# Lottery and Sprint: Generate a Board Game with Design Sprint Method on
AutoGPT

Maya Grace Torii  0000-0003-4025-9212
<EMAIL_ADDRESS>University of TsukubaTsukubaJapan ,
Takahito Murakami  0000-0003-2077-9747
<EMAIL_ADDRESS>University of TsukubaTsukubaJapan
and  Yoichi Ochiai  0000-0002-4690-5724<EMAIL_ADDRESS>University
of TsukubaTsukubaJapan

(2023; 2023-06-22; 2023-08-03)

###### Abstract.

In this paper, we introduce ”Lottery and Sprint”, a board game creation
methodology that cooperates human design intuition with the structured Design
Sprint framework executed by the AutoGPT system. By aligning AI-driven
processes with human creativity, we aim to facilitate a collaborative game
design experience. A user study is conducted to investigate the playability
and enjoyment of the generated games, revealing both successes and challenges
in employing systems like AutoGPT for board game design. Insights and future
research directions are proposed to overcome identified limitations and
enhance computational-driven game creation.

board game design, auto game creation, design sprint, generative AI

††copyright: acmlicensed††price: 15.00††doi:
10.1145/3573382.3623706††journalyear: 2023††submissionid:
playcomp23int-p1013-p††isbn: 979-8-4007-0029-3/23/10††conference: Companion
Proceedings of the Annual Symposium on Computer-Human Interaction in Play;
October 10–13, 2023; Stratford, ON, Canada††booktitle: Companion Proceedings
of the Annual Symposium on Computer-Human Interaction in Play (CHI PLAY
Companion ’23), October 10–13, 2023, Stratford, ON, Canada††ccs: Human-
centered computing Interactive systems and tools††ccs: Applied computing Arts
and humanities

##  1\. Introduction

Creating novel, enjoyable and effective board games typically requires a
detailed understanding of game mechanics, player engagement, and strategic
balance (Eck et al., 2017). Inexperienced individuals often face challenges in
designing board games that cater to various play styles, objectives, and
constraints (Mochocki and Milewski, 2018). To address this, we present the
”Lottery and Sprint” method. This approach allows human designers to work with
the AutoGPT system—an AI model based on Generative Pre-trained Transformers
(GPT)—wherein the Design Sprint framework is embedded into the AutoGPT’s
prompt mechanism. Through this method, the collaboration between human and AI
leads to the creation of innovative and balanced board games.

This research explores the application of the AutoGPT system in board game
design, detailing our ”Lottery and Sprint” approach. The process begins with
AutoGPT generating a variety of game designs in a metaphorical ”lottery”,
allowing users to select a favored design. Once a game is chosen, it enters a
customized Design Sprint (Knapp et al., 2016) process to refine and improve
the game concept. The central research question of this study is whether the
AutoGPT system, operating within the proposed framework, can effectively
create board games that are enjoyable, playable, and captivating, even for
individuals with limited experience in game design.

Our investigation encompasses a review of related works. We then present the
system design using AutoGPT and customized Design Sprint. This is followed by
a user study evaluating the generated board games in terms of playability and
entertainment by Creativity Support Index (CIS) and qualitative analysis.

##  2\. Related Works

Game creation tools have advanced significantly in recent years, with
techniques like Monte Carlo tree search enabling automated game content
generation (Chaslot et al., 2008). However, these systems often focus on
automation and may not effectively support human designers’ creative ideation.
For instance, GDL (Thielscher, 2017) employs complex formalisms that could
hinder non-experts’ intuitive expression of ideas (De Giacomo et al., 2020),
while Monte Carlo tree search in Regular Boardgames (Kowalski et al., 2019)
primarily targets AI techniques rather than assisting designers. As a result,
there is room for improved human-computer collaboration in the game design
process.

In this context, AI-driven game design has introduced new possibilities for
collaborative game creation using AI (Khalifa et al., 2019).

The rapid growth and advancements in GPT and Large Language Models (LLMs) have
expanded creative processes across various domains (Yang et al., 2023a). For
example, Buncho employs GPT-2 to enhance writers’ enjoyment in crafting novels
(Osone et al., 2021), and similar capabilities have been applied to adjust
video game level difficulty (Todd et al., 2023).

Considering these developments, AutoGPT, an open-source project for automatic
task decomposition and problem-solving, emerges as a promising candidate for
creative endeavors (Significant-Gravitas, 2023). Yang et al. conducted a
comprehensive benchmark study on AutoGPT agents in real-world scenario
simulations (Yang et al., 2023b). Building upon this foundation, our study
aims to explore AutoGPT’s potential in board game design through the ”Lottery
and Sprint” method. This approach seeks to facilitate collaboration between
human designers and AI, with the goal of creating engaging and playable board
games for individuals with limited design experience while maintaining a
rigorous analytical perspective.

##  3\. System Overview

Figure 1. Overview of the ”Lottery and Sprint” process, which is inspired by
the Design Sprint framework. Step 1, the Lottery phase (Prompt 1), consists of
(1) G1: AutoGPT surveying online board games, (2) G2: AutoGPT creating board
games based on user constraints, (3) G3: Reviewing and refining the generated
games, and (4) G4: Updating the board game according to the refining process.
This process follows a cyclical pattern, inspired by the flexibility and
iterative aspects of Design Sprint, and continues until the generated rules
are bug-free. Then, in G5, the single user (GM) selects a game from the
generated options then gives feedback to AutoGPT (Prompt 2). In Step 2, the
process follows a similar pattern, with (1) G1: GM reading user feedback in
detail, (2) G2: AutoGPT regenerating the selected game based on feedback
provided by the GM, (3) G3: Reviewing and refining the game again, and (4) G4:
Updating the board game according to the refining process. This cycle repeats
until the game is ready for test play (G5) with multiple user groups (GM and
players). In Step 3 and onward, the process repeats the pattern inspired by
Design Sprint as in Step 2, with feedback from test play being incorporated
into the game by the GM, leading to further refinement and updates with the
help of AutoGPT.

Figure 2. Using the creation prompt, 15 board games were randomly generated in
the same condition. The generated text was then evaluated through a TF-IDF
test, followed by Cosine Similarity, and the results are presented.

Using the creation prompt, 15 board games were randomly generated in the same
condition. The generated text was then evaluated through a TF-IDF test,
followed by Cosine Similarity, and the results are presented.

###  3.1. Design Sprint and Its Integration

Design Sprint is a time-bound, structured process that aims to solve complex
problems through design, prototyping, and testing ideas with users. Developed
by Google Ventures, it is a five-day process where each day is dedicated to a
specific phase: Understand, Diverge, Converge, Prototype, and Test (Knapp et
al., 2016; Araújo et al., 2019). In the context of the AutoGPT board game
creation system, the principles of Design Sprint are integrated into the
architecture, enhancing the efficiency and effectiveness of the game design
process. This integration can be mapped onto the four key stages of the
system’s architecture as follows:

1\. Research existing board games (Understand and Diverge): In this stage, the
system surveys existing board games, gathering insights and generating ideas,
similar to the Understand and Diverge phases of Design Sprint. The system
learns about diverse rule sets, game balance, and strategic depth to inform
its game design process. 2\. Create a board game with constraints (Converge
and Prototype): This stage corresponds to the Converge and Prototype phases of
Design Sprint. The system uses the insights gathered from the research stage
to create a new board game based on given materials and constraints, akin to
building a prototype for testing. 3\. Reflect on game design (Test): The Test
phase of Design Sprint is mirrored in this stage, where the system evaluates
the created game based on usability, rule clarity, design consistency,
strategic balance, and enjoyment. This serves as an internal evaluation
mechanism for the system’s output. 4\. Update and iterate (Iterate): The final
stage of the AutoGPT system emphasizes iteration, a core principle of Design
Sprint. Informed by the reflections from the previous stage, the system
refines and updates the game to ensure that all materials, instructions, and
game elements are clear and consistent.

###  3.2. Prompts

In the AutoGPT board game creation system(Fig. 1)
111https://github.com/DigitalNatureGroup/Lottery-and-Sprint, the four key
stages of the architecture are supported by two types of prompts, creation
prompts and feedback prompts. These prompts guide the AutoGPT system during
the design process and are divided according to their role within the system
architecture. Creation Prompts play a role in the first step of the system
architecture: they survey, create an initial draft of board game rules,
reflect on the created game, and update the game based on that reflection.

Prompt 1: Creation prompts

⬇

(G1) In order to make an interesting board game, survey board games focusing
on basic various rules of board game, how other games are keeping game
balance, and how other games exist.

(G2) Create a board game with the material and constraint written in text file
”<board game material and constraint>.txt”. Output the game explanation
including all information to play the game with Title, Materials, setup,
gameplay rules, game ending conditions, board layout and design, the unique
point, the enjoyable point and strategy to win the game as text file so you
can memorize or refer any time.

(G3) Step-by-step reflect the game you have created and consider if there are
any failure in following the materials and constraints, difficult instructions
for human to keep track while playing the game(usability in games), unclear
rule instruction for situation potentially to occur(rule design), failure in
rule, inconsistent in between the winning strategy and intention of the
game(board game design and game balance, game theory) and export your result
in text file. If there are not any problems, write so. use_gpt4 for better
reflection.

(G4) Reading the reflection, updating the game. Check if the game is perfect
and ready to play by taking the step in (G3).

Feedback Prompts are involved in the later stages of the system architecture.
In these stages, the system processes user feedback and updates the game
accordingly: it reads the feedback, recreates the board game, reflects on the
game, and then updates it.

Prompt 2: Feedback prompts

⬇

(G1) Read the board game rule which was created and written in ”<Target board
game>.txt”, material and constraints defined in ”<board game material and
constraint>.txt”.

(G2) You have feedback from players of the game in this file ”<feedback>.txt”.
Read the file and recreate the game. output and save the reflection to the
text file so you can refer any time.

(G3) Step-by-step reflect the game you have created by considering the
materials used (only defined materials should be used), rules or instructions
to be very clear for first time players to play, clear winning conditions.
output and save the reflection to the text file so you can refer any time.

(G4) Reading the reflection, updating the game by cooperating with the
reflections and output and saving the updated game to a text file so you can
refer any time. Make sure to include all the rules in the text file.

###  3.3. System Output and Evaluation

As defined in the prompts (Prompt 1), the system generates board game rules
including elements such as title, materials, setup, gameplay rules, game
ending conditions, etc11footnotemark: 1.

Our analysis shown in Fig. 2 evaluate the diversity of game rules generated by
the lottery phase using the AutoGPT system. We generated 15 different game
rules and analysed their similarity using NLTK for morphological analysis
(Loper and Bird, 2002) and the TF-IDF method for text vector representation
(Singhal et al., 2001).

By using cosine similarity, which ranges from 0 to 1, to compare the TF-IDF
representations of the game rules, we were able to evaluate the diversity of
the generated content over randomness. Our results show a mean cosine
similarity of 0.59, a standard deviation of 0.08, with minimum and maximum
cosine similarity values of 0.39 and 0.76 respectively. These values suggest
that there is at least a reasonable level of diversity in the proposed
designs, given the limitations of the data and constraints provided to users,
such as the limited number of players and items available.

##  4\. User Study

Figure 3. The user study design: Step1-board game preference survey
Step2-select a board game generated and give first feedback to the AutoGPT
system. Step3-playtest the game the first time and answer post-game play
survey. Step4-game master gives second feedback to the system. Step5-playtest
the game the second time and answer post-game play survey, final survey and
board game preference survey.

The user study design: Step1-board game preference survey Step2-select a board
game generated and give first feedback to the AutoGPT system. Step3-playtest
the game the first time and answer post-game play survey. Step4-game master
gives second feedback to the system. Step5-playtest the game the second time
and answer post-game play survey, final survey and board game preference
survey.

###  4.1. User Study Design and Methodology

The user study was designed as a two-game experiment. Each game consisted of a
playtest and a survey. Participants wrote feedback in the format of listing
”problems” and ”requirements” as a set in their native language, and the study
conductor then translated it into English for the AutoGPT system. The overall
view of the user study is explained in Fig. 3.

At the beginning and end of each phase, participants completed a survey
designed to gather quantitative and qualitative data on the AutoGPT system’s
usability, the playability of the generated games, and the overall enjoyment
and engagement of the games.

In the user study, 12 participants were grouped into three teams of four: one
volunteer Game Master (GM) and three players per team. The participants, with
ages ranging from 20 to 27 years and a balanced representation of both
genders, were recruited from the University of Tsukuba. Participants had
varied board game experiences, ranging from less than a year to over a decade,
with play frequencies from a few times a year to once a month. The GMs,
responsible for guiding sessions, chose a board game from three generated
rules, provided system feedback, and supervised playtesting. The players were
responsible for playtesting the game, providing feedback, and participating in
the discussion.

###  4.2. Data Collection

Data was collected through board game preference surveys, post-gameplay
surveys, final surveys, Creativity Support Index (CSI) (Carroll et al., 2009)
for Game Masters (GMs), and user study conductor observations. This approach
allowed us to gather insights into the usability, playability, and enjoyment
of the AutoGPT-generated games. Surveys were administered to gauge
participants’ initial expectations and subsequent impressions. Post-gameplay
surveys covered various aspects of game experience, while the final survey
allowed participants to reflect on their overall experience. The CSI (Carroll
et al., 2009) was used to evaluate the level of creativity support provided by
the AutoGPT system to the GMs. GM’s feedback, survey responses, and conductor
observations contributed to the data collection regarding the system’s
usability and game development process.

##  5\. Result and Discussion

Figure 4. (A) Quantitative results of “post-gameplay survey” GM- Q1 Overall,
the game was what I expected. Q2 the game rule was understandable. Q3 the rule
documents were written for easy to graphic design. Q4 The game had enough
strategic elements. Q5 the length of the game was appropriate. Q6 the rules of
the game were fair. Q7 the game had the originality. Q8 the choice of theme
for the game was what I expected. Q9 there was appropriate interaction with
other players. Q10 I could prepare a contingent event. Q11 There was adequate
player control elements in the game. (B) Player- Q1 I enjoyed. Q2 the game
rule was understandable. Q3 the rule documents were written for easy to
graphic design. Q4 the length of the game was appropriate. Q5 The game had
enough strategic elements. Q6 the rules of the game were fair. Q7 the game had
the originality. Q8 the theme of the game was great. Q9 The game had
interaction with other players. Q10 The game had contingent events. Q11 the
game had adequate control over what I could control in the game. Q12 I want to
play the game again. Q13 I want to play games with friends and family. (C)
Quantitative results of “final survey” GM- Q1 The board games output by GPT
were fascinating. Q2 the board games output by GPT were as expected. Q3
comparing the first and second sessions, GPT’s improved board game was as
expected. (D) Player- Q1 the board games were fascinating Q2 Comparing the
first and second sessions, the board game was improved. (E)Board of the board
game drawn by the participant for playtest.

Figure 5. (A)quantitative results of “board game preference survey”
Questions:Q1 Concepts Q2 Board graphic design Q3 Game themes Q4 Game strategy
Q5 Communications Q6 speed of progression Q7 Easy rule Q8 Randomness Q9
Controllable many objects Q10 Fairness rule. p ** ¡ 0.05. and (B) Average of
weighted factors and total CSI result.

(A)“board game preference survey” Questions:Q1 Concepts Q2 Board graphic
design Q3 Game themes Q4 Game strategy Q5 Communications Q6 speed of
progression Q7 Easy rule Q8 Randomness Q9 Controllable many objects Q10
Fairness rule. p ** ¡ 0.05. and (B) Avg. CSI test result.

Overall Assessment of the AutoGPT System: The results of the final survey
showed that participants provided positive evaluations of the AutoGPT board
game creation system. In the Game Master (GM) category, high scores were
recorded for Q1 (Median: 4.00, SD: 0.58), Q2 (Median: 4.00, SD: 0.00), and Q3
(Median: 4.00, SD: 0.00). Similarly, in the player category, high scores were
observed for Q1 (Median: 4.00, SD: 0.87) and Q2 (Median: 4.00, SD: 1.45) in
the final survey. These high scores indicate that the generated games met
participants’ expectations and that the AutoGPT system was effective in
creating board games that were generally well-received (Fig. 4).

The system scored an average of 77.58 (n=3) on CSI scores ranging from 0
(worst) to 100 (best). The highest factor was Collaboration with an average of
30.67, followed by Exploration with an average of 27.67. Collaboration
evaluates if the system allowed other people to work with the user easily, and
Exploration evaluates if the user could explore many different ideas, options,
designs, or outcomes. Expressiveness had the lowest score of 5.29, evaluating
if the user was able to be very expressive and creative while doing the
activity. This result is supported by a comment from the final GM survey,
where the respondent was asked to identify which aspects of the system they
found convenient or supportive and any difficulties they encountered: ”I can’t
predict what kind of instructions make what kind of difference.”

Comprehensibility of Game Rules: In the post-gameplay survey, both GMs and
players reported increased median scores for the understandability of game
rules in the second game compared to the first game. GMs showed an increase in
median score from 2.00 in the first game to 3.00 in the second game (SD1:
1.00, SD2: 1.00), and players displayed an increase in the median score from
2.00 to 3.00 as well (SD1: 1.33, SD2: 0.97). However, this improvement could
be attributed to participants playing the game twice with similar rules. It
remains unclear whether the generated game itself enhanced its
comprehensibility (Fig. 4). Several participants provided qualitative
comments, with one noting, ”It would be better if the content was a bit more
detailed.” The lack of clarity in the generated rules was also identified as a
potential factor affecting comprehensibility.

Strategic Elements and Fairness: The evaluation of strategic elements revealed
a discrepancy between GM and player perceptions, with GMs reporting a higher
score in the second game, increasing the median score from 3.00 to 4.00 (SD1:
1.53, SD2: 0.58). In contrast, players’ evaluations remained unchanged, with
their median score staying at 3.00 in both the first and second game (SD1:
1.00, SD2: 1.27). This suggests that the improvement in strategic elements,
based on GM input and expectations, did not translate into an enhanced
experience for the players. Similarly, evaluations of fairness improved for
GMs, with an increase in the median score from 2.00 in the first game to 4.00
in the second game (SD1: 1.73, SD2: 2.08); however, player evaluations did not
exhibit a significant change between the first and second games (Median1:
3.00, Median2: 3.00, SD1: 1.27, SD2: 1.23). This suggests that only GMs were
satisfied with the improvements in fairness while players did not share the
same sentiment (Fig. 4). Player feedback, such as ”The game balance isn’t
fully adjusted,” supports this observation. Even with GMs having the authority
to adjust rules, ensuring universally accepted fairness was challenging.

Originality and Game Theme: The AutoGPT system successfully delivered novel
and original game concepts, as evidenced by consistently high scores provided
by both GMs and players. In the post-gameplay survey, GMs recorded median
scores of 4.00 for both games (SD1: 0.58, SD2: 0.00), and players reported
median scores of 4.00 for both games as well (SD1: 1.24, SD2: 0.87). In
contrast, evaluations of game theme showed differing opinions between GMs, who
perceived a positive improvement, and players whose assessment declined in the
second game (Fig. 4). GMs offered qualitative comments such as, ”It’s
unpredictable how instructions will alter the game,” and ”It brings ideas
humans might not immediately consider.” These remarks highlight the system’s
unique ability to introduce originality in game creation.

Contingency and Board Graphic Design: The decrease in player evaluations of
contingency in the post-gameplay survey, with the median score dropping from
4.00 in the first game to 3.00 in the second game (SD1: 1.13, SD2: 1.30), and
the increased importance placed on contingency in the board game preference
survey indicate that contingent or random events play a crucial role in the
overall enjoyment of generated board games (Fig. 5).

Furthermore, high-quality graphic design may not be as essential to board game
enjoyment as initially thought (Fig. 4A,B,D). The evaluation of board graphic
design in the ”board game preference survey” (Q2) changed, with the median
score decreasing from 4.50 (SD1: 0.89) to 4.00 (SD2: 1.22)(Fig. 5). Despite
its simplicity, the hand-drawn board provided an engaging gaming experience.
This finding aligns with Kieran Hicks et al.’s observation that visual
embellishments do not always improve the user experience (Hicks et al., 2019).

##  6\. Limitation and Future Work

Our study of the AutoGPT system for board game creation showed promising
results. However, there are some limitations to address, and we suggest
potential areas for future research. First, the system had difficulty creating
complete game board designs, often resulting in issues with overlapping
components and missing elements. Future work could focus on improving the
graphical capabilities of the AutoGPT system for better board designs. Second,
sometimes the system generated games with contradictory rules or materials not
allowed by the given constraints. To make the games more playable, future
research should focus on refining the system’s ability to understand
constraints and provide clear rules. Lastly, the limited working memory of the
AutoGPT system prevented it from researching existing board games thoroughly.
Enhancing its working memory by developing database system could help the
system better understand existing games and create more engaging designs.

##  7\. Conclusion

In conclusion, our study demonstrated the potential of the AutoGPT system in
generating engaging and original board games. By addressing limitations and
refining the system’s capabilities, it can better match players’ expectations
and deliver satisfying gaming experiences. Future work can focus on enhancing
the system for various game types, contributing to the advancement of human-
computer interaction in the gaming industry. Overall, the AutoGPT system shows
promise in revolutionizing the game creation process and offering enjoyable,
well-rounded board games for diverse audiences and gaming preferences.

###### Acknowledgements.

The manuscript was drafted using OpenAI ChatGPT and GPT-4. The AI generated
text was read, revised and proofed carefuly by the authors. This work was
supported by Grant-in-Aid for JSPS Fellows Grant Number JP23KJ0265.

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# Terminating a common envelope jets supernova impostor event with a super-
Eddington blue supergiant

Tamar Cohen Department of Physics, Technion, Haifa, 3200003, Israel;
<EMAIL_ADDRESS><EMAIL_ADDRESS>Noam Soker
Department of Physics, Technion, Haifa, 3200003, Israel;
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We conducted one-dimensional stellar evolutionary numerical simulations to
build blue supergiant stellar models with a very low-envelope mass and a
super-Eddington luminosity of $10^{7}L_{\odot}$ that mimic the last phase of a
common envelope evolution (CEE) where a neutron star (NS) accretes mass from
the envelope and launches jets that power the system. Common envelope jets
supernovae (CEJSNe) are CEE transient events where a NS spirals-in inside the
envelope and then the core of a red supergiant (RSG) star accretes mass and
launches jets that power the transient event. In case that the NS (or black
hole) does not enter the core of the RSG the event is a CEJSN-impostor. We
propose that in some cases a CEJSN-impostor event might end with such a phase
of a blue supergiant lasting for several years to few tens of years. The
radius of the blue supergiant is about tens to few hundreds’ solar radii. We
use a simple prescription to deposit the jets’ energy into the envelope. We
find that the expected accretion rate of envelope mass onto the NS at the end
of the CEE allows the power of the jets to be as we assume, $10^{7}L_{\odot}$.
Such a low-mass envelope might be the end of the RSG envelope, or might be a
rebuilt envelope from mass fallback. Our study of a blue supergiant at the
termination of a CEJSN-impostor event adds to the rich variety of transients
that CEJSNe and CEJSN-impostors might form.

supernovae: general; transients: supernovae; stars: jets; stars: neutron;
binaries (including multiple): close

##  1 Introduction

In a common envelope jet supernova (CEJSN) event a neutron star (NS) or a
black hole (BH) spiral-in inside the envelope of a red supergiant (RSG) star
and then inside its core, accrete mass and launche jets (e.g., Papish et al.
2015; Soker & Gilkis 2018; Gilkis et al. 2019; Grichener & Soker 2019a; López-
Cámara et al. 2019, 2020; Soker 2021; Grichener 2023). The jets explode the
star and power the transient event. In this study we deal with the CEJSN
impostor scenario (Gilkis et al., 2019), where the NS/BH spirals-in inside the
envelope of the RSG but does not merge with, and hence does not destroy, the
core (Gilkis et al., 2019).

The two opposite jets that the NS/BH launches as it accretes mass from the
envelope collide with the outer envelope and with circumstellar material that
early jets and/or the common envelope evolution (CEE) process have removed.
The post-shock jet’s gas and the post-shock ambient gas form a ‘cocoon’ (e.g.,
López-Cámara et al. 2019). Photons diffusing out from the cocoon form a bright
transient event, a CEJSN (or CEJSN-impostor) event (e.g., Schreier et al.
2021) that mimics a peculiar core collapse supernova (CCSN; e.g., Soker,
Grichener, & Gilkis 2019). Examples include the unusual gamma-ray burst GRB
101225A that Thöne et al. (2011) explain by a NS that merged with a helium
star and the enigmatic supernovae iPTF14hls (Arcavi et al., 2017) and SN
2020faa (Yang et al., 2021) that might be be CEJSN events (e.g., Soker &
Gilkis 2018). Adopting the CEJSN scenario and processes from these earlier
studies, Dong et al. (2021) followed these studies and suggested the CEJSN
scenario for the luminous radio transient VT J121001+495647. CEJSNe might be
the sites of some extreme-condition processes, like a fraction of the
r-process nucleosynthesis in the Universe (Grichener & Soker, 2019a, b;
Grichener, Kobayashi, & Soker, 2022; Grichener & Soker, 2022) and the site of
some fraction of the very high-energy neutrinos in cases of a BH companion
that spirals-in inside the RSG envelope (Grichener & Soker, 2021). Even before
entering the CEE the NS/BH might accrete mass at a high rate and launch jets
that produce very high-energy neutrino (Sridhar, Metzger, & Fang 2022; they
term this phase hypernebula, Sridhar & Metzger 2022).

The large energy of the jets that the NS/BH launches in CEJSNe is the main
difference from many other binary interaction routes with and without CEE
(e.g., Han et al. 2020) and from the much less powerful jets that a main
sequence companion might launch in a CEE (e.g., Shiber, Schreier, & Soker
2016). The high energy-jets also distinguish the CEJSN scenario from related
NS/BH scenarios of CEE without jets (e.g., Thorne & Zytkow 1977; Fryer &
Woosley 1998; Zhang & Fryer 2001; Barkov & Komissarov 2011; Thöne et al. 2011;
Chevalier 2012; Schrøder et al. 2020).

The peculiar transient AT2018cow (Prentice et al., 2018) and the CEJSN-
impostor scenario for AT2018cow-like fast blue optical transients (FBOTs)
motivate our study. AT2018cow-like FBOTS are bright transients, which might be
brighter than superluminous CCSNe (e.g., Margutti et al. 2019), and with a
rise time of only few days (e.g., Prentice et al. 2018; Perley et al. 2019).
Their kinetic energy is $\simeq 10^{51}-10^{52}{~{}\rm erg}$, they possess
velocity components of up to $\gtrsim 0.1c$ (e.g., Coppejans et al. 2020),
they might show hydrogen spectral lines (e.g., Margutti et al. 2019), they
might display rapid X-ray variability (e.g., Pasham et al. 2021; Yao et al.
2022), have a dense circumstellar material (CSM; e.g., Nayana & Chandra 2021;
Bright et al. 2021), and tend to occur in star-forming galaxies (e.g.,
Prentice et al. 2018).

There are several scenarios for AT2018cow-like FBOTs (e.g., Liu et al. 2018;
Fox & Smith 2019; Kuin et al. 2019; Lyutikov & Toonen 2019; Margutti et al.
2019; Quataert, Lecoanet, & Coughlin 2019; Yu et al. 2019; Leung et al. 2020;
Mohan, An, & Yang 2020; Piro & Lu 2020; Uno & Maeda 2020; Kremer et al. 2021;
Xiang et al. 2021; Chen & Shen 2022; Gottlieb, Tchekhovskoy, & Margutti 2022).
Soker, Grichener, & Gilkis (2019) proposed the polar CEJSN channel for
AT2018cow. In this scenario early jets remove most of the envelope material
along the polar directions, allowing late jets to expand almost freely to
large distances, accounting for the fast rise and decline of FBOTs and for the
$>0.1c$ velocities.

Soker (2022) considers the polar CEJSN-impostor scenario for AT2018cow, i.e.,
a scenario where the NS companion does not destroy nor enter the core of the
RSG. According to the CEJSN-impostor scenario many FBOTs become the
progenitors of binary compact objects mergers (NS/BH-NS/BH merger). In this
study we adopt this scenario that has four main phases (for schematic
illustrations see figures in Soker 2022). (1) A RSG star expands and tidally
interacts with a NS/BH companion. The interaction enhances the mass loss rate
of the RSG star to form a CSM. (2) Just before the onset of a CEE, months to
few years before explosion, the NS/BH launches jets as it accretes mass. The
jets inflate two opposite lobes at $\approx 100{~{}\rm AU}$ inside the CSM.
(3) The binary system enters a CEE that ejects most of the envelope in a dense
equatorial outflow. (4) The CEE ends with a circumbinary disk around the
binary system of the NS/BH and the core of the RSG star. The NS/BH accretes
mass from the circumbinary disk and launches energetic jets. The collision of
the jets with the CSM in the two lobes powers an FBOT event. In a later time
(long after the FBOT event) the core remnant of the RSG collapses to form a
NS/BH in a CCSN explosion.

In this study we consider another possible phase of evolution after the FBOT
event, where the NS and the core orbit each other inside a a newly rebuilt
very low-mass common envelope. Namely, after the RSG envelope contracted to
leave the NS outside the envelope at the termination of the main CEE phase,
further evolution of the core and/or accretion of fallback material, most
likely from the circumbinary disk before it disappears or from equatorial
fallback beyond the circumbinary disk, caused the RSG envelope to re-expand
and engulf the NS again. This leads to a second CEE phase, now with a very low
mass envelope. We note that a secular evolution (stellar evolution that does
not include extra ingredients) does not lead by itself to a second CEE phase
with a NS, but rather to a stable mass transfer, Case BB mass transfer (e.g.,
Tauris, Langer, & Podsiadlowski 2015; Tauris et al. 2017; Vigna-Gómez et al.
2018). To form a second CEE phase requires additional process(es), that might
be rare. One such effect might be the same effect that causes some massive
stars to experience outbursts years to months before they explode, i.e.,
strong core activity. Another possibility is that the rapid mass removal by
the NS in the first CEE phase, due to the jets, causes the envelope to over-
contracts. The envelope then expands in a time scale of months which is the
thermal time of the outer envelope zones alone. Soker & Bear (2023) find that
the rapid removal of mass from a $7M_{\odot}$ star might reduce its radius to
$2.1R_{\odot}$ and that within about year it re-expands to $2.5R_{\odot}$ with
an expansion rate that rapidly decreases. We leave it to future studies to
find the most likely scenario to form a second CEE.

The envelope density is too low to force the NS and the core to merge and
practically the additional spiralling-in is negligible. . However, the
accretion energy onto the NS, via jets, powers this common envelope system for
several years to tens of years. The motivation to propose this phase of
evolution comes from the observation by Sun et al. (2022) who detect a
luminous source at $\simeq 2-3$ years after the outburst of AT2018cow that is
consistent with a hot, $T>5\times 10^{4}{~{}\rm K}$, and luminous,
$L>10^{7}L_{\odot}$, black body radiation source. In a very recent study Chen
et al. (2023a) also study and discuss this very hot black-body source. (for
other recent studies of AT2018cow see, e.g., Zhang et al. 2022; Chen et al.
2023b; Sun et al. 2023). We describe this scenario in section 2. In section 3
we simulate a spherically symmetric stellar model with energy deposition that
mimics the powering jets, and in section 4 we calculate the power of the jets
to show self-consistency. We summarize this study is section 5.

##  2 The scenario

Sun et al. (2022) reported the detection of a luminous source $\simeq 2-3$
years after the outburst of AT2018cow that is consistent with a hot,
$T>5\times 10^{4}{~{}\rm K}$, and luminous, $L>10^{7}L_{\odot}$, black body
radiation source. Chen et al. (2023a) derive black-body temperatures and radii
at $t\simeq 703{~{}\rm days}$ and $t\simeq 1453{~{}\rm days}$ of
$(R,T)\simeq(14.5R_{\odot},3\times 10^{5}{~{}\rm K})$ and
$(R,T)_{\simeq}(30R_{\odot},6.4\times 10^{4}{~{}\rm K})$, respectively. We
note that they disfavour a stellar source for this emission.

We examine the following scenario in the frame of the CEJSN impostor scenario
for FBOTs (Soker 2022; section 1) to form such a source. While the CEJSN
impostor scenario refers to the powering of the event in the first several
months of the main outburst, in this study we refer to a later phase. In the
CEJSN impostor scenario the powering of the main outburst event is by a NS
that accretes mass from a post-CEE circumbinary disk after the RSG envelope
contracts to below the core-NS separation. This circumbinary disk survives for
a timescale of $\tau_{\rm CBD}\approx 1-100~{}{\rm day}$ (eq. 4 in Soker
2022). In this study we consider a later phase that occurs after the
circumbinary disk does not exit anymore, namely, at about a year and later.

The basic additional assumption is that there is a low-mass re-expanded common
envelope remnant that rebuilds itself in a time scale of several months.
Either the CEE did not remove the entire RSG envelope such that after several
months of further evolution of the RSG core the RSG envelope re-expands and
engulfs the NS for the second time, or after several months fallback material
forms a low-density common envelope that re-engulfs the NS. We discussed these
possibilities in section 1. We will not study them in the present paper. This
paper should motivate the search for scenarios that allow a second CEE with a
low-mass rebuilt envelope.

The NS that is re-engulfed by the re-expanded RSG envelope, orbits now, for
the second time, inside the low density envelope remnant of the RSG star.
Therefore, the NS launches jets while accreting mass from the envelope. These
jets collide with the envelope and power the common envelope remnant. For a
spherical photosphere with a uniform temperature the radius of the RSG remnant
star is $R=42(L/10^{7}L_{\odot})^{1/2}(T/50000{~{}\rm K})^{-2}R_{\odot}$. It
is actually a blue supergiant at this phase.

For the rest of the paper we consider this late phase of evolution that
follows the main FBOT outburst. In this phase the re-expanded low-mass RSG
envelope engulfs the NS and a second phase of CEE takes place.

We consider a model where instead of a spherical uniform photosphere the star
is larger and the jets heat only small polar caps. For example, for a stellar
radius of $R_{\ast}\simeq 120R_{\odot}$ an uniform effective temperature of
$T_{\rm s}\simeq 30000{~{}\rm K}$ gives $L_{\ast}=10^{7}L_{\odot}$. Because we
are limited to spherically symmetric stellar models by the numerical code
(section 3) we cannot simulate the hot caps. Therefore, the stellar model
represents a star where the zones around the equator are cooler than $T_{\rm
s}$ while two hot cups spanning an angle from the pole to about $20^{\circ}$
from the pole are at $T_{\rm cap}=50000{~{}\rm K}$. Larger stars with even
smaller caps are also possible.

In the CEJSN impostor scenario the powering of $L>10^{7}L_{\odot}$ comes from
a NS that accretes mass via an accretion disk and launches two opposite jets.
We follow Grichener, Cohen, & Soker (2021) and Hillel, Schreier, & Soker
(2022) and take for the power of the jets that the NS launches the expression

$\dot{E}_{\rm 2j}=\zeta\frac{GM_{\rm NS}\dot{M}_{\rm BHL,0}}{R_{\rm NS}},$ (1)

where for the analytical Bondi-Hoyle-Lyttleton accretion rate we take

$\dot{M}_{\rm BHL,0}=\pi\rho(a)v_{\rm NS}(a)\left[\frac{2GM_{\rm
NS}}{v^{2}_{\rm NS}(a)}\right]^{2},$ (2)

and where we take for the relative velocity of the NS and the envelope,
$v_{\rm NS}$, the Keplerian velocity of the NS inside the envelope at its
orbital radius $a$. For the NS mass and radius we take $M_{\rm
NS}=1.4M_{\odot}$ and $R_{\rm NS}=12{~{}\rm km}$, respectively, throughout
this study.

Note that according to the CEJSN impostor scenario during the main FBOT
outburst, which lasts for up to several months, the NS accretes mass from a
post-CEE circumbinary disk. Here we study a second CEE phase that occurs after
the main FBOT outburst, where the low-mass RSG envelope engulfs the NS for a
second time. The NS is deep inside the re-expanded envelope, and so we
consider the BHL accretion process.

Three factors determine the parameter $\zeta$ inside a common envelope (see
Grichener, Cohen, & Soker 2021 and Hillel, Schreier, & Soker 2022 for
derivation and simulations): the ratio of the accretion rate obtained in
numerical simulations to that of the analytical BHL value $\dot{M}_{\rm BHL}$,
$\xi\simeq 0.1-0.5$ (e.g., Livio et al. 1986; Ricker & Taam 2008; Chamandy et
al. 2018; Kashi, Michaelis, & Kaminetsky 2022), the effect of the jets in
further reducing the the mass accretion rate (the jet feedback mechanism)
$\chi_{\rm j}\simeq 0.1-0.2$ (Grichener, Cohen, & Soker 2021; Hillel,
Schreier, & Soker 2022), and the ratio of the jets’ power to the total
accretion power $\eta\simeq 0.1$ (e.g., Schrøder et al. 2020), as neutrinos
carry most of the accretion energy. These values give

$\zeta\equiv\eta\chi_{\rm j}\xi\approx 10^{-3}-10^{-2}.$ (3)

Lower values are also possible, e.g., due to lower values of $\xi<0.1$ that
some simulations claim for (e.g., MacLeod & Ramirez-Ruiz 2015a; MacLeod &
Ramirez-Ruiz 2015b). Hillel, Schreier, & Soker (2022) simulated values of
$\zeta\leq 10^{-4}$, partly due to numerical limitations.

As we will show in section 3, a typical density at several solar radii from
the core is $\simeq 10^{-7}{~{}\rm g}{~{}\rm cm}^{-3}$. In our scenario the NS
accretes mass from the envelope only as it orbits the core at several solar
radii. The core mass of the models we build later is $M_{\rm core}\simeq
6.5M_{\odot}$. Substituting the mass and radius of the NS that we use here and
scaling by the other variables, equation (1) reads

$\displaystyle\begin{aligned} \dot{E}_{\rm 2j}&=2.95\times
10^{7}\left(\frac{\zeta}{0.001}\right)\left(\frac{M_{\rm core}+M_{\rm
NS}}{8M_{\odot}}\right)^{-3/2}\\\
&\times\left(\frac{a}{10R_{\odot}}\right)^{3/2}\left(\frac{\rho(a)}{10^{-7}{~{}\rm
g}{~{}\rm cm}^{-3}}\right)L_{\odot}.\end{aligned}$ (4)

The mass accretion rate by the NS is

$\displaystyle\begin{aligned} \dot{M}_{\rm NS}&=1.2\times
10^{-4}\left(\frac{\xi\chi_{\rm j}}{0.01}\right)\left(\frac{M_{\rm
core}+M_{\rm NS}}{8M_{\odot}}\right)^{-3/2}\\\
&\times\left(\frac{a}{10R_{\odot}}\right)^{3/2}\left(\frac{\rho(a)}{10^{-7}{~{}\rm
g}{~{}\rm cm}^{-3}}\right)M_{\odot}{~{}\rm yr}^{-1}.\end{aligned}$ (5)

We conclude that in principle a NS in a CEE inside the low-mass envelope
remnant of a RSG that accretes mass and launches jets might power the central
source that Sun et al. (2022) and Chen et al. (2023a) observe at the location
of AT2018cow. At this phase the star is a blue supergiant. One outcome is that
this luminosity is much above the Eddington luminosity of the system, leading
to a rapid evolution. We discuss this point in section 5.

We end by noting that the spiralling-in process of the NS-core binary system
inside the re-expanded low-mass envelope is limited to supply the energy for a
time period of $\tau_{\rm SI}<(GM_{\rm core}M_{\rm NS}/2R_{\rm
core})/10^{7}L_{\odot}$. In our model the radius of the core is $R_{\rm
core}\simeq 0.6R_{\odot}$ (after envelope removal it expands somewhat) and we
find $\tau_{\rm SI}\lesssim 3{~{}\rm yr}$. It is possible that a later
evolutionary phase when the core is smaller can in principle support such a
powering, but we consider it less likely because the density that is required
to force the NS to spiral in implies an accretion energy as well. In any case,
the re-expanded envelope is of low mass and the spiralling-in timescale is
much longer than several years. This low density envelope, which leads to a
very slow spiralling-in, is still sufficient to allow powerful accretion.

##  3 Numerical procedure

We evolve stellar models with a zero age main sequence (ZAMS) mass of $M_{\rm
ZAMS}=20M_{\odot}$ and with two different values of the initial metalicity
$Z=0.0016$ and $Z=0.02$, with version r22.05.1 of the stellar evolution code
Modules for Experiments in Stellar Astrophysics (mesa; Paxton et al. 2011,
2013, 2015, 2018, 2019).111 The default capabilities of mesa relay on the MESA
EOS that is a blend of the OPAL (Rogers & Nayfonov, 2002), SCVH (Saumon et
al., 1995), FreeEOS (Irwin, 2012), HELM (Timmes & Swesty, 2000), PC (Potekhin
& Chabrier, 2010), and Skye (Jermyn et al., 2021) EOSes. Radiative opacities
are primarily from OPAL (Iglesias & Rogers, 1996), with low-temperature data
from Ferguson et al. (2005) and the high-temperature, Compton-scattering
dominated regime by Poutanen (2017).

At an age of $t_{\rm S}=9.68\times 10^{6}{~{}\rm yr}$ of the low-metalicity
($Z=0.0016$) model when the RSG core mass is $M_{\rm core,S}=6.5M_{\odot}$,
the RSG radius is $R_{\rm S}=225R_{\odot}$, and its luminosity is $L_{\rm
S}=2.6\times 10^{5}L_{\odot}$ we remove most of its hydrogen-rich envelope and
leave a small remnant envelope mass $M_{\rm PS}$. This numerical rapid mass
removal mimics the initial phase of the CEJSN impostor event when the NS
spirals from the RSG surface to deep inside its envelope, a process that
strips most of the hydrogen-rich envelope. We conduct the same procedure for
the high metalicity ($Z=0.02$) stellar model at the age of $t_{\rm
S}=8.85\times 10^{6}$ when the RSG core mass is $M_{\rm core,S}=6.4M_{\odot}$,
its radius is $R_{\rm S}=229R_{\odot}$, and its luminosity is $L_{\rm
S}=1.8\times 10^{5}L_{\odot}$.

After numerically (artificially) removing the mass we mimic the energy that
the jets that the NS launches deposit to the envelope at a power of
$\dot{E}_{2j}=0.99\times 10^{7}L_{\odot}$. Together with the nuclear burning
in the core the total luminosity of the star at thermal equilibrium is
$L_{\ast}\simeq 10^{7}L_{\odot}$. We deposit the energy to the envelope from
an inner radius $R_{\rm Din}$ to an outer radius $R_{\rm Dout}$, and with a
constant power per unit mass. This energy deposition prescription is not
unique, but we take it to reasonably describe the energy that the jets deposit
in the envelope as they propagate out starting from several solar radii from
the core of the giant.

The envelope rapidly expands to a stellar radius of $R_{\ast}\gtrsim
1000R_{\odot}$. Due to the large radius and high luminosity the wind mass loss
rate (physical mass loss rate, not an artificial one) is high and the low-mass
envelope losses mass and contracts on a timescale of hundreds of years (the
mass loss rate in mesa is based on Glebbeek et al. 2009 ,Nugis & Lamers 2000
and Vink, de Koter, & Lamers 2001). We present the post-stripping envelope
mass $M_{\rm env}$ as function of the stellar radius $R_{\ast}$ for five cases
in Fig. 1. In Fig. 2 we present the evolution of $M_{\rm env}$ and of
$R_{\ast}$ with time $t_{\rm PS}$. The post-stripping time $t_{\rm PS}$ is
measured from the end of rapid numerical mass removal. The mass evolution of
the high metalicity models is much faster than the low-metalicity models. This
is expected as the mass loss rate of the high metalicity models is much higher
according to the mass loss prescription.

Figure 1: Post-stripping envelope mass as function of stellar radius for five
cases. The inset shows the post-stripping mass $M_{\rm env}$ and the
metalicity $Z$ in the form $(M_{\rm PS},Z)$. The thin and thick lines are for
cases with $Z=0.0016$ and $Z=0.02$, respectively. Solid lines represent cases
where we inject the energy to the envelope from $R_{\rm Din}=1R_{\odot}$ to
the stellar surface $R_{\rm Dout}=R_{\ast}$. The red-dashed line represent the
case where we inject the energy between $R_{\rm Din}=1R_{\odot}$ and $R_{\rm
Dout}=0.5R_{\ast}$. The post-stripping time $t_{\rm PS}=0$ corresponds to the
end of the numerical rapid mass removal. The early phase from numerical mass
removal and until the radius shrinks to $R_{\ast}\simeq 300R_{\odot}$ is not
physical and results from the numerical procedure to build the stellar model.
Figure 2: The evolution with time of post-stripping envelope mass (upper
panel) and the stellar radius (lower panel) after the numerical rapid mass
removal. Meaning of lines is as in Fig. 1. We are only interested in the
models after the stellar radius is $R_{\ast}<300R_{\odot}$.

Our goal with the above numerical procedure is to build a stellar model with a
luminosity of $L_{\ast}\simeq 10^{7}L_{\odot}$ and a radius of $R_{\ast}\simeq
40-300R_{\odot}$, as we described in section 2. In reality there is no mass
removal and then energy deposition, but rather the jets that the NS launches
deposit energy during the entire CEE. Therefore, in what follows we deal only
with the post-stripping evolutionary phase after the stellar radius becomes
$R_{\ast}<300R_{\odot}$.

##  4 Stellar properties

As we discuss in section 2 we want to present the possibility of a stellar
model with a radius of $R_{\ast}\simeq 40-300R_{\odot}$ and a luminosity of
$L_{\ast}\simeq 10^{7}L_{\odot}$. We here present the evolution of five such
stellar models that we built as we describe in section 3. To facilitate
comparison of the different models we set the time in what follows to be $t=0$
when the stellar radius is $R_{\ast}(t=0)=300R_{\odot}$.

In Fig. 3 we present the evolution of the stellar radii and luminosities with
time. We see that the rate of radius decline depends on the metalicity and on
the energy deposition prescription. The luminosities are negligibly smaller
than $10^{7}L_{\odot}$, showing that during the relevant phase of evolution
the envelope does not add nor remove much energy as its mass and radius
decrease.

Figure 3: Comparing the evolution with time of the stellar radii (upper panel)
and luminosities (lower panel) of the 5 models that we study here, focusing on
the stellar shrinking from $R_{\ast}=300R_{\odot}$ to $R_{\ast}=40R_{\odot}$.
Meaning of lines is as in Fig. 1. We arbitrarily set the time here to be $t=0$
when the stellar radii are $R_{\ast}(t=0)=300R_{\odot}$.

However, even the high metalicity models (thick lines) and the model where we
deposit energy in the inner half radius of the envelope (dashed line) might
hold an inflated luminous envelope for several years, or even longer, if some
ejected envelope gas falls back to the star. This would imply a practically
lower mass loss rate.

In Fig. 4 we present the density profiles of the five models when the stellar
radius of each model is $R_{\ast}\simeq 150R_{\odot}$. We use the density
$\rho(r)$ at a radius $r$ to calculate the power of the two jets that a NS at
radius $r$ launches according to equation 4. In Fig. 5 we present the power of
the two jets as function of the NS location and for $\zeta=0.001$. We note
that the value of $\zeta\simeq 10^{-3}-10^{-2}$ (equation 3) is highly
uncertain, but is expected to be in this range as we discuss section 2.

Figure 4: The density profiles of the five models when their radii are about
similar at $R_{\ast}\simeq 150R_{\odot}$. Meaning of lines is as in Fig. 1.
Figure 5: The power of the two jets that the NS launches according to equation
(4) with $\zeta=0.001$ as function of the NS location in the envelope for the
density profiles that we present in Fig. 4.

The graphs in Fig. 5 show that with $\zeta$ in the expected range the power of
the jets can be as we take in the simulations $\dot{E}_{\rm 2j}\simeq
10^{7}L_{\odot}$. In that respect our simulations are self-consistent
according to the scenario we propose here.

##  5 Discussion and Summary

We conducted stellar evolutionary simulations as we described in section 3 to
build blue supergiant stellar models with a luminosity of $\simeq
10^{7}L_{\odot}$ and radii of $R_{\ast}\simeq 40-300R_{\odot}$. The low-mass
envelope and very high luminosity mimic a possible final phase of the CEJSN-
impostor scenario that we propose here (section 2). This last phase of the
CEJSN-impostor event (that does not happen in all cases) lasts for few years
to several tens of years. During this time the NS accretes mass from the low-
mass envelope and launches jets that power the system according to equation
(4).

The main findings of our simulations are as follows (section 4). (1) We could
build a stellar model with a low envelope mass of $M_{\rm env}\simeq
0.05-0.1M_{\odot}$, a radius of $R_{\ast}\simeq 40-300R_{\odot}$, and
luminosity of $L_{\ast}\simeq 10^{7}L_{\odot}$ (Fig. 4). (2) The envelope can
last for few years to tens of years (Fig. 3). This time period can be longer
even if the blue supergiant accretes fallback material, implying a practically
lower mass loss rate. In an extreme case the main CEE might remove the entire
envelope, and a later fallback process rebuilds the low-mass envelope (we did
not simulate this case here). In any case, the luminosity is highly super-
Eddington and therefore this stellar model cannot last for a long time. (3)
The jets’ power that we assume in building the envelope in the models,
$\dot{E}_{\rm 2j}=0.99\times 10^{7}L_{\odot}$, is consistent with the expected
jets’ power in the CEJSN-impostor scenario as we showed in Fig. 5. For a given
envelope structure, the jets’ power depends on the location of the NS in the
envelope $r$, and on the efficiency parameter $\zeta$ (equation 3).

We mentioned in section 2 that our first motivation to study this final CEJSN-
impostor phase came from the finding of Sun et al. (2022) of a luminous source
$\simeq 2-3$ years after the outburst of the FBOT AT2018cow ( also Chen et al.
2023a). Sun et al. (2022) argue that it is consistent with a hot, $T>5\times
10^{4}{~{}\rm K}$, and luminous, $L>10^{7}L_{\odot}$, black-body radiation
source. However, it is not clear that this scenario is compatible with the
properties of AT2018cow. The reason is that AT2018cow has X-ray variability
that suggests that we do see the central engine. An optically thick giant
envelope obscures the central engine.

Nonetheless, we showed that in some cases the final phase of CEJSN-impostor
events might be a blue supergiant with low-mass envelope and high-luminosity
that is powered by a mass-accreting NS.

## Acknowledgements

We thank Aldana Grichener and Dmitry Shishkin for very helpful discussions and
comments and an anonymous referee for improving the manuscript. This research
was supported by a grant from the Israel Science Foundation (769/20).

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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