s. Yabushita Statistical tests of a periodicity hypothesis for crater formation rate - II

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1 Mon. Not. R. Astron. Soc. 279, (1996) Statistical tests of a periodicity hypothesis for crater formation rate - II s. Yabushita Department of Applied Mathematics and Physics, Kyoto University, Kyoto 606, Japan Accepted 1995 October 3. Received 1995 October 2; in original form 1993 September 27 ABSTRACT A statistical test is made of the periodicity hypothesis for crater formation rate, using a new data set compiled by Grieve. The criterion adopted is that of Broadbent, modified so as to take into account the loss of craters with time. Small craters (diameters ~ 2 km) are highly concentrated near the recent epoch, and are not adequate as a data set for testing. Various subsets of the original data are subjected to the test and a period close to 30 Myr is detected. On the assumption of random distribution of crater ages, the probability of detecting such a period is calculated at 50, 73 and 64 per cent respectively for craters with D > 2 km (N = 49), for those with 10 ~D > 2 km (N =31) and for large craters [D> 10 km, (N = 18)] (where N is the number of craters). It is thus difficult to regard the detected period as being significant based on statistical argument alone. It is pointed out that a similar period is associated with geomagnetic reversals and the climatic variation as revealed by the deep ocean (j 18 0 spectrum. Key words: methods: statistical - comets: general - Solar system: general. 1 INTRODUCTION Distributions in the ages and the diameters of impact craters provide important information on the nature and dynamics of impacting bodies. Since Alvarez & Muller (1984) claimed to have detected a periodicity in the rate of crater formation (derived period ~ 28 Myr), the problem has been investigated by a number of authors. Rampino & Stothers (1986) used 65 craters of known ages, and supported a period of close to 30 Myr, while Grieve et al. (1985) took a more cautious approach towards the hypothesis and argued that a mere time-series analysis was insufficient to establish the periodicity. In a recent paper, the present author (Yabushita 1991, Paper I hereafter) adopted a criterion proposed by Broadbent (1955, 1956) and applied it to the data sets of craters in Rampino & Stothers (1986) and in Grieve (1987). It has been found that the craters as a whole appear to yield a periodicity of ~ 30 Myr, and that small craters (diameters < 10 km) better satisfy the criterion adopted for periodicity. Since then, nearly 30 craters have either been found or have had their ages determined (Grieve 1991, 1993). Also, some of the ages of craters of the earlier data set have been better determined. It therefore seems worthwhile to carry out a statistical test to see how the hypothesis of periodicity is affected by the new set of data. Although the earlier investigation adopted the Broadbent criterion, one feature which may invalidate a straightforward application of the criterion is that old craters are lost owing to weathering and tectonic activity. This is seen in the smaller number of recognized craters as one goes back in time. This aspect will need to be taken into account, and for this purpose a new technique will be adopted which is a modification ofthe Broadbent criterion. An account thereof will be presented in Section 3. It will be seen that, although a period of close to 30 Myr is obtained from the testing, it appears difficult to substantiate the periodicity hypothesis using the presently available set of craters. 2 DATASET The data sets used earlier are as follows. First, the set used by Rampino & Stothers (1986) consists of 65 craters. The diameters are not explicitly given. The set used by Grieve (1987) and Yabushita (Paper I) contains 101 craters, with given diameters and ages. On the other hand, a recent compilation of Grieve (1993) contains 141 craters. Diameters are given for all of the craters, but for many of them only upper or lower bounds are given for the ages. Some of these have upper limits of less than a million years. Since the period to be derived is in the vicinity of 30 Myr, no significant error would be caused by ascribing ages of ~ 0.1 Myr to them. We may summarize the data set of Grieve (1993) as 1996 RAS

2 728 S. Yabushita follows: (i) there are 141 craters; (ii) of these, there are 106 craters of known ages and diameters; and (iii) there are 35 craters for which only upper or lower age limits are given. As to the probable errors in the given ages, those with small ages have small probable errors, while those with large ages often have large errors. In the following, statistical testing is conducted by adopting the 106 craters. Of these, 101 have ages of less than 600 Myr. Those with ages ~ 600 Myr are scarce and are not adopted. Table 1 provides basic statistics of craters used in the following for statistical testing. Next we consider the loss of craters with time. As referred to in the Introduction, the statistical testing would be a rather straightforward matter if the distribution of the crater ages was more or less uniformly random. As one can see from Fig. 1, which gives a histogram of the distribution of their ages, the number of craters decreases as one goes back in time. It is thus important to specify the rate of decay (or loss) before carrying out statistical testing. First we note that small craters are lost quite rapidly, as one may easily see from Table 1. There are only two craters with diameters D:::; 2 km and ages ~ 25 Myr. Of 19 craters with D < 1 km in the adopted Grieve data set there is none with age ~ 25 Myr. This fact shows that small craters are quite inadequate for deriving any significant period. It is thus natural to classify the craters into large (D > 10 km) and intermediate (10 ~D > 2 km) ones, and to discard small ones (D :::; 2 km) in the statistical testing. Now, the loss of craters with time naturally leads one to introduce what may be called the decay constant. On the assumption that the cratering is uniformly random in 0:::; t < 600 Myr, the decay constant, a, may be defined by the expression N(t)&=b exp (-at)& (2.1) where N(t) & is the expected number of craters in the interval & and b have been determined by the least-squares fit method and are presented in Table 1. Clearly, the constant b gives the expected number of craters in the first bin (0:::; t < 25 Myr). By comparing the numerical value of b with the observed number of craters in the first bin, it is possible to discuss whether the present epoch is in a socalled comet shower or not. The observed numbers of craters are 13 (D > 2 km), 7 (D > 10 km), and 6 (10 ~ D > 2 km), while the corresponding values of bare 7.9,6.8 and 1.5 respectively. It seems that there is no significant excess of large craters (D> 10 km) in the latest epoch, while there is a considerable excess of young intermediate craters. This result may be compared with Stothers (1988), who from the discussion of large craters argued for the current Solar system being in an intense comet shower. On the other hand, intermediate craters are present in considerable numbers in the latest epoch. If this is genuine, one is led to conclude that the present Solar system is in an epoch of comet shower. Finally, there still remains the problem of errors associated with crater ages. Some of the crater ages compiled by Grieve (1993) contain large probable errors. Since the data set and subsets thereof yield periods of close to 30 Myr, as will be shown, we tentatively adopt only those craters with errors equal to or less than 10 Myr. This limitation significantly reduces the number of craters available, but in a discussion such as the present one, which is of controversial character, it seems desirable to derive conclusions adopting conservative data sets. 3 CRITERION The criterion which was adopted in Paper I to test the periodicity hypothesis is that of Broadbent (1955, 1956). A brief account thereof, and its modification, are given here. Let P be the period and (J. the latest epoch (of crate ring) to be determined, and t l' t 2'.., t N be the ages ascribed to the craters. If there were an exact periodicity, the ages should equal (J. + nip k, where ni is an integer (0, 1, 2,... ). One may measure deviation from an exact periodicity by the sum (3.1) where the integers n i are specified so as to satisfy the inequality Table 1. Statistics of the craters used for testing periodicity hypothesis. Regardless of D D~2km D>lOkm 1O~D>2km Regardless of D D>2km D>lOkm 1O~D>2 km D~lkm D~2km No. of craters Crater age~ < 600 Myr D (average of D) km Craters with age ~ 25 Myr No. of craters 15 (km) Fraction of small craters age < 1 Myr age < 5 Myr Decay const. (Myel) 0.105/ / /25 age~25 Myr bin eq. (2.1) (Myr-l) 7.9/25 6.8/25 1.5/25

3 n 10 5 Figure 1. Histogram distribution of crater ages compiled by Grieve (1993). Small craters (D:s 2 km) are not included. The hatched part represents large craters (D > 10 km), whilst the rest corresponds to those with 10"?D > 2 km. i=1,2,..., N. By carrying out a number of numerical experiments, Broadbent proposed that for the periodicity (quantum hypothesis) to be statistically significant, the following inequality should hold true: IN (--- 1 S2) >1 '\In 3 d 2 ' P whered=-. 2 If tl> t 2,.., tn were uniformly randomly distributed, the probability of the above inequality being satisfied would be less than The criterion is thus empirical. Now, a direct application of this criterion would not be justified when it is known beforehand that the ages of known craters are not uniformly distributed. What one may call the decay constant of craters has been determined in the previous section. Having determined the decay constant, it remains to modify the Broadbent method so as to take into account the effect of crater loss with time. Basically, this consists of obtaining the probability distribution of the minimal values of sip on the assumption that t l, t 2,., tn are randomly distributed and estimating whether the calculated minimal value of sip from the crater data set is attained only by rare chances. First, we generate a series of random numbers between 0 and 600 Myr which obey an exponential distribution. These are denoted by Tl> T 2,, TN' For assumed values of IX and P, calculate the sum (,z 1 N -=- '\' {T-(IX+nP)V p 2 NP2.'--' " 1=1 and minimize the sum by varying IX and P. For a given set of T l, T 2,, TN the minimum value of (lip can also be calculated. By carrying out a large number of similar calcu- Test of periodicity on crater formation rate 729 lations, it is possible to find out the distribution of the minimum values of (lip. The cumulative distribution of (lip can then be used to estimate how likely is the calculated sip value on the assumption of random distribution of crater ages. In the present work, 400 random samplings have been used to obtain the cumulative distribution of (lip. 4 STATISTICAL TEST 4.1 Craters with D :::;; 2 km removed One may start the discussion of the periodicity or otherwise of crater age by considering the set of craters with diameters D > 2 km; those with D :::;; 2 km are inadequate for use in the testing of the periodicity hypothesis. As seen from Table 1, there are 74 craters in the data set of Grieve with ages <600Myr andd>2km. There still remains the problem of errors associated with crater ages. Here, and in the following, those with errors greater than 10 Myr are removed from the data set. One is then left with 49 craters. Fig. 2( a) gives the probability distribution, p, against the minimum deviation (lip on the assumption that the crater ages are random variables sampled from an exponential distribution, exp ( -at), where the decay constant a has been determined in Section 2. The distribution of minimum (lip has been estimated for each case by carrying out 400 samplings. We point out that minimal values of (lip were obtained by varying IX between - PI2 andpl2 and P between 10 and 50 Myr, because earlier discussions of the periodicity all yielded P close to 30 Myr. On the other hand, in order to obtain the minimum value of sip, we have varied IX between - PI2 and PI2 and P between 10 and 50 Myr. For the data set under consideration (D > 2 km and age < 600 Myr), two solutions have been obtained. One is IX = - 6 Myr and P = 31 Myr and the other is IX=4 Myr and P=30 Myr. The variation of sip with P is given in Fig. 2(b). As may be seen, the minimum value of sip is The corresponding probability,p, on the nullhypothesis ofrandom distribution is 0.5 (50 per cent) as may be seen from Fig. 2(a). For a null-hypothesis to be rejected, it is customary to set the significance level close to 10 per cent or smaller. If, therefore, the craters with D > 2 km can be taken as an unbiased record of impact events, one is led to the conclusion that, although a periodicity at 30 Myr is detected, it cannot be regarded as statistically significant. 4.2 Intermediate craters (10"?D > 2 km) When craters with age uncertainties greater than 10 Myr are removed, there are 31 craters with diameters between 2 and 10 km. To obtain the minimum value of sip, we have varied IX between - PI2 and PI2 and P between 10 and 50 Myr. As for the subset lo"?d > 2 km, two solutions have been found. One i~ IX = - 6 Myr and P = 31 Myr and the other is IX=4 Myr and P=30.5 Myr. The behaviour of sip which yields these solutions is given in Fig. 3(a), while the cumulative probability distribution of (lip appropriate to this subset is given in Fig. 3(b). The minimum value of sip is 0.226, and the corresponding value of p is It seems clear that the detected periods (30 and 30.5 Myr) are not to be taken as statistically significant.

4 730 S. Yabushita 4.3 Large craters (D > 10 km) Finally, we consider large craters, characterized by diameters > 10 km. There are 18 such craters with ages < 60 Myr and age errors less than 10 Myr. The decay constant for them is a =0.165 x 25 Myr-'. Considerable care is needed in dealing with these craters. A formal plot of sip in (IX, P) space yielded a period at P=27.5 Myr and IX= - 2 Myr, where s/p=0.205 (see Fig. 4a). However, a far lower value (sip=0.163) is obtained for IX = 5 Myr and P = 49.5 Myr. Thus, without any previous dis- 1.0,...---, ~-r, cussions of the periodicity hypothesis, one would be tempted to regard this period as more significant than the period close to 30 Myr. Since all of the earlier investigations of the periodicity hypothesis discussed periods in the neighbourhood of 30 Myr, we discard the period at 49.5 Myr. The cumulative distribution (lip is thus obtained by varying IX between - 10 and 10 Myr and P between 15 and 40 Myr. The cumulative probability of (lip is plotted in Fig. 4(b). On the other hand, the minimum value of sip calculated from the data set of Grieve (1993) is at P=27.5 Myr, as may be seen from Fig. 3(b). The corresponding value of p is 0.64 (or 64 per cent). Thus, the derived period of P=27.5 Myr may not be regarded as significant. The results obtained here are summarized in Tables 2 and 3. p O.35r ,.---~--,---_r --. N=49 O. 201L:-O :2:l=-O :3~O, l P (myr) Figure 3(a). The same as Fig. 2(b), except that craters with 10 ~ D > 2 km are adopted. The solid line shows IX = - 6 Myr and the dotted line 1X=4 Myr. alp r r-----,r----r---r-----,--:::,.----, Figure 2(a). Probability distribution of (lip on the assumption that crater ages obey a random exponential distribution. N = 49 is the number of craters with ages < 600 Myr. Vertical bars indicate uncertainties owing to a finite number (400) of random samplings. The dot denotes the value of sip calculated from crater data. p 0.5 N=31 O.201L:-O :2'-:cO----L-----:3:'-:O l P (myr) Figure 2(b). sip is plotted against P for the values of IX which give the minimal values. The solid line denotes IX = - 6 Myr; the dotted line denotes IX = 4 Myr. Craters with D > 2 km are taken into account trip 0.25 Figure 3(b). The same as Fig. 2(a), except that the decay constant and the number of craters are those for craters with lo~d >2km RAS. MNRAS

5 sip 0.25 Test of periodicity on crater formation rate 731 Table 2. Detected periods from the crater data. No. of craters Period 0( sip (Myr) (Myr) Regardless of D D~2km O~D>2km D>lOkm ~0-----L------::2~0-----L------::"=----I P (myr) Figure 4(a). The same as Figs 2(b) and 3(a), except that the craters with D > 10 km are taken into account. 0(= - 2 Myr r r--~-_r :::::=_---r---..., Table 3. Probability of the detected period being consistent with the assumption of random (exponential) distribution (ages <600Myr) No. of craters sip Probability Regardless of D < D>2km O~D>2km D>lOkm p 0.5 and P=31 Myr, while the cumulative probability distribution of (lip (calculated by adopting the decay constant for craters with D > 2 km) at is found to be less than From the point of view of statistical testing, the detected period might appear significant. However, in the presence of many young small craters, sip may take on a small value regardless of the periodicity in the time series: so the small value of sip here obtained should not be taken as evidence supporting the periodicity hypothesis ~/P 0.24 Figure 4(b). The same as Figs 2(a) and 3(b), except that the decay constant and the number of craters are those for D > 10 km. The dot corresponds to the minimal value of sip. 4.4 Small craters taken into account As has been shown in Section 2 (Table 1), there is an overwhelming abundance of small (D:;;; 2 km) and young (age < 25 Myr) craters so that small craters may not be adequate ; for the purpose of testing the periodicity hypothesis. It is, however, interesting to see how the result obtained so far is modified when they are taken into account. When we exclude those craters with probable errors > 10 Myr, there are 75 craters. The minimum value of sip (=0.195) is obtained for 0( = - 8 Myr and P = 31 Myr and for 0( = 2 Myr 4.5 Test of the hypothesis Based on the results presented in the present section, we now briefly discuss if the periodicity hypothesis can be substantiated. The problem being considered is, is it possible to regard the observed ages of craters, t H t 2'..., t N as samples taken from a population which obeys a random exponential distribution? On the null-hypothesis of the random distribution, the probability of (minimum) (lip taking on the observed minimum value sip or less has been found at less than , 0.5 and 0.73 and 0.64 for the craters regardless of diameters, for those with D ~ 2 km, for those with 10 ~ D > 2 km and for those with D ~ 10 km, respectively. In testing a statistical hypothesis, it is customary to place the level of significance at 10 per cent. If the customary level is adopted, one would be led to conclude that the craters with D > 2 km are randomly distributed in time while if smaller ones (D:;;; 2 km) are included, it may be possible to regard the adopted criterion for periodicity hypothesis not contradictory to the statistical data. The result is true for craters with ages of less than 600 Myr as well as for those with ages < 240 Myr. One must, however, note why small craters appear to satisfy the adopted criterion for periodicity. As shown in Table 1, there is an excess of small craters with age < 25 Myr. Bearing this in mind, equation (3.1) shows that if many craters are clustered at t = 0(, a small value of sip will

6 732 S. Yabushita be obtained and this will certainly make the adopted criterion easily satisfied. It has been confirmed that small craters alone do not exhibit periodicity, as might be expected. 5 SUMMARY AND DISCUSSIONS An investigation has been carried out to see if the crater ages compiled by Grieve (1993) exhibit a periodicity, which a number of authors claim to have detected. The method adopted is a modified form of the Broadbent (1955, 1956) method, and loss of craters with time has been taken into account. It has been found that craters with diameters > 2 km can safely be regarded as random samples taken from a population which obeys an exponential distribution, although a period close to 30 Myr is detected. When small craters are included, it may be possible to regard the adopted criterion for the periodicity at P ~ 30 MJyr as being satisfied, but this is due to the excess of young small craters. It must therefore be concluded that on the crater data alone, it is not possible to substantiate the periodicity hypothesis. It seems worthwhile, however, to point out that there are geological, climatological and palaeontological data which support some types of periodic phenomena. These are: a periodicity in the extinction rate of marine fauna of about 27 Myr periodicity (Rampino & Caldeira 1992), a variation of deep sea water temperature (Shakleton & Imbrie 1990), and the frequency distribution of geomagnetic reversals (Napier 1989). As to the geomagnetic reversal frequencies, the method adopted here may be applied to their distribution and a period at 30 Myr has been confirmed (Yabushita 1996), although its significance remains uncertain owing to a long-term trend in the distribution. These phenomena appear to exhibit periodicities, and the similarity of the period detected in the crater data, although difficult to substantiate by the statistical argument, to these periods may be taken as a whole to indicate periodic external disturbances. ACKNOWLEDGMENT The author wishes to thank RAP Grieve for providing a recent set of data on craters, and a referee for constructive comments. This work was supported by grant no of the Ministry of Science, Culture and Education, Japan. REFERENCES Alvarez W., Muller R A., 1984, Nat, 308, 718 Broadbent S. R, 1955, Biometrika, 42, 45 Broadbent S. R, 1956, Biometrika, 43, 32 Grieve R A. F., 1987, Ann. Rev. Earth planet. Sci., 15, 245 Greive R A. F., 1991, Meteoritics, 26, 175 Grieve R A. F., 1993, Compilation of crater data obtainable on application to Grieve Grieve R A. F., Sharpton V. L., Goodacre A. K., Garvin J. B., 1985, Earth Planet. Sci. LeU., 76, 1 Napier W. M., 1989, in Clube S. V. M., ed., Catastrophes and Evolution. Cambridge Univ. Press, Cambridge Rampino M. R., Caldeira K., 1992, in Clube S. V. M., Yabushita S., Henrard J., eds, Dynamics and Evolution of Minor Bodies with Galactic and Geological Implications. Kluwer, Dordrecht Rampino M. R, Stothers R B., 1986, in Smoluchowski R, Bahcall J. N., Mathews M. S., eds, The Galaxy and the Solar System. Univ. Ariwna Press, Tucson Shackleton N. J., Imbrie J., 1990, Climactic Change, 16,217 Stothers R B., 1988, Observatory, 108, 1 Yabushita S., 1991, MNRAS, 250, 481 (Paper I) Yabushita S., 1996, in Rickman H., Valtonen M., eds, Small bodies in the solar system and their interaction with the planets. Kluwer, Dordrecht, in press

A statistical test for correlation between crater formation rate and mass extinctions

Mon. Not. R. Astron. Soc. 282,1407-1412 (1996) A statistical test for correlation between crater formation rate and mass extinctions Maki Matsumoto * t and Hirota Kubotani* t Department of Physics, Ochanomizu