The $\Sigma-{\sf \textit D}$ relation for planetary nebulae

Abstract

We present an extended analysis of the relation between radio surface brightness and diameter -the so-called Σ − D relation for planetary nebulae (PNe). We revise our previous derivation of the theoretical Σ − 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:for the first and second phases of evolution and Σ ∝ 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 Σ − 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 β ≈ 2. However, we obtain one empirical Σ − 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.

Radio planetary nebulae in the Small Magellanic Cloud

“…For larger values of slope, the sensitivity effect should force the measured slope to be little more than 2. Our measured value of β is expected to be subject to the flattening effect, however, we measure a β = 2.7 ± 0.4 which is consistent with the Galactic data analysed by Urošević et al (2009).…”

“…Our examination indicates that they are actually PN mimics. The exclusion of the identified mimics results in the excellent agreement with the Galactic measurements that Urošević et al (2009) reported.…”

“…Amnuel et al (1984) 44 Galactic PN calibrators with distances of less than 0.7 kpc to determine the multiplier, A, and the exponent, β in the equation Σ = A · D −β (β = 2.07 ± 0.19) with a strong correlation coefficient (r) of 0.86. Urošević et al (2009) used the interacting stellar winds model of Kwok (1994) to derive a theoretical Σ − D model for optically thin PNe. They noted that the majority of PNe will evolve from optically thick to optically thin but the time it takes for this to occur is dependent on many factors.…”

“…Their empirical fits to various Galactic data sets from the literature with 5 GHz RC surface brightness and optically measured diameters result in values of β from approximately 1.4 to 2.6. The data set with the best correlation coefficient Urošević et al (2009) used was Galactic data from PNe at a distance of < 1 kpc and found to be β = 2.61±0.21 with r = −0.97. With this distance limit there should be no Malmquist bias in their results.…”

The Σ – D relation for Galactic planetary nebulae: Application of orthogonal fitting procedure

“…Here we calibrate radio surface brightness to diameter (Σ − D) relation. Urošević et al (2009) derived theoretical Σ − D in the common power law form using the interacting stellar winds model (Zhang & Kwok 1993;Kwok 1994). Σ = S/Ω and D = θ d, where S is the source flux density, Ω is source occupied solid angle, θ is the source angular radius and d is the distance to the source.…”