Smithsonian

Contributions to Astrophysics

VOLUME 2, NUMBER 9

ON TWO PARAMETERS USED IN THE

PHYSICAL THEORY OF METEORS

by LUIGI G. JACCHIA

SMITHSONIAN INSTITUTION

Washington, D. C.

1958
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Washington 25, D. O. - Price 15 cents
On Two Parameters Used in the Physical

Theory of Meteors1

by Luigi G. Jacchia

The fragmentation index

The "fragmentation index" x w&s introduced

by the author in an earlier paper (Jacchia,

1955a) as a convenient measure of the progressive departure of the observed meteor deceleration from single-body theory. Accelerations

vT were computed from the drag equation

with atmospheric densities pT as given by the

Rocket Panel Atmosphere (Rocket Panel, 1952)

and masses, m, obtained from the integration

of the light-intensity curve. When these

theoretical accelerations were compared with

the observed accelerations voht, it was found

that the ratio ifobs/vT increased for most meteors

in the course of the trajectory; in the section

of the meteor trajectory which yielded usable

decelerations, the increase was nearly proportional to (ma—m)/m; i. e., the ratio of the

meteor's loss in mass from its original value mmt

to its instantaneous value m. We therefore

introduced a mass-loss parameter s defined as

(2)

and defined the "fragmentation index" as

Strictly speaking, the instantaneous mass at

the time t should be proportional to the integral,

from t to oo f of Iv~ndt, where / is the instantaneous brightness of the meteor, and n is the

power of the velocity, which may be different

for meteors of different luminosity groups.

In practice v varies only slightly, in percent, in

the course of the trajectory, and we can replace

the masses by the corresponding integrals of

the luminosity curves. We can thus write

=J"

Idt; s=l

In this paper, / will represent the photographic

intensity of the meteor, in units of the intensity

of a zero-magnitude star.

A correlation between x and meteor brightness

was pointed out in an earlier paper (Jacchia,

1955a); later the question (Jacchia, 1955b)

arose whether a maximum was observable for x

within the range of brightness of the SuperSchmidt cameras. As a measure of meteor

brightness a parameter e was used, defined as

«=log10 (- tf.Vlog.o Em +18.49, (5)

where r is the adopted value of the coefficient of

luminous efficiency for meteors. If a conversion to actual meteor magnitudes is desired, use

can be made of the following empirical formula,

which was derived from photographic meteors

with smooth light curves:

m=1.0X10" (6)

1

 Work carried ont In part under U. 8. Army Office of Ordnance Research Contract DA-10-020-ORD-2556, with Harvard University.

* Smithsonian Astrophysical Observatory; Harvard College Observatory.

181
182 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS TOL.I

Here IPM is the photographic intensity of the

meteor at maximum light, expressed in units of

a zero-magnitude star, and ZR is the zenith

distanced the meteor radiant. The velocity v

is"expressed in cm/sec. The exponent of Ipn is

larger than unity, which means that fainter

meteors have relatively steeper light curves

than bright meteors—a fact which was adduced

by Jacchia (1955a) as further evidence of

fragmentation. For meteors exhibiting flares

the value of Em is, of course, somewhat smaller

than that computed from (6) if the peak

magnitude reached in a flare is taken as 7^.

Values of x computed from decelerations are

tabulated as a function of E in table 1 for a total

of 420 meteors, 369 photographed with the

Super-Schmidt cameras and 51 with small

cameras. The data are plotted in figure 1.

-11.0

-\\2 -

-11.4 -

-11.6 -

-11.8 -

-12.0

FIGURE 1. Mean values of x and a- as a function of meteor brightness. In abscissae the parameter «=logio^oD+18.49, where E

is the integrated photographic intensity of the meteor in units of a zero-magnitude star. The confidence marks represent standard

deviations of the means.
TWO PARAMETERS IN PHYSICAL THEORY OF METEORS

TABLE 1.— The fragmentation index x<wa function of the integrated meteor brightness e.

183

Integrated

brightness, <

(logarithmic

scale)

17.11 to 17.49

17.60

17.75

18.00

18.25

18.60

19.00

19.60

20.00

20.60

21.00

17.74

17.99

18.24

18.49

18.99

19.49

19.99

20.49

20.99

21.83

Mean

*

17.36

17.64

17.87

18.11

18.35

18.68

19.18

-

-

-

Super-Schmidt cameras

Mean

X

+.35

+.34

+.36

+.31

+.26

+.28

+.15

-

-

-

Median

X

+.32

+.34

+.31

+.24

+.24

+.24

(+. 14)

-

-

-

-

No.

obs.

20

64

97

71

51

45

18

2

-

_

-

Small cameras

Mean

*

-

-

-

-

18.80

19.35

19.74

20.29

20.75

21.50

Mean

X

-

-

-

-

+.08

+.16

+.05

-.02

-.10

+.01

No.

obs.

-

-

-

-

2

9

12

14

8

(5

Mean

c

17.36

17.64

17.87

18.11

18.35

18.69

19.24

19.74

20.29

20.75

21.50

All cameras

Mean

X

+.35

+.34

+.36

+.32

+.26

+.27

+.15

+.05

-.02

-.10

+.01

Standard deviation

of mean

±.06

.03

.03

.03

.03

.04

.03

.04

.04

.06

.05

of 1 obs.

±.26

.23

.30

.23

.23

.24

.14

.14

.16

.17

.11

No.

obs.

20

64

97

71

51

47

27

14

15

8

6

Although it appears quite clear that an upper

limit for x is reached at c= 17.5, it is not possible

to decide whether this corresponds to a peak or

to a plateau; probably this question must

remain unsolved until means are found to

obtain accurate decelerations for much fainter

meteors. The standard deviation of x is considerably larger for fainter meteors than for

brighter meteors, reflecting the fact that the

relative amount of fragmentation varies greatly

among fainter meteors.

Tables 2 and 3 give mean values of x for

groups of va and cos ZR. A clear correlation

exists between x and cos ZR, which can be

neatly expressed by the linear equation

mean x=+0.50—0.27 cos ZR. (7)

This correlation is reasonable if we consider

that the lifetime of individual fragments is

longer when cos ZR is smaller, so that fragments

become scattered over a greater length on the

trail (this is clearly shown by the fact that

terminal blending increases as cos ZR decreases).

The correlation between x and meteor velocity is not so clear-cut. From an inspection of

the data it is obvious that high-velocity meteors

have, on the average, higher values of x than

slow meteors, but there is little evidence of a

smooth correlation.

The evidence even seems to suggest that

meteors with va <40 km/sec have a uniform

average value of x near +0.30, while meteors

with vm >40 km/sec have an average x close to

+0.38. Since the means of individual velocity

groups are likely to be strongly influenced by

major showers, and it has been shown by Jacchia (1955b, 1956) that the fragmentation

index differs greatly for different showers,

another set of means was prepared for x in

TABLE 2.—The fragmentation index x <*s o function of

meteor velocity vm.

Meteor velocity

outside atmosphere, »OO

(km/sec)

11. 5 to 20

20 to 25

25 to 30

30 to 35

35 to 40

40 to 50

50 to 60

60 to 72. 8

All meteors

Mean x

+.30

+.31

+.31

+.22

+.29

+.41

+.36

+.39

No.

obs.

60

58

61

46

37

38

24

45

Sporadic

meteors

Mean x

+.30

+.31

+.30

+.25

+.29

+.38

+.36

+. 40

No.

obs.

60

52

49

30

13

22

20

32

TABLK 3.—Fragmentation index x <*« <* function of zenith

distance.of meteor radiant, cos ZR.

Cos ZB

0. 1 to 0. 2

0. 2 to 0. 3

0. 3 to 0. 4

0. 4 to 0. 5

0. 5 to 0. 6

0. 6 to 0. 7

0. 7 to 0. 8

0. 8 to 0. 9

0. 9 to 1. 0

Mean x

+ 46

+ 38

+ 47

+ 44

+.28

+ .32

+ .30

+.26

+.25

No.

obs.

4

8

28

35

30

78

72

70

44
184 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

function of vn, for sporadic meteors only (table

2). Although the number of meteors in some of

the velocity groups is then greatly changed, the

picture regarding x remains essentially unaltered. Since the apparent velocity of 40

km/sec corresponds very closely to the gap

between the meteors of the Jupiter family and

the long-period meteors, the possibility is not

excluded that there is some difference in fragmentability between the two groups of meteors.

A similar difference in behavior between the two

groups has been found by the writer while

examining the beginning heights of meteors,

and will form the subject of another paper.

The coefficient a

As was previously shown (Jacchia, 1955b), the

coefficient a of the mass equation

TO (8)

shows a strong dependence on meteor brightness. The cause should be clear: I IE is a

photometric quantity and refers to the sum of

all the fragments, while vi is a dynamic quantity and refers to the larger fragments only.

On the whole a is a little easier to determine

than XJ because it involves only the first time

derivative of the velocity, while x involves the

second derivative. The determination of a

from equation (8) has two weak points: 1) it

involves the instantaneous meteor brightness /

and therefore is affected by the vagaries of the

light curve. A two-magnitude flare will cause

a six-fold local increase in the value of <r; 2)

/ is the instantaneous brightness, while v is a

value of the deceleration obtained from observations over a long portion of the meteor

trajectory, often over all of it, so the two quantities do not, strictly speaking, correspond to

each other. Whenever possible, a should be

determined from the integral of (8), i. e., from

the equation

lnlMcV-**2), (8a)

with the use of two separate points (subscripts

1 and 2) on the meteor trajectory.

For all Super-Schmidt meteors, "instantaneous" values of a were determined for each

observed deceleration; in addition, equation

(8a) was used whenever more than one deceleration was available on the meteor trajectory.

A mean value of a was derived for each meteor

on the basis of all these determinations; those

derived from equation (8a) were given greater

weight in calculating the mean.

The observational results relative to <r as a

function of e are collected in table 4 and shown

graphically in figure 1. It appears that <r

follows the same general trend as \, although

the plateau corresponding to fainter meteors

seems to be a little broader. It is interesting

to note, however, that the standard deviation

of a does not seem to increase with decreasing

c, as is the case with x- I n other words, for

faint meteors we observe a large range in x»

from the largest values all the way down to

zero, while we observe only large values of <r,

Integrated

brightness, «

(logarithmic

scale)

17.11 to 17.49

17.50

17.75

18.00

18.25

18.50

19.00

19.50

20.00

20.50

21.00

17.74

17.99

18.24

18.49

18.99

19.49

19.99

20.49

2a 99

21.83

TABLE 4.—The coefficient a

x

Super-Schmidt cameras

Mean «

17.36

17.64

17.86

18.11

18.36

18.68

19.18

19.59

20.41

Mean

log<r

—11.11

—11.16

-11.16

-11.18

—11.24

-11.22

-11.33

-11.67

-11.88

No.

obs.

21

59

92

66

50

43

18

2

1

Mean

17.87

18.82

19.30

19.72

20.25

20.75

21.46

as a function of the integrated brightness, *.

Small cameras

Mean

« logo-

-11.05

-11.28

-11.34

-11.49

-11.60

-11.77

-11.71

No.

obs.

1

11

16

22

17

11

8

Mean c

17.36

17.64

17.86

18.11

18.36

18.71

19.24

19.71

20.26

20.75

21.46

All cameras

Mean

log a

-11.11

—11.16

-11.16

-11.18

—11.24

-11.23

-11.33

-11.50

-11.61

-11.77

-11.71

Standard deviation

of mean

±.05

.04

.02

.03

.03

.03

.04

.05

.05

.07

.13

o fl

obs.

±.21

.28

.22

.22

.20

.23

.25

.24

.23

.24

.37

No.

obs.

21

59

93

66

50

54

34

24

18

11

8

• See equation (8), above.
TWO PARAMETERS IN PHYSICAL THEORY OP METEORS 185

-11.0

-11.1 -

-11.2-

-11.3 -

11.4 -

-11.5

V (km/sec.)

70 80

FIGURE 2. Mean values of log «r in function of meteor velocity. The confidence marks represent standard deviations of the means.

Super-Schmidt meteors only.

which do not overlap with the smaller values

characteristic of bright meteors. This shows

that the variatior of <r is due not only to meteor

fragmentation, but must also have a true systematic component connected with meteor size.

Mean values of log <r taken for 8 velocity

groups (table 5, figure 2), for Super-Schmidt

meteors only, show a clear dependence on

velocity. The variation is nearly linear with

log »„; the slowest meteors («<a<20 km/sec)

yield values of <r which are nearly twice as

large as those of the fastest meteors (»co>60

km/sec).

In the course of the meteor trajectory a

changes but little, although there is a general

tendency for it to decrease a little from the

earlier to later portions of the meteor trajectory.

Values of d log <rjds were determined for 111

meteors which had yielded multiple decelerations. The mean of all these determinations

is mean ^^=-0.07 9 ±0.016 (m. e). The

standard deviation of a single determination

was ±0.17. Thirty-nine meteors gave a positive value of d log a/ds, 69 gave a negative value,

and three gave a value of zero.

To examine this trend more closely, all the

1115 determinations of log «r from SuperSchmidt meteors (an average of a little over 3

determinations per meteor) were arranged in

increasing order of s and means were taken

within equidistant groups. The results, given

in table 6 and in figure 3, show that the mean

observed range in log a in the reliable part of

TABLE 5.—Super-Schmidt meteors: the coefficient <T > in function of meteor velocity v.

Velocity, v

(km/sec)

11. 5 to 20

20 to 25

25 to 30

30 to 35

35 to 40

40 to 50

50 to 60

60 to 72. 8

Mean v

17. 1

22.5

27.3

31.8

36.7

44.3

56.3

65.3

Mean

log v

-11.04

-11. 11

-11. 18

-11.24

-11.22

-11.29

-11.36

-11.30

Standard deviation

of mean

±.03

.03

.03

.03

. 0 4

.04

.04

.04

of 1 obs.

±.19

.20

.21

. 19

. 2 6

.24

. 19

.27

No.

obs.

55

56

56

46

37

40

22

43

1

 See page 184, equation (8).
186 SMITHSONIAN CONTRIBUTIONS TO ASTROPHTSIC8

-11.0

-112 -

-11.4-

— . T .

: <*#*'

i

2 s - 1

I

J J l *

i

0

T

1

I /

+i + 2

FIGURE 3. Mean variation of <r in the course of the meteor trajectory, derived from Super-Schmidt meteors only. In abscissae the

mass-loss parameter s. The open circle at /=> —0.055 represents the weighted mean of all 1115 observed values of a; the slanting

straight line through it has a slope equal to the mean value of d log aids determined directly from individual meteors. The confidence marks represent standard deviation of the means.

the trajectory is only 0.14. A decrease in a is

observed from «=—0.2 to «=-fl.0, which

accounts for the negative mean of all the

values of d log aids. The lower values of a

for large negative values of « may be accounted

for by the fact that only for long, bright meteors,

which have smaller values of a (fig. 1), can

decelerations be determined so early ii the

trajectory. The rise in a for «>+1.0 is

extremely uncertain, being based on very few

observations. If real, it could be due to the

presence of a residual mass, which the numerical

integration of the light-intensity curve fails to

take into account.

The larger values of the standard deviation

for a in the interval between s=0 and s=+0.3

reflect the fact that most of the irregularities

and individual peculiarities of meteor light

TABLE 6.—Super-Schmidt meteors: the coefficient a in function of the mass-loss parameters.

Mass-loss

parameter «

-1.64 to -1. 4

-1. 4

-1. 2

-1. 0

-0. 8

-0. 7

-0. 6

-0. 5

-0. 4

-0. 3

-0. 2

- 0 . 1

0.0

0. 1

0.2

0.3

0.4

0.5

0.6

0.7

0.9

1. 1

1.3

1.5

to -1. 2

to -1. 0

to -0. 8

to -0. 7

to -0. 6

to -0. 5

to -0. 4

to -0 . 3

to -0. 2

to -0 . 1

to -0 . 0

to 0.1

to 0.2

to 0.3

to 0.4

to 0.5

to 0.6

to 0.7

to 0.9

to 1.1

to 1.3

to 1.5

to 1.79

Mean

8

-1.51

-1.26

-1.06

-0.86

-0.75

-0.65

-0.54

-0.4 4

-0.3 4

-0.24

-0.15

-0.05

+ 0.04

+ 0. 15

+ 0.24

+0.34

+ 0.44

+0.54

+ 0.64

+0.78

+ 0.96

+ 1. 19

+ 1.41

+ 1.61

Mean

log <r

-11.4 0

-11 . 25

-11.2 9

-11 . 18

-11.2 3

-11 . 18

-11.2 3

-11.2 1

-11 . 18

-11 . 18

-11 . 19

-11.2 2

-11 . 14

-11.2 0

-11.3 3

-11.2 6

-11 . 24

-11 . 28

-11 . 27

-11.3 4

-11.3 1

-11.3 4

-11.2 0

-11 . 16

Standard deviation

of mean

db. 05

.06

.07

.04

.07

.04

.03

.03

.02

.02

. 02

.02

.03

.05

.05

.06

.03

.04

.03

.05

.04

.06

.03

.14

of 1 obs.1

±.09

.20

.23

. 22

. 22

.20

. 19

.23

. 23

.20

.23

.25

. 27

.30

.31

.36

. 21

.25

. 21

.30

. 18

.18

.07

.32

No.

obs.

4

12

12

25

27

25

51

79

83

139

144

111

84

43

33

33

49

39

42

41

18

11

7

5

' Of average weight.
TWO PARAMETERS IN PHYSICAL THEORY OF METEORS 187

curves occur around maximum light or shortly References

afterwards. JACCHIA, L. G.

The author is indebted to Mr. Robert E. 1955a. Astrophys. Journ., vol. 121, p. 521.

Briggs for assembling the observational mate-

 1955b

- A8*TM. Journ., vol. 60, p. 165.

. ,

SB

 , , , , • * 1956. Sky and Telescope, vol. 15, p. 207.

rial on punched cards and preparing it for a ROCKET PANEL

statistical analysis. 1952. Phys. Rev., vol. 88, p. 1027.

Abstract

Observational results are given for two useful meteor parameters. The first is the "fragmentation index" x»

which was introduced by the author in an earlier paper to measure the progressive departure of the observed deceleration of a fragmenting meteor body from the deceleration expected on the basis of single-body theory. The second

parameter is the coefficient a of the "mass equation" m/m=avi>.

The fragmentation index x is found to vary with the brightness and the angle of incidence of the meteors. A

variation with velocity is indicated, but it is probably discontinuous and due to the cosmic distribution of meteors.

The coefficient <r of the mass equation shows a variation with brightness which nearly parallels that of x- The

variation of <r with velocity is quite regular. There is but little variation of a in the course of the meteor trajectory.