11institutetext: Astrophysikalisches Institut Potsdam, An der Sternwarte 16,
D-14482 Potsdam, Germany
11email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# The evolution of planetary nebulae

VI. On the chemical composition of the metal-poor PN G135.9+55.9††thanks:
Based in part on observations collected at the Centro Astronómico Hispano
Alemán (CAHA), operated jointly by the Max-Planck-Institut für Astronomie and
the Instituto de Astrofisica de Andalucia (CSIC).

C. Sandin  R. Jacob  D. Schönberner  M. Steffen  M. M. Roth

(Received February 5, 2009 / Accepted December 22, 2009)

The actual value of the oxygen abundance of the metal-poor planetary nebula PN
G135.9+55.9 has frequently been debated in the literature. We wanted to
clarify the situation by making an improved abundance determination based on a
study that includes both new accurate observations and new models. We made
observations using the method of integral field spectroscopy with the PMAS
instrument, and also used ultraviolet observations that were measured with
HST-STIS. In our interpretation of the reduced and calibrated spectrum we used
for the first time, recent radiation hydrodynamic models, which were
calculated with several setups of scaled values of mean Galactic disk
planetary nebula metallicities. For evolved planetary nebulae, such as PN
G135.9+55.9, it turns out that departures from thermal equilibrium can be
significant, leading to much lower electron temperatures, hence weaker
emission in collisionally excited lines. Based on our time-dependent
hydrodynamic models and the observed emission line $[{O\textsc{iii}}]\,\lambda
5007$, we found a very low oxygen content of about 1/80 of the mean Galactic
disk value. This result is consistent with emission line measurements in the
ultraviolet wavelength range. The C/O and Ne/O ratios are unusually high and
similar to those of another halo object, BoBn-1.

###### Key Words.:

ISM: planetary nebulae: general – ISM: planetary nebulae: individual (BoBn-1,
PN G135.9+55.9) – Hydrodynamics

††offprints<EMAIL_ADDRESS>
##  1 Introduction

The stellar-like object SBS 1150+599A from the Second Byurakan Survey (Balayan
1997) has been spectroscopically identified by Tovmassian et al. (2001,
hereafter T01; ) to be an old planetary nebula (PN) of the Galactic halo and
was thereafter renamed PN G135.9+55.9. The same authors perform an abundance
study based on photoionization models and come to the conclusion that this
particular object has the lowest oxygen abundance known so far for planetary
nebulae, viz. below about 1/100 of the solar value.

This object appears to be a challenge for any detailed spectroscopic analysis
since very few emission lines are detectable in the optical wavelength range.
Because of a lack of suitable lines, it is impossible to make a direct plasma
diagnostic, and it is also difficult to constrain photoionization models such
that meaningful abundances will emerge. The analysis is, moreover, hampered
since this object is faint, with a mean $\text{H}\beta$ surface brightness of
only $\sim\\!5\\!\times\\!10^{-16}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}\,\text{arcsec}^{-2}$ (adopting a
diameter of 5″ and a total $\text{H}\beta$ flux of $1.9\\!\times\\!10^{-14}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$; see Table 1), making it difficult
to accurately measure lines close to the 1% level of $\text{H}\beta$
(corresponding to $\simeq\\!2\\!\times\\!10^{-16}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$).

These observational difficulties have led to a dispute about whether PN
G135.9+55.9 is as extremely metal-deficit as T01 claim. A critical point in
this discussion is the determination of the stellar temperature from the
ionization balance, using the line ratio of $[{Ne\textsc{v}}]\,\lambda 3426$
and $[{Ne\textsc{iii}}]\,\lambda 3869$ in the nebula. The line strength of
$[{Ne\textsc{iii}}]\,\lambda 3869$ is uncertain, but decisive when fixing the
effective temperature ($T_{\text{eff}}$) of the ionizing source. Richer et al.
(2002, hereafter R02) and Jacoby et al. (2002, hereafter J02) find a rather
low $T_{\text{eff}}\\!\simeq\\!100\,000$ K, hence low oxygen abundances of
less than 1/100 solar from the weak $[{O\textsc{iii}}]\,\lambda 5007$ line,
the only oxygen line that is observed in the optical.

Tovmassian et al. (2004, hereafter T04) conclude, by means of optical and
ultraviolet (UV) spectra (FUSE), that the central ionizing source of PN
G135.9+55.9 must be a very hot ($T_{\text{eff}}\\!\approx\\!120\,000$ K) pre-
white dwarf of rather low mass ($\approx\\!0.55$ $\text{M}_{\odot}$), which
resides in a short-period binary system with a more massive companion
($P\\!=\\!3.92$ h, Napiwotzki et al. 2005). Recently Tovmassian et al. 2007,
based on X-ray observations, have estimated that the temperature of this
massive component is very high, $T_{\text{eff}}\\!\simeq\\!170\,000\,$K, while
the less massive optical-UV component is cooler,
$T_{\text{eff}}\\!\simeq\\!58\,000\,$K.

Péquignot & Tsamis (2005, hereafter PT05), moreover, critically examine the
situation and come to the conclusion that all evidence favors a higher
temperature of the central source, viz. $T_{\text{eff}}\\!\simeq\\!130\,000$
K, and that the claimed strength of $[{Ne\textsc{iii}}]\,\lambda 3869$ most
likely is wrong. The oxygen abundance would then be 1/30–1/15 solar and not as
extreme as previously estimated. Jacoby et al. (2006, hereafter J06) also find
a higher stellar temperature, using the much stronger UV emission lines of
highly ionized nitrogen and carbon to constrain $T_{\text{eff}}$. Stasińska et
al. (2005, hereafter S05) include UV lines in their study (likely using the
same HST-STIS data as J06) and conclude that the oxygen abundance is lower
than what PT05 find, 1/130–1/40.

Our aim was to better measure the nebular emission line spectrum, which is why
we performed new observations using the integral-field spectrograph PMAS. With
such a spectrum we could determine a reliable line strength, or an upper
limit, of $[{Ne\textsc{iii}}]\,\lambda 3869$. We begin in Sect. 2 with a
description of our observations and data processing, we then present our
results in Sect. 3. We describe the physical setup of our radiation
hydrodynamic models, how we estimate abundances and compare observations with
our models, in Sect. 4. In Sect. 5 we discuss non-equilibrium effects and
compare our abundances with values of previous studies found in the
literature. We close the paper with our conclusions in Sect. 6.

##  2 Observations and data reduction

Our observations were made with the 3.5 m telescope at Calar Alto using the
lens array (LARR) integral field unit (IFU) of the PMAS instrument (Roth et
al. 2005). The V600 grating was used to cover the spectral interval 3490–5150
Å; at a dispersion of 0.81 Å pixel-1 and a resolving power of ${R=1340}$. The
wavelength range was chosen to cover the Balmer lines $\text{H}\beta$ and
H$\gamma$, as well as [O iii]$\,\lambda\lambda\,4959,\,5007$, He ii$\,\lambda
4686$, and [Ne iii]$\,\lambda 3869$. The LARR IFU, furthermore, holds
$16\\!\times\\!16$ separate fibers, where each fiber represents a spatial
element on the sky. We used the 0$\aas@@fstack{\prime\prime}$5 sampling mode
where every pointing with the IFU covers an area of ${8\arcsec\times
8\arcsec=64\arcsec^{2}}$ on the sky. Any number of spatial elements can be co-
added to create a final spectrum.

Two 2700 s exposures, which were both centered on PN G135.9+55.9, were taken
at an airmass of 1.09–1.12 on 2007 February 13. Two 2700 s exposures were also
taken at an airmass of 1.12–1.19 on 2007 February 14. These two latter
exposures were offset by 5″ E from the first two exposures. One additional
1800 s exposure was taken at an airmass of 1.09 and an offset of 5″ W in the
second night. Weather conditions were less than optimal, and the seeing was
1$\aas@@fstack{\prime\prime}$4–1$\aas@@fstack{\prime\prime}$7\. In addition to
the science exposures continuum and arc lamp flatfields were taken, as well as
spectrophotometric standard-star exposures of G191-B2B. In order to correct
for a varying fiber-to-fiber transmission sky flats were taken at the
beginning of the second night.

Reducing the data we used the tool P3d_online that is a part of the PMAS P3d
pipeline (Becker 2002; Roth et al. 2005). At first the bias level was
subtracted and cosmic-ray hits removed (cf. Sandin et al. 2008). Second, a
trace mask was generated from an internal continuum calibration lamp exposure,
identifying the location of each spectrum on the CCD along the direction of
cross-dispersion. Third, a dispersion mask for wavelength calibration was
created using an arc-exposure. In order to minimize effects due to a
significant flexure in the instrument all continuum and arc lamp exposures
were taken within two hours of the respective science exposure. Fourth, a
correction to fiber-to-fiber sensitivity variations was applied by dividing
with an extracted and normalized sky flat-field exposure. In this process the
data was changed from a CCD-based format to a row-stacked-spectra format. As a
final step flux calibration was performed in iraf using standard-star
exposures. We did not correct for differential atmospheric refraction since
we, due to poor spatial resolution and high seeing, have co-added the flux of
147 spatial elements. These elements fully cover the object with an area of
$36.75\arcsec^{2}$. In order to avoid the periodic line shift of stellar
absorption lines of the central star (CS) we made a second spectrum where we
did not add the nine spatial elements, which were the closest to the CS at
$\lambda_{\text{H}\beta}$ (we also used fewer elements on the outer boundary);
this latter spectrum covers an area of $30.75\arcsec^{2}$.

$\text{H}\beta$ is blended by the helium Pickering line
${He\textsc{ii}}\,\lambda 4859$, and the resulting line intensity is hereby
about 5% too high (cf. J02; PT05). The other Balmer lines are likewise blended
by other helium Pickering lines. Since we did not include the CS in our final
spectrum we have not corrected for underlying stellar absorption as PT05 do.
The discussion of the dereddening performed by R02 remains inconclusive, with
the result of a somewhat uncertain and very low extinction towards PN
G135.9+55.9. Also T04 find a low interstellar extinction towards this object
($E(B\\!-\\!V)\\!=\\!0.04$) that is mainly based on the $N_{\text{H\,I}}$
column density they derive from FUSE spectra. Owing to the large differences
of the line strengths found by different observers (see below), however, any
(rather small) correction due to dereddening appears unimportant at this point
of our discussion.

##  3 Results for PN G135.9+55.9

We made all line fits using the IFU analysis package ifsfit (Sandin, in prep.)
that was initially developed for Sandin et al. (2008). We used a second order
polynomial to fit the continuum and Gaussian curves to fit emission lines.
Because of less favorable weather conditions in the second night we only used
the co-added spectrum of the two exposures of the first night, totaling an
exposure time of 5400 s, see Fig. 1.

The intensities of all object lines we measured are presented in Table 1,
together with the values of R02, J02, and T01. We present both raw values, and
values where we subtracted the contribution of He ii Pickering lines from the
Balmer lines. Note that errors of our intensity measurements decrease for
redder emission lines, reflecting the better signal-to-noise of the spectrum
in the redder region. Since the first night was not photometric our value of
the integrated flux of $\text{H}\beta$ is very likely somewhat uncertain.
Nevertheless, we calculated a spatially integrated flux of $\text{H}\beta$,
using the spectrum where the CS is included, of
$1.92\\!\times\\!10^{-14}\,$$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$ (Table
1), and this value compares well with values given in the literature. R02 find
values ranging from $1.0\,\ldots\,2.6\\!\times\\!10^{-14}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$, and J02 evaluate a total
$\text{H}\beta$ flux of $1.5\\!\times\\!10^{-14}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$. While J02 correct their flux
estimate for the region that is not covered by the slit, the varying values of
R02 seem to be a result of the width of the slit they use.

We did not detect $[{Ne\textsc{iii}}]\,\lambda 3869$ (cf. the upper inset of
Fig. 1). Since the signal-to-noise of our spectrum appears to be better than
in previous studies, with five new measured lines in the nebula (see Table 1),
it is questionable if anyone has detected $[{Ne\textsc{iii}}]\,\lambda 3869$.

In order to calculate an upper detection limit of emission lines in our data
we proceeded as follows. For a certain wavelength we assumed that we can
detect a line with the maximum flux of $\sigma$ above the continuum, where
$\sigma$ is the error of the measured flux. We calculated the intensity $I$ of
this limiting line by integrating over a triangle of width 7Å, and
$I\\!=\\!7/2\times\sigma$. At the three wavelengths
$\lambda\\!=\\!3790,\,3860,\,\text{and}\,5010\,$Å we measured
$\sigma\\!=\\!6.17,\,4.70,\,\text{and}\,1.46\\!\times\\!10^{-17}\,$$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}\,\text{\AA}^{-1}$.
Dividing the resulting intensities with the intensity of $\text{H}\beta$ we
got the following limiting line ratios ($\times\\!100$): 1.1
($\lambda\\!=\\!3790\,\AA$), 0.85 ($\lambda\\!=\\!3860\,\AA$), and 0.27
($\lambda\\!=\\!5010\,\AA$). Hence, we consider $0.01\text{H}\beta$ as an
upper limit of the line strength of $[{Ne\textsc{iii}}]\,\lambda 3869$.

Figure 1: The co-added blue spectrum of PN G135.9+55.9 using 123 spatial
elements, including the sky. The upper and lower panels show the wavelength
ranges 3640–4350 Å and 4320–5030 Å, respectively. The insets in the upper and
lower panels show a close-up of the spectrum where H8 and a potential
$[{Ne\textsc{iii}}]\,\lambda 3869$, and $[{O\textsc{iii}}]\,\lambda\lambda
4959,\,5007$ are found, respectively. Gray thick lines in the inset panels
show the line fits. The wavelength ranges of the inset panels are indicated in
the spectrum with horizontal lines. The positions of four emission lines of
telluric origin are also indicated (Hg). The diffuse emission features at,
e.g., $\lambda\\!\simeq\\!4510\,$Å and $\lambda\\!\simeq\\!4980\,$Å are also
of telluric origin. For further details see Sect. 3. Table 1: Flux
measurements of PN G135.9+55.9

Emission line |  $\lambda_{0}$ [Å] | PMAS${}_{\text{raw}}$ | PMAS${}_{\text{corr.}}$ | R02 (SPM1) | R02 (CFHT) | J02${}^{\text{a}}$ | T01 | $I(\text{case B})$
---|---|---|---|---|---|---|---|---
$[{O\textsc{iii}}]$ | 5006.84 | 3.36 | (0.33) | 3.48 | (0.35) | 2.7 | (1.3) | 2.87 | (0.85) | 3 | (1) | 3.1 |
$[{O\textsc{iii}}]$ | 4958.92 | 1.27 | (0.29) | 1.33 | (0.30) |  |  |  |  |  |  |  |
$\text{H}\beta$ | 4861.32 | 100.00 | (0.49) | 100.00 | (0.49) | 100.0 | (2.1) | 100.0 | (1.7) | 100 |  | 100 | 100.0
${He\textsc{ii}}$ | 4685.65 | 78.72 | (0.62) | 82.11 | (0.64) | 76.1 | (2.3) | 78.6 | (1.5) | 77 | (3) | 92 |
${He\textsc{ii}}$ | 4541.59 | 2.96 | (0.39) | 3.09 | (0.40) |  |  |  |  |  |  |  |
H$\gamma$ | 4340.45 | 45.49 | (0.60) | 45.40 | (0.63) | 41.0 | (1.6) | 42.05 | (0.98) | 39 | (3) | 56 | 47.6
${He\textsc{ii}}$ | 4199.83 | 2.48 | (0.59) | 2.59 | (0.61) |  |  |  |  |  |  |  |
H$\delta$ | 4101.74 | 23.81 | (0.64) | 23.69 | (0.66) | 21.4 | (2.8) | 20.6 | (1.2) | 17 | (2) | 30 | 26.6
H7 | 3970.07 | 9.70 | (0.73) | 10.11 | (0.76) | 10.0 | (2.5) | 6.11 | (0.64) | 4 | (1) | 11 | 16.4
H8 | 3889.06 | 8.63 | (0.89) | 9.00 | (0.93) |  |  | 2.92 | (0.73) |  |  |  | 10.8
$[{Ne\textsc{iii}}]$ | 3868.80 | $<\\!1$ |  | $<\\!1$ |  |  |  | 1.04 | (0.52) | 15 | (7) |  |
H9 | 3835.40 | 8.22 | (0.99) | 8.6 | (1.0) |  |  |  |  |  |  |  | 7.5
H10 | 3797.91 | 3.4 | (1.2) | 3.3 | (1.2) |  |  |  |  |  |  |  | 5.8
$F$($\text{H}\beta$)${}^{\text{b}}$ |  | 1.924 | (0.009) |  |  | 1.82 | (0.03) | 2.55 | (0.03) | 1.47 | 1.19 |
${}^{\text{a}}$ The line ratios are corrected for an extinction value of
$E(B-V)\\!=\\!0.03$ and for a $\sim\\!5\%$ contribution of
${He\textsc{ii}}$-lines to Balmer lines.
${}^{\text{b}}$ This is the total flux in emission measured for
$\text{H}\beta$ in units of $10^{-14}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$, systematic errors are not
considered in the error estimate.

Note.— Errors are given in parentheses. The emission line names and the rest
wavelength $\lambda_{0}$, and Case B line ratios for the Balmer lines to
$\text{H}\beta$, are given in Cols. 1, 2, and 9. We present our raw values in
Col. 3 and values corrected for the contribution of ${He\textsc{ii}}$
Pickering lines to the Balmer lines in Col. 4. Columns 5–8 give the values
presented by R02 (SPM1), R02 (CFHT), J02, and T01, respectively.

##  4 Data analysis using hydrodynamic models

Consequences for estimates of abundances in the context of photoionization
models and the assumption of thermal equilibrium are poorly known. In a study
based on hydrodynamical models using time-dependent ionization Schönberner et
al. (2009, hereafter Paper VII; also see ) demonstrate that metal-poor nebulae
with low densities are prone to deviations from thermal equilibrium, because
heating by photoionization is no more balanced by _line_ cooling only. In
extreme cases the electron temperature is also controlled by _expansion_
cooling, which results in _lower_ electron temperatures compared to the
(standard) equilibrium case by up to 30%, although ionization is still close
to equilibrium. Differences are insignificant when solar abundances are used
instead (Perinotto et al. 1998).

In this section we combine our observed line strengths and UV-data obtained
with HST-STIS (Jacoby, priv. comm.) with outcome of our time-dependent
hydrodynamic models in order to estimate abundances. The UV-spectrum we used
is publicly available and can be retrieved from the HST archive (see proposal
9466, PI: Garnavich); it is also published by J06 (see Fig. 1 therein).

At first we present our modeling approach in Sect. 4.1, thereafter our model
sample in Sect. 4.2 and a discussion of how we determine our abundances in
Sect. 4.3.

###  4.1 Physical properties of our time-dependent models

Our one-dimensional radiation hydrodynamic (RHD) models of envelopes of PNe
are described in detail in Perinotto et al. (1998, 2004, also see references
therein). We emphasize that our models calculate ionization, recombination,
heating, and cooling time-dependently at every time step. The cooling function
is composed of the contribution of all considered ions, and for every
individual ion up to 12 ionization stages are taken into account. Physical
input parameters to the models include properties of the coupled CS model,
element abundances, and the density and velocity structures of matter in the
envelope.

In calculating the models the wind of the CS was used as input at the inner
boundary of the grid. The full model evolution was thereafter followed across
the Hertzsprung-Russell diagram for about 15 000 years until (partly)
recombination sets in close to the turn-around point (at an age of about 10
000 yr), and into subsequent stages of re-ionization due to advanced
expansion. An important feature of our models is that we, at any time, can
switch off all time-dependent terms and simultaneously fix the density
structure and the radiation field. The models are thereafter evolved until
they settle into equilibrium, we are then able to study differences between
dynamical and static models (also see Paper VII). We refer to these models as
equilibrium models, in comparison to the dynamical models. We calculated
surface brightnesses, emission line profiles and strengths of individual lines
using a supplementary code that is based on a version of Gesicki et al.
(1996).

###  4.2 Properties of our model sample

A definite mass estimate for the ionizing source of PN G135.9+55.9 is still
lacking. T04 identify the nucleus of PN G135.9+55.9 as a close binary, which
hampers their analysis, using non-LTE model atmosphere spectra, to derive the
mass of the ionizing star. They fit the photospheric Balmer lines of selected
optical spectra to obtain the surface gravity $\log g$. Assuming an effective
temperature of $T_{\text{eff}}\\!=\\!120\,000\,$K, and that the companion does
not contribute to the optical emission, a comparison with stellar evolutionary
tracks indicate a mass of $0.88\,M_{\odot}$ that they infer as unrealistically
high.

T04 support the final estimate of 0.55–$0.57\,M_{\odot}$ considering only the
Population II characteristics of the object (not being in the Galactic plane,
having a high radial velocity and being metal-poor), and its relatively high
kinematic age that the authors consider a rough estimate for the post-AGB age.
In order to be an evolved CS, which still is in a pre-white dwarf stage, a
kinematic age of $t_{\text{kin}}\\!=\\!16\,000\,$yr demanded a mass range that
low (see Fig. 9 in T04). The claim of Tovmassian et al. (2007) that the binary
core of PN G135.9+55.9 consists of one component of $0.565\,\text{M}_{\odot}$
and $T_{\text{eff}}\\!\simeq\\!58\,000\,$K, and a second component of
$0.85\,\text{M}_{\odot}$ and $T_{\text{eff}}\\!\simeq\\!170\,000\,$K, agrees
poorly with their previous results. For instance, the hydrogen and helium
lines will hardly be fitted by an object where
$T_{\text{eff}}\\!=\\!58\,000\,$K. The ionization of the nebula is likewise
inconsistent with an ionizing source where $T_{\text{eff}}\\!=\\!170\,000\,$K,
as PT05 show.

PT05 set $T_{\text{eff}}$, the object distance and luminosity, and select
models using a range of masses (0.583–0.600$M_{\odot}$) based also on
arguments using a kinematic age. Conclusions, which are based on kinematic
ages ($R_{\text{xxx}}/V_{\text{yyy}}$), are questionable when they are used as
evolutionary ages. Errors emerge from uncertain distances, the confusion of
matter (Doppler) velocities with structure (shock) velocities, inappropriate
(i.e. unrelated) combinations of $R_{\text{xxx}}$ and $V_{\text{yyy}}$, and
neglecting the expansion history of the object (Schönberner et al. 2005a, also
see Fig. 33 in Paper VII). The subscripts (xxx and yyy) denote that various
combinations of $R$ and $V$ could be used in order to obtain a kinematic age.
For $R$ one could use the outermost radius ($R_{\text{out}}$), the rim radius,
or the radius of any other substructure. For $V$ one could, moreover, use
corresponding differential changes ($\dot{R}_{\text{xxx}}$, although this
quantity is rarely available), or alternatively spectroscopic velocities such
as the half-width half-maximum (HWHM), HW10%M of spatially unresolved
profiles, or any velocity component of a decomposition of a spectrum of a
spatially resolved profile.

Because the properties and evolutionary history of PN G135.9+55.9 are
uncertain we restricted our analysis to one stellar evolutionary track as
input for our RHD models, and used a post-AGB (CS) model of
$0.595\,M_{\odot}$. Using a similar approach as T04 we only considered one
stellar component when evolving the nebula, and assumed that the influence
from the companion is weak. The stellar wind from the CS is calculated
according to Marten & Schönberner (1991). Its dependence on the metallicity
(in this case C, N, and O) is approximately accounted for by correction
factors; we used $\dot{M}\\!\propto\\!Z^{0.69}$ (Vink et al. 2001) and
$v_{\infty}\\!\propto\\!Z^{0.13}$ (Leitherer et al. 1992), whereby
$L_{\text{wind}}\\!=\\!0.5\dot{M}v_{\infty}^{2}\\!\propto\\!Z^{0.95}$. The CS
radiates as a black body. For the AGB wind we, moreover, adopted a power-law
density profile $\rho\\!\propto\\!r^{-\alpha}$ ($\alpha\\!=\\!3$–3.25, cf.
Sect. 4.3) and a constant outflow velocity
$v\\!=\\!10\text{km}\,\text{s}^{-1}$ (cf. Schönberner et al. 2005b, hereafter
Paper II). We used the radial domain
$4.0\times\\!10^{14}\\!\leq\\!r\\!\leq\\!2.8\\!\times\\!10^{18}\,\text{cm}$.
The models are normalized such that $n\\!=\\!10^{5}\,\text{cm}^{-3}$ at
$r\\!=\\!3\times 10^{16}\,$cm. The abundances are based on scaled values of
mean Galactic disk PNe abundances ($Z_{\text{GD}}$; this abundance
distribution is first quoted by Perinotto et al. 1998) for nine elements, see
Table 2; except for carbon and nitrogen $Z_{\text{GD}}$ is close to solar. The
abundance values $\epsilon_{i}$ are specified in (logarithmic) number
fractions relative to hydrogen, i.e.
$\epsilon_{i}\\!=\\!\log\,(n_{i}/n_{\text{H}})\\!+\\!12$. Abundance
distributions, which are similar to $Z_{\text{GD}}$, are used by for example,
Perinotto (1991), Kingsburgh & Barlow (1994), Exter et al. (2004), and Hyung
et al. (2004). We summarize all model parameters and properties in Table 4.
Selecting metallicities we made use of the set of six models of Paper VII that
we refined with four additional models (these additional models are marked
with the prefix ⋆): $3Z_{\text{GD}}$, $Z_{\text{GD}}$, $Z_{\text{GD}}/3$,
$Z_{\text{GD}}/10$, ${}^{\star}Z_{\text{GD}}/15$,
${}^{\star}Z_{\text{GD}}/20$, ${}^{\star}Z_{\text{GD}}/25$,
$Z_{\text{GD}}/30$, ${}^{\star}Z_{\text{GD}}/60$, and $Z_{\text{GD}}/100$.

Table 2: Mean Galactic disk element abundance distribution ($Z_{\text{GD}}$) | H | He | C | N | O | Ne | S | Cl | Ar
---|---|---|---|---|---|---|---|---|---
$Z_{\text{GD}}$ | 12.00 | 11.04 | 8.89 | 8.39 | 8.65 | 8.01 | 7.04 | 5.32 | 6.46

In order to illustrate differences between dynamical and equilibrium models we
present line strength ratios of three collisionally excited oxygen lines in
the UV, optical, and infrared wavelength ranges in Fig. 2. Representing the
model evolution the line ratios are shown as a function of the CS effective
temperature $T_{\text{eff}}$. The figure shows that lines of dynamical models
are weaker than those of equilibrium models for $T_{\text{eff}}\ga
50\,000\,$K. The reason is that dynamical models have a lower electron
temperature (cf. Sect. 3.1.3 and Fig. 16, both in Paper VII). We do not
consider young PNe, where $T_{\text{eff}}\\!<\\!50\,000\,$K; in such objects
the ionization equilibrium is disturbed for a short time, with the consequence
that the ionization is somewhat overestimated in the equilibrium models.

Discrepancies are larger with a lower metallicity, compare the thin lines with
the thick lines in Fig. 2, and shorter wavelengths, compare the dotted lines
with the dashed lines. When we compare the line intensities at
$T_{\text{eff}}\\!\simeq\\!130\,000\,$K we see that, as expected, the
difference is the largest for the UV-line ${O\textsc{iv}}]\,\lambda\lambda
1402\\!+\\!1405$ that is up to 250% stronger in the equilibrium case of the
$Z_{\text{GD}}/100$ sequence. The optical line $[{O\textsc{iii}}]\,\lambda
5007$ is stronger by about 70%. The infrared line is the least affected, it is
up to about 30% stronger in thermal equilibrium.

Because differences in line strengths depend strongly on the input density and
the velocity structure, these values should only be regarded as indicative.
Nevertheless, non-equilibrium effects introduce an ambiguity to the
determination of electron temperatures, hence elemental abundances, which is
neglected when using standard photoionization codes (cf. Sect. 5).

Figure 2: Comparison between emission line strengths of dynamical (dyn) and
equilibrium (eq) nebular models around a $0.595\,\text{M}_{\odot}$ central
star evolving across the H-R diagram. The two shown model sequences use
abundances $Z_{\text{GD}}/10$ (thick lines) and $Z_{\text{GD}}/100$ (thin
lines). We show the dyn/eq line strength ratios
(I${}_{\text{eq}}/$I${}_{\text{dyn}}$) of three collisionally excited oxygen
lines, viz. ${O\textsc{iv}}]\,\lambda\lambda 1402\\!+\\!1405$ (dotted lines),
$[{O\textsc{iii}}]\,\lambda 5007$ (solid lines), and
$[{O\textsc{iv}}]\,\lambda 26\,\mu$m (dashed lines), as a function of
$T_{\text{eff}}$. The evolution is only traced until maximum $T_{\text{eff}}$
is reached in order to avoid confusion. See Fig. 9 for absolute values on the
intensities. For further details see Sect. 4.2.

###  4.3 Estimating abundances for PN G135.9+55.9

A full abundance analysis of PN G135.9+55.9 is beyond the scope of this work,
but as an example we determine abundance estimates that are based on our
hydrodynamical models. Our model grid is not large enough to allow iteration
of all parameter dimensions. The primary goal of this study is instead to find
a model that agrees reasonably with the observational quantities, and then use
this _best-match_ model to elaborate on the influence of non-equilibrium
effects.

We used the following four criteria to find such a best-match model: emission
line strengths should match observed values, the model H$\alpha$ emission line
profile should match the observed line profile (such as Fig. 1 of Richer et
al. 2003), the model H$\alpha$ surface-brightness distribution should match
the observed distribution (such as Fig. 3 of R02), and both distance estimates
from the $\text{H}\beta$ flux of the object and from the corresponding
apparent size, which is obtained from the surface-brightness distribution,
should be in fair agreement. In addition to these properties that apply to the
nebula, the visual magnitude of the central star is also indicative of the
distance. However, since we have only used one single stellar track (with a
stellar mass $M\\!=\\!0.595\,M_{\sun}$) such a visual magnitude is difficult
to match properly. The emission lines and their line strengths that we used in
our study, are shown in Cols. 1–3 of Table 3.

Table 3: Observational vs. modeled line strengths

ID |  $\lambda_{0}$ [Å] | observed | model | ratio
---|---|---|---|---
|  |  |  | dyn | eq | eq/dyn
${N\textsc{v}}$ | 1238+1242 | 426${}^{\text{a}}$ | (40) | 330 | 697 | 2.11
${N\textsc{iv}}]$ | 1486 | 87${}^{\text{a}}$ | (30) | 124 | 276 | 2.23
${C\textsc{iv}}$ | 1548+1550 | 660${}^{\text{a}}$ | (50) | 668 | 1319 | 1.97
${C\textsc{iii}}]$ | 1906+1909 | 55${}^{\text{a}}$ | (30) | 31 | 64 | 2.06
${O\textsc{iv}}]$ | 1402+1405 |  $<\\!37$${}^{\text{a}}$ |  | 6 | 13 | 2.17
$[{O\textsc{iii}}]$ | 5007 | 3.5${}^{\text{b}}$ | (0.4) | 3.3 | 4.4 | 1.33
$[{O\textsc{iv}}]$ | 26 $\mu$m | – |  | 26 | 30 | 1.15
$[{Ne\textsc{iv}}]$ | 2422+2425 | 25${}^{\text{a}}$ | (12) | 38 | 62 | 1.63
$[{Ne\textsc{v}}]$ | 3426 | 86${}^{\text{a}}$ | (9) | 78 | 119 | 1.52
$[{Ne\textsc{iii}}]$ | 3869 |  $<\\!1$${}^{\text{b}}$ |  | 0.4 | 0.5 | 1.25
$[{Ne\textsc{v}}]$ | 14 $\mu$m | – |  | 90 | 102 | 1.13
${He\textsc{ii}}$ | 4686 | 82${}^{\text{b}}$ | (1) | 82 | 80 | 0.98
${}^{\text{a}}$ Jacoby (priv. comm.), ${}^{\text{b}}$ this paper

Note.— All values are specified in units of $\text{H}\beta\\!=\\!100$. Columns
1 & 2 give the emission line names and the rest wavelength $\lambda_{0}$.
Measured values of PN G135.9+55.9 are given in Col. 3, with errors in
parentheses. Corresponding values of dynamic (dyn) and equilibrium (eq)
models, and their ratio, are given in Cols. 4–6 (cf. Sect. 4.3).

Since we have not been able to measure $[{Ne\textsc{iii}}]\,\lambda\,3869$ we
cannot determine $T_{\text{eff}}$ using that line. Instead we proceeded as
follows. Using the existing measurements of the two neon lines
$[{Ne\textsc{iv}}]\,\lambda\lambda\,2422\\!+\\!2425$ and
$[{Ne\textsc{v}}]\,\lambda\,3426$ we determined a range of effective
temperatures where these measurements match our model emission line strengths
(see Fig. 10). The result is shown in Fig. 3a. The upper limit of the
temperature range corresponding to the $[{Ne\textsc{iv}}]$-line is set by the
maximum stellar temperature possible, in this case
$\max(T_{\text{eff}})\\!=\\!146\,870\,$K. Note that the extent of the two
regions of the effective temperatures of $[{Ne\textsc{iv}}]$ and
$[{Ne\textsc{v}}]$ that do not overlap, depends on both observations and model
initial conditions. We extrapolated both regions in order to find the nearest
point of intersection ($Z_{\text{Ne}}$, $T_{\text{eff}}$) that we then used as
two of the initial values when iterating our models. From this procedure we
found $T_{\text{eff}}\\!=\\!138\,000\,$K.

For carbon, nitrogen and oxygen we evaluated line strengths at
$T_{\text{eff}}$ for all model sequences (Figs. 7–9); the variation of model
line strengths with stellar effective temperature is moderate at
$T_{\text{eff}}\\!=\\!138\,000\,$K (see below). We show the results in Fig.
3b, together with power law fits in relevant regions. By a comparison with the
observational data we then found a first set of object abundances. For sulfur,
chlorine and argon we simply reduced the respective mean Galactic disk value
using the mean scaling value that we found for carbon, nitrogen, oxygen, and
neon.

Figure 3: For each model sequence panel a) shows the range of $T_{\text{eff}}$
that corresponds to the observed line strengths of
$[{Ne\textsc{iv}}]\,\lambda\lambda 2422,\,2425$ and $[{Ne\textsc{v}}]\,\lambda
3426$, compare the plotted values with the neon line strengths in Fig. 10. The
value of the effective temperature of our first custom model is indicated with
an open circle ($\circ$; $T_{\text{eff}}\\!=\\!138\,000\,$K). In panel b) each
sub-panel shows the line strength of all models (but 3Z) that are all
evaluated at $T_{\text{eff}}\\!=\\!138\,000\,$K, for four different emission
lines. The horizontal dashed lines indicate the observed values. In both
panels error intervals are indicated with gray regions. All axes are
logarithmic. For further details see Sect. 4.3.

In a final step, during the creation of our first custom model, we adjusted
the helium abundance. Figure 11 shows that model line strengths of helium are
nearly independent of metal abundances. The same figure also shows that the
variation of each model sequence with the effective temperature is small for
evolved objects (where $T_{\text{eff}}\\!>\\!10^{5}\,$K). We reduced the
initial helium abundance by one third, as is suggested by the ratio of model-
to-observed line strength. With this set of initial abundances we then
calculated additional custom models with modified abundances in order to
converge on the observed line strengths. In this process we iterated element
abundances, which were the closest to the observed values, accounting for the
size of the error bars of the different lines. Setting the carbon abundance we
first gave ${C\textsc{iv}}\,\lambda\lambda 1548\\!+\\!1550$ a stronger weight
than ${C\textsc{iii}}]\,\lambda\lambda 1906\\!+\\!1909$, since its error is
smaller (also see Fig. 3b).

In the course of our iterative calculations we found that the model
$\text{H}\beta$ flux did not match the apparent size of the object at a common
distance. In order to achieve a less extended model we modified the power law
density distribution of the AGB wind by replacing $\alpha\\!=\\!3$, that is
used in all other models of this study, with $\alpha\\!=\\!3.25$; keeping the
density normalization as before (see Sect. 4.1). By this approach changes to
resulting model emission line strengths should be small. In Paper II we show
that a steeper density gradient results in faster expansion rates of the
leading shock of the shell, i.e. the outer edge of the PN. Therefore our new
choice of $\alpha$ may appear counterintuitive when forming a geometrically
smaller PN. The size of a PN, however, depends on the expansion velocity
integrated over the entire expansion period. Early on a high circumstellar
density at the inner edge of the model domain (due to our choice of a power
law) prevents nebular matter from becoming rapidly ionized. The formation of a
D-type ionization front is thereby delayed as the acceleration starts later
with a steeper density gradient, and the overall PN lifetime thereby becomes
shorter. In the case of $\alpha\\!=\\!3.25$ a higher expansion velocity cannot
compensate for the simultaneous shortening of the expansion period.

Figure 4: This figure shows the evolution of emission line strengths of our
best-match model, as a function of the effective temperature,
$T_{\text{eff}}$. Using the ten observed emission lines of Table 3 we show
model-to-observed line ratios (solid lines). The gray regions mark the
corresponding error bars. Note that only upper limits were observed for
${O\textsc{iv}}]\,\lambda\lambda 1402,\,1405$ and $[{Ne\textsc{iii}}]\,\lambda
3869$. For further details see Sect. 4.3. Table 4: Summary of our best-match
model parameters

Parameter | PN G135.9+55.9 | $Z_{\text{GD}}$
---|---|---
Spectrum | black body |
Stellar mass, $M$ | 0.595 $\text{M}_{\odot}$ |
Model age, $t$ |  $8982\,$yr |
Stellar effective temperature, $T_{\text{eff}}$ | 138 049 K |
Stellar luminosity, $L$ | $2994\,\text{L}_{\odot}$ |
Central star wind, $L_{\text{wind}}=0.5\dot{M}v^{2}_{\infty}$ |  $\dot{M}\propto Z^{0.69}$, $v_{\infty}\propto Z^{0.13}$ |
AGB wind |  $\rho\propto r^{-3.25}$, $v\\!=\\!10\,\text{km}\,\text{s}^{-1}$,
| $n_{r=3\times 10^{16}\,\text{cm}}\\!=\\!1\\!\times\\!10^{5}\,\text{cm}^{-3}$
Abundances, $\epsilon_{i}\\!=\\!\log\,(n_{i}/n_{\text{H}})+12$: |  |
He | 10.88 | 11.04
C |  7.90 |  8.89
N |  7.47: |  8.39
O |  6.74 |  8.65
Ne |  6.96 |  8.01
S | [5.94] |  7.04
Cl | [4.22] |  5.32
Ar | [5.36] |  6.46
Distance, $d$ | 18 kpc |
Visual magnitude, $m_{\text{V}}$ | 19.5 mag |
Nebular density, $\langle n_{\text{e}}\rangle$ | 65 $\text{cm}^{-3}$ |
Nebular temperature, $\langle T_{\text{e}}\rangle$ | 21 100 K |
Nebular $\text{H}\beta$-luminosity, $L(\text{H}\beta)$ | 0.193 $\text{L}_{\odot}$ |
Model HWHM velocity, $V_{\text{HWHM}}$ | $41.8\,\text{km}\,\text{s}^{-1}$ |

Note.— The element abundances $\epsilon_{i}$ are used as input in the
calculation of our radiation hydrodynamic models; the values of S, Cl, and Ar
are not fitted, but only scaled. $Z_{\text{GD}}$ denotes the mean abundance
distribution in the Galactic disk (cf. Sect. 4.1).

The abundance distribution of our best-match model, and all relevant model
properties, is given in Table 4, and resulting emission line strengths are
given in Col. 4 of Table 3. We also show model-to-observed line strength
ratios in Fig. 4. This figure illustrates a weak dependence with effective
temperature at values about $T_{\text{eff}}\\!\simeq\\!138\,000\,$K. Since
most lines depend only weakly temperatures about this value, the precise value
of $T_{\text{eff}}$, for say $138\,000\\!\pm\\!5000\,$K, is uncritical to the
ionization structure. We could not achieve a simultaneous agreement for the
two nitrogen lines. The high value of 426 for ${N\textsc{v}}\,\lambda\lambda
1238\\!+\\!1242$ could not be reached with any of our models (Fig. 8), which
is why the nitrogen abundance of our best-match model should be considered
approximate. ${O\textsc{iv}}]\,\lambda\lambda 1402\\!+\\!1405$ can,
furthermore, hardly be identified in the spectrum of J06. We consider the
value of 37 a very conservative upper limit – compare with the value of $87\pm
30$ for ${N\textsc{iv}}]\,\lambda 1486$. A possible blending with
${Si\textsc{iv}}\,\lambda\lambda 1394\\!+\\!1403$ should not be excluded.

Figure 5: This figure shows the basic physical properties of our best fit
model at the stellar parameters $L\\!=\\!2\,985\,\text{L}_{\odot}$ and
$T_{\text{eff}}\\!=\\!138\,152\,$K, at an age of $t\\!=\\!8\,988\,\text{yr}$.
The four panels show: a) the radial structure of of the particle density
(thick line) and the gas velocity (dotted line), b) radial structures of the
electron temperature for the dynamical model (solid line) and the equilibrium
model (dashed line), c) a comparison between the H$\alpha$surface-brightness
distribution of the model at a assumed distance of $18.3\,$kpc and the
observational data of R02 (filled/open squares = semi-major/semi-minor axis,
see text for details) with the actual H$\alpha$-image (dotted line) and an
image that results with a seeing of $1\aas@@fstack{\prime\prime}5$ (solid
line), and d) a comparison between the H$\alpha$ emission line profile of the
model (solid line) and the observation of Richer et al. (2003, see Fig. 1 and
slit 7; open circles $\circ$). The simulated profile was additionally
broadened by a Gaussian with FWHM$=\\!26\,\text{km}\,\text{s}^{-1}$, in order
to be consistent with their observation. The observed left wing is due to
emission of ${He\textsc{ii}}\,\lambda 6560$. For further details see Sect.
4.3.

In Fig. 5 we show structural and kinematic properties of our best-match model;
the CS has evolved to $t\\!=\\!8982\,$yr, $L\\!=\\!2994\,\text{L}_{\odot}$,
and $T_{\text{eff}}\\!=\\!138\,049\,$K (Table 4). The mean electron density is
$\langle n_{\text{e}}\rangle\\!=\\!65\,\mbox{cm}^{-3}$. The density (Fig. 5a)
shows a gradual decline with increasing radius. Neither the density nor the
H$\alpha$ surface-brightness structure (Fig. 5c) show a distinct double shell
morphology. The same panel compares the observed data of R02 (see Fig. 3
therein) with the outcome of our model, using a distance of
$d\\!=\\!18.3\,$kpc. We show the real structure with a central dip that is due
to the hot bubble, and the structure under the seeing conditions of the
observations ($1\aas@@fstack{\prime\prime}5$). A comparison of our observed
$\text{H}\beta$ flux, $F(\text{H}\beta)\\!=\\!1.924\times 10^{-14}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$, and the
$\text{H}\beta$-luminosity of the model,
$L(\text{H}\beta)\\!=\\!0.193\,\text{L}_{\odot}$, yields a distance of
$d\\!=\\!17.95\,$kpc – that agrees well with the one of the surface-brightness
structure. We adopt $d\\!=\\!18\,$kpc as a final value of the distance. In our
models we use exclusively black bodies for the CSs. Applying the distance
estimate of $d\\!\approx\\!18\,$kpc to the bolometric luminosity of the model,
$L\\!=\\!2\,994\,\text{L}_{\odot}$, yields
$m_{\text{V}}\\!=\\!19.5\,$mag111R02 measure $m_{\text{V}}\\!=\\!17.9\,$mag.
Assuming this value instead our best-match model would shift to a
corresponding distance of only $d\\!=\\!8.6\,$kpc. This discrepancy in our
third distance estimate can be explained by our selected stellar mass (one
single track of 0.595 $\text{M}_{\odot}$), which should also be iterated in
order to achieve a better agreement. Of course, our model cannot reproduce the
actual intensity if the nucleus really is a double degenerate..

Throughout the nebula the matter velocity gradient is positive with increasing
radius (Fig. 5a), reaching a maximum velocity of
$v\\!\simeq\\!65\text{km}\,\text{s}^{-1}$. In the adjacent (radiative) shock
layer at $12.5\\!\la\\!r\\!\la\\!14.5\\!\times\\!10^{17}\,\text{cm}$ the
velocity is about constant. The simulated emission line profile (Fig. 5d)
resembles the long slit Echelle spectral analysis of Richer et al. (2003, see
Fig. 1, slit 7). Our one-dimensional model cannot be fully applied to this
slightly non-spherical PN and its asymmetric line profiles. The observed HWHM
velocity, $V_{\text{HWHM}}\\!=\\!42.5\,\text{km}\,\text{s}^{-1}$, is, however,
well matched by our model with
$V_{\text{HWHM}}\\!=\\!41.8\,\text{km}\,\text{s}^{-1}$.

In Fig. 5b we show the radial electron temperature structure. The temperature
peak behind the outer shock does not contribute to the mean temperature due to
low ion densities in that region. We will in the following subsection discuss
the consequences of non-equilibrium conditions for the electron temperature
and, consequently, for the strengths of collisionally excited lines.

##  5 Discussion

In order to study differences in the outcome of our time-dependent and static
(equilibrium) models we should, ideally, also calculate a corresponding best-
match equilibrium model by the same procedure we used to find the best-match
dynamical model. As such an approach is extremely time-consuming we instead
compare our outcome with literature values, which are all based on standard
photoionization codes (these correspond to our equilibrium models). We present
our abundances anew in Table 5 together with a compilation of literature
values, which are derived using observed emission lines. Errors of individual
estimates are specified where such values are provided. Differences between
estimates and physical assumptions of different sources are large in general.
Since we focus on understanding general trends of values, and not on providing
final abundances, we have not estimated errors of our values. This is also
difficult to do with our models where abundances are input parameters, and not
the outcome.

Table 5: Literature compilation of abundances estimates for PN G135.9+55.9

Ref. | spectral | $T_{\text{eff}}$ | $\langle T_{\text{e}}\rangle$ | He | C | N | O | Ne | C/O | N/O | Ne/O
---|---|---|---|---|---|---|---|---|---|---|---
| domain | $[10^{3}\text{K}]$ | $[10^{3}\text{K}]$ |  |  |  |  |  |  |  |  |  |  |  |  |  |
T01 | O | 150 | – |  |  |  |  |  | 6.3 | (0.5) |  |  |  |  |  |  |
J02 | O | 100 | 17.6 | 10.82 |  |  |  |  | 6.93 |  | 7.47 |  |  |  |  | 3.47 |
R02 | O | 100 | 30 | 10.9 |  |  |  |  | 6.15 | (0.35) | 6.35 |  |  |  |  | 0.5 | (0.3)
PT05 | O | 130 | 30 | 10.91 |  |  |  |  | $7.5\phantom{0}$ | (0.3) | 6.65 |  |  |  |  | 0.14 | (0.14)
S05 | O+U | – | – |  | 7.51 | (0.15) | 6.87 | (0.19) | 6.85 | (0.25) | 6.6 | 4.7 | (1.1) | 1.05 | (0.15) | 0.65 | (0.35)
J06 | O+U | 130 | 30${}^{\text{a}}$ | 10.87 | 7.58 |  | 6.94 |  | 7.18 |  | 6.66 | 2.5 |  | 0.58 |  | 0.30 |
this work | O+U | 138 | 21.1 | 10.88 | 7.90 |  | 7.47: | 6.74 |  | 6.96 | 14 |  | 5.4: |  | 1.7 |
$Z_{\text{GD}}$ |  |  |  | 11.04 | 8.89 |  | 8.39 |  | 8.65 |  | 8.01 | 1.74 |  | 0.55 |  | 0.23 |
BoBn-1 |  |  |  | 11.05 | 8.85 |  | 8.00 |  | 7.83 |  | 7.72 | 10.5 |  | 1.48 |  | 0.78 |
${}^{\text{a}}$ Jacoby (priv. comm.) |  |  |  |  |  |  |  |

Note.— The table only includes estimates that are based on observed emission
lines. Columns 1–4 specify the source reference, the wavelength range (O –
optical, and U – UV), the stellar effective temperature ($T_{\text{eff}}$) and
the mean electron temperature ($\langle T_{\text{e}}\rangle$) used in the
study. Columns 5–9 give element abundances using the same units as in Table 2.
In Cols. 10–12 we also give abundance ratios relative to oxygen. Uncertainties
are, were provided, given in parentheses. A colon indicates an uncertain value
of our best-match model. The abundances of our best-match model are given in
the row marked _this work_. In the last two rows we, for comparison, give the
mean abundance distribution of the Galactic disk ($Z_{\text{GD}}$; Table 2)
and the halo-PN BoBn-1 (the values of this object are taken from Howard et al.
1997). For further details see Sect. 5.

Previous studies of PN G135.9+55.9 present improvements to different parts of
the abundance analysis. PT05 provide a thorough model analysis, without making
own observations, where they account for several physical issues, which were
not addressed previously. Notably, they study differences in models assuming
case B vs. non-case B photoionization, they use different sets of collisional
recombination coefficients, and use a stellar atmosphere model of the CS, in
addition to the commonly used black-body model. Due to lack of data they
calibrate their models using only observational data in the visual wavelength
range, and therefore they cannot calculate precise values for the abundances
of carbon and nitrogen. The main conclusion of PT05 is that the oxygen
abundance of previous studies is too low. S05 and J06, moreover, add UV lines,
which are sampled with HST-STIS, to their linelist, and can thereby constrain
the abundances of carbon and nitrogen better than PT05 (S05 also announce IR
observations using SPITZER). S05 also argue that PT05 use too high abundances
for carbon and nitrogen, and therefore have to use a higher oxygen abundance
than is necessary; S05 find a lower value on the oxygen abundance than PT05,
which is in better agreement with estimates of previous studies.

In agreement with all previous studies, except for PT05, we found that the
oxygen abundance of PN G135.9+55.9 is very low. It is difficult to make a
meaningful, more detailed, comparison between our abundances and those of T01,
J02, and R02 since we use different stellar effective temperatures (mainly).
PT05 are the latest authors who base their analysis on only optical emission
lines. The very thorough analysis these authors make is found to be of small
use as there is a strong disagreement between the predicted UV-line
intensities of their best model (that agrees best with model M1, cf. Table 3
in PT05) and observed UV line strengths (Table 3); the difference is (with the
exception of ${C\textsc{iii}}]\,\lambda 1908$) in every case about a factor
two. It is in this context worth mentioning that the electron temperature they
adopt is about 9000 K higher than in our study, resulting in a different
ionization structure (see below). J06 use a similar electron temperature as
PT05 in their photoionization models, which is why their results should be
affected to a similar degree. Furthermore, although S05 include UV-intensities
in their analysis they do not provide any temperatures at all and it is
impossible to make a meaningful comparison with their abundances.

Compared to the mean abundances of the Galactic disk our values for PN
G135.9+55.9 are 1/1.45 (He), 1/9.8 (C), 1/8.3 (N), 1/81 (O), and 1/11 (Ne).
Compared to the total mean metallicity of the Galactic disk our value is 1/13.
Our abundance estimates relative to oxygen are in better agreement with the
values of another halo PN, BoBn-1. In this case our values of C/O, N/O, and
Ne/O are 33–260% higher, although PN G135.9+55.9 is considerably more depleted
of metals.

Figure 6: In panel a) we show the ratio between the emission line strength
values of our best-match model (dyn) and the observed values (cf. Table 3 and
Sect. 4.3). For all, but the two nitrogen lines,
${N\textsc{v}}\,\lambda\lambda 1238\\!+\\!1242$ and ${N\textsc{iv}}]\,\lambda
1486$, the model values and observations agree within error bars. In panel b)
we show emission line strengths of the thermally relaxed best-match model (eq)
relative to the dynamical model (dyn) for all line listed in Table 3. The
vertical dashed lines in both panels indicate limits of different spectral
domains.

In Fig. 6 we illustrate differences between line ratios of our best-match
dynamic model and its corresponding equilibrium model, plotted as a function
of wavelength; we present the same data in Table 3. Fig. 6a shows that the
observed line ratios are satisfactorily matched by the model (Sect. 4.3).
Equilibrium-to-dynamic model line ratios are shown in Fig. 6b for all lines
that we used with our best-match model (also including IR lines; at the
assumed temperature $T_{\text{eff}}\\!\simeq\\!138\,000\,$K, compare with Fig.
2 for the models $Z_{\text{GD}}/10$ and $Z_{\text{GD}}/100$). In this case
differences can be larger than 100% in the EUV, about 50% in the UV, about
20–30% in the optical wavelength range, and about 10–20% in the infrared
wavelength range. A hint of the importance of accurate line ratios is given in
Fig. 3b. If a model line ratio changes by a factor two it is easily seen that
different abundances are required to match the change. Although the non-linear
response to the nebula of the full model is complex, which is why plots such
as Fig. 3b are unsuitable when making quantitative estimates of abundances.

Differences in line ratios between dynamical and equilibrium models occur as a
consequence of a different sensitivity of collisionally excited lines to the
electron temperature. The mean temperature in the nebular region of the two
models are $\langle T_{\text{e}}\rangle_{\text{dyn}}\\!=\\!21\,100\,$K and
$\langle T_{\text{e}}\rangle_{\text{eq}}\\!=\\!25\,100\,$K, compare the two
radial structures in Fig. 5b, the difference is significant. The electron
temperature of our evolved metal-poor models is determined by line cooling and
expansion cooling. It is worth noting that although the oxygen abundance lies
closer to $Z_{\text{GD}}/100$ the mean model abundance is closer to
$Z_{\text{GD}}/10$, and it is this higher abundance that determines the
physical structure of the object (see Fig. 2 and Paper VII, Figs. 15 and 16).
In an observational study using the full wavelength range (EUV–infrared),
where measurement errors are sufficiently small, there should be significant
problems determining abundances of metal-poor objects using models that are
unable to account for dynamical effects.

##  6 Conclusions

PN G135.9+55.9 is an extraordinary object as it is a metal-poor PN with the
lowest oxygen abundance known. In order to clarify contradictory abundance
determinations of PN G135.9+55.9 in the literature we made a new study of this
object using a two-fold approach. At first we re-observed the nebula and could
measure a more accurate spectrum in the visual wavelength range than has been
done so far. Unlike previous observational studies we could only measure an
upper limit of $[{Ne\textsc{iii}}]\,\lambda 3869$ of $0.01\text{H}\beta$,
although we measured five new lines in the nebula. We therefore chose to base
our estimate of the stellar effective temperature using supplementary UV-data
of Jacoby (priv. comm.).

In the second part of our study we used a newly calculated set of our
radiation hydrodynamic models in order to determine abundances and study the
influence of time-dependent effects. In this case such effects are found to be
important, causing lower electron temperatures in the nebula. Resulting line
strengths of dynamical models are lower than in the corresponding equilibrium
models (these models are relaxed after all time-dependent terms are set to
zero). Consequently, different abundances are required to match line strengths
when using either approach. We found that it is only possible to make a self-
consistent abundance determination using a dynamical model that is constrained
using measurements in the entire wavelength range. Our final set of abundances
of the five most abundant elements is: 1/1.45 (He), 1/9.8 (C), 1/8.3 (N), 1/81
(O), and 1/11 (Ne), all with respect to the mean abundance of objects in the
Galactic disk ($Z_{\text{GD}}$). The total metallicity is $Z_{\text{GD}}/13$.
Additionally, using our single $0.595\,M_{\odot}$ evolutionary track we found
an effective temperature of PN G135.9+55.9 of
$T_{\text{eff}}\\!\simeq\\!138\,000\,$K, and a distance of $d\\!=\\!18\,$kpc.
This distance is about the double value assumed by PT05, but agrees well with
the value of T04. The mean electron temperature of our dynamical model is
$\langle T_{\text{e}}\rangle\\!=\\!21\,000\,$K; this is 4000 K lower than the
value of the corresponding equilibrium model.

Although we believe that our approach provides a most significant improvement
when determining abundances of metal-poor objects our modeling can be improved
to provide more accurate values. At first one could consider to iterate more
dimensions of the parameter space, such as e.g. the mass of the central star
and properties of the AGB wind. Three additional suggestions for such
improvements that are all considered by PT05 are: using non-CaseB radiative
transfer, replacing the black-body model of the central star with a model
atmosphere, and using improved collision rates. Observationally an accurate
multi-wavelength study including the infrared wavelength range, such as is
announced by S05, will help to constrain the models further. Last, but not
least, it is important to clarify the parameters and evolutionary history of
the ionizing central star(s) unambiguously.

###### Acknowledgements.

C. S. acknowledges support by DFG grant SCHO 394/26. We thank G. Jacoby both
for providing us with UV data prior to their publication, and for providing us
feedback on a late version of the manuscript.

## References

  * Balayan (1997) Balayan, S. K. 1997, Astrofizika, 40, 153
  * Becker (2002) Becker, T. 2002, PhD thesis, Univ. Potsdam
  * Exter et al. (2004) Exter, K. M., Barlow, M. J., & Walton, N. A. 2004, MNRAS, 349, 1291
  * Garnavich & Stanek (1999) Garnavich, P. M. & Stanek, K. Z. 1999, JAAVSO, 27, 79
  * Gesicki et al. (1996) Gesicki, K., Acker, A., & Szczerba, R. 1996, A&A, 309, 907
  * Howard et al. (1997) Howard, J. W., Henry, R. B. C., & McCartney, S. 1997, MNRAS, 284, 465
  * Hyung et al. (2004) Hyung, S., Pottasch, S. R., & Feibelman, W. A. 2004, A&A, 425, 143
  * Jacoby et al. (2002) Jacoby, G. H., Feldmeier, J. J., Claver, C. F., et al. 2002, AJ, 124, 3340 (J02)
  * Jacoby et al. (2006) Jacoby, G. H., Garnavich, P. M., Bond, H. E., et al. 2006, in Planetary Nebulae in our Galaxy and Beyond, ed. M. J. Barlow & R. H. Méndez, IAU Symp., 234, 431 (J06)
  * Kingsburgh & Barlow (1994) Kingsburgh, R. L. & Barlow, M. J. 1994, MNRAS, 271, 257
  * Leitherer et al. (1992) Leitherer, C., Robert, C., & Drissen, L. 1992, ApJ, 401, 596
  * Marten & Schönberner (1991) Marten, H. & Schönberner, D. 1991, A&A, 248, 590
  * Napiwotzki et al. (2005) Napiwotzki, R., Tovmassian, G., Richer, M. G., et al. 2005, in Planetary Nebulae as Astronomical Tools, ed. R. Szczerba, G. Stasińska, & S. K. Gorny, AIP Conf. Series, 804, 173
  * Péquignot & Tsamis (2005) Péquignot, D. & Tsamis, Y. G. 2005, A&A, 430, 187 (PT05)
  * Perinotto (1991) Perinotto, M. 1991, ApJS, 76, 687
  * Perinotto et al. (1998) Perinotto, M., Kifonidis, K., Schönberner, D., & Marten, H. 1998, A&A, 332, 1044
  * Perinotto et al. (2004) Perinotto, M., Schönberner, D., Steffen, M., & Calonaci, C. 2004, A&A, 414, 993
  * Richer et al. (2003) Richer, M. G., López, J. A., Steffen, W., et al. 2003, A&A, 410, 911
  * Richer et al. (2002) Richer, M. G., Tovmassian, G., Stasińska, G., et al. 2002, A&A, 395, 929 (R02)
  * Roth et al. (2005) Roth, M. M., Kelz, A., Fechner, T., et al. 2005, PASP, 117, 620
  * Sandin et al. (2008) Sandin, C., Schönberner, D., Roth, M. M., et al. 2008, A&A, 486, 545
  * Schönberner et al. (2005a) Schönberner, D., Jacob, R., & Steffen, M. 2005a, A&A, 441, 573
  * Schönberner et al. (2005b) Schönberner, D., Jacob, R., Steffen, M., et al. 2005b, A&A, 431, 963 (Paper II)
  * Schönberner et al. (2005c) Schönberner, D., Jacob, R., Steffen, M., & Roth, M. M. 2005c, in Planetary Nebulae as Astronomical Tools, ed. R. Szczerba, G. Stasińska, & S. K. Gorny, AIP Conf. Ser., 804, 269
  * Schönberner et al. (2009) Schönberner, D., Jacob, R., Sandin, C., & Steffen, M. 2009, A&A, submitted (Paper VII)
  * Stasińska et al. (2005) Stasińska, G., Tovmassian, G. H., Richer, M. G., et al. 2005, in From Lithium to Uranium: Elemental Tracers of Early Cosmic Evolution, ed. H. V., P. François, & F. Primas, IAU Symp., 228, 323 (S05)
  * Tovmassian et al. (2007) Tovmassian, G., Tomsick, J., Napiwotzki, R., et al. 2007, in Asym. Planetary Nebulae IV, ed. R. L. M. Corradi, A. Manchado, & N. Soker, I.A.C. electronic publ., 515
  * Tovmassian et al. (2004) Tovmassian, G. H., Napiwotzki, R., Richer, M. G., et al. 2004, ApJ, 616, 485 (T04)
  * Tovmassian et al. (2001) Tovmassian, G. H., Stasińska, G., Chavushyan, V. H., et al. 2001, A&A, 370, 456 (T01)
  * Vink et al. (2001) Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2001, A&A, 369, 574

##  Appendix A Intensity evolution of our RHD models

For each model sequence we mention in Sect. 4.2 we show the intensity
evolution of all emission lines of Table 3 in Figs. 7-11. Solid lines show
dynamic models and dotted lines equilibrium models (for those sequences where
they were calculated). The abscissa is in every case the stellar effective
temperature $T_{\text{eff}}$. Horizontal dashed lines mark observed values for
PN G135.9+55.9, and gray shaded regions mark corresponding error intervals.
Additionally we show the evolution of the two infrared lines,
$[{O\textsc{iv}}]\,\lambda 26\,\mu$m and $[{Ne\textsc{v}}]\,\lambda 14\,\mu$m,
despite a lack of currently existing observational data.

Figure 7: The line strength evolution of different carbon lines as a function
of $T_{\text{eff}}$ for all models. Figure 8: The line strength evolution of
different nitrogen lines as a function of $T_{\text{eff}}$ for all models.
Figure 9: The line strength evolution of different oxygen lines as a function
of $T_{\text{eff}}$ for all models. Figure 10: The line strength evolution of
different neon lines as a function of $T_{\text{eff}}$ for all models. Figure
11: The line strength evolution of ${He\textsc{ii}}\,\lambda\,4686$ as a
function of $T_{\text{eff}}$ for all models.