SMITHSONIAN

CONTRIBUTIONS

to

ASTROPHYSICS

Smithsonian Institution

Astrophysical Observatory

Volume 8, Number 5

On the Luminous Efficiency

of Meteors

By Franco Verniani

Washington, D.C.

Smithsonian

Contributions to Astrophysics

VOLUME 8, NUMBER 5

ON THE LUMINOUS EFFICIENCY

OF METEORS

by FRANCO VERNIANI

SMITHSONIAN INSTITUTION

Washington, D.C.

1965
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On the Luminous Efficiency of Meteors1

FRANCO VERNIANI 2

Introduction

By far the most precise astronomical data for

meteor orbits have been obtained by use of

photographic techniques (Jacchia and Whipple,

1961). Photographic data also make it possible

to evaluate physical quantities, including those

that cannot be determined by other means of

investigation, such as Jacchia's fragmentation index. Such basic quantities as the

masses of meteoroids are, however, still very

uncertain, although the photographic technique, by providing a light curve, is much

more helpful than other means of investigation.

The large uncertainty results from the lack of

knowledge of the luminous efficiency of meteors.

The purpose of this work is to extract, from

the Harvard photographic meteors, direct

information concerning this parameter.

Since the luminous intensity / of a meteor is

generally accepted to be proportional to the

kinetic energy of the ablated atoms,8 the

luminosity equation is usually written in form

m and v being the mass and velocity of the

body; T, the luminous efficiency; and -rr' the

rate of the mass loss in atomic (or molecular)

form. The gas-cap radiation will be neglected

because the meteors that we will refer to have

been photographed in blue emulsion, and there

is experimental evidence that such radiation is

important only in the red and the infrared

regions (Millman and Cook, 1959).

The proportionality between / and -JT> i.e.,

»This work was supported In part by grant NsO 87-60 from the

National Aeronautics and Space Administration.

« On leave: Centro Nazionale per la Fisica dell'Atmosfera del O.N.R.,

Roma, Italia.

1

 The photographed spectra of meteors confirm that the kinetic energy

lost by deceleration of the body does not contribute to the light.

between the luminous intensity and the rate of

ablation of free atoms, is quite clear. In contrast, the proportionality between / and tf2

seems much harder to understand. It is

actually easy to envisage a very intricate

dependence of /, i.e., of T, on the velocity v.

Let us consider the spectrum of a meteor with

a given velocity. If we progressively increase

the velocity, the intensity of the single lines

will change; some low-excitation lines will fade

out; new lines will appear; and certainly the

total emitted light will not be a simple function

of the velocity. It is evident, moreover, that

the luminous intensity depends on the spectral

range that one considers. In the following we

will refer to a visual luminous intensity Iv and

to a photographic luminous intensity /,,,

defined by the sensitivity of the blue emulsion

generally used in Harvard photographic studies.

Accordingly, we will have two different luminous efficiencies TV and TV, defined by

and

1

1 ,

dm

It

dm

If

(2)

(2a)

If we assume, as is generally accepted, that

the dependence of the luminous efficiency Qn the

velocity can be expressed by a power law of

the form T~O", and if we introduce the color

index of the meteors C, which is defined as the

difference Mp—Mt between the photographic

and the visual magnitudes, we get

(7=-2.5 log

 J-f =-2. 5 log ^

-2.5(np -n,) log c. (3)

The symbols are self-explanatory.

141
142 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Jacchia (1957a) found that Ois independent

of the meteor velocity; this result has been

confirmed by the analysis of the complete set

of data, precisely reduced by Jacchia, concerning Super-Schmidt material (Jacchia, Verniani

and Briggs, unpublished). We therefore have

observational evidence that the exponents nv

and n, are equal—a result that is far from

obvious. This being so, in the following n

will refer both to n, and nv. Jacchia's results

show also that C decreases in absolute value

with the brightness. Hence the ratio TOPITOV

must be a function of the brightness and

consequently of the mass of the meteors.

Knowing the values of C as a function of MP

allows us to determine T9 as a function of rv.

For meteors brighter than Mv—— 2, we can

safely assume C= —1.9; then T,=0.17TP .

For meteors with Mv between —2 and +1.5

Jacchia's results can be approximately described

by the linear equation

so that

(4)

(5)

Since at present only Opik (1933,1955 a and b)

has studied theoretically the luminous efficiency

in the visual range, his work has necessarily been

the only basis for the computation of meteoric

masses. In fact, Whipple (1938) approximated

Opik's results for bright meteors by assuming

T,=TOVV with rM=8.5X10~10 scm"1 in the spectrum range between 4500 and 5700 A. All

Harvard meteoric masses have been computed

with this value and with the linear dependence

of T on v. Both Tot and the exponent n are very

uncertain. Opik (1958) writes that his basic

work of 1933 is "a semi-empirical, semi-theoretical approach on the basis of a combination

of classical and quantum-mechanical principle,"

but Thomas and Whipple (1951) have considered it only "a rough approximation." No

attempt will be made here to summarize the

work of Opik; the number of assumptions he

introduced to get his results is such that even

a Trough estimate of the errors involved in the

final results is impossible. Only a comparison

with direct experimental results can afford such

an estimate. In the revision of his own work

Opik (1955a) confirmed n=l for bright meteors

and reduced the value rot to about 60 percent of

the preceding value. The uncertainty involved

in TOV may, however, be as large as two orders

of magnitude CWhipple and Hawkins, 1959).

The determination of masses through the motion of meteoric trains led to a value of rot about

220 ±150 times less than Opik's original value

(Cook, 1955). Conversely, the extrapolation

of recent results of McCrosky (1961) and McCrosky and Soberman (1962) on artificial

meteors suggests an intermediate value between

the two extremes, as does a work on a few

clearly recognized asteroidal meteors (Cook,

Jacchia, and McCrosky, 1963). A theoretical

study on meteor density (Verniani, 1962) leads

to the conclusion that Opik's value cannot be

wrong by orders of magnitude. We will later

examine more closely the results obtained with

artificial pellets and with the asteroidal meteors

in searching for the range in which TOP should

lie and for its most probable value.

Current ideas, summarized by McKinley

(1961), on the dependence of r on velocity

are that n= l applies only to the brighter

photographic meteors; for fainter meteors r

should vary more slowly than the first power

of v, perhaps even as a negative power of v for

fainter dustballs. In fact Opik (1955b), obtained theoretically for dustballs n=—0.88 and

then assumed the round relation rr = scm"1,

v

and Jacchia (1957b) found for faint

Super-Schmidt meteors TV independent of

velocity. This result was obtained from the

fitting of the atmospheric profile computed by

means of meteor decelerations to the standard

atmosphere of that time (Minzner and Bipley,

ARDC 1956), which had a slope different

enough from the present U.S. Standard

(1962) to produce a non-negligible error in n.

A re-analysis of the Super-Schmidt meteors,

using the new standard atmosphere, shows

that n cannot be far from 1 (Jacchia, Verniani,

and Briggs, unpublished). Unfortunately, the

results are only slightly sensitive to a variation

of n. The results of the present work show

clearly that TP is independent of the brightness of meteors. A dependence of the luminous

efficiency on the magnitude or on the mass

should not be easy to justify theoretically,

although it seems that the spectrum of meteors
LUMINOUS EFFICIENCY OF METEORS 143

changes with the brightness in such a way

that the emission in the red appears to become

important for fainter meteors. Ceplecha

(1959) has suggested this spectrum change

to explain the results on the color index that

he obtained by using panchromatic emulsions,

which attain maximum sensitivity for a wavelength equal to 6700 A. Davis' (1963) photoelectric measurements confirmed this suggestion. A variation of r with brightness was

also predicted by Kallmann (1955), who

concluded that the luminous efficiency should

be proportional to a power of the duration of

the individual meteors. Obviously such an

hypothesis has absolutely no physical meaning.

Recently Ananthakrishnan (1960, 1961), trying to remove the disagreement between the

theoretical beginning and end heights and the

measured heights of the photographic meteors

reduced by Hawkins and Southworth (1958),

introduced the hypothesis that TP varies along

the trail of an individual meteor in proportion

to the atmospheric density pa. Evidently this

assumption has no physical basis either. The

emitted light must be proportional to pa because

of the proportionality between p0 and the rate

at which atoms leave the body and are responsible for the emission of light. Ananthakrishnan's hypothesis means that the emitted light

is proportional to pi, and this is absurd; of

course the observational results confirm that

such an hypothesis is wrong. Incidentally, the

above-mentioned discrepancy between the

theoretical and the observed heights is without

any doubt an effect of fragmentation, as

Hawkins and Southworth correctly stated on

the basis of Jacchia's (1955) fundamental

finding. That the experimental heights of the

Super-Schmidt meteors seem to fit the curve

computed with r~pav better than the theoretical

curve based on T~~V is solely the result of the

upward shifting of the meteoric heights produced by the fragmentation. This has the

same effect as would increasing luminous

efficiency with po. Moreover, when Ananthakrishnan shows that the less fragmenting

part of the Hawkins-Southworth meteors are

not so far removed from the theoretical curve

as the Draconids of Jacchia, Kopal, and Millman

(1950), he finds only the very well known fact

that the fragmentation of the members of the

Draconid shower is extreme. The fragmentation index x of Draconids is near to 2, while the

average x for all the Super-Schmidt meteors is

0.25 and that of the group selected in his paper

is certainly even smaller.

Ceplecha (1958) and Ceplecha and Padev&t

(1961) have attempted to evaluate n by using

Harvard data on small-camera meteors (Jacchia,

1952) and on Super-Schmidt meteors (Hawkins

and Southworth, 1958). Their results are n=

—0.33 ±0.61 for small-camera meteors and

n=— 2.8 ±0.2 for Super-Schmidt meteors.

These values of n appear unacceptable. There

are several reasons for this:

1. Ceplecha used the theoretical relationship between the intensity at maximum light

Im and the mass outside of the atmosphere mm.

When there are flares—as there frequently are

in small-camera bright meteors—or diffuse

fragmentation, as in Super-Schmidt meteors,

that relation ceases to be meaningful.

2. He wrote such a relation, following

Levin (1956), in a form very similar to that of

Herlofson (1948), which is known to be a rough

approximation even in the cases in which the

theory could be successfully used (Verniani,

1961).

3. In the same equation Ceplecha eliminated the mass ma by making use of another

equation of Levin (1956), obtained with the

assumption that the velocity of the meteoroid

does not change during the entire path. Moreover, that equation contains the atmospheric

density of the end point of the meteor. Such a

value is often meaningless because of the very

irregular behavior of meteors in the last part

of their recorded path, as a result of crambling

and fragmentation (Jacchia, 1955). It may

differ from the corresponding theoretical value

by an order of magnitude.

4. The Harvard small-camera material,

especially the earlier Massachusetts meteors,

is very inhomogeneous with regard to the

accuracy of the data. Therefore a good analysis can be done not only by rejecting those

meteors for which the time of appearance is

poorly determined or the mass is not accurately

known, but also by using a suitable system of

weights, as Jacchia did in his 1952 analysis,

for taking into account the accuracy of the
144

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

computed decelerations. This accuracy varies

remarkably from one meteor to another.

5. Ceplecha used the Rocket Panel atmosphere, which now is known to be in error

both in slope and in density values at meteoric

heights.

6. He found n by plotting a certain

computed quantity versus log v*. The theoretical slope of the expected straight line is

9+n. Even if the other strong causes of error

were absent, the small value of n in comparison

with 9 clearly makes his determination highly

unreliable, because of the non-negligible fluctuations of the computed quantities caused

by (a) experimental errors; (b) variations in

the drag coefficient with height and velocity;

(c) irregularities in the shape of the meteoroids; and (d) fluctuations from the standard

atmosphere.

7. The masses of Hawkins-South worth

meteors are averages of individual values given

by each of the two plates of the same meteor

and computed with relationship (48) of

Hawkins' (1957) paper on the method of reduction of short-trail meteors. Often there was

a large difference between the two individual

values. Although on the average those masses

represent a fair approximation, they are much

less precise than the masses Jacchia computed

by integrating the light curve for small-camera

meteors and for his selected sample of long-trail

Super-Schmidt meteors (Jacchia, physical

data not yet published). This difference can

be another reason why the exponent Ceplecha

found for Super-Schmidt faint meteors is so

much further off the mark than that of smallcamera meteors. The principal reason is the

basically different degree of fragmentation

among the two samples of meteors.

Levin (1956) made a statistical verification of the

theory of meteors by writing /OT~7>O>t cos

 yZR

CZB=zenith angle of the meteor path) and

by computing a, ft and y by least squares for

Harvard small-camera data. The masses Levin

used were those computed by Jacchia on the

basis of T,~y; therefore, according to singlebody theory, the value of 0 must be 4. The

result, 0=3.98±0.27 and a=1.00±0.05, confirmed that for bright meteors the classic

theory seems adequate to describe accurately

the average behavior of the phenomenon. On

the other hand the computed value of y was

completely different from the expected value of

1. Levin and Majeva (1963) did the same kind

of verification on the Super-Schmidt meteors reduced by Hawkins and Southworth, but the

result was quite different. They divided those

meteors into two groups, according to the

different degree of fragmentation estimated from

the beginning and end heights. The group of

less-fragmenting meteors gave 0=3.67 ±0.08,

while the other group, containing meteors that

experienced more severe fragmentation, gave

/3=2.67±0.10. These deviations from theory

can be easily understood with the aid of some

of the preceding remarks on Ceplecha's work.

They show that one cannot extract any information on n by following that route. As a

curiosity, in the above-mentioned book Levin,

by using the masses computed with Opik's value

of rot, determines the average visual magnitude

of a meteor having mm = \ g, #a> = 10 kms"1.

Then, making use of this value, he reverses his

procedure to find the value of T09. It is peculiar that the result he gets is completely different

from Opik's value.

I shall conclude this introduction with some

brief remarks on the units in which the luminous

intensity (and consequently the luminous

efficiency) is expressed. In many papers the

luminous intensity is measured in ergs; in this

case the relation between the visual magnitude

Mv of a meteor and the corresponding intensity

/„ has been given by Opik (1958) as

M,=24.30—2.5 log It, (6)

where the constant 24.30 is determined by

taking into account the sensitivity curve of the

human eye; it corresponds to the sun's stellar

magnitude —26.72 and to the energy distribution of the solar radiation. This system, in

which r is a dimensionless quantity, seems

quite natural to the physicist, but it is not

convenient for practical use, especially in

photographic work. Therefore Jacchia (1948)

introduced a system that takes as unit of

intensity that of a star of magnitude zero. In

this system T has dimensions zero mag g"1 cm~2s

3

.

In the following we will use Jacchia's system,

which has the clear advantage of being closer
LUMINOUS EFFICIENCY OF METEORS 145

to the observed quantities, i.e., to the magnitudes. In Jacchia's system we have

and

Mp=-2.5logIP

M,= -2.5log/,.

(7)

(7a)

From equations (6) and (7a) it follows immediately that the logarithmic difference between

r, expressed in cgs and in Jacchia's units is

9.72. Therefore we have

(log O(5plk = —9.07 (cgs)

(log Oop l k = —18.79 * (Jacchia's units).

For bright meteors, to which a color index

correction —1.8 is applicable, we have

(log = -18.07.

In addition to Jacchia's units for Iv and TP, we

will use cgs units for all other quantities.

Procedure used for determining n

The drag equation is usually written in the form

pai?, (8)

where a is the deceleration of the meteor; pmf

its density; y, the drag coefficient; and A, the

shape factor. The integration of equation (2)

yields

tB being the time at which the meteor ends.

We assume that the terminal mass can be neglected; this should be correct in most cases

(Jaccbia, 1948; Verniani, 1959). Equation (9)

may be written

mrp=l^EIvdt=2^ (10)

where E= I Ipdt is the integrated brightness,

and v is a value of v at some instant between t

and tB. The symbol TV refers here to the same

« The value used in Harvard's work Is —18.91. The difference of 0.12

results from the different form of equation (6), which was previously

established by Opik (1937) as: M,=24.6-2.5 log /..

velocity. Eventually, by eliminating m between equations (8) and (10), we obtain

TP _2E / a V (ID

In writing equation (11), we admit the equality

of the "dynamic" mass derived from the drag

equation and of the "photometric" mass

obtained by integrating the light curve. Such

an assumption is valid only for meteors that

ablate without fragmenting. Therefore, for

most of the Super-Schmidt meteors, which

crumble during their flight in the atmosphere,

such an assumption is not right. It is well

known that the photometric mass takes into

account the light emitted from all fragments

and therefore may also be considered reliable

for fragmenting meteors. In contrast, the

dynamic mass corresponds only to the mass of

the larger fragments to which the measured

deceleration refers. We must conclude that

in the general case, when fragmentation is

present, equation (11) overestimates rp\p*m. We

will see under "Data and results" (p. 148ff.),

however, that it is always possible to make

use of equation (11) by taking into account the

effects of fragmentation by means of Jacchia's

fragmentation index xThe photographic data allow the determination of the velocity v, deceleration a, integrated brightness E and height z. The U.S.

Standard Atmosphere (1962) gives pa as a

function of z. The Super-Schmidt meteors

have several decelerations each; we will see

that the most reliable for our purpose are those

near to the beginning of the light. Therefore,

for each meteor only one value v has been computed. We assumed v to be equal to the value

of v corresponding to m=){mm, i.e., to the

value zero of Jacchia's mass-loss parameter 8.

Inspection of the light curves and of the observational velocity curves shows that vs=0 represents a very adequate approximation of v.

The difference between v and va is generally

very small. It is of the order of 1 or 2 percent

for slow meteors and goes down to 0.1 or 0.2

percent for the fast ones.

The shape factor A is unknown. According

to Opik (1958), rotation, vibration, and oscillation of meteors during their flight smooth
146 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

their shape so that we can ignore the possible

variations of A during the meteor life and

/Q_\l/3

accept the value 4=(— J =1.21 corresponding to a spherical shape. This value is usually

employed in Harvard's work. The effect of

fragmentation makes useless any search for a

better approximation. We shall discuss this

effect later.

The last quantity to be considered is the

drag coefficient 7. This quantity is not well

known. The results yielded by an interpolation

formula worked out by Baker (1959 a and b) on

the basis of the theory of Baker and Charwat

(1958) for the drag coefficient in the transitional

flow region do not agree with recent experimental results (Maslach and Schaaf, 1962).

Baker's formula shifts the transition from the

free molecular flow to continuum flow to

heights greater than the reality. A. F. Cook

(1963, private communication) has given

another expression for 7 in the form

7=0.55[l+8.3X10-5(rPa)-1/2], (12)

where r, the equivalent radius of the pellet,

and pa are expressed in cgs units. This equation

applies only to compact bodies, like meteorites

and asteroidal meteors.5 The use of equation

(12) for the Super-Schmidt meteors shows that

many of these are still in free molecular flow,

although the hypothesis that these meteors are

single compact bodies is certainly wrong for

most of them. For better consideration of the

effect of fragmentation, we will assume that

all the Super-Schmidt meteors are in free

molecular flow. We will therefore use 7=7^=

1.1, the value generally assumed for satellites

in free molecular flow (Cook 1959; Jacchia

1964). This value is also used in the general

analysis of the physical data of the SuperSchmidt meteors (Jacchia, Verniani and Briggs,

unpublished).

If we make the extreme assumption that

equation (12) may also be applied to the SuperSchmidt meteors, considered as single compact

bodies, we obtain from the observational data

• In equation (12) y tends to Infinity when p«_*0. Therefore, when

equation (12) yields a value greater than TT=1.1, it is understood that y

must be taken equal to yr.

a value of n only slightly smaller than that

obtained by assuming 7=7^. The difference is

about 15 percent, which is very reassuring,

since it means that if 7 is not exactly what we

have chosen, the reliability of the final result

is not affected.

The data employed for the analysis are those

of the 413 Super-Schmidt meteors precisely

reduced by Jacchia (astronomical data published by Jacchia and Whipple, 1961; Jacchia,

physical data not yet published). The old Harvard small-camera data have been used separately, although their accuracy and homogeneity is much poorer than that of the SuperSchmidt, to provide a basis of comparison for

larger masses and lower heights. For these

meteors, which are generally very bright, we

have used Cook's expression (12) for 7.

The quantities involved in the computation

of rpjp\ are variously affected by observational

errors. The velocities and the heights are

quite reliable. The error in the velocity for

about half of the Super-Schmidt meteors is of

the order of 0.1 percent. The small-camera

meteors are also generally accurate in v to

better than 1 percent. The decelerations, on

the other hand, are not so reliable; the probable

error may differ widely from meteor to meteor.

This is particularly true for small-camera

meteors, whose probable errors range from

0.01 to 10 times the value of the deceleration

itself. Also, the reliability of the probable

error varies greatly, depending on the number

of breaks used to compute the deceleration.

The integrated brightness is generally more

reliable when t is near the beginning time tBf

because of the uncertainty in the interpolation

to zero of the intensity curve at the end of the

detectable trajectory. The lower reliability of

the values of rv\p% computed from data near

the end of the trajectory is augmented by the

effects of fragmentation, which affects equation

(8) by increasing the cross-sectional area A.

Hence it is clear that, to do a correct analysis,

we must use a suitable system of weights, which

will take all these factors into account. In

his analysis of the small-camera data for determining the atmospheric density as a function

of the height Jacchia (1952) solved the same

problem. Therefore I have used here a slightly

modified form of the weight function introduced
LUMINOUS EFFICIENCY OF METEORS 147

by Jacchia. This modified formula has been

also used in the general analysis of the SuperSchmidt physical data, which will be published

soon.

For all meteors the quantity

(proportional to pmrm) has been computed

from each deceleration. Then a weighted

average has been taken for every meteor, by

using a weight p defined as

when

*) = i U + erf (2.5 log s-2.1)]; (14)

p.e. is the probable error in the deceleration a;

and N is the number of shutter breaks used

in the least-squares solution (which contains

3 unknowns) for determining a. Then, by

means of the weighted average values of

yApavPa~l, I have computed rv\p\ for each

meteor to which a weight w=(^p)1/2 has been

assigned for the final analysis. I have put w

into this form because, since the results obtained for meteors with many decelerations are

somewhat more reliable than those obtained

for meteors with only two decelerations, we

must attribute a larger weight to them. Taking w=Xp would overweight the slow meteors,

which are generally longer, and the final

analysis would be biased. Calculations with

different weights have been done to investigate how much the chosen system of weights

affects the final results. The result was that

the analysis of the Super-Schmidt meteors

is only slightly affected by the system of

weights. This was to be expected because

of the general reliability and homogeneity of

Jacchia's Super-Schmidt material. For smallcamera meteors, the inaccuracy in the determination of the deceleration varies the weights

p from less than 0.01 to about 9, and the choice

of the weight system is, conversely, very

important.

The quantity rPjpm varies with the velocity v

of the meteor. We shall attribute this variation uniquely to T, i.e., we shall assume that the

meteor density pm does not depend systematically on the velocity. Of course there is no

reason why the meteor density should vary

745-685—65 2

continuously with the velocity. Nevertheless,

we must consider two different circumstances

that, if ignored, would affect the conclusions,

perhaps remarkably. The first is the possibility that some among the very slow meteors

may be of asteroidal origin; in this case the

value of Tp/pm in the region of the lowest

velocities would be smaller and our determination of n overestimated. It is easy enough to

obviate this difficulty by taking out of the

analysis all the meteors that show a difference

from the average larger than the expected

scatter in the values. Naturally, some meteors

will be clearly suspected of having different

densities, but it is a difficult task to decide

whether or not certain others should be

excluded.

The second circumstance to keep in mind

is Jacchia's (1958) discovery of the difference

in fragmentability between the meteors of

the Jupiter family and the long-period meteors.

This difference leads us to think that the

different fragmentation may correspond to a

difference in density between the two groups.

This hypothesis is corroborated by the fact

that long-period meteors begin to appear at a

height that on the average is about 4 km

greater than that of short-period meteors

(Jacchia, 1963). Therefore the sporadic

meteors have been divided in two groups

according to their aphelion distance in order

to search for a difference in the averages of

the Tp/pm- Such a difference has actually been

found, and we will see that taking it into

account changes n appreciably, because one

group is concentrated toward the low velocities

and the other, toward the high velocities.

Obviously this difference could be caused

either by a variation of TV resulting from

differences in chemical composition or by

different densities or by a variation of

both these qualities. However, the well-established current ideas on the cometary

origin of meteors and on Whipple's (1950, 1951)

icy-comet model lead us to conclude that the

difference must be ascribed to the density only.

In fact, according to Whipple's theory, shortperiod comets evaporate faster, which means

that short-period meteors should come from the

inner core of their parent comets, while the

long-period meteors should come from more
148

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

external layers, subjected during the comet's

life to a smaller pressure. We can therefore

expect that short-period meteors have a greater

solidity than long-period ones. Thus, it is

easy to understand a difference in density

among meteors with much different orbital

characteristics, while it is difficult to find

reasons for justifying a difference in composition.

We should expect a nonsystematic spread in

the values of the density pm among sporadic

meteors. Taken together with the irregularity

in the shape of the bodies, the observational

errors and the effects of fragmentation, which

will be discussed in detail later, this spread must

give a wide scatter in the individual values of

TPlfhn- In spite of this scatter, all of whose

causes are random and unpredictable, we may

expect to arrive at correct conclusions by taking

averages of TTjp2

m in groups with a large number

of meteors in the same range of velocities or

better, with a least-squares solution for all the

data.

Data and results

The value of n.—For the determination of n by

the least-squares method it is convenient to put

the dependence of TP on v in a logarithmic8

form:

log ^|=log ^

Pm r'm

log t;, (15)

where v is v. Of the 413 original Super-Schmidt

meteors, 12 are not available for our purpose

because some basic physical data are missing or

because their precision is lower. Taking out

all the meteors having a clearly anomalous log

rjpm leads us to eliminate from the analysis

40 meteors (10 percent of the total number),

36 of which are sporadic. Thus the analysis

concerns 361 meteors, 247 sporadic and 114

shower. The least-squares method gives for the

sporadic meteors n= 1.52±0.15. This result,

however, is not to be considered final because

of the difference in density between long-period

meteors and those of the Jupiter family. Table

1

 7 contains the unconnected average values of

• In the following, log always Indicates decimal logarithms.

»In table 1, as In the following tables, the numerical values are sometimes given, for mathematical purposes only, with one digit more than

those really significant.

TABLE 1.—Sporadic Super-Schmidt meteors: log rD/p^ as a function of meteor velocity (no correction applied for

the difference in density between short-period and long-period meteors).

Velocity, v

(105 cms-1)

10-15

15-20

20-25

25-30

30-35

35-40

40-45

45-50

50-55

55-60

60-65

65-70

70-75

log v

6. 130

6. 245

6.347

6.436

6.505

6.572

6.633

6.678

6.720

6.760

6.791

6. 829

6.854

log Tp/pi

-11.898

-11.377

-11.397

-11.227

-10. 996

-10. 753

-10.730

-10.806

-10. 544

-10.465

-10 . 887

-10.934

-10.534

S.d. of

mean

±0. 311

0.095

0. 105

0. 104

0. 101

0.217

0. 160

0. 150

0.230

0. 139

0. 198

0. 164

0.454

COB

-0.7 3

-0.4 2

- 0 . 51

— 0. 47

-0.2 0

-0.4 6

-0.4 6

-0.0 7

- 0 . 27

- 0 . 17

-0.4 1

-0.2 6

+ 0. 06

S.d. of

mean

±0. 13

0.07

0.06

0.07

0.09

0. 13

0.08

0. 18

0. 16

0. 16

0.09

0.07

0.21

no.

10

52

45

40

24

10

12

8

6

10

13

12

5

w

4. 82

4 96

4 74

4 37

5. 09

3.69

3. 73

4 67

3. 17

3. 34

2. 82

2. 70

3. 23

The least-squares solution for mean values gives ra= 1.51 ±0.15.

Correlation coefficient between mean values of log v and log rvfp\ = 0.90.

Results of the general least-squares solution and mean values of the basic quantities: n = 1.52±0.15; log T0Vlpm

= -20.99±0.99; a.d. = 0.53; no. of meteors=247; log t;=6.433± 0.008; log TP/P* = -11.182 ± 0.031; «„ =-0.40±0.02;

tc=4.35, where *„ =log Ea, and a.d.=average deviation of individual value of log rjp^ from the least-squares solution.
- 1 0 •

o

-12"

LUMINOUS EFFICIENCY OF METEORS

—\ 1 1

149

Q<7o.u.

Q>7a.u.

6.0 6.2 6.4 6.6

—h6.8

Log v

FIGURE 1.—Super-Schmidt meteors: log TP/P^ as a function of meteor velocity for meteors

in short-period (Q<7 a.u.) and long-period (Q>7 a.u.) orbits. The confidence marks

represent standard deviations of the means. The number of meteors of each group is

written near the points. (See table 2.)

log T pipit for groups of increasing velocity. These

values are given only as a reference for comparison with the final data. Before correcting

the data, we computed two other least-squares

solutions for estimating how much the chosen

form of the weight w could affect the analysis.

Two extreme cases have been considered. In

the first, all the weights w have been put equal

to 1; in the other, individual decelerations

instead of individual meteors have been used.

The results are both quite close to n=1.5, which

confirms that the definition of the weight w

affects neither the final results nor their

reliability.

Let us now examine how to introduce a

correction for the difference in density between

meteors of the Jupiter family and long-period

meteors. By dividing the sporadic meteors

into two groups, one having the aphelion distance Q<7 a.u. and the other having Q>7

a.u., we get the results listed in table 2 and

plotted in figure 1. These results show a mean

difference of about 0.30 between the values of

l°g Tp/pl, f°r tn e tw 0 groups. This means that

the average density of the short-period meteors

is about 1.4 times the density of long-period

meteors. The reduction of the values of log

Tp/pm to the same velocity has been done

with successive approximations, which lead to

a value of n close to 1. If we compute two

separate least-squares solutions for the two

groups, we do not obtain reliable results.

The reasons for this are very clear: (a) The

range in velocity becomes too small for both

groups; and (b) the statistical fluctuations of

the data become important, because the amount

of data is diminished. These fluctuations are

clearly illustrated in figure 1, which shows

that they are particularly important for the

long-period meteors, which have a very large

scatter. If we remove the extreme-velocity

groups from the small-aphelion-distance

meteors, we get a slope very close to n=l.

Figure 1 also illustrates that the slope n=l

does not contradict the very irregular trend of

the long-period meteors. We could correct

the results by reducing the average values of

velocity groups to the density of the meteors
150 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

with #< 7 a.u. and in this way get the final

value of n. In order to study the residuals,

however, it is better also to correct the individual data by subtracting 0.30 from all the

values of log TV\(U belonging to meteors with

Q>7a.u.

After the reduction of all meteors to the

average density of the group of the Jupiter

family, the least-squares method gives

n=1.01±0.15; (16)

- f = —17.80±0.98. (17)

The error in TOP/PI is quite large, because of

the large extrapolation involved from the

meteor velocities' range to v—1 cms"1. The

actual error in rv in the range 10-70 kms"1

in which we are interested is, however, small

enough. The least error in the determination

TABLE 2.—Sporadic Super-Schmidt meteors divided in two groups, short-period (Q<7 a.u.) and long-period (Q>7 a.u.);

log rp/pm as a function of meteor velocity for each group.

Velocity, v

(10* cms-1)

10-15

15-20

20-25

25-30

30-35

35-40

40-45

45-50

50-55

55-60

60-65

65-70

70-75

no.

9

51

40

32

15

4

3

1

log v

6. 124

6.245

6.346

6.433

6.508

6.569

6. 621

Q<7 a.u.

lOg Tp/pJ!

-12. 065

-11 . 384

— 11. 418

-11.324

-11.081

-11 . 090

-10.616

S.d. of

mean

±0.301

0. 097

0. 113

0. 124

0. 119

0.387

0. 161

no.

1

1

5

8

9

6

9

7

6

10

13

12

5

log v

6. 353

6.447

6.499

6.573

6.637

6.680

6.720

6.760

6.791

6.829

6.854

Q>7 a.u.

log rp/pl

-11.262

-10. 896

-10.857

— 10. 576

-10. 769

-10.810

-10. 544

-10.465

-10.887

-10.934

-10.534

S.d. of

mean

±0.290

0. 127

0. 181

0. 253

0. 210

0. 171

0. 230

0. 139

0. 198

0. 164

0. 454

Mean values of basic quantities for short-period (Q<7 a.u.) and long-period meteors (Q>7 a.u.):

(1) Q<7a.u.:no.=155;log»=6.338±0.007; log rp/P

2

m= -11.368±0.039; €„ = -0.49±0.02.

Q>7 a.u.: no.=92; log v=6.621 ±0.013; log rjp2

m= —10.803±0.041; «„ = -0.22±0.03.

By reducing the values of log Tvjp2

m to equal velocity v' by successive approximations, we obtain

(log rp/p^)Q>7-(log Tp/p^,)0<7=0.28±0.12.

(2) The overlapping velocity region is 20-45 kms"1. For meteors having velocities between these two limits we have:

Q<7a.u.: no.=94; log i>=6.417;logTp/p^=-11.295±0.047;eoD = -0.50±0.02.

Q>7 a.u.: no.=37; log v=6A99; logrvlp2

m= —10.878±0.062; «„ =—0.26±0.05.

Then: (log Tvlp2

m)Q>7-(\og TP/P

2

JQ<7=0.33±0.12.

From the results listed in (1) and (2), I have assumed:

Gog Tp/p^,)Q>7-(log

Q > 7

v=vt

)O<7 =0.30 .

(3) For low-fragmentation meteors (|x|<0.2), in the overlapping velocity range:

Q<7 a.u.: no.=33; log t>=6.462; log TP / P ^=-11.167±0.082 .

Q^>7 a.u.: no. = 18; log v= 6.512; log Tvlp2

m=

Then: aogrp/p2,)0>7 -(log rvfP

2

m)Q<7 =0.25±0.17.

lxl<0.2 lxl<0.2
LUMINOUS EFFICIENCY OF METEORS 151

of log Tp/pm is that corresponding to the centroid

of the distribution, for which we have

log rP/p2

m=~ 11.281 ±0.030;

log w=6.433± 0.008. (18)

For each other point, the error er should be

given by

«r=0.03 + 0.15X|logv-6.433|. (19)

In the following, however, it is convenient to

adopt ft = 1.0, so that the error in log rp/pl, will

be constant and equal to 0.03. Equation (17)

now reads

log ^ = -17.71 ±0.03. (17a)

By sorting the sporadic meteors into groups

according to their velocity and computing the

mean values of log Tp/p2

m, we obtain the

results shown in tables 3 and 4 and plotted in

figures 2 and 3, respectively. Since the

residuals do not show any other definite

dependence on velocity, the power law Tv=Topv

n

is sufficiently close to reality. It is important

to remember that the individual intensity in

each wavelength is a function of velocity and

that our rv corresponds to integrated light,

which must depend on v in some complicated,

almost certainly irregular manner. In order to

have a check on the correction for the Q effect

on the values of log Tvlp2

m, we have arranged

10 equally populated groups of sporadic meteors

in order of increasing Q and have computed the

averages of the residuals for each of them.

The correlation coefficient between the average

residuals and Q is —0.28, which shows a practical

absence of correlation, because we are dealing

with averages and not with individual values,

for which the large scatter could lower the

correlation coefficient. When Q increases, the

residuals tend to pass from negative to positive

values for meteors with Q<7 a.u., but this

cannot affect the results in any way. In fact,

for meteors of the Jupiter family, there is no

correlation between v and Q.

Figure 4 shows the average values of log

Tp/pn for each shower; the same data are listed

with more details in table 5. As I have said

before, it is unlikely that different showers have

appreciably different composition to account

for variations in TV; conversely, their different

ages and distances from the sun may account for

differences in density. Thus, the differences

(l°g

 TP/PII) shower (log Tp/p%) sporadic, » = » shower

are attributed only to a difference in density.

With this assumption the ratios, contained in

table 5, of the density of each shower to

the average density of the sporadic meteors

with Q < 7 a.u. have been computed. Northern and Southern Taurids, Southern i-Aquarids

and 5-Aquarids, Perseids, Lyrids and <r-Hydrids have density values close to that of

the sporadic meteors of the Jupiter family.

Conversely, Quadrantids, Orionids, K-Cygnids

and a-Capricornids have about the same density as the sporadic long-period meteors. As

the reader can see from the indicative values of

Q listed in table 5, however, some of the

showers with Q > 7 a.u. have approximately

the same density as the sporadic meteors with

Q <C 7 a.u. and vice versa. The members of

the Geminid shower have a density four times

larger than that of the sporadic meteors. This

confirms a result that Jacchia obtained in his

earlier work. The extremely low density of

Draconids is also confirmed, but the figures in

that case do not have much meaning because

the effects of fragmentation are too large to be

properly taken into account. The results confirm that each shower has its own peculiarities,

as Jacchia found from his analysis of points

near the beginning of meteors.

Effects of fragmentation.—Let us now discuss

the effects of fragmentation on the computed

values of log rpfp^. Jacchia (1955) found that

the anomalously large increases of the observed

decelerations of the Super-Schmidt meteors are

explained by progressive fragmentation. He

introduced, as a measure of the phenomenon,

the fragmentation index x, defined as

aT

(20)

where aou is the observed deceleration (a in the

preceding equations); a? is the deceleration

computed from the drag equation by using the

photometric mass; and s is the mass-loss

parameter, defined as

(21)
152

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 3.—Sporadic Super-Schmidt meteors: log TP/P£ as a function of meteor velocity: mean values of log

groups of different v (all values of log rjpl, reduced to the density of short-period meteors).

Velocity, v

(10s cms"1)

10-20

20-30

30-40

40-50

50-60

60-73

log»

6.227

6.387

6.520

6.653

6.745

6.818

lOg Tp/pi

-11.469

-11.373

-11.073

-11.009

-10.794

-11 . 138

S.d. of

mean

±0.095

0.073

0.089

0. 115

0. 118

0. 130

Coo

-0.47

-0.4 9

-0.2 6

-0.2 8

-0.2 1

-0.26

S.d. of

mean

±0.06

0.04

0.08

0.10

0. 12

0.07

no.

62

85

34

20

16

30

to

4.94

4. 57

4.68

4. 10

3. 28

2. 84

p^ for 6

The least-squares solution for mean values affords: n=0.93±0.17.

Correlation coefficient between mean values of log v and log TP/P1 = 0.87.

Results of the general least-squares solution and mean values of the basic quantities: n= 1.01 ±0.15; log TOp/pJ

= — 17.80±0.98;a.d.=0.53;no. of meteors=247; log t>=6.433±0.008; log Tp/pi= -11.281 ±0.030; «0O = —0.40± 0.02;

u>=4.35; a.d.=average deviation of the individual value of log rv\p\ from the least-squares straight line.

TABLE 4.—Sporadic Super-Schmidt meteors: log rjp^ as a function of meteor velocity (all values reduced to the density

of short-period meteors).

Velocity, v

(108 cms-1)

10-15

15-20

20-25

25-30

30-35

35-40

40-45

45-50

50-55

55-60

60-65

65-70

70-75

log v

6.130

6.245

6.347

6.436

6.505

6.572

6.633

6.678

6.720

6.760

6.791

6.829

6.854

lOg Tp/pi

-11.931

-11.383

-11.438

-11.295

— 11. 110

-10. 949

— 10.954

-11 . 076

-10. 844

-10.765

-11 . 187

-11.234

— 10. 834

S.d. of

mean

± 0. 296

0.095

0. 105

0. 101

0.098

0.204

0. 169

0. 155

0. 230

0. 139

0. 198

0. 164

0. 454

-0.7 3

-0.42

- 0 . 51

-0.47

-0.2 0

-0.4 6

-0.4 6

-0.0 7

-0.27

- 0 . 17

-0.4 1

-0.26

+ 0.06

S.d. of

mean

±0. 13

0.07

0.06

0.07

0.09

0.13

0.08

0. 18

0. 16

0. 16

0.09

0.07

0. 21

no.

10

52

45

40

24

10

12

8

6

10

13

12

5

w

4.82

4.96

4.74

4.37

5.09

3.69

3.73

4. 67

3. 17

3.34

2. 82

2.70

3.23

The least-squares solution for mean values affords: TI=1.00±0.15 .

Correlation coefficient between mean values of log v and log rp/p^=0.81 (the decrease of the

correlation coefficient with respect to the unconnected data of table 1 is caused by the decrease of

the value of n; such a decrease does not mean a worse correlation).
LUMINOUS EFFICIENCY OF METEORS 153

E

i

-10-

-II-

-12-

1 h

10

h1—

T

t

4 5

h -

I

2 4

no

H

I •

10 Al2 T

H

— ' 1

lit^>~-—1

13

 L

1

6.0 6.2 6.4 6.6

Log v

6.8

FIGURE 2.—Sporadic Super-Schmidt meteors: log rv/p^ as a function of meteor velocity.

(See table 4.)

-10-

en

3

-12 "

6.0

• All meteors

x Low-fragmentation meteors only (IXk0.2)

1 1 1—

6.2 6.4 6.6

Log v

6.8

FIGURE 3.—Sporadic Super-Schmidt meteors: log rvlp\ vs. log v for all meteors and for

meteors having |x|<0.2 only. (See tables 3 and 6.)
154 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS
LUMINOUS EFFICIENCY OF METEORS 155

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T45-6S5-65-
156 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

The fragmentation index x is not easy to determine because it involves the second-time

derivative of the velocity. Since x is, however,

the quantity that best describes the fragmentation, it is very helpful for correcting data. Its

meaning lies in its being approximately constant

for each meteor during the detectable part of

its flight. If fragmentation were absent we

would find, in the limits of observational error,

the same value of rp/p4 for whatever point of

the trajectory the decelerations introduced in

equation (11) refer to. But the fragmentation

introduces a variation in the computed values

of rv]pn. In fact, by using equations (11)

and (20) we can easily find the relation between

the values of log rv\p^ computed in two different points of the trajectory corresponding to

s and s0:

(log -S0). (22)

We can immediately see the order of magnitude

of the variation introduced by the fragmentation if we consider a meteor for which several

decelerations have been determined and for

which x=0.25, the mean value for sporadic

meteors. The difference As between the values

of s corresponding to the first and to the last

determination of the deceleration is about 1.

Therefore the fragmentation changes log TP/p2

m

by 0.75. In case of larger x the effect is, of

course, larger. It is clear that all the present

data should be reduced to the value of s corresponding to the beginning of fragmentation.

Unfortunately, this value is not known, but we

could refer the values of log Tp/p2

m to a value

of 8 near the beginning of the meteors. This

value may be found by calibrating the results

by means of the nonfragmenting meteors.

This is not necessary for the purpose of this

work, because the use of the weights p, which

attribute greater importance to the decelerations in the first part of the trail, when fragmentation has not yet played its role, also

satisfactorily accomplishes this task. In fact,

for the purpose of empirically finding out the

effect of fragmentation on the actual weighted

values of log rvjpn, all the sporadic meteors

with |x|<0.2 have been selected and processed

by the least-squares method. The solution,

after the usual correction for Q, is n=1.24 ±0.22

and log rp/p2

m = -11.214±0.040 for log v=

6.421 ±0.011. The mean values of log rp/p2

m

as functions of velocity are listed in table 6

and plotted in figure 3 (p. 153) for comparison

with the results of all the sporadic meteors. The

value 1.24 for n agrees, within the limits of the

error, with the value 1.01 afforded by the general

solution; it is worth noting that if we remove

from the general solution the slowest meteors

(10<C#<O5 kms"1), for which the average value

of log rv\pm is smaller by about 0.3 than the

expected one,8 we would get a slightly smaller

value of n. As a compromise between this

value and that afforded by the low-fragmentation meteors, n= 1 is really the best value for n.

The agreement for the value of log Tv\pm in the

centroid of the distribution is actually very

good. For log t>=6.42 we get —11.213 and

— 11.268. Although we might expect a larger

difference and in the opposite direction, as we

shall see, TV depends on the degree of fragmentation, insofar as meteors that do not fragment

seem to be more efficient in producing light.

Anyway, the consistency of these results show3

the reliability of Jacchia's weight system, which

was used throughout.

We must remark that there are cases in which

neither x

 no r

 the weight system is sufficient to

find out the correct value of log Tp/p2

m by using

equation (11). This is true when a meteor

breaks up into NF fragments of approximately

equal mass before it is detected or before the

earliest deceleration is measured. This is the

case of the so-called "abrupt-beginning

meteors," which appear suddenly at or very

near the maximum light, and of the very

short meteors, whose light curve is practically

reduced only to a flare. In these cases the

value of log rv\pm is overestimated, and it is

easy to see that the difference between the

correct and the computed value is log N?.

Fortunately, the number of these kinds of

meteors is small, and most of them are among

those rejected from the analysis because of the

clearly anomalous value of log TV\P\. Consequently, the reliability of the results does

j

 This is because of the presence In the group of some meteors that,

although they do not clearly have the characteristics of the asteroldal

meteors, appear to have a larger density. We cannot even exclude the

possibility that the luminous efficiency decreases more rapidly than

T~P for meteors slower than 15 kms-' (Jacchia, 1949). This last possibility seems, however, to be very unlikely.
LUMINOUS EFFICIENCY OF METEORS 157

TABLE 6.—Sporadic Super-Schmidt meteors with fragmentation index \x\<^0.2: log rjpl, as a function of meteor velocity-

(all values of log Tv/pm reduced to the density of short-period meteors).

Velocity, v

(105 cms-1)

10-20

20-30

30-40

40-50

50-60

60-73

log v

6. 220

6. 394

6. 522

6. 665

6.740

6. 819

log r

-11 .

-11 .

-11 .

-11 .

-10.

-10.

JA

360

318

001

229

742

680

S.d. of

mean

±0. 130

0. 094

0. 131

0. 148

0. 182

0.230

e

-0.37

-0.46

-0.22

-0.02

- 0 . 04

- 0 . 17

S.d. of

mean

±0.09

0.06

0. 10

0. 16

0. 15

0. 12

no.

30

49

20

8

7

11

to

5.66

4. 69

4.77

4.95

3.67

2. 87

Basic results: n=1.24±0.22; log TOP/PI= —19.18± 1.41; a.d. = 0.51; no. of meteors=125; log t>=6.421±0.011;

log TP/pi=-11.214±0.040;«oo = -0.33±0.028; u>=4.74.

not seern jeopardized by the effects of fragmentation.

It is very important to note that the broad

scatter among the individual values of log

rp/p2

m is not the result of the fragmentation.

In fact, the average deviation of an individual

value from the least-squares general solution is

0.53, while the corresponding average deviation

for low-fragmentation meteors is 0.51. The

average |x| for this last group is 0.08; the average individual difference from the average value

of s is ~0.2.9 Therefore the average value of

3xAs is 0.05, i.e., one tenth of the actual scatter.

»/ l..., /:

(;= x ) p*< \ =number of decelerations of each meteor IEven taking into account the relative inaccuracy

with which the x's are known, the discrepancy

remains very large. Several factors may account for such a large spread, among which is

the existence of groups of sporadic meteors

having different density. On the other hand,

the mean of the average deviations of the individual values of log rp\pm for shower meteors

is about 0.4. This .shows clearly that the

difference in density is not the only cause of

the scatter. To explain it, we must remember

that the actual error in the values of the deceleration is much larger than the internal error

previously considered.

As a matter of fact, the decelerations of this

sample of Super-Schmidt meteors have another

TABLE la.—Sporadic Super-Schmidt meteors: mean values of basic data for three groups, equally populated, of different

brightness.

no.

log v

log TV/PI

(lOg TJPI) lo, ,_J.40

X

log as

(lOgTp/p^)*lol,_,.40

x-0.22

lo«»—11.11

i 0.15

Faintest

83

-0.86 ±0.01

6. 355 ±0.012

-11 . 341 ±0.056

-11.30 ±0.07

0.24 ±0.05

-11 . 15 ±0. 04

— 0. 06 ±0. 02

-11 . 35 ±0. 14

Intermediate

82

-0.5 1 ±0.01

6. 452 ±0. 013

-11 . 362 ±0.050

-11 . 41 ±0. 06

0. 29 ± 0. 04

-11 . 17 ±0.04

— 0. 13 ±0.03

-11.40 ±0. 13

Brightest meteors

82

0.07 ±0.02

6. 482 ±0. 015

-11 . 165 ±0.047

-11.25 ±0.06

0. 13 ±0. 03

-11 . 24 ±0. 03

-0.25 ±0. 04

-11 . 31 ±0. 11

*The reduction of the log Tv/pl to standard values of x, log <r, and s is done by means of the correlations discussed

on pages 156, 161, and 162.
15

8 SMITHSONIA

N CONTRIBUTIONS TO ASTROPHYSICS

"8

Brightest meteors Intermediate Faintest

i

-dCO-H8

-6CO-H*fac». 60 _OobO_O6cT3 CO-H8 TJ CO-Hal>

bfi _ObC_O6a"CCO-H8 CO-H(•bCwo6c

Velocity, v (105 cms-')

20 6.242 -11. 25±0. 15 0. 06±0. 07

28 6.386 -11. 31±0. 12 -0. 10±0. 07

12 6.509 -11. 13±0. 14 0. 18±0. 06

7 6.666 -11. 08±0. 18 0. 09±0. 13

6 6.750 -10. 82±0. 18 0. 22±0. 12

10 6.832 -10. 96±0. 28 0. 08±0. 10

83 6.444 —11. 19±0. 07 0. 04±0. 04

n=0. 57 ±0. 24

21 6.239 -11. 55±0. 17 -0. 62±0. 03

29 6.393 -11. 60±0. 13 -0. 61±0. 02

11 6.529 -11. 05±0. 14 -0. 40±0. 05

6 6.652 -11. 07±0. 29 -0. 35±0. 04

5 6. 729 - 10. 85±0. 29 -0. 37±0. 06

10 6.811 -11. 67±0. 14 -0. 34±0. 02

82 6.435 -11. 44±0. 08 -0. 53±0. 02

n=0. 95±0. 27

21 6.196 -11. 65±0. 17 -0. 94±0. 0328 6.382 -11. 21±0. 13 -0. 90±0. 03

11 6.528 -11. 00±0. 21 -0. 80±0. 04

7 6.636 -10. 84±0. 16 -0. 76±0. 05

5 6. 758 -10. 67±0. 16 -0. 75±0. 06

10 6.807 -10. 82±0. 11 -0. 60±0. 05

82 6.416 -11. 23±0. 08 -0. 86±0. 02

n=l. 59 ±0. 25

10-20

20-30

30-4040-5050-60

60-73

Global
LUMINOUS EFFICIENCY OF METEORS 159

o

U-4 ©

60 60

O 'C

2-S

8 g

"g S3

— O a

X!

2 5

£

o

a

T
160 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

source of uncertainty: the flutter that affected

the rotation of the camera shutters. When the

trail covered two or more cycles of the flutter,

it was possible to eliminate its effect fairly well,

but for shorter trails a correction was impossible

(Whipple and Jacchia, 1957). Moreover, the

elongated and fuzzy form of the dashes, particularly when wake or blending or both together

are present, makes it impossible to know exactly to what the deceleration refers. It can

never be too much emphasized that very often

in meteor physics the quantities determined

from the observations do not correspond strictly

with those that enter the equations of the

theory.

Another source of error in rPlp2

m is the atmospheric density. We should not forget that local

transient fluctuations of the atmospheric density may appreciably affect the results, since

Tvlpm is inversely proportional to the third

power of pa.

Investigation of the dependence of T on mass.—

Let us now examine the problem of the dependence of the luminous efficiency on the mass of

meteoroids. By assuming that TP depends on

some power P of the initial mass, we can write

By using equation (2) we get:

m-2P P

(23)

(24)

where — =n. Equation (24) shows that

we can investigate the dependence of TP on mm

by studying the correlation between TP and the

total integrated brightness Ea. This is true

even if we assume that TV is proportional to the

power P of the instantaneous mass. Therefore,

Jacchia's Super-Schmidt sporadic meteors have

been arranged in three equally populated groups

in order of increasing em (€ro=log Ea). The

results, which are listed in table 7 (a and b),

clearly show that rv does not depend on the

meteor mass. We cannot check whether the

exponent n keeps the same value when the

brightness changes by using the least-squares

method for these three groups, because each

group corresponds to a different mean velocity

and the actual spread in v becomes too small.

Only the intermediate group has roughly the

same velocity distribution over the entire

sample. Most of the meteors that belong to

the faintest group are very slow, while the

brightest group contains a great part of the

fast meteors. It is possible, however, to

arrange the sporadic meteors in three groups of

different brightness but of the same distribution

in velocity. The results obtained with the

least-squares method show that n decreases as

we go from the faint meteors to the bright ones.

This result disagrees with the current opinions

of McKinley (1961), quoted above, and with

Opik's theory, according to which n should

decrease with brightness. The differences

between n=l, as found for all the sporadic

meteors together, and the single values corresponding to each group of brightness could,

however, hardly be said to be significant, since

they are of the same order as the probable

error involved in these determinations. Moreover, the average deviations of one individual

value of log Tp/p2

m from the least-squares

straight lines are just as large as that found in

the general solution involving all the sporadic

meteors. Study of figure 5, in which average

values of log rp\pm taken in intervals of 10 kms"1

for each group are plotted, clearly shows that

the variations of n given by the least-squares

method certainly result more from accidental

fluctuations than from a real difference in the

behavior of T, as a function of v. But the

final proof of the independence of n from the

brightness is afforded by the analysis of

the small-camera meteors. These meteors are

much brighter than the Super-Schmidt meteors,

since their average E*, is about 100 times the

average £"„ of the Super-Schmidt meteors. But

the value of the exponent n of the small-camera

meteors turns out to be very close to 1, as we

will see later.

The possibility of a dependence of TP on the

mass m^ has also been studied by means of the

residuals A, defined as the difference between

the values of log Tp/p2

m obtained by the observational data and those obtained by the leastsquares solution. Table 8 reports the average

values of A for 10 groups of 25 meteors in order

of increasing brightness. These results are also
LUMINOUS EFFICIENCY OF METEORS 161

plotted in figure 6. It is clear once more that

TP does not depend on the mass.

TABLE 8.—Sporadic Super-Schmidt meteors: residuals

A as a junction of integrated brightness.

Cm

-1.049

- 0 . 854

- 0 . 756

- 0 . 664

- 0 . 568

-0.447

- 0 . 329

- 0 . 169

+ 0. 019

+ 0. 454

S.d. of

mean

± 0. 025

0.007

0.005

0.004

0.007

0.005

0.008

0.008

0.013

0.045

A*

0.046

- 0 . 051

0.070

0.022

- 0 . 302

0.009

- 0 . 153

0.093

0. Ill

0.093

S.d. of

mean

±0. 147

0. 130

0. 146

0.137

0. 126

0. 118

0. 156

0. 114

0. 124

0. 121

no.

25

25

25

25

25

25

25

25

25

22

*A=(log TJPI,) ob.. — (log Tp/pjS,) u«.t-.ao.t...

The luminous efficiency as a function of fragmentation.—The results given in the section

on effects of fragmentation lead us to investigate whether there is some correlation

between the luminous efficiency and the degree

of fragmentation of meteoroids. According to

their fragmentation index x> sporadic meteors

have been sorted into 10 groups, and the

weighted averages of the residuals A have been

computed. The results are listed in table 9

0.4-

and plotted in figure 7. A weak correlation

appears between A and x- The correlation

coefficient is —0.73. An empirical equation

correlating mean values of A and x> obtained

by the least-squares method, is

A=(0.11±0.04)-(0.50±0.11)x. (25)

One would expect that meteors with a large x

had lower density. If this were so, their log Tv\f?m

should be larger than the average. The

results show just the opposite: A is positive

for meteors with low x and negative for meteors

with a high degree of fragmentation. Therefore equation (25) may be interpreted as an

actual dependence of the luminous efficiency

on the degree of fragmentation of the meteors,

such that a compact body is more efficient in

producing light than a porous, crumbling one.

The possible decrease of the density with

increasing x tends to mask the effect, so that

the value 4 of the ratio of the luminous efficiencies at the two extremes has to be considered as underestimated. It is worth noting

that the average value of 8 does not vary

systematically with x, so that even if we reduce

the residuals to the average values of s, we

arrive at practically the same results. The

same is true also if we take the data uncorrected

for the difference in density between short- and

long-period meteors.

I 1 1 1 1 1 1 1

-1.0 -0.8 -O.6 -0.4 -0.2 0 0.2 0.4

FIGUBE 6.—Sporadic Super-Schmidt meteors: average residuals A for groups of meteors of

different brightness «„, A=(log rv(p*)ob.— (log rjp^) i,,lt.,a,IIel. (See table 8.)
162 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

0.4-

FIGURE 7.—Sporadic Super-Schmidt meteors: average residuals A for groups of meteors of

different fragmentation. Straight line from a least-squares solution, A= (log Tp/p^)ob,—

Gog TJPI) i...t-.qu.re.. (See table 9.)

An explanation of the dependence of TP on

X could be found in the different density of the

vapors in the meteoric coma. Low-fragmentation meteors should have a denser coma and

radiation arising from the collisions among the

molecules of the coma itself. Conversely, the

radiation of the most crumbling meteors should

be caused by collisions between air molecules

and ablated meteor atoms. On the basis of

such an explanation we should, however, also

expect some dependence of TP on the mass. We

could also explain the dependence by a difference of composition among the sporadic

meteors such that low-fragmentation meteors

contain a larger percentage of materials having

a higher radiation efficiency than crumbling

meteors do. This does not, however, seem very

likely, as I pointed out for shower meteors in

an earlier section. I must also remark that the

residuals A are strongly affected by all the

causes of error previously mentioned (observational errors, fluctuations in the atmosphere,

irregularities in the shape, differences in density

of the meteoroids, and so on) and that some error is

also present in the values of x, as is shown by

the number of meteors having x<0. Therefore the validity of equation (25) must be

considered with caution.

A clearer correlation also exists between

mean values of A and log <r, <r being the coefficient of the mass equation

m (26)

The results, contained in table 10, are plotted

in figure 8. The regression line is

A=(-7.03±1.45) —(0.63±0.13) log a, (27)

TABLE 9.—Sporadic Super-Schmidt meteors: residuals

A as a function of the fragmentation index xX

- 0 . 122

- 0 . 018

0.036

0.096

0.149

0.222

0.309

0.383

0.532

0.909

S.d. of

mean

±0.014

0.003

0.004

0.004

0.003

0.005

0.005

0.005

0.015

0.052

A

0.082

0.281

0. 016

0. 153

- 0 . 197

0.116

- 0 . Ill

0. 091

— 0. 232

- 0 . 368

S.d. of

mean

±0. 132

0.082

0. 120

0. 132

0. 146

0. 140

0. 164

0. 112

0. 118

0. 124

no.

25

25

25

25

25

25

25

25

25

22

Correlation coefficient between mean values of A

and x= —0. 73.
LUMINOUS EFFICIENCY OF METEORS

H h

163

-10.8 -11.0 -11.2 -11.4 -11.6

Log oFIGUHE 8.—Sporadic Super-Schmidt meteors: average residuals A vs. log <r. Straight line

from a least-squares solution. A=(log rjp^)ob,— (log rp/p£) ie..t-.au«r,.. (See

table 10.)

and the correlation coefficient is —0.75. Also

in this case the correlation is not appreciably

changed by reducing the residuals to the

average value of s, because s does not vary

systematically with log a. This correlation

is similar to the preceding one between A and

X. In fact, the most efficiently radiating

meteors are those with smaller values of

a, i.e., values near to those pertaining to

compact bodies. This correlation must, however, also be taken with caution, for substantially the same reasons advanced for the

preceding one. Log a is known with a better

accuracy than x, but its physical meaning is

questionable for fragmenting meteors.

Independence of the luminous efficiency from

the atmospheric density.—As I stated in the

introduction, there are no theoretical reasons

for a proportionality between the luminous

efficiency TP and the atmospheric density

pa. This can also be proved directly by the

observational data. In fact, for a large

minority of the Super-Schmidt meteors of

the present sample, several decelerations were

determined at heights that are enough different

to allow us to see if TP depends on pa and how.

Obviously, when dealing with meteors showing

a non-negligible degree of fragmentation, one

must apply a correction 3x(s-s0), as discussed

TABLE 10.—Sporadic Super-Schmidt meteors: residuals

A as a function of log <r.

log a

-10.785

-10.937

-11.040

-11.099

-11 . 150

-11 . 204

-11.251

-11 . 304

— 11.406

-11.633

S.d. of

mean

±0.016

0.007

0.005

0.003

0.003

0.003

0.002

0.004

0.007

0. 032

A

- 0 . 317

- 0 . 207

0.035

- 0 . 184

—0. 094

- 0 . 051

0.200

0. 141

0.287

0.078

S.d. of

mean

±0. 155

0. 146

0. 145

0. 146

0. 128

0. 138

0. 125

0.108

0. 105

0.086

no.

24

25

25

25

25

25

25

25

25

22

Correlation coefficient between mean values of A

and log <r= — 0.75.

above. In table 11 are listed values of log

Tphm computed by individual decelerations

at different heights for each of several randomly

chosen sporadic meteors having x close to or

equal to zero. The table also contains the

data concerning a few other meteors that

experienced fragmentation in varying degrees,

to the extent of showing in real cases how the

correction Sx(s-s0) works. As we expected,

there is absolutely no dependence of TP on pa,

and the hypothesis advanced by Ananthakrishnan is completely meaningless.
164 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 11.—Values of log rjpl, computed by single decelerations at different heights for each of several sporadic SuperSchmidt meteors.

Trail no.

9030

v=23.7 kms"1

x=0.00

5572

p=16.4kms-1

X=0.00

12361

t>=22.1 kms"1

x=0.01

11973

»=44.7 kms"1

•v— n 01

5532

»=22.5 kms"1

X=0.01

7161

u=48.8 kms-1

x=— 0.01

5195

v=28.4 kms"1

X=0. 02

8640

r=55.4kms"1

Y = 0 02

7216

v=31Akms-1

x=0. 06

Height

(km)

90.01

89.64

87.78

86. 86

82. 24

82. 23

87. 35

87. 22

82. 86

82.75

79. 88

79. 80

93. 13

92. 53

91. 43

91. 12

90.40

90. 22

88. 08

87.85

98.83

98. 30

97.38

96. 84

91. 04

90.80

87.03

86. 68

85. 07

84 87

80.46

105. 78

105. 25

102. 74

102. 31

101. 57

100. 90

96.64

94. 79

96. 52

95.97

88.73

88. 63

106. 14

105. 65

101. 44

101. 05

99.23

98. 88

98. 13

97. 02

94.70

93. 93

lOg Pa

(gcm~3)

-8.5 0

-8.4 7

-8.3 3

-8.2 5

-7.8 8

-7.8 8

-8.29

-8.2 8

-7.9 3

-7.92

-7.69

-7.69

-8.76

- 8 . 71

-8.6 1

-8.59

-8.5 3

-8.52

-8.3 5

-8.3 3

-9.22

- 9 . 18

- 9 . 11

-9.07

-8.5 8

- 8 . 56

-8.2 6

-8.2 3

- 8 . 10

-8.0 9

-7.7 4

-9.7 3

-9.69

-9.5 1

- 9 . 47

-9.42

-9.37

-9.0 4

-8.9 0

-9.02

-8.9 9

-8.4 0

- 8 . 39

—9.75

-9.72

- 9 . 41

- 9 . 38

- 9 . 24

- 9 . 22

- 9 . 18

- 9 . 07

-8.8 9

-8.8 3

s

- 0 . 22

- 0 . 18

0.04

0.38

0.80

0.80

-0.56

- 0 . 54

0.07

0.08

0.71

0.75

-0.54

- 0 . 42

- 0 . 19

- 0 . 13

0.03

0.07

0.90

1.07

-0.5 3

-0.47

-0.3 8

-0.3 2

0.53

0.57

- 0 . 39

-0.3 5

- 0 . 15

- 0 . 12

0.67

-0.8 5

-0.79

-0.5 1

-0.46

-0.3 8

-0.3 1

0. 17

0.44

-0.4 2

-0.3 4

0.68

0. 69

-0.3 3

- 0 . 28

0.31

0.35

0.80

0.88

-0.9 4

-0.8 0

-0.5 4

-0.4 5

Plate

ST

SL

SL

ST

SL

ST

ST

ss

ss

ST

ST

SS

SK

SL

SK

SL

SK

SL

SL

SK

ST

SL

ST

SL

SL

ST

ST

SL

ST

SL

SL

SL

ST

ST

SL

ST

SL

ST

SL

SL

ST

SL

ST

SK

SL

SK

SL

SK

SL

ST

SL

ST

SL

lOg Tp/pi

-11 . 42

-11.31

-11.28

-11.43

-11.38

-11.27

-11 . 10

-11 . 16

-11.21

-11.34

-11.02

-11 . 21

-10.93

-10. 80

-10. 80

-10.80

-10.91

-10. 87

-10.99

-10.80

-11.02

-11 . 20

-10.95

-11 . 12

-11.20

-10.98

-11.85

-11.70

-11.76

-11 . 68

-11.76

-10.80

-10.74

-10. 71

-10.68

-10.50

-10. 63

-10.84

-10.74

-10.64

-10.70

-10.64

-10.53

-10. 13

-10.33

-10. 17

-10.29

-10. 18

-10.06

-11 . 11

-10.95

-10.86

-10.80

V

7.7

7.7

6.9

6. 1

3.5

3.4

8.5

7.8

5.6

5.3

3.5

3. 4

8. 8

8. 5

7.8

7.6

6.9

6. 7

3. 1

2. 7

8.6

8.4

8.4

8. 1

3.6

3.7

8.0

7.9

7.5

7.4

3.7

9.0

9. 1

8.7

8.6

7.9

8.2

6. 2

5.0

7.0

7. 2

2.2

2. 0

8.2

7.8

5. 1

4. 4

3.3

2.8

9.4

9.3

8.8

8.6

(log Tp/pi) _o

-10.94

-10. 81

-10.76

-10.72
LUMINOUS EFFICIENCY OF METEORS 165

TABLE 11.—Values of log rp/p^ computed by single decelerations at different heights for each of several sporadic SuperSchmidt meteors—Continued

Trail no.

7216—Continued

4464

t>=20.7kms-i

AY =. \J*\Jt 0 07

7946

v= 17.8 kms"1

v=0 08

6949

r=26.7kms-i

v—0 12

X—u. 14

12577

t;=29.2 kms"1

x=0. 20

4702

t>=31.5 kms-1

x=0.97

10414

v=15. 6kms-J

x= -0.10

Height

(km)

91.35

90.31

86.47

85.31

94.08

93. 43

85.45

84. 24

80. 72

79.04

77.39

75.48

74.76

77.49

76.44

65.78

65. 69

59. 40

58.94

55.56

55. 21

51.51

51. 05

91.68

91. 10

88.63

88.40

84. 00

83. 50

79.92

79.76

94.62

94.47

91. 92

91. 63

89. 70

89. 54

85. 58

85.42

93.67

89. 53

89. 12

85. 44

88.46

86.62

77.04

76. 47

69. 39

69. 26

63. 80

63.58

logPo

(gcm-3)

- 8 . 61

- 8 . 53

- 8 . 22

- 8 . 13

- 8 . 83

- 8 . 78

- 8 . 14

- 8 . 04

-7.76

- 7 . 64

- 7 . 52

- 7 . 40

-7.35

- 7 . 53

-7.46

-6.72

-6.72

- 6 . 48

- 6 . 46

-6.2 8

-6.26

-6.07

- 6 . 04

- 8 . 64

-8.59

-8.39

-8.37

-8.02

-7.98

-7.69

-7.68

-8.8 8

-8.87

-8.66

- 8 . 64

-8.48

-8.47

- 8 . 15

- 8 . 13

-8.8 1

-8.46

-8.4 3

- 8 . 14

-8.38

-8.2 3

- 7 . 50

-7.46

-7.0 3

-7.02

-6.72

-6.7 1

s

- 0 . 17

0.05

0.68

0.93

- 1 . 42

-1.34

- 0 . 51

-0.40

-0.0 3

0. 16

0. 39

0.99

1. 42

- 1 . 05

-0.94

- 0 . 22

-0.2 1

0.20

0.22

0.49

0. 52

1. 13

1.28

-0.87

-0.8 1

-0.59

-0.57

- 0 . 04

0.03

0.59

0. 62

- 0 . 60

- 0 . 59

-0.34

-0.32

- 0 . 11

- 0 . 08

0.53

0. 56

-0.7 5

- 0 . 11

-0.07

0. 51

- 1 . 64

-1.3 3

-0.32

- 0 . 27

0.39

0.41

1. 19

1.25

Plate

SL

ST

ST

SL

ST

SS

ST

SS

ST

SS

ST

SS

ST

SL

SK

SL

SK

SK

SL

SK

SL

SK

SL

SL

ST

SL

ST

SL

ST

SL

ST

SL

SK

SL

SK

SK

SL

SK

SL

SS

SS

KB

SS

SL

SK

SL

SK

SL

SK

SL

SK

log Tp/pi

-10.74

-10.75

-10.77

-10. 69

-11.28

-11 . 04

-10.88

-11 . 02

-11.20

-10.94

-10.70

-10.55

-10.51

-14. 00

-13 . 87

-14. 20

-14. 21

-14. 07

-14. 01

-13.80

-13 . 72

-13. 12

-13. 01

— 12.26

-12.43

-12. 12

-12. 19

-12.04

-12. 15

-11.83

-11.67

-11.40

-11.42

-11 . 17

-11.22

-11 . 07

— 11. 14

-10.8 1

-10 . 56

— 13.03

-11.04

-10 . 58

— 9. 39

— 11. 18

-11.25

-11 . 52

-11 . 52

-11.87

-11.90

-11 . 90

-11.90

V

7.7

7.2

4.0

3. 1

8. 1

8.8

8.4

8.3

6. 5

6. 1

4.6

2.7

1.4

9.4

4.9

7.9

7.8

6.2

6. 1

4.7

4.8

2.4

2. 1

9.3

9. 2

8.9

8.9

7. 1

6.6

4.2

4.0

8.4

8.3

8.3

8.2

7.3

6.8

4.3

4. 1

2. 1

5.5

4.8

1.3

9.7

7.9

8. 1

7.8

5.4

5. 1

2.5

2.3

dog TP/PI)._O

-10.71

-10.74

-10.89

-10 . 86

-10.98

-10 . 76

-10 . 78

-10.94

-11.20

-10.97

-10. 78

-10.76

-10.81

-13.75

-13 . 64

-14. 15

-14 . 16

-14. 12

-14.06

-13 . 92

-13.84

-13 . 39

-13.32

-11.95

-12. 14

-11.91

-11.98

-12.03

-12 . 16

-12.04

-11.89

-11.04

-11.06

-10.97

-11.03

-11 . 01

-11 . 09

-11 . 13

— 10.90

-10.85

-10.72

— 10. 38

-10.87

— 11. 67

-11 . 65

-11.62

— 11. 60

— 11. 75

— 11. 78

— 11. 54

-11.53
166 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Very recently Rajchl (1963) has found that

the luminosity coefficient rov decreases during

the meteor flight, being proportional to (1 +

pav)~x- Rajchl reached this conclusion by

analyzing only 8 meteors and by using some

approximated equations to which we may

apply the same criticisms as to Ceplecha's

work in point (2) of the introduction. The

present results show that this dependence

also does not exist.

Results for small-camera meteors.—While shower

meteors are less than one-third of all SuperSchmidt material, they are the majority among

small-camera meteors; therefore the number of

sporadic meteors available for the present

analysis is small. This is also the result of the

lack of accuracy in the decelerations of many

of them, which often leads to weights smaller

than 1. Because of their poor reh" ability,

meteors having w<Cl have been removed from

the final analysis. Therefore, only 44 sporadic

meteors are available for the analysis. The

basic results are listed in table 12 (a and b).

The difference in density between long-period

meteors and those of the Jupiter family is also

present among these meteors, as is shown in

the table. The consequent difference in the

average values of log rp/pm is estimated to be

0.25 (for Super-Schmidt meteors it was 0.30).

After the correction, the least-squares solution

gives

n=0.9±0.5, (28)

and the centroid of the distribution is

logr,/p* = -11.69±0.08

log 0=6.39 ± 0.02. (29)

The accuracy of the determined value of n is

very poor, but this value is very close to that

obtained for the Super-Schmidt meteors. For

log y=6.39 the value of log rv\pm for SuperSchmidt meteors is —11.32 ±0.04, with a

difference of 0.37 ±0.12 from the value (29) for

small-camera meteors. By dividing the

sporadic small-camera meteors into two groups,

so that the first group contains meteors having

Ceo not exceeding that of the brightest SuperSchmidt meteor, and reducing the results to

log y=6.39, we get:

log r,/pu=-11.16±0.12; €<D=0.6±0.1 (30)

log Tvlpn= — ll.86drO.ll; «. = 1.8±0.1. (31)

The difference between the two groups is quite

clear. The faint meteors have about the same

value of log rp/p^ as the Super-Schmidt (the

difference is equal to the probable error),

while the bright meteors appear much more

dense. The result is

(Pm)brtght meteors—2(Pm) faint meteors. (32)

The group of fainter small-camera meteors has

an average brightness near to that of the

Super-Schmidt meteors of the last of the 10

groups in table 8 (p. 161), which do not show

any difference from the fainter Super-Schmidt

meteors. The maximum tm for the sporadic

Super-Schmidt meteors is 0.94. Therefore we

are led to conclude that the gap in density is

close to eOD = l. It corresponds roughly to a

photographic magnitude —2.7 or to a visual

magnitude —1, which is just the limit of the

fireballs.

The value of rov.—The observational data allow

the determination of the ratio rvjp*m, but it is

impossible to separate these two parameters

except in the few cases in which we have complete evidence that the meteors are of asteroidal

origin. Only one of the discussed samples,

meteor no. 1242, has been recognized to be

certainly asteroidal by Cook, Jaccbia, and

McCrosky (1963), who found evidence that its

composition was similar to that of meteoritic

stone (pm=3.4 gcm~3). This meteor has 4

decelerations that allow values of log rP/p*

very close to one another. The weighted

average is

log rv/pl=-U.l 15 ±0.076

log 0=6.037.

(33)

The meteor does not show progressive fragmentation. Its value of x is —0.10, so that a

reduction of all the decelerations to the same

value of s does not change the value of equation

(33). The accuracy of the decelerations is very

good. The average weight p is 5.9, and w is

about 5. By assuming pOT=3.4 gem"3 and

7&=1.0, we get

log TOJ>= — 19.09 ±0.08. (34)
LUMINOUS EFFICIENCY OF METEORS

TABLE 12a.—Sporadic small-camera meteors: basic results.

167

All together

Meteors with Q<7 a. u.

Meteors with (?>7 a. u.

Meteors with «„,>!

Meteors with «„, < 1

no.

44

30

13

26

18

log t>±p.e.

6. 389 ±0.019

6. 324 ±0.017

6. 584 ±0.034

6. 378 ±0.028

6. 413 ±0.022

log Tp/pi±p.e.

-11 . 630 ±0.084

—11. 779 ±0. 096

-11 . 264 ±0. 140

-11 . 868±0. 101

— 11. 132±0. 115

«»±p.e.

1. 43 ±0. 08

1.47 ±0.09

1. 32 ±0. 16

1.82 ±0.08

0.63 ±0.07

w

3.66

4.03

2.88

4 18

2.90

Least-squares solution for all meteors: n= 1.15 ±0.5 (no correction applied for the difference of density between

short-period and long-period meteors).

Least-squares solution for all meteors, after reduction of all data to the density of short-period meteors: n=0.90

±0.5.

Although the quality of meteor no. 1242 is

very good, it is obviously unsafe to rely on a

value obtained from only one meteor. Fortunately, the recent experiences with artificial

meteors have led to other estimates of rov.

Let us examine briefly the results of these

works. McCrosky (1961) reports the results

of the firing of two shaped charges from an

Aerobee rocket at a height of 80 km. The

liners inserted in the explosive were made of

aluminum. From laboratory evidence McCrosky assumed that 2 percent (0.11 g) of the

mass was ejected with an initial speed of 14.2

kms"1. From the observed light curve, assuming the linear dependence of T on v given by

Opik for bright meteors, he found

'OB aluminum — "Ai u otul , \"^V

which seems in perfect agreement with Opik's

earlier value (8.5X10"10 scm"1). Of course

these two values cannot be compared directly,

since aluminum is not a major constituent of

meteors. Although the correction cannot be

accurately computed, McCrosky was able to

work out a lower limit for TOV. He figured out

that a maximum of 83 percent of the radiation

could be produced by the oxidation of aluminum. Moreover, he assumed that iron and

aluminum had equal efficiency per atom and

that all the meteor radiation is caused by the

iron, which, following Opik (1958), he estimated

to be 15 percent of the meteor material. The

final result was

(36)

85 times less than Opik's value and about three

times larger than Cook's value estimated on

the basis of the motion of meteoric trains.

Since McCrosky used a color-index correction

— 1.8 for passing from photographic to visual

magnitude, we can go back to TOV, which in

Jacchia's units turns out to be

log TOP=— 20.0. (37)

Since this value is considered an extreme lower

limit, McCrosky warns that a value 100 times

larger (—18.0) is not precluded by the results

of his experiment.

More recently McCrosky and Soberman

(1962) have produced an iron meteor at a

height of about 70 km by accelerating to a

velocity of about 10 kms"1 a stainless-steel

pellet (m ~ 2.2 g) by an air-cavity charge

attached to the nose of a 7-stage Trailblazer

rocket. The primary purpose of the experiment was to determine the meteor luminous

efficiency. The result is

.teei=8X 10~19 0 mag (38)

The estimated error involved is such that we

may write, in logarithmic form,

(log rop)8talnle,»steel=-18.10±0.1, (39)

which represents a very accurate result. Three

corrections are necessary, however, to convert

the value of equation (38) to a luminosity

coefficient valid for meteoric material. The

first of these is caused by the presence in the

pellet of about 20 percent chromium. Me-
168 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 126.—Sporadic small-camera meteors: log rjpl, as a function of meteor velocity.

Velocity, v

(105 cms-1)

10-20

20-27

27-40

40-70

no.

11

15

10

8

log v

6. 191

6.354

6.506

6.734

log T,

-12.

— 11 .

-11 .

— 11.

JA

105

310

652

305

S.d. of mean

0.226

0.204

0.206

0.341

Gog Tp/pDeorr.

-12. 105

-11.359

-11.708

-11.540

(a

1.

1.

1.

1.

,±S.d.

2±0. 2

6±0. 2

5±0. 3

4±0. 2

w

4. 45

3. 61

3.67

2.64

(log oorr. is the mean value of log Tp/pjj, after reduction of all data to density of short-period meteors.

Crosky and Soberman conclude that the

luminous efficiency of chromium should be

roughly equal to that of iron, so that the

value of equation (38) should also be true for

iron. They warn, however, that the probable

extremes are —18 and —18.54. The second

correction results from the possibility that the

pellet is not completely ablated at the end of

the detected luminosity. Fortunately, after a

detailed analysis of the ablation experienced by

the pellet, McCrosky and Soberman find that

no significant correction is necessary for terminal mass in their case.

The last and most important correction is

for the composition of meteoroids. There is

observational evidence from the spectra of

low-velocity meteors that practically all the

light seen on a blue emulsion is caused by iron

emission. Therefore the luminous efficiency

of iron would give that of meteoric material

if the iron content in meteoroids were known.

Although the composition of iron and stony

meteorites is well known, no truly reliable

information is available on the composition of

cometary meteoroids, which are by far the most

common among meteors. McCrosky's extreme

estimates are —20.10 and —18.80, which

correspond to relative iron abundances of 1

percent and of 20 percent, with the warning

that these should be lower limits because other

elements will contribute, though little, to the

luminosity. It is not unreasonable at the

present stage to assume that the iron percentage

in cometary meteors is equal to that of meteoritic stone, i.e., 15.4 percent (Opik, 1958).

In this case the result is

A detailed discussion for three asteroidal

meteors, one iron and two stone, has been done

by Cook, Jacchia, and McCrosky (19G3). The

density of these meteors being known, they

were able to find the value of TOP: log TOJ>=

—18.37 for stone. They removed the discrepancy from the value of equation (40) by

adopting for the three meteors 7-4=0.92

instead of 7-4=0.6. If we assume that the

radiation of Al 0 is negligible for meteors, the

value (37a) becomes —19.23, which agrees

fairly well with the preceding results. In any

case, the reliability of this value has to be

considered much lower than that obtained with

the Trailblazer. The value of equation (40)

has also been accepted by Whipple (1963) in

his recent paper on meteoroids and penetration.

By putting the value of equation (40) into

equation (17a), we find the average density of

the Super-Schmidt sporadic meteors with Q<CJ

a.u. The result,

pm=0.25 gem"3, (41)

log r o p= —18.91. (40)

is in good agreement with the value obtained

theoretically by Whipple (1955) on the basis of

his well-known cometary model. He found, in

fact, that pm must lie between 0.2 and 0.4 gcm~3.

If we reverse the procedure and assume pm=

0.3±0.1 gem"3, we get from equation (17a)

log r o p= —18.76 with the extreme values

— 18.51 and —19.11.

Another experiment with a Trailblazer

(McCrosky, 1963, private communication) has

confirmed the previous result, thus enforcing

the reliability of the method. It seems that

the efficiency of chromium in producing light

is larger than that of iron. In this case the

value of equation (40) should be slightly
LUMINOUS EFFICIENCY OF METEORS 169

decreased. In view of the uncertainties still

present, the author proposes as a best estimate

for log TOP the round value log TOV= — 19.0.

The masses of meteoroids must therefore be

computed by using

r p=lX10- 1V, (42)

with v expressed in cms"1 and Iv in units of zero

magnitude. The error in the mass resulting

from the intrinsic error of TOV may be estimated

to be of the order of 2 or even larger. From the

preceding discussion it is clear that it is more

likely that equation (42) gives overestimated

values of rp than vice versa. The error in

relative masses, caused by the uncertainty in

the adopted value of n, between 10 and 70

kms"1 does not exceed 20 percent.

If we express the luminous efficiency TV in

physical units as the ratio between the energy

radiated in the blue-emulsion spectral range and

the kinetic energy of the parent meteoroid, we

have

rp=5.25X10-10». (43)

Therefore rv turns out to be 0.002 for a velocity

of 40 kms"1.

It is interesting to observe that the dependence of TP on the velocity appears to be linear,

as Opik predicted in his earlier work (1933) for

bright meteors. The linear dependence is,

however, also valid for fainter meteors, and

here Opik's theory fails. The present best

estimate of the coefficient of the luminous

efficiency TOP is about 8 times smaller than

Opik's 1933 value

 10

 and only 5 times smaller

than the value he proposed in 1955.

On the basis of equation (42) we can evaluate

the mass of a zero visual magnitude meteoroid

at a given velocity. After converting visual

into photographic magnitude by means of

equation (4), we may use the following formula,

empirically derived by Jacchia (1958) from

photographic meteors with smooth light curves:

(44)

v cos ZR

where Ivm is the photographic intensity at

maximum light, and ZR is the zenith angle of

the meteor path. The integrated brightness

Em and the meteor mass outside the atmosphere

me are related [see equation (10)] by

ma,=—-v (45)

At r=40 kms * and cos Za=0.6 (mean value

of cos ZR for the Super-Schmidt meteors) mm

is 0.84 gram.

The ionizing efficiency.—It appears that at

present the uncertainty concerning the ionizing

efficiency # of meteors is an order of magnitude

larger than the uncertainty affecting the luminous efficiency rp. It therefore seems reasonable to deduce the ionizing efficiency from the

present results on TP and from experimental

data. As a result of their program of combined

radio and photographic observations of meteors,

Davis and Hall (1963) were able to determine

the ratio Ip\q_ for 7 meteors, q being the electronic line density of the meteor trail. Their

result was log Ip/q=— 3.96±0.14 at »=32.2

kms""1.

The ionizing efficiency is defined by an

equation analogous to equation (1):

IQ=—Td

v

^~j~' (46)

/„ being the power going into the production of

electrons. The electronic line density is given

by

j8 being the probability of ionization of an

ablated meteor atom of mass /x. The terms /3

and TQ are related by

2 *

(48)

io Harvard meteoric masses are, however, only 6.6 times smaller than

the values computed on the basis of equation (42). See footnote, page 145.

where $ is the mean ionization potential; <t»~7

ev (Cook and Mill man, 1955).

From equations (2) and (46), we have

^=^ 3£ - (49)
170 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

The average mass n of a meteor atom is

3.8X10"23 g (Opik, 1958). Therefore, at r=32.2

kms"1, we obtain

0=0.010; T4 =5.6X10" 4.

The errors in TP and Ip!q cause an uncertainty

in 0 of a factor of 3. The values of ra and /8

deduced by Davis and Hall (1963) were based on

Opik's late evaluation of the luminous efficiency and are consequently about three times

larger.

A preliminary result of a study in progress

on the dependence on velocity of the ionizing

efficiency suggests j8~y4. Accordingly, we

have, in cgs,

0=9.5 • 10"2V.

Acknowledgments

I am indebted to Benno Benima for preparing programs for the IBM 7094 computer

covering the numerical computations of this

work. I am especially grateful to Drs. Luigi G.

Jacchia and Allan F. Cook for many helpful

discussions and suggestions in the course of

this work. I also acknowledge useful comments

on the original manuscript by Drs. Richard E.

McCrosky and Richard B. Southworth.

Particular thanks are due to Dr. Jacchia for

permitting the use of the unpublished data on

his Super-Schmidt meteors.

References

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1960. Light curves of meteors. Nature, vol. 187,

pp. 675-676.

1961. Anomalies in the light curves of meteors resulting from fragmentation. Nature, vol.

190, pp. 896-897.

BAKER, R. M. L., Jr.

1959a. The transitional aerodynamic drag of meteorites. Astrophys. Journ., vol. 129, pp.

826-841.

1959b. Effect of accommodation on the transitional

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BAKER, R. M. L., Jr., and CHARWAT, A. F.

1958. Transitional correction to the drag of a sphere

in free molecule flow. Phys. Fluids, vol. 1,

pp. 73-81.

CEPLECHA, ZD.

1958. On the composition of meteors. BAC, vol. 9,

pp. 154-159.

1959. On the colour index of meteors. BAC, vol.

10, p. 39.

CEPLECHA, ZD., and PADEVET, V.

1961. The beginning of rapid evaporation of meteors of different dimensions. BAC, vol.

12, pp. 191-195.

COOK, A. F.

1955. On the constants of the physical theory of

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Abstract

The ratio of the photographic luminous efficiency TV to the square of the density pm of about 400 Super-Schmidt

meteors, precisely reduced by Jacchia, has been computed directly from the observational data. Fragmentation

has been taken into account in the computation.

After critically reviewing previous papers on the subject, the author discusses his assumptions and the procedure

he employed. The theory allows the evaluation of the quantity rjp^. It is possible to determine TP only in the

few cases in which the meteors show positive evidence of being of asteroidal origin. The correct dependence of rp

on v is found, after allowing for the different mean densities of meteors in short-period and long-period orbits, since

the meteor density is independent of the velocity v. The author shows that a power law TP=TOV v

n

 best represents

TP as a function of v. The detailed behaviour of TP may actually be extremely complicated, since most of the meteor

light comes from emission lines whose intensity may not vary uniformly or even smoothly with velocity. The exponent n turns out to be 1.0±0.15 for both faint and bright meteors. The best small-camera meteors have been used

for comparison. The value of TOP comes out to be about 1X10~JO 0 mag g~' cm"1 s4, which agrees very well with

the value inferred by McCrosky and Soberman from the results of artificial meteors. The masses computed at

Harvard on the basis of Opik's original theory therefore appear to be about 6 times smaller than those obtained by

using the value of the luminous efficiency derived in the present paper. At a velocity of 40 kins"1 the photographic

luminous efficiency, expressed in physical units as the ratio of the energy radiated in the blue spectral range to the

kinetic energy of the meteoroid, is about 2X 10~3. At the same velocity the mass of a zero visual magnitude meteoroid

is roughly 0.8 g. The ionizing probability 0 of an ablated meteor atom can be estimated from rp and from some

results of Davis and Hall (1963) on combined radio and photographic observations of meteors; at t;=32 kms-1 /9

turns out to be 0.01.

The analysis shows no dependence of TP on mass or brightness and fails to detect any appreciable change of TP

in the course of a meteor trajectory. Moreover, the results show that TP is independent of atmospheric density.

Luminous efficiency does seem to depend on fragmentation in the sense that nonfragmenting meteors appear to

produce light more efficiently than those that crumble easily. The results obtained for small-camera meteors are in

general agreement with those obtained for Super-Schmidt meteors, but it appears that fireballs must have a density

about twice that of fainter meteors. The mean relative density of meteors in the main recurrent showers has been

also determined. Each shower appears to have its peculiar density. Jacchia obtained a similar result from an analysis of points near the beginning of meteor trajectories.

172