## GALPROP obtained LIS using modulated spectra from HelMod

The local interstellar spectra (LIS) are input cosmic ray intensities for any modulation models. Fluxes are assumed isotropically distributed at heliosphere boundary, in a steady-state configuration. The LIS is usually computed using numerical model accounting for the propagation in galaxy. The heliosphere modulation is the added using, e.g.,the modulation module available here for on line use or download.

Recently, Della Torre et al. 2017, Boschini et al 2017b deduced LIS's for protons, helium and antiprotons using the most recent experimental results combined with GALPROP model for propagation in galaxy and HelMod for the propagation in heliosphere.

HelMod and GALPROP were combined to provide a single framework and run to reproduce a comprehensive set of observations of CR species collected in different time periods, from 1997 up to 2015.

The numerical table of computed LIS is available as supplementary material of Boschini et al 2017b.

In Della Torre et al. 2017, Boschini et al 2017a,b authors proposed an analytical expression for proton and helium nuclei LIS's as a function of the rigidity:

\begin{equation}

J_{LIS}(P)\times P^{2.7} =\left\{

\begin{array}{ll}

\sum_{i=0}^5 a_i P^{i}, &P\le1\ {\rm GV},\\

b + \frac{c}{P} + \frac{d_1}{d_2+P} + \frac{e_1}{e_2+P} + \frac{f_1}{f_2+P} + g P, & P>1 \ {\rm GV},

\end{array}

\right.

\end{equation}

where \(a_i, b, c, d_i, e_i, f_i, g\) are the numerical coefficients summarized as follow:

The accuracy of the low-energy expression is 2% in the range \(0.2 \) GV\( < R < 1\) GV for the proton LIS. The high-energy part reproduces the numerical proton LIS calculated with GALPROP with an accuracy of \(\sim\)9\% for \(0.45\) GV \( < R < 1\) GV (i.e., \(E_{\rm kin}>0.11\) GeV), where the constraints from Voyager 1 are wider than 10%, and of \(2\)% for \(R>1\) GV. The discrepancies with respect to AMS-02 data at higher energies expressed in standard deviations are virtually zero, with \(0.5\sigma\) around \(1-2\) GV at most. In the case of Helium, the low-energy expression is valid in the range of \(0.15\ {\rm GV} < R < 2 \) GV, i.e., approximately between 3 MeV/nucleon and 450 MeV/nucleon. At higher rigidities \(1.5\) GV\( < R < 2\times 10^{4}\) GV (i.e., \(>0.3\) GeV/nucleon), it reproduces the He LIS calculated with GALPROP with an accuracy of 2%.

The derived expressions are virtually identical, to \(<1-2\)%, to numerical solutions in over 5 orders of magnitude energy interval including the spectral flattening at high energies, and are based on Voyager 1, AMS-02, and CREAM-I data.

In addition to the Proton and Helium nuclei LIS, Boschini et al 2017a,b provides analytical fits to the derived LIS. The fit to the antiproton LIS provides an accuracy of 2-3% for 3 GV \(<R<1000\) GV and 10% for 1.5 GV \(<R<3\) GV:

\begin{equation}

F(R)\times R^{2.7} = -\frac{168}{31.1+R^{2}} +\frac{13600}{22200+R^{2}}, \quad R>1\ {\rm GV},

\end{equation}

while the average accuracy of AMS-02 data is about 10%-20%.