EFFECTS OF REACCELERATION AND SOURCE GRAMMAGE ON SECONDARY COSMIC - PowerPoint PPT Presentation

PHYSICS and ASTROPHYSICS of COSMIC RAYS OHP Saint Michel l'Observatoire, France EFFECTS OF REACCELERATION AND SOURCE GRAMMAGE ON SECONDARY COSMIC RAYS SPECTRA 28/11/19 - Virginia Bresci

2 GALACTIC DISK h d ≪ H ,Li.. GALACTIC HALO n H ≪ n d

THE TRANSPORT EQUATION ∂ z [ D α ∂ z ] + v A I α ∂ I α − ∂ ∂ z + 2 h d n d v α σ α I α δ ( z ) − ∂ p [ ( I α ] = dt ) α ,Ion. 3 v A A α p 3 ∂ F α − 2 ∂ p δ ( z ) + 2 h d δ ( z ) ∂ dp = A α p 2 Q α ( p ) δ ( z ) Galactic z-direction Halo 2h d Disc R …What experiments really measure: d I α ( z , E k ) dE k = v α ( p ) F α ( z , p ) p 2 dp 3

THE TRANSPORT EQUATION Spatial Diffusion ∂ z [ D α ∂ z ] I α − ∂ ∂ I α + v A ∂ z + 2 h d n d v α σ α I α δ ( z ) − ∂ p [ ( I α ] = dt ) α ,Ion. 3 v A A α p 3 ∂ F α − 2 ∂ p δ ( z ) + 2 h d δ ( z ) ∂ dp = A α p 2 Q α ( p ) δ ( z ) Galactic z-direction Halo 2h d Disc R …What experiments really measure: d I α ( z , E k ) dE k = v α ( p ) F α ( z , p ) p 2 dp 4

THE TRANSPORT EQUATION Advection ∂ I α ∂ z [ D α ∂ z ] + v A I α − ∂ +2 h d n d v α σ α I α δ ( z ) − ∂ z ∂ p [ ( I α ] = dt ) α ,Ion. 3 v A A α p 3 ∂ F α − 2 ∂ p δ ( z ) + 2 h d δ ( z ) ∂ dp = A α p 2 Q α ( p ) δ ( z ) Galactic z-direction Halo 2h d Disc R …What experiments really measure: d I α ( z , E k ) dE k = v α ( p ) F α ( z , p ) p 2 dp 5

THE TRANSPORT EQUATION Spallation reactions ∂ I α ∂ z [ D α ∂ z ] + v A I α − ∂ +2 h d n d v α σ α I α δ ( z ) − ∂ z ∂ p [ ( I α ] = dt ) α ,Ion. 3 v A A α p 3 ∂ F α − 2 ∂ p δ ( z ) + 2 h d δ ( z ) ∂ dp Adiabatic expansion Ionization of the medium = A α p 2 Q α ( p ) δ ( z ) Galactic z-direction Halo 2h d Disc R …What experiments really measure: d I α ( z , E k ) dE k = v α ( p ) F α ( z , p ) p 2 dp 6

THE TRANSPORT EQUATION ∂ z [ D α ∂ z ] + v A I α − ∂ ∂ I α ∂ z + 2 h d n d v α σ α I α δ ( z ) − ∂ p [ ( I α ] = dt ) α ,Ion. 3 v A A α p 3 ∂ F α − 2 ∂ p δ ( z ) + 2 h d δ ( z ) ∂ dp = A α p 2 Q α ( p ) δ ( z ) Source term Galactic z-direction Halo 2h d Disc R …What experiments really measure: d I α ( z , E k ) dE k = v α ( p ) F α ( z , p ) p 2 dp 7

SOURCE TERM 8 Primary nuclei Secondary CRs ∂ z [ D ( p ) ∂ f ( z , p ) ] − u ∂ f ( z , p ) ∂ Created in interactions between + ∂ z ∂ z primaries - ISM: Advection Diffusion 3 ( dz ) p ∂ f ( z , p ) + η n 1 u 1 + 1 du Nuclei δ ( p − p inj ) δ ( z ) = 0 4 π p 2 ∂ p inj A α p 2 Q α ( p ) = ∑ 2 h d n d v α ′ � ( E k ) σ α ′ � α I α ′ � ,0 ( E k ) Compression Injection at the shock surface α ′ � > α − s inj ( p inj ) f 0 ( p ) = s η n 1 p Antiparticles 4 π p 3 + ∞ d σ α ′ � , j p n d , j ∫ p 2 Q ¯ p = 2 h d v ¯ dE k , α ′ � I α ′ � ,0 ( E k , α ′ � ) dE k ,¯ p E th Power law in momentum, slope s=3r/ (r-1) depends ONLY on the compression ratio r=u1/u2 -> 4 No dependence upon diffusion

TRANSPORT EQUATION: THE HIGH ENERGY LIMIT 9 ∂ p [ ( I α ] = ∂ z [ D α ∂ z ] + v A dt ) α ,Ion. I α ∂ I α 3 v A A α p 3 ∂ F α − ∂ ∂ z + 2 h d n d v α σ α I α δ ( z ) − 2 ∂ p δ ( z ) + 2 h d δ ( z ) ∂ dp = A α p 2 Q α ( p ) δ ( z ) Primary CRs Secondary CRs D α ∝ E δ D α ∝ E δ k Secondary to Primary k p 2 Q α ′ � ∝ I α ( E k ) ∝ E − s ′ � − δ p 2 Q α ∝ E − s ′ � (nuclei) Ratios: k k I α ′ � ( E k ) I α ( E k ) ∝ X α ( E k ) ∝ E − δ k “GRAMMAGE”

RECENT AMS-02 OBSERVATIONS 10 Helium Boron Boron over Carbon ( R = A E 2 k + 2 m p c 2 E k ) Z

NON-LINEAR EFFECTS • Super-Alfvenic streaming of cosmic rays instability growth of Alfvén waves (e.g. Blasi+ 2012 PRL 109,061101) • Transition from this regime to Galactic turbulence generates a break in the diffusion coefficient just around 200 GV : PURPLE VS CYAN LINE: δ 1 = δ 2 δ 1 ≠ δ 2 NOT ENOUGH ENOUGH (Blasi 2017)

A. REACCELERATION EFFECT: 12 The same shock waves at supernovae explosions that accelerate primary CRs in the first place are expected to pick up and re-energize any other charged particle in the upstream above the threshold of injection DIFFERENT BOUNDARY CONDITION: f ( −∞ , p ) = g ( p ) SOURCE TERM NOW ? A α p 2 Q α ( p ) = A α p 2 f 0 ( p ) V SN ℛ SN − s s p inj ( p inj ) p ′ � ( p ) f 0 ( p ) = s η n 1 p + s ∫ dp ′ � p ′ � g ( p ′ � ) π R 2 4 π p 3 d p 0 AS BEFORE.. ..RE-ACCELERATION TERM (NOT FOR SECONDARIES) (EVERYONE)

A. REACCELERATION EFFECT: 13 The same shock waves at supernovae explosions that accelerate primary CRs in the first place are expected to pick up and re-energize any other charged particle in the upstream above the threshold of injection DIFFERENT BOUNDARY CONDITION: f ( −∞ , p ) = g ( p ) SOURCE TERM NOW ? A α p 2 Q α ( p ) = A α p 2 f 0 ( p ) V SN ℛ SN − s s p inj ( p inj ) p ′ � ( p ) f 0 ( p ) = s η n 1 p + s ∫ dp ′ � p ′ � g ( p ′ � ) π R 2 4 π p 3 d p 0 AS BEFORE.. ..RE-ACCELERATION TERM (NOT FOR SECONDARIES) (EVERYONE)

A. REACCELERATION EFFECT: 14 • Distribution of SNRs that CRs found in the Galactic disk as a function of their radius in the S-T phase dt ( r SN ) P ( r SN ) dr SN = K P T Max ≈ 3 × 10 4 yr ; T Max • Maximum energy cut-off E Max ( t ) ≈ 100 ( TeV = 100 ( − 4 − 2 t ST ) r ST ) 5 r SN t TeV

A. REACCELERATION EFFECT: 15 • Distribution of SNRs that CRs found in the Galactic disk as a function of their radius in the S-T phase dt ( r SN ) P ( r SN ) dr SN = K P T Max ≈ 3 × 10 4 yr ; T Max • Maximum energy cut-off E Max ( t ) ≈ 100 ( TeV = 100 ( − 4 − 2 t ST ) r ST ) 5 r SN t TeV e − p / p Max ( r SN ) − s s p inj ( p inj ) p ′ � ( p ) f 0 ( p ) = s η n 1 p + s ∫ dp ′ � p ′ � I ( i − 1) ( p ′ � ) 4 π p 3 p 0

A. REACCELERATION EFFECT: 16 • Distribution of SNRs that CRs found in the Galactic disk as a function of their radius in the S-T phase dt ( r SN ) P ( r SN ) dr SN = K P T Max ≈ 3 × 10 4 yr ; T Max • Maximum energy cut-off E Max ( t ) ≈ 100 ( TeV = 100 ( − 4 − 2 t ST ) r ST ) 5 r SN t TeV e − p / p Max ( r SN ) − s s p inj ( p inj ) p ′ � ( p ) f 0 ( p ) = s η n 1 p + s ∫ dp ′ � p ′ � V SN = ∫ I ( i − 1) ( p ′ � ) r Max 33 π r 11/2 Max − r 11/2 P ( r SN )4 SN dr SN = 20 4 π p 3 ST ¯ 3 π r 3 p 0 r 5/2 Max − r 5/2 r ST ST AVERAGE VOLUME

A. REACCELERATION EFFECT: 17 • Distribution of SNRs that CRs found in the Galactic disk as a function of their radius in the S-T phase dt ( r SN ) P ( r SN ) dr SN = K P T Max ≈ 3 × 10 4 yr ; T Max • Maximum energy cut-off E Max ( t ) ≈ 100 ( TeV = 100 ( − 4 − 2 t ST ) r ST ) 5 r SN t TeV e − p / p Max ( r SN ) − s s p inj ( p inj ) p ′ � ( p ) f 0 ( p ) = s η n 1 p + s ∫ dp ′ � p ′ � V SN = ∫ I ( i − 1) ( p ′ � ) r Max 33 π r 11/2 Max − r 11/2 P ( r SN )4 SN dr SN = 20 4 π p 3 ST ¯ 3 π r 3 p 0 r 5/2 Max − r 5/2 r ST ST AVERAGE VOLUME

B. SOURCE GRAMMAGE 18 Particles up to TeV/n are typically confined for a time yr T SN ≈ 3 × 10 4 E k ∼ inside the sources A non-negligible production of secondaries might come from interactions occurring inside the SNR BEFORE the escape of primaries: CONTRIBUTION FROM PRIMARY sV SN T SN ℛ PARTICLES THAT ARE STILL LOCATED Q src , α = v α r ( s ) n src, j × π R 2 INSIDE THE SOURCES d 2 − s dE k , α ( p inj, α ′ � ) + ∞ d σ α ′ � , j p ′ � ( E ′ � k , α ′ � ) × A α ′ � K α ′ � ∫ dE ′ � k , α ′ � E th d σ α ′ � , j NB: for spallation processes ≡ σ α , α ′ � δ ( E ′ � k , α ′ � − E k , α ) dE k , α

RESULTS

RESULTS Model without Reacceleration Model with Reacceleration Model with Reacceleration before solar modulation is included Model without Reacceleration Model with Reacceleration Model with Reacceleration before solar modulation is included Model without Reacceleration Model with Reacceleration Model with Reacceleration before solar modulation is included VB, E. Amato, P. Blasi, G. Morlino, MNRAS 488 , 2068–2078 (2019)

RESULTS Models without Reacceleration Models without Reacceleration & source grammage Models with Reacceleration Models with Reacceleration & source grammage Models with Reacceleration before solar modulation is included 21

RESULTS Models without Reacceleration Models without Reacceleration & source grammage Models with Reacceleration Models with Reacceleration & source grammage Models with Reacceleration before solar modulation is included 22

RESULTS Models without Reacceleration Models without Reacceleration & source grammage Models with Reacceleration Models with Reacceleration & source grammage Models with Reacceleration before solar modulation is included 23