be observed for all radioactive ratios, independently of the model chosen. 

The resulting marginalised posterior PDFs of r$_{h}$ are given in Figs. 8 and 9 (lower right) for models II and III, respectively. The addition of an underdensity in the local interstellar medium is preferred by the data in the best-fit model III, but it is disfavoured in model II. The most prob- able values for r$_{h}$ range from 90 pc for the $^{36}$Cl/Cl ratio to 140 pc for the the $^{26}$Al/$^{27}$Al ratio, and the overall fit points to a most probable radius of 130 +10 − $_{20}$pc. 

These results confirm and extend the slightly differ- ent analysis of Donato et al. (2002), who found that for model III, the best-fit values for r$_{h}$ was ∼ 80 pc (see also Appendix B). 

## 5.1.3. Envelopes of 68% CL 

Confidence contours (for any combination of the CR fluxes) corresponding to given confidence levels (CL) in the χ 2 dis- tribution can be drawn, as detailed in Appendix A and Sect. 5.1.4 of Paper I. From the MCMC calculation based on the B/C + $^{10}$Be/$^{9}$Be + $^{26}$Al/$^{27}$Al + $^{36}$Cl/Cl constraint, we select all sets of parameters for which the χ 2 meets the 68% confidence level criterion. For each set of these pa- rameters, we calculate the B/C and the three isotopic ra- tios. We store for each energy the minimum and maximum value of the ratio. The corresponding contours (along with the best-fit ratio) for models II (standard DM, red) and III (standard and modified DM, blue) are drawn in Fig. 10. To ease the comparison with the data, all results correspond to IS quantities (the approximation made in the demodu- lation procedure, see Paper I, is negligible with respect to the experimental error bars). 

We see that the present data already constrain very well the various ratios for the standard DM. The difference be- tween the results of models II and III are more pronounced at high energy (effect of δ ), as seen from the B/C ratio be- yond 10 GeV/n. All contours are pinched around 10 GeV/n, which is a consequence of the energy chosen to renormalise the flux to the data in the propagation code. In princi- ple, the source abundance of each species may be set as an additional free parameter in the fit (Paper I), but at the cost of the computing time. The three isotopic ratios ( $^{10}$Be/$^{9}$Be$^{26}$, Al/$^{27}$Al, and $^{36}$Cl/Cl) provide a fair match to the data for all models, considering the large scatter and possible inconsistencies between the results quoted by vari- ous experiments. In particular, for $^{10}$Be/$^{9}$Be, new data are necessary to confirm the high value of the ratio measured at ∼ GeV/n energy. The envelope for the modified DM is quite large at 

high energy, because the uncertainty in r$_{h}$ is responsi- ble for a larger scatter in the other parameters. The two standard DM contain non-overlapping envelopes beyond GeV/n energies. This means that to disentangle the mod- els, having measurements of the above isotopic ratios in the 1 − 10 GeV/n may be more important than just having more and higher quality data at low energy. 

## 5.1.4. General dependence of L with δ (for r$_{h}$ = 0 ) 

To investigate the difference in the results obtained from models II and III, we fit B/C and the three isotopic ratios for different values of δ (a similar trend with L is obtained 

if just one isotopic ratio is selected). The analysis relies on the Minuit minimisation routine to quickly find the best-fit values, as described in Maurin et al. (2010). The evolution of the parameters with δ is shown on the left side of Fig. 11. The bottom panel shows the evolution of χ 2 $_{min}$/ d.o.f, where we recover that the best-fit δ for model II (dashed-blue line) lies around δ ≈ 0 . 2, whereas that for model III (solid- black line) lies around δ ≈ 0 . 8. As already underlined, the contribution to the χ 2 value is dominated by the B/C con- tribution because as discussed in Appendix C, the values of transport parameters that reproduce the B/C ratio are expected to remain within a narrow range. This explains what is observed in the various panels showing these com- binations. For model II, we emphasise that for δ ≳ 0 . 5, the best-fit value for V$_{a}$ is zero (Model II becomes a pure diffusion model). 

The most important result, given in the top panel, is for L as a function of δ , where any uncertainty in the determi- nation of δ translates into an uncertainty in the determina- tion of L . When a Galactic wind is considered (Model III, black-solid line), the correlation between L and δ is stronger than for model II (no wind). There is no straightforward explanation of this dependence. The flux of the radioactive isotope can be shown to be N $^{rad}$(0) ≈ $_{hq/}$√$_{Kγτ}$$_{0}$ (e.g., Maurin et al. 2006). Since secondary fluxes should match the data regardless of the value for δ , this implies that the ratio $^{10}$Be/$^{9}$Be depends only on √$_{Kγτ}$$_{0}$$_{. At the same time,}$ to ensure that the secondary-to-primary ratio is constant, we must maintain a constant L/K . The difficulty is that the former quantity is a constant at low rigidity where the iso- topic ratio is measured, whereas the latter quantity should remain as close a possible to the B/C data over the whole energy range. Hence, all we can say is that the variation in L with δ is related to the variation in K$_{0}$/L with δ , as shown in the second figure (left panel) of Fig. 11. 

We note that all the calculations in the paper are based on the W03 (Webber et al. 2003) fragmentation cross- sections. The impact of using the W03 set or the GAL09 set (provided in the widely used GALPROP package $^{2}$) on the determination of the halo size L is shown as thin-dotted lines (left panel, same figure) $^{3}$. Any difference existing be- tween these two sets of production cross-sections has no impact on the best-fit value for L : thin-dashed curves (ob- tained with GAL09 cross-sections) almost match the thick- solid curves (obtained with W03 cross-sections). For other ratios, the effect of the GAL09 cross-sections is always the same, so it is not discussed further. 

## 5.1.5. General dependence of L with δ (for r$_{h}$ ̸ = 0 ) 

We repeat the analysis with the underdensity r$_{h}$ as an ad- ditional free parameter. The dependence of L and r$_{h}$ on the diffusion slope δ is shown in the right panel of Fig. 11. A comparison between the left and the right panel shows that the combinations of parameters K$_{0}$/L , V$_{a}$$_{/}$√$_{K}$$_{0}$$_{, and}$ V$_{c}$ are almost unaffected by the presence of a local bubble; the χ $^{2}$/ d.o.f is also only slightly affected. 

For δ ≲ 0 . 2, r$_{h}$ is consistent with 0 for both model II (diffusion/reacceleration) and model III (diffu- 

