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

Oh, an empty article!

You can get started by double clicking this text block and begin editing. You can also click the Insert button below to add new block elements. Or you can drag and drop an image right onto this text. Happy writing!

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.


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