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The existence of cosmic ray sidereal anisotropies of galactic and solar origins with energies lower than 104 GeV and their modulation caused by the presumed behavior pattern of the heliomagnetosphere and of its neighboring gaseous matter in interstellar magnetic field

Abstract

It is shown that the sidereal anisotropy (SiA) of cosmic rays (CRs) with energies smaller than 104 GeV consists of three kinds: one (GA) is of galactic origin from the direction ΦG (αG = 0 hr; δG = −20°), and the other two (tail-in TA and nose-in HA) are of solar origin from the respective directions ΦT (αT = 6 hr; δT −24°) and ΦH (αH = 18 hr; δH > 0°) and supposed to be produced by the acceleration of CRs on the tail and nose boundaries of the heliomagnetosphere (HMS). This conclusion was arrived at in 1995 after a long-term delay since the beginning of CR observations in the early 20-th century. This delay was mainly due to the inconsistency among observations caused by the belief that the sidereal anisotropy must be unidirectional in space. The inconsistency has been solved at least qualitatively by the discovery of GA and TA. These anisotropies, including also HA, are subject, respectively, to their proper solar modulations in the HMS characterized by a polarity reversal every 11 years of the solar polar magnetic field and solar activity dependence with an 11-year periodicity. By using these modulation patterns, the origins of the three anisotropies have been determined. TA and HA thus determined inversely produce the following kinds of evidence and problems in the HMS: (1) the structure of the HMS, (2) acceleration of CRs on the boundary of the HMS, (3) CR Lens Effect of the HMS for the sharp concentration of TA and HA, (4) the proper motion (VHMS) of the HMS relative to neighboring stars, (5) the proper motion of interstellar gaseous matter (including the magnetic field) relative to neighboring stars, and (6) the existence of the Subordinate HMS surrounding the HMS for the explanation of the duality of the motion of the HMS and also of the absence of the Compton-Getting (C-G) effect on the HMS. The present paper not only presents a brief summary of the studies of CR sidereal anisotropy made by many researchers during the 20th century leading to the present understanding, but also presents some problems to open up a new vista of the future.

1. Introduction

The sidereal anisotropy SiA of CRs has been inferred indirectly from observations of the sidereal daily variation (SiD) on the ground ever since the discovery of CRs (e.g. Forbush, 1937a, b, 1939). Recently, due to greatly improved observational methods, the direct observation of SiA has become possible in the high-energy region by observations with cosmic ray and gamma-ray detectors (e.g. Amenomori et al.,2004, 2005; Guillian et al., 2007; Abdo et al., 2008; Abbasi et al., 2010). This method is, however, not applicable to SiA in the lower energy region as cosmic rays cannot maintain their direction hindered by the influence of the heliomagnetic and geomagnetic fields. In such a low-energy region, SiD, with the frequency 366 cycles/yr must be detected under the following difficult condition that it is always superposed by the solar daily variation (SoD) with the adjacent frequency of 365 cycles/yr produced inside the HMS and disturbed by the daily variations of the Earth’s atmospheric pressure, temperature and wind velocity. If SoD is constant, its yearly average in units of sidereal time, called the spurious sidereal daily variation (SSiD), becomes zero and does not affect SiD. Practically, however, as SoD is supposed to be almost always disturbed by various kinds of noises, the elimination of SSiD is indispensable. In addition to such indirect influences upon SiD, some of the HMS-phenomena, which directly influence SiD, have been discovered successively during a long elapsed time, such as the polarity reversal every 11 years of the solar polar magnetic field. It is not an exaggeration to say that the history of the study of SiD in the low-energy region (≤104 GeV) has been the history of the study of the direct and indirect influences upon SiD. In the following, we present a brief summary of the study of SiA of CRs made by many researchers with a dedicated attention to the direct and indirect influences in order to reach a better understanding of the nature of SiA and the features associated with it.

2. Influence of SSiD on SiD

The attempt to eliminate SSiD from the observed SiD of CRs has continued ever since the observation of SiD began in the early 20th century. Besides the well-known atmospheric influence on SiD (e.g. Myssowsky and Tuwim, 1928; Sekido,1943; Duperier, 1944, 1949; Olbert, 1953; Jacklyn, 1954; Trefall, 1955; Dorman, 1957; Wada, 1960), some of the elimination methods are presented in advance to simplify the following discussion on SiD.

Type-1.—The East-West difference method.

An atmospheric correction method was proposed by Elliot in 1952, which uses two CR telescopes at a station pointing, respectively, east- and west-ward with the same zenith angle. The elimination method is illustrated by the configuration of the solar diurnal harmonic vectors SoD1(E) and SoD1(W) with the same amplitude observed by the telescopes as shown in Fig. 1. The two vectors at the left side do not contain the atmospheric effect. The phase difference \(\Delta \phi\) between the two vectors is the geographical longitudinal difference and the direction of the anisotropy is 12 hr in this case. If the unknown vector C due to the atmospheric effect common to the two telescopes is superposed on SoD1(E) and SoD1(W), the resultant vectors \({{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{r}}({\rm{E}})\) and \({{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{r}}({\rm{W}})\) are expressed on the right side. The direction of the anisotropy is obtained by rotating the phase of the difference vector \(({{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{r}}({\rm{E}}) - {{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{r}}({\rm{W}}))\) clockwise by 90°. The unknown vectors C, SoD1(E) and SoD1(W) are derived by constructing the isosceles triangle with the base \(|{{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{r}}({\rm{E}}) - {{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{r}}({\rm{W}})|\) and the top angle \(\Delta \phi\) as shown in the figure. This method is suitable for the elimination of the atmospheric effects as far as the effects contained in the two telescopes are approximately equal to each other. This method was applied to the observation of the sidereal diurnal harmonic vector SiD1 by the Ottawa Horizontal Muon Array (OHMA), (Bercovitch and Agrawal, 1981; Bercovitch, 1984). This method was also applied to \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{o}}({\rm{E}})\) and \({{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{o}}({\rm{W}})\) observed with two directional air shower telescopes (E 104 GeV) at Mt. Norikura (36°N; 138 ° E) which succeeded in detecting SoD1 from the direction of the 6 hr local solar time, which is produced by the CG effect due to the Earth’s rotation around the Sun relative to the interplanetary cosmic-ray gas (Compton and Getting, 1935; Nagashima et al., 1989).

Fig. 1
figure1

Elimination of SSiD by east- and west-ward CR telescopes with the same zenith angle (Elliot, 1952).

Type-2.-The Farley-Storey method.

This method was proposed by Farley and Storey in 1954, and eliminates the influence of the annual variation of the amplitude of \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}\) from the observed \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\) by the following frequency modulation method (FMM). If the amplitude \(|{{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}|\) with a frequency of 365 cycles/yr is subject to an annual modulation with a frequency of 1 cycle/yr, the modulated \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}\) can be decomposed into three kinds of variation \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}},{{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) and \({\bf{A}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) with their respective frequencies 366, 365 and 364 cycles/yr as shown on the respective harmonic diagrams Si, So and ASi in Fig. 2. \({\bf{A}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\), called the anti-sidereal diurnal variation, is a newly-defined variation in the new time scale called the anti-sidereal time tASi (0 hr-24 hr) and normalized to the other times tSi and tSo at the September equinox. The vectors shown on ASi- and Si-harmonic diagrams express, respectively, \({\bf{A}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) and \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) with the same amplitude and the respective phases at the September equinox. These vectors, after a parallel transformation to the top of \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) on the So-diagram, are axis-symmetric to each other with respective to the \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\)-axis and rotate, respectively, clockwise and anti-clockwise with the angular velocity \(\omega (2\pi /{\rm{yr}})\), starting from the September equinox, and the resultant of the three vectors at a time expresses the annual variation of the original \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}\). The undetectable \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\), supposed to be contained in the observed \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\), can be eliminated by using the vector derived from the axis-symmetric inversion of \({\bf{A}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) with respect to the \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\)-axis. This elimination method has been called the Farley-Storey correction method (Farley and Storey, 1954; Jacklyn, 1962). The application of this method to observations was made frequently in those days though it was not certain whether or not the seasonal modulation of \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}\) would belong to the type stated in the above. However, this method will be used later successfully to find the annual variation of \({{\bf{S}}_{\bf{i}}}{{\bf{D}}_{\bf{1}}}\) (instead of \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}\)) produced by SiA, which is supposed to keep its direction \((\alpha ,\delta)\) through a year, but vary its amplitude with the maximum and minimum, respectively, at the Earth’s orbital location, where the Sun has the same or opposite direction to the source of SiA, and a significant seasonal variation would also be expected because of the Sun’s strong magnetic field. In this case, the annual variation of SiD1 is decomposed into three vectors \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{f}},{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) and \({\bf{E}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) expressed respectively on the harmonic diagrams So,Si and ESi. The newly-introduced \({\bf{E}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) is called the extended sidereal diurnal variation with the frequency 367 cycles/yr and will be used later for the determination of the annual variation of SiA (Nagashima et al., 1998, called hereafter Ref. 3).

Fig. 2
figure2

The Farley-Storey method for the elimination of \({\bf{S}}{{\bf{S}}_{\bf{i}}}{\bf{D}}\) (Farley and Storey, 1954).

Type-3.—Correction for stationary solar anisotropy (CSSA).

In this section, a method is applied that is effective for the elimination of SSiD produced by the annual variation of the stationary axis-symmetric SoA with the direction along the heliomagnetic field (9 hr–21 hr). The annual variation of \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}\) is produced from that of the second term of the spherical harmonic series of SoA owing to the Earth’s rotation around the Sun. \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}\) is decomposed into the three vectors \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}},{{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) and \({\bf{A}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) by the FMM. \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) and \({\bf{A}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) at the geographic latitude (γ), shown on the respective harmonic dials in Fig. 3, have the same amplitude proportional to \(|\cos {\rm{\lambda }} \cdot {\rm{sin \lambda }}|\) and reverse their directions for the sign change of \({\rm{sin \lambda }} \ {{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) can be obtained by rotating the corresponding \({\bf{A}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) anticlockwise by 90° and used for the elimination of the influence of the annual variation of \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}\) from the observed \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\) (Nagashima and Ueno, 1971). This correction method is called CSSA hereafter. Or, if two stations in the N- and S-hemispheres are conjugate to each other with respect to |γ|, pc (the geomagnetic cut-off rigidity) and Em (the CR median energy), the corrected \({{\bf{S}}_{\bf{i}}}{{\bf{D}}_{\bf{1}}}\) multiplied by a factor 2 can be obtained by adding the observed \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}({\rm{N}})\) without decomposing the three vectors \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}},{{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) at each station, as \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}({\rm{N}})\) is equal to —\({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}({\rm{S}})\). This method is called the simplified CSSA.

Fig. 3
figure3

The CSSA method for the elimination of SSiD produced by stationary SoD (Nagashima and Ueno, 1971).

It is noted here that although the 2nd term of the spherical harmonic series of SoA in the above was introduced for the explanation of the observed solar semi-diurnal variation, without any theoretical verification of the existence of such a distribution in space in those days, it seemed to be useful also for an explanation of the SSiD. The existence of the term has recently been confirmed theoretically (Munakata and Nagashima, 1986) on the basis of the CR diffusion-convection theory in the HMS (Gleeson and Axford, 1967; Jokipii and Kopriva, 1979). It is also noted that the CSSA method shown in Fig. 3 was obtained in 1971 on the assumption that the solar modulation axis coincides with the solar magnetic dipole axis. The rigorous treatment of the solar modulation in considering the difference of the axes was made by Tatsuoka and Nagashima in 1985 (Nagashima et al., 1983b; Tatsuoka and Nagashima, 1985) and the decomposed vectors were obtained as follows (Nagashima et al., 1985). where \({\eta _2}\) is the magnitude of the second term of the spherical harmonics. The corrections for those data after 1985 have been based on the revised data shown in the above. Several examples of the correction effect are shown in the following.

Table 1
  • Ex. 1—Some correction effects, shown in Fig. 4, indicate that almost all the \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\)s with the phase in the 4th quadrant (18 hr–24 hr), observed by telescopes pointing to the northern hemisphere, change their phases to the 1st quadrant (0 hr–6 hr) indicating that the \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\) contain a very large SSiD. In Figs. 4(a) and 4(b), the process of the correction method is shown.

  • Ex. 2—The correction would produce a serious influence on the interpretation of the sidereal anisotropy. The following is another example. In 1966, the bi-directional sidereal anisotropy model was presented by Jacklyn (1966), which was the first concrete anisotropy model ever derived from the observations of sidereal diurnal \(({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}})\), and semi-diurnal \(({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{2}}^{\bf{o}})\), variations by the underground muon telescopes at the northern and southern stations Budapest (47°N) and Hobart (43°S) in the period 1959–1962. The outline of his result is explained by the schematic diagram of the observation in Figs. 5(a) and (b). In it, the resultant of the observed \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\) and \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{2}}^{\bf{o}}\) expressed by dotted curve shows a sharp peak at 18 hr in the N-hemisphere and 6 hr in the S-hemisphere, suggesting that some concentrated CRs are arriving, respectively, from opposite directions probably along a straight line. This is called bi-directional anisotropy, being in accord with the L. Davis model for a pitch angle anisotropy of intensity (Davis, 1954). This model was soon afterwards reconfirmed by directional observations in the N- and S-hemispheres, making use of the Cerenkov detecting telescope at Nagoya (γ = 35°) (Sekido et al., 1968, 1971), and also by the ion-chamber data at the northern and southern stations Cheltenham (γ = 39°N) and Christchurch (γ = 44°S), (Nagashima et al., 1968). However, the \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}{\rm{s}}\) shown in Figs. 5(a) and (b) were not corrected by the CSSA method which was not known in those days. The real SiD1 has been obtained later by adding the conjugate vectors \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}({\rm{N}})\) and \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}({\rm{S}})\) according to the simplified CSSA, as shown in Fig. 5(c). SiD1 is common in both hemispheres and the resultant of SiD1 with \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{2}}^{\bf{o}}({\rm{N}})\) or \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{2}}^{\bf{o}}({\rm{S}})\) produces a sharply concentrated variation with a phase of 6 hr common in both hemispheres, asshown in Fig. 5(d), suggesting the disappearance of the essential properties of the bidirectional anisotropy model.

    On looking back to 1971, we notice that we abandoned at that time the most important principle that sidereal anisotropy should be studied by analyzing at least the sidereal diurnal and semi-diurnal variations and then composing them following the procedure which successfully discovered the bi-directional anisotropy. If this principle has been applied at that time, we would have noticed immediately after the disintegration of the bi-directional anisotropy that the diurnal and semi-diurnal variations at Budapest and Hobart had a common peak at 6 hours, in the N- and S-hemispheres, suggesting the existence of a sharply concentrated uni-directional anisotropy from 6 hours, as shown in Fig. 5(d). Because of this failure in 1971, almost 25 years passed before its discovery in 1995 (Ref. 3). It is further noted here that, during this lost period, only one paper was presented, by Cutler and Groom (1991), concerning the study of the sidereal anisotropy by composing the 1st, 2nd and 3rd harmonic components, and a considerably interesting image of the anisotropy was obtained (cf. Ref. 3).

  • Ex. 3—The simplified CSSA method applied to the conjugate stations Cheltenham (γ = 39°N) and Christchurch (44° S) is presented in the following. The long-term averages of \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\) ’s at these stations are shown in Fig. 6, together with that at the equatorial station Huancayo (γ = 12° S), (cf. Ref. 3). In it, the north-south difference of \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\)’s at Cheltenham and Christchurch is clearly seen. In those days of the analysis (1976), this difference was misinterpreted by Nagashima and Mori (1976) as being due to some north-south asymmetric sidereal vectors being superposed on a common sidereal vector, in spite of the earlier findings of their real origin in 1971 by Nagashima and Ueno. Fortunately, the decomposed vectors \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}({\rm{N}})\) and \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}({\rm{S}})\) shown in Fig. 6 can be identified with those derived from the simplified CSSA method at the two conjugate stations Cheltenham and Christchurch. The persistent difference between SiD1’s at these stations is shown by their phase dif-ference classified into morning (0–12 hr) and evening (12–24 hr) sides by the open and solid circles in Fig. 7. This asymmetry is very stable as can be seen also in the summation dial of the yearly vectors \(\Sigma 2 \times {{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{f}}({\rm{N}})\) shown in Fig. 8(a). On the other hand, SiD1 almost coincides with \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\) at Huancayo, indicating the negligible influence of \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) at the equatorial station Huancayo (cf. Fig. 6). It is emphasized that these two vectors show remarkable yearly fluctuations, especially in the period 1948–1957, of the positive polarity (P-) state of the solar polar magnetic field shown in Fig. 8. Especially, the solar activity minimum period in the P-state seems to be a special epoch for the appearance, or the phase change, of the sidereal anisotropy, which occurs every 22 years. The occurrences of the event have been successively observed in 1954 (Conforto and Simpson, 1957), in 1975–1976 (Swinson, 1976; Nagashima et al., 2010) and in 1996 (Nagashima et al., 2010).

  • Ex. 4—The simplified CSSA was applied also to SiD1’s and ASiD1’s at the conjugate neutron monitor stations in the N- and S-hemispheres which were artificially made by averaging respectively the \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\)s and \({\bf{A}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\)s of almost all the data of 473 station-years in the northern stations and 147 station-years in the southern stations in the period of 1958–1979. The averaged \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\) and \({\bf{A}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\) are shown in Fig. 9(a) with the labels N or S. Their sum and difference between the N- and S-hemispheres express respectively \({{\bf{S}}_{\bf{i}}}{{\bf{D}}_{\bf{1}}},{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}},{\bf{A}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) and those residuals shown in Figs. 9(b) and 9(c). \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) and \({\bf{A}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) in Fig. 9(c) are almost the same in magnitude and their phases are, respectively, in the proper directions of 18 hr and 0 hr in the N- and S-hemispheres, justifying inversely the simplified CSSA method applied to the artificial conjugate stations. The phase of the real SiD1 is 6.8 ± 0.3 hr and almost coincides with that observed at Hobart (γ 43°S; Em 184 GeV) as shown in Fig. 4(a). However, the phases at London (γ 52°N; 60 m.w.e.) and those in Figs. 4(a) and 4(b), deviate toward the mid-night side from 6 hrs. The deviation (\(\Delta \phi \)) increases with the increase of cosmic-ray energy and reaches at the maximum (\(\Delta \phi \) = 6 hr) in the energy region 104 GeV observed by air showers at Mt. Norikura (36°N, 133°E) (cf. figure 9 in Ref. 3). Such an energy dependence was found in a wide range of rigidities (90–660 GeV) with the OHMA in the period 1976–1984 by Bercovitch (1984), as shown in Fig. 10.

Fig. 4
figure4

Elimination of \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) from \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\) by using \({\bf{A}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) according to the CSSA method. The corrected vector \({{\bf{S}}_{\bf{i}}}{{\bf{D}}_{\bf{1}}}\) is on the right. (a) Underground CR intensities at London (L) and Hobart (H), (Nagashima et al., 1985); (b) those at Sakashita (S), Misato (M) and Nagoya (N) at sea level; (c) those at Embudo (E), Socorro (S) and Bolivia (B). The telescopes point vertical (V), North (N) and South (S) at each location and only the observed vectors and, on the right, the corrected vectors, as shown. The amplitude scale is in percent.

Fig. 5
figure5

Schematic diagrams for the explanation of the bi-directional sidereal anisotropy presented by Jacklyn (1966) which used the underground muon telescopes at Budapest (47°N) and Hobart (43°S) and its disappearance by the CSSA correction method.

Fig. 6
figure6

SiD1 ’s with the error circles observed with the ion chambers at Cheltenham (\(({\rm{\lambda }} = 39^\circ {\rm{N;}}\phi = 283^\circ {\rm{E; }}{E_m} = 67 \ {\rm{GeV}})\)) and Christchurch (44°S; 173°E; 67 GeV) during 1937–1958 and at Huancayo (12°S; 285°E; 62 GeV) during 1936–1959, cited from Nagashima and Mori (1976). These results were derived from the data published by Lange and Forbush (1948, 1957) and Beach and Forbush (1961).

Fig. 7
figure7

Time series of phase of \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\) without CSSA (Nagashima and Mori, 1976). Solid circle: Morning side (0–12 hr); Open circle: Evening side (12–24 hr); NAG, ITA, MIS and TAK (Ichinose and Murakami, 1976); EMB, SOC and BOL (Swinson, 1976); HOB (Humble and Fenton, 1975; Fenton and Fenton, 1976); CHE, HUA and CHR (Lange and Forbush, 1948, 1957; Beach and Forbush, 1961).

Fig. 8
figure8

Summation diagrams \(\Sigma {{\bf{S}}_{\bf{i}}}{{\bf{D}}_{\bf{1}}}\) (on the right) and \(2\Sigma {{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}({\rm{N}})\) in Fig. 8(a) and of the yearly vectors, and \(\Sigma {{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}({\rm{HUA}})\) in Fig. 8(b), of the yearly vectors in Fig. 6.

Fig. 9
figure9

The elimination of SSiA from \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\) by the modified CSSA method, applied to the conjugate \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}({\rm{N}})\) and \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}({\rm{S}})\) artificially produced by averaging, respectively, almost all the northern and southern neutron monitor data (Nagashima et al., 1983a, 1984). Black triangles, observed total vectors; open triangles, expected spurious; black circle, corrected vector; open circle, resulting spurious.

Fig. 10
figure10

Energy dependence of SiD1 observed Ottawa Horizontal Muon Array (Bercovitch, 1984).

In spite of such differences in phase in the low- and high-energy regions, the coincidences in phase observed by the neutron monitor (Ex. 4, 20 GeV, 1983), ion-chamber (Ex. 3, 67 GeV, 1976) and the underground muon telescope (Ex. 2, 184 GeV, 1966) seem to suggest the existence of some sidereal anisotropy in the low-energy region in space from α = 6 hours (Nagashima et al., 1983a, 1984). But this was not accepted unanimously at that time for the following reasons: (1) the difference of the phases in the low- and high-energy regions mentioned above cannot be explained by the difference between CR deflections produced by the single anisotropy in the HMS, (2) the signal of the anisotropy in the low-energy region (20 GeV) would be difficult to maintain its shape until its arrival at the Earth because of disturbance in the HMS and therefore the observed SiD1 might be some residual of SSiD, and (3) the CSSA-method was not unanimously accepted in those days. Therefore, it took about another 20 years for the confirmation of such an anisotropy.

Although all the events corrected by the CSSA method do not directly connect with the image of sidereal anisotropy, they will be used to specify the discussion for the determination of the anisotropy, without any explanation of their complicated derivation process.

3.SiA in the HMS

Since the discovery of the polarity reversal of the solar polar magnetic field (Babcock, 1959; Howard, 1974a, b), its influence has been discussed actively in considerations of cosmic-ray intensity variations (e.g. Moraal, 1976; Nagashima, 1977 and references therein; Kuzmin et al., 1977; Nagashima and Morishita, 1980a, b; Kota and Jokipii, 1983; Nagashima, 1990 and references therein; Nagashima et al., 1991; Belov, 2000, references therein; Duldig, 2000 and references therein) and the cosmic-ray solar diurnal variation (Duperier, 1946; Thambyahpillai and Elliot, 1953; Gleeson and Axford, 1967; Kota, 1975; Moraal, 1976; Jokipii and Kopriva, 1979; Mori et al., 1981; Bieber and Pomerantz, 1983; Munakata and Nagashima, 1986; Nagashima et al., 1986 and references therein). The observational data of the cosmic-ray sidereal anisotropy, however, were limited in those days, but the theoretical expectation of the polarity dependence has been proposed by many authors in considering the interaction between interstellar and interplanetary magnetic fields, (e.g. Davis, 1954,1955; Sarabhai and Subramanian, 1966; Schatten and Wilcox, 1969; Marsden et al., 1976; Nagashima and Mori, 1976; Davies et al., 1977; Benko et al., 1979; Krainev, 1981). On the other hand, another approach to study SiD1 has been made by putting special emphasis on the behavior of the CR orbits arriving at several specific stations from a specified SiA, (e.g. Speller et al., 1972; Marsden et al., 1976; Davies et al., 1979). As part of further progress of such studies, the general formulation of the solar modulation of the axis-symmetric SiA from an arbitrary direction was presented (Nagashima et al., 1982 Nagashima and Morishita, 1983, called hereafter Refs. 1 and 2; Yasue et al., 1985). According to this formulation, SiD1 and SiD2 produced by this modulation can be expressed respectively by the linear combination of the vectors selected from the eight basic vectors in the geographic coordinate depending on the direction and constituent of SiA. Hereafter, this formulation of the modulation is called the modulation model. The most characteristic feature of the modulation is that SiD1 produced by SiA from the direction of the equatorial region is greater in the N- state than in the P-state, while, from the direction of the polar region, it is greater in the P-state than the N-state, or becomes almost equal in both states.

As regards observations, on the other hand, Cini-Castagnoli et al. (1975) first pointed out that a remarkable phase change from 18 hr to 3–6 hr was observed in the sidereal variation only in the Northern hemisphere for the transition from the N-state to the P-state in 1968/69. These stations were London (52°N; pm = 261 GV), Socorro (34°N; 280 GV) and Torino (45°N; 291 GV). They attributed the phase shift to the change of cosmic-ray orbital deflection in the HMS. This suggestion was supported by Gavril’yev et al. (1977), based on the observation at Yakutsk (62°N; 67 GV) in 1958–1975. Ichinose and Murakami (1976) also confirmed the occurrence of the shift at Takeyama (35°N; 212 GV), Misato (36°N; 145 GV), Tokyo (36° N; 71 GV) and Nagoya (35° N; 60 GV). They pointed out, however, that these phases were changeable from year to year and another marked phase shift, which occurred in 1973/74, was not attributable to the polarity reversal proposed by Cini-Castagnoli et al. According to the modulation model, one can find one vector out of the eight basic vectors which produces the shift from 18 hr to 3– 6 hr in the northern hemisphere for the transition from the N- to the P-state (1968/69). However, the vector is north-south asymmetric and must produce also a shift inversely from 6 hr to 18 hr in the southern hemisphere against the observation (cf. Ref. 2). The polarity dependence was further studied by using the data during a long-term period in order to increase the statistical accuracy. Some of the results are shown in Figs. 11(a), 11(b) and 11(c). The polarity dependence of neutrons in Fig. 11(a), which was obtained by the same method applied to the previous event in Fig. 9, shows some phase difference between the two vectors but does not show any amplitude modulation against the modulation model. The other two cases in Fig. 11(b) and 11(c) also do not show any clear difference in their amplitudes (Nagashima et al., 1985). Furthermore, the multidirectional OHMA mentioned previously could not observe the polarity dependence of the amplitude in the wide energy region (90–660 GeV) although their observation period (1976–1984) was not so long (cf. figures 3 to 8 in Bercovitch, 1984). The discrepancy between the observation and the modulation model concerning the polarity dependence was a serious question in those days. In addition to this difficult problem, there was another difficulty as follows.

Fig. 11
figure11

Polarity dependence of SiD1. (a) Neutron in the N-state (n) (1961–1969) and the P-state (1970–1978). (b) London (L) (52°N, 60 m.w.e.) in the N-state (1961–1969; 1981–1982) and the P-state (1970–1978); Hobart (H) (43°S, 36 m.w.e.) in the N-state (1958–1969; 1981–1982) and in the P-state (1970–1980). (c) London equatorially pointing telescope (E) (421 GV) in the N-state (1981–1982) and in the P-state (1970–1978).

The sidereal anisotropies observed at various energy regions did not show a definite direction as pointed out in the previous section. This is called the phase anomaly, hereafter. This anomaly has not been eliminated by the CSSA, nor by an increase of observations and statistical accuracy. The energy dependence of the phases observed at individual stations is shown in Fig. 12(cf. Ref. 3). These data are selected from many observations in the past on the condition that they finished CSSA, or that this was unnecessary as their telescopes were directed towards the equatorial region. In the low-energy region, the phase is 6 hr observed by neutrons and the underground muon detector at Hobart (cf. Figs. 9 and 10) and, with an increase of energy, it shifts towards 0 hr observed by the air showers at Mt. Norikura (36°N, 1.5 × 104 GV), (Nagashima et al, 1989). Such a phase anomaly cannot be explained by the magnetic deflection produced by the modulation model (Ref. 2). The anomaly could be observed, even at a single station, by the OHMA in the energy region (90–660 GV) under the same condition as shown in Fig. 10.

Fig. 12
figure12

Energy dependence of the phase of the observed SiD1 cited from Ref. 3. All the SiD1’s except several in the high-energy region are those of the vertical intensities corrected by the CSSA method, or those for the equatorially directed telescopes which do not require the above correction owing to a negligible influence. BO, Bolivia; SO, Socorro (Swinson and Nagashima, 1985); MI, Misato; HO, Hobart; BU, Budapest; YA, Yakutsk; LO, London (Nagashima et al., 1985 and references therein); SA, Sakashita (Ueno et al., 1990); LS, London (Thambyahpillai, 1983); LP, Laipootah; MV, Matsushiro (Mori et al., 1993); OT, Ottawa (Bercovitch and Agrawal, 1981); MS, Matsushiro (Mori et al., 1989); UT, Utah (Cutler and Groom, 1991); PO, Poatina (Fenton and Fenton, 1975;Humble et al., 1985); HK, Hong Kong (Lee and Ng, 1987); BM, Baksan (Andreyev et al., 1987); BA, Baksan (Alexeenko et al., 1981); NO, Norikura (Nagashima et al., 1989); DA, Norikura, directional air showers (Nagashima et al., 1989) and MU, Musala (Gombosi et al., 1975).

In addition to these problems, the following most important and difficult one must be added. In the northern hemisphere, the phase \({\phi _1}\) of the first harmonic vector SiD1 of the sidereal anisotropy SiA varies from 0 hr to 6 hr depending on the energy; on the other hand, the phase \({\phi _2}\) of the second harmonic vector SiD2 is 6 hr regardless of Em (Nagashima and Mori, 1976; Fujii et al., 1984; Mori et al., 1989; and references therein). On the contrary, in the southern hemisphere, \({\phi _1}\) and \({\phi _2}\) are all 6 hrs regardless of the energy. The solution for these problems will be discussed in the next section.

In relation to these phenomena, it would be worthwhile to mention that the anomalous enhancement of SiD1 in the equatorial zone was observed at Sakashita (540 GeV) by Ueno et al. (1984) and at Matsushiro (220 m.w.e.) by Mori et al. (1989, 1993), which was not certain whether it is due to the North-South asymmetric distribution, or the equatorial anomaly. Although the anomaly failed to be explained by the unidirectional anisotropy, it is conducive to the determination of GA and TA that follows.

4. GA and TA of Three Kinds of Sidereal Anisotropy in the HMS

A clue to understanding those difficult problems mentioned in the previous section has been obtained by a casual finding of the seasonal variation of SiD1 with \({\phi _1}\)(tSi) (α = 6 hours) at Hobart (Nagashima et al., 1995a, b, c). As mentioned previously, the seasonal variation of SiD1 (366 cycles/yr) is expressed by the variations with the sideband frequencies 365 and 367 cycles/yr, called the solar \(({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{f}})\) and the extended sidereal \(({\bf{E}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}})\) variations. It was found that the SiD1 in question contains \({\bf{E}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) with the phase of tESi = 12 hours, suggesting the existence of a seasonal variation of some sidereal anisotropy SiA responsible for SiD1. The seasonal variation of SiA must be of the Farley-Storey type as the declination \({{\rm{\delta }}_{{{\rm{S}}_{\rm{i}}}}}\) of SiA is constant, contrary to that \(({{\rm{\delta }}_{{{\rm{S}}_{\rm{o}}}}})\) of SoA of solar origin. The relative configuration between \({\bf{E}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}}\) and the average vector SiD1 is shown in Fig. 13 (cf. Fig. 2 and cf. Ref. 3). The configuration can be explained if SiA is subject to a seasonal variation with its maximum and minimum, respectively, at the December solstice (α = 6 hours) and the June solstice (α = 18 hours) when the Earth is located where the Sun has the same, or opposite, direction to the source of SiA, and a significant seasonal variation would also be expected because of the Sun’s strong magnetic field. The dashed vector in Fig. 13 is another anticipated sideband vector \(({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{f}})\) of SiA. Although it cannot be observed explicitly by the disturbances of a large SoD1, it nevertheless disturbs SoD1, like the seasonal variation \(({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{f}})\) of \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}\) disturbs SiD1 mentioned earlier. It will be shown later that the influence of the disturbance is not negligibly small.

Fig. 13
figure13

Schematic representation of the seasonal variation of SiD1 and φi(tSi)(α = 6 hours) at Hobart (cf. Type 2.—the Farley-Storey method). WS: Winter Solstice; SS: Summer Solstice.

As pointed out earlier, the evidence of the existence of an anisotropy similar to SiA in the above should have been recognized in the period when the bi-directional anisotropy observed at Budapest and Hobart was reported (cf. Fig. 5). In fact, the \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{1}}^{\bf{o}}\) and \({{\bf{S}}_{\bf{i}}}{\bf{D}}_{\bf{2}}^{\bf{o}}\) observed, respectively, at the two stations finally produced a sharply-concentrated SiD, with the phase of 6 hr common in both hemispheres supposed to have been produced by a SiA (cf. Fig. 5(d)). The SiA thus derived by the two different methods as above is also supported by the previous observations with neutron monitors and ion-chambers shown in Figs. 6 and 9(b) and also by the observations (1983–1993) at Mawson (68°S, 63°E) with the telescope pointing in the same direction as the Hobart telescope (43°S), which showed the same sidereal diurnal vector as Hobart (Jacklyn, 2000). These evidences inevitably require another anisotropy mainly responsible for the sidereal variations observed in the northern hemisphere. This idea was the key to finding two kinds of cosmic-ray sidereal anisotropy of galactic and solar origins (Nagashima et al., 1995a, b, c). In order to confirm the existence of these two kinds of anisotropy, the latitude and energy dependence of the SiD(tSi)’s at several stations are shown on an hourly basis in Fig. 14(a) (cf. Ref. 3). At the top of Fig. 14(a), the air shower observation of SiD(tSi) at Mt. Norikura is also shown for comparison, which is compatible with that observed at Baxan (Alexeenko and Navarra, 1985). In Fig. 14(b) on the right side, some un-corrected SiDo(tSi)’s and their associated anti-sidereal variations ASiD(tAS)’s, observed with three telescopes looking, respectively, in geographic latitudes γ > 0°, 0° and <0°, are plotted in order to show the influence of SoD(tSo). In Fig. 14(b), SiDo(tSi)SAK, with positive γ, has two peaks near 18 and 6 hours. Once the correction is applied, the larger peak at 18 hours vanishes into a flat plateau, but the smaller one is rather enlarged as shown in Fig. 14(a). The same can be said of \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{o}}}}})_{{\rm{NAG}}}^{\rm{V}}\) (cf. Fig. 14(a)). On the other hand, in the case of the equatorially pointing SSAK telescope, the SiD(tSi)’s before, and after, the correction do not differ much from each other owing to an almost flat \({\rm{A}}{{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{\rm{AS}}}}{\rm{)}}\) as expected, and show a good similarity with \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}_{{\rm{NAG}}}^{\rm{V}}\). At the southern station Hobart (43°S), on the other hand, the correction does not change the form of \({{\rm{S}}_{\rm{i}}}{\rm{D}}^\circ {\rm{(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) so drastically as in the N-hemisphere but only reduces its peak at 6 hours and flattens the minimum.

Fig. 14
figure14

(a) Latitude and Energy dependence of \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\), suggesting the existence of two kinds of sidereal anisotropies \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and T\({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) respectively from α = 0 hr and 6 hr and (b) the three of them without CSSA, together with their associated \({\rm{A}}{{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{\rm{AS}}}}{\rm{)}}\), to show their influence on \({{\rm{S}}_{\rm{i}}}{\rm{D}}^\circ {\rm{(}}{{\rm{t}}_{{\rm{Si}}}}{\rm{)}}\).

The sidereal variations in Fig. 14(a) are characterized by narrow peaks and/or deep valleys, whose centers are almost at 6 and 12 hours, respectively. The peaks almost disappear for the northernmost pointing NNAG telescope, while the valleys lose their similarity to others for the southernmost VHOB telescope. This suggests that the peaks and valleys are of a different origin. It is emphasized that all the valleys, except at Hobart, are very similar to that of the air showers (F3) with almost no phase difference. Furthermore, it cannot be emphasized too much that the entire time profile of \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}_{{\rm{NAG}}}^{\rm{N}}\) for the northernmost looking NNAG-telescope almost completely coincides in shape with that of \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}_{{\rm{NOR}}}^{{\rm{F3}}}\) as can be seen by their superposition in Fig. 15. This implies that the sidereal anisotropy, observed by the air showers with the median energy (Em 104 GeV), can be observed even at very low energies (Em 60 GeV) without any deformation in shape and with almost no phase difference from the air showers. Other \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)s}}\) except for \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}_{{\rm{HOB}}}^{\rm{V}}\) also coincide with that of the air showers fairly well except for the appearance of the peak of another origin, as can be seen by the examples of the superposed curves in Fig. 15. A sidereal variation of galactic origin, which has a broad plateau and a narrow valley with its minimum near 12 hours, is characterized by its constituent first (24 hours) and second (12 hours) harmonic vectors GD1 and GD2, with their respective phases 0 hours and 6 hours, and is called the G variation or G(tSi), hereafter. According to the previous analysis of G(tSi) of the air showers (Nagashima et al., 1989), the space distribution of the galactic anisotropy responsible for \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) can be expressed by a deficit flux confined in a narrow cone with the half opening angle \(\chi {\rm{ = 57}}^\circ \) from the direction of \({\Phi _{\rm{G}}}({\alpha _{\rm{G}}} = 12\) hours and \({\delta _{\rm{G}}} = 20^\circ \)). On the other hand, the variation, characterized by a narrow peak with a maximum at 6 hours, can be separated from \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) by a similar process shown in Figs. 1516. The result is shown in Fig. 16.

Fig. 15
figure15

Similarity of the air showers (light line) due to GA with the \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) (dark line) except for the sharply-concentrated variation due to TA, produced by adjusting their magnitude but without their phase change.

Fig. 16
figure16

T(tSi) due to TA obtained according to the preceding procedure shown in Fig. 15.

This variation is characterized by its constituent first and second harmonic vectors TD1 and TD2 with their common phase of 6 hours, and is called the T-variation, or \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\), whose origin is called T-anisotropy or TA. It is emphasized that as the sidereal diurnal variation SiD1 is the resultant of GD1 and TD1, its phase varies from 0 hours to 6 hours depending on the contribution of the two kinds of component, while the sidereal semi-diurnal variation SiD2 has a constant phase of 6 hours as GD2 and TD2 have a common phase of a constant value of 6 hours regardless of their energies. It is noteworthy that these variations make it possible to derive GD1 and TD1 inversely from the observed SiD1 by taking its projections respectively along the 0 hr-axis and the 6 hr-axis of the harmonic dial. Some examples of the energy dependence of the amplitude of GD1 and TD1 is shown in Fig. 17, which is derived from those SiD1 ’s whose phases are shown in Fig. 12 (Ref. 3).

Fig. 17
figure17

Energy dependencies of the amplitude GD1 (0 hr) and TD1 (6 hr) derived from the observed SiD1 ’s whose phases are shown in Fig. 12 (Ref. 3). The symbols are the same as in Fig. 12. The amplitude for \(\lambda \ne 0^\circ \) is converted into that at the equator by multiplying a provisional correction factor \(1/\cos {\rm{\lambda }}\) although its latitude dependence cannot be exactly expressed by cos γ as it contains the north-south asymmetric term (sin γ).

Nagashima et al. (1998) were indicating that at energies less than 104 GeV the amplitude of TA was decreasing with increasing energy, giving strength to the conclusion that TA was of solar origin. Remarkably, since that time it has been reported from a number of advanced experiments that TA continues to exist at energies in the multi-TeV region, raising new questions regarding the evidence for the solar origin of TA.

According to the previous analysis (cf. Ref. 3), the space distribution of the T-anisotropy (or TA) responsible for these \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) can be expressed by a directional excess flux confined in a cone with a half opening angle \({\rm{(}}{\chi _{\rm{T}}}{\rm{)}}\) from the direction \({\Phi _{\rm{T}}}({\alpha _{\rm{T}}} \sim 6{\rm{hr }},{\delta _{\rm{T}}} \sim - 24^\circ).{\Phi _{\rm{T}}}\) approximately coincides with the heliomagnetotail direction \(({\alpha _{{\rm{TP}}}} = 6{\rm{hr, }}{\delta _{{\rm{TP}}}} = - 29.2^\circ)\) inferred from the direction of proper motion of the solar system (Campbell and Moore, 1928), although it is emphasized that this is different from the inferred tail direction \(({\alpha _{{\rm{TN}}}} = 4.8{\rm{hr, }}{\delta _{{\rm{TN}}}} = 15^\circ \sim 17^\circ)\)) opposite to the relative motion of the solar system to the neutral gas (Ajello, 1978; McClintock et al., 1978). This difference will be discussed later.

The space distribution of TA is shown on the map in Fig. 18, together with that of GA (cf. Nagashima et al., 1989). In Fig. 18, the excess cone of TA, within which all the directional excess flux is confined, is shown bounded by the thick line. A thinner line bounds the deficit cone of GA, in which all the directional deficit flux is confined. As can be seen in the figure, a telescope directed to the region of \(\delta \ {\rm{ > }} \ \sim {\rm{44}}^\circ \) or \(\delta < - 37^\circ \) can observe only one of GA and TA as it sweeps either the deficit cone only, or the excess cone. In fact, the air shower observation \(({E_m}\; \sim \;7 \times {10^4}\;{\rm{GeV}})\) at Liawenee (42°S, 147°E; height = 1060 m), Tasmania, Australia and also the underground (depth 357 m.w.e.) at Poatina, Tasmania could not find GD1 (Fenton et al., 1990, 1995). On the other hand, a telescope directed to the intermediate region of \( \sim 44^\circ \; > \;\delta \; > \; - 37^\circ \) successively sweeps the excess and deficit cones and would observe a complicated α-distribution of the variational intensity as shown in Fig. 14(a). It is easily understandable that such GA and TA can solve at least qualitatively all the previous problems except that of the absence of the polarity dependence at SiD1 which will be resolved after the HMS modulation of GA and TA has been determined. It is further noted here that the discovery of GA and TA was soon supported by joint observations in the northern and southern hemispheres at Matsushiro (37°N), (cf. Mori et al., 1981, 1989, 1993) and at Poatina (42°S), (Hall et al., 1998, 1999).

Fig. 18
figure18

The distribution of the tail-in (TA) and galactic (GA) anisotropies on the equatorial coordinate grid (Ref. 3). The excess cone of TA, within which all the directional excess flux is confined, is shown bounded by the thick line including the south pole. A thinner line bounds the deficit cone of GA, in which all the directional deficit flux is confined (cf. Nagashima et al., 1989). PM, direction of proper motion of the solar system; RM, direction of relative motion of the system to the neutral gas; (thin line), galactic equator.

5.Modulation of Anisotropies in the HMS

The HMS modulation of \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) shown in Fig. 19 has been obtained by the multi-directional muon telescopes at the underground station Sakashita (36°N, 138° E, 331 GeV), (Nagashima et al, 2004; Ueno et al, 1984). The latitude distribution of \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) in Fig. 19(a) indicates clearly the coexistence of \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\). The peak at 6 hours expresses \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and the step-type variation on the left side of \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) expresses a part of \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\). \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) increases toward the southern hemisphere while the variation of \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) is against \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\). The most typical coexistence of \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) is observed by the S- and SW-component telescopes. These variations are shown in Fig. 19(b) after being classified into four groups AP, AN, QP and QN in which P and N indicate P- and N-states, and A and Q are the Active and Quiet periods of solar activity. In the figure, \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\), which appears as a part of the plateau, has the following magnitude relation between four periods, which is called the modulation pattern hereafter.

$${\rm{G}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}):{\rm{QN}} > {\rm{AN}} > {\rm{QP}} > {\rm{AP}},\,{\rm{for}}\,330 \sim 650\,{\rm{GeV}},$$
(1)

. \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) is larger in the Q-period than the A-period. This gives a firm basis on the galactic origin of \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) as the galactic CR-flux suffers less disturbance in a Q-period than an A-period during its passage through the HMS, and is larger in the N-state than in the P-state. According to the modulation model (Refs. 1 and 2), this polarity dependence can be realized only when the direction of GA is near the equatorial plane of the HMS. This inferred direction is in good accord with that (\(({\delta _{\rm{G}}} = 20^\circ)\)) estimated from the space distribution of \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) pointed out previously (cf. Fig. 18). On the contrary, \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) with a sharp peak at 6 hours has a modulation pattern which is almost quite the opposite to that of G(tSi) as follows, (cf. Fig. 19).

$${\rm{T}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}):{\rm{QN}} < {\rm{AN}} \sim {\rm{QP}} < {\rm{AP}},\,{\rm{for}}\,330 \sim 650\,{\rm{GeV}}.$$
(2)

\({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) is larger in an A-period than in a Q-period. This cannot be realized by the galactic anisotropy. The only possible solution is to assume the acceleration of galactic CRs proportional to the solar activity in the HMS. Furthermore, \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) is larger in the P-state than the N-state, especially in the AP-period showing quite opposite polarity dependence to that of \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) (cf. Fig. 19(b)). This would draw an inference for the location of the acceleration of CRs in the HMS, as will be discussed later together with other kinds of evidence. Whatever the mechanism of the origin of TA may be, however, it is clear that the modulation pattern in Eq. (2) is quite the reverse of that of GA in Eq. (1) as can be clearly seen in Fig. 19(b), and, as a result, the polarity-dependencies of GA and TA would always compensate each other. This would be the reason why the observation of the polarity dependence of SiD1 did not show that expected from GA of the modulation model.

Fig. 19
figure19

Solar modulation of \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) observed by the underground muon telescopes (NE, NW, SW, ... S3) at Sakashita in the period (1971–2000), (Nagashima et al, 2004). (a) The latitude dependence of \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\). (b) The polarity (P or N) and solar activity (A or Q) dependence of the two components ‘sW’ and ‘s’ in (a), which shows the most clear coexistence of \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\), are separated, respectively, into four periods of AP, QP, AN and QN. The magnitudes of \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) show, respectively, the proper modulation patterns shown by Eqs. (1) and (2).

The existence of \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) in the lower energy region was inferred from TD1 in Fig. 9 observed by neutron monitors (Nagashima et al., 2005). In order to study the modulation pattern of \({\rm{T}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}),{{\rm{S}}_{\rm{i}}}{\rm{D}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) were obtained by using neutron monitor data at 74 stations (WDC-C2 Center) in the period 1960–2000. Their average \(\langle {{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\rangle \) for all the stations, for all the years, was obtained by neglecting the difference of geomagnetic deflection angles at each station, as shown in Fig. 20(a). The histogram of the maximum phases (tmax) of the yearly-averaged first harmonic component SiD1’s of \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) for all the stations during the entire period is shown in Fig. 20(b) (cf. Nagashima et al., 2005). It is clear that the sharp peak of \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) at 6 hours exactly corresponds to the \({\rm{T}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\). In addition, \({{\rm{S}}_{\rm{i}}}{\rm{D}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) contains another variation which is almost submerged beneath \({\rm{T}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) and can be narrowly recognized by the unnatural distribution in the time interval of \(15\; < \;{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}\; \le \;21\) hours (cf. Fig. 20(a)). This distribution is supposed to be produced by the superposition of the individual yearly harmonic components with their maximum in the above time interval \((15\; < \;{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}\; \le \;21\;{\rm{hr)}}\) as can be seen in the histogram in Fig. 20(b). The variation is a maximum at 18 hours and is called \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) hereafter. The modulation pattern of \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) is shown in Fig. 20(c). \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) in the AP-period seems to be almost purely \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and the largest in all the periods, while others are supposed to become smaller by the interaction with \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\), as can be seen by the time shift of the minimum intensity from 12 hours to 24 hours and also by the appearance of smaller peaks in the QP- and QN-periods. \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) has a modulation pattern nearly equal to that at the underground station Sakashita as

$${\rm{T}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}):{\rm{QN}} < {\rm{AN}} \le {\rm{QP}} < {\rm{AP}},\,{\rm{for}}\, \sim 20\,{\rm{GeV}}.$$
(3)

The modulation pattern for \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) cannot be obtained because of the lack of its explicit form in Fig. 20(c). \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\), however, can be clearly observed at some stations in some times together with \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) as shown in Fig. 21. It has a sharp peak at α18 hours as expected and is called hereafter the nose-in anisotropy. \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) are unstable and easily reduce their magnitudes by their mutual interaction, resulting in the averaged small variations shown in Figs. 20 and 21.

Fig. 20
figure20

(a) The average of yearly \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) of neutron intensities observed at 74 stations in the period of 1960–2000. (b) Histogram of the maximum phase of yearly \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\). (c) The modulation patterns of \({{\rm{S}}_{\rm{i}}}{\rm{D(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\).

Fig. 21
figure21

Rise and fall of \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) of the neutron intensity with their maximum phases, respectively, at 6 hours and 18 hours, (Nagashima et al, 2005).

Although \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) was first found by neutron monitors, its separation from \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) was more clearly seen in the muon telescopes at Nagoya (35°N; 137°E) as shown in Fig. 22(a) (Nagashima and Fujii, 2006). The variation with a large peak at α6 hours expresses \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and that of the small one with a maximum at α18 hours expresses \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\). Among them, the S-component (5°N, 64 GeV), which is characterized by the most remarkable \({\rm H}({{\rm t}_{\rm S}}_{\rm i})\), is separated into the four groups as shown in Fig. 22(b). Their modulation pat-

$${\rm{T}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}):{\rm{QN}} \sim {\rm{AN}} < {\rm{QP}} < < {\rm{AP}},\;\,{\rm{for}}\,60 \sim 100\,{\rm{GeV}},$$
(4)
$${\rm{H}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}):{\rm{QN}} < {\rm{AN}} > {\rm{QP}} > {\rm{AP}},\;\,{\rm{for}}\,60 \sim 100\,{\rm{GeV}}.$$
(5)

\({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) is larger in an A-period than in a Q-period, similarly to those in Eqs. (2) and (3), but, in the present case, it is extremely larger in an AP-period than in a QP-period. This is due to the following reason. As can be seen in Fig. 23, the yearly decomposed variations of \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) in the respective AP- and QP-periods show quite different variations. In a QP-period, all the variations show one or two peaks at α6 and/or α18 hours which can be regarded as being due to \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and/or \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\). In the case of two peaks, the variations are generally small. This is partly due to the interaction between \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) with the mutually opposite maximum phases. On the other hand, almost all \({{\rm{S}}_{\rm{i}}}{\rm{D}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}){\rm{s}}\) in Fig. 23(a) show a large single peak of \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) hours, indicating that an AP-period activates \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and/or inactivates \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\). On the other hand, \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) in Fig. 22(b) shows the largest enhancement in an AN-period, being different from \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) in a QN-period and from \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) in an AP-period. These differences will give a hint as to the origin of the anisotropies.

Fig. 22
figure22

(a) Latitude dependence of \({{\rm{S}}_{\rm{i}}}{\rm{D}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) observed by muon telescopes (S3, S2, S, N) at Nagoya in the period of 1971–2003 to show the existence of \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) with the maximum phase at 18 hr, and (b) the modulation pattern of \({{\rm{S}}_{\rm{i}}}{\rm{D}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) of S-component in (a) which shows clearly the existence of

Fig. 23
figure23

Yearly variations of \({{\rm{S}}_{\rm{i}}}{\rm{D}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) in (a) AP and (b) QP periods in Fig. 22(b), showing the enhancement of \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) in the AP-period due to less influence of \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) at tSi = 18 hr.

Now, as for the direction of HA, the declination \({\delta _{\rm{H}}}\) has been estimated to be in the range of 21 ° α 41 ° by an analysis of muon data (cf. Nagashima and Fujii, 2006). For reference, one set of data used for the estimation of \({\delta _{\rm{H}}}\) is the \({{\rm{S}}_{\rm{i}}}{\rm{D}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\)’s in Fig. 24(a). \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) in these \({{\rm{S}}_{\rm{i}}}{\rm{D}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) first appears in the E-telescope in the equatorial region as a tiny protuberance at α18 hours, then gradually increases toward the northern regions and finally reaches its maximum at the W-component. The latitude distribution of the magnitude of \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) is shown in Fig. 24(b). In this data, the maximum appears at \({\delta _{\rm{H}}} \sim 40^\circ \).

Fig. 24
figure24

(a) The rise and fall of \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) with its maximum at 18 hr, and (b) the rigidity dependence of the magnitude of \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\), (Nagashima and Fujii, 2006).

It is emphasized here that \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) seems to be the same as \({{\rm{C}}_{\rm{g}}}{\rm{(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) to be produced by the C-G effect by the motion of the HMS, as their phases are considered to be the same as each other (18 hr). But, they differ in the following points. First of all, \({{\rm{C}}_{\rm{g}}}{\rm{(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) is a broad sinusoidal diurnal variation with a declination distribution of cos δ-type and has a flat energy spectrum (Gleeson and Axford, 1967). On the other hand, \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) shows the sharply-concentrated diurnal variation with an N–S asymmetric δ-distribution (cf. Fig. 24) and has a soft energy spectrum which almost disappears in the energy region ≥300 GeV (cf. Fig. 19). Furthermore, \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) is greater in an AN-period than a QN-period (cf. Eq. (5)). This is against the galactic origin hypothesis as Cg(tSi) must be smaller in the A-period than the Q-period like the behavior of the modulation pattern of \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\), (cf. Eq. (1)). Therefore, the C-G effect of CRs cannot be observed in an energy region less than 104 GeV, as pointed out previously (Ref. 3).

Finally, in addition to \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\), Fig. 22(b) contains also \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\). \({{\rm{S}}_{\rm{i}}}{\rm{D}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) in a QN-period in the figure shows a straight line with its center at 0 hr and with the level higher than the lowest value at 12 hr. This line, called a g-line, is supposed to express a part of \({\rm{G}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) whose whole figure would be a trapezium bounded by the line through the following four points; g1 and g2 are the intersection points between the g-line and \({{\rm{S}}_{\rm{i}}}{\rm{D}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) near the 12 hours, and b1 and b2 are minimum points of \({{\rm{S}}_{\rm{i}}}{\rm{D}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) at 12 hours. Such a flat g-line disappears in AN, QP and AP-periods, but the influence of \({\rm{G}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) in a QN-period remains as a small dip at 0 hrs in the variation averaged for all the periods. The whole picture of the \({\rm{G}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) can be seen in Fig. 25 which shows the averages of six selected years in AN- and QN-periods. \({\rm{G}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) appears as a typical plateau-type variation with a minimum at 12 hours in the equatorial regions and decreases towards the northern latitude with the gradual increase of \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) with a maximum at 18 hours. It is noteworthy that similar enhancements of \({\rm{G}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) in the N-state have been observed also by the ion-chambers at Godhaven (69°N, 53°W; 67 GeV; 1936–1959; Lange and Forbush, 1948, 1957) and at Yakutsk (62°N, 130°E; 67 GeV; 1953–1980; Shafer and Shafer, 1984), (cf. Nagashima and Fujii, 2006).

Fig. 25
figure25

The averages of \({{\rm{S}}_{\rm{i}}}{\rm{D}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) for 6 years (1981, ’83, ’84, ’87, ’88, ’89) in AN and QN periods, selected from those in Fig. 22(a). \({\rm{G}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}})\) can be clearly seen, especially in the low latitude region.

6. Origin of GA, TA and HA, and the Related Discussion

6.1 Origin of GA, TA and HA

The three anisotropies GA, TA and HA mentioned earlier are subjected, respectively, to different modulation patterns as shown in Eqs. (1)(5) and in Table 1. The patterns clearly indicate that GA could be explained at least qualitatively by a dependence on the polarity state and on the solar activity expected from the modulation model (Refs. 1 and 2) and that it arrives from galactic space near the equatorial plane of the HMS \({\rm{(}}\delta \sim {\rm{0}}^\circ {\rm{)}}\), which is in good agreement with the direction \({\Phi _{\rm{G}}}({\alpha _{\rm{G}}} = 0\;{\rm{hr}}\) and \({\delta _{\rm{G}}} = - 24^\circ)\) shown in Fig. 18. On the other hand, TA and HA are supposed to be of a non-galactic origin as they produce, respectively, larger variations \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) in the A-period than in the Q-period. The most plausible source of TA and HA would be the acceleration of galactic cosmic rays on the HMS boundary with the nose and tail, being composed of the aggregation of point sources with their main axes perpendicular to the boundary surface. As TA and HA are produced by a curved surface source, unlike the plane source of GA, the modulation model for GA cannot be applied to TA and HA. However, the CRs in the HMS which can arrive at the Earth from the high latitude boundary \({\rm{(|}}\lambda |\; \gg \;{\rm{0}}^\circ {\rm{)}}\) in the P-state, and from the equatorial region \({\rm{(}}\lambda \sim {\rm{0}}^\circ {\rm{)}}\) in the N-state, do not drastically change their behavior in spite of the deformation from the spherical boundary to a comet-like structure. This behavior can explain the modulation patterns of \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) assuming that:

  1. (1)

    The structure of the HMS is surrounded by the nose-and tail-boundaries, and has a wavy neutral sheet at the heliomagnetic (HM) equatorial plane which divides the HM polarity states in the N- and S-hemispheres.

  2. (2)

    CRs are assumed to be accelerated uniformly all over the boundaries depending on the solar activity.

Table 1 Modulation pattern of sidereal anisotropy by GA, TA, and HA.

The polarity dependence of \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) that is,

$$\vert {\rm{T}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}){\vert _{\rm{P}}} > \vert {\rm{T}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}){\vert _{\rm{N}}}$$
(6)

would be produced by the following process. The accelerations of CRs responsible for \(\left| {{\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}} \right|\) P are limited on the northern-most and southern-most narrow tail boundary regions. In the P-state, all the CRs having been ejected from the specified region in the above are deflected towards the Sun. If the source region is far away from the Sun, some selected CRs could arrive at the Earth through the wavy neutral sheet and others would escape from the HMS through the northern and southern boundaries (cf. Fig. 26). Such northern- and southern-most regions are called Rb hereafter. The flux of TA responsible for \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) is the congress of those selected CRs. The sharply-concentrated distribution from the direction of α = 6 hr would be determined by two factors: the width of the narrow boundary region, and the amplitude of the wavy neutral sheet. For the transition from the P-state to the N-state, the CRs ejected from the boundaries change their direction towards the tail-end side and cannot be observed at the Earth. Instead, alternative flux from the area (Ste) of the tail-end near the central line of the neutral sheet can arrive at the Earth along the wavy neutral sheet after automatically forming the sharply concentrated flux (TA) from the direction of α= 6 hr, similar to the distribution in the P-state. In this case, some additional contribution to the flux might be expected from the two side-boundaries of the tail near the sheet. But this contribution is likely to be negligible, as almost all the CRs from the side-boundaries far away from the Sun escape from the HMS across the other side-boundary without changing their direction. Such a process to formulate the sharply-concentrated flux from α= 6 hr is called hereafter the CR Lens Effect by the HMS.

Fig. 26
figure26

Cosmic-Ray-Lens Effect of HMS which focuses CRs along the direction of the neutral sheet of HMS. Arrowed curves express the CR trajectories in P- and N-states. Dashed line is the wavy neutral sheet.

The polarity dependence of Eq. (6) can be realized if the HMS tail is extended long enough to achieve the CR flux from the total sum of Rb larger than that of the tail-end region (Re) near the central line of the neutral sheet.

On the other hand, the polarity dependence of \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) shown by Eq. (7) is observed in the head-side region of the HMS where the CR trajectories are given by the axis-symmetric reversal of those in Fig. 26 with respect to the solar magnetic polar axis.

$$\vert {\rm{H}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}){\vert _{\rm{P}}} > \vert {\rm{H}}({{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}){\vert _{\rm{N}}}.$$
(7)

As the movement of CRs in the region is the same as before, all the statements in the case of TA are applicable also in the present case. Therefore, the polarity dependence in Eq. (7) can be satisfied if the nose-boundary length is shortened so that the CR flux from the total sum of Rb becomes smaller than that from the head region (Rh) near the central line of the neutral sheet. It is emphasized here that the direction of motion of the HMS in space, in other words the direction of the nose and tail of the HMS, can be determined automatically by the HMS modulation of HA and TA without any aids, such as the motion of the HMS relative to the neighboring stars (Campbell and Moore, 1928).

6.2 Direction of nose head of the HMS in space

The nose direction \({\rm{(}}{\Phi _{{\rm{HMS}}}}{\rm{)}}\) of the HMS can be determined by observation on the basis of the model. As pointed out earlier, the T- and H-anisotropies are observed in the respective directions \({\Phi _{\rm{T}}}({\alpha _{\rm{T}}} = 6\,{\rm{hr,}}\,{\delta _{\rm{T}}} = - 24^\circ)\) and \({\Phi _{\rm{H}}}({\alpha _{\rm{H}}} = 18\,{\rm{hr,}}\,{\delta _{\rm{H}}} = 24^\circ \sim 41^\circ)\), both of which could be considered to constitute a straight line through the Sun; that is, the center line of the neutral sheet of the HMS. Therefore, the motion of the HMS would be in the direction \({\Phi _{{\rm{HMS}}}}({\alpha _{{\rm{HMS}}}} = 18\,{\rm{hr,}}\,{\delta _{{\rm{HMS}}}} = 24^\circ)\) and should produce the C-G effect of CRs from the direction of 18 hr. Contrary to this expectation, the C-G effect could not be observed (Nagashima et al., 1998). The non-existence of the effect is of importance with regard to the interaction between the HMS and the interstellar gaseous matter (IGM), as will be discussed in the next section.

The motion of the HMS is further supported by the seasonal variation of \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) which shows the maximum flux at the December solstice as pointed out earlier (cf. Fig. 13). This flux produces the maximum \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) with the phase of \({\alpha _{\rm{T}}} = 6\) hr and can also be observed on the Earth as the solar diurnal variation \({{\bf{S}}_{\rm{o}}}{\bf{D}}_1^{\rm{T}}\) with the maximum phase at 0 hr local solar time \({\rm{(}}{{\rm{t}}_{{{\rm{S}}_{\rm{o}}}}}{\rm{)}}\) at the December solstice \(({\alpha _{\rm{T}}} = 6\,{\rm{hr}})\). This phase \({\rm{(}}{{\rm{t}}_{{{\rm{S}}_{\rm{o}}}}}{\rm{ = 0}}\,{\rm{hr)}}\) is opposite to the daytime phase of SoD1 due to SoA caused by the CR diffusion convection in interplanetary space (Gleeson and Axford, 1967; Munakata and Nagashima, 1986). If \({{\bf{S}}_{\rm{o}}}{\bf{D}}_1^{\rm{T}}\) is larger than SoD1, the observed phase will be reversed towards the night side, suggesting the existence of the seasonal variation of T-anisotropy. The reversal has happened during several months of the winter season around the December solstice in the solar activity minimum period (1995/96) in the P-state when the enhancement of TA and the depressions of GA and SoA were observed (cf. Table 1). The magnitude of the reversal, and its duration periods, depend on the latitude of the stations. The total magnitude of the reversals at each neutron monitor station in Fig. 27 shows a clear north-south asymmetry with the maximum in the southern hemisphere, which means that the declination \(({\delta _{\rm{T}}})\) of the T-anisotropy is negative; in other words, that \(({\delta _{{\rm{HT}}}})\) of the nose direction \(({\Phi _{{\rm{HT}}}})\) is positive (Nagashima et al., 2010). It is further noted that the reversal has been observed in the period by the muon telescopes (60 GeV) at Nagoya (cf. Nagashima et al., 2010).

Fig. 27
figure27

The latitude distribution of the magnitudes R(j)’s of the total sum of D(j, y)’s in the respective reversal periods, where D(j, y) shows the monthly vector SoD1 of neutrons in the text whose phase is toward the night sides in 1995–1996. P: south pole, Te: Terre Adelie, Ke: Kerguelen, He: Hermanus, P: Potchefstroom, Ts: Tsumeb, Ha: Haleakala, C: Climax, N: Newark, R: Rome, J: Jungfraujoch, Ki: Kiel, Y: Yakutsk, O: Oule, A: Apatity, I: Inuvik, Th: Thule (Nagashima et al., 2010).

6.3 The duality of the motion of solar system in space

As above, the direction of the motion \(({\bf{V}}_{{\bf{HMS}}}^{\bf{C}})\) of the solar system inferred from the HMS modulation of the TA and HA is \(\Phi _{{\rm{HMS}}}^{{\rm{C}}}({\alpha _{{\rm{HMS}}}} = 18\,{\rm{hr,}}\,{\delta _{{\rm{HMS}}}} = 24^\circ)\) and coincides with \(\Phi _{{\rm{HMS}}}^{{\rm{S}}}\) of the motion of the HMS relative to the neighboring stars (Ajello, 1978; McClintock et al., 1978), (cf. Table 2). On the other hand, the corresponding direction of the motion (Vnp) of the HMS relative to neutral particles observed by the space probe is \({\Phi _{{\rm{np}}}}({\alpha _{{\rm{np}}}} = 16.2\,{\rm{hr;}}\,{\delta _{{\rm{np}}}} = - 15^\circ \sim - 17^\circ)\) and clearly different from \(\Phi _{{\rm{HMS}}}^{\rm{C}}\) The difference is likely to be attributable to the inaccurate observation by the CR charged particles, because the motion of the neutral and ionized gases in space would have been in an equilibrium state by their mutual interaction over the long-term in the past, and also the motion of the neutral gas is supposed to be more easily observable without any influence of interstellar- and helio-magnetic fields. The judgement on this matter seems to be correct in so far as one adheres to the conventional C-G effect for the relative motion of the HMS to the ionized and neutral particles moving freely in space. The physical state of the interstellar space, however, is supposed to be different from this assumption and the interstellar gaseous matter (IGM) containing the ionized and neutral particles, together with the magnetic field (BISM), is supposed to be a good electromagnetic conductor and moving itself in some direction in space. It will be shown in the following that this assumption explains the duality of \({\bf{V}}_{{\bf{HMS}}}^{\bf{C}}\) and Vnp observed, respectively, by the CRs and the neutral particles.

Table 3 Table 2

As pointed out previously, the direction \(\Phi _{{\rm{HMS}}}^{\rm{C}}\) of the motion \(({\bf{V}}_{{\bf{HMS}}}^{\bf{C}})\) of the HMS inferred from the sidereal diurnal variation of CRs almost exactly coincides with that \((\Phi _{{\rm{HMS}}}^{\rm{S}})\) of the motion \(({\bf{V}}_{{\bf{HMS}}}^{\bf{S}})\) of the HMS relative to the neighboring stars and, therefore, \({\bf{V}}_{{\bf{HMS}}}^{\bf{C}}\) is renamed \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\) for convenience (cf. Table 2). This coincidence indicates that Vnp is not due to the motion of the HMS itself, but due to the flow \({\bf{V}}_{{\bf{np}}}^{{\bf{HMS}}}\) of the neutral particles relative to the HMS with the phase \(\Phi _{{\rm{np}}}^{{\rm{HMS}}}(\alpha _{{\rm{np}}}^{{\rm{HMS}}} = 4.2\,{\rm{hr;}}\,\delta _{{\rm{np}}}^{{\rm{HMS}}} = 15^\circ \sim 17^\circ)\). \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\) and \({\bf{V}}_{{\bf{np}}}^{{\bf{HMS}}}\) are coplanar as shown in Fig. 28. Their relative configuration is obtained in polar coordinates (r, δ, ϕ) on the assumption that \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\) is nearly equal to \({\bf{V}}_{{\bf{HMS}}}^{\bf{S}}\) (cf. Table 2). \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\) expresses the motion of the HMS relative to the neighboring stars, while \({\bf{V}}_{{\bf{np}}}^{{\bf{HMS}}}\) is relative to the HMS and can be transformed to a new vector \({\bf{V}}_{{\bf{IGM}}}^{\bf{S}}\) relative to the stars, as shown in Fig. 28. \({\bf{V}}_{{\bf{IGM}}}^{\bf{S}}\) expresses the motion of the interstellar gaseous matter (IGM) relative to the stars and \({\bf{V}}_{{\bf{np}}}^{{\bf{HMS}}}\) is interpreted as the transformed \({\bf{V}}_{{\bf{IGM}}}^{\bf{S}}\) observed on the moving frame \(({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}})\) of the HMS. \({\bf{V}}_{{\bf{IGM}}}^{\bf{S}}\) is determined in the polar coordinates (r, δ, α) as shown in Table 2, and the relative configuration among \({\bf{V}}_{{\bf{IGM}}}^{\bf{S}},\,{\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\) and \({\bf{V}}_{{\bf{np}}}^{{\bf{HMS}}}\) are specified as shown in Fig. 28. As the IGM is electromagnetically a good conductor, it would interact with the HMS. The magnetic field BIGM in IGM would be pushed or dragged by the HMS and would be strengthened and forced to move with the HMS. In addition to this interaction, there is another interaction of the HMS with IGM which is moving vertically towards the southern hemispherical boundary of the HMS (cf. Fig. 28). However, their direct interaction would be reduced as IGM gradually changes its direction toward \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\) with the approach to the HMS by the dragging effect of the HMS. Therefore, the interaction region remains almost axis-symmetrically with respect to the direction of \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\). Such an interaction region surrounding the whole HMS is called hereafter the Subordinate HMS (SuHMS) of the HMS, whose outer boundary becomes beyond recognition as the magnetic field (BSUH) inside the SuHMS gradually reduces its strength toward the outside and tends to BIGM at the boundary. It is noted that the present interaction cannot produce the so-called bow shock in front of the HMS contrary to the prediction by Venkatesan and Badruddin (1990), because IGM has no velocity component of \({\bf{V}}_{{\bf{IGM}}}^{\bf{S}}\) parallel to \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\) (cf. Fig. 28) and is only passively pushed and/or dragged by the HMS. The SuHMS could play a buffer role to reduce the direct interaction between the interstellar CRs and the HMS, if its width is larger than the gyration radius of CRs. Therefore, CRs with low energies confined in a stationary state in IGM would lose their ability to produce the C-G effect against the HMS by an interaction with the SuHMS, showing good agreement with the undetectable C-G effect by CRs. Such a low-energy region would be extended at least up to 104 GeV as the C-G effect could not be found by the air showers at Mt. Norikura (cf. Ref. 3). In high-energy regions, however, the C-G effect could be expected to be observed by the motion \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\) with 20 km/s (not Vnp) according to the process of the conventional C-G effect, as CRs could arrive at the HMS without any influence of BIGM and BSUH.

Fig. 28
figure28

Motion \({\bf{V}}_{{\bf{IGM}}}^{\bf{S}}\) of the interstellar gaseous matter (IGM) relative to the neighboring stars (S) inferred from the motion \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\) of HMS relative to S observed by CRs and the motion \({\bf{V}}_{{\bf{np}}}^{{\bf{HMS}}}\) of the neutral particles (np) relative to the HMS.

On the other hand, the neutral particles in the IGM could not recognize the existence of the SuHMS owing to their long scattering mean free path in comparison with the scale of the SuHMS. Therefore, the particles are able to freely pass through the SuHMS and the HMS with the velocity \({\bf{V}}_{{\bf{IGM}}}^{\bf{S}}\) and produce the motion \({\bf{V}}_{{\bf{np}}}^{{\bf{HMS}}}\) observed in the solar system (cf. Fig. 28).

As above, the introduction of the two proper motions \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\) and \({\bf{V}}_{{\bf{IGM}}}^{\bf{S}}\) relative to the neighboring stars could explain the duality of the motion of the HMS in space and prove also the absence of the C-G effect of CRs at low energies due to the motion of the HMS by the interaction of the SuHMS surrounding the HMS. Furthermore, the interaction of the SuHMS with the HMS can contribute to the acceleration of CRs at the boundary of the HMS required for the origin of TA and HA.

Finally, it is noteworthy that the existence of the motion of IGM relative to the neighboring stars has been found by the duality of the motion of the solar system in space probably for the first time.

6.4 Influence of TA and HA on SoA

As pointed out previously, the seasonal variation of TD1 produces two sideband variations \({{\bf{S}}_{\rm{o}}}{\bf{D}}_1^{\rm{T}}\) and \({\bf{E}}{{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{T}}\) (cf. Fig. 13). \({\bf{E}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_1^{\bf{T}}\) was found by the observation at Hobart and has contributed to the discovery of GA and TA. The other sideband variation \({{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{T}}\) with the phase 0 hr local solar time cannot be observed by the disturbance of SoD1 of the solar origin due to the diffusion convection of CRs inside the HMS (cf. Gleeson and Axford, 1967), but can be estimated from \({\bf{E}}{{\bf{S}}_{\bf{i}}}{\bf{D}}_1^{\bf{T}}\) by the Farley-Storey method (cf. Fig. 2), and eliminated from the solar diurnal variation \({{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{r}}(= {{\bf{S}}_{\bf{o}}}{{\bf{D}}_1} + {{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{T}})\) As another method, the influence of \({{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{T}}\) on \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_1}\) can be directly observed by the relative configuration of \({{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{T}}\) to \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_1}(j)\) and \({{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{r}}(j)\) at \(j = X,\,Y\) and Z stations as shown on the solar diurnal harmonic coordinates in Fig. 29. In the figure, \({{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{T}}\), being assumed to be constant at 0 hours by neglecting the geomagnetic deflection for simplicity, shortens all the vectors SoD1(j)s and advances or retards their phases φ(j)s in the 3rd or 2nd quadrant. However, \({{\bf{S}}_{\bf{o}}}{{\bf{D}}^{\bf{r}}}(j)\) might be produced by a change of the direction \({\phi _{{{\rm{S}}_{\rm{o}}}{\rm{A}}}}\) and of the rigidity spectrum f(p)dp of SoA defined by

$$\matrix{{f(p)dp = {f_0}{p^\gamma }dp\quad {\rm{for}}\;p \le {p_u},} \cr { = 0\quad \quad {\rm{for}}\,p > {p_u},} \cr }$$
(8)

where the exponent γ and the upper cut-off rigidity pu are parameters. Under such circumstances, the existence of \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{T}}\) will be shown by the following process. Among the three unknown parameters \({\phi _{{{\rm{S}}_{\rm{o}}}{\rm{A}}}},\gamma \) and pu of \({{\rm{S}}_{\rm{o}}}{\rm{A}}\) can be substituted by \(\phi ({\rm{INU}})\) of \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}({\rm{INU}})\) observed at the INUVIK station \((\lambda = 68^\circ {\rm{N,}}\,\varphi = 226^\circ {\rm{E}})\) as follows,

$${\Phi _{{{\rm{S}}_{\rm{o}}}{\rm{A}}}}(\gamma ,{p_u}) = \phi ({\rm{INU}}\vert \gamma ,{p_u}) + \sim {9^ \circ },$$
(9)

as the longitudinal geomagnetic deflection angles of CRs at the station are independent of p and given by −#x00B0; (cf. Inoue et al., 1983). Therefore, γ and pu can be determined by the relative positions between \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}(j)\) and \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}({\rm{INU}})\) on the harmonic dial; in other words, by the following two quantities:

$${\rm{R}}(j) = \vert {{\bf{S}}_{\bf{o}}}{{\bf{D}}_1}(j)\vert /\vert {{\bf{S}}_{\bf{o}}}{{\bf{D}}_1}({\rm{INU}})\vert ,$$

and

$$\Delta \phi (j) = \phi (j) - \phi ({\rm{INU}}).$$
(10)

These observed quantities can be compared respectively with the theoretical values obtained for various γ and pu (cf. Yasue et al., 1982; Fujimoto et al., 1984). Figure 30 shows the \(\Delta \phi \) diagram in which the theoretical R and \(\Delta \phi \) are drawn as a function of γ and pu by the net area. If the observed point is inside the area, \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}(j)\) can be explained by SoA with \({\phi _{{{\rm{S}}_{\rm{o}}}{\rm{A}}}},\gamma \) and pu at the point. The diagram on the left side of Fig. 30 is the distribution at the neutron monitor station Hermanus (3.4°S, 19°E; cut-off rigidity pc = 4.9 GV) during the period (1965–2000), and on the right that at the muon station at Nagoya (35°N, 137°E; pc = 11 . 5 GV). The number attached to each point expresses the observation year in the ’70s and ’90s in the P-state. As \({\Phi _{{{\rm{S}}_{\rm{o}}}{\rm{A}}}}\) is dependent on the polarity and solar activity, showing near 18 hours in the AN-period and 15 hours in the QP-period, \(\phi (j)\) also shows the same dependence (cf. Munakata and Nagashima, 1986). As SoD1 (Hermanus) and SoD1 (Tokyo) in the above vary their phases, respectively, from about 18 hr to 12 hr and 12 hr to the morning side by the dependence, they are typical examples for j = X and Z in Fig. 29. \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{r}}\) (Nagoya) of the Z-type is converted into the right outside point of the net area on the \(\Delta \phi \) diagram in Fig. 30. Almost all the points in the P-state deviate from the net area and the deviation increases with the decrease of solar activity and becomes a maximum in the solar activity minimum period (1976, ’77, ’96 and ’97). Some of these points would be able to be included in an enlarged net area by increasing γ and reducing pu, but it is very difficult to do so as such a large γ and small pu cannot be accepted by the other analyses of the solar diurnal variation (cf. Nagashima et al., 2010). The points in the N-state, on the other hand, are almost on the net area. Therefore, the influence of \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{T}}\) on SoD1 cannot be confirmed, except for the following three points in the minimum period of solar activity (1986, ’87 and ’88), which show very small R’s, being isolated from the other group (cf. Fig. 30). These points would be due to the influences of \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{T}}\) as \(\phi (j)\) in the N-state is retarded toward 12 hours from the middle of the 3rd quadrant by the softening of f(p)dp with a decrease of the solar activity and, as a result, SoD1(j) of the X-type changed to the Y-type and reduced only its magnitude without changing its phase (cf. Figs. 29 and 30). Similar phenomena to those at Nagoya were observed also at Huancayo (γ = 12.0°S, pc = 13.0 GV), Tokyo (35.8°N, 11.5 GV) and Beijing (40.1°N, 8.9 GV).

Fig. 29
figure29

Influence of three kinds of SoD1 (X, Y and Z) caused by the seasonal variation \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{T}}\) of TA. \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{r}} = {{\bf{S}}_{\bf{o}}}{{\bf{D}}_1} + {{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{T}}\).

Fig. 30
figure30

R-\(\Delta \phi \) diagram at Hermanus (34°S, 19°E; 4.9 GV) during 1965–2000 on the left and at Nagoya (35°N, 137°E; 11.5 GV) during 1971–2000 on the right (Nagashima et al., 2010). Any point in the net area can be explained by the solar anisotropy SoD1 with parameters γ and pu at the point. The number attached to each point expresses the observation year in the ’70s and ’90s in the P-state.

On the contrary, SoD1(Hermanus) belongs to the X-type and almost all the characteristic points on the \(\Delta \phi \) diagram are in the left-outside region of the net area. Such a deviation from the net area is not attributable to statistical errors, as almost all the deviations are distributed in the left-side area with respect to the iso-pu line of 103 GV. It is also noted that this area is forbidden, not only for the points derived from SoA with the present spectrum form but also for those with any reasonable spectra as far as SoA is an unidirectional form. Therefore, the distribution of the points is of the X-type and clearly indicates the existence of \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{T}}\). However, there are two exceptional points similar to the 3 exceptional cases at Nagoya, which show very small R’s in the solar activity minimum periods (1996, ’97) as SoD1(Hermanus) of the X-type changed to the Y-type due to a phase shift toward 12 hours for a decrease of solar activity. It is emphasized that the appearance of these Y-type \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{r}}\)s at the two stations were in the same Q-period but occurred in different polarity states. Hermanus observed them in the N-state, while Nagoya observed them in the P-state. This is due to the phase shift with decrease of solar activity from 18 hr to 12 hr at Hermanus in the P-state and from 15 hr to 12 hr at Nagoya in the N-state, showing a good agreement with the solar modulation of the phase of SoD1 mentioned earlier (Munakata and Nagashima, 1986).

Such an influence of \({{\bf{S}}_{\bf{o}}}{{\bf{D}}^{\bf{T}}}\) on the \({{\bf{S}}_{\bf{o}}}{{\bf{D}}_{\bf{1}}}\) can be produced also by the seasonal variation \({{\bf{S}}_{\bf{o}}}{{\bf{D}}^{\bf{H}}}\) with the phase \({\phi _{\rm{H}}}\) of 0 hr local solar time caused by the seasonal variation of HD1 (cf. Nagashima et al., 2010). Therefore, the resultant variation \({{\bf{S}}_{\bf{o}}}{\bf{D}}_{\bf{1}}^{\bf{r}}\) is expressed by

$${{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{r}} = {{\bf{S}}_{\bf{o}}}{{\bf{D}}_1} + {{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{T}} + {{\bf{S}}_{\bf{o}}}{\bf{D}}_1^{\bf{H}}.$$
(11)

It is emphasized that the influence of the sidereal anisotropies on the solar diurnal variation is not negligibly small.

7. Conclusion

The existence of CR sidereal anisotropies GA and TA was discovered in 1995 by the analysis of the observation of sidereal diurnal variations at the underground stations Hobart and Sakashita, and the ground station Nagoya, being incited by the casual finding of the seasonal variation of the sidereal diurnal variation at Hobart and also assisted by the acknowledged elimination method for the spurious sidereal variation produced by the CR solar anisotropy (cf. Table 1). The discovery enabled a unified explanation to be offered at least qualitatively for those conflicting observed phenomena of the sidereal variations in the past.

The sidereal diurnal variations \({\rm{G(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) and \({\rm{T(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) produced by GA and TA and another one \({\rm{H(}}{{\rm{t}}_{{{\rm{S}}_{\rm{i}}}}}{\rm{)}}\) by HA (cf. Table 1) are subject, respectively, to the three kinds of HMS modulation characterized by the polarity reversal of the solar polar magnetic field with 22-years periodicity and the solar activity dependence with 11-years periodicity (cf. Table 1). These modulation patterns together with the theoretical modulation model of CRs in the spherical HMS can determine their origins; one (GA) is of galactic origin from α = 0 hr and others (TA and HA) are, respectively, the sharply-concentrated CR fluxes along the neutral sheet of the HMS from α = 6 hr and 18 hr. The sharp concentration of the anisotropies can be produced by applying the CR-Lens-Effect of the HMS to the accelerated CRs on the boundary of the HMS with the short nose and the long tail together with the wavy neutral sheet.

The duality of the motion of the HMS observed, respectively, by the CRs and the neutral particles suggests the existence of the proper motion VIGM of the interstellar gaseous matter (IGM) including the magnetic field. \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\) produces the electromagnetic interaction between the HMS and IGM and forms the disturbed region surrounding the HMS, called the Subordinate HMS (SuHMS), which prevents the C-G effect expected to be caused by the relative motion between the HMS and the low energy CRs in IGM. In the high-energy region, however, as CRs can freely pass through the IGM and the SuHMS, the C-G effect can be observed in the direction of \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\). On the other hand, as the neutral particles cannot recognize the existence of the SuHMS owing to their long scattering mean free path compared with the scales of the SuHMS, the particles can pass through the SuHMS with the velocity VIGM and produce the motion of Vnp observed on the HMS. As above, the solar modulation of CR sidereal anisotropies are intimately correlated with the behavior pattern of the heliomagnetosphere (HMS) and also with the CR solar diurnal variation produced by the diffusion-convection process inside the HMS. One of the most remarkable results of the present analysis is that the motion \(({\bf{V}}_{{\bf{IGM}}}^{\bf{S}})\) of the interstellar gaseous matter (IGM) relative to the neighboring stars could be estimated for the first time by using the duality of the velocities \({\bf{V}}_{{\bf{HMS}}}^{{\bf{CS}}}\) and \({\bf{V}}_{{\bf{np}}}^{{\bf{HMS}}}\).

However, the sidereal anisotropies of CRs and their related phenomena are not completely solved yet, as they are based on basic assumptions such as the acceleration of CRs on the boundary of the HMS, the motion of IGM in space, the formation of the SuHMS surrounding the HMS and so on. It is necessary to confirm these assumptions theoretically and experimentally not only for the sidereal anisotropies of CRs but also for the further development of the interaction between the HMS and the interstellar medium.

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Acknowledgments

The authors express their sincere appreciation for the long-term continuous observations of cosmic ray diurnal variations performed with the neutron monitor network in the world by many researchers, with the ion chambers at Godhaven, Cheltenham, Huancayo and Christchurch by the Late Dr. S. E. Forbush and his colleagues, and at Yakutsk by the Late Professor Emeritus Yu. G. Shafer, Dr. G. B. Shafer and their colleagues, with the underground muon telescope at Cambridge, Hobart by the Late Drs. A. G. and K. B. Fenton, Dr. J. E. Humble and their colleagues, with the muon telescopes at Mawson, Antarctica by one of the authors (R. M. J.), Drs. J. E. Humble, M. L. Duldig and their colleagues and expeditionists, with the underground muon telescopes at Misato and Matsushiro by one of the authors (S. M.) and his colleagues, with the muon telescopes on the ground at Nagoya and the underground station at Sakashita by the Late Professors Emeritus Y. Sekido, K. Murakami and H. Ueno, the Late Dr. K. Fujimoto and their colleagues, with the air shower observation at Mt. Norikura (the University of Tokyo) by the Late Professor H. Ueno and his colleagues at Nagoya University and, finally, for the positive support for the data analysis of the World Data Center C2 for Cosmic Rays by the Late Dr. Y. Miyazaki, Dr. M. Wada, one of the authors (I. K.) and their colleagues. Without these observations and the analysis, the present result could not be obtained.

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Nagashima, K., Morishita, I., Fujii, Z. et al. The existence of cosmic ray sidereal anisotropies of galactic and solar origins with energies lower than 104 GeV and their modulation caused by the presumed behavior pattern of the heliomagnetosphere and of its neighboring gaseous matter in interstellar magnetic field. Earth Planet Sp 64, 613–639 (2012). https://doi.org/10.5047/eps.2012.02.004

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Key words

  • Cosmic rays (CR)
  • sidereal anisotropy
  • heliomagnetosphere (HMS)
  • interstellar matter (ISM)
  • Compton-Getting effect
  • motion of the HMS
  • motion of ISM
  • acceleration of CRs
  • cosmic ray lens effect