Mem. S.A.It. Vol. 82, 863
SAIt 2011c Memoriedella
A Markov Chain Monte Carlo technique to sample transport and source parameters of
Galactic Cosmic Rays
A. Putze1, B. Coste2, L. Derome2, F. Donato3, and D. Maurin2
1 The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden, e-mail: [email protected]
2 Laboratoire de Physique Subatomique et de Cosmologie, Universit´e Joseph Fourier Grenoble 1, CNRS/IN2P3, Institut Polytechnique de Grenoble, 53 avenue des Martyrs, Grenoble, 38026, France
3 Department of Theoretical Physics and INFN, via Giuria 1, 10125 Torino, Italy
Abstract. We implemented a Markov Chain Monte Carlo (MCMC) technique within the USINE propagation package to estimate the probability-density functions for cosmic-ray transport and source parameters within an 1D diffusion model. From the measurement of the B/C and3He/4He ratios as well as of radioactive cosmic-ray clocks, we calculate their probability density functions, with a special emphasis on the halo size L of the Galaxy and the local underdense bubble of size rh. We also derive the mean, best-fit model parameters and 68% confidence intervals for the various parameters, as well as the envelopes of isotopic ratios. Additionally, we verify the compatibility of the primary fluxes with the transport parameters derived from the B/C analysis before deriving the source parameters. Finally, we investigate the impact of the input ingredients of the propagation model on the best- fitting values of the transport parameters (e.g., the fragmentation cross sections) in order to estimate the importance of the systematic uncertainties. We conclude that the size of the diffusive halo depends on the presence/absence of the local underdensity damping effect on radioactive nuclei. Moreover, we find that models based on fitting B/C are compatible with primary fluxes. The different spectral indices obtained for the propagated primary fluxes up to a few TeV/n can be naturally ascribed to transport effects only, implying universality of elemental source spectra. Finally, we emphasise that the systematic uncertainties found for the transport parameters are larger than the statistical ones, rendering a phenomenological interpretation of the current data difficult.
Key words.methods: statistical – cosmic rays
1. Introduction
The transport equation of Galactic cosmic rays (GCRs) is described in standard text-
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book. It generally contains a source term (stan- dard or exotic source/production), a diffusion term, and possibly convection, energy gain and losses, and catastrophic losses (fragmentation and decay). A full numerical treatment is gen- erally required to solve the transport equation.
864 A. Putze et al.: An MCMC technique to sample GCR parameters However, analytical (or semi-analytical) solu-
tions may be derived assuming a simplified de- scription of the spatial dependence of the in- gredients (e.g. the gas distribution) and some transport parameters. Such a semi-analytical solution has been implemented in the USINE propagation code1used below.
In the first paper Putze et al. (2009) of a series (Putze et al. 2009, 2010, 2011; Coste et al. 2011), we implemented a Markov Chain Monte Carlo (MCMC) to estimate the prob- ability density function (PDF) of the trans- port and source parameters. This allowed us to constrain these parameters with a sound sta- tistical method, to assess the goodness of fit of the models, and as a by-product, to pro- vide 68% and 95% confidence level (CL) en- velopes for any quantity we are interested in (e.g., B/C and3He/4He ratios, primary cosmic- ray fluxes). The analysis was performed for a diffusion model, by considering constraints set by radioactive (on the halo size of the Galaxy) and primary nuclei (on the CR source parame- ters).
We summarise below the results we ob- tained in the diffusion model (normalisation K0 and slope δ of the diffusion coefficient), with a halo size L, with minimal reacceleration (Alfv´enic speed Va), a constant Galactic wind perpendicular to the disc plane Vc(Jones et al.
2001), and a possible central underdensity of gas (of a few hundreds of pc) around the solar neighbourhood rh(Donato et al. 2002).
2. Transport parameters
Secondary species in GCRs are produced dur- ing the CR journey from the acceleration sites to the solar neighbourhood, by means of nu- clear interactions of heavier primary species with the interstellar medium. Hence, they are tracers of the CR transport in the Galaxy.
Therefore, secondary-to-primary ratios, such as B/C and3He/4He, are suitable quantities to constrain the transport parameters for species Z ≤ 30. For the parameter estimation of the above described transport equation (K0, δ, Vc
and Va), the usage in the past has been based
1 http://lpsc.in2p3.fr/usine
(GeV/n) Ek/n
10-1 1 10 102 103
B/C ratio
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
=250 MV) - Garcia-Munoz et al. 1987 Φ
IMP7-8 (
=225 MV) - Lukasiak 1999 Φ Voyager1&2 (
=225 MV) - De nolfo et al. 2006 Φ
ACE 97-98 (
=250 MV) - Engelmann et al. 1990 Φ
HEAO-3 (
=200 MV) - Swordy et al. 1990 Φ
Spacelab (
=425 MV) - Ahn et al. 2008 Φ CREAM 04 (
Model III (SA) /d.o.f=1.47) χ2
=0.86, δ (
Model II (SA) /d.o.f=4.73) χ2
=0.23, δ
=-1.30) ( ηT Model III/II (
/d.o.f=2.26) χ2
=0.51, δ (
=-2.61) ηT Model I/0 (
/d.o.f=3.29) χ2
=0.61, δ ( TOA
=225 MV Φ
=250 MV Φ
||
Fig. 1.Best-fit of the B/C ratio for different con- figurations: Model II (Vc = 0; red line), III (K0, δ, Vc, Va , 0; black line), I/0 (Vc = Va = 0, ηT , 1; light grey line), and III/II (Vc = 0, ηT , 1; dark grey line). A more detailed description of this figure can be found in Maurin et al. (2010).
mostly on a manual or semi-automated (hence partial) coverage of the parameter space (e.g.
Jones et al. 2001). More complete scans were performed, e.g., in Maurin et al. (2001), based on a grid analysis. The Markov Chain Monte Carlo is a step further to optimise the explo- ration of the parameter space.
Figure 1 illustrates the best-fit B/C ratio found from a χ2 analysis, for different prop- agation models. In the following, we will fo- cus on the full convection/reacceleration con- figuration of the diffusion model (Model III = {K0, δ, Vc, Va}). This specific configuration leads to a rather large value of the diffusion slope δ (∼ 0.86) since it includes convection (Vc∼ 18.8 km/s) (see Putze et al. 2010).
Figure 2 illustrates the power of the MCMC technique, which allows to retrieve the PDF of the parameters (here for a fixed halo size of L = 4 kpc). The symbols cor- respond to the best-fit values obtained from different parameterisations of the production cross-sections. Their spread illustrates the fact that systematic uncertainties in the calcula- tion are already of the same order or larger than the statistical ones (i.e., the width of the PDFs). See Maurin et al. (2010) for more de- tails. This will be an issue with the forthcom- ing high-precision measurements of AMS-02 on the ISS.
Most secondary-to-primary ratios have A/Z ∼ 2, and in that respect,3He/4He is unique since it probes a different regime and allows to address the issue of the ‘universality’ of prop-
A. Putze et al.: An MCMC technique to sample GCR parameters 865
16 18 20 22
0 0.5 1
[km/s]
Vc
16 18 20 22
0.7 0.8 0.9 1
0.7 0.8 0.9 1
0 5
10 δ
16 18 20 22
0.005 0.01
0.7 0.8 0.9 1
0.005 0.01
0.005 0.01
0 200 400
600 K0 [kpc2/Myr]
16 18 20 22
30 40 50
0.7 0.8 0.9 1
30 40 50
0.005 0.01
30 40 50
30 40 50
0 0.1
0.2 Va [km/s]
Best-fit parameters W03 WKS98 S01 GAL09 HE-bias: x = +0.05 HE-bias: x = +0.02 HE-bias: x = -0.02 HE-bias: x = -0.05
Fig. 2. PDF of the transport parameters as ob- tained in Putze et al. (2010), along with the best- fit values for different production cross-section sets:
Webber et al. (2003, W03), black stars; Webber et al. (1998, WKS98), red dots; Soutoul (2001, S01), red squares; Galprop code v50.1p (2009, GAL09), red triangles; high-energy (HE) bias on W03, grey symbols. A more detailed description of this figure can be found in Maurin et al. (2010).
[GeV/n]
Ek/n
10-2 10-1 1 10 102
Secondary-to-primary ratio (TOA)
0 0.05 0.1 0.15 0.2
0.25 Balloon72 (400 MV) Balloon73 (500 MV) Balloon77+Voyager (400 MV) Balloon89 (1400 MV) IMP7+Pioneer10 (540 MV) ISEE3 (740 MV) ISEE3(HIST) (500 MV)
Voyager87 (360 MV) IMAX92 (750 MV) SMILI-I (1200 MV) SMILI-II (1200 MV) AMS-01 (600 MV) BESS98 (700 MV) CAPRICE98 (700 MV)
= 700 MV) φ Model II with prior (
= 700 MV) φ Model III with prior (
= 700 MV) φ Model II (
= 700 MV) φ Model III ( He (TOA) He/4 3
Fig. 3. Top-of-atmosphere (TOA, modulated at 700 MV) envelopes at 95% CL for the secondary- to-primary3He/4He ratio for Model II and III with and without a prior on the source parameters.
agation histories. In our analysis of recent data (Coste et al. 2011), which is shown in Fig. 3 to- gether with the envelopes at 95% CL, we find that the constraints obtained with3He/4He on the transport parameters are competitive with those set by the B/C ratio. Moreover, the high abundance of the 3He/4He data make them a powerful alternative to the standard B/C anal- ysis, so that the measurement of1H,2H,3He, and4He should be a prime target for AMS-02.
40 60 80 100 120 140 160
0 0.01 0.02 0.03 0.04
0.05 L [kpc]
9Be 10Be/
27Al 26Al/
Cl/Cl 36
Cl/Cl Al + 36 Al/27 Be + 26 Be/9 10
h = 0 r
0 20 40 60 80 100 120
0 0.01 0.02 0.03 0.04 0.05
0.06 L [kpc]
9Be 10Be/
Al Al/27 26
Cl/Cl 36
Cl/Cl Al + 36 Al/27 Be + 26 Be/9 10
≠ 0 rh
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0
5 10 15 20 25
[kpc]
rh
Fig. 4.Marginalised posterior PDFs of L for rh= 0 (top panel) and L and rh(bottom panels). The four curves result from the combined analysis of B/C plus separate or combined isotopic ratios of radioac- tive species.
3. Halo sizeLfrom radioactive nuclei The typical distance travelled in a diffusive process in a finite time t is given by lrad∼ √
Kt, where K is the diffusion coefficient. For a β- decay unstable secondary species, it means that at low energy (using a typical value for the dif- fusion coefficient), these nuclei cannot travel farther than a few hundreds of parsecs: they do not feel the halo size L and are only sensitive to the diffusion coefficient K. In principle, this lifts the degeneracy between K0 and L (seen from the analysis of the stable secondary-to- primary ratio). However, things are not as sim- ple when we have a closer look at the hundred- of-parsec scale. It happens that there is no tar- get to produce these species in the solar neigh- bourhood: we live in a local bubble. This lack of targets affects their flux (Donato et al. 2002).
We find (see Putze et al. 2010) that the diffu- sion model including reacceleration and con- vection favours an underdense bubble of rh = 120+20−20 pc (consistent with direct observations of the local cavity) and L = 8+8−7 kpc using
10Be/9Be data (see Fig. 4). The results from the combined analysis of B/C plus separate or combined isotopic ratios of radioactive species are shown in Fig. 4.
866 A. Putze et al.: An MCMC technique to sample GCR parameters
2.3 2.4
0 20 40
2.3 2.4
0 20 40
α Model III
AMS 01 BESS 98 BESS-TeV protons helium
0 0.1
0 10 20
0 0.1
0 10 20
αp 4He - α
Fig. 5.Left panel: PDF of the source slope α for p (unfilled histograms) and He (hatched histograms).
Right panel: PDF for αHe− αp. The colour code cor- responds to three experimental data used: AMS-01 (solid black line), BESS98 (dashed red lines), and BESS-TeV (dash-dotted blue lines).
R (GV)
1 10 102 103 104
p/He
10
=1300 MeV) φ ATIC-1 (
=650 MV) φ CAPRICE 94 (
500 MV)
≈ φ Webber et al. (
=600 MV) φ AMS-01 (
=600 MV) φ IS (thick) vs TOA (thin,
σ=0
He, α
p= α
=90 mb σHe
=30 mb, σp He, α
= αp
=90 mb σHe
=30 mb, σp
=0.03, αHe p- α
Rδ β-2
∝ R, K
∝ dQ/dP
Fig. 6.p/He ratio as a function of rigidity. The lines show toy-model calculations with the destruction rate σ set to 0 (solid lines), set to its required value (dotted lines), plus a difference in the spectral in- dex of p and He (dashed lines) for interstellar (IS, thick lines) and top-of-atmosphere (TOA, thin lines) fluxes.
4. Source parameters
The primary flux data alone are unable to se- lect any particular propagation model, as the transport and source parameters are degenerate (Putze et al. 2011). We thus have to fit simul- taneously the primary fluxes and secondary-to- primary ratios. Below, we adopt a different ap- proach: we select only a few propagation mod- els (fitting well the B/C ratio and shown in Fig. 1), and restrict the parameter space to the source parameter space only.
The most important parameter is the source slope α. Figure 5 shows the PDF of α for the best-measured primary species, i.e. p and He,
as well the difference of slope between these two species: αp − αHe is consistent with 0.
Actually, the shape of the p/He ratios, when plotted as a function of the kinetic energy per nucleon, is driven by the solar modulation ef- fect (not shown here). But the same ratio plot- ted as a function of the rigidity minimises the effect of the solar modulation, and is well adapted to probe the values of αp− αHe. This is illustrated in Fig. 6, where a simple toy model was used to reproduce the experimental data. A difference of spectral index between p and He is also not supported by the ratio, in agreement with the direct fits to p and He fluxes.
We have also studied heavier primary nu- clei spectra, whose relevant destruction rate on the ISM increases roughly with atomic num- ber. The different spectral indices for the prop- agated primary fluxes up to a few TeV/n can be ascribed to transport effects only, implying universality of elemental source spectra. See Putze et al. (2011) for more details.
5. Conclusions and perspective We have shown that the use of an MCMC tech- nique is very powerful to study the source and transport parameters. The AMS-02 experiment should provide data with an unprecedented ac- curacy, and the MCMC technique will be a key tool to interpret them.
Acknowledgements. A. P. is grateful for financial support from the Swedish Research Council (VR) through the Oskar Klein Centre.
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