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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 lowerdensity interstellar medium. Applying the new empirical relation to estimate distances of Galactic SNRs results in a dramatically changed distance scale.

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Here, we compare the sunspot counts and the number of sunspot groups (SGs) with variations of total solar irradiance (TSI), magnetic activity, Ca II K-flux, faculae and plage areas. We applied a time series method for extracting the data over the descending phases of solar activity cycles (SACs) 21, 22 and 23, and the ascending phases 22 and 23. Our results suggest that there is a strong correlation between solar activity indices and the changes in small (A, B, C and H-modified Zurich Classification) and large (D, E and F) SGs. This somewhat unexpected finding suggests that plage regions substantially decreased in spite of the higher number of large SGs in SAC 23 while the Ca II K-flux did not decrease by a large amount nor was it comparable with SAC 22 and relates with C and DEF type SGs. In addition to this, the increase of facular areas which are influenced by large SGs, caused a small percentage decrease in TSI while the decrement of plage areas triggered a higher decrease in the magnetic field flux. Our results thus reveal the potential of such a detailed comparison of the SG analysis with solar activity indices for better understanding and predicting future trends in the SACs.