D. Maurin et al.: Systematic uncertainties on the cosmic-ray transport parameters Table 9. Influence of biasing HEAO-3 data. HEAO-3 K best 0 × 10 2 δ best V best c V best a χ $^{2}$/d.o.f HE bias x (kpc $^{2}$Myr − $^{1}$) (km s − $^{1}$) (km s − $^{1}$) 0: +0 . 10 4.97 0.28 . . . . . . 22.5 0: +0 . 05 4.50 0.34 . . . . . . 25.2 0: + 0 . 00 4.08 0.40 . . . . . . 28.8 0: − 0 . 05 3.69 0.46 . . . . . . 33.0 I: +0 . 10 0.45 0.81 13.4 . . . 9.77 I: +0 . 05 0.44 0.87 13.5 . . . 10.2 I: + 0 . 00 0.42 0.93 13.5 . . . 11.2 I: − 0 . 05 0.40 0.99 13.6 . . . 12.5 II: +0 . 10 10.1 0.17 . . . 56.6 4.54 II: +0 . 05 9.95 0.20 . . . 64.8 4.27 II: + 0 . 00 9.76 0.23 . . . 73.1 4.73 II: − 0 . 05 9.56 0.27 . . . 81.5 5.79 III: +0 . 10 0.31 0.85 19.1 30.0 1.32 III: +0 . 05 0.42 0.83 19.0 34.8 1.16 III: + 0 . 00 0.48 0.86 18.8 38.0 1.47 III: − 0 . 05 0.50 0.90 18.7 40.0 2.18 Ek/n (GeV/n) 1 10 2 10 3 10 B/C ratio 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 IS =250 MV) - Garcia-Munoz et al. 1987 IMP7-8 ( =225 MV) - Lukasiak 1999 Voyager1&2 ( =225 MV) - De Nolfo et al. 2006 ACE 97-98 ( =250 MV) - Engelmann et al. 1990 HEAO-3 ( =200 MV) - Swordy et al. 1990 Spacelab ( =425 MV) - Ahn et al. 2008 CREAM 04 ( Biased HEAO-3 data x = +0.10 x = +0.05 x = -0.05 Fig. 5. Demodulated B/C data used in this paper (de- noted dataset F). Biased HEAO-3 data are shown as solid, dashed, and dotted lines. ferent classes of models: it strongly depends on the presence or absence of convection and, to a lesser extent, on reac- celeration. Nevertheless, for all models, the typical scatter in the best-fit values is a factor of 2 for K$_{0}$ , ∼ 50% for V$_{a}$ , ∼ 10% for δ , and ∼ 5% for V$_{c}$ . Figure 4 illustrates that un- certainties in some input parameters (here the production cross-sections) provide larger SystUnc on the transport pa- rameters than their StatUnc calculated using the MCMC technique (taken from Paper II). As for the other effects inspected in previous sections, Model III always provide δ larger than ∼ 0 . 6. Model II provides values of δ in the correct range for a Kolmogorov spectrum, but it never does better than Model III in terms of the goodness-of-fit. It is obvious that most of the results derived in this pa- per rely on the confidence we place in the HEAO-3 data, since they are the most constraining data to date. Their very small error bars at high energy could over-constrain δ . Several other studies use large error bars for high en- ergy B/C HEAO-3 points, contradicting the prescription given in the original paper (Engelmann et al. 1990), and this could bias their results. As an exercise, we allowed for an energy bias in these data. This approach is disputable so we do not wish to defend it strongly. We inspected whether a small bias in the data strongly affects the best-fit values of the transport parameters. The HEAO-3 data are biased above E$_{c}$ , using ( E$_{k}$$_{/}$$_{n}$/E$_{c}$ ) x for E$_{k}$$_{/}$$_{n}$ > E$_{c}$ , with E$_{c}$ = 2.9 GeV/n, as shown in Fig. 5. As in previous sections, we fit Models 0, I, II, and III on dataset F (as shown in Fig. 5), where the HEAO- 3 data have been replaced by the biased ones. The results have been gathered in Table 9. Intuitively, in a similar fash- ion as for the bias in the cross-sections, we expect that a higher value of the B/C data at high energy (positive bias) leads to a smaller δ . This is what is observed for all models. The effect of the bias on δ is at its maximum for the pure diffusion model (Model 0), where Δ δ ∼ 0 . 2. For Models II and III, the effect is less pronounced owing to the pres- From the analysis of Paper II, the StatUnc found for the various transport parameters typically fall in the range 5% − 10% (see Fig. 4). The SystUnc generated by each input we varied are summarised below. The guiding questions for this paper were the following. First, do uncertainties in the input ingredients lead to systematic uncertainties ( SystUnc ) on the derived trans- port parameters greater than their statistical uncertainties ( StatUnc )? Second, can we reproduce the various values of δ given in the literature, and which one should be preferred? We elaborate on the answers below before concluding. Finally, for completeness, we checked that the effect of the solar modulation parameters (we took Δ φ ± 50 MV for the HEAO-3 data) was negligible on all the derived trans- port parameters. ence of reacceleration, with a change of only Δ δ ∼ 0 . 1. However, Model III (convection and reacceleration), which is statistically preferred over Model II, still points to un- comfortably high values for δ . The only way out would be to have a non-systematic energy effect in the data. For in- stance, the result on the fit to the recently published AMS- 01 data (Tomassetti & AMS-01 Collaboration 2009) leads to δ ≈ 0 . 5 (see Paper II). Note 9. Models 0, I, II, and III for L = 4 kpc, where HEAO-3 data are biased using the formula ( E$_{k}$$_{/}$$_{n}$/E$_{c}$ ) x for E$_{k}$$_{/}$$_{n}$ > E$_{c}$ , with E$_{c}$ = 2.9 GeV/n and x = +0 . 1 , +0 . 05 , 0 ., − 0 . 05. – A change of x % in the gas surface density Σ$_{ISM}$ trans- lates into a change of x % for all transport parameters but δ , which is unaffected. Such a simple scaling can be used for comparing models in the literature, as different Σ$_{ISM}$ are generally used. Depending on how confident we are in the measurement of Σ$_{ISM}$, we may conclude 9 9.1. SystUnc and StatUnc: which ones dominate? 9. Summary and discussion 8. Influence of an energy bias in HEAO-3 B/C data