SDE method for cosmic rays modulation in the heliosphere statistical error and solution uniqueness

The cosmic ray modulation in the heliosphere models comes to a new phase of their evolution, where the still higher number of models will be public with published source code. At the dawn of this period, we want to address two topics, the statistical error of their results and the uniqueness of their solutions.

We present a method for the evaluation of statistical error for the numerical stochastic differential equation method, which is probably the most used method to numerically solve Parker's equation. We defined a limit of statistical error, for which we present a method to estimate the number of particles needed to be simulated to reach this limit. The estimation of statistical error from a scan of parametric space of two currently available models with public code, for the SOLARPROP model and Geliosphere 2D model, is presented.

We present a test of the uniqueness of the solution of Parker's equation for 1D and 2D models of heliospheric modulation. Namely for the 1D B-p model and Geliosphere 2D model. The dependence of solution uniqueness on the selected model's input parameters is presented and discussed.