A. Putze et al.: An MCMC technique to sample transport and source parameters of Galactic cosmic rays. II. studies (Moskalenko et al. 2001)—the posterior PDF of L extends from 25 to 85 kpc for model III (most probable value at 46 kpc). In terms of statistics  the best-fit model is stil model III  for which the χ $^{2}$/d.o.f is 1.41. Modified DM ( r$_{h}$ = 0 ): the presence of a local bubble re- sults in an exponential attenuation of the local radioactive flux  see Sect. 2.3 and Eq. (10). We thus expect to have a different best-fit parameter for L in that case. The re- sulting posterior PDFs of L and r$_{h}$ and the correlations to the propagation parameters for this modified DM are given in Figs. 6 and 7 for models II and III respectively. The most probable values are gathered in Table 7 (third and last lines). As expected  the local bubble radius r$_{h}$ is negatively correlated with the Galactic halo size L . The effect is more striking for model III  where the favoured range for L ex- tends from 1 to 50 kpc. The most probable value is 8 +8 − $_{7}$kpc for a local bubble radius r$_{h}$ = 120 +20 − $_{20}$pc. The χ $^{2}$/d.o.f of this configuration is 1.28  instead of 1.41 for the standard DM. The improvement to the fit is statistically significant according to the Fisher criterion. The situation for model II is different. The halo size L is already small for the standard configuration r$_{h}$ = 0. Adding the local bubble radius r$_{h}$ to the fit decreases the most probable value of L only slightly to 4 +1 − $_{1}$kpc and the measured value of r$_{h}$ is compatible with 0 pc. In addition  the χ $^{2}$/d.o.f is 3.69 and hence poorer than for the configu- ration without the local bubble feature. In this model (dif- fusion/reacceleration  no convection)  a local underdensity is not supported. We repeat the analysis for the remaining isotopic ratios. The resulting marginalised posterior PDFs of the Galactic halo size L and the local underdensity r$_{h}$ are given in Figs. 8 and 9 for models II and III  respectively. The correlation plots with the transport parameters are similar to those of Figs. 6 and 7 and are not repeated. Standard DM ( r$_{h}$ = 0 ): as for the $^{10}$Be/$^{9}$Be ratio (red- dotted line)  L is well constrained in model II at small val- ues for the $^{26}$Al/$^{27}$Al (green-long dashed-dotted line)  and $^{36}$Cl/Cl (blue dashed-dotted line) ratios  covering slightly different but consistent ranges from 4 to 14 kpc. The width of the estimated PDFs increases when moving from the $^{10}$Be/$^{9}$Be ratio to the $^{36}$Cl/Cl ratio  due to the decreasing accuracy of the data. In the same way  the adjustment to the data becomes poorer  as expressed by the increase in χ $^{2}$/d.o.f from 3.59 to 4.09. Used alone  the radioactive ratio $^{10}$Be/$^{9}$Be constrains the most precisely the halo size L   but the constraints obtained with the other radioactive ratios are completely compatible within the 2 σ range. The most likely value of L is ascertained when the three radioactive ratios are fitted simultaneously (black solid line). The best-fit model is model III  where the overall cov- ered halo size range extends from 20 to 140 kpc. The most probable value found for L with 68% confidence level (CL) errors is 62 +7 − $_{10}$kpc. Modified DM ( r$_{h}$ = 0 ): the resulting marginalised poste- rior PDFs of L are shown in Figs. 8 and 9 (lower left) for models II and III  respectively. Again  the extracted PDFs for all radioactive ratios are completely compatible for both models. As described above  the decrease in L is more pro- nounced for model III than for model II. This decrease can Fig. 9. Same as in Fig. 8  but for model III. Fig. 8. Model II: marginalised posterior PDFs of the Galactic geometry parameters for the standard DM ( r$_{h}$ = 0  top panel) and for the modified DM ( r$_{h}$ = 0  bot- tom panels). The four curves result from the combined analysis of B/C plus isotopic ratios of radioactive species: B/C+ $^{10}$Be/$^{9}$Be (red dotted line)  B/C+$^{26}$Al/$^{27}$Al (green long dashed-dotted line)  and B/C+ $^{36}$Cl/Cl (blue dashed- dotted line). The black solid curve represents the extracted PDF resulting from a simultaneous fit of B/C plus all three isotopic ratios. All PDFs are smoothed. 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 L [kpc] Be 9 Be/ 10 Al 27 Al/ 26 Cl/Cl 36 Cl/Cl 36 Al +  27 Al/ 26 Be +  9 Be/ 10 Model II  = 0 h r 2 4 6 8 10 12 14 16 18 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 L [kpc] Model II  0 h r 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 2 4 6 8 10 12 14 16  [kpc] h r Be 9 Be/ 10 Al 27 Al/ 26 Cl/Cl 36 Cl/Cl 36 Al +  27 Al/ 26 Be +  9 Be/ 10 40 60 80 100 120 140 160 0 0.01 0.02 0.03 0.04 0.05 L [kpc] Be 9 Be/ 10 Al 27 Al/ 26 Cl/Cl 36 Cl/Cl 36 Al +  27 Al/ 26 Be +  9 Be/ 10 Model III  = 0 h r 0 20 40 60 80 100 120 0.01 0.02 0.03 0.04 0.05 0.06 L [kpc] Be 9 Be/ 10 Al 27 Al/ 26 Cl/Cl 36 Cl/Cl 36 Al +  27 Al/ 26 Be +  9 Be/ 10 Model III  0 h r 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 5 10 15 20 25  [kpc] h r 11 5.1.2. Results and comparison with fits to $^{26}$Al/$^{27}$Al and $^{36}$Cl/Cl