1. THE TIME-DEPENDENT ONE-ZONE HADRONIC MODEL A. Mastichiadis University of Athens

2. ...in collaboration with Stavros Dimitrakoudis – UoA Maria Petropoulou – UoA Ray Protheroe – University of Adelaide Anita Reimer – University of Innsbruck

3. THE HADRONIC MODEL: PHYSICAL PROCESSES Courtesy of R.J. Protheroe leptonic hadronic

4. KEY IDEAS AND ASSUMPTIONS Spherical source of radius R containing magnetic field B. Injection of high energy protons – proton energy and luminosity are free parameters. Protons cool by (i) synchrotron – source of soft photons (ii) photopair (Bethe-Heitler) (iii) photopion (neutral/charged) Model (ii) from MC results of Protheroe & Johnson (1996) and (iii) from SOPHIA code (Stanev et al 2000). Leptonic processes as usual – see leptonic modeling. Study the simultaneous evolution of protons and secondaries through time-dependent, energy conserving kinetic equations. Analogous to SSC leptonic model (i.e. all soft photons are produced from physical processes within the source) with addition of hadronic processes. No external photons / no electron injection (pure hadronic model).  keep free parameters to a minimum (R, B, γ p, L p ) Continuation of AM, Protheroe and Kirk (2005)

5. AIMS 1. Study photon spectral formation simultaneously with neutrino spectra. 2. Calculate efficiencies (e.g. photon luminosity/proton luminosity) 3. Study expected time signatures (as in leptonic models). 4. Study potential supercriticalites (Stern et al. 1995, Kirk & AM 1992) Self consistent approach allows it while they go unnoticed when different approaches are employed (e.g. ‘ready’ proton distribution function).

6. PROTON INJECTION PROTON DISTRIBUTION FUNCTION PROTON LOSSES PROTON ESCAPE ELECTRONS- POSITRONS PHOTONS OBSERVED SPECTRUM Leptonic processes

7. Protons: injection Electrons: Photons: Neutrinos: Neutrons: Bethe-Heitler proton triplet ssa γγ annihilationphotopion synchrotron pair production

8. SECONDARY PRODUCTION RATES – PROTON ENERGY LOSSES photopion photopair Loss timescale

9. INJECTION OF SECONDARΥ ELECTRONS - RESULTING PHOTON SPECTRA Bethe-Heitler electrons photopion electrons γγ electrons R = 3e16 cm B = 1 G γ p = 2e6 l p = 0.4 electrons photons S. Dimitrakoudis et al., in preparation

10. PHOTON AND NEUTRINO SPECTRA Power law proton injection injected protons photons neutrinos neutrons Photon spectra --different proton indices p Neutrino spectra --different proton indices p p=1.5 p=2.5 p=1.5 p=2.5 S. Dimitrakoudis et al., in preparation

11. INCREASING THE PROTON INJECTED ENERGY R = 3e16 cm B = 1 G l p = 0.4 γ p = 3e5γ p = 3e6γ p = 3e7

12. INCREASING THE PROTON INJECTED LUMINOSITY log lp B=1 G Proton injected luminosity is increased by a factor 3 linear quadratic linear quadratic onset of supercriticality R=3e16 cm highly non-linear

13. THE STEP TO SUPERCRITICALITY SUPERCRITICAL REGIME B=10 GR=3e16 cm In all cases the proton injection luminosity is increased by 1.25 SUBCRITICAL REGIME I Time-dependent transition of photon spectra from the subcritical to the supercritical regime

14. PAIR PRODUCTION – SYNCHROTRON LOOP PROTONS  BETHE-HEITLER PAIRS | | | SYNCHROTRON PHOTONS |_______________|________ | escape Loop operational if two criteria, feedback and marginal stability, are simultaneously satisfied  protons supercritical  Pairs/photons exponentiate in the source n γ ~ n e ~ exp(st) supercritical regime Kirk & AM 1992 s=1 s=0 Feedback criterion Marginal stability criterion

15. AUTOMATIC PHOTON QUENCHING Stawarz & Kirk 2007 Petropoulou & AM 2011 Injected gamma-ray luminosity Escaping luminosity gamma-rays soft photons Gamma-rays can be self-quenched: Non-linear network of photon-photon annihilation electron synchrotron radiation Operates independently of soft photons Poses a strong limit on the gamma-ray luminosity of a source SEE MARIA’S POSTER quenched gamma-rays ‘reprocessed’ gamma-rays --------------. l γ,crit

16. PROTONS  GAMMA-RAYS – escape | | | quenching | | (self-generated) | SOFT PHOTONS |_____________| Soft photons from γ-ray quenching pump proton energy  proton losses  more secondaries  more γ-rays Exponentiation starts when γ-rays enter the quenching regime. PHOTON QUENCHING AND PROTON SUPERCRITICALITY γ-ray quenching regime γ-ray quenching regime Photon spectra for various monoenergetic proton injection energies just before supercriticality

17. A MAP OF PROTON SUPERCRITICALITIES quenching- πγ induced cascade PPS Loop quenching- BH induced cascade SUBCRITICAL REGIME inefficient proton cooling B=10 GR=3e16 cm

18. TIME VARIATIONS SUBCRITICAL REGIMESUPERCRITICAL REGIME p-syn (~n) e-syn of Bethe-Heitler pairs (~n^2) amplitude Lorentzian variation

19. CONCLUSIONS One-zone hadronic model –Accurate secondary injection (photopion + Bethe Heitler) –Time dependent - energy conserving PDE scheme Five non-linear PDE – c.f. leptonic models have only two First results for pure hadronic injection - Low efficiencies - γ-rays steeper than neutrino spectra - Quadratic time-behaviour of radiation from secondaries (similar to synchrotron – SSC relation of leptonic models) - Inherently non-linear (c.f. ‘Compton catastrophe’ in leptonic models) - Strong supercriticalities exclude sections of parameter-space used for modeling AGNs CONCLUSION

20. CONCLUSIONS First results for pure hadronic injection - Low efficiencies - γ-rays steeper than neutrino spectra - Quadratic time-behaviour of radiation from secondaries (similar to synchrotron – SSC relation of leptonic models) - Inherently non-linear (c.f. ‘Compton catastrophe’ in leptonic models) - Strong supercriticalities exclude sections of parameter-space used for modeling AGNs

21. CONCLUSIONS Hadronic models are inherently non-linear and they can have interesting behaviour. Only way to investigate their properties fully is through energy conserving equations which couple protons to secondaries. Approach where protons are modeled through a distribution function can be erroneous. Strong supercriticalities exclude large sections of parameter- space used for modeling AGNs.