1 Introduction

When entering the heliosphere, galactic cosmic rays (GCRs) are subjected to the solar modulation effect which causes a significant modification in the energy spectrum of their flux in comparison with the local interstellar spectrum (LIS) outside the heliosphere. To understand solar modulation, it is crucial to model the transport processes of GCRs in the solar wind and its embedded magnetic field. The main processes are diffusion, drift, advection and adiabatic deceleration. All these processes are time dependent and follow the quasiperiodical 11-year solar cycle. Solar modulation is very important in GCR physics and heliophysics Moraal (2013); Potgieter (2013). Modeling the temporal evolution of GCRs in interplanetary space is also important for assessing radiation risks and hazards in long-duration crewed space missions. In this respect, the recent high-precision and time-resolved data from AMS-02 Aguilar (2018a, 2018b) and PAMELA Adriani (2013); Martucci (2018) experiments offer a unique possibility to study the GCR modulation over a long period of time.

2 Methodology

2.1 The model

We implemented a 2D description of the heliosphere, modeled as a spherical bubble centered on the Sun. The wind flows radially from the Sun, with a speed \(V_{sw}(r,\theta ,t)\) that depends on helioradius r, heliolatitude \(\theta \), and time t (Fiandrini et al. 2021). The solar wind drops to a subsonic speed across the termination shock \(r_\mathrm{{TS}}=85\) AU, and vanishes at the heliopause \(r_\mathrm{{HP}}=122\) AU. The Earth is placed in the equatorial plane, at \(r_{0}=\)1 AU. The interplanetary magnetic field (IMF) vec B is wound up in a rotating spiral, where its angular aperture depends on the wind speed. Similarly, the heliospheric current sheet (HCS) is modeled on that structure. The HCS is a rotating layer which divides the IMF into two hemispheres of opposite polarity. The angular size of the HCS amplitude depends, in particular, on the tilt angle \(\alpha \) between magnetic and solar rotational axis. The tilt angle is time dependent. It ranges from \(\sim {10}^{\circ }\) during solar minimum (flat HCS) to \(\sim {80}^{\circ }\) during maximum and reversal (wavy HCS). Measurements of the tilt angle are provided by the Wilcox Solar Observatory (WSO), since the 1970s to date, on a 10-day basis Hoeksema (1995).

The transport of GCRs in the heliosphere is described by the Parker equation  1:

$$\begin{aligned} \frac{\partial f}{\partial t} = \nabla \cdot [{\textbf{K}}^{S}\cdot \nabla f ] - (\vec{\textbf{V}}_{sw} + \vec{\textbf{V}}_D) \cdot \nabla f + \frac{1}{3}(\nabla \cdot \vec{\textbf{V}}_{sw})\frac{\partial f}{\partial (ln R)} \end{aligned}$$

where f is the phase space density of GCR particles, \(R=p/Z\) is their rigidity (momentum/charge ratio), \({\textbf{K}}^{S}\) is the symmetric part of the diffusion tensor, \(\vec{\textbf{V}}_{sw}\) is the solar wind speed, and \(\vec{\textbf{V}}_{D}\) is the drift speed

$$\begin{aligned} \vec{\textbf{V}}_{D}= \frac{\beta {R}}{3}\nabla \times \frac{\vec{\textbf{B}}}{B^2}. \end{aligned}$$

The GCR flux \(J=J(t,R)\) is given by \(J=\frac{\beta {c}}{4\pi }n\), where \(\beta {c}\) is their speed and \(n=4{\pi }R^{2}f\) is their number density. In this work, we solved Eq. 1 by means of the stochastic differential equation method in steady-state conditions (\(\partial /\partial {t}=0\)) Strauss and Effenberger (2017),

The diffusion of GCR particles arises from their scattering off the small-scale irregularities of the turbulent IMF. Drift motion is caused by gradient and curvature of the regular component of the IMF, and in particular across HCS. Diffusion and drift can be formally incorporated in the diffusion tensor \({\textbf{K}}\) as symmetric and antisymmetric parts, respectively: \({\textbf{K}}={\textbf{K}}^S+{\textbf{K}}^A\), with \(K_{ij}^S = K_{ji}^S\) and \(K_{ij}^A = -K_{ji}^A\). However, in Eq. 1, drift is explicitly accounted by the \(\textbf{V}_{D}\)-term, and thus only the symmetric part of the diffusion tensor appears in the \({\textbf{K}}\)-term Moraal (2013). The \({\textbf{K}}^{S}\) tensor can be also split into parallel and perpendicular diffusion \(K_{\parallel }\) and \(K_{\perp }\), where we assume \(K_{\perp }= \xi K_{\parallel }\), with \(\xi \cong \,0.02\) Giacalone and Jokipii (1999). The corresponding mean free paths are \(\lambda _{\parallel }\) and \(\lambda _{\perp }\), respectively, such that \(K_{\parallel } = \beta c \lambda _{\parallel }/3\), where \(\beta =v/c\) is the particle speed. A large compilation of observational on the parallel mean free path in the \(\sim \) 0.5 MV - 5 GV rigidity range was reported in Palmer (1982). The mean free path, however, is rigidity and time dependent. From the condition of cyclotron resonance, the scattering of GCRs occurs when their Larmor radius \(r_{L}=r_{L}(R)\) is comparable with the typical size of the irregularities \(\hat{\lambda }\). From the condition \(r_{L} \sim \hat{\lambda }\), it turns out that GCRs with rigidity R resonate at wave number \(k_\mathrm{{res}} \sim 1/R\). The IMF irregularities follows a distribution of the type \(w(k) \propto k^{-\eta }\), which is the spectrum of interplanetary turbulence expressed in terms of wave numbers \(k=2\pi /\lambda \). An important parameter is the index \(\eta \), on which different regimes can be distinguished for the IMF power spectrum Kiyani et al. (2015). The resulting rigidity dependence of the diffusion mean free path (or coefficient) is \(\lambda _{\parallel } \sim R^{2-\eta }\). To account for different regimes in the IMF power spectrum Kiyani et al. (2015), the mean free paths are often parameterized as a double power-law function of the particle rigidity. For the parallel component, we have adopted the following description:

$$\begin{aligned} \lambda _{\parallel } =K_{0} \left( \frac{B_0}{B}\right) \left( \frac{R_0}{R}\right) ^{a} \times \left[ \frac{(R/R_0)^h + (R_k/R_0)^h }{1 + (R_k/R_0)^h} \right] ^{\frac{b-a}{h}}, \end{aligned}$$

where \(R_{0}\equiv \) 1 GV sets the rigidity scale, \(B_{0}\) is the local value of the IMF B at \(r_{0}=\) 1 AU, and the normalization factor \(K_{0}\) is given in units of \(10^{23}\) \(cm^{2}s^{-1}\). The spectral indices a and b set the slopes of the rigidity dependence of \(\lambda _{\parallel }\) below and above \(R_{k}\), respectively. The parameter h sets the smoothness of the transition. The perpendicular component follows from \(\lambda _{\perp }\equiv \xi \lambda _{\parallel }\), with the addition of small corrections in the polar regions Heber (1998).

2.2 The key parameters

In general, all parameters entering Eq. 3 might be time-dependent Manuel et al. (2014). We identify a minimal set of diffusion parameters as \({K_{0}, a, b}\). These parameters and their temporal dependence will be determined using time-resolved GCR proton data from AMS-02 and PAMELA Adriani (2013); Aguilar (2018a); Martucci (2018). Along with diffusion parameters, we define a minimal set of heliospheric parameters \(\{\alpha , B_{0}, A\}\) that describe the time-dependent conditions of the heliosphere in a given epoch: the HCS tilt angle \(\alpha \), the local IMF intensity \(B_{0}\), and its polarity A. Magnetic polarity is defined as the sign of the IMF in the outward (inward) direction from the Sun’s North (South) pole. To obtain the solution of Eq. 1 for a given GCR species, the LIS has to be specified as boundary condition. For GCR protons, our LIS model is obtained by Galactic propagation calculations and GCR flux data Feng et al. (2016); Tomassetti (2012, 2015a); Tomassetti et al. (2018). The data, used to constrain the GCR propagation model, are from the Voyager-1 spacecraft at \(\sim \) 100 -500 MeV of kinetic energy Cummings (2016), and from AMS-02 experiment at \(E\sim \) 100 GeV – 2 TeV Aguilar (2018a, 2015a, 2015b). Our proton LIS agrees fairly well with other recent models Boschini (2017); Corti (2019); Tomassetti et al. (2017); Tomassetti (2015b, 2017a, 2017b). A compilation of proton LIS models is shown in Fig. 1, along with the data from Voyager-1 and AMS-02.

Fig. 1
figure 1

Compilation of proton LIS models from various works: long-dashed red (Tomassetti et al. 2017), dot-dashed black (Corti 2019) dotted blue (Tomassetti 2017a), dotted green (Boschini 2017), dashed pink (Tomassetti et al. 2018), solid orange line (Tomassetti 2017b). Data are from Voyager-1 Cummings (2016) and AMS-02 (Aguilar 2015a)

2.3 The analysis

We determine the time-dependent GCR diffusion parameters by means of a statistical inference on the monthly measurements of the GCR proton fluxes reported by AMS-02 and PAMELA Adriani (2013); Aguilar (2018a); Martucci (2018). For every month, however, the heliospheric parameters have to be specified as well. They are evaluated from observations of the WSO observatory (\(\alpha \), A) and by in situ measurements of the ACE space probe (\(B_{0}\)). For a given epochs t, a backward moving average is calculated within a time window \([t-\Delta {T}, t]\), with \(\Delta {T}=6-12\) months. This ensures that the average values \(\hat{\alpha }\), \({\hat{A}}\), and \({\hat{B}}_{0}\) reflect the average IMF conditions sampled by GCRs arriving Earth at the epoch tFiandrini et al. (2021); Tomassetti et al. (2017). Hence, for each epoch, the diffusion parameters \(K_{0}\), a, and b can be determined with a global fit on the GCR proton measurements from AMS-02 and PAMELA. In practice, to make the fit, we have built a 6D parameter grid where each node corresponds to a parameter configuration \(\vec{\textbf{q}}=\) (\(\alpha \), \(B_0\), A, \(K_0\), a, b). The grid has 938,400 nodes. With the stochastic method, the GCR proton spectrum \(J_{m}(E, \vec{\textbf{q}})\) was calculated for each node of the grid at several values of kinetic energies between 20 MeV and 200 GeV. The simulation was highly CPU consuming. It required the simulation of 14 billion pseudo-particle trajectories, backward-propagated from Earth to the heliopause and then re-weighted according to their LIS. Once the proton grid was fully sampled, the parameters were determined as follows. From measured fluxes \(J_{d}(E,t)\) made at epoch t, the model calculation \(J(E,\vec{\textbf{q}})\) was evaluated as function of its parameters. The heliospheric parameters \({\hat{\alpha },\hat{B_{0}},{\hat{A}}}\) were kept fixed at their evaluation at epoch t. For a GCR flux measurements \(J_{m}(E,t)\), as function of energy E and observed at epoch t, the diffusion parameters are determined by the minimization of the function:

$$\begin{aligned} \chi ^{2}(K_{0},a,b) = \sum _{i} \frac{\left[ J_{d}(E_{i},t) - J_{m}(E_{i}, \vec{\textbf{q}}) \right] ^{2}}{\sigma ^{2}(E_{i},t)}, \end{aligned}$$

where the errors are given by \(\sigma ^{2}(E_{i},t) = \sigma _{d}^{2}(E_{i},t) + \sigma _{mod}^{2}(E_{i},t)\). The errors account for various contributions: experimental uncertainties in the data, theoretical uncertainties of the model, and uncertainties associated with the minimization procedure.

Fig. 2
figure 2

One-dimensional projections of the \(\chi ^{2}\) surfaces as function of the transport parameters \(K_{0}\), a, and b evaluated with CR proton flux data in two epochs: April 2014, corresponding to solar maximum (pink dashed lines, from AMS-02), and March 2009 corresponding to solar minimum (blue solid lines, from PAMELA)

The projections of the \(\chi ^{2}\) surfaces calculated for two flux measurements are illustrated in Fig. 2 as function of the GCR diffusion parameters \(K_{0}\), a, and b. The figure shows two distinct epoch of solar minimum (March 2009) and solar maximum (April 2014). The best-fit parameter is shown in each curve together with its uncertainty band. The data come from PAMELA (March 2009) and AMS-02 experiment (April 2014). It can be seen that the parameters \(K_{0}\) and b are tightly constrained by the AMS-02 data. The parameter a is sensitive to low-rigidity data and thus it is better constrained by PAMELA. In general AMS-02 gives larger \(\chi ^{2}\)-values in comparison with PAMELA, but the convergence of the fit is overall good.

3 Results and discussion

Along with the two epochs of Fig. 2, the fits have been performed for the whole time-series of CR proton flux measurements of AMS-02 and PAMELA. The AMS-02 time series consists in 79 proton fluxes measured on 27-day basis between May 2011 and May 2017. The PAMELA series are 83 proton fluxes measured on 27-day basis between June 2006 and January 2014. Their data, provided on monthly basis, cover large fractions of the solar cycles 23 and 24. With the least-square minimization described in Sect. 2.3, we obtained time-series of best-fit diffusion parameters \({K_{0}, a, b}\), along with their uncertainties. These results are shown in Fig. 3.

Fig. 3
figure 3

Best-fit results for the diffusion parameters \({K_{0}, a, b}\) obtained with the monthly flux measurements of CR protons made by PAMELA (blue triangles) and AMS-02 (pink circles). The greenish band indicates the magnetic reversal epoch. During this period, the IMF polarity is not well defined

The fit covers epochs of solar activity that include minimum, maximum, and IMF reversal. From the fit, we found that the diffusion parameters show a distinct time dependence that associated with solar activity (Fiandrini et al. 2021). The parameter \(K_{0}\) is in anti-correlation with the monthly sunspot number, which can be understood easily within the force field model, as the modulation parameter is \(\phi \propto 1/K_{0}\) (Tomassetti 2017a). Smaller \(K_{0}\) values imply slower diffusion and a more significant modulation effect, i.e., a stronger suppression of the low-energy GCR flux. In contrast, larger \(K_{0}\) values imply faster GCR diffusion, therefore causing a minor modification of the LIS. We found, for instance, a positive correlation between \(K_{0}(t)\) and the flux \(J(E_{0},t)\) evaluated at a given reference energy (Fiandrini et al. 2021). Our finding are in good agreement with other works (Corti 2019; Manuel et al. 2014; Tomassetti et al. 2017). In all these works, the GCR transport is dominated by parallel diffusion.

Fig. 4
figure 4

Envelope of the diffusion mean free paths \(\lambda _{\parallel }\) as function of GCR rigidity inferred in the examined period. The pink and blue bands correspond to the AMS-02 and PAMELA dataset, respectively. Note that the two bands are largely overlapped. The shaded green box corresponds to the so-called Palmer consensus on the mean value of \(\lambda _{\parallel }\) at rigidity below 5 GV(Palmer 1982)

We also find that the parameter b shows a remarkable time dependence, reflecting the connection between solar variability and the spectrum of magnetic turbulence in the inertial range. In this range, the associated spectral index evolves from \(\eta = 0.74\pm 0.08\) at solar minimum to \(\approx {1.3}\pm 0.15\) during the solar maximum. This shows that the IMF turbulence is subjected to variations Horbury et al. (2005); Vaisanen et al. (2019). Regarding the parameter a, we found a milder temporal dependence, i.e., a nearly constant spectral index of \(\eta =0.79\pm 0.13\). In both ranges, our findings are in agreement with direct measurements of the IMF power spectrum Kiyani et al. (2015).

Once the full time-series of the diffusion parameters \({K_{0}, a, b}\) is reconstructed, from their best-fit values of Fig. 3 it is possible to calculate the time- and rigidity-dependent diffusion mean free path \(\lambda _{\parallel }(t,R)\) using Eq. 3. The result is shown in Fig. 4, where we plot the envelope of all mean free paths as function of the GCR rigidity inferred in the examined periods from AMS-02 (pink circles) and PAMELA (blue triangles). The two bands are largely overlapped. The resulting mean free path for parallel diffusion in good accordance with the so-called Palmer consensus on the observations of \(\lambda _\parallel \), shown in the figure as green shaded box (Palmer 1982).