# Zero-Shot Federated Learning with New Classes for Audio Classification

###### Abstract

Federated learning is an effective way of extracting insights from different
user devices while preserving the privacy of users. However, new classes with
completely unseen data distributions can stream across any device in a
federated learning setting, whose data cannot be accessed by the global server
or other users. To this end, we propose a unified zero-shot framework to
handle these aforementioned challenges during federated learning. We simulate
two scenarios here – 1) when the new class labels are not reported by the
user, the traditional FL setting is used; 2) when new class labels are
reported by the user, we synthesize Anonymized Data Impressions by calculating
class similarity matrices corresponding to each device’s new classes followed
by unsupervised clustering to distinguish between new classes across different
users. Moreover, our proposed framework can also handle statistical
heterogeneities in both labels and models across the participating users. We
empirically evaluate our framework on-device across different communication
rounds (FL iterations) with new classes in both local and global updates,
along with heterogeneous labels and models, on two widely used audio
classification applications – keyword spotting and urban sound classification,
and observe an average deterministic accuracy increase of $\sim$4.041% and
$\sim$4.258% respectively.

Index Terms: keyword spotting, urban sound classification, federated learning,
new class identification, zero-shot learning, on-device learning

##  1 Introduction

Deep learning for audio classification is a broad research area with
applications like Keyword Spotting (KWS), urban sound identification, etc. KWS
is an important application for detecting keywords of importance to specific
users, which could be used as voice commands to on-device personal assistants
such as Amazon’s Alexa, Apple’s Siri, etc. [1]. Urban environment sound
classification is another interesting application particularly in context-
aware computing, urban informatics [2]. The emergence of deep neural networks
have conveniently alleviated problems of creating shallow (hand-picked)
features to achieve state-of-the-art performance in such acoustic
classification tasks [3, 4]. With the recent compute capabilities vested in
resource-constrained devices, there is a huge research focus on audio
classification using on-device deep learning [5, 6].

Figure 1: Architecture of proposed Federated Learning framework with new
classes streaming in across different users.

Such applications require characterization of insights across numerous user
devices for personalization, and collaborative on-device deep learning becomes
necessary. Federated Learning (FL) is a decentralized method of training
neural networks by securely sharing model updates with a server without the
need to transfer sensitive local user data [7, 8]. On-device federated
learning has been an active area of research addressing challenges on secure
communication protocols, optimization, privacy-preserving networks, etc. [9,
10]. However, handling new/unseen classes in local devices and training them
in an FL setting for the global model to possess characteristics of the new
classes is a challenging task, since data transfer from local device to server
and vice versa is not feasible. Moreover, the new class information of one
user is not known among the other users as well, hence the new classes could
be similar or different between the users. In addition, there are multiple
statistical heterogeneities like model heterogeneities (ability of end-users
to architect their own local models), label heterogeneities and non-IIDness
across various communication rounds/FL iterations (disparate data and label
distributions across devices).

One way of handling model heterogeneities and independence in a federated
learning setting is by using knowledge distillation [11] with a common student
model architecture on each local device [12]. Label and model heterogeneities
are handled in an inertial Human Activity Recognition scenario in [13].
Federated learning for keyword spotting [14], and new class learning and
identification in various speech recognition settings are addressed in [15,
16]. [17] proposes a new augmentation technique to reduce false reject rates
and addresses algorithmic constraints in FL-KWS training to label examples
with no visibility. However, the scope of our proposed work is different in
the nature that it primarily addresses identification and similarity detection
of new labels in a zero-shot manner when heterogeneous label and model
distributions exist across various FL iterations and users. To the best of our
knowledge, none of the papers discuss new label identification in FL settings
with statistical heterogeneities for audio classification.

Our scientific contributions are:  (1) A framework with zero-shot learning
mechanism by synthesizing Anonymized Data Impressions from class similarity
matrices to identify new classes for keyword spotting and urban sound
detection in on-device FL settings.  (2) Provide two scenarios for label
acquisition – when class label is reported by user, and when class label is
not, and propose unsupervised clustering to identify and differentiate between
newly reported classes.  (3) Handling statistical heterogeneities such as
heterogeneous distributions in labels, data and models across devices and FL
iterations.

##  2 Our Approach

In this section, we discuss the problem formulation of new classes in FL, and
our proposed framework (Algorithm 1) to handle the same. The overall
architecture is given in Figure 1.

###  2.1 Problem Formulation

We assume the following scenario in federated learning. Suppose there are $M$
nodes (devices) in the FL network, holding private local data
$\mathcal{D}_{i}=\\{x_{i,j},y_{i,j}\\}$ where $i$ is the FL iteration and $j$
is the user index. Each node consists of public data
$\mathcal{D}_{0}=\\{x_{0},y_{0}\\}$. The public data is assumed to be present
across the global and all local users as discussed in [12] to handle the
various statistical (model) heterogeneities which is a common phenomena in FL.
The overall label-set of public dataset is $Y=\\{y_{0}\\}$, which represent
its unique labels. We re-purpose this public dataset as test set and do not
expose it to local models during FL training iterations, but expose only
during testing for consistency. Our work’s main contribution is to propose a
framework to identify new labels across different users without transferring
private data in FL setting. We also assume each user can stream data with new
labels at any iteration which does not belong to public label-set $Y$, i.e.
$\mathbf{y}_{i,j}\not\in Y$. In other words, the global user has no idea of
these new labels.

Algorithm 1 Our Proposed Framework

Input: Public Dataset $\mathcal{D}_{0}\\{x_{0},y_{0}\\}$, Private Datasets
$\mathcal{D}_{m}^{i}$, Total users $M$, Total iterations $I$, LabelSet $l_{m}$
for each user, Overall Public LabelSet $Y$,

Output: Trained Model scores $f_{G}^{I}$

Initialize $f_{G}^{0}=\mathbf{0}$ (Global Model Scores)

for $i=1$ to $I$ do

for $m=1$ to $M$ do

Build: Model $\mathcal{D}_{m}^{i}$ and predict
$f_{\mathcal{D}_{m}^{i}}(x_{0})$

Local Update:

Choice 1: New classes are not
reported$f_{\mathcal{D}_{m}^{i}}(x_{0})=f_{G}^{I}(x_{0}^{l_{m}})+\alpha
f_{\mathcal{D}_{m}^{i}}(x_{0})$, where $f_{G}^{I}(x_{0}^{l_{m}})$ are global
scores of $l_{m}$ with $m^{th}$ user,
$\alpha=\frac{len(\mathcal{D}_{m}^{i})}{len(\mathcal{D}_{0})}$

Choice 2: New classes are reportedTrain a new model with $\mathcal{D}_{0}$ and
$\mathcal{D}_{m}^{i}$ (new data) together, and send weights of the last layer
($\mathbf{W}_{m}^{i}$) to global user.

end for

Global Update:

Choice 1: No user reports new classesUpdate label wise
$f_{G}^{i+1}=\displaystyle\sum_{m=1}^{M}\beta_{m}f_{\mathcal{D}_{m}^{i}}(x_{0})$,
where $\beta=\begin{cases}1&\text{If labels are unique}\\\
\text{acc}(f_{\mathcal{D}_{m}^{i+1}}(x_{0}))&\text{if labels are not
unique}\end{cases}$ where $\text{acc}(f_{\mathcal{D}_{m}^{i+1}}(x_{0}))$ is
the accuracy metric, defined by the ratio of correctly classified samples to
total samples for a given local model.

Choice 2: Any user reports new classesCreate Data Impressions (DI) for each
user $m$ with weights $\mathbf{W}_{m}^{i}$ (Section 2.2). Average DI of all
users with new classes, $\mathbf{X}^{i}=\sum_{m\in
M_{S_{k}}}\mathbf{X}_{m}^{i}$, where $M_{S_{k}}$ is set of users with new
label $k$.Perform k-medoids clustering on $\mathbf{X}^{i}$ across $M_{S_{k}}$.
Number of clusters = Number of new labels ($l_{new}$). Update public dataset
with new DI ($\mathbf{X}^{i}$),
$\mathcal{D}_{new}=\mathcal{D}_{0}\bigcup\mathbf{X}^{i}$, add $l_{new}$ to
$l_{m}$ and $Y$.

end for

###  2.2 Anonymized Data Impressions

The main challenge/objective is to detect similar labels across different
users in FL heterogeneous settings without the knowledge of local user data.
This necessitates us to construct anonymized data without transferring raw
sensitive data, and identify new class similarities on the anonymized data. We
motivate our framework from the creation of Data Impressions (DI) using zero-
shot learning as proposed in [18] to compute Anonymized Data Impressions. Let
us assume a model $\mathcal{M}$ with input $\mathbf{X}$ and output
$\mathbf{y}$, where $\mathbf{X}\in\mathcal{R}^{M\times N}$ is the set of
features and $\mathbf{y}\in\mathcal{R}^{M}$. Now, the anonymized feature set
$\bar{\mathbf{X}}$, which has same properties of $\mathbf{X}$, can be
synthesized in two steps:

Table 1: Model Architectures (filters in each layer), Labels and Audio frames
per FL iteration across user devices for both datasets. Note the disparate
model architectures and labels across users.

| User_1 | User_2 | User_3 | Global_User
---|---|---|---|---
Architecture |  | 2-Layer CNN
---
(16, 32)
Softmax Activation
| 3-Layer CNN
---
(16, 16, 32)
ReLU Activation
| 3-Layer Depth-Separable CNN
---
(16, 16, 32)
ReLU Activation
–
Keywords | {Yes, No, Up, Down} | {Up, Down, Left, Right} | {Left, Right, On, Off} |  | {Yes, No, Up, Down, Left, Right,
---
Left, Right, On, Off}
| Keyword Frames
---
per iteration
| {200-300, 200-300,
---
200-300, 200-300}
| {200-300, 200-300,
---
200-300, 200-300}
| {200-300, 200-300,
---
200-300, 200-300}
| {300*8} = 2400
---
Sounds |  | {air conditioner, car horn,
---
children playing}
| {children playing, dog bark,
---
drilling }
| {drilling, engine idling,
---
gun shot, jackhammer}
| {air conditioner, car horn, children playing,
---
dog bark, drilling, engine idling, gun shot, jackhammer}
| Sound Frames
---
per iteration
| {40-50, 40-50, 40-50}
---
| {40-50, 40-50, 40-50}
---
| {40-50, 40-50,
---
40-50, 40-50}
| {50*8} = 400
---

(a) Sample Softmax Values: The first step is to sample the softmax values from
the Dirichlet distribution [19]. The Class Similarity Matrix (CSM) is created
which contains important information on how similar the classes are to each
other. If the classes are similar, we expect the softmax values are
concentrated over these labels. CSM is obtained by considering the weights of
the model’s last layer. Typically, any classification model has the final
layer as fully-connected layer with a softmax non-linearity. If the classes
are similar, we find similar weights between connections of the penultimate
layer to the nodes of the classes [18]. The Class Similarity Matrix is
constructed as,

$\displaystyle
C(i,j)=\frac{\mathbf{w}_{i}^{T}\mathbf{w}_{j}}{||\mathbf{w}_{i}||||\mathbf{w}_{j}||}$
(1)

where $\mathbf{w}_{i}$ is the vector of weights connecting the previous layer
nodes to the class node $i$. $\mathbf{C}\in\mathcal{R^{K\times K}}$ is the
Class Similarity Matrix for $K$ classes. We then sample the softmax values as,

$\displaystyle\text{Softmax}=Dir(K,C)$ (2)

where $C$ is concentration parameter which controls the spread of softmax
values over class labels.

(b) Creating Anonymized Data Impressions: Let
$\mathbf{Y}^{k}=[\mathbf{y}_{1}^{k},\mathbf{y}_{2}^{k},\cdots,\mathbf{y}_{N}^{k}]\in\mathcal{R}^{K\times
N}$ be the N softmax vectors corresponding to class $k$, sampled from
Dirichlet distribution from previous step. Once we obtain the softmax values,
we compute the synthesized data features (Data Impressions) by solving the
following optimization problem using model $\mathcal{M}$ and sampled softmax
values $\mathbf{Y}^{k}$

$\displaystyle\bar{\mathbf{x}}=\arg\underset{\mathbf{x}}{\min}L_{CE}(\mathbf{y}_{i}^{k},\mathcal{M}(\mathbf{x}))$
(3)

To solve this optimization problem, we initialize the input $\mathbf{x}$ to be
random input and iterate until cross-entropy loss ($L_{CE}$) minimization.
This process is repeated for all $K$ categories. In this way, anonymized data
impressions are created for each class without the visibility of original
input data. We use the TensorFlow framework [20] for all our experiments.

(a) GKWS - Local Update

(b) GKWS - Global Update

(c) US8K - Local Update

(d) US8K - Global Update

Figure 2: Local-Global update accuracies across 10 users and 30 FL iterations
for both datasets with new classes and heterogeneities.

###  2.3 Proposed Framework

There are three steps in our proposed framework (Algorithm 1).

(a) Build: Each local user creates their own model with their local private
data for a specific iteration.

(b) Local Update: In this step, if new classes are not reported, we perform
simple weighted $\alpha$-update [21], where $\alpha$ governs the contributions
of new and old models across FL iterations shown in Algorithm 1 Choice 1. If
new classes are reported, we train the new class data along with public
dataset, and send the new model weights to global user (Choice 2).

(c) Global update: In this step, if no user reports new classes, we perform
label-based averaging using the parameter $\beta$, which governs contributions
of overlapping labels using corresponding test accuracies (Choice 1). If user
reports new classes, we create Anonymized Data Impressions (DI) for new
classes followed by unsupervised clustering using k-medoids with motivations
from [22] (Choice 2).

Typically, statistical heterogeneities are widely observed in practical FL
settings, hence Choice 1 handles heterogeneities in local and global update
steps [13], while Choice 2 handles new classes in our proposed framework.

##  3 Experiments and Results

We simulate our experiments using Raspberry Pi 2 as our user device with
Google Speech Commands (GKWS) [23] and UrbanSound8K (US8K) [2] datasets across
10 FL iterations/communication rounds using our proposed framework. In GKWS,
we choose the keywords: Yes, No, Up, Down, Left, Right, On, Off, Stop and Go,
and perform regular Mel-frequency Cepstral Coefficients (MFCC) extraction as
performed in [1], with sampling frequency of 14400 HZ. The MFCC data is
divided into 20 windows and each window is of size 50 ms. US8K, an
environmental sound dataset, consists of 10 classes of sound events: air
conditioner, car horn, children playing, dog bark, drilling, engine idling,
gun shot, jackhammer, siren and street music. We perform similar preprocessing
as performed in GKWS for US8K as well.

Public Dataset: We create a Public Dataset ($D_{0}$) with 2400 audio frames
for GKWS (8 keywords with 300 frames each), and 400 audio frames for US8K (8
sounds with 50 frames each) as shown in Table 1. $D_{0}$ is visible to both
global and local users in each FL iteration, and is updated with data
synthesized for unseen/new classes only – Anonymized Data Impressions.

We initially consider eight labels with an initial Public Dataset in both
datasets before streaming new classes (Table 1). We simulate two scenarios for
testing just our zero-shot framework – 1) new classes only (homogeneous) with
limited users and FL iterations (3 users and 10 FL iterations) for effective
analysis of results, 2) new classes with statistical heterogeneities in both
labels and models as performed in [13], (10 users and 30 FL iterations). This
exhibits near-real-time model heterogeneities as shown in Table 2, and
effective convergence.

New Classes: We introduce two new/unseen labels {Stop, Go} for GKWS and
{Siren, Street music} for US8K across four FL iterations and two users. In the
homogeneous case, for GKWS, we induce 400 samples each with Stop class in
iteration 4 for both User 1 and User 2, and 500 samples each with Stop in User
1 iteration 8 and Go in User 2 iteration 8. Similarly, we induce 50 samples
each with Siren class in iteration 4 for both User 1 and User 2, and 50
samples each with Siren in User 1 iteration 8 and Street music in User 2
iteration 8. This is the FL scenario with new classes without any
heterogeneities. We also discuss similar FL scenarios with statistical
heterogeneities.

Table 2: Details of heterogeneities - model architectures (filters) and new
classes changing across FL iterations and users for both datasets.

Iteration | New Model | New Class
---|---|---
User 1 Iteration 6 |  | 3-Layer ANN (16, 16, 32)
---
ReLU Activation
-
User 1 Iteration 8 |  | 1-Layer CNN (16)
---
Softmax Activation
-
User 2 Iteration 4, 6 |  | 3-Layer CNN (16, 16, 32)
---
Softmax activation
Stop/Siren
User 3 Iteration 5 |  | 4-Layer CNN (8, 16, 16, 32)
---
Softmax activation
-
User 4 Iteration 3, 7 | - | Go/Street Music
User 6 Iteration 3, 5 | - | Stop/Siren
User 9 Iteration 4 | - | Stop/Siren

(a) Label Heterogeneities: In every FL iteration, we consider a random number
of audio frames generated between 200-300 samples per label for GKWS, while
40-50 samples per label for US8K. We split these labels across three users
such that labels can either be unique or overlapping across users. We also
simulate non-IIDness across FL iterations with disparities in both labels and
distributions in data (statistical heterogeneities).

(b) Model Heterogeneities: We consider the three model architectures as shown
in Table 1 motivated from [1, 24], and also change model architectures,
filters and activation functions across FL iterations in addition to label
heterogeneities with new classes (Table 2). The user iterations are chosen at
random.

(a) GKWS - PCA - Different Class

(b) GKWS - PCA - Same Class

(c) US8K - PCA - Different Class

(d) US8K - PCA - Same Class

Figure 3: PCA (with 2 dimensions) of k-medoids unsupervised clustering with
new classes, with same and different classes for both datasets.

###  3.1 Discussion on Results

Table 3: Local-global update accuracies (%) for both datasets across 3 users
with just new classes, without heterogeneities.

| GKWS | US8K
---|---|---
User | Local | Global | Increase | Local | Global | Increase
User 1 | 89.684 | 93.166 | 3.482 | 76.526 | 80.214 | 3.688
User 2 | 91.888 | 95.28 | 3.391 | 75.272 | 77.944 | 2.672
User 3 | 91.517 | 94.727 | 3.211 | 77.61 | 81.838 | 4.228
Average | 91.03 | 94.391 | 3.361 | 76.469 | 80 | 3.529

From Table 3, we can observe that there is an accuracy increase in the FL
scenario with just new classes (without heterogeneities) in corresponding
global updates for all three users than their respective local update
accuracies for both datasets in spite of new classes streaming in. The average
local-global accuracy increase across all 10 FL iterations and 3 users is
$\sim$3.361% and $\sim$3.529% respectively for GKWS and US8K. Similarly, we
can also observe that with our proposed framework, the final global accuracies
(with convergence after all FL iterations) even with new classes and
heterogeneities are 96.541% and 82.498% (Figure 5) which are much higher than
their respective local update accuracies. The corresponding local-global
update accuracies across 30 FL iterations and 10 users are shown in Figure 2.
The class similarity matrix of different classes for GKWS is showcased in
Figure 5, which elucidates the misclassifications. We can also infer that the
clusters effectively formed with k-medoids are equal to the number of new
classes, which are visualized using Principal Component Analysis (PCA) in two-
dimensions. The new classes can either be different or same across user
devices (Figure 3), and these classes are correctly mapped back to the
respective end-user devices. The new labels are then finally added to the
overall label set while the corresponding averaged data impressions are added
to the public dataset.

Figure 4: Class Similarity Matrix for GKWS.

Figure 5: Final accuracies (%) for 10 users, 30 FL iters.

Update | GKWS | US8K
---|---|---
Local | 92.5 | 78.24
Global | 96.541 | 82.498
Increase | 4.041 | 4.258

###  3.2 On-Device Performance

Raspberry Pi 2 (900MHz quad-core ARM Cortex-A7 CPU with 1GB RAM) is used for
evaluating our proposed FL framework as it has similar hardware and software
(HW/SW) specifications to predominant contemporary IoT/mobile devices. The
computation times are identical for both datasets due to similar
preprocessing. The size of the models used are also 520 kB, 350 kB, 270 kB
respectively for user architectures mentioned in Table 1.

Table 4: Computation Times with Raspberry Pi 2 Process | Time
---|---
| Training time per epoch
---
in an FL iteration ($i$)
$\sim$1.2 sec
Inference time |  $\sim$11 ms

##  4 Conclusions

This paper presents a novel framework for handling new labels in a federated
learning setting. We propose a zero-shot learning framework by synthesizing
Anonymized Data Impressions from Class Similarity matrices to learn new
classes across different user devices. We also account for heterogeneities in
labels and models across different FL communication rounds, and systematically
analyze the results for two widely used audio classification applications –
keyword spotting and urban sound classification. We further demonstrate the
effectiveness and scalability of our proposed FL framework by simulating our
experiments on-device using a Raspberry Pi 2.

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11institutetext: Department of Physics and Astronomy, University of Denver,
2112 E Wesley Ave., Denver, CO 80208, USA
11email<EMAIL_ADDRESS>22institutetext: Okayama Observatory, Kyoto
University, Honjo, Kamogata, Asakuchi, Okayama, 719-0232, Japan
22email<EMAIL_ADDRESS>
# On the N/O abundance ratio and the progenitor mass
for the most luminous planetary nebulae of M 31

Toshiya Ueta (植田 稔也) 11 Masaaki Otsuka (大塚 雅昭) 22

(Received August 12, 2022; Accepted October 9, 2022)

###### Abstract

Context. Plasma diagnostics are the bases of investigation into the physical
and chemical properties of line-emitting gaseous systems.

Aims. To perform plasma diagnostics properly, it is essential to correct the
input spectrum for extinction properly. This is simply because determining the
degree of extinction is dependent on the physical properties of the line-
emitting gas. Hence, both extinction correction and plasma diagnostics have to
be performed simultaneously and self-consistently.

Methods. By comparing the results of analyses performed for a sample of nine
bright planetary nebulae in M 31 with and without the fully iterative self-
consistent simultaneous extinction correction and plasma diagnostics, we
demonstrate how a seemingly benign initial assumptions for the physical
conditions of the line-emitting gas in extinction correction would compromise
the results of the entire analyses in terms of the extinction, electron
density/temperature, and ionic/elemental abundances.

Results. While the electron density/temperature are relatively immune to the
imposed inconsistent assumptions, the compromised extinction would cause
systematic offsets in the extinction-corrected line strengths/spectrum, which
consequently would impose adverse effects on the resulting ionic and elemental
abundances, and other inferences made from the incorrect results.

Conclusions. We find that this M 31 PN sample simply represents those around
the high-mass end of the mass range for low-mass planetary nebula progenitor
stars as expected from the existing theoretical models. It appears that the
suspicion raised in the previous study – these PNe being anomalously nitrogen
overabundant for the expected progenitor mass range – is simply caused by the
apparent underestimate in extinction that originates from the imposed
inconsistent assumptions in extinction correction. In a larger context, the
results of plasma diagnostics in the literature without seeking simultaneous
self-consistency with extinction correction have to be handled cautiously.
Ideally, such previous results should be re-evaluated by seeking simultaneous
self-consistency.

###### Key Words.:

Methods: data analysis – Techniques: spectroscopic – planetary nebulae:
individual: M1687, M2068, M2538, M50, M1596, M2471, M2860, M1074, M1675 –
circumstellar matter – ISM: abundances – dust, extinction

##  1 Introduction

In a recent article “On the most luminous planetary nebulae of M 31,” Galera-
Rosillo et al. (2022, hereafter GR22) have presented the results of plasma
diagnostics of planetary nebulae (PNe) in the Andromeda Galaxy (M 31). The
analyses were based on optical spectra of nine bright PNe (the brightest four,
plus five others as control cases) taken with the OSIRIS instrument at the
10.4 m Gran Telescopio Canarias (GTC) supplemented by archival spectra. Their
study was aimed at investigating the physico-chemical properties of these PNe
and the progenitor mass of the central stars expected to be at the tip of the
PN luminosity function (PNLF; Jacoby 1980).

The nine bright PNe in M 31 in question have been found to have achieved the
maximum temperature as suggested by the post-AGB evolutionary tracks for stars
with initial mass of $\sim$1.5 M⊙, providing spectroscopic constraints for the
stellar progenitors that define the PNLF cutoff for M 31 and for other similar
star-forming galaxies. However, these PNe have also been found with the N/O
abundance ratio that is 1.5 to 3 times larger than predicted for such stars
(of $\sim$1.5 M⊙ initial mass), indicating possible limitations for the
existing theoretical models.

Meanwhile, our recent examination on a specific PN case has indicated that,
unless both extinction correction and plasma diagnostics are performed
simultaneously and self-consistently, the results of the subsequent abundance
analyses can be off by tens of percents (Ueta & Otsuka, 2021). For emission-
line objects, the degree of extinction, $c({\rm H}\beta)$,111The base-10
power-law index $c(\lambda)$ as in $I(\lambda)=I_{0}(\lambda)\times
10^{-c(\lambda)}=I_{0}(\lambda)\times 10^{-c({\rm H}\beta)(1+f(\lambda))}$
referenced at H$\beta$. Here, $I(\lambda)$ and $I_{0}(\lambda)$ are the
observed (attenuated) and intrinsic (unattenuated) specific flux at $\lambda$
and $f(\lambda)$ refers to the adopted extinction law. is typically estimated
by comparing the observed (attenuated) diagnostic line flux ratio (usually of
a Balmer H line pair, most often the H$\alpha$-to-H$\beta$ line flux ratio)
with its theoretical (unattenuated) counterpart.

This is a classic chicken-and-egg problem, because the theoretical H line
ratio – the basis for extinction correction – is actually dependent on the
electron density ($n_{\rm e}$) and temperature ($T_{\rm e}$) of the line
emitting gas, which are the very quantities to be determined via plasma
diagnostics that must be performed after extinction is properly corrected.
Therefore, to achieve the maximum self-consistency, the theoretical H line
ratio used in extinction correction must be updated simultaneously and
iteratively with $n_{\rm e}$ and $T_{\rm e}$ obtained in plasma diagnostics
until they converge to the optimum values. However, such an approach has been
seldom practiced in the literature for some unknown reasons, most likely to
alleviate the volume of non-linear numerical calculations when computational
resources were still scarce (Ueta, 2022). At any rate, it is simply incorrect
to perform extinction correction just once with some assumed $n_{\rm e}$ and
$T_{\rm e}$ before performing plasma diagnostics.

In the recent M 31 work by GR22, the H$\alpha$-to-H$\beta$ line ratio was
fixed to 2.86 (equivalent to assuming $T_{\rm e}=10^{4}$ K and $n_{\rm
e}=10^{3}$ cm-3) in deriving the extinction, $c({\rm H}\beta)$, irrespective
of what the subsequent plasma diagnostics suggested for $n_{\rm e}$ and
$T_{\rm e}$. Here, we opt to demonstrate how this widely adopted initial
assumption – the fixed H$\alpha$-to-H$\beta$ line ratio – would influence the
results of extinction correction and plasma diagnostics as well as any
subsequent inferences based on the results of these analyses.

##  2 Analyses

We adopt the measured (uncorrected) line fluxes of these nine PNe in M 31 as
presented by GR22 (from their Table A.1). Regretfully, these fluxes are given
in only three significant figures and without measurement uncertainties. As
for uncertainties, GR22 stated that “total errors were determined from the
quadratic propagation of the measured statistical errors of the spectra, which
in general decrease from blue toward red, plus an estimated additional 3 % of
the measured flux in order to account for systematic errors, which include
continuum determination, flux calibration and, although less important,
wavelength calibration uncertainties.” Thus, without exact uncertainties to
quote, we choose to assume 5 % uncertainty across the spectrum.

For the rest of the analyses, we follow the same procedure as outlined by GR22
as long as they are described. The apparent deviation is adopting the self-
consistent H$\alpha$-to-H$\beta$ line ratio in extinction correction according
to the updated $n_{\rm e}$ and $T_{\rm e}$ from plasma diagnostics in order to
guarantee simultaneous self-consistency between extinction correction and
plasma diagnostics. Both extinction correction and plasma diagnostics are
performed with PyNeb (Luridiana et al., 2015) using the same atomic parameter
set (PYNEB_18_01, as summarized in Tables 4 and 6 by GR22). Uncertainties are
assessed statistically by computing 1500 Monte Carlo simulations while
allowing input line fluxes with Gaussian uncertainties.

Also, we assume a two-component nebula with high- and low-excitation regions
characterized by the cut-off ionization potential (IP) of 17 eV. Transitions
above this IP cut-off are considered of high-excitation and computed $n_{\rm
e}$ and $T_{\rm e}$ derived from the [O iii] $\lambda$4363/$\lambda$4959 and
[Ar iv] $\lambda$4711/$\lambda$4740 diagnostic line ratios, while those below
(of low-excitation) are calculated with the [N ii] $\lambda$5755/$\lambda$6548
and [S ii] $\lambda$6716/$\lambda$6731 diagnostic line ratios. A minor
exception adopted by GR22 was that they used $T_{\rm e}$([O iii]) even for
low-excitation regions when $T{\rm e}$([N ii]) resulted with uncertainties
greater than $\sim$2,000 K (for M2068 and M1675). As shown below, we do not
make any such exceptions because uncertainties are below 1,200 K in our
analyses.

##  3 Extinction Correction

To verify that the results of our analyses can be compared squarely with those
by GR22, we first emulate only a single run of extinction correction and
plasma diagnostics as was performed by GR22. This means that $T_{\rm e}$ and
$n_{\rm e}$ are fixed in extinction correction at $10^{4}$ K and $10^{3}$
cm-3, respectively (equivalent to adopting the fixed theoretical
H$\alpha$-to-H$\beta$ line ratio of 2.86) under the same extinction law
(Cardelli et al., 1989) and the total-to-selective extinction ratio, $R_{\rm
V}$, of 3.1. Then, $T_{\rm e}$ and $n_{\rm e}$ are iterated only in plasma
diagnostics. In other words, plasma diagnostic would find optimum $T_{\rm e}$
and $n_{\rm e}$ via iteration, but only after diagnostic lines are extinction-
corrected with $c({\rm H}\beta)$ based on the pre-fixed $T_{\rm e}=10^{4}$ K
and $n_{\rm e}=10^{3}$ cm-3 (hence, the optimum $T_{\rm e}$ and $n_{\rm e}$
are most likely inconsistent with the presumed $T_{\rm e}$ and $n_{\rm e}$).
With this emulated procedure established, the only difference is the adopted
input flux uncertainties. As shown in Table 1, our $c({\rm H}\beta)$ values
(“Emulated”) in this verification are generally consistent with those obtained
by GR22 (within $94.0\pm 6.5$ %). Thus, we consider that our calculations do
reproduce the results obtained by GR22 reasonably well, and hence, all
comparisons that follow are indeed valid.

Table 1: The extinction $c({\rm H}\beta)$ toward nine bright PNe in M 31. Listed are values as computed by GR22, GR22-emulated by us (Emulated), and fully iterated by us (Full). Also listed are the GDRE values. PN | GR22 | Emulated | Full | GDRE
---|---|---|---|---
M1687 | 0.20$\pm$0.06 | 0.20$\pm$0.03 | 0.24$\pm$0.02 | 0.26$\pm$0.06
M2068 | 0.13$\pm$0.06 | 0.15$\pm$0.02 | 0.37$\pm$0.02 | 0.25$\pm$0.07
M2538 | 0.00$\pm$0.00 | 0.06$\pm$0.01 | 0.35$\pm$0.10 | 0.04$\pm$0.01
M50 | 0.19$\pm$0.06 | 0.20$\pm$0.03 | 0.28$\pm$0.02 | 0.11$\pm$0.01
M1596 | 0.17$\pm$0.06 | 0.18$\pm$0.02 | 0.20$\pm$0.02 | 0.10$\pm$0.03
M2471 | 0.04$\pm$0.06 | 0.09$\pm$0.01 | 0.34$\pm$0.04 | 0.07$\pm$0.01
M2860 | 0.10$\pm$0.06 | 0.12$\pm$0.01 | 0.54$\pm$0.08 | 0.08$\pm$0.01
M1074 | 0.11$\pm$0.06 | 0.12$\pm$0.01 | 0.21$\pm$0.02 | 0.31$\pm$0.06
M1675 | 0.23$\pm$0.06 | 0.23$\pm$0.03 | 0.54$\pm$0.05 | 0.26$\pm$0.06

In Table 1, we also quote the $c({\rm H}\beta)$ values according to the
NASA/IPAC Infrared Science Archive Galactic Dust Reddening and Extinction
(GDRE) service (based on the SDSS spectra; Schlafly & Finkbeiner
2011).222https://irsa.ipac.caltech.edu/applications/DUST/ The GDRE values are
the mean $c({\rm H}\beta)$ value within a sampling circle of 5 arcmin radius
toward each of the nine M 31 PNe, accounting for extinction from all but the
circumstellar component for each PN, i.e., the interstellar component in both
the Milky Way and M 31 and the intragalactic component between the Milky Way
and M 31. Thus, the GDRE value may overestimate the M 31 interstellar
component because it accounts for extinction along the line of sight even
beyond the target PN. At any rate, our fully iterated $c({\rm H}\beta)$ values
(“Full”) turn out generally greater than the GDRE values, corroborating the
presence of non-zero circumstellar extinction component that varies a lot from
object to object (from 0.46 to 0.1, corresponding to 65 to 20 % attenuation).
This means that individual circumstellar $c({\rm H}\beta)$ value cannot just
be neglected or generically assumed: it must be computed for each object self-
consistently. At a minimum, extinction correction with fixed $n_{\rm e}$ and
$T_{\rm e}$ appears already dubious.

On the whole, direct comparisons between the GR22 values and our fully
iterative results show that $c({\rm H}\beta)$ by GR22 was more than 50 %
underestimated ($52.5\pm 31.2$ %) with individual variations from 12.6 to 88.3
%. Because $c(\lambda)$ is the base-10 power-law index varying non-linearly
across the spectrum, the impact of offsets in $c(\lambda)$ is hard to gauge
unless actually calculated. The extinction-corrected line fluxes based on the
fully iterative $c({\rm H}\beta)$ are listed in Tables 3 and 4, along with the
original-to-revised ratio for comparison. We see that the GR22 fluxes were
underestimated (not sufficiently extinction-corrected) by up to 20 % at the
shortest [O ii] $\lambda$3727 line and overestimated (excessively extinction-
corrected) by up to 59 % at the longest [Ar iii] $\lambda$7751 line.

##  4 Plasma Diagnostics: Physical Conditions

The resulting optimized $n_{\rm e}$ and $T_{\rm e}$, after full iteration of
extinction correction and plasma diagnostics, are summarized in Table 5.
Convergence is achieved in less than or equal to 4 iterations in all cases.
The ratio column shows comparisons between the GR22 values to ours. No PN
shows more than two-$\sigma$ deviations after one full iteration in both
$T_{\rm e}$([O iii]) and $T_{\rm e}$([N ii]). This stability of $T_{\rm e}$
seems to stem from the fact that the $T_{\rm e}$-diagnostic is not only
insensitive to $n_{\rm e}$ (which is why it works as the $T_{\rm e}$
diagnostic), but also varies only slightly in $T_{\rm e}$ in the $n_{\rm
e}$–$T_{\rm e}$ diagnostic plane for the given diagnostic line ratio.

The $n_{\rm e}$([S ii]) values agree remarkably well ($100\pm 3$ % agreement),
neglecting two very deviant cases (substantially overestimated by 177 and
235,% for M2068 and M1675, respectively). For these two deviant PNe, GR22
found very large uncertainties (not explicitly defined). So do we with about
50 % uncertainties. Meanwhile, the $n_{\rm e}$([Ar iv]) values were
overestimated by GR22 ($141.2\pm 24.8$ %). This seems to have been caused by
the $c({\rm H}\beta)$ underestimates in mitigating He i $\lambda$4713 line
contamination in the [Ar iv] $\lambda$4711 line flux, as will be described
below.

Given the overall rough consistency among the $n_{\rm e}$ and $T_{\rm e}$
values obtained with and without fully iterative calculations, especially the
rather invariant $T_{\rm e}$ results, one may be tempted to forgo seeking
convergence in extinction correction. In determining $n_{\rm e}$ and $T_{\rm
e}$ in plasma diagnostics, one tends to think of it as “fitting” for which the
optimum values are found from a free excursion in the $n_{\rm e}$-$T_{\rm e}$
space. In reality, however, the applicable range of $n_{\rm e}$ is also rather
restricted, because the $n_{\rm e}$-diagnostic curve varies from one limiting
ratio to another within a narrow range of $n_{\rm e}$ given the choice of the
$n_{\rm e}$-diagnostic line.

In the $n_{\rm e}$-$T_{\rm e}$ space, given the measured line ratios of the
adopted diagnostic lines, where the solution exists (i.e. where diagnostic
curves intersect) is already set. Hence, what “fitting” does in the $n_{\rm
e}$-$T_{\rm e}$ space is to find the optimally closest ($n_{\rm e}$, $T_{\rm
e}$) point to the intersection of the diagnostic curves within measurement
uncertainties. In this sense, it is expected that plasma diagnostics with the
measured line ratios would yield more or less the same $n_{\rm e}$ and $T_{\rm
e}$ values, irrespective of the rigorousness in the preceding extinction
correction. A true exploration of the $n_{\rm e}$-$T_{\rm e}$ space can only
be done by allowing the measured diagnostic line ratios vary according to
$c({\rm H}\beta)$. This relative insensitivity of $n_{\rm e}$ and $T_{\rm e}$
against plasma diagnostics probably fostered a sense of “lessez-faire” in the
community to the extent that the theoretical H$\alpha$-to-H$\beta$ line ratio
of 2.86 is referred to as “canonical” when there is no canonicity whatsoever
to this value. As we will see below, omitting iteration in extinction
correction is not at all a viable option, because the resulting $c({\rm
H}\beta)$ offset would seriously impact the rest of the plasma diagnostics.

##  5 Plasma Diagnostics: Ionic Abundances

As for calculating ionic abundances with PyNeb, we again follow GR22 as much
as possible even though sometimes the procedure was not explicitly described.
The derived ionic abundances for all the input collisionally excited lines and
He recombination lines are summarized in Tables LABEL:tab:ion1 and
LABEL:tab:ion2, even though GR22 presented only a subset of lines in their
Table A.2.

Theoretically speaking, ionic abundances derived from the same ionic
transitions should turn out identical given the adopted $n_{\rm e}$ and
$T_{\rm e}$. In the literature, this is not necessarily accomplished.
Discrepancies among derived ionic abundances for a particular ionic species
are often attributed to local temperature (and density) fluctuations in the
line-emitting gas. This attribution is actually very odd, especially when the
fixed “canonical” H$\alpha$-to-H$\beta$ line ratio of 2.86 is adopted (i.e.,
the uniform $n_{\rm e}$ at $10^{3}$ cm-3 and $T_{\rm e}$ at $10^{4}$ K are
imposed). If we are to believe varying $n_{\rm e}$ and $T_{\rm e}$ in the
target nebula along the line of sight, assuming the “canonical” uniformity in
$n_{\rm e}$ and $T_{\rm e}$ in extinction correction is equivalent to
injecting inconsistency into the analyses in the first place. This is even
worse in analysing 2-D plasma diagnostics with line emission maps, in which
spatial variation is surely expected by default. At any rate, when ionic
abundances derived from multiple lines of an ionic specie vary, some sort of
averaging needs to be done to define a representative value. GR22 took the
straight average, while we take the uncertainty-weighted mean.

Our derived ionic abundances show good self-consistency nearly across the
board: the derived abundances for most of the ionic species all agree within
uncertainties. In all major transitions to be used in calculating the total
elemental abundances (He+, He2+, O+, and O2+) with the help of the ionization
correction factors (ICFs; Delgado-Inglada et al. 2014), the derived ionic
abundances are consistent with each other except for O+ from M1675, for which
the [O ii] $\lambda$7320 line was not measured among the trio of [O ii] lines
at 3727, 7320, and 7330 Å. Among this O+ trio, the [O ii] $\lambda$3727
line333The [O ii] $\lambda$3727 line is actually a blend of two lines at 3726
and 3729 Å. In our calculations, it is treated as a blend. is located at the
blue-end of the detector bandwidth, while the [O ii] $\lambda$7330 is at the
red-end. Not knowing which is more accurate between the two, we simply take
the uncertainty-weighted mean as others.

GR22 obtained O+ abundances that were very discrepant (see their Table A.2).
The O+ $\lambda$3772 abundance came out to be less than half of the O+
$\lambda$7320/7330 abundances, except for the recurring anomalous case of
M2068 (for which the former was 2.5 times greater). There is a simple
explanation as to why the O+ $\lambda$3772 abundance came out much smaller:
the underestimated $c({\rm H}\beta)$. Here, it is reminded that what counts is
a ratio of given line flux relative to H$\beta$ flux. The [O ii] $\lambda$3772
is on the short-ward of H$\beta$, and hence, the extinction-corrected flux is
underestimated (i.e., not corrected enough by the underestimated $c({\rm
H}\beta)$). On the other hand, the [O ii] $\lambda$7320/7330 are on the long-
ward of H$\beta$, and hence, the extinction-corrected flux is overestimated
(i.e., not reduced enough by the underestimated $c({\rm H}\beta)$). As a
result, the underestimated $c({\rm H}\beta)$ leads to under/over-estimated
abundances on the opposite sides of the reference H$\beta$ wavelength (as seen
in Tables LABEL:tab:ion1 and LABEL:tab:ion2). The O+ abundance adopted by GR22
appears to have been biased toward an overestimate, as the [O ii]
$\lambda$7320/7330 lines are much farther away from H$\beta$ than the [O ii]
$\lambda$3772 line.

The He+ and He2+ abundances by GR22 suffered from the same issue, because the
He+ abundance was adopted from the He i $\lambda$5876 (hence, tended to be
overestimated) and the He2+ abundance from the He i $\lambda$4686 (hence,
tended to be underestimated). This trending is also seen in Tables
LABEL:tab:ion1 and LABEL:tab:ion2. These observations reveal that anomalies in
abundance analyses would arise not from offsets in the $n_{\rm e}$ and $T_{\rm
e}$ values, but rather from the possible under/over-estimate by the
underestimated $c({\rm H}\beta)$ on either side of the reference wavelength of
H$\beta$ at 4861 Å. GR22 also used the corrected He+ and He2+ fluxes to
constrain $T_{\rm eff}$ in comparison with theoretical models. Hence, the
impact of an $c({\rm H}\beta)$ offset can be seen at many different places in
the analyses.

As for the rest of the observed lines, we find derived abundances somewhat
discrepant for Ar2+ (7136/7751 Å), Cl2+ (5518/5538 Å), and S+ (4069/4076 Å).
The [Ar iii] $\lambda$7751 line is located at the red-end of the bandwidth and
appears relatively strongly affected by the atmospheric absorption (by the O2
band around 8000 Å, verified in the OSIRIS 2-D spectrum itself). The [Cl iii]
lines are intrinsically weak (about $\sim$1 when $I({\rm H}\beta)=100$). As
for the [S ii] lines, GR22 referred the [S ii] $\lambda$4076 line as the [S
ii] $\lambda$4071 line, and hence, the line identification might have been
compromised. As these ionic abundances are not directly involved in the
subsequent elemental abundance calculations, we leave them as they are.
However, these ionic abundances, if incorrect, will affect the corresponding
elemental abundances in the end. At any event, there always seem some
reasonable explanations as to why the derived abundances are discrepant.

However, there is more subtle but involved complications in the [Ar iv]
$\lambda$4711/4740 lines, which play an important role in the present analyses
as the high-excitation $n_{\rm e}$-diagnostic lines. The [Ar iv] $\lambda$4711
is contaminated by the neighboring He i $\lambda$4713 line, whose strength
needs to be estimated by scaling the He i $\lambda$5876 measurement (the
strongest, hence, the most reliable of all the detected He lines) at the
corresponding $n_{\rm e}$ and $T_{\rm e}$. Thus, the [Ar iv] $\lambda$4711
measurement is unavoidably underestimated when the He i $\lambda$5876
measurement is overestimated by the underestimated $c({\rm H}\beta)$. Hence,
the [Ar iv] $\lambda$4711 line flux corrected for the He i $\lambda$4713
contamination by GR22 was most likely doubly affected by this mechanism. This
indeed artificially reduced the [Ar iv] $\lambda$4711/$\lambda$4740 ratio for
GR22, forcing the resulting $n_{\rm e}$ become larger by $141.2\pm 24.8$ %.

By the same token, the [O iii] $\lambda$4363/$\lambda$4959 $T_{\rm
e}$-diagnostic ratio obtained by GR22 was at least mildly affected by the
underestimated $c({\rm H}\beta)$: the ratio was artificially reduced, forcing
the resulting $T_{\rm e}$ become smaller by $96.5\pm 2.0$ % (Table 5), which
is still an underestimate. Conversely, other $n_{\rm e}$ and $T_{\rm e}$
diagnostics did not seem to suffer from the underestimated $c({\rm H}\beta)$,
because in these cases with the [N ii] $\lambda$5755/$\lambda$6548 and [S ii]
$\lambda$6716/$\lambda$6731 lines for the low-excitation region the lines are
on the same side of H$\beta$ and not far from each other. Thus, the effects of
the underestimated $c({\rm H}\beta)$ were marginalised.

##  6 Plasma Diagnostics: Elemental Abundances

As the final step of plasma diagnostics, the total elemental abundances are
computed from the derived ionic abundances in terms of $A({\rm
X})=12+\log({\rm X}/{\rm H})$ (Table LABEL:tab:abun). First, we emulate GR22
by adopting the same ICFs (Delgado-Inglada et al., 2014) with the derived
ionic abundances of O+ and O2+, N+, Ne2+, S+, Ar2+, and Cl2+. However, there
are unused ionic species such as S2+, Ar3+, and Ar4+ in the analyses. In fact,
it is rather strange that the Ar3+ abundance was not used by GR22, because [Ar
IV] was adopted as the $n_{\rm e}$ diagnostic. Because the Ar abundance can
work as an important metallicity indicator, there is no reason not to adopt
all three measured Ar ionic species to reduce reliance/uncertainty of the ICF.
Hence, we also compute the total elemental abundances by adopting all observed
ionic abundances. In this case, the adopted ICFs are still based on those
suggested by Delgado-Inglada et al. (2014) except for Ar, for which we assume
${\rm Ar}={\rm Ar}^{2+}+{\rm Ar}^{3+}+{\rm Ar}^{4+}$.

In the ICF formulation, to yield the total elemental abundance, the sum of the
observed ionic abundances for a specific element has to be scaled to account
for the unobserved ionic species. These scaling factors are empirically
defined as functions of He and O ionic abundances (He+, He2+, O+, and O2+;
Delgado-Inglada et al. 2014). This means that the reliability of the observed
ionic abundances depends on the derived abundances of these He and O ionic
species.

The O elemental abundance is based on the observed O+ and O2+ ionic
abundances. Both of the O+ and O2+ abundances are based on three lines, [O ii]
lines at 3727, 7320, and 7330 Å and [O iii] lines at 4363, 4959, and 5007 Å,
respectively. As these lines are distributed on both sides of H$\beta$ at 4861
Å, the effects of the underestimated $c({\rm H}\beta)$ was most likely
marginal even in the results by GR22. Indeed, the median O abundance among the
nine PN sample obtained by GR22 was $8.63\pm 0.09$, while ours is $8.59\pm
0.08$: fairly consistent with relatively small uncertainties. This does not
necessarily mean that the O abundance is insensitive to the $c({\rm H}\beta)$
discrepancy. The present PN sample is of high-excitation, and hence, their O
abundance is relatively less dependent on the more uncertain low-excitation O+
abundance. If targets are of low-excitation, the relative importance of the
more uncertain O+ abundance would be greater, and hence, the O abundance would
have been compromised by the $c({\rm H}\beta)$ discrepancy.

On the other hand, the N elemental abundance is based solely on the observed
N+ ionic abundance. All three [N ii] lines at 5755, 6548, and 6583 Å are
located on the red-side of H$\beta$. Hence, their line fluxes and ionic
abundances were most likely overestimated by GR22 because of the
underestimated $c({\rm H}\beta)$. The median N elemental abundance obtained
was $8.27\pm 0.38$ by GR22, while ours is $7.99\pm 0.35$: this is a factor of
1.9 difference. While the two are statistically indifferent given the
relatively large deviation, the difference in the elemental abundance amounts
to nearly 90 %. When both of the N and O elemental abundances are combined to
assess the N/O abundance ratio, the values come out to be $0.40\pm 0.30$ (by
GR22) and $0.28\pm 0.16$ (by us). Again, these are statistically indifferent,
but the absolute difference is an overestimate at nearly 30 %.

It was further argued by GR22 that the derived N/O ratio for the M 31 PN
sample was 1.5–3 times greater than expected for PNe of the $\sim$1.5 M⊙
initial mass based on comparisons with theoretical models (Karakas & Lugaro,
2016; Miller Bertolami, 2016; Ventura et al., 2018), even referring to
possible limitations of the existing models. However, we can simply interpret
this as another consequence of the underestimated $c({\rm H}\beta)$, which
artificially reddened the whole spectra of target PNe.

Table 2: Comparison of the best-fit luminosity ($L_{\ast}$ in $\log(L_{\ast}/L_{\odot})$), surface temperature ($T_{\rm eff}$ in $\log(T_{\rm eff})$), and initial mass ($M_{i}$ in $M_{\odot}$) of the central star for the M 31 PN sample, based on CLOUDY model fitting constrained by the extinction-corrected line fluxes by GR22 (the solar metallicity of $Z=0.02$ assumed for all) and us (an appropriate metallicity in the range of $Z=0.003-0.009$ informed from abundance analyses adopted for each PN; Otsuka & Ueta in prep.). | GR22 |  | Otsuka & Ueta
---|---|---|---
PN | $L_{\ast}$ | $T_{\rm eff}$ | $M_{i}$ |  | $L_{\ast}$ | $T_{\rm eff}$ | $M_{i}$
M1687 | 3.66 | 5.06 | 1.48 |  | 3.87 | 5.01 | 1.9
M2068 | 3.62 | 5.05 | 1.34 |  | 4.04 | 5.01 | 2.4
M2538 | 3.65 | 5.11 | 1.56 |  | 4.03 | 5.07 | 2.4
M50 | 3.59 | 5.13 | 1.42 |  | 3.88 | 5.08 | 2.0
M1596 | 3.49 | 5.22 | 1.76 |  | 3.75 | 5.11 | 1.6
M2471 | 3.50 | 5.20 | 1.70 |  | 3.82 | 5.10 | 1.7
M2860 | 3.51 | 5.08 | 1.20 |  | 3.97 | 5.03 | 2.2
M1074 | 3.50 | 5.03 | 1.12 |  | 3.76 | 5.00 | 1.4
M1675 | 3.60 | 5.10 | 1.39 |  | 4.05 | 5.05 | 2.6

Table 2 lists the best-fit luminosity ($L_{\ast}$), surface temperature
($T_{\rm eff}$), and initial mass ($M_{i}$) of the central star, obtained via
CLOUDY model fitting by GR22 (with the $c({\rm H}\beta)$ underestimate, and
the fixed solar metallicity of $Z=0.02$ assumed for all PNe) and by us
(without the $c({\rm H}\beta)$ underestimate, and an appropriate metallicity
in the range of $Z=0.003-0.009$ informed from abundance analyses adopted for
each PN; Otsuka & Ueta, in prep.). As has been discussed above, the apparent
$c({\rm H}\beta)$ underestimate imposed false reddening in the previous
analyses. Hence, the revised models naturally suggest greater $L_{\ast}$ (by a
factor of two on average). Correspondingly, the expected initial progenitor
mass for the nine-PN sample comes out to be greater.

Especially, the brightest four PNe in M 31 (M1687, M2068, M2538, and M50) are
now appropriately found to be the most massive among the low-mass progenitors
(1.9 – 2.4 M⊙ with the average of 2.2 M⊙, as opposed to 1.3 – 1.6 M⊙ with the
average of 1.5 M⊙). For such relatively higher-mass progenitors, we would
indeed expect comparatively enhanced N/O abundance ratios. However, they are
by no means N over-abundant as in Type I PN (Peimbert & Torres-Peimbert 1983).
Therefore, the present results from fully iterative self-consistent extinction
correction and plasma diagnostics are in reasonable agreement with what is
predicted by the existing theoretical models. There is no need for a greater
amount of N from low-mass progenitors.

##  7 Concluding Remarks

Plasma diagnostics determine the physico-chemical conditions of line-emitting
objects in terms of $n_{\rm e}$ and $T_{\rm e}$ as well as ionic and elemental
abundances. Because $c({\rm H}\beta)$ is directly influenced by $n_{\rm e}$
and $T_{\rm e}$, $n_{\rm e}$ and $T_{\rm e}$ must be determined self-
consistently via an iterative search for convergence through both extinction
correction and plasma diagnostics. In the present exercise, we have
demonstrated, for a sample of nine bright PNe in M 31, how inconsistently
assumed $n_{\rm e}$ and $T_{\rm e}$ in extinction correction can affect the
results of the subsequent plasma diagnostics.

If $c({\rm H}\beta)$ is not iteratively updated, and given the measured line
fluxes, the resulting $n_{\rm e}$ and $T_{\rm e}$ values are more or less
fixed as soon as the $n_{\rm e}$\- and $T_{\rm e}$-diagnostic lines are
selected. If $c({\rm H}\beta)$ is not determined properly, observed line
fluxes above and below the reference H$\beta$ wavelength can be over/under-
corrected depending on their wavelengths. Consequently, diagnostic line ratios
can be erroneously amplified/reduced especially when the lines involved are
taken from both sides of H$\beta$. This practically means that the derived
ionic and elemental abundances can be unreliable, even though the resulting
$n_{\rm e}$ and $T_{\rm e}$ values may still appear reasonable. Moreover, the
results of model calculations would be equally compromised, especially when
such wrongly extinction-corrected lines are used as constraints.

For the present case of the nine bright PN sample in M 31, we have been able
to attribute the differences between the previous and present results to
$c({\rm H}\beta)$ that was underestimated more than 50 % in the previous
analyses. It was because the H$\alpha$-to-H$\beta$ line ratio was assumed to
be fixed at 2.86 (equivalent to assuming $n_{\rm e}=10^{3}$ cm-3 and $T_{\rm
e}=10^{4}$ K) in extinction correction even when the final $n_{\rm e}$ and
$T_{\rm e}$ were different. We have also shown that the underestimated $c({\rm
H}\beta)$ inflicted inconsistencies in the derived ionic and elemental
abundances as well as the subsequent photoionization model calculations. In
the end, we have established that this bright M 31 PN sample represents the
high-mass end of the low-mass PN progenitor stars of less than solar
metallicities. Hence, no anomalous N overabundance has been found to suspect
any irregularities in the existing evolutionary models as was hinted at with
the previous analyses.

More specifically, the N/O abundance ratio can be significantly affected in
plasma diagnostics based on optical spectra, if $c({\rm H}\beta)$ is not
determined properly. For the present case, the previously suspected N
overabundance was simply caused by the underestimated $c({\rm H}\beta)$. The
empirical N abundance is based almost exclusively on the [N ii] lines around
6000 Å (5755 and 6548/83 Å), on the much redder side of the reference
wavelength at H$\beta$. Therefore, the [N ii] line strengths, and hence, the N
abundance can be artificially inflated when $c({\rm H}\beta)$ is
underestimated. In the mean time, the O abundance, based on multiple lines of
multiple ionic species scattered across the optical spectrum on either side of
H$\beta$, is relatively insensitive to the apparent $c({\rm H}\beta)$
underestimate. For example, the N/O ratio is often used as an indicator of the
initial stellar mass based on predictions made by theoretical models (e.g.
Karakas & Lugaro 2016; Miller Bertolami 2016; Ventura et al. 2018). Hence, the
erroneously estimated N/O ratio can lead to a variety of wrong conclusions
beyond the physico-chemical conditions of the target PNe. This issue is of
course not isolated in PNe and can happen in any line-emitting objects.
Therefore, caution must be exercised when quoting abundances from the
literature.

To summarize, the general lessons learned are;

  * •

The quality of the input spectra is of course important. Especially, line
fluxes should be free from any anomalies such as the atmospheric dispersion,
sky emission/absorption, etc.

  * •

Extinction correction and plasma diagnostics should be performed
simultaneously and self-consistently through full iteration, with a careful
choice of diagnostic lines.

  * •

It is best to incorporate as many lines as possible in the analyses,
especially from either side of the reference wavelength of extinction
correction (typically at H$\beta$).

  * •

The adverse effects of not performing extinction correction and plasma
diagnostics self-consistently do not necessarily incur in the resulting
$n_{\rm e}$ and $T_{\rm e}$, but would be more likely in the resulting ionic
and elemental abundances via diagnostic line ratios compromised by the wrongly
derived $c({\rm H}\beta)$.

  * •

The incorrectly determined $c({\rm H}\beta)$ would systematically redden or
blue the input spectrum and affect the outcomes of any subsequent analyses.

  * •

Abundances in the literature need to be treated with care, unless plasma
diagnostics were performed self-consistently in conjunction with extinction
correction. It may be worthwhile to re-evaluate abundances via self-consistent
extinction correction and plasma diagnostics, especially when anomalies are
reported in the previous results.

###### Acknowledgements.

This research made use of the NASA/IPAC Infrared Science Archive Galactic Dust
Reddening and Extinction (GDRE) service and PyNeb, a toolset dedicated to the
analysis of emission lines (Luridiana et al., 2015). T.U. was supported
partially by the Japan Society for the Promotion of Science (JSPS) through its
invitation fellowship program (FY2020; L20505). M.O. was supported by JSPS
Grants-in-Aid for Scientific Research(C) (JP19K03914 and 22K03675). Authors
thank the anonymous referee, whose inputs helped to clarify some critical
points in the manuscript.

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##  Appendix A Supplemental Materials

Table 3: Extinction-Corrected Line Fluxes and Original-to-Revised Ratios for
the Brightest Four PNe in M 31

Line | M1687 | Ratio | M2068 | Ratio | M2538 | Ratio | M50 | Ratio
---|---|---|---|---|---|---|---|---
$[$O ii$]$ 3727 | 21.0 $\pm$ 2.3 | 0.97 | 50.2 $\pm$ 9.4 | 0.83 | 66.2 $\pm$ 12.1 | 0.77 | 53.4 $\pm$ 7.2 | 0.94
H10 3798 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 4.4 $\pm$ 0.8 | 0.77 | 5.8 $\pm$ 0.8 | 0.94
H9 3835 | $\cdots$ | $\cdots$ | 6.1 $\pm$ 1.1 | 0.85 | 8.9 $\pm$ 1.6 | 0.78 | 5.8 $\pm$ 0.8 | 0.94
$[$Ne iii$]$ 3868 | 172.7 $\pm$ 18.6 | 0.97 | 162.0 $\pm$ 30.5 | 0.85 | 119.3 $\pm$ 21.8 | 0.79 | 126.3 $\pm$ 16.9 | 0.94
H8 3889aa$a$Blend with He i 3889. | 0.1 $\pm$ 1.6 | $\cdots$ | 2.6 $\pm$ 2.8 | $\cdots$ | 5.7 $\pm$ 2.8 | $\cdots$ | 2.4 $\pm$ 2.0 | $\cdots$
$[$Ne iii$]$ 3967bb$b$Blend with H7 3970 and He i 3965. | 50.7 $\pm$ 6.7 | 0.97 | 49.8 $\pm$ 11.3 | 0.81 | 25.3 $\pm$ 6.4 | 0.66 | 33.9 $\pm$ 5.9 | 0.92
He i 4026 | 6.1 $\pm$ 0.7 | 0.97 | 2.8 $\pm$ 0.5 | 0.87 | 3.8 $\pm$ 0.7 | 0.81 | 2.5 $\pm$ 0.3 | 0.95
$[$S ii$]$ 4069 | 0.0 $\pm$ 0.0 | $\cdots$ | 4.3 $\pm$ 0.8 | 0.88 | $\cdots$ | $\cdots$ | 3.9 $\pm$ 0.5 | 0.96
$[$S ii$]$ 4076 | 0.0 $\pm$ 0.0 | $\cdots$ | 2.9 $\pm$ 0.5 | 0.88 | $\cdots$ | $\cdots$ | 1.1 $\pm$ 0.2 | 0.96
H6 4101cc$c$Blend with He ii 4100. | 29.3 $\pm$ 3.1 | 0.98 | 30.2 $\pm$ 5.7 | 0.88 | 30.1 $\pm$ 5.5 | 0.83 | 27.1 $\pm$ 3.6 | 0.95
H5 4340dd$d$Blend with He ii 4338. | 48.5 $\pm$ 5.2 | 0.99 | 50.0 $\pm$ 9.4 | 0.92 | 50.7 $\pm$ 9.3 | 0.88 | 46.2 $\pm$ 6.2 | 0.97
$[$O iii$]$ 4363 | 27.0 $\pm$ 2.9 | 0.99 | 15.3 $\pm$ 2.9 | 0.92 | 19.1 $\pm$ 3.5 | 0.88 | 17.7 $\pm$ 2.4 | 0.98
He i 4472 | 6.0 $\pm$ 0.6 | 0.99 | 6.0 $\pm$ 1.1 | 0.94 | 4.7 $\pm$ 0.9 | 0.91 | 5.1 $\pm$ 0.7 | 0.98
He ii 4686 | 2.9 $\pm$ 0.3 | 1.00 | 3.9 $\pm$ 0.7 | 0.97 | 12.5 $\pm$ 2.3 | 0.96 | 14.6 $\pm$ 1.9 | 0.99
$[$Ar iv$]$ 4711ee$e$Blend with He i 4713. | 1.9 $\pm$ 0.3 | 0.99 | 1.5 $\pm$ 0.4 | 0.91 | 2.7 $\pm$ 0.6 | 0.93 | 2.4 $\pm$ 0.4 | 0.98
$[$Ar iv$]$ 4740 | 5.3 $\pm$ 0.6 | 1.00 | 4.1 $\pm$ 0.8 | 0.98 | 3.4 $\pm$ 0.6 | 0.97 | 4.4 $\pm$ 0.6 | 0.99
H$\beta$ 4861ff$f$Blend with He ii 4859. | 100.0 $\pm$ 10.5 | 1.00 | 100.0 $\pm$ 18.8 | 1.00 | 100.0 $\pm$ 18.1 | 1.00 | 100.0 $\pm$ 13.1 | 1.00
He i 4922 | 0.9 $\pm$ 0.1 | 1.01 | 1.4 $\pm$ 0.3 | 1.01 | 1.0 $\pm$ 0.2 | 1.02 | 0.7 $\pm$ 0.1 | 1.01
$[$O iii$]$ 4959 | 599.5 $\pm$ 76.7 | 1.00 | 506.0 $\pm$ 116.2 | 1.02 | 461.1 $\pm$ 101.9 | 1.03 | 543.0 $\pm$ 86.9 | 1.01
$[$O iii$]$ 5007 | 1774.9 $\pm$ 227.0 | 1.01 | 1492.0 $\pm$ 342.5 | 1.03 | 1360.8 $\pm$ 300.6 | 1.04 | 1606.6 $\pm$ 256.9 | 1.02
$[$N i$]$ 5198 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.5 $\pm$ 0.1 | 1.01
He ii 5411 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1.4 $\pm$ 0.3 | 1.12 | 1.4 $\pm$ 0.2 | 1.03
$[$Cl iii$]$ 5518 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.4 $\pm$ 0.1 | 1.04
$[$Cl iii$]$ 5538 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.8 $\pm$ 0.1 | 1.03
$[$N ii$]$ 5755 | 2.0 $\pm$ 0.3 | 1.02 | 2.3 $\pm$ 0.5 | 1.11 | 1.0 $\pm$ 0.2 | 1.18 | 2.1 $\pm$ 0.3 | 1.04
He i 5876 | 16.9 $\pm$ 2.1 | 1.02 | 15.7 $\pm$ 3.6 | 1.12 | 13.0 $\pm$ 2.9 | 1.19 | 15.6 $\pm$ 2.5 | 1.04
$[$O i$]$ 6300 | 7.5 $\pm$ 0.9 | 1.03 | 6.3 $\pm$ 1.4 | 1.16 | 5.0 $\pm$ 1.1 | 1.25 | 9.1 $\pm$ 1.4 | 1.06
$[$S iii$]$ 6312gg$g$Blend with He ii 6310. | 2.1 $\pm$ 0.3 | 1.03 | 1.7 $\pm$ 0.4 | 1.17 | 1.7 $\pm$ 0.4 | 1.26 | 1.6 $\pm$ 0.3 | 1.07
$[$O i$]$ 6363 | 2.7 $\pm$ 0.3 | 1.03 | 2.0 $\pm$ 0.5 | 1.16 | 2.0 $\pm$ 0.4 | 1.26 | 2.9 $\pm$ 0.5 | 1.06
$[$N ii$]$ 6548 | 10.7 $\pm$ 1.3 | 1.04 | 21.9 $\pm$ 5.0 | 1.18 | 12.5 $\pm$ 2.7 | 1.29 | 23.3 $\pm$ 3.7 | 1.07
H$\alpha$ 6563hh$h$Blend with He ii 6560. | 277.3 $\pm$ 34.6 | 1.03 | 241.7 $\pm$ 54.7 | 1.18 | 216.3 $\pm$ 47.2 | 1.29 | 268.1 $\pm$ 42.1 | 1.07
$[$N ii$]$ 6583 | 27.0 $\pm$ 3.4 | 1.03 | 63.5 $\pm$ 14.4 | 1.18 | 36.3 $\pm$ 7.9 | 1.29 | 67.2 $\pm$ 10.5 | 1.07
He i 6678ii$i$Blend with He ii 6683. | 3.6 $\pm$ 0.5 | 1.03 | 3.7 $\pm$ 0.8 | 1.20 | 2.9 $\pm$ 0.6 | 1.31 | 3.4 $\pm$ 0.5 | 1.07
$[$S ii$]$ 6716 | 1.3 $\pm$ 0.2 | 1.03 | 1.6 $\pm$ 0.4 | 1.20 | 2.0 $\pm$ 0.4 | 1.31 | 2.4 $\pm$ 0.4 | 1.07
$[$S ii$]$ 6730 | 2.3 $\pm$ 0.3 | 1.03 | 3.2 $\pm$ 0.7 | 1.20 | 3.4 $\pm$ 0.7 | 1.31 | 4.5 $\pm$ 0.7 | 1.07
$[$Ar V$]$ 7005 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
He i 7065 | 11.0 $\pm$ 1.4 | 1.04 | 8.0 $\pm$ 1.8 | 1.23 | 6.7 $\pm$ 1.5 | 1.36 | 8.7 $\pm$ 1.4 | 1.08
$[$Ar iii$]$ 7136 | 9.4 $\pm$ 1.2 | 1.05 | 13.1 $\pm$ 2.9 | 1.23 | 7.6 $\pm$ 1.6 | 1.36 | 13.8 $\pm$ 2.1 | 1.08
He i 7281 | 0.8 $\pm$ 0.1 | 1.05 | 0.5 $\pm$ 0.1 | 1.25 | 1.1 $\pm$ 0.2 | 1.40 | 0.7 $\pm$ 0.1 | 1.09
$[$O ii$]$ 7320 | 6.6 $\pm$ 0.8 | 1.04 | 3.5 $\pm$ 0.8 | 1.25 | 2.9 $\pm$ 0.6 | 1.40 | 5.8 $\pm$ 0.9 | 1.09
$[$O ii$]$ 7330 | 6.5 $\pm$ 0.8 | 1.04 | 3.4 $\pm$ 0.8 | 1.25 | 3.0 $\pm$ 0.6 | 1.40 | 5.2 $\pm$ 0.8 | 1.09
$[$Ar iii$]$ 7751 | 1.8 $\pm$ 0.2 | 1.05 | 1.9 $\pm$ 0.4 | 1.29 | 1.3 $\pm$ 0.3 | 1.47 | 2.5 $\pm$ 0.4 | 1.10

444Contaminating He i and He ii lines scaled from He i 5876 and He ii 4686,
respectively.

Table 4: Extinction-Corrected Line Fluxes and Original-to-Revised Ratios for
the Other Five PNe in M 31

Line | M1596 | Ratio | M2471 | Ratio | M2860 | Ratio | M1074 | Ratio | M1675 | Ratio
---|---|---|---|---|---|---|---|---|---|---
$[$O ii$]$ 3727 |  65.0$\pm$5.3 | 0.98 |  78.7$\pm$13.7 | 0.80 |  28.7$\pm$7.5 | 0.73 |  20.8$\pm$ 1.9 | 0.93 |  17.8$\pm$4.7 | 0.80
H10 3798 | $\cdots$ | $\cdots$ |  5.4 $\pm$0.9 | 0.81 |  4.3 $\pm$1.1 | 0.73 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
H9 3835 |  8.3$\pm$0.7 | 0.98 |  6.0 $\pm$1.0 | 0.81 |  5.1 $\pm$1.3 | 0.74 |  7.2 $\pm$ 0.7 | 0.93 |  5.3 $\pm$1.4 | 0.81
$[$Ne iii$]$ 3868 |  127.5$\pm$10.4 | 0.98 |  141.9$\pm$24.6 | 0.82 |  115.6$\pm$30.2 | 0.74 |  118.8$\pm$ 10.7 | 0.93 |  112.4$\pm$29.5 | 0.81
H8 3889aa$a$Blend with He i 3889. |  0.0$\pm$1.0 | $\cdots$ |  10.8$\pm$3.1 | $\cdots$ |  5.5 $\pm$4.0 | $\cdots$ |  3.7 $\pm$ 1.5 | $\cdots$ |  3.1 $\pm$3.7 | 4.01
$[$Ne iii$]$ 3967bb$b$Blend with H7 3970 and He i 3965. |  38.7$\pm$4.0 | 0.97 |  44.4$\pm$9.3 | 0.77 |  45.7$\pm$14.3 | 0.67 |  35.6$\pm$ 4.2 | 0.90 |  32.7$\pm$11.0 | 0.73
He i 4026 |  5.6$\pm$0.5 | 0.98 |  2.7 $\pm$0.5 | 0.84 |  4.6 $\pm$1.2 | 0.78 |  1.6 $\pm$ 0.2 | 0.94 | $\cdots$ | $\cdots$
$[$S ii$]$ 4069 |  3.9$\pm$0.3 | 0.98 |  3.6 $\pm$0.6 | 0.85 |  3.5 $\pm$0.9 | 0.78 |  2.1 $\pm$ 0.2 | 0.94 | $\cdots$ | $\cdots$
$[$S ii$]$ 4076 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |  1.9 $\pm$0.5 | 0.79 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
H6 4101cc$c$Blend with He ii 4100. |  27.3$\pm$2.2 | 0.99 |  31.3$\pm$5.5 | 0.85 |  32.8$\pm$8.6 | 0.79 |  26.0$\pm$ 2.3 | 0.94 |  27.1$\pm$7.2 | 0.85
H5 4340dd$d$Blend with He ii 4338. |  47.0$\pm$3.8 | 0.99 |  50.8$\pm$8.9 | 0.90 |  51.7$\pm$13.7 | 0.85 |  46.2$\pm$ 4.1 | 0.96 |  51.7$\pm$13.8 | 0.90
$[$O iii$]$ 4363 |  20.1$\pm$1.6 | 0.99 |  18.9$\pm$3.3 | 0.90 |  11.6$\pm$3.1 | 0.86 |  17.5$\pm$ 1.6 | 0.96 |  13.9$\pm$3.7 | 0.90
He i 4472 |  4.3$\pm$0.4 | 0.99 |  4.1$\pm$0.7 | 0.92 |  6.1 $\pm$1.6 | 0.89 |  5.4 $\pm$ 0.5 | 0.97 |  4.7 $\pm$1.3 | 0.92
He ii 4686 |  35.4$\pm$2.8 | 1.00 |  33.6$\pm$5.8 | 0.97 |  7.7 $\pm$2.1 | 0.95 |  2.7 $\pm$ 0.2 | 0.99 |  12.8$\pm$3.4 | 0.97
$[$Ar iv$]$ 4711ee$e$Blend with He i 4713. |  6.4$\pm$0.5 | 1.00 |  4.2$\pm$0.8 | 0.96 |  3.1 $\pm$0.9 | 0.90 |  1.7 $\pm$ 0.2 | 0.96 |  3.3 $\pm$1.0 | 0.94
$[$Ar iv$]$ 4740 |  8.1$\pm$0.6 | 1.00 |  4.8$\pm$0.8 | 0.98 |  4.5 $\pm$1.2 | 0.97 |  3.8 $\pm$ 0.3 | 0.99 |  4.5 $\pm$1.2 | 0.98
H$\beta$ 4861ff$f$Blend with He ii 4859. |  100.0$\pm$7.7 | 1.00 |  100.0$\pm$17.0 | 1.00 |  100.0$\pm$26.7 | 1.00 |  100.0$\pm$ 8.7 | 1.00 |  100.0$\pm$26.7 | 1.00
He i 4922 | $\cdots$ | $\cdots$ |  0.7$\pm$0.2 | 1.03 |  1.3 $\pm$0.4 | 1.02 | $\cdots$ | $\cdots$ |  2.4 $\pm$0.8 | 1.02
$[$O iii$]$ 4959 |  575.3$\pm$54.3 | 1.02 |  499.8$\pm$103.7 | 1.03 |  477.6$\pm$156.1 | 1.03 |  476.0$\pm$ 50.8 | 1.01 |  510.6$\pm$167.0 | 1.02
$[$O iii$]$ 5007 |  1717.7$\pm$161.9 | 1.01 |  1491.1$\pm$309.3 | 1.04 |  1411.8$\pm$461.7 | 1.04 |  1429.6$\pm$152.4 | 1.01 |  1535.9$\pm$502.5 | 1.03
$[$N i$]$ 5198 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |  2.1 $\pm$0.7 | 1.07
He ii 5411 |  3.1$\pm$0.3 | 1.03 |  2.6$\pm$0.5 | 1.11 |  0.5 $\pm$0.2 | 1.15 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$[$Cl iii$]$ 5518 |  0.6$\pm$0.1 | 1.02 | $\cdots$ | $\cdots$ |  0.3 $\pm$0.1 | 1.16 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$[$Cl iii$]$ 5538 |  1.2$\pm$0.1 | 1.03 | $\cdots$ | $\cdots$ |  0.4 $\pm$0.1 | 1.18 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$[$N ii$]$ 5755 |  2.5$\pm$0.2 | 1.03 |  0.9 $\pm$0.2 | 1.16 |  0.8 $\pm$0.3 | 1.21 |  0.5 $\pm$ 0.1 | 1.05 |  2.3 $\pm$0.8 | 1.15
He i 5876 |  14.1$\pm$1.3 | 1.03 |  10.9$\pm$2.2 | 1.17 |  13.5$\pm$4.4 | 1.23 |  16.1$\pm$ 1.7 | 1.05 |  13.9$\pm$4.6 | 1.16
$[$O i$]$ 6300 |  7.0$\pm$0.6 | 1.03 |  6.6 $\pm$1.4 | 1.22 |  2.2 $\pm$0.7 | 1.31 |  3.6 $\pm$ 0.4 | 1.07 |  7.7 $\pm$2.5 | 1.20
$[$S iii$]$ 6312gg$g$Blend with He ii 6310. |  4.0$\pm$0.4 | 1.03 |  1.3 $\pm$0.3 | 1.24 |  1.3 $\pm$0.4 | 1.32 |  2.1 $\pm$ 0.2 | 1.07 |  2.2 $\pm$0.7 | 1.22
$[$O i$]$ 6363 |  2.2$\pm$0.2 | 1.04 |  2.0 $\pm$0.4 | 1.22 |  0.9 $\pm$0.3 | 1.32 |  1.2 $\pm$ 0.1 | 1.07 |  2.4 $\pm$0.8 | 1.22
$[$N ii$]$ 6548 |  36.4$\pm$3.3 | 1.04 |  14.7$\pm$3.0 | 1.24 |  9.0 $\pm$3.0 | 1.35 |  5.9 $\pm$ 0.6 | 1.08 |  29.0$\pm$9.5 | 1.24
H$\alpha$ 6563hh$h$Blend with He ii 6560. |  276.2$\pm$25.5 | 1.04 |  229.0$\pm$47.0 | 1.25 |  211.0$\pm$69.2 | 1.36 |  265.3$\pm$ 27.6 | 1.08 |  230.3$\pm$75.6 | 1.24
$[$N ii$]$ 6583 |  105.8$\pm$9.7 | 1.04 |  42.4$\pm$8.6 | 1.25 |  24.8$\pm$8.1 | 1.36 |  17.1$\pm$ 1.8 | 1.08 |  84.9$\pm$27.8 | 1.24
He i 6678ii$i$Blend with He ii 6683. |  3.8$\pm$0.4 | 1.04 |  2.2 $\pm$0.5 | 1.29 |  2.9 $\pm$0.9 | 1.39 |  3.5 $\pm$ 0.4 | 1.08 |  4.2 $\pm$1.4 | 1.26
$[$S ii$]$ 6716 |  6.2$\pm$0.6 | 1.04 |  2.6 $\pm$0.5 | 1.27 |  1.5 $\pm$0.5 | 1.38 |  1.2 $\pm$ 0.1 | 1.08 |  3.4 $\pm$1.1 | 1.26
$[$S ii$]$ 6730 |  10.6$\pm$1.0 | 1.03 |  4.4 $\pm$0.9 | 1.27 |  2.6 $\pm$0.8 | 1.39 |  1.9 $\pm$ 0.2 | 1.08 |  7.1 $\pm$2.3 | 1.26
$[$Ar V$]$ 7005 |  1.0$\pm$0.1 | 1.04 |  0.4 $\pm$0.1 | 1.30 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
He i 7065 |  6.7$\pm$0.6 | 1.04 |  5.2 $\pm$1.1 | 1.30 |  5.1 $\pm$1.7 | 1.45 |  8.5 $\pm$ 0.9 | 1.09 |  5.5 $\pm$1.8 | 1.30
$[$Ar iii$]$ 7136 |  22.4$\pm$2.0 | 1.04 |  9.5 $\pm$1.9 | 1.31 |  9.9 $\pm$3.2 | 1.46 |  7.6 $\pm$ 0.8 | 1.10 |  11.3$\pm$3.7 | 1.31
He i 7281 | $\cdots$ | $\cdots$ |  0.5 $\pm$0.1 | 1.32 |  0.3 $\pm$0.1 | 1.51 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$[$O ii$]$ 7320 |  4.4$\pm$0.4 | 1.04 |  3.4 $\pm$0.7 | 1.34 |  1.5 $\pm$0.5 | 1.50 |  3.7 $\pm$ 0.4 | 1.10 | $\cdots$ | $\cdots$
$[$O ii$]$ 7330 |  4.0$\pm$0.4 | 1.04 |  3.1 $\pm$0.6 | 1.34 |  1.4 $\pm$0.5 | 1.50 |  3.7 $\pm$ 0.4 | 1.10 |  2.6 $\pm$0.8 | 1.33
$[$Ar iii$]$ 7751 |  4.2$\pm$0.4 | 1.05 |  2.0 $\pm$0.4 | 1.39 |  1.8 $\pm$0.6 | 1.59 |  1.4 $\pm$ 0.2 | 1.12 |  2.2 $\pm$0.7 | 1.43

555Contaminating He i and He ii lines scaled from He i 5876 and He ii 4686,
respectively.

Table 5: Derived $n_{\rm e}$ and $T_{\rm e}$ for both high- and low-excitation regions from fully iterative extinction correction/plasma diagnostics with original-to-revised ratios. | M1687 | Ratio | M2068 | Ratio | M2538 | Ratio
---|---|---|---|---|---|---
$T_{\rm e}$([O iii]) | 12940$\pm$1420 | 0.983 | 11250$\pm$1310 | 0.945 | 12870$\pm$1550 | 0.942
$n_{\rm e}$([Ar iv]) | 22340$\pm$3510 | 1.386 | 18020$\pm$2530 | 1.970 |  14430$\pm$1230 | 1.563
$T_{\rm e}$([N ii]) | 25650$\pm$1690 | 0.842 | 13020$\pm$1110 | 0.795 | 12600$\pm$1580 | 0.947
$n_{\rm e}$([S ii]) |  14720$\pm$1320 | 0.977 | 11010$\pm$5120 | 1.771 |  13340$\pm$1880 | 1.002
| M50 | Ratio | M1596 | Ratio | M2471 | Ratio
$T_{\rm e}$([O iii]) | 11670$\pm$1320 | 0.984 | 12030$\pm$1340 | 0.992 | 12380$\pm$1380 | 0.954
$n_{\rm e}$([Ar iv]) |  19780$\pm$1690 | 1.364 |  15410$\pm$1100 | 1.116 |  13640$\pm$1930 | 1.312
$T_{\rm e}$([N ii]) | 13030$\pm$1670 | 0.979 | 11770$\pm$1450 | 0.994 | 11390$\pm$1420 | 0.959
$n_{\rm e}$([S ii]) |  15980$\pm$2120 | 1.081 |  13460$\pm$1030 | 1.019 |  13220$\pm$1870 | 1.000
| M2860 | Ratio | M1074 | Ratio | M1675 | Ratio
$T_{\rm e}$([O iii]) | 10660$\pm$1310 | 0.939 | 12110$\pm$1340 | 0.976 | 11000$\pm$1270 | 0.959
$n_{\rm e}$([Ar iv]) |  16000$\pm$1070 | 1.527 | 15590$\pm$2480 | 1.438 |  15170$\pm$1050 | 1.720
$T_{\rm e}$([N ii]) | 13780$\pm$1560 | 0.910 | 13120$\pm$1480 | 0.977 | 11380$\pm$1860 | 1.002
$n_{\rm e}$([S ii]) |  13440$\pm$1890 | 0.994 |  12750$\pm$1710 | 1.006 | 10330$\pm$4680 | 2.351

4]

Table 6: Ionic Abundances and Original-to-Revised Ratios for the Brightest Four PNe in M 31 Line/Ion | M1678 | Ratio | M2068 | Ratio | M2538 | Ratio | M50 | Ratio
---|---|---|---|---|---|---|---|---
He I 4026 | 20.763$\pm$2.247($-2$) |  | 11.255$\pm$2.102($-2$) |  | 15.521$\pm$2.767($-2$) |  | 10.098$\pm$1.378($-2$) |
He I 4472 | 8.840$\pm$0.978($-2$) |  | 11.040$\pm$2.065($-2$) |  | 8.908$\pm$1.617($-2$) |  | 9.438$\pm$1.258($-2$) |
He 4922 | 4.428$\pm$0.650($-2$) |  | 9.240$\pm$2.113($-2$) |  | 7.155$\pm$1.589($-2$) |  | 4.986$\pm$0.860($-2$) |
He I 5876 | 6.472$\pm$0.862($-2$) |  | 8.701$\pm$2.094($-2$) |  | 8.093$\pm$1.820($-2$) |  | 9.086$\pm$1.473($-2$) |
He I 6678 | 4.779$\pm$0.660($-2$) |  | 7.319$\pm$1.766($-2$) |  | 6.457$\pm$1.423($-2$) |  | 7.084$\pm$1.193($-2$) |
He I 7065 | 11.290$\pm$1.567($-2$) |  | 12.201$\pm$3.179($-2$) |  | 13.152$\pm$3.011($-2$) |  | 14.545$\pm$2.673($-2$) |
He I 7281 | 4.671$\pm$0.663($-2$) |  | 3.841$\pm$0.982($-2$) |  | 9.850$\pm$2.139($-2$) |  | 6.262$\pm$1.130($-2$) |
He+(adopted) | 6.472$\pm$0.862($-2$) | 1.45 | 8.701$\pm$2.094($-2$) | 1.22 | 8.093$\pm$1.820($-2$) | 1.14 | 9.086$\pm$1.473($-2$) | 1.08
He II 4686 | 2.513$\pm$0.271($-3$) |  | 3.299$\pm$0.603($-3$) |  | 1.071$\pm$0.191($-2$) |  | 1.235$\pm$0.160($-2$) |
He II 5411 | $\cdots$ |  | $\cdots$ |  | 1.525$\pm$0.331($-2$) |  | 1.478$\pm$0.235($-2$) |
He2+(adopted) | 2.513$\pm$0.271($-3$) | 0.99 | 3.299$\pm$0.603($-3$) | 0.97 | 1.071$\pm$0.191($-2$) | 0.95 | 1.235$\pm$0.160($-2$) | 0.98
$[$O I$]$ 6300 | 1.001$\pm$0.177($-6$) |  | 4.826$\pm$1.636($-6$) |  | 4.247$\pm$1.121($-6$) |  | 7.030$\pm$1.587($-6$) |
$[$O I$]$ 6363 | 1.144$\pm$0.197($-6$) |  | 4.964$\pm$1.675($-6$) |  | 5.216$\pm$1.361($-6$) |  | 7.021$\pm$1.536($-6$) |
O0(adopted) | 1.064$\pm$0.265($-6$) |  | 4.893$\pm$2.342($-6$) |  | 4.639$\pm$1.763($-6$) |  | 7.025$\pm$2.208($-6$) |
$[$O II$]$ 3727 | 1.180$\pm$0.228($-5$) |  | 4.077$\pm$0.961($-5$) |  | 1.516$\pm$0.377($-5$) |  | 2.541$\pm$0.494($-5$) |
$[$O II$]$ 7320 | 1.497$\pm$0.288($-5$) |  | 1.644$\pm$0.435($-5$) |  | 1.025$\pm$0.317($-5$) |  | 2.614$\pm$0.564($-5$) |
$[$O II$]$ 7330 | 1.819$\pm$0.359($-5$) |  | 1.944$\pm$0.539($-5$) |  | 1.309$\pm$0.403($-5$) |  | 2.882$\pm$0.625($-5$) |
O+(adopted) | 1.405$\pm$0.514($-5$) | 1.50 | 2.017$\pm$1.185($-5$) | 2.43 | 1.250$\pm$0.637($-5$) | 1.82 | 2.653$\pm$0.976($-5$) | 0.74
$[$O III$]$4363 | 2.830$\pm$0.552($-4$) |  | 3.582$\pm$0.881($-4$) |  | 2.168$\pm$0.618($-4$) |  | 3.457$\pm$0.721($-4$) |
$[$O III$]$4959 | 2.841$\pm$0.440($-4$) |  | 3.621$\pm$0.904($-4$) |  | 2.164$\pm$0.544($-4$) |  | 3.411$\pm$0.612($-4$) |
$[$O III$]$5007 | 2.818$\pm$0.439($-4$) |  | 3.554$\pm$0.852($-4$) |  | 2.123$\pm$0.533($-4$) |  | 3.377$\pm$0.600($-4$) |
O2+(adopted) | 2.830$\pm$0.832($-4$) | 1.08 | 3.584$\pm$1.523($-4$) | 1.26 | 2.150$\pm$0.981($-4$) | 1.25 | 3.410$\pm$1.120($-4$) | 1.09
$[$Ne III$]$3868 | 7.234$\pm$1.091($-5$) |  | 1.100$\pm$0.235($-4$) |  | 5.093$\pm$1.191($-5$) |  | 7.525$\pm$1.266($-5$) |
$[$Ne III$]$3967 | 7.026$\pm$1.165($-5$) |  | 1.133$\pm$0.279($-4$) |  | 3.599$\pm$1.022($-5$) |  | 6.687$\pm$1.298($-5$) |
Ne2+(adopted) | 7.137$\pm$1.597($-5$) | 1.05 | 1.114$\pm$0.365($-4$) | 1.05 | 4.232$\pm$1.570($-5$) | 0.15 | 7.116$\pm$1.813($-5$) | 1.08
$[$Ar III$]$7136 | 4.436$\pm$0.602($-7$) |  | 8.313$\pm$1.879($-7$) |  | 3.605$\pm$0.828($-7$) |  | 8.087$\pm$1.278($-7$) |
$[$Ar III$]$7751 | 3.627$\pm$0.505($-7$) |  | 4.842$\pm$1.105($-7$) |  | 2.637$\pm$0.603($-7$) |  | 6.036$\pm$0.967($-7$) |
Ar2+(adopted) | 3.962$\pm$0.786($-7$) | 1.21 | 8.313$\pm$1.879($-7$) | 1.38 | 2.973$\pm$1.024($-7$) | 1.88 | 6.783$\pm$1.603($-7$) | 1.35
$[$Ar IV$]$4711 | 4.329$\pm$0.791($-7$) |  | 4.741$\pm$1.307($-7$) |  | 3.771$\pm$0.904($-7$) |  | 5.584$\pm$1.046($-7$) |
$[$Ar IV$]$4740 | 5.503$\pm$0.757($-7$) |  | 6.534$\pm$1.298($-7$) |  | 4.504$\pm$1.001($-7$) |  | 6.824$\pm$1.100($-7$) |
Ar3+(adopted) | 4.942$\pm$1.095($-7$) | 1.13 | 5.644$\pm$1.842($-7$) | 1.29 | 4.100$\pm$1.349($-7$) | 1.18 | 6.173$\pm$1.518($-7$) | 1.10
$[$N I$]$ 5198 | $\cdots$ |  | $\cdots$ |  | $\cdots$ |  | 5.038$\pm$1.593($-7$) |
N0(adopted) | $\cdots$ |  | $\cdots$ |  | $\cdots$ |  | 5.038$\pm$1.593($-7$) |
$[$N II$]$5755 | 0.921$\pm$0.177($-6$) |  | 7.854$\pm$3.040($-6$) |  | 4.302$\pm$1.291($-6$) |  | 7.741$\pm$1.942($-6$) |
$[$N II$]$6548 | 1.089$\pm$0.169($-6$) |  | 7.851$\pm$2.317($-6$) |  | 4.440$\pm$1.024($-6$) |  | 7.905$\pm$1.503($-6$) |
$[$N II$]$6583 | 9.275$\pm$1.463($-7$) |  | 7.631$\pm$2.234($-6$) |  | 4.381$\pm$1.038($-6$) |  | 7.743$\pm$1.447($-6$) |
N+(adopted) | 9.752$\pm$2.853($-7$) | 3.42 | 7.763$\pm$4.427($-6$) | 2.55 | 4.385$\pm$1.948($-6$) | 1.48 | 7.803$\pm$2.850($-6$) | 1.12
$[$S II$]$4069 | $\cdots$ |  | 2.395$\pm$0.827($-7$) |  | $\cdots$ |  | 1.631$\pm$0.346($-7$) |
$[$S II$]$4076 | $\cdots$ |  | 9.046$\pm$2.852($-7$) |  | $\cdots$ |  | 3.738$\pm$0.835($-7$) |
$[$S II$]$6716 | 2.689$\pm$0.596($-8$) |  | 1.608$\pm$0.690($-7$) |  | 1.137$\pm$0.309($-7$) |  | 1.684$\pm$0.472($-7$) |
$[$S II$]$6730 | 3.016$\pm$0.507($-8$) |  | 1.778$\pm$0.669($-7$) |  | 1.263$\pm$0.306($-7$) |  | 1.873$\pm$0.425($-7$) |
S+(adopted) | 2.879$\pm$0.783($-8$) | 3.72 | 1.695$\pm$0.961($-7$) | 4.86 | 1.201$\pm$0.435($-7$) | 1.70 | 1.788$\pm$0.635($-7$) | 1.40
$[$S III$]$6312 | 1.607$\pm$0.271($-6$) |  | 2.212$\pm$0.547($-6$) |  | 1.410$\pm$0.372($-6$) |  | 1.791$\pm$0.338($-6$) |
S2+(adopted) | 1.607$\pm$0.271($-6$) | 1.08 | 2.212$\pm$0.547($-6$) | 1.42 | 1.410$\pm$0.372($-6$) | 1.55 | 1.791$\pm$0.338($-6$) | 1.14
$[$Cl III$]$5518 | $\cdots$ |  | $\cdots$ |  | $\cdots$ |  | 2.890$\pm$0.612($-8$) |
$[$Cl III$]$5538 | $\cdots$ |  | $\cdots$ |  | $\cdots$ |  | 5.903$\pm$1.065($-8$) |
Cl2+(adopted) | $\cdots$ |  | $\cdots$ |  | $\cdots$ |  | 5.903$\pm$1.065($-8$) | 1.10
Table 6: continued. 666The adopted notation of the abundance (Xi+/H+),
$\alpha\pm\beta(-\gamma)$, means $(a\pm b)\times 10^{-\gamma}$.

5]

Table 7: Ionic Abundances and Original-to-Revised Ratios for the Other Five PNe in M 31 Line/Ion | M1596 | Ratio | M2471 | Ratio | M2860 | Ratio | M1074 | Ratio | M1675 | Ratio
---|---|---|---|---|---|---|---|---|---|---
He I 4026 | 23.120$\pm$1.809($-2$) |  | 11.248$\pm$1.879($-2$) |  | 18.550$\pm$4.793($-2$) |  | 6.538$\pm$0.640($-2$) |  | $\cdots$ |
He I 4472 | 8.334$\pm$0.666($-2$) |  | 8.003$\pm$1.358($-2$) |  | 11.377$\pm$3.056($-2$) |  | 10.259$\pm$0.917($-2$) |  | 8.809$\pm$2.349($-2$) |
He 4922 | $\cdots$ |  | 5.163$\pm$1.124($-2$) |  | 9.027$\pm$2.900($-2$) |  | $\cdots$ |  | 16.353$\pm$5.232($-2$) |
He I 5876 | 8.992$\pm$0.858($-2$) |  | 7.056$\pm$1.429($-2$) |  | 7.953$\pm$2.537($-2$) |  | 9.992$\pm$1.134($-2$) |  | 8.478$\pm$2.789($-2$) |
He I 6678 | 8.691$\pm$0.829($-2$) |  | 4.976$\pm$1.059($-2$) |  | 6.145$\pm$1.971($-2$) |  | 7.610$\pm$0.811($-2$) |  | 9.243$\pm$3.021($-2$) |
He I 7065 | 14.203$\pm$1.721($-2$) |  | 11.479$\pm$2.509($-2$) |  | 8.858$\pm$2.903($-2$) |  | 16.511$\pm$2.064($-2$) |  | 9.937$\pm$3.446($-2$) |
He I 7281 | $\cdots$ |  | 5.235$\pm$1.116($-2$) |  | 2.179$\pm$0.755($-2$) |  | $\cdots$ |  | $\cdots$ |
He+(adopted) | 8.992$\pm$0.858($-2$) | 0.99 | 7.056$\pm$1.429($-2$) | 1.13 | 7.953$\pm$2.537($-2$) | 1.33 | 9.992$\pm$1.134($-2$) | 0.98 | 8.478$\pm$2.789($-2$) | 1.19
He II 4686 | 2.993$\pm$0.232($-2$) |  | 2.887$\pm$0.495($-2$) |  | 6.490$\pm$1.687($-3$) |  | 2.260$\pm$0.199($-3$) |  | 1.063$\pm$0.283($-2$) |
He II 5411 | 3.323$\pm$0.309($-2$) |  | 2.843$\pm$0.581($-2$) |  | 5.745$\pm$1.910($-3$) |  | $\cdots$ |  | $\cdots$ |
He2+(adopted) | 2.993$\pm$0.232($-2$) | 1.00 | 2.887$\pm$0.495($-2$) | 0.95 | 6.490$\pm$1.687($-3$) | 0.94 | 2.260$\pm$0.199($-3$) | 0.99 | 1.063$\pm$0.283($-2$) | 0.97
$[$O I$]$6300 | 7.494$\pm$1.150($-6$) |  | 7.906$\pm$1.885($-6$) |  | 1.429$\pm$0.513($-6$) |  | 2.691$\pm$0.410($-6$) |  | 9.044$\pm$3.703($-6$) |
$[$O I$]$6363 | 7.230$\pm$1.135($-6$) |  | 7.610$\pm$1.823($-6$) |  | 1.873$\pm$0.655($-6$) |  | 2.801$\pm$0.423($-6$) |  | 8.722$\pm$3.592($-6$) |
O0(adopted) | 7.360$\pm$1.615($-6$) |  | 7.753$\pm$2.622($-6$) |  | 1.598$\pm$0.832($-6$) |  | 2.745$\pm$0.589($-6$) |  | 8.878$\pm$5.159($-6$) |
$[$O II$]$3727 | 2.064$\pm$0.325($-5$) |  | 1.922$\pm$0.415($-5$) |  | 1.512$\pm$0.446($-5$) |  | 1.154$\pm$0.200($-5$) |  | 0.778$\pm$0.225($-5$) |
$[$O II$]$7320 | 2.025$\pm$0.361($-5$) |  | 1.529$\pm$0.397($-5$) |  | 1.254$\pm$0.473($-5$) |  | 1.217$\pm$0.208($-5$) |  | $\cdots$ |
$[$O II$]$7330 | 2.320$\pm$0.413($-5$) |  | 1.777$\pm$0.466($-5$) |  | 1.492$\pm$0.551($-5$) |  | 1.547$\pm$0.268($-5$) |  | 2.366$\pm$0.868($-5$) |
O+(adopted) | 2.116$\pm$0.637($-5$) | 1.28 | 1.733$\pm$0.740($-5$) | 2.23 | 1.417$\pm$0.852($-5$) | 0.63 | 1.266$\pm$0.393($-5$) | 1.74 | 2.366$\pm$0.868($-5$) | 0.63
$[$O III$]$4363 | 3.326$\pm$0.576($-4$) |  | 2.672$\pm$0.632($-4$) |  | 3.954$\pm$1.247($-4$) |  | 2.716$\pm$0.469($-4$) |  | 3.928$\pm$1.205($-4$) |
$[$O III$]$4959 | 3.298$\pm$0.410($-4$) |  | 2.629$\pm$0.597($-4$) |  | 4.004$\pm$1.357($-4$) |  | 2.708$\pm$0.356($-4$) |  | 3.805$\pm$1.263($-4$) |
$[$O III$]$5007 | 3.296$\pm$0.405($-4$) |  | 2.618$\pm$0.588($-4$) |  | 3.948$\pm$1.342($-4$) |  | 2.722$\pm$0.357($-4$) |  | 3.878$\pm$1.260($-4$) |
O2+(adopted) | 3.303$\pm$0.815($-4$) | 1.05 | 2.638$\pm$1.049($-4$) | 1.20 | 3.968$\pm$2.280($-4$) | 1.29 | 2.715$\pm$0.688($-4$) | 1.10 | 3.872$\pm$2.153($-4$) | 1.17
$[$Ne III$]$3868 | 6.843$\pm$0.868($-5$) |  | 6.891$\pm$1.385($-5$) |  | 0.967$\pm$0.273($-4$) |  | 6.223$\pm$0.795($-5$) |  | 8.376$\pm$2.335($-5$) |
$[$Ne III$]$3967 | 6.938$\pm$0.977($-5$) |  | 7.092$\pm$1.630($-5$) |  | 1.269$\pm$0.421($-4$) |  | 6.190$\pm$0.923($-5$) |  | 7.969$\pm$2.705($-5$) |
Ne2+(adopted) | 6.885$\pm$1.308($-5$) | 1.02 | 6.975$\pm$2.139($-5$) | 0.96 | 1.057$\pm$0.502($-4$) | 0.88 | 6.209$\pm$1.218($-5$) | 1.02 | 8.202$\pm$3.574($-5$) | 0.98
$[$Ar III$]$7136 | 1.231$\pm$0.132($-6$) |  | 4.932$\pm$1.010($-7$) |  | 7.022$\pm$2.328($-7$) |  | 4.079$\pm$0.475($-7$) |  | 7.454$\pm$2.496($-7$) |
$[$Ar III$]$7751 | 0.954$\pm$0.102($-6$) |  | 4.241$\pm$0.890($-7$) |  | 5.269$\pm$1.717($-7$) |  | 3.221$\pm$0.381($-7$) |  | 6.064$\pm$1.920($-7$) |
Ar2+(adopted) | 1.057$\pm$0.167($-6$) | 1.24 | 4.543$\pm$1.346($-7$) | 1.58 | 5.887$\pm$2.892($-7$) | 2.04 | 3.557$\pm$0.610($-7$) | 1.34 | 6.581$\pm$3.149($-7$) | 1.64
$[$Ar IV$]$4711 | 1.139$\pm$0.141($-6$) |  | 6.364$\pm$1.318($-7$) |  | 0.819$\pm$0.259($-6$) |  | 3.970$\pm$0.627($-7$) |  | 7.553$\pm$2.353($-7$) |
$[$Ar IV$]$4740 | 1.253$\pm$0.149($-6$) |  | 7.174$\pm$1.356($-7$) |  | 1.003$\pm$0.278($-6$) |  | 5.007$\pm$0.610($-7$) |  | 9.099$\pm$2.519($-7$) |
Ar3+(adopted) | 1.193$\pm$0.205($-6$) | 1.04 | 6.758$\pm$1.891($-7$) | 1.14 | 0.905$\pm$0.380($-6$) | 1.21 | 4.503$\pm$0.875($-7$) | 1.13 | 8.273$\pm$3.447($-7$) | 1.15
$[$Ar V$]$7005 | 1.258$\pm$0.152($-7$) |  | 4.875$\pm$1.115($-8$) |  | $\cdots$ |  | $\cdots$ |  | $\cdots$ |
Ar4+(adopted) | 1.258$\pm$0.152($-7$) | 1.07 | 4.875$\pm$1.115($-8$) | 1.47 | $\cdots$ |  | $\cdots$ |  | $\cdots$ |
$[$N I$]$5198 | $\cdots$ |  | $\cdots$ |  | $\cdots$ |  | $\cdots$ |  | 4.409$\pm$2.318($-6$) |
N0(adopted) | $\cdots$ |  | $\cdots$ |  | $\cdots$ |  | $\cdots$ |  | 4.409$\pm$2.318($-6$) |
$[$N II$]$5755 | 1.499$\pm$0.287($-5$) |  | 6.370$\pm$1.671($-6$) |  | 2.439$\pm$0.917($-6$) |  | 1.878$\pm$0.368($-6$) |  | 1.420$\pm$0.651($-5$) |
$[$N II$]$6548 | 1.502$\pm$0.190($-5$) |  | 6.565$\pm$1.404($-6$) |  | 2.665$\pm$0.890($-6$) |  | 1.908$\pm$0.245($-6$) |  | 1.397$\pm$0.515($-5$) |
$[$N II$]$6583 | 1.489$\pm$0.185($-5$) |  | 6.447$\pm$1.379($-6$) |  | 2.454$\pm$0.822($-6$) |  | 1.881$\pm$0.242($-6$) |  | 1.381$\pm$0.522($-5$) |
N+(adopted) | 1.496$\pm$0.391($-5$) | 1.06 | 6.470$\pm$2.582($-6$) | 1.38 | 2.517$\pm$1.520($-6$) | 1.63 | 1.891$\pm$0.504($-6$) | 1.36 | 1.396$\pm$0.981($-5$) | 1.20
$[$S II$]$4069 | 1.676$\pm$0.220($-7$) |  | 1.660$\pm$0.327($-7$) |  | 11.030$\pm$3.083($-8$) |  | 6.943$\pm$0.854($-8$) |  | $\cdots$ |
$[$S II$]$4076 | $\cdots$ |  | $\cdots$ |  | 66.702$\pm$19.455($-8$) |  | $\cdots$ |  | $\cdots$ |
$[$S II$]$6716 | 4.107$\pm$0.809($-7$) |  | 1.804$\pm$0.455($-7$) |  | 7.198$\pm$2.648($-8$) |  | 5.674$\pm$0.998($-8$) |  | 4.394$\pm$2.122($-7$) |
$[$S II$]$6730 | 4.605$\pm$0.629($-7$) |  | 2.012$\pm$0.454($-7$) |  | 8.048$\pm$2.793($-8$) |  | 6.386$\pm$0.842($-8$) |  | 4.947$\pm$2.177($-7$) |
S+(adopted) | 4.418$\pm$1.025($-7$) | 1.20 | 1.908$\pm$0.643($-7$) | 1.58 | 7.600$\pm$3.849($-8$) | 1.92 | 6.090$\pm$1.306($-8$) | 1.52 | 4.664$\pm$3.040($-7$) | 1.19
$[$S III$]$6312 | 4.161$\pm$0.575($-6$) |  | 1.223$\pm$0.291($-6$) |  | 2.161$\pm$0.764($-6$) |  | 2.047$\pm$0.298($-6$) |  | 3.263$\pm$1.118($-6$) |
S2+(adopted) | 4.161$\pm$0.575($-6$) | 1.08 | 1.223$\pm$0.291($-6$) | 1.47 | 2.161$\pm$0.764($-6$) | 1.72 | 2.047$\pm$0.298($-6$) | 1.16 | 3.263$\pm$1.118($-6$) | 1.42
$[$Cl III$]$5518 | 4.619$\pm$0.633($-8$) |  | $\cdots$ |  | 2.763$\pm$1.015($-8$) |  | $\cdots$ |  | $\cdots$ |
$[$Cl III$]$5538 | 9.073$\pm$1.108($-8$) |  | $\cdots$ |  | 4.145$\pm$1.461($-8$) |  | $\cdots$ |  | $\cdots$ |
Cl2+(adopted) | 9.073$\pm$1.108($-8$) | 0.90 | $\cdots$ |  | 3.213$\pm$1.780($-8$) | 1.91 | $\cdots$ |  | $\cdots$ |
Table 7: continued. 777The adopted notation of the abundance (Xi+/H+),
$\alpha\pm\beta(-\gamma)$, means $(a\pm b)\times 10^{-\gamma}$.

6]

Table 8: Total elemental Abundances and Original-to-Revised Ratios, with the
Solar Abundances for Comparison. ICFs follow those by Delgado-Inglada et al.
(2014)
---
Elem. | M1687 | Ratio | M2068 | Ratio | M2538 | Ratio | M50 | Ratio | Z⊙ |
He | 10.83$\pm$0.06 | 1.43 | 10.95$\pm$0.10 | 1.22 | 10.96$\pm$0.09 | 1.13 | 11.01$\pm$0.06 | 1.06 | 10.92$\pm$0.02 |
N | 7.32$\pm$0.24 | 2.52 | 8.17$\pm$0.39 | 1.38 | 7.94$\pm$0.34 | 0.99 | 8.07$\pm$0.26 | 1.59 | 7.85$\pm$0.12 |
O | 8.48$\pm$0.12 | 1.10 | 8.58$\pm$0.17 | 1.33 | 8.40$\pm$0.18 | 1.26 | 8.60$\pm$0.13 | 1.05 | 8.73$\pm$0.07 |
Ne | 7.87$\pm$0.07 | 1.04 | 8.06$\pm$0.09 | 1.08 | 7.74$\pm$0.10 | 0.95 | 7.91$\pm$0.07 | 1.00 | 8.15$\pm$0.10 |
S | 6.05$\pm$0.22 | 2.88 | 6.68$\pm$0.40 | 1.29 | 6.56$\pm$0.32 | 1.14 | 6.64$\pm$0.25 | 1.06 | 7.15$\pm$0.03 |
Cl | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 5.05$\pm$0.09 | 1.19 | 5.23$\pm$0.06 |
Ar | 5.59$\pm$0.08 | 1.91 | 5.91$\pm$0.10 | 2.09 | 5.48$\pm$0.15 | 2.93 | 5.85$\pm$0.10 | 2.20 | 6.50$\pm$0.10 |
Elem. | M1596 | Ratio | M2471 | Ratio | M2860 | Ratio | M1074 | Ratio | M1675 | Ratio
He | 11.08$\pm$0.03 | 0.98 | 11.00$\pm$0.07 | 1.08 | 10.94$\pm$0.13 | 1.30 | 11.01$\pm$0.05 | 0.97 | 10.98$\pm$0.12 | 1.18
N | 8.47$\pm$0.20 | 0.87 | 8.11$\pm$0.30 | 0.76 | 7.88$\pm$0.44 | 3.29 | 7.63$\pm$0.21 | 0.88 | 8.42$\pm$0.42 | 2.16
O | 8.62$\pm$0.10 | 1.07 | 8.54$\pm$0.16 | 1.23 | 8.63$\pm$0.24 | 1.25 | 8.46$\pm$0.11 | 1.12 | 8.64$\pm$0.23 | 1.14
Ne | 7.91$\pm$0.06 | 1.03 | 7.93$\pm$0.09 | 0.95 | 8.01$\pm$0.12 | 0.94 | 7.82$\pm$0.06 | 1.03 | 7.96$\pm$0.12 | 0.94
S | 7.10$\pm$0.18 | 0.73 | 6.69$\pm$0.27 | 0.77 | 6.49$\pm$0.49 | 3.11 | 6.42$\pm$0.19 | 1.62 | 7.07$\pm$0.47 | 0.96
Cl | 5.30$\pm$0.06 | 0.87 | $\cdots$ | $\cdots$ | 4.85$\pm$0.25 | 2.22 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
Ar | 6.08$\pm$0.07 | 1.95 | 5.73$\pm$0.13 | 2.22 | 5.77$\pm$0.21 | 3.53 | 5.54$\pm$0.07 | 2.10 | 5.83$\pm$0.21 | 2.77
Using all available Xi+ abundances with appropriate ICFs
Elem. | M1687 | Ratio | M2068 | Ratio | M2538 | Ratio | M50 | Ratio | Z⊙ |
He | 10.83$\pm$0.05 | 1.41 | 10.96$\pm$0.09 | 1.20 | 10.96$\pm$0.08 | 1.12 | 11.01$\pm$0.06 | 1.07 | 10.92$\pm$0.02 |
N | 7.36$\pm$0.22 | 2.32 | 8.16$\pm$0.39 | 1.42 | 7.83$\pm$0.35 | 1.27 | 8.00$\pm$0.25 | 1.84 | 7.85$\pm$0.12 |
O | 8.49$\pm$0.12 | 1.08 | 8.60$\pm$0.17 | 1.30 | 8.38$\pm$0.18 | 1.32 | 8.59$\pm$0.14 | 1.07 | 8.73$\pm$0.07 |
Ne | 7.87$\pm$0.06 | 1.04 | 8.06$\pm$0.09 | 1.07 | 7.74$\pm$0.10 | 0.95 | 7.91$\pm$0.07 | 1.01 | 8.15$\pm$0.10 |
S | 6.50$\pm$0.09 | 1.03 | 6.64$\pm$0.13 | 1.42 | 6.46$\pm$0.15 | 1.45 | 6.55$\pm$0.10 | 1.29 | 7.15$\pm$0.03 |
Cl | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 5.05$\pm$0.09 | 1.20 | 5.23$\pm$0.06 |
Ar | 5.96$\pm$0.07 | 0.81 | 6.16$\pm$0.09 | 1.19 | 5.86$\pm$0.11 | 1.23 | 6.12$\pm$0.07 | 1.18 | 6.50$\pm$0.10 |
Elem. | M1596 | Ratio | M2471 | Ratio | M2860 | Ratio | M1074 | Ratio | M1675 | Ratio
He | 11.08$\pm$0.03 | 0.98 | 11.00$\pm$0.07 | 1.08 | 10.94$\pm$0.13 | 1.29 | 11.01$\pm$0.05 | 0.97 | 10.98$\pm$0.13 | 1.19
N | 8.40$\pm$0.19 | 1.02 | 7.99$\pm$0.30 | 1.01 | 7.74$\pm$0.50 | 4.56 | 7.71$\pm$0.19 | 0.74 | 8.33$\pm$0.45 | 2.65
O | 8.63$\pm$0.10 | 1.05 | 8.53$\pm$0.17 | 1.25 | 8.65$\pm$0.23 | 1.21 | 8.47$\pm$0.10 | 1.11 | 8.65$\pm$0.22 | 1.12
Ne | 7.91$\pm$0.06 | 1.02 | 7.93$\pm$0.09 | 0.96 | 8.01$\pm$0.12 | 0.93 | 7.82$\pm$0.06 | 1.02 | 7.95$\pm$0.11 | 0.95
S | 6.99$\pm$0.07 | 0.96 | 6.48$\pm$0.12 | 1.24 | 6.65$\pm$0.21 | 2.12 | 6.61$\pm$0.08 | 1.04 | 6.85$\pm$0.16 | 1.57
Cl | 5.31$\pm$0.07 | 0.86 | $\cdots$ | $\cdots$ | 4.83$\pm$0.27 | 2.33 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
Ar | 6.38$\pm$0.05 | 0.99 | 6.07$\pm$0.08 | 1.02 | 6.20$\pm$0.16 | 1.33 | 5.92$\pm$0.06 | 0.87 | 6.18$\pm$0.14 | 1.22