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# Lepto-hadronic Unified Model of $$\gamma$$-ray Supernova Remnants

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# 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

## 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 $$\frac{F_{TeV}}{F_{GeV}}$$

## 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