Lecture Plan: 1) Cosmic Ray acceleration- accelerated spectrum, - - PowerPoint PPT Presentation

Andrew Taylor

Lecture Plan:

1) Cosmic Ray acceleration- accelerated spectrum, efficient accelerators, nuclei friendly 2) Cosmic Ray proton + nuclei interaction rates in extragalactic radiation fields 3) Cosmic Ray propagation through Galactic and extragalactic magnetic fields

PROBLEMS PROBLEMS

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When E-2 and when not?

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Shock Acceleration

E1, µ1 E0

1, µ0 1

E0

1, µ0 2

E2, µ2 E2 = E1 ✓1 + βµ1 1 + βµ2 ◆ Strong shock wave propagating at supersonic velocity (sound speed depends

• n temperature)

upstream

u1

downstream

u2

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(energy gain) (advection downstream) Energy Number

Fermi Acceleration (more)

SNRs have vsh~103 km s-1 so β~10-2

∆E E = 4v 3c = 4 3β ∆N N = −4v 3c = −4 3β E1 = ✓ 1 + 4 3β ◆ E0 N1 = ✓ 1 − 4 3β ◆ N0 En = ✓ 1 + 4 3β ◆n E0 Nn = ✓ 1 + 4 3β ◆n N0

So n~1/β crossings are needed before the particle population is significantly altered

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Energy Number

Fermi Acceleration (more)

β~10-2

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Fermi Acceleration (more)

So,

∆N ∆E = N0 E0 ✓1 − 4β/3 1 + 4β/3 ◆n ≈ N0 E0 (1 + 4β/3)−2n ≈ N0E0E−2

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Stochastic Acceleration/Propagation

DxxDpp ≈ β2

scatp2

βscat βscat βscat βscat βscat βscat

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Random Walks

x t f(x, t) = γ(t + 1)/[γ([t − x]/2 + 1)γ([x + t]/2 + 1)]/(2t) f(x, t) ≈ e−x2/(2t) [π/(t/2)]1/2 γ(t + 1) = t! γ(t + 1) = Z ∞ xte−xdx

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Random Walks

∆E E ∝ β

Spatial spread: Momentum spread:

dN dx ∝ e−x2/4Dxxt dN dp ∝ e−(ln p)2/4(Dpp/p2)t dN dx ∝ e−x2/4c2tscatt dN dp ∝ e−(ln p)2/4(t/tacc)

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Gamma-Ray Probes of Particle Acceleration –Flaring Blazars (Mrk 501)

astro-ph/1107.1879, Tramacere et al.

No energy losses

10

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Fermi (Second Order) Acceleration

Acceleration Radiative Losses Escape Source term

∂f ∂t = rp ·  (Dpprpf) p τloss(p)f

• f

τesc(p) + Q p2

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Stochastic Particle Acceleration- Random Walk Result (Spatial)

∂2f ∂r2 + 2 r ∂f ∂r = δ(r) f = r−α −α(−α − 1) − 2α = 0 α(α − 1) = 0 r · (Dxxrf) = δ(r)

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Stochastic Particle Acceleration- Random Walk Result (Momentum)

Steady state No losses Delta injection

∂f ∂t = rp ·  (Dpprpf) p τloss(p)f

• f

τesc(p) + Q p2

For

f = p−α Dpp ∝ pq

and

q = 2 ∂2f ∂p2 + (2 + q) p ∂f ∂p − 4τacc τesc f p2 = δ(p) α2 − 3α − 4τacc τesc = 0

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Stochastic Particle Acceleration- Random Walk Result (Momentum)

α2 − 3α − 4τacc τesc = 0 α = 3 2 ± ✓4τacc τesc + 9 4 ◆1/2 τacc τesc = 1 f = dN d3p = p−4

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Fermi (First Order) Acceleration Time

upstream downstream

tacc = E ∆tcycle ∆Ecycle

Transport of particles in each region is dictated by competition between diffusion and advection

tdiff = R2 Dxx tadv = R vadv

Balancing these timescales

tresid = Dxx (cβsh)2

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Fermi (First Order) Acceleration Time

tacc = E ∆tcycle ∆Ecycle tresid = Dxx (cβsh)2 ∆tcycle = Dxx (c2βsh)

However, during the time it takes advection to dominate over diffusion, the particle will have crossed the shock times

1/β

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Fermi (First Order) Acceleration Time

tacc = E ∆tcycle ∆Ecycle ∆tcycle = Dxx (c2βsh) ∆Ecycle = Eβsh

E1, µ1 E0

1, µ0 1

E0

1, µ0 2

E2, µ2 E2 = E1 ✓1 + βµ1 1 + βµ2 ◆

tacc = Dxx (cβsh)2 = tscat β2

sh

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Fermi (Second Order) Acceleration Time

tacc = E ∆tscat ∆Escat ∆Escat = Eβ2

scat

tacc = tscat β2

scat

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Efficient Accelerators….what means efficient?

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Particle Acceleration in AGN

tacc = η Rlar cβ2

Rlar = β η R Maximum energy (Hillas criterion)

• tesc. =

R2 ηcRlar

AM Hillas (1984)

Rlar(E, B) = ✓ E 10 EeV ◆ ✓1 mG B ◆ 10 pc

Andrew Taylor AM Hillas (1984)

Compactness of UHECR Sources: Proton/Nuclei Synchrotron Losses

η ≈ 1

assumed in above plot

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Particle Acceleration with Cooling

tacc = η Rlar cβ2

tcool = 9 8πα ✓ me Esync

γ

◆ tlar Esync

γ

≈ 9 4η−1β2 me α Maximum synchrotron energy tells us how efficient accelerator is!

η < 103

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Emission Site?

Where are the misaligned (X)HBLs?

Cen A

Hardcastle et al. (1103.1744) ~2 kpc

η < 103

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Future Probes- Cutoff Region

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0.001 0.01 0.1 1 0.001 0.01 0.1 1 10

Eγ

bgdN/dEγ bg

Eγ

bg

10-3 10-2 10-1 100 101 102 103 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101

EγdN/dE Eγ

total

Eγ dN dEγ tot = Z ✓Eγ E2

e

◆ dN dEγ ✓Eγ E2

e

◆ Ee dN dEe dEe

Possibility to probe cutoff region

10-1 100 101 102 10-3 10-2 10-1 100 101

EedN/dEe Ee

total

Esync

γ

= Γ2

e

✓ B Bcrit ◆ me Bcrit = 4 × 1013 G

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Nuclei Friendly Accelerators

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UHECR Air Showers

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UHECR Air Showers

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Composition Measurements by the PAO

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Nuclei Transmutation Within their Source Time [Myr]

0 1500 3000 4500

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IMPLICATIONS for UHECR Sources

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Photo-disintegration threshold:

Since, , where , ergo....

IMPLICATIONS for UHECR Sources

A similar expression holds for TeV photon transparency

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Since, Only heavily sub-Eddington power objects need apply! IF magnetic + photon luminosity are in equipartition: Requiring, to ensure safe passage.

IMPLICATIONS for UHECR Sources

OVERALL MESSAGE: Compact Sources Disfavoured

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Are there Any Candidate Sources Left?

Accelerators of Iron nuclei

Hillas Diagram LHC

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Are there Any Candidate Sources Left?

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Example Candidate UHECR Source

(a Nuclei Friendly Environment)

Diagram taken from Ferrari -1998

Stochastic Acceleration in Radio Lobes:

for

General PROBLEM for Large Accelerators- ACCELERATION TIME

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Can Centaurus A's Radio Lobes Accelerate UHECR?

(me Yes, but requires: where

βA > 0.1 βA = 1 c B p4πmpnp

astro-ph/0903.1259, O’Sullivan et al.

36

(δB/B0)2 = 1

1D 3D

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Diffusion Coefficient

• From resonant scattering between particles and

magnetic field perturbations

With Larmor radius RL B0 + δB(k) resonance for k ~ RL

• 1

Since

• Bohm -> q=1
• Kolmogorov -> q=5/3
• Kraichnan -> q=3/2
• Hard-sphere -> q=2

P(k) ∝ k−q

Probability to scatter off resonant mode within Larmor period

∝ p2−q Dxx β = ⌧ B2 (δB(k))2

• RL =

RL kP(k) Dpp ∝ pq p2 Dpp ∼ Dxx β2

scat

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Particle Transport Equation

• Cut-offs arise naturally in the general solution of the

transport equation for particles

Acceleration Radiative Losses Escape Source term

∂f ∂t = rp ·  (Dpprpf) p τloss(p)f

• f

τesc(p) + Q p2

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Cut-off Shape

• Interplay of acceleration and cooling defines the value of the

cut-off of the primary particles:

• In the following, demonstrations for this result will be shown

for the case of stochastic acceleration scenarios. However, in reality, this result is more general, holding also for shock acceleration scenarios. [see Schlickeisser et al. 1985, Zirakashvili et al. 2007, Stawarz et

• al. 2008]

dN dEe ∝ E−Γ

e

e−(Ee/Emax)βe βe = 2 − q − r

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A Simple Case- q=1, only escape

• Bohm diffusion (q=1) + only escape results in simple

exponential cutoff.

• Some simplifications to the transport equation:

Steady state No losses Delta injection

∂f ∂t = rp ·  (Dpprpf) p τloss(p)f

• f

τesc(p) + Q p2

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A Simple Case (II)- q=1, only escape

• Rearranging the terms (and explicitly stating the

dependences from p of the parameters):

Cutoff comes from balancing 1st and 3rd term Recall generally, βe = 2 − q − r

q = 1, r = 0, → βe = 1

f ∝ Ae−p/pτ τesc(p) ∝ p−1

(Note- energy losses for the case will not alter this result)

r = 0

1 p2 ∂ ∂p ✓ p2D0 p p0 ∂f ∂p ◆ − f τesc(p) = δ(p), ∂2f ∂p2 + 3 p ∂f ∂p − ✓ 1 D0τ0 ◆ f = δ(p)

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Intuitive Insights into Cut-off Shape Origin

For

τ(x) = τ∗ f ∝ e−x/xτ

For

f ∝ e−(x/xτ )2

For

τ(x) = τ∗(x/x∗)2 τ(x) = τ∗(x/x∗)−2 f ∝ const.

r · (Dxxrf) f τ(x) = δ(r)

Consider the steady-state case of diffusion (constant diffusion coefficient) of particles into an absorbing medium

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End of First Lecture

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Shock Acceleration

E1, µ1 E0

1, µ0 1

E0

1, µ0 2

E2, µ2 E2 = E1 ✓1 + βµ1 1 + βµ2 ◆

upstream

u1

downstream

u2

E2 = Γ2E1(1 − βµ1)(1 + βµ0

2)

µ0 = µ − β 1 − βµ E2 = Γ2E1(1 − βµ1) ✓ 1 + β ✓ µ2 − β 1 − βµ2 ◆◆

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Random Walks

From Stirling’s formula

f(x, t) = γ(t + 1)/[γ([t − x]/2 + 1)γ([x + t]/2 + 1)]/(2t) γ(x) ≈ (x/e)x π1/2 γ(x + 1) ≈ (2πx)1/2(x/e)x f(x, t) ≈ tte−t [(t − x)/2](t−x)/2[(t + x)/2](t+x)/2e−t

log[f(x, t)] ≈ t [ 1

2(t − x)(log t/2 − x/2t)] + [ 1 2(t + x)(log t/2 + x/2t)]

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Particle Acceleration with Cooling

tcool = 9 8πα ✓ me Esync

γ

◆ tlar

σTUBcrit hc (mec2)2 = (2π/3)α dEe cdt = 4 3Γ2

eσTUB

tcool = 9 8πα h Ee UBcrit. UB tcool = Ee dt dEe

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Particle Acceleration with Cooling

tcool = 9 8πα ✓ me Esync

γ

◆ tlar

tcool = 9 8πα h Ee UBcrit. UB Esync

γ

= Γ2

e

✓ B Bcrit ◆ me tlar = 2πEe eBc = Γe ✓Bcrit B ◆ h me

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Intuitive Insights into Cut-off Shape Origin

Consider the steady-state case of diffusion (constant diffusion coefficient) of particles into an absorbing medium For

τ(x) = τ∗ f ∝ e−x/xτ

For

f ∝ e−(x/xτ )2

For

τ(x) = τ∗(x/x∗)2 τ(x) = τ∗(x/x∗)−2 f ∝ const.

r · (Dxxrf) f τ(x) = δ(r)

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Intuitive Insights into Cut-off Shape Origin

For

Dxx ∂2f ∂x2 + Dxx 2 x ∂f ∂x − f τ(x) = 0

τ(x) = τ∗ f ∝ e−x/xτ

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Cut-off Shape- Electrons & Photons

f(x) = Z ∞ e−(x/y2)e−yβe dy

Ee Ee dN dEe Eγ = γ2

e

✓ B Bcrit ◆ me

0.001 0.01 0.1 1 0.001 0.01 0.1 1 10

Eγ

bgdN/dEγ bg

Eγ

bg

∝ E−1

e e−(Ee/Ecut)βe

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 10-3 10-2 10-1 100 101 102 103 βe=1, βγ=1/3

EγdN/dE Eγ

total Analytic Ee0 Ee1 Ee2 Ee3 Ee4 Ee5

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Integrand-

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 0.01 0.1 1 10 100

βe=1

y*e-(x/y2+yβe) y

x=0.1 x=1 x=10 x=100 x=1000 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 0.01 0.1 1 10 100

βe=2

y*e-(x/y2+yβe) y

x=0.1 x=1 x=10 x=100

y2 ✓ yβe − 1 βe ◆ = 2x βe y2 ≈ ✓2x βe ◆

2 βe+2

x y2 ≈ x

βe βe+2

e

− ⇣

x y2 +yβ⌘

βγ = βe βe + 2

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Cut-off Shape- Emission Dependence

• Different emission processes dictate different

relation between electrons and gamma rays e.g.

• Synchrotron/IC Thomson:
• SSC:
• IC (Klein Nishina)

Good measurement of gamma ray cut-off can give insight on the cut-

• ff region of primary

electrons

βγ = βe βe + 2 βγ = βe βe + 4

dN dEe ∝ E−Γ

e

e−(Ee/Emax)βe dN dEγ ∝ E−Γ

γ e−(Eγ/Emax)βγ

βγ = βe

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Observation of Cut-offs in Gamma-ray Spectra

• Test case- Vela Pulsar (brightest source)
• Note- MCMC method used to explore

‘good-fit’ region. This has the benefit of being stable on the landscape being explored

dN dEγ ∝ E−Γ

γ e−(Eγ/Emax)βγ

Romoli et al., Astropart.Phys. 88 38-45 (2017)

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MCMC Parameter Constraints

dN dEγ ∝ E−Γ

γ e−(Eγ/Emax)βγ

False minima

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Observation of Cut-offs in Gamma-ray Spectra

• Brightest AGN Flare-

3C 454 Nov 2010

• Indicating a cut-off value of the primary

particles around 1 GeV

• Caveats:
• Values obtained on a 7 days

integration (for statistics)

• Spectrum variable during the flare
• > superposition effects?

Romoli et al., Astropart.Phys. 88 38-45 (2017)

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Observation of Cut-offs in Gamma-ray Spectra

• 2nd Brightest AGN Flare-

3C 279 June 2015

Romoli et al., Astropart.Phys. 88 38-45 (2017)

Values obtained on a 3 days integration Note- X-ray observations during flare indicated that Γ = 1.17 ± 0.06

βγ = βe βe + 2

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3C 279 June 2015 Flare- Temporal Evolution

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Prospects for CTA (South)

• Study using the expected CTA

performance

• Fermi data integrated over 3 days
• Constraint on

parameter at 10% level obtained during only 0.5 hr flare!

βγ

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HESSI and HESSII Eras

H.E.S.S. Phase I: 2002-2012 ▪ 4 telescopes of 12m ▪ 100 GeV - 100 TeV H.E.S.S. Phase II: 2012-++ ▪ Addition of CT5 to the array: 28m ▪ ~30 GeV - 100 TeV

CT5 allows E < 100 GeV measurements — best for:

• High redshift AGN + GRBs
• EBL studies at large z

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Can We Do Better Already? Fermi + H.E.S.S.II Fit

• Joint fit of Fermi-LAT data

(9 hours centred on HESSII

• bs.) taken on night 2
• βγ = 0.34+0.32

−0.14

(HESSII data taken from ICRC2017 Presentation) HESS PRELIMINARY

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An Aside…..

The pp Cross-Section

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Cut-Offs for Primary and Secondaries

For spectra of the form, and the cutoff regions may be fit with a function of the form where

Kafexhiu et al., Phys.Rev. D90 12, 123014 (2014)

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π Spectra for Tp

th<Tp<1 GeV

Kafexhiu et al., Phys.Rev. D90 12, 123014 (2014)

Note- Kamae description has artificially high threshold (~0.5 GeV)

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• Optimal level of statistics (bright low energy transients, plenty of photons)
• Retrieve the primary particle spectrum (using the most up-to-date cross sections)

Constraining the Particle Spectra in Solar flares

If we try to fit high energy cut-off, strong degeneracy exists with the spectral index

dN dT = dN dT

• T=T0

✓ T T0 ◆−α dN dT = dN dT

• T=T0

✓ T T0 ◆−α e−(T/Tcut)

Fermi-LAT data

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This degeneracy can be broken by the lower energy emission detected by GBM, which nuclear de- excitation is expected to contribute/dominate

• Optimal level of statistics (bright low energy transients, plenty of photons)
• Retrieve the primary particle spectrum (using the most up-to-date cross sections)

Constraining the Particle Spectra in Solar flares

dN dT = dN dT

• T=T0

✓ T T0 ◆−α dN dT = dN dT

• T=T0

✓ T T0 ◆−α e−(T/Tcut)

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Future Sources to be Probed.....GRB CTA (South)

• Evolution of spectra during flare
• Detection of as yet undetected

VHE transients (eg. GRB)

• Detection of unexpected new VHE

transient phenomena

Energy (keV)

10

2

10

3

10

4

10

5

10

6

10

7

10

/s)

2

(erg/cm

ν

F ν

−7

10

−6

10

−5

10

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/s)

(erg/cm

F ν

10

10

Time−integrated photon spectrum (3.3 s − 21.6 s) Energy (keV)

10

10

10

10

10

10

/s)

(erg/cm

F ν

10

10

10

10

a b c d

Recent HESSII GRB Upper Limits

From Ackermann et al. 2011

10-3 10-2 10-1 100 101 102 1011 1012 1013

τ E [eV]

z=0.5 z=1 z=2 τ=2

HESS PRELIMINARY

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Integrand

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 0.01 0.1 1 10 100

βe=1

y*e-(x/y2+yβe) y

x=0.1 x=1 x=10 x=100 x=1000 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 0.01 0.1 1 10 100

βe=2

y*e-(x/y2+yβe) y

x=0.1 x=1 x=10 x=100

y2 ✓ yβe − 1 βe ◆ = 2x βe y2 ≈ ✓2x βe ◆

2 βe+2

x y2 ≈ x

βe βe+2

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Integrand

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 0.01 0.1 1 10 100

βe=1 βe=1

y*e-((x/y2)βe/(βe+2)+yβe)

y

x=0.1 x=1 x=10 x=100 x=1000

y

2βe βe+2

✓ yβe − 1 βe ◆ = ✓ 2 2 + βe ◆ x

βe βe+2

y2 ≈ x

2 βe+4

x y2 ≈ x

βe+2 βe+4

✓ x y2 ◆

βe βe+2

≈ x

βe βe+4

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10-6 10-5 10-4 10-3 10-2 10-1 100 101 10-3 10-2 10-1 100 101 102 103 βe=1 βγ=1/3

EγdN/dE Eγ

total 1.2*exp(-(x/0.64)**0.37) Ee0 Ee1 Ee2 Ee3 Ee4 Ee5 10-6 10-5 10-4 10-3 10-2 10-1 100 101 10-3 10-2 10-1 100 101 102 βe=2 βγ=1/2

EγdN/dE Eγ

total 1.2*exp(-(x/0.8)**0.58) Ee0 Ee1 Ee2 Ee3 Ee4 Ee5

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A Further Aside…..

pN Interactions

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Multi-MeV Gamma-Ray Production Cross-Sections

There are also multiple channels by which multi-MeV gamma-ray emission can be produced from non- thermal electrons:

• Secondary Bremstrahlung
• Secondary Annihilation in flight
• Primary Bremstrahlung

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Multi-MeV Gamma-Ray Production Cross-Sections

There are multiple channels by which multi-MeV gamma-ray emission can be produced from non- thermal protons:

• Nuclear Line Emission
• a+B → B*
• Nuclear Line Continuum:
• statistical photons
• direct photons
• pre-equilibrium processes
• Hard Photon Emission

(nuclear Bremstrahlung)

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Other Bright Sources Seen By Fermi

• Gamma ray emission during flaring events
• Most probable scenario, magnetic

reconnection in Solar Corona

Emission of gamma rays! Most important channels:

• De-excitation of atomic nuclei (low energy)
• Decay of neutral pions π0⃗γγ (high energy)

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π Spectra for Tp

th<Tp<1 GeV

Kafexhiu et al., Phys.Rev. D90 12, 123014 (2014)

Note- Kamae description has artificially high threshold (~0.5 GeV)

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γ-ray Spectra for Tp

th<Tp<1 GeV

Kafexhiu et al., Phys.Rev. D90 12, 123014 (2014)

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Stochastic Particle Acceleration- Random Walk Result (Spatial)

Spherically symmetric case:

r · (Dxxrf) = δ(r) 1 r2 ∂ ∂r ✓ r2 ∂ ∂rf ◆ = δ(r) u = rf 1 r ∂2u ∂r2 = δ(r)

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Stochastic Particle Acceleration- Random Walk Result (Spatial)

1 r ∂2u ∂r2 = δ(r) u = Ar + B f = A + B r

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Radiative Loss Timescale

• Relativistic particle will loose its energy on a timescale

that depends of the different processes

Synchrotron: r = -1 Inverse Compton (Thomson): r = -1 Inverse Compton (K.N): r = 1

τcool(E) ∝ Er

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Radiative Loss Timescale

Synchrotron: r = -1 Inverse Compton (Thomson): r = -1 Inverse Compton (K.N): r = 1

τcool(E) ∝ Er

10 100 1000 0.01 0.1 1 10

σeγ [mb] b

Thomson regime Klein Nishina regime

Eγ ≈ γ2

e

✓ B Bcrit ◆ me = bEe Eγ = ✓ b 1 + b ◆ Ee

1. 2.