content_image/413826.jpg
this model so is excluded. For Model II, almost any value of δ can be reached. 

6.3. Summary for the diffusion coefficient effect 

Similars to the effect of the source parameter η$_{S}$ , varying the parameter η$_{T}$ on a wide range i) does not to allow δ ≲ 1 / 2 to reached for Model III, and ii) strongly affects δ for Model II. For the latter, the best-fit η$_{T}$ leads to δ ≈ 0 . 4, slightly more than the Kolmogorov spectrum of turbulence. 

From a statistical point of view, Model III is again pre- ferred, except in the region − 3 ≲ η$_{T}$ ≲ − 1 . 5 where V$_{c}$ drops to zero, so that it is equivalent to Model II. For η$_{T}$ ≲ − 2 . 75, V$_{a}$ also drops to zero, so that all models are close to the purely diffusive case (Model 0), which favours δ ∼ 1 / 2. As before, one of the interesting features of Model II (reac- celeration only) is its versatility. As the range of allowed η$_{T}$ remains unspecified to some extent, Model II does not point to any specific value of δ . It can accommodate values as low as 1 / 3, going through 1 / 2 and even higher values, if a sharp turn-off (negative values of η$_{T}$ ) exists in the diffusion coefficient (e.g., Ptuskin et al. 2006). 

7. Influence of the production cross-sections 

Many reaction channels are required to calculate the pro- duction of secondary species. Semi-analytical formulae, semi-empirical approaches, or even fit to the data are cur- rently used in the literature to obtain the full set of pro- duction cross-section in CR physics. We show first the dif- ferent best-fit values obtained for different sets available in the literature. We then inspect what would be the effect of energy-biased cross-sections on the transport parameter derivation. 

7.1. Using different sets of fragmentation cross-sections 

We use four different sets of cross-sections. Among them, the most up-to-date are W03 (Webber et al. 2003) and GAL09 (from the GALPROP code $^{1}$). The former is based on the semi-empirical approach of Webber and coworkers, initiated in the 90’s (Webber et al. 1990). The latter set takes advantage of the former approach, renormalising some cross-sections to selected data. The two others, WKS98 (Webber et al. 1998) and S01 (private communication of Aim´ e Soutoul, 2001), are a bit outdated, but they serve as an illustration here. 

The effect of these different sets on the parameter de- termination is shown in Table 7. The scatter in the values of the different parameters is not the same depending on the model considered. WKS98 requires less convection than any other set, and GAL09 needs more reacceleration than the others. Otherwise, it is difficult to find clear trends. For instance, the W03 and GAL09 sets give very similar results for Model I, but not as similar as for Models II and III. 

If we discard the S01 set (which is based on an un- published preliminary analysis), we can conclude that the SystUnc on the transport parameters related to the choice of the production cross-section is ≲ 20%. This figure is sim- ilar to the uncertainty quoted for the parameterisation of the cross-section themselves ( ∼ 10 − 20%). 

D. Maurin et al.: Systematic uncertainties on the cosmic-ray transport parameters 

<img src='content_image/413826.jpg'>

Note 7. Models 0, I, II, and III for L = 4 kpc. The four base sets of cross-sections are WKS98 from Webber et al. (1998), S01 from a pri- vate communication of Aim´e Soutoul (2001), W03 from Webber et al. (2003), and GAL09 from the GALPROP code v50.1p (2009). 

7.2. Influence of a systematic energy bias in W03 cross-sections 

A striking result of the updated cross-section formula- tion of W03 (Webber et al. 2003) compared to WKS98 (Webber et al. 1998) was a systematic energy bias (smaller cross-sections at higher energy). To address the effect of a possible residual bias, we allow for a systematic energy bias in all production cross-sections, either at low energy (LE) or at high energy (HE). Biased sets are obtained from the reference set W03 by applying a factor ( E$_{k}$$_{/}$$_{n}$/ 10) − x below 10 GeV/n (LE bias), and ( E$_{k}$$_{/}$$_{n}$ ) x above 1 GeV/n (HE bias, but left constant above 10 GeV/n). This is illustrated for the $^{12}$C → ( $^{10}$B+$^{11}$B) cross-section, as shown in Fig. 3. As seen from this figure and the experimental data, biases x as large as 0 . 05 are marginally consistent with the data at low energy, but are acceptable at high energy. 

In Table 8, we only report the results of the high-energy biases on the best-fit values of the transport parameters. We checked that the result of a low-energy or a high-energy bias gives the same trends. Quite naturally, there is a significant correlation of the bias with the transport parameters. We note that if δ increases, all the other transport parameters K$_{0}$ , V$_{a}$ , and V$_{c}$ decrease. In principle, we would expect that δ increases if the bias is positive: larger cross-sections at high-energy produce more secondaries at high-energy, re- quiring a larger δ to match the same B/C data. This is observed for Models 0 and II. However, the reverse effect is observed for Models I and III, i.e., for models with convec- tive wind. 

Such an exercise has limitations since it is not realistic to expect such systematic energy biases for all production channels. However, it gives some trend and further confirms that Model III cannot be reconcile with small δ . 

