Table 6. Best-fit model parameters on B/C data: 1D versus 2D DM ( L = 4 kpc). Table 7. Most probable values for models II and III for the free parameters of the local bubble radius r$_{h}$ and/or the Galactic halo size L (constrained by B/C and $^{10}$Be/$^{9}$Be data). from the relationships between DMs and the LBM (e.g., Maurin et al. 2006). From the B/C analysis point of view, it implies that even if we are unable to reach conclusions about the value of δ (see Maurin et al. 2010), once this value is known, all other transport parameters are well con- strained. The conclusions obtained for the 1D DM naturally hold for the 2D DM. We recall that the main difference between the 1D and 2D geometry is that i) the spatial distribution of sources, which was constant in 1D, is now q ( r ); and ii) the Galaxy has a side-boundary at a radius taken to be R = 20 kpc. As a check, we first used the 2D solution (presented in Appendix A.2) with R = 20 kpc, but set q ( r ) to be constant. The best-fit parameters were in agreement with those obtained from the 1D solution. We present in Table 6 the best-fit parameters for models II and III for L = 4 kpc in the 2D solution where q ( r ) follows the SN remnant distribution of Case & Bhattacharya (1998). The values for the 1D solution are also reported for the sake of comparison. The main difference is in the value of K$_{0}$ , which varies by ∼ 10% and also affects V$_{a}$ (by means of the ratio V$_{a}$$_{/}$√$_{K}$$_{0}$$_{, which is left unaffected). This is consistent}$ with the variations found by Maurin et al. (2002). We now attempt to lift the degeneracy between the halo size and the normalisation of the diffusion coefficient, using radioactive nuclei. The questions that we wish to address are the following: i) With existing data, how large are the uncertainties in L for a given model? ii) Do radioactive nu- clei provide different answers for models with different δ ? iii) Is the mean value (and uncertainty) for L obtained from a given isotopic/elemental ratio consistent with or stronger constrained than that obtained from another measured iso- topic/elemental ratio? iv) How does the presence of a local underdense bubble (modelled as a hole of radius r$_{h}$ , see Sect. 2.3) affect the conclusions? Until now, almost all studies have focused on the isotopic ratios of $^{10}$Be/$^{9}$Be, $^{26}$Al/$^{27}$Al, $^{36}$Cl/Cl, and $^{54}$Mn/Mn. An alternative, discussed in Webber & Soutoul (1998), is to consider the Be/B, Al/Mg, Cl/Ar, and Mn/Fe ratios. The advantage of considering these elemental ratios is that they are easier to measure than isotopic ratios, and thus provide a wider energy range to which we can fit the data. Taking ratios such as Be/B maximises the effect of radioactive decay, since the numerator represents the de- caying nucleus and the denominator the decayed nucleus. However, the radioactive contribution is only a fraction of the elemental flux, and HEAO-3 data were found to be less The marginalised posterior PDFs of L and r$_{h}$ and the cor- relations between these new free parameters and the prop- agation parameters of models II and III are given in the Figs. 6 and 7, respectively. The most probable values of the parameters are gathered in Table 7. For all configurations, the diffusion slope δ and the Galactic wind V$_{c}$ are unaffected by the addition of the free parameters L and r$_{h}$ . The B/C fit is degenerate in K$_{0}$/L and V$_{a}$$_{/}$√$_{K}$$_{, so that the values of}$ K$_{0}$ and V$_{a}$ vary as L varies. For model III, their evolution follows the relations given in Fig. 5. This implies that there is a positive cor- relation between K$_{0}$ and V$_{a}$ , and K$_{0}$ and L , as seen from Figs. 6 and 7. The uncertainty in the diffusive halo size L is smaller for II than for III. This is a consequence of the inclusion of the constant wind, which decreases the resolu- tion on K$_{0}$ from 2% (Model II) to 10% (Model III)—see e.g., Table 2 or 7—hence broadening the distribution of L . Below, the results for the standard DM—for which r$_{h}$ is set to be 0—and those for the modified DM—for which r$_{h}$ is left as an additional free parameter—are discussed separately. This allows us to emphasise the impact of r$_{h}$ constraining that the isotopic ratios in Webber & Soutoul (1998). Below, we consider and compare the constraints from both the isotopic ratios and the elemental ratios. The data used are described in Appendix D.2. We discard $^{54}$Mn because it suffers more uncertainties than the oth- ers in the calculation (and also experimentally) due to the electron capture decay channel. The free parameters for which we seek the PDF are the four transport parameters { K$_{0}$, δ, V$_{c}$, V$_{a}$ } , plus one { L } or two geometrical param- eters { L, r$_{h}$ } , depending on the configuration considered. The main results of this section are thus in identifying the PDF of L for the standard DM, and the PDFs of both L and r$_{h}$ for the modified DM. We start with a simultaneous fit to B/C and $^{10}$Be/$^{9}$Be, for both model III (diffusion/convection/reacceleration), and model II (diffusion/reacceleration), the latter being fre- quently used in the literature. 5.1. PDFs of L and r$_{h}$ using isotopic measurements 5.1.1. Simultaneous fit to B/C and $^{10}$Be/$^{9}$Be 5. Results for radioactive species (free halo size L ) 9 Model K best 0 × 10 2 δ best V best c V best a χ $^{2}$/d.o.f Data (kpc $^{2}$Myr $^{$_{−}$1}$) (km s $^{$_{−}$1}$) (km s $^{$_{−}$1}$) 1D II-F 9.74 0.23 . . . 73.1 4.68 2D II-F 8.56 0.24 . . . 68.6 4.67 1D III-F 0.48 0.86 18.9 38.0 1.47 2D III-F 0.42 0.86 18.7 35.5 1.46 K 0 × 10 2 δ V c V a L r h (kpc $^{2}$Myr $^{$_{−}$1}$) (km s $^{$_{−}$1}$) (km s$^{$_{−}$1}$) (kpc) (pc) II 8 . 6 +0 $^{.}$2 $_{−}$0 $^{.}$2 0 . 239 +0 $^{.}$005 $_{−}$0 $^{.}$007 · · · 69 +2 $_{−}$2 [4] · · · II 13 +2 $_{−}$2 0 . 234 +0 $^{.}$006 $_{−}$0 $^{.}$005 · · · 84 +5 $_{−}$6 5 . 2 +0 $^{.}$7 $_{−}$0 $^{.}$6 · · · II 11 +2 $_{−}$3 0 . 235 +0 $^{.}$008 $_{−}$0 $^{.}$004 · · · 77 +8 $_{−}$11 4 +1 $_{−}$1 3 +70 $_{−}$3 III 0 . 41 +0 $^{.}$04 $_{−}$0 $^{.}$07 0 . 86 +0 $^{.}$04 $_{−}$0 $^{.}$03 18 . 7 +0 $^{.}$4 $_{−}$0 $^{.}$3 35 +2 $_{−}$2 [4] · · · III 6 . 1 +0 $^{.}$8 $_{−}$0 $^{.}$8 0 . 86 +0 $^{.}$03 $_{−}$0 $^{.}$05 19 . 4 +0 $^{.}$4 $_{−}$0 $^{.}$3 135 +11 $_{−}$10 46 +9 $_{−}$8 · · · III 0 . 8 +1 $_{−}$0 $^{.}$7 0 . 86 +0 $^{.}$03 $_{−}$0 $^{.}$04 18 . 7 +0 $^{.}$5 $_{−}$0 $^{.}$4 55 +31 $_{−}$21 8 +8 $_{−}$7 120 +20 $_{−}$20 A. Putze et al.: An MCMC technique to sample transport and source parameters of Galactic cosmic rays. II.