# Modeling Sustainable Resource Management using Active Inference

Mahault Albarracin 1 Université du Québec à Montréal, Montréal, Canada 1
0000-0001-2345-6789 Ines Hipolito 2Macquarie University, Sydney, Australia 2
0000-0002-3456-7890 Maria Raffa 3IULM University, Milan, Italy 3
0000-0003-4567-8901 Paul Kinghorn 4Department of Informatics and Engineering,
University of Sussex, Brighton, UK 4 0000-0004-5678-9012 0000-0001-2345-6789
0000-0002-3456-7890 0000-0003-4567-8901 0000-0004-5678-9012

###### Abstract

Active inference helps us simulate adaptive behavior and decision-making in
biological and artificial agents. Building on our previous work exploring the
relationship between active inference, well-being, resilience, and
sustainability, we present a computational model of an agent learning
sustainable resource management strategies in both static and dynamic
environments. The agent’s behavior emerges from optimizing its own well-being,
represented by prior preferences, subject to beliefs about environmental
dynamics. In a static environment, the agent learns to consistently consume
resources to satisfy its needs. In a dynamic environment where resources
deplete and replenish based on the agent’s actions, the agent adapts its
behavior to balance immediate needs with long-term resource availability. This
demonstrates how active inference can give rise to sustainable and resilient
behaviors in the face of changing environmental conditions. We discuss the
implications of our model, its limitations, and suggest future directions for
integrating more complex agent-environment interactions. Our work highlights
active inference’s potential for understanding and shaping sustainable
behaviors.

###### Keywords:

Active Inference and Sustainability and Generative model.

Modeling Sustainable Resource Management using Active Inference

Mahault Albarracin1, Ines Hipolito2, Maria Raffa3, Paul Kinghorn4

1Université du Québec à Montréal, Montréal, Canada
2Macquarie University, Sydney, Australia
3IULM University, Milan, Italy
4Department of Informatics and Engineering, University of Sussex, Brighton, UK

##  1 Introduction

For the past decade, we have shown that the Free Energy Principle can serve as
a foundational concept in predicting and modeling the present and future
behaviors of a system. Under this principle, the behavior of a system aims to
maintain equilibrium and sustaining life through the minimization of free
energy [Parr et al., 2022, Parr and Friston, 2019, Stubbs and Friston, 2024].
Systems, particularly biological ones, act to minimize the difference between
their representation of the world, encoded in internal states and the external
environment. By reducing this discrepancy, as quantified by free energy, the
system achieves a state of balance and effectively adapts to its surroundings
[Friston et al., 2017, 2023, Parr et al., 2023]. We can thus understand it as
a measure of uncertainty or surprise, used such that agents are driven to more
predictable and stable states. Using free energy, systems can slowly make
adjustments and adaptations, and thus maintain homeostasis in a changing
environment [Ramstead et al., 2018, Kirchhoff et al., 2018, Karl, 2012, Da
Costa et al., 2023, Pezzulo et al., 2024]. This modeling approach has been
used for various types of systems, from neural processes and cognitive
functions to broader ecological and social dynamics [Friston et al., 2010, Da
Costa et al., 2024, Solymosi and Schulkin, 2024, Albarracin et al., 2024a,
Matsumura et al., 2023, Montgomery and Hipólito, 2023, Ramstead et al., 2020,
Pezzulo et al., 2024].

While it is a widely held assumption that all systems will invariably minimize
free energy (FE), this is not always a simple linear process. To understand
this, we have to consider the system’s goals and constraints. These goals can
sometimes result in behaviors that do not align perfectly with immediate free
energy minimization. This is partly what can make a system, given a specific
scale of measurement, somewhat unsustainable. Not all systems are capable of
effectively minimizing free energy. They may indeed have constraints in their
structure or function. Think of certain pathological conditions impeding a
system’s ability to minimize free energy efficiently. These conditions can
lead to maladaptive behaviors or states that deviate significantly from what
would be predicted by the FEP. For example, constraints in structure or
function: in neurological disorders such as schizophrenia, the brain’s ability
to minimize free energy can be impaired. As it distorts connectivity, it may
also alter perceptions and thoughts - no longer fully related to the external
world. Someone with Schizophrenia can struggle to reduce uncertainty about its
environment, resulting in maladaptive behaviors [Friston et al., 2016,
Harikumar et al., 2023, Zarghami et al., 2023].

Free energy minimization can also be influenced by external perturbations and
environmental factors. The very nature of the environment is unpredictable
dynamics, which temporarily disrupt a system’s meta-stable states. The system
transiently increases free energy as it adapts to new conditions. Albarracin
et al. (2024) explore how systems must deal with external shocks and stresses
(perturbations) to maintain sustainability, resilience and well-being. They
suggest that resilience means absorbing shocks and stresses from the
environment, while sustainability requires enduring capacity to stay
resilient, but without causing a loss of resilience of the environment super-
system. In this paradigm, external perturbations are central to developing
better strategies to maintain well-being across system strata. Since these
perturbations can be unpredictable, the temporality of strategies can change.
Long-term strategies can weather slight increases in free energy temporarily
to achieve more stable and favorable conditions in the future. This is the
case we will be testing and presenting in this paper: long-term strategies
involving temporary increases in free energy. When an agent learns that it
does not have to satisfy its greed immediately, even if it is very hungry,
because the aim is to maintain a balance between itself and the environment
(such as a room with food) over time, the agent can resist the urge for
immediate gratification and managing its resources judiciously. And thus, the
agent can endure short-term discomfort (increased free energy) to ensure long-
term stability and sustainability.

To do so, we test out two cases, detailed in the Methods section. Case 1 acts
as a baseline scenario, and involves a static environment where the agent
decides whether to eat food or not. In Case 2, the environment is dynamic, and
the agent must learn to moderate its consumption behavior over time. Food
increases when the agent does not eat, introducing a dynamic aspect to the
environment. This study is important for two reasons. First, we must
understand adaptive strategies to properly predict when systems will achieve
long-term stability and sustainability. It will help us predict when systems
choose to balance short-term needs with long-term goals. But this will also
help us identify potential vulnerabilities, such that we can identify areas
where intervention may be needed to prevent collapse or dysfunction. The FEP
dictates that behavior should align with minimizing free energy. But we have
to better understand the variability in paths where this principle isn’t
consistently upheld at a given scale of measurement.

##  2 Methods

Figure 1: The agent’s generative model encodes beliefs about the causal
structure of the environment and how its actions affect the state of the
world. The true state of the environment is represented by two hidden state
factors - the availability of food (s1) and the agent’s satiety (s2). The
prior preference C matrix specifies the agent’s innate drives or goals, in
this case a strong preference for being satiated. The starting conditions are
specified by the initial state distribution, D. Here, food is initially
present but the agent is not satiated. The agent has two observation
modalities - the presence of food (o1) and its own satiety level (o2). The
agent can select between two actions at each time step - "eat" or "do not
eat". We have two hidden state factors: food left and satiety. For the "do not
eat" action, for Case 1 the B matrix is an identity matrix, as this action
does not change the state, while for Case 2 it changes, since not eating leads
to an increase in available food. When the agent chooses the "eat" action, if
food is present, the states will transition by reducing "food left" by 1 (down
to a minimum of 0) state and by increasing "satiety" by 1 (to a maximum of 2).

###  2.1 Case 1: Static environment

We build this simulation using the PyMDP package, by Conor Heins and
colleagues [Heins et al., 2022]. In Case 1, we consider a static environment
where the agent’s goal is to maintain satiety by deciding whether to eat the
available food. The generative model for Case 1 (detailed in Table 1, and
visualized in figure 1) includes hidden states for food availability and agent
satiety, observations that directly correspond to these hidden states, and
actions to eat or not eat. The Likelihood Matrix ($\mathbf{A}$) assumes an
identity mapping between hidden states and observations, meaning that the
agent directly observes the true states of food availability and its own
satiety. Mathematically, this is represented as $P(o_{t}\mid
s_{t},\mathbf{A})=\text{Cat}(\mathbf{A})$, where $\mathbf{A}$ is an identity
matrix. This means the probability of an observation given a state is 1 if
they correspond and 0 otherwise:

$P(o_{t}=i\mid s_{t}=j)=\begin{cases}1&\text{if }i=j\\\ 0&\text{if }i\neq
j\end{cases}$

The agent performs variational inference to optimize an approximate posterior
$Q(s_{t})$ over hidden states at each timestep, using the expected log
likelihood of observations $\mathbb{E}[\log P(o_{t}\mid
s_{t})]=Q(s_{t})^{T}\log\mathbf{A}$.

The Transition Matrix ($B$) specifies that the "eat" action leads to satiety
when food is present, while food remains constantly available regardless of
the agent’s actions. The "don’t eat" action leads to hunger.The transition
likelihood $B$ is represented as a set of matrices $B[f]$, one for each hidden
state factor $f$, with dimensions $S_{f}\times S_{f}\times U_{f}$, where
$S_{f}$ is the number of levels for factor $f$ and $U_{f}$ is the number of
control states or actions for that factor.

The entry $B[f][i,j,k]$ represents the probability of transitioning from state
$j$ to state $i$ for factor $f$, given action $k$: $P(s_{t+1}^{f}=i\mid
s_{t}^{f}=j,u_{t}^{f}=k)$. In this case, the "eat" action ($k=0$) would have a
high probability of leading to the "satiated" state ($i$) when the current
state is "food available" ($j$), while the "don’t eat" action ($k=1$) would
likely lead to the "hungry" state.

Importantly, the transition matrices $B[f]$ are assumed to be conditionally
independent across factors, meaning that the next state of factor $f$ only
depends on the current state and action for that factor, and not on the states
of other factors: $P(s_{t+1}^{f}\mid s_{t}^{f},u_{t}^{f})=P(s_{t+1}^{f}\mid
s_{t},u_{t})$. This simplifies the computation of the joint transition
probability.

The Preference Vector ($C$) encodes a strong preference to observe satiation
and food present. The agent’s goals and preferences are represented as a prior
distribution over observations, $P(o_{1:T})$. The $C$ vector encodes these
preferences as a categorical distribution, where higher values correspond to
preferred observations. The agent aims to maximize the probability of sampling
these preferred observations.

The Initial State Distribution ($D$) is not specified, so that it is a uniform
distribution where each state has an equal probability of being the initial
state. In the PyMDP framework, the initial state distribution is represented
as a categorical distribution over hidden states at the first timestep,
$P(s_{1}\mid D)=\text{Cat}(D)$. If not specified, it defaults to a uniform
distribution, assigning equal probability to all possible initial states.

During the generative process, the agent interacts with the environment, and
its actions affect the state transitions according to the generative process.
If the agent chooses not to eat, the state of the environment remains
unchanged. If the agent chooses to eat and food is present, the agent becomes
satiated, but food remains available due to the static nature of the
environment. We then instantiate the simulation loop. First, the agent
performs state inference based on the current observation, evaluating policies
to maximize expected free energy, and selecting an action that minimizes free
energy and aligns with its preferences. The selected action is applied to the
environment, resulting in state transitions and new observations, and the loop
continues with the agent updating its beliefs, inferring policies, and
selecting actions to achieve its goals. In the extension to Case 1, we
introduce variations to test the agent’s adaptability. In Case 1.1, we set
incorrect A and B matrices, introducing flawed perceptions and beliefs about
state transitions. This extension is intended for us to validate that the
behavior of the agent is in fact predicated on its appropriate appraisal of
the environment.

Component |  Values |  Description
---|---|---
Hidden States |  Food availability: present (1), absent (0) |  Represents the true state of food availability in the environment
|  Agent satiety: hungry (0), satiated (1) |  Represents the true state of the agent’s satiety
Observations |  Observed food availability |  Corresponds directly to the food availability state
|  Agent’s perceived satiety |  Corresponds directly to the agent satiety state
Actions |  Eat (1), Don’t Eat (0) |  The actions available to the agent
Likelihood Matrix (A) |  Identity mapping |  Assumes the agent directly observes the true states
Transition Matrix (B) |  "Eat" (1) leads to satiety (1) when food is present (1) |  Specifies the state transitions based on the agent’s actions
|  "Don’t Eat" (0) leads to hunger (0) |
Preference Vector (C) |  Strong preference for satiated (1) and food present (1) |  Encodes the agent’s goals and drives its behavior
Initial State Distribution (D) |  Uniform Distribution |  Sets the starting conditions for the simulation
Table 1: Components of the generative model for Case 1

###  2.2 Case 2: Dynamic Environment

In Case 2 (laid out in table 2), we extend our model to a dynamic environment
where the agent’s actions have consequences on the availability of food
resources over time. The goal is to study how the agent adopts a sustainable
behavior, balancing its immediate need for satiety with the long-term
availability of food. The generative model for Case 2 (Figure 1) builds upon
the previous model by introducing more granularity in the states and
observations, allowing for a wider range of behaviors and interactions between
the agent and the environment. Both the observations and hidden states are
expanded to have three levels each: food left (0: none, 1: some, 2: abundant)
and satiety (0: not satiated, 1: somewhat satiated, 2: fully satiated). In
this model, we assume that the agent directly observes the true environmental
states with some variations across different levels of food availability and
satiety. In this dynamic environment, the transitions depend on both the
current state and the action taken by the agent. If the agent does not eat,
food availability increases over time, while if the agent eats, food
availability decreases or remains depleted. For the satiety state, if the
agent does not eat, satiety decreases over time, while if the agent eats,
satiety increases. In Case 2, the preferences are designed to balance between
maintaining satiety and ensuring a sustainable food supply, encouraging the
agent to maximize its satiety while also considering the long-term
availability of food resources. Specifically, the agent has a strong
preference for being satiated, while flat preference over food left. The agent
interacts with the dynamic environment over multiple time steps, updating its
beliefs and actions based on the observed states and the changing dynamics of
the environment, and it learns to not eat even if it is not fully satiated.
The agent is initialized with the generative model specified in Case 2, using
an extended policy length to plan multiple time steps ahead and anticipate
future consequences. In the simulation loop, the environment starts with food
fully available and the agent being half satisfied. The agent can plan over
multiple time steps (policy length of 3), and thus has the opportunity to
balance immediate consumption with long-term sustainability. The extended
policy length allows the agent to anticipate future states and avoid greedy
behavior that could lead to starvation. The agent’s policies are restricted to
ensure consistent and sustainable actions across all time steps for both
observation modalities.

Component |  Values |  Description
---|---|---
Hidden States |  Food left: none (0), some (1), abundant (2) |  Represents the true state of food availability in the environment
|  Agent satiety: not satiated (0), somewhat satiated (1), fully satiated (2) |  Represents the true state of the agent’s satiety
Observations |  Observed food availability |  Corresponds to the food availability state with some variability
|  Agent’s perceived satiety |  Corresponds to the agent satiety state with some variability
Actions |  Eat (1), Don’t Eat (0) |  The actions available to the agent
Likelihood Matrix (A) |  High probability of correct observations, lower for adjacent states |  Defines the probability of observations given the true hidden states
Transition Matrix (B) |  “Eat” (1): food left decreases, satiety increases |  Specifies the state transitions based on the agent’s actions and current state
|  “Don’t Eat” (0): food left increases, satiety decreases |
Preference Vector (C) |  Strong preference for satiety. Balances maintaining satiety and sustainable food supply |  Encodes the agent’s goals and drives its behavior
Initial State Distribution (D) |  Uniform Distribution |  Sets the starting conditions for the simulation
Policy Length |  3 time steps |  Allows the agent to plan ahead and consider long-term effects
Table 2: Components of the generative model for Case 2

In Case 2.1, we extend the dynamic environment setup from Case 2 by
introducing a learning mechanism for the agent. The key change is that the
agent now starts with a random B matrix and updates it based on its
experiences in the environment. The agent starts with a random B matrix
instead of a predefined one, which will be updated as the agent interacts with
the environment. The B matrix is initially random, and the agent updates this
matrix based on observed transitions between states. The A matrix remains the
same as in Case 2, mapping the hidden states to observations, and the C
vector, representing the agent’s preferences over observations, remains
unchanged from Case 2. To learn, the agent starts with a randomly initialized
B matrix, which does not initially capture the correct state transitions. The
random initialization is done using a Dirichlet distribution to ensure valid
probability values. At each time step, the agent receives observations, infers
states, infers policies, and samples actions, similar to Case 2. After
executing an action and receiving the next observation, the agent updates its
B matrix. The agent notes the transition from the previous state to the
current state given the action taken. The B matrix is updated using a learning
rate to adjust the probabilities of the observed transitions. For states that
depend on a single factor, the transition probability is updated directly by
increasing the probability of the observed transition by the learning rate.
For states that depend on two factors (e.g., satiety depends on both food left
and previous satiety), the transition probabilities are updated based on the
dependencies specified. After updating, the B matrix is normalized to ensure
that the probabilities sum to 1, maintaining a valid probability distribution.
We also introduce several extended case variations for Case 2 (with and
without learning) to explore the agent’s behavior and performance under
different conditions. We test the agent’s robustness by initializing the B
matrix with incorrect values set to very high (1) or very low (0). The agent’s
performance is expected to degrade, demonstrating the importance of accurate
transition models, and avoiding inertia. We examine the agent’s behavior when
it has different prior preferences by modifying the C vector to represent a
strong preference for food being present. The agent prioritizes actions that
ensure food is present, potentially at the expense of satiety. We test the
agent in an environment where food increases at a slower rate (0.5 units per
step, compared to 1 unit per step previously) when not eating and decreases at
a faster rate (1 unit per step) when eating. Satiety decreases faster when not
eating (0.2 units per step, compared to 1 previously) and increases at a
different rate when eating (0.8 units per step, compared to 1 previously). The
agent needs to adapt its strategy to account for these specific changes in the
environment dynamics. Its performance may be lower compared to Case 2 due to
the increased difficulty in balancing food and satiety levels, as the food
depletes more quickly when eating and satiety decreases more rapidly when not
eating. We finally assess the impact of planning horizon on the agent’s
performance by comparing agents with different planning horizons (1 time step
vs. 3 time steps). Agents with a longer planning horizon are expected to
perform better, as they can anticipate future states more effectively and make
decisions that lead to more sustainable resource management.

##  3 Results

In Case 1, the agent is in a static environment where food is always
available, and its task is to maintain satiety by deciding whether to eat. The
agent consistently chooses to eat at every time step, reflecting its
understanding that food is always available and that eating maximizes its
satiety (Appendix, figure 3, first row). Food availability remains constant
throughout the simulation, as expected in a static environment where food does
not deplete (Appendix, figure 3, second row). The agent’s satiety increases as
it eats and remains at a high level, indicating successful learning and
adaptation to maintain its internal state optimally (Appendix, figure 3, third
row). This setup demonstrates the agent’s ability to perform optimally in an
environment with constant resources. In Case 1.1, we introduce errors in the A
and B matrices to test the agent’s resilience and adaptability when its
internal model does not accurately represent the environment. The agent’s
actions show a more erratic pattern, reflecting confusion or uncertainty due
to the incorrect matrices. Despite the confused matrices, food availability
remains constant as in the standard case (Appendix, figure 4, second row). The
agent’s satiety fluctuates more compared to the standard case, indicating that
the agent’s ability to maintain a consistent internal state is impaired by the
incorrect perception and planning models. This case shows how deviations from
accurate environmental models can affect an agent’s behavior and performance,
leading to less optimal decisions. With Case 1 results, we have shown that the
agent has a degree of validity, and that it does in fact show how the agent
reacts to model fitness, and chooses the best actions relative to its own
survival. In Case 2, the environment is dynamic, with food depleting when
eaten and replenishing if not consumed. The agent must balance its eating
behavior to avoid starvation and resource depletion. It is equipped with a
strong preference over satiety = 2, and flat preference over food left
(Appendix, figure 5). Over multiple runs with a policy length of 3 time steps,
the agent tends to avoid eating, leading him to die of starvation (Appendix,
figure 6 left), or eats too much, leading him to death as well (Appendix,
figure 6 right) as his food gets depleted. In Case 2 with learning, the agent
starts with a randomly initialized B matrix and updates it through
interactions with the dynamic environment. The agent’s actions fluctuate
regularly between "Eat" and "Do Not Eat," suggesting that it has learned a
strategy to balance its actions, so that it manages to survive the whole time
of the run and keeps satiety between 0 and 1 (Appendix, figure 7). The
survival time plot for each run shows that the agent consistently survives for
the maximum number of time steps after the initial learning phase, indicating
that it quickly learns an effective strategy to avoid starvation and maintain
survival (Figure 2).

Figure 2: Case 2, showing the survival time of agents with both learning and
no learning when the agent starts with a random B matrix. The survival time
plot for each run averaged over 10 agents, shows that the agents can quickly
learn to survive by acting in a sustainable way.

Compared to Case 2 without learning, the case with learning shows the agent’s
ability to learn from its interactions with the environment and develop more
effective strategies for survival and resource management. The extended Case 2
variations explore the agent’s behavior and performance under different
conditions, such as incorrect transition models, altered prior preferences,
and varying planning horizons. With an incorrect B matrix, where the values
are set to extremes from the start (1 and 0, rather than lower probabilities),
the agent consistently chooses to eat in every time step, leading to
suboptimal behavior and eventual starvation (Appendix, figure 8, left). Under
certain conditions, the agent was unable to learn, being stuck in the inertia
of its transitions. But overall, with learning enabled, the agent was able to
pull itself out of high values and was able to survive - highlighting the
value of plasticity to get out of bad bootstraps.

In the case of strong preferences on both states (high satiety and high food
left - without learning), the agent initially chooses to eat but then stops
eating as food becomes scarce, demonstrating the influence of strong
preferences on the agent’s actions (Appendix, figure 9, left). This leads the
agent to die over most of its runs quite quickly. With learning enabled
(Appendix, figure 9, right), the agent is able to balance its actions again
and can survive longer, balancing its preferences and the environmental
demands.

When the environment rate of change changes (Appendix, figure 10), the agent’s
performance declines compared to the previous case, but learning still
provides a significant advantage. With learning enabled, the agent adapts its
strategy - eating less often to conserve food, maintaining higher average food
levels, and sustaining satiety more effectively. This allows the agent to
consistently survive the full run timesteps when learning, while it only
survives around 3 timesteps without learning. Although the tougher environment
dynamics make it more challenging, as the agent must plan in a different way
and possibly over longer timescales, the agent demonstrates an impressive
ability to adjust its policy through learning to match the new rate of change
in food and satiety levels. Learning is critical for the agent to find the
right balance and survive in this more complex scenario.

With a policy length of 1, the agent does very poorly without learning, dying
basically after the first step. With learning, it takes the agent a little bit
of time to learn how to survive, but it eventually does. Its actions are a
little more erratic, but it does find a short term strategy (Appendix, figure
11). However, even with this short term strategy, it is unable to survive for
very long, truly highlighting the need to focus on longer term strategies.

##  4 Discussion

Our sustainable agent demonstrates how active inference can give rise to
sustainable resource management strategies at the level of an individual
agent. The agent’s behavior emerges from the interaction between its model of
the world, prior preferences, and the environmental dynamics. It seeks to
optimize for immediate needs (e.g., hunger) and long-term outcomes (e.g.,
consistent food availability), learning to balance consumption and resource
replenishment to promote sustainability. Our findings align with our previous
formalization of sustainability, resilience, and well-being within the active
inference framework.

In Case 1, the static environment allowed the agent to exhibit inertia,
maintaining a consistent consumption pattern without considering long-term
resource availability. While this behavior was adequate for the given context,
it lacked the flexibility needed for sustainable outcomes in more dynamic
environments. Case 2 introduced environmental variability, requiring the agent
to demonstrate elasticity and plasticity. The agent’s ability to adapt its
eating habits in response to changing food availability exemplifies
elasticity, as it temporarily endures increases in free energy (i.e., hunger)
to ensure long-term stability. The agent’s capacity to learn and update its
model of the world based on new information reflects plasticity, enhancing its
resilience in the face of environmental shifts. The agent’s adaptive behavior
in Case 2 reflects resilience, as it adjusts its actions to maintain well-
being under changing resource availability. The dynamic coupling between agent
and environment in the study of sustainable resource management was critical,
even at the level of a single agent. In the extended cases, we can see the
issues with inertia, and the possibility for even adaptive agents to get stuck
in difficult policies.

The agent’s actions optimized its own well-being and contributed to the
resilience of the environment by preventing complete resource depletion. This
reciprocal relationship between the agent and its environment is a fundamental
aspect of sustainability, as the generative models of different layers in a
hierarchical system are inherently linked through niche construction
Albarracin et al. [2024b]. However, the model’s simplicity also reveals its
limitations. The single-agent, single-resource setup does not capture the
complex interdependencies and feedback loops present in real-world systems.
Future research should explore multi-agent scenarios with competing interests
and shared resources, as well as environments with multiple, interconnected
resource types and more sophisticated replenishment dynamics. Additionally,
the model does not consider the possibility of permanent resource depletion,
which would require conditioning the environment’s survival on the maintenance
of certain values. In the future, we need to incorporate this aspect to
understand the long-term implications of resource management strategies. To
further advance the application of active inference in sustainable resource
management, future work should focus on integrating network theory and
dynamical systems theory to model and quantify the interdependencies between
resources and their impact on overall system sustainability. Optimizing
precision or learning rates could also help foster the elastic and plastic
resilience necessary for long-term sustainability and abundance. We would need
to explore this avenue further. Our paper presents a proof-of-concept model
demonstrating how active inference can inform sustainable resource management
at the individual level. We consider the relationship between agent and
environment to highlight the importance of resilience, adaptability, and long-
term planning in achieving sustainable outcomes. While the model’s simplicity
limits its direct applicability to real-world systems, it provides a
foundation for future research exploring the complex dynamics of sustainable
resource management through the lens of active inference.

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##  5 Appendix 1 - Figures

Fig. 3: Case 1. The three plots above show the expected behavior. At time step
0, food is available (food left = 0), and the satiety level is low (satiety =
0). Due to the agent’s strong preference for high level of satiety, it keeps
eating at subsequent time steps and the satiety increases. Since the
environment is static, the food is always present. Fig. 4: Case 1.1 - where
the agent is given incorrect A and B matrices, introducing errors in its
perception and beliefs about state transitions. The top plot shows the agent’s
actions over time. The pattern is more erratic compared to the standard Case
1, as the agent is confused. The middle plot shows the food left observations.
Food availability remains constant at 1 throughout the simulation since the
environment is static. The bottom plot shows the agent’s satiety over time.
Satiety level fluctuates more. This indicates that the agent’s ability to
maintain a stable, high satiety state is impaired by the incorrect perception
and planning models.

Fig. 5: Prior preference for Case 2. The agent’s preferences are changed so
that, unlike case 1 and 1.1 it no longer has a preference over food left. Its
only non-uniform preference is to have a preference over satiety.

Fig. 6: Case 2 - Dynamic Environment. Without learning, the agent either does
not eat (as shown in the three plots on the left) or eats too much and
therefore allows food in the environment to go to 0 (as shown in the three
plots on the right). As a result, the agent dies. Fig. 7: Case 2 - Dynamic
Environment. Example run with learning on and policy length = 3. With this
depth of policy, the agent is able to plan further in time, and with learning
it manages to survive for the whole length of the run. Over 10 time steps the
agent is able to plan its behaviour so that it never reaches satiety = 0, and
always has food left. Fig. 8: Example run from Case 2 without learning enabled
on the left and with learning enabled on the right, but starting with an
extreme B matrix setting (probabilities set to 1 or 0, on the left three plots
and middle three plots, and high but non-extreme values on the right). The
agent dies quickly, just as the randomly set values of the B matrix in plot 6,
and is able to learn on the right. Fig. 9: Case 2 with strong prior
preferences on both high satiety and high food left states. On the three top
left plots, the agent has no learning, and on the top right, the agent has
learning. On the bottom, we can see that survival time is vastly different
with and without learning, as the preferences affect the behavior of the
agent. Fig. 10: Case 2 in a changing environment where food and satiety change
at different time rates. The three top left plots show the results without
learning off, and the three top right plots show the results with learning on.
The bottom plot represents the comparison between the survival time over 10
time steps. Food increases at a slower rate (0.5 units per step) when not
eating and decreases at a faster rate (1 unit per step) when eating. Satiety
decreases faster when not eating (0.2 units per step) and increases at a
different rate when eating (0.8 units per step).  Fig. 11: Case 2 example runs
runs with policy length = 1, left plot without learning, right plot with
learning, and survival time on the bottom.


# Determining the Shape, Size, and Sources of the Zodiacal Dust Cloud
Using Polarized Ultraviolet Scattered Sunlight

Geoffrey Bryden Jet Propulsion Laboratory, California Institute of Technology,
4800 Oak Grove Dr., Pasadena, CA 91109, USA  Neal J. Turner Jet Propulsion
Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena,
CA 91109, USA  Petr Pokorný Department of Physics, The Catholic University of
America, Washington, DC 20064, USA  Astrophysics Science Division, NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA  Center for Research and
Exploration in Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771,
USA  Youngmin Seo Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA  Brian Sutin Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr.,
Pasadena, CA 91109, USA  Virginie Faramaz Steward Observatory, University of
Arizona, Tucson, AZ 85721, USA  Keith Grogan  Jet Propulsion Laboratory,
California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109,
USA  Amanda Hendrix Planetary Science Institute, Tucson, AZ 85719 USA
Bertrand Mennesson Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA  Susan Terebey
California State University Los Angeles, Los Angeles, CA 90032, USA

###### Abstract

The solar system’s Zodiacal Cloud is visible to the unaided eye, yet the
origin of its constituent dust particles is not well understood, with a wide
range of proposed divisions between sources in the asteroid belt and Jupiter
Family comets. The amount of dust contributed by Oort Cloud comets is
uncertain. Knowledge of the Zodiacal Cloud’s structure and origins would help
with NASA’s aim of characterizing potentially Earth-like planets around nearby
stars, since the exo-Earths must be studied against the light scattered from
extrasolar analogs of our cloud. As the only example where the parent bodies
can be tracked, our own cloud is critical for learning how planetary system
architecture governs the interplanetary dust’s distribution. Our cloud has
been relatively little-studied in the near-ultraviolet, a wavelength range
that is important for identifying potentially-habitable planets since it
contains the broad Hartley absorption band of ozone. We show through radiative
transfer modeling that our cloud’s shape and size at near-UV wavelengths can
be measured from Earth orbit by mapping the zodiacal light’s flux and linear
polarization across the sky. We quantify how well the cloud’s geometric and
optical properties can be retrieved from a set of simulated disk observations,
using a Markov chain Monte Carlo analysis. The results demonstrate that
observations with sufficient precision, covering a set of fields distributed
along the ecliptic and up to the poles, can be used to determine the division
between asteroidal, Jupiter Family, and Oort Cloud dust components, primarily
via their differing orbital inclination distributions. We find that the
observations must be repeated over a time span of several months in order to
disentangle the zodiacal light from the Galactic background using the Milky
Way’s rotation across the sky.

zodiacal dust, circumstellar matter, planets and satellites: composition

††journal: PASP††software:  astropy (Astropy Collaboration et al., 2022),
emcee (Foreman-Mackey et al., 2013)

##  1 Introduction

The Zodiacal Cloud (ZC) is a continuously replenished structure composed of
dust grains and meteoroids, stretching from a few solar radii from the Sun
(Stenborg et al., 2021) to locations beyond the Kuiper belt (Poppe et al.,
2019). Due to radiative (Burns et al., 1979) and collisional processes (Grun
et al., 1985) that remove dust on relatively short ($<$Myr) timescales, the
Zodiacal Cloud presents a recent imprint of the dust generating activity of
various source populations. It is continuously evolving as dust particles and
meteoroids are released through collisions among and sublimation of their
asteroidal and cometary parent bodies, and subsequent collisions between the
dust particles and meteoroids themselves.

The major reservoirs of source bodies in the solar system range from main-belt
asteroids and Jupiter-family comets (JFCs) in the inner solar system to
Halley-type and Oort-Cloud comets, and Kuiper-belt objects (Nesvorný et al.,
2011; Pokorný et al., 2014). As the locations and inclinations of these
reservoirs imprint different spatial characteristics on dust within the ZC,
the dust carries crucial information about the relative abundances of parent
bodies and the evolution of the Zodiacal Cloud (Hahn et al., 2002).

The distinct dynamical imprints in the ZC from the separate sources arise from
their different dynamical histories. Dust grains are released with orbital
elements similar to their parent bodies, which vary significantly between
source populations that are concentrated toward the ecliptic plane (asteroids
and Jupiter-family comets) and those that follow an isotropic distribution
(Oort cloud comets). Once released, in addition to solar and planetary
gravity, dust grains are also influenced by solar radiation, the solar wind,
and the solar system magnetic field (Burns et al., 1979). The radiative forces
act on dust via two different effects: (1) radiation pressure pushes the
particles radially away from the Sun, counteracting the Sun’s gravity enough
to blow out the smallest grains from the system, and (2) Poynting-Robertson
(PR) drag effectively pushes against the particles’ motion, causing their
orbits to spiral in towards the Sun. While interplanetary electric and
magnetic fields are important for the dynamics of nanometer-scale particles
(Czechowski & Mann, 2010), they are a minor force on the grains larger than a
few microns which provide most of the optical cross-section of the ZC.

While the smallest grains ($\lesssim$1 µm, depending on composition) can be
quickly blown out of the solar system, larger grains remain orbiting the Sun
for hundreds to millions of years, undergoing often-complex dynamical
evolution. Every particle born on a bound orbit is eventually removed by
either: (1) evaporation close to the Sun after spiraling down under the
influence of PR drag, (2) collision with a planet or large asteroid, (3)
ejection from the solar system after a close encounter with a planet, or (4)
destruction in a collision with another particle. Smaller grains have shorter
lifetimes under PR drag, while larger particles ($\gtrsim$100 µm) are likely
to be fragmented in collisions with other particles before they spiral down
very far (Grun et al., 1985; Wyatt, 2005). Hence the Zodiacal Cloud needs to
be continuously replenished and the sources of its resupply leave imprints in
the orbits of the particles making up the cloud.

The sources of dust in the inner solar system – main-belt asteroids and
Jupiter-family comets – produce components of the ZC that are close to the
ecliptic, arriving at Earth with both eccentricity and inclination low
(Nesvorný et al., 2010, 2011; Pokorny et al., 2019; Koschny et al., 2019). The
sources of dust originating in the outer solar system, on the other hand, form
a more isotropic component of the ZC, with orbital inclinations ranging from
0∘ to 180∘ and relatively high eccentricities (Nesvorný et al., 2011; Pokorný
et al., 2014). There are two general reasons for this dichotomy. The first is
simply that each grain is created with an inclination similar to its parent
body, and the inner solar system source populations (the Jupiter-family comets
and asteroids) are generally on low-inclination orbits confined within 30∘ of
the ecliptic (Nesvorný et al., 2017), whereas the outer solar system source
populations include the broad scattered disk and the Oort Cloud with their
higher inclinations (Vokrouhlický et al., 2019). Second, dust grains
originating from inner solar system sources do not need to overcome Jupiter’s
gravitational barrier, while outer solar system particles must pass by Jupiter
to enter the inner solar system and are often dynamically scattered (and even
ejected) in the process. The Jupiter barrier effectively protects the inner
solar system from Kuiper-belt particles (Poppe, 2016).

Knowing the relative abundances of dust from the different populations of
source bodies would be a substantial advance in our understanding of the ZC’s
structure and evolution, yet these abundances remain poorly constrained by
existing observations. While models of asteroid collisions matched to the IRAS
dust bands find that the majority of zodiacal dust is asteroidal in origin
(Dermott et al., 2002), an analysis of cometary dust production and dynamical
evolution yields the opposite conclusion, that most dust comes from Jupiter-
family comets (Nesvorný et al., 2010). The same analysis finds that up to 10%
of the dust could be from long-period comets, a possibility supported by
several recent observations suggesting nearly-isotropic components in the
optical and infrared sky backgrounds (Sano et al., 2020; Lauer et al., 2021;
Korngut et al., 2022).

We consider in this paper whether new observations from a dedicated space-
based telescope can resolve this issue, determining the relative fractions of
asteroidal vs. cometary dust and detecting for the first time the dust from
Oort Cloud comets, should it exist as a significant component. Fortunately the
Earth’s location at 1 au is in a critical “sweet spot” for probing the
dynamics of different ZC populations and revealing the sources and structure
of the ZC; all major ZC source populations have semi-major axes larger than 1
au, which allows detection of meteoroids from all sources in different stages
of dynamical evolution. Meanwhile, the Earth is far enough from the Sun that
many particles survive the journey to 1 au without being destroyed in
collisions, unlike the particles impacting Mercury (Grun et al., 1985; Pokorný
et al., 2018).

In the following, we first develop a model for the sky brightness that
includes contributions from both the Zodiacal Cloud (§2) and background
Galactic sources (§3). We use this sky model to generate synthetic
observations as might be obtained with a small Earth-orbiting satellite (§4).
We then apply Markov-chain Monte Carlo analysis to the observations to
retrieve the Zodiacal Cloud’s parameters, as a function of qualities of the
observations including the signal-to-noise ratio, number of fields observed,
and number of epochs (§5). Finally, we summarize the findings in (§6).

##  2 Zodiacal Cloud Model

We consider a model for the Zodiacal Cloud that is matched to current
observational constraints. We do not trace its physical evolution – dust
creation through asteroid collisions and comet sublimation; subsequent orbital
migration – instead adopting a set of phenomenological parameters to describe
the disk structure and its radiative properties.

###  2.1 Spatial Distribution

Our geometric model of the disk is based on an all-sky map of the dust’s
thermal emission observed by the Cosmic Background Explorer (COBE) mission’s
Diffuse Infrared Background Experiment (DIRBE) infrared instrument (Silverberg
et al., 1993). Kelsall et al. (1998) fit the zodiacal dust component of this
COBE/DIRBE map across wavelengths from 1.25 to 240 µm, constraining the
overall shape of the Zodiacal cloud. While the Kelsall model provides detailed
measurement of some disk substructure including three dust bands and an Earth-
trailing blob of enhanced emission, it does not distinguish between the
different physical components of the disk. In particular, Kelsall et al.
(1998) do not provide any explicit constraints on a spherical component to the
disk (as would be supplied by Oort cloud comets). Instead they model the
vertical disk profile as a 3-parameter piece-wise exponential (dashed line in
Figure 1).

Figure 1:  Dust from asteroids and Jupiter-family comets is concentrated in
the disk midplane, following Gaussian distributions with H/R = 0.15 and 0.25
for the two components (their combined vertical profile is shown as a dotted
line). Including dust from Oort Cloud comets (solid line) increases the dust
density away from the midplane. This three-component model for the dust is
more consistent with Kelsall et al. (1998) profile fit to COBE’s all-sky map
of thermal emission.

We consider a disk model with three components, each with its own vertical
profile. The asteroidal and Jupiter-family comet components are modelled as
Gaussians with scale heights of 0.15 and 0.25 times the radius, respectively,
matching the distribution of dust produced by the known asteroids and Jupiter-
family comets, which have average inclinations of 9.0∘ and 13.5∘ respectively
(Moskovitz et al., 2022). The dust from these two components is concentrated
toward the ecliptic plane, much more so than the gradual dropoff with height
seen in the Kelsall model (dotted vs. dashed lines in Figure 1). For
consistency with the DIRBE observations – and because Oort Cloud dust must
exist at some level – we include a third, spherically symmetric dust
component. For the fraction of dust in each component we adopt a 70/20/10
split in surface density between Jupiter family comets, asteroids, and Oort
cloud comets, with the total surface density at 1 au set to match the Kelsall
et al. (1998) optical depth of $8\times 10^{-8}$. While the combined vertical
profile (solid line in Figure 1) is a better match to the Kelsall model,
additional Oort Cloud dust would be needed for a perfect match. Instead we
limit the model to a 10% fraction of Oort Cloud dust as a conservative guess
for this key unknown parameter. Note that the split between the three
components is in terms of vertically integrated optical depth; the Oort Cloud
dust only contributes 4% of the volume density in the disk midplane
(5.6$\times 10^{-9}$ au-1, cf. the total midplane density of 1.4$\times
10^{-7}$ au-1, in terms of cross sectional area per volume).

For the radial distribution of the dust, we again follow Kelsall et al.
(1998), who find a gradually declining dust surface density moving away from
the Sun (proportional to orbital radius to the $-$0.34 power). For the
spherically-symmetric Oort-Cloud component, we assume a steeper falloff in
volume density $\rho\propto r^{-2.5}$. We apply an outer cutoff at 3.3 au, as
observed by in-situ photometry from the Pioneer 10 and 11 spacecraft as they
traveled to the outer Solar System (Mann et al., 2006).

The overall density distribution in our model is then

$\rho_{\rm ZC}(R,z)=\tau_{\rm asteroid}\,R^{\gamma_{\rm
asteroid}}\,e^{-\frac{1}{2}(z/H_{\rm asteroid})^{2}}+\tau_{\rm
JFC}\,R^{\gamma_{\rm JFC}}\,e^{-\frac{1}{2}(z/H_{\rm JFC})^{2}}+\rho_{\rm
Oort}\,R^{\gamma_{\rm Oort}}\,\,\,\,[\rm{for}\,\,R<R_{\rm out}]$ (1)

with the parameters described and the default values summarized in Table 1.

Note that in addition to the nine geometric parameters prescribing the dust
density distribution, there are three additional parameters describe the dust
scattering properties (§2.2) and another four to fit the smooth part of the
Galactic background (§3). Each model thus has a total of 16 parameters to be
determined from the synthetic observations as described in (§4).

Table 1: Zodiacal Cloud Model Parameters  Parameter | Description | Value | Motivation
---|---|---|---
$R_{\rm out}$ | outer edge | 3.3 au | Pioneer 10,11 in-situ photometry (Mann et al., 2006)
Asteroidal component
$\tau_{\rm asteroid}$ | optical depth | $1.6\times 10^{-8}$ | 20% of $\tau_{\rm total}$ (Kelsall et al., 1998)
$\gamma_{\rm asteroid}$ |  $\tau$ radial exponent |  $-$0.34 | COBE/DIRBE thermal emission (Kelsall et al., 1998)
$H/R_{\rm asteroid}$ | disk height ratio | 0.15 | known asteroid distribution (Moskovitz et al., 2022)
Jupiter-family comet component
$\tau_{\rm JFC}$ | optical depth | $5.6\times 10^{-8}$ | 70% of $\tau_{\rm total}$ (Kelsall et al., 1998)
$\gamma_{\rm JFC}$ |  $\tau$ radial exponent |  $-$0.34 | COBE/DIRBE thermal emission (Kelsall et al., 1998)
$H/R_{\rm JFC}$ | disk height ratio | 0.25 | known comet distribution (Moskovitz et al., 2022)
Oort-cloud comet component
$\rho_{\rm Oort}$ | density at 1 au | 0.56 $\times 10^{-8}$ | 10% of $\tau_{\rm total}$ (Kelsall et al., 1998)
$\gamma_{\rm Oort}$ |  $\rho_{\rm Oort}$ radial exponent |  $-$2.5 | sublimation-based estimate

###  2.2 Dust Scattering Properties

The focus of this paper is the sunlight scattered and polarized by zodiacal
dust at near-ultraviolet wavelengths around 280 nm; thermal emission from the
dust at infrared wavelengths is not considered. Because the ZC is optically-
thin, calculating the scattered light flux as observed from Earth orbit is a
straightforward integration along each chosen line-of-sight. To calculate the
amount and direction of polarization, we keep track of the light scattered
parallel and perpendicular to the scattering plane containing the Sun, dust
particle, and observer.

For the scattered light’s angular dependence we use the functional form that
Hong (1985) fit to the zodiacal light at optical wavelengths. The zodiacal
dust is both strongly forward-scattering and significantly back-scattering, so
it is not well-approximated by a single Henyey-Greenstein scattering phase
function,

$\Phi(\theta)=\frac{1-g^{2}}{[1+g^{2}-2g\cos(\theta)]^{3/2}}$ (2)

where $\theta$ is the scattering angle and $g\in\left[-1,1\right]$ governs the
anisotropy, ranging from $-1$ for complete back-scattering to $+1$ for
complete forward-scattering (Henyey & Greenstein, 1941). Instead, Hong fit the
scattered intensity versus angle with a sum of three Henyey-Greenstein
scattering functions (Table 2). In modeling below, we fit the overall dust
albedo and the degree of forward scattering ($g$ for the largest-weight Hong
component) for the scattering of radiation polarized parallel and
perpendicular to the scattering plane. The remaining scattering parameters are
fixed at the values Hong (1985) determined by fitting the optical zodiacal
light.

Table 2: Assumed Dust Scattering Properties

Polarization Direction | Albedo | $g$ |  Weight
---|---|---|---
Perpendicular | 0.43 |  $+$0.68 | 0.636
|  |  $-$0.14 | 0.361
|  |  $-$0.84 | 0.003
Parallel | 0.34 |  $+$0.72 | 0.694
|  |  $-$0.28 | 0.299
|  |  $-$0.81 | 0.006

Note. — Adopted from the Hong (1985) fit to the Zodiacal Cloud.

Figure 2 shows the scattering phase function and the net polarization produced
by these assumed dust properties. The difference between the parallel and
perpendicular scattering efficiencies results in a net linear polarization,
but the direction and magnitude of this polarization varies with scattering
angle. For $\sim$90∘ scattering angles, light is preferentially scattered with
polarizations perpendicular to the scattering plane (dashed line in the left
panel of Figure 2), resulting in strong net polarization (more than 20%).
Looking closer to or farther away from the Sun, the polarization is both
weaker and rotated; the negative values in the right panel of Figure 2
indicate polarization parallel to the scattering plane, rather than
perpendicular.

Figure 2:  Left: The net polarization fraction for zodiacal dust is shown as a
function of scattering angle, based on the Hong (1985) fit to the Zodiacal
Cloud at optical wavelengths. At $\sim$90∘ scattering angles, light is
preferentially scattered with polarizations perpendicular to the scattering
plane, while the reverse is true close to 0∘ and 180∘. Right: Scattering at a
right angle produces strong ($>$20%) polarization perpendicular to the
scattering plane. The polarization is weaker for nearly forward or backward
scattered light; the negative values there indicate polarization parallel to
the scattering plane.

Figure 3 shows the flux and polarization for our ZC model alone, without the
background Milky Way and without any instrumental noise. The zodiacal flux
varies smoothly across the sky, with emission in the ecliptic plane a factor
of $\sim$3-4 brighter than the ecliptic pole, except where it rises
dramatically close to the Sun, consistent with published flux tables as a
function of ecliptic angle (Leinert et al., 1998). The polarization map has
more structure, varying across the sky in both magnitude and direction, from a
peak of 18% polarization when viewing at angles roughly perpendicular to the
Sun down to $-$2% polarization (negative value indicates polarization parallel
to the scattering plane, rather than perpendicular) when pointing nearly away
from the Sun.

Figure 3: Total intensity (top) and polarization (bottom) within a near-UV
bandpass, for a model with just the zodiacal light and no Galactic
contamination. The polarization vectors (black lines) are concentric around
the Sun, with expected polarization fraction ranging up to nearly 20% for
light scattered at $\sim$90∘ angles.

##  3 Galactic Background

Emission from the Zodiacal Cloud can be contaminated by other sources in the
sky, most significantly by the background of Galactic stars and interstellar
dust. We include this source of noise in our models based on measurements from
the Galaxy Evolution Explorer (GALEX) of the Galactic background at 231 nm,
close to the 280-nm wavelength of interest here. Although GALEX was a pointed
mission, its large (1.2∘ diameter) field of view enables the creation of
nearly all-sky UV maps at resolution as fine as 6′ (Murthy et al., 2010;
Hamden et al., 2013; Murthy, 2014). Murthy (2014) also provide a catalog of
GALEX’s 44,843 observations with $\sim$233 million lines of individual
measurements across the sky. At each of these locations they provide the FUV
(153 nm) and NUV (231 nm) flux of the diffuse Galactic background. Binning
these data by Galactic latitude yields an overall trend of higher emission in
the Galactic plane, roughly following a cosecant law (Murthy et al., 2010). To
obtain a model for the Galactic emission that can be subtracted from the
observations to determine the zodiacal light contribution, we approximate the
GALEX-measured trend with a cosecant offset to the south and also apply a
maximum so the cosecant does not reach infinity in the Galactic plane:

$F_{Galactic}(\ell)\approx F_{0}+F_{\rm
slope}\left(\frac{\sqrt{C_{0}^{2}+1}}{\sqrt{C_{0}^{2}+\rm{sin}^{2}(\ell-\ell_{\rm
offset})}}-1\right)$ (3)

where $\ell$ is the Galactic latitude, $F_{0}$ = 630 photon cm-2 s-1 Sr-1 Å-1
is the minimum at the Galactic poles, $F_{\rm slope}$ = 500 photon cm-2 s-1
Sr-1 Å-1 is the slope, $\ell_{\rm offset}$ is a 3∘ shift of the peak to the
south, and $C_{0}$ = 0.13 adds a cap to the peak emission. Note that this flux
is considerably higher than Murthy (2014) because we are fitting the median
flux, rather than calculating a minimal floor to the background. Figure 4
shows the binned GALEX data as points, while the dashed line shows our fit.

Figure 4: GALEX observations of the NUV Galactic background (open circles,
with error bars showing the dispersion between pointings) find much higher
emission toward the Galactic midplane, where star-forming regions are
concentrated. While the overall trend can be fit by a modified cosecant
(dashed line; Equation 3) the emission varies significantly at each latitude.

Figure 5: Total intensity within a near-UV bandpass for the Galactic
background, based on observations by GALEX. Two dates are shown: Jan 1 (left)
and Apr 1 (right). In our Sun-centered reference frame, the Galactic plane
(dotted curve) shifts from right to left across the sky during the year, while
the Zodiacal Cloud remains fixed.

Both the overall background flux and its dispersion increase toward the
Galactic midplane. To fully account for the effect of this Galactic structure
on observations of the Zodiacal Cloud, we include all of the observed
structure directly into our simulations. We condense the Murthy (2014) catalog
into 1-deg2 bins, resulting in an all-sky map of the Galactic background with
the same angular resolution as our simulated observations. Note that as the
Earth orbits the Sun, the Galactic contribution rotates behind the Zodiacal
dust; Figure 5 shows the Galactic background at two epochs – Jan 1 and Apr 1 –
illustrating this shift. Repeating observations of the same heliocentric
fields over the course of 6 months, as described in the following section,
makes it possible to disentangle the two contributions, greatly reducing the
deleterious effects of Galactic contamination.

Figure 6 shows the flux and polarization for our combined model, including
both zodiacal dust and Galactic background.

Figure 6: Total intensity (top) and polarization (bottom) within a near-UV
bandpass, for our full model, including Galactic background contamination
based on observations by GALEX. The horizontal rows of squares along the
ecliptic plane and vertical rows of squares along a meridian 90∘ from the Sun
show the locations of simulated observations (§4), where each square is a
3∘x3∘ field of view. Observations are excluded within 30∘ of the Sun (dashed
circle). In this Sun-centered reference frame the Galactic plane (dotted
curve) shifts from right to left across the sky during the year, while the
Zodiacal Cloud remains fixed; this image corresponds to observations made on
Jan 1.

##  4 Simulated Observations

Our intention is to mimic a set of observations that could realistically be
obtained with a small space telescope such as the proposed Polarized Zodiacal
Light Experiment (PoZoLE; Turner et al., 2022). While small satellites cannot
compete with flagship missions in terms of angular resolution and sensitivity
to point sources, they are well-suited to diffuse light measurements such as
those considered here, where the large pixels and large field of view
compensate for the smaller collecting area.

Table 3: Assumed Mission Parameters

Parameter | Assumed Performance
---|---
Simultaneous bandpasses | 2
UV bandpass | 267 – 293 nm
UV throughput | 10%
Optical bandpass ($B$ band) | 380 – 490 nm
Optical throughput | 25%
Mission lifetime | 6 months
Number of observing epochs | 10
Number of pointings per epoch | 220
CCD pixels | 1024 x 1024
Field of view | 3∘ x 3∘
Patch size | 1∘ x 1∘
Flux precision | 1% per patch
Polarization precision | 0.01 per patch
Solar exclusion angle | 30∘

Note. — Throughputs include losses from the waveband filter, a dual-band
dichroic, the polarization splitter, and the detector efficiency.

We assume a 3∘ x 3∘ field of view, broken down in 1∘ x 1∘ patches,
corresponding to $\sim$10′′ pixels on a 1K$\times$1K CCD. This large field of
view maximizes the photons collected while offering sufficient resolution to
mask out nearby UV-bright stars (GALEX found about $\sim$200 stars per square
degree with NUV magnitudes from 12 to 19). Infrared and optical surveys anyway
show little structure in the ZC on smaller angular scales (Kelsall et al.,
1998; Koschny et al., 2019).

One pointing is made for each Earth orbit, with the $\sim$45-minute
integration time split between two orthogonal polarization angles aligned with
the scattering plane. Uniquely determining the linear polarization generally
requires measurements at three angles, but here two are enough since the
zodiacal light’s polarization direction is known from the scattering geometry
(Koschny et al., 2019). The minimum surface brightness of the zodiacal light
(at the ecliptic pole) is $\nu I_{\nu}$ = 42 nW m-2 Sr-1 in the NUV band and
315 nW m-2 Sr-1 in B band (Leinert et al., 1998). For our chosen UV bandpass
(267-293 nm) and instrument throughput (10%, including losses from the
waveband filter, a dichroic, the polarization splitter, and the detector
efficiency), this surface brightness can be detected with signal-to-noise
ratio (SNR) of hundreds for each polarization direction. We conservatively
assume a SNR of 100 for the flux measurement in each patch, enabling the
polarization fraction to be measured to within 0.01.

The focus of these observations is the overall distribution of dust as a
function of orbital radius and height above the ecliptic, rather than the
narrow bands of dust associated with asteroid families (Grogan et al., 2001).
It is therefore sufficient to observe fields along two great circles, one on
the ecliptic plane probing radial gradients and the other along the meridian
90∘ from the Sun probing gradients out of the plane. Each pointing along the
two circles is separated by the 3∘ field of view, resulting in 120
observations along the vertical meridian and another 100 observations around
the ecliptic plan, after excluding the region within 30∘ of the Sun.

It is critical that the observations extend over several months, such that the
Galactic background shifts in our heliocentric-coordinates frame and can then
be subtracted out; we assume a 6 month mission lifetime. This duration allows
13 repeats of the 220-pointing observing pattern (at one Earth orbit per
pointing); we conservatively assume 10 such observing epochs spaced over 6
months. If it is possible to observe twice per orbit around the Earth,
flipping the telescope 180∘ every half orbit, then twice as many epochs would
be attainable.

While the focus of this paper is on UV observations, it would be possible to
use the same observatory to measure the Zodiacal Cloud at optical wavelengths.
We have included the UV throughput loss due to a dual-band dichroic in our SNR
estimate, but otherwise to not consider optical observations in our modeling
effort. Combining UV with optical would provide a dust color along each line-
of-sight, which could assist in associating the dust with various source
populations. Lab measurements of dust grains from the stony S-type asteroid
Itokawa Park et al. (2015) and carbonaceous asteroid Ryugu Nakamura et al.
(2022), for example, show compositions consistent with the parent bodies’
remotely-sensed colors Lederer et al. (2005); Tatsumi et al. (2020).

Table 3 summarizes the key parameters we adopt for the nominal observing
platform. Based on these assumptions, we calculate a complete set of synthetic
observations, apply random photometric noise, and quantify how well the disk
parameters are retrieved via a Markov chain Monte Carlo method (emcee;
Foreman-Mackey et al., 2013). The results are described in the following
section.

##  5 Results

The goal of this section is to quantify how well we can extract the underlying
disk parameters from the simulated observations. This is mainly a
straightforward Markov chain Monte Carlo (MCMC) analysis, with the same
forward model (§2,§3) used both to create the observations and to compute
likelihoods within the MCMC sampler. Our prior information is uninformative
(flat, uniform), but bounds are applied where quantities that cannot
physically go below zero are forced to be positive. The likelihood function is
a standard $\chi^{2}$ integral of data-model offsets (data minus model divided
by uncertainty, squared).

The Galactic background, however, requires special consideration. We fit the
smooth part of the background with four parameters, as described in §3 – the
minimum at the Galactic pole, the slope increasing toward the midplane, the
offset from the midplane, and a cap on the midplane maximum. On top of this
overall trend there is Galactic substructure – the main source of noise in our
fits. While this noise is non-Gaussian, potentially introducing bias to the
fit, we find that the majority of the scatter in the background can be
adequately described by a symmetric 1-$\sigma$ uncertainty as a function of
Galactic latitude. We gauge these uncertainties by the dispersion in our
GALEX-derived background map (1-$\sigma$ ranges shown as dotted lines in
Figure 4). Rather than throw out the regions most contaminated by Galactic
structure (e.g. everything within 20∘ of the Galactic midplane), we include
all observations in our likelihood function to test the robustness of our
fitting procedure.

Table 4: Parameter Fitting Results

Parameter | Description | Input Value | Retrieved Value | Uncertainty
---|---|---|---|---
$\tau_{\rm comet}$ | JFC optical depth | $5.6\times 10^{-8}$ | $5.3\times 10^{-8}$ | $0.3\times 10^{-8}$
$\tau_{\rm asteroid}$ | asteroid optical depth | $1.6\times 10^{-8}$ | $1.4\times 10^{-8}$ | $0.2\times 10^{-8}$
$\rho_{\rm oort}$ | Oort-cloud dust density | 5.6 $\times 10^{-9}$ | 5.4 $\times 10^{-9}$ | 0.6 $\times 10^{-9}$
$H/R_{\rm comet}$ | JFC disk height | 0.25 | 0.25 | 0.01
$H/R_{\rm asteroid}$ | asteroid disk height | 0.15 | 0.11 | 0.03
$\gamma_{\rm comet}$ | JFC $\tau$ radial exponent |  $-$0.34 |  $-$0.34 | 0.01
$\gamma_{\rm asteroid}$ | asteroid $\tau$ radial exponent |  $-$0.34 |  $-$0.33 | 0.02
$s_{\rm oort}$ |  $\rho_{\rm oort}$ radial exponent |  $-$2.5 |  $-$2.4 | 0.1
$R_{\rm out}$ | outer edge | 3.3 au | 3.2 au | 0.1 au
$A$ | dust albedo | 0.38 | 0.41 | 0.02
$g_{\rm scat,}\perp$ | perp. scattering coeff. | 0.68 | 0.68 | 0.01
$g_{\rm scat,}\parallel$ | para. scattering coeff. | 0.72 | 0.72 | 0.01
$F_{0}$ | background at Gal. pole | 630 | 628 | 3
$F_{\rm slope}$ | background slope | 500 | 491 | 6
$\ell_{\rm offset}$ | shift in background |  $-$3∘ |  $-$2.9∘ | 0.1∘
$C_{0}$ | background cap | 0.13 | 0.13 | 0.01

Note. — The three groups of parameters are described in §2.1 (dust spatial
distribution), §2.2 (dust scattering properties), and §3 (Galactic
background).

Figure 7: Posterior distributions are shown for each parameter (upper
histograms) with the 1-$\sigma$ ranges marked by dotted lines and compared
against the input value (blue lines). While most parameters exhibit
independent, Gaussian-like distributions, cross-correlations reveal some
covariance, e.g. the two scattering coefficients whose ratio determines the
level of polarization ($g_{\rm scat,}\perp$ and $g_{\rm scat,}\parallel$) are
highly correlated.

Overall we find that the methodology works, retrieving the key Zodiacal Cloud
parameters to sufficient accuracy to disentangle the contributions from the
various dust source body populations. The precisions achieved for the model
parameters are listed in Table 4, with the full set of parameter cross-
correlations is shown in Figure 7.

We comment here on individual parameters of interest:

  * •

Three components of the Zodiacal Cloud: The most critical measurement is the
relative amounts of dust from the three families of sources bodies. Figure 8
shows the posterior distributions and correlations between the fractional
abundances of the three types of dust. As expected, there is some inverse
correlation between the Jupiter-Family comet and asteroidal components (the
two that are close to the ecliptic plane), as there is the potential for some
interchange between the two densities whil still matching the total column
density. More surprising is that there a similar relationship between the
Oort-cloud and asteroidal components. Most importantly, most difficult
component to measure – the spherically symmetric Oort Cloud component – is
measured to within 1 percentage point (10$\pm$1%; SNR=10).

  * •

Height of the two disk components: The aspect ratios of the two disk
components are measured with 0.01 and 0.03 precision (for the cometary and
asteroid components). Recall that the observations are spaced uniformly along
and vertical meridian. The lower precision on the asteroid component suggests
that concentrating the measurements closer to the ecliptic midplane might be
warranted.

  * •

Radial exponent to the disk: Repeat observations in the ecliptic plane result
in very good measurement of the radial gradient of the disk surface density.
The exponent to the radial power-law (nominally -0.34) is measured to within
0.02.

  * •

Outer edge: While the outer edge of the Zodiacal Cloud is also measured with
quite good precision (0.1 au), this could be improved if future modeling
efforts included in situ radiometry or particle counts by missions traveling
to the outer solar system (e.g. Mann et al., 2006), thereby providing a strong
prior constraint on the outer edge.

Figure 8:  Our primary goal is to measure the optical depth of three
contributors to zodiacal dust at 1 au – Jupiter-family comets, asteroids, and
Oort-cloud comets. The correlations between the retrieved values are shown
here. Starting with 70/20/10% allocations for the three components (blue lines
show these “truth” values), each value is retrieved to within 2 percentage
point of the total. The Oort cloud component (the most tenuous contributor) is
detected with SNR=10.

The specifics of the above results are of course dependent on the assumed
mission parameters (Table 3). A shorter mission with fewer observing epochs
will not perform as well, both because of less overall integration time but
also because multiple epochs are needed for the Galactic background to rotate
relative to the Zodiacal Cloud. To test this, we have performed a range of
trade studies, measuring the response as we vary individual parameters. As
expected, changing the measurement SNR gives a more or less linear change in
the retrieval uncertainties, while changing the number of observations gives a
square-root(N) dependence. Figure 9 shows how the precision on the Oort-cloud
dust density changes depending on the number of observing epochs, against
consistent with a simple square-root(N) dependence. While most mission
parameters have a similar effect on each of the retrieved model parameter
uncertainties, the solar exclusion angle is an exception. Excluding a large
zone (while keeping the total number of observations constant) decreases the
precision of all measurements, but usually with a rather flat dependency, e.g.
increasing the nomimal angle from 30∘ to 80∘ only doubles the uncertainty for
$\rho_{\rm Oort}$ (left panel of Figure 10) and most other model parameters.
The solar exclusion angle, however, has a very strong impact on our ability to
determine the radial distribution of dust. For the same change in angle (from
30∘ to 80∘) the uncertainty on the radial power-law exponent increases by
nearly a factor of 5 (right panel of Figure 10) when we can no longer track
the material close to the Sun.

Figure 9: Our ability to measure model parameters depends on the input mission
parameters, such as the number of observing epochs. As one would expect, the
uncertainty in the Oort-cloud dust density ($\rho_{\rm Oort}$) improves as
more observations are made. Our nominal mission has 10 epochs (solid circle).

Figure 10: Constraints placed on two mode parameters – $\rho_{\rm Oort}$ and
the exponent of the radial power-law – are shown as a function of the number
of the solar exclusion angle. While most parameters (including $\rho_{\rm
Oort}$) have a relatively mild dependence on this angle, the radial exponent
is uniquely sensitive to it. Our nominal mission has a solar exclusion angle
of 30∘ (solid circles).

##  6 Summary & Discussion

We find that the sky coverage and photometric precision from our nominal
mission are sufficient to determine the Zodiacal Cloud’s size and shape, as we
have demonstrate by synthetically observing radiative transfer models of the
cloud and recovering the models’ input parameters using a Markov-chain Monte
Carlo analysis. The cloud’s shape is governed by the orbital inclinations of
the component dust particles and so measuring the shape enables distinguishing
asteroidal from short-period cometary sources and determining whether Oort
Cloud comets contribute a nearly spherically-symmetric dust component.

Revisiting the same heliocentric-coordinates fields after the Milky Way has
rotated behind the zodiacal light provides an effective strategy for
mitigating the effect of Galactic background light on Zodiacal Cloud
observations. When we synthetically observe the same heliocentric field 10
times spanning at least 6 months, a tenuous spherically-symmetric (Oort Cloud)
component to the dust is clearly detected above the Galactic backgrounds.
NASA’s premier UV observatory, the Hubble Space Telescope, can measure the
zodiacal light (Kawara et al., 2017) but cannot feasibly revisit enough of the
sky over multiple epochs to precisely remove the Galactic background.
Furthermore, Hubble cannot map in polarized UV light to best gauge the cloud’s
size and shape, nor can Hubble safely look as close to the Sun as needed to
determine the cloud’s appearance inside 1 au.

While the immediate application of the observations considered here would be
to further our understanding of the origin and evolution of our own planetary
system, the results would also have implications for observations of planetary
systems around other stars, in particular for a flagship mission to directly
image Earth-like planets. NASA preparation for a such mission involves
estimating the frequency of Earth-size planets (from Kepler; Bryson et al.,
2021) and measuring the expected noise level from exo-Zodiacal dust (from
LBTI; Ertel et al., 2020). The analogous dust clouds veiling target exoplanets
may be brighter than our own; the Large Binocular Telescope Interferometer
(LBTI) survey of nearby stars found a median exozodiacal brightness of 3 times
our Zodiacal Cloud’s, though with large uncertainty (Ertel et al., 2020).
Further characterization of exozodiacal dust is needed, to assist in
interpreting observations of candidate exo-Earths. There are three specific
issues that the observations considered in this paper can address: 1) the
color of the dust at UV wavelengths (related to composition), 2) the
likelihood of a forward-scattering halo inducing “pseudo-zodi” noise (related
to dust geometry and scattering properties), and 3) the capture efficiency of
Earth-mimicking dust blobs (related to the orbits of the dust particles).

The first issue (UV characteristics of the dust) arises because of the
importance of ozone. Detection of an Earth-like planet requires a signature of
potential habitability. Among the strongest such signatures is the Hartley
absorption band of ozone (O3) at near-UV wavelengths. The Hartley band is so
strong it produces a spectral feature even when atmospheric oxygen (O2) levels
are too low for detecting signs of O2 itself (The LUVOIR Team, 2019). The band
is strong enough to pick out at modest spectral resolutions (R $\simeq$ 6;
Gaudi et al., 2020) and will be among the first signatures within reach as our
technical capabilities advance. Ozone’s importance means that a four-meter-
class space telescope without a near-UV coronagraph would fail to measure the
Earth-like fraction among exoearth candidates (Checlair et al., 2021). Even
with a near-UV coronagraph, detection of ozone depends on the UV albedo of the
dust, with this noise source potentially setting the required size of the
telescope (Stark et al., 2019). The UV described in this paper would provide a
direct measure of UV-optical color of dust explicitly linked to source bodies
in the solar system and could thereby serve as a reference point for
predictions of exozodiacal dust emission at UV wavelengths.

The second issue concerns the so-called “pseudo-zodi” – an effect where dust
in the outer regions of a planetary system can be brighter than dust in the
habitable zone (Stark et al., 2015). This noise source is not due to the
amount of outer dust, but rather its geometry (disk-like, edge-on) and its
scattering properties (strongly forward scattering). Improved understanding of
the distribution and optical properties of dust in the solar system will help
address the frequency of pseudo-zodi effects around nearby stars, currently
estimated at 50% as common as standard exozodiacal emission (Stark et al.,
2015).

Lastly there is the third issue of dust clumps in exozodiacal disks, which
might be mistaken for planets. While clumps can arise from recent collisions,
the timescale for removal is relatively short, sometimes disappearing within a
year (Melis et al., 2012). The greater concern for observations of mature
planetary systems is the presence of steady-state dust blobs created by the
planets themselves as they capture inspiraling dust. The Earth, for example,
induces a $\sim$10% enhancement of surface density in the dust trailing its
orbit (Kelsall et al., 1998). Depending on the baseline level of dust and
planet mass, such enhancements can emit more flux than the planet itself
(Stark & Kuchner, 2008). The difficulty in interpreting any observed blobs
comes from not knowing the orbital elements of the dust; dust on near circular
orbits is captured more efficiently than eccentric particles. So while
improvements in models of dust evolution (e.g. inclusion of dust collisions;
Stark & Kuchner, 2009) have improved their fidelity, not knowing the source
bodies for the dust leads to an inherent uncertainty in the shape and optical
depth of each simulated dust blob. Measurements of our Zodiacal Cloud’s
structure can be used to validate models of dust sources and transport into
the solar system’s habitable zone, so that the models become more reliable
when applied to exozodiacal clouds far away.

Part of this work was carried out at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and
Space Administration (80NM0018D0004). P.P. was supported by the NASA Solar
System Workings award No. 80NSSC21K0153. ©2023. All rights reserved.

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