The shape of the power spectrum of cosmic ray at ground level up to 7 × 10−3 Hz

Planer.

Space

Sci., Vol. 23, pp. 1603 to 1609.

Pcr~attton

Press,

1975.

Printed

in Northern

Ireland

THE SHAPE OF THE POWER SPECTRUM OF COSMIC AT GROUND LEVEL UP TO 7 x 1O-3Hz M. R. A’ITOLINI,

RAY

S. CECCI-DNI and I. GUIDI

Laboratorio Te. S.R.E./C.N.R.,

via de’ Castagnoli 1, Bologna, Italy and

M. G&LI

Istituto di Fisica dell’Universit8, Bologna, Italy (Received 3 April 1975)

power spectral density of cosmic ray fluctuation observed at ground level during the years 1966-1968 has been calculated. In order to obtain the correct shape of the spectrum, the Fast Fourier Transform method with a triangular data window was used and corrections were made for uncorrelated errors and aliasing effects. When ignoring the Earth rotation peaks, the spectral index a, for a sample of polar, middle latitude and equatorial stations, is N -1.96 in the frequency range 3 x 1O-T-1O-4Hz. A possible break around IO+ Hz, if existing, would be, on the whole, barely significant as CLwould change from N -1.96 to -2.10. There are indications that beyond lo-* Hz up to 7 x lo+ Hz the spectrum continues with a ~-2.

Abstract-The

1. INTRODUCTION

Fluctuations in the cosmic ray (CR) intensity inside and outside the maguetosphere, over the frequency range IO-?-lw Hz, using statistical techniques have only recently been investigated (Dhanju and Sarabhai, 197Oa, b; Dhanju, 1970; Martinic, 1971; Owens and Jokipii, 1972; Fujii et al., 1973; Attolini et al., 1974a) and a theoretical model for their production by the interaction of galactic CR with the irregularities of the interplanetary magnetic field (IMF) has been proposed (Owens, 1974; Owens and Jokipii, 1974). One of the basic features of this theory is the power spectral density of CR fluctuation intensity, which is related to the power spectra of the IMF. It is then evident that precise determination of the shape of the CR power spectral density (p.s.d.) is f~damental. We have recently published (Attolini et al., 1974a) some results about the bilateral p.s.d. of CR at ground level for the years 1966-68 for stations at equatorial, middle and polar latitudes with a resulting power law dependence on frequency with the exponent around -1.80. Such spectra were computed using the autocovariance method with Hamming lag window, prewhitening and recolouring. The side lobes of the spectral window, although reduced to less than O*S%, still decrease with only the inverse of the frequency distance from the center of the window. Then, with this method, some bias might still have been present in the results and the

negative power law index of the p.s.d. might have been greater than 1.80. In this paper we present an analysis of the CR p.s.d. computed using the Fast Fourier Transform (FFT) method, with a triangular data window. This calculation is more reliable than the previous one because it enables a spectral index up to -4 to be detected. Furthermore, spectral density results obtained by other authors beyond l(5-4 Hz (Dhanju and Sarabhai, 1970b; Dhanju, 1970; Fujii ef al., 1973) have also been reconsidered and compared together with a p.s.d. computed for a 9 day period of 6 min data from a neutron monitor station. 2. CALCUJiATIONS OF THE CR POWER SPECTRA We have used hourly CR intensity data from stations listed in Table 1. For the calculation of the biIatera1 p.s.d. we considered the natural logarithms of the hourly total counting, which approximate the fractional variation, in order to compare directly the absolute value of the amplitudes (Attolini et al., 1974a). The number of data points was N = 3 x 213 = 24,576, corresponding to about 3 yr of hourly data. The triangular data window 4(t) for N data that we have used is given by:

1603

4(t)

= {l - ]2r/(N c 1) - 11) for

d,(t) = 0 t=1,2,...,lV

for

1G IQ N c
t>N,

M. R. Arro~m,

1604

S. CJ~ccm,

I. Gum

and M. GALLI

TABLE 1

STATIONS

Vertical cut-off (W

Bologna

5.22

Kiel

2.29

Leeds

2.20

Calgary

1.09

7O

1.02

Chacal%@a

13.10

loo **__

0.01

@cleonic *

0.65

0.13

51°

0.!4

0.12

270°

n

1.20

0.09

270’

I

0.90

0.10

318O W-W*

I)

2.15

0.07

*

5.80

0.04

335O

II

0.77

0.11

15*

1.14

81’

. . (r;h;

0.05

.._“_

DeepRiver

counts/h Xl06

6.00

e-m_

..-_-

8-=

Component

Total ionking

c*_-

Sulphur Mtn.

Alert

Median asymptotic longitude*

Median asymptotic latitude *

*J.B.H@cer:Strircture of a. Forbush Decrease,Ph,D.Thesis,UnlversitY Of Calgary (1967). whose discrete transform

is

3. NUMERICAL RESULTS removed, from the original data, the steady diurnal wave, because such a component is meaningless in the p.s.d. since it would appear in the spectra as a spike proportional to the number of data used su~rimposed to the peak of the diurnal frequency band (Attolini er al., 1973). The amplitudes and phases of the subtracted diurnal waves, that were computed by Fourier analysis, are reported in TabIe 2.t In Fig. 1 we present p.s.d. in iog~it~ic variation units (I.v.u.) for middle latitude stations listed in Table 1, before applying the corrections for aliasing and uncorrelated errors; as one can see, due to these two effects, the spectra curl up at the high frequencies. Figure 2 shows the results from the same data of Fig. 1 after correction for these effects. In Fig. 3 we show the corrected spectra obtained from the data of an equatorial neutron station (~hacaltaya), a middle latitude station (Deep River), a polar neutron station (Alert) and from the data of the total ionizing component of Bologna. For the sake of an easier comparison, in all figures the spectra have been shifted, in the log-log scale, downwards (+) or upwards (-) about the overall mean of the corrected vaIues, in the frequency range 3.05 x lo-’ to 1.43 x 10+ Hz. ‘For some relevant frequency values, the correct numerical amplitudes of the p.s.d. are reported in Table 3. First we

which gives a frequency window that decreases with about the fourth power of the frequency. Thejth estimate PI of the p.s.d. is given by the mean of mj consecutive (l/N)-equispaced periodograms Iv;n) as 1 fil4 where

ICf1= 1~,E~,y(t)d~(t)ll~~N,

here x, indicates a series of N dara xt and FBindicates the discrete Fourier transform operation. The results were averaged over an integer multiple of I/40 of a decimal logarithmic unit in such a way that, on the logarithmic scale of frequency, the distance between various harmonics of terrestrial rotation, (i.e. T = 24 hr, 12 hr, 8 hr, 6 hr) would be an integer number of such frequency intervals. The random error on each of these estimates is then given by: 63 S Pjfd;;;;. Using a method described elsewhere (AttoIini

ef

al., 1974a, 1974b) we have eliminated the bias due to the tutcorrelated errors and to the aliasing effect.* * At thispoint we want to stress that,if theabovecorrections are not made, about 80 % of the resulting spectrum, in the frequency linear scale, wilf be strongly biased.

t A detailed analysis on the nature of this component will be presented elsewhere.

1605

Cosmic ray power spectrum at ground level TABLET Ampli+.wb of the stea&y aiwm (73 -149.72 0.153 162.95 0.253

statioml

Boloenr, Kiel lmda CW

0.270 0.247

135.13 12.70

salphur ntn DoopBiver

0.268 0.263

9.03 66.85

cboorl*a Aloti ‘

0.285 0.024

88.75 101.48

.

. . I

“’

I

.

HZ

. i

FIG.2. COMPARISON BETWEEN BILATERAL P.S.D. CORERRORS, FOR RXTJ3D FOR MEASURRMWT AND ALUSING SOM6 hUDDLELATITUDE NEUTRON STATIONSHOURLYDATA, FORTHEYRARS 1966-68.

EC--- w &--

P-

F=$_)

The values have been shifted towards their overall average in the log-log representation; in order to obtain the true values in this representation, add (+) or subtract (-) the quantity PL-P for Leeds, PPP for Kiel, PDB-P for Deep River, Pas-P for Calgary, and ParP for Sulphur Mountain. The solid line represents the power law least square fit of the points as explained in the text for a’ (see also Table 4).

4. DISCUSSION

OF TI-IR RRSULTS AND CONCLUSIONS

Hz

FIG.1. P.S.D.OFTHEHOURLYDATAOF

MIDDLRLA~E

196668. The curves have been shifted in the log-log representation; in order to obtain the true values in this re resentation add (+) or subtract (-) the quantity Pt- % for Leeds, PpP for Kiel, PDx-P for Deep River, PwP for Calgary and PsrP for Sulphur Mountain. NEUTRONSTATIONSDUUNGTEIBYRARS

8

First of all we notice that the diurnal peak in Alert p.s.d. is not clearly evident and that the whole spectrum follows a power law of exponent a = -1.96 f 0.02 from 35 x lO-‘Hz (- 1~133 days) to 1.43 x l@ Hz (= lc/2 hr). All theotherstations show the Earth rotation effect in a much greater extent, and appear to follow asymptotically the same slope as Alert from 1Cr6to 1.43 x l(P Hz.

4.75 10

1.47 ?O

7.34

1.02

6.24 10-l

3.38 10-l

1.22 10-l

1.38 10-l

4.87 10'~

5

5

7

72

17

24

3'

'9

57

72

40

108

8,l IO-'

1.3 1o-6

2.2 1o-6

3.6 l0'6

5.0 10-6

7.1 1O-6

a.9 10-6

"1.16 1O-5

1.7 10-6

2.1 1oe5

+2+32 'O+

2.53 1o'3

5.08 1O-3

1.13 10-2

3.99 10'2

4.36 10'2

1.18 10-l

9.66 JO-*

1.79 10"

1.03

3.61 'Q"

6.04 10-t

1.29

i.8'

7.16

1.58 lo

3.94 10

1.81 102

~~LP~URMTN

1.99 1 .Ol 6.27 10;' 5.74 10-l 1.97. 1.67 to-' 1.49 10-l 2.03 10-l 4.52 t0-2 4.98 10*~ 7.23 10-2 4.8' ioS3

2.08 I..08 6.10 10" 4.j3 10-l 2.14 2.02 10-l 1.33 10-l 1.79 10-l 5.04 lo-* 5.71 1o-2 1.25 lo'2 6.10 lo'3

k

1967-1968-1969

2.45 1O-3

8.24

7.29

2.13 lO'3

1.34 '0

1.46 10

1;99 lo2 4.52 10

1o’t

KIEt

4.96 10

2.07

DEEP RIVER

* These frequenciescorrespondto 1,2,3 cycles/day

2.64 lO-3

6.57 15-3

870

290

10-4

8.5 1O-5

1.35

1.20 vx2

94

5.5 w-6

4.82 'oe2

59

1.90 10"

1.25

s3.47 vx5

3.2 lO*!j

2.35 lo2

4

3.5 10-7

1.62

CAla~R~

frequency mj

2.24 1O-3

5.30 1o'3

1.39 10'2

5.87 low2

4.82 'II"*

1.42 '0-l

1.39 10-l

1.72 lo-'

2.97

5.37 10-l

6.36 10-l

1.12

1.93.

a.39

1.52 '0

4.26 '0

2.07 lo2

LEEOS

5.57 10-3 2.46 lO-3

5.37 lo-3 2.35 lO-3

9.20 10-4

1.84 1O-2

7.07 10-4

2.25 'O-2 2.51 '0'2

6.92 lo-'

2.05 10-l 2.39 to-2

9.76 lO-2 5.23 1o-2

2.84 la-' 5.45 'a"

1.16 10"' 1.06 10-l

3.81 'o-1 2.49

10

10

9.12 5.04 10"

1,50

6.64 lo-' 4.01 10"'

8.76 6.39

2.01

5.15

‘f-43

'0

la

$:

C~ACALTAYA

6.56 5.94

1.85

9.84

aOLOGNA

1.55 w3

2.85 tos3

5.59 10-3

1.92 'O-2

1.94 'o*2

4.30 BY2

9.68 1O-2 5.05 1o-2

2.84 10-l 2.03 la-"

1.04 10-l 4.55

1.47

a.76 Y.90

2.41 10

1.34 102

ALERT

3.4 1o-2

5.9 10'2

1.0 10-l

1.3 10-l

9.6 IO-*

1.6 '0-l

1.3 10-f 1.2 10-l

2.3 10-l

13 76’

2.4 10" 2.0 10-l

2.9 '0"

4.5 3,8 10"' 10-l

4.5 10'1

5.0 10"'

"j/pj

TABLE 3. P.s.D.VALUES OF SOME POINTS OF GIGS. 2 AND 3: WE TRUE VALUES OF THE P.&D. FOR EACH STATION CAN BE OBTAINED M~~PLYING~VALUESOFCOL~NSFROM 3 ~010~~ THE FACTORS 1.271,2.223,0.783,0.656,0.690,0.224,2.431~~ 1.343;RESPECXWLY

0 :: f:

F

6 a

9 8

I!2 ."

i

,j

%

P

K

Cosmic ray power spectrum at ground level

1607

different spectral indices al and a2 by least square fitting the periodograms separately, in the two frequency intervals before and after lO+ Hz as shown in Table 4. The results are as follows: (a) for ail stations, a’ ranges between -1.90 and -2.00 with mean equal to -1.95. For middle latitude stations the range is -1.92 to 1.98 with a mean value of - 1.94. (b) For all stations a1 has a mean equal to - 1.92 (in the range -1.78 to -2@6) and a2 has a mean value -2-13 (with a range -1.90 to -2=30), while for only middle latitude stations a1 ranges from -194 to -2.06 with mean -1.97 and az has a mean value of -2.10 having a range from -2GO to -2.25. From these figures one can conclude that the difference between eciand crBappears, if existing at all, barely significant. This nearly constant slope of the spectrum could be an occasional feature, i.e. due to

COMPARISON BETWEEN BILATSRAL P.S.D. CORRECTED FOR MEASUREMSNT AND ALIASING ERRORS FOR AN EQUACORIAL NEUTRON STATION, A WDLE LATITUDE NEUTRON STATION, A POLAR NEUTRON STATION, AND A STATION FOR THE TOTAL IONIZING ~~~, FOR THE YEARS 196648.

FIN. 3.

In order to obtain the true values add (+) or subtract (-) the uantity &-p for Chacaltaya, &E-P for Deep River, p,-% for Alert, && for Poiogna. The solid line represents the power law least square fit of the points as explained in the text for TV’(see also TabIe 4). In Table 4 we have summarized the results on the spectral index of all the stations considered, under two different hypotheses: (a) For all stations the spectral index a’ is constant from 1W7 Hz to beyond 1.43 x lo-* Hz (as it is for the Alert spectrum), but it is perturbed around the Earth rotation frequencies. a’ is calculated by least square fitting, to a straight line in a log-log plot, the periodo~~s co~espond~g to all frequency points in the intervals (3.05 x 1W7 to 9.46 x 10W6)Hz and (1.13 x 1O-Qto l-43 x 104) Hz. (b) The spectra for non-polar stations have a break at about f = 1O-6 Hz. In this case we get two

UP TO 7 X l&=* Hi. FIO. 4. P.&n. OFCR m points refer to (a) Chacaltaya (this paper), (0) Dhanju and Sarabbai (197Ob), (0) Dhanju (1970), (+> Fujii et al. (1973), (a) Jungfraujoch (this paper). The solid line reoresents a function nrouortional to zf-*. * 1 ‘ The

M. R. ATTOLINI,S. CECCHINI,I. Gum and M. GALLI

1608

TABLE4 Spectral index

a’ 1.59 10-5 - 2.00 10-5

3.05 10-7 - 9.46 10-6

Frequency intervals

3.01 10-5 - 3.37 10-5

1.13 10'4 - I.43 1o'4

(Hz)

3.57 10-5 - 1.43 1o-4

3.05 10-7 - 9.46 10-6

Dee: River

- 1.920 + .042

-2.25

-1.936 + .006

Leeds

- 1.937 + .051

-2.00

-1.935 + .017

Kiel

- 1.907 + ,062

-2.05

-1.907 + ,009

Calgary

- 1.965 + .036

-2.15

-1.894 + ,007

Sulphur Mtn.

- 2.045 + .036

-2.05

-1.926 + ,006

Chacaltaya

- 1.812 + ,053

-2.25

-2.067 2 .Oll

Bologna

- 1.841 + .072

-2.30

-2.086 + ,021

Alert

- 1.867 _t.,043

-1.05

-1.926 + ,019

-1.91

-2.11

-1.96

-1.95

-2.10

-1.92

Mean Values All stations Mid-lat. stations only

Alert (3.05 IO-'the superposition

1.43 10m4) Hz

: a’=

-1.964 f 0.018

of the spectrum of the worldwide variations to that of the scintillations treated by Owens (1974) and Owens and Jokipii (1974), the peaks of the diurnal harmonics being a function of the Earth rotation and of the spread of the asymptotic cone of arrival of the particles. Two more points can be made, looking at our results: (1) All non-polar stations show a peak corresponding to the non-steady diurnal variation; its amplitude is different for every station and might depend on the dispersion in longitude of the asymptotic cone. Its width indicates the presence of trains of diurnal variation lasting about a week, as already found by Attolini et al. (1973). (2) From Figs. 2 and 3 we see that the shape of the spectra is independent of the coupling coefficients of the counters; this fact indicates that the variational energy spectrum of the fluctuations does not depend on their frequency. At low frequencies (w lo-’ Hz), where the variations of nonscintillation type (i.e. Forbush decreases and 11 yr modulation) are predominant, the spectra seem to bend and then to rise again cf < lo-’ Hz). At higher frequencies there are indications that the spectra will continue with the same spectral index of N -2.00 up to 7 x 10” Hz. We have in

fact reconsidered the results obtained by Dhanju and Sarabhai (197Oa, 1970b), Dhanju (1970) and Fujii et al. (1973). Dhanju and Sarabhai (1970a) and Dhanju (1970) presented an extensive analysis on the data of the giant p-meson telescope of Chacaltaya with a counting rate of ~10~ countslsec, using the counts integrated over 2 min interval, and 1 min interval counts (Dhanju and Sarabhai, 1970b). We corrected their results for the Poissonian error which is the cause of the flattening of their spectra at f > 3 x 1VHz and plotted them in Fig. 4. As can be seen one obtains values that follow a power law with spectral index -1.5 rather than -2 (as would result with the extrapolation of the points of the neutron data from the same station in Figure 4). The difference is probably due to the spectral window used by the authors which allows a considerable power leak towards the higher frequencies. The results of Fujii et al. (1973), that we present in the same Fig. 4, may be biased for the same reason, and the spectral index of - 1.7 as quoted by the authors, may well be consistent with a real one around -2. In Fig. 4 we report also the p.s.d. obtained from a 9 day period of 6 min data containing the CR storm of September 1974 from the neutron monitor of

Cosmic ray power spectrum at ground level Jungfraujoch. As one can see the behaviour of the spectrum, particulariy in the low frequency range, is consistent with a power law -f”. The higher absolute values of the spectrum is evidently caused by the stormy conditions. Further investigations on the shape of the CR p.s.d. and its interpretation on the basis of the IMF structure in stationary and perturbed conditions of the interplanetary medium are now under way. Finally we want to point out that the triangular shape of the diurnal peak and the possible break in the spectrum indicate, in our opinion, that the theory of Owens (1974) and Owens and Jokipii (1974) is correct. Nevertheless, at this point, it is clear, that any interpretation of the shape of the spectra on the basis of that theory is only possible if one takes into account the response function, the cut-off rigidity, the spread of the asymptotic cone, the Earth rotation effect, the pitch angle distribution and the separation of the CR intensity fluctuations into a scintillation and non-scintillation part.

1609

~ck~w~e~e~~t-we are grateful to the colleagues whose data have been used in this work. REFERENCES

Attolini, M. R., Guidi, I. and Galli, M. (1973). 13th Znt. Co@ on CR, Conf. Papers 2, 795, Denver. Attolini, M. R., Cecchini, S. Guidi, I. and Galli, M. (1974a). Left. Nuovo Cirn. 2, 391. Attolini, M. R., Cecohini, S., Guidi, I. and Galli, M. (1974b). Lett. Nuovo Cim., 2,386. Dhanju, M. S. (1970). Plazzet.Space Sci. 18, 1719. Dhanju, M. S. and Sarabhai, V. (1970a). Acta Phys. Acad. Sci. Hang. 29, (21, 237.

Dhanju, M. S, and Sarabhai, V. (1970b). J.geophys. Res. 75, 1975. Fujii, Z., Mori, S., Yasue, S. and Nagashima, K. (1973). 13th Znt. Co@ on CR, Conf. Papers 2, 783, Denver. Martinic, N. J. (1971). 12th Znt. Co@ on CR, Conf. Papers 5,1959, Hobart. Owens, A. J. (1974). J. geophys. Res. 79, 895. Owens, A. J. and Jokipii, J. R. (1974). J. geophys. Res. 79,907.