. As γ becomes very small these surfaces can be represented as ellipsoids. We may now evaluate the distances ρ(0), ρ and ρ(π) at which ne=ne.

It is now necessary to substitute the expressions

into equations (20), (21), and (22) and solve for Δ_1ρ and Δ_2ρ. We neglect Δ_2ρ in the terms involving γ and both Δ_1ρ and Δ_2ρ in terms involving γ^2. The solutions are:

The corresponding surface where ne=ne is a prolate ellipsoid for small values of γ with a ratio of major to minor axes of 1+Δ_2ρ/ρ(π). The ratio of power reflected laterally to that reflected axially will exceed unity by an amount we shall call E. Then it follows that

Case II: Small recombination rate.

If we neglect the term , the lowest order solution is given by the equation,

where we have applied the boundary condition that ne(0)(r, z) vanishes as z becomes positively infinite. Solutions in closed form can be derived only in the limits, as the optical distance from the meteor goes either to zero or to infinity. The latter case is of no current interest, so we shall consider only the former, for which we have the solution:

or, in terms of ρ and θ,

The corresponding solution for the surface where the electron density equals the critical density for the radar is:

The first approximation can be derived in the usual way by substitution of solution (27) into the neglected term, whence by integration we obtain the equation,

The maximum value of ne occurs at zm, where

at which point the approximation is breaking down. The corresponding value of θ is θm where

This equation shows that all contours of constant electron density have a maximum width at an angle θ=θm as indicated in figure 2.

FIGURE 2.—Coordinate system (ρ, θ) and regions of solution for Case II and Case III.

Case III: Small photo-ionization rate.

Integration is straightforward and exact, and yields the equation:

where F(r) is a function of z to be determined from the initial conditions. This solution applies to the tail of the reflecting surface as shown in figure 2. If we ignore a transition zone and use a boundary at θ=θm then the initial conditions are given by Case II, equation (27). These assumptions determine F(r), and the density, ne, is then given by the expression

where zm≡r cot θm.

Combination of Cases II and III: An isosurface of electron density is found in the range 0≤θ≤θm from equation (29). From equation (34) it follows that in the range θm≤θ≤π the isosurface is given by the equation:

Equations (29), (32), and (35) apply only in the limit as σn_0ρ goes to zero.

Application to the observations

Meteor I appears to be best treated as an example of Case I, where the shape of the target is a prolate ellipsoid, elongated in the direction of motion of the meteor. We have, from equation (21), for the radius of curvature of the ends of the ellipsoid, the expression:

The ratio of axial to lateral power reflected from the meteor head has been defined in Case I as (1+E) where from equation (25) we see that

From equations (36) and (37) we may eliminate the exponential term to obtain:

which yields the equation,

For Meteor I we deduce from figure 1 that at 19h 8m E.S.T., the radius of curvature of the ends of the ellipsoid ρ=10^4 cm. From the change in signal strength as the meteor crossed the line of sight we deduce E= 0.3. We may take V= 6.0×10^6 cm sec–1 and n_0=1.5×10^13cm–3. As extreme assumptions we may take σ= 0 or 2.75×10–17cm^2, the maximum value found by Weissler and Lee (1952). The corresponding limits on the optical distance are 0≤σn_0ρ≤ 8.25 where ρ has been doubled to allow for refraction (Manning, 1953). From equation (39) we find that the recombination coefficient α is approximately 3×10–5cm^3sec–1.

Meteor II appears to be best treated as an example of Cases II and III combined. The curvature of the isosurface parallel to the direction of flight is given by the equation,

while the radius along the axial direction is given by the expression,

A preliminary estimate of the situation is best made from equations (29), (32), and (35). For θm≤θ≤π, rM is infinite. To avoid this difficulty we use the value obtained as θ approaches θm but is less than θm. Then

The observations give at 17h 59m , ρ(0) = 2.4×10^3cm, rMrm=1.03×10^4cm, V=3.5×10^6cm sec–1, n_0=2.5×10^12cm–3. In this case for σ 2.75×10–l7cm^2 the corresponding limit on the optical distance is σn_0ρ≤0.165. Then (sin θm/θm)=0.39 and 4πσn_0=ρ(0)ne=6.14×10^10cm–2 where ρ(0) has been increased by a factor of 2 to allow for refraction. From equation (32) we find α≃0.9×10–5cm^3sec–1.

Comparison of the two estimates suggests that α≃2×10–5cm^3sec–1. This definitely eliminates N_2 as the ionized constituent, since the experiments of Bialecke and Dougal (1958) indicate for N_2 that α=9×10–7cm^3sec–1, at the height of the meteor. A revised estimate of (sin θm/θm)2=0.35 is indicated by these results, but this corresponds to a value of θm only slightly greater than π/2 so it appears that numerical integrations of the kinetic equation (14) are required to fully interpret the observations of this meteor.

Some conclusions can be drawn concerning the ionizing radiation from Meteor I. By equations (17) and (36),

If we now differentiate with respect to σ and equate the result to zero, we obtain at the minimum value of the relation

Thus at maximum radius ρ=3.3×10^4cm, H=77 km, n_0=1.4×10^14cm–1. For the minimum value of meteor radiation, , σ=2.2× 10–19cm–2 which occurs very close to the ionization limit at 1030 A (Weissler and Lee, 1952). Then =2.5×10^22sec–1. The total emission of ionizing photons from the meteor is at the rate 16π^2 and is ≥4.0×10^24sec–1. The total power of the ionizing radiation is 16π^2 hv which is ≥8×10^13ergs sec–1.

Conclusions and discussion

From this work we may conclude that the meteoric head echo can be accounted for by photo-ionization of molecular oxygen caused by radiation from the meteor. The required recombination coefficient is approximately 2×10–5cm^3sec–1. Even if O_2+ is not formed directly it would appear as the end product of electron exchange with all other ions which could be formed in the meteor region except nitric oxide. Miller (1957) has shown that nitric oxide is present only as a trace constituent and could not produce the head echo because of its low recombination coefficient (Bialecke and Dougal 1958) and other factors.

For the Perseid meteor, with visual magnitude –5.7, we have been able to estimate the total ultraviolet power required to produce the ionization in the meteor head. It is ≥8×10^13ergs sec–1.

References

BIALECKE, E. P., AND DOUGAL, A. A.

1958. Pressure and temperature variation of the electron-ion recombination coefficient in nitrogen. Journ. Geophys. Res., vol. 63, pp. 539–546.

BROWNE, I. C., AND KAISER, T. R.

1953. The radio echo from the head of meteor trails. Journ. Atmosph. Terr. Phys., vol. 4, pp. 1–4.

CHANDRASEKHAR, S.

1950. Radiative transfer. 393 pp.

GREENHOW, J. S.

1952. Characteristics of radio echoes from meteor trails. III. The behavior of the electron trails after formation. Proc. Phys. Soc. London, ser. B., vol. 65, pp. 169–181.

HAWKINS, G. S.

1955. Meteor ionization and its dependence on velocity. Astrophys. Journ., vol. 124, pp. 311–313.

HEY, J. S.; PARSONS, S. J.; AND STEWART, G. S.

1947. Radar observations of the Giacobinid meteor shower, 1946. Monthly Notices Roy. Astron. Soc. London, vol. 107, pp. 176–183.

MANNING, L. A.

1953. The strength of meteoric echoes from dense columns. Journ. Atmosph. Terr. Phys., vol. 4, pp. 219–225.

MCKINLEY, D. W. R.

1955. The meteoric head echo. In Kaiser, ed., Meteors. Journ. Atmosph. Terr. Phys., special suppl., vol. 2, pp. 65–72.

MCKINLEY, D. W. R., AND MILLMAN, P. M.

1949. A phenomenological theory of radar echoes from meteors. Proc. Inst. Rad. Eng., vol. 37, pp. 364–375.

MILLER, L. E.

1957. “Molecular weight” of air at high altitudes. Journ. Geophys. Res., vol. 62, pp. 351–365.

MILLMAN, P. M.

1950. Meteoric ionization. Journ. Roy. Astron. Soc. Canada, vol. 44, pp. 209–220.

MINZNER, R. A., AND RIPLEY, W. S.

1956. The ARDC model atmosphere, 1956. Air Force Surveys in Geophys., TN–56–204, No. 86, p. 15.

WEISSLER, G. L., AND LEE, P.

1952. Absorption coefficients of oxygen in the vacuum ultraviolet. Journ. Opt. Soc. America, vol. 42, pp. 200–203.

Abstract

In this paper the authors examine the hypothesis that the meteoric head echo is due to ultraviolet ionizing radiation from the meteor, and the subsequent dissociative recombination of the molecular ions and electrons produced. It is shown that molecular oxygen is probably the ionized constituent. The hypothesis is examined quantitatively and a recombination coefficient of approximately 2×10–5cm^3sec–1 for molecular oxygen ions is deduced.