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

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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.

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

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

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.

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

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}. \[\label{eq: proto_spec} f_{had}(E, R, t)=\int_0^t\frac{Q_0 E^{-\alpha}}{4\pi^{3/2}R_{esc}R\sqrt{4D(E)(t-\tau)}} \left( e^{-\frac{(R-R_{esc})^2}{4D(E)(t-\tau)}} - e^{-\frac{(R+R_{esc})^2}{4D(E)(t-\tau)}}\right) d\tau\]

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