The Σ - D relation for planetary nebulae

We present an extended analysis of the relation between radio surface brightness and diameter - the so-called Sigma - D relation for planetary nebulae (PNe). We revise our previous derivation of the theoretical Sigma - D relation for the evolution of bremsstrahlung surface brightness in order to include the influence of the fast wind from the central star. Different theoretical forms are derived: Sigma proportional to D-1 for the first and second phases of evolution and Sigma proportional to D-3 for the final stage of evolution. Also, we analyzed several different Galactic PN samples. All samples are influenced by severe selection effects, but the Malmquist bias seems to be less influential here than in the supernova remnant (SNR) samples. We derived empirical Sigma - D relations for 27 sample sets using 6 updated PN papers from which an additional 21 new sets were extracted. Twenty four of these have a trivial form of beta approximate to 2. However, we obtain one empirical Sigma - D relation that may be useful for determining distances to PNe. This relation is obtained by extracting a recent nearby (< 1 kpc) Galactic PN sample.