# Decomposing the Fundamentals of Creepy Stories

Sakshi Goel, Haripriya Dharmala  Yuchen Zhang
University of Southern California
{sakshigo<EMAIL_ADDRESS>Keith Burghardt
USC Information Sciences Institute
<EMAIL_ADDRESS>
###### Abstract

Fear is a universal concept; people crave it in urban legends, scary movies,
and modern stories. Open questions remain, however, about why these stories
are scary and more generally what scares people. In this study, we explore
these questions by analyzing tens of thousands of scary stories on forums
(known as subreddits) in a social media website, Reddit. We first explore how
writing styles have evolved to keep these stories fresh before we analyze the
stable core techniques writers use to make stories scary. We find that writers
have changed the themes of their stories over years from haunted houses to
school-related themes, body horror, and diseases. Yet some features remain
stable; words associated with pseudo-human nouns, such as clown or devil are
more common in scary stories than baselines. In addition, we collect a range
of datasets that annotate sentences containing fear. We use these data to
develop a high-accuracy fear detection neural network model, which is used to
quantify where people express fear in scary stories. We find that sentences
describing fear, and words most often seen in scary stories, spike at
particular points in a story, possibly as a way to keep the readers on the
edge of their seats until the story’s conclusion. These results provide a new
understanding of how authors cater to their readers, and how fear may manifest
in stories.

##  1 Introduction

“The story had held us, round the fire, sufficiently breathless, but except
the obvious remark that it was gruesome…I remember no comment uttered till
somebody happened to say that it was the only case he had met in which such a
visitation had fallen on a child.” This is the opening line to Henry James’s
“The Turning Of The Screw” (James, 2008) and summarizes a key and counter-
intuitive aspect of scary stories: they can be both revolting and captivating;
they capture a fundamental emotion, fear (Adolphs, 2013), but in doing so,
provide entertainment. The closest analogy might be how spicy foods cause pain
yet are tasty. We understand what makes food spicy, and can tailor a meal’s
spice to fit one’s palate, yet it is less clear how to generate fear in a
story because we do not know the ingredients that make a story scary. This is
a special case of a more general issue, in which we do not yet fully
understand how stories elicit particular emotions, i.e., what emotion the
reader should feel, in contrast to detecting the emotion a writer expresses
(Alhuzali and Ananiadou, 2021; Strapparava and Mihalcea, 2008).

In this paper, begin to understand how stories make readers scared by
quantifying patterns in scary stories. First, we analyze the topics associated
with scary stories, and find writers are making fewer stories about houses and
more stories about human bodies, school, and diseases, potentially to stay
topical. We then develop a method to quantify more universal patterns in scary
stories by measuring the percentage of times a word appears in scary and non-
scary stories, a metric which we call Scary Story Token Prevalence (SSToP).
Words with high SSToPs, such as _creature_ or _silhouette_ , are more likely
to be human-like words, suggesting scary story writers intentionally use words
that are semi-human. Finally, we analyze where writers use high SSToP words or
words expressing fear within a story. We find that stories can be decomposed
into modes where the high SSToP or fearful words appear in regular intervals,
and combinations of these modes are used to develop scary story patterns.
Overall, these results lend insight into how stories express fear, and
patterns writers exploit to make scary stories.

Our contributions can be summarized as follows:

  * •

we analyze dynamic patterns of scary stories to understand what themes have
become popular or fell out of favor.

  * •

we analyze static patterns, such as the relationship between scary story nouns
like “human” to capture what words writers use and why they might use those
words

  * •

we capture how scary stories are written by quantifying the common scary story
modes. We find some modes are particularly common, which offers clues about
how writers make a scary story.

##  2 Related Work

###  2.1 Uncanny Valley

What makes something scary is most similar to previous work on the uncanny
valley effect. Research on the uncanny valley began with Jentsch and Freud who
coined the term _unheimlich_ (literally, “unhomely”) when referencing human
uncanniness and eeriness (Freud, 1919; Jentsch, 1906). The uncanny valley also
came into prominence among human-computer interaction (HCI) and human-robot
interaction (HRI) researchers with a paper by Mori (1970). Researchers were
inspired by Mori to learn what can make a robot unsettling (Broadbent, 2017)
because willingness to work with robots can depend on their likability
(Destephe et al., 2015). An objective definition of uncanny (what makes
something human-like) or valley (what is and is not unsettling) is still,
however, a fiercely debated topic (Mathur and Reichling, 2016; Langer and
König, 2018; Mathur et al., 2020; Palomaki et al., 2018; MacDorman and
Ishiguro, 2006; Ho and MacDorman, 2017). Moreover, the qualitative shape of
the uncanny valley varies significantly between studies (Mathur and Reichling,
2016; Langer and König, 2018; Mathur et al., 2020; Palomaki et al., 2018;
MacDorman and Ishiguro, 2006; Ho and MacDorman, 2017). Finally, these works
(with rare exception (McAndrew and Koehnke, 2016)) do not explore non-image
media, such as text. Open questions, such as how to make less-scary text, or
how make scarier ghost stories, have also been under-explored.

###  2.2 Emotion Detection

The fear detection model we create is related to previous work on emotion
detection. Emotions are fundamental to humans across cultures (Plutchik,
1982), and are typically split into eight components: terror (fear),
admiration, ecstacy, vigilance, rage, loathing, grief, and amazement. Research
on detecting these emotions in text has a long history (Acheampong et al.,
2020; Chen and He, 2022), but this research recently came to prominence with
text embedding-based methods that can accurately detect emotion in sentences.
Detection methods have recently exploited advances in Bi-LSTMs and attention
for higher accuracy (c.f., (Cai and Hao, 2018; Alhuzali and Ananiadou, 2021))
and have proven useful for chatbots, for example, which are used in a range of
applications (Chen and He, 2022). Emotion detection is closely related to
sentiment analysis, such as VADER (Hutto and Gilbert, 2014), or the
hedonometer, studied by Dodds et al., (Dodds et al., 2011; Reagan et al.,
2016). Both methods aim to detect the positivity or negativity of words or
sentences, and has been used to understand story arcs (Reagan et al., 2016) or
tweet sentiment Dodds et al. (2011); Hutto and Gilbert (2014).

##  3 Data

###  3.1 Fear Training Data

We use multiple datasets to train a fear-detection model, where each dataset
has sentences labeled as expressing fear or not. The datasets were gathered
and converted into a binary classification problem with a label 1 or 0
representing if the sentence does or does not express fear, respectively. The
datasets used for modeling are described in Table 1. This is one of the most
complete collections of data to analyze fear in text. Alike to previous
research (Bostan and Klinger, 2018), we collect a range of emotion data, but
in contrast to that work, some of our data are unique and only available upon
request from the authors (Aman and Szpakowicz, 2007). This diverse dataset is
trained with a model we describe in the Metrics section. In total, there are
70K labeled sentences, with roughly 7K of these sentences labeled “fear.”

Dataset |  Description
---|---
ISEARs (Scherer and Wallbott, 1994) |  Obtained from cross-cultural studies in 37 countries and contains sentences annotated for joy, sadness, fear, anger, guilt, disgust and shame emotions
WASSA 2017 (Mohammad and Bravo-Marquez, 2017) |  Constructed from tweets and annotated for joy, sadness, fear, and anger emotions
Affect Data (Alm, 2008) |  Constructed from Tales and classified into angry, fearful, happy, sad, disgusted and surprised emotions
Emotion Stimulus No Cause Dataset (Ghazi et al., 2015) |  Data developed from FrameNets’ annotated data for emotion lexical unit, each sentence is marked with an emotion tag.
CrowdFlower Twitter Sentiment Analysis Emotion Data (CrowdFlower, 2021) |  Constructed from tweets and classified into 13 emotions (sadness, relief, anger, worry, enthusiasm etc.)
Emotion-Annotated Dataset (Aman and Szpakowicz, 2007) |  Constructed from blog data sentences classified into happiness, sadness, fear, anger, surprise and disgust.
DailyDialog Dataset (Li et al., 2017) |  Human-written text sentences tagged with emotions and communication intention
Emotion Hashtag Corpus (Mohammad and Kiritchenko, 2015) |  Tweets annotated with emotions based on the emotion-word hashtag in the tweet text
Table 1: Datasets used to construct fear model. Figure 1: The fear model
framework. (a) The neural network used to predict fear in sentences (5,840,850
parameters total). (b) The receiver operative characteristic for the neural
network model.

###  3.2 Reddit Data

We extract stories from subreddits using the Python wrapper for the Pushshift
API (https://psaw.readthedocs.io). Stories are collected within two broad
genres: scary stories and non-scary stories. For the scary stories, our data
comes from the subreddits r/CreepyPasta, r/ShortScaryStories, and r/NoSleep,
and for non-scary stories we selected the subreddits r/TIFU, r/Self,
r/StoriesAboutKevin, r/Confession, and r/Confessions, which are common story-
writing subreddits. All stories are collected from the start of the subreddit
until June, 2020. These data are used to train FastText embeddings (Joulin et
al., 2016), and to find SSToP scores. After cleaning, described below, there
are in total 123K stories.

###  3.3 Data cleaning

To clean data for fear detection, we remove all characters except English
alphabets and spaces. This ensures that the fear model’s vocabulary is only
English words, and any numbers/HTML markup is removed. Next, we split phrases
written in the camel case (a frequent occurrence in the Reddit and Tweet
datasets due to hashtags) into its constituent words. (e.g. #EndThisTyranny
split to ’end this tyranny’). Finally, we convert all strings to lowercase and
remove all English stopwords (collected from NLTK corpus), to ensure that the
model doesn’t falsely learn fear from universally occurring stopwords.

###  3.4 Metrics

####  3.4.1 Fear Confidence

To determine the confidence that a word or sentence expresses fear we train a
model whose architecture is described in Fig. 1a. A Bidirectional GRU (Bi-GRU)
is used because it is capable of incorporating contextual information in the
forward and reverse direction in contrast to a standard GRU or LSTM. A Global
Average Pooling layer was used after the Bi-GRU layer and a dropout layer of
50% was used to regularize the dense layer just before the two-node prediction
layer. A batch size of 64 was used with categorical cross-entropy loss and an
Adam optimizer.

To train this model, we split the meta-dataset into 3 parts (where each
datapoint is assigned at random): 75% for training, 10% for validation and 10%
for testing. The original training dataset contained 55.8K sentences, of which
5.5K were sentences expressing fear. To reduce imbalance, we up-sample the
fear sentences to create 100.5K sentences for training. Validation and test
sets are separate sets of 7.0K sentences without up-sampling or down-sampling
(roughly 700 sentences in each set were labeled _fear_). Tensorflow Keras is
used for the tasks of tokenization, sequencing and padding of the three
datasets. The tokenizer is only fit on the training data. The vocabulary size
was 56.5K. Each sentence is then converted into a sequence and zero-padded to
a length of 100. The model ROC curve is shown in Fig. 1b, which shows high
micro- and macro-averaged AUC (0.96 and 0.85, respectively).

We compare this model to logistic regression models with L1 or L2
regularization. Text is tokenized and fed through FastText (Joulin et al.,
2016) embeddings trained on the scary and non-scary story corpora. All
embedded words in a story are then averaged to create a document embedding.
When training the logistic regression models, the training data is down-
sampled so that each class is of equal size. The comparisons against the
baselines are shown in Table 2. We see significant improvements over the
baseline models.

Model | Accuracy | F1 | ROC-AUC
---|---|---|---
|  | (Macro Avg) |
Neural Net | 0.92 | 0.76 | 0.85
Logistic (L1) | 0.71 | 0.60 | 0.84
Logistic (L2) | 0.74 | 0.61 | 0.84
Majority | 0.90 | 0.57 | 0.5
Table 2: Model performance for neural network-based fear model and logistic
regression baselines. Bottom row are metrics for majority class null model, in
which we guess all test data is not scary (the majority class).

####  3.4.2 Scary Story Token Prevalence

SSToP is defined as the odds ratio between a word’s frequency in scary and
non-scary stories

$c(word)=\frac{f_{scary}(word)}{f_{normal}(word)}$ (1)

Where $f_{X}(word)$ is the word frequency across all posts and subreddits
within the genre $X$. To check the robustness of this method, we re-calculate
$c(word)$ such that non-scary word frequencies were calculated from each of
the non-scary subreddits (creating five separate metrics) and results are
qualitatively the same.

High SSToP Words | SSToP | Low SSToP Words | SSToP
---|---|---|---
(Common in Scary Stories) |  | (Uncommon in Scary Stories) |
eyeless | 100 | rself | 0.0018
scarecrows | 108 | fwb | 0.0021
cryptids | 110 | superjail | 0.0022
skinwalkers | 130 | 19f | 0.0023
headstones | 137 | tldr | 0.0025
Table 3: Examples of words with high and low SSToP scores. Skin-walkers, which
have a high SSToP score, are a harmful witch in Navajo culture.

To calculate this metric for each word, we first select all stories which
contain at least 500 words. This helps ensure that the lengths of the stories
are approximately uniform and allows us to calculate story modes described
later. The stories are preprocessed by tokenizing and removing punctuation. We
utilize a dictionary to keep track of the number of times each word occurs.
This is calculated for both scary and non-scary stories and Eq. 1 is applied
to determine a SSToP value for each word. Since Reddit stories are not checked
for correct grammar or spelling, there are several unique names or words which
may be misspelt. Because the occurrence of each of these words does not span
broadly throughout the entire subreddit, SSToP cannot accurately assign a
value to these unique terms. Therefore, we only consider words which have
occurred at least 100 times in the entire corpora of scary and non-scary
stories.

###  3.5 Generating Story Modes

The main motivation behind creating story modes is to identify patterns
writers exploit to make scarier stories. Some stories may, for example, get
scary towards the end while other stories may start out scary and end scary,
but have a less-scary segment in the middle. Therefore, while we generate a
model to detect whether fear is expressed in sentences, it is the pattern of
fear, the modes, that we study to better understand how readers might be
provoked to experience fear.

To construct these story modes, we first segment each story into ten points
based on the percentage of story covered (0%, 10%,…,90%). We then calculate
the fear confidence of a sentence located at that point in the story or
average the SSToP score for 50 words after that point in the story. After the
metrics are sampled, we construct a matrix, $\mathbf{M}$, such that columns
represent the 10 sampled points, and rows represent each story. We then apply
singular value decomposition to calculate the story modes (Reagan et al.,
2016), such that $\mathbf{M}=\mathbf{V\Sigma U}=\mathbf{WU}$. The matrix
$\mathbf{U}$ represents the modes while $\mathbf{W}$ represents the
coefficients of each mode used to reconstruct each story, alike to the
coefficients of a Fourier transform. The largest element in each row of
$\mathbf{W}$ represents the mode that best aligns with each story, but in
general several modes are needed to reconstruct a story.

###  3.6 Word similarity to human

To determine how similar a noun is to _human_ , we first embed nouns using
FastText. Human synonyms, namely human, mortal, person, soul, Homo Sapien,
earthling, higher animal, and living person (common synonyms of human that are
gender neutral) are averaged together into one _human_ vector to avoid noise
associated with the embedding of a single word. This follows similar work on
gender bias in word embeddings, where the first principle component of a
collection of words are used to construct a he-she vector, in order to reduce
noise associated with each individual pair of words (Bolukbasi et al., 2016).
Next we determine what words in the word embeddings are nouns using Python’s
TextBlob library (https://textblob.readthedocs.io). If NN, PN, or NP are in
the TextBlog tags for a word, the word is categorized as a noun. This method
helps us avoid finding the unrealistic similarity between, e.g., a verb and a
noun. Finally, we determine distance between words using Euclidean distance.
Other distance metrics, such as Manhattan distance (Aggarwal et al., 2001),
show similar results.

##  4 Results

Figure 2: Scary story statistics. (a) The number of stories each year (inset:
the number of stories written each hour in GMT time). (b) LDA topics over
time, which shows a significant drop in the house-like topic but an increase
in body-related and school-related topics in recent years. (c) The proportion
of disease-related stories over time.

We show data statistics in Fig. 2. This figure demonstrates when writers post
their stories and what they discuss. First, Fig. 2a shows the frequency of
stories over time, which rapidly increases to over 20,000 stories a year
before dropping slightly. In the inset, we see that users are least likely to
post their stories at 12 p.m. GMT, or around 5 a.m. EST, perhaps because most
writers might be from America and do not post when they just wake up. Next, we
use Latent Derichlet Allocation (with lemmatized words and all stop words
removed) to find ten story topics, the largest of which are shown in Fig. 2b.
We notice that the house topic (with words like “door" or “window”) drops off
in frequency while topics related to bodies (with words like “bodies" or
“blood”) and school (with words like “phone” or “school”) become increasingly
popular. These results suggest long-term trends in what writers consider
topical for their stories, possibly because writers focus on different topics
as they age or they believe readers find school or body horror more
interesting. Finally, we show the frequency of disease-like stories, in which
“lockdown,” “infect,” “viru,” or “diseas” lemmas are within the story,
increase dramatically in 2020 (likely due to COVID-19 being in the news), but
this follows a long-term trend, in which disease-like stories have become
increasingly popular. We also notice a small peak in their prevalence in
mid-2014, when the disease Ebola became topical.

Next, we show in Fig. 3 that a word’s SSToP score varies with its similarity
to human. Words seen in scary stories tend to be more similar to human, which
is reminiscent of the uncanny valley, in which people find semi-human objects
more unsettling. A number of plausible reasons for these patterns exist. For
example, scary story authors use semi-human words (such as “devil” or “clown”)
in an attempt to unsettle readers, or they are discussing human-related
topics, or they avoid words that are static and boring, such as cars, which
are far from “human” in latent embedding space. Whatever the reason, this
pattern gives new insights into how writers make scary stories.

Figure 3: Scary story writers use more human-like words. Mean SSToP score
(fraction of words in scary stories) in red versus noun similarity to “human.”
Error bars are standard errors. Figure 4: Story modes in r/NoSleep. (a) 3
lowest story modes for fear (modes are blue lines), (b) 3 lowest story modes
for SSToP scores (modes are red lines), and (c) histograms of each story mode
for the respective metrics. Gray lines in (a-b) are example stories that best
match those respective modes (they have the largest W coefficients for those
modes).

Finally, we analyze how scary story authors develop story modes. In Fig. 4, we
show story modes of fear confidence and SSToP score in the r/NoSleep subreddit
(14K stories in total). Blue lines in Fig. 4a correspond to the three lowest
modes for fear confidences, red lines in Fig. 4b are the three lowest modes
for SSToP scores. Gray lines correspond to 20 stories which have the highest
coefficients in the corresponding $\mathbf{W}$ matrix for a given mode. We
normalize the story modes by dividing by the largest $\mathbf{W}$ coefficient
in each row. These figures show that stories use high SSToP words, such as
“shadow” or “creature” at very particular points in the story, presumably at
story climaxes or key scenes. Stories also express fear at particular points
in the story as well, presumably to keep the audience thrilled or afraid until
the end of the story.

In Fig. 4c, we show the frequency of the largest mode associated with each
story. We see that many stories are in the lowest SSToP mode. Scary story
authors therefore tend to use scary words in the center of the story, perhaps
to set up scenes. Surprisingly, however, there are a variety of modes used for
fear, and a combination of these modes often appear in stories. Furthermore,
the fear and SSToP modes rarely align. Namely, the largest SSToP modes for a
given story have a low correlation with the top fear modes (Spearman
correlation $=0.06$, p-value $<10^{-10}$). Finally mode 9 is rarely the
largest mode because it corresponds to a story without any change (the mode is
a flat line).

##  5 Conclusions

In conclusion, we analyze text patterns to better understand how writers make
scary stories. The results suggest that writers tend to be topical or change
the story topic to fit the mood of readers. Moreover, scary story authors tend
to use human-like words, which is reminiscent of the uncanny valley in which
people find human-like images more unsettling. In addition, authors create
stories whose modes may help scare readers. Overall, our work uncovers
patterns associated with text that elicits fear, but we believe more work will
be needed to understand how these patterns may cause fear to be elicited.

##  6 Limitations

There are a number of limitations in our work. While we uncover patterns
associated with scary stories, more work is needed to understand whether or
how these patterns create fear in readers. This task will require analyzing
reader reactions in a controlled study. In addition, the relationship between
nouns and words like “human” have not been tested on alternative text
embedding models, such as GLoVe (Pennington et al., 2014), or attention-based
embeddings, such as BERT (Devlin et al., 2019). Finally story motifs we
uncover should be compared against alternative methods (Reagan et al., 2016).

##  7 Ethical Considerations

All results come from publicly available and anonymized data, therefore the
study poses minimal ethical risk.

##  8 Acknowledgements

This project was developed at the USC Center for Knowledge-Powered
Interdisciplinary Data Science (CKIDS) DataFest 2020. We also thank the USC
Graduates Rising in Information and Data Science (GRIDS) for their help. Code
to reproduce some of our major results are shown here:
https://github.com/KeithBurghardt/Creepypasta.

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# The orbital evolution of the tidally stripped star and disk-driven stable
mass transfer for QPEs in GSN069

Di Wang Department of Physics, Huazhong University of Science and Technology,
Wuhan 430074, China Di Wang<EMAIL_ADDRESS>
###### Abstract

The origin of the quasi-periodic eruptions (QPEs) is possibly mass loss at the
periastron of a body moving around the supermassive black hole (SMBH) in a
high eccentric orbit. Such a tidally stripped star is expected to radiate
gravitational wave thereby leading to shrinkage of the periastron distance,
and thus will eventually be disrupted by the SMBH in the previous studies.
This scenario predicts a gradually increasing mass transfer, contradicting the
observed decay in the intensity of the QPEs in GSN 069. In this paper, we
first revisited the orbital evolution of the stripped star. The effect of the
mass transfer finally dominates the orbital evolution, resulting in the
stripped star finally escaping the SMBH rather than being disrupted by it.
Then we suggested a model of a tidal stripped WD moving inside an accretion
disk for QPEs in GSN 069. The drag force by the disk can effectively reduce
mass transfer and thus explain the observed decay in the intensity of the QPEs
in GSN 069. The disk is likely a fallback disk of the tidal disruption event
in GSN 069. Considering the evolution of its accretion rate, the increase in
the intensity of the latest eruption can also be explained.

##  1 Introduction

A white dwarf (WD) orbiting around a supermassive black hole (SMBH) with high
eccentricity may lose mass each time through the periastron. Such a tidally
stripped WD is not only expected to emit periodic electromagnetic wave signals
(Zalamea et al., 2010), but also the target of space-based gravitational wave
detectors such as Laser Interferometer Space Antenna (LISA) (Babak et al.,
2017) and TianQin (Luo et al., 2016). The first discovered quasi-periodic
eruptions (QPEs) from GSN 069 are suggested to possibly originate from a
tidally stripped WD in a high eccentric orbit (Miniutti et al., 2019; Wang et
al., 2022; King, 2020).

Due to gravitational wave radiation decreasing the periastron distance, the
mass loss of the tidally stripped WD gradually increases until finally the WD
is tidally disrupted by the SMBH (Zalamea et al., 2010; Wang et al., 2022).
However, the intensity of QPEs in GSN 069 persistently decays (Miniutti et
al., 2023b), which is inconsistent with the previous prediction. King (2022)
points out that the mass transfer stability requires the angular momentum
transfer from the disk to the orbiting star, he suggested that orbital
resonance with the disk possibly could accomplish this, but further discussion
was lacking.

GSN 069 also has a tidal disruption event(TDE) that has lasted more than a
decade (Shu et al., 2018; Sheng et al., 2021). This TDE was recently found to
have re-brightened, and while it was brightening, the QPEs disappeared,
probably due to being covered by radiation from the TDE Miniutti et al.
(2023b). When radiation from the TDE faded again, the QPEs reappeared and were
stronger than in the last outburst before they disappeared (Miniutti et al.,
2023a). Such behavior implies that the evolution of mass transfer may be
related to the accretion rate of the accretion disk.

The origin of this TDE is likely the disruption of a common envelope of the
binary containing the current orbiting body before it is captured by the SMBH
(Wang, 2023). This means that the fallback disk of the TDE is nearly coplanar
with the orbit of the orbiting body that produced the QPEs. In this situation,
the drag force by the disk may significantly affect the secular orbital
evolution of the orbiting body.

In this paper, we first revisited the secular evolution of the tidally
stripped star with high eccentricity taking into account the effect of the
mass transfer on the orbit, which is ignored in the previous studies. We found
that the effect of mass transfer significantly changes the criterion for
unstable mass transfer. The mass transfer for WDs is no longer dynamically
stable, while it remains dynamically stable for main sequence stars.
Furthermore, the effect of the mass transfer dominates the secular orbital
evolution, which leads to the fate of the stripped star escaping the SMBH
rather than being disrupted by it.

While the origin of QPEs is still debated, there are various other models
(Chen et al., 2022; Zhao et al., 2022; Wang et al., 2022; Krolik & Linial,
2022; Lu & Quataert, 2023; Xian et al., 2021; Suková et al., 2021; Linial &
Sari, 2023; Linial & Metzger, 2023). In this paper, we focus on the stripped
WD model in a high eccentric orbit for GSN 069 (King, 2020, 2022). We
calculated the secular effect of drag force by the disk on the orbit for GSN
069. We found that the drag force can transfer the angular momentum of the
disc to the stripped WD, and then stabilize the mass transfer. This effect is
strongly dependent on the accretion rate, which can be estimated from the
luminosity of the TDE. The derived change in mass transfer rate is in very
good agreement with the observed intensity decay of QPEs in GSN 069. The
increase in the intensity of the QPEs after the TDE re-brightened can also be
explained by this model.

This paper is organized as follows. In Section 2, we revisited the secular
orbital evolution of the stripped star under the effect of gravitational wave
radiation and mass transfer. In Section 3, we discuss the effect of drag force
by the accretion disk on the orbit for QPEs in GSN 069. We made discussions in
section 4 and a summary in Section 5.

##  2 Evolution with mass transfer and gravitational radiation

We consider a WD of mass $M$ and radius $R$ in a highly eccentric orbit around
a SMBH of mass $M_{h}$. When it reaches the periastron, the stripped mass can
be obtained by calculating the mass of the matter in the WD outside Roche lobe
(Chen et al., 2023)

$\frac{\Delta M}{M}\simeq
4.8\left[1-\left(M/M_{\mathrm{ch}}\right)^{4/3}\right]^{3/4}\left(1-\frac{\beta_{0}}{\beta}\right)^{5/2}.$
(1)

Here $\beta\equiv r_{t}/r_{p}$ is impact factor, where
$r_{t}=R(M_{h}/M)^{1/3}$ is the disrupted radius and $r_{p}$ is the periastron
radius. The Chandrasekhar mass $M_{\mathrm{ch}}$ is $1.44M_{\odot}$ and
$\beta_{0}$ is about 0.5. Mass transfer occurs only if $\beta$ is greater than
$\beta_{0}$. And the WD is disrupted when $\beta$ is greater than 1.

Differentiating it gives

$\frac{\dot{\Delta M}}{\Delta
M}=\frac{1-2\left(M/M_{\mathrm{ch}}\right)^{4/3}}{1-\left(M/M_{\mathrm{ch}}\right)^{4/3}}\frac{\dot{M}}{M}+\frac{5\beta_{0}}{2(\beta-\beta_{0})}\frac{\dot{\beta}}{\beta}.$
(2)

The latter term is usually dominant because $\beta$ is close to $\beta_{0}$
when mass transfer is not extremely high. Setting the mass-radius index
$\zeta$ which gives mass-radius relation $R\propto M^{\zeta}$, the
$\dot{\beta}$ can be obtained by (King, 2022)

$\frac{\dot{\beta}}{\beta}=\frac{2\dot{M}}{M}\left(\frac{5}{6}+\frac{\zeta}{2}-q\right)-\frac{2\dot{J}}{J}+\frac{\dot{e}}{1+e}.$
(3)

Here $q=M/M_{h}$ is the mass ratio, $J$ is the total orbital angular momentum
and $e$ is the orbital eccentricity.

###  2.1 Mass transfer

The mass-radius relation of the WD is (Paczynski, 1971)

$R=9\times
10^{8}\left[1-\left(\frac{M}{M_{\mathrm{ch}}}\right)^{4/3}\right]^{1/2}\left(\frac{M}{M_{\odot}}\right)^{-1/3}\mathrm{~{}cm},$
(4)

which is adopted in the derivation of Eq. (1) (Chen et al., 2023). So $\zeta$
for the WD is about -1/3 for low mass, and more negative when mass is close to
$M_{\mathrm{ch}}$. And it is always greater than -5/3 so that the first term
in Eq. (3) is negative, which can suppress mass transfer. King (2022) claim
that this term is dominant on short timescales so that the mass transfer is
dynamically stable. However, he ignored the $\dot{e}$ caused by the mass
transfer, which is111Wang et al. (2022) had considered the effect of the mass
transfer on the orbit too. Their formula is different from this because they
mistake the instantaneous mass transfer rate in Sepinsky et al. (2007b) for
the secular mass transfer rate thus underestimate this effect. (Sepinsky et
al., 2007b)

$\langle\dot{e}\rangle_{\dot{M}}=-2(1+e)\frac{\dot{M}}{M}(1-q).$ (5)

Substituting it into Eq. (3) gives

$\left\langle\frac{\dot{\beta}}{\beta}\right\rangle_{\dot{M}}=\frac{\dot{M}}{M}\left(\zeta-\frac{1}{3}\right).$
(6)

So the self-driven mass transfer for the WDs is unstable. Noted that Eq. (5)
show that orbit is more eccentric under mass transfer, and the change of the
semi-major axis (Sepinsky et al., 2007b),

$\left\langle\frac{\dot{a}}{a}\right\rangle_{\dot{M}}=-2\frac{1+e}{1-e}\frac{\dot{M}}{M}(1-q),$
(7)

is positive. So unstable mass transfer would lead to the eventual escape of
the WD, contrary to the prediction that gravitational radiation would lead to
the disruption of the WD (Zalamea et al., 2010; Wang et al., 2022).

###  2.2 Gravitational radiation

The change of orbit caused by the gravitational radiation is (Peters, 1964)

$\left\langle\frac{\dot{J}}{J}\right\rangle_{GW}=-\frac{32}{5}\frac{G^{3}MM_{h}M_{t}}{c^{5}a^{4}\left(1-e^{2}\right)^{5/2}}\left(1+\frac{7}{8}e^{2}\right),$
(8)
$\left\langle\frac{\dot{a}}{a}\right\rangle_{GW}=-\frac{64}{5}\frac{G^{3}MM_{h}M_{t}}{c^{5}a^{4}\left(1-e^{2}\right)^{7/2}}\left(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}\right),$
(9)
$\langle\dot{e}\rangle_{GW}=-\frac{304}{15}e\frac{G^{3}MM_{h}M_{t}}{c^{5}a^{4}\left(1-e^{2}\right)^{5/2}}\left(1+\frac{121}{304}e^{2}\right).$
(10)

Here $G$ is the gravitational constant, $M_{t}=M_{h}+M$ is total mass and $c$
is the speed of light. Substituting Eq. (8) and (10) into Eq. (3) yields

$\left\langle\frac{\dot{\beta}}{\beta}\right\rangle_{GW}=\frac{64}{5(1+e)}\frac{G^{3}MM_{h}M_{t}}{c^{5}a^{4}\left(1-e^{2}\right)^{5/2}}\left(1-\frac{7}{12}e+\frac{7}{8}e^{2}+\frac{47}{192}e^{3}\right),$
(11)

which is positive. So both mass transfer and gravitational radiation enhance
mass transfer, but gravitational wave radiation reduces $a$ and $e$, which is
the opposite of the effect of mass transfer.

###  2.3 Orbital Evolution for the WD

The effect of the gravitational radiation is stronger than the effect of the
mass transfer when the mass transfer rate is low. As the mass transfer rate
gets higher, the effect of mass transfer becomes dominant. While factor
$1/(1-e)$ evolves to very high values, the effect of gravitational radiation
becomes smaller. Because the periastron $r_{p}=a(1-e)$ is constant under mass
transfer, $\langle\dot{r_{p}}\rangle_{\dot{M}}=0$, which make the effect of
gravitational radiation proportional to $(1-e)^{1/2}$ for $\dot{a}/a$ and
$(1-e)^{3/2}$ for $\dot{J}/J$ and $\dot{e}$ when $e\approx 1$.

The mass transfer rate $|\dot{M}|$ doesn’t grow endlessly. Because $\dot{M}$
is related not only to $\Delta M$ but also to the orbital period $P_{b}$,
$\dot{M}=-\Delta M/P_{b}$. And the $a$ grows dramatically at high $e$, as does
the $P_{b}$. When the growth rate of $P_{b}$ is greater than the growth rate
of $\Delta M$, $|\dot{M}|$ will start to decrease until the WD escapes. And
the critical condition can be obtained by

$\frac{\ddot{M}}{\dot{M}}=\frac{\dot{\Delta M}}{\Delta
M}-\frac{3}{2}\frac{\dot{a}}{a}\approx\left[\frac{5\beta_{0}}{2(\beta-\beta_{0})}\left(\zeta-\frac{1}{3}\right)+3\frac{1+e}{1-e}\right]\frac{\dot{M}}{M}=0.$
(12)

So the WD finally escapes with eccentricity $e_{f}\gtrsim 1$, then the total
mass it loses can be obtained from Eq. (5)

$-\frac{\Delta
M}{M}\approx\frac{e_{f}-e_{i}}{2(1+e_{i})(1+q)}\approx\frac{1-e_{i}}{4(1+e_{i})},$
(13)

where $q\ll 1$ and $e_{i}$ is the initial eccentricity while the effect of the
mass transfer is dominant. The WD loses at most 1/4 of its total mass and can
survive the mass loss.

Here we show an example of the evolution under mass transfer and gravitational
radiation in Fig. 1. We adopt possible parameters of GSN 069:
$P_{b}=9\mathrm{~{}hr}$, $M_{h}=5\times 10^{5}M_{\odot}$ and
$M=0.21M_{\odot}$. Then $e=0.968$ can be obtained from Eq. (4) with
$\beta\approx\beta_{0}$. The initial $\beta$ is 0.50001. The orbital
parameters and the mass and radius of the WD are updated each time the WD
passes through the periastron. We use $\langle\dot{r_{p}}\rangle$ and
$\langle\dot{e}\rangle$ to evolve orbit, because $\langle\dot{a}/a\rangle$ is
too high to properly represent orbital changes in the above method when $1-e$
is very small. Evolution will stop when $e$ is greater than 1.

Figure 1: The evolution of orbit parameters and WD’s mass. The first panel
from top to bottom shows the evolution of $1-e$ and orbital period $P_{b}$.
The second one is the evolution of the average mass transfer rate and mass of
the WD. And the last one represents the evolution of mass loss at periastron.
The dashed line represents the critical time that the effect of the mass
transfer becomes dominant. For each parameter, the critical times are not the
same but very close. Here we use the critical time of $e$ for all panels. The
last 10 orbits are plotted as scatter plots and the rest as line plots.

As shown in Fig. 1, the mass loss is rapidly enhanced after the effects of
mass transfer dominate. When the resulting increase in orbital period exceeds
the increase in mass loss, $\Delta M$ becomes stable. During this process, the
eccentricity keeps growing rapidly until it exceeds 1 and the WD escapes from
the SMBH. The period of the last few orbits can vary significantly.

###  2.4 Orbital Evolution for the main-sequence star

If $\zeta$ of the stripped star is greater than 1/3, such as a main-sequence
star, then the mass transfer will stabilize itself, and
$\langle\dot{\beta}\rangle$ will evolve towards 0. Once this happens,
$\langle\dot{\beta}\rangle=0$ will remain. Since the timescale of the mass
transfer evolution is larger than that of the orbital parameter evolution,
$\dot{M}$ can be able to adjust itself rapidly to balance with the effects of
gravitational radiation in the secular evolution. That is, it always satisfies

$\left\langle\frac{\dot{\beta}}{\beta}\right\rangle=\left\langle\frac{\dot{\beta}}{\beta}\right\rangle_{\dot{M}}+\left\langle\frac{\dot{\beta}}{\beta}\right\rangle_{GW}\approx
0.$ (14)

Figure 2: Comparison of $\langle\dot{a}\rangle$ and $\langle\dot{e}\rangle$
caused by the mass transfer and gravitational radiation with
$\langle\dot{\beta}\rangle=0$. Here $\eta$ is taken as 2/3.

While these two effects are balanced on $\langle\dot{\beta}\rangle$, they are
not balanced on $\langle\dot{a}\rangle$ and $\langle\dot{e}\rangle$. As shown
in Fig. 2, the effect of the mass transfer is always dominant when
$\langle\dot{\beta}\rangle=0$, but it is of the same magnitude as GW. Thus,
the evolution of a stripped main-sequence is similar to that of a WD, but on a
much longer timescale, the gravitational radiation timescale, because
$\dot{M}$ is driven by gravitational radiation.

Figure 3: Similar as Fig. 1, but for main-sequence star. The last panel from
the top also shows the evolution of $\langle\dot{\beta}\rangle$ of
gravitational radiation and mass transfer.

We show an example in Fig. 3. The initial orbital parameters are
$M_{h}=5\times 10^{5}M_{\odot}$, $M=0.1M_{\odot}$, $\beta=0.50004$ and
$P_{b}=40\mathrm{~{}hr}$. The radius is obtained by
$R=1.06(M/M_{\odot})^{0.945}R_{\odot}$ (Demircan & Kahraman, 1991). Then the
eccentricity can be obtained $e=0.912$.

As shown in Fig. 3, the effect of mass transfer is initially weak and
$|\dot{M}|$ increase rapidly driven by gravitational radiation, and then
reaching balance $\langle\dot{\beta}\rangle=0$. At this point, the evolution
of the orbit is dominated by the effects of mass transfer, with $r_{p}$
remaining almost constant while $e$ grows. Since
$\langle\dot{\beta}/\beta\rangle_{GW}$ scales as $(1-e)^{3/2}$, $|\dot{M}|$
coupled to it will keep decreasing. Eventually, like WDs, main sequence stars
escape from SMBH, but on much longer timescales.

There is also a special case where, if the initial $|\dot{M}|$ and $e$ are
both high, $e$ may evolve to greater than 1 before the $|\dot{M}|$ has had
time to be reduced to balance state. In addition, in the final stages of
evolution, when the orbital period is growing rapidly, there is a possibility
that the star may be perturbed by other gravitational sources away from the
black hole and thus escape early.

###  2.5 Comparison with the observation of QPEs

QPEs from GSN 069 were rapidly decaying and disappeared as TDE brightened
again (Miniutti et al., 2023b). Then, Miniutti et al. (2023a) found that it
had reappeared and was brighter than before. Orbital evolution under mass
transfer and gravitational radiation does not explain these variations, the
$\Delta M$ always increased rapidly or is constant, as shown in Fig. 1.

##  3 Evolution driven by the accretion disk

###  3.1 Drag force by the accretion disk

The drag force caused by the accretion disk can be expressed as

$\mathbf{a}=-\frac{\mathbf{v}_{rel}}{\tau_{F}}.$ (15)

Here $\mathbf{v}_{rel}=\mathbf{v}-\mathbf{v}_{k}$ is the relative velocity of
the star with respect to the local gas, where $\mathbf{v}$ is the speed of
star and the keplerian velocity $\mathbf{v}_{k}$ is the velocity of the local
gas. The timescale $\tau_{F}$ of the drag force, which is considered as
hydrodynamic force or dynamical friction. The hydrodynamic timescale is
(Szölgyén et al., 2022)

$\tau_{H}=\frac{2M}{C_{D}\pi R^{2}\rho_{g}v_{rel}}.$ (16)

Here $C_{D}$ is related to the shape of the star, and equals 1 for a sphere,
and $\rho_{g}$ is the density of the local gas. The timescale of the dynamical
friction is (Ostriker, 1999)

$\tau_{D}=\frac{v_{\mathrm{rel}}^{3}}{4\pi G^{2}M\rho_{g}F(\mathcal{M})},$
(17)

where $F(\mathcal{M})$ is

$F(\mathcal{M})=\begin{cases}\frac{1}{2}\ln\left(\frac{1+\mathcal{M}}{1-\mathcal{M}}\right)-\mathcal{M},&0<\mathcal{M}<1,\\\
\frac{1}{2}\ln\left(1-\frac{1}{\mathcal{M}^{2}}\right)+\ln\left(\frac{r}{R}\right),&\mathcal{M}>1.\end{cases}$
(18)

Here $\mathcal{M}=v_{\mathrm{rel}}/c_{s}$ is the Mach number where $c_{s}$ is
the local sound speed. And $r$ is the characteristic size, and we adopted the
distance from the central black hole.

The change of the specific angular momentum $h=\sqrt{G(M_{h}+M)a(1-e)}$ caused
by the drag force can be obtained by

$\frac{\dot{h}}{h}=-\frac{1}{h\tau_{F}}\mathbf{r}\times\mathbf{v}_{rel}=\frac{1}{\tau_{F}}\left(\frac{\chi}{\sqrt{1+e\cos
f}}-1\right),$ (19)

where $f$ is the true anomaly, and $\chi$ is 1 for the prograde star and -1
for the retrograde star. And the change of the specific energy
$K=-G(M_{h}+M)/2a$ is

$\frac{\dot{K}}{K}=-\frac{1}{K\tau_{F}}\boldsymbol{v}\cdot\boldsymbol{v}_{rel}=\frac{2}{\tau_{F}}\left(\frac{(1+2e\cos
f+e^{2})-\chi(1+e\cos f)^{3/2}}{1-e^{2}}\right).$ (20)

The secular evolution can be obtained by averaging them over an orbital
period. Then the secular evolution of orbital parameters are

$\left\langle\frac{\dot{a}}{a}\right\rangle_{disk}=-\left\langle\frac{\dot{K}}{K}\right\rangle,$
(21)
$\left\langle\dot{e}\right\rangle_{disk}=-\frac{1-e^{2}}{e}\left(\left\langle\frac{\dot{h}}{h}\right\rangle+\frac{1}{2}\left\langle\frac{\dot{K}}{K}\right\rangle\right),$
(22)

and the change of $\beta$ is

$\left\langle\frac{\dot{\beta}}{\beta}\right\rangle_{disk}=-\frac{1+e}{e}\left\langle\frac{\dot{h}}{h}\right\rangle-\frac{1-e}{2e}\left\langle\frac{\dot{K}}{K}\right\rangle.$
(23)

While near the periastron, the velocity of the orbiting star is greater than
the Keplerian velocity and is therefore slowed down, as it moves away from the
periastron, its velocity becomes less than the Keplerian velocity and is
therefore accelerated. Since the latter interaction time is longer, star will
likely gain angular momentum from the accretion disk and thus reduce the
$\beta$.

Eq. (23) can also be used to calculate instantaneous $\dot{\beta}/\beta$ for
different $f$, as shown in Fig. 4. Even for a star near the periastron,
$\dot{\beta}$ is still positive, so the drag force by the disk can always
suppress mass transfer.

Figure 4: $\dot{\beta}/\beta$ with true anomaly for different eccentricity.
For all eccentricity, $\dot{\beta}/\beta$ is always negative and equal to zero
while $f=0$.

###  3.2 Application on GSN 069

####  3.2.1 Stable mass transfer

Figure 5: Timescales with true anomaly for the hydrodynamic force and
dynamical friction for a WD with $M=0.21M_{\odot}$, $P_{b}=9\mathrm{~{}hr}$
and $e=0.95$. The upper panel is for the prograde star and the lower panel is
for the retrograde star. Solid lines and dashed lines represent the timescales
of the hydrodynamic force and dynamical friction, respectively. Different
colors represent the different accretion rate of the disk.

We consider an $\alpha$ standard disk with $\alpha=0.1$ and $M_{h}=5\times
10^{5}\mathrm{~{}M_{\odot}}$. The density and sound speed can be calculated
for a given accretion rate $\dot{M}_{Acc}$ in the unit of Eddington accretion
rate $\dot{M}_{Edd}$ (radiation efficiency $\eta=0.1$). As shown in Fig. 5,
the hydrodynamic force is always stronger than the dynamical friction in all
orbital phases, thus dominant on the effect by disk. We only consider the
hydrodynamic force in the following calculation.

Figure 6: Secular change of orbital parameters caused by hydrodynamic force
for a prograde WD with $M=0.21M_{\odot}$ and $P_{b}=9\mathrm{~{}hr}$ under
different accretion rate. The top panel shows the change of $\beta$, specific
angular momentum, and specific orbital energy caused by drag force, it also
shows the effect on $\beta$ caused by gravitational radiation for comparison.
The dashed black line represents the point with $\dot{M}_{Acc}$ of GSN 069 and
$e=0.95$, it predicts $\dot{\beta}/\beta\approx-3.5\times
10^{-4}\mathrm{~{}yr}^{-1}$(see texts). The bottom panel represents the change
of the $a$ and $e$ relative to $\beta$.  Figure 7: As in Fig. 6, but for a
retrograde WD. The sign of $\dot{\beta}/\beta$, $\dot{h}/h$,
$(\dot{a}/a)/(\dot{\beta}/\beta)$, and $\dot{e}/(\dot{\beta}/\beta)$ is
changed. To show value of $-\dot{e}/(\dot{\beta}/\beta)$ in the logarithmic
axis, we add 1 on them, because they change sign at high $\dot{M}_{Acc}$.

The secular evolution of orbital parameters is shown in Fig. 6. For varies
$\dot{M}_{Acc}$ and $e$, a prograde WD always gains angular momentum and loses
energy from the accretion disk, which decreases $\beta$, $a$ and $e$. A
retrograde WD shows different evolution, as shown in Fig. 7. The angular
momentum is transferred from it to the accretion disk, which makes $\beta$
increase. The energy is still transferred from the WD to the accretion disk.
They make $\beta$ increase and $a$ still decrease. And $e$ continues to
decrease at low $\dot{M}_{Acc}$ but increases at high $\dot{M}_{Acc}$.

The different dependence of the parameter evolution on $\dot{M}_{Acc}$ at high
and low $\dot{M}_{Acc}$ is because the inner region of the accretion disk is
dominated by radiation pressure at high $\dot{M}_{Acc}$ and gas pressure at
low $\dot{M}_{Acc}$. Therefore, the disk density decreases with increasing
$\dot{M}_{Acc}$ for radiation pressure dominance resulting in a decrease in
drag force, while the disk density increases with increasing $\dot{M}_{Acc}$
for gas pressure dominance increasing drag force.

A prograde WD is preferred for GSN 069. The WD may be captured from a binary
system by the Hill mechanism (Hills, 1988; Wang et al., 2022). The accretion
disk is likely to be a fallback disk from a TDE, which is possibly a
disruption of the common envelope of the binary system (Wang, 2023). In this
scenario, the WD is prograde.

Here we adopt the results in Miniutti et al. (2023b). Assuming radiation
efficiency $\eta=0.1$, the accreted mass is about $3.7\times
10^{-8}\mathrm{~{}M_{\odot}}$, corresponding to $\dot{M}/M\approx-1.7\times
10^{-4}\mathrm{yr}^{-1}$ and $\beta\approx 0.5005$ with
$M=0.21\mathrm{~{}M_{\odot}}$. The QPEs intensity is roughly halved per year,
corresponding to $\dot{\Delta M}/\Delta M=-0.5\mathrm{~{}yr}^{-1}$ in Eq. (2),
which requires $\dot{\beta}/\beta=-2\times 10^{-4}\mathrm{~{}yr}^{-1}$.

The change of $\beta$ induced by mass transfer is
$\langle\dot{\beta}/\beta\rangle_{\dot{M}}\approx 1.1\times
10^{-4}\mathrm{~{}yr}^{-1}$ with $\zeta=-1/3$. The bolometric luminosity of
TDE is about $10^{44}\mathrm{~{}erg/s}$ corresponding to $\sim
1.6\dot{M}_{Edd}$. Assuming $e=0.95$, it predicts
$\langle\dot{\beta}/\beta\rangle_{disk}\approx-3.5\times
10^{-4}\mathrm{~{}yr}^{-1}$ as shown in Fig. 6. Both of them give
$\dot{\beta}/\beta\approx-2.4\times 10^{-4}\mathrm{~{}yr}^{-1}$, which is
consistent with the required value. Consequently, the hydrodynamic drag force
can explain the decaying of the intensity of QPEs from GSN 069.

####  3.2.2 History and future of evolution

Early in the TDE, there is a high accretion rate of the fallback disk,
resulting in weak drag force by the disk. The evolution of $\beta$ is dominant
by the mass transfer or gravitational wave radiation. Thus, the intensity of
QPEs kept increasing during this stage. QPEs can be observed when the
intensity of QPEs is higher than TDE. It is consistent with the fact that QPEs
in GSN 069 do not accompany the TDE at the same time, but appear suddenly
during the decay of the TDE (Miniutti et al., 2023b).

As the accretion rate decreases, drag force by the disk begins to dominate
$\dot{\beta}$, and at that point the intensity of the QPEs begins to decrease.
As the TDE rebrighten, QPEs disappeared, and the increased accretion rate led
to a weaker drag force and thus an increase in the intensity of the QPEs. It
is consistent with the recent observation, which shows that QPEs in GSN 069
reappeared with a higher intensity than before they disappeared (Miniutti et
al., 2023a).

As the accretion rate continues to decrease, the effect of the drag force
enhances and the intensity of QPEs will keep decreasing. After switching from
radiation pressure dominance to gas pressure dominance in the inner region of
the disk, the drag force decreases with decreasing accretion rate. When
$\dot{\beta}$ is dominated by gravitational wave radiation, the mass transfer
begins to enhance. When $\dot{\beta}$ is dominated by mass transfer, the
unstable mass transfer leads to WD escape from the SMBH.

While the accretion rate decreases, the outer radius of the fallback disk may
change (Shen & Matzner, 2014). The orbital semi-long axis of the stripped WD
may be larger than the outer radius of the disk. In this situation, the drag
force by the disk acts only on that side of the periastron. The secular
evolution of the orbital parameters can be obtained by considering drag force
only at $-\kappa\pi<f<\kappa\pi$ when averaging them over an orbit. Here
$\kappa$ ranges from 0 to 1. As shown in Fig. 8, the secular evolution of
orbital parameters is significantly affected by $\kappa$, especially for
$\dot{\beta}/\beta$. When $\kappa<0.5$, the drag force only acts near the
periastron, thus the angular momentum of the stripped WD is transferred to the
disk as expected.

Figure 8: Distribution of secular change of orbital parameters for different
$\dot{M}_{Acc}$ and $\kappa$. The top, middle and bottom panels represent
$\dot{h}/h$,$\dot{K}/K$ and $\dot{\beta}/\beta$, respectively. They are
obtained with the drag force acting only when true anomaly $f$ is within
$-\kappa\pi$ to $\kappa\pi$. The solid line in the top panel represents the
$\dot{h}=0$.

Even though the change in the outer radius of the disk affects the evolution
of $\beta$, the tendency for the effect of the drag force by the disk to
diminish is constant. Therefore, $\beta$ will start to grow when the effect of
gravitational wave radiation exceeds the drag force by the disk. Finally, the
stripped WD will escape from SMBH after entering self-unstable mass transfer,
as described by Section 2.

##  4 Discussion

###  4.1 Unconservative mass transfer

It is also possible for material to be lost through the Lagrangian point L2,
and this is more likely to happen for systems with high eccentricity (Sepinsky
et al., 2007a). This lost material may form a circumbinary binary disk, which
may significantly affect the orbital evolution(Dermine et al., 2013). I will
not discuss this scenario here, but rather the case where this material leaves
the binary system altogether.

Differentiating the total angular momentum gives

$\frac{\dot{J}}{J}=\frac{\dot{M_{h}}}{M_{h}}+\frac{\dot{M}}{M}-\frac{\dot{M_{t}}}{2M_{t}}+\frac{1}{2}\frac{\dot{e}}{1+e}+\frac{1}{2}\frac{\dot{r}_{p}}{r_{p}}.$
(24)

Assuming the mass transfer occurs only instantaneously at the periastron, then
$\dot{r}_{p}=0$. The fraction of mass that is accreted to the central black
hole is defined as $\gamma=-\dot{M}_{h}/\dot{M}$. The mass lost by the system
also carries away angular momentum. Defining that the ratio of its specific
angular momentum to that of the stripped star is $\lambda$, the change of the
total angular momentum is $\dot{J}/J=\lambda\dot{M}_{t}/qM_{t}$. Then Eq. (24)
becomes

$\langle\dot{e}\rangle_{\dot{M}}=-\mathcal{F}(\gamma,q,\lambda)(1+e)\frac{\dot{M}}{M},$
(25)

where

$\mathcal{F}(\gamma,q,\lambda)=\frac{2+2(\gamma-1)\lambda+(1-\gamma)q-2\gamma
q^{2}}{1+q}.$ (26)

By $\dot{r}_{p}=0$, we have

$\left\langle\frac{\dot{a}}{a}\right\rangle_{\dot{M}}=\frac{\langle\dot{e}\rangle_{\dot{M}}}{1-e}=-\mathcal{F}(\gamma,q,\lambda)\frac{1+e}{1-e}\frac{\dot{M}}{M}.$
(27)

When $\gamma=1$, they reduce to Eq. (5) and (7).

Assuming that half the mass is lost from L2 and their specific angular
momentum is the same as that of the stripped star, i.e., $\gamma=0.5$ and
$\lambda=1$, Eq. (25) becomes

$\left\langle\dot{e}\right\rangle_{\dot{M}}=-\frac{1+0.5q-q^{2}}{1+q}\frac{\dot{M}}{M}(1+e)\approx-(1+e)\frac{\dot{M}}{M}\left(1-\frac{q}{2}\right).$
(28)

It is about half of what it is at $\gamma=1$, as is
$\langle\dot{a}/a\rangle_{\dot{M}}$. The change of $\beta$ now is

$\left\langle\frac{\dot{\beta}}{\beta}\right\rangle_{\dot{M}}\approx\frac{\dot{M}}{M}\left(\zeta-\frac{1}{3}-\frac{q}{2}\right),$
(29)

which is approximately same as Eq. (6) for $q\ll 1$.

In this scenario, the effect of mass transfer is about 2 times the effect of
the gravitational radiation in the balance state by comparing it with the case
of $\gamma=1$ as shown in Fig. 2. So the effect of mass transfer is still
dominant on the orbital evolution in the balance state for the main-sequence
star.

On the other hand, in the presence of a large-scale disk, material passing
through L2 is injected into the disk and then accreted by the black hole.
Therefore, the loss of matter through L2 does not affect the previous results.

###  4.2 Extreme mass ratio inspirals

Extreme mass ratio inspirals are the potential gravitational wave sources, and
they are expected to form when compact objects gradually inspiral via
gravitational wave radiation after being captured in a high eccentricity orbit
(Amaro-Seoane, 2018). Our results show that if the mass transfer is triggered
during this process, the inspiral will be terminated and thus fail to become a
gravitational wave source.

###  4.3 On the other QPEs

GSN 069 is not the only one galaxy found to have QPEs (Giustini et al., 2020;
Song et al., 2020; Arcodia et al., 2021; Chakraborty et al., 2021), but is the
only one with both TDE and QPEs (Shu et al., 2018; Sheng et al., 2021;
Miniutti et al., 2023b). The orbiting WD is likely captured from a binary via
the Hills mechanism (Hills, 1988; Wang et al., 2022), while the TDE possibly
originates from a disruption of a common envelope (Wang, 2023). In this
situation, the fallback disk is coplanar with the orbit of the common
envelope, which is almost coplanar with the orbit of the captured WD. Thus,
the orbiting star moves inside the disk for GSN 069.

As for the other QPEs sources, the situation may be different. The pre-
existing disk may not be present, or the pre-existing disk may be inclined to
the orbit of the stripped star. In the former case, the orbital evolution is
the same as discussed in Section 2. In the latter case, the effect of the disk
on the orbit will be different from that discussed in Section 3. For example,
Linial & Quataert (2024) found the hydrodynamic drag force received by the
star as it passes through the inclined disk can make the orbital period
decrease.

The TDE in GSN 069 makes it unique. On the one hand, it may be the only source
whose accretion disk is coplanar with the orbit of the stripped star. On the
other hand, the accretion rate of its disk is dominated by the TDE and
therefore evolves rapidly. These two points lead to a complicated orbital
evolution, making the QPEs in GSN 069 an excellent sample to use to study the
interaction between the disk and the star moving inside it.

##  5 Summary

By accounting for the effect of the mass transfer, the evolutionary path of
the stripped star is different from the evolutionary only with gravitational
wave radiation. The criterion of the unstable mass transfer is changed and the
fate of the stripped star is to escape the SMBH rather than be disrupted by
it. In this case, however, mass transfer remains unstable.

The TDE in GSN 069 may be a disruption of the common envelope before the WD
was captured, so the orbiting WD is likely moving inside the fallback disk.
The hydrodynamic drag force by the disk can effectively reduce $\beta$ and
stabilize the mass transfer, which can explain the observed decay in the
intensity of QPEs. After the TDE re-brightened, the mass transfer increases
due to the weakening of the drag force caused by the reduced density of the
inner zone of the accretion disk, which is consistent with the observed
increase in the intensity of the latest QPEs.

As the accretion rate of the fallback disk gradually decreases, the drag force
by the disk will decrease until it is negligible. Then the mass transfer will
start to be unstable eventually the WD will escape the SMBH. Continued
observations of GSN 096 will help us to understand the interaction of the disk
with the star inside it and to test our model. By accounting for the effect of
the mass transfer, the evolutionary path of the stripped star is different
from the evolutionary only with gravitational wave radiation. The criterion of
the unstable mass transfer is changed and the fate of the stripped star is to
escape the SMBH rather than be disrupted by it. In this case, however, mass
transfer remains unstable.

The TDE in GSN 069 may be a disruption of the common envelope before the WD
was captured, so the orbiting WD is likely moving inside the fallback disk.
The hydrodynamic drag force by the disk can effectively reduce $\beta$ and
stabilize the mass transfer, which can explain the observed decay in the
intensity of QPEs. After the TDE re-brightened, the mass transfer increases
due to the weakening of the drag force caused by the reduced density of the
inner zone of the accretion disk, which is consistent with the observed
increase in the intensity of the latest QPEs.

As the accretion rate of the fallback disk gradually decreases, the drag force
by the disk will decrease until it is negligible. Then the mass transfer will
start to be unstable eventually the WD will escape the SMBH. Continued
observations of GSN 096 will help us to understand the interaction of the disk
with the star inside it and to test our model.

This work was supported by the National Key Research and Development Program
of China (No. 2020YFC2201400) and National SKA Program of China
(2020SKA0120300).

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