ABSTRACT

Techniques are described which use observed magnitude distributions for the determination of the correction factors necessary to reduce observed stream and sporadic rates to uniform, ideal sky conditions (ZS7.0). These techniques are applied to meteor magnitude distributions (sporadic, Quadrantid, Perseid and Geminid) observed by the author. For sporadic meteors, it is found that observed rates decrease significantly faster with deteriorating ZS than is commonly assumed ( » 15 - 17% rate decrease per 0.5 magnitude deterioration in ZS, versus the canonical 10%). The expected systematic sporadic/stream differences are evident in the derived correction factors. The determination of relative perception is a straightforward by-product of the adopted data reduction technique.

I. INTRODUCTION

Differing sky conditions and perception are two well known factors which complicate the comparison of simultaneous visual meteor rates obtained by different observers. This paper deals primarily with the first of these difficulties and empirically determines the correction factors necessary to reduce observed meteor rates to uniform sky conditions. A common assumption for the relation between sky darkness and observed meteor rates is that each 0.5 magnitude deterioration in the faintest stars visible (ZS) causes a 10% decrease in the observed meteor rates (stream and sporadic). This has never been quantitatively demonstrated to be true, to this author's knowledge.

Further, the use of the same 'sky-condition-rate-correction-factors' (hereafter, SCRCF's) for both sporadic and stream meteor rates is a simplification which undoubtedly leads to systematic errors in reduced and published rates. As an extreme example of this problem, consider a stream which produces only magnitude -1 meteors. The observed rates for such a stream will be independent of sky conditions for any reasonable observing situation (ZS³ 4.0). Thus, use of identical SCRCF's for sporadics, a large percentage of which are faint meteors, and a stream such as described above, is ridiculous. Yet, on a lesser scale, this assumption of identical stream and sporadic SCRCF's constitutes current practice.

The purpose of this paper is to describe and apply a method which allows the determination of consistent and separate sets of SCRCF's for sporadic and stream meteors. Section II describes the sporadic data, reduction methods and results. In Section III these results are applied to derive the SCRCF's for several of the major meteor streams. Section IV summarizes the conclusions of this study.

II. PROCEDURE : SPORADICS

a) The Observations

The sporadic data used in this study consists of 5549 sporadic meteor magnitude estimates made by the author over the period 1968-1979 under ZS5.0 - 7.0 sky conditions. The use of a single observer's data eliminates the need to consider perception.

Table-1 lists the observed sporadic magnitude data. Each row contains data covering a 0.5 magnitude ZS interval, the centers of which are ZS5.5, 6.0, 6.5 and 7.0, respectively. The ZS5.5 row also contains fifty six meteors observed under ZS5.0 - 5.2 skies. Since the author estimates the ZS to the nearest 0.1 magnitude while observing, the values in column-1 should not be expected to coincide exactly with the center of the ZS range in that row.

b) Reduction Methods

There are two reduction methods which permit the straightforward calculation of the desired sporadic SCRCF's. Method-l uses sporadic meteor rates observed during the same local hour under differing sky conditions on approximately the same date. The constraints of identical local hour and similar date (± 1 - 2 weeks ) are necessary because of the significant diurnal and annual variation in sporadic meteor rates. As an illustration of method-1, consider the observations below:

                LOCAL    SPORADIC
    DATE        HOUR       RATE        ZS

8-29/30-1973     1-2     10 ± 3.2     6.0

8-27/28-1978     1-2     15 ± 3.9     7.0

This data implies that the ratio of observed sporadic rates under ZS6.0 skies to those under ZS7.0 skies is

Note that to reduce the uncertainty in F(6.0,7.0) to 10% requires (0.28/0.067)² » 17 independent pairs of observations satisfying the rigid requirements already noted. Only those persons who have been regularly observing meteors for many years can expect to derive meaningful results by this method.

The second reduction method, the one which is adopted here, uses observed sporadic magnitude distributions to calculate the sporadic SCRCF's. The validity of this method, in the present context, rests on two assumptions: (1) that the sporadic magnitude distribution does not markedly vary throughout the year and (2) that the bright sporadic meteor rate is independent of sky conditions (ZS5.0 - 7.0).

c) Results

Close examination of the data in Table-1 has revealed no marked variations of the sporadic meteor magnitude distribution throughout the year. Assumption - 1 thus poses no major problems in the following analysis. The present data is inadequate, however, to search for small variations (£ 5%).

Qualitatively, assumption - 2 is common sense. Quantitatively, we determine the faintest magnitude meteors (magnitude m₁) for which this assumption is valid. Magnitude m₁ is found by considering the quantity

where n(m_i )is the number of meteors of magnitude m_i and / n (m < m_i) is the number of meteors brighter than magnitude m_i. If assumption - 2 is valid at magnitude m then the calculated ratio, r, is constant for each sporadic magnitude distribution in Table-1, within the observational errors. The desired limit, m_l, is thus the faintest magnitude for which the ratio r in independent of ZS. The data used in this analysis indicate that for the range of sky conditions considered here an optimum choice is m₁ = +1.

To calculate the sporadic SCRCF's, each of the sporadic magnitude distributions in Table-1 is normalized to the same number of meteors with m £ +1. The total number of meteors of all magnitudes in such 'bright-normalized' magnitude distributions will increase for fainter ZS since darker skies increase the observable number of faint meteors. Hereafter, 'bright' meteors are defined to have m £ +1; faint meteors, m ³ +2.

Table-2 lists the data from Table-1 with each magnitude distribution normalized to the same number of bright meteors, arbitrarily chosen to 100.0. Below each value is its estimated uncertainty (random error only) assuming counting statistics and standard error propagation formulae. Note that column-9 is identical to column-8 except that the ZS6.94 distribution has been approximately corrected to a true ZS7.00 distribution by a simple linear extrapolation.

The sporadic SCRCF's are calculated relative to an ideal ZS7.0 sky using the data in column-9 of Table - 2

These values are listed in column-10 and the final ZS to which they refer is given in column-11. Figure-1 is a plot of F(ZS,7.0) versus ZS. The line through the points in Figure-1 is a weighted (1 /s _i²) least squares fit to the data. Using this fit and normalizing the result to F = 1.00 at ZS7.00 yields the final sporadic SCRCF's listed in Table-3.

Tables-2 and 3, together with Figure-1, demonstrate very strikingly the importance of excellent sky conditions for obtaining high rates. We cannot expect raw observed rates obtained by different observers under different sky conditions to agree very well, even if the persons involved have similar perception. Note that the data presented here indicate a larger decrease ( » 15 - 17%) in observed sporadic rates, per 0.5 magnitude deterioration in ZS, than the 10% decrease which is commonly assumed.

This steep dependence of the SCRCF's on ZS dictates that observers must be very careful in their estimation of the ZS. The author has found AAVSO type-a variable star charts to be ideal for determining ZS since each of these charts generally include the magnitudes for ten or more stars in range 5.0 £ m £ 7.5.

Early in this study, the author noted that the ZS6.0 and 6.5 sporadic magnitude distributions from his first two years of observing (1968-9) were inconsistent with all later data at these ZS. The reason for these discrepancies probably lies in a combination of incorrect estimation of the ZS and inaccurate meteor magnitudes. These early, discrepant magnitude distributions are not included in this analysis. With the above noted exception, for which there is a reasonable explanation, all data included in Table-1 is internally consistent within the observational errors. There has been no significant change in the author's perception over the period of the observations.

III. APPLICATION TO STREAMS

Given the entirely empirical derivation of the sporadic SCRCF's , it is now possible to calculate the SCRCF's for any meteor stream given a single magnitude distribution for the stream at any ZS. The important point is that the number of magnitude +4 sporadic meteors seen under ZS5.5 skies, relative to ZS7.0 (» 127/247, from Table-2), for example, is independent of the fact that these meteors are sporadics. Thus, simultaneous observations of equal duration, by persons of identical perception, under ZS5.5 and ZS7.0, would yield 127 and 247 magnitude +4 stream meteors, respectively, within the bounds of observational error (ie, counting statistics).

Therefore, a single magnitude distribution, preferably observed under ZS7.0 skies, together with the ZS/magnitude calibration provided by Table-2, allows a complete and consistent determination of the SCRCF's for any stream. This approach of calculating stream SCRCF's using a single ZS magnitude distribution is adopted owing to the lack of a sufficient stream data base from a variety of sky conditions.

The observed stream magnitude distribution is preferred to be from ZS7.0 skies since data obtained under such ideal conditions contains the most statistically significant results for the faint stream meteor population. A precise knowledge of the faint meteor content of a stream is critical to a meaningful evaluation of the stream SCRCF's. In principle, one could, for example, base the calculations on a ZS5.5 distribution. Such an approach would, however, lead to significantly larger errors in the results.

Table-4 lists, in a format similar to Table-1, magnitude distributions observed by the author under ZS7.0 skies for the Perseids (1974, 1975, 1978), Geminids (1975, 1977) and Quadrantids (1973, 1976). Table-5 lists the bright-normalized form for each of these stream magnitude distributions.

The bright-normalized sporadic magnitude distributions of Table-2 contain the calibration necessary to derive the stream SCRCF's from the data given in Table-5. We calculate, from the Table-2 data, the factors

Here, n(m,7.0) and n(m,ZS) represent, respectively, the number of meteors of magnitude m observed under skies of limiting magnitude 7.0 and ZS. Before proceeding to apply these f(m,ZS) to the calculation of the stream SCRCF's, the f(m,ZS) are smoothed by least-squares fitting the data at each magnitude. After fitting, all values are renormalized to f(m,7.0) = 1.00 . Figure-2 illustrates the magnitude +4 data and least-squares fit.

Table-6 lists the final adopted f(m,ZS) values. The estimated errors are given beneath each value. Note that the values of f(m,5.0) represent an extrapolation of the available data and, as such, should be treated cautiously.

To calculate the stream SCRCF's, the results of Table-6 are used together with equation-4, inverted to solve for the n(m,ZS). The total number of meteors at each ZS, N(ZS), is derived and the SCRCF's are then calculated using equation-3.

Table-7 illustrates the complete application of this method to the ZS7.0 Geminid data of Table-5. The final SCRCF's derived for the Quadrantids, Perseids and Geminids, according to the method described above, are listed in Table-8. Also included in Table-8, for comparison, are the sporadic SCRCF's determined in Section III. Even though the random errors in the results of Table-8 are rather large (» 10%), the expected systematic stream/sporadic differences are obvious.

Separate sets of SCRCF's are truly feasible only for those streams with reasonably high rates (³ 10/hour). The application of incorrect SCRCF's to rates from low-level streams will not lead to significant systematic errors. The accurate determination of the SCRCF's for low level streams is a difficult task since many observing hours are required to obtain. enough data (³ 200 meteors) to make such calculations worthwhile.

An interesting by-product of bright-normalized magnitude distributions is that a comparison of such distributions obtained bv two different observers, at the same ZS, permits a derivation of the relative perception of the two observers. Thus, in the notation of Table-2:

(5) p(Smith,Adams) = N'(Smith,ZS)/N'(Adams,ZS),

where p is the perception coefficient of Smith relative to Adams. Magnitude distributions from any common ZS may be used for this purpose. Relative perception should be independent of ZS. Note, however, that the magnitude distributions which are compared should each contain meteors of the same type, ie, either sporadics or meteors from the same stream.

A final note of caution regarding the use of rate correction factors is in order. Never report corrected meteor rates without also reporting the raw observed rates, the corrections used, their source, a description of the manner in which they were applied, and an adequate description of the sky conditions during the observations (ie, the ZS and percentage cloudiness). Failure to follow these fundamental rules of scientific reporting results in the production of meaningless and uninterpretable data.

The application of large correction factors (³ 1.5) to raw, observed rates is a practice which should be nothing more than a last resort. In addition, a careful estimation of the errors in the corrected rates is of paramount importance in interpreting the results. These cautions are particularly applicable to observational descriptions of minor stream activity.

IV. SUMMARY AND CONCLUSIONS

The sky condition rate correction factors (SCRCF's) for sporadic meteors, as a function of sky quality (ZS), have been determined. Relative to an ideal ZS7.0 sky (Fº 1.00), the sporadic SCRCF's are found to be 0.836 ± 0.086, 0.672 ± 0.065 and 0.509 ± 0.051 for ZS6.5, 6.0 and 5.5, respectively. These SCRCF's permit the reduction of observed sporadic meteor rates to uniform sky conditions.

Separate sets of SCRCF's are desirable for each stream with rates greater than 10 per hour since the magnitude distributions of meteor streams are usually significantly different from the sporadic distribution. The application of the sporadic SCRCF's to the reduction of observed stream rates would lead to serious systematic errors in the corrected stream rates.

A method has been developed for the calculation of the SCRCF's for any stream. The execution of this method requires only a statistically significant magnitude distribution for the stream, observed at any ZS (preferably ZS7.0). This technique has been applied to the derivation of the SCRCF's for the Quadrantid, Perseid and Geminid meteor streams. Table-8 lists the results of these calculations. The expected systematic stream/sporadic differences are evident. The techniques employed in this analysis also permit a straightforward determination of relative perception.

Future work will include an extension of the calculation of SCRCF's to other major streams (eg, the April Lyrids, Eta Aquarids, etc...), and an attempt to reduce the random errors in the already determined SCRCF's. Other topics which must be examined are perception corrections and the applicability of the standard zenithal-hourly-rate (ZHR) formula. In addition, the author hopes to review the historical activity of the major meteor streams with an emphasis on observations in the last century.

TABLE -1: RAW SPORADIC MAGNITUDE DATA
<Zs>	£ +1	+2	+3	+4	+5	+6	TOTAL
5.51	143	151	237	181	33	0	745
6.12	175	194	387	290	151	11	1208
6.59	150	166	354	340	246	56	1312
6.94	230	235	574	567	446	182	2284

 

 

TABLE -2: NORMALIZED SPORADIC MAGNITUDE DATA
ZS	£ +1	+2	+3	+4	+5	+6	N(ZS)	N'(ZS)	F	ZS
5.51	100.0 ± 11.8	105.6 12.3	165.7 17.5	126.6 14.2	23.1 4.5	0.0 ----	521.0 28.6	521.0 28.6	0.514 .051	5.51
6.12	100.0 ± 10.7	110.9 11.6	221.1 20.1	165.7 15.9	86.3 9.6	6.3 2.0	690.3 31.7	690.3 31.7	0.682 .065	6.12
6.59	100.0 ± 11.5	110.7 12.5	236.0 23.0	226.7 22.2	164.0 17.0	37.3 5.8	874.7 40.4	874.7 40.4	0.864 .086	6.59
6.94	100.0 ± 9.3	123.9 11.0	249.6 19.5	246.5 19.3	193.9 15.7	79.1 7.8	993.0 35.6	1012.8 36.3	1.000 .088	7.00

 

 

TABLE -3 : FINAL SPORADIC RATE CORRECTION FACTORS
Zs	F(ZS,7.0)
7.00	1.000 ± 0.088
6.50	0.836 ± 0.086
6.00	0.672 ± 0.065
5.50	0.509 ± 0.051

 

 

TABLE -4: OBSERVED STREAM MAGNITUDE DISTRIBUTIONS
Stream Name	ZS	£ +1	+2	+3	+4	+5	+6	Total
Quadrantids	7.00	68	45	54	48	33	4	252
Perseids	7.00	253	161	201	155	98	29	897
Geminids	7.00	120	98	200	161	131	44	754

 

 

TABLE -5: NORMALIZED STREAM MAGNITUDE DISTRIBUTIONS
Stream Name	ZS	£ +1	+2	+3	+4	+5	+6	N(ZS)
Quadrantids	7.00	100.0 ± 17.1	66.2 12.7	79.4 14.5	70.6 13.3	48.5 10.3	5.9 3.1	370.6 30.9
Perseids	7.00	100.0 ± 8.9	63.6 6.4	79.4 7.5	61.3 6.3	38.7 4.6	11.5 2.3	354.5 15.6
Geminids	7.00	100.0 ± 12.9	81.7 11.1	166.7 19.2	134.2 16.2	109.2 13.8	36.7 6.5	628.5 34.0

 

 

TABLE -6: FINAL ADOPTED f(m,ZS) FACTORS
ZS	£ +1	+2	+3	+4	+5	+6
5.0	1.000 ± .150	.796 .117	.542 .071	.318 .043	----	----
5.5	1.000 ± .150	.864 .127	.656 .086	.488 .066	.107 .022	----
6.0	1.000 ± .142	.910 .125	.770 .092	.659 .081	.404 .055	----
6.5	1.000 ± .148	.954 .137	.886 .111	.329 .104	.702 .092	.308 .057
7.0	1.000 ± .131	1.000 .126	1.000 .110	1.000 .111	1.000 .115	1.000 .139

 

 

TABLE -7: GEMINID NORMALIZED MAGNITUDE DISTRIBUTIONS AND SKY CONDITION RATE CORRECTION FACTORS.
Zs	£ +1	+2	+3	+4	+5	+6	N(ZS)	F(ZS,7.0)
5.0	100.0 ± 19.8	65.0 13.0	90.4 15.8	42.7 7.7	----	----	298.1 29.5	0.474 .053
5.5	100.0 ± 19.8	70.6 14.1	109.4 19.1	65.4 11.9	11.7 2.8	----	357.1 33.2	0.568 .061
6.0	100.0 ± 19.2	74.3 14.4	128.4 21.3	88.4 15.2	44.1 8.2	----	435.2 36.4	0.692 .069
6.5	100.0 ± 19.6	77.9 15.4	147.7 25.1	111.3 19.4	76.7 14.0	11.3 2.9	524.9 42.8	0.835 .082
7.0	100.0 ± 12.9	81.7 11.1	166.7 19.2	134.2 16.2	109.2 13.8	36.7 6.5	628.5 34.0	1.000 .077

 

 

TABLE -8: FINAL STREAM AND SPORADIC SKY CONDITION RATE CORRECTION FACTORS
ZS	QUADRANTIDS	PERSEIDS	GEMINIDS	SPORADICS
5.0	.589 ± .091	.601 .061	.474 .053	----
5.5	.672 ± .099	.680 .066	.568 .061	.509 .051
6.0	.775 ± .107	.776 .069	.692 .069	.672 .065
6.5	.885 ± .120	.882 .077	.835 .082	.836 .086
7.0	1.000 ± .118	1.000 .062	1.000 .077	1.000 .088

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