- LETTER
- Open Access
Estimation of index of power law rigidity spectrum of cosmic rays using effective rigidity of multidirectional muon detector
- G. Kalugin^{1}Email author and
- K. Kabin^{1}
https://doi.org/10.1186/s40623-015-0318-8
© Kalugin and Kabin. 2015
- Received: 12 May 2015
- Accepted: 27 August 2015
- Published: 15 September 2015
Abstract
We develop a new approach for calculating index of power law for variations in the differential rigidity spectrum of cosmic rays registered by Nagoya multidirectional muon detector. Our approach is based on effective rigidity which we define for the entire instrument, not just for its individual channels. Our definition of effective rigidity can be straightforwardly extended to other multichannel instruments and networks of instruments. As examples, we calculate the power law index under quiet conditions and during 12 Forbush decrease events. We also discuss sensitivity of the Nagoya muon detector to different values of the power law index and different rigidities.
Keywords
- Cosmic rays
- Multidirectional muon detector
- Coupling coefficients
- Effective rigidity
Findings
Introduction
The ability of cosmic ray (CR) particles to penetrate magnetic fields and reach the top of the Earth’s atmosphere is controlled by their rigidity R which is defined as the particle’s momentum multiplied by the speed of light per unit charge. The rigidity is usually measured in gigavolts (GV). As CRs propagate through the solar system, they interact with the interplanetary magnetic field (IMF) and their fluxes are modulated by its intensity and direction. Fluxes of CRs with different rigidities change differently since lower rigidity particles are more affected by the changes in the magnetic field. Thus, variations δ D(R)/D(R) in the differential rigidity spectrum D(R) of CRs are related to changes in power spectral density of IMF turbulence (Wawrzynczak and Alania 2009) and thereby related to fundamental properties of the solar wind and IMF, which are important for space weather and astrophysical applications (Dorman 2006). Variations δ D(R)/D(R) are often well approximated by a power law (Dorman 1974; Fujimoto et al. 1984).
Characteristics of channels of the Nagoya muon detector
Channels | N _{0},10^{6} counts/h | σ,% | R _{c}, GV | R _{m}, GV | γ _{m} |
---|---|---|---|---|---|
V | 2.76 | 0.06 | 11.5 | 59.4 | – |
N30 | 1.25 | 0.09 | 12.9 | 64.6 | −1.13 |
E30 | 1.20 | 0.09 | 16.2 | 66.7 | −0.97 |
S30 | 1.23 | 0.10 | 11.3 | 62.6 | – |
W30 | 1.26 | 0.09 | 9.4 | 61.8 | – |
NE39 | 0.58 | 0.14 | 17.9 | 72.0 | −1.03 |
SE39 | 0.58 | 0.14 | 15.0 | 69.3 | −1.14 |
SW39 | 0.60 | 0.13 | 9.2 | 65.6 | – |
NW39 | 0.62 | 0.13 | 11.0 | 66.6 | −1.68 |
N49 | 0.61 | 0.13 | 12.9 | 83.0 | −1.24 |
E49 | 0.58 | 0.14 | 21.0 | 88.3 | −1.10 |
S49 | 0.60 | 0.13 | 10.9 | 80.5 | −1.39 |
W49 | 0.62 | 0.13 | 9.3 | 79.3 | −1.55 |
N64 | 0.18 | 0.27 | 11.0 | 105.0 | −1.20 |
E64 | 0.17 | 0.28 | 25.1 | 113.7 | −1.10 |
S64 | 0.18 | 0.27 | 10.8 | 103.7 | −1.25 |
W64 | 0.18 | 0.27 | 8.7 | 103.0 | −1.29 |
In this paper, we assume a power law for variations δ D(R)/D(R) in the differential rigidity spectrum of CRs registered by the Nagoya muon detector and develop a new approach for finding its power law index based on the effective rigidity of the detector. We present the calculations using this method for cases corresponding to both quiet and disturbed conditions. We also discuss the concept of the effective rigidity of the detector and its implications on the sensitivity of the instrument.
Coupling coefficients
The values of \(c_{0,i}^{0}\) are computed by Fujimoto et al. (1984) under assumption (1) and are available as a table for γ=−2,−1.5,−1,−0.5,0 and 0.5 and for R _{ u }=30 GV,50 GV,100 GV,200 GV,500 GV and 1000 GV. For convenience of our analysis, we interpolate the γ-dependence of these data points with hyperbolas. We find that the standard deviation of this approximation is less than 0.01.
We also note that in our calculations, we analyze only the isotropic part of CR intensity. However, the Nagoya muon detector measurements have also been used for studies of anisotropy of CRs, for example, by Kozai et al. (2014) who derived the yearly mean values for the three-dimensional anisotropy and the modulation parameters and found it to be consistent with those derived from the Global Muon Detector Network data.
Effective rigidity
Therefore, the effective rigidity of a muon detector channel depends on both the characteristics of the detector and the considered event (through values of γ and R _{ u }).
We note that the effective rigidity defined above is a completely different concept from the effective cutoff rigidity used, for example, by Cooke et al. (1991) to describe the magnetospheric transmissivity (Kudela and Usoskin 2004) and penumbral structure of the CRs. This effective cutoff rigidity typically has a much lower value than the effective rigidity described here.
This, however, is not the only possible choice for the definition of the effective rigidity of a detector; another definition with different weights is discussed later.
In these formulas 〈R _{ eff }〉, R _{u}, p, and r are in gigavolts; q is the dimensionless coefficient. The error of this approximation is increasing with R _{u} but does not exceed 2 %. As an example, fitting for R _{u}=1000 GV is shown in Fig. 1 by solid line.
Calculating the power law index using the effective rigidity
which is similar to (2).
Therefore, if 〈k〉 is computed from data, the corresponding value of γ is determined from Eq. 7. To avoid temperature and noise effects, we can use 24-hour moving average intensity recorded in the i-th directional channel (e.g. Okazaki et al. 2008).
We first demonstrate computing γ under quiet geomagnetic conditions using day of the year (DOY) 223/2011 as an example. For this day, the planetary index K _{p} (Rostoker 1972) did not exceed \(1\frac {2}{3}\), and for the previous 3 days K _{p} did not exceed 3 (NGDC site 2015). For the quiet conditions, we define intensity variation ν _{0,i } as relative changes in count rates referred to the minimum value of counts during the day. Taking averages over a day, we obtain, using (5), 〈ν _{0}〉=0.025 %. Since ν _{0,1}=0.026 % for this day, we get 〈k〉=0.96 and, finally, γ=−0.3, which agrees well with the typical values of the perturbation spectrum index under quiet conditions (Kane, 1963). In addition, once spectrum index is known, the amplitude of the variation can be found using Eq. 6. Specifically, from Eq. 3, 〈R _{ eff }〉=72.45 GV for γ=−0.3 and, thus, the amplitude averaged over the day is a _{0}=0.046 %.
Rigidity spectrum index for 12 Forbush decreases
No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Year | 2003 | 1998 | 2005 | 2001 | 2004 | 2000 | 2000 | 2001 | 2005 | 1998 | 2001 | 2001 |
DOY | 302 | 268 | 135 | 118 | 22 | 160 | 195 | 86 | 254 | 312 | 310 | 229 |
〈k〉 | 0.88 | 0.86 | 0.89 | 0.88 | 0.85 | 0.87 | 0.86 | 0.87 | 0.85 | 0.84 | 0.83 | 0.81 |
n | −1.07 | −1.04 | −0.89 | −0.91 | −1.11 | −1.09 | −1.11 | −0.98 | −1.13 | −1.28 | −1.33 | −1.23 |
γ _{ σ } | −1.45 | −1.66 | −1.36 | −1.45 | −1.71 | −1.53 | −1.65 | −1.58 | −1.74 | −1.82 | −1.89 | −2.0 |
γ _{k} | −0.80 | −0.99 | −0.70 | −0.77 | −1.07 | −0.88 | −1.00 | −0.91 | −1.10 | −1.23 | −1.33 | −1.49 |
\(\tilde {\gamma }_{\mathrm {k}}\) | −0.81 | −1.01 | −0.73 | −0.81 | −1.06 | −0.89 | −0.99 | −0.93 | −1.09 | −1.18 | −1.27 | −1.46 |
It is, however, possible to modify the averaging procedure in formula (3) by replacing weights \(\sigma _{i}^{-2}\) with k _{ i } which is the ratio of FD amplitude measured by i-th channel to that measured by the vertical channel, see (Kalugin and Kabin 2015). Choosing k _{ i } values as weights is similar to the approach used by Plainaki et al. (2007) for study of ground level enhancement events. The values of the spectral index calculated using this approach are shown in Table 2 as γ _{k}. The difference between n and γ _{k} is smaller than before and for half of the events, particularly, for events No. 2, 5, 8, 9, 10, and 11, it does not exceed 7 %. Thus, for disturbed conditions, this may be a better choice than (3) for defining the average effective rigidity of a detector.
We also found that the curves obtained with using weights k _{ i } for different events are very close to each other. Therefore, we computed the mean of these curves shown in Fig. 2 by solid line. The values of the spectral index obtained using this average curve for all events are shown in the last line of Table 2; the difference between γ _{k} and \(\tilde {\gamma }_{\mathrm {k}}\) does not exceed 5 %.
The standard deviation for these fits to Eq. 7 is less than 0.002. Approximation (8) worked well for all the events listed in Table 2 and can be expected to be adequate for other FD events as well.
Additional remarks
In this section, we discuss some additional properties of the effective rigidity which, while are not directly connected with the problem of computing the power law index, may be interesting for CR studies using mutidirectional instruments.
First, we note that the value of the median rigidity of the vertical channel, which is the most statistically significant one, is 59.4 GV. The effective rigidity 〈R _{ eff }〉 reaches this value at γ=−0.5. Thus, the muon detector is expected to be most sensitive to CR variations with spectrum index close to −0.5 or, equivalently, when the normalized averaged intensity is close to 0.94. Since the derivative of 〈R _{ eff }〉 increases with γ, the Nagoya muon detector is less sensitive to CR variations with small spectrum index.
The non-monotone behavior of the normalized effective rigidities is explained by the interplay between the coupling functions and cutoff rigidities of different channels. The only channels for which R _{ e f f,i }/R _{ e f f,1} monotonously increases with γ are S30, W30, and SW39. These three channels are closest to the vertical channel in terms of measured fluxes and the shape of the coupling functions and are characterized by low cutoff rigidities. The fact that the behavior of channels S30, W30, and SW39 is different from all the others suggests that in some future studies of CRs using the Nagoya muon detector, these channels may be combined with the vertical channel, while the other are treated separately. This argument can be used, for example, to define identical channels to increase the statistical significance of certain measurements (Alaniya et al. 1975). Similar considerations for a detector with many solid angle bins may be used for regrouping data recorded in fine solid angle bins into data in coarse bins (Subramanian et al. 2009).
Conclusions
In this paper, we discussed possible definitions of the average effective rigidity for the Nagoya muon detector, which extend the corresponding concept of effective rigidities for individual channels. Connections between the effective rigidity and the differential rigidity spectrum make it a useful data analysis tool. We assumed a power law for the spectrum and develop a new approach for finding the exponent of this power law and estimating the isotropic part of variations in CR intensity using data from the muon detector. We find that this exponent can be easily and effectively calculated using effective rigidity which depends on the CR fluxes measured by the instrument as well as the applicable coupling coefficients. As an illustration of this method, we calculated γ for quiet and disturbed conditions using the Nagoya multidirectional muon detector. We also suggest the approximate formulas for the effective rigidity which can simplify this calculation even further.
In addition, we show that certain characteristics of directional channels S30, W30, and SW39 of the Nagoya muon detector are close to that of the vertical channel and different from all the others and therefore, in some future studies, these channels may be combined with the vertical channel, while the others are treated separately. We also showed that the Nagoya muon detector is expected to be the most sensitive to CR spectrum variations characterized by the spectral index close to −0.5.
Finally, we note that effective rigidity of any multichannel instrument detecting high-energy CRs can be defined similarly to that described in the paper. However, this extension is limited for detectors registering low-energy CRs, such as neutron monitors because the asymptotic directions of low energy CRs strongly depend on the level of the geomagnetic activity and local time of observations (Kudela and Usoskin 2004; Kudela et al. 2008). In particular, change in cutoff rigidity during Forbush decreases significantly depends on the location of neutron monitor (Flückiger et al. 1986; Katsoulakos et al. 2013). Thus, the approach described in this paper can have limits to application to network of such instruments as neutron monitors.
Declarations
Acknowledgements
We are thankful to the team of the Solar-Terrestrial Environment Laboratory for providing data from the Nagoya muon detector (STEL site 2015) and to the team of the National Geophysical Data Center (NGDC site 2015) for providing indices of the magnetic activity. We appreciate Dr. K. Munakata for discussions of various issues associated with the Nagoya muon detector. We are also grateful to anonymous reviewers and editor for valuable comments and remarks which significantly improved the paper. The work was supported by the National Sciences and Engineering Research Council of Canada.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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