A. Putze et al.: An MCMC technique to sample transport and source parameters of Galactic cosmic rays. II. Fig. 1. Sketch of the model: sources and interactions (in- cluding energy losses and gains) are restricted to the thin disc ∝ 2 hδ ( r ). Diffusion K and convection V$_{c}$ transport nu- clei in both the disc (half-height h ) and the halo (half-height L ). The Galaxy radial extension is R . The local bubble is featured to be a cavity of radius r$_{h}$ in the disc devoid of gas. where V$_{a}$ is the Alfv´ enic speed in the medium. The source term S j is a combination of i) primary sources q $^{j}$( r , E ) of CRs (e.g., super- novae), ii) secondary fragmentation-induced sources ∑ m$_{k}$>m$_{j}$ k n$_{ISM}$ ( r ) vσ $^{k}$→j frag ( E ) N $^{k}$( r , E ), and iii) secondary decay-induced sources ∑ k N $^{k}$( r , E ) / ( γτ $^{k}$→j 0 ). In partic- ular, the secondary contributions link one species to all heavier nuclei, coupling together the n equations. However, the matrix is triangular and one possible approach is to solve the equation starting from the heavier nucleus (which is always assumed to be a primary). The Galaxy is modelled to be a thin disc of half-thickness h , which contains the gas and the sources of CRs. This disc is embedded in a cylindrical diffusive halo of half-thickness L , where the gas density is assumed to be 0. CRs diffuse into both the disc and the halo independently of their position. A constant wind V$_{c}$ perpendicular to the Galactic plane is also considered. This is summarised in Fig. 1 (see next section for the definition of r$_{h}$ ). We use the δ ( z ) approximation introduced in Jones (1979), Ptuskin & Soutoul (1990), and Webber et al. (1992). Considering the radial extension R of the Galaxy to be either infinite or finite leads to the 1D version or 2D version of the model, respectively. The corresponding sets of equations (and their solutions) obtained after these sim- plifications are presented in Appendix A. These assump- tions allow for semi-analytical solutions of the problem, as the interactions (destruction, spallations, energy gain and losses) are restricted to the thin disc. The gain is in the computing time, which is a prerequisite for the use of the MCMC technique, where several tens of thousands of models are calculated. These semi-analytical models re- produce all salient features of full numerical approaches (e.g., Strong & Moskalenko 1998), and they are useful for systematically studying the dependence on key parame- ters, or some systematics of the parameter determination (Maurin et al. 2010). We note that most of the results of the paper are based on the 1D geometry (solutions only depend on z ), which is less time-consuming than the 2D one in terms of computing time $^{1}$. The parameter degeneracy is also more easily extracted and understood in this case (Jones et al. 2001; Maurin et al. 2006). Nevertheless, the results for the 2D geometry are also reported, as it has been used in a series of studies inspecting stable nuclei (Maurin et al. 2001, 2002), β -radioactive nuclei (Donato et al. 2002), standard anti-nuclei (Donato et al. 2001, 2008, 2009) and positrons (Delahaye et al. 2008a). It has also been used to set constraints on dark matter annihilations in anti- nuclei (Donato et al. 2004), and positrons (Delahaye et al. 2008b). The reader is referred to these papers, and espe- cially Maurin et al. (2001) for more details and references about the 2D case. Our model does not take into account all the observed ir- regularities of the gas distribution, such as holes, chimneys, shell-like structures, and disc flaring. The main reason is that as far as stable nuclei are concerned, only the average grammage crossed is relevant when predicting their flux (which motivates LBM). As such, the thin-disc approxima- tion is a good trade-off between having a realistic descrip- tion of the structure of the Galaxy and simplicity. However, the local distribution of gas affects the flux calculation of radioactive species (Ptuskin et al. 1997; Ptuskin & Soutoul 1998; Donato et al. 2002). We consider a radioactive nucleus that diffuses in an unbound volume and decays with a rate 1 / ( γτ$_{0}$ ). In spherical coordinates, appropriate to describe this situation, the diffusion equa- tion reads The solution for the propagator G (the flux is measured at r = 0 for simplicity) is Secondary radioactive species, such as $^{10}$Be, originate from the spallations of the CR protons (and He) with the ISM. We model the source term to be a thin gaseous disc, except in a circular region of radius r$_{h}$ at the origin. In the δ ( z ) approximation (see Fig. 1) and in cylindrical coordinates, where Θ is the Heaviside function. The flux of a radioac- tive species is thus given by (we rewrite the propagator in cylindrical coordinates) The ratio of the flux calculated for a cavity/hole r$_{h}$ to that of the flux without hole ( r$_{h}$ = 0) is 1 The 2D solution is based on a Bessel expan- sion/resummation [see Eq. (A.14)]. For each Bessel order, an equation similar to that for the 1D geometry needs to be solved. Nine Bessel orders are in many cases enough to ensure convergence (Maurin et al. 2001), but at least 100 orders are required in the general case, which multiply the computing time by roughly the same amount. N$_{r}$$_{h}$ N$_{r}$$_{h}$$_{=0}$ = exp ( − r$_{h}$ √$_{Kγτ}$$_{0}$ ) = exp ( − r$_{h}$ l$_{rad}$ ) . (10) N ( r = z =0) ∝ ∫ ∞ 0 ∫ $^{+}$∞ −∞ G ( √ r ′ $^{2}$+ z ′ $^{2}$) Q ( r ′ , z ′ ) r ′ dr ′ dz ′ . (9) Q ( r, z ) ∝ Θ( r − r$_{h}$ ) δ ( z ) , (8) G ( r $^{′}$) ∝ e − r ′ $_{/}$√$_{Kγτ}$ 0 r ′ . (7) − K △ r G + G γτ$_{0}$ = δ ( r ) . (6) 2.3. Radioactive species and the local bubble 2.2. Geometry of the Galaxy and simplifying assumptions 3 c V c V c V constant wind Convective K(E) Diffusive Halo L 2h r$_{h}$ z R (z=r=0)