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  • Lepto-hadronic Unified Model of \(\gamma\)-ray Supernova Remnants

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    Properties of observed SNRs

    Known GeV/TeV SNRs and bimodal age distribution

    Leptonic v.s. Hadronic

    Useful References

    • Dwarkadas (2013) has similar idea of SNR hadronic \(\gamma\)-ray evolution; also mature SNR evolution.

    • Cristofari et al. (2013) calculated the flux evolution, but doesn’t compare with observations

    • We need progenitor mass v.s. bubble size; ambient density distribution; thin shell baryonic mass bushed by wind; SNR evolution in wind environment; electon propagation and cooling; magnetic field strength evolution at SNR shock, etc.

    Model

    Progenitor of SNR and environment

    \label{sec:progenitor} \(M_{prog}\): Bubble size (\(R_{bub}\)); Density distribution; SNR energy

    SNR evolution

    \label{sec:SNRevo} (Dwarkadas et al., 1998)

    Electron energy spectrum evolution and propagation

    Magnetic Field Evolution

    Magnetic field \(B(t)\propto t^{\alpha}\). Assuming magnetic flux conservation: \(B\times A = \rm const.\), we get \(B\propto R^{-2}\propto t^{-4/5}\) for Sedov case. Incorporating \(V\propto t^{-3/5}\), the relationship between magnetic energy density and velocity becomes \(E_B \propto B^2 \propto V^{8/3}\), well belongs to the range 2 to 3.

    Electron Energy Density Evolution

    Injection: Energy Density of Electrons (normalization: const on time, determined by \(K_{ep}\)), spectral index (-2), maximum energy (acceleration, escape, loss).

    Diffusion: One-zone or space-dependence

    Energy losses

    Energy Continuum Equation: \[\frac{\partial}{\partial t} N(\gamma, t) = Q_{inj}(\gamma, t) - \frac{\partial}{\partial \gamma}[\dot{\gamma}N(\gamma, t)] \sim Q_{inj}(\gamma, t) - \frac{N(\gamma, t)}{\tau_{loss}}\]

    Proton diffusion and emission

    Consider all baryonic target is gathered in a thin shell with size comparable to the projenitor bubble size (\(R_{bub}\)), see Section \ref{sec:progenitor}. CR protons are accelerated by SNR shock with time-dependent size evolution, see Section \ref{sec:SNRevo}. \[\label{eq: proto_spec} f_{had}(E, R, t)=\int_0^t\frac{Q_0 E^{-\alpha}}{4\pi^{3/2}R_{esc}R\sqrt{4D(E)(t-\tau)}} \left( e^{-\frac{(R-R_{esc})^2}{4D(E)(t-\tau)}} - e^{-\frac{(R+R_{esc})^2}{4D(E)(t-\tau)}}\right) d\tau\]

    Prediction of SNR statistics

    keV/GeV/TeV time-evolution

    Also the flux ratio of keV-to-GeV GeV-to-TeV: \(\frac{F_{keV}}{F_{GeV}}\) or \(\frac{F_{TeV}}{F_{GeV}}\)

    GeV/TeV spectral index

    Detectablity of GeV/TeV SNRs by Fermi and HESS with current SN rate and IMF

    References

    1. P. Cristofari, S. Gabici, S. Casanova, R. Terrier, E. Parizot. Acceleration of cosmic rays and gamma-ray emission from supernova remnants in the Galaxy. 434, 2748-2760 (2013). Link

    2. V. V. Dwarkadas. Exploring the \(\gamma\)-ray emissivity of young supernova remnants - I. Hadronic emission. 434, 3368-3377 (2013). Link

    3. V. V. Dwarkadas, R. A. Chevalier. Interaction of Type IA Supernovae with Their Surroundings. 497, 807-823 (1998). Link