Elsevier

Advances in Space Research

Volume 62, Issue 10, 15 November 2018, Pages 2859-2879
Advances in Space Research

Propagation of cosmic rays in heliosphere: The HelMod model

https://doi.org/10.1016/j.asr.2017.04.017Get rights and content

Abstract

The heliospheric modulation model HelMod is a two dimensional treatment dealing with the helio-colatitude and radial distance from Sun and is employed to solve the transport-equation for the GCR propagation through the heliosphere down to Earth. This work presents the current version 3 of the HelMod model and reviews how main processes involved in GCR propagation were implemented. The treatment includes the so-called particle drift effects – e.g., those resulting, for instance, from the extension of the neutral current sheet inside the heliosphere and from the curvature and gradient of the IMF –, which affect the transport of particles entering the solar cavity as a function of their charge sign. The HelMod model is capable to provide modulated spectra which well agree within the experimental errors with those measured by AMS-01, BESS, PAMELA and AMS-02 during the solar cycles 23 and 24. Furthermore, the counting rate measured by Ulysses at ±80° of solar latitude and 1–5 AU was also found in agreement with that expected by HelMod code version 3.

Introduction

Modulated omni-directional intensities of galactic cosmic rays (GCRs) were observed during different phases of solar activity using both balloon flights (for instance, see Boezio et al., 1999, Menn et al., 2000, Haino et al., 2004, Shikaze et al., 2007, Abe et al., 2008, Abe et al., 2016) and space-borne missions (e.g., see Alcaraz et al., 2000d, Alcaraz et al., 2000c, Alcaraz et al., 2000a, Alcaraz et al., 2000b, Aguilar et al., 2002, Aguilar et al., 2007, Aguilar et al., 2014, Aguilar et al., 2015b, Aguilar et al., 2015a, Aguilar et al., 2016, Adriani et al., 2009a, Adriani et al., 2009b, Adriani et al., 2010, Adriani et al., 2011, Adriani et al., 2013, Adriani et al., 2015, Adriani et al., 2016), in particular during the latest solar cycles. The increased performance of on-board spectrometers was and is currently enabling to enhance the accuracy of the observed spectra. Thus, it was opening the way to a better understanding of processes related to the transport of GCRs through the Heliosphere – the so-called modulation effect – and, ultimately, to the capability (a) to unveil local interstellar spectra (LIS) of GCR species(e.g., see Bisschoff and Potgieter, 2014, Bisschoff and Potgieter, 2016, Della Torre et al., 2017a, Boschini et al., 2017 and also references therein), (b) to investigate their generation, acceleration and diffusion process within the Milky Way (e.g., see Boella et al., 1998, Strong et al., 2007, Evoli et al., 2008, Putze et al., 2009), and, in turn, (c) to possibly untangle features due to new physics – i.e., dark matter (e.g., see Bottino et al., 1998, Cirelli and Cline, 2010, Ibarra et al., 2010, Salati, 2011, Weniger, 2011 and references therein) – or additional astrophysical sources so far not taken into account (e.g., see Chang et al., 2008, Abdo et al., 2009, Adriani et al., 2009a, Cernuda, 2011, Mertsch and Sarkar, 2011, Della Torre et al., 2015, Rozza et al., 2015 and references therein).

Among space missions currently observing GCRs, AMS-02 – on-board of the International Space Station since May 2011 – is continuously providing data with unprecedented measurement accuracy. In fact, this spectrometer allowed one to determine the most accurate differential intensities of protons (Aguilar et al., 2015b), helium nuclei (Aguilar et al., 2015a), antiprotons (Aguilar et al., 2016), electrons and positrons (Aguilar et al., 2014). The high precision of these experimental data together with those from Ulysses spacecraft (e.g., see Simpson et al., 1992, Simpson et al., 1996, Heber et al., 1996, Ferrando et al., 1996, De Simone et al., 2011, Gieseler and Heber, 2016) constitute a challenge for any modulation model of the inner part of heliosphere. Actually such a treatment has to reproduce the observed GCR spectra transported – during different solar activity phases – down to Earth and, also, outside the ecliptic plane at distances from Sun ranging from about 1–5 AU. In fact, the observations made using Ulysses spacecraft allowed one to determine both the latitudinal and radial dependence of GCR intensity. Furthermore, the data taken during Ulysses fast latitudinal scan exhibited a latitudinal dependence on (i) the charge sign of the GCR species (i.e., protons and electrons, which are the dominant positively and negatively charged component, respectively), (ii) solar activity and (iii) polarity of the interplanetary magnetic field (IMF). It is worth to remark that these data may, in addition, allow a better understanding of space radiation environment close to Earth, thus extending our capability of predicting radiation hazards for astronauts and device damages in space missions (e.g., see Leroy and Rancoita, 2007, Golge et al., 2015 and Chapters 7 and 8 of Leroy and Rancoita, 2016).

In this work we present the version 3 of 2-D heliospheric modulation (HelMod) model – i.e., a two dimensional treatment dealing with the helio-colatitude and radial distance from Sun (Gervasi et al., 1999, Bobik et al., 2012, Bobik et al., 2013b) – currently employed to solve the transport-equation for the GCR propagation through the heliosphere down to Earth. The relevant GCR propagation processes are described in Sections 2 Heliospheric propagation of cosmic rays, 3 Effective Heliosphere Parameters and LIS’s in order to illustrate how they are implemented in his latest version of HelMod model. Furthermore, details on Monte Carlo technique used to solve the stochastic integration are treated in Sections 4 The Monte Carlo Code, 5 Comparison with observations during solar cycles 23–24, and allow one a better understanding on HelMod capabilities to deal with solar modulation, within the inner part of the heliosphere. At present, the model only treats GCRs with energies 0.5 GeV/nucleon, thus modulation effects occurring in the outer heliosphere – i.e., beyond the termination shock (TS) (see, e.g., Langner et al., 2003, Langner and Potgieter, 2004, Bobik et al., 2008, Potgieter, 2008, Florinski and Pogorelov, 2009, Luo et al., 2013, Senanayake and Florinski, 2013) – are not accounted for. It has to be pointed out that the HelMod model is capable of describing the current large set of observation data, which were collected during solar cycles 23 and 24 with the occurrence of two solar minimum. For this purpose, the model includes the so-called particle drift effects – e.g., those resulting, for instance, from the extension of the neutral current sheet inside the heliosphere and from the curvature and gradient of the IMF –, which affect the transport of particles entering the solar cavity as a function of their charge sign. These effects are particularly relevant when IMF exhibits a well-defined large-scale structure or this latter is still relevant. In fact, at the solar minimum and when the solar activity is not too far from such a condition, GCR modulated intensities exhibit a dependence on charge sign (e.g., see Garcia-Munoz et al., 1986, Clem et al., 1996, Clem et al., 2000, Boella et al., 2001). In fact, the IMF polarity reversal causes charge sign dependent modulation effects, for instance, those observed in particle over anti-particle intensities ratio at rigidities lower than about 10–20 GV (e.g., Adriani et al., 2016). These effects are treated in the Parker transport equation through the terms including the drift velocity. The analysis on Ulysses out-of-ecliptic observations (e.g., see Simpson, 1996, Simpson et al., 1996, Heber et al., 1996, Heber et al., 1998, Heber et al., 2008, Ferrando et al., 1996, De Simone et al., 2011, Gieseler and Heber, 2016) provided, so far, a unique point of view highlighting the presence of latitudinal gradients in the spatial distribution of GCRs, during period of low solar activity, i.e., when the combination of particle charge (q) and solar magnetic polarity (A) is positive (qA>0); while a more uniform distribution of GCRs in the inner part of the heliosphere occurs for qA<0.

As discussed in Bobik et al., 2011b, Bobik et al., 2012, the model exhibits a smooth time dependence introduced by the parameters – related to solar activity and adopted within the model itself, as described in Section 2.3 –, which are averaged over time durations corresponding to Carrington rotations, i.e., (a) solar wind speed (Vsw, see Section 2.2), (b) tilt angle (αt, see Section 3) of the neutral current sheet, and (c) diffusion parameter (K0, see Section 2.3). Furthermore, it has to be remarked that the solar wind usually takes one year or even more to reach the border of heliosphere. The above parameters – usually determined at 1 AU – are transferred to describe the properties of any distant heliospheric sector, according to the time required by the solar wind to reach such a region from Sun (see discussion in Section 3).

In the present article, the LIS fluxes are those derived in Della Torre et al. (2017a) and Boschini et al. (2017) by means of GALPROP v55. In that article leptons were not yet treated, i.e., no electron LIS is available yet using the new GALPROP version. Thus, the discussion on the modulated spectra obtained using HelMod model is mostly restricted to comparisons with experimental data regarding protons and helium nuclei. As discussed in Della Torre et al. (2017a) and Boschini et al. (2017), the LIS, presented in Section 3, accommodate both the low energy interstellar CR spectra measured by Voyager 1 and the high energy observations by BESS, Pamela, AMS-01, and AMS-02 over solar cycles 23–24.

Finally, we have to remark that Engelbrecht and Burger (2013) exploited an ab initio approach for a three dimensional steady state GCR modulation model, in which the effects of turbulence on both the diffusion and drift of these cosmic-rays are treated in a self-consistent description; Strauss et al. (2013) uses a hybrid modeling approach incorporating the plasma flow from a magnetohydrodynamics model with the particle transport; and, finally, Vos and Potgieter (2016, and reference therein) computed spatial gradients and absolute flux variations for GCR protons in the heliosphere for solar minimum. Although these models provide encouraging results, they still depends on parameters whose time evolution is not yet measurable or fully understood. So far, the found agreement among the modulated spectra from HelMod code and experimental data collected over a long period (e.g., see Bobik et al., 2012, Bobik et al., 2013b, Della Torre et al., 2017a, Boschini et al., 2017) motivated the choice, in HelMod, to reduce the complexity of diffusion process using a unique time dependent variable, as described in Section 2.2.

Section snippets

Parker equation

The cosmic rays propagation trough the heliosphere was treated by Parker (1965), who demonstrated that – in the framework of statistical physics – the random walk of cosmic ray particles is a Markoff process, describable by a Fokker–Planck equation (FPE). In his original formulation, Parker’s transport-equation was expressed in terms of particle density for unit space and energy, i.e., U(x,T) (e.g., see Jokipii and Parker, 1970, Fisk, 1971, Bobik et al., 2012 and also Sections 8.2–8.2.5 of

Effective Heliosphere Parameters and LIS’s

One of the success of HelMod model with respect to other available Monte Carlo codes for heliospheric propagation – like, (e.g., SolarProp–Gaggero et al., 2014 and HelioProp–Kappl, 2016)– is its reduced number of free parameters necessary for the description of modulation mechanisms. These parameters – as described later along the present section – are related to quantities determined from observations and need, in turn, to be tuned in order to obtain a comprehensive set of modulated spectra,

The Monte Carlo Code

For most applications Parker’s transport equation [Eqs. (1), (3)] has been solved using numerical methods, because its intrinsic complexity.

The traditional approach to solve multi-dimensional partial differential equations makes use of numerical integration methods such as the finite difference technique (e.g., see Jokipii and Kopriva, 1979, Kota and Jokipii, 1983, Potgieter and Moraal, 1985, Burger and Hattingh, 1995) or as the standard implicit difference technique (e.g., see Fisk, 1971, Kota

Comparison with observations during solar cycles 23–24

The current HelMod model provided modulated differential intensity for protons, helium nuclei and antiproton for low and high solar activities (as discussed in Section 1). In this article, we focus on HelMod results regarding protons and helium, whose LIS’s were recently investigated in Della Torre et al. (2017b,a) and Boschini et al. (2017). The current parametrization is also suited to reproduce the high energy behavior of the measured spectra (e.g., see Figs. 3 and 5 in Della Torre et al.,

Conclusions

The CR propagation inside the heliosphere was initially treated – almost 60 years ago – by Parker (1965) who provided a general theoretical framework for the heliospheric modulation through the interplanetary medium. Since then, continuous advances allowed a deeper and deeper understanding of the general properties of the IMF affecting particle motion. However, the description of the transport mechanisms occurring still needs further refinements for allowing a comprehensive treatment along the

Acknowledgements

This work is supported by ASI (Agenzia Spaziale Italiana) under contract ASI-INFN I/002/13/0 and ESA (European Space Agency) contract 4000116146/16/NL/HK.

We acknowledge the NMDB database (www.nmdb.eu), supported under the European Union’s FP7 programme (contract No. 213007) for providing data. The data from McMurdo were provided by the University of Delaware with support from the U.S. National Science Foundation under grant ANT-0739620. Finally, we acknowledge the use of NASA/GSFC’s Space

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