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

Abstract

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. Program summaryProgram Title: CudaHelioCommanderLightCPC Library link to program files:https://doi.org/10.17632/cptpkwphn4.1Developer's repository link:https://github.com/gladiatus55/CudaHelioCommanderLightLicensing provisions: GPLv3Programming language: C#, .NET 6.0Nature of problem: Despite the decades-long history of cosmic ray modulation in the heliosphere models, the topics of the statistical error of their results, and the uniqueness of their solutions were in published articles addressed only in a very limited way. A precise description of the method for statistical error evaluation, with examples applied to available models with published source code, is needed. The uniqueness of the solution of Parker's equation was not addressed in published literature.Solution method: The estimation of statistical error was evaluated for two currently available models with public code, the SOLARPROP model and Geliosphere 2D model, by a scan of their parametric space. The dependence of solution uniqueness on the selected model's input parameters was also done by scanning their parametric space.