# Simple Open Stance Classification for Rumour Analysis

Ahmet Akera,b Leon Derczynskia Kalina Bontchevaa
Department of Computer Science, University of Sheffielda
Department of Information Engineering, University of Duisburg-Essenb
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Stance classification determines the attitude, or stance, in a (typically
short) text. The task has powerful applications, such as the detection of fake
news or the automatic extraction of attitudes toward entities or events in the
media. This paper describes a surprisingly simple and efficient classification
approach to open stance classification in Twitter, for rumour and veracity
classification. The approach profits from a novel set of automatically
identifiable problem-specific features, which significantly boost classifier
accuracy and achieve above state-of-the-art results on recent benchmark
datasets. This calls into question the value of using complex sophisticated
models for stance classification without first doing informed feature
extraction.

##  1 Introduction

Stance detection is the problem of classifying the attitude taken by an author
in a short piece of text. Typical stances include showing support, denying,
commenting on or querying an existing claim or fact. Knowing the stance that
authors hold in response to claims, e.g. in online commentary, gives useful
insights. It can reveal rumours and fake news claims as the discourse around
them is monitored Procter et al. (2013). Stance reflects how certain authors
are of a claim’s veracity Biber (2006), which enables the effective detection
of potential false rumours Lukasik et al. (2015). Stance also reveals how
online populations react to business and political news.

This paper addresses the general-purpose, or open stance classification task.
This is distinct from target-specific stance classification, as in Augenstein
et al. (2016) and Mohammad et al. (2016), which focus on stances towards
known, pre-determined targets. In the latter task, the target has already been
extracted, from e.g. conversational cues. Target-specific stance
classification is suited to situations where the target is already known, such
as analyses of a specific product or political actor. In contrast, the open
stance classification task is appropriate in emerging news or novel contexts,
such as working with online media or streaming news analysis.

Open stance classification is often applied in rumour resolution. Since
attitudes in discourse around a claim are indicative not only of the
controversiality of the claim, but also can act as a proxy for its veracity,
it is reasonable to consider the application of open stance detection for
rumour analysis. Indeed, many approaches to rumour and fake news analysis rely
on this signal Derczynski et al.
(2017)111http://approximatelycorrect.com/2017/01/23/is-fake-news-a-machine-
learning-problem/. In veracity analysis, the claim is already known, and the
goal is to gather observations and analyse crowd reaction in order to resolve
the claim. Instead of being concerned with specific targets, we apply non-
targeted – open – stance analysis to messages replying to a claim, where the
target may vary but the high-level rumour topic rumour remains the same.

Our simple approach to open stance classification implements common features
used in stance classification reported by related work (e.g. bag-of-words,
named entities, user activity information, URL presence). We extend this with
problem-specific features (which we refer to as the AF features) designed to
capture how users react to tweets and express confidence in them. Our results
show adding these features gives significantly higher performance on benchmark
datasets, compared to recent state-of-the-art systems. We make our classifier
freely available on the PHEME software
repository.222https://gate.ac.uk/wiki/pheme-stance.html

The outline of the paper is as follows. First we describe related work
(Section 2) and then introduce our method along with the classification
techniques used and features extracted (Section 3). Next, Section 4 describes
our experimental set-ups, followed by results in Section 5. We report on
feature analysis in Section 6, prior to concluding the paper (Section 7).

##  2 Related Work

The first study that tackles automatic stance classification is that of
Qazvinian et al. Qazvinian et al. (2011). With a dataset containing 10K tweets
and using a Bayesian classifier and three types of features categorised as
“content”, “network” and “Twitter specific memes”, the authors achieved an
accuracy of 93.5%. Similar to them, Hamidian and Diab Hamidian and Diab (2015)
perform rumour stance classification by applying supervised machine learning
using the dataset created by Qazvinian et al. Qazvinian et al. (2011).
However, instead of Bayesian classifiers, the authors use J48 decision tree
implemented within the Weka platform Hall et al. (2009). The features from
Qazvinian et al. Qazvinian et al. (2011) are adopted and extended with time-
related information and the hastags themselves, instead of the content of the
hashtag as used by Qazvinian et al. Qazvinian et al. (2011). In addition to
the feature categories introduced above, Hamidian and Diab Hamidian and Diab
(2015) introduce another feature category, namely “pragramatic”. The pragmatic
features include named entity, event, sentiment and emoticons. The evaluation
of the performance is casted as either 1-step problem containing a 6 class
classification task (not rumour, 4 classes of stance and not determined by the
annotator) or 2-step problem containing first a 3 class classification task
(non-rumour, rumour, and not determined), followed by a 4 class classification
task (stance classification). The two step approach achieves better
performance with 82.9% F-1 measure, compared to 74% with the 1-step approach.
The authors also report that the best performing features were the content
based features and the worst performing ones – the network and Twitter
specific features. In their most recent paper, Hamidian and Diab Hamidian and
Diab (2016) introduce the Tweet Latent Vector (TLV) approach that is obtained
by applying the Semantic Textual Similarity model proposed by Guo and Diab Guo
and Diab (2012). The authors compare the TLV approach to their own earlier
system, as well as to the original features of Qazvinian et al. Qazvinian et
al. (2011) and show that the TLV approach outperforms both baselines.

Liu et al. Liu et al. (2015) use a rule-based method and show that it
outperforms the approach reported by Qazvinian et al. Qazvinian et al. (2011).
Zeng et al. Zeng et al. (2016) enrich the feature sets investigated by earlier
studies by features derived from the Linguistic Inquiry and Word Count (LIWC)
dictionaries Tausczik and Pennebaker (2010). Lukasik et al. Lukasik et al.
(2016) investigate Gaussian Processes as rumour stance classifier. For the
first time the authors also use Brown Clusters to extract the features for
each tweet. Unlike researchers above, Lukasik et al. evalute on the rumour
data released by Zubiaga et al. Zubiaga et al. (2016b), where they report an
accuracy of 67.7%. This result is achieved when the classifier is trained on
$n-1$ rumours and tested on the nth rumour. However, the authors achieve
substantially better results when a small proportion of the in-domain data
(data from the nth rumour) is also included in the training (68.6% accuracy).
Performance scores differ substantially from those in the studies described
above, given that Lukasik et al. Lukasik et al. (2016) tackled classification
of stance in new rumours that differ from those in the training set.

Subsequent work has also tackled stance classification for new, unseen
rumours. Zubiaga et al. Zubiaga et al. (2016a) moved away from the
classification of tweets in isolation, focusing instead on Twitter
’conversations’ Tolmie et al. (2015) initiated by rumours, as part of the
Pheme project Derczynski and Bontcheva (2014). They looked at tree-structured
conversations initiated by a rumour and followed by tweets responding to it by
supporting, denying, querying or commenting on the rumour.

Rumour stance classification for tree structured conversations has also been
studied in the RumourEval shared task at SemEval 2017 Derczynski et al.
(2017). Subtask A there consisted of stance classification of individual
tweets discussing a rumour within a conversational thread as one of support,
deny, query, or comment. Eight participanting teams submitted results to this
task. Most of the systems viewed this task as a 4-way single tweet
classification task, with the exception of the best performing system by
Kochkina et al. Kochkina et al. (2017), as well as the systems by Wang et al.
Wang et al. (2017) and Singh et al. Singh et al. (2017). The winning system
addressed the task as a sequential classification problem, where the stance of
each tweet takes into consideration the features and labels of the preceding
tweets. The system by Singh et al. Singh et al. (2017) takes as input pairs of
source and reply tweets, whereas Wang et al. Wang et al. (2017) addressed
class imbalance by decomposing the problem into a two step classification task
– first distinguishing between comments and non-comments and then classifying
non-comment tweets as one of support, deny or query. Half of the systems
employed ensemble classifiers, where classification was obtained through
majority voting Wang et al. (2017); García Lozano et al. (2017); Bahuleyan and
Vechtomova (2017); Srivastava et al. (2017). In some cases the ensembles were
hybrid, consisting both of machine learning classifiers and manually created
rules, with differential weighting of classifiers for different class labels
Wang et al. (2017); García Lozano et al. (2017); Srivastava et al. (2017).
Three systems used deep learning, with Kochkina et al. Kochkina et al. (2017)
employing LSTMs for sequential classification, Chen et al. Chen et al. (2017)
using convolutional neural networks (CNN) for obtaining the representation of
each tweet, assigned a probability for a class by a softmax classifier and
García Lozano et al. García Lozano et al. (2017) using CNN as one of the
classifiers in their hybrid conglomeration. The remaining two systems by
Enayet and El-Beltagy Enayet and El-Beltagy (2017) and Singh et al. Singh et
al. (2017) used support vector machines with a linear and polynomial kernel
respectively.

##  3 Method

###  3.1 Data

In our experiments we used two different data sets: RumourEval dataset
Derczynski et al. (2017) and the PHEME dataset Zubiaga et al. (2016b). In the
PHEME dataset the authors identify rumours associated with events, collect
conversations sparked by those rumours in the form of replies and annotate
each of the tweets in the conversations for stance. These data consist of
tweets from 5 different events: Ottawa shooting, Ferguson riots, Germanwings
crash, Charlie Hebdo and Sydney siege. Each dataset has a different number of
rumours where each rumour contains tweets marked with stance annotations:
“supporting”, “questioning”, “denying” or “commenting”. A summary of the data
is given in Table 1.

Dataset | Rumours | S | D | Q | C
---|---|---|---|---|---
Ottawa shooting | 58 | 161 | 76 | 64 | 481
Ferguson riots | 46 | 192 | 83 | 94 | 685
Charlie Hebdo | 74 | 236 | 56 | 51 | 710
Sydney siege | 71 | 89 | 4 | 99 | 713
Table 1: PHEME Data: Counts of tweets with supporting (S), denying (D),
questioning (Q) and commenting (C) labels in each event collection.

The RumourEval dataset is derived from the PHEME dataset, however, for the
purpose of the RumourEval shared Task A the data has a given split into
training and testing. This provides an established basis for evaluation. The
training data draws from stories in 2014–2016, from the earlier PHEME dataset.
The evaluation split covers two new stories, both from 2016: first, the
disappearance of Marina Joyce, a British Youtube personality, who was rumoured
to have been abducted in July 2016. There was significant speculation in
social media, and the case was brought to a concrete resolution as the police
investigated and posted an open public response. The second story was that
Hillary Clinton had pneumonia during mid-September 2016. The prevalence and
spread of this story could be tracked easily, and it emerged in a short space
of time, though among background noise of speculative, unsubstantiated claims
about her and her opponent’s health. More details about this dataset can be
obtained from the SemEval website333http://alt.qcri.org/semeval2017/task8/.

In keeping with prior work Zeng et al. (2016); Lukasik et al. (2016); Zubiaga
et al. (2016a), our experiments assume that incoming tweets already belong to
a particular rumour, e.g. a user is tracking tweets related to a certain
rumour. For each new tweet, features are extracted into a feature vector,
which is then used to assign each tweet its stance towards the rumour.

###  3.2 Classifiers

We experiment with three different, well known machine learning classifiers:
(1) a decision tree, J48; (2) Random Forests Breiman (2001); and (3) an
Instance Based classifier (K-NN). For the Random Forest we use 50 trees (-I
50). Pruning is enabled for J48. Finally we run the Instance Based classifier
with -I -K 10 settings.

###  3.3 Features

Prior work on stance classification investigated various features which can be
categorized into linguistic, message-based, and topic-based categories Mendoza
et al. (2010); Qazvinian et al. (2011); Hamidian and Diab (2015); Liu et al.
(2015); Zeng et al. (2016); Lukasik et al. (2016); Zubiaga et al. (2016a). The
following list summarizes the features adopted in this work.

  * •

BOW (Bag of words): For this feature we first create a dictionary from all the
tweets in the out-of-domain dataset. Next each tweet is assigned the words in
the dictionary as features. For words occurring in the tweet the feature
values are set to the number of times they occur in the tweet. For all other
words “0” is used.

  * •

Brown Cluster: Brown clustering is a hard hierarchical clustering method and
we use it to cluster words in hierarchies. It clusters words based on
maximising the probability of the words under the bigram language model, where
words are generated based on their clusters Liang (2005). In previous work, it
has been shown that Brown clusters yield better performance than directly
using the BOW features Lukasik et al. (2015). Brown clusters are obtained from
a bigger tweet corpus that entails assignments of words to brown cluster ids.
We used 1000 clusters, i.e. there are 1000 cluster ids. All 1000 ids are used
as features however only, ids that cover words in the tweet are assigned a
feature value “1”. All other cluster id feature values are set to “0”.

  * •

POS tag: The BOW feature captures the actual words and is domain dependent. To
create a feature that is not domain dependent, we added Part of Speech (POS)
tags as additional feature. Similar to the BOW feature we created a dictionary
of POS tags from the entire corpus (excluding the health data) and used this
dictionary to label each tweet with it – binary, i.e. whether a POS tag is
present.444We also experimented with frequencies of POS tags, i.e. counting
how many times a particular POS tag occurs in the tweet. The counts then have
been normalized using mean and standard deviation. However, the frequency
based POS feature negatively affected classification accuracy, so it has been
omitted from the feature set. However, instead of using just single POS tags,
we created sequences containing bi-gram, tri-gram and 4-gram POS tags. Feature
values are the frequencies of POS tag sequences occurring in the tweet.

  * •

Sentiment: This is another domain-independent feature. Sentiment analysis
reveals the sentimental polarity of the tweet such as whether it is positive
or negative. We used the Stanford sentimentSocher et al. (2013) tool to create
this feature. The tool returns a range from 0 to 4 with 0 indicating “very
negative” and 4 “very positive”. First, we used this as a categorical feature
but turning it to a numeric feature gave us better performance. Thus each
tweet is assigned a sentiment feature whose value varies from 0 to 4.

  * •

NE: Named entity (NE) is also domain independent. We check for each tweet
whether it contains Person, Organization, Date, Location and Money tags and
for each tag present, “1” is added, or a “0” otherwise.

  * •

Reply: This is a binary feature, which is assigned “1” if the tweet is a reply
to a previous one, or a “0” otherwise. Tweet reply information is extracted
from the tweet metadata. Again this feature is domain independent.

  * •

Emoticon: We created a dictionary of emoticons using
Wikipedia555https://en.wikipedia.org/wiki/List_of_emoticons. In Wikipedia
those emoticons are grouped by categories, which we use as a feature. If any
emoticon from a category occurs in the tweet, we assign for that category
feature the value “1” – otherwise “0”. Again similar to the previous features
this feature is domain independent.

  * •

URL: This is again domain independent. We assign the tweet “1” if it contains
any URL, or “0” otherwise.

  * •

Mood: Mood detection analyses textual content using different view points or
angles. Mood detection is performed using the tool from Celli et al. (2016),
which analyses tweets from five different angles: amused, disappointed,
indignant, satisfied and worried. For each of this angles it returns a value
from -1 to +1. We use the different angles as the mood features and the
returned values as the feature value.

  * •

Originality score: This is the count of tweets the user has produced, i.e.
“statuses count” in the Twitter API.

  * •

isUserVerified(0-1): Whether the user is verified or not.

  * •

NumberOfFollowers: Number of followers the user has.

  * •

Role score: This is the ratio between the number of followers and followees
(i.e. NumberOfFollowers/NumberOfFollowees).

  * •

Engagement score: the number of tweets divided by the number of days the user
has been active (number of days since the user account creation till today).

  * •

Favourites score: The “favourites count” divided by the number of days the
user has been active.

  * •

HasGeoEnabled(0-1): User has enabled geo-location or not.

  * •

HasDescription(0-1): User has description or not.

  * •

LenghtOfDescription in words: The number of words in the user description.

  * •

averageNegation: We determine using the Stanford parser Chen and Manning
(2014) the dependency parse tree of the tweet, count the number of negation
relation (“neg”) that appears between two terms and divide this by the number
of total relations.

  * •

hasNegation(0-1): Tweet has negation relationship or not.

  * •

hasSlangOrCurseWord(0-1): A dictionary of key
words666www.noswearing.com/dictionary is used to determine the presence of
slang or curse words in the tweet.

  * •

hasGoogleBadWord(0-1): Same as above but the dictionary of slang words is
obtained from Google.777http://fffff.at/googles-official-list-of-bad-words

  * •

hasAcronyms(0-1): The tweet is checked for presence of acronyms using a
acronym dictionary.888www.netlingo.com/category/acronyms.php

  * •

averageWordLength: Average length of words (sum of word character counts
divided by number of words in each tweet).

  * •

hasQuestionMark(0-1): The tweet has “?” or not.

  * •

hasExclamationMark(0-1): The tweet has “!” or not.

  * •

hasDotDotDot(0-1): Whether the tweet has “…” or not.

  * •

numberOfQuestionMark: Count of “?” in the tweet.

  * •

NumberOfExclamationMark: Count of “!” in the tweet.

  * •

numberOfDotDotDot: Count of “…” in the tweet.

  * •

Binary regular expressions applied on each tweet: .*(rumor?—debunk?).*, .*is
(that—this—it) true.*, etc. In total there are 10 features covering regular
expressions.

This work extends the features above, with new additional problem-specific
features (AF features). AF features score the level of confidence in a tweet.
We compute scores for surprise (surpriseScore (SS)), doubt (doubtScore (DS)),
certainty (noDoubtScore (NDS)) and support (supportScore (SPS)) towards
rumourous tweets. For each of these features a list of typical words is
collected. We use this list to compute a cumulative vector using word2Vec
Mikolov et al. (2013). For each word in the list, we obtain its word2Vec
representation, add them together and finally divide the resulting vector by
the number of words to obtain the cumulative vector. Similarly a cumulative
vector is computed for the words in the tweet excluding acronyms, named
entities and URLs. We use cosine to compute the angle between those two
cumulative vectors to determine each of the scores. Our word embeddings
comprise the vectors published by Baroni et al. (2014). The full list of tweet
confidence AF features is as follows:

  * •

surpriseScore (SS): cosine between embedding of tweet content and the list of
surprise words, e.g. “surprise”, “wonder”, etc.

  * •

doubtScore (DS): cosine between embedding of tweet content and the list of
doubt words, e.g. “doubt”, “uncertain”, etc.

  * •

noDoubtScore (NDS): cosine between embedding of tweet content and the list of
certainty words, e.g.“surely”, “sure”, etc.

  * •

supportScore (SPS): cosine between embedding tweet content and the list of
support words, e.g. “support”, “confirm”, etc.

Furthermore, the following two AF features are included:

  * •

initialTweetSim (ITS) captures tweets that tend to support rumours. Every
rumour is initiated by a tweet. We compute the cosine similarity based on
word2Vec of the tweet being classified to the first tweet in the rumour
thread. If the tweet is just a simple re-retweet of the initial tweet, this is
taken as an evidence that the tweet is supportive of that tweet.

  * •

isQuestion (IQ) indicates whether a tweet starts with an interrogative. The
feature is binary and aims to capture questioning tweets.

##  4 Experimental Setup

Classifier |  All features |  w.o. AF
---|---|---
Decision tree |  74.16 |  72.25
Random Forest |  79.02 |  76.54
IBk |  75.59 |  73.02
Baseline-Turing |  78.4 |  –
Table 2:  Accuracy scores of different stance classifiers for the RumourEval dataset. The baseline is the best performing system in the SemEval evaluation Turing. classifier |  Ottawa shooting |  Ferguson riots |  Charlie Hebdo |  Sydney siege |  macro mean
---|---|---|---|---|---
IBk |  70.31* |  72.35 |  78.33 (ref) |  75.44 |  74.10
Decision tree |  76.28 (ref) |  75.20 (ref) |  78.21 |  80.01 (ref) |  77.42
Random Forest |  69.39* |  69.16 |  74.57 |  74.49 |  71.90
Baseline - GP |  62.28 |  64.31 |  70.66 |  65.04 |  65.57
Baseline - HP |  67.77 |  68.44 |  72.93 |  68.59 |  69.43
Table 3:  Accuracy scores for different stance classifiers on the PHEME dataset. * indicates a significant difference to (“ref”) scores for each column of the table respectively as indicated by the paired t-test with $p<0.001$. classifier |  Ottawa shooting |  Ferguson riots |  Charlie Hebdo |  Sydney siege |  macro mean
---|---|---|---|---|---
IBk / AF |  69.26 |  69.54 |  77.09 |  73.28 |  72.29
J48 / AF |  75.62 |  74.85 |  77.05 |  79.21 |  76.68
Random Forest / AF |  67.87 |  68.31 |  75.40 |  72.57 |  71.03
Table 4:  Accuracy scores of different stance classifiers on the PHEME dataset with AF features removed. Features |  Accuracy
---|---
All features |  79.02
All without AF |  76.54
All without ITS |  78.55
All without SS |  77.59
All without SPS |  78.16
All without DS |  78.36
All without NDS |  77.59
All without IQ |  78.64
Table 5:  Contribution of each AF feature. Accuracy scores are for the Random
Forest classifier on RumourEval data set with each feature removed in turn.

###  4.1 Baselines

On the RumourEval dataset we run different classifiers (see Section 3.2). We
compare the performance of these classifiers against the best-performing
system from the RumourEval challenge, namely Turing Kochkina et al. (2017).

We also run all the classifiers from the RumourEval dataset on the PHEME
dataset. The results are compared against the following baseline systems
reported on the PHEME dataset:

  * •

Gaussian Processes (GP) reported by Lukasik et al. (2015).

  * •

Hawkes Processes (HP) reported by Lukasik et al. (2016). HPs make use of both
temporal and textual information of tweets.

###  4.2 Training-Testing Settings

We have two different settings. In the first setting we use the SemEval
training data to train the models and apply on the testing data. In the second
setting we perform training and testing on the PHEME dataset. For the PHEME
dataset, we follow the leave one out (LOO) strategy taken by Lukasik et al.
Lukasik et al. (2016) to construct the training and testing data. In LOO n-1
rumours (all tweets within these rumours) are used for training and the
resulting model is tested on the nth rumour. Finally, results are macro-
averaged.

##  5 Results

As shown in Table 2 (column two of the table) the best performing learner on
the RumourEval dataset is the Random Forest classifier. It achieves the
accuracy of 79.02, higher than any participating system in the RumourEval Task
A.999The results are reported in
http://alt.qcri.org/semeval2017/task8/index.php?id=results

The results on the PHEME dataset are shown in Table 3. Overall the best
performing classifier is the J48 decision tree learner. The difference in
accuracy scores between the classfiers is tested for significance using paired
t-test (p$<$0.001). J48 is only significantly better than IBk and J48 for the
Ottawa shooting event type. In the remaining event types, J48 performs better,
but not significantly better than IBk and Random Forest.

All classifiers J48, IBk and Random Forest, however, outperform the GP and HP
baselines on all event types 101010Although the significance test could not be
run for the baselines, as the single data point values are not available, the
proportion of difference in accuracy and the fact that it is the same data
sets let us assume that the classifiers significantly outperform the
baselines..

What these results demonstrate is that simpler classifiers, such as J48 and
Random Forest can outperform significantly more sophisticated machine learning
methods (GPs and HPs in this case, and LSTMs in the RumourEval case), thanks
to the additional knowledge captured in the rich feature set. In contrast, for
example, the GP and HP models relied primarily on BOW and Brown clustering
features.

##  6 Feature Analysis

The results described in Section 5 are based on features reported by related
work, enhanced by us with AF features (Section 3.3). We repeat the experiments
with AF features removed from the feature set, in order to quantify the extent
of their contribution.

For the RumourEval dataset the results are shown in column 3 of Table 2. The
omission of the AF features leads to a performance decrease for all
classifiers. The accuracy scores also fall below that of the SemEval winner
Turing – the state-of-the-art system on the RumourEval dataset.

The results on the PHEME dataset are shown in Table 4. The exclusion of the AF
features leads to an overall drop in performance when compared to the same
classifiers in Table 3. However, these differences are not significant and the
classifiers with AF features removed still perform at least as well as the GP
and HP baselines (for the event type Ferguson riots), or outperform the
baselines (for all other event types).

Table 5 shows the accuracy scores of the Random Forest stance classifier, the
best performing system on the RumourEval dataset when each AF feature is
removed in turn. The results indicate that each AF feature contributes to the
accuracy boost in stance classification. The highest accuracy loss results
from removing the surprise (SS) and certainty (NDS) scores and the least –
when the isQuestion (IQ) feature is removed. None of the AF feature removals
cause a significant drop in accuracy. However, the loss is significant
($p<0.0001$) when all AF features are removed.

Both RumourEval and PHEME dataset evaluations show that the AF features play
an important role in terms of achieving higher accuracy for tweet-based stance
classification. They also show the importance of task or problem-specific
feature engineering and point out that it is possible with some feature
engineering effort to outperform state-of-the-art techniques that are
typically considered more powerful and sophisticated than traditional learning
methods.

##  7 Conclusion

This paper tackled the problem of stance classification of tweets towards
rumours. In our approach we use a simple classification approach, combining
common features reported by related studies with our novel AF features, to
boost overall accuracy. Our results show that this approach leads to
significantly better results on both RumourEval and PHEME datasets compared to
current state-of-the-art systems. Furthermore, our results show that the
omission of the AF features proposed in this work leads to significantly lower
performance. Adding AF to the feature set causes our approach to outperform
the best performing system on the RumourEval dataset. These results show the
importance of task- or problem-oriented feature engineering.

The proposed features are content based and work on text level. In our future
work we plan to investigate features that are able to capture communication
behaviours between users. We also plan to apply stance information as a
feature in rumour veracity classification.

## Acknowledgments

This work was partially supported by the European Union funded COMRADES (grant
agreement No. 687847) and PHEME projects (grant agreement No. 611223), as well
as an EPSRC career acceleration fellowship (EP/I004327/1).

## References

  * Augenstein et al. (2016) Isabelle Augenstein, Tim Rocktäschel, Andreas Vlachos, and Kalina Bontcheva. 2016.  Stance detection with bidirectional conditional encoding.  In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing. Association for Computational Linguistics, Austin, Texas, pages 876–885.  https://aclweb.org/anthology/D16-1084.
  * Bahuleyan and Vechtomova (2017) Hareesh Bahuleyan and Olga Vechtomova. 2017.  UWaterloo at SemEval-2017 Task 8: Detecting Stance towards Rumours with Topic Independent Features.  In Proceedings of SemEval. ACL.
  * Baroni et al. (2014) Marco Baroni, Georgiana Dinu, and Germán Kruszewski. 2014.  Don’t count, predict! a systematic comparison of context-counting vs. context-predicting semantic vectors.  In Proceedings of ACL. pages 238–247.
  * Biber (2006) Douglas Biber. 2006.  Stance in spoken and written university registers.  Journal of English for Academic Purposes 5(2):97–116.
  * Breiman (2001) Leo Breiman. 2001.  Random forests.  Machine learning 45(1):5–32.
  * Celli et al. (2016) Fabio Celli, Arindam Ghosh, Firoj Alam, and Giuseppe Riccardi. 2016.  In the mood for sharing contents: Emotions, personality and interaction styles in the diffusion of news.  Information Processing & Management 52(1):93–98.
  * Chen and Manning (2014) Danqi Chen and Christopher D Manning. 2014.  A fast and accurate dependency parser using neural networks.  In Proceedings of EMNLP. pages 740–750.
  * Chen et al. (2017) Yi-Chin Chen, Zhao-Yand Liu, and Hung-Yu Kao. 2017.  IKM at SemEval-2017 Task 8: Convolutional Neural Networks for Stance Detection and Rumor Verification.  In Proceedings of SemEval. ACL.
  * Derczynski and Bontcheva (2014) Leon Derczynski and Kalina Bontcheva. 2014.  PHEME: Veracity in Digital Social Networks.  In Proceedings of the UMAP Workshops.
  * Derczynski et al. (2017) Leon Derczynski, Kalina Bontcheva, Maria Liakata, Rob Procter, Geraldine Wong Sak Hoi, and Arkaitz Zubiaga. 2017.  SemEval-2017 Task 8: RumourEval: Determining rumour veracity and support for rumours.  In Proceedings of SemEval. ACL.
  * Enayet and El-Beltagy (2017) Omar Enayet and Samhaa R. El-Beltagy. 2017.  NileTMRG at SemEval-2017 Task 8: Determining Rumour and Veracity Support for Rumours on Twitter.  In Proceedings of SemEval. ACL.
  * García Lozano et al. (2017) Marianela García Lozano, Hanna Lilja, Edward Tjörnhammar, and Maja Maja Karasalo. 2017.  Mama Edha at SemEval-2017 Task 8: Stance Classification with CNN and Rules.  In Proceedings of SemEval. ACL.
  * Guo and Diab (2012) Weiwei Guo and Mona Diab. 2012.  Modeling sentences in the latent space.  In Proceedings of ACL. Association for Computational Linguistics, pages 864–872.
  * Hall et al. (2009) Mark Hall, Eibe Frank, Geoffrey Holmes, Bernhard Pfahringer, Peter Reutemann, and Ian H Witten. 2009.  The weka data mining software: an update.  ACM SIGKDD explorations newsletter 11(1):10–18.
  * Hamidian and Diab (2015) Sardar Hamidian and Mona T Diab. 2015.  Rumor Detection and Classification for Twitter Data.  In Proceedings of SOTICS.
  * Hamidian and Diab (2016) Sardar Hamidian and Mona T Diab. 2016.  Rumor identification and belief investigation on twitter.  In Proceedings of NAACL-HLT. pages 3–8.
  * Kochkina et al. (2017) Elena Kochkina, Maria Liakata, and Isabelle Augenstein. 2017.  Turing at SemEval-2017 Task 8: Sequential Approach to Rumour Stance Classification with Branch-LSTM.  In Proceedings of SemEval.
  * Liang (2005) Percy Liang. 2005.  Semi-supervised learning for natural language.  Ph.D. thesis, Massachusetts Institute of Technology.
  * Liu et al. (2015) Xiaomo Liu, Armineh Nourbakhsh, Quanzhi Li, Rui Fang, and Sameena Shah. 2015.  Real-time rumor debunking on twitter.  In Proceedings of CIKM. ACM, pages 1867–1870.
  * Lukasik et al. (2015) Michal Lukasik, Trevor Cohn, and Kalina Bontcheva. 2015.  Classifying tweet level judgements of rumours in social media.  arXiv preprint arXiv:1506.00468 .
  * Lukasik et al. (2016) Michal Lukasik, P. K. Srijith, Duy Vu, Kalina Bontcheva, Arkaitz Zubiaga, and Trevor Cohn. 2016.  Hawkes processes for continuous time sequence classification: an application to rumour stance classification in twitter.  In Proceedings of the 54th Meeting of the Association for Computational Linguistics. Association for Computer Linguistics, pages 393–398.
  * Mendoza et al. (2010) Marcelo Mendoza, Barbara Poblete, and Carlos Castillo. 2010.  Twitter under crisis: can we trust what we rt?  In Proceedings of the workshop on social media analytics. ACM, pages 71–79.
  * Mikolov et al. (2013) Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. 2013.  Distributed representations of words and phrases and their compositionality.  In Advances in neural information processing systems. pages 3111–3119.
  * Mohammad et al. (2016) Saif M Mohammad, Svetlana Kiritchenko, Parinaz Sobhani, Xiaodan Zhu, and Colin Cherry. 2016.  Semeval-2016 task 6: Detecting stance in tweets.  Proceedings of SemEval 16\.
  * Procter et al. (2013) Rob Procter, Jeremy Crump, Susanne Karstedt, Alex Voss, and Marta Cantijoch. 2013\.  Reading the riots: What were the police doing on twitter?  Policing and society 23(4):413–436.
  * Qazvinian et al. (2011) Vahed Qazvinian, Emily Rosengren, Dragomir R Radev, and Qiaozhu Mei. 2011.  Rumor has it: Identifying misinformation in microblogs.  In Proceedings of EMNLP. pages 1589–1599.
  * Singh et al. (2017) Vikram Singh, Sunny Narayan, Md Shad Akhtar, Asif Ekbal, and Pushpak Bhattacharya. 2017.  IITP at SemEval-2017 Task 8: A Supervised Approach for Rumour Evaluation.  In Proceedings of SemEval.
  * Socher et al. (2013) Richard Socher, Alex Perelygin, Jean Y Wu, Jason Chuang, Christopher D Manning, Andrew Y Ng, and Christopher Potts. 2013.  Recursive deep models for semantic compositionality over a sentiment treebank.  In Proceedings of EMNLP. volume 1631, page 1642.
  * Srivastava et al. (2017) Ankit Srivastava, Rehm Rehm, and Julian Moreno Schneider. 2017.  DFKI-DKT at SemEval-2017 Task 8: Rumour Detection and Classification using Cascading Heuristics.  In Proceedings of SemEval. ACL.
  * Tausczik and Pennebaker (2010) Yla R Tausczik and James W Pennebaker. 2010.  The psychological meaning of words: Liwc and computerized text analysis methods.  Journal of language and social psychology 29(1):24–54.
  * Tolmie et al. (2015) Peter Tolmie, Rob Procter, Mark Rouncefield, Maria Liakata, and Arkaitz Zubiaga. 2015.  Microblog analysis as a programme of work.  arXiv preprint arXiv:1511.03193 .
  * Wang et al. (2017) Feixiang Wang, Man Lan, and Yuanbin Wu. 2017.  ECNU at SemEval-2017 Task 8: Rumour Evaluation Using Effective Features and Supervised Ensemble Models.  In Proceedings of SemEval. ACL.
  * Zeng et al. (2016) Li Zeng, Kate Starbird, and Emma S Spiro. 2016.  #unconfirmed: Classifying rumor stance in crisis-related social media messages.  In Proceedings of ICWSM.
  * Zubiaga et al. (2016a) Arkaitz Zubiaga, Elena Kochkina, Maria Liakata, Rob Procter, and Michal Lukasik. 2016a.  Stance classification in rumours as a sequential task exploiting the tree structure of social media conversations.  In Proceedings of COLING.
  * Zubiaga et al. (2016b) Arkaitz Zubiaga, Maria Liakata, Rob Procter, Geraldine Wong Sak Hoi, and Peter Tolmie. 2016b.  Analysing how people orient to and spread rumours in social media by looking at conversational threads.  PLoS ONE 11(3):1–29.  https://doi.org/10.1371/journal.pone.0150989.


# A Note on the ”Various Atmospheres over Water Oceans on Terrestrial Planets”
with a One-Dimensional Radiative-Convective Equilibrium Model

Tetsuya Hara<EMAIL_ADDRESS>Department of Physics, Kyoto Sangyo
University, Kyoto 603-8555, Japan  Anna Suzuki<EMAIL_ADDRESS>Department of Physics, Kyoto Sangyo University, Kyoto 603-8555, Japan
Masayoshi Kiguchi<EMAIL_ADDRESS>Kindai University Research
Institute for Science and Technology, HigashiOsaka 577-8502, Japan  Akika
Nakamichi<EMAIL_ADDRESS>Institute of General Education, Kyoto
Sangyo University, Kyoto 603-8555, Japan

###### Abstract

It has been investigated the possibility of the various atmospheres over water
oceans. We have considered the H2 atmosphere and He atmosphere concerning to
N2 atmosphere over oceans. One of the main subjects in astrobiology is to
estimate the habitable zone. If there is an ocean on the planet with an
atmosphere, there is an upper limit to the outgoing infrared radiation called
the Komabayashi-Ingersoll limit (KI-limit). This limit depends on the
components of the atmospheres. We have investigated this dependence under the
simple model, using the one-dimensional gray radiative-convective equilibrium
model adopted by Nakajima et al. (1992). The outgoing infrared radiation
($F_{IRout}$) with the surface temperature ($T_{s}$) has shown some peculiar
behavior. The examples for H2, He, and N2 background gas for H2O vapour are
investigated. There is another limit called the Simpson-Nakajima limit (SN-
limit) mainly composed of vapour. This steam limit does not depend on the
background atmosphere components. Under super-Earth case ($g=2\times$9.8
m/s2), several cases are also calculated. The KI-limit dependence on the
initial pressure is presented. The various emission rates by Koll & Cronin
(2019) are investigated.

Keywords: Hycean Worlds $-$ Exoplanets atmospheres $-$ H2 atmospheres $-$ He
atmospheres $-$ Komabayashi-Ingersoll limit $-$ Simpson-Nakajima limit.

##  I Introduction

It has been pointed that hydrogen could play an important role in the early
stages of terrestrial planet history, when the disk gas is almost hydrogen and
He at first (Sekiya, Nakazawa, & Hayashi 1981; Pierrehumbert & Gaidos 2011;
Koll & Cronin 2019). It has been made-up word ’Hycean’ referred to H2 rich
atmospheres over the massive oceans (Madhusdhan et al. 2021). As hydrogen is
light, it is said to escape to outer space (Sekiya, Nakazawa & Hayashi 1980),
and it will take a long time to decrease to He dominant atmosphere (Hu et al.
2015).

Here we investigate cases of background gas with H2 atmospheres over oceans.
We also investigate cases with He atmospheres over oceans as well as N2
atmospheres over oceans. It is taken the One-Dimensional Radiative-Convective
Model adopted by Nakajima et al. (1992) for the gray atmosphere, which has
been followed and investigated by many researchers (eg. Bressler & Shaviv
2015). The atmosphere is assumed to consist of the non-condensible component
(mainly N2, or H2, He) and condensible component (eg. H2O). For the
stratosphere, radiative equilibrium is assumed. About the radiative transfer,
the atmosphere is considered to be transparent to solar radiation in the
optical range. In the infrared range, the absorption coefficient is taken to
be constant and independent of wavelength which is said to be a gray
atmosphere. The radiation transfer equation is integrated by using the
Eddington approximation.

There are two important limits such as ”Komabayashi-Ingersoll limit” (KI-
limit; Komabayashi 1967; Ingersoll 1969) which is called ”dilute limit” by
Koll & Cronin (2019) and ”Simpson-Nakajima limit” (SN-limit; Simpson 1927;
Nakajima et al. 1992; Goldblatt et al. 2013) which is called ”steam limit” by
Koll & Cronin (2019). KI-limit is the upper limit for the emission from the
top of the atmosphere in the Infrared radiation ($F_{IRtop})$ which depends on
the component of the atmospheres and gravity.

SN-limit is the approximation limit of $F_{IRtop}$ for H2O atmosphere, which
becomes the dominant component of the atmosphere in the high approximate
temperature ($T\geq 500\sim$ 600 K), so it does not depend on the component of
the background atmosphere. SN-limit depends only on the gravity in this paper
for $\simeq$ 294 W/m2 for $g$= 9.8m/s2 and $\simeq$340 W/m2 for $g=2\times$9.8
m/s2, respectively.

For H2 background atmospheres, it is characteristic that $F_{IRtop}$ increases
with surface temperature $T_{s}$ at the first time, however, it seems to reach
the upper limit, then decrease down and again increase to reach the second
saturated limit denoted as the SN-limit (Goldblatt & Watson 2012). This
feature is noticed by Suzuki (2017) and is named as ”Souffl$\acute{e}$ Effect”
by Koll & Cronin (2019). One of the purpose of this work is to investigate the
various limits and mechanism of these features.

We calculate cases of the surface accelerating gravity $g$=9.8 m/s2 and 2
$\times$ 9.8 m/s2. The latter corresponds to almost ten times of Earth mass,
called super-Earth (Madhusudhan et al. 2021). Although it is pointed out that
molecular hydrogen interacts strongly with infrared radiation via collision-
induced absorption (CIA) in the high pressure situation (Wordsworth &
Pierrehumbert 2013), we have not considered such effects for H2 and N2 here.
We assume that background gases except H2O are transparent.

The only assumed different part among H2, He and N2 is the mean molecular
weight $\bar{m}$ of the background gas, given by

$\tilde{m}=m_{n}x_{n}+m_{v}x_{v}.$ (1)

The parameter $m_{n}$ and $m_{v}$ are the molecular (or atomic) weight of the
non-condensible and condensable (vapour) components, respectively. And the
parameters $x_{n}$ and $x_{v}$ are the mole fractions of the non-condensible
and condensible components, respectively.

In Sect. II, the method of the model is outlined. It is applied to H2
atmosphere in Sect. III and applied to He atmosphere in Sect. IV. In Sect. V,
the KI-limit dependence on the initial pressure is investigated. The results
and discussion are deployed in Sect VI..

##  II One-Dimensional Radiative-Convective Model

We have followed the method of Nakajima et al. (1992) for the One-Dimensional
Radiative-Convective Model. Under the tropopause, the adiabatic lapse rate is
adopted where water vapour is saturated. It is assumed that the saturation
water vapour pressure $p^{*}$ is derived under the Clausius-Clapeyron
relationship and given by

$p^{*}(T)=p^{*}_{o}\exp\left(-\frac{l}{RT}\right),$ (2)

where $T$ and $R$ are temperature, the gas constant, respectively. The $l$,
and $p^{*}_{o}$ are the latent heat of condensible component ($l=43655$J mol-1
), and the constant for water saturation curve ($p^{*}_{o}=1.4\times 10^{11}$
Pa).

For the troposphere, it is assumed to be pseudoadiabatic lapse rate for the
temperature gradient with pressure as

$\left(\frac{\partial T}{\partial
p}\right)=\frac{\frac{RT}{pc_{pn}}+\frac{x_{v}^{*}}{x_{n}}\frac{l}{pc_{pn}}}{x_{n}+x_{v}^{*}\frac{c_{pv}}{c_{pn}}+\frac{x_{v}^{*}}{x_{n}}\frac{l^{2}}{RT^{2}c_{pn}}},$
(3)

where $c_{pv}$, and $c_{pn}$ are the mole specific heat at constant pressure
of condensable and non-condensible components, respectively.

Nakajima et al. have assumed that the molecular weights are the same for
condensible and non-condensible substances for simplicity. It is checked that
the above equation (3) is applicable to the different molecular weights and
the derivation of the above equation is commented in Appendix A.

Given the non-condensible pressure $p_{n\ 0}$ and temperature at the surface
$T_{s}$, $T(p)$ is obtained up to the tropopause of the atmosphere. There
appears to be a height where the net convergence becomes positive in the upper
levels of the atmosphere. The position of the tropopause is taken to be there.

###  II.1 Upper limit (KI-limit)

One of the main problems in astrobiology is to find out the habitable zone
where liquid water could exist. The injection from the central emitting object
(Sun) increases when the planet comes near to the center. On the other hand,
there is an upper limit that the planet could emit radiation. It depends on
the atmospheric component and this paper investigates of the dependence under
a simple model.

The optical depth $\tau$ is defined as

$\tau=\int_{z}^{\infty}\kappa\rho dz.$ (4)

The only assumed different part of H2 and He from N2 is the mean molecular
weight of the background gas. The different point is the following equation
about the optical depth $\tau$ (Nakajima et al. 1992)

$d\tau=(\kappa_{v}x_{v}m_{v}+\kappa_{n}x_{n}m_{n})\frac{dp}{\bar{m}g},$ (5)

where $p$ is the pressure and $g$ is the acceleration of gravity. As we assume
$\kappa_{n}$ =0, the effective factor $\bar{m}/(x_{v}m_{v})$ for gravity has
changed due to the mole fraction, and mean molecular weight.

In the stratosphere where $T$ and $x_{n}$ are almost constant, the integration
of Eq. (5) becomes

$\tau=(\kappa_{v}x_{v}m_{v}+\kappa_{n}x_{n}m_{n})\frac{p}{\bar{m}g}.$ (6)

The stratosphere is in a radiative equilibrium and the temperature structure
is given as

$\pi B=\frac{1}{2}F_{IRtop}\left(1+\frac{3}{2}\tau\right),$ (7)

where $B=\sigma T^{4}/\pi$ is the blackbody radiation intensity. At the
tropopause, we have

$\frac{1}{2}F_{IRtop}\left(1+\frac{3}{2}\tau_{tp}\right)=\sigma T_{tp}^{4},$
(8)

and

$\tau_{tp}=\kappa_{v}p^{\ast}(T_{tp})\frac{1}{g}\frac{m_{v}}{\bar{m}}.$ (9)

There is a maximum value of $F_{IRtop}(=F_{KI-limit})$ which will satisfy the
above two equations which is dnoted Komabayashi-Ingersoll limit (Nakajima et
al. 1992).

##  III Applied to H2 background atmosphere

It is investigated the gray calculations for cases of background gases with H2
for $g=9.8$m/s2. The relationships between $T_{s}$ and $F_{IRtop}$ for several
initial $p_{H2\ 0}$ cases are presented in Fig. 1.

Figure 1: The relationships between $T_{s}$ and $F_{IRtop}$ for several
initial $p_{H2\ 0}(=10^{4},10^{5},10^{6},2\times 10^{6},5\times 10^{6},$ and
$10^{7}$ Pa) cases of H2.

For H2 background atmosphere, it is characteristic that $F_{IRtop}$ increases
with surface temperature $T_{s}$ at the first time, however, it seems to reach
the upper limit, then decrease down and again increase to reach the second
saturated limit denoted as the SN-limit (Goldblatt & Watson 2012).

The first upper limit is due to the dominant H2 background atmosphere, where
the optical depth is unity at the gas temperature $T\simeq 260\sim 280$ K as
shown in Figs. 2 and 3 where the case of $P_{H2\ 0}=10^{6}$ Pa is presented.
The decrease of $F_{IRtop}$ after the first upper limit is due to the
temperature decrease of the gas temperature corresponding to the region where
the optical depth is about unity (later we call it as unity optical depth).
After the lower limit of $F_{IRtop}$, it increases due to the temperature
increase of the gas temperature corresponding to the unity optical depth which
is shown in Fig. 2 and magnified in Fig. 3, where the detailed feature could
be seen. The $F_{IRtop}$ then increases to the SN-limit as shown in Fig. 1.

###  III.1 Moist tropospheric asymptotic limit (SN-limit)

There is a limit of $F_{IRtop}$ that the initial $p_{0}$ of the background
atmosphere is small or null where the atmosphere is consisted almost only of
saturated H2O vapour. This is called the ”Moist tropospheric asymptotic
limit”, ”Saturated vapour limit”, ”SN-limit” (Goldblatt & Watson 2012), or
”steam limit” (Koll & Cronin 2019). The second upper limit is the same with
the only water vapour atmosphere, being the case of saturated vapour pressure
($\simeq$ 294 W/m2 for $g=9.8$m/s2 and $\simeq$ 340 W/m2 for super-Earth in
Fig. 10).

When the surface temperature is enough high, the mole fraction of water vapour
is almost unity and optical depth increases. When the optical depth is smaller
than unity, the outgoing flux comes from the bottom of the Hydrogen
atmosphere. However, when optical depth is greater than unity, the outgoing
flux comes from the place where the optical depth is almost unity from the top
of the atmosphere ( UOD: unity optical depth). Then the outgoing flux is
almost the same as that of a pure H2O atmosphere.

Figure 2: The relationships between $T_{s}$ and optical depth $(\tau)$ of
$p_{H2\ 0}=10^{6}$ Pa for several $T_{s}$ cases.

The relationships between surface temperature $T_{s}$ and optical depth
$(\tau)$ of $p_{H2\ 0}=10^{6}$ Pa for several $T_{s}$ cases are presented in
Fig. 2. The cases of $T_{s}=250$K, 300K, 350K, 400K, 450K, 500K, 550K, and
600K are presented by red, dark-red, brown, orange, purple, light-blue, blue,
and green colour curve, respectively, The unit optical depth ($\tau=1$) is
also presented by the red line and the magnified figure is presented in Fig.
3.

Figure 3: The relationships between $T_{s}$ and optical depth $(\tau)$ of
$p_{H2\ 0}=10^{6}$ Pa for several $T_{s}$ cases are presented. It is the
magnified figure of Fig. 2 to explain the change of $F_{IRtop}$ for different
of $T_{s}$.  Figure 4: The relationships between the temperature of UOD (Unity
optical depth) and $T_{s}$ of $p_{H_{2}\ 0}=10^{6}$ and $10^{5}$ Pa are
presented by blue and red curves, respectively. The cases for N2 atmospheres
of $p_{N_{2}\ 0}=10^{6}$ and $10^{5}$ Pa are also presented for reference by
light blue and orange curves, respectively.

The luminosity is mainly radiated at around $\tau\simeq 1$ and the intensity
is related to the local temperature of $\tau\simeq 1$. The first bump of
$p_{H2\ 0}=10^{6}$ in Fig. 1 is appeared at $T_{s}=300\sim 350$K and then the
luminosity decreases around $T_{s}=400\sim 450$K and then increases around
$T_{s}=500\sim 600$K. The detailed features could be seen in Fig. 3 as the
line of $\tau=1$ is crossed by each $T_{s}$ curve.

The features could be understood in Fig. 4, where the relationship between the
temperature at UOD (Unity optical depth) and $T_{s}$ of $p_{H_{2}\ 0}=10^{6}$
Pa is presented by blue curves. The luminosity decreases around $T_{s}=400\sim
450$K and then increases around $T_{s}=500\sim 600$K. The case of $p_{H_{2}\
0}=10^{5}$ Pa is shown there by red curve. The cases for N2 atmospheres of
$p_{N_{2}\ 0}=10^{6}$ and $10^{5}$ Pa are also presented for reference by
light blue and orange curves, respectively. For the case that the total
optical depth is smaller than unity, the temperature of UOD is assumed to be
equal to $T_{s}$.

Figure 5: The relationships between the height of UOD and $T_{s}$ of
$p_{H_{2}\ 0}=10^{6}$ and $10^{5}$ Pa are presented by blue and red curves,
respectively. The cases for N2 atmospheres of $p_{N_{2}\ 0}=10^{6}$ and
$10^{5}$ Pa are also presented for reference by light blue and orange curves,
respectively. The height is zero for the case that the total optical depth is
smaller than unity.

In Fig. 5, the relationships between the height of UOD and $T_{s}$ of
$p_{H_{2}\ 0}=10^{6}$ is presented by blue and curve. The height of UOD has
increased to $\sim 350$ km, where the temperature of UOD has decreased to
$\sim$ 260 K (see Fig. 4). The case of $p_{H_{2}\ 0}=10^{5}$ Pa is shown there
by red curve. The cases for N2 atmospheres of $p_{N_{2}\ 0}=10^{6}$ and
$10^{5}$ Pa are also presented for reference by light blue and orange curves,
respectively. The height of UOD is taken to be zero for the case that the
total optical depth is smaller than unity.

In the low temperature limit, the lapse rate is assumed to be the dry adiabat,
$dT/dz=-g/c_{p}$, as described in Pierrehumbert (2010) and Appendix B

$H_{T}^{dilute}=\frac{c_{p}T}{g}\propto 1/M,$ (10)

where $c_{p}$ is the specific heat per unit mass and proportional to the
inverse of the molecular weight $M$. In Appendix B, it is investigated the
analytical expression derived by Koll & Cronin (2019) for the OLR limit values
of the steam limit and dilute limit. In Appendix C, it is tried to derive Eq.
(33) in Koll & Cronin (2019) for suspected typos.

It should be noted that at $T_{s}\sim$ 400 K for $p_{H_{2}\ 0}=10^{6}$ Pa the
height of UOD has increased to $H_{T}^{UOD}\simeq 350$ km in H2, whereas
$H_{T}^{UOD}\simeq 40$ km in N2. The main difference must be due to the low
molecular weight of H2 relative to N2.

The increase of the UOD height is related to the features that $F_{IRtop}$
increases with surface temperature $T_{s}$ at first time then decreases down
and again increase to reach the second saturated limit. Thr first $F_{IRtop}$
increase is due to the increase of $T_{s}$ and the lower value of the water
vapoure (Figs. 6 and 7). The $F_{IRtop}$ upper limit is due to the KI-limit.
The $F_{IRtop}$ decrease is due to the increase of the water vapour and the
optical depth (Figs. 6, 7, and 8). The UOD height increase is related to the
decrease the UOD temperature (Figs. 4 and 5). The second saturated limit is
due to the increase and saturation limit of the water vapoure (Figs. 4
$\sim$8).

Figure 6: The relationships between temperature and pressure for several
$T_{s}$ cases of $p_{H_{2}\ 0}=10^{6}$ Pa.

The relationships between temperature and pressure of $p_{H2\ 0}=10^{6}$ Pa
for several $T_{s}$ cases are calculated in Fig. 6 and the relationships
between mole fraction and pressure are shown in Fig. 7. In Fig. 8, the
relationships between mole fraction and optical depth are presented.

Figure 7: The relationships between the mole fraction of the water vapour and
the pressure of $p_{H_{2}\ 0}=10^{6}$ Pa for several $T_{s}$ cases.  Figure 8:
The relationships between the mole fraction and the optical depth for several
$T_{s}$ cases of $p_{H_{2}\ 0}=10^{6}$ Pa.

For H2 atmosphere, optical depth $\tau$ at tropopause is given by Eq. (9)
where the factor $m_{v}/\bar{m}$ increases about 10 from N2 to H2 atmosphere
which causes the decrease of KI-limit ($\simeq$ 420 W/m2 (N2) to $\simeq$ 257
W/m2 (H2), as shown in Fig. 9).

The difference between N2 and H2 background atmosphere is the difference of
the mean molecular weight which corresponds to the change of the optical
depth. For H2 atmosphere, the optical depth has increased. Even for the same
surface temperature, the temperature at optical depth $\tau=1$ decreases for
H2 atmosphere. Then the outgoing radiation has decreased for the H2 atmosphere
compared to the N2 atmosphere.

If H2 pressure increases, the outline of the graph has moved to the right-hand
which is shown in Fig. 1. If the surface temperature increased to the right-
hand direction, there is a possibility that the ocean would evaporate.

The relationships between $T_{s}$ and $F_{IRtop}$ for several $p_{H_{2}\ 0}$
and $p_{N_{2}\ 0}$ cases are presented in Fig. 9. Some H2 results in Fig. 1
are included in Fig. 9. The cases of N2 are almost the same given in Nakajima
et al. 1992, except that the molecular weight of N2 is taken 28 (it is taken
18 in Nakajima et al. (1992) for its simplicity), then KI-limit has increased
from $\simeq$ 385 W/m2 to $\simeq$ 420 W/m2.

The cases of H2 for super-Earth ($g=9.8\times 2$ m/s2) are shown in Fig. 10.

Figure 9: It is compared the gray calculations for cases of background gases
with H2 and N2 for $g=9.8$m/s2. The relationships between $T_{s}$ and
$F_{IRtop}$ for several initial $p_{H2\ 0}$ and $p_{N2\ 0}$ cases are
presented. Some H2 results in Fig. 1 are included. The arrow of $P_{H2O}$
shows the pure H2O case and SN-limit ($\simeq$ 295 W/m2). Figure 10: For
super-Earth cases with H2 atmosphere, the relationships between $T_{s}$ and
$F_{IRtop}$ for several initial $p_{H2\ 0}$ cases are presented.  Figure 11:
For super-Earth cases of background gases with H2 and N2, the relationships
between $T_{s}$ and $F_{IRtop}$ in several initial $p_{H2\ 0}$ cases are
presented. The arrow of $P_{H2O}$ shows the case of $p_{H2\ 0}=0$ Pa which
means only H2O atmosphere component corresponding to SN-limit.

In Fig. 11, super-Earth cases of background gases with H2 and N2 are presented
for the relationships between $T_{s}$ and $F_{IRtop}$ in several initial
$p_{H2\ 0}$ cases. The arrow of $P_{H2O}$ shows the case of $p_{H2\ 0}=0$ Pa
which means only H2O atmosphere component corresponding to the SN-limit.

##  IV Applied to ${\rm He}$ background atmosphere

The situations of He background atmosphere are similar to the case of H2
background atmosphere except the treatment of the mean molecular weight and
the specific heat of He as $c_{p}=2.5$ R, being mono atomic gas, compared to
that of H2, as $c_{p}=3.5$ R, being dipole atomic gas.

Figure 12: For He background gases of $g=9.8$m/s2, the relationships between
$T_{s}$ and $F_{IRtop}$ for several initial $p_{He\ 0}$ cases are presented.
The value of KI-limit for He atmosphere is $\simeq$ 291 W/m2 which is almost
the same value of the SN-limit $\simeq$ 294 W/m2.

In Fig. 12, the results of He atmosphere are similar to those of H2 cases in
Fig. 1. The first upper limit of $p_{He\ 0}=10^{6}$ and $10^{7}$ are the KI-
limit of He atmosphere. Then they decrease down and increase approximately to
the SN-limit as the cases of H2 background gas as presented in Fig. 1.

It must be noticed that the KI-limit for He atmosphere is $\simeq$ 291 W/m2
which is almost the same value of the SN-limit $\simeq$ 294 W/m2. It seems to
be a problem that the SN-limit is greater than the KI-limit, however it is not
a problem because the components of the background atmospheres have changed
from He to H2O. It is almost the same for H2 atmosphere cases in Fig. 1.

It is pointed out by Koll & Cronin (2019) that H2 is the only background gas
for which the dilute runaway (KI-limit) lies below the steam limit (SN-limit).
However as shown in Fig 12, He is also the background gas for which the dilute
limit lies below the steam limit.

Figure 13: The relationships between $T_{s}$ and optical depth $(\tau)$ for
several $T_{s}$ cases of $p_{He\ 0}=10^{6}$ Pa are presented.

The relationships between $T_{s}$ and optical depth $(\tau)$ of $p_{He\
0}=10^{6}$ Pa for several $T_{s}$ cases are presented in Fig. 13. The $T_{s}$
=250K, 300K, 350K, 400K, 450K, 500K, 550K, and 600K are presented by red,
dark-red, brown, orange, purple, light-blue, blue, and green colour curves,
respectively, The unit optical depth ($\tau=1$) is also presented in red line.

Figure 14: The relationships between $T_{s}$ and optical depth $(\tau)$ of
$p_{He\ 0}=10^{6}$ Pa for several $T_{s}$ cases. It is the magnified figure of
Fig. 13.

Fig. 14 is the magnified figure of Fig. 13 to explain the change of
$F_{IRtop}$ for different of $T_{s}$. The luminosity is mainly radiated at
around $\tau\simeq 1$ and the intensity is related to the local temperature of
$\tau\simeq 1$. The first bump of $P_{He\ 0}=10^{6}$ Pa in Fig. 12 is appeared
at $T_{s}\simeq 300\sim 350$ K and then the luminosity decreases around
$T_{s}\simeq 400\sim 450$ K and then increases around $T_{s}\simeq 500\sim
600$ K.

Figure 15: The relationships between the temperature of UOD (Unity optical
depth) and $T_{s}$ of $p_{He\ 0}=10^{6}$ and $10^{5}$ Pa.

The features could be understood in Fig. 15, where the relationship between
the temperature at UOD (Unity optical depth) and $T_{s}$ of $p_{He\ 0}=10^{6}$
Pa is presented by dark blue curves. The luminosity decreases around
$T_{s}=400\sim 450$K and then increases around $T_{s}=500\sim 600$K. The case
of $p_{He\ 0}=10^{5}$ Pa is shown there by dark red curve. The cases for H2
atmospheres of $p_{H_{2}\ 0}=10^{6}$ and $10^{5}$ Pa are also presented by
blue and red curves, respectively. The cases for N2 atmospheres of $p_{N_{2}\
0}=10^{6}$ and $10^{5}$ Pa are also presented for reference by light blue and
orange curves, respectively. For the case that the total optical depth is
smaller than unity, the temperature of UOD is assumed to be equal to $T_{s}$.

In Fig. 16, the relationships between the height of UOD and $T_{s}$ of $p_{He\
0}=10^{6}$ is presented by dark blue curve. The height of UOD has increased to
$\sim 250$ km, where the temperature of UOD has decreased to $\sim$ 270 K (see
Fig. 15). The case of $p_{He\ 0}=10^{5}$ Pa is shown there by dark red curve.
The cases for H2 atmospheres of $p_{N_{2}\ 0}=10^{6}$ and $10^{5}$ Pa are also
presented by blue and red curves, respectively. The cases for N2 atmospheres
of $p_{N_{2}\ 0}=10^{6}$ and $10^{5}$ Pa are also presented for reference by
light blue and orange curves, respectively. The height of UOD is taken to be
zero for the case that the total optical depth is smaller than unity.

The relationships between temperature and pressure of $p_{He\ 0}=10^{6}$ Pa
for several $T_{s}$ cases are presented in Fig. 17, and the relationships
between mole fraction and pressure are shown in Fig. 18 The relationships
between mole fraction and optical depth are calculated in Fig. 19.

Figure 16: The relationships between the height of UOD and $T_{s}$ of $p_{He\
0}=10^{6}$ and $10^{5}$ Pa are presented by dark blue and dark red curves,
respectively. The cases for H2 atmospheres of $p_{H_{2}\ 0}=10^{6}$ and
$10^{5}$ Pa and the cases for N2 atmospheres of $p_{N_{2}\ 0}=10^{6}$ and
$10^{5}$ Pa are also presented for reference. Figure 17: The relationships
between temperature and pressure of $p_{He\ 0}=10^{6}$ Pa for several $T_{s}$
cases with He.  Figure 18: The relationships between mole fraction and
pressure of $p_{He\ 0}=10^{6}$ Pa for several $T_{s}$ cases with He Figure 19:
For He in $g=9.8m/s^{2}$, the relationships between mole fraction and optical
depth of $p_{He0}=10^{6}$ Pa for several $T_{s}$ cases are presented.  Figure
20: The relationships between $T_{s}$ and $F_{IRtop}$ for several $p_{He\ 0}$
and $p_{N2\ 0}$ cases of background gases with He and N2.

To compare the cases of N2, the results of N2 and He are presented in Fig. 20
for $g$=9.8m/s${}^{2}.$ The relationships between $T_{s}$ and $F_{IRtop}$ for
several $p_{He\ 0}$ and $p_{N2\ 0}$ cases are presented there. Several cases
of $p_{He\ 0}$ in Fig. 12 are also included here. The case of pure H2O
atmosphere is shown by black curve.

For super-Earth in $g=9.8\times 2$ m/s2, it is investigated the gray
calculations for cases of background gases with He in Fig 21. The
relationships between $T_{s}$ and $F_{IRtop}$ for several $p_{He\ 0}$ cases
are presented. The case of pure H2O atmosphere is shown by black curve. It is
noticed that KI and SN-limits are different from Earth cases in Fig. 12.

It is investigated the gray calculations for cases of background gases with N2
and He for super-Earth case in Fig. 22. The relationships between $T_{s}$ and
$F_{IRtop}$ for several $p_{N2\ 0}$ and $p_{He\ 0}$ cases are presented there.
Several cases of $p_{He\ 0}$ in Fig. 21 are also included here.

Figure 21: For super-Earth in $g=9.8\times 2$ m/s2, the relationships between
$T_{s}$ and $F_{IRtop}$ for several $p_{He\ 0}$ cases are presented. Figure
22: For super-Earth case of background gases with He and N2, the relationships
between $T_{s}$ and $F_{IRtop}$ for several initial $p_{He\ 0}$ and $p_{N2\
0}$ cases are presented.

##  V KI-limit dependence on the initial pressure

It is noticed that the KI-limit depends on the initial pressure of the
background component as well as $MW$ (molecular weight).

Figure 23: The relationship between $\tau_{tp}$ and $T_{tp}$ of He atmosphere
with $F_{IRtop}$ as a parameter for $p_{He\ 0}=10^{6}$ Pa.

In Fig. 23, the relationship between the optical depth $\tau_{tp}$ and the
temperature $T_{tp}$ is presented of He atmosphere for $F_{IRtop}$ as a
parameter with 250, 291, and 384 W/m2 in $p_{He\ 0}=10^{6}$ Pa. The coloured
curves represent Eq. (8) for $F_{IRtop}$ and the almost straight blue curve
represents Eq. (9), where the suffix ”$tp$” means the value at the tropopause
(Nakajima et al. 1992).

Figure 24: The relationship between $T_{tp}$ and $F_{IRtop}$ for $p_{He\
0}=10^{6}$ Pa. The KI-limit of this model for He atmosphere is $\simeq$ 291 W
m-2.

The relationship between $T_{tp}$ and $F_{IRtop}$ for $p_{He\ 0}=10^{6}$ Pa is
shown in Fig. 24. The KI-limit of this model in He atmosphere is $\simeq$ 291
W m-2 for $p_{He\ 0}\simeq 10^{6}$ Pa.

Figure 25: The relationship between $T_{s}$ and $F_{IRtop}$ for $p_{He\
0}\simeq 10^{3}$ Pa is presented. The KI-limit of this model for He atmosphere
is $\simeq$ 298 W m-2, which is different from $\simeq$ 291 W m-2 for $p_{He\
0}\simeq 10^{6}$ Pa.

In Fig. 25, the relationship between $T_{tp}$ and $F_{IRtop}$ for $p_{He\
0}=10^{3}$ Pa is presented where the KI-limit is $\simeq$ 298 W/m2, which is
different from $\simeq$ 291 W/m2 for $p_{He\ 0}=10^{6}$ Pa. The KI-limit of He
atmosphere is 291 $\sim$ 298 W/m2 for $p_{He\ 0}\simeq 10^{3}\sim 10^{6}$ Pa,
which is shown in Fig. 26 as $MW=4$ case for He. It shows that the KI-limit
depends on the initial pressure $p_{He\ 0}$ of the background gas.

Figure 26: The KI-limit dependence on molecular weight for fixed initial
pressure of the background gases. The relationships between molecular weight
and $F_{IRtop}$ for each initial pressure of the background gases is
presented. Molecular weight of H2, He, and N2 are shown as 2, 4, and 28,
respectively.

In Fig. 26, it is calculated the KI-limit dependence on molecular weight for
fixed initial pressure $p_{n\ 0}$ of the background gases. The relationships
between molecular weight and $F_{IRtop}$ for each initial pressure of the
background gases is presented. For example, in the case that the value of
molecular weight is 18 where the molecular weight of every molecule is the
same with water H2O, $F_{IRtop}$ is the same for any initial pressure $p_{n\
0}$. On the other hand, in the He background atmosphere the mean molecular
weight (MMW) for $p_{He\ 0}\simeq 10^{-2}\sim 10^{1}$ Pa, the KI-limit is
$\simeq 390$W/m2 which is almost the same of $MW=18$, because the dominant
component of the atmosphere is H2O. For $p_{He\ 0}\simeq 10^{4}\sim 10^{6}$
Pa, KI-limit of $MMW=4$ is $\simeq$ 291 W/m2 where the dominant component of
the atmosphere is He. Even for fixed molecular weight of the background
atmosphere, $MMW$ changes due to the increase of H2O compared to the initial
component pressure. The KI-limit depends on the initial component pressure
$p_{n\ 0}$.

About the saturation vapour pressure on temperature, we take Eq. (2), so the
results in Fig. 26 seem to be independent on any peculiar temperature.

##  VI Results and Discussion

We have presented the results of the outgoing radiation for the different
components of background gas under a simple model with gray approximation
adopted by Nakajima et al. 1992. The main results of the paper are the
following:

1\. It is studied the possibility of the various atmospheres over oceans.

2\. H2 atmospheres as well as He are investigated. To our concern, the
treating

He atmosphere in this style seems to be rather new.

3\. The strange features are found for H2 atmosphere as well as He that
$F_{IRtop}$ increases

to the first upper limit, then decreases and increase again to the second
limit :

named ”Souffl$\acute{e}$ effect” by Koll & Cronin (2019).

4\. The first upper limit is the Komabayashi-Ingersoll limit (KI-limit), which
is not stated

explicitly by Koll & Cronin (2019).

5\. The KI-limit value depends on the molecular weight and the initial
pressure of

the background atmosphere (Sect. V).

6\. The second limit is the Simpson-Nakajima limit (SN-limit) which is the
atmosphere

mainly composed by vapour, called ’steam limit’ (Koll & Cronin 2019).

7\. ”Souffl$\acute{e}$ effect” is analysed by taking the UOD (Unit optical
depth) temperature

which has decreased and the UOD height which has increased.

8\. H2 atmosphere as well as He is the non-condensible background component
where

the KI-limit is lower than the SN-limit.

9\. The mixed gas of the relative realistic mixed atmosphere (H${}_{2}\simeq
72\%\ \&$ He $\simeq 28\%)$

is investigated. The first upper limit is $\sim 267$W/m2 (Sect. VI).

10\. The various approximate limits derived by Koll & Cronin (2019) are
investigated

and considered its applicability (Appendix B).

From the above consideration, it becomes clear that it must be considered the
atmospheric components of the extra terrestrial planets if one want to
investigate ”Habitable Zone”. We would like to expect much more observations
further to find out the life trace in extra-terrestrial planets (Hill et al.
2022).

####  VI.0.1 Injection flux vs. Distance from a parent star

To apply the above results to the planet formation, it is better to consider
the solar constant (1364 W/m2) for Earth orbit where the gravitational
acceleration as 9.8m/s2. Taking the spherical mean for the injected flux
($\times$ 1/4), the flux becomes 341 W/m2.

As the solar luminosity is almost 70$\%$ of present value at 4.6 Gy ago
(Bahcall, Pinsonneault, & Basu 2001), the flux is $\simeq$ 239 W/m2. If we
take the albedo $\simeq$ 0.3 which is the corresponding value of Earth at
present, the solar injection flux decreases to $\simeq$ 167 W/m2.

The KI-limit for H2 atmosphere is $\simeq$ 257 W/m2 which is greater than the
above $\simeq$ 239 W/m2 and $\simeq$ 167 W/m2 values. Then it is stable for H2
atmosphere for assumed planets in Earth orbit. Even for super-Earth, the KI-
limit for H2 atmosphere is $\simeq$ 295 W/m2 which is greater than the above
$\simeq$ 239 W/m2and $\simeq$ 167 W/m2 values. Then it is stable for H2
atmosphere for assumed super-Earth planets in Earth orbit.

The KI-limit for He atmosphere is $\simeq$ 291 W/m2 which is greater than the
above $\simeq$ 239 W/m2 and $\simeq$ 167 W/m2 values. Then it is stable for He
atmosphere for the assumed planets in Earth orbit. Even for super-Earth, the
KI-limit for He atmosphere is $\simeq$ 328 W/m2 which is greater than the
above $\simeq$ 239 W/m2 and $\simeq$ 167 W/m2 values. Then it is stable for He
atmosphere for the assumed super-Earth planets in Earth orbit.

If the orbit has decreased to 0.5 Au, the injection flux increased four times
as $\simeq$ 956 (=239$\times$ 4) W/m2 and $\simeq$ 668 W/m2. Then it becomes
unstable for Earth and super-Earth planets, because the situations are over
the KI-limit and SN-limit. They will be in a greenhouse runaway and/or moist
greenhouse situation.

Figure 27: Injection flux vs. distance from a parent star.

In Fig. 27, the injection flux versus the distance from parent star is shown.
The star is taken for a G and an M host star by red and green line,
respectively. For an M-type star the luminosity is taken with 1.3 % of the Sun
(Pierrehumbert & Gaidos 2011). G star is normalized 167 W/m2 at 1AU. Three
dilute limits for H2 ($\simeq$ 257 W/m2), He ($\simeq$ 291 W/m2), and N2
($\simeq$ 420 W/m2) atmospheres and the steam limit ($\simeq$ 295 W/m2) are
presented for reference. The steam limit is shown by black.

Under this model, it is found that the OLR upper limit decreases in the H2
rich atmosphere for the decrease of the mean molecular weight. Then it becomes
clear that it is possible to become the runaway situation for super-Earth in
the distance smaller than 1AU from the G type star (Sun). Even if OLR has not
reached the upper limit, there is a possibility to become dry up the ocean due
to the high surface temperature of the greenhouse effect.

If there are extra planets with H2 atmosphere over ocean and a life such as
photosynthetic bacteria of the class Cyanobacteria (indicated by Pierrehumbert
& Gaidos, 2011), it must be an unstable situation for the chemical reaction
between H2 and O2. It could be a stable atmosphere with He and O2. We would
like to estimate the diffusion time of H2 through the initial atmosphere
mainly composed H2 and He (Sekiya, Nakazawa & Hayashi 1980; Wordworth 2012; Hu
et al. 2015; Pahlevan et al. 2022).

####  VI.0.2 Mixed background gases with H2 and He

Figure 28: It is investigated the gray calculations for mixed background gases
with H2 (weight X$\simeq 0.72$) and He (weight Y$\simeq 0.28)$ for
$g=9.8$m/s2.

In Fig. 28, the gray calculations for mixed background gases with H2 (weight
X$\simeq 0.72$) and He (weight Y$\simeq 0.28)$ for $g=9.8$m/s2 are presented.
This mixed gas is considered to be the primordial gas from which Sun and Earth
are formed. The relationships between $T_{s}$ and $F_{IRtop}$ for several
initial $p_{\ 0}(=10^{4},10^{5},10^{6},$ and $10^{7}$ Pa) cases for Earth are
shown. The first upper limit for this atmosphere is $\sim$ 267 W/m2), which
has increased from H2 of $\sim$ 257 W/m2 (see Fig. 1). The second limit (steam
limit) is the same of H2 and He as $\sim$ 294 W/m2.

####  VI.0.3 Various uncertainty

There are various uncertainty factors to estimate the relationship between the
surface temperature $T_{s}$ and outgoing infrared radiation at the top of the
atmosphere $F_{IRtop}$. Our results do not apply directly to any real planet
history because of large uncertainties in our calculation due to the absence
of clouds and the use of a one-dimensional model. In order to determine
quantitatively, it seems to be necessary to evaluate the parameters such as
albedo, effects of clouds, and relative humidity, circulation of the
atmosphere over the surface of the planet (Manabe & Wetherald 1967, Zsom et
al. 2013, Manabe & Broccoli 2020). It will be better to estimate the
distribution of those parameters for the greenhouse effects and under the
increase of solar luminosity.

a. gray opacity approximation

If one considers more than gray opacity approximation, one has to treat
opacity, including the line by line treatments, Lorentz factor, pressure
effect, random approximations, and further (Pierrehumbert 2010; Seager 2010),
which make the physical understanding to be complicated.

##  Appendix A Derivation of the equation (3) in relation to Pierrehumbert
(2010)

In this appendix, it is explained that the equation of (4) in Nakajima et al.
(1992) (Eq. (3) in this paper) is equivalent to the equation of (2.33) in
Pierrehumbert (2010). The notation is followed to Pierrehumbert (2010) and
equation number is shown as (Pie. 2.33). As stated before, it is checked for
the situation for the different molecular weights of condensible and non-
condensible substances (Nakajima et al. have assumed the same for simplicity).

The equation in (Pie. 2.33) is written as

$\frac{d\ {\rm ln}\ T}{d\ {\rm ln}\
p_{a}}=\frac{R_{a}}{c_{pa}}\frac{1+\frac{L}{R_{a}T}r_{sat}}{1+\left(\frac{c_{pc}}{c_{pa}}+\left(\frac{L}{R_{c}T}-1\right)\frac{L}{c_{pa}T}\right)r_{sat}},$
(A.1)

where subscript $a$ and $c$ are related to non-condensible and condensible
substance, respectively. Other notations are the following.

A partial pressure of non-condensible substance is described as $p_{a}$ which
is related to total pressure $p$ as $p=p_{a}+p_{c}$ where $p_{c}$ is the
partial pressure of condensible substance (water vapour). Then
$dp=dp_{a}+dp_{c}$. It is introduced saturation assumption that $p_{c}$ is
replaced by $p_{c,sat}$. Using Clausius-Clapeyron to re-write $dp_{c,sat}$
that

$\frac{dp_{c.sat}}{dT}=\frac{L}{R_{c}T^{2}}p_{c,sat},$ (A.2)

where $R_{c}$ is the gas constant for the substance which is condensing
(water) as shown in (Pie. 2.25) and $L$ is the latent heat associated with the
transportation to the more condensed phase.

Then $R_{a}d\ {\rm ln}\ p_{a}$ of the second term in (Pie. 2.32) becomes as

$R_{a}d\ {\rm ln}\
p_{a}=\frac{R_{a}}{p_{a}}(dp-\frac{Lp_{c,sat}dT}{R_{c}T^{2}}).$ (A.3)

So the second term in (Pie. 2.32) is written as

$\displaystyle-(1+\frac{L}{R_{a}T}r_{sat})\frac{R_{a}}{p_{a}}(dp-\frac{Lp_{c,sat}}{R_{c}T^{2}}dT)$
$\displaystyle=-(1+\frac{L}{R_{a}T}r_{sat})\frac{R_{a}}{p_{a}}dp+(1+\frac{L}{R_{a}T}r_{sat})\frac{R_{a}}{p_{a}}\frac{Lp_{c,sat}}{R_{c}T^{2}}dT.$

The term of $dT$ is moved to the first term of (Pie. 2.32).

Then the equation of (Pie. 2.32) becomes for taking $\partial Q=0$ as

$\left[{\tilde{c}}_{pa}+({\tilde{c}}_{pc}+(\frac{L}{R_{c}T}-1)\frac{L}{T})r_{sat}+(1+\frac{L}{R_{a}T}r_{sat})\frac{R_{a}Lp_{c,sat}}{p_{a}R_{c}T}\right]d\
{\rm ln}\ T$ (A.4)
$=\left[(1+\frac{L}{R_{a}T}r_{sat})\frac{R_{a}p}{p_{a}}\right]d\ {\rm ln}\ p.$
(A.5)

Then taking $r_{sat}=\epsilon p_{c,sat}/p_{a},\ \epsilon=M_{c}/M_{a},\
p_{a}=\rho_{a}RT/M_{a},\ p_{c}=p_{c,sat}=\rho_{c}RT/M_{c},\
x_{n}=p_{n}/p=p_{a}/p,$ and $x_{c}=p_{c}/p,\ R_{a}=R/M_{a},\ R_{c}=R/M_{c},\
{\it l}=LM_{c},\ c_{pa}={\tilde{c}}_{pa}M_{a},\ c_{pc}={\tilde{c}}_{pc}M_{c},$
where $M_{a}$ and $M_{c}$ are molecular weight of the substanc a (=n: non-
condensible background substance, eg. H2, He, N2, and air: suffix ’a’ comes
from Pierrehumbert and ’n’ comes from Nakajima et al.), and c (=v: condensible
substance, eg. water vapour), the above equations have changed to the
followings.

Using $(R_{a}Lp_{c})/(p_{a}R_{c}T)=(M_{c}p_{c}L)/(M_{a}p_{a}T)=(r_{sat}L)/T$,
the square bracket of Eq. (A. 4) becomes as

$\displaystyle\left[\cdots\right]$ $\displaystyle=$
$\displaystyle\left[{\tilde{c}}_{pa}+({\tilde{c}}_{pc}+(\frac{L}{R_{c}T}-1)\frac{L}{T})r_{sat}+(1+\frac{Lr_{sat}}{R_{a}T})\frac{LR_{a}p_{c,sat}}{p_{a}R_{c}T}\right]$
$\displaystyle=$
$\displaystyle\left[{\tilde{c}}_{pa}+({\tilde{c}}_{pc}+\frac{L}{R_{c}T}\frac{L}{T})r_{sat}-\frac{L}{T}r_{sat}+r_{sat}\frac{L}{T}+\frac{Lr_{sat}}{R_{a}T}\frac{r_{sat}L}{T}\right]$
$\displaystyle=$
$\displaystyle\left[{\tilde{c}}_{pa}+{\tilde{c}}_{pc}r_{sat}+\frac{l^{2}}{M_{c}^{2}}\frac{M_{c}}{R}\frac{1}{T^{2}}\frac{M_{c}}{M_{a}}\frac{p_{c}}{p_{a}}+\frac{l^{2}M_{a}}{M_{c}^{2}RT^{2}}\frac{M_{c}^{2}}{M_{a}^{2}}\frac{p_{c}^{2}}{p_{a}^{2}}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{M_{a}}\left[c_{pa}+c_{pc}\frac{x_{v}}{x_{n}}+\frac{l^{2}}{RT^{2}}\frac{x_{v}}{x_{n}}+\frac{l^{2}}{RT^{2}}\frac{x_{v}^{2}}{x_{n}^{2}}\right]$
$\displaystyle=$
$\displaystyle\frac{c_{pa}}{M_{a}x_{n}}\left[x_{n}+x_{v}\frac{c_{pc}}{c_{pa}}+\frac{x_{v}}{x_{n}}\frac{l^{2}}{RT^{2}c_{pa}}\right],$

where we use $x_{n}+x_{c}=1$.

The square bracket of Eq. (A. 5) becomes as

$\displaystyle\left[\cdots\right]$ $\displaystyle=$
$\displaystyle\frac{R_{a}}{x_{n}}\left[(1+\frac{Lr_{sat}}{R_{a}T})\right]$
$\displaystyle=$
$\displaystyle\frac{R_{a}}{x_{n}}\left[1+\frac{l}{RT}\frac{x_{c}}{x_{n}}\right].$

Considering $d\ {\rm ln}\ T/d\ {\rm ln}\ p=\frac{p}{T}\frac{\partial
T}{\partial p}$, the equation (Pie. 2.33) becomes as

$\left(\frac{\partial T}{\partial
p}\right)=\frac{\frac{RT}{pc_{pn}}+\frac{x_{v}^{*}}{x_{n}}\frac{l}{pc_{pn}}}{x_{n}+x_{v}^{*}\frac{c_{pc}}{c_{pn}}+\frac{x_{v}^{*}}{x_{n}}\frac{l^{2}}{RT^{2}c_{pn}}},$
(A.6)

where we neglect the differences between partial derivative and total
derivative, and take $x_{c}=x_{v}^{*}$ and $c_{pa}=c_{pn}$. When

Then the derivation of equation (4) in Nakajima et al. is equivalent to the
equation of (Pie. 2. 33).

It is also equivalent to the equations of (3.7) and (3.14) in Houghton (1977)

$dp=-g\rho_{a}(1+\xi)dz$ (A.7)

and

$-\frac{dT}{dz}=\frac{g}{{\tilde{c}}_{pn}}\frac{(1+Lp_{c,sat}M_{c}/pRT)(1+(p_{c,sat}\epsilon/p)}{1+(\epsilon
p_{c,sat}/p{\tilde{c}}_{pn})(\tilde{c}_{pc}+(dL/dT)-L/T)+(\epsilon
p_{c,sat}L^{2}M_{c}/{\tilde{c}}_{pn}pRT^{2})}.$ (A.8)

If the term $dL/dT-L/T$ in the denominator is neglected for its small values,
the above two equations becomes equivalent to the equation (3) in this paper
and equation of (4) in Nakajima et al. (1992).

##  Appendix B Trial to derive the limit values

###  B.1 MMW matters (Koll & Cronin 2019)

First, taking the water vapour scale height
$H_{v},\rho_{v}=\rho_{v,0}e^{-{z/H_{v}}}$, the optical depth is

$\tau=\int_{0}^{\infty}\kappa_{v}\rho_{v}dz=\kappa_{v}\rho_{v,0}\int_{0}^{\infty}e^{-z/H_{v}}dz=\frac{H_{v}}{l_{0}},$
(B.1)

where $l_{0}=1/(\kappa_{v}\rho_{v,0})$ (in general, the subscript $v$ will
denote quantities related to water vapour).

Using the ideal gas law for water vapour, $\rho_{v}=e^{*}/(R_{v}T)$ where
$e^{*}$ is the saturation vapour pressure, and the Clausius-Clapeyron
relation, $d$ ln $e^{*}/d$ ln $T=L_{v}/(R_{v}T)$ where $L_{v}$ is the latent
heat of vapourization:

$(-H_{v})^{-1}=\frac{1}{\rho_{v}}\frac{d\rho_{v}}{dz}=\frac{1}{e^{*}}\frac{de^{*}}{dz}-\frac{1}{T}\frac{dT}{dz}=\left(\frac{1}{e^{*}}\frac{de^{*}}{dT}-\frac{1}{T}\right)\frac{dT}{dz},$
(B.2)

and

$=\left(\frac{L_{v}}{R_{v}T}-1\right)\frac{1}{T}\frac{dT}{dz}\simeq\frac{L_{v}}{R_{v}T}\frac{1}{T}\frac{dT}{dz}=\frac{L_{v}}{R_{v}T}{H_{T}}^{-1},$
(B.3)

where $L_{v}/R_{v}T\gg 1$ is used.

Here it is denoted $H_{T}=(-1/T\times dT/dz)^{-1}$ as the temperature scale
height, which is related to the water vapour scale height as

$H_{v}=\frac{R_{v}T}{L_{v}}H_{T},$ (B.4)

then it is accepted $H_{T}\gg H_{v}$ for water vapour and a wide range of
other condensible gases.

In the low temperature limit, the lapse rate is assumed to be the dry adiabat,
$dT/dz=-g/c_{p}$, and

$H_{T}^{dilute}=\frac{c_{p}T}{g}.$ (B.5)

At high temperature the total pressure is dominated by the saturation vapour
pressure of water and the temperature scale height is given by

$H_{T}^{steam}=-T\frac{dz}{dT}=-T\frac{dz}{de^{*}}\frac{de^{*}}{dT}=T\frac{1}{\rho_{v}g}\frac{L_{v}e^{*}}{R_{v}T^{2}}=\frac{L_{v}}{g},$
(B.6)

where the hydrostatic equation, $de^{*}/dz=-\rho_{v}g$, is used.

It should be noted that at a temperature of 400 K with an Earth-like gravity,
$H_{T}^{dilute}\simeq 350$ km in H2, whereas $H_{T}^{dilute}\simeq 40$ km in
N2. The main difference is due to the low molecular weight of H2 relative to
N2 (see Fig. 5).

###  B.2 Trial to derive the steam and dilute limit values

Koll & Cronin (2019) have tried to derive an approximate analytical expression
for the OLR limit values of the steam limit (SN-limit) and dilute limit (KI-
limit). They have approximated the saturation pressure with temperature as

$p^{*}(T)=p^{*}_{0}\left(\frac{T}{T_{0}}\right)^{\gamma},$ (B.7)

taking the Clausius-Clapeyron relation d ln $p^{*}/$d ln $T\simeq
L_{v}/(R_{v}T)=\gamma$. They have expressed the relation between optical depth
and temperature as a power law (see Koll & Cronin 2019)

$\tau(T)=\kappa
H_{T}\frac{p^{*}_{0}}{L_{v}}\times\left(\frac{T}{T_{0}}\right)^{\gamma},$
(B.8)

where $T_{0}$ is a reference temperature close to the interest range.

OLR is said to be equal to

${\rm OLR}=\sigma T_{s}^{4}e^{-\tau}+\int^{\tau}_{0}\sigma
T(\tau^{\prime})^{4}e^{-\tau^{\prime}}d\tau^{\prime},$ (B.9)

the first term can be neglected as optical depth increasing and the upper
limit of the integral is replaced with infinity

$\displaystyle{\rm OLR}_{\infty}\simeq\int_{0}^{\infty}\sigma
T(\tau)^{4}e^{-\tau}d\tau,$ (B.10)
$\displaystyle=\sigma{T_{0}}^{4}\int_{0}^{\infty}\left(\frac{\tau}{\tau_{0}}\right)^{4/\gamma}e^{-\tau}d\tau,$
(B.11)
$\displaystyle=\sigma{T_{0}}^{4}{\tau_{0}}^{-4/\gamma}\int_{0}^{\infty}\tau^{4/\gamma}e^{-\tau}d\tau,$
(B.12) $\displaystyle{\rm
OLR}_{\infty}\simeq\sigma{T_{0}}^{4}\times\frac{\Gamma(1+4/\gamma)}{\tau_{0}^{4/\gamma}}.$
(B.13)

Here $\Gamma$ is the gamma function, defined by
$\Gamma(s)=\int_{0}^{\infty}x^{s-1}$exp$(-x)dx$, and $\tau_{0}$ denotes for
steam limit

$\tau_{0}=\frac{\kappa_{v}p^{*}(T_{0})}{g},$ (B.14)

and for dilute limit

$\tau_{0}=\frac{\kappa_{v}p^{*}(T_{0})}{g}\frac{c_{p}T_{0}}{L_{v}},$ (B.15)

where it is taken $c_{p}=7R/2$ for H2 and N2 diatomic molecule and
$c_{p}=5R/2$ for He monoatomic molecule. The latent heat $L_{v}$ is
tentatively taken, for the moment, as $2265-4.186\times(T_{0}-373)$ (J/g),
where 2265 (J/$g$) is the vapourization energy per gram at 373K and 4.186
(J/($g\cdotp$ K)) is the specific heat of water, approximately.

Figure 29: Dilute limit dependence on $T_{0}$ for H2, He, and N2 atmospheres
given by Eqs. (B.13) and (B.15) are presented.

Dilute limit dependence on $T_{0}$ for H2, He, and N2 atmospheres given by
Eqs. (B.13) and (B.15) are presented by blue, green, and orange curve in Fig.
29, respectively. For the steam limit given by Eqs. (B.13) and (B,14) is
presented by red curve in Fig. 29. They are almost constant or monotonically
decreasing with $T_{0}$, however the value of $\sigma T_{0}^{4}$ has changed
which is shown for reference by purple curve in Fig. 29.

Taking $T_{1}=T_{0}-\Delta T$ and $T_{2}=T_{0}+\Delta T$, $\gamma$ for $T_{0}$
is taken as a mean value of $\gamma_{i}$ as $(\gamma_{1}+\gamma_{2})/2$ where

$\gamma_{i}=\frac{L}{R}\times\left(\frac{1/T_{i}-1/T_{0}}{{\it ln}T_{0}-{\it
ln}T_{i}}\right),$ (B.16)

being i (=1, 2) and $\Delta=25$. Ten times of $\gamma$ is presented in Fig. 29
in black curve as $10\times\gamma$ and some $\gamma$ values are shown in Table
I.

It must be noticed that He dilute limit (green line) lies below the steam
limit (red line) beyond $T_{0}\simeq 330$ K, which is not indicated by Koll &
Cronin 2019.

For taking $T_{0}\simeq 300$ K where $\gamma\simeq 17.55$, OLR $\simeq 316$
W/m2 for steam limit (calculated result is $\simeq 294$ W/m2 (Fig. 1); then
316/294$\simeq$1.075: the difference is $\simeq$ 7.5%). The dilute limit of H2
atmosphere is $\simeq 280$ W/m2 (calculated result is $\simeq 257$ W/m2 (Fig.
1), then 280/257 $\simeq$ 1.089: the difference is $\simeq$ 8.9 %). The dilute
limit of He atmosphere is $\simeq 328$ W/m2 (calculated result is $\simeq$ 291
W/m2 (Fig. 12), then 328/291 $\simeq$ 1.127: the difference is $\simeq$ 13 %).
The dilute limit of N2 atmosphere is $\simeq 511$ W/m2 (calculated result is
$\simeq 420$ W/m2 (Fig. 9), then 511/420 $\simeq$ 1.217: the difference is
$\simeq$ 22 %).

Although they have shown some limiting values, it seems to be a little bit
difficult to accept that the limiting values could approximately represent the
calculated values. The absolute limiting values seems to have some differences
(dilute limit of N2 differs about 22 %) for $T_{0}\simeq$ 300K and (dilute
limit of H2 is only about 32 % (differs 68 %) for $T_{0}\simeq$ 500K. They are
shown in Fig. 29 and particular values are presented in Table I.

Let us consider the limiting values from the different point. The relative
differences of the limiting values could be understood by the above Eq.
(B.14).

1\. The difference of the accelerating gravity: if $g$ has changed to $\alpha
g$ by factor $\alpha$, $\tau_{0}$ has changed to $\tau_{0}/\alpha$ and OLR∞
has changed by factor $\alpha^{4/\gamma}$. Taking $\alpha=2$ and $\gamma\simeq
17.55$, it becomes $2^{0.2279}\simeq 1.1711$. The steam limit has changed from
$\simeq$ 294 W/m2 (Fig. 1) to $\simeq$ 340 W/m2 (Fig. 10) which is almost the
same as $340/294\simeq 1.1564(1.1711/1.1564\simeq 1.0128$). The difference is
within 1.3%. . The dilute limit for H2, it has changed from $\simeq$ 257 W/m2
(Fig. 1) to $\simeq$ 295 W/m2 (Fig. 10) where 295/257 $\simeq$1.1479
(1.1711/1.1479 $\simeq$1.0202). The difference is within 2.1 %. The dilute
limit for He, it has changed from $\simeq$ 291 W/m2 (Fig. 12) to $\simeq$ 328
W/m2 (Fig. 21) where 328/291 $\simeq$ 1.12714 (1.1711/1.2714 $\simeq$ 0.9211).
The difference is within 7.9 %. The dilute limit for N2, it has changed from
$\simeq$ 420 W/m2 (Fig. 9) to $\simeq$ 484 W/m2 (Fig. 22) where 480/420
$\simeq$ 1.14286 (1.1711/1.14286 $\simeq$ 1.0247). The difference is within
2.5 %.

2\. The difference of the mean molecular weight: $c_{p}$ is the specific heat
per gram related to the specific heat per mol $\hat{c}_{p}=c_{p}\times MW$
where $MW$ is the molecular weight. Then if the molecular weight has increased
by factor $\beta$, $\tau_{0}$ has decreased by factor $\beta$. The $\tau_{0}$
has changed to $\tau_{0}/\beta$ and OLR∞ has changed by factor
$\beta^{4/\gamma}$. Taking $\beta=2$ from H2 to He and $\gamma\simeq 17.55$,
it becomes $2^{0.22792}\simeq 1.1711$. The dilute limit has changed from
$\simeq$ 257 W/m2 (Fig. 1) to $\simeq$ 291 W/m2 (Fig. 12) which is almost the
same as 291/257 $\simeq 1.132(1.17114/1.1323\simeq 1.0343$). The difference is
within 3.5 %.

Taking $\beta=14$ from H2 to N2 and $\gamma\simeq 17.55$, it becomes
$14^{0.22792}\simeq 1.8248$. The dilute limit has changed from $\simeq$ 257
W/m2 (Fig. 1) to $\simeq$ 420 W/m2 (Fig. 9) which is almost the same as
420/257 $\simeq 1.63424(1.8248/1.63424\simeq 1.117$) . The difference is
within 12 %.

Taking $\beta=7$ from He to N2 and $\gamma\simeq 17.55$, it becomes
$7^{0.222792}\simeq 1.5582$. The dilute limit has changed from $\simeq$ 291
W/m2 (Fig. 12) to $\simeq$ 420 W/m2 ( Fig. 9)which is almost the same as
420/291 $\simeq 1.4433(1.5582/1.4433\simeq 1.0796).$ The difference is within
8.0 %.

It is a rather good approximation to estimate the OLR∞ by the variation of
gravity (differences are within 7.9 %) and molecular weight (differences are
within 12 %).

Table 1: Values of $\gamma$, the steam limit (SL), dilute limit of H2, He and N2 for T0 are presented. R1, R2, R3 and R4 show the ratio of the approximate value to the calculated value.  $T_{0}$ K | $\gamma$ |  SL |  R1 |  H${}_{2}\ $ |  R2 |  He |  R3 |  N${}_{2}\ $ |  R4
---|---|---|---|---|---|---|---|---|---
300 | 17.55 | 316 | 1.078 | 280 | 1.089 | 328 | 1.126 | 511 | 1.216
350 | 15.03 | 286 | 0.975 | 233 | 0.905 | 280 | 0.961 | 469 | 1.117
400 | 13.15 | 238 | 0.813 | 176 | 0.684 | 217 | 0.746 | 393 | 0.935
450 | 11.68 | 187 | 0.639 | 123 | 0.480 | 156 | 0.537 | 304 | 0.725
500 | 10.51 | 140 | 0.479 | 81.2 | 0.316 | 106 | 0.363 | 222 | 0.528

In the Table I the particular values of $\gamma$, the steam limit and the
dilute limit for of H2, He and N2 are shown, where SL, H2, He and N2 represent
steam limit, dilute limit of H2, He and N2, respectively. R1, R2, R3 and R4
show the ratio of the approximate value to the calculated value, as SL/293,
H2/257, He/ 291 and N2/420, respectively. For example in $T_{0}\simeq 300$ K,
the difference of the steam limit is $\simeq$ 7.8 %. The difference of the
dilute limit of H2 is $\simeq$ 8.9 %. If $T_{0}$ has changed to 500 K, the
differences have increased much further.

From the values in the Table I, the approximate analytical expressions seems
to be applicable in the temperature $T_{0}\simeq 300\sim 350$ K where the
differences are within and around 20 % (not bad). For $T_{0}\geq 400$ K, the
differences are worse than 20 %. The reason is not so clear for the moment.

##  Appendix C Try to derive Eq. (33) in Koll & Cronin (2019)

We have tried to derive the following three equations in Koll & Cronin (2019),

$\displaystyle\frac{dp}{dz}{|}_{steam}$ $\displaystyle=$
$\displaystyle-\bar{\rho}g,\hskip 250.38434pt({\rm KC}\ 32)$
$\displaystyle\approx$
$\displaystyle\rho_{\nu}g\times\left(1+\frac{R_{d}-R_{\nu}}{R_{d}}\frac{p_{d}}{e^{*}}\right),\hskip
142.26378pt({\rm KC}\ 33)$ $\displaystyle H_{T}$ $\displaystyle=$
$\displaystyle\frac{L_{\nu}}{g}\left(1+\frac{R_{d}-R_{\nu}}{R_{d}}\frac{p_{d}}{e^{*}}\right),\hskip
156.49014pt({\rm KC}\ 34)$

where (KC 32) represents Eq. (32) in Koll & Cronin (2019) and so on.

Using $p=p_{\nu}+p_{d}$=$e^{*}+p_{d}$ and $x_{d}=p_{d}/p$, the equation of (KC
32) becomes

$\displaystyle\frac{dp}{dz}{|}_{steam}$ $\displaystyle=$
$\displaystyle\frac{dp}{dz}-\frac{dp_{d}}{dz}\approx-(\rho_{\nu}+\rho_{d})g+(\rho_{\nu}+\rho_{d})x_{d}g$
$\displaystyle=$
$\displaystyle-\rho_{\nu}g(1-x_{d})\left(1+\frac{\rho_{d}}{\rho_{\nu}}\right)=-\rho_{\nu}g\left(1-\frac{1}{1+p_{d}/e^{*}}\frac{p_{d}}{e^{*}}\right)\left(1+\frac{p_{d}}{e^{*}}\frac{R_{\nu}}{R_{d}}\right)$
$\displaystyle\approx$
$\displaystyle-\rho_{\nu}g\left(1+\left(\frac{R_{\nu}}{R_{d}}-\frac{1}{1+p_{d}/e^{*}}\right)\frac{p_{d}}{e^{*}}\right)$
$\displaystyle\approx$
$\displaystyle-\rho_{\nu}g\left(1+\left(\frac{R_{\nu}-R_{d}}{R_{d}}\right)\frac{p_{d}}{e^{*}}\right),$

where we use the relation
$p_{\nu}=e^{*}=\rho_{\nu}R_{\nu}T,p_{d}=e^{*}=\rho_{d}R_{d}T$, and
$p_{d}/e^{*}\ll 1$. It must be noticed that signatures in and out of the
parentheses are different from (KC 33). We believe that those are typos.

Using the Clausius-Clapeyron relation, $d\,{\rm ln}\,e^{*}/d\,{\rm
ln}\,T=L_{\nu}/(R_{\nu}T)$, Eq. (KC 34) becomes as

$\displaystyle H_{T}$ $\displaystyle=$
$\displaystyle-T\frac{dz}{dp}\frac{dp}{dT}\approx-T\frac{dz}{de^{*}}\frac{de^{*}}{dT}\approx-\frac{T}{\rho_{\nu}g}\left(1+\frac{R_{\nu}-R_{d}}{R_{d}}\frac{p_{d}}{e^{*}}\right)^{(-1)}\frac{L_{\nu}e^{*}}{R_{\nu}T^{2}}$
$\displaystyle\approx$
$\displaystyle\frac{L_{\nu}}{g}\left(1-\frac{R_{\nu}-R_{d}}{R_{d}}\frac{p_{d}}{e^{*}}\right)=\frac{L_{\nu}}{g}\left(1+\frac{R_{d}-R_{\nu}}{R_{d}}\frac{p_{d}}{e^{*}}\right).$

where it must be noticed that the signature of the term in the parenthsis is
exact of Eq. (KC 34).

## References

  * (1)
  * (2) [] When the altitude rises above the sea level and the pressure drops, the liquid precipitates, but since the liquid does little work, the approximation here is to omit the contribution of the liquid from the system. Therefore, since the reversibility is lost from the equation dS = 0, it is described as quasiadiabat.
  * (3) [Betts, H. C., Puttick, M. N., Clark, J. W., et al.] 2018, Nature Ecology & Evolution, 2, 1556,
  * (4) [Bressler, S., & Shaviv, G.,] 2015, Astronomical Rev. 11, 41.
  * (5) [Goldblatt C., Robinson T.D., Zahnle K. J., & Crisp D.] 2013, Nature Geosci. 6, 661.
  * (6) [Goldblatt C., & Watson A. J.] 2012, Phil. Trans. R. Soc. A370, 4197.
  * (7) [Hara T., & Suzuki A.] 2021, SCIREA J. Physics, 6, 154, (https://doi.org/10.54647/physics14381; arXiv: 2009.04040),
  * (8) [Hill L. M., Bott K., Dalba A. P., Fetherolf T., Kane R. S., Kopparapu R., Li Z., & Osberg C.] 2022, arXiv:2210.02484v1.
  * (9) [Houghton J. T.] 1977, The Physics of Atmospheres, Cambridge Univ. Press, Cambridge, UK.
  * (10) [Hu R., Seager S., & Yung Y. L.] 2015, ApJ, 807, 8.
  * (11) [Ingersoll A. P.] 1969, J. Atmos. Sci. 26, 1191.
  * (12) [Kasting J.F.] 1988, Icarus, 74, 472.
  * (13) [Koll D.D.B., & Cronin T. W.] 2019, ApJ. 881,120.
  * (14) [Komabayashi M.] 1967, J. Meteor. Soc. Japan, 45, 137.
  * (15) [Madhusudhan N., Piette A.A.A., & Constantinou S.] 2021, arXiv:2108.10888v2.
  * (16) [Manabe S., & Broccoli A. J.] 2020, Beyond Global Warming: How Numerical Models Revealed the Secrets of Climate Change, Princeton Univ. Press, Princeton, NJ.
  * (17) [Manabe S., & Wetherald R.T.] 1967, J. Atmos. Sci., 24, 241.
  * (18) [Nakajima S., Hayashi Y., & Abe Y.] 1992, J. Atmospheric. Scie. 23, 2256.
  * (19) [Pahlevan K, Schaefer L., Elkins-Tanton L., Desch J. S, & Buseck R. P.] 2022. arXiv: 2209.10635.
  * (20) [Pierrehumbert R.T.] 2010, Exoplanet Atmosphere: Physical Process, Cambridge Univ. Press, Cambridge, UK.
  * (21) [Pierrehumbert R., $\&$ Gaidos, E.] 2011, ApJ, 734, L13.
  * (22) [Seager S.] 2010, Exoplanet Atmosphere: Physical Process, Princeton Univ. Press, Princeton, NJ.
  * (23) [Sekiya M., Nakazawa K., & Hayashi C.] 1980, EPSL, 50, 197.
  * (24) [Sekiya M., Nakazawa K., & Hayashi C.] 1981, Prog. Theor. Phys. 66, 1301.
  * (25) [Simpson, G. C. ] 1927, Mem. R. Meteorol. Soc. 11, 69.
  * (26) [Suzuki A.] Part of the work was postered at Quy Nhon, Vietnam (Dec. 11-17, 2016). 2017, Master thesis at Kyoto Sangyo University in Japanese.
  * (27) [Wordsworth R.,] 2012, Icar, 219, 267.
  * (28) [Wordsworth R., Pierrehumbert R., & Del Genio A.D.] 2013, Science 339, Issue 6115, 64.
  * (29) [Zsom A., Seager S., de Witt J., & Stamenkovic V.] 2013, ApJ, 778, 109.