The limiting primary rigidity of cosmic ray diurnal anisotropy

0032-0633’93 $6.00 +o.oo kc. 1993 Pergamon Press ttd

P&et. Spur SC+.,Vol. 41, No. 2. pp. 105-l il. 1993 Printed in Great Britain.

The ~irn~~i~~primary rigidity of cosmic ray diurnal a~i~~~rupy H. S. Ah~uwa~ia and I. S. Sahbah

Department of Physics and Astronomy. The University ofNew Mexico. Albuquerque,

New Mexico 8713’t.t 156, U.S.A.

Rcceivcd in final form 2X July IQ92

Our calculations of the upper cut-&primary rigidity (&). applicable to the solar diurnal anisotropy, have been extended to cover the declining phase of the solar activity cycIe 21. The calculated values of R, now encompass a complete solar magnetic cycle (196% 1987). We found that the values of & were very large for the 1982-I 985 period. almost twice as large as those observed previously for the periods of maximum solar activity. The largest value occurred in 1983. Also, the value obtained at the solar activity minimum year of 1986 was about twice as large as that obtained in 1965. Our results arecompared with those reported by others. Abstract.

We have shown that a careful analysis of the diurnal variation data (Riker and Ahfuwalia. 1987a). obtained with a variety of detectors located all around the world. provides valuable insights about the physical states of the heiiosphere (Ahluwalia, 1988a.b) and the transport of the charged primaries therein (Ahluwalia and Riker. 1987b; Riker and Ahluwalia. 1987b). Elsewhere (Ahluwalia and Rikcr. 19873)” we have described how the amplitude (,4 1of the anisotropy vector can be derived from the observed diurnal variation amplitude ([I,) obtained with a given dc~cctor (i). The upper

cut-off primary rigidity (R,) plays an i~npor~~nt role in this transfhrmation. Primaries with li > R, do not tontribute to the observed modulation of cosmic rays. Our earlier work covered the period 1%?-I979 (Ahlu~aiia and Erickscn. 1970. 1971 ; Ahluwalia and Riker. I987a). We found that ii, changes in a systematic manner over a

activity cycle. Furthermore. we noted that when solar activity is low. R, has a smaller value (R, < 50 GV) and when activity is high, R, a 100 GV (Riker it ul..

solar

1989). As a result of these analyses, we are able to understand why the amplitude of the diurnal variation is zero for the vertical underground muon telescope (UMT) at Socorro ft06.6’W 34X), located at a depth of 74 meters of water equivalent (MWE), during the lower solar activity period of 1975-1977. This aiso led us to conclude that the mechanism proposed by Erdos and Kota (1979) is of limited value in understanding some aspects of the observed long-term changes in the parameters applicable to the diurnal variation (Ahluwaiia. 1977). However, we did not understand the result of Ueno et d. fl985), who found that the average value of R, = 150 GV for the underground muon telescope at Sakashita ft38”E 36”N, 80 MWE) for the period 1978-1980. and that the average value of R, = 270 CV during the 1981-1983 period. Our results indicate that R, =r 100 GV for the 1978-1979 period. Ueno et af, offered no explanation as to why such large values of R, are acceptable. These circumstances motivated us to extend our earlier analysis to cover the period 19X0-1987.

Characteristics

of detectors

We have used the available data obtained with the neutron monitor (NM) at Deep River (symbol DR) and the vertical underground muon telescopes (UMTs) Iocated at selected depths. Data from three UMTs located at Embudo (27.3 MWE), Socorro (74 MWE) and Sakashita (X0 MWE) are used here. For reference, these three stations are assigned the symbols of EMB, SOC and SAK, respectively. The effective threshold rigidities (R,) for these four detectors are. respectivety, I. 1 GV, I9 GV, 45 CV and 49 GV. The median primary rigidities of response (R,,) for these detectors cover a broad range given by 16 GV G I?, < 331 GV. The median primary asymptotic latitude f&f of viewing for Deep River NM is about I&, and that for the UMTs at the three sites is given by 2, c 35 . North of the ecliptic plane. The instrLlmenta~ response characteristics for the four

106

H. S. Ahluwalia and I. S. Sabbah : The limiting primary rigidity of cosmic ray diurnal anisotropy

ASYMPTOTIC

LONGITUDE

(DEGREES)

Fig. 1. Variational coefficients are plotted, as a function of the asymptotic longitudes of response. for Deen River neutron monitor (la) and the underground muon telescopes at Embudo (1b), Socorro (Ic) and Sakashita (Id). See text ‘for details

detectors may be further compared by examining the corresponding cones of acceptance and the variational coe~cients (Rao rr ul., 1963). A FORTRAN program for computing the asymptotic coordinates and the variational coefficients for a given simulation of the geomagnetic field, for any given detector at a given location, is provided by McCracken et ul. (1962). Variational coefficients for a large number of NMs. operating at different global sites, have been computed by McCracken et al. (I 965) and Shea et al. (1968), for a range of values of the power-law spectral index /I of the primary variational spectrum. The value /j = 0.0 is applicable to the observed diurnal variation (Ahluwalia and Riker, 1987a). These authors have used geomagnetic coordinates for the computations. We have used their FORTRAN program. but have used the geographic asymptotic longitudes and seven discrete zenith angles for the entry of the primaries into the atmosphere given by lnoue et al. (1983) to compute the variational coefticients for Deep River NM and the UMTs at Embudo. Socorro and Sakashita. Coupling functions used are those given by Lockwood and Webber (1967) for the NM and Murakami ti II/. (1979) for the UMTs. In Fig. 1. we have plotted the variational coefficients (in percent) as a function of the asymptotic longitudes for Deep River NM (Fig. la) and the UMTs at Embudo (Fig. 1b). Socorro (Fig. Ic) and Sakashita (Fig. Id). The station meridian is shown by the dashed vertical line. One can see that the Deep River NM samples primaries from within a narrow range of asymptotic longitudes about 40. East of its meridian. while the UMTs at the three locations respond to the primaries incident from a wider range of asymptotic longitudes. We therefore anticipate that the amplitude and time of maximum of the diurnaf variation will be different at the four stations. as is observed. Variational coefficients thus provide us with the means of carrying out quantitative calculations. to derive information about the characteristics ofthe diurnal anisotropy. The method used by 11s in the past (Ahluwalia and Riker. 1987a) relies on the fact that the bulk of the counting rate of NMs and UMTs is contributed by particles incident from the vertical direction. So. in comparing the signals recorded by different detectors, we compare their

response in the vertical direction. An elaborate variance analysis is then carried out. The results obtained by using our simpler method compare favorably with those obtained by using the method of the variationaf coefficients. Recently, Yasue et al. (1982) have computed the variational coefficients applicable to the global network of neutron monitors, as a function of /I and R,. Also, Fujimoto et nl. (1984) have made a similar contribution, with regard to the muon detectors located at the surface and at underground sites. Although the variational coefficients for the NMs have been available for a long time (McCracken et al., !965), they were computed for a single value of R, = 500 GV. Now that the said limitation has been removed. they may be used more extensively to quantitatively relate the characteristics of cosmic ray space anisotropies with the observed temporal variations in the counting rates of detectors. We have used them to derive the results presented in this study. This makes it possible to make a direct comparison between our results and those of others. Relative amplitudes (a,), derived from the variational coefficients (Rae et crf., 3963). may be used to convert the observed amplitude of diurnal variation into that of anisotropy in free space. Also, geomagnetic bending correction for the cosmic ray flux, for a given detector, may be computed as a function of &. One may use 2, to compare the response of different detectors to a given change in R,. In Fig. 2, we have plotted the relative amplitude of the diurnal variation (a,) as a function of R, for the NMs at Deep River and Huancayo (Yasue et al.. 1982), and the UMTs at Embudo (EMB), Socorro (SOC) and Sakashita @AK), as well as the ion chamber (IC) at Cheltenham/Fredricksburg (CH), given by Fujimoto et a!. (1984). The response of the detectors tends to level off at R, = 500 GV, i.e. there is a very small increase in CI,, corresponding to a relatively large increase in the primary rigidity, to which modulation extends. The following features may be noted. (1) The response of the neutron monitor at Deep River (Ip, = 16 GV) tends to flatten out at lower primary rigidities compared to Huancayo NM (R, = 46 GV), IC

H. S. Ahluwalia

0.001

and I. S. Sabbah : The limiting primary

107

rigidity of cosmic ray diurnal anisotropy

(arbitrarily) that R, = 50 GV in performing computations. We note that, for Deep River NM, the value of c(, increases by about 9% in going from 50 to 100 GV. For other detectors, the change is appreciable, well outside the experimental errors. For Huancayo NM and IC, the changes are 20% and 30%, respectively. However. for Embudo UMT the change is 42%. The wrong choice is absolutely disastrous for the data obtained with the UMTs at Socorro and Sakashita, because there is a factor of 10 change in LX,for these detectors. In other words. NMs located at high latitudes are not very sensitive to change in R,, but those at low latitudes (such as Huancayo) are. The same is true of muon detectors, such as the shielded ion chambers (Ahluwalia and Dessler, 1962) and the underground muon telescopes.

1

10

30 UPPER

50 CUT-OFF

100

200

RIGIDITY

500 GV (Rc)

Fig. 2. Values of the relative amplitudes (a,) are plotted as a function of the upper cut-off primary rigidity (R,) for neutron monitors at Deep River (DR) and Huancayo (HU) and the underground muon telescopes at Embudo (EMB), Socorro (SOC) and Sakashita (SAK). Dashed curve is for Chel-

tenham/Fredricksburg

(CH) shielded ion chamber (IC)

(R,, = 67 GV) and the UMTs at Embudo (R, = 134 GV), Socorro (R, = 299 GV) and Sakashita (R, = 331 GV). (2) A figure of merit (FOM) may be defined. It may be used to devise a strategy to select certain detectors to determine the value of R, at any given time, using the annual mean amplitudes of the observed solar diurnal variation. We use a ratio of the amplitudes obtained with a pair ofdetectors (Ahluwalia and Ericksen, 1970 ; Ahluwalia and Riker, 1987a) and a complex variance analysis is performed. A simple measure of FOM may be the primary rigidity (R’) at which LX,has a value equal to, say, 75% of its value at 500 GV. For Socorro and Sakashita UMTs. R’ 2 300 GV. and for Embudo UMT, R’ z 200 GV. For IC. R’ z 100 GV. for Huancayo NM, R’ z 64 GV, while for Deep River NM. R’ z 30 GV. Below these values, the contributions to tl, per unit increase in R are from substantial to significant. So, if the value of R, > 30 GV. the strategy requires us to choose the pair consisting of Deep River NM or equivalent and any of the other detectors, depending upon what our guess is for the approximate value of R,. Of course, the best choice would be the UMT at Socorro or Sakashita or equivalent for R, < 300 GV. One has to keep in mind that the amplitude of the diurnal variation is small at Socorro and Sakashita. so the error in the ratio is large. Also. one should remember that the choice of IC or equivalent surface level muon detector requires us to apply temperature correction (Ahluwalia. 1992a). Our prior experience indicates that R, < 100 GV. So. we have always used the pair consisting of Deep River NM and Embudo UMT (Ahluwalia and Ericksen, 1970 : Ahluwalia and Riker. 1987a). This choice may not be satisfactory if R, z 200 GV. Such a situation has been encountered recently and is discussed later. (3) We may emphasize the need to be extremely careful in this matter by citing another feature from Fig. 2. Suppose R, = 100 GV for a certain year. However. we assume

Inverse problem method

In this paper, we present another method of calculating R,. It is similar to the method used for solving a wide variety of problems in geophysics and aeronomy (Rogers, 1976). The amplitude (ai) of the observed diurnal variation for a detector (i) may be written as : R~6D(R) ~

a, = i R,,

W,(R) dR.

D(R)

where W,(R) is the coupling function applicable to the detector and hD(R)/D(R) is the variational spectrum of the primaries responsible for the diurnal anisotropy (A) It is defined as : bD(R)

AR” cos i

D(R)

0

-1

if R < R.C’ if R > R,,

(2)

where E.is the asymptotic latitude of viewing for the detector. We may rewrite equation ( 1) as :

where : K,(R) = W,(R) cos E.

(4)

may be called the kernel of equation (3) which is very similar to the equation used in solving problems involving remote sensing (Rodgers, 1976). We may therefore choose from a wide variety of mathematical techniques employed in solving inverse problems. A set of values of the amplitudes a,. obtained with different detectors having different kernels (K,), are required to determine the appropriate solution F(R). The restriction is that the kernels must neither completely overlap, nor be completely detached. This requirement in turn. limits the choice of the set of amplitudes u, to carry out the inversion successfully. Figure 3 shows a plot of the kernels for Deep River (DR). Embudo (EMB) and Socorro (SOC). The relation between the kernels for the three detectors is just about right to obtain the optimum solution F(R). Several techniques have been used for treating different types of inverse problems (Rodgers. 1976 and references therein). It should be noted that the function F(R) is retrievable only

108

Ii. S. Ahluwalia and I. S. Sabbah : The limiting primary rigidity of cosmic ray diurnal anisotropy

Fig. 3. Values of the kernels (K,) arc plotted as a function of the primary rigidity (R) for Deep River neutron monitor and the underground muon telescopes at Embudo and Socorro

over a finite range of primary rigidities (R). This is the range over which the kernels provide valuable information. We investigate the variation of the ratio of the kernels as a function of the integration parameter (R). According to equation (3). if the ratio between any two kernels is constant, no information is added by the new measurements, since they could be predicted accurately from an earlier set of measurements by multiplying the right-hand side by this constant. The ratio of the kernels is plotted. for several pairs of detectors, in Fig. 4. The foilowing points may be noted. (I) Of the three UMTs, the ratios of the kernels for the pairs SAK, SOC and SOC, EMB become nearly constant at primary rigidities R > 75 GV. Thus, they are not useful if R, 2 75 GV. For a similar reason. the ratio of the kernels for all sea-level neutron monitors (not shown), taken two at a time, are also not useful for solving the inverse problem if R, > 50 GV. (2) The ratio of the kernels involving Deep River NM and any of the UMTs keeps increasing (nearly linearly) over a wide range of primary rigidities (R). For example, the ratio of kernels for DR and EMB increases steeply with R. over values of R < 100 GV, while the ratio for DR and SOC increases sharply with R, over the values of R < 150 GV. We arc therefore well advised to use the ratios of the kernels for Deep River NM and Socorro UMT to obtain the solution F(R). and thereby the value of R,, if we expect that R, < 150 GV. For values of R, < 100 GV. we may continue to use the ratios of the kernels for DR and EMB or equivalent. We also note that the ratio of kernels for DR and EMB does increase up to

100

200 300 PRIMARY

400 500 600 RIGIDITY, R (GV)

700

800

Fig. 4. Ratios of the kernels (K,) are plotted for various pairs of detectors. as a function of the primary rigidity (R)

150 GV, and that for DR and SOC up to 250 GV, though not as steeply as at lower values of R. These may be the limiting values that determine, in practice. which particular ratio of amplitudes one uses to compute &. These inferences are consistent with those drawn from our discussion of Fig. 2, which depicts the relative amplitude of diurnal variation (ct,) for different detectors. These insights do not come as a surprise to us. In the past, we have always used the ratios of the amplitudes of DR and EMB to determine R, (Ahluwalia and Ericksen, 1970; Ahluwalia and Riker, 1987a). We noted early on that the muon deteectors are more sensitive to changes in R,, while high-latitude neutron monitors are more sensitive to changes in /I (Ahluwalia and Ericksen, 1970). Our choice of the Deep River NM for this purpose was dictated by its very high counting rate and therefore superior statistics. In a sense. we lucked out. However, the diurnal amplitudes have smaller values for detectors located deeper underground, so the ratios have larger errors associated with them.

Cosmic ray data Elsewhere (Ahluwalia. 19751, we have described our method of processing the data obtained with Embudo UMT, and the various corrections which are applied to it. We reject days on which ground-level enhancements are observed at Deep River, as well as those for which less than 21 h of data are available. Corrected data for Embudo UMT and Deep River NM are subjected to harmonic analysis to determine the amplitude and the phase of the solar diurnal variation for each year. The errors ( tcr) are derived from the observed scatter of the daily vectors from the annual mean vector. For each .vear, the variance is calculated for each of the harmonic coefficients. The error in the annual mean diurnal amplitude is calculated by using the general formula for error propagation (Taylor. 1982). In Fig. 5a. we have plotted the annual mean amplitudes for the two detectors. as well as the annual mean sunspot numbers. for the 19791987 period. The data for 1979 are included for the sake of a tie-in with our previous study (Ahluwalia and Riker. 1987a.b). The errors, calculated from the observed scatter in the data for each year as described above, are indicated by flags for Deep River NM. and by the size of the open circles for Embudo UMT. One can recognize the epochs of sunspot maximum (1979), minimum (1986) and solar polar field reversal (shaded area). The polarities of the held in the Northern Hemisphere, both before (N) and after (S) the reversal, are shown. The amplitudes for both detectors have larger values when solar activity is high (1979-1980) and smafler values when it is low (l984-1987); ?lte arn~~~~f~~efor Et~l~ii~l~~b’M7‘ is zer’ct ji,r I%&lM7 when experime~ia~ crro~s f 5 31-j WC rcdcetz irrto accuunt. Also, note that amplitudes for Embudo UMT have larger values for the 1982.-- 1985 period. when solar activity is declining rapidly. The amplitudes for Deep River NM are also large for two of those years f 1984 1985). Previously, we suggested that these subsidiary maxima may be related to the epochs of the high-speed solar wind streams (Ahluwalia et al., 1990),

H. S. Ahfuwafia and I. S. Sabbah : The limiting primary rigidity of cosmic ray diurnal anisotropy

109

faf

YEARS YEARS

Fig. 5. (a) The annual mean amplitudes of the diurnaf variation obtained with Deep Rivet neutron monitor (@) and Embudo underground muon telescope (0) are ptotted for 1979-1987. Annuaf mean values of sunspot numbers (-- x --) are also plotted. Epoch of solar polar field reversal in 1980 is represented by the shaded area. Before the reversal, the solar polar field at the Northern Hemisphere points away from the Sun. (b) The annual mean amplitudes, corrected for the orbital effect, are plotted for the 1979-1987 period for the underground muon telescope (UMT) at Embudo and Socorro and for the 1979-1984 period for UMT at Sakashita. See text for details

which appear shortfy after the reversaf ofsolar polar fields. However. we note that the magnitude ofthe i~t~~lanetary magnetic field (IMF) is also quite large for this period (Slavin et cd.. 1986). Recently, we have shown that a correlation exists between the magnitude of the IMF and the limiting primary rigidity (&), applicable to the diurnal anisotropy (Ahluwalia, 1992b). Even so, the physical significance ofthe ~n~errclat~onshi~s among IMF, solar wind and cosmic ray diurnai variation parameters are sdtt not fully understood. In Fig. 5(b), we have plotted the annual mean amplitudes for the UMTs at Socorro and Sakashita, as well as for Embudo (for the sake of comparison). The Fourier coefficients for Socorro UMT were supplied to us by D. B. Swinson and those for Sakashita by H. Ueno. The corresponding errors ji>r euch _~?ar tz’ere a/so ~ff~~~~~i~~ hr rlre?~ Typicaf errors are shown for different years. for clarity. The three data sets have been corrected for the CommonGetting orbital effect (Ahtuwalia and Ericksen, 1970). The amplitudes for Socorro UMT also have larger values for the 1982-1985 period. They are even larger than the amplitude at the solar activity maximum in 1979, by as much as a factor of 2. One notes that the errors are large. but even so, &heincrease is remarkable. The same is true of the Sakashita annual mean amplitudes for the t 982-1984 period, for which the data are available to us. As mentioned before, a subsidiary maximum exists for this period in the Embudo data. Ueno et crl. (1985) have labeled this increase as “anomalous”. Nagashima rr uI. ( 1987) suggest that the origin of this effect may lie in an interaction (undefined by them) between the primary cosmic rays and the solar magnetic dipole moment. We show that the observed large amptitudes are characterized by larger values of R,, which in turn are correlated with the larger magnitudes of the IMF (Ahluwaiia, i 99%). Also, note that the magnitudes of the amplitude are greatly reduced for the UMTs at Embudo and Socorro when the solar activity becomes very low in the 1986-1987

period. We must emphasize the fact that, although the corrected amplitudes are reduced, they are still finite. The reader is reminded that before the orbital effect correction is applied to the Embudo UMT data, the observed amplitude is zero. This, however, is not the case for Socorro UMT, where the amplitude has a finite value for the 1986-1987 period, even before the correction is applied. This situation may be compared to that obtainable for the 1975-1977 period, when the diurnal amplitude for Socorro UMT is zero, but is not for the Embudo UMT. We estimated a vaiue of R, d 45 GV (Ahluwalia and Riker, 1987a) for this epoch. Thus we may expect a larger value of R, for the 1986-1987 period. This might sound like a contradiction, considering the data for Embudo UMT for the corresponding period. However, it is not. as we will discuss. We have investigated why the observed amplitude for Embudo UMT is negligibfy small for the 1986-1987 period, when such is not the case for Socorro UMT. For this purpose. we have examined the monthly mean values of the amplitudes obtained with Deep River NM, as well as with the UMTs at Embudo, and Mawson, Australia. The two UMTs have similar characteristics, although their median asymptotic latitudes of response (2,) are different. We find that for Deep River NM, monthly mean amplirudes become quite small during the last trimester of 1986 and the first trimester of 1987, but they do remain finite. On the other hand, during the same time period, the monthly mean vectors obtained with UMTs at Embudo and Mawson not only become small, but also undergo a systematic rotation on the harmonic dial, forming a closed loop. The aggregate contribution is thereby reduced to a very smaft vatue for this time period. This is not true of the Socorro UMT data. A similar (unexplained) effect may have been observed during the deep solar activity minimum (1954-l 955) period. with the NMs at Huancayo and Climax and with the surface muon telescopes at Rome (Conforto and Simpson, 1957), as well as with the shielded ion chambers at Huancayo and Cheltenham (Venkatesan

H. S. Ahluwalia

and I. S. Sabbah : The limiting primary

YEARS

Fig. 6. Calculated values of R, are plotted for 1965-1987. In each case, the value of R, is in the center of the bin for the year. Diffcrcnt pairs of diurnal amplitudes are used to calculate Rc for a given year. Note that values of R, are large when solar activity is high (1968-lY70, 1979-1980) and small when the activity is

low (1965. 1976-3977). Also, note the exceptions when R, is very large for 1982--l 985 and for the solar activity minimum year of 1986. Epochs of solar polar field reversals (shaded areas). See text for details

are also

and Dattner, 19.59). This part of our investigation incomplete and will be reported later.

shown

is still

Upper cut-off primary rigidity To be able to define the anisotropy vector in free space using diurnal variation amplitudes and phases, we have to know the value of the upper cut-off primary rigidity (I?,) applicable for a given time period. We now proceed to compute the values of R, for the 1980-1987 period, thereby extending our earlier calculations (Ahluwalia and Riker. I987a) to a complete solar magnetic cycle. In Fig. 6, we have plotted the values of R, determined in two different ways. In each case, the annual mean value for R, is plotted in the center of the bin for the year. One may note the following features. (1 f The continuous curve. with error bars, represents the values of R, determined from the ratios of the diurnal amplitudes obtained with Deep River NM and Embudo UMT. for the corresponding years. Values of R, are determined by taking the ratios of the observed amplitudes, after correcting them for the Conlpton-Getting orbital effect. The results for 1980--19X5 are shown, along with those reported earlier for the 1965- 1979 period (Ahluwalia and Riker. 1987a). Note that the values of R, are not computed for the lY8& 19x7 period. because the observed values of diurnal amplitudes at Embudo are insignificant at i 2tr level. as discussed above. (2) The values of R, derermtned from the ratios of the diurnal amplitudes obtained with Deep River NM and Socorro UMT are also given (dashed lines with x ‘s). The ratios are taken after correcting the observed amplitudes for the Compton-Getting orbital eH‘cct. Note that the values of R,are not calculated for the 197S--1977 period. because the observed amplitude of the diurnal variation at Socorro is insignificant (within I 3cr) for this period (Ahluwalia and Riker. 198721). It is interesting to see that the values of R, for the 196X 1974 period. determined

rigidity of cosmic ray diurnal

anisotropy

from Socorro UMT in a similar manner, are in agreement with those calculated from Embudo UMT data, if one keeps in mind that the errors for the Socorro UMT data are larger. This is also true for the 1978-1980 period. So, unlike Ueno et al. (1985), we still maintain that R,z 100 GV rather than 150GV for 1978-1980. Afterwards, the values of R,determined from Socorro UMT data diverge sharply from those calculated from Embudo UMT data ; the former are much higher. Two important conclusions may then be drawn. First, the value of I& exceeds IO0GV after 1980 ; R, = 175?;,’ GV in 1982 and reaches its highest value of 195) 10 GV in 1983. Ueno er ai. (1985) report a value of R, = 300 GV for Sakashita UMT for 1982. However, we note that Kudo et al. (1987) give a value of IX0 GV and 178 GV for 1982 and i 983. respectively, in good agreement with our estimates. Second, for 1984 and 1985 we have R, = 1X71:,8 and Rc = 1401i’! GV compared with R, = 90:;'GV and R, = 672; GV, respectively, determined from the ratio aDR/aEMB. The lack of agreement between the two sets of values of R, is surprising, but may indicate that aDR/aEMB ratios are not appropriate to use if R, exceeds 100 GV by a significant amount. However, this is consistent with our prior insight, which was derived from a comparison of the corresponding kernels. We note that DR and EMB ratios of kernels may not be appropriate to use if R, exceeds 100 GV by a significant amount. (3) The dark. thick line in the diagram indicates that for the 1965-1980 period, it is acceptable to use & z 50 GV when solar activity is low (1965. 1976-1977) and R cz 100 GV when activity is high (1966-1975, 197% 1981). This rule of thumb breaks down after 1981. Even so, it is appropriate to use R, = 100GV for NM and Embudo UMT data for the 1982-1984 period, within the errors in the observations. Socorro UMT data indicate that the mean value of R, for the 1986-1987 period is given by R, = 128 +A” GV, which agrees with the value of about 120 GV quoted by Fujii and Ueno (I 990) for the data obtained with Sakashita UMT. It should be pointed out that the values of R,plotted in Fig. 6 are systemati~aliy larger (by about 10%) than those cdiculated by considering the detector response in the vertical direction (Ahluwalia, 1997b.c). (4) There is a tendency for the value of R, to be lower immediately after the epochs of the solar polar field reversals (shaded area). Also, values of R,computed from the ratio of the amplitudes for Deep River NM and Embudo UMT exhibit a noticeable periodicity of 5-6 years. We note that there are subsidiary maxima in the annual mean diurnal variation for the 1973-1975 and 1982-1985 periods, when the values of R, are larger in Fig. 6. These are also the epochs of high-speed solar wind streams (Ahluwalia and Riker, t987a ; Ah~uwalia et al., 1990 ; Kozlov cz al., 1990). For the latter epoch, the annual mean magnitude of the 1MF is also very large. In fact, it has been shown that R,and the IMF are correlated (Ahluwalia. 1992b). Diurnal anisotropy In addition to the determination of R,values depicted in Fig. 6 using the method of the ratios of the diurnal

H. S. Ahluwaiia and 1. S. Sabbah : The ~irn~tingprimary rigidity ofcosmic ray diurnal anisotropy amplitudes, an independent set of R, values were also calculated by seeking the solution for F(li) in equation (3). Inverse rnet~l~ds suggested by Backus and Gilbert (1970) and by Chahine (I 977 and references therein) were used for this purpose. The calculated values of R, are found to be in good agreement with those obtained by considering the ratios of the diurnal amplitudes obtained with Deep River NM and Socorro UMT. The mathematical and numerical details of the calculations, when inverse methods are used, will be described elsewhere. However, we would like to quote some pertinent results obtained from these calculations. For example, for 1980 we find that /? = 0.0, R, = 89 GV and the amplitude of the anisotropy is given by A = 0.42%. We find that in general, the solution F(fz) may be represented analytically as follows :

(5)

where Ac( < depend upon the primary rigidity (R) in some characteristic manner, so as to lend some physical si~nj~c~Iilcc to the concept of R, (Ahluwatia and Rikcr. 1%7a). Clearly. the magnitude of the IMF will have to hc considered in such a discussion. The search for the physical justification research.

of&

remains

an important

goal ofour

fit

1962). Their own crude estimates support this fact. They state, ‘*.. . the variations seen at these stations (HuanCayo~Cheitenham) are similar in kind to those seen at lower rigidities. Therefore, we see no reason to believe that their origin is different.” The point is that, although the origin of the variations may not be different. the quantitative estimates of the physical parameters involved may be misleading, in that any inherent rigidity dependence of the parameters may be seriously compromised. We have shown in this paper that the value of Rl, is significantly greater than 100 CV for the 1982-1985 period. Also. it varies over a wide range during a solar magnetic cycle. Estimates of the physical parameters made using UMT data, but ignoring these temporal variations in R, are likely to be erroneous. If we apply the argument of Bieber and Chen to the data obtained with the UMTs, we finds that if the real cut-off is at 100 GV. then assuming R, to be 50 GV will re&uce the value of CI,(see Fig. 2) for Socorro by a,fbctor qfnhout 10. For the case of R, = 200 GV, the value of c1i will increase h_v a factor oJ’ about 3. For Embudo UMT, the corresponding &cturs are about 42% and 50%. So. Bieber and Chen’s argument turns out to be faftacious when applied to UMT data. The need for extreme care in these matters is absolutely essential. In fact. we are now investigating the rigidity dependence of the transport phrameters, using NM and UMT telescope data. They will be reported elsewhere.

Summary

We have shown that the calculated values of I?, vary over a wide range during a solar magnetic cycle (1965-1987). In particular. we find that R, is signi~cantly greater than 100 GV for the 1982-1987 period. This is, indeed, the first time that such large values of R, have ever been reported and confirmed. The physical significance of this result has not been established clearly. We find the surprising result that R, is not equal to SO GV at the solar minimum of 19Xft. as our prior experience would have indicated. It is possible that a similar situation exists For prior solar magnetic cycles. This experience suggests that a greater value should be attached to the observations made at depths exceeding 70 MWE than has been realized before. We have provided a rational basis for understanding the fact that the calculated values of R, appear to depend upion the median primary rigidity of response (R,) of the dttcctors whose data are used for the ~lculations. If R, exceeds IiN CV by a significant amount, UMT data from Socorro or equivalent should be used along with Deep River NM data.

future

Recently, Bieber and Chen f t991) have anatyzcd NM and shielded ion chamber (KY) data at Chrltenham over a Ion: time period by assuming that R, = 100 GV. They have ignored the temporal variations of R, repot-led by us and others. As we have pointed out eariier (and Bcibcr and Chen agree) NMs located at high latitudes are not very sensitive to changes in R,. but those at equatorial locations (such as Huancayu) are, as are also the muon detectors. such as the shielded ion chambers ~~hIuwaiia and DessIcr.

.,tc,ii,iirtr,~~,~kl[,~?r~~~?~.s. We appreciate the advice received from S%L’@ICII P. Hucstis ol‘ the Geology Dcpartmcnt regarding the ~ISL‘ OFinversion tcchniqtfcs. WC are indebted to D. 3. Swinson :tnd H. ticno For the data from underground muon telescopes :it So~orro and Sakashita. rcspcctively. We arc g’atcful to the rckrcc for helpful comments. This research is supported by NASc\ Grant No. NAGW-1468 to HSA. One of us (ISS) is very pratcful lo the Pcacc Fellowship Program for Egypt, which was fundcti by rhe U.S. AFcncy for International Development and the Gnvcmmcnt of F;gypt. Additional support from the Unjvcrsity of New Mexico fs also a~knowied~~d by ISS.

H. S. Ahluwalia and I. S. Sabbah : The limiting primary rigidity of cosmic ray diurnal anisotropy

112

Ahluw~ii~~ H. S. and Dessfer, A. J.+ Sofar diurnal

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Obscrvcd solar diurnal variation of cosmic rays during the period : 1979-87. 21st ICC!?. Adelaide. 6,303, 1990. Ahluwalia, H. S., Meteorological and environmental contributions to the counting rate of a muon telescope underground. iitrlt IC’CR, Miinchcn. 12,4215, 1975. Ahluwali~, H. S., Long term changes in the parameters ofcosmic ray daily harmonics. 15rl7 ICCR, Plovdiv. 11,398, 1977. Ahluwalia, H. S.. Is there a twenty year wave in the diurnal anisotropl of cosmic rays? Geopi~,ss. Rcs. Lcrtl. 15, 287, 19XXa. Ahiuwalia. H. S., The regimes of the East-West and the radial anisotropies of cosmic rays in the hciiosphere. Pfan~t. Space Sci. 35, 1451, IY88b. Ahtuwatia, H. S., Halt cvclc cf%cts in cosmic ray East-West anisotropy and IMF. C&@I:S. &.r. Lerr. (submitted), 1992a. Ahiuwalia, H. S., A correlation between IMF and the limiting prmtary rigidity for the cosmic ray diurnal anisotropy. Gt~“ph~x R<*s.Leti. 19. 633. I99?b. Ah~uwa~i~. H. S.. Solar magnetic cycle effects in cosmic rays. P~-oc. Isr S@LTZP $rrtrp.. Liblice. f, 26. 199% Backus, G. E. and Gilbert, J. F., Uniqueness in the inversion of inaccurate gross Earth data. PKi Ttw~s. R. Sot. Lad. A266, 123. 1970. Bieber, J. W. and Chen, J., Cosmic ray diurnal anisotropy, 1936XX: implicalions for drift and modulation thcorics. .-lsrrop,zw. f. 372, 30 I. 199 I. Chahine, M.. ~c~lcraliz~It~o[l inwrse

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