Theoretical $Sigma$-$D$ Relations for Shell-Type Galactic Supernova Remnants

Relations between radio surface brightness ($Sigma$) and diameter ($D$) of supernova remnants (SNRs) are important in astronomy. In this paper, following the work Duric & Seaquist (1986) at adiabatic phase, we carefully investigate shell-type supernova remnants at radiative phase, and obtain theoretical $Sigma$-$D$ relation at radiative phase of shell-type supernova remnants at 1 GHz. By using these theoretical $Sigma$-$D$ relations at adiabatic phase and radiative phase, we also roughly determine phases of some supernova remnant from observation data.