# GPT-FinRE: In-context Learning for Financial Relation Extraction using Large
Language Models

Pawan Kumar Rajpoot<EMAIL_ADDRESS>MUST ResearchP.O. Box
1212BangaloreKarnatakaIndia and  Ankur Parikh<EMAIL_ADDRESS>UtilizeAI ResearchBangaloreKarnatakaIndia

(2023)

###### Abstract.

Relation extraction (RE) is a crucial task in natural language processing
(NLP) that aims to identify and classify relationships between entities
mentioned in text. In the financial domain, relation extraction plays a vital
role in extracting valuable information from financial documents, such as news
articles, earnings reports, and company filings. This paper describes our
solution to relation extraction on one such dataset REFinD. The dataset was
released along with shared task as a part of the Fourth Workshop on Knowledge
Discovery from Unstructured Data in Financial Services, co-located with SIGIR
2023. In this paper, we employed OpenAI models under the framework of in-
context learning (ICL). We utilized two retrieval strategies to find top K
relevant in-context learning demonstrations / examples from training data for
a given test example. The first retrieval mechanism, we employed, is a
learning-free dense retriever and the other system is a learning-based
retriever. We were able to achieve 3rd rank overall (model performance and
report). Our best F1-score is 0.718.

relationship extraction, gpt, in context learning, text tagging, REFinD, KDF,
SIGIR, CEIL, ICL, In Context Learning, GPT NEO, Finance, Large Language Model

††copyright: acmcopyright††journalyear: 2023††conference: The 4th Workshop on
Knowledge Discovery from Unstructured Data in Financial Services; July 27,
2023; Taipei, Taiwan

##  1\. Introduction

Figure 1. Relation Extraction example, here both organizations are connected
with ”acquired by” relation.

The emergence of large language models (LLMs) such as GPT-3 (Brown et al.,
)(Thoppilan et al., ) represents a significant advancement in natural language
processing (NLP). These models have expertise in variety of domains and hence
they can be used as it is in multiple NLP tasks. Traditionaly language models
use separate pre-training-and fine-tuning pipelines (Devlin et al., ) (Beltagy
et al., ) (Raffel et al., ) (Lan et al., ) (Zhuang et al., ) where fine-tune
stage follows pre-training. Models are fine-tuned on a task-specific dataset
in a fully-supervised manner. More recently a new paradigm known as in-context
learning (ICL) (Brown et al., )(Min et al., ) is being used which formulates
an NLP task such that LLMs make predictions by learning from demonstrations.
These demonstrations are presented to the LLMs in the context prompt itself.

Under the framework of ICL, LLMs achieve remarkable performance rivaling
previous fully-supervised methods even with only a limited number of
demonstrations provided in the prompt in various tasks such as solving math
problems, commonsense reasoning, text classification, fact retrieval, natural
language inference, and semantic parsing (Brown et al., ) (Min et al., ) (Shin
et al., ). Recently, ICL based approach(Wan et al., ) is utilized for Relation
Extraction (RE) task. RE seeks to identify a semantic relationship between a
given entity pair mentioned in a sentence, which is the central task for
knowledge retrieval requiring a deep understanding of natural language. The
approach achieves improvements over not only existing GPT-3 baselines, but
also on fully-supervised baselines. Specifically, it achieves SOTA
performances on the Semeval and SciERC datasets, and competitive performances
on the TACRED and ACE05 datasets.

Retrieval of examples to demonstrate is a key factor in the overall
performance on these pipelines. LLMs can relate to the presented ”to be
predicted” data point more if the contextual examples predicted are similar to
it. More relevant examples help us to leverage more out from LLMs both in
terms of improvement in performance and less hallucination as examples can
demonstrate model not to hallucinate in some cases.

In this paper, we employed GPT-3.5 Turbo and GPT-4 under the framework of ICL
for the relation extraction task on REFinD dataset. We utilized two retrieval
strategies to find top K relevant in-context learning demonstrations /
examples from training data for a given test example. The first mechanism we
have employed is a learning-free dense retriever. The other system we have
utilized is a learning-based retriever (Rubin et al., ).

##  2\. Preliminary Background

###  2.1.  Task Definition

As per the challenge ”Relation Extraction is the task of automatically
identifying and classifying the semantic relationships that exist between
different entities in a given text.” This shared task is a part of ”Knowledge
Discovery from Unstructured Data in Financial Services” (KDF) workshop which
is collocated with SIGIR 2023.

Let C denote the input context and e1 in C, e2 in C denote the pair of entity
pairs. Given a set of predefined relation classes R, relation extraction aims
to predict the relation y in R between the pair of entities (e1, e2) within
the context C, or if there is no predefined relation between them, predict
y=”no relation”.

###  2.2. Data

The dataset (Kaur et al., ) released with this task is the largest relation
extraction dataset for financial documents to date. Overall REFinD contains
around 29K instances and 22 relations among 8 types of entity pairs. REFinD is
created using raw text from various 10-X reports (including 10-K, 10-Q, etc.
broadly known as 10-X) of publicly traded companies obtained from US
Securities and Exchange Commission. Figure-2 shows different entity types and
relations exist between them.

Figure 2. REFinD dataset relation and entity types.

###  2.3. In Context Learning

In-context learning (ICL) refers to one of the core emergent abilities (Wei et
al., ) that infers new tasks from context. We use the terms ’in-weights
learning’ and ’in-context learning’ from prior work on sequence models (Brown
et al., ) to distinguish between gradient-based learning with parameter
updates and gradient-free learning from context, respectively. Formally, each
training instance is first linearized into an input text x = (x1…xn ) and an
output text y = (y1…yn), where for all tokens x1…xn, y1…yn in V and V is the
vocabulary set of the LM. Given a new test input text x-test, in-context
learning defines the generation of output y as y-test $\sim$ PLM(y-test —
x1,y1,…,xk,yk, x-test), where $\sim$ refers to decoding strategies (e.g.,
greedy decoding and nuclear sampling (Li et al., )), and each in-context
example ei= (xi,yi) is sampled from a training set D. The generation procedure
is especially attractive as it eliminates the need for updating the parameters
of the language model when encountering a new task, which is often expensive
and impractical. Notably, the performance of ICL on downstream tasks can vary
from almost random to comparable with state-of-the-art systems, depending on
the quality of the retrieved in-context examples (Rubin et al., ) (Liu et al.,
) (Wu et al., ).

Retriever |  LLM |  F1-Score
---|---|---
`KNN with openAI embeddings` |  GPT 3.5 Turbo (Examples: 5 retrieved + 5 random per possible relation) |  0.643
`KNN with openAI embeddings` |  GPT 4 (Examples: 5 retrieved + 5 random per possible relation) |  0.697
`EPR with GPT-Neo-2.7B` |  GPT 4 (Examples: 2 retrieved + 3 random per possible relation) |  0.703
`EPR with GPT-Neo-2.7B` |  GPT 4 (Examples: 5 retrieved + 4 random per possible relation) |  0.718
Table 1. Our performance on test data with different combinations of retriever
and LLM

##  3\. GPT-FinRE

GPT-RE is formalized under the ICL framework, using GPT models as shown in
Figure-3.

###  3.1. Prompt Construction

We construct a prompt for each given test example,which is fed to the GPT
models. Each prompt consists of the following components.

Task Description and Predefined Classes : We provide a succinct overview of
the RE task description and the subset of predefined classes R, denoted by O.
This subset is all possible relations exist between entity types of e1 and e2.
The model is explicitly asked to output the relation, which belongs to the O.
Otherwise, the model will output ”no relation”.

K-shot Demonstrations : In the demonstration part, we reformulate each example
by first showing the input prompt x-demo = Prompt(C, e1, e2) and the relation
label y-demo.

Test Input : Similar to the demonstrations, we offer the test input prompt
x-test, and GPT models are expected to generate the corresponding relation
y-test.

###  3.2. Retrieval Systems

We have employed two retrieval strategies to find top K relevant in-context
learning demonstrations / examples from training data for a given test
example.

####  3.2.1.  KNN with OpenAI Embeddings

Since ICL demonstrations closer to the test sample in the embedding space
result in more consistent and robust performance (Liu et al., ). We utilized
the KNN to retrieve the most similar examples in the training set as the few-
shot demonstrations for each test example. As this learning-free dense
retriever relies on the choice of the embedding space, we used OpenAI
embeddings (text-embedding-ada-002) to obtain example representations. For
similarity search, we used FAISS tool (Johnson et al., ).

####  3.2.2.  EPR (Efficient Prompt Retrieval)

This learning-based dense retriever is trained to retrieve a better singleton
in-context example (Rubin et al., ) , and Top-K most similar examples are
selected in the inference stage. This method for retrieving prompts for in-
context learning uses annotated data and a LM. Given an input-output pair, it
estimates the probability of the output given the input and a candidate
training example as the prompt, and labels training examples as positive or
negative based on this probability. It then trains an efficient dense
retriever from this data, which is used to retrieve training examples as
prompts at test time. Due to limited access to OpenAI, we have used the gpt-
neo-2.7B model (Gao et al., ) as our choice of LM.

Figure 3. GPT-FinRE pipeline flow

####  3.2.3. Random Class Examples

Along with KNN / EPR based examples, we also added K examples randomly for
each possible class between two entity types to add more variety in the final
prompt.

##  4\. Experiments

Due to limited access and cost associated with OpenAI, We performed 4 primary
experiments on the test dataset. We tried various rule based heuristics to
improve the F1-score, but it didn’t work as expected. We used retriever
implementations from 111https://github.com/HKUNLP/icl-ceil .

##  5\. Results

The results are shown in Table-1. Our best F1-Score is 0.718. We got 4th
position in the shared-task. We find that GPT 4 performs better than GPT 3.5
Turbo. We also find that learning based retriever (EPR) outperforms learning-
free retriever (KNN with OpenAI embeddigs).

##  6\. Future Work

In future we want to utilize GPT 4 for EPR. We also want to use different
retrieval approaches such as Compositional Exemplars for In-context Learning
(CEIL)(Ye et al., ).

##  7\. Conclusion

This work explores the potential of GPT + ICL on Financial Relation Extraction
(REFinD dataset). We used two retrieval mechanisms to find similar examples:
(1) KNN with OpenAI Embeddings (2) EPR. We tried two different GPT models: (1)
GPT 3.5 Turbo and GPT 4. The experimental results show that GPT 4 with
learning based retriever EPR is giving the best F1-Score of 0.718.

## References

  * (1)
  * (2) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_. Association for Computational Linguistics, Minneapolis, Minnesota, 4171–4186.  https://doi.org/10.18653/v1/N19-1423
  * (3) Jeff Johnson, Matthijs Douze, and Hervé Jégou. 2019\.  Billion-scale similarity search with GPUs.  _IEEE Transactions on Big Data_ 7, 3 (2019), 535–547.
  * (4) Iz Beltagy, Kyle Lo, and Arman Cohan. 2019.  SciBERT: A Pretrained Language Model for Scientific Text. In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_. Association for Computational Linguistics, Hong Kong, China, 3615–3620.  https://doi.org/10.18653/v1/D19-1371
  * (5) Zhenzhong Lan, Mingda Chen, Sebastian Goodman, Kevin Gimpel, Piyush Sharma, and Radu Soricut. 2020\.  ALBERT: A Lite BERT for Self-supervised Learning of Language Representations.  arXiv:1909.11942 [cs.CL]
  * (6) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2020.  Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer.  arXiv:1910.10683 [cs.LG]
  * (7) Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. 2020\.  Language models are few-shot learners.  _Advances in neural information processing systems_ 33 (2020), 1877–1901.
  * (8) Leo Gao, Stella Biderman, Sid Black, Laurence Golding, Travis Hoppe, Charles Foster, Jason Phang, Horace He, Anish Thite, Noa Nabeshima, et al. 2020\.  The Pile: An 800GB Dataset of Diverse Text for Language Modeling.  _arXiv preprint arXiv:2101.00027_ (2020).
  * (9) Richard Shin, Christopher H. Lin, Sam Thomson, Charles Chen, Subhro Roy, Emmanouil Antonios Platanios, Adam Pauls, Dan Klein, Jason Eisner, and Benjamin Van Durme. 2021.  Constrained Language Models Yield Few-Shot Semantic Parsers.  _CoRR_ abs/2104.08768 (2021).  arXiv:2104.08768 https://arxiv.org/abs/2104.08768
  * (10) Liu Zhuang, Lin Wayne, Shi Ya, and Zhao Jun. 2021\.  A Robustly Optimized BERT Pre-training Approach with Post-training. In _Proceedings of the 20th Chinese National Conference on Computational Linguistics_. Chinese Information Processing Society of China, Huhhot, China, 1218–1227.  https://aclanthology.org/2021.ccl-1.108
  * (11) Jiachang Liu, Dinghan Shen, Yizhe Zhang, Bill Dolan, Lawrence Carin, and Weizhu Chen. 2021\.  What Makes Good In-Context Examples for GPT-3?  _CoRR_ abs/2101.06804 (2021).  arXiv:2101.06804 https://arxiv.org/abs/2101.06804
  * (12) Xiang Lisa Li, Ari Holtzman, Daniel Fried, Percy Liang, Jason Eisner, Tatsunori Hashimoto, Luke Zettlemoyer, and Mike Lewis. 2022\.  Contrastive decoding: Open-ended text generation as optimization.  _arXiv preprint arXiv:2210.15097_ (2022).
  * (13) Romal Thoppilan, Daniel De Freitas, Jamie Hall, Noam Shazeer, Apoorv Kulshreshtha, Heng-Tze Cheng, Alicia Jin, Taylor Bos, Leslie Baker, Yu Du, YaGuang Li, Hongrae Lee, Huaixiu Steven Zheng, Amin Ghafouri, Marcelo Menegali, Yanping Huang, Maxim Krikun, Dmitry Lepikhin, James Qin, Dehao Chen, Yuanzhong Xu, Zhifeng Chen, Adam Roberts, Maarten Bosma, Vincent Zhao, Yanqi Zhou, Chung-Ching Chang, Igor Krivokon, Will Rusch, Marc Pickett, Pranesh Srinivasan, Laichee Man, Kathleen Meier-Hellstern, Meredith Ringel Morris, Tulsee Doshi, Renelito Delos Santos, Toju Duke, Johnny Soraker, Ben Zevenbergen, Vinodkumar Prabhakaran, Mark Diaz, Ben Hutchinson, Kristen Olson, Alejandra Molina, Erin Hoffman-John, Josh Lee, Lora Aroyo, Ravi Rajakumar, Alena Butryna, Matthew Lamm, Viktoriya Kuzmina, Joe Fenton, Aaron Cohen, Rachel Bernstein, Ray Kurzweil, Blaise Aguera-Arcas, Claire Cui, Marian Croak, Ed Chi, and Quoc Le. 2022\.  LaMDA: Language Models for Dialog Applications.  arXiv:2201.08239 [cs.CL]
  * (14) Ohad Rubin, Jonathan Herzig, and Jonathan Berant. 2022\.  Learning To Retrieve Prompts for In-Context Learning. In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_. Association for Computational Linguistics, Seattle, United States, 2655–2671.  https://doi.org/10.18653/v1/2022.naacl-main.191
  * (15) Sewon Min, Xinxi Lyu, Ari Holtzman, Mikel Artetxe, Mike Lewis, Hannaneh Hajishirzi, and Luke Zettlemoyer. 2022.  Rethinking the Role of Demonstrations: What Makes In-Context Learning Work?  _arXiv preprint arXiv:2202.12837_ (2022).
  * (16) Jiacheng Ye, Zhiyong Wu, Jiangtao Feng, Tao Yu, and Lingpeng Kong. 2023.  Compositional Exemplars for In-context Learning.  (2023).  arXiv:2302.05698 [cs.CL]
  * (17) Zhen Wan, Fei Cheng, Zhuoyuan Mao, Qianying Liu, Haiyue Song, Jiwei Li, and Sadao Kurohashi. 2023.  Gpt-re: In-context learning for relation extraction using large language models.  _arXiv preprint arXiv:2305.02105_ (2023).
  * (18) Jerry Wei, Jason Wei, Yi Tay, Dustin Tran, Albert Webson, Yifeng Lu, Xinyun Chen, Hanxiao Liu, Da Huang, Denny Zhou, et al. 2023\.  Larger language models do in-context learning differently.  _arXiv preprint arXiv:2303.03846_ (2023).
  * (19) Simerjot Kaur, Charese Smiley, Akshat Gupta, Joy Sain, Dongsheng Wang, Suchetha Siddagangappa, Toyin Aguda, and Sameena Shah. 2023\.  REFinD: Relation Extraction Financial Dataset.  _arXiv preprint arXiv:2305.18322_ (2023).
  * (20) Zhiyong Wu, Yaoxiang Wang, Jiacheng Ye, and Lingpeng Kong. 2023\.  Self-Adaptive In-Context Learning: An Information Compression Perspective for In-Context Example Selection and Ordering.  arXiv:2212.10375 [cs.CL]


# The Atacama Cosmology Telescope: Millimeter Observations of a Population of
Asteroids
or: ACTeroids

John Orlowski-Scherer Department of Physics, McGill University, 3600 Rue
University, Montréal, QC, H3A 2T8, Canada  Ricco C. Venterea Department of
Astronomy, Cornell University, Ithaca, NY 14853, USA  Nicholas Battaglia
Department of Astronomy, Cornell University, Ithaca, NY 14853, USA  Sigurd
Naess Institute of Theoretical Astrophysics, University of Oslo, Norway  Tanay
Bhandarkar Department of Physics and Astronomy, University of Pennsylvania,
209 South 33rd St, Philadelphia, PA, 19104, USA  Emily Biermann Department of
Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15213, USA
Erminia Calabrese School of Physics and Astronomy, Cardiff University, The
Parade, Cardiff, Wales CF24 3AA, UK  Mark Devlin Department of Physics and
Astronomy, University of Pennsylvania, 209 South 33rd St, Philadelphia, PA,
19104, USA  Jo Dunkley Joseph Henry Laboratories of Physics, Jadwin Hall,
Princeton University, Princeton, NJ 08544, USA  Department of Astrophysical
Sciences, Peyton Hall, Princeton University, Princeton, NJ USA 08544  Carlos
Hervías-Caimapo Instituto de Astrofísica and Centro de Astro-Ingeniería,
Facultad de Física, Pontificia Universidad Católica de Chile, Av. Vicuña
Mackenna 4860, 7820436 Macul, Santiago, Chile  Patricio A. Gallardo Kavli
Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637,
USA  Matt Hilton Wits Centre for Astrophysics, School of Physics, University
of the Witwatersrand, Private Bag 3, 2050, Johannesburg, South Africa
Astrophysics Research Centre, School of Mathematics, Statistics, and Computer
Science, University of KwaZulu-Natal, Westville Campus, Durban 4041, South
Africa  Adam D. Hincks David A. Dunlap Department of Astronomy and
Astrophysics, University of Toronto, 50 St. George St., Toronto ON M5S 3H4,
Canada  Specola Vaticana (Vatican Observatory), V-00120, Vatican City State
Kenda Knowles Centre for Radio Astronomy Techniques and Technologies,
Department of Physics and Electronics, Rhodes University, P.O. Box 94,
Makhanda 6140, South Africa  South African Radio Astronomy Observatory, 2 Fir
Street, Observatory 7925, South Africa  Yaqiong Li Department of Physics,
Cornell University, Ithaca, NY 14853, USA  Kavli Institute at Cornell for
Nanoscale Science, Cornell University, Ithaca, NY 14853, USA  Jeffrey J
McMahon  Department of Astronomy and Astrophysics, University of Chicago,
Chicago, IL 60637, USA  Kavli Institute for Cosmological Physics, University
of Chicago, Chicago, IL 60637, USA  Department of Physics, University of
Chicago, Chicago, IL 60637, USA  Enrico Fermi Institute, University of
Chicago, Chicago, IL 60637, USA  Michael D. Niemack Department of Physics,
Cornell University, Ithaca, NY 14853, USA  Department of Astronomy, Cornell
University, Ithaca, NY 14853, USA  Lyman A. Page Joseph Henry Laboratories of
Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA  Bruce
Partridge Department of Physics and Astronomy, Haverford College, 370
Lancaster Ave, Haverford, PA 19041, USA  Maria Salatino Department of Physics,
Stanford University, Stanford 94305 CA USA  Kavli Institute for Particle
Astrophysics and Cosmology, Stanford CA, 94305 USA  Jonathan Sievers
Department of Physics, McGill University, 3600 Rue University, Montréal, QC,
H3A 2T8, Canada  Cristóbal Sifón Instituto de Física, Pontificia Universidad
Católica de Valparaíso, Casilla 4059, Valparaíso, Chile  Suzanne Staggs Joseph
Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton,
NJ 08544, USA  Alexander van Engelen School of Earth and Space Exploration,
Arizona State University, Tempe, AZ 85287, USA  Cristian Vargas Instituto de
Astrofísica and Centro de Astro-Ingeniería, Facultad de Física, Pontificia
Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 7820436 Macul,
Santiago, Chile  Eve M. Vavagiakis Department of Physics, Cornell University,
Ithaca, NY 14853, USA  Edward J. Wollack NASA Goddard Space Flight Center,
8800 Greenbelt Road, Greenbelt, MD 20771 USA

###### Abstract

We present fluxes and light curves for a population of asteroids at millimeter
(mm) wavelengths, detected by the Atacama Cosmology Telescope (ACT) over
$18,000$ deg2 of the sky using data from 2017 to 2021. We utilize high cadence
maps, which can be used in searching for moving objects such as asteroids and
trans-Neptunian Objects (TNOs), as well as for studying transients. We detect
160 asteroids with a signal-to-noise of at least $5$ in at least one of the
ACT observing bands, which are centered near $90$, $150$, and $220$ GHz. For
each asteroid, we compare the ACT measured flux to predicted fluxes from the
Near Earth Asteroid Thermal Model (NEATM) fit to WISE data. We confirm
previous results that detected a deficit of flux at millimeter wavelengths.
Moreover, we report a spectral characteristic to this deficit, such that the
flux is relatively lower at $150$ and $220$ GHz than at $90$ GHz.
Additionally, we find that the deficit in flux is greater for S-type asteroids
than for C-type.

Asteroids (72), CMB (322), mm Astronomy (1061)

##  1 Introduction

The study of asteroids is critical for understanding the formation history of
the solar system (e.g., Michel et al., 2015) as they compose leftover material
from the coalescence of the solar system. Asteroids have long been studied in
the optical to infrared (IR) wavelengths, which encompass both the reflected
and emitted peaks of asteroid emission. The Wide-field Infrared Survey
Explorer (WISE) has detected hundreds of thousands of asteroids and measured
their key features, including their sizes, emissivities at IR wavelengths, and
temperatures (e.g., Mainzer et al., 2011; Masiero et al., 2011). These
properties give clues about their material composition, and hence
understanding them at multiple frequencies is key. Observations at the
millimeter and sub-millimeter (mm and sub-mm) wavelengths supplement those
made at IR frequencies (e.g., Conklin et al., 1977; Johnston et al., 1982;
Viikinkoski et al., 2015). Emission in the mm and sub-mm is thermal in nature
and originates from the depth of the attenuation length (i.e., several mm to
cm) in the regolith, the unconsolidated surface of the asteroid. On the other
hand, emission in the IR is primarily reflected light and originates much
nearer the surface (Campbell & Ulrichs, 1969).

Flux measurements of asteroids made in the sub-mm and mm have consistently
found lower emission than expected from models fit to IR and optical data
(Johnston et al., 1982; Webster et al., 1988). Historically, the reduced flux
at these wavelengths has been interpreted as a drop in effective emissivity
due to scattering of photons as they pass through the regolith (Redman et al.,
1992). However, there is a growing body of evidence that the reduced flux
actually arises from significantly lower temperatures than expected within the
regolith compared to its surface (Keihm et al., 2013).

Progress on resolving this issue has been hindered by the lack of systematic
surveys of asteroids in the mm and sub-mm. The field has long relied on
targeted observations of asteroids, which require significant observatory
resources (see e.g.; Müller & Barnes, 2007; Chamberlain et al., 2007b). Cosmic
microwave background (CMB) experiments, which survey wide areas of the sky in
the mm, offer the promise of large, well-calibrated catalogs of asteroids.
Observations of asteroids with survey instruments have the advantage that they
are ‘free’ – only an analysis is required to extract their fluxes from
existing data, and no new observations are required. This is of great
advantage in systematizing the study of asteroids in the mm and sub-mm.

Recently, the South Pole Telescope (SPT; Carlstrom et al., 2011) team reported
flux measurements for a trio of main-belt asteroids (Chichura et al., 2022).
For two of the three asteroids they detected, the flux measurements were
consistent with predictions derived from WISE observations of unitary
emissivity; for the third, the measured emissivity was $\epsilon=0.66\pm
0.11$. SPT is located at the South Pole, and as such relatively few main-belt
asteroids pass through its observing field, and those that do are only
observable for a relatively short period of time. Observations from mid-
latitude telescopes offer a better view of the ecliptic and correspondingly
have more potential for asteroid observations.

In this paper we present an analysis of over $100$ asteroids extracted from
observations made by the Atacama Cosmology Telescope (ACT) between $2017$ and
$2021$ from its location in Chile. We compare the fluxes measured by ACT to
models calibrated with WISE data. We call the WISE model minus ACT data the
“model difference.” We confirm that, in general, the model difference is
negative, i.e., we observe a mm flux deficit. We study the dependence of the
model difference on wavelength and asteroid class in a manner that has not
previously been possible due to the relative scarcity of targeted
observations.

This paper is structured as follows. In Section 2, we provide an overview of
the ACT telescope, the WISE data set, and the data processing pipeline used to
perform our analysis. In Section 3, we summarize the analysis used to
investigate the model difference as well as to create light and phase curves.
In Section 4, we present the results of that analysis. In Section 5, we
interpret these results with a particular view towards what they may mean for
the regolith composition of asteroids. Finally, in Section 6, we summarize and
consider the opportunities presented by upcoming experiments to expand on this
work.

##  2 Data

###  2.1 The Atacama Cosmology Telescope

The Atacama Cosmology Telescope (ACT) was a $6$ m off-axis Gregorian telescope
located in the Atacama Desert in Chile (Fowler et al., 2007; Thornton et al.,
2016) that was primarily used to make survey observations of the CMB from 2007
to 2022. ACT had three generations of receivers, most recently the Advanced
ACTpol receiver (AdvACT, Henderson et al., 2016; Ho et al., 2017; Choi et al.,
2020). ACT observed the sky in six bands, f030, f040, f090, f150, and f220,
and f280 (Li et al., 2021). Of these, only the data from f090, f150, and f220
are used in this analysis. These bands are centered at approximately $90$,
$150$, and $224$ GHz, respectively, corresponding to diffraction-limited
resolutions of $2.0\arcmin$, $1.4\arcmin$, and $1.0\arcmin$.

###  2.2 Depth-1 Maps

ACT observed the sky by scanning back and forth at constant elevation,
allowing the sky to pass through the observation track. Depth-1 maps are a
single constant elevation scan deep, such that no point on the sky is observed
more than once as it drifts through the telescope beam. They have the useful
property that each pixel can be time-stamped with an accuracy of the time it
takes a sky coordinate to drift through a detector array around 4 minutes.
Since the asteroids do not move significantly on that timescale, we can
accurately stack (Section 3.1) and phase-fold (Section 3.4) the asteroid maps.
Depth-1 maps are made using the same maximum-likelihood framework as the
normal ACT sky maps (see Dünner et al. 2013, Naess et al. 2020, and Aiola et
al. 2020), except that the conjugate gradient iteration used to invert the
map-making equation is cut short, after 100 steps instead of 600, since the
slower-converging large angular scales are irrelevant for the point-like
objects that make up ACT’s time-variable sky (point sources generally
converage after 10 steps, e.g., Marsden et al., 2014). As with the normal
maps, each frequency of each of ACT’s dichroic detector arrays is mapped
separately, resulting in a total of 29,175 depth-1 maps used in this work. All
depth-1 maps were considered in this paper, although since not every depth-1
map contains an asteroid not all of them are actually used in the analysis.
These depth-1 maps are part of the ACT data release 6 (DR6), and the exact
depth-1 map-making procedure will be detailed in the upcoming DR6 paper. These
maps will be of great use generally in searching for transient objects in the
mm (e.g., Li et al., 2023) as well as characterizing the variability of bright
sources.

###  2.3 WISE

The most up-to-date IR measurements of asteroid fluxes come from the WISE
satellite (Wright et al., 2010), and were analyzed by the Near Earth Object
WISE (NEOWISE) group (Mainzer et al., 2011). WISE observed the whole sky in
four IR bands centered at $3.4$, $4.6$, $12$, and $22\,\mu$m, with $5\sigma$
point source sensitivities of $0.08$, $0.11$, $1$, and $6$ mJy, respectively.
We use the preliminary results from Mainzer et al. (2011); the IR sensitivity
relative to the asteroid flux is much higher than the same in the mm, thus the
preliminary release is sufficient for our purposes. These observations can be
used to estimate the asteroid size, sub-solar temperature (which is the
temperature at the hottest point of the asteroid), and emissivity. There are a
number of models for asteroid fluxes (see e.g., Mommert et al., 2018, for an
overview), the most common of which is the Near Earth Asteroid Thermal Model
(NEATM; Harris, 1998). This model is used by the NEOWISE team and it can be
used to generate predictions for the asteroid fluxes at the ACT frequencies.

Figure 1: Summary of the data analysis pipeline used in this paper. Starting
from the ACT time-ordered-data (TODs), we first construct depth-1 maps as
described in Section 2.2. We then use the JPL Horizons Query service provided
by astroquery to obtain asteroid positions as a function of time, and then
extract stamps from each depth-1 map at the location of each asteroid within
it. For a given asteroid, we then stack the stamps, normalizing for the
observational geometry and distance, as detailed in Section 3.1. Next, for
each asteroid observed with high signal-to-noise by ACT, we use WISE IR
observations of that asteroid, in combination with the ATM software package,
to compute predictions from IR data for the flux at ACT’s observing
frequencies (Section 3.2). The differences between these predictions and the
observed fluxes constitute the model difference. The model differences for (4)
Vesta are shown in Figure 3.

##  3 Methods

To compute our measured flux, we first extract small maps centered on a given
asteroid from the depth-1 maps, which we refer to as stamps. We then stack
those stamps to obtain the asteroid flux. We also use WISE data in combination
with the Asteroid Thermal Modeling software package (ATM; Moeyens et al.,
2020) to make predictions for the asteroid flux at the ACT observing
frequencies. We then compare the observations to the predictions. The details
of this workflow are given in Section 3.1, and a pictorial summary is shown in
Figure 1.

###  3.1 ACT Stacking

Figure 2: Top: Stacked matched-filtered depth-1 maps of (4) Vesta showing the
net flux at f090 (left), f150 (center), and f220 (right). The procedure for
making these maps is described in Section 3.1. The map center is indicated by
a black cross, while the beam full-width half-max at each frequency is
indicated by a green circle. Note the faint blue ring surrounding (4) Vesta at
each frequency; this is ringing from the matched-filter used in the
construction of the stamps (Section 2.2). For this plot, we combined data
across arrays using a pixel-by-pixel inverse-variance weight. This is not how
we combined arrays in our analysis (Section 3.1), and is not an accurate
method of capturing the noise properties of the maps. These images are for
illustration only and were not used in scientific analysis. In our analysis,
we do not combine maps across arrays. We instead only combined the central
flux estimates in an inverse-variance way, including the $1\%$ bias discussed
in Section 3.1.
Bottom: Light curve for (4) Vesta, generated as described in Section 3.3. The
three colored dashed lines with shading represent the average (with $95\%$
uncertainty) expected flux at each frequency based on WISE observations
(Section 3.2), scaled according to Equation 1. The data lie below the WISE
model line indicating that there is a deficit of mm flux. To show that the
modeling scheme specified by Equation 2 is likely correct, we fitted the data
at each frequency to the model curve times an overall amplitude, indicated by
the dash-dotted line. The data points cluster around these best fit curves,
indicating that our intrinsic relative flux modeling is likely correct. This
fit amplitude can be compared to the ratio of the amplitude of the stacked
maps to the WISE expected fluxes as an internal consistency check. They are in
excellent agreement for all frequencies.

We measure the flux of individual asteroids using a stacking method. For each
asteroid, we consider each matched-filtered depth-1 map that contains the
asteroid. We localize the asteroid within the map at the time of observation
using the JPL Horizons service,111https://ssd.jpl.nasa.gov/horizons/ accessed
using astroquery (Ginsburg et al., 2019). We take a $20^{\prime}\times
20^{\prime}$ square stamp with $0.5^{\prime}$ resolution centered at that
position and tangent to the plane of the sky using the Pixell software
suite.222https://github.com/simonsobs/pixell The asteroids are point sources
in our maps;333The largest asteroids are $\sim 1000$ km in diameter, with
closest approach of $\sim 1$ AU, yielding an angular size $\lesssim
0.01^{\prime}$, much lower than our resolution. however, using larger stamps
helps us filter out stamps with undesirable properties, for example those
where the asteroid is near the edge of the map. To maximize the signal-to-
noise (S/N), we apply a matched-filter to the depth-1 maps before cutting out
and stacking the stamps. The filter removes both the large scales dominated by
atmospheric noise and scales smaller than the beam. Given a sky map $m$ with
noise covariance $N$ and beam matrix $B$, we form the matched-filter flux map
$f$ as $f=\rho/\kappa$, with $\rho=B^{T}N^{-1}m$ and
$\kappa=diag(B^{T}N^{-1}B)$. The associated flux uncertainty map is
$1/\sqrt{\kappa}$. These are part of the standard ACT DR6 depth-1 release and
are described in more detail in the upcoming ACT DR6 map paper. Our estimate
for the asteroid’s flux $F$ in a single depth-1 map is simply $f$ evaluated at
the asteroid’s location.

We normalize the observed flux by the expected flux in the Rayleigh-Jeans
limit of the Standard Thermal Model (STM) (Lebofsky et al., 1986)

$\displaystyle F_{i}$
$\displaystyle=F_{0}\left(\frac{d_{\text{earth},i}}{1\text{AU}}\right)^{-2}\left(\frac{d_{\text{sun},i}}{1\text{AU}}\right)^{-1/2}10^{-0.004\alpha_{i}}$
(1) $\displaystyle\equiv F_{0}W_{i},$

where $d_{\text{earth},i}$ and $d_{\text{sun},i}$ are the Earth and Sun
centered distance, and $\alpha_{i}$ is the Sun-asteroid-Earth phase angle in
degrees, all at the time of the observation of stamp $i$. Here, $F_{i}$ is the
observed asteroid flux and $F_{0}$ is the normalized asteroid flux, evaluated
at $d_{\text{earth}}=d_{\text{sun}}=1\text{AU}$. This weighting accounts for
the specific viewing geometry and distance at the time of observation for a
particular stamp. We then form the maximum-likelihood stacked flux estimate
($F_{stack,0}$) from the normalized stamps ($F_{stamp,i}$) as;

$\displaystyle F_{stack,0}$
$\displaystyle=(W^{T}N^{-1}W)^{-1}W^{T}N^{-1}F_{stamp,i}$ (2)
$\displaystyle=(\sum_{i}W_{i}N^{-1}_{ii}F_{stamp,i})/(\sum_{i}W_{i}^{2}N^{-1}_{ii})$

for $N$ the noise covariance matrix and $W$ the same as Equation 1. Measuring
the flux at the center of $F_{stack,0}$ at the center pixel of the stack gives
us a maximum likelihood estimate of the normalized flux. This flux can then be
scaled by a particular Earth-asteroid distance, Sun-asteroid distance, and
Sun-asteroid-Earth phase angle using Equation 1 to compare with WISE
observations (see Sec. 2.3). Since the instrument beam is not yet well
characterized for the daytime data, we only use nighttime data, specifically
data from observations between 11pm and 11am coordinated universal time. As a
consistency check for this method, we use the same pipeline to create flux
maps for Uranus and compared them to dedicated scans of Uranus which are used
for ACT calibration (Hajian et al., 2011; Hasselfield et al., 2013).
Additionally, we computed the fractional difference in flux between arrays at
the same frequency for all asteroids. We then combined this fractional
difference in an inverse-variance weighted sense to get an average fraction
difference between arrays at the same frequency. In both cases, the observed
discrepancy is $<1\%$ between various arrays at the same frequency. This is
consistent with the precision of the overall ACT calibration relative to
Planck, which is of order $1\%$ (Aiola et al., 2020), as well as with
dedicated scans of Uranus (Hajian et al., 2011; Dünner et al., 2013;
Hasselfield et al., 2013). As a final consistency check, we compare the flux
values from stamps of Uranus to estimates of its flux made directly from time-
ordered-data (TODs); these agree within uncertainties. To be conservative,
when stacking fluxes between arrays we add a $1\%$ systematic uncertainty term
to the final flux. This systematic term is subdominant to the statistical one
for all but four asteroids. A summary plot for the asteroid (4) Vesta is shown
in Figure 2. It includes stacked flux maps as described in this section, as
well as a light curve (Section 3.3).

###  3.2 ATM Predictions

We use the ATM package, an open-source software which fits NEATM models to
WISE data. The ATM package accurately reproduces the NEOWISE results, and also
includes all the WISE data that are required to compare with ACT measured
fluxes. The ATM package includes notebooks for fitting NEATM models to WISE
data. Within these there are a number of prescriptions for treating the
emissivities in the various WISE bands (see Moeyens et al., 2020, Chapter 3
for details of these prescriptions). We used the NEOWISE model, wherein the
albedo is a free parameter, for comparison with ACT fluxes. When comparing our
measured mm fluxes to the predictions from ATM, we scale the mm fluxes to the
orbital configuration at the time of WISE observations using Equation 1. Each
WISE observation of an asteroid typically includes $\sim 4-6$ individual
exposures, which are spaced much less than a day apart. We evaluate Equation 1
at the median time of each observation. The correction for the differing
observation times is much less than $1\%$ and so it is not included in our
analysis. An example comparison of ATM/WISE predictions and ACT observations
is shown in Figure 3.

Figure 3: Spectral energy distribution (SED) for the NEOWISE NEATM models as
computed by ATM for the asteroid (4) Vesta, fit to observations in all four
WISE bands (four left-most points). The SED has been scaled by the wavelength
to the fourth power, so that blackbody regions are easy to distinguish as flat
lines. The ACT data from stacking (Section 3.1) are the three right-most
points; they are not included in the fit. The zoom shows the ACT deficit, i.e.
that the ACT points lie bellow the blue line at all frequencies The mm ACT
fluxes are significantly deficient compared to the NEATM predictions ($6.8$
and $10.4\sigma$ at f150 and f220, respectively). Moreover, the deficiency is
larger at shorter wavelengths. This pattern is consistent across our
observations as detailed in Section 4.1.

###  3.3 Light Curves

To generate the light curves, we use the stamps described at the beginning of
Section 3.1. Instead of stacking them, we take the flux value from the center
of each stamp and arrange them according to the time of observation of each
stamp. We also use fluxes from the NEATM model, scaling each model flux based
on the Earth and Sun centered distance and Sun-asteroid-Earth phase angle
according to Equation 1. There is an associated error for each flux in the
stamps, which we use to generate the flux error bars. Figure 2 provides an
example light curve for (4) Vesta along with the NEATM scaled flux. The
modulation in this light curve is apparent across all frequency bands and is a
consequence of the change in observational geometry and distance.

###  3.4 Phase Curves

Figure 4: Schematic figure showing the angular relations between the Sun,
Earth, and asteroid used to compute the asteroid phase. The faint vertical
line represents the reference direction at $t_{0}$ at which time $\Psi_{0}=0$,
which was chosen to be January 1, 1970. The Sun-Earth distance $r_{e}$ and
Sun-Asteroid distance $r_{a}$, and Sun-Asteroid-Earth interior angle $\alpha$
are all computable from the ephemerides of the asteroid and Earth. Figure is
not to scale.

Phase curves are light curves that we fold in time by some frequency in order
to detect periodic behavior in the asteroid flux. Specifically, we generate
phase curves from the light curves in order to provide information about flux
variations as a function of the asteroid sub-Earth longitude; that is, the
line of longitude which intersects a line drawn from the center of the
asteroid to the center of the Earth (see e.g., Chamberlain et al., 2007b). We
refer to this as the phase of the asteroid. Figure 4 shows the relevant
geometry. From the Figure, the Sun-Asteroid-Earth signed interior angle,
$\alpha$, plus the difference between the observed phase, $\Psi_{0}$, and the
rotational phase, $\Psi$, plus the asteroid longitude444Note this is not the
ecliptic longitude, but the longitude relative to some reference time at which
$\Psi_{0}\equiv 0$. We have chosen this time to be midnight on January 1,
1970. $\phi_{A}$ must be $\pi$. Rearranging, in radians we have;

$\Psi=\Psi_{0}-\alpha-\phi_{A}+\pi$ (3)

Computing $\Psi_{0}$ requires the rotational period of the asteroid, while
computing $\alpha$ and $\phi_{A}$ requires the asteroid ephemerides, both of
which we acquire from the Small-Body Database.555https://ssd-
api.jpl.nasa.gov/doc/sbdb.html There is one small correction to Equation 3;
the finite light travel time means that the phase at the time of observation
is not the phase at the time the light was emitted from the asteroid. The
Earth-Asteroid distance can change by several light-minutes, and this induces
a non-negligible wobble in the phase. Since the light travel time is known, we
subtract the light travel time from the time of observation. We then compute
$\Psi_{0}$ modulo $2\pi$ and divide by $2\pi$ to obtain the dimensionless
phase $p$.

In order to combine S/N across multiple arrays and frequencies, we fit the
relative flux. For each observation $i$, we normalize the observed flux
$F_{i}$ from Equation 1 by multiplying by the appropriate factor to account
for the observational distance and geometry (i.e. the weighted flux, Equation
1), then dividing by the average of the weighted flux at that frequency and
array:

$F_{i,norm}=\frac{(F_{i}W_{i})}{(\sum_{N}F_{j}W_{j})/N}$ (4)

In addition to phase folding the light curves, we also plot the normalized
flux and a best fit sinusoidal function for the phase curves. Our curve fit is
a function of phase $p$ and is defined as follows:

$F(p)=A_{1}\sin{(4\pi p+\phi_{1})}+A_{2}\sin{(2\pi p+\phi_{2})}+\delta\\\ $
(5)

where we fit for the amplitudes $A_{1}$ and $A_{2}$, phase factors $\phi_{1}$
and $\phi_{2}$, and offset $\delta$. Since we do not resolve the asteroids, we
expect the primary modulation of the asteroid signal to be the observational
cross section of the asteroid. Viewing the asteroid from one perspective and
$180\deg$ from that perspective produces the same observational cross-section.
Therefore, the phase curve frequency is twice the frequency of rotation; this
is the $A_{1}$ term. We include the additional $A_{2}$ term to account for any
potential modulation due to surface variations, such as low emissivity patches
on the asteroid surface.

We use the MCMC implementation from emcee (Foreman-Mackey et al., 2013) to fit
Equation 5 to the data with flat, uninformative priors on all
parameters666Specifically, $|A_{1}|,|A_{2}|,|C|\leq 10$ and
$-2\leq\phi_{1},\phi_{2}\leq 2$. We restricted our analysis to the 40 highest
S/N asteroids.

To determine the validity of the fits, we compared them to a pure constant fit
using an F-test. An F-test can be used to compare nested models (Allen, 1997),
where one model is a proper subset of the other. This allows us to compare the
two sine fit to the constant fit. To perform the F-test, we construct the
F-statistic:

$F=\frac{\Delta\chi^{2}/\Delta\text{DoF}}{\chi_{nested}^{2}/\text{DoF}_{nested}}$
(6)

for $\Delta\chi^{2}$ and $\Delta\text{DoF}$ the change in $\chi^{2}$ and
degrees of freedom between the nested and base model, and $\chi_{nested}^{2}$
and $\text{DoF}_{nested}$ the $\chi^{2}$ and degrees of freedom of the nested
model. The F-statistic can then be converted into a p-value by using the
F-distribution with appropriate number of degrees of freedom, which we then
convert into a $\sigma$ value for convenience. Of our $40$ asteroids, we
detect variation in the phase curve of the form given by Equation 5 at
$>5\sigma$ for two of them, (6) Hebe and (15) Eunomia. Those fits are shown in
Figure 7.

Aside from variation of the form given by Equation 5, we consider how well a
given phase curve is described by a constant fit. To do so, we fit the phase
curves to a constant value in the same manner as for the sine fits. Following
Andrae et al. 2010, we compare the resulting normalized residuals to a
Gaussian distribution using the one-sided Kolmogorov-Smirnov (KS) test. The
same test was performed with the normalized residuals of the binned data.

##  4 Results

###  4.1 Model Difference: Deficit

We applied the method outlined in Section 3 to $1200$ of the expected
brightest asteroids. Of those 1200 asteroids, we detect 160 at $5\sigma$ in
the stack in at least one band, meaning that
$F_{\text{freq}}/\sigma_{\text{freq}}>5$ for one of the three frequency bands.
Similarly, we detect $65$ asteroids at $5\sigma$ in all bands. We detect $218$
asteroids at $5\sigma$ when combining the detection significance across all
three ACT bands. By this we mean combining the detection significance in each
band in an inverse-variance manner. Using our entire catalog for which there
are WISE data, we compared the measured flux at each frequency to the
predictions from the model fit to WISE data (Section 3.2); this is the model
difference. When the model difference is positive, we refer to it as an
excess; when it is negative, a deficit. Here and throughout, the term larger
is always used in an absolute sense, i.e., further from 0. We found a
consistent deficit at each frequency, with the deficit increasing with
frequency. We considered the deficit for asteroids with WISE modeling with S/N
$>5$ ($173$ asteroids) at the relevant frequency band, combining deficits via
inverse-variance weighting. Defining the relative model difference to be:

$\Delta F_{90}\equiv(F_{90,ACT}-F_{90,{\it WISE}})/F_{90,{\it WISE}}$ (7)

we find that the average relative model difference to be $(-4.4\pm 0.6\pm
1.0)\%$ at f090, $(-23.0\pm 0.4\pm 1.0)\%$ at f150, and $(-28.8\pm 0.5\pm
1.0)\%$ at f220, where the first uncertainty is $1\sigma$ statistical and the
second is the $1\%$ systematic gain uncertainty.777The deficit can be
interpreted as an effective emissivity. However, due to the non-thermal
distortion in the emission evidenced by the difference in deficit at various
wavelengths, we do not do so in this paper. This spectral dependence of the
deficit is systematic in the sense that individual asteroids tend to have a
larger deficit at f220 than at f090 (see Figure 5). For asteroids with S/N
$>3$ in each ACT frequency band, $89$ out of $95$ have $\Delta F_{090}>\Delta
F_{220}$, and $91$ out of $95$ have $\Delta F_{090}>\Delta F_{150}$.
Relatedly, $23$ such asteroids have $\Delta F_{090}>\Delta F_{220}$ at
$2\sigma$ and $10$ have so at $3\sigma$. No asteroids with S/N $>3$ have
$\Delta F_{090}<\Delta F_{150}$ or $\Delta F_{090}<\Delta F_{220}$, even at
the $1\sigma$ level.

Figure 5: ACT measured asteroid fluxes versus their WISE modeled counterparts
for asteroids with S/N $>3$ at f220 and f090. Points to the left of $x=0$ have
a deficit at f090 relative to the WISE predictions, and those that are below
$y=0$ have a deficit at f220. Points with S/N $>5$ are in bold. The dashed
line shows $\Delta F_{90}=\Delta F_{220}$ indicating an equal deficit at f090
and f220. 89 out of 95 asteroids with S/N $>3$ at f220 and f090 have a larger
relative deficit at f220 than at f090, equivalent to lying in the lower right
half of this plot.

###  4.2 Model Difference: Excess

Although the majority of the asteroids’ model differences are $\leq 0$, a
small number have statistically significant (S/N $>3$) excess emission. Three
asteroids ((511) Davida, (423) Diotima, and (611) Valeria) have excess
emission at f090, two ((423) Diotima and (784) Pickeringia) have excess
emission at f150, and only one, ((423) Diotima,) has excess emission at f220.
We inspected the light curves and individual stamps for each of these
asteroids, as well as the ATM fits. In the case of (611) Valeria, the excess
at f090 was found to be due to a single exposure where the asteroid was lying
in the extended emission of a very bright point source. We instituted a check
for this sort of scenario and the excess flux at f090 of (611) Valeria
disappeared. In the case of the other asteroids, we found nothing out of the
ordinary. In the context of the full sample, that (511) Davida and (784)
Pickeringia have $3\sigma$ excess flux in one frequency band is consistent
with the statistical expectation. A $3\sigma$ outlier is about 1-in-300, and
we have about $200$ asteroids with $3\sigma$ significant fluxes in any
frequency band. (423) Diotima, on the other hand, is hard to accept as
statistical coincidence. The light curves, individual stamps, and ATM fits for
(423) Diotima all look normal, and the excess is statistically significant in
each band, $3.5\sigma$ at f090, $3.3\sigma$ at f150 and $3\sigma$ at f220.
Moreover, the fluxes follow the same behavior observed in other asteroids,
wherein the flux at f090 ($125\pm 36\%$ excess) is higher than at f150 ($53\pm
16\%$) or f220 ($66\pm 22\%$). Notably, (423) Diotima, a C-type asteroid (see
Section 5.3), has very low albedo (Masiero et al., 2011).

Table 1: Relative model difference for various asteroid classes. Note that in all bands the S-type asteroids have a larger deficit than the C-type asteroids. The M- and P-type asteroids also have generally large deficits while X-types are low deficit, although the low number of asteroids detected in each of these classes makes it difficult to say for certain. Note that both (511) Davida and (423) Diotima are are C-type asteroids with high S/N; these two alone contribute significantly to the relatively low deficit at 90 GHz for this class of asteroids. The number of asteroids with total S/N $>5\sigma$ in each class is given in parentheses. The “All” row is all asteroids with total S/N $>5\sigma$ when combining across all bands, and includes the $1\%$ systematic error, as the statistical error is $\lesssim 1\%$. Class ($\\#$) | $\%\Delta\mathrm{F}_{090}$ | $\%\Delta\mathrm{F}_{150}$ | $\%\Delta\mathrm{F}_{220}$
---|---|---|---
All (173) | $-4.4\pm 1.6$ | $-23.0\pm 1.4$ | $-28.8\pm 1.5$
C (67) | $-1\pm 3$ | $-20.1\pm 1.1$ | $-29.1\pm 1.8$
S (26) | $-20\pm 3$ | $-39.7\pm 1.4$ | $-39\pm 2$
M (6) | $-23\pm 17$ | $-28\pm 7$ | $-32\pm 12$
X (10) | $30\pm 19$ | $-7\pm 8$ | $3\pm 15$
P (9) | $-9\pm 8$ | $-31\pm 3$ | $-30\pm 6$

###  4.3 Model Difference by Class

Under the assumption that the measured model differences are due to the
composition of the regolith, we considered them as a function of asteroid
class using the Tholen classification scheme (Tholen, 1984), which
differentiates asteroids into $14$ types based on their spectra and albedo. By
far the most common types of asteroids are C-types, which are dark and
carbonaceous, and S-types, which tend to be more silicaceous. We used the
Small-Body Database to obtain Tholen types for the asteroids detected by ACT.
For the purposes of this section, we restricted our analysis to asteroids with
unambiguous Tholen classifications, total S/N $>5$ across all bands, and WISE
data. This results in $67$ C-type asteroids and $26$ S-type asteroids.
Interestingly, from our full sample (i.e., without a S/N cut), $109$ asteroids
were C-type while $168$ were S-type, indicating that the C-type asteroids on
average have higher mm flux than the S-type. We believe this is the first
detection of this phenomenum in the mm. We also detected $10$ X-type, $9$
P-type and $6$ M-type asteroids. The P- and M-type asteroids are defined by
their reddish spectra (Tholen, 1984), and are distinguished from one another
by their albedos, with P-types being of low albedo and M-types of moderate
albedo. The X-type asteroids have the reddish spectra characteristic of M- and
P-types, but do not have albedo measurements. For each class of asteroid, we
computed the inverse-variance weighted average model difference at each of
f090, f150, and f220. These are summarized in Table 1. We also show plots of
the distributions of model differences in Figure 6.

Figure 6: Plots showing the inverse-variance weighted average and population
distribution of the model difference (Equation 7) at f090 (left), f150
(middle), and f220 (right) for various asteroid classes. The inverse-variance
weighted mean and $1\sigma$ error on the mean are shown in black, while the
colored violin-plots show the approximate unweighted population distribution.
All asteroids in this plot have unambiguous types from the Small-Body Database
and have a total S/N $>5$ combining all frequencies. Note that the deficit for
C-type asteroids is systematically smaller than for S-type asteroids,
indicating that their composition is impacting the mm flux of those asteroids
as compared to their IR flux.

###  4.4 Phase Curves

Figure 7: Phase curves for (15) Eunomia and (6) Hebe, the two asteroids for
which we make a tentative detection of phase variation. The orange line
indicates the best fit of the two sine model from Equation 5. The green bands
indicate the individual observations, while the blue points are the inverse-
variance weighted binning of those green points into 20 evenly spaced bins.
The models are fit to the individual data points; the binned points have been
plotted to guide the eye. One data binned data point for (6) Hebe has been
omitted as it only contained one individual data point.

We detect phase variation of the form given by Equation 5 for two asteroids at
S/N greater than $5\sigma$ as determined by an F-test. Those asteroids are
(15) Eunomia ($8.1\sigma$) and (6) Hebe ($7.5\sigma$). The best fit parameters
for (15) Eunomia are $A_{1}=0.14^{+0.03}_{-0.03}$,
$\phi_{1}=0.32^{+0.12}_{-0.17}$, $A_{2}=0.04^{+0.04}_{-0.06}$,
$\phi_{2}=0.6^{+0.3}_{-1}$, and $\delta=0.89^{+0.02}_{-0.02}$. For (6) Hebe,
the best fit parameters are $A_{1}=0.10^{+0.06}_{-0.06}$,
$\phi_{1}=0.7^{+0.4}_{-0.3}$, $A_{2}=0.09^{+0.07}_{-0.06}$,
$\phi_{2}=0.9^{+0.7}_{-0.3}$, and $\delta=0.95^{+0.04}_{-0.04}$. Those models
are shown in Figure 7.

Of the asteroids for which we did not detect sinusoidal phase modulation,
three showed statistically significant variation from constant flux. For (4)
Vesta, we reject the null hypothesis of constant flux with phase at
$5.3(3.5)\sigma$ for the binned and unbinned data, for (7) Iris we reject it
at $4.5(2.5)\sigma$, and for (511) Davida we reject it at $3.3(4.1)\sigma$.
For these asteroids, the phase curves show variation in a more complicated
manner than the simple sinusoidal variation given by Equation 5. For
reference, the phase curve for (4) Vesta is shown in Figure 8.

Figure 8: Phase curve for (4) Vesta, one of the three asteroids whose phase
curve is statistically inconsistent with constant flux and which is not well
described by sinusoidal variation of the form given by Equation 5. The green
lines are the individual data points while the blue points are inverse-
variance weighted bins of the green points. The black line is the best fit
constant to the green data point. It is statistically inconsistent with the
green data at $3.5\sigma$ and with the blue binned data at $5.3\sigma$.

##  5 Discussion

###  5.1 Asteroid Fluxes

On the whole, our observations of asteroids with ACT confirm previous findings
of a flux deficit in the mm (Johnston et al., 1982; Webster et al., 1988). Our
f090 deficit of $-4.5\pm 0.6\%\pm 1.0\%$ is less than the deficits found by
other works, which tend towards $-25\%$ (Johnston et al., 1982; Webster et
al., 1988). On the other hand, our f150 ($-23.0\pm 0.4\%\pm 1.0\%$) and f220
deficits ($-28.9\pm 0.6\%\pm 1.0\%$) are in line with the literature. Since
the deficits we quote are averaged across all asteroids, we do not necessarily
expect them to exactly agree with observations of any one individual asteroid.
Further, the decrease in flux with decreasing wavelength suggests that the
material properties of the regolith change with depth, as has been suggested
by e.g., Keihm et al. (2013). However it is not possible to determine which
material property, the emissivity or the temperature, changes without joint
modeling of the ACT and WISE fluxes. We are currently working on extending the
ATM software package to do so, and plan to present them in a future paper.
Note that we observe the relative deficit to be decreasing with wavelength.
Because longer wavelengths probe deeper into the regolith, this would require
either the emissivity or the temperature to be increasing with depth. In
general, one would instead expect the temperature to drop with increasing
depth into the regolith; however, without full joint modeling we cannot rule
out emissivity variations as a source of the mm flux deficit. It may be that
both variations in emissivity and temperature are involved in producing the
deficit. Finally, it is somewhat counter-intuitive that the deficits are
decreasing with wavelength; the shorter wavelength (f150/f220) observations do
not probe the regolith as deeply as the f090 observations, and they are closer
in frequency to the WISE observations to which the ATM was fit. On both counts
one would expect this to make the short wavelength observations more similar
to the IR, but the opposite is observed.

###  5.2 Asteroid Flux Comparisons

Given the large number of asteroids ACT detected, it is possible to directly
compare our flux measurements with previous studies.

####  5.2.1 SPT

We directly compare our observations of (324) Bamberga, (13) Egeria, (22)
Kalliope to those made by SPT in Chichura et al. (2022), specifically to the
effective emissivity listed in Table 1 of that paper. We present the
comparison in Table 2. The ACT f090/f150 bands are slightly different than the
SPT 95 and 150 GHz channels. While this difference is small, we compare the
effective emissivities, not the fluxes, which should be less sensitive to band
differences. There resulting difference in the effective emissivities due to
the differing band centers between ACT and SPT is relatively small,
particularly in comparison to the uncertainties on the SPT results. In general
these two works agree, with none of the asteroids showing a statistically
significant difference in effective emissivity between the two studies. There
is a tendency for the ACT measurements to have lower flux than the SPT
measurements, but it is not statistically significant.

Table 2: Comparison of ACT effective emissitivy measurements to SPT effective
emissivity measurements for the three asteroids significantly detected by SPT.
The results are statistically consistent.

.  Asteroid ACT f090 SPT 95 GHz ACT f150 SPT 150 GHz (324) Bamberga $1.2\pm
0.2$ $1.4\pm 0.3$ $0.93\pm 0.09$ $1.1\pm 0.1$ (13) Egeria $0.77\pm 0.13$
$<1.2$ $0.75\pm 0.06$ $0.90\pm 0.09$ (22) Kalliope $0.29\pm 0.25$ $<0.47$
$0.64\pm 0.10$ $0.66\pm 0.11$

####  5.2.2 Kitt Peak Antenna – (1) Ceres

Webster et al. (1988) used the Kitt Peak Antenna to make a number of
observations of (1) Ceres at $89.5$ and $227$ GHz (Webster et al., 1988). Due
to the differences in observing frequency, we use the prescription for
converting to brightness temperature888The brightness temperature is the
temperature a blackbody would have to be to produce the observed flux at the
observing frequency, and cannot be directly compared to the sub-solar
temperatures that are estimated by the ATM fit. used in Webster et al. (1988),
which is described in Johnston et al. (1982). Our results, summarized in Table
3, are in reasonably good agreement, given the large uncertainties on the
Webster et al. (1988) results. Note that our f090 brightness temperature is
significantly higher than that at f220; this is simply another way of stating
that the relative deficit at f220 is higher than at f090.

Table 3: Comparison of ACT flux measurements to Kitt Peak observations of (1) Ceres. The brightness temperature is computed following Equation 3 in Johnston et al. (1982). We improve the uncertainty by about a factor of 2 at $90$ GHz and about an order of magnitude at $220$ GHz. Frequency (GHz) |  $\mathrm{T}_{\mathrm{B,n,ACT}}$ (K) |  $\mathrm{T}_{\mathrm{B,n,KP}}$ (K)
---|---|---
90 | $190\pm 10$ | $170\pm 20$
220 | $148\pm 2$ | $156\pm 25$

####  5.2.3 ATCA – (4) Vesta and (9) Metis

Müller & Barnes (2007) made observations of the asteroids (4) Vesta and (9)
Metis using the Australia Telescope Compact Array (ATCA) at $93$ and $95.5$
GHz. They included the time of observations, and so we were able to convert
our flux measurements to their observing geometry and distance. We used the
median time of observation to compute observing geometry and distance. The
results change by $<0.5\%$ when using the two extremal times. Their observing
frequencies are slightly different than ACT’s, but from their work the
difference in flux between $93$ and $95.5$ GHz flux was $0.2$ mJy, far less
than the uncertainty on those measurements, so that comparisons to ACT are
reasonable. These comparisons are given in Table 4. We find a somewhat higher
flux for (4) Vesta, by $1.5\sigma$. Moreover, we confirm their “tentative”
result that the effective emissivity of (4) Vesta increases with wavelength.
We find the effective emissivity to be $0.88\pm 0.04$ at f090, $0.70\pm 0.03$
at f150, and $0.63\pm 0.04$, where we have combined the statistical
uncertainty and $1\%$ systematic bias. Our effective emissivity is
significantly higher than that found in Müller & Barnes (2007), despite our
measured fluxes being statistically consistent. This is due to differences in
our modeled emission from (4) Vesta. Müller & Barnes (2007) undertook a much
more detailed modeling of the expected thermal emission from (4) Vesta than we
did, and so it should be considered the more accurate estimate. Note that our
observed increase in relative flux with wavelength could also be explained by
a rapidly rising temperature under the surface.

We also find higher emission at f090 for (9) Metis, although again this
difference is only significant at $\sim 2\sigma$. Unfortunately we do not have
WISE observations of (9) Metis, and so were unable to generate a NEATM model.
As a rough comparison, we take the f150 fluxes and scale them to f090 and f220
using a simple $\nu^{2}$ scaling. This results in a expected flux of $28.1\pm
1.3$ mJy at f090 as compared to a measured $40\pm 3$ mJy, and an expected
$168\pm 14$ mJy as compared to a measured $150\pm 10$ mJy at f220, suggesting
that (9) Metis has a reddish spectrum. However, due to the lack of a NEATM
model to compare to, this result should be considered tentative.

Table 4: Comparison of ACT flux measurements to ATCA observations of (4) Vesta and (9) Metis. We have used the median time of observation for ATCA to account for the observational geometry and distance. The units on all fluxes are mJy. In general the ACT fluxes are somewhat higher, although the difference is not statistically significant. Asteroid | $\mathrm{F}_{\mathrm{ACT},90}$ | $\mathrm{F}_{\mathrm{ATCA},93}$ | $\mathrm{F}_{\mathrm{ATCA},95.5}$
---|---|---|---
(4) Vesta | $162\pm 8$ | $147.3\pm 3.8$ | $147.5\pm 4.8$
(9) Metis | $40\pm 3$ | $32.4\pm 1.1$ | $34.6\pm 1.2$

###  5.3 Asteroid Fluxes by Class

As shown in Table 1 and Figure 6, the model difference (as computed in Section
3.2) is larger in S-type asteroids than in C-type ones. It is possible that
this difference in deficits is caused by the different regolith properties of
C-type (mostly carbon and ice) and S-type (mostly silicacious; Binzel et al.,
1989; Bus & Binzel, 2002) asteroids. Alternatively, it may be that the C-type
asteroid spectra are better described as gray-bodies than those of S-type
asteroids, so that the ATM modeling is more accurate for the C-type asteroids.
However, there is good evidence that both types of asteroids are accurately
modeled as gray-body (Moeyens et al., 2020). Finally, comparing the flux model
differences at each frequency, there is evidence that the mm spectrum differs
between C- and S-type asteroids (Section 4.3). The ratio of the mean f150 to
f220 model difference for S-type asteroids is
$\%\Delta\mathrm{F}_{150}/\%\Delta\mathrm{F}_{220}=1.02$ while for C-type
asteroids it is $0.70$. Further, the ratio of the f090 to f150 model
difference for S-type asteroids is
$\%\Delta\mathrm{F}_{090}/\%\Delta\mathrm{F}_{150}=0.50$ while for C-type
asteroids it is $0.06$. We do not report uncertainties on these numbers since
they are derived from the population means, and hence the error on the mean. A
change in the spectral shape at mm wavelengths suggests at least a partially
physical origin for the difference in C- versus S-type fluxes. A difference in
the spectral shape could also potentially illuminate the physical mechanism,
presumably in the regolith, sourcing that difference, and hence could indicate
differences in regolith composition between C- and S-type asteroids.
Observations of asteroids at higher frequencies, such as $280$ or $350$ GHz,
could shed further light on this. The upcoming Simons Observatory (SO; Ade et
al., 2019) and Fred Young Submillimeter Telescope (FYST; CCAT-Prime
Collaboration et al., 2023) will both provide that potential.

Due to the low number of asteroids with high S/N in each class, we do not
interpret the variation in model difference between M-, P- and X-type
asteroids. .

Figure 9: Scatter plot of the ecliptic latitude of spin axis versus maximum
ellipticity for the 40 highest S/N asteroids in our sample. For two asteroids
((554) Peraga and (375) Ursula) we were unable to find shape parameters, and
hence they are omitted. The S/N scale has been restricted to a maximum of 80
to increase the dynamic contrast. (15) Eunomia and (6) Hebe have been labeled.
(15) Eunomia is a noticeable outlier, while (6) Hebe lies nearer to the center
of the population.

###  5.4 Phase Curves

(15) Eunomia and (6) Hebe have phase curves that are well described by
sinusoid variation of the form of Equation 5, with
$\chi^{2}_{\text{red}}=1.02$ and $0.99$, respectively. The best fit phase
curve for (15) Eunomia indicates that the variation in emission is dominated
by $A_{1}$, or half period term. Variation due to modulation of the asteroid
cross-sectional area will lead to a large value for $A_{1}$, and indeed (15)
Eunomia is quite non-spherical, with ellipticity parameters $a/b=1.47,b/c=1.0$
(De Angelis, 1995), where $a$ is the longest semi-axis, $b$ the intermediate,
and $c$ the shortest. On the other hand, for the more spherical asteroid (6)
Hebe ($a/b=1.1,b/c=1.2$, Marsset et al., 2017), we find $A_{1}\simeq A_{2}$,
where $A_{1}$ and $A_{2}$ are the double and single frequency terms as defined
in Equation 5.

While we do detect phase variation for two asteroids in our sample, it is
notable that we do not detect variation for other, higher S/N asteroids. While
in general the S/N of phase curves increases with the S/N of the asteroid, the
asteroid S/N is not the only determining factor. Firstly, as discussed above,
the variation in asteroid cross-sectional area, and hence the asteroid shape,
drives the $A_{1}$ variation term. This term is dominant in the phase curves
of most asteroids (although not all, see e.g., Chamberlain et al., 2007a), so
that variation is difficult to detect for highly spherical asteroids, even if
they are high S/N. Secondly, the orientation of the rotational axis can
suppress phase variation. As an extreme example, when the spin axis of an
asteroid is pointed toward Earth, we observe no phase variation. In general,
asteroids with highly inclined rotations will have their phase variation
suppressed at some periods in their orbit. Finally, while sine functions do
approximate the phase variation due to cross-sectional area, they do not
necessarily describe all phase curves well. Surface features can easily cause
other forms of variation, such as a step in emissivity, which would not be
well described by a sine wave. Prior mm observations of (4) Vesta have shown
highly irregular light curves, for example (Müller & Barnes, 2007).

We performed a literature review for our 40 highest S/N asteroids, finding
their shape parameters and spin-pole alignments. A scatter plot of these
parameters, as well as our S/N, is shown in Figure 9. (15) Eunomia is a
noticeable outlier, with high S/N, high ellipticity, and a spin-axis that lies
close to perpendicular to the ecliptic. (6) Hebe, on the other hand, is above
average in S/N and spin-axis orientation, but more moderate ellipticity. More
extreme objects than (6) Hebe are detected. Since we cannot cleanly explain
our phase curve selection, we only tentatively claim detections for (15)
Eunomia and (6) Hebe.

Moreover, three asteroids have phase curves that are neither well described as
constant nor sinusoidal. More in-depth analysis, perhaps in conjunction with
existing observations such as Redman et al. (1992), are needed to understand
the origin of this variation.

##  6 Conclusion

We report flux measurements of asteroids in the millimeter using ACT. We
detect $218$ asteroids at $5\sigma$ significance when combining significance
across all three ACT bands, $160$ asteroids at $5\sigma$ significance in at
least one band, and $66$ asteroids at $5\sigma$ significance in all bands. We
confirm a deficit in mm flux as compared to expectations from IR measurements
of those asteroids. Moreover, we detect a statistically significant spectral
shape in the deficits, wherein the flux deficits are systematically larger at
f150 and f220 than at f090. This suggests a more complicated source for the
flux deficit than a simple change in effective emissivity with respect to the
IR effective emissivity. However, we cannot determine the source conclusively
without joint modeling of the IR and mm fluxes.

Additionally, the relative mm flux as compared to WISE expectations is higher
for C-type asteroids than for S-type. The spectrum of relative flux is also
different between C- and S-type asteroids, with S-type asteroids having a flat
spectrum between f150 and f220, while that of C-type asteroids falls in the
same range. Both of these observations suggest compositional differences in
the regoliths of C- versus S-type asteroids. Although Planck measured Zodiacal
emission from families of asteroids (Planck Collaboration et al., 2014), this
is the first systematic study of asteroid fluxes in the mm as a function of
their class.

While we do not offer an explicit physical interpretation for these relative
flux measurements, we do confirm the existence of a mm flux deficit, and we
report a spectral and asteroid-class dependence of this deficit which suggests
a physical origin in the regolith.

We produce light curves for detected asteroids, as well as phase curves. For
two asteroids, (15) Eunomia and (6) Hebe, we detect statistically significant
variation in the phase curves of the sinusoidal form given by Equation 5.
However, we cannot adequately explain why phase variation in some asteroids is
detected but variation in other asteroids is not. As such, caution should be
exercised in ascribing any physical significance to that variation.

Looking to the future, the SO Large Aperture Telescope (SO LAT; Zhu et al.,
2021; Parshley et al., 2018) will provide a significant increase in both
sensitivity and frequency coverage over ACT, adding $280$ GHz observations.
The Fred Young Submillimeter Telescope (FYST) will complete a wide-field
survey fully overlapping with the SO LAT survey area using its Prime-Cam
(CCAT-Prime Collaboration et al., 2023) instrument to observe at $220$, $280$,
$350$, $410$, and $850$ GHz, adding to our ability to examine the spectra of
asteroids in the sub-mm. Finally, looking to the far future, the proposed
CMB-S4 (Abazajian et al., 2016) and CMB-HD (The CMB-HD Collaboration et al.,
2022) experiments would both represent over an order of magnitude improvement
in sensitivity, and hence of asteroid detection. More sensitive observations
over a wider wavelength range will further increase our ability to
characterize the regolith composition of asteroids through their mm fluxes.

##  7 Acknowledgments

This work was supported by the U.S. National Science Foundation through awards
AST-1440226, AST0965625 and AST-0408698 for the ACT project, as well as awards
PHY-1214379 and PHY-0855887. The development of multichroic detectors and
lenses was supported by NASA grants NNX13AE56G and NNX14AB58G. ACT operates in
the Parque Astronómico Atacama in northern Chile under the auspices of the
Comisión Nacional de Investigación Científica y Tecnológica de Chile
(CONICYT), now La Agencia Nacional de Investigación y Desarrollo (ANID).
Colleagues at AstroNorte and RadioSky provide logistical support and keep
operations in Chile running smoothly.Computing for ACT was performed using the
Princeton Research Computing resources at Princeton University, the National
Energy Research Scientific Computing Center (NERSC), and the Niagara
supercomputer at the SciNet HPC Consortium. SciNet is funded by the CFI under
the auspices of Compute Canada, the Government of Ontario, the Ontario
Research Fund–Research Excellence, and the University of Toronto.

John Orlowski-Scherer acknowledges support from the Trottier Space Institute
Fellowship. This work was supported by a grant from the Simons Foundation (CCA
918271, PBL). Ricco C. Venterea acknowledges funding and support from the
Nexus Scholars Program. Nick Battaglia acknowledges the support from NSF grant
AST-1910021 and NASA grants 21-ADAP21-0114 and 21-ATP21-0129. Erminia
Calabrese acknowledges support from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 849169). Adam D. Hincks acknowledges support from the Sutton
Family Chair in Science, Christianity and Cultures and from the Faculty of
Arts and Science, University of Toronto. Yaqiong Li is supported by the KIC
Postdoctoral Fellowship. Cristóbal Sifón acknowledges support from the Agencia
Nacional de Investigación y Desarrollo (ANID) through FONDECYT grant no.
11191125 and BASAL project FB210003. Eve M. Vavagiakis acknowledges support
from NSF award AST-2202237.

## References

  * Abazajian et al. (2016) Abazajian, K. N., Adshead, P., Ahmed, Z., et al. 2016, arXiv e-prints, arXiv:1610.02743, doi: 10.48550/arXiv.1610.02743
  * Ade et al. (2019) Ade, P., Aguirre, J., Ahmed, Z., et al. 2019, J. Cosmology Astropart. Phys, 2019, 056, doi: 10.1088/1475-7516/2019/02/056
  * Aiola et al. (2020) Aiola, S., Calabrese, E., Maurin, L., et al. 2020, J. Cosmology Astropart. Phys, 2020, 047, doi: 10.1088/1475-7516/2020/12/047
  * Allen (1997) Allen, M. P. 1997, Testing hypotheses in nested regression models (Boston, MA: Springer US), 113–117, doi: 10.1007/978-0-585-25657-3_24
  * Andrae et al. (2010) Andrae, R., Schulze-Hartung, T., & Melchior, P. 2010, arXiv e-prints, arXiv:1012.3754, doi: 10.48550/arXiv.1012.3754
  * Binzel et al. (1989) Binzel, R., Gehrels, T., & Matthews, M. 1989, Asteroids II, Asteroids II No. v. 4 (University of Arizona Press)
  * Bus & Binzel (2002) Bus, S. J., & Binzel, R. P. 2002, Icarus, 158, 146, doi: 10.1006/icar.2002.6856
  * Campbell & Ulrichs (1969) Campbell, M. J., & Ulrichs, J. 1969, J. Geophys. Res., 74, 5867, doi: 10.1029/JB074i025p05867
  * Carlstrom et al. (2011) Carlstrom, J. E., Ade, P. A. R., Aird, K. A., et al. 2011, PASP, 123, 568, doi: 10.1086/659879
  * CCAT-Prime Collaboration et al. (2023) CCAT-Prime Collaboration, Aravena, M., Austermann, J. E., et al. 2023, ApJS, 264, 7, doi: 10.3847/1538-4365/ac9838
  * Chamberlain et al. (2007a) Chamberlain, M. A., Lovell, A. J., & Sykes, M. V. 2007a, Icarus, 192, 448, doi: 10.1016/j.icarus.2007.08.003
  * Chamberlain et al. (2007b) Chamberlain, M. A., Sykes, M. V., & Esquerdo, G. A. 2007b, Icarus, 188, 451, doi: 10.1016/j.icarus.2006.11.025
  * Chichura et al. (2022) Chichura, P. M., Foster, A., Patel, C., et al. 2022, ApJ, 936, 173, doi: 10.3847/1538-4357/ac89ec
  * Choi et al. (2020) Choi, S. K., Hasselfield, M., Ho, S.-P. P., et al. 2020, J. Cosmology Astropart. Phys, 2020, 045, doi: 10.1088/1475-7516/2020/12/045
  * Conklin et al. (1977) Conklin, E. K., Ulich, B. L., Dickel, J. R., & Ther, D. T. 1977, in IAU Colloq. 39: Comets, Asteroids, Meteorites: Interrelations, Evolution and Origins, ed. A. H. Delsemme, 257–261
  * De Angelis (1995) De Angelis, G. 1995, Planet. Space Sci., 43, 649, doi: 10.1016/0032-0633(94)00151-G
  * Dünner et al. (2013) Dünner, R., Hasselfield, M., Marriage, T. A., et al. 2013, ApJ, 762, 10, doi: 10.1088/0004-637X/762/1/10
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Fowler et al. (2007) Fowler, J. W., Niemack, M. D., Dicker, S. R., et al. 2007, Appl. Opt., 46, 3444, doi: 10.1364/AO.46.003444
  * Ginsburg et al. (2019) Ginsburg, A., Sipőcz, B. M., Brasseur, C. E., et al. 2019, AJ, 157, 98, doi: 10.3847/1538-3881/aafc33
  * Hajian et al. (2011) Hajian, A., Acquaviva, V., Ade, P. A. R., et al. 2011, ApJ, 740, 86, doi: 10.1088/0004-637X/740/2/86
  * Harris (1998) Harris, A. W. 1998, Icarus, 131, 291, doi: 10.1006/icar.1997.5865
  * Hasselfield et al. (2013) Hasselfield, M., Moodley, K., Bond, J. R., et al. 2013, ApJS, 209, 17, doi: 10.1088/0067-0049/209/1/17
  * Henderson et al. (2016) Henderson, S. W., Allison, R., Austermann, J., et al. 2016, Journal of Low Temperature Physics, 184, 772, doi: 10.1007/s10909-016-1575-z
  * Ho et al. (2017) Ho, S.-P. P., Austermann, J., Beall, J. A., et al. 2017, in Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VIII, ed. W. S. Holland & J. Zmuidzinas, Vol. 9914, International Society for Optics and Photonics (SPIE), 991418, doi: 10.1117/12.2233113
  * Johnston et al. (1982) Johnston, K. J., Seidelmann, P. K., & Wade, C. M. 1982, AJ, 87, 1593, doi: 10.1086/113249
  * Keihm et al. (2013) Keihm, S., Kamp, L., Gulkis, S., et al. 2013, Icarus, 226, 1086, doi: 10.1016/j.icarus.2013.07.005
  * Lebofsky et al. (1986) Lebofsky, L. A., Sykes, M. V., Tedesco, E. F., et al. 1986, Icarus, 68, 239, doi: 10.1016/0019-1035(86)90021-7
  * Li et al. (2021) Li, Y., Austermann, J. E., Beall, J. A., et al. 2021, IEEE Transactions on Applied Superconductivity, 31, 1, doi: 10.1109/TASC.2021.3063334
  * Li et al. (2023) Li, Y., Biermann, E., Naess, S., et al. 2023, arXiv e-prints, arXiv:2303.04767, doi: 10.48550/arXiv.2303.04767
  * Mainzer et al. (2011) Mainzer, A., Bauer, J., Grav, T., et al. 2011, ApJ, 731, 53, doi: 10.1088/0004-637X/731/1/53
  * Marsden et al. (2014) Marsden, D., Gralla, M., Marriage, T. A., et al. 2014, MNRAS, 439, 1556, doi: 10.1093/mnras/stu001
  * Marsset et al. (2017) Marsset, M., Carry, B., Dumas, C., et al. 2017, A&A, 604, A64, doi: 10.1051/0004-6361/201731021
  * Masiero et al. (2011) Masiero, J. R., Mainzer, A. K., Grav, T., et al. 2011, ApJ, 741, 68, doi: 10.1088/0004-637X/741/2/68
  * Michel et al. (2015) Michel, P., DeMeo, F. E., & Bottke, W. F. 2015, in Asteroids IV (University of Arizona Press), 3–10, doi: 10.2458/azu_uapress_9780816532131-ch001
  * Moeyens et al. (2020) Moeyens, J., Myhrvold, N., & Ivezić, Ž. 2020, Icarus, 341, 113575, doi: 10.1016/j.icarus.2019.113575
  * Mommert et al. (2018) Mommert, M., Jedicke, R., & Trilling, D. E. 2018, AJ, 155, 74, doi: 10.3847/1538-3881/aaa23b
  * Müller & Barnes (2007) Müller, T. G., & Barnes, P. J. 2007, A&A, 467, 737, doi: 10.1051/0004-6361:20066626
  * Naess et al. (2020) Naess, S., Aiola, S., Austermann, J. E., et al. 2020, J. Cosmology Astropart. Phys, 2020, 046, doi: 10.1088/1475-7516/2020/12/046
  * Parshley et al. (2018) Parshley, S. C., Niemack, M., Hills, R., et al. 2018, in 2018SPIE10700, ed. H. K. Marshall & J. Spyromilio, Vol. 10700, International Society for Optics and Photonics (SPIE), 1292 – 1304, doi: 10.1117/12.2314073
  * Planck Collaboration et al. (2014) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2014, A&A, 571, A14, doi: 10.1051/0004-6361/201321562
  * Redman et al. (1992) Redman, R. O., Feldman, P. A., Matthews, H. E., Halliday, I., & Creutzberg, F. 1992, AJ, 104, 405, doi: 10.1086/116248
  * The CMB-HD Collaboration et al. (2022) The CMB-HD Collaboration, :, Aiola, S., et al. 2022, arXiv e-prints, arXiv:2203.05728, doi: 10.48550/arXiv.2203.05728
  * Tholen (1984) Tholen, D. J. 1984, PhD thesis, University of Arizona
  * Thornton et al. (2016) Thornton, R. J., Ade, P. A. R., Aiola, S., et al. 2016, ApJS, 227, 21, doi: 10.3847/1538-4365/227/2/21
  * Viikinkoski et al. (2015) Viikinkoski, M., Kaasalainen, M., Ďurech, J., et al. 2015, A&A, 581, L3, doi: 10.1051/0004-6361/201526626
  * Webster et al. (1988) Webster, W. J., Johnston, K. J., Hobbs, R. W., et al. 1988, AJ, 95, 1263, doi: 10.1086/114722
  * Wright et al. (2010) Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868, doi: 10.1088/0004-6256/140/6/1868
  * Zhu et al. (2021) Zhu, N., Bhandarkar, T., Coppi, G., et al. 2021, ApJS, 256, 23, doi: 10.3847/1538-4365/ac0db7