SMITHSONIAN

CONTRIBUTIONS

to

ASTROPHYSICS

Smithsonian Institution

Astrophygical Observatory

Volume 10, Number 3

Meteor Masses

and

Luminosity

by Franco Verniani

Washington, D. C.

Smithsonian

Contributions to Astrophysics

VOLUME 10, NUMBER 3

METEOR MASSES AND LUMINOSITY

by FRANCO VERNIANI

SMITHSONIAN INSTITUTION

Washington, D.C.

1967
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FEED L. WBIPPLE, Director,

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Smithsonian Institution.

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

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Meteor Masses and Luminosityl

Franco Verniani *

Introduction

A comprehensive study of the luminous efficiency of meteors was made by the author (Verniani, 1964a, hereafter referred to as Paper I)

using the photographic material gathered over

many years under the Harvard Photographic

Meteor Project. Assuming, as was customary,

a dependence of the luminous efficiency r, on

the meteor velocity v of the form T9—T0PV

U

,

the author found n=1.0 with a probable error

of ±0.15, and T# , = 1XK)- W 0 mag g~l cm"1

sec4 with an uncertainty of the order of a factor

of 2 or more. The results described in Paper I

showed no dependence of the photographic

luminous efficiency r, either on the mass or on

the brightness of meteors and no appreciable

systematic change of r, in the course of the

meteor trajectories.

At the time that this study was carried out,

in 1963, the general analysis of the atmospheric

trajectories of the precisely reduced meteors

was in its early stages. By the time the general

analysis was completed by Jacchia, Verniani,

and Briggs (1965), the knowledge acquired

suggested the possibility of improving the

accuracy of the luminous efficiency, both in

its coefficient TOP and in its exponent n. The

new determinations of r*, and n will be discussed separately in the following sections.

1

 This work was supported in part by Grant NSR09-015-033 from the National Aeronautics and Space

Administration.

1

 Smithsonian Astrophysical Observatory, Cambridge,

Mass., on leave from Centro Nazionale per la Fisica

dell'Atmosfera e la Meteorologia del C.N.R., Rome,

Italy.

The velocity dependence of the luminous

efficiency

The photographic luminous efficiency T, is

defined, as in Paper I, by the so-called luminosity equation:

j =z__\Tf^dmi (ls

where /, is the luminous flux3 in the wavelength range of the blue plate (roughly 0.38M

to 0.55M), and m is the meteor mass.

The drag equation, expressing the conservation of momentum in the interactions of the

meteoroid with the molecules of the atmosphere, is

m Pay> \* /

where a and pM are the deceleration and the

density of the meteor, respectively, y is the

drag coefficient, A is the shape factor, and

pa is the atmospheric density. By integrating

the luminosity equation and combining the

result with the drag equation, we obtain

2 E ( a V,

>\A*J

where

E=

5» \yAPat?J (3)

is the integrated brightness, v is a value of v at

some instant between t and tB, and tB is the

* In Paper I, the quantity Iv was referred to as the

photographic luminous intensity, a term generally

accepted in the literature on meteors. The new name—

flux instead of intensity—is in agreement with the

recommendations of the Commission for Symbols,

Units, and Nomenclature of the International Union

of Pure and Applied Physics (Wolfe, 1962).

181
182 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS YOU to

instant at which the meteor luminosity ceases

to be detectable on the photographic plate. The

observational data allow the determination of

the velocity v, deceleration o, integrated brightness E, and height z; the U.S. Standard Atmosphere 1962 (1962) gives p, as a function

of 2. For a given meteor, v varies only slightly

with t. It was shown in Paper I that the velocity

at m=mm/2 represents an adequate approximation of v. Most of the meteors photographed

with the Super-Schmidt cameras have probably

experienced free-molecule flow (see Paper I and

Jacchia, Verniani, and Briggs, 1965), and

therefore justify the use of the constant value

1.1 for 7. The shape factor is unknown. In

Paper I we adopted A=1.21, the value for a

sphere, as used in all earlier Harvard works.

For any shape other than spherical and with

random orientation, A will have a somewhat

larger value. Here, following Jacchia, Verniani,

and Briggs (1965), we shall assume A= 1.5.

Equation (3) allows us to compute r,/pjj, from

the observational data. This equation, however, is valid only when the "dynamic" mass

appearing in the drag equation is equal to the

"photometric" mass obtained from the integration of the light curve. When fragmentation

occurs, the values of rjpm computed from

equation (3) require a correction, since the

dynamic mass becomes smaller than the photometric mass. It was shown in Paper I that the

relation between the values of r^p\ computed

at two different points of the trajectory is

(log TJ,/pi).= «-« O ), (4)

where x is the fragmentation index introduced

by Jacchia (1955) and defined by

X=-j- log —>

Ja & a

(5)

a being the observed deceleration, aT the deceleration computed from the drag equation and

the photometric mass, and 8 the mass-loss

parameter s=log [(m»/m) — 1]. Let us consider

a meteor for which it was possible to determine several decelerations at different points

of its trajectory. The difference between the

values of 8 corresponding to the first and the

last deceleration is of the order of 1. An average

value of x is 0.25. Consequently, for a typical

long meteor, the fragmentation causes the

value of rjp'm computed from equation (3)

with the last deceleration to be larger by a

factor of 6 than the value computed with the

first deceleration. Equation (4) would allow us

to reduce all the values of T,/p2

m computed

from equation (3) to the point of beginning of

fragmentation, if only we could know where

this point is for each meteor. Since, generally,

this point is not known, no correction was

applied in Paper I to the values of rjpl, computed from equation (3). Although it was

shown empirically in Paper I that the system

of weights * employed in the analysis greatly

reduced the effects of fragmentation and allowed

a correct determination of the exponent n, it

is clear that the use of a direct correction would

improve the accuracy of the determination of n.

In the general analysis of Jacchia, Verniani,

and Briggs (1965), the meteor decelerations

were used to derive atmospheric densities in

order to study their seasonal fluctuations. The

atmospheric densities, computed from the drag

equation and the photometric masses, had to

be corrected for the fragmentation effects by

reducing them to a given point in the trajectory. Using the least-squares method, Jacchia,

Verniani, and Briggs (1965) showed that the

scatter in the differences between the atmos-

* Velocities and heights are quite accurate, even in

the wont cases. Decelerations, on the contrary, are

not so well determined and their accuracy varies from

meteor to meteor (see Jacchia, Verniani, and Briggs,

1965). As a consequence, in the determination of *•„/<£

for each meteor, a weight pt was given to each deceleration pertaining to the meteor. The weight p,- is

defined by

where p.e.< is the probable error of the deceleration ait

n< is the number of segments used in the computation

of a,-, and f(x) is defined by

(2.51og*-2.1)J. (7)

The factor (,mjmmy

t

 is introduced to decrease the

importance of decelerations near the end of trajectories,

where fragmentation might be so severe as to make a

correction meaningless. In our analysis we have assigned a weight w— (Z<Pi)xn to the value of rvlt?m for

each meteor. A thorough discussion on this system of

weights is given in Paper I.
METEOR MASSES AND LUMINOSITY 183

pheric densities computed from meteor decelerations and the densities given by the U.S.

Standard Atmosphere 1962 (1962) was minimized for so=O. It is clear that the same value

8o=O applies for the correction of the values

of TF/pJ,. We have then

(log rp/pi),.0=(log rp/p3J,-3X8. (8)

In the analysis described in this section,

values of (log rjpjj reduced to «=0 are used

throughout.

Of the 413 original precisely reduced SuperSchmidt meteors, 12 did not allow the computation of the fragmentation index. We were left

with 401 meteors, 281 sporadic and 120 shower

meteors. Since each shower has its own peculiar

density pm> the sporadic meteors only can be

used in the determination of n. In Paper I, 33

sporadic meteors having clearly anomalous

values of r,/p7

m were eliminated from the

analysis. Some of those meteors showed either

an abrupt beginning or a very short, flarelike

light curve. For both these kinds of meteors,

the use of the fragmentation index is generally

not sufficient to correct the value of rjp\ (see

Paper I). In the present analysis we have

eliminated from the sporadic sample all abruptbeginning meteors (19), all flarelike meteors

(10), all meteors with x^0.7 (25), all meteors

with log <r> —10.6 (4) or without any computed

value of o- (1) (the ablation coefficient a is

defined by the mass equation ar—m/mvb), the

asteroidal Meteor No. 7946, and a meteor

having IP<1 . We were left with 220 sporadic

meteors.

The values of Gog T,/P^,_0 range, in cgs

units, from —9.68 to —13.55; the weighted

average is —11.44 and the standard deviation

of 1 observation is ±0.80. Since the scatter is

so large, it is clear that we can reach a better

accuracy in the determination of n if we use

only the best part of the data. Some of the

scatter, due to the differences in velocity, may

be removed by the provisional assumption that

n = l in order to compute the values of (log

TOP/P2I)»-O- These quantities range from —16.52

to —20.04; the average is —17.88 and the

standard deviation of 1 observation is ±0.75.

We will discard all those meteors whose values

of (log T,,/P^),.O differ from the average by

more than 1.5 times the standard deviation of

1 observation; i.e., more than ±1.12. We are

thus left with 189 meteors whose values of

flog T0PIP>J,.O range from —16.78 to —19.01.

In this manner, by eliminating only 31 meteors,

or less than 15 percent of the material, we

considerably reduce the scatter, from a total

range of 3.52 to one of 2.23. Also the standard

deviation of 1 observation is reduced, from

±0.75 to ±0.52. As far as the Cog T,/P£),_0

values are concerned, they range now from

— 10.04 to —12.84, where the standard deviation of 1 observation is ±0.56.

Before proceeding to the determination of n,

we shall consider in some detail the causes of

the large scatter found in the values of

(log r9fpD,m0. The error A in one individual

value of Gog T,//^),. O can be easily expressed

as a function of the uncertainties in the quantities used for its determination, as:

(9)

on the assumption that the errors in the various

parameters are independent. The known accuracy of the data yields the following rough

estimates:

£-0.05; f=0.01; £=0.1; ^=<U5;

^=0.1 ; ^-0.15 ; ^-0.003 ; x=0.2;

As=0.05; «=—0.2; Ax=0.1.

By substituting the preceding values into equation (9), we obtain A ==0.77, in good agreement

with the standard deviation of 1 observation

equal to ±0.80. This shows that although the

individual observational quantities are fairly

accurate, our lack of direct knowledge of A and

7, the variability of the atmospheric density at

a given height, and the propagation of the

errors cause the large scatter in the values of

(JP/PO*-O> which is a quantity far remote from

the directly observed data.

The dependence of Gog TP/P2J$,0 on velocity

for our 189 meteors is shown in table 1. Meteors
184

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

in short-period orbits are known to be denser

than those in long-period orbits (Paper I).

Since there is a correlation between velocity and

type of orbit, we must account for the consequent variation of pm with velocity before we

can determine the value of n. A parameter

that describes conveniently the type of orbit

is the aphelion distance Q. Table 2 shows mean

values of (log T^P^^O for 7 groups of meteors

in order of increasing Q. The values of some

groups are directly comparable, since their

mean velocity is approximately the same; it is

immediately apparent that, when Q<5 a.u.,

the density increases sharply as Q decreases.

We see also that the density of meteors having

Q greater than about 10 a.u. is approximately

constant.

In order to compare all groups, we have to

assume a provisional value of n and then proceed by successive approximations. In the last

column of table 2, we reduced the values of

(log Tp/p'Jt.o to log t?=6.35 by assuming provisionally n=l. Dividing the meteors in different groups of increasing Q, we find that the

transition from the region where pm decreases

with increasing Q to the region where pm is

roughly constant occurs between 5 and 6 a.u.

We can very well approximate the dependence

of (log Tp/p*m)tm0 on Q simply with two straight

lines:

(log Tp/pi)..0=-12.252+0.18Q Q<5A a.u.

(10)

(log r,/pl)..o=-l 1.280, Q>5A a.u. (11)

TABLE 1.—Mean values of (log T»/P*).H> and other physical characteristics at a function of velocity v for 189 sporadic

Super-Schmidt meteors

Velocity

range (10*

cm sec-1)

10-20

20-25

25-30

30-35

35-45

45-55

55-65

65-73

All

log »±s.d.

(cm see-1)

6.228±0.008

6.345±0.004

6.436±0.005

6.506 db 0.004

6.603±0.008

6.694 ±0.008

6.777 ±0.005

6.838 ±0.004

6.446±0. 014

(logT,,//£)_o±s.d.

(0 mag g-'em'sec1)

-11.477± 0.093

—11.469± 0.097

-11.372± 0.088

-11.201 ±0.094

— 11.023 ±0.142

-10.872± 0.107

-10.864± 0.130

-11.018 ±0.135

-11 . 288± 0.041

c±s.d.

-0.40±0.08

-0.51±0.07

-0.44±0.08

-0.20±0.10

-0.48±0.08

-0.07±0.14

-0.27±0.10

-0.14±0.11

- 0 37±0 03

log <r±8.d.

(cm"lsecl)

-11.09±0.03

-11.15±0.03

-11.24±0.03

-11.28±0.05

-11.29±0.04

-11.43±0.05

-11.38±0.04

-11 . 31±0.05

—11.22±0. 05

X±s.d.

0.12±0.03

0.17 ±0.04

0.13 ±0.03

0.07±0.04

0.26±0.05

0.14 ±0.05

0.24 ±0.05

0.26±0.06

0.15 ±0.06

No.

obs.

40

33

34

21

18

12

18

13

189

Wt.

5.16

4.92

4.66

5.30

3.84

3.96

3.10

3.02

4.50

TABLE 2.—Mean values of (log r-o and other physical characteristics as a function of the aphelion distance Q

for 189 sporadic Super-Schmidt meteors

Range

of the

aphelion

distance

Q (a.u.)

0 - 3

3 - 4

4 - 5

5 - 7

7- 15

15-100

100-oc

Q±s.d.

(a.u.)

2.27±0.24

3. 59 ±0.06

4.57±0.05

5. 72±0. 08

9.87±0.46

33.7 ±3.0

(log TP/pi)^n±8.d.

(0 mag g-* cm*

aet?)

-11.77±0.06

-11.60±0.11

-11 . 46±0.10

—11. 25±0. 08

-11 . 05±0.06

-11.10±0.11

-10.86±0.10

log v±s.d.

(cm sec"1)

6. 36±0. 06

6.39 ±0.02

6. 33 ±0.02

6.37±0.02

6. 53 ±0.03

6.64±0.03

6. 71 ±0.03

log«r±s.d.

(cm~2 sec2)

-11.13±0.08

-11.17±0.04

-11.17±0.04

-11.19±0.03

-11.26±0.04

-11.32±0.03

-11 . 31±0.06

No.

obs.

10

24

38

42

27

29

19

Wt.

4.7

4.4

4.9

5.0

4.5

3.8

3.7

(lOg Tp/pJ,),-0, t-t»

(0 mag g-* cm*

sec1)

-11.78

-11.64

-11.43

-11.27

-11 . 23

-11.38

-11.22

Note: logo'=6.350.
METEOR MASSES AND LUMINOSITY 185

By reducing, in turn, the mean values of

(log T,/p2

m),»0 of the groups of increasing

velocity to the density of long-period meteors

(i.e., with Q>5A a.u.), we obtain the values of

table 3, plotted in figure 1. It is apparent from

figure 1 that a straight line fits well all the

points, with the exception of the very last

one for meteors with v^>65 km sec"1. A leastsquares fit for all points yields n=0.87, with a

standard deviation of ±0.14. The residual of

the last point (0.18) is about 4 times greater

than the average and twice as large as the

next largest residual. Therefore, it is clear that

a better fit for the velocity range 10 to 65

km sec"1 is obtained by disregarding the last

point, corresponding to a mean velocity of

69 km sec"1. The result is

n=0.99±0.12. (12)

TABLE 3.—Mean values of (log TP/PJ,),H) as a function of

velocity, reduced to the density p'm of the long-period

meteors

Velocity

range

(10» cm

8CC-')

1O-20

20-25

25-30

30-35

35-45

45-55

55-65

65-73

log v

(cm

sec"1)

6.228

6.345

6.436

6.506

6.603

6.694

6.777

6.838

Qog rp/pi).-<).,„,-,;

(0 mag g-$ cm4

see1)

— 11.345

-11.327

-11.216

-11.090

-10.963

-10.872

-10 . 864

-11.018

No.

obs.

40

33

34

21

18

12

18

13

Total

wt.

206.3

162.3

158.4

111.3

69. 1

47.6

55.8

39.3

-10.6

-11.6

15 70

FIGURE 1.—Mean values of (log rp/p^),_o, reduced to the

density of long-period meteors, as functions of velocity for

189 sporadic Super-Schmidt meteors.

The doubt may arise that the provisional

value of n used to find the dependence of pm on

Q affects its final determination. To investigate

this point, we have repeated the analysis starting with n=0 . The first approximation yields

n=0.6; proceeding with successive approximations we finally obtain the same value as in

equation (12).

We have so far neglected the possible dependence of (log TP/P^),_O on mass, fragmentation index, and ablation coefficient. The following results justify our procedure. It was

shown in Paper I that a dependence of TP on

mass would produce a dependence on the

brightness of the meteor. As a measure of the

brightness, we will use

Table 4 lists average values of (log Tp/p2

m),.o

for groups of meteors in order of increasing c .

It is immediately apparent that Tp/p2

m is independent of 6« and consequently of mass. The

result, together with equation (12), enables us

to compute meteor masses as

T0P J-m

(13)

The independence of TP on mass is confirmed

by table 5, which shows mean values of

(log Tp/p2

m),.o for groups of meteors with increasing mass. We have assumed provisionally,

from Paper I, the value T O P=1X10" 1 9 0 mag

g"1 cm"3 sec4.

In Paper I, a correlation was found between

TP and the fragmentation index x in the sense

that nonfragmenting meteors appeared more

efficient in producing light than easily crumbling

meteors. Most of this correlation was due to the

group of meteors with the highest degree

of fragmentation (average x=0.91). The present

analysis, limited to meteors with x^O-7, does

not reveal any correlation between TP and x»

as can be seen in table 6. It appears, however,

from the results listed in table 7 that some

correlation exists between (log Tp/p2

m),^0 and

the ablation coefficient er. Though we could

attribute this dependence either to TP or to

pm, it appears much more likely that TV is responsible. In fact, should we assume the op-
186 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

 yoL

- "

TABLE 4.—Mean values of (log rjpi),-, as a function of the brightness «. for 189 sporadic Super-Schmidt meteors

(0 mag)

a

U

2

1

i

M

•e

-1.4 0 to

— 0.90 to

-0.7 0 to

— 0.50 to

— 0.30 to

0. 00 to

-0.90

-0.70

-0.50

-0.30

-0.00

1.00

•it 00

 C

-H©

8

-1.05±0.03

-0.80±0.01

-0.61±0.01

-0.40±0.01

-0.14±0.01

0.32±0.05

•

0

t

-11.

-11.

-11.

-11.

-11.

-11.

47 ±0.14

34 ±0. 10

24±0.10

37 ±0.10

13 ±0.08

27 ±0.10

<r

"O 8

£

i?

6.34±0.04

6. 37 ±0.03

6. 48 ±0.03

6. 46 ±0.03

6. 52 ±0.03

6. 46±0.04

- 1 1

- 1 1

- 1 1

- 1 1

— 11

- 1 1

•6 3

mm

s

-

. 19±0.07

. 14 ±0.03

.23±0.03

. 16±0. 04

.29 ±0.04

.26±0. 04

.1

• c

jt,p

1©

-11 .

-11 .

-11 .

-11 .

- 1 1

-11 .

26

21

14

23

08

20

• 5

li

""* B

-11.2 6

-11.2 3

-11.2 7

-11.3 3

-11.2 5

-11.3 0

o

6

55

19

36

36

34

32

32

Note: log v' — 6.350. p'm is the density of the long-period meteors (Q>5.4 a. u.).

J

+C9

/ , dt, where /„ is the absolute photographic flux in units of 0-mag stars.

TABLE 5.—Mean values of (log rp/pi),_o as a function of the mass mm for 189 sporadic Super-Schmidt meteors

Mass range

(g)

0.01- 0.1

0.1 - 0.2

0.2 - 0.5

0.5 - 1.0

1.0 -34

mm±s.d.

(g)

0.05±0.00

0.15 ±0.00

0.34±0.01

0.72±0.02

4.8 ±1.3

Gog rJpUr-o

±s.d.

(0 mag g~l

cm* se<r)

-10.94 ±0.07

-11.18±0.08

-11.44±0.11

-11.42±0.09

-11.39±0.10

log v± s.d.

(cm sec"1)

6.68±0.02

6.50±0.02

6.44±0.02

6.36±0.02

6.30±0.02

log a ± s.d.

(cm-* seo»)

-11.36±0.0 3

-11.23±0.0 2

-11.18±0.03

-11.17±0.0 4

-11.16±0.0 4

0og rjt?m)^.

(0 mag g->

cm* sec*)

-10.9 2

-11.09

-11.23

-11.3 0

-11.29

(logf,/pi).-o.

(0 mag g-»

cm* sec1)

-11.2 5

-11.2 5

-11.3 2

-11.3 1

-11.2 3

No.

obs.

47

36

35

38

33

Note: log v'=6.350. p'm is the density of the long-period meteors (Q>5.4 a.u.).

posite, i.e.f pm is a function of <r, the results of

table 7 would show that meteors of higher

density have larger values of a and therefore

undergo a mass-loss process more anomalous

with respect to the single-body theory than

those meteors that are less dense and more

crumbly. We are therefore led to admit, as we

did in Paper I, that rop increases as <r decreases;

this means that meteors having light curves in

better agreement with the classical theory

produce light more efficiently. The determination of the true dependence of (log TP //4),. O

on <r is made more difficult by the correlation

between a and velocity. Once again we have

to use a method of successive approximations.

The final results, listed in the last columns of

table 7, are plotted in figure 2. It is now clear

that a part, although small, of the dependence

of (log Tp/p^,_0 on velocity is actually due to

the dependence of TOP on a. As far as the applications are concerned, this can be safely

ignored, since it is the complete dependence of

T, on v that counts. Should we, however, wish

to distinguish between the true dependence on

v and that due only to <r, we find roughly:

r,'^Vi7V-0138. (14)

The probable errors of the two exponents are,

respectively, ±0.09 for the v exponent and

±0.12 for the a exponent.
NO. 3 METEOR MASSES AND LUMINOSITY 187

TABLE 6.—Mean values of (log TP!P\,),^O as a function of the fragmentation index \for 189 sporadic Super-Schmidt meteors

Fragmentation

index

range

X < 0

0. 0-0. 1

0. 1-0. 2

0. 2-0. 3

0. 3-0. 4

0. 4-0. 7

X±s.d.

- 0 . OS ±0.01

0.04 ±0.00

0. 14±0. 01

0.24+0.01

0. 34±0. 01

0.51 ±0.02

(lOg Tp/pl),-0

±8.d.

(0 mag g~3

cm4 sec3)

-11.25±0.08

-11.35±0. 09

-11.25±0. 11

-11.32±0. 14

-11 . 31 ±0.13

-U.24±0 . 09

log r±s.d.

(cm sec~')

6. 43 ±0.03

0.43 ±0.03

6. 42 ±0.03

6. 43 ±0.04

6. 48 ±0.04

G. 51 ±0.04

log <r±S.d.

(cm"! sec2)

-11 . 19±0. 03

-11.24±0.03

-11.22±0.04

-11 . 21 ±0.04

-11.25±0. 05

-11 . 19±0.03

(lOg Tp/pi),_0,

Pm~fm

(0 mag g"3

cm4 sec3)

-11 . 10

-11.22

-11.14

-11.2 1

-11.18

-11.17

flog TP/pi),_o.

(0 mag g~3

cm4 sec3)

-11.2 5

-11.30

-11.22

-11.30

-11.31

-11.3 3

Mo.

obs.

39

36

32

24

29

29

Note: log t/ = 6.350. p'm is the density of the long-period meteors (Q>5.4 a.u.).

TABLK 7.—Mean values of (log TP /P1),_0 a« a function of the ablation coefficient <r for 189 sporadic Super-Schmidt

vieleors

Ablation coefficient

range (cm~* sec*)

-10 . 60 to -11.00

-11 . 00 to -11.15

-11 . 15 to —11.30

-11 . 30 to -11.45

-11 . 45 to -12.05

mm

t!a

*

-10. 88 ±0.02

-11.08±0.01

-11. 22±0.01

-11.36±0.01

-11.57±0.03

J

" e j <*>

*?£

bc£.

-11.67±0. 12

-11.46±0. 09

-11. 19±0. 07

-11.09±0. 07

-11. 12±0. 10

6. 29 ±0.03

6. 38 ±0.02

6. 45 ±0.02

6. 53±0. 03

6. 59 ±0.04

<Q

be ^

_O ^^

-11.48

-11.34

-11.10

-11.02

-11.03

jf

* °

iT« bC

b0 S

o

-11.41

-11.37

-11.19

-11.20

-11.27

'E

 S

I "bC

r* be

-2: °*

be S

o

-11.43

-11.37

-11 . 18

-11 . 16

-11.22

m

O

o

25

45

62

32

25

i i

1

i

25

I

45

1

• 32

62

.

25

— -114 -

-108 — 11.0 -11.2 -11.4 -11.6

log <T (cm 2 sec 2)

FIGURE 2.—Mean values of (log Tp/p^),_o, reduced to constant velocity and to the density of long-period meteors,

as functions of the ablation coefficient a for 189 sporadic

Super-Schmidt meteors.

Since <r varies only slightly with Q, we would

not expect that the use of equation (14) to re263-404—67 2

duce the values of (log TP/p^),.o to the constant

density of meteors with long-period orbits

would make any difference in the final value of

n. A check has nevertheless been made and the

result has shown that no change in the final

value of n occurs. The value n=1.0 can therefore be considered well established. The standard deviation ±0.12 corresponds to a probable

error of ±0.08, one-half of the probable error

of the determination made in Paper I. In view

of the irregular dependence of rv on a (see

fig. 2), the constant exponent of a in equation

(14) provides only a rough approximation; for

practical applications it is advisable to use only

the apparent dependence of TV on v as expressed

by rp=Topv.

The luminous efficiency coefficient TOP

In Paper I the author suggested the adoption
188 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

of the value rop= 1X10"19 0 mag g"1 cm"8 sec* for

computing meteor masses by means of equation (13). This value, extensively used since

then, was based only on the data of one asteroidal meteor (Harvard Meteor No. 1242) and

on the extrapolation of the results from the

artificial meteors of McCrosky and Soberman

(1963). To these we can add now the results

from two other asteroidal meteors, namely,

Harvard Meteors No. 7946 and No. 19816.

Meteor No. 7946 provides the most reliable

data. The complete set of observational data

for this meteor has been recently published by

Jacchia, Verniani, and Briggs (1965). These

authors have also discussed the evidence for

its asteroidal origin. Here we shall recall only

that the deceleration of this meteor was determined at 10 different points of its trajectory

and that a single fragment became detached

from the main body a little before midtrajectory

and was followed on the photograph for 0.1

sec. Up to the point when the fragment became

detached (at s=—0.07), the meteor did not

show any sign of progressive fragmentation

and x is zero. After the detachment of the

fragment, however, progressive fragmentation

appeared to take place and x increased appreciably as is shown in figure 3, where the logarithm of the ratio a/aT is plotted against the

mass-loss parameter s. Two straight lines

provide a very good fitting of the observational

points. The first, horizontal (x=0), fits from

the beginning to s^—0.05; the second, from

«^—0.05 to the end, with a slope x=0.31.

Since x is zero in the first hah* of the trajectory,

the reduction of the values of TP/P^ to a given

value s0 of s (of course, so< —0.05) has a full

physical meaning.

FIGURE 3.—The decimal logarithm of the observed-totheoretical deceleration ratio, ajar, plotted versus the massloss parameter s for Meteor No. 7946. The value of a? has

been computed assuming arbitrarily log (TO,^3/P!I) = —19.5.

This asteroidal meteor is compact and fairly

large: the use of 7= 1.1 would not be justified at

all. We have computed its 7 from A. F. Cook's

(1966) formula

7=0.5o[l + S.3X10-J(rpa)-1'

2

], (15)

where r, the equivalent radius of the meteoroid,

and pa are expressed in cgs units. The radius r

was computed from the observed brightness,

assuming TOP= 1X 10~19 0 mag g"1 cm"8 sec4 and

pm=3.5 g cm"3. Since r is proportional to

(TOppm)~1/d, the term r-lli~(Toppn)

xlt is quite insensitive to the uncertainty in roppm. It can be

shown that an error in roppm of a factor of 10

produces an error in 7 of only 10 percent. The

values of log (TOP/13/P1)» reduced to «=s 0 (*o^

—0.05), range from —19.67 to —19.52. The

arithmetical average is —19.606, with a standard deviation of ±0.012. Weights have not been

used, since the accelerations are all very good

and the corrections for fragmentation can be

computed accurately also in the last part of the

trajectory. The terminal mass mB of this meteor

is completely negligible, as can be seen by use

of the integrated form of the mass equation:

(16)

The data concerning Meteor No. 19816 have

been published by Cook, Jacchia, and McCrosky

(1963). The value of the total integrated brightness, not given in the publication, has been

furnished by McCrosky (1966) in the form

Tlf(m.-mE) = 1.446X10"18 0 mag cm"1 sec4.

Eight deceleration solutions were computed for

this meteor. "During the final portion of the

trajectory, two particles are in evidence": the

splitting occurs when s is approximately —0.1.

"The deceleration history of the separate particles indicates that they have nearly comparable masses," and therefore account must be

taken of the increase of the cross-sectional area

by a factor of 21/J when two particles are present. "It appears that the terminal mass is

negligible, as would be expected for a meteor

with an initial velocity of 21 km sec"1." (The

quotations are from Cook, Jacchia, and

McCrosky, 1963.) The values of log a/aT versus

s, plotted in figure 4, show an irregular pattern.

Contrary to what was expected, figure 4 shows
METEOR MASSES AND LUMINOSITY 189

signs of more progressive fragmentation at the

beginning than at the end. By forcing a straight

line fit to the points of figure 4, we obtain

X —0.09 ±0.04. Without any reduction to a

given value of s, the values of log (T0PA

J

/P^)

range from —19.88 to —19.00, with an average

of —19.37 and a standard deviation (of the

mean) of ±0.11. If we reduce the values of

log (TorA'/Pm)

 t o

 « = 0, the new values range

from —19.77 to —19.11; the average is —19.39

±0.09. It is evident, however, that in this case

the reduction to s = Q has little meaning, since

even in the early portion of the trajectory x

appears to be positive. [Should we reduce all

the values of log (to^A'/pl) to 8 — — 1, i.e., to a

point close to the actual beginning of the meteor, we would obtain log (rOI^48

/pi0 — — 19.65]

The result obtained from this meteor is far

more uncertain than the one obtained from

Meteor No. 7946.

The observational data for Meteor No. 1242

were originally published by Jacchia (1949),

who computed the meteor deceleration at four

points of the trajectory. Using these data and

neglecting the terminal mass, we found in

Paper I log (TOP A*/p2J — —19.91. Cook, Jacchia,

and McCrosky (1963) did not use Jacchia's

four original solutions, but recomputed the

deceleration at six different points. By assuming

the quantity (jov y'A'/pQ to be constant along

the trajectory, grouping the first three epochs

and the last three epochs, and using average

values for each group, they found that the

terminal mass of the meteor should have been

about one-half its initial value. However, if we

compute the terminal mass by means of equation (16) by assuming vB=Q km sec"1

 and by

using successive approximations, we find a value

of the order of one-tenth the initial mass.

Should we assume for vB a smaller value, e.g.,

VJI=4 km sec"1 as suggested by McCrosky and

Soberman (1963), mB might be even smaller.

Cook, Jacchia, and McCrosky (1963) used the

ARDC 1959 atmosphere (Minzner, Champion,

and Pond, 1959); the same calculation with the

U.S. Standard Atmosphere 1962 (1962) yields

mElm,a, — 0.44. However, since figure 5 shows

that the first acceleration is somewhat anomalous, it seems reasonable to repeat the computation of Cook, Jacchia, and McCrosky

(1963) with only the other five accelerations.

The result is mB/ma, = 0.74. Should we discard

also the last deceleration, we would obtain a

negative value for the terminal mass. In

conclusion, the method used by Cook, Jacchia,

and McCrosky (1963) appears to yield results

too much dependent on individual accelerations,

and therefore seems less reliable than the

direct approach of equation (16). We shall

adopt m^/m,,,=0.1. The values of log (T0PA'/P2J

computed from Jacchia's original data range

from —19.61 to —20.05 cgs units, with an

average of —19.86; those computed from the

data of Cook, Jacchia, and McCrosky (1963)

range from —19.62 to —20.03, with an average

of —19.81. The mean of all 10 values is

— 19.83 ±0.05. The values of log a/aT plotted

in figure 5 show that a reduction to a given

value of s is not necessary. If we had adopted

mBjma,=Q.Z, the mean value of log (T0J^iJ/Pm)

would have been —19.62.

Cook, Jacchia, and McCrosky (1963) also

took into consideration Halliday's (1960) iron

meteor (known as spectrum no. 210) and

obtained a value of the luminous efficiency

O.I

o

-O.I

-0.2

1

_

0

-

1

-0.5

1 1

—

• DATA FROM JACCHIA (1949)

o DATA FROM COOK etol (1963)

• o

o

o

• 0

1 1

0 0.5

s=iog(^r-n

FIGURE 4.—The decimal logarithm of the observed-to-theoretical deceleration ratio, ajar, plotted versus the mass-loss

parameter / for Meteor No. 19816. The value of a r has been

computed assuming arbitrarily log (ro,/fl/pJm)=—19.5.

FIGURE 5.—The decimal logarithm of the observed-totheoretical deceleration ratio, ajar, plotted versus the massloss parameter s for Meteor No. 1242. The value of <ir has

been computed assuming arbitrarily log (TOP^'/P1*) = —19.5 .
190 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

coefficient for iron. However, since the deceleration and the light curve of that meteor were not

available, they had to compute the total

emitted light from the panchromatic magnitude, by using a simple approximation of the

classical theory, quite similar to Herlofson's

(1948) treatment. They also computed upper

and lower limits for the radius of the particle,

making use again of the approximate theory and

with the help of several assumptions. Consequently, the uncertainty in the value of the

luminous efficiency that can be obtained from

spectrum no. 210 is much larger than that for

the meteors previously discussed. We will not

use spectrum no. 210 at all for our determination of rop. Ceplecha, Rajchl, and Sehnal (1959)

published complete data for a very bright

meteor (absolute visual magnitude ~—6) whose

orbit, in the author's words, is typically asteroidal. The completeness of the published data

allows us to compute as many as 20 values of

log (T0PA

3

/PI,). The meteor shows a value of x

very close to zero. Unfortunately, the value of

log (ropA'/pm) is —17.87, typical of cometary

meteors.

Summing up the results obtained in this

section, we see that the best value for log

(TopA

a

/p2

m) is —19.6. In view of the exceptional

reliability of the result from Meteor No. 7946,

we estimate the uncertainty as being of the

order of ±0.1. Stony meteorites greatly outnumber the irons. It is therefore statistically

very likely that our three asteroidal meteors

were made of stone. The density of stony

meteorites ranges generally from 3.3 to 3.9 g

cm'3, with an average of about 3.6 g cm"3

(Wood, 1963). The most reasonable average

value for the shape factor A is 1.5, with an

uncertainty of the order of 10 percent. Consequently, we obtain log TOP= — 19.02 with an

uncertainty of ±0.15. This result confirms for

the asteroidal meteors the value proposed by

the author in Paper I:

rOJ,= 1.0Xl0-19 0 mag g"1 cm"3 sec4, (17)

but the uncertainty is reduced to 30 or 40 percent of the value itself. This value is also in good

agreement with the value extrapolated by

McCrosky and Sobennan (1963) from their

artificial meteors.

The question naturally arises, how does the

value of TOP obtained for asteroidal meteoroids

apply to cometary particles, the most common

bodies by far, but also the least known?

Whipple (1963) does not explicitly raise the

question but implicitly assumes for comet ary

meteors the same value found for asteroidal

stone. In view of the lack of direct information,

this seems to be the only reasonable procedure,

since the abundance of nonvolatile elements in

the earth and in the sun, which is representative

of cometary materials, is roughly the same as

the abundance found in stony meteorites, which

are representative of asteroidal stones.

In conclusion, we have for asteroidal stones,

and probably also for coinetary meteors:

(18)

in cgs and 0-mag units. In normal cgs units,

equation (18) is equivalent to (see Paper I):

(19)

where TV expresses the ratio between the energy

(in ergs) radiated in the blue-emulsion spectral

range by the atoms ablated from the meteoroid

in a time interval At and the kinetic energy lost

by the same atoms.

Equation (18) confirms the result obtained

by Jacchia, Vemiani, and Briggs (1965) that

the initial mass of a zero visual magnitude

meteoroid (color index= — 1.86) with a velocity of 40 km sec"1

 and a zenith angle of

50° is 0.76 g. The visual luminous efficiency

can be obtained from equation (18) and the

color index curve given in Jacchia, Verniani, and

Briggs' (1965) figure 12.2. The ionizing efficiency Tj, found by Verniani and Hawkins

(1964) on the basis of the photographic luminous efficiency of Paper I, is also confirmed.

In cgs units it is

Tt=6X10-17v*, (20)

with an uncertainty of about 50 percent

in the coefficient and 20 percent in the velocity

exponent.

Meteor densities

Once the mass scale has been established by

determining the luminous efficiency, we can
METEOR MASSES AND LUMINOSITY 191

easily evaluate the meteor densities. The density distribution for the 220 sporadic meteors

discussed in a previous section is shown in

figure G. The weighted mean of log pm is

— 0.558 ±0.025, corresponding to a logarithmic

mean density of 0.28 g cm"3, with a standard

deviation of G percent. The mean weighted

density is (0.42 ±0.04) g cm"8, but it is not a

representative average because of the effect of

the large individual values of pm. The weighted

median is 0.23 g era"8. The arithmetic averages

are very close to the weighted ones.

We have seen in an earlier section that

because of the several uncertainties in the

computations and the error propagation, the

standard deviation expected in the individual

value of (log TP/P2,),-O is about ±0.8, in good

agreement with the observed standard deviation of one observation. This agreement means

that the presence of a large scatter either in the

true values of TP or in those of pm is unlikely.

We already know, however, from the discussion

above that the density pn is a function of the

aphelion distance Q, the average density of a

meteor with Q=2 a.u. being about 50 percent

larger than the average density of a meteor

with Q=4 a.u. The mean logarithmic density

r

i— '—|

i

-1.0 , 0.0

log p (g cm"3)

1.0

as a function of the aphelion distance is shown

in table 8 and plotted in figure 7. Figure 7

shows clearly that the logarithmic spread in

log pm due to the orbital dependence is of the

order of only 0.1. Consequently, it is not large

enough to affect the observed spread due to the

error propagation; the total scatter is the square

root of the sum of the squares of each error.

For this reason, since the errors due to atmospheric variability, changes of shape, and

observational inaccuracies are random, we

would expect a Gaussian distribution for log pm

centered around the logarithmic mean. Figure 6

shows, on the contrary, that the log pm distribution extends itself much more on the highdensities side. In figure 8 the same meteors

have been divided according to their aphelion

distance, and the density distribution is shown

separately for meteors with Q<5.4 a.u. and

<2>5.4 a.u. It is immediately apparent that the

density distribution of the long-period meteors

is Gaussian. A x-square test confirms that the

distribution obtained is fully compatible with a

Gaussian curve centered around the logarithmic

mean and with the dispersion fixed by the

observed standard deviation. The mean

(weighted) logarithmic density for these longperiod sporadic meteors is 0.21 g cm"8

, with

a standard deviation of 7 percent. The median

is also 0.21 g cm"3.

The distribution of the short-period meteors

is different. The low- and middle-density regions

TABLE 8.—The mean logarithmic density as a function

of the aphelion distance, Q, for £20 sporadic SuperSchmidt meteors

FIGURE 6.—The density distribution for 220 sporadic SuperSchmidt meteors.

Range

of Q

(a.u.)

0.9-3

3-4

4^5

5-6

6-10

10-30

30-100

100-c»

All

Mean Q

(a.u.)

2.20

3.63

4.56

5.41

7.24

16.27

44.84

*766

Mean log p«±

s.d. (g cm-1)

-0.184 ±0.099

-0.381 ±0.068

-0.515 ±0.060

-0.690 ±0.055

-0.649 ±0.045

-0.632± 0.061

-0.672 ±0.082

-0.690 ±0.069

-0.558 ±0.025

S.d. of

1 obs.

±0.40

±0.37

±0.40

±0.30

±0.24

±0.32

±0.38

±0.31

±0.37

No.

obs.

16

30

45

31

29

27

21

21

220

•Harmonic mean.
192 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

0

-0 1

i2 -06

-0.8

-1.0

1 1 r~

1.6

i"

-

-

1 1 1

1

1

1 1

31 I 2

1 1

1

-

_

-

I

Q(o.u)

-

1—

—1

1

Q < 5 4 o u.

"

l 0

 logf (gem*)

 0 0

NO

FIGURE 7.—The density of 220 sporadic Super-Schmidt

meteors, as function of the aphelion distance Q.

are quite similar to the corresponding ones in

the distribution of long-period meteors. Conversely, on the high-densities side it extends

considerably, and it looks as though there is an

overlapping of two Gaussian distributions, one

centered around log pm=—0.7, as for the longperiod meteors, and the other centered around

log Pm=0.1. We are thus led to the conclusion

that a small group of short-period meteors may

have densities of the order of 1 g cm"3. Figure 9

shows the density distributions for three subgroups of short-period meteors. It is immediately apparent that the group with the smallest

aphelion distances (Q<3 a.u.) includes the

largest percentage of higher density meteors:

50 percent of the total with log pm>—0.2. The

other two subgroups contain a much smaller

percentage of high-density meteors (around

20 or 25 percent); it is apparent also that the

average density of the majority group is shifted

systematically toward higher values when we

pass from a subgroup with a greater Q to one

1

1

—

' —

1

0 > 54 o u.

-1.0 log/> (gem'3

)

FIGURE 8.—The density distribution for two groups of sporadic

Super-Schmidt meteors, respectively, in short- and longperiod orbits.

with a smaller Q. The latter fact shows that the

dependence of the mean density on Q (fig. 7)

is due only in part to the varying importance

of the higher density group.

Among the 108 meteors with an aphelion

distance smaller than 5.4 a.u., about 30 meteors

are in the high-density group. The true consistence of this high-density group cannot be

estimated simply by these figures. In fact,

this sample of meteors is far from being random.

In selecting the meteors to be reduced, Jacchia

used the basic criterion that they should yield

excellent decelerations, a fact that strongly

favored the inclusion of long-trail meteors, and

thus enhanced the probability of inclusion of

high-density meteors (see Jacchia, Verniani,

and Briggs, 1965). It is nevertheless important

to have established the probable existence of a

small percentage of meteors in very short orbits
METEOR MASSES AND LUMINOSITY 193

30

t- 20

z

<z

°- 10

0

1

4<Q< 5.4o.u.

-

-

-

1 j 1

4 0

3 0

20

10

0

1 1

3<£0<4au. _

-

-

-

I—;—'

-

-

i L

3 0

20

10

0

1 1

Q < 3ou.

• r__n -

FIGURE 9.—The density distribution for three subgroups of

short-period sporadic meteors.

having much higher densities than the great

majority of all cometary meteors.

Among the original sample of 413 meteors,

123 belonged to well-established showers. Nineteen of these were disregarded for the following

reasons: Six had abrupt beginnings; three were

flarelike; three did not allow the computation

of x; two did not allow the computation of

o-; four had too anomalous values of x> snxd

finally, Meteor No. 5180, a member of the

Northern Taurids, had physical characteristics

drastically different from those of the meteors

of the shower itself. Table 9 gives the results

about the average density of each individual

shower. For most of them, although the data

are very scanty, the average density is near the

average density for sporadic meteors. The most

important exception, as is already well known

(Jacchia, 1952; Verniani, 1964a; Jacchia, Verniani, and Briggs, 1965) are the Geminids. On

the average, they are about four times more

dense than the sporadic meteors. At the other

extreme, the Draconids have an extremely

small density. In figure 10, we have plotted the

density distributions for Geminids, Southern

Taurids and a Capricornids, i.e., the only

showers with less scanty data. While the latter

shows a distribution more or less similar to that

of long-period sporadic meteors, the Geminids

have a distribution strongly resembling that

of the high-density group of meteors in very

short orbits.

A summary of the results concerning shower

-1.0

60

UJ

o

r

x. 20

0

a CAPRICORNIDS

-

1 1

_

-

! i

-1.0 0.0 1.0

60

40 -

1

-

—1

1

-

1

-1.0

(gem3)

1.0

FIGURE 10.—The density distribution for three meteor

showers: the Geminids, a Capricornids, and Southern

Taurids.
194

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 9.—Average densities for Super-Schmidt shower meteors

Note: In the second column, pm is the weighted logarithmic mean.

Showers

Geminids

Southern Taurids

a Capricornids

Quadrantids

Perseids

5 Aquarids

Southern i Aquarids

Orionids

K Cygnids

Northern Taurids

o Hydrids

Lyrids

Northern t Aquarids

Draconids

Virginids

Leonids

v Aquarids

Mean log p,,±8.d.

(g cm-»)

0.03±0.09

-0.56±0.04

-0.85±0.07

- 0 . 70±0.14

- 0 . 54±0. 07

- 0 . 57±0.07

-0.52±0.10

-0.60±0.08

-O.78±O. 08

- 0 . 58±0. 04

- 0 . 40±0. 18

| -0.41 ±0.13

-0.20±0. 45

; —

I -0.13

-0.21

-0.25

I

Pm

(g cm-1)

1.06

0.28

0. 14

0.20

0.29

0.27

0.30

0.25

0. 17

0.26

0.40

0.39

0.63

<0. 01

-0. 7

-0. 0

~0. 6

Median pm

(g cm-1)

1.14

0.25

0. 16

0.17

0.32

0.27

0.32

0.23

0.17

0.27

0.58

0.30

0. 63

—

—

—

—

Mean Q± s.d.

(a.u.)

2.6±0.0

3. 4±0. 2

5. 5±0. 7

5. 0±0. 2

59± 11

5. 3 ±0.3

4.7±0.6

66±19

5. 3±0. 4

4. 7±0. 7

56±31

51 ±19

3. 6±0. 6

5.6±0. 1

4.4

24

26

No.

obs.

20

18

12

9

8

7

5

4

4

4

3

3

2

2

1

1

1

and sporadic meteors is given in table 10. On

the average, there is no difference between

sporadic and shower meteors. The mean density

for all sporadic meteors (0.28 g cm"1) compares

very well with the density found by the author

(Verniani, 1964b) for 284 faint Super-Schmidt

meteors, 0.30 g cm"1, when we correct for the

shape factor, which was then assumed to be 1.21

instead of 1.5. Preliminary results for meteors

detected by radar methods, having an average

mass 10* times smaller than those discussed in

this paper, indicate a somewhat larger value for

the density, of the order of 0.8 g cm"1 (Verniani

and Hawkins, 1965; Verniani, 1966).

TABLE 10.—Summary of the densities of S22 Super-Schmidt meteors

All meteors

Shower meteors

Shower meteors, Geminids excluded

Sporadic meteors

Sporadic meteors with Q<5.4 a.u.

Sporadic meteors with Q>5.4 a.u.

Sporadic low-density meteors

Sporadic high-density meteors

Mean log p»±s.d.

(g cm~3)

-0.533± 0.021

-0.474 ±0.037

- 0 . 590± 0.029

— 0.558 ±0.025

-0.458 ±0.039

-0.671 ±0.029

-0.673 ±0.019

0.140± 0.024

Pm

(g cm~»)

0.29

0.34

0.26

0.28

0.35

0.21

0.21

1.38

Median pm

(g cm-*)

0.25

0.27

0.24

0.23

0.26

0.20

0.21

1.46

No.

obs.

322

102

82

220

108

112

189

31

Note: In the second column, pm is the weighted logarithmic mean.
METEOR MASSES AND LUMINOSITY 195

References

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Abstract

In 1964 the author published a study of the meteor luminous efficiency based on Harvard's photographic data.

This paper presents a more sophisticated analysis of the same data since it takes better into account the effects of

fragmentation. On the whole, 1964 results are confirmed, but the accuracy is greatly improved. The luminous efficiency

TP can be expressed as a function of velocity simply as T P = 1.0X 10-" v, with v in cm sec"1 and the luminosity expressed

in units of 0-mag stars. The uncertainty is ±0.1 in the velocity exponent and about 40 percent in the coefficient. It is

confirmed that, in the mass range 10~J—30 g, the luminous efficiency does not depend either on mass or on brightness,

nor does it show any appreciable variation in the course of the trajectory. The coefficient, rBV— IX 10~19 0 mag g"1

cm"1 sec4, has been established by means of the data on three asteroidal meteors. Once the mass scale has been established, meteor densities have been determined. Most meteors, both sporadic and shower, have an average density in

the vicinity of 0.2 g cm"1. It appears, however, that a small group of sporadic meteors in very short orbits exists, with

higher densities of the order of 1 g cm-*. Because of selection effects, it is impossible to estimate the actual consistency

of this group; it is clear, however, that it must represent a very small minority of cometary meteors.