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The 2D Heliosphere Modulation Monte Carlo Code :: HelMod

- cyan - diffusion : Small Scale Magnetic Field irregularities effect
- blue - convection : Presence of the solar wind moving out from the Sun
- red - drift effect
- green - energetic loss : Due to adiabatic expansion of the solar wind

where U is the number density of cosmic rays per unit interval of particle kinetic energy T (the so-called differential density). K is symmetric part of the diffusion tensor, V_SW is solar wind speed and drift velocity v_D is determined by the antisymmetric part of the diffusion tensor.

Stochastic Differential Equations (SDE)

As implemented in the HelMod code version 1.5, the current approach exploits a Monte Carlo technique to determine the number density U using the set of the approximated stochastic differential equations (SDEs) for a 2-D approximation (radial distance and co-latitude).

For
a) an IMF described by the standard Parker field
b) both solar wind and drift velocity in the region of WHCS radially directed (e.g., Vsw,r = Vsw and vHCS,r = vHCS ), the SDEs approximated in terms of the increments ∆r, ∆μ(θ), ∆T and ∆t [with μ(θ) ≡ cos(θ)] are





Heliosphere is divided to 15 regions, each one equivalent to the average of solar activity in periods before the experiment. Parameters in each region are
  • Diffusion parameter
  • Tilt angle of the Neutral Sheet
  • Magnetic Field Magnitude at Earth
  • Solar Wind Speed


The effective heliosphere is divided in 15 regions, each one referred from 1 to 14 solar rotation before the simulated period (upper panel). Example of heliosphere division for June 1998 (bottom panel).

Used Heliospheric Magnetic Field (HMF) - Parker field + Jokipii & Kota, 1989; Langner, 2004

Details on HelMod modulation code, and how to compute the SDE, could be found in [Bobik et al. Ap.J. 2012, 745:132].



Input parameters

Used input parameters in HelMod 1.5.






Results

Next figures show differential intensity determined with the HelMod code (continuous line) compared to the experimental data of AMS-1998; BESS-1997,1998,1999,2000 and 2002, the dashed line is the Local Interstellar Spectrum (LIS).