Semi-analytical approaches and recent results from MCMC ......December 6, 2011. Semi-analytical...

Semi-analytical approachesand

recent results from MCMC anduncertainty studies

Antje Putze for the USINE team

The Oskar Klein Centre for Cosmoparticle PhysicsStockholm University

December 6, 2011

Semi-analytical approaches Constraints on CR parameters and uncertainty studies Conclusion

Outline

1 Semi-analytical approaches

2 Constraints on CR parameters and uncertaintystudies

3 Conclusion

GALPROP workshop Antje Putze OKC Analytical approach & recent results 2/28


Cosmic rays in the interstellar medium

[Igor Moskalenko]

nuclei and antinuclei(antiproton,antideuteron)

electrons andpositrons

γ-rays (hadroniccomponent)



CR propagation: the transport equation

q

∂N∂t = q︸︷︷︸

sources

+ ~∇ · (K̂xx~∇N − ~VcN)︸︷︷︸

spatial diffusion & convection

+ ∂∂pp

2Kpp∂∂p

1p2 N︸︷︷︸

momentum diffusion

− ∂∂p

[∂p∂t N −

p3

(~∇~Vc

)N]

︸︷︷︸energy losses

− 1τsN︸︷︷︸

spallation

− 1τrN︸︷︷︸

radioactive decay



CR propagation: different approaches

Leaky-Box model

diffusion − ~∇·(K̂xx~∇N)⇐⇒ N

τescescape

uniform source and CR distribution

Diffusion model

cylindrical geometry of the Galaxywith

a Galactic disc

a diffusive halo

q



CR propagation: semi-analytical solutions at 1D

Simplifying assumptions

isotrope diffusion coefficient

constant galactic windperpendicular to the disc

q

1D equation w/o energy losses/gains, convec-tion, radioactive decay(−K d2

dz2+ 2hnvσiδ(z)

)Ni = 2hδ(z)Qi

with Qi = qi +∑j>i

nvσijNj

Solution

Ni (z ≶ 0) = N0

(1± z

L

)with N0 =

2hQi (0)2KL + 2hnvσi

Full solutions of the transport equation can be found in [Putze et al., A&A (2010)]



B/C

Computational time: < 1 s on 1 CPU



Transport equation for e±

Time-dependent transport equation

D̂N = ∂tN −∇·{K (E )∇N}+ ∂E

{dE

dtN}

= Q(E , x, t)

−dE

dt= b(E ) energy losses (next slide)

Time-dependent solution: time-dependent Green function

D̂Gt = δ3(x− xs)δ(E − Es)δ(t − ts)

Gt(t,E , x← ts ,Es , xs) =δ(∆t −∆τ)

b(E )

exp{−(x− xs)2/λ2

}(πλ2)3/2

λ = e± propagation horizon

∆τ = energy loss time



Energy losses

Full relativistic energy losses (black body parameterisation)

Parametrisation of locally averaged ISRF energy density distribution



Distant and local sources

Source distribution

Injection spectrum

global (individual) distribution fordistant (local, i.e. d < 2 kpc)SNRs and pulsars

Q(E ) = Q0E−γ exp

(E

Ec

)Source catalogue

Green SNR catalogue

ATNF pulsar catalogue



e+ + e− flux and positron fraction



Hadronic component of the Galactic γ-ray diffuse emission

Φγ =

∫los

ds∑CR

∑A

nA(~x) ECRA (~x ,E )

ECRA =

∫ ∞Tmin

dT

{dσ

dE(CR[T ] + A→ γ[E ])× ΦCR(~x ,T )

}Retropropagation

get ΦCR everywhere along the los from ΦexpCR (�,E )

What are the ingredients?

CR source distribution, CR spectrum at Earth, CR propagation model,π0 production cross-sections ([Huang et al. (2007)]), gas maps (3D [Pohl et al. (2008)])



Skymap at 30 GeV

Computational time: ∼ 1 min on 1 CPU



Local emissivity



The USINE propagation code

Slide from D. Maurin



1 Semi-analytical approaches

2 Constraints on CR parameters and uncertaintystudies

3 Conclusion



Phenomenology of nuclear cosmic rays

Why study cosmic-ray propagation?

Study of the standard cosmic-ray astrophysics;

Indirect search for dark matter.

Propagation models:USINE

q

=⇒

Statistical tools:MCMC

⇐=

Experimental data:AMS

used for the first time in the context of cosmic-ray physics [Putze et al., A&A

(2009–2011)].



Diffusion Model

USINE — semi-analytical propagation code [D. Maurin]

Diffusion model with minimal reacceleration, constant Galactic wind

q

Galaxy is divided into two zones:

1 a thin disc of size h;

2 a diffusive halo of size L� h.

K (R) ∝ K0Rδ

Q(R) ∝ qR−α

nd = n, nh = 0

Free parameters

5 transport par.: K0 [kpc2/Myr], δ, Vc and Va [km/s], L [kpc].

2 source par.: q in (m3 s GeV/n)−1, α



Stable nuclei: constraining transport parameters – B/C

[Putze et al., A&A 516 (2010), A66]

Configuration with Vc and Va

preferred:L = 4 kpc fixed

Vc = 18.8+0.3−0.3 km/s

δ = 0.86+0.04−0.04

K0 = 0.0046+0.0008−0.0006 kpc2/Myr

Va = 38+2−2 km/s

Kolmogorov spectral index(δ = 1/3) disfavoured.



Stable nuclei: constraining transport parameters – 3He/4He

B/C:

abundant;

elementalseparation.

3He/4He:

very abundant;

isotopicseparation.

[GeV/n]k/nE-210 -110 1 10 210

Seco

ndar

y-to

-pri

mar

y ra

tio

(TO

A)

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)φModel II with prior ( = 700 MV)φModel III with prior (

= 700 MV)φModel II ( = 700 MV)φModel III (

He (TOA)4He/3

[Coste et al., A&A, accepted for publication]

Data Vc δ K0 × 102 Va

(km s−1) - (kpc2 Myr−1) (km s−1)

3He/4He 17.3+0.3−0.4 0.79+0.05

−0.04 0.50+0.05−0.06 40+2

−3

B/C 18.9+0.3−0.4 0.86+0.04

−0.04 0.46+0.08−0.06 38+2

−2



Radioactive secondaries: constraining the halo size L

40 60 80 100 120 140 1600

0.01

0.02

0.03

0.04

0.05

L [kpc]Be9Be/10

Al27Al/26

Cl/Cl36

Cl/Cl36Al + 27Al/26Be + 9Be/10

Model III

= 0hr

0 20 40 60 80 100 1200

0.01

0.02

0.03

0.04

0.05

0.06L [kpc]

Be9Be/10

Al27Al/26

Cl/Cl36

Cl/Cl36Al + 27Al/26Be + 9Be/10

Model III

0≠ hr

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

5

10

15

20

25

[kpc]hr

[Putze et al., A&A 516 (2010), A66]

Results with 10Be/9Be data

without local bubble:L = 46+9

−8 kpc

with local bubble:L = 8+8

−7 kpc, rh = 120+20−20 pc

Local bubble

underdensity in the LISM modelled as a hole in the disc

exponential decrease of radioactive fluxes [Donato et al., A&A 381 (2002), 539]



Primary nuclei: constraining source parameters

=⇒

Model δ Vc Va χ2/d.o.f(km s−1) (km s−1)

II 0.23 0. 73.2 4.73

III 0.86 18.8 38.0 1.47

I/0 0.61 0. 0. 3.29

III/II 0.51 0. 45.4 2.26[Maurin et al., A&A 516 (2010), A67]



Primary nuclei: p and He

[Putze et al., A&A 526 (2011), A101]

[Putze et al., A&A 526 (2011), A101]

α well constrained between 2.2 and 2.5 (independent of model and data)asymptotic regime (γasymp = α + δ) not reached

same results for heavier nuclei



Uncertainties: cross sections

[Maurin et al., A&A (2010)] [Delahaye et al., A&A (2011)]

Large influence!



Uncertainties: halo size

[Maurin et al., A&A (2010)]

The value of δ depends on the dif-fusion coefficient description. [Delahaye et al., A&A (2011)]

Drastic influence of L on γ-ray emissivity!



Uncertainties: gas maps

[Delahaye et al., A&A (2011)]

Drastic influence: 200% mainly due to the different XCO profiles anddifferent treatments of the Magellanic Clouds and Messier 31.



Conclusion

Semi-analytical approaches

allows description of “realistic” phenomenological models

fast computation

=⇒ Parameter scans and uncertainty studies

“What you get depends on what you put in”

Theoretical uncertainties on CR propagation and γ-ray production by π0

decay is far from negligible

We are only scratching the surface

Need for more data, at higher energies, and more precise theoretical models


Semi-analytical approaches and recent results from MCMC ......December 6, 2011. Semi-analytical...

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