The H\alpha surface brightness – radius relation

# The Hα surface brightness – radius relation: a robust statistical distance indicator for planetary nebulae

David J. Frew, Q.A. Parker and I.S. Bojičić
Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China
Department of Physics and Astronomy, Macquarie University, NSW 2109, Australia
Australian Astronomical Observatory, P.O. Box 296, Epping, NSW 1710, Australia
E-mail: djfrew@hku.hk
Accepted ; Received ; in original form
###### Abstract

Measuring the distances to Galactic planetary nebulae (PNe) has been an intractable problem for many decades. We have now established a robust optical statistical distance indicator, the H surface brightness – radius or  relation, which addresses this problem. We developed this relation from a critically evaluated sample of primary calibrating PNe. The robust nature of the method results from our revised calibrating distances with significantly reduced systematic uncertainties, and the recent availability of high-quality data, including updated nebular diameters and integrated H fluxes. The  technique is simple in its application, requiring only an angular size, an integrated H flux, and the reddening to the PN. From these quantities, an intrinsic radius is calculated, which when combined with the angular size, yields the distance directly. Furthermore, we have found that optically thick PNe tend to populate the upper bound of the trend, while optically-thin PNe fall along the lower boundary in the  plane. This enables sub-trends to be developed which offer even better precision in the determination of distances, as good as 18 per cent in the case of optically-thin, high-excitation PNe. This is significantly better than any previous statistical indicator. We use this technique to create a catalogue of statistical distances for over 1100 Galactic PNe, the largest such compilation in the literature to date. Finally, in an appendix, we investigate both a set of transitional PNe and a range of PN mimics in the  plane, to demonstrate its use as a diagnostic tool. Interestingly, stellar ejecta around massive stars plot on a tight locus in  space with the potential to act as a separate distance indicator for these objects.

###### keywords:
techniques: photometric – circumstellar matter – stars: distances – ISM: bubbles – H ii regions – planetary nebulae: general.
pagerange: The H surface brightness – radius relation: a robust statistical distance indicator for planetary nebulaeLABEL:lastpagepubyear: 2002

## 1 Introduction

One of the greatest difficulties still facing the study of planetary nebulae (PNe) in our own Galaxy has been the problem of determining accurate distances to them. Due to the wide range of effective temperatures and bolometric luminosities seen in their ionising stars, they are not suitable as standard candles111However the well-known PN luminosity function (PNLF) works as an effective distance indicator for an ensemble of luminous PNe (see Ciardullo 2012, for a recent review)., nor can their expanding PNe be used as standard rulers. Indeed, the most reliable distances are for PNe located in external galaxies, such as M 31 and the Large and Small Magellanic Clouds (e.g. Jacoby & De Marco 2002; Reid & Parker 2006). This problem has led to the application of a range of secondary distance methods for Galactic PNe, which we will evaluate as part of this work. For reviews of the older Galactic distance scales, the reader is referred to the works of Minkowski (1965), Gurzadyan (1970), Smith (1971) and Liller (1978). The PN distance-scale problem was nicely summarised by Ciardullo et al. (1999, hereafter CB99) who stated that “it is unfortunately less obvious how one could devise a new ‘grand unification’ calibration that simultaneously handles both the lower surface brightness objects that prevail among the nearby nebulae and the brighter PNe that dominate samples like those in the Galactic bulge and extragalactic systems. We leave this daunting task to future workers.

So far accurate primary distances (with uncertainties 10%) are known for less than one per cent of the more than 3400 Galactic PNe that have so far been catalogued (Parker et al. 2015, in preparation), of which the most accurate come from trigonometric parallaxes of their central stars (CSPNe; Benedict et al. 2003, 2009; Harris et al. 2007). Generally speaking, distance estimates to the bulk of PNe are statistical in nature and rely on quantities which have a large observed dispersion (e.g. Cahn, Kaler & Stanghellini 1992, hereafter CKS; Stanghellini, Shaw & Villaver 2008, hereafter SSV). Uncertainties in the Galactic PN distance scale have been significant, up to factors of three or more (e.g. Zhang 1995, hereafter Z95; Van de Steene & Zijlstra 1995; CB99; Napiwotzki 2001; Phillips 2002; SSV). This uncertainty severely hampers attempts to derive meaningful physical quantities for most Galactic PNe. Almost every quantity of interest, including nebular radii, masses, luminosities and dynamical ages, and the luminosities and masses of their CSPNe, depends on accurate knowledge of their distances, as do all statistical determinations of the PN scale height, space density, and formation rate (Ishida & Weinberger 1987).

In this paper we develop and calibrate a new optical statistical distance indicator, the H surface brightness – radius relation ( relation hereafter). Here we address the problem posed by CB99, and our results show that the controversy surrounding the long-running PN distance scale problem has finally been put to rest. Our technique is relatively simple in its application, requiring an angular size, an integrated H flux, and the reddening of the PN. From these quantities, an intrinsic radius is calculated, which when combined with the angular size, yields the distance directly. We have chosen H as the most optimum emission-line, firstly as it best represents the nebular ionized mass, and secondly because a number of narrowband H imaging surveys have recently become available, from which large numbers of accurate integrated fluxes, diameters, and surface brightnesses can be determined. These include the SuperCOSMOS H-alpha Survey (SHS; Parker et al. 2005; Frew et al. 2014a), the INT Photometric H-Alpha Survey (IPHAS; Drew et al. 2005), the VST Photometric H-Alpha Survey (VPHAS+; Drew et al. 2014), and the lower-resolution Southern H-Alpha Sky Survey Atlas (SHASSA; Gaustad et al. 2001) and Virginia-Tech Sky Survey (VTSS; Dennison, Simonetti & Topasna 1998).

Our paper is arranged as follows: in §2 we review the various distance methods that have been used in the literature, while we compile a sample of critically-assessed primary distances in §3, which underpins our new relation. In §4 we describe the  relation in detail, and discuss the increase in accuracy obtained by using specialised sub-trends. We also examine the theoretical basis for the relation in this section. We present our catalogue of  distances in §5 (presented in full as an online supplement), and in §6 we investigate the dispersion of the relation, before comparing our final mean distance scale with previous work in §7. This work refines the distance scales presented by Frew (2008; hereafter F08), and the earlier preliminary results given by Pierce et al. (2004), Frew & Parker (2006, 2007), and Frew, Parker & Russeil (2006). We present our conclusions and suggestions for future work in §8, including a discussion of the data expected from the recently launched GAIA astrometric satellite, and how our  relation will remain complimentary to that well into the future. Finally, in an appendix, we investigate both a set of transitional PNe and a range of PN mimics in the  plane, to test its use as a diagnostic tool. Preliminary results show it to have great promise.

## 2 Previous Statistical Methods

The last few decades have seen a wide range of techniques used to measure PN distances, both primary methods which generally have the highest accuracy, and statistical (secondary) methods, which can have considerable uncertainties (of factors of two or more), even if appropriately calibrated. In this section we briefly review the standard statistical techniques previously used in the literature. The reader is referred to the review of Smith (2015) for a fuller discussion of the limitations and biases of each distance technique.

The classical Shklovsky method was the first statistical method to be applied that had any claim to veracity. It assumed a constant ionised mass (typically 0.2 M) for the PN shell and was first applied by Minkowski & Aller (1954) and Shklovsky (1956). Osterbrock (1960) applied this method to NGC 3587 and O’Dell (1962) used newly-determined H fluxes to derive an early distance scale, based on emission theory and the assumption of constant ionised mass; several calibrating nebulae were used to determine the mean ionised mass for PNe. This was followed by the work of Abell (1966), using ‘photored’ fluxes for over 90 evolved PNe, before being further developed by Cahn & Kaler (1971). This distance scale was later utilised by Kaler (1983), Shaw & Kaler (1989), and Kaler, Shaw & Kwitter (1990). Other Shklovsky scales have used the observed proper motions of the central stars, in combination with assumptions regarding their space motions (e.g. O’Dell 1962) to fix the zero point. Cudworth (1974) undertook a statistical calibration of the PN distance scale using a large set of uniformly obtained proper motions, obtaining one of the longest scales to date. However, as these are constant-mass scales, distances to the youngest compact PNe and the largest evolved PNe were in general overestimated and underestimated respectively.

In the simplest terms, and assuming a constant ionised mass, the nebular radius () increases as the PN evolves, and the mean electron density () falls in sympathy. If the mean electron density can be determined from measurements of [O ii] or [S ii] doublet intensities, the intrinsic nebular radius can be calculated. Comparing this to the angular size of the PN leads directly to a distance via simple trigonometry. Variations on this technique, by assuming an ionised mass derived from a set of calibration objects at known distance and using the observable electron density and H flux to infer a distance, have been utilised by Kingsburgh & Barlow (1992) and Kingsburgh & English (1992). A more novel method has been utilised by Meatheringham, Wood & Faulkner (1988), who found that Magellanic Cloud (MC) PNe fall on fairly tight plane in dynamical age – density – excitation-class space. For a sample of Galactic PNe the dynamical age was estimated from the observed electron density and excitation class, and once the expansion velocity is measured, the intrinsic radius can be inferred. Comparing this number with the angular size leads directly to a distance.

An equally common approach in the literature is a variable-mass derivation of the Shklovsky method, as it is now known that PNe have a range of ionised masses, and the standard method can be inaccurate for highly evolved PNe with more massive shells (e.g. Buckley, Schneider & van Blerkom 1993). An initial method was developed by Daub (1982), who empirically related the ionised mass to an optical thickness parameter, derived from the observed 5 GHz (6 cm) radio flux density (), and the angular radius, (in arcsec). The thickness parameter , is defined as:

 T=log(θ2/F5) (1)

A value of = 3.65 (corresponding to = 0.12 pc) was found to separate optically-thick from optically-thin PNe, which were assumed to have a constant mass at large radii. This approach was re-calibrated by CKS, based on a refined set of nebulae with primary distance estimates, setting the thick-thin transition at = 3.13 (corresponding to = 0.09 pc). The ionized mass was determined using:

 logM={T−4    for T<3.13−0.87    for T>3.13} (2)

The intrinsic radius (in pc) was then calculated from the following expression

 logr=0.4logM+0.2T−1.306 (3)

Finally the distance, (in pc), was determined from the well-known formula:

 D=206,265 rθ (4)

Recently SSV re-calibrated the CKS scale using updated Galactic distances as well as data for a large set of LMC and SMC PNe, where the thick-thin transition was now determined to be at = 2.1, or a smaller radius, = 0.06 pc. The SSV scale has been commonly used to date. We will compare our distance results with their work in §7.

Other statistical approaches use an ionised mass that is a continuous function of linear radius, as estimated from the surface brightness (e.g. Maciel & Pottasch 1980; Pottasch 1980, 1984). In general terms the ionised mass–radius relation can be expressed as:

 Mion∝rβ (5)

where is a power-law index determined through observation. While Maciel & Pottasch (1980) found = 1, other authors derived significantly different values for (see Milne 1982; Pottasch 1984; Kwok 1985; Zhang 1995), to be further discussed in §4.4. For more detailed discussions of this point, the reader is referred to Kwok (1985, 1993) and Samland et al. (1993).

A natural variant of the relationship is the brightness temperature–radius () relationship. Again the primary observables are the 5 GHz radio flux, or an equivalent radio or optical Balmer-line flux, and the angular radius, from which a surface brightness can be calculated. Various versions in the radio domain have been proposed by Amnuel et al. (1984), Van de Steene & Zijlstra (1994, 1995), Buckley & Schneider (1995), Zhang (1995), Bensby & Lundström (2001), Phillips (2002, 2004b) Uros̆ević et al. (2009) and Vukotić et al. (2009), amongst others. The 5 GHz brightness temperature, (in K), is defined as:

 Tb=c22πkν2F5θ2   ≃18.3F5θ2 (6)

Based on a set of calibrating nebulae with known distances, an expression for the distance can then be derived, of the form:

 logD=a−blogθ−clogF5 (7)

where , and are empirically determined constants. Relations of this form were used by Zhang (1995), Van de Steene & Zijlstra (1995), and Bensby & Lundstrom (2001), with relatively small (10%) differences in the proportionality constants derived in each study. Schneider & Buckley (1996) took an alternative approach, since they considered a single power-law inadequate to handle both young and old PNe. They fit a second-order polynomial to their calibration sample. However, with the exception of the youngest PNe, a single power law is a reasonable fit to the range of surface brightness seen in PNe, from compact nebulae down to the very faintest objects dissolving into the interstellar medium (ISM). Also, in an attempt to develop a novel robust approach for distance scale calibration, Vukotic et al. (2014) utilized the calibrating sample from SSV. Instead of using the usual fitting procedure they calculated the density of the data points in the fitting plane which resulted in probability distributions of diameter values for selected values of surface brightness. A comparison of some of these radio-based distance scales with our  distance scale is given in §7.

Another potential distance technique is based on the subset of PNe which have central stars still evolving left along the constant luminosity track in the theoretical Hertzsprung-Russell (HR) diagram. If a canonical central star mass of 0.6 M (or a similar value) is assumed, and a temperature of the CSPN can be determined, then an absolute visual magnitude can be predicted using an appropriate bolometric correction (e.g. Vacca, Garmany & Shull 1996). If accurate reddening-corrected photometry is available, then a distance directly follows. Note that the resultant distance scale depends on the adopted mean CSPN mass. Mal’kov (1997, 1998) seems to be the first to mention such a technique, but did not apply it, and it was first utilised (using bolometric magnitudes) by Phillips (2005b). A related approach is to assume a constant absolute magnitude (i.e. a standard candle) for a homogenous sub-sample of CSPNe. Phillips (2005a) took this approach for a set of CSPNe on the cooling track in evolved PNe but there appears to be a significant spread (2 mag) in the absolute magnitudes of the CSPNe in old PNe (see F08), meaning the technique needs to be applied with caution.

Other statistical methods assume a standard ruler technique such as the angular size of the waists in Type I bipolar PNe, assuming these all have a similar intrinsic diameter (Phillips 2004a), but this approach was criticised by Frew et al. (2006). Similarly, Gurzadyan (1970) used the angular diameter of the He ii Strömgren zone at the centre of optically-thick PNe to estimate a distance. However the systematics are not well quantified, and the method saw little application owing to the wide variety of intrinsic diameters, structural parameters, and excitation classes seen in PNe. Finally there are also methods based on mid-infrared (MIR) fluxes, obtained either from IRAS (Tajitsu & Tamura 1998) or MSX data (Ortiz et al. 2011). These generally utilized an assumed dust mass, scaling the distances according to the observed MIR fluxes.

## 3 Calibration of a new statistical distance scale

CB99 stressed the importance of deriving a statistical calibration that simultaneously handles both luminous PNe and the demographically common evolved, faint PNe. These represent a population that are usually avoided as calibrating objects, and this may be the reason for the systematic offsets that have plagued the various statistical distance scales in the past (e.g. Pottasch 1996). Previously, Stanghellini et al. (2002) found a relationship between H surface brightness and radius for a sample of LMC PNe, and Jacoby et al. (2002) outlined the potential of using an  relation as a distance indicator. Such a relation is analogous to the radio relationships that have been the basis of many previous statistical distance scales (see §2).

Independently, we came to the same conclusion regarding the benefits of using an  relation as a distance indicator, based on a sample of Galactic PNe (see Pierce et al. 2004). Our new relation also has the added benefit of including the most extreme PNe at the very bottom of the PNLF, which have traditionally been selected against in the radio regime (Zhang & Kwok 1993; CB99). We chose to use the H emission-line (rather than the radio continuum) owing to the recent increase in reliable H fluxes available for Galactic PNe. In particular, Frew et al. (2013, hereafter FBP13) and Frew et al. (2014a) have presented accurate H fluxes for about 1300 PNe in total, a significant fraction for the first time. However, a disadvantage in using the brighter H flux instead of H is that a correction for the [N ii]contribution is often required, though if done correctly the derived H integrated flux is accurate (see the discussion in FBP13). Drawing on our new database of fundamental parameters for PNe (Parker et al., in prep.), which includes fluxes, extinctions, emission-line ratios and angular diameters, the  relation has been calibrated across the full range of surface brightness seen in PNe, from young, high-density, luminous objects like NGC 7027 through to some of the faintest known PNe such as TK 1 (Ton 320).

It is crucially important that the sample be as free from systematic bias as possible. Earlier authors have diluted the precision of their calibrating sample by including PNe with poorly known distances, or by not weighting the individual distance estimates to the PN calibrators with appropriate uncertainties (cf. Bensby & Lundström 2001; Phillips 2002, 2004b). Furthermore, more than one study has inadvertently included H ii regions, symbiotic outflows, and other mimics as ‘PN calibrators’, which add significant noise to the derived relationship. We have used a range of diagnostic tools to remove these contaminants (Frew & Parker 2010), so our approach does not suffer from the same issues.

### 3.1 A Critical Evaluation of Primary Methods and Distances

Unfortunately, published primary distances are of widely varying quality, but a number of primary methods have been used with varying degrees of success; for earlier reviews, see Acker (1978) and Sabbadin (1986). These techniques include direct trigonometric parallaxes of the CSPN (Harris et al. 2007; Benedict et al. 2009), or a photometric or spectroscopic parallax determined for a physical companion to the CSPN (Bond & Ciardullo 1999; CB99). The analysis of eclipsing binary CSPNe (e.g. Bell, Pollacco & Hilditch 1994) is potentially one of the most accurate to constrain PN distances, and the membership of a PN in a star cluster of known distance is also a highly promising technique, especially for the future (see Parker et al. 2011).

A description of the primary distance methods used to define the Galactic calibrating sample are briefly described in the following subsections. Individual PN distances are tabulated in each section, and a critical assessment of their associated uncertainties also follows. These literature distances have been carefully examined, and in many cases revised with better systematics, and we also include several new kinematic and extinction distance determinations derived as part of this work. We then present a final set of calibrating distances in §4.2, which has allowed an  relationship to be defined over six decades in log H surface brightness. It should also be emphasised that no statistical distances from other studies have been used as calibrators for our  relation (cf. Bensby & Lundström 2001; Ortiz et al. 2011).

#### 3.1.1 Trigonometric distances

Direct trigonometric parallaxes have been measured for more than a dozen nearby CSPNe, either from the ground (e.g. Harris et al. 1997, 2007), the Hipparcos satellite (Acker et al. 1998; van Leeuwen 2007), or the Hubble Space Telescope (HST; Benedict et al. 2003, 2009). The ground-based US Naval Observatory (USNO) CCD parallaxes of Harris et al. (2007) form an homogenous sample of accurate distances for several nearby PNe, and Smith (2015) has shown that they form a reliable, internally consistent dataset. Owing to uncertain systematics, we have not used the ground-based data from Gutiérrez-Moreno et al. (1999). The Hipparcos parallaxes (van Leeuwen 2007) have also been shown to be problematic (e.g. Smith 2015), especially for compact PNe where subtle surface brightness variations across the PN may have had an undue influence on the astrometric reductions, therefore the Hipparcos parallaxes have not been used as calibrating data (cf. F08). Finally, we also adopt the distance to the young, compact nebula K 3-35 (Tafoya et al. 2011), determined using VLBI Exploration of Radio Astrometry (VERA) array observations of a bright water maser in the nebula222Maser trigonometric distances for several pre-PNe are discussed in Vickers et al. (2015)..

Note that the trigonometric method is susceptible to the so-called Lutz-Kelker (L-K) bias (Lutz & Kelker 1973; Smith 2003, 2006; Francis 2014) which causes measured parallaxes to be systematically greater than their actual values in a statistical sense, and is broadly related to the Trumpler-Weaver bias (Trumpler & Weaver 1953). As emphasised by van Leeuwen (2007) and Francis (2014), the L-K bias is a sample statistical correction, and has not been applied to individual distances. In the future, the number of trigonometric parallaxes for CSPNe will be revolutionised with the results from the Gaia satellite (Perryman et al. 2001; Bailer-Jones 2002). This point will be further discussed in our conclusions. Table 1 summarises the determinations taken from the literature.

#### 3.1.2 Photometric distances

This method estimates a spectroscopic or photometric parallax for a companion star of normal spectral type. The advantage of using this method was noted early on by Minkowski & Baum (1960) and Cudworth (1973, 1977). The archetype is the well-studied, high-excitation PN, NGC 246 (Bond & Ciardullo 1999) and the method has been applied to a number of more distant PNe with wide binary companions, mostly by CB99. Still other binary systems are dominated by the companion star, usually a B or A main-sequence star, or a cooler giant or subgiant, and for these a spectroscopic parallax is also feasible (e.g. Longmore & Tritton 1980). Absolute magnitudes have been taken from De Marco et al. (2013) for main sequence stars, and Schmidt-Kaler (1982) or Jaschek & Gómez (1998) for the evolved stars.

The binary associations evaluated by CB99 have been re-evaluated here using better estimates for the reddening, derived both from unpublished spectroscopic data and from all available CSPN photometry (see F08; De Marco et al. 2013). Furthermore, none of CB99’s possible or doubtful associations have been considered (cf. F08), and of the probable associations, the distance of K 1-27 has been rejected. This distance, based on the companion being a white dwarf (WD) which was fit to the cooling sequence, is only a quarter of a newly calculated gravity distance (see Table 5), derived from the data presented by Reindl et al. (2014). If the companion to K 1-27 is in turn an unresolved dM / WD pair, the true colour of the WD would be bluer and hence the luminosity larger. Alternatively, though with low probability, the companion is a background quasar. This object is further noted in §4.3.4.

We have revised the luminosity class of the companion of NGC 7008 to IV (from V as assumed by CB99). Using = for the G8 star (Schmidt-Kaler 1982), the distance is 900 pc adopting the C99 reddening, or 970 pc using our revised value. Now the ionising star is no longer underluminous as it was using the original distance. We also determine a revised distance of 3.0 kpc to the barium K-type giant in WeBo 1, based on the same arguments as Bond et al. (2003). However, we adopt a larger stellar mass of 4 , based on the nebula’s type I chemistry as inferred from the observed [N ii]/[S ii] ratio (see Fig. 10 of Smith, Bally & Walawender 2007). Table 2 summarises the distance determinations.

Some other companion-dominated systems are not used as calibrators owing to the uncertain luminosity class of the cool star; examples include Abell 70 (Miszalski et al. 2012), Abell 82 (CB99), Hen 3-1312 (Pereira 2004), K 1-6 (Frew et al. 2011), and IC 972 (Douchin et al. 2014). In other cases, the identification of the central star is in doubt (e.g. RCW 21; Rauch et al. 1999), or the object is unlikely to be a true PN, such as the nearby bowshock nebula Abell 35 (F08; Ziegler et al. 2012b).

#### 3.1.3 Eclipsing / Irradiated Binaries

Eclipsing binaries are fundamental astrophysical yardsticks, but the analysis of the small sample of eclipsing binary CSPNe has led to few distance determinations to date (Pollacco & Bell 1993, 1994; Bell et al. 1994). Close binary CSPNe showing a large irradiation (reflection) effect can also be used, such as DS 1 (Drilling 1985), and LTNF 1 around BE UMa (Liebert et al. 1995; Ferguson et al. 1999). These methods are partly model dependent however, but offer great promise if the systematics are well understood. Unfortunately, eclipsing CSPNe are rather uncommon, but a recent very accurate distance for the double-lined binary in SuWt 2 has been obtained by Exter et al. (2010). Table 3 summarises the adopted calibrating distances.

We also note the bipolar object Hen 2-428, which has recently been suggested to contain a super-Chandrasekhar, double-degenerate nucleus (Santander-García et al. 2015), indicating this is a potential Type Ia supernova progenitor. However, this interpretation has been questioned by Garcia-Berro, Soker & Althaus (2015). We note that a short model distance of 1.4 kpc is derived from the analysis of Santander-García et al. (2015), making the surrounding nebula very underluminous as well as the central star’s luminosity discrepant with standard post-AGB evolutionary tracks (Garcia-Berro, Soker & Althaus 2015). Our mean  distance of 2.7 kpc suggests the evolutionary interpretation of Garcia-Berro et al. may be more likely.

#### 3.1.4 Cluster Distances

Physical membership of a PN in an open or globular star cluster provides an accurate distance, and is an important key that can help to unlock many of the problems facing PN research (Parker et al. 2011). At present the number of Galactic PNe thought to be genuine members of clusters is small; a few at best in open clusters, with four Galactic globular clusters currently thought to contain PNe (Jacoby et al. 1997). Historically, NGC 2438 was assumed to be a member of the young open cluster M 46 (NGC 2437), but Kiss et al. (2008) showed that they were unrelated.333Vickers et al. (2015) summarised the evidence showing that the bipolar, symbiotic-like outflow OH 231.8+4.2 is a bona fide member of this cluster. Additionally, NGC 2818 was thought to be physically associated with the open cluster of the same name (e.g. Tifft, Connolly & Webb 1972), but Mermilliod et al. (2001) claimed the objects were unrelated. However, recent work by Vázquez (2012), has shown that the PN velocity is consistent with membership. In the meantime, PHR J1315-6555 was shown by Parker et al. (2011) to be a physical member of the intermediate-age open cluster ESO 96-SC04. The compact object NGC 6741 has been suggested to be a possible member of Berkeley 81 (Sabbadin et al. 2005), but while the distances are comparable, the radial velocity of the cluster is 8km greater (Sabbadin et al. 2005; Magrini et al. 2015), suggesting non-membership of the PN. It should be noted that the recent increases in numbers of both Galactic PNe and open clusters (Dias et al. 2002) have increased the probability of positional coincidences between these two classes of object. Lists of coincidences between clusters and PNe have been given by Kohoutek (2001) and Majaess, Turner & Lane (2007), and recently two more possible associations (Abell 8 and Hen 2-86) have been presented in the literature (Turner et al. 2011; Moni Bidin et al. 2014). The currently suggested associations are discussed individually below.

NGC 2818:   Tifft et al. (1972) argued that NGC 2818 was a member of the open cluster of the same name, and this became accepted as a valid association. Dufour (1984) and Pedreros (1989) also assumed a physical association, but gave conflicting distances to the cluster. However, Mermilliod et al. (2001) obtained accurate velocities for 12 cluster red giants to obtain a mean velocity of = kms, significantly different to the PN velocity of kms (Meatheringham et al. 1988), suggesting a line-of-sight coincidence. More recently, Vázquez (2012) reanalysed the complex kinematics of the nebula, finding a systemic heliocentric velocity (27 km) in closer agreement with the open cluster, suggesting membership, with which we now concur. The cluster distance of 3.0 kpc is derived from the reddening and distance modulus given by Mermilliod et al. (2001), in turn based on a deep colour-magnitude diagram from Stetson (2000).

PHR J1315-6555:  Parker et al. (2011) undertook a detailed study of the physical association between this bipolar PN and the intermediate-age open cluster ESO 96-SC04 (AL 1). Majaess et al. (2014) refined the distance to the cluster to  kpc, which we have adopted herein.

BMP J1613-5406:  This evolved bipolar PN is a likely member of the Cepheid-hosting open cluster NGC 6067, based on positional coincidence and close agreement in radial velocities. A full account of this very interesting association will be published separately (Frew et al., in preparation).

Abell 8:  Bonatto, Bica & Santos (2008) have identified a new intermediate-age open cluster in the field of this faint, round PN, giving a reddening of () = 0.29 0.03 and a distance, = 1.7 0.1 kpc. Based on their similar radial velocities, Turner et al. (2011) argued that this is a real association. However, there are difficulties with this assessment. Using the integrated flux of (H) = (FBP13), an average reddening, () = 0.51 0.09 (Kaler 1983; Ali 1999; Phillips, Cuesta & Kemp 2005), and a diameter of 60 (Abell 1966), the PN plots well below other optically thick PNe of similar surface brightness in  space. We conclude that the PN is either a cluster non-member or that the cluster distance is significantly in error. Owing to these uncertainties, we have not used Abell 8 as a primary calibrator.

Hen 2-86:  Moni Bidin et al. (2014) suggested this compact PN was a likely member of NGC 4463, primarily based on the similarities in their radial velocities. However the reddening to the PN, () = 1.3 – 1.5, is much greater than the cluster value, () = 0.42. Those authors suggested the PN shows high internal reddening, but the amount would be greater than any other PN reliably measured to date (see Phillips 1998). Owing to this discrepancy, we prefer not to use Hen 2-86 as a primary calibrator.

Globular cluster PNe:  Both Pease 1, also known as K 648 (Buell et al. 1997; Alves, Bond & Livio 2000) and the peculiar H-deficient nebula GJJC 1 (Cohen & Gillett 1989; Borkowski & Harrington 1991) are bona fide members of their respective globular clusters, M 15 (NGC 7078) and M 22 (NGC 6656).  Pease 1 has been imaged with HST and has good estimates of its angular size (Alves et al. 2000) and integrated flux which qualify it to be a primary calibrator. Jacoby et al. (1997) conducted an extensive search for PN candidates in Galactic globular clusters, finding two new examples, JaFu 1 in Palomar 6 and JaFu 2 in the luminous cluster NGC 6441. JaFu 2 is a certain member of NGC 6441, but JaFu 1 was a less convincing candidate, owing to its large separation from the core of Pal 6 (though still within the tidal radius), and its radial velocity being only marginally consistent with membership. However, a new cluster velocity, = +181 3 kms (Lee, Carney & Balachandran 2004), greatly increases the membership probability. JaFu 1, JaFu 2 and Ps 1 are all adopted as primary calibrators.

#### 3.1.5 Model Atmosphere (Gravity) Distances

This is potentially a powerful method to determine spectroscopic distances directly for the CSPN (cf. Heap 1977). It aims to determine the stellar effective temperature and the surface gravity based on an NLTE model atmosphere analysis (e.g. Méndez et al. 1988; Napiwotzki 2001). In principle, it is an elegant method, albeit partly model dependent. It appears most published distances have systematic errors, with the greatest observational uncertainty being the determination of the surface gravity, expressed as log  (e.g. Pottasch 1996; Rauch et al. 2007). The other observables are the visual magnitude and reddening. From these data, the surface flux, mass and intrinsic radius of the star can be inferred, and using the reddening-corrected magnitude, a distance can be directly determined. The distance is derived using the following equation (Méndez et al. 1988):

 D2=3.82×10−9 McF⋆g 100.4V0 (8)

where is the distance in kpc, is the stellar (core) mass in solar units, is the monochromatic Eddington flux in units of erg cms Å at 5480 Å (Heber et al. 1984), is the surface gravity in cm s and is the extinction-corrected visual magnitude. In turn the Eddington flux can be suitably approximated by the following linear equation if the effective stellar temperature, (in K), is known (Cazetta & Maciel 2000):

 F⋆=1.85×104 T⋆ − 9.97×107 (9)

Nonetheless there are caveats to this approach, and a number of criteria have been employed to minimise any bias in the adopted distance scale. Because it is often difficult to simultaneously fit a model atmosphere to all the Balmer lines in the optical spectrum of a hot WD (the Balmer line problem) due to the incomplete treatment of metal opacities in the models (e.g. Werner 1996), there can be significant errors in the effective temperature and the surface gravity, though modern analyses consider more detailed treatments of the metal lines (e.g. Gianninas et al. 2010). The problem had also been noted by Pottasch (1996) who found that the log  values derived from the often-used H line profile are often systematically too low (see also Rauch et al. 2007).

Indeed, several independent lines of evidence point to problems with some of the published determinations, especially some of the older ones (see Pottasch 1996; Smith 2015). More specifically, the log  values are often underestimated, especially at low to moderate surface gravities. This is illustrated in Fig. 2 of Napiwotzki (1999), where the mean mass of CSPNe with log  6.0 is considerably less than the mean mass of the higher gravity objects, indicating a systematic underestimation of the gravities. As a further example, the gravity distances derived from the Lyman-line data of Good et al. (2004) are in better agreement with the USNO trigonometric distances, than the Balmer-line determinations, and in turn, the older Balmer determinations of Napiwotzki (1999, 2001). As another consistency check, the mean mass of an ensemble of DAO WDs (see table 5 of Good et al. 2005) using the Lyman method agrees better than the Balmer method with the canonical WD average mass of 0.60  (e.g. Tremblay, Bergeron & Gianninas 2011; Kleinman et al. 2013). Yet despite recent advances in NLTE modelling, systematic errors in the determination of the surface gravity persist. Traulsen et al. (2005) give a surface gravity for the CSPN of the Helix nebula, as log  = 6.3 (in cgs units). The resulting distance of 780 pc is way outside the error bar of the recent trigonometric distance of 216 pc (Benedict et al. (2009). Even for the well-studied star LS V +46 21, the CSPN of Sh 2-216, there remains an unexplained discrepancy between the recent spectroscopic distance of Rauch et al. (2007) and the well-determined parallax distance from Harris et al. (2007).

Pauldrach, Hoffmann & Méndez (2004) have taken a different approach, also based on model atmospheres. The mass and radius of the CSPN are calculated from the mass loss rate, and the terminal wind velocity , as estimated from a fit to the spectral lines. However, very high masses were determined for some CSPNe, near the Chandrasekhar limit, and the resulting very large distances have not been supported by other methods (see the discussion of Napiwotzki 2006). They have not been considered further.

In order to derive appropriately weighted mean gravity distances (in cases where two or more NLTE analyses exists in the literature), all suitable and log  determinations have been compiled to be used in conjunction with updated reddening values and visual magnitudes (e.g. F08; De Marco et al. 2013) to calculate a new, internally consistent data-set. Preference has been given to the most recent analyses. Table 5 gives the various PN central stars and the resulting gravity distances derived using equations 8 and 9 above. The stellar mass (needed for the equation 8) has been determined from the log  diagram (not shown) from a comparison with the evolutionary tracks of Blöcker (1995) and Vassiliadis & Wood (1994), interpolating linearly if necessary. Our new distances may differ somewhat from values published prior, due to slight differences between our adopted magnitudes, reddenings, and temperatures, and individual determinations found in the literature.

#### 3.1.6 Expansion parallaxes

A potentially powerful technique is the expansion parallax method, where the PN’s angular expansion in the plane of the sky over a suitably long time period is compared to the shell’s radial velocity, based on either optical or radio data;  Terzian (1997) and Hajian (2006) provide reviews of the technique. We have decided that the expansion parallaxes based on older, ground-based, optical photographs (e.g. Chudovicheva 1964; Liller 1965; Liller et al. 1966) are not of sufficient quality to be useful. Several PNe have distance estimates based on multi-epoch Very Large Array (VLA) 6 cm radio observations (Masson 1986, Hajian, Terzian & Bignell 1993, 1995; Hajian & Terzian 1996; Kawamura & Masson 1996), and are potentially far more accurate than the older optical determinations. Other distance determinations are given by Christianto & Seaquist (1998), Guzmán, Gómez & Rodríguez (2006), Guzmán-Ramirez et al. (2009), and Guzmán et al. (2011). Precise HST optical parallaxes, also based on multi-epoch nebular images, have become available in the last decade (Reed et al. 1999; Palen et al. 2002; Li, Harrington & Borkowski 2002; Hajian 2006) which promise to have a significant impact on the local PN distance scale. Furthermore, Meaburn et al. (2008) and Boumis & Meaburn (2013) have used the proper motions of fast-moving outer optical knots (assuming ballistic motion) to derive distances for NGC 6302 of 1170 140 pc, and KjPn 8 of 1800 300 pc, respectively, though the extended nebula of KjPn 8 does only as a rough integrated H flux available, so has been excluded as a calibrator (but see §A.1).

While expansion parallaxes were thought to be a relatively simple, yet powerful method, it has become apparent that there are serious sources of systematic error in the technique which need to be considered before reliable distances can be determined. Firstly, the majority of PNe are aspherical, so various corrections for prolate ellipsoidal geometries have been applied (e.g. Li et al. 2002), and secondly, the angular expansion rate on the sky (a pattern velocity) was assumed to be equal to the spectroscopically measured gas velocity. However, these are usually not identical in nature. Mellema (2004) modelled the jump conditions for both shocks and ionization fronts, and found that the pattern velocity is typically 30% larger than the matter velocity, hence the calculated distances are too short by this amount. Schönberner, Jacob & Steffen (2005b), using 1-D hydrodynamical modelling, also found that the pattern velocity is always larger than the material velocity. These authors found that the necessary correction factor ranged between 1.3 and 3.0, depending on the evolutionary state of the CSPN. That such biases in expansion distances do exist is provided by the study of the symbiotic nebula Hen 2-147 (Santander-García et al. 2007). These authors found that the expansion parallax method gave a distance of 1.5 0.4 kpc, a factor of two lower than the distance of 3.0 0.4 kpc obtained from the period-luminosity (P-L) relationship for the central Mira variable. Correcting for the jump condition described earlier, these authors find = 2.7 0.5 kpc, in much better agreement with the P-L distance.

Following Mellema (2004), the exact value of the correction factor depends upon the shock’s Mach number444The Mach number is defined as = /, where is the velocity of the object relative to the ambient gas and is the sound velocity in the gas. (), given by:

 M=(γ+1)(u1−u0)+[(γ+1)2(u0−u1)2+16a20]1/24a0 (10)

where is the adiabatic index (for isothermal shocks555Mellema (2004) shows that the isothermal case is justified as most PNe (at least the ones which have had expansion parallaxes determined), have relatively high densities and slow shocks., = 1), is the pre-shock velocity of the gas (taken to be 13 km, noting that the correction factor is only weakly dependent on the exact value), is the spectroscopically derived expansion velocity, and is the pre-shock sound speed ( = 11.7 km for nebular gas at 10 K, following Mellema 2004). The correction factor , is then found from equation (4) of Mellema (2004), viz:

 R=(γ+1)Mu0+(γ+1)M2a0(γ+1)Mu0+2(M2−1)a0 (11)

The ratio tends to unity for high values of , that is high spectroscopic expansion velocities. Several PNe with optical expansion parallaxes have bright rims with attached shells, and so the rim can be considered to be shock bounded (Mellema 2004), and not indicative of an ionization front. However, the very youngest PNe (e.g. Vy 2-2) need to be modelled as expanding (D-type) ionization fronts surrounded by neutral material (see also Schönberner et al. 2005b). In this case the correction factor is more difficult to evaluate (Mellema 2004) but has been applied to BD+303639. He obtains = 1.3 0.2 kpc, in agreement with the distance from Schönberner et al. (2005b). The most recent distance for this PN comes from the detailed analysis by Akras & Steffen (2012), who give = kpc, which we adopt here.

We have applied a numerical correction to all expansion distances taken from the literature to account for the jump condition, unless it had been specifically taken into account, or the distance is based on the ballistic motion of high-proper motion features. In addition, unpublished HST expansion parallaxes were kindly provided by A. Hajian (2006, pers. comm.; see also Hajian 2006), that were also utilised by F08 and Smith (2015). An example is given here (the southern PN, NGC 5882) to show how the correction factor () is calculated. For this object, the new (uncorrected) expansion distance is = 1.32 0.2 kpc, with the additional note that the [N ii] and [O iii] images give the same distance. The [N ii] and [O iii] expansion velocities (Hajian et al. 2007) are also similar, with a mean of 25 km. Correcting for the jump condition, and assuming an isothermal shock ( = 1) following Mellema (2004), equations 10 and 11 can be used to estimate a correction factor, = 1.3 0.1. The corrected distance is = 1.72 kpc, and a distance uncertainty of 25% has been assumed. Table 6 provides expansion distances compiled from the literature, including the unpublished data from Hajian (2006), except for the kinematically complex objects NGC 6326 and NGC 7026 (e.g. Clark et al. 2013).

#### 3.1.7 Distances from Photoionization Modelling

Relatively accurate distance determinations using a self consistent treatment of spatiokinematic and photoionization modelling is a comparatively recent development. The development of powerful 2-D and 3-D photoionization codes (e.g. Ercolano et al. 2003) allows the self-consistent determination of the PN structure, central star characteristics, and distance, once accurate spectrophotometric line mapping, narrowband imaging, and kinematic data are available. This technique as applied to individual PNe (e.g. Monteiro et al. 2004; Schwarz & Monteiro 2006; Monteiro et al. 2011) holds promise, with all recent determinations summarised in Table 7. However, we have not utilised the distance for Mz 1 (Monteiro et al. 2005), owing to the lack of a reliable CSPN magnitude needed for modelling. Additionally, Bohigas (2008) presented photoionization models for 19 PNe, deriving two distances per object by comparing the model output with the observed H flux and the angular size respectively. We only used PNe which had the model distances consistent to better than 25%, with the additional requirement that the input parameters agreed with those in our database (Parker et al., in prep.). Only two PNe matched these requirements: JnEr 1 and K 3-72.  In Table 7, we present the photoionization model distances for 16 calibrating PNe.

#### 3.1.8 Kinematic distances

Kinematic distances can be determined for a restricted sample of PNe, namely those with little or no peculiar motion with respect to the local standard of rest. In other words, the PN partakes of nearly circular orbital motion around the Galaxy. The technique uses the position on the sky and the measured radial velocity of the PN to infer a distance (e.g. Corradi & Schwarz 1993; Corradi et al. 1997; Phillips 2001), assuming a model for the Galactic rotation curve. The approach can also be used for any neutral hydrogen in the foreground of the PN which causes an absorption line at 21 cm in the radio spectrum. Thus the distance for the absorbing cloud can be determined, which is a lower limit to the distance of the PN (e.g. Pottasch et al. 1982; Gathier, Pottasch & Goss 1986; Maciel 1995). This limit in some cases constrains the distance quite well. In this work an updated Galactic rotation curve slightly different to the IAU standard has been utilised: the adopted values are = 240 km, and = 8.3 kpc (Brunthaler et al. 2011). A flat rotation curve in the range of  kpc has also been assumed. For the cases where there is a kinematic ambiguity, the overall interstellar extinction proved useful in determining that the near distance was the only solution in each case.

Only a few kinematic determinations have been adopted as calibrating data. Type I PNe (Peimbert 1978; Kingsburgh & Barlow 1994), which are produced from higher-mass progenitor stars, are in general the only objects for which this approach is valid, where we assume these objects have a low peculiar velocity relative to its local ISM. Their peculiar velocity is assumed to be equal to the velocity dispersion of main sequence stars of spectral types B3–A0, = 15 km (Cox 2000), as such stars, with main sequence masses of 3–4  are the plausible progenitors for Type I PNe (cf. Karakas et al. 2009). This uncertainty dominates the error budget for each distance determination, especially as most have accurate systemic velocities. Table 8 summarises the best currently available distances (or limits) utilising this technique. The radial velocities were taken from the references given in the table, and were all converted to the LSR frame. Two distance determinations for non-Type I PNe are described in more detail below.

HFG 2 (PHR J0742-3247).  This high-excitation, optically-thin nebula was discovered by Fesen, Gull & Heckathorn (1983), and later confirmed by Parker et al. (2006). The 17th-mag central star is ionizing part of an extended H ii region of dimensions 7 4. That the source of ionization is the CSPN is shown by spectroscopically detectable [O iii] emission in the nebulosity immediately closest to the PN (F08). We adopt a revised H flux from Frew et al. (2014a) to calculate the surface brightness. A CO detection to the H ii region is reported by Brand et al. (1987), and the measured LSR velocity, +23.5 km leads to a distance for the PN of 2.1 0.6 kpc.

NGC 6751.  This is another example of an ambient H ii region ionised by a hot CSPN (Chu et al. 1991), in this case an early [WO] type. A revised kinematic distance of  kpc has been determined from the radial velocity data presented by Clark et al. (2010). See that reference for further details.

#### 3.1.9 Extinction Distances

. Individual extinction distances can be determined for PNe by comparing their observed extinctions with stars in the immediate vicinity of the PN at a range of distances that bracket the PN’s distance (Lutz 1973; Kaler & Lutz 1985; Gathier et al. 1986). While the method has the advantage of making no assumptions about the PN, it has proved difficult to calibrate in practice (Saurer 1995; Giammanco et al. 2011). The extinction is usually determined from the observed Balmer decrement of the nebular shell (e.g. Kimeswenger & Kerber 1998; Giammanco et al. 2011; Navarro, Corradi & Mampaso 2012), or by measuring the apparent colours of the CSPN, and assuming an intrinsic value for the colour index (see De Marco et al. 2013) to get the reddening directly (Weston, Napiwotzki & Sale 2009). In general, extinction distances have been taken from the literature only if the PN is within 4 of the Galactic plane (cf. Phillips 2006), which as an example, excludes all the distances from Martin (1994). At greater latitudes, the extinction distances for more remote PNe can be greatly underestimated as it is effectively outside the main dust layer of the disk (see the discussion by Phillips 2006). Furthermore, distance determinations based on average extinction-distance diagrams or their equivalents (e.g. Acker 1978; Pottasch 1984; Napiwotzki 2001) have been excluded as calibrating data owing to the potentially low precision of the method.

The distance uncertainties for the various literature determinations are rather inconsistent, with some being little more than rough estimates. If the nominal uncertainty on an individual extinction distance is less than 25%, it has been reset to that value here. While individual distances have rather large errors, the method as a whole is not expected to be biased to a short or long scale, provided that a substantial number of PNe are used as calibrators and no high-galactic latitude PNe are included. However, extinction distances to compact PNe might be overestimated if internal dust is significant (e.g. Ciardullo & Jacoby 1999; Giammanco et al. 2011), and the effect has been seen in young PNe like NGC 7027 (Navarro et al. 2012). Nevertheless, most PNe seem to show little or no internal extinction due to intrinsic dust (F08), verified from the observed blue colours of the CSPNe in evolved objects at high latitudes, such as NGC 246 and NGC 7293 (see Bond & Ciardullo 1999; Landolt & Uomoto 2007; F08; De Marco et al. 2013). Table 9 gives a summary of the adopted extinction distances, taken from the references listed following the table.

#### 3.1.10 Miscellaneous Distance Methods

This section includes a small but varied set of distances obtained using methods other than those described above, as summarised in Table 10. For the historically observed final-flash CSPNe, we have assumed for visual maximum a luminosity of 5000  and a bolometric correction of zero (i.e. = ). The peak visual brightness for V605 Aql (Abell 58), FG Sge (Hen 1-5), and V4334 Sgr (Sakurai’s object = DPV 1) has been taken from Duerbeck et al. (2002), van Genderen & Gautschy (1995), and Duerbeck et al. (2000), respectively. An independent distance to FG Sge based on pulsation theory has been obtained by Mayor & Acker (1980). In addition, the classical nova V458 Vul is located inside a faint planetary nebula, which was flash-ionized by the nova outburst. Wesson et al. (2006) has described the various distance determinations to this object, which all agree within the uncertainties.

As a further example, Wareing et al. (2006) modelled the morphology of the strongly asymmetric object Sh 2-188 to determine the relative velocity in the plane of sky that best reproduces the observed PN/ISM interaction. Combining this transverse velocity with a measured proper motion of the CSPN leads directly to a distance. Lastly, Eggen (1984) has determined a convergent parallax to NGC 7293 based on its assumed membership of the Hyades moving group. While this distance is consistent with the trigonometric distance from Table 1, we have used the latter owing to its much smaller uncertainty.

### 3.2 The Bulge Sample

We also use a restricted set of Galactic bulge objects as an adjunct to our calibration process (included in Table 11). To constrain Bulge membership and exclude foreground disk objects, we applied constraints on the flux and diameter as is usual. We further constrained the sample using the observed radial velocities, taken primarily from the compilation of Durand et al. (1998). We further assumed that Bulge PNe had  km. While this approach excludes many bona fide Bulge PNe, it has the benefit of excluding the vast majority of foreground disk interlopers, which would add noise to the relation. Integrated fluxes were taken from the sources discussed previously, and angular dimensions were mostly taken from Tylenda et al. (2003), Ruffle et al. (2004) and Kovacevic et al. (2011), and we have adopted the distance to the Galactic centre of  kpc from Brunthaler et al. (2011). However, owing to the substantial line of sight distance through the Bulge, and the fact that the Bulge sample may not be symmetrically located around the Galactic centre, we have only given half-weight to these PNe in our final calibration.

### 3.3 The Extragalactic Sample

PNe in the nearest satellite galaxies of the Milky Way are resolved with HST, and have the advantage of an accurately known distance. F08 showed that the  relation for the Galactic sample is consistent within the uncertainties with the  relation seen for MC PNe. In contrast to F08, we have now used these PNe in our analysis, enlarging our calibrating sample by a factor of two. We adopt distances of  kpc ( = 18.49) for the LMC (Pietrzyński et al. 2013) and  kpc ( = 18.95) for the SMC (Graczyk et al. 2014), adopting line-of-sight depths of 1.0 kpc and 2.0 kpc respectively. Similarly, we use three PNe belonging to the Sagittarius dSph galaxy (e.g. Zijlstra et al. 2006) as calibrating nebulae. We adopt a distance to this system of  kpc, for consistency with the complementary analysis of Vickers et al. (2015).666There is a moderately bright PN in the Fornax dSph galaxy at a distance of  kpc (Kniazev et al. 2007), but no HST imagery is available for it, and a second peculiar H-deficient PN in its globular cluster Hodge 5 (Larsen 2006), but like GJJC 1 in M 22, this exhibits no H emission. We should note that there is a significant line-of-sight depth to the SMC (Haschke, Grebel & Duffau 2012, and references therein), but evidently a much smaller depth for the main body of the SMC (Graczyk et al. 2014), which contains most of our calibrating PNe. There is considerable potential for a depth effect to be found in the Sgr dSph system as well, since none of the three PNe are located near the centre of the galaxy. Thus we have given half-weight in our final calibration to the PNe in the latter system.

## 4 The SHα–r relation

The  relation requires only an angular size, an integrated H flux, and the reddening to the PN. From these quantities, an intrinsic radius is calculated, which when combined with the angular size, yields the distance. Recall that the  relation has better utility than the equivalent [O iii] and [N ii] relations (Shaw et al. 2001; F08), as it includes both bright objects and the most senile PNe over a broad range of excitation, and best reflects the underlying ionised mass. The [N ii] relation, especially, is strongly influenced by abundance variations between objects, and furthermore, there is negligible [N ii] emission in the PNe of highest excitation. The H relation is also preferred to the equivalent H relation, as at a minimum, H fluxes are a factor of approximately three brighter. As mentioned above, a number of high-quality H imaging surveys have recently become available, which have also allowed the determination of accurate integrated H fluxes for a significant fraction of Galactic PNe.

Overall, the inclusion of additional calibrating PNe and the use of refined input data (fluxes, extinctions, and angular dimensions) have led to a slight improvement of the distance scale with respect to F08; the present mean scale is about four per cent longer, and more in agreement with the independent theoretical tracks computed by Jacob, Schönberner & Steffen (2013). While some previous authors (e.g. Schneider & Buckley 1996) have suggested that a single power-law is inadequate to handle both young and old PNe, we find that a linear  relation is applicable as a robust distance method, excluding only the very youngest optically-thick PNe and transitional objects.

### 4.1 Fundamental observables

#### 4.1.1 Angular Dimensions

For the brighter Galactic calibrating PNe, the angular dimensions have been taken from Tylenda et al. (2003) and Ruffle et al. (2004) if available. These works quote major and minor axes at the 10% level of the peak surface brightness isophote, which is a standard adopted throughout this work where feasible. Note that the adopted dimensions are for the main PN shell, which encloses the rim, or primary shock, but does not include any faint outer halo(s) if present (e.g. Corradi et al. 2003; Frew et al. 2012). Major and minor dimensions for most of the largest PNe have been determined here anew, based on available digital broadband red or H + [N ii] images at the same isophote level. These were primarily taken from the SHS, SSS, and IPHAS surveys with some recent images from the POPIPLAN survey (Boffin et al. 2012) also utilized. For compact Galactic PNe, we utilised HST images if available, either from the literature (e.g. Sahai et al. 2007; Gesicki et al. 2014; Hsia et al. 2014) or from the Hubble Legacy Archive.7 The dimensions of compact PNe derived from ground-based measurements were corrected using a PSF deconvolution if needed (e.g. Ruffle et al. 2004). We then calculated geometric mean diameters and radii for each PN. The uncertainties have been adopted directly from the relevant references if present, or calculated from inverse variances if more than one determination is available.

For the LMC and SMC PNe we adopt the major and minor axial dimensions from Shaw et al. (2001), Stanghellini et al. (2002, 2003) and Shaw et al. (2006), based on HST imagery. For consistency with the sample of Galactic calibrating objects, the angular dimensions at the 10 per cent brightness contour have been used from these references, rather than the ‘photometric radii’, encompassing 85% of the total flux, defined by Stanghellini et al. (1999). For the three calibrating PNe belonging to the Sagittarius dSph galaxy, we adopted the dimensions from Zijlstra et al. (2006).

The isophote method is best suited for elliptical and round PNe. However, some highly evolved PNe strongly distorted by interaction by the ISM have been treated differently. In these cases a strict application of the 10 per cent isophote rule may only give dimensions of the bright interacting rim, a typical example being Sh 2-188 (Wareing et al. 2006). In this case an isophote which includes the non-interacting part of the main shell is used to give the overall dimensions of the object. Similarly, the dimensions for some evolved bipolar PNe are sometimes hard to define, and are dependent on the exact orientation of the ‘waist’. In most cases these are relatively large PNe, so the subjective effect of choosing an appropriate contour has only a relatively small percentage change on the overall dimensions of the nebula. Figure 1 shows how the major and minor axes have been determined for three PNe of differing morphological types.

#### 4.1.2 Integrated Fluxes

For Galactic PNe, the integrated H fluxes and their uncertainties are mostly adopted from Kohoutek & Martin (1981), Dopita & Hua (1997), Wright, Corradi & Perinotto (2005), and FBP13, for the brighter objects, or from F08 and Frew et al (2014a) for a few of the largest and most evolved PNe. For the LMC and SMC PNe we adopt the H fluxes and associated uncertainties from Shaw et al. (2001), Stanghellini et al. (2002, 2003) and Shaw et al. (2006), supplemented with data from Reid & Parker (2010b). For the PNe belonging to the Sagittarius dSph galaxy, we average the integrated H fluxes from Ruffle et al. (2004), Zijlstra et al. (2006) and FBP13.

Note that the integrated fluxes for less-evolved PNe, especially those measured with photoelectric photometry through large apertures or from CCD surveys of limited resolution may include some or all of any faint surrounding AGB halo. We expect this to be a minor effect, as the typical halo surface brightness is a factor less than the main shell, while the surface area of the halo is an order of magnitude larger than the main shell (see Corradi et al. 2003). This means that on average, only about one per cent of the total flux resides in a typical AGB halo. Moreover, since the calibrating sample includes several PNe with surrounding haloes, there should be little error in the application of our method to other objects.

#### 4.1.3 Extinction Constants

The logarithmic extinction constants, , for Galactic PNe are widely scattered in the literature. Extensive data compilations include CKS, Tylenda et al. (1992), Condon & Kaplan (1998), Ruffle et al. (2004), Giammanco et al. (2011), Kovacevic et al. (2011), and FBP13. The extinction constants are usually determined from the Balmer decrement, as derived from optical spectroscopy, or by comparing the Balmer and radio continuum fluxes. To be as homogenous as possible, we re-calculated the radio–H extinctions using the radio data and methods given in Bojičić (2010) and Bojičić et al. (2011a, 2011b, and references therein).

However, since the extinctions for many faint PNe were either previously unknown, or unreliable, new values were determined where applicable. Similarly, extinctions for brighter PNe were re-derived from published H/H ratios adopting a Howarth (1983) reddening law. For the H/H ratios we adopted an average of the data presented in Acker et al. (1992), the Catalog of Relative Emission Line Intensities Observed in Planetary Nebulae (ELCAT) compiled by Kaler, Shaw & Browning (1997), the extensive database of 2000 spectra taken as part of the MASH survey and related programmes, supplemented with data taken from more recent papers in the literature, including Torres-Peimbert & Peimbert (1977), Kohoutek & Martin (1981), Gutiérrez-Moreno, Cortes & Moreno (1985), Shaw & Kaler (1989), Dopita & Hua (1997), Tsamis et al. (2003, 2008), Liu et al. (2004), Wright, Corradi & Perinotto (2005), Zhang et al. (2005), and Wang & Liu (2007). Other papers were outlined in Frew et al. (2013). For higher-latitude objects that still had poor-quality data, we utilised the reddening data from Schlafly & Finkbeiner (2011) as a cross-check. Finally, for PNe with adequate central star data, () values for the central stars have been calculated following F08, De Marco et al. (2013) and Douchin et al. (2014), using available photometry from the literature.

An average of measurements from several independent sources should be fairly representative of the extinction for each PN. The extinction uncertainties have been adopted directly from the relevant references if present, or calculated from inverse variances if more than two independent values are available. We plan to publish the individual extinction determinations separately, in Version 1 of our global MASPN Database (see Parker et al. 2015).

For the extragalactic PNe we calculate the extinction constants from the flux data presented by Shaw et al. (2001, 2006), Stanghellini et al. (2002, 2003), Ruffle et al. (2004), Zijlstra et al. (2006), and Reid & Parker (2010b), adopting a minimum value from Schlafly & Finkbeiner (2011) if the calculated extinction is less than this. The H and H logarithmic extinction constants, and , are related to the reddening following the Howarth (1983) extinction law:

 cβ=1.45 E(B−V)cα=0.99 E(B−V) (12)

The H extinction coefficient was added to the observed logarithmic H flux to get the reddening-corrected flux for each PN. The intrinsic H surface brightness888To convert a log flux per steradian to a log flux per square arcsec, subtract 10.629 dex. in units of erg cms sr was then calculated from the angular geometric radius () and reddening-corrected flux, using the formula:

 SHα=FHα4πθ2 (13)

### 4.2 Final Calibrating Sample

Nearly 30 Galactic PNe have distances based on more than one primary method. For these PNe, a weighted average distance has been calculated based on the quoted uncertainties of each individual distance determination. For consistency, individual distances were combined within each method first (after removing outlying data points using a 2 cut). These were then combined with distances from other primary methods weighted by inverse variances to determine the final weighted distance, using:

 Dw=∑ni=1wiDi∑ni=1wi (14)

where [] are the individual distance estimates, with associated weights [] determined from the inverse variances, . The uncertainty of the weighted mean distance was calculated (following FBP13) as:

 σDw=(V1V21−V2n∑i=1wi(Fi−¯Fw)2)0.5 (15)

where and .

Finally, for each calibrator, the linear radius was determined from the angular radius and the adopted distance using Equation 4.

This approach is quite robust to any error in the angular dimensions, because this flows through to both the surface brightness and the radius. For example, a 20 per cent uncertainty in each angular dimension (40 per cent uncertainty in the calculated surface brightness) leads to only a 10 per cent uncertainty in the distance. Similarly, owing to the form of the  relation, an uncertainty of 20 per cent in the H flux leads to only a 5 per cent error in the computed radius, i.e. the PN distance. Hence, scatter introduced into the  relation due to observational uncertainties in the angular dimensions, fluxes or extinctions are generally minor compared to the uncertainties in the distances of the calibrating PNe, or the dispersion in the relation due to cosmic scatter (see below). However, for highly reddened PNe, the uncertainty in the surface brightness is dominated by the extinction uncertainty which can reach 0.3 dex in some cases, leading to an additional uncertainty of 20 per cent in the distance..

Table 11 gives the relevant observational and derived data for the full calibrating sample of 322 PNe. These range from the very nearest objects out to PNe at the distance of the SMC (0.13  60 kpc). The columns in Table 11 consecutively give the PN designation, common name, adopted distance (in kpc), the method of distance determination, a simplified morphological code (E = elliptical, B = bipolar, R = round, A = Asymmetric) after Parker et al. (2006), the major and minor dimensions in arcseconds, the adopted value (in mag), the reddening-corrected H surface brightness (in cgs units per steradian), and the logarithm of the nebular radius (in pc).

The general form of the relationship between surface brightness and radius is expected to be a power law, with constants and describing the slope and zero point respectively, viz:

 logSHα=γlogr−δ (16)

We use an ordinary least-squares (OLS) bisector fit (Isobe et al. 1990) to represent the full calibrating sample, since observational errors are present in both the nebular fluxes and diameters (i.e. the surface brightness) which are independent of the errors on the distances, and hence the physical radii. The justification for this approach was discussed by Isobe et al. (1990) and Feigelson & Babu (1992). Disk PNe with formal uncertainties in the distance of less than 10% have been given double weight in the calculation of the coefficients. All other PNe have been assigned unit weight, except for the Bulge objects, assigned half weight. The best fit based on our full sample of 332 PNe is represented by the equation:

 logSHα=−3.63(±0.06)logr−5.34(±0.05) (17)

with a Pearson correlation coefficient, . The slope is steeper than the r law previously found for LMC and SMC PNe by Shaw et al. (2001) and Stanghellini et al. (2002), primarily due to the different treatment of the PN dimensions by those authors.

The overall impression of the  relation (Fig. 2) is a well behaved linear trend, but with a shallower gradient at small radii, discussed further in §4.3.3. There may be a flattening of the slope at the very bottom of the locus, but this needs to be confirmed with more data. The origin of the radius dependence at large radii may be due to more uncertain distances combined with lower quality H fluxes for the very largest PNe. It is also possible that some of the very oldest PNe may be ‘re-brightened’ by an interaction with the ISM (see Wareing 2010). The most discrepant objects Sh 2-188, Sh 2-216, and WeDe 1 are within 100 pc of the Galactic mid-plane or less than the dust scale height (Spitzer 1978). The surface brightness of these PNe might be enhanced by mass augmentation from the ISM, especially close to the Galactic plane, or alternatively by shock excitation for fast moving PNe (Wareing et al. 2006), which would lead to these objects lying above the power law derived from the total calibrating sample. Indeed the optical spectrum of Sh 2-188, shows extraordinarily strong [S ii] lines for a PN (Rosado & Kwitter 1982), suggesting shock excitation is important in this object.

### 4.3 PN subsamples in the SHα–r plane

We recommend applying the mean  trend (Equation 17) for all PNe that have a spectroscopic signature that does not allow classification as definitively optically-thick (§4.3.1) or optically-thin (§4.3.2), and for other PNe for which the required optical spectroscopy is currently lacking. The calibrating PNe in the  relation represent the full range of properties manifested by PNe, such as morphological type, excitation class, ionised mass, metallicity, and central star luminosity, so we have hopefully circumvented the thorny problem of Malmquist bias999Malmquist bias is present when the intrinsic (cosmic) dispersion of a sample of objects is significant. In other words, if a sample of objects (stars, PNe or galaxies, for example) is flux-limited, then only the most luminous objects are selected at large distances, so there is an observed increase in the average luminosity of a flux-limited sample as distance increases. (Malmquist 1924). Having new and revised data available for these calibrators also provides the opportunity to investigate the presence of any sub-trends within the relation. Table 12 provides a summary of the equation coefficients for the most important subsets of calibrating nebulae. Excluding the very youngest PNe, the observed power-law slope of the  relation is between and , depending on the subset used. The small offset between the Galactic disk and extragalactic samples is due to one or more of Malmquist bias (the extragalactic sample is flux and surface brightness limited), systematic errors in measuring PN diameters (more difficult for extragalactic PNe), and possibly progenitor mass and metallicity differences (e.g. Jacob et al. 2013) between the different galaxies.

Owing to the relative difficulty of morphologically classifying PNe from two-dimensional images (e.g. Kwok 2010; Chong et al. 2012), we do not formally calculate different sub-trends for the various morphological classes, but only provide a visual breakdown by class, seen in the right panel of Figure 3. Canonical bipolar PNe and elliptical PNe with bipolar-cores tend to populate the upper part of the broad trend in the  plane. Elliptical PNe without bipolar cores are more uniformly spread, while spherical PNe tend to plot beneath the mean trend-line, at moderate to large radii. To help alleviate the problem of cosmic scatter, we now subdivide the full ensemble of PNe into different subsets based on spectroscopic criteria, discussed in the following sections.

#### 4.3.1 Optically-thick PNe

These PNe have relatively strong low-excitation features such as the [N ii], [O ii] and [S ii] lines. We follow Kaler & Jacoby (1989) and Jacoby & Kaler (1989) in defining an optically-thick PN as having the reddening-corrected ratios /(H) 1.5 and/or /(H) 1. Using only those calibrators that meet these spectroscopic criteria to define the relation, the optically-thick (or ‘long’) trend is given by the equation:

 logSHα=−3.32(±0.12)logr−4.97(±0.08) (18)

Many optically-thick bipolar PNe also have Type I chemistries, using the Kingsburgh & Barlow (1994) definition. A subset of 45 known Type I PNe was extracted from the overall calibration sample, all but one of which is morphologically bipolar, and the coefficients are given in Table 12. The resulting relation is statistically indistinguishable from the general optically-thick trend, which is preferred.

#### 4.3.2 Optically thin PNe

These PNe are the spectroscopic opposites of the optically-thick PNe, and are defined as PNe having very weak or absent low excitation lines of [N ii], [O ii] and [S ii] (cf. Kaler 1981; Frew et al. 2014c). Formally we define optically-thin PNe as having the line ratio /(H) 0.1. The [O ii] and [S ii] emission lines are similarly weak to absent. A subset of high-excitation (HE) objects have the same [N ii] criterion, but also have (Heii) 0.75 (H), and relatively strong emission lines of other high-excitation species, such as [O iv], [Ar iv], [Ar v], and [Ne v] (cf. F08). Representative examples of the latter group include NGC 1360 (Goldman et al. 2004) and the more evolved object MWP 1, which is invisible on deep [N ii] images (see Tweedy & Kwitter 1996). Note that the nebular excitation class (e.g. Dopita & Meatheringham 1991; Reid & Parker 2010a) does not map closely with our definition of optical depth, so has not been investigated further.

Most HE PNe appear to have CSPNe still on the nuclear burning track close to the turnaround point or ‘knee’ in the HR diagram. These PNe are optically thin to the H i continuum and usually to the He ii continuum as well, and consist essentially of a He Strömgren zone, i.e. (He ii) (H) (Köppen 1979; Torres-Peimbert et al. 1990). The ionization parameter is high, and their spectroscopic uniformity reflects the systematically lower ionised masses of these nebulae. Consequently, these PNe plot near the lower bound of the overall  locus. However, their CSPNe are spectroscopically heterogeneous, with both H-rich and H-deficient nuclei, and at least two belong to the born-again class (e.g. Guerrero et al. 2012 and references therein). This is suggestive that several evolutionary scenarios may produce low-mass PNe (Frew & Parker 2007, 2010, 2012). The optically-thin (or ‘short’) trend should only be used for PNe that meet the spectroscopic criteria described above. It is represented by the equation:

 logSHα=−3.75(±0.11)logr−5.73(±0.07) (19)

HE PNe typically have either round or elliptical morphologies, sometimes with amorphous filled centres, though some are strongly axisymmetric objects associated with post-common envelope nuclei (De Marco 2009; Corradi et al. 2011). Indeed many post-common envelope PNe are optically-thin following our definition, from which Frew & Parker (2007) and F08 suggested that these PNe have systematically lower ionised masses in the mean, typically only 0.1 . Curiously, known close-binary PNe show a somewhat restricted range of H surface brightness (   erg cmssr) compared to the full observed range for all PNe ( +0.2 to  erg cmssr). In other words, the PNe of highest surface brightness are rarely observed to host close-binary nuclei. This has been traditionally interpreted as a selection effect (e.g. Bond & Livio 1990), but may instead be pointing to a physical effect in that post-CE PNe are born “old”, with moderate surface brightnesses at best, and with preferentially lower-mass CSPNe. To address this problem, a more detailed statistical study of these PNe is planned for a future paper in this series.

#### 4.3.3 Compact high-SB PNe

The overall impression of the  relation is that of a shallower gradient at small radii. This was noted by F08, but is more apparent with the revised calibrating sample from this work, albeit demonstrated mostly by the Cloud and Bulge sub-samples. To investigate this, we subdivided the calibrating sample into two groups on the basis of intrinsic radius, separated at log  = ( = 0.04 pc). A bisector fit to the compact PN sample () is given by:

 logSHα=−2.74(±0.51)logr−4.15(±0.80) (20)

with a markedly lower correlation coefficient of . This slope is shallower than the gradient we observe for the full calibrating sample. However, as compact PNe in the Galaxy tend to have a lower surface brightness for a given radius than those observed to date in the Magellanic Clouds, likely due to selection effects, we recommend against using this relation at this point. Alternatively, the youngest, dustiest PNe may be amenable to having distances calculated via the SED technique (e.g. Vickers et al. 2015), further described in §A.2.

#### 4.3.4 Subluminous PNe

We find evidence for a small heterogeneous group of peculiar, subluminous PNe that fall 3 below the main  locus, based on the primary distance estimates tabulated here. These are RWT 152, HbDs 1, K 1-27, HaWe 13, Hen 3-1357 (the Stingray nebula), and the central core of KjPn 8 (discussed in §A.1). The first two appear to have low-mass H-normal stars and may represent a population of objects largely overlooked in current surveys, though there is some evidence that HbDs 1 might be a wisp of ionized ISM (Frew et al., in preparation). However RWT 152 appears to have the typical morphology of a PN; its flux and diameter data have been taken from Pritchet (1984) and Aller et al. (2014) respectively. K 1-27 has a rare O(He) CSPN (Reindl et al. 2014a) and its discrepant nature has been discussed previously by Frew & Parker (2010). HaWe 13 has a ionizing star on either a post-RGB or post-EAGB track, based on the parameters given in Table 5, and its morphology appears to be consistent with it being produced via a common-envelope interaction (e.g. Hall et al. 2013). A further object, Hen 3-1357, has been argued to be the product of a post-EHB pathway (see Reindl et al. 2014b), being a young, compact nebula at an unusually short distance of 1.5 kpc, calculated via the gravity method. For these reasons, none have been used as calibrating nebulae, but are plotted in Fig. 8 for illustrative purposes.

### 4.4 The physical basis for the SHα–r relation

Detailed photoionization modelling of the  relation and its relationship to central star evolutionary tracks (Kwok 1985; Van de Steene & Zijlstra 1995; Jacob et al. 2013), is beyond the scope of this paper but making some simple assumptions from emission theory, we can relate the observed gradient of the  relation to other parameters such as the ionized mass and electron density. For an uniform spherical nebula of radius , the integrated flux emitted by the H recombination line is given by:

 FHα=(r33D2)hνHαnenpαeffHα (21)

where and are the electron and proton densities respectively, and is the distance to the PN (see Hua & Kwok 1999). In practice, PNe are not homogenous, and a volume filling factor is used to take this into account. Various values are presented in the literature, but a consensus value, = 0.3, is often adopted (Boffi & Stanghellini 1994; cf. Gathier 1987; Pottasch 1996).

In the absence of extinction, the nebular H surface brightness is given by:

 SHα=ϵ3n2erhνHααeffHα (22)

while the nebular ionized mass, is calculated with the following expression:

 Mion=4π3npμmHεr3 (23)

where is the mean atomic mass per hydrogen atom. Combining equations 21 and 23, the ionized mass can be expressed in terms of the angular radius, , and the H flux as:

 Mion=4πμmH(3hνHαxeαeffHα)1/2ε1/2θ3/2D5/2F1/2Hα (24)

where  1.16 (Hua & Kwok 1999). Simplifying, equation 24 can be finally expressed in terms of the distance:

 Mion/M⊙=0.035ε1/2θ3/2D5/2F′1/2Hα (25)

where now, is the nebular H flux in units of 10 erg cms, is in arcmin and in kpc. Since, from equation 22, the surface brightness , and since , we can simplify to:

 SHα ∝ M2ionr−5 (26)

A natural consequence of the interacting stellar winds (ISW) model (Kwok, Purton & FitzGerald 1978; Kwok 1982) is that the mass of a PN shell increases with age, due primarily to the expansion of the ionization front within the nebula, as well as the snow-plow effect when the PN becomes evolved (Villaver, Manchado & García-Segura 2002). Hence, unlike the Shklovsky method which assumes constant ionised mass, PNe manifest an observable mass-radius relation. Recalling that from equation 5, we can also write:

 SHα∝r2β−5 (27)

Now the Shklovsky constant-mass assumption ( = 0) predicts a power law (Seaton 1968), which is much steeper than observed. Using the set of calibrating PNe defined here, the observed mean relation predicts a value for = 0.7, somewhat smaller than earlier determinations (e.g. Daub 1982; Milne 1982; Kwok 1985), which we attribute to this study including the most evolved PNe with very faint central stars. Since the temperature and luminosity of the ionizing star change markedly during the evolution of the PN, this has a direct influence on the index (Perinotto et al. 2004). Yet despite our simplifying assumptions, it is quite remarkable that a simple linear relationship essentially defines the full population of PNe in the  plane, excluding the very youngest objects.

Both Kwok (1985, 1993) and Samland et al. (1993) showed that errors in statistical distances increase rapidly as 2.5, at which point the method becomes degenerate; i.e. there is no dependence of surface brightness on radius. Since observationally, the value of is much less than this, we conclude that the various  relations in the literature are valid if calibrated correctly, with the only disadvantage being the observed cosmic scatter.

Our mean  scale is fully consistent with the theoretical evolutionary tracks of Jacob, Schönberner & Steffen (2013). These tracks were generated from the hydrodynamical nebular models of Perinotto et al. (2004) and Schönberner et al. (2005a) along with the CSPN, using the evolutionary tracks for the latter from Blöcker (1995) and Schönberner (1981). The nebular radius and surface brightness for PNe with a range of core masses were then over-plotted on the  plane in Fig. 5. The agreement is very good between these tracks and the observational data, with a slight offset owing to the slightly differing definition of angular size between the studies (see the discussion of Jacob et al. 2013). Note that the evolutionary models do not extend to the lowest surface brightness owing to constraints in computational time. Our results (see §5) show that with care, the mean-scale distances derived here have comparable accuracy to most direct methods currently in use, and significantly better than any other statistical distance indicator published in the literature to date (Jacob et al. 2013; Ali et al. 2015; Smith 2015).

We further note that Smith (2015) identified and discussed the scale error at large radii that affects the SSV distance scale. Recall that SSV used a constant mass assumption for all PNe larger than a radius of 0.06 pc. Following Van de Steene & Zijlstra (1995), the optical thickness parameter of SSV is related to the brightness temperature by the following expression:

 logTb=−T+4.86 (28)

In the general plane, the two power laws have slopes of 1.64 (for thick PNe) and 5.0 for thin PNe; i.e. a constant mass trend. However the observational data (Fig. 5) appear to rule out a constant mass trend at large radii, at least for the PNe discovered and observed to date. Similar scale errors afflicted the earlier studies of Daub (1982) and CKS, which had the optically thick/thin boundary at somewhat larger nebular radii of 0.12 pc and 0.09 pc respectively.

## 5 The Distance Catalogue

Table 13 provides a catalogue of  distances for over 1100 Galactic PNe, published in its entirety as an online supplement. The columns consecutively give the PN G identifier, the usual name, the adopted geometric diameter in arcsec, the adopted reddening and its uncertainty, the method used to determine the reddening, the logarithm of corrected H surface brightness and its uncertainty, the logarithm of the computed radius (in pc), and the resulting mean statistical distance in kiloparsecs (kpc) and its uncertainty. The next two columns provide either a short (optically thin) or long (optically thick) distance if applicable. The last column lists any notes, including if the PN is a calibrator for the relation. The mean-trend distance is given for all objects, which can be conveniently used for future statistical comparisons with sets of primary distances or other secondary distance scales. If an alternative distance is given, then this is the preferred distance to be used for studies of individual properties.

The inferred radius is derived from the adopted reddening-corrected surface brightness. Typical uncertainties in this parameter are calculated from the quadratic sum of the individual uncertainties in the angular size (actually the surface area), the integrated H flux and the extinction; typical logarithmic uncertainties are respectively 0.04 dex, 0.02 dex and 0.02 dex for a bright well observed PN (e.g. NGC 2022 or NGC 3242), ranging to 0.10 dex, 0.10 dex and 0.04 dex for a large asymmetric PN like Sh 2-188 (recall Fig. 1). However, for highly reddened PNe, the uncertainty in the surface brightness is dominated by the extinction uncertainty, which can reach 0.3 dex in the worst cases. This is a contributing reason to our decision to give Galactic Bulge PNe reduced weight as calibrating objects.

The distances given in Table 13 supersede any  distances previously published (Pierce et al. 2004; Frew et al. 2006b, 2011; F08; Viironen et al. 2009, 2011; Bojičić et al. 2011b; Corradi et al. 2011) using earlier calibrations of the  relation, though in all cases the differences in distances are less than five per cent.

## 6 Intrinsic Dispersion of the SHα–r relation

The  relation is a robust statistical distance indicator for all PNe, and especially for those for which no primary distance technique is available. In the fist instance, a measure of the dispersion of the technique can be evaluated by comparing the distances of the PNe in the calibrating sample with the distances derived for these PNe from the mean  relation. The calculated distances have a dispersion of  28 per cent across the full range of intrinsic diameter. In figure 6 we refine this approach, by plotting individual PNe using the high- and low-trend statistical distances separately. Using the relation for optically thick PNe only, a 28 per cent dispersion is similarly obtained. In addition, using the ‘short’ trend for optically-thin PNe gives a small resulting dispersion of only  18 per cent. This 1 dispersion is considerably better than any previous statistical distance indicator, validating the use of sub-trends based on spectroscopic criteria. The dispersion in the thick relation is higher than in the shorter thin relation, and close inspection shows that for a few bipolar PNe, the thick relation appears to be less accurate than the mean trend. This may be due in part to the difficulty of accurately measuring the angular sizes of many bipolar PNe, but is also likely that the bipolar PNe are a heterogeneous group. It appears likely that bipolar nebulae may be produced by both high-mass single progenitors as well as lower-mass close-binary stars (e.g. De Marco 2009). SSV also find that their distance scale does not work well for bipolar PNe.

The observed dispersion includes a convolution of the uncertainties in both the calibrating distances and the statistical distances. In order to gauge the uncertainties of each primary technique, the distances for individual PNe were compared with the adopted  distances. Table 14 shows the results, which reveal that the gravity, kinematic, and extinction distance methods have the greatest uncertainties, unsurprisingly given the discussion in § 3.1. The problems with the gravity method have already been discussed. The kinematic method was primarily applied to Type I PNe, but it seems even these can sometimes have significant peculiar velocities, meaning that the technique should be used with caution. The extinction method, while powerful in the sense that it can be applied to many PNe, is problematic, and care should be taken to avoid using PNe that are found in fields with significant differential extinction over small spatial scales (Giammanco et al. 2011).