The radio surface brightness to diameter relation for galactic supernova remnants: sample selection and robust analysis with various fitting offsets

In this paper we present new empirical radio surface brightness-to-diameter (Σ - D) relations for supernova remnants (SNRs) in our Galaxy. We also present new theoretical derivations of the Σ - D relation based on equipartition or on constant ratio between cosmic rays and magnetic field energy. A new calibration sample of 60 Galactic SNRs with independently determined distances is created. Instead of (standard) vertical regression, used in previous papers, different fitting procedures are applied to the calibration sample in the log Σ - log D plane. Non-standard regressions are used to satisfy the requirement that values of parameters obtained from the fitting of Σ - D and D - Σ relations should be invariant within estimated uncertainties. We impose symmetry between Σ - D and D - Σ due to the existence of large scatter in both D and Σ. Using four fitting methods which treat Σ and D symmetrically, different Σ - D slopes β are obtained for the calibration sample. Monte Carlo simulations verify that the slopes of the empirical Σ - D relation should be determined by using orthogonal regression, because of its good performance for data sets with severe scatter. The slope derived here (β = 4.8) is significantly steeper than those derived in previous studies. This new slope is closer to the updated theoretically predicted surface brightness-diameter slope in the radio range for the Sedov phase. We also analyze the empirical Σ - D relations for SNRs in the dense environment of molecular clouds and for SNRs evolving in lower-density interstellar medium. Applying the new empirical relation to estimate distances of Galactic SNRs results in a dramatically changed distance scale.