# A Markov Chain Monte Carlo technique to sample transport and source parameters of Galactic cosmic rays 

## II. Results for the diffusion model combining B/C and radioactive nuclei 

1 The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, SE- 10691 Stockholm, Sweden 2 lpsc 

Laboratoire de Physique Subatomique et de Cosmologie ( ), Universit´ e Joseph Fourier Grenoble 1, CNRS/IN2P3, Institut Polytechnique de Grenoble, 53 avenue des Martyrs, Grenoble, 38026, France 3 lpnhe 

Laboratoire de Physique Nucl´ eaire et des Hautes Energies ( ), Universit´ es Paris VI et Paris VII, CNRS/IN2P3, Tour 33, Jussieu, Paris, 75005, France 4 

Dept. of Physics and Astronomy, University of Leicester, Leicester, LE17RH, UK 5 iap 

Institut d’Astrophysique de Paris ( ), UMR7095 CNRS, Universit´ e Pierre et Marie Curie, 98 bis bd Arago, 75014 Paris, France 

Received / Accepted 

A. Putze 1 , $^{2}$, L. Derome$^{2}$, and D. Maurin3 , 4 , 5 

## ABSTRACT 

Context. Ongoing measurements of the cosmic radiation (nuclear, electronic, and γ -ray) are providing additional insight into cosmic-ray physics. A comprehensive picture of these data relies on an accurate determination of the transport and source parameters of propagation models. Aims. 

A Markov Chain Monte Carlo method is used to obtain these parameters in a diffusion model. By measuring the B/C ratio and radioactive cosmic-ray clocks, we calculate their probability density functions, placing special emphasis on the halo size L of the Galaxy and the local underdense bubble of size r h . We also derive the mean, best-fit model parameters and 68% confidence level for the various parameters, and the envelopes of other quantities. Methods. 

The analysis relies on the USINE code for propagation and on a Markov Chain Monte Carlo technique pre- viously developed by ourselves for the parameter determination. Results. +0 $^{.}$04 

The B/C analysis leads to a most probable diffusion slope δ =0 . 86 $_{−}$0 $^{.}$04 for diffusion, convection, and reacceler- ation, or δ = 0 . 234 +0 $^{.}$006 $_{−}$0 $^{.}$005 for diffusion and reacceleration. As found in previous studies, the B/C best-fit model favours the first configuration, hence pointing to a high value for δ . These results do not depend on L , and we provide simple functions to rescale the value of the transport parameters to any L . A combined fit on B/C and the isotopic ratios ( $^{10}$Be/$^{9}$Be, $^{26}$Al/$^{27}$Al, $^{36}$Cl/Cl) leads to L = 8 +8 $_{−}$7 kpc and r h = 120 +20 $_{−}$20 pc for the best-fit model. This value for r h is consistent with direct measurements of the local interstallar medium. For the model with diffusion and reacceleration, L = 4 +1 $_{−}$1 kpc and r h = 3 +70 $_{−}$3 pc (consistent with zero). We vary δ , because its value is still disputed. For the model with Galactic winds, we find that between δ = 0 . 2 and 0.9, L varies from O (0) to O (2) if r h is forced to be 0, but it otherwise varies from O (0) to O (1) (with r h ∼ 100 pc for all δ ≳ 0 . 3). The results from the elemental ratios Be/B, Al/Mg, and Cl/Ar do not allow independent checks of this picture because these data are not precise enough. Conclusions. 

We showed the potential and usefulness of the Markov Chain Monte Carlo technique in the analysis of cosmic-ray measurements in diffusion models. The size of the diffusive halo depends crucially on the value of the dif- fusion slope δ , and also on the presence/absence of the local underdensity damping effect on radioactive nuclei. More precise data from ongoing experiments are expected to clarify this issue. 

Key words. Methods: statistical – ISM: cosmic rays 

## 1. Introduction 

Almost a century after the discovery of cosmic ra- diation, the number of precision instruments devoted to Galactic cosmic ray (GCR) measurements in the GeV-TeV energy range is unprecedented. The GeV γ - ray diffuse emission is being measured by the Fermi satellite (The Fermi-LAT Collaboration 2009), while the TeV diffuse emission is within reach of ground ar- rays of Cerenkov Telescopes (e.g., Hess , Aharonian et al. 2006; Milagro , Abdo et al. 2008). The high-energy spec- 

trum of electrons and positrons uncovered some sur- prising and still debated features ( Atic , Chang et al. 2008; Fermi , Abdo et al. 2009; Hess , Aharonian et al. 2008, 2009; Pamela , Adriani et al. 2009a; Ppp-bets , Torii et al. 2008). For nuclei, many experiments (satellites and balloon-borne) have acquired data, that remain to be published ( Cream , Ahn et al. 2008; Tracer , Ave et al. 2008; Atic , Panov et al. 2008; Pamela ). Anti-protons are also being measured ( Pamela , Adriani et al. 2009b) and are targets for future satellite and balloon experiments ( Ams-02 , Bess -Polar). Anti-deuteron detection should be achieved in a few years ( Ams-02 , Choutko & Giovacchini 

