The Cosmic Rays (CRs) propagation inside the heliosphere is a very complex physical process, well described by the transport equation presented by Parker, in 1965. The heliosphere is a spheroid shaped structure centered around the Sun, moving across the Local Interstellar Medium (LISM). While travelling from the outside interstellar medium through the inner heliosphere up to the Earth, the particles undergo convection, diffusion and energy changes as they move in the irregular magnetic fields originated from the Sun. Interplanetary Magnetic Field (IMF) fluctuations are due to the outward flowing Solar Wind (SW) plasma, which encapsulates the IMF within the boundary of the heliosphere, called Heliopause (HP). At the so called Termination Shock (TS) the SW speed drops from supersonic to subsonic. The extremely turbulent region from the TS to the HP is called Heliosheath (HS), which, in particular for low energy CRs, is responsible for a significant part of the modulation.

Voyager 1 probe has provided for the first time in situ observations of CRs flux along the full extension of the heliosphere. The energy modulated spectra for Galactic Cosmic Rays (GCRs) such as H, He, C, O, Mg and Si ions are computed and compared with the Voyager 1 data for a fixed date, taking into account the main isotopes for each species. The TS and the HP crossings for Voyager 1 are respectively on 16 December 2004 and on 25 August 2012. Therefore, in order to fully exploit the Voyager 1 data the simulations are performed during the time period 2000 - 2016. Thus, the study of the diffusion inside the HS region is brought on by the means of GCRs time spectra.

The Parker transport equation describing the convection, diffusion and adiabatic loss processes that CRs are subjected to inside the heliosphere is a Fokker Planck type equa- tion, which can be solved by implementing a Monte Carlo approach with a set of Stochastic Differential Equations (SDEs). HelMod is a code which implements such a method in a backward in time approach, meaning the injected particles are considered to travel from the Earth up to the HP with the SW flowing inward. The code returns the modulated energy spectrum of the particles inside the heliosphere, starting from the LISM outside the heliopause.

The purpose of the work is to study the diffusion coefficient behaviour inside the heliosheath, where the problem is treated in a one dimensional way because of the com- plexity of the region. This simplification is not applied in the inner heliosphere, where the physics is described by HelMod in a two dimensional approach. In order to derive the diffusion coefficient k functional form in this region, time spectra in a ∼ 0.5 - 2 GV rigidity range for the chosen isotopes are examined. To compute the time modulated spectra the coefficient is supposed to reproduce a heuristic model having a rigidity (P) dependence of the form ∼ Pγ, where γ varies from 0.1 to 3.6 throughout the work. Six species are taken into account together for the study of temporal spectra in order to better constraint the chosen diffusion coefficient functional form.

Two models well represent the final result for the diffusion coefficient behaviour: a power law with γ = 1.4 and a linear model (γ = 1), respectively. By performing some tests on the two possible results, the power law is slightly preferred.A brief comparison with the theoretical models proposed in the literature is also presented.

Organized by: Edoardo Giangrandi