Non-parametric determination of H and He interstellar fluxes from cosmic-ray data

Abstract

Top-of-atmosphere (TOA) cosmic-ray (CR) fluxes from satellites and balloon-borne experiments are snapshots of the solar activity imprinted on the interstellar (IS) fluxes. Given a series of snapshots, the unknown IS flux shape and the level of modulation (for each snapshot) can be recovered. We wish (i) to provide the most accurate determination of the IS H and He fluxes from TOA data alone, (ii) to obtain the associated modulation levels (and uncertainties) while fully accounting for the correlations with the IS flux uncertainties, and (iii) to inspect whether the minimal force-field approximation is sufficient to explain all the data at hand. Using H and He TOA measurements, including the recent high-precision AMS, BESS-Polar, and PAMELA data, we performed a non-parametric fit of the IS fluxes $J^{\rm IS}_{\rm H,~He}$ and modulation level $\phi_i$ for each data-taking period. We relied on a Markov chain Monte Carlo (MCMC) engine to extract the probability density function and correlations (hence the credible intervals) of the sought parameters. Although H and He are the most abundant and best measured CR species, several datasets had to be excluded from the analysis because of inconsistencies with other measurements. From the subset of data passing our consistency cut, we provide ready-to-use best-fit and credible intervals for the H and He IS fluxes from MeV/n to PeV/n energy (with a relative precision in the range [2-10\%] at 1$\sigma$). Given the strong correlation between $J^{\rm IS}$ and $\phi_i$ parameters, the uncertainties on $J^{\rm IS}$ translate into $\Delta\phi\approx \pm 30$~MV (at 1$\sigma$) for all experiments. We also find that the presence of $^3$He in He data biases $\phi$ towards higher $\phi$ values by $\sim 30$~MV. The force-field approximation gives an excellent ($\chi^2/$dof$=1.02$) description of the recent high-precision TOA H and He fluxes.