# The Sigma-D Relation for Planetary Nebulae: Preliminary Analysis

## TL;DR

This paper investigates the Sigma-D relation for planetary nebulae, deriving a theoretical model, analyzing observational data, and concluding that the empirical relation is not reliable for distance estimation.

## Contribution

The paper presents a theoretical derivation of the Sigma-D relation for planetary nebulae and evaluates its empirical usefulness, addressing biases present in previous studies.

## Key findings

- Theoretical Sigma-D relation derived for planetary nebulae.
- Updated Galactic PNe sample shows minimal Malmquist bias.
- Empirical Sigma-D relation is not effective for distance determination.

## Abstract

An analysis of the relation between radio surface brightness and diameter, so-called Sigma-D relation, for planetary nebulae (PNe) is presented: i) the theoretical Sigma-D relation for the evolution of bremsstrahlung surface brightness is derived; ii) contrary to the results obtained earlier for the Galactic supernova remnant (SNR) samples, our results show that the updated sample of Galactic PNe does not severely suffer from volume selection effect - Malmquist bias (same as for the extragalactic SNR samples) and; iii) we conclude that the empirical Sigma-D relation for PNe derived in this paper is not useful for valid determination of distances for all observed PNe with unknown distances.