Neutron monitors and muon detectors for solar modulation studies: Interstellar flux, yield function, and assessment of critical parameters in count rate calculations
Introduction
After the discovery of cosmic rays (CR) by Hess in 1912, ground-based CR detectors located at various latitudes, longitudes and altitudes, played a major role to determine the CR composition and spectrum (see Stoker (2009) for a historical perspective). From the 50’s, networks of neutron monitors (Simpson, 2000) and muon telescopes (Duldig, 2000) were developed. They provide today one of the most valuable data to inspect time variations of the integrated CR flux in the 10–100 GeV/n range.
The formal link between these variations and the Sun activity was established in the mid-fifties, by means of a transport equation of CR fluxes in the solar cavity (Parker, 1965, Jokipii, 1966). In the 80’s, the effect of particle drift was shown to be responsible to a charge-sign dependent modulation (Potgieter, 2013), following the Sun polarity cycle.1 However, the Force-Field approximation (Gleeson and Axford, 1967, Gleeson and Axford, 1968) has remained widely used thanks to its simplicity: this approximation, used in this work, has only one parameter .
Several strategies have been developed for time series reconstruction of the modulation level , and/or CR TOA fluxes at any time (of interest for many applications):
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Using spacecraft measurements (e.g., Davis et al., 2001, Buchvarova et al., 2011, Buchvarova and Draganov, 2013): it is the most direct approach, but the time coverage is limited to a few decades with a poor sampling;
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Comparison of calculated and measured count rates in ground-based detectors (Usoskin et al., 1999, Usoskin et al., 2002, Usoskin et al., 2005, Usoskin et al., 2011): it covers a larger period (60 years), with a very good time resolution (a few minutes)2;
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Extracting relationships between the modulation level and solar activity proxies, based on empirical (Badhwar and O’Neill, 1994, Badhwar and O’Neill, 1996, O’Neill, 2006, O’Neill, 2010) or semi-empirical (Nymmik et al., 1992, Nymmik et al., 1994, Nymmik et al., 1996, Nymmik, 2007, Tylka et al., 1997, Ahluwalia, 2013) approaches.
All these strategies provide a satisfactory description of CR fluxes, though some fare better than others (for comparisons, see Buchvarova and Velinov, 2010, Mrigakshi et al., 2012, Zhao and Qin, 2013, Matthia et al., 2013). Note also that empirical methods are expected to provide effective and less meaningful values for (O’Neill, 2006).
In this paper, we focus on the second strategy, for a systematic study of the main uncertainties affecting the calculation of expected count rates in NM and muon detectors. This requires the description of the atmosphere and of the ground-based detector responses to incoming CRs (e.g., Clem and Dorman, 2000). The various uncertainties, described in the Dorman, 1974, Dorman, 2004, Dorman, 2009 textbooks, are generally discussed separately in research articles (uncertainty on the yield function, geomagnetic rigidity cutoff, seasonal effects…), and not propagated to the modulation parameter. For this reason, we believe it is useful to recap and gather them in a single study, re-assess which ones are the most important, and link these uncertainties to the expected level of variation/uncertainty they imply on the modulation level . The complementarity (different uncertainties and time coverage) of NM count rates and TOA CR flux measurements to obtain time-series of the solar modulation parameters is left to a second study.3
The paper is organised as follows: we start with a general presentation of the ingredients involved in the count rate calculations (Section 2), and discuss a new fit for the IS fluxes (Section 3). We then detail the calculation of the propagation in the atmosphere, providing a new yield function parametrisation (Section 4). Combining these inputs allows us to link the count rate variation with the solar modulation parameter, and to study the various sources of uncertainties (Section 5). The final ranking of the uncertainties in terms of both count rates and concludes this study (Section 6).
Section snippets
From IS fluxes to ground-based detector count rates
A ground-based detector at geographical coordinate measures, at time t, a count rate per unit interval , from the production (from CRs) of secondary particles in the atmosphere (atmospheric shower):with , and i running on CR species:
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is the transmission function in the geomagnetic field, which depends on the detector location and can vary with time (Section 5.2.2);
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is the yield function at altitude h
Determination of IS fluxes: from H to Ni
Due to the interplay between the CR relative abundances and the yield function, the most important primary CR contributors to the count rates are protons, heliums, and heavier nuclei (in a small but non negligible fraction). In recent studies, in addition to proton and helium, the contribution of species heavier than He is accounted for as an effective enhancement of the He flux (Webber and Higbie, 2003, Usoskin et al., 2011).
In order to assess the uncertainties (on count rate calculations)
Atmospheric propagation, yield function, and detectors
When entering the Earth atmosphere, CRs initiate cascades of nuclear reactions involving primary energetic particles (mainly hydrogen and helium but also heavier nuclei) and atmospheric nuclei such as oxygen or nitrogen. The so-called Extensive Air Showers (EAS) generate secondary particles along their path, to be detected by ground-based instruments.
In this section, we discuss the generation of secondary particles (Section 4.1) as an input to provide a new yield function parametrisation
Count rates: variations and uncertainties
In this section, count rates are calculated from Eq. (1), which involves the yield function , the modulated fluxes for all CR species i, and the geomagnetic transmission . To validate our code, we compare count rate variations (vs ) to existing latitude surveys (Section 5.1). We then propagate, on count rates, IS flux and yield function uncertainties (Section 5.2.1), and geomagnetic transmission function uncertainties (Section 5.2.2). We conclude the section with
Conclusions: count rate variation and uncertainty iso-contours in the plane
We have made a detailed study of count rates (and uncertainties) for neutron monitors and detectors, as a function of the rigidity cut-off and the modulation level , in the context of the force-field approximation.
Acknowledgements
D. M. thanks K. Louedec for useful discussions on the Auger scaler data, A. L. Mishev and I. G. Usoskin for clarifications on their yield function, and P. M. O’Neill for providing his BO11 flux model. We thank the anonymous referee for her/his careful reading that helped correcting several mistakes in the text.
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