The Sigma - D relation for planetary nebulae

TL;DR

This paper refines the theoretical $ ext{Σ-D}$ relation for planetary nebulae by including the effects of stellar winds and analyzes multiple Galactic PN samples to derive empirical relations, identifying a potentially useful distance indicator.

Contribution

The paper introduces an extended theoretical model for the $ ext{Σ-D}$ relation incorporating stellar wind effects and provides empirical relations from diverse samples, highlighting a potentially practical distance estimation method.

Findings

Theoretical $ ext{Σ-D}$ relations vary across evolutionary stages.

Most empirical relations have a slope around 2.

A specific empirical relation from a nearby PN sample may aid distance measurements.

Abstract

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: $\Sigma \propto D^{-1}$ for the first and second phases of evolution and $\Sigma\propto D^{-3}$ for the final stage of evolution. Also, we analyzed several different Galactic PN samples. All samples are influenced by severe selection effects, but 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 $\beta \approx 2$. 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.