1. Gamma-ray Evidence for Galactic Cosmic Rays from SNRs Chuck Dermer
(formerly,
Determining the Interstellar Cosmic-Ray Spectrum
from the Diffuse Galactic Gamma-Ray Emissivity )
Chuck Dermer
Naval Research Laboratory, Washington, DC
joint work with
Andy Strong (MPE), Justin Finke (NRL), Nicola Mazziotta (Bari),
Armen Atoyan (Concordia), Gauri Powale (UCB)
discussions with
T. Kamae, E. Orlando, F. Stecker, L. Tibaldo
Gamma-ray emission from interstellar gas can probe Galactic proton and helium spectra down to ~400 MeV, while solar modulation restricts the use of direct cosmic-ray measurements to energies above 10 GeV. We use the latest gas emissivity measurements by the Fermi LAT to determine the flux of the local interstellar cosmic-ray spectrum. Particular attention is paid to uncertainties in our knowledge of the hadronic gamma-ray production processes. A further complication is the flux of bremsstrahlung emitted by primary cosmic-ray electrons and lepton secondaries, which is constrained by synchrotron emission at low energies and direct measurements at high energies. We use the results to verify whether the momentum spectrum is a power-law over the energy range accessible by gamma-ray experiments, and whether there is evidence from the gamma-ray spectrum above ~20 GeV for the hardening of the proton and helium spectra above ~200 GeV reported by PAMELA. Implications for the physics of Galactic cosmic-ray production and propagation are discussed.
Charles Dermer

2. Break in Cosmic-Ray Proton Spectrum
Neronov et al. (2012) claim to infer the presence of a break in the cosmic-ray proton spectrum at 9(±3) GeV from the g-ray emissivity of molecular clouds ~
1. Power-law momentum spectrum gives kinematic break when plotted as kinetic energy
2. Including uncertainties in nuclear production cross sections increase c2n
3. Accurate treatment of bremsstrahlung
4. Break in spectrum reported before (e.g., Strong et al. 2000; Delahaye et al. 2011)
Gamma-ray emission from interstellar gas can probe Galactic proton and helium spectra down to ~400 MeV, while solar modulation restricts the use of direct cosmic-ray measurements to energies above 10 GeV. We use the latest gas emissivity measurements by the Fermi LAT to determine the flux of the local interstellar cosmic-ray spectrum. Particular attention is paid to uncertainties in our knowledge of the hadronic gamma-ray production processes. A further complication is the flux of bremsstrahlung emitted by primary cosmic-ray electrons and lepton secondaries, which is constrained by synchrotron emission at low energies and direct measurements at high energies. We use the results to verify whether the momentum spectrum is a power-law over the energy range accessible by gamma-ray experiments, and whether there is evidence from the gamma-ray spectrum above ~20 GeV for the hardening of the proton and helium spectra above ~200 GeV reported by PAMELA. Implications for the physics of Galactic cosmic-ray production and propagation are discussed.
Neronov, Semikoz, & Taylor (2012)
Charles Dermer

3. Outline
What is the local interstellar cosmic-ray spectrum inferred from emissivity measurements?
Direct detection >10 GeV/nuc; g-ray detection > 400 MeV/nuc ~ ~
1. Fermi-LAT emissivity measurements
2. Uncertainties in nuclear production cross sections
3. Shock-acceleration spectrum: power-law in momentum
4. Fits to emissivity spectrum  cosmic-ray spectrum
5. Evidence from g-ray observations of SNRs
6. Implications for the theory of cosmic-ray origin
Gamma-ray emission from interstellar gas can probe Galactic proton and helium spectra down to ~400 MeV, while solar modulation restricts the use of direct cosmic-ray measurements to energies above 10 GeV. We use the latest gas emissivity measurements by the Fermi LAT to determine the flux of the local interstellar cosmic-ray spectrum. Particular attention is paid to uncertainties in our knowledge of the hadronic gamma-ray production processes. A further complication is the flux of bremsstrahlung emitted by primary cosmic-ray electrons and lepton secondaries, which is constrained by synchrotron emission at low energies and direct measurements at high energies. We use the results to verify whether the momentum spectrum is a power-law over the energy range accessible by gamma-ray experiments, and whether there is evidence from the gamma-ray spectrum above ~20 GeV for the hardening of the proton and helium spectra above ~200 GeV reported by PAMELA. Implications for the physics of Galactic cosmic-ray production and propagation are discussed.
If SNRs accelerate the Galactic hadronic cosmic rays, is this consistent with the inferred local interstellar cosmic-ray spectrum?
Charles Dermer

4. Fermi LAT Emissivity Measurements
Abdo et al., Ap J, 703, 1249, 2009
4 Aug 2008 – 31 Jan 2009 (6 mos)
LAT observations in third quadrant (200o < ℓ < 260o, 22o < |b| < 60o)
No known molecular clouds, point
sources subtracted; low ionized H N(HII)~1-2×1020cm-2
Flux(E>100 MeV) ph cm-2 s-1
6 months of data 4 Aug 2008 – 31 Jan 2009
Residual 100 MeV - 10 GeV γ-ray intensity exhibits linear correlation with N(HI)
Measured integral γ-ray emissivity: (1.63±0.05)×10−26 ph(>100 MeV) s−1sr−1 H−atom−1
with an additional systematic error of ~10%.
Dermer

5. 3 Year Fermi-LAT Emissivity Spectrum
Ref.: J.-M. Casandjian, Fermi Symposium, Monterey, Gamma 2012, Heidelberg paper in preparation
Template mapping, after subtracting point and extended sources and isotropic emission
Dispersion correction at low energies
How to explain the spectrum? Cosmic rays colliding with gas in the Galaxy
p + p, p+ a, etc.  g + X, dominated by ° 2
cosmic-ray e--e+ bremsstrahlung, Compton emission

6. Deriving Local Interstellar Cosmic Ray Spectrum
Fermi LAT emissivity
Interpretation/Theory
Nuclear cross sections
Interstellar CR Spectrum
Analysis
Software
Comparison with Solar Modulation Studies
Electron spectrum

7. Uncertainties in Nuclear Production Physics
p+p → g + X (mostly through p+p → p0 → 2g)
Isobar + Scaling Model
Fireball /Fermi (1950) Statistical Theory
Resonance Baryon Excitation (Stecker 1968)
Feynman Scaling (Stephens & Badhwar 1981; Blattnig et al. 2000)
Hybrid Model (Dermer 1986)
Diffractive Effects + Scaling Violations
1. D(1236) 2. N(1600) 3. Diffractive
4. Non-diffractive/scaling (Kamae et al. 2005, 2006)
Monte Carlo Event Generators Kamae et al. (PYTHIA); Huang et al. (2007: DPMJET-III); Kelner et al. (2006; SIBYLL); Kachelriess & Ostapchenko (2012: QGSJET-II)
Cross Section Enhancement
p+a, a+p a+a, p+C, …; nuclear enhancement factor
k = 1.45 – ( Mori 1997, 2009)
But…spectral differences
Comparison of Different Models:
30% uncertainty at Eg < 100 MeV and < 15% uncertainty at Eg > 1 GeV
Now examining MC FLUKA code to evaluate gamma-ray event yield

8. Inclusive Cross Sections for Kamae et al. Model
Kamae et al.: Tp > GeV 8

9. Comparison of Dermer (1986) and Kamae et al. (2006)
Total
D(1232)
Nondiff
N(1600) 9

10. Comparison of Dermer (1986) and Kamae et al. (2006)
Total
N(1600)
D(1232)
Nondiff 10

11. Comparison of Dermer (1986) and Kamae et al. (2006)
Total
D(1232)
Nondiff
N(1600) 11

12. Comparison of Emissivity Calculations

13. Measured Cosmic-Ray Spectrum
Naïve Theoretical Expectations:
1st order Fermi shock spectrum
Test particle limit
Strong shock
Steepening due to escape
Power-law momentum spectrum makes break in kinetic energy representation (index  –s/2 at Tp << mpc2, index  –s at Tp << mpc2)
Search for deviations from cosmic ray flux from power-law in momentum
Dermer (2012)
1. uncertainties in nuclear production pion^0 decay gamma rays
preclude ruling out cosmic ray spectral models. These uncertainties
invalidate naïve reduced chi^2 methods as in Neronov et al. (2012)
2. cosmic-ray spectrum proportional to power-law in momentum
gives adequate fit to the gamma-ray data and makes a kinematic
break when plotted vs. kinetic energy/nucleon
3. Deviations from a power-law in momentum at low energies (<<10 GeV)
require better p-p nuclear production data at low energies to establish;
p-He, alpha-He processes are also important for metallicity and shadowing
corrections, as well as Solar flare calculations involving soft spectra.
4. Hardening of cosmic-ray spectrum seen with PAMELA and ATIC
should show up at ~20-30 GeV/nuc in the gamma-ray spectrum.
5. Deviations from momentum power-law reveal interesting transport
physics, e.g., transition in diffusion coefficient from self-generated waves,
and advection (see recent paper by Blasi et al.). But a lower energy break
or deviation in the cosmic-ray spectrum is, in my opinion, still hypothetical,
and not confirmed by data due to inadequacies of nuclear modeling.
Charles Dermer

14. Fits to g-ray Emissivity
Photon Spectrum
Protons only; ions treated through nuclear enhancement factor k
(cm2-s-sr-GeV)-1
Gives adequate fit to data within uncertainties of nuclear physics
1. uncertainties in nuclear production pion^0 decay gamma rays
preclude ruling out cosmic ray spectral models. These uncertainties
invalidate naïve reduced chi^2 methods as in Neronov et al. (2012)
2. cosmic-ray spectrum proportional to power-law in momentum
gives adequate fit to the gamma-ray data and makes a kinematic
break when plotted vs. kinetic energy/nucleon
3. Deviations from a power-law in momentum at low energies (<<10 GeV)
require better p-p nuclear production data at low energies to establish;
p-He, alpha-He processes are also important for metallicity and shadowing
corrections, as well as Solar flare calculations involving soft spectra.
4. Hardening of cosmic-ray spectrum seen with PAMELA and ATIC
should show up at ~20-30 GeV/nuc in the gamma-ray spectrum.
5. Deviations from momentum power-law reveal interesting transport
physics, e.g., transition in diffusion coefficient from self-generated waves,
and advection (see recent paper by Blasi et al.). But a lower energy break
or deviation in the cosmic-ray spectrum is, in my opinion, still hypothetical,
and not confirmed by data due to inadequacies of nuclear modeling.
Dermer (2012)
Charles Dermer

15. Fits to g-ray Emissivity
Photon Spectrum
Protons only; ions treated through nuclear enhancement factor k
(cm2-s-sr-GeV)-1
Gives adequate fit to data within uncertainties of nuclear physics
Exceeds CR flux between 10 GeV and 4 TeV
(black solid curve)
Cosmic Ray Spectrum
1. uncertainties in nuclear production pion^0 decay gamma rays
preclude ruling out cosmic ray spectral models. These uncertainties
invalidate naïve reduced chi^2 methods as in Neronov et al. (2012)
2. cosmic-ray spectrum proportional to power-law in momentum
gives adequate fit to the gamma-ray data and makes a kinematic
break when plotted vs. kinetic energy/nucleon
3. Deviations from a power-law in momentum at low energies (<<10 GeV)
require better p-p nuclear production data at low energies to establish;
p-He, alpha-He processes are also important for metallicity and shadowing
corrections, as well as Solar flare calculations involving soft spectra.
4. Hardening of cosmic-ray spectrum seen with PAMELA and ATIC
should show up at ~20-30 GeV/nuc in the gamma-ray spectrum.
5. Deviations from momentum power-law reveal interesting transport
physics, e.g., transition in diffusion coefficient from self-generated waves,
and advection (see recent paper by Blasi et al.). But a lower energy break
or deviation in the cosmic-ray spectrum is, in my opinion, still hypothetical,
and not confirmed by data due to inadequacies of nuclear modeling.
Charles Dermer

16. Fits to g-ray Emissivity
Photon Spectrum
Protons only; ions treated through nuclear enhancement factor k
(cm2-s-sr-GeV)-1
Gives adequate fit to data within uncertainties of nuclear physics
Exceeds CR flux between 10 GeV and 4 TeV
(black solid curve)
Cosmic Ray Spectrum
1. uncertainties in nuclear production pion^0 decay gamma rays
preclude ruling out cosmic ray spectral models. These uncertainties
invalidate naïve reduced chi^2 methods as in Neronov et al. (2012)
2. cosmic-ray spectrum proportional to power-law in momentum
gives adequate fit to the gamma-ray data and makes a kinematic
break when plotted vs. kinetic energy/nucleon
3. Deviations from a power-law in momentum at low energies (<<10 GeV)
require better p-p nuclear production data at low energies to establish;
p-He, alpha-He processes are also important for metallicity and shadowing
corrections, as well as Solar flare calculations involving soft spectra.
4. Hardening of cosmic-ray spectrum seen with PAMELA and ATIC
should show up at ~20-30 GeV/nuc in the gamma-ray spectrum.
5. Deviations from momentum power-law reveal interesting transport
physics, e.g., transition in diffusion coefficient from self-generated waves,
and advection (see recent paper by Blasi et al.). But a lower energy break
or deviation in the cosmic-ray spectrum is, in my opinion, still hypothetical,
and not confirmed by data due to inadequacies of nuclear modeling.
Fit to CR proton flux above 10 GeV (purple dashed curve), shock spectrum below
Underproduces g-ray emissivity
But,….no electron emissions
Charles Dermer

17. Cosmic Ray Electrons from Radio Observations
Injection index = 1.6 below 4 GeV
Derive ambient spectrum using Fermi-LAT electron spectrum above 7 GeV, and synchrotron spectrum at lower energies
Synchrotron Tb index = 2.4 – 2.6 below a few GHz  electron index = 1.8 – 2.2 below a few GeV
Injection cannot be too hard (1.3 – 1.6), or underproduce directly measured electrons
Compare with GALPROP propagation model at lower energies using parameters from GALPROP modeling of CR secondary to primary nuclei
For more details, see poster: Diffuse radio emission from the Galaxy, Implication for cosmic rays and magnetic fields, E. Orlando & A. Strong
LIS
Strong, Orlando, & Jaffe (2011)
Charles Dermer

18. Synchrotron from Cosmic Ray Electrons
Injection index = 1.6 below 4 GeV
Injection index = 1.6 below 4 GeV
Require break in electron injection spectrum from ~1.6 to 2.5 at ~4 GeV to fit synchrotron, so hard injection spectra
Implies less Solar modulation than usually assumed
Alternate approach of Casandjian: derive proton and helium spectra from emissivity, heliospheric fluxes using Solar modulation model, and synchrotron
LIS
Strong, Orlando, & Jaffe (2011)
Charles Dermer

19. Bayesian Model Analysis
1. Explicit scan of model parameters
2. Spectra using posterior averaging
3. No use made of modulation approximations
4. Base models on momentum spectra n(p)
5. Express problem in matrix form
q(E) = MH(E,p) nH (p) + ML (E,p) nL (p)
q(E) = emissivity
E = gamma-ray energy
p = momentum
nH (p) = proton, Helium spectrum
nL (p) = electron + positron spectrum
MH = Hadronic production matrix
ML = Leptonic production matrix
6. Free parameters in the most general case:
protons:
1. proton break momentum 2. proton index gp1 below break 3. proton index gp2 above break 4. proton normalization
electrons:
5. electron break momentum
6. electron index ge1 below break
(constrained by synchrotron)
Fix electron index above break and normalization to Fermi > 20 GeV electron spectrum.
Charles Dermer

20. Fits to Emissivity Spectrum with Break
Cross sections (1) Kamae et al. (2006) at low energies; Kachelriess & Ostapchenko (2012) > 20 GeV
Fermi LAT data: Casandjian (2012)
PRELIMINARY.
total
Derived
CR spectrum
hadron
PAMELA
electron
Note Solar modulation
BESS-TeV
(Left) Data: Local gamma-ray emissivity measured by Fermi-LAT (Casandjian 2012)
Model: Hadronic (red), bremsstrahlung (green).
Total (yellow, with one standard deviation range). (Right) Cosmic-ray proton momentum spectrum derived from emissivity.
Displayed as a power-law in momentum, with low-energy break.
Analysis includes bremsstrahlung contribution and synchrotron constraints on electrons. One standard deviation range.
Dashed red (green) curve is PAMELA (BESS-TeV) cosmic-ray proton data
Note agreement at high energies and modulation at low energies.

21. Effects of Cross Section on Fits to Emissivity
Cross sections (2) Dermer (1986): Stecker isobar model at low energies; Stephens & Badhwar at high energies
Fermi LAT data: Casandjian (2012)
PRELIMINARY
total
Derived
CR spectrum
hadron
PAMELA
electron
Solar modulation
BESS-TeV
Cross sections: (1) (2)
1. proton break momentum 6.5 (±2.1) GeV 6.7 (±2.5) GeV 2. proton index gp1 below break 2.4 (0.1) (0.1) 3. proton index gp2 above break 2.9 (0.1) (0.1) 4. proton normalization (0.1) (0.1)
5. electron break momentum GeV GeV
6. electron index ge1 below break 1.8 (0.1) (0.1)
Kamae-Kachelriess cross sections: pbreak_gp1_norm_gp2_ge1_elebreak_6pars_parset31__options12_Kamae_Kachelriess_JMC_2012. pHe_break mean=6.52e+03 MeV var=2.12e+03 MeV pbreak_gp1_norm_gp2_ge1_elebreak_6pars_parset31__options12_Kamae_Kachelriess_JMC_2012. gp1 mean=2.37 var=0.117 pbreak_gp1_norm_gp2_ge1_elebreak_6pars_parset31__options12_Kamae_Kachelriess_JMC_2012. pHe_norm mean=1.26 var= pbreak_gp1_norm_gp2_ge1_elebreak_6pars_parset31__options12_Kamae_Kachelriess_JMC_2012. gp2 mean=2.86 var= pbreak_gp1_norm_gp2_ge1_elebreak_6pars_parset31__options12_Kamae_Kachelriess_JMC_2012. ge1 mean=1.84 var= pbreak_gp1_norm_gp2_ge1_elebreak_6pars_parset31__options12_Kamae_Kachelriess_JMC_2012. ele_break mean=1.52e+04 var=8.37e+03
Dermer-only cross-sections:
pbreak_gp1_norm_gp2_ge1_elebreak_6pars_parset33__options12_Dermer_JMC_2012. pHe_break mean=6.73e+03 var=2.55e+03 pbreak_gp1_norm_gp2_ge1_elebreak_6pars_parset33__options12_Dermer_JMC_2012. gp1 mean=2.46 var=0.13 pbreak_gp1_norm_gp2_ge1_elebreak_6pars_parset33__options12_Dermer_JMC_2012. pHe_norm mean=1.44 var= pbreak_gp1_norm_gp2_ge1_elebreak_6pars_parset33__options12_Dermer_JMC_2012. gp2 mean=2.8 var= pbreak_gp1_norm_gp2_ge1_elebreak_6pars_parset33__options12_Dermer_JMC_2012. ge1 mean=1.86 var= pbreak_gp1_norm_gp2_ge1_elebreak_6pars_parset33__options12_Dermer_JMC_2012. ele_break mean=1.48e+04 var=8.47e+03
Labels parameters
PRELIMINARY
Charles Dermer

22. SNR Origin of Galactic Cosmic Rays
Fermi LAT 3 yr
> 1 GeV sky
Galactic disk with volume:
Escape from disk of Galaxy:
Cosmic ray energy density:
Cosmic-ray power:
SNRs provide
6 months of data 4 Aug 2008 – 31 Jan 2009
Dermer

23. Independent Calculation
Power in ions:  3.0e40 erg/s
power law index:  2.3
minimum injected momentum (protons)= 1.0e-4
Power in electrons:  2.0e38 erg/s
power law index:  1.6 below the break; 2.5 above the break (same as Strong, Orlando, & Jaffe 2011)
break at 4 GeV (momentum p = 7.84e3)
minimum injected momentum (electrons) = 5.0e-2
nuclear enhancement factors are 2 for both bremsstrahlung and pi0 production.
Power in ions:  3.0×1040 erg/s
power law index:  2.3
minimum injection momentum (protons) bg = 10-4
Power in electrons:  2.0×1038 erg/s
power law index: 1.6 below the break; 2.5 above the break (same as Strong, Orlando, & Jaffe 2011)
break at 4 GeV (momentum bg = 7840)
minimum injection momentum (electrons) bg = 5.0e-2
nuclear enhancement factors are 2 for both p0 production and bremsstrahllung
(calculations by J. Finke)
Charles Dermer

24. Cosmic-ray synchrotron spectrum
Magnetic field: 5 mG

25. Power-law injection for protons and electrons
Here is the latest fit with the new escape timescale. There is only a single power-law injected for the electrons here, with index 2.3 and power 7.3e38 erg/s. The protons are injected with index 2.26 and power 7.9e39 erg/s. I think all the other parameters are the same as before. The fit to the synchrotron is worse than before, and the emissivity is fit worse than before at lower energies. It looks like the flat escape timescale at lower energies doesn't make a sharp enough break in the electrons at lower energies.
Electrons injected with index 2.3 and power 7.3×1038 erg/s.
Protons injected with index 2.26 and power 7.9×1039 erg/s.
Charles Dermer

26. From Supernova Explosion to Cosmic Rays
Directed explosion kinetic energy is dissipated into heat and nonthermal particles. Curves 1, 2, and 3 give
Rate at which kinetic energy is swept into shocked shell
Rate at which kinetic energy is lost by shell deceleration
Total rate of change in kinetic energy
Injection problem: Energy dissipation efficiency h:
proportional of total swept-up kinetic energy deposited in cosmic rays.
In slow cooling regime, h total swept-up kinetic energy
Dermer

27. Multi-zone Leptonic Model Fit to RX J1713.7-3946
SED inconsistent with hadronic model with power-law proton spectrum
Good fit to g-ray spectrum with leptonic model requires a multiple-zone model: entire remnant + compact knots
Magnetic field in both regions ~16 mG
Predicts radio-emitting knots
6 months of data 4 Aug 2008 – 31 Jan 2009
Finke & Dermer, ApJ, 2012
Dermer

28. Two-zone Model for Tycho
dot-dashed
Zone 1, dashed
Zone 2, Compton
Single zone model unphysical for
first-order Fermi shock acceleration
Simplest two-zone model explains spectra
with hybrid electron bremsstrahlung + Compton scattering model
B1 = 100 μG and B2 = 34 μG
Ee.1 = 4.4 × 1047 erg and Ee.2 = 4.2 × 1048 erg
Atoyan & Dermer, ApJL, 2012
Atoyan et al. 2000

29. SNR g-Ray Luminosities
Dermer & Powle, A&A, submitted
g-ray luminosities consistent with secondary nuclear or bremsstrahlung production with ~10% efficiency of proton and electron acceleration
~10% efficiency for electrons inconsistent with electron/proton ratio in the cosmic rays
Increase in g-ray luminosities at ~Sedov age; decrease at > 104 yr
6 months of data 4 Aug 2008 – 31 Jan 2009
Dermer

30. Summary
Must account for uncertainties in nuclear production and leptonic bremsstrahlung g rays to test or rule out CR spectral models using reduced c2 methods contra Neronov et al. (2012)
Cosmic-ray spectrum proportional to power-law in momentum gives adequate fit to the g-ray data and makes a kinematic break when plotted vs. kinetic energy/nucleon
Deviations from a power-law in momentum at low energies (<<10 GeV) require good p-p nuclear production data at low energies to establish; p-He, a-He processes are also important for metallicity corrections
Simplest spectral model is power-law in momentum; good fit to emissivity with s=2.75, but implies CR spectrum larger than observed at >10 GeV, also neglects electron bremsstrahlung
More detailed analysis with bremsstrahlung and synchrotron consistent with low-energy break in proton spectrum at ~6 GeV, as expected from injection momentum power-law modified by propagation with path length distribution inferred from B/C ratio
Spectral analysis of Tycho and RX J1713 SNRs gives inconclusive evidence for cosmic-ray proton/ion acceleration
Statistics of g-ray SNRs most easily explained by cosmic-ray hadrons
Cosmic ray spectrum implied by g-ray emissivity in accord with origin of cosmic rays by acceleration at supernova remnant shocks
g-ray observations demonstrate level of absolute Solar modulation on interstellar cosmic-ray spectrum

31. LAT Emissivity Spectrum of Molecular Clouds
Molecular Clouds as GCR Detectors
High Galactic latitude Gould Belt Clouds
August 4, 2008 – July 15, 2011, P6_v11
Subtract point sources from 2 yr catalog
Flux ~ M/D2 as inferred from CO maps
Perform spectral analysis
Molecular Cloud Spectra Consistent
“Passive” Cosmic-ray Detectors
Break in Photon Spectrum at ~2 GeV
Invert for ISM CR spectrum
Neronov, Semikoz, & Taylor (2012)

32. Spectra of Galactic Cosmic Rays

33. SNR Gamma-Ray Hardness with Age