1. Yutaka Fujita (Osaka U.) Fuijta, Takahara, Ohira, & Iwasaki, 2011, MNRAS, in press (arXiv:1105.0683)

2. Tokyo Osaka 200 km Fukushima Epicenter Tsunami Fukushima nuclear power plants 11 March 2011 CRISM 2011 Yutaka Fujita

3.  Expanding supernova remnants (SNRs) are believed to be the main source of cosmic-rays (CRs) in the Galaxy  CRs are accelerated at the shock of SNRs Shock SNR CR particle CRISM 2011 Yutaka Fujita

4.  Gamma-rays have been observed from molecular clouds around SNRs  Molecular clouds may be illuminated by protons escaped from the SNRs (pp-interaction)  CR escape (e.g. Ptuskin & Zirakashvili 2005; Lee et al. 2008; Gabici et al. 2009; Reville et al. 2009 ; Caprioli et al. 2010; Ohira et al. 2010; Drury 2010; Casanova et al. 2010; Ellison & Bykov 2011) Aharonian et al. (2008) SNR W 28 Gamma-ray emission from the SNR W28 Molecular Clouds Gamma-ray Map

5.  Typical diffusion coefficient for CRs in the Galaxy  D(E ~ 1 TeV) ~ 10 29 cm 2 s -1  Diffusion time for E~1 TeV  L c 2 /(6 D) ~ 100 yr  Cloud size: L c ~ 15 pc  Much smaller than the time since active CR acceleration stopped (~10 4 yr) W 28 (Fujita et al. 2009) Gamma-rays are mainly produced through pp- interaction and pion decay (blue) Cloud SNR CRISM 2011 Yutaka Fujita

6.  Gamma-rays appear to have been emitted for a long time (~10 4 yr)  Diffusion coefficient in and/or around clouds must be much smaller than the typical value in the Galaxy (~1%; Torres et al. 2008; Fujita et al. 2009)  If not, gamma-rays should have disappeared in 100 yrs.  In the following, it is assumed that the diffusion coefficient around clouds is small  We consider protons as CRs CRISM 2011 Yutaka Fujita

7.  Generation of plasma waves by CRs (e.g. Wentzel 1974)  Waves scatter CRs  Decrease of the diffusion coefficient  Wave amplification rate (Skilling 1975)  Waves interact with CRs with E f : CR distribution function CRISM 2011 Yutaka Fujita

8.  Fujita, Ohira, & Takahara (2010)  We put CRs at the shock of a model SNR  CR spectrum is given by a simple model  We calculated the diffusion of the CRs around the SNR and the amplification of Alfvén waves at the same time  For diffusion, we performed Monte Carlo simulations  We treated the motion of CR particles as random walks CRISM 2011 Yutaka Fujita

9.  Distribution of CRs with E = 1 TeV around an SNR  Black: t = 10 3 yr  Blue: t = 2.2  10 3 yr  Red: t = 10 4 yr  Marks: CRs  Circles: Size of the SNR  Most CRs do not escape from the periphery of the SNR for at least 10 4 yrs  Wave amplification delays diffusion CRISM 2011 Yutaka Fujita

10.  In Fujita et al. (2010), CR acceleration at the shock and their diffusion into the interstellar space were treated separately.  However, both of the processes follow the same transport equation.  They should be treated seamlessly.  In this study, we treat CR acceleration (very close to the shock front) and their propagation into the interstellar space (far away from the shock front) at the same time. CRISM 2011 Yutaka Fujita

11.  We cannot assume a plane geometry  We assume spherical symmetry  CR particles  : diffusion coefficient w : velocity of the background gas Q : source term CRISM 2011 Yutaka Fujita

12.  Background gas  We solved these equations using the model of Berezhko et al. (1994) CRISM 2011 Yutaka Fujita

13.  Wave amplification (Skilling 1975, Bell 1978)   : wave energy density  Diffusion coefficient for CRs   0 : background gas density  B 0 : background magnetic field CRISM 2011 Yutaka Fujita

15.  Amplification of waves  Waves grow even at r ~ 2 R s (for waves resonant with CRs with pc ~ 20 TeV)  However, the way they grow is somewhat different between pc = 1 TeV and 20 TeV  : wave energy R s : shock radius t 0 : end time of free expansion Dashed: initial (t = 0.55 t 0 ) Solid: t = 10 t 0 Shin Instep Wave energy density 2 R s CRISM 2011 Yutaka Fujita

16.  Evolution of particle distribution  CRs with pc ~ 1 TeV escape into the “instep”, but those with pc ~ 20 TeV are confined in the “shin”  The difference affects the growth of the wave energy For 0.55t 0 < t < t 0 (t 0 : end time of free expansion) t CRISM 2011 Yutaka Fujita

17.  It depends on the initial diffusion coefficient  The initial diffusion coefficient away from the shock front is the one in the interstellar space in the Galaxy  Wave energy density is  ISM  E /  ISM  E 0.5   ISM is smaller for CRs with smaller E  Smaller energy CRs tend to go further away from the shock front if the distance is represented in the units of  Bohm / V s  They can escape into the “instep” (Gabici et al. 2009) CRISM 2011 Yutaka Fujita

18.  Spectra at t = 10 t 0 (t 0 : end time of free expansion)  If diffusion is always Bohm everywhere (dotted), low energy CRs cannot go away from the shock front  If waves do not grow (dashed), low energy CRs can escape from the shock front  If waves grow (solid), the results are between the two  Wave growth significantly affects CR spectra away from the shock front Solid: wave growth Dotted: always Bohm Dashed: no wave growth (initial values are kept) CRISM 2011 Yutaka Fujita

19.  We have investigated the escape of CR protons accelerated at an SNR and their diffusion in the surrounding ISM.  We solved a transport equation from the vicinity of the shock front to the region far away from the front.  We also considered the amplifications of Alfvén waves.  We found that the amplification of the waves reduces the diffusion coefficient on a scale of the SNR.  The initial (=ISM) value of diffusion coefficient is important.  Gamma-ray emission form escaped CRs could be significantly affected by wave amplification. CRISM 2011 Yutaka Fujita