SMITHSONIAN

CONTRIBUTIONS

to

ASTROPHYSICS

VOLUME 5

SMITHSONIAN

INSTITUTION

WASHINGTON

D.C.

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 A and $ 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

ihe 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).

iii
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.

IV
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

For sale bj the Superintendent of Documents, U.S. Government Printing Office

Washington 25, D.C. • Price 30 cents
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
The Meteoric Head Echo1

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 me by McKinley and Millman

(1949) and at a frequency of 60 me 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

• The research in this document was supported Jointly by tbe U.8.

Army, Nary, and Air Force under contract with tbe Massachusetts

Institute of Technology.

* Harvard College Observatory, Cambridge, Mass.

• Boston University, Boston, Mass.,and Harvard College Observatory.

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.

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):

(1)

M= 39.8—2.5 log10 q, (2)

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:

l
SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

G: Antenna gain over an isotropic radiator, 3.0.

c: Noise level in the receiver, 2X10-" watts.

P: Transmitter power, 3X105 watts.

X: Radar wavelength, 9.2 meters.

nc: Critical electron density, 1.28X10Ism-»

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:

G

2P\2

ira2

(3)

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

Meteor I

100

90

80

70

50

40

30

20

10

1.0 1.5 2.0 2.5 3.0 3.5

Time in seconds

Meteor I I

-

-

1

ft

\l

1 /

1/

,f

;' 1

: 1 / l ,-'

•' I/'

\

\

\

/

\

\

\

V

4.0 4.5 5.0 6.5

/ \ A Using curve A

/ \ f (Ott»w)

\

•AUsing curve B y \

(Amprior) »

44 45 46 47 48 49 50 51 52 53

Tim* in seconds

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

of time.

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.

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, p, from the center of the

meteoric coma as follows:

Op

(4)

where Iydvdu is the intensity of the radiation

through the solid angle da between frequency

v and v-\-dv\ aT is the absorption cross section;

n is the number of molecules per unit volume;

and S, 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

(5)

where n, 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, v0 is the

threshold frequency for ionization, dot is an

elementary solid angle, a is the volume recombination coefficient of the ions, and the

symbol (b denotes integration over all directions. The term under the integral sign indicates that within a given solid angle, du, the

intensity of photons (number per cm2) produces

ionizations at a rate n(<r,I,lhv) over the range

of frequency, dp.

Finally, if n0 is the initial number of oxygen

molecules per unit volume before passage of
THE METEORIC HEAD ECHO

the meteor, the conservation of particles re- where </* is the average photon intensity at

quires that unit distance from the meteor in the absence

of absorption:

n=n (6) y*=\ zpz. (i2) o—ne—~

where n'o 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:

, S,=0. (7)

The first expression is justified because

that is to say, a small fraction of

the oxygen molecules is ionized. The source

function, S,, 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

,=I? exp (—< (8)

where /* dv du> is the intensity along the direction of integration in the absence of absorption.

The average intensity, J,, is given by the integration

J*

= -f exp

—<r,nop), (9)

where J* denotes the average intensity at unit

distance from the meteor in the absence of absorption. We have assumed that p 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=n0 into equation (5) yields the equation,

<T,J,

exp (—<r,7iop)'dp—

(10)

We will now introduce a mean absorption

cross section a defined as follows:

C jrJ* exp (—<r,nop)

(11)

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,

Tin ?Vn ?im

Dt~dc (13) Hz

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 -jjf=Q' The kinetic equation then becomes

r

e x p

 i-<

(14)

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.

X

exp [— (aJV)nt

2

.

Neglecting the term, bnjdz, we may write

the lowest order solution as:

(0) (r z \ _

(15)

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

form
SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. S

•,2)+<£2(r,z)] (16) It is now necessary to substitute the expressions

is now sought. The following result is obtained:

P(O) = P (|J—A,p+A2p,

(17)

(rz)=^ex (--Znop}fl-y[l+-w3p]x

p \ 2 / \ L 2 J

/i _ \ , 3 ,r. i _, 2 2i^

cos 0 exp ^ 2 <TMoP )+ 2 VM 1"~ J2

 <^ww> J X

cos20 exp (<rnop) r» (18)

where a change has been made to polar coordinates p and 0,

r (19)

p=-\lr

2

-{-z2, cos 0 =

So long as y is small, the term with y as a

factor indicates that the surfaces of equal

density, nt, are spheres with centers along the z

axis. The third term with y

2

 as a factor in

these surfaces introduces an elongation. As y

becomes very small these surfaces can be represented as ellipsoids. We may now evaluate

t h e distance s p(0),p (^ ) an d p(x) a t which

nt=nt.

At 0=0,

f 1- "1

exp I —7^rnep(0)

L 2 J

ne=p- P(0)

At 0=| ,

f

exp |- - anoP ^

(20)

(21)

exp |- 2

exp [

l ~ ^«» exp (22)

(23)

into equations (20), (21), and (22) and solve for

Aip and A2p. We neglect A2p in the terms

involving 7 and both Aip and A2p in terms

involving y

2

. The solutions are:]

=7P (| ) exp I ^ «

n

I ^ oP Q (24)

A2p =2

 y2p (I) exP I 7n°p

 (|) J

(25)

The corresponding surface where ne=ne is a

prolate ellipsoid for small values of 7 with a

ratio of major to minor axes of 1-\-A2PIP(T).

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.

exp

If we neglect the term (^JnJ, the lowest order

solution is given by the equation,

exp [ -

(26)

where we have applied the boundary condition

that n,(0)(r,2) 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:

= 4 T

Vr arccot(- (27)
or, in terms of p and 0,

THE METEORIC HEAD ECHO

Case III: Small photo-ionization rate.

(28)

 -tJSnoS'W+W exP <-«

The corresponding solution for the surface

where the electron density equals the critical

density for the radar is:

* a

sin 0

(29)

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,

<r?lo / * r / 2 \

wi1)(r,z)=47r—^-— arccot(-)—

^4TT — ^ — I arccot2wd« • (30)

The maximum value of n, occurs at zm, where

at which point the approximation is breaking

down. The corresponding value of 6 is Bm

where

sin

V

2

(32)

This equation shows that all contours of constant electron density have a maximum width

at an angle 8=0m as indicated in figure 2.

CASE m

FIGURE 2.—Coordinate system (p, 8) and regions of solution

for Case II and Case III.

Integration is straightforward and exact, and

yields the equation:

nj»(r,«)- (33)

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 d=6M 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

n.(r,z)=

V

(34)

where zm=r cot 0m.

Combination of Cases II and III: An isosurface of electron density is found in the range

O<0<0M from equation (29). From equation

(34) it follows that in the range 0«,<0<x the

isosurface is given by the equation:

(35)

Equations (29), (32), and (35) apply only in

the limit as <rngp 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:

=^- exp 1—2 anopi (36)
6 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

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

A2p 6

— = z 7

p 4

l—i;Zn0p+-o2n

2

oP2

, -> o r- i

2

 :1- ex p [<rnoP].

"2

(37)

From equations (36) and (37) we may eliminate

the exponential term to obtain:

1 "* _ I ^ ~2 _ 2 2

1—— erWoP+o <f»oP

(38)

1 + - anop

which yields the equation,

F /S-^n0p+a2n

2

0p

2

 l_

2(2+anoP) E (39)

For Meteor I we deduce from figure 1 that at

19» 8m 3*75 E.S.T., the radius of curvature of

the ends of the ellipsoid p=104 cm. From

the change in signal strength as the meteor

crossed the line of sight we deduce 2?= 0.3.

We may take V= 6.0X10* cm sec"1 and

wo=1.5X1013cm~3. As extreme assumptions

we may take <?=0 or 2.75X10"17cm2, the

maximum value found by Weissler and Lee

(1952). The corresponding limits on the optical

distance are 0<<m0p< 8.25 where p has been

doubled to allow for refraction (Manning,

1953). From equation (39) we find that the

recombination coefficient a is approximately

3X10-5cm3sec-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,

d /dr/dO\

-LjEL^fL f4l\ de _je\dzlde)=

rM~~dz2~d0 \dz) dz~ dz/dd ~~

(dz/dd) (d2r/dd2) - (dr/dO) (d2

(dz/dd)3 (40)

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 dm<8<ir, ru is infinite. To avoid this

difficulty we use the value obtained as 0 approaches dm but is less than 8m. Then

rM=4v

-•••vr/- Jm—sin 0m co s 9m)

3

Vne sin 48m(l-8m s'm 9, cos 9m)

Vne \sin 0,

(42)

(43)

rMrm

Po Vsui e n-6m cos 0m) *

The observations give at I7h 59m 47*25, p(0) =

2.4X103cm, rArrra=1.03X104cm, F=3.5X10flcm

sec'1, n0=2.5X1012crn~3. In this case for

a 2.75X10"l7cm2 the corresponding limit on

the optical distance is anop<0.165. Then

(sin 9m/dm) =0.39 and 4iran0J

r

*!V=p (0)wc=6.14

X1010cm"2 where p(0) has been increased by

a factor of 2 to allow for refraction. From

equation (32) we find a^0.9X10~8cm3sec~1.

Comparison of the two estimates suggests

that a=si2X10~scm3sec~1. This definitely eliminates N2 as the ionized constituent, since the

experiments of Bialecke and Dougal (1958)

indicate for N2 that a=9X10~7cm3sec~1, at the

height of the meteor. A revised estimate of

(sin 9m/9m)

2=0.35 is indicated by these results,

but this corresponds to a value of 0m only

slightly greater than x/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),

filt=ancp

2

 exp

(45)

If we now differentiate with respect to <r and

equate the result to zero, we obtain at the

minimum value of J* the relation

rm=p sec (41) (46)
THE METEORIC HEAD ECHO

Thus at maximum radius p=3.3X104cm,

H=77 km, n0=1.4X1014cm-1. For the minimum value of meteor radiation, ^*,"5=2.2

X10~19cm"2 which occurs very close to the

ionization limit at 1030 A (Weissler and Lee,

1952). Then £/*=2.5X1022sec"1. The total

emission of ionizing photons from the meteor

is at the rate 16**^/* and is >4.0X1024sec~1.

The total power of the ionizing radiation is

v which is >8X1013ergs 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

2X10~8cm3sec~1. Even if O2

+

 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

References

BIALECKE, E. P., AND DOTJGAL, A. A.

1958. Pressure and temperature variation of the

electron-ion recombination coefficient in

nitrogen. Journ. Geophys. Res., vol. 63,

pp. 539-546.

BEOWNE, 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.

CHANDBASEKHAB, S.

1950. Radiative transfer. 393 pp.

GBEENHOW, 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.

MCKINLET, D. W. R.

1955. The meteoric head echo. In Kaiser, ed.,

Meteors. Journ. Atmosph. Terr. Phys.,

special suppl., vol. 2, pp. 65-72.

MCKINLET, 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.

MlNZNER, R. A., AND RlPLET, 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 2X10~5cm3sec~1 for molecular oxygen ions is

deduced.

».!. SOVflMHT MIMTIH* OFFICIillfO