# Open Player Modeling: Empowering Players through Data Transparency

Jichen Zhu* and Magy Seif El-Nasr* University of California at Santa Cruz
Santa Cruz, CA, USA
<EMAIL_ADDRESS>* Both authors contributed equally to the paper. IT
University of Copenhagen
Copenhagen, Denmark
<EMAIL_ADDRESS>
###### Abstract

Player modeling is an area that has received much interest over the past
decade in Games, where applications investigating its utility in
personalization, player segmentation and interest understanding. Despite the
recent growing recognition that transparency is important to users, most
existing player models remain to be blackbox to players. In this paper, we
synthesize ideas, from the research communities around Intelligent User
Interface and Learning Science, about giving interactive systems the ability
to show users the models of themselves. We thus present a new emerging
research area called Open Player Modeling. We define its design space and
present exciting open problems that the games research community can explore.
We conclude the paper with a case study and discuss the potential value of
this approach.

###### Index Terms:

eXplainable AI, Player Modeling, Transparency

††publicationid: pubid: 978-1-7281-4533-4/20/$31.00 ©2021 IEEE

##  I Introduction

As Artificial intelligence (AI) and machine learning (ML) are increasingly
adopted in a wide range of everyday digital products, there is a rapidly
growing demand to make the underlying algorithmic processes and models more
transparent. As a response, new research areas such as eXplainable AI (XAI)
and Interactive Machine Learning (IML) have emerged to investigate how to make
AI and ML models less opaque and more interpretable [fails2003interactive,
Gunning2017, amershi2014power]. Researchers have made progress to open the
blackbox of AI and ML algorithms in a wide range of domains, including image
recognition, natural language processing, and personalized apps. However, how
to apply current XAI and IML techniques to end-user products remains an open
research problem [stumpf2009interacting, kulesza2013too].

In computer games, many AI techniques are integrated into modern video games,
such as AI opponents and procedural content generation. Among them, an active
and core research area is player modeling [yannakakis2013player]. Player
models allow human analysts to gain further insights into different player
types, which can help different stakeholders in the game development process
[el2016game]. Player modeling can also be used to adapt the games’ content and
experience based on individual players’ needs and preferences
[Kantharaju2018AIIDE], enabling developers to add recommendation systems, or
support eSports coaching or feedback systems that are tuned to the specific
players’ skills and preferences.

Despite the wide adoption of player modeling in games, the research of
explanations or transparent player models is still nascent. Typically player
models are hidden from regular players and only accessible to the game AI
and/or game analysts and researchers who are experts in interpreting
behavioral data. With the exception of a few preliminary attempts to make game
AI more transparent to players [hooshyar2020transparent] or game designers
[Zhu2018], there is a lack of systematic effort to investigate opening the
blackbox of player models and its potential benefits. In this paper, we
synthesize existing work around the theme of making player models more
transparent to players and identify open problems.

In this paper, we propose a new research area — Open Player Modeling (OPM) —
to make computational models of players accessible and more transparent to the
players themselves. This would involve answering questions such as WHAT data
or models would be useful for players, HOW would such data be communicated,
and WHEN to shown the open models to players in the gameplay experience.
Further, making such models transparent would also help explain game AI system
decisions, especially when it comes to recommendations and personalization
based on player modeling. This may help engender players’ trust in game AI,
improve the player experience, and make game-based interventions more
effective.

While we draw connections to prior work in open models outside games (e.g.,
intelligent user interfaces, intelligent tutoring systems), we identify the
key reasons why the domain of computer games poses unique challenges to
developing open models. Our core position is that open player modeling is a
promising research area that can potentially empower players to learn from
their own data, increase their trust in AI, and enhance the effectiveness of
some games as interventions. For example, in educational games, OPM can be
used to increase players’ reflection on their learning strategy and self-
regulate learning more effectively. In e-Sports, OPM can be used to enhance
the experience of spectators and to help players learn from other players’
gameplay strategies. And similar to interactive Machine Learning [Fails2003],
in some cases making player models open is the first step towards allowing
players to edit their own models, thus improving model accuracy. In this
paper, we illustrate the usefulness of OPMs through a case study showing how
an OPM can be applied to an educational game on parallel programming. In the
context of educational games, we argue that OPM can shed light on the learning
and problem-solving process which takes on a more constructionist approach.

The contribution of this paper is the following:

  * •

We identify Open Player Modeling as a promising new research area by
synthesizing related research on Player Modeling, Intelligent User Interfaces
(IUI), and eXplainable AI (XAI), which are currently unconnected.

  * •

We provide a theoretical framework to map out the design/problem space of open
player modeling and identify key open problems.

  * •

We offer a case study of how OPM can be used to enhance players’ experience
and capabilities.

We hope this paper can help position the discussion of this new area within
the broader context of XAI and Games. Similar to recent movements in XAI and
interactive machine learning [amershi2014power], making player modeling more
transparent to players can improve their trust, enjoyment, and performance.

##  II Related Work

User modeling is commonly used in many areas for data mining, data analytics,
and automatic content adaptation. The domain of games is no exception. Player
modeling can be used to detect, model, and predict human player
characteristics which are manifested through cognitive, affective, and
behavioral patterns [yannakakis2013player]. It is essential to game analytics
[el2016game], personalized adaptive games [zhu2020player], recommendation
systems [chen2018q] and player experience research. However, the vast majority
of user and player models are hidden from humans or designed for trained
analysts [nguyen2015glyph]. In this section, we discuss existing work that
attempts to make them more open to end users and players.

###  II-A Open User/Student Modeling

Most research effort on showing users computational models of themselves has
so far concentrated around the Intelligent User Interfaces community. Most
notably, research on Open User Modeling [Brusilovsky2011] aimed to make the
process and products of user models explicit and available to users
themselves. Existing research found that open user models can improve a user’s
understanding of the system [Knijnenburg2012, Gretarsson2010] and encourage
reflection on user data and actions [bull1995extending, bull2004supporting,
Gretarsson2010]. When applied to the context of web-based e-learning, studies
showed that open user modeling, sometimes referred to as open student
modeling, helped students improve self-reflection and the meta-cognitive
processes [Hsiao2013, Hsiao2017, Hsiao2012, Hsiao2012a, Law2017, Kump2012,
Brusilovsky2011]. In addition to individual open models, recently, researchers
started to explore the social aspects of open user modeling. Motivated by
social comparison theory [suls1991social], Hsiao and Brusilovsky [Hsiao2017]
proposed what they called an open social student models by making a student’s
open model visible to other students going through the same learning
experience. In a user study, they observed that students “spent more time on
[the system], attempted more self-assessment quizzes, and explored more
annotated examples” (p. 10).

Open player modeling extends the main idea behind open user/student modeling
to computer games. The majority of existing work on open user/student modeling
is based on adaptive web applications where hyperlinks are the primary source
of user interaction. Because player experience and player interactions with
games tend to be more complex than with web applications, so are the data used
to build player models. Games can represent complex dynamic systems and
players tend to interact with games in longer duration, higher frequency, and
more varied interaction types (e.g., navigation, puzzle-solving, and narrative
apprehension). For example, gameplay interactions in educational games
potentially contain rich data that can be used to depict learners’ strategy
[valls2015exploring], knowledge acquisition process [baker2007modeling],
affect [conati2002modeling], etc. We argue that, due to this increased
complexity, open player models face unique challenges in terms of how to open
the models and how to make the resulting information useful to the players.

###  II-B Open Models in Games

While the concept of open player models is nascent, there have been a few
examples of work in different sub-areas within games that started to explore
the idea of visualizing players’ behavior. Most relevant, researchers recently
proposed the idea of Transparent Player Models [hooshyar2020transparent]. They
adapted research in Open Learner Models into educational games and proposed to
use learning analytics and visualization to expose learner models to players.
While similar to [hooshyar2020transparent], our work encompasses a broader
scope of player modeling, including MOBA games, health games, and other
serious games, as well as educational games. Further, our work is situated in
games research, specifically looking at player modeling as part of the player
experience and questioning how players can benefit from viewing models of
their gameplay data.

Figure 1: Simulation results for the network.

Traditional player models, a component of the AI systems in games, are used
directly by the game system to, for example, adapt the gameplay experience.
They are typically not designed to be open to humans. As game user research
develops, post-hoc player models start to be developed and visualized for user
researchers, game analysts, game business teams, and game developers. Since
these models are intended to be used by highly trained experts, not the
average players themselves, they are often highly technical and do not focus
on general usability and user experience.

The most developed applications of open player models for non-technical
stakeholders are dashboards of player performance. In their most
straightforward forms, they were broadly used in modern computer games to
display player performance stats, such as win rate, quality of solutions in
puzzle games, and average speed in racing games. More recently, with the
increased popularity of eSports, more sophisticated dashboards have been
developed. Such dashboards have also been extended to be used to show
spectators stats of what is happening within the match [koomen2020esports,
kokkinakis2020dax, charleer2018real, kriglstein2020part]. Example dashboards
are shown in Fig. 1. Recently there has been an increase in companies emerging
in this area, e.g., SenpAI and Mobalytics, where they focus on developing such
dashboards, as well as embedding other AI and player modeling systems, such as
recommendation systems targeting different kinds of MOBA games.

On the research side, there has been several works investigating the
development of dashboards for various games (e.g., [charleer2018real],
[kokkinakis2020dax], [koomen2020esports], [robinson2017all]). For example,
Charleer et al. [charleer2018real] sought to explore the effect of dashboards
on spectators’ experience and how they were used to gain insight into the
game. They ran surveys to study the value of the different metrics to help
design their dashboards. Based on this study they developed specific
dashboards geared towards the different games studied and evaluated their
usefulness. They found that the dashboards were useful in that they
facilitated analysis and interpretation, but some participants wanted access
to data as they liked to dig into the data themselves and took pride in doing
so. Further, researchers discussed issues of trust and warned about the
complexity of the visualizations as well as issues of cognitive load.

Another interesting example is the work of Robinson et al. [robinson2017all],
where they developed a tool called ‘All the Feels’ (ATF), where they
investigated displaying streamer’s physiological data, specifically heart
rate, skin conductance, and facial emotion recognition, to streamers. The
results of their evaluation show an increased enjoyment, where participants
mentioned that they quite often viewed the ATF tool. However, 82.3% found it
distracting to some degree. They reported that compared to the stream without
the ATF, ATF produced higher consistent viewership time. But the main
contribution is in the fact that most spectators felt more connected to the
streamer when streams were viewed with ATF than without. Post interviews with
the streamer also showed no issues with exposing their emotions that way
through the tool.

##  III Open Player Modeling

We propose the new term Open Player Modeling to capture a range of practices
of showing players computational representations of themselves (i.e., player
models) in the context of play. These models can vary in their computational
complexity — for example, ranging from players’ in-game performance through
descriptive statistics (e.g., average speed) to cognitive, affective, or
behavioral models of player characteristics (e.g., learning preferences in
educational games).

Figure 2 shows the key components of OPM. First, player’s log data is
processed through a player modeling module to produce a player model which is
then interpreted and then shown to the user. In some cases, human experts are
involved to provide labeling and domain knowledge to the modeling process.
Next, these models are explained and communicated to the players and other
stakeholders (e.g., game analysts) through texts, visualization, or other
medium. In cases of complex player models with low interpretability, such as
deep neural network-based player models, extra effort is needed in producing
explanations to the underlying models before they can be communicated to the
players.

Figure 2: Components of Open Player Modeling

###  III-A Open Player Modeling as a New Research Area

As shown above, there are pockets of knowledge about open models in different
research communities. So is OPM simply an extension of these prior works of
OLM and OUM in a different domain, or is it fundamentally different from them?
Below, we argue that there are three key aspects of OPM that make it the
latter and worth further research investigation.

First, player data is typically more complex than the type of user data
applied to OLM and OUM research. As summarized above, most existing OLM and
OUM research are conducted based on how users interact with hypermedia
applications. In comparison, players interact with games with higher frequency
and longer duration. A 2020 international
report111https://www.limelight.com/resources/white-paper/state-of-online-
gaming-2019/#spend, last accessed on January 31, 2021. show that games around
the world spend an average of 7.11 hours per week playing games. It also
reports that mobile phones remain the primary gaming platform, indicating that
gameplay takes place in a wide range of contexts. Furthermore, the report
shows that although gamers typically play for an average of 1.37 hours at a
time, many of them play for longer sessions. For instance, U.S. players’
average longest consecutive play sessions are 5.10 hours. Globally the
equivalent figure is 4.31 hours. All the above statistics indicate the size
and complexity of player data.

Second, due to this data complexity, player modeling techniques are different
from those used in other open models. Due to the fact that play experience
often involves time and virtual space, most player models often involve a way
to deal with the time and space continuum. Some models aggregate over these
dimensions [drachen2014comparison], while others specifically look into
sequence analysis methods that span time and space (e.g., [sifa2018profiling,
kleinman2020and, pfau2018towards, min2014deep]).

Third, most player models will integrate some way of measuring or integrating
preference, emotions, and engagement (some examples of these models have been
discussed in Yanakakis et al.’s paper and book [yannakakis2013player,
yannakakis2018artificial]), which are all important factors in modeling game
experiences. These are important for personalizing the play experience and
also for developing game content. Due to the influence of game characters on
team performance within a game like DoTA or League of Legends, such aspects
will also need to be modeled (e.g., [chen2018art]). Such types of models have
been exemplified in many of the MOBA AI types of recommendation systems,
including in Senpai.gg and Mobalytics.gg, which recommend items, characters,
or other aspects of gameplay that are based on player state modeling.

Fourth, how players interact with OPM may be different from other open models,
imposing new research questions for open models in the context of play.
Players’ desire to improve their in-game performance and win the game may
provide them with the extra motivation to use OPM. As we can see from the
relatively simple OPMs such as player stats and scoreboards, open player
models can be used as indicators of player performance. OPMs designed for
reflection can thus be perceived by players as an essential tool to analyze
their gameplay and come up with ways to improve their performance. This can
create a broad appeal for OPMs and increase their adoption. Compared to target
users of other open models, gamers are more familiar with viewing information
about their activities. Many games already use player stats to communicate how
well each player is doing. In some complex games, players have to make sense
of a wide range of stats and use them to inform gameplay decisions. Having
this background means players may be more prepared to make use of more
sophisticated open models in ways they may not be outside games. It thus makes
OPM an exceptionally suitable domain to investigate eXplainable AI problems of
communicating complex AI decisions to end-users.

###  III-B Design Space of Open Player Modeling

This section maps the problem space of OPM along two main dimensions: the
nature of the player models and their level of openness.

####  III-B1 Types of Player Models

There are different ways that different models in the literature have been
categorized. For example, Yanakakis et al. [yannakakis2013player] categorized
computational player models in terms of their computational structure defining
them as model-based techniques, techniques that embed a theoretical or expert
model that is used to derive the player model, model-free techniques, are
those that construct a model based on data without a specific knowledge or
theory, and the last type of techniques are those that are hybrid and mix
model-based and model-free techniques. Another approach to categorize models
is the approach borrowed from statistics or engineering, where models are
categorized into: (a) descriptive, where data is used to show or describe what
has happened, this can be done at different levels of abstraction, (b)
predictive, where data is used to predict an outcome or a state, (c)
diagnostic, which extends a descriptive model to specifically answer specific
questions, and (d) prescriptive extends predictive models to prescribe what to
do next.

In this paper, we propose a different view of player models. Specifically, we
look at models from the players’ perspective and categorize them into the
following:

  1. 1.

Prediction Models: models that offer a guess on a state of the game or the
player [yannakakis2013player]. Some of these models may target the prediction
of an outcome, e.g., skill level or win/loss (e.g., [kang2020sports,
lan2018player, norouzzadeh2017predicting]. Other models may predict players’
state emotions (e.g., [yannakakis2013player]) or churn (e.g.,
[castro2015churn, tamassia2016predicting]). Other models may predict players’
next action or a goal based on observed behavior [min2014deep,
harrison2014survey, harrison2011using]. These models can show the players
these predictions pre-, during, or post-play.

  2. 2.

Reminder Models: models that offer the player a certain reminder. These models
tend to be context-aware and use expert knowledge to show specific information
to players based on their play and the data analyzed. A good example of this
model can be seen in the Senpai.gg tool.

  3. 3.

Recommendation Models: These models recommend items, characters, or specific
elements to player pre- or during play (e.g., [chen2018q, chen2018art,
bertens2018machine].

  4. 4.

Descriptive Models: These models tend to describe a system or a phenomenon,
constraints, complexity, and interaction. In the context of games, an example
of a descriptive problem-solving model provides the sequence of behaviors
taken to solve a puzzle in a level (e.g., [ahmad2019modeling,
kleinman2020and]). Some may include prompts for reflection or learning. Models
can vary in terms of the abstraction level that they encompass.

####  III-B2 Level of Openness

We classify OPMs into the following levels of openness based on how much of
player modeling process is revealed to the players and whether players can
modify their models.

  1. 1.

Open Model Outcome has the lowest level of openness. It communicates to
players the result of the player models, without detailed information of how
the models were calculated from the players’ gameplay behavior. For example,
an OPM of an adaptive game similar to Left4Dead may communicate to the player
that its model of her is that she is a highly-skilled player and therefore the
game is sending her more difficult enemies. Notice this type of OPM does not
contain information about what player metrics it uses to compute the user
model and how. It focuses on showing the outcome of the model. The main
benefit of this type of OPMs is its simplicity. Especially when players are
engaging in time-critical gameplay, such as combat, extra information of the
model can be overwhelming.

  2. 2.

Open Model Process reveals the computational process through which the outcome
of player models are calculated from the input of player behavior. For
instance, this type of OPM can use visualization to display the clusters of
players based on their gameplay and show a player where her gameplay is
situated in them. It can also be used to explain certain predictions the game
AI made (e.g., probability of winning a game). The main benefit of this type
of OPMs is that they can help players understand WHY and HOW they are modeled
in a particular way.

  3. 3.

Editable Open Models affords the highest level of openness. In addition to
making the player model open to the players, they allow players to modify
their own models. When designed well, editable models can improve the accuracy
of the models by allowing players to make corrections. For instance, editable
OPMs can be used in adaptive games for physical re-habitation patients. The
game can model a player’s joint limitations based on her movements in the
gameplay and other information (e.g., age) in order to generate targeted
exercises. However, it there is an error in the player model, it could likely
lead to pain and additional injury. Through Editable Open Models, players can
see their own models and make appropriate corrections so that the game can
provide the most suitable exercises. Since Editable Open Models allow players
to direct modify the player models, there is a higher requirement for the
players to understand how the models function and therefore provide the
correct editions. Additional design considerations are also needed to align
players’ interest with the model accuracy.

###  III-C Open Problems

This section discusses the main open problems that are key to open player
modeling.

####  III-C1 Make Player Models Explainable

Opening player models to players requires the underlying player models to be
explainable, especially for the ones with Open Model Process and Editable Open
Models. The game AI community has developed a large body of work to model
players, but the vast majority of them are used as a blackbox. A key open
problem for OPM research is how to open the technical blackbox of player
modeling. For player models that use AI modeling techniques that have low
interpretability (e.g., deep learning), how can we still offer insights to
players about why the player model classified them in a certain way. For other
models, how can we choose the appropriate level of abstraction to reduce the
noise of low-level actions and highlight the main trends that are useful to
differentiate among the different players?

####  III-C2 Communicate Open Models to Players

Once we have an explainable player model, research is needed to effectively
communicate the model to players. Currently, information visualization has
been a useful tool to convey player stats. As open player models can contain
more complex information, it opens up new research problems for visualization
and multi-modal explanations (e.g., visualization, text, and voice). What is
the information that the players want to know about their own models? How can
we visualize high-dimensional player data and gameplay trace? For OPMs of open
model outcome, how can we help players to establish the connection between
their gameplay actions and the model outcome? For OPMs of open model process,
how can we provide relevant information without overwhelming the player? And
for editable open models, can we provide sufficient scaffolding so that the
players can provide more informed edits to the model?

####  III-C3 Design Player-AI/Model Interaction in the Eco-System of Games

Compared with other eXplainable AI domains, a unique aspect of OPM is that it
needs to be embedded in the rich player interaction in computer games. This
allows us to investigate new OPM and explainable AI research questions in a
wide variety of contexts. For example, we can explore when and how to provide
OPM information to players in the entire player experience. In open learner
modeling literature, the models are often displayed at the end of a
pedagogical module for reflection.However, in educational games, it is
possible that OPM can be used to provide guidance and encourage reflection
during the gameplay. How can we address different player needs at various
stages of the game and incorporate OPM accordingly? Can we even align the
gameplay incentives with OPM so that players are motivated to provide accurate
information to the editable player models?

##  IV Case Study

Below we discuss a case study and position it within the dimensions discussed
above. In particular, we discuss our ongoing work of a case study that uses
descriptive models of player actions and focuses on Open Model Outcome OPM to
encourage player reflection in an educational game. Note this work is still
ongoing and the validation of this model has yet to be measured.

###  IV-A Context: Learning Game called Parallel

Parallel is a single-player 2D puzzle game, designed to teach parallel and
concurrent programming core concepts, especially non-determinism,
synchronization, and efficiency, to Computer Science undergraduate students
[zhu2019programming, ontanon2017designing] (Figure 3). In each level, arrows
(the equivalent of threads) move along a pre-determined set of tracks (a
program). A player places semaphores (wait function in parallel programming)
and buttons (signal) in order to direct arrows to pick up packages and deliver
them to the designated delivery points. In addition, the game contains
directional switches to represent conditional statements. In essence, the
player designs a synchronization mechanism to coordinate threads executing in
parallel. Once the player successfully delivers all required packages, she
wins and moves to the next level.

Because the arrows move at random speeds in each simulation, the puzzles in
Parallel are non-deterministic. The game allows learners to test their
solutions with one configuration of arrow movement schedule at a time before
they submit their final solution. At that point, the game will systematically
check if the solution works in all configurations. The complexity of the
puzzles can scale up from simple exercises of one concept to well-known
parallel programming challenges, such as the classic Smokers Problem and the
Dining Philosophers Problem [downey2009little]. This means that, as the game
progresses, the problem-solving space for each level is large. A level can
require long sequences of player actions, thus making it difficult for players
to reflect without additional scaffolding.

Figure 3: A screenshot of the game Parallel. The two arrows represent threads.
Players need to place semaphores and signals to direct arrows, which carry
packages and move along predefined tracks, to the designated delivery points.

From an in-class user study [zhu2020understanding], we observed the difficulty
for students to think in a multi-threaded way. Specifically, players tend to
assume the same problem-solving strategy even when they encountered problems.
It often leads to stagnation in learning. Based on this observation, we
hypothesize that enabling players to see their own problem-solving patterns
through OPM, especially in the context of others’, can encourage reflection,
motivate them to explore other problem-solving strategies in the game, and
hence enhance learning.

###  IV-B Open Player Modeling for Parallel

####  IV-B1 Player Model Type

We believe that the use of OPM here can potentially enable players to reflect
on their own problem-solving patterns and explore new ways when comparing them
with other players’. Here we decided to use a descriptive model, where low-
level gameplay actions, such as mouse clicks and button presses, are
abstracted into higher-level actions such as level test, placing a semaphore,
linking a semaphore to a signal, etc., which focuses on decision-making and
problem-solving strategies. A key challenge here is that, directly showing
players’ every gameplay action will be overwhelming in terms of the number of
actions as well as the complexity of the action sequences. Yet the established
technique of showing aggregated player stats hides necessary information about
problem-solving strategies. This challenge creates a rich domain for OPM to
explore how to model player action sequences and how to make them open and
transparent to players for reflection [villareale2020reflection].

####  IV-B2 Level of Openness

Since our goal is to improve players’ reflection and learning, we intend our
player model to be open in a way that players can use to compare their
gameplay/problem-solving process with others’. We decided to explore Open
Model Outcome as the foundation type of OPM. In our case, the model is the
individual players’ problem-solving processes, expressed as sequences of
abstracted player actions during a level.

We use visualization, through the Glyph visualization system, to communicate
the OPM to the players. Glyph [nguyen2015glyph], developed to display traces
of players’ sequences of actions, supports comparison across many players. It
is composed of a dual-view interface that shows data from two related
perspectives: a state graph and a sequence graph. A state graph shows a
sequence of abstract behaviors and abstracted states. As shown in Figure 4
Left, this state graph representation is a node-link graph. The nodes
represent different game states — a game state is a context of the game where
a player action is taken. The links between these nodes are actions that a
player took to get from one state to another.

To facilitate comparison of individual action sequences, we augmented the
state graph with a synchronized sequence graph showing the popularity and
similarity of sequence patterns exhibited by users see Figure 4 Right. Each
node in the sequence graph represents a full play trace, the size is an
indication of popularity, i.e. how many players made these actions. Further,
the distance between each node provides a visual representation of
similarity/dissimilarity.

For this particular level, the Glyph-based OPM revealed three distinctive ways
that players in our dataset have solved it. The sequence graph (Figure 4
Right) shows three distinct clusters of the gameplay sequences. Since distance
here is used to show how different the sequences are from each other, the
three clusters indicate three distinct solutions. A player can locate her
gameplay sequence and see how it relates to others.

Figure 4: A screenshot from Glyph where a user highlighted four different
sequences or patterns for solving this game level. These patterns are
highlighted in red, blue, green, orange shown in the sequence graph (right)
with the corresponding traces shown in the state graph (left).

To inspect this even further, the visualization system allows users to
interact with the graphs, highlighting different sequences to inspect the
decision-making steps in detail. Figure 4 shows such rendering of four
different paths towards solutions for the level. This system then demonstrates
an example Open Player Model - visualizing a descriptive process-oriented
Model. One can thus imagine a player who may be struggling with a level to
inspect the data from another player who was successful and follow through the
sequence: “added a wait statement for red arrow”, “tested and failed”,
“adjusted the wait statement for the red arrow”, “tested and passed”.
Visualizing such patterns can give the player a way to inspect their own
actions, reflect on them and also compare them to others. The specificity of
the design and the visualization will depend on the model used.

We are currently tackling several open problems. First, we are investigating
whether Glyph-based visualizations are interpretable for regular players. It
is possible that players may find the state graph too difficult to understand.
Through user studies, we will test and refine appropriate techniques that are
informative and understandable to our target player group. There needs to be a
balance between the expressivity of the visualization and its
understandability. Second, we are also exploring the type of interactions and
features needed to make our OPM useful for reflection. For example, we
anticipate using annotations to allow users to leave notes or reflections on
detailed problem-solving processes. Further, we also plan to use AI techniques
to guide attention to similar sequences or add filtering options. These
interactions will need to be tested through user studies. Third, we also need
to investigate the type of player model to construct for an effective OPM.
This is due to the fact that OPMs are tightly coupled with how explainable the
resulting visualization is to the players. As mentioned above, there is a
tension between models that capture sufficient information about problem-
solving patterns (tend to be complex player models) and models that can be
easily understood by players (tend to be simple player models).

##  V Discussion

While the case study above specifically showcased one example in the dimension
space of possible OPMs, there are many other examples that one can imagine for
Parallel. These include recommendation models where the system recommends
actions or strategies to players based on their models and level. We can also
imagine models that give players reminders about specific aspects of the level
to pay attention to, again given players’ models. These types of models are
different from the descriptive models described above and can be effective for
learning. However, they also need to be equipped with more explanations for
the information presented. These could include different ways to establish the
openness of the model. We can imagine, for example, a system that explains the
process by which the model is developed, i.e. Open Model Process. We can also
imagine that sometimes such models need to be edited or adjusted due to low
accuracy or issues with the model, in such cases, designers can investigate
the use of editable models as a level of openness.

Researchers can develop different types of OPMs given the framework we
outlined through adjusting the model’s type and level of openness. Each point
in this space of OPMs requires further research and investigations in terms of
their effectiveness and use given the game and goal of the game and the player
model. We hope these dimensions can help establish and further expand the work
on player modeling to deliver better game mechanics as well as ways to explore
engendering trust and fairness in the process of player modeling.

##  VI Conclusion

In conclusion, we proposed a new research area of Open Player Modeling. By
opening the blackbox of player models, this approach can potentially increase
player trust in game AI and improve their gameplay experience. OPM also opens
up new research opportunities for the games research community, such as
interpretable player models, visualization of complex models, and interactive
machine learning for player models.


# Following the Lithium: Tracing Li-bearing Molecules Across Age, Mass, and
Gravity in Brown Dwarfs

Ehsan Gharib-Nezhad NASA Ames Research Center, Moffett Field, CA 94035, USA
Mark S. Marley Lunar and Planetary Laboratory, University of Arizona, Tucson,
Arizona 85721, USA  Natasha E. Batalha NASA Ames Research Center, Moffett
Field, CA 94035, USA  Channon Visscher Chemistry $\&$ Planetary Sciences,
Dordt University, Sioux Center, IA. 51250, USA  Center for Extrasolar
Planetary Systems, Space Science Institute, Boulder, CO 80301, USA  Richard S.
Freedman  SETI Institute, Mountain View, CA 94035, USA  NASA Ames Research
Center, Moffett Field, CA 94035, USA  Roxana E. Lupu  BAER Institute/NASA Ames
Research Center, Moffett Field, CA 94035, USA

###### Abstract

Lithium is an important element for the understanding of ultracool dwarfs
because it is lost to fusion at masses above $\sim 68\,M_{\rm J}$. Hence, the
presence or absence of atomic Li has served as an indicator of the nearby
H-burning boundary at about $75\,M_{\rm J}$ between brown-dwarfs and very low-
mass stars. Historically the “Lithium test”, a search for the presence and
strength of the Li line at 670.8 nm, has been a marker if an object has a
substellar mass with stellar-like spectral energy distribution (e.g., a late-
type M dwarf). While the Li test could in principle also be used to
distinguish masses of later-type L-T dwarfs, Li is predominantly no longer
found as an atomic gas, but rather a molecular species such as LiH, LiF, LiOH,
and LiCl in their cooler atmospheres. L- and T-type brown dwarfs are also
quite faint at 670 nm and thus challenging targets for high resolution
spectroscopy. But only recently have experimental molecular line lists become
available for the molecular Li species, allowing molecular Li mass
discrimination. In this study, we generated the latest opacity of each of
these Li-bearing molecules and performed thermochemical equilibrium
atmospheric composition calculation of the abundance of these molecules.
Finally, we computed thermal emission spectra for a series of radiative-
convective equilibrium models of cloudy and cloudless brown dwarf atmospheres
(with $T_{\rm eff}=$ 500–2400 K, and $\log g$=4.0, 4.5, 5.0) to understand
where the presence or absence of atmospheric lithium-bearing species is most
easily detected as a function of brown dwarf mass and age. After atomic Li,
the best spectral signatures were found to be LiF at $10.5-12.5\,\micron$ and
LiCl at $14.5-18.5\,\micron$. LiH also shows a narrow feature at $\sim
9.38\,\micron$.

††thanks: NASA Postdoctoral Fellow
corresponding author<EMAIL_ADDRESS>
##  1 Introduction

More than 25 years ago, the observation of the first brown dwarfs, Teide 1 and
Gliese 229B, ushered a new era in astronomy (Rebolo et al., 1995; Oppenheimer
et al., 1995; Nakajima et al., 1995). Since then, many models and observations
have been done to understand the chemistry and physics of brown dwarf
atmospheres, their atmospheric evolution with age, mass, and effective
temperature (e.g., Saumon & Marley, 2008; Chabrier et al., 2000; Burrows et
al., 1997), as well as their connection to the planet and stellar main
characteristics (e.g., Marley & Robinson, 2015; Zhang, 2020). Luminosity, age,
mass, radius, surface gravity, atmospheric evolution, effective temperature,
and cloud/aerosol formation are among the main active open questions waiting
to be addressed to better understand these objects.

One indicator used to distinguish between stars and brown dwarfs is the
detection of the atomic lithium resonance line at 670.8 nm, which was proposed
by Rebolo et al. (1992) and aptly named the “lithium test”. Atomic Li is
converted to He through fusion at central temperatures of $\sim 2-3\times
10^{6}\,{\rm K}$. Thus objects which are not massive enough to reach such
temperatures in their interior do not burn Li. Nelson et al. (1993) and Pozio
(1991) calculated a transition mass of $62-65\,M_{\rm J}$ for the Li burning.
Magazzu et al. (1993) showed that the required mass to burn Li by half is 84
and $M=63\ M_{\rm J}$ for 60 Myr and 250 Myr objects, respectively. Similarly,
Chabrier et al. (1996) modeled the depletion of Li along with mass and age,
and found that the initial lithium burns by a factor of 2 over $\sim$0.26 Gyr
and by a factor of 100 over $\sim 1\,{\rm Gyr}$ for 64 $M_{\mathrm{J}}$
object. In addition, Bildsten et al. (1997) provided an analytical formations
to calculate the Li depletion as a function of age, mass, radius, and
luminosity of objects within 52–105$M_{\rm J}$. By comparison, current
calculations place the mass limit for H burning at $74\,M_{\mathrm{J}}$ (e.g.,
Marley et al., 2021, submitted). Because of the happy near coincidence of
these two masses, the detection of Li has become a straightforward diagnostic
test of most brown dwarfs; even though, consumption of Li in the most massive
brown dwarfs leads to their confusion with the lowest mass stars (Basri,
2000).

Interestingly Rebolo et al. (1996) further validated Teide 1 as a true brown
dwarf through the detection of atomic Lithium and application of the Lithium
test. Since then, several studies have implemented the Li test to confirm the
true detection of brown dwarfs and also determine their ages and masses (e.g.,
Martín et al., 1999). Age is difficult to ascertain, however, and systematic
studies of the Li strength of a sample of $\sim$150 late-M to L dwarfs by
Kirkpatrick et al. (2008) did not reveal any strong correlation with age.

In general, the Li test has been challenging to apply to all types of brown
dwarfs. First, moderately large telescopes are required with a resolution
$\lambda/\Delta\lambda\geqslant 1200$ and signal-to-noise ratio (SNR) of $\sim
50$ to detect the Li resonance line at 670.8 nm (Kirkpatrick et al., 2008;
Martín et al., 1999)111 Note that lower SNR and resolution is required to
record the Li resonance line in L dwarfs. For example, (Martín et al., 2018)
showed that SNR$>$15 and $\lambda/\Delta\lambda\geqslant 600$ are sufficient
to detect the Li 6708 Åabsorption in L-type Hyades BD candidates. Second, for
the mass interval between this minimum Li-burning mass and the hydrogen-
burning minimum mass $\sim 74\ M_{\rm J}$, absent atomic Lithium is ambiguous
as a star/brown dwarf discriminant. This is particularly troublesome in the
effective temperature range occupied by both young massive brown dwarfs and
the coolest, lowest mass stars.

For objects cooler than any star at the bottom of the main sequence there is
no question that an object is substellar (e.g., the atmospheric methane
detected in Gl 229B, clearly indicated a substellar atmosphere (Oppenheimer et
al., 1995)). However, the Li test still has relevance even at these cool
effective temperatures. Gravity signatures for brown dwarfs are currently
lacking and beyond general rules-of-thumb such as the triangular-shaped H band
as a signpost of low gravity in L dwarfs (Kirkpatrick et al., 2008), there are
no straightforward markers of mass in brown dwarfs. As a result, many L and
T-type dwarfs have uncertain masses. This is primarily due to the well known
degeneracy between age, mass, and gravity.

Specific examples of objects with divergent mass estimates include Gliese 229
B (late-T, $\sim$1000 K) for which Brandt et al. (2020), by combining radial
velocities, imaging, and astrometry methods, derived a dynamical mass of
$\sim~{}70\pm 5M_{\rm J}$, whereas early evolutionary models suggested a
younger age and noticeably lower mass of $20-55M_{\mathrm{J}}$ (Marley et al.,
1996; Allard et al., 1996). HD 19467B is another T dwarf (T5-T7, $\sim$1050 K)
with high uncertainty in its modeled mass and age, which are
$74^{+12}_{-9}M_{\mathrm{J}}$ and $\sim 9-11$ Gyr (Maire et al., 2020).
Another T-dwarf example is GJ 570D (T7.5, $\sim$800 K) with a calculated mass
of 15–72 $M_{\mathrm{J}}$ (Geballe et al., 2001). This degeneracy problem is
not only limited to T dwarfs. HD 7672B and HD 4747B are two examples of
massive L dwarfs (L4.5 and L8, $\sim$1700 K) with large uncertainties in their
mass and age (Liu et al., 2002; Crepp et al., 2016). For objects such as
these, a discrete mass discriminant would be particularly valuable.

Note that the computed thermal evolution tracks of brown dwarfs (e.g., Marley
et al., 2021) predict that objects massive enough to burn Li would not cool
below 900 K in the age of the universe. However as some of the above mass
estimates attest, a few objects, like Gl 229 B, have reported apparently
problematic masses. Having an independent mass indicator would greatly
illuminate such cases.

To the best of our knowledge, there has been only one study before by Weck et
al. (2004) to investigate the LiCl signature for dwarfs with $T_{\rm
eff}$=900–1500 K. However, the main goals of their study were to provide the
LiCl $ab\,initio$ line list and consider the brown dwarf effective temperature
range over which LiCl could be detected. In comparison, we here assess a
broader range of atmospheric temperatures and include all Li species for which
there is opacity data available.

We have organized this article as follows; In § 2, we give an account of the
equilibrium chemistry in M to T dwarfs, and the chemistry of the dominant Li-
bearing molecules is explained. In § 3, we describe the calculation of
absorption cross-section (or opacity) data for LiH, LiF, and LiCl molecules.
The degeneracy in the determination of brown dwarf masses is addressed in §4.
Following that, § 5 presents a set of synthetic thermal fluxes for brown
dwarfs with effective temperature in a range of 500–2400 K, and for surface
gravity ($\log g$) of 4.0, 4.5 and 5.0. In this section, the presence and
spectral location of LiH, LiF, and LiCl molecules in the cloudy and clear
atmospheres is discussed. In the end, § 5.3 comments on the non-detection of
atomic Li in T dwarf observed fluxes even if modeling spectra predict its
presence. Summary and future works are presented in § 6.

##  2 Lithium Chemistry

Thermochemical equilibrium simulations are performed to calculate the
abundance of the following species in the temperature-pressure space at
75–6000 K and 10-6–3000 bar: Li(g), LiOH, LiH, LiF, LiCl, e-, H2, H, H+, H-,
H2-, H2+, H3+, H2O, CH4, C2H2, C2H4, C2H6, CO, CO2, NH3, N2, HCN, OCS, PH3,
H2S, TiO, VO, FeH, CrH, SiO, MgH, He, Na, K, Rb, Cs, Fe, graphite, Li(s).
Since we are exploring objects in which lithium is not expected to burn, we
used the protosolar abundance of Li is used in this study. We adopt the
protosolar Li abundance from Lodders (2003) (i.e., A(Li)=3.28), which is
derived from meteoritic CI chondrites (carbonaceous meteorites of Ivuna type).
CI chondrites are the most primitive chondrites because they have not
chemically fractionated and their relative abundances are considered to be
similar to protosolar abundance. Therefore, they are enriched in elements like
lithium.

Figure 1 illustrates the Volume Mixing Ratios (hereafter, VMR) of Li-bearing
species predicted by equilibrium chemistry as a function of temperature (left)
and pressure (right). There are three main thermal regions in the VMR-$T$-$P$
parameter space as defined by these species. First, hot atmospheres with
$T_{\rm eff}$ in $\sim$1800–2800 K (e.g., M-L type dwarfs) in which monoatomic
Li is the dominant gas in $\sim 10^{-3}$–100 bar. Second, warm atmospheres
with $T_{\rm eff}$ in $\sim$900-1800 K (e.g., L-T type dwarfs) where LiCl and
LiOH dominate. The third region, with T${}_{\rm eff}\leqslant$1000, is
characterized by LiF and LiCl as the favored molecules.

Figure 2 illustrates a broad view of the presence and abundance of Li species
in the temperature-pressure space. In this phase diagram, Li, LiCl, LiOH, and
LiF are shown by yellow, green, blue, and red shading. For example, gaseous Li
is the most stable form at $T>1400$ K. At each area, dashed and dotted contour
lines represent 50$\%$ and 20$\%$ of that species with respect to the total
protosolar lithium abundance ($\sim 3.5\times 10^{-9}$) in the atmosphere. The
solid and dashed black curves are the $T(P)$ profiles (for $\log g$=5) for
cloudy and cloudless (Marley et al., 2021) models, respectively. We aim to
understand whether these different Li species domains can be recognized in
brown dwarf spectra and if they could serve as mass indicators.

##  3 Absorption Cross-Sections

Radiative transfer modeling of synthetic thermal emission requires accurate
and complete absorption cross-section data (hereafter, ACS) of Li (g), LiH,
LiF, and LiCl. ACS data of a given absorber consists of several million lines
with unique position, intensity, and width. The width is sensitive to the
temperature and pressure of broadening gases, and it is controlled by the
convolution of Doppler and Lorentz profiles. The knowledge of pressure
broadening coefficients for the Lorentz profile is instrumental and is
dependent on $J$ quantum numbers. The choice-of-broadening for the background
gases in this study is 85$\%$ H2 and 15$\%$ He by number. Because the H2 and
He pressure-broadening Lorentz coefficients for these species have not been
calculated or measured, we have estimated pressure-broadening data for these
Li-broadening molecules in question.

In our recent paper by Gharib-Nezhad et al. (2021), we provided a detailed
discussion on how to generate these ACS data and their pressure-broadening
coefficients. The generated ACS data of atomic Li(g), LiH, LiF, and LiCl
covers 1460 temperature and pressure grid points, and range of 75–4000 K and
10-6–3000 bar, respectively. The line lists used to generate these ACS data
are the most accurate and up-to-date, which are provided in Table 1. Note that
the LiOH line list is not available, and hence we could not generate the LiOH
ACS data.

Figure 3 depicts the weighted ACS$\times$VMR for Li, LiH, LiF, and LiCl for
800, 1600, and 2600 K at $P=$10 bar. The dominant spectral features LiH, LiF,
and LiCl are approximately located at $\sim$9 – 10 $\micron$, 10.5 – 14
$\micron$, and 14.5 – 19 $\micron$, respectively. Although the atomic Li is
not stable at $T<1000$ K, its ACS value is a few order-of-magnitude greater
than its molecular forms. Hence, the Li absorption is predominant in the
optical region and, as we will show, is present even after molecular species
become predominant in cooler atmospheric layers. At $T<$2000 K, LiCl and LiF
become predominant features in the infrared.

Table 1: Summary list of opacities: molecules, temperature, pressures, and
their line lists

Absorber |  $\lambda$ [$\mu$m] | Method | Reference
---|---|---|---
LiCl | 2.2–330 | laboratory | (Bittner & Bernath, 2018)
LiF | 5.5–330 | laboratory | (Bittner & Bernath, 2018)
LiH | 5.5–330 | ab initio | (Coppola et al., 2011)

Note: We computed the ACS over the pressure and temperature ranges of
10-6–3000 bar and 75-4000 K. See §3 for more details.

##  4 Degeneracy on age-mass-gravity space: Challenges

Figure 4 presents in black the Sonora-Bobcat solar-metallicity evolution
tracks for isolated 10 to 83 $M_{\rm J}$ dwarfs in $\log g-T_{\rm eff}$
parameter space from Marley et al. (2021). Green curves show isochrones from
0.04 to 10 Gyr. This figure also shows the Li depletion region (yellow) where
atomic Li is depleted by fusion. The presence and overlap of LiF, LiCl, and
LiH molecules are highlighted as well.

Our understanding of brown dwarfs is shaped by the degeneracy in the $T_{\rm
eff}-t-g-M-R-L$ space (i.e., effective temperature, age, surface gravity,
mass, radius, luminosity). The $T_{\rm eff}$ and $L$ parameters may be
estimated from observational spectra (e.g., Saumon & Marley, 2008; Cushing et
al., 2005). In addition, objects with masses 20–80 $M_{\mathrm{J}}$ and
age$\geqslant$0.2 Gyr have radii within a range of 0.7–1.3 $R_{\mathrm{J}}$
(e.g., see figure 3 by Burrows et al., 2001). As a result, age, gravity, and
mass are the three puzzling parameters as younger, lower mass objects can have
comparable luminosity and spectra to older, higher mass objects.

There have been attempts to disentangle this $M-t-g$ degeneracy by employing
atomic and molecular spectral features as a probe. For instance, McGovern et
al. (2004) have proposed to estimate the gravity and age by comparing the
absorption features of NaI, KI, CsI, RbI, TiO, VO, CaH, and FeH at $\sim
0.7-1.25$ $\micron$. Following that, Kirkpatrick et al. (2008) used the
strength and the presence-or-absence of gravity-sensitive features of NaI, KI,
CaH, TiO/VO ratio, and Hα emission to determine the youth and gravity of a
large sample of late-M through L dwarfs. Allers & Liu (2013) also used FeH,
VO, KI, and H-band continuum shape to probe the gravity of several ultracool
M5–L7 dwarfs.

##  5 Modeling M-L-T dwarf Atmospheres: Cloudy and clear

In this work we aim to explore the utility of the Li-bearing species to help
disentangle age, mass, and gravity in brown dwarfs by providing a well defined
mass boundary. Figure 4 highlights which Li species would denote which regions
and where the absence of Li would signify a high mass. Details on the
evolutionary and atmospheric models are provided in §5.1, following by
discussion on the results in §5.2 and 5.3.

###  5.1 Modeling Details

Here we utilize the Sonora-Bobcat generation of atmosphere models computed for
brown dwarfs with clear atmospheres by Marley et al. (2021). These are
supplemented by a few additional cloudy models computed specifically for this
study, following the same methodology and the cloud treatment described in
(Ackerman & Marley, 2001). For each model of interest we compute a high
resolution spectrum using PICASO, an open source radiative transfer code
(Batalha et al., 2019; Batalha,
2019)222https://natashabatalha.github.io/picaso/.

We used our generated opacities for Li (g), LiH, LiCl, and LiF as well as
opacities from (Freedman et al., 2014) (see Table 2 in Lupu et al., 2014, for
details) for other species mentioned in §2. We did not investigate LiOH due to
the lack of a line lists, although theoretical ab intio studies suggest that
the LiOH main rotation-vibration bands are located at 2.6, 10.8, 31.3
$\micron$ (e.g., Bunker et al., 1989; Koput, 2013). The evolution calculation
is from Marley et al. (2021).

###  5.2 Impact of Li, LiCl, LiF, LiH on Brown-dwarf Spectra

We investigate the influence of Li species on the emergent spectra of
candidate brown dwarfs with protosolar abundances. We consider effective
temperatures ($T_{\rm eff}$) of 500, 1000, 1500, 2000, and 2400 K and $\log
g=4.0,4.5,$ and 5 to span the range of conditions of interest where Li species
may be detected. The cloudy models are computed for this study for all the
aforementioned $T_{\rm eff}$ but with $f_{\mathrm{sed}}=3$ and $\log g=5$. The
sedimentation parameter, $f_{\mathrm{sed}}$, controls the particle size and
vertical extent of the cloud layer (Ackerman & Marley, 2001). Aluminium Oxide
(Al2O3), forsterite (Mg2SiO4(s)), and Fe(l,s)) cloud species are included. The
location of the modeled objects on the evolutionary model space is indicated
with yellow circles in figure 4.

The presence, absence, and strength of the Li-species spectral features on
emergent spectra are assessed for different effective temperatures and surface
gravities to see if they can productively be used as a high mass indicator in
brown dwarfs. For each case, we perform the model twice by in turn including
(ON) and excluding (OFF) each molecule or atom in our calculation of the
emergent thermal spectrum. This was done to isolate the effect of each
species. The results for cloudy and cloudless models are illustrated in
Figures 5 (for $\log g$=5) and 6 (for $\log g$=4.0, 4.5 and 5). In Figure 5,
the top panel represents the 500–2400 K synthetic thermal fluxes with Li, LiH,
LiF, and LiCl ON (solid) and OFF (dashed). The bottom panel shows the relative
flux ratio, which is the difference between $F^{\rm ON}$ and $F^{\rm OFF}$
normalized by $F^{\rm OFF}$ (hereafter, $\mathcal{F}_{\rm R}\equiv\frac{F^{\rm
OFF}-F^{\rm ON}}{F^{\rm OFF}}$). In Figure 6, the $\log g$=4.0, 4.5 and 5 are
shown with dashed-dotted, dotted, and dashed lines, respectively, which
includes both ON and OFF cases.

Our focus is to ascertain whether the presence or absence of each Li species
under different atmospheric effective temperatures and gravities can be
detected in the synthetic emergent spectra. Below we discuss each species in
turn in order to better interpret these spectra and how they relate to the
object’s mass.

Atomic Li is the predominant form of Li species at high temperatures. The Li
resonance line is shown in the left panel at 0.667–0.675 $\micron$. Since Li
has a large opacity (see Figure 3), its absorption is predicted even at
$T_{\rm eff}<1000$ K in both cloudy and cloudless models. Given the fact that
Li absorption is narrow and in a faint part of the spectrum, it requires a
resolution of $R\geqslant 1200$ and signal-to-noise (S/N) of $>50$ to be
detected (Kirkpatrick et al., 2008; Martín et al., 1999). In particular, for T
dwarfs, large telescopes and a significant amount of exposure time are
required.

In the cloudy model (Fig. 5), the Li $\mathcal{F}_{\rm R}^{Li}$ ratio has the
largest value which is $\sim 100\%$ at the line center for $T_{\rm eff}$=1500
K. In contrast, this ratio decreases to $\sim~{}20\%$ for $T_{\rm eff}$=500 K,
and this might be why Li has not been observed in late-type T dwarfs (see §5.3
for more details). Note that this Li $\mathcal{F}_{\rm R}$ ratio is about 3–4
order-of-magnitude larger than LiH, LiF, and LiCl, which shows the significant
difference between presence and absence of atomic Li in the model spectra.

In the clear model (Fig. 6), the Li $\mathcal{F}_{\rm R}$ ratio for $T_{\rm
eff}$=1500–2000 K is $100\%$ for all surface gravities ($\log g$=4.0, 4.5, 5)
at the line center. The behavior of the $\mathcal{F}_{\rm R}^{\rm Li}$ value
with the wavelength in the wings (i.e., $d\mathcal{F}_{\rm R}^{\rm
Li}/d\lambda$) could also be an indicator of the gravity because
$d\mathcal{F}_{\rm R}^{\rm Li}/d\lambda$ is larger for $T_{\rm eff}=2400$ and
500 K in models with smaller surface gravity. Comparing the $\log g$ of 4.0,
4.5 and 5 in Figure 6 shows the negligible impact that surface gravity has on
the Li line core. This change in the wings with surface gravity is appeared in
the figure.

Comparing cloudy and clear models for Li shows some remarkable change in the
Li $F^{\rm OFF}$ and $F^{\rm ON}$ fluxes, and slightly in $\mathcal{F}_{\rm
R}^{\rm Li}$ ratio. Clouds have the largest impact on atmospheres with $T_{\rm
eff}$ in 1000–2000 K, although the cloudy models still exhibit the atomic Li
line.

LiH is the molecular form of Li at high temperatures and its spectral feature
is located at 7–10 $\micron$, but its $\mathcal{F}_{\rm R}^{\rm LiH}$ ratio is
greatest at $\sim$9.36 $\micron$ according to our results. The
$\mathcal{F}_{\rm R}$ ratio at $T_{\rm eff}>$1000 K is $\sim 0.001-0.005\%$
which would be challenging to detect. In the cloudy model, the highest
$\mathcal{F}_{\rm R}^{\rm LiH}$ ratio is for $T_{\rm eff}$=1500 K, while this
changes to 2000 K for the clear model. Assessing both cloud and cloudless
models and different surface gravities suggests that LiH is not sensitive to
gravity and so could not be employed as a gravity indicator. However, LiH
region (black area) in Figure 4 shows that its detection can be an indicator
of $M<65M_{\rm J}$ and temperature of $\sim$1700–2500 K. Moreover, LiH is
present in young objects with $t<1.0$ Gyr.

The LiF absorption feature is most prominent in the spectral range of
10.5–12.5 $\micron$. As it is predicted by thermochemical equilibrium, LiF
mostly forms at $\sim$800–1200 K, and its $\mathcal{F}_{\rm R}^{\rm LiF}$
ratio is almost $\sim 0.005-0.5\%$ for T–L type dwarfs with $T_{\rm eff}$ of
900–1600 K. According to the cloudy and clear models, LiF can only be detected
in atmospheres with $T_{\rm eff}$ in 1000–1500 K. In clear model, $\log g$=4.0
has the greatest $\mathcal{F}_{\rm R}^{\rm LiF}$. According to the
evolutionary $T_{\rm eff}-g-t-M$ parameter space (Fig. 4), the detection of
LiF is a signature of an object with $M\leq 65M_{\rm J}$ and $t\leq 10$ Gyr.
Lower surface gravity in this space leads to a higher $\mathcal{F}_{\rm
R}^{\rm LiF}$ ratio, and hence lower mass and younger age.

The predominant LiCl feature is located at 14.5–18.5 $\micron$. This molecule
is present for $T_{\rm eff}$=1000–2100 K, and its $\mathcal{F}_{\rm R}^{\rm
LiCl}$ ratio is $\sim 0.005-0.5\%$. The feature is most detectable at $T_{\rm
eff}$=1500–2100 K. Both cloudy and clear models show that the LiCl feature and
its $\mathcal{F}_{\rm R}^{\rm LiCl}$ ratio are sensitive to surface gravity.
$\mathcal{F}_{\rm R}^{\rm LiCl}$ is strongest for the lowest gravity case
considered, $\log g=4.0$. Such objects would be young and low mass (i.e,
$M\leq 15M_{\rm J}$ and $t\leq 0.1$ Gyr). LiCl is less detectable in higher
gravity objects. Figure 4 illustrates the relation between the detection of
these species with mass, gravity, and age. Note that both LiCl and LiF
features are present in the range $T_{\rm eff}=\sim 900-1600~{}K$, and their
$\mathcal{F}_{\rm R}$ ratio is higher for objects with lower surface gravity.

Weck et al. (2004) investigated the signature of LiCl on dwarfs with $T_{\rm
eff}$=900–1500 K and $\log g$=3.0, 4.0, and 5.0 using the PHOENIX code with
solar abundance. They concluded that the strongest $\mathcal{F}_{\rm R}$ LiCl
features are expected in T dwarfs with $T_{\rm eff}$=1200 K and $\log g$=3.0.
According to the evolutionary models, only very young objects with age$<$0.01
Gyr and mass$<10~{}M_{\rm J}$ could have $T_{\rm eff}$=1200 K and $\log
g=3.0$. In comparison to their work, our main focus here is the study all Li
species but for older objects with greater surface gravity and so with greater
mass (i.e., $5M_{\rm J}\leq M\leq 65M_{\rm J}$). We also assess the impact of
clouds on the detection of the various Li species using the latest and most
accurate their opacity data.

In another study, Pavlenko et al. (2007) calculated the atomic Li resonance
abundance in LP 944-20 M-dwarf atmosphere using VLT/UVES and AAT/SPIRAL
observed spectra. They used DUSTY and COND model atmospheres from Allard et
al. (2001) with solar metallicity, and included Li, LiOH, LiH, LiF, LiBr, and
LiCl species, and obtained $\log N({\rm Li})=3.25\pm 0.25$. They found that
atomic Li is predominant Li-bearing species in objects with $T_{\rm
eff}$=1800–2400 K with $\log g=4.5$.

###  5.3 Comments on Atomic Li line in T dwarfs

Figure 7 presents a set of observed emergent spectra for different T dwarfs,
encompassing Gliese 570D (T8), 2MASS 1503+2525 (T5.5) (Burgasser et al.,
2003b), and 2MASS 0755+2212 (T5) (Burgasser et al., 2002). These T dwarfs
apparently do not show any atomic Li absorption in their spectra. In this
figure, SDSS 0423-0414 (SDSS J042348.57-041403.5) is obtained from Kirkpatrick
et al. (2008), and is the only object in these spectra that shows Li
absorption with a pseudo equivalent width measurement of 11 Å. Note, this
object is a peculiar one because its optical part is classified as L7.5 while
the Near-Infrared part is T0 (Vrba et al., 2004). We also showed our synthetic
spectra from the clear (dotted) and cloudy (dashed) models to highlight the
presence of atomic Li from our modeling work.

The key difference between the observed and modeled spectra is the atomic Li
line at 6708 Å. At low temperatures, atmospheric thermodynamic simulations
expect Li to predominantly be found in the atomic form only at high $P$
(Lodders, 1999) (Figure 1). Nevertheless our grids of atmospheric cloudy and
clear models predict the presence of the atomic Li line even at $T_{\rm
eff}\simeq$600 K (e.g., Allard et al., 2001; Burrows et al., 2002). However,
the observed T-dwarf spectra do not show any sign of Li detection at $T_{\rm
eff}\leqslant$900 K (e.g., Kirkpatrick et al., 2000; Burgasser et al., 2003a).
Theoretically this cannot simply be due to Lithium burning having removed Li
as objects massive enough to burn Li are not expected to cool below 900 K in
the age of the universe (e.g., Marley et al., 2021), although dynamical masses
for at least some T dwarfs come close (e.g., Sahlmann et al., 2020; Brandt et
al., 2020).

Other possible explanations for the predicted atomic Li not being found in the
data include incorrect Li thermochemistry data and clouds obscuring the
deeper, higher pressure regions where atomic Li resides. The accuracy of
thermochemical data for key reactions such as LiOH+H =Li+H2O and LiH+H=Li, as
well as the impact of condensation on the abundance of Li gas needs further
investigation. Greater cloud opacity in the optical than considered here might
also obscure the Li line. Additionally, since this Li line is located in the
faint part of the optical spectrum, higher S/N ratio spectra should be
acquired to more robustly test if the feature is indeed missing. The detection
of Li at 670.8 nm at $T_{\rm eff}<$1000 K deserves further observational and
theoretical attention.

##  6 conclusion and Future Work

Since the discovery of the first brown dwarfs, numerous massive cool dwarfs
have been detected. However ascertaining precise masses have been a challenge
to the degeneracy between mass, gravity, and age for these objects. Therefore,
the calculated mass of many ultracool dwarfs including Gl 229B and Gl 510D is
still under debate and spans a wide range.

Traditionally, the Li test has been used to distinguish between low mass stars
and brown dwarfs with similar spectral types. At lower temperatures such as
L-T dwarfs, LiF and LiCl molecules form. Because LiF and LiCl features are
located in regions where brown dwarfs are brighter than 670 nm, they could in
principle be used as mass indicators at these lower effective temperatures.

In this article, we investigated the spectral signatures of LiCl, LiF, and LiH
as well as atomic Li using cloudy and cloudless atmospheric models. We showed
that LiCl is present at an effective temperature of $\sim$1000–2000 K in
cloudy atmospheres and $\sim$1500–2000 K in clear atmospheres. Hence, the LiCl
spectral features should be found in L-T type dwarfs with masses below 65
$M_{\rm J}$. The lack of LiCl at these $T_{\rm eff}$s would be an indicator of
larger masses with concomitant older ages and larger surface gravities. The
LiF feature at 10.5–12.5 $\micron$ is another indicator of masses lower than
65 $M_{\rm J}$ and $T_{\rm eff}\leqslant 1600~{}K$. LiH is a stable molecule
at high temperatures (e.g., early-L–M dwarfs), however its signature is
confined to a narrow spectral interval of $9.35-9.4~{}\micron$. We could not
assess the contribution of LiOH because of the lack of appropriate opacity
data. In ultracool T dwarfs such as Gl 229B with a large mass uncertainty, the
detection of LiF and LiCl would confirm the object has mass is lower than 68
$M_{\mathrm{J}}$.

The relative flux ratio, $\mathcal{F}_{\rm R}$, of LiF and LiCl is very small,
thus the detection of these species requires high SNR spectra in the mid-IR.
However, their strongest spectral bands are found in the relative bright
region of the infrared thermal with less overlap with other spectral features
than the atomic Li line in the optical, making them potentially favorable mass
indicators. The $JWST$ MIRI instrument covers the wavelength range of
$5-28\micron$, and its spectrograph will enable medium-resolution spectroscopy
with $\lambda/\Delta\lambda\sim 2000-3700$, and so MIRI might be capable of
detecting these Li features.

The high-resolution cross-correlation technique could be another method to
detect LiH, LiF, and LiCl molecules. In this method, the number of detected
lines is crucial to increasing the signal-to-noise ratio, and so a large
number of transitions of these molecules would be important. However, current
ground-based telescopes such as $VLT$ CRIRES+ (Follert et al., 2014) only
reach up to 5 micron. Atomic Li lines at 600–900 nm could be detected using
$CAHA3.5$ CARMENES (520–1710 nm, R$>$80,000) (Quirrenbach et al., 2010), and
the $Subaru$/High Dispersion Spectrograph (298–1016 nm, R$>$90,000) (Noguchi
et al., 2002).

Another potential avenue for fundamental research is to improve the
thermodynamic data in order to understand the reason for the apparent non-
detection of the Li 670.8 nm line in T-dwarf thermal fluxes. Knowing the
kinetic data between all Li-bearing reactions as well as their condensation
would be another topic to advance for future work.

##  7 Supplementary data

The pre-generated absorption cross-sections (ACS) data from this study will be
uploaded in ZENODO upon acceptance.

##  8 Acknowledgements

We would like to thank Dr. Adam Burgasser for his assistance regarding the
observational T-dwarf spectra. EGN acknowledge Research Computing at Arizona
State University for providing HPC resources that have contributed to the
research results reported within this paper. EGN acknowledge support from HST-
AR-15796 to generate Li-species opacities. We also acknowledge the ExoMol team
for their continued production of large line lists and pertinent data critical
to high-temperature atmospheric modeling. EGN’s research was supported by an
appointment to the NASA Postdoctoral Program at the NASA Ames Research Center,
administered by Universities Space Research Association under contract with
NASA.

##  9 Software

We used the publicly available $ExoCross$ code (Yurchenko et al., 2018)
developed by the ExoMol group to generate the cross sections. To perform the
radiative-transfer modeling of brown dwarfs, PICASO tool was used (Batalha et
al., 2019; Batalha, 2019). Brown-dwarf thermal structures and cloud
compositions were used from Sonora cloudless and cloudy models (Marley et al.,
2018, 2021).

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Figure 1: Volume Mixing ratio (VMR) plots of Li(g), LiF, LiCl, LiH, and LiOH
with respect to temperature (left) and pressure (right). At high temperatures
($T\gtrsim 1500\,\rm K$), gaseous Li is the most abundant Li-bearing species.
At moderate temperatures ($1500\,{\rm K}\lesssim T\lesssim 1000\,\rm K$), both
LiCl and LiOH will be the most stable form of Li-bearing species, depending on
the pressure. At low pressures and low temperatures ($P\lesssim 0.01\,\rm bar$
and $T\lesssim 1000\,\rm K$), LiF could also be abundant. Figure 2: Li phase
diagram: temperature-pressure diagram showing the mole fraction of atomic
gaseous Li (yellow), LiH (black), LiOH (blue), LiCl (green), and LiF (red).
For each area, the dotted and dashed contours represent the 20% and 50% of the
species with respect to the total Lithium abundance in both solid and gas
phases (i.e., the protosolar value $\sim 3.5\times 10^{-9}$). The solid and
dashed black curves are the $T(P)$ profiles (for $\log g$=5) for cloudy and
cloudless (Marley et al., 2021) models, respectively. As a brown dwarf cools,
the dominant Li-bearing species in the observable atmosphere (generally
$P<10\,\rm bar$) varies. Figure 3: Smoothed absorption cross-sections (ACSs)
of atomic Li, LiH, LiF, and LiCl multiplied by its volume mixing ratios (VMR)
for $P=10$ bar. The resonance Li line at 670.8 nm has a large ACS value than
its other molecular forms, and so its ACS$\times$VMR value is larger than
others even at low temperatures. In contrast, LiCl and LiF are the two
dominant molecules at low temperatures and their spectral features are clearly
distinguishable. At $1600\,\rm K$, the fundamental band of LiH at $\sim
8.5\,\rm\micron$ is dominant. In this study, their latest line lists are used
to compute their opacities and also their pressure-broadening coefficients are
revised. See Table 1 for more details.  Figure 4:  Evolution tracks (black,
moving right to left with increasing age) showing gravity as a function of
effective temperature for substellar-mass objects. Masses (black, in
$M_{\mathrm{J}}$) and ages (green, in Gyr) are labeled. Yellow circles denote
individual radiative-convective models shown in the other figures ($T_{\rm
eff}=$ 500, 1000, 1500, 2000, $2400\,\rm K$, and surface gravity of $\log
g=4.0,4.5\,\rm{and}\,5.0$ for cloudless models. The Sonora-Bobcat evolution
grid models are adopted from Marley et al. (2021). Objects in the yellow area
are massive and old enough to burn protosolar Li atoms. LiH, LiCl, and LiF
spectral features become apparent in the black, blue, and red areas,
respectively. For the strength of their features, see Figs. 5–6. Figure 5:
Overview of model spectra as computed for cloudy models with various $T_{\rm
eff}$ in 500-2400$\,\rm K$ range with $\log g=5.0$ and $f_{\rm sed}=3$. The
top row of panels shows the computed spectra as computed with ($F^{\rm ON}$,
solid) and without ($F^{\rm OFF}$, dashed) Li, LiH, LiF, and LiCl. Note the
difference in the vertical axis between Li and the other species; in contrast
to Li-bearing molecules, atomic Li shows an easily apparent difference between
its $F^{\rm ON}$ and $F^{\rm OFF}$ fluxes. The lower row of panels shows the
relative flux ratio, which is the difference between $F^{\rm ON}$ and $F^{\rm
OFF}$ normalized by $F^{\rm OFF}$ ($\mathcal{F}_{\rm R}\equiv\frac{F^{\rm
OFF}-F^{\rm ON}}{F^{\rm OFF}}$). All spectra are smoothed for clarity. LiF and
LiCl could be an indicator of dwarfs with $M<65M_{\rm J}$ at
$T_{\mathrm{eff}}$ ranges of 1000–1500 K and 1000–2000 K, respectively. Their
absence in the observational spectra would be an indicator of higher mass.
Figure 6:  Similar to Figure 5, but for cloudless spectra and different
$T_{\rm eff}$ in 500–2400 K range with $\log g$=4.0 (dashed-dotted), $\log
g$=4.5 (dotted) and 5.0 (dashed). (Top panels) The presence ($F^{\rm ON}$,
solid) and absence ($F^{\rm OFF}$, dashed) of Li, LiH, LiF, and LiCl. As with
the cloudy model cases, atomic Li shows a noticeable difference between its
$F^{\rm ON}$ and $F^{\rm OFF}$ spectra. The bottom three rows show the
relative flux ratio, which is the difference between $F^{\rm ON}$ and $F^{\rm
OFF}$ normalized by $F_{\rm OFF}$ ($\mathcal{F}_{\rm R}\equiv\frac{F^{\rm
OFF}-F^{\rm ON}}{F^{\rm OFF}}$) for $\log g$=4.0, 4.5 and 5. All spectra are
smoothed for clarity. The spectral features of LiF and LiCl are stronger at
lower surface gravity. LiH is not sensitive to gravity and its
$\mathcal{F}_{\rm R}$ is lower than other Li species. LiF and LiCl are most
detectable in dwarfs with $M<65M_{\rm J}$ at $T_{\mathrm{eff}}$ ranges of
1000–1500 K and 1000–2000 K, respectively. Their absence in the observational
spectra would be an indicator of higher mass.  Figure 7: Comparing a set of
observed T dwarf fluxes with synthetic spectra from Sonora cloudy and cloud-
free models ($\log g$=5). Theoretical evolution models predict the detection
of atomic Li 6708 $\AA$, whereas many T-dwarf spectra have not shown any Li
signature. For example, observed fluxes for Gliese 570D, 2MASS 0559-1404,
2MASS 0755+2212, 2MASS 1503+2525 do not show any Li absorption. Note, the
Near-Infrared part of the spectrum is T0, while the optical part of the SDSS
0423-0414 (SDSS J042348.57-041403.5) flux is classified as L7.5 and so it
shows the Li line with EW=11 Å(Kirkpatrick et al., 2008; Vrba et al., 2004)
(spectra are from Burgasser et al. (2003a) and Kirkpatrick et al. (2008)).