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In this paper we present new empirical radio surface brightness-to-diameter({\Sigma} - D) relations for supernova remnants (SNRs) in our Galaxy. We alsopresent new theoretical derivations of the {\Sigma} - D relation based onequipartition or on constant ratio between cosmic rays and magnetic fieldenergy. A new calibration sample of 60 Galactic SNRs with independentlydetermined distances is created. Instead of (standard) vertical regression,used in previous papers, different fitting procedures are applied to thecalibration sample in the log {\Sigma} - log D plane. Non-standard regressionsare used to satisfy the requirement that values of parameters obtained from thefitting of {\Sigma} - D and D - {\Sigma} relations should be invariant withinestimated uncertainties. We impose symmetry between {\Sigma} - D and D -{\Sigma} due to the existence of large scatter in both D and {\Sigma}. Usingfour fitting methods which treat {\Sigma} and D symmetrically, different{\Sigma} - D slopes {\beta} are obtained for the calibration sample. MonteCarlo simulations verify that the slopes of the empirical {\Sigma} - D relationshould be determined by using orthogonal regression, because of its goodperformance for data sets with severe scatter. The slope derived here ({\beta}= 4.8) is significantly steeper than those derived in previous studies. Thisnew slope is closer to the updated theoretically predicted surfacebrightness-diameter slope in the radio range for the Sedov phase. We alsoanalyze the empirical {\Sigma} - D relations for SNRs in the dense environmentof molecular clouds and for SNRs evolving in lower-density interstellar medium.Applying the new empirical relation to estimate distances of Galactic SNRsresults in a dramatically changed distance scale.

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