# Requirements Analysis and Management for Benefiting Openness

Johan Linåker  Department of Computer Science
Lund University, Lund, Sweden
<EMAIL_ADDRESS>Krzysztof Wnuk  Department of Software Engineering
Blekinge Institute of Technology, Karlskrona, Sweden
<EMAIL_ADDRESS>
###### Abstract

Requirements Engineering has recently been greatly influenced by the way how
firms use Open Source Software (OSS) and Software Ecosystems (SECOs) as a part
of their product development and business models. This is further emphasized
by the paradigm of Open Innovation, which highlights how firms should strive
to use both internal and external resources to advance their internal
innovation and technology capabilities. The evolution from market-driven
requirements engineering and management processes, has reshaped the
understanding of what a requirement is, and how it is documented and used. In
this work, we suggest a model for analyzing and managing requirements that is
designed in the context of OSS and SECOs, including the advances and
challenges that it brings. The model clarifies how the main stages of
requirements engineering and management processes can be adjusted to benefit
from the openness that the new context offers. We believe that the model is a
first step towards the inevitable adaptation of requirements engineering to an
open and informal arena, where processes and collaboration are decentralized,
transparency and governance are the key success factors.

Keywords: Open Innovation, Open Source Software, Software Ecosystem,
Requirements Engineering, Requirements Management, Software product management

##  I Introduction

Software-intensive firms continuously have to face new challenges in order to
sustain their competitive advantage. Known factors that they have learned to
identify, and to a large degree control over the years include frequent
technology changes [3] and shifting market needs [23]. During the past decade,
the advent of various forms of openness, such as Software Ecosystems (SECOs)
[12] and Open Source Software (OSS) has become pivotal parts of many firms’
product development and business models. This has pushed them into a new
unknown context where they need to “learn how to play poker as well as chess”
[5] by searching for, experimenting with, and ultimately using externally
generated innovation. This new open context may be further explained by the
Open Innovation (OI) model, which highlights how firms should strive to use
both internal and external resources to advance their internal innovation and
technology capabilities [22].

Requirements Engineering (RE) has primarly focused on internal (from within
the firm) stakeholder interaction in regards to activities such as analysis,
research and development [28]. Market-Driven Requirements Engineering [23] and
crowdsourcing [11] brought more focus on external stakeholders. OI has not
only blurred the boundaries between internal and external contexts, but also
greatly extended the scope of requirements activities to entire ecosystems or
open communities. Moreover, OI brought the support for both the adoption of
externally acquired innovation and the active commercialization of internally
generated innovations that are not aligned with the current business model
(e.g. via licensing or sale) [5]. As a result, and due to strong relationship
between requirements engineering and value creation [3], requirements
engineering for OI need to be reshaped to better “sustain innovation” [13]. As
pointed out by Rohrbeck et al. [24] value should no longer only be created for
customers but must also be captured for partners and suppliers in increasingly
more complex and collaborating ecosystems.

In our previous work, we investigated OI in a large organization that recently
transitioned to an OI model by abandoning the development of a purely
proprietary code base for their software product and making use of an OSS
project (referred as a ”platform”) as a source of innovation (in both
knowledge and technology) [28]. The platform is not only the main component of
the firm’s product, but is also available to, or used by any other player
within the SECO. The focus on previous investigation was on identification of
requirements management [10] and decision making challenges associated with
the adoption of OI.

In this vision paper, we propose a model for managing requirements that helps
to benefit from openness and responds to previously outlined challenges. In
particular, we believe that the model contributes in requirements elicitation,
analysis, release planning and commercialization of internally generated
requirements that are not aligned with the current products or business
models.

This paper is structured as follows. Section II brings background and related
work while Section III outlines and explains our model. We conclude the paper
in Section IV and discuss future work direction.

##  II Background

Here we present how OI connects to Software Engineering, and how RE
differentiates in an open versus a closed context.

###  II-A Software Engineering in Open Innovation

The OI model is commonly described with the use of a funnel [5], see Fig. 1.
The funnel itself represents the software development process. This may in
turn be illustrated either as an iterative process, or just as one single
development cycle (e.g. from RE, design, implementation, test, to release).
The funnel is full of holes representing openings between the firm’s
development process and the open environment, in our case, an OSS ecosystem.
The arrows going in and out represent transactions and exchange of knowledge
between the two, such as feature requests or bug reports, design proposals or
feature road maps, feature implementations or bug fixes, test cases, complete
sets of code as in plugins, components, platforms or even products. From the
RE perspective this includes activities such as elicitation, prioritization
and release planning, which are performed interactively between the internal
and the open environment.

Figure 1: The OI model illustrated with interactions between the firm (funnel)
and its external collaborations. Adopted from Chesbrough [5].

The knowledge exchange illustrated in Figure 1 may be bi-directional, i.e. it
can go into the development process from the open environment (outside-in), or
from the development process out to the open environment (inside-out). On the
left-hand side the arrows represent the intent and business model, which
encompasses and motivates the development process. On the right-hand side the
arrows represent the output from the development process, with some degree of
innovation. This output may be a new or improved product or service from which
the company may capture or create value with the help of their business model.
Also on the right-hand side are the bulls-eye marks, which represents the
current but also new alternative markets to which the firm’s business model
could deliver the product or service produced from their development process.

###  II-B Requirements Engineering in Open Innovation

Fig. 2 presents RE in the context of OI. An example software-intensive firm is
represented by a funnel, connecting to the OI model. The open environment, or
a OSS ecosystem, with which the firm interacts, is represented by the cloud.
Other actors, which for example may include individuals, NPOs and other firms
are also represented by a funnel each. We further need to differentiate
between an internal and external RE process. With internal, we refer to the
process connected to a firm’s internal software product development. The
external concerns the one used in an OSS ecosystem with which the firm, and
all other actors, interact with.

Figure 2: The requirements engineering context influenced by the OI paradigm.

Looking at OSS ecosystems, requirements practices are often informal and
overlapping [25]. Requirements are commonly asserted through transparent
discussions and suggestions by the OSS project’s developers and users, often
together with prototypes or proof-of-concepts [2, 8]. Assertion may also be
done post-hoc, simultaneously as the requirement realization [7, 8]. These
assertions are specified and managed in what Scacchi refers to as informalisms
[25], e.g. reports in an issue tracker, messages in a mailing list, or commits
in a version control system. Through social interaction facilitated by the
infrastructure persisting the informalisms, requirements are further enriched
and validated [26, 7, 8]. Prioritization is commonly conducted by ecosystem
maintainers overseeing the project management, though care is often taken to
the opinions of other developers and users (e.g. through votes and comments
[16]). Ernst & Murphy refers to this lightweight and evolutionary process of
requirements refinement as Just-In-Time (JIT) requirements (illustrated by the
circular arrows inside the OSS ecosystem in Fig. 2), compared to the more
traditional upfront requirements characterized by heavy processes and tool
support [7]. Further, Alspaugh & Scacchi contrasts how OSS RE steps away from
what they refer to as Classical Requirements, characterized as having a
central repository, with requirements defined in the problem space, describing
the product of need, along with processes for examining the requirements for
completeness and consistency [2]. For better consistency, we choose to re-
label JIT and OSS RE as Open RE, and that described as traditional upfront,
and classical requirements as Closed RE.

Release-planning in OSS ecosystems is often employed using either a feature-
based and time-based strategy [21]. The feature-based performs a release when
a set goals have been fulfilled, e.g. a certain set of features has been
implemented. The time-based performs releases according to a preset schedule.
Hybrid versions has also used [29]. Earlier work has reported a favor towards
the time-based strategy [20, 29]. With fixed release dates and intervals,
firms can better adapt their internal plans so that additional patchwork and
differentiating features may be added in time for product shipment to market
[20, 21]. Other issues associated with time-based release strategy include:
rushed implementations, workload accumulation, outdated software and delayed
releases [20].

Open RE, compared to Closed RE, can be seen as being informal to different
degrees, e.g., to what level requirements are analyzed and managed [7].
Requirements are often decentralized and distributed over multiple sources,
often with a limited tracing. Influence and participation in the work and
decision-making are also distributed. Discussions and steering documents are
all public and transparent for anyone to see, or participate. Collaboration
and negotiation about requirements are key, as consensus often is needed to
make certain decisions [1]. These distinctions between Open and Closed RE
visualize a process and knowledge gap that commonly exist between firms and
OSS ecosystems on the context of OI [22, 28]. This gap need to be addressed to
the firms can increase their benefits from ecosystem participation (both
monetary and non-monetary), ecosystems can get more active players and
enriched discussions and influence and customer can receive better products
with shorter time to market and greater degree of innovation. This is what we
aim to address with the construction of the Requirements Analysis and
Management for Benefiting Openness (RAMBO) model. The model focuses on the
interaction and overlap between the internal RE process of the focal firm,
with that of its connected OSS ecosystem, to better manage the challenges
implied by OI.

##  III Requirements Analysis and Management for Benefiting Openness (RAMBO)

Figure 3: The Requirements Analysis and Management for Benefiting Openness
(RAMBO) model.

Figure 3 outlines the RAMBO model. The model is divided into five phases
outlined in the subsections that follow: 1) requirements elicitation, 2)
requirements screening and pre-analysis, 3) Open Innovation potential
analysis, 4) spin off analysis and 5) realization and contribution analysis.
In the context of this work we follow the definition of Requirements
Management (RM) provided by Hood et al. as “the set of procedures that support
the development of requirements including planning, traceability, impact
analysis, change management and so on… and the sum of the interfaces between
requirements development and all other systems engineering disciplines such as
configuration management and project management.” [10]. We also recognize the
role of requirements management in supporting managing requirements between
products and projects.

###  III-A Step 1: Elicitation

Requirements Elicitation in OI (marked with 1 in Figure 3) differs greatly
from previously known and published contexts. During requirements elicitation,
a firm that operates in OI needs to quickly and properly identify which of the
sources of requirements to use, and for what purpose. OI still contains the
”classical“ internal and MDRE requirements sources, but is extended by the
open sources of requirements.

The internal requirements sources (typical for bespoke RE) are characterized
by a small and well known set of customers or stakeholders whose wishes or
constraints are important and taken into consideration. Apart from that, any
firm that sells a product on an open market needs to elicit requirements from
the MDRE sources through a balance of market-push and technology-pull [23].
Wishes from customers, both known and unknown, are elicited through more
traditional methods (e.g., interviews) complemented with approaches such as
market analysis. New functionality and innovative requirements are added
internally as R&D progresses and the product evolves.

Regarding open sources, such as an OSS ecosystem, requirements are both
market- and technology-driven as in MDRE. A difference is that in an OSS
ecosystem, it usually is the end-users who are actively suggesting new
requirements as well as driving the innovation forward by co-developing the
products. However, active elicitation may still be needed, but through new
approaches such as participation in discussions facilitated by the ecosystem
infrastructure (e.g., mailing-lists, issue-trackers and IRC-channels) and at
off-line events (e.g., user-conferences and hackathons). Crowdsourcing as a
requirements elicitation technique is also highly recommended here as it
increases the quality and comprehensiveness and even the economic feasibility
of requirements elicitation [11]. Moreover, requirements elicited from the
crowd rather accurately represent the needs and pains of that crowd, allowing
to skip a large part of requirements validation process and providing quick
feedback. Stakeholder discovery is also greatly supported by crowdsourcing
that provides opportunities to create user personality clusters [9] and filter
their opinions about the products and requested requirements. This greatly
supports user feedback analysis and helps in defining new requirements.
Finally, the identification, analysis and prioritization of present
stakeholders is key to find those relevant, and can bring valuable
requirements that increase profitability.

Embracing and supporting creativity is important as this stage as new ideas
that both fit into the current product portfolio and are way beyond it are
much appreciated. The illumination and verification parts of creativity
workshops can greatly support requirements discovery and idea generation that
can be further detailed in OI contexts [19]. Automated support for creativity
is highly desirable here as the number of unfamiliar connections between
familiar possibilities of requirements is high and can overload requirements
analysts [4].

As with MDRE, the number of requirements grows fast in an OSS ecosystem,
creating potential problems in regard to managing quality aspects such as
dependencies and complexity. Another aspect is that prioritization and
throughput of requirements suffers due to the large repositories. Specific for
the OI context however, is that the requirements are spread out in several
decentralized repositories (informalisms [25]), often very unstructured and
expressed both through a variety of implementations and natural language
descriptions.

###  III-B Step 2: Pre-Screening and Analysis

After all relevant sources (Open, MDRE and internal if relevant) are
identified and prioritized, the requirements originating from these sources
need to be screened. Requirements screening in OI is significantly challenging
due to the following reasons: 1) they are spread out in several decentralized
repositories, 2) they are unstructured and 3) everyone has access to them and
can already be working on their implementation, 4) consideration often has to
be taken to others’ opinion (e.g., in a meritocracy), and 5) many of the
requirements are actually bug fixed, Wrong Usage of a Product (WUP) (user of
customer misunderstood how to use the product) or Problems that have
Workarounds (PW) and can be solved without additional implementations.

Therefore, the main part of the screening process is the decision if the
analyzed piece of information is a requirements or should be a requirement.
Here, we would like to stress that RE in OI should adopt a broad definition of
what a requirement is, which also includes needs for stakeholders that are not
relevant for a given product but could be relevant for a spin-off product.
This implies that the product boundaries or constraints are greatly removed
and make a vast majority of requirements potentially interesting or relevant.
Therefore, RE in OI requires a lightweight and efficient method for quickly
investigating the relevance and novelty of potential requirements, along with
the potential cost and contribution strategy in consideration. Previous work
by Maalej and Nabil [18] provides promising results for classifying app
reviews as bug reports, feature requests, user experiences and ratings.
Laurent et al. [15] proposed a method for automated requirements triage that
cluster incoming stakeholder requests into hierarchical feature sets. We argue
that their work should be extended by adding OI-specific concerns, e.g.
contribution potential and realization options.

The challenge here remains in analyzing incoming requirements with openness in
mind, e.g. taking the broader perspective than suggested by Davis effort,
dependency and important analysis, multiple release planning [6] or impacted
business goals suggested by Laurent et al. [15]. In this broader perspective,
a requirements analysis should calculate the probability of completion if
someone else from the ecosystem contributes in co-development or analyze
optimistic or pessimistic scenarios with or without other contributions within
the same OSS ecosystem.

###  III-C Step 3: OI Analysis

When relevant requirements are identified and screened, their OI potential
need to be analyzed. This third step of the RAMBO model represents a
significant difference between MDRE and RE in OI. In MDRE, requirements are
analyzed through the lens of the current product portfolio, along with the
stakeholders, their needs and future plans. Incremental innovation has higher
changes to be selected as radical ideas or ideas not associated with
significant stakeholders are considered risky or unprofitable. This was
considered as a challenge in our previous investigations and resulted in many
promising ideas been rejected [28]. RE in OI needs to take a broader
perspective and consider these “not fitting” requirements as potential spin-
off ideas. This addressed the well-known ”PARC Problem” experienced by Xerox
[5]; the inability to assess and capture value for (technology) innovations
that were not directly related to Xerox products. As quite often not all
requirements can be implemented in the next release due to low priorities and
lack of resources, the potential waste of unimplemented ideas can be shared
with other ecosystem players. Therefore, step 3 of the model investigates if
the potential requirements fit into the current product portfolio or future
plans, or not. Even if some requirements do not fit into the current
strategies, they should not stay “locked” internally in the requirements
database or be down-prioritized by more urgent requirements, but made open and
shared with other players who may find them more relevant for their offerings.

###  III-D Step 4: Spin Off Analysis

Requirements that are considered as not the best fit for the current product
portfolio, can and should be shared with other ecosystem players. Moreover,
potential spin-offs should be discussed and executed to benefit from potential
novelty and innovation included in those requirements. This step often
involves making alliances with other ecosystem players or spinning-off start-
up companies that can develop and monetize these ideas. The important factor
in this step is also to make an effort on keeping the key personnel away from
leaving the firm together with the new ideas. This step corresponds to the
inside-out part of the knowledge exchange in the OI model, see Figure 1. Open
requirements and increased transparency in release plans are important at this
stage. For example, assuming that two ecosystem players are working jointly on
a feature that is going to provide substantial benefits for the customers,
there should to be a discussion and agreement about release plans
synchronization so that both players can benefit from the feature. These
release plans should also be synchronized with previously agreed contribution
strategy for the feature so that maintenance costs can be directly minimized
via commoditization. Finally, ideas may also be dropped if considered not
interesting enough or economically viable.

###  III-E Step 5: Realization and Contribution Analysis

Requirements considered as a good match for the current product strategy need
to be analyzed from the realization and contribution perspectives. First, the
Architecture Impact Analysis (AIA) need to be performed to understand the
impact of new requirements on the current architecture and potential changes
that need to be made during the implementation. The involvement of software
architect as a technical counterpart of software product manager is important
at this stage to ensure the alignment between product requirements and
architecture and create a stepwise architectural evolution that fits with the
product roadmaps [17].

Second, the Implementation Possibilities (IP) (adaption of OSS solution,
differentiation based on commodity offered by OSS, or fully own
implementation) are analyzed and considered. Each of the three options need to
be analyzed from the potential value and long-term cost perspective. The risk
associated with fully own implementation is a significant change to the common
code base and several patches. Moreover, the OSS ecosystem may in the near
future come up with a solution to exactly the same requirement that may be
equally as good, or even better than the firm’s own. Finally, Contribution
Options (CO) need to be considered, and the scope and time line of
contributions need to be set. Both realization and analysis of contribution
options require the following two elements: 1) a good understanding of the
value from both customer and internal business perspectives [14] and 2) an
understanding of the commoditization process for a given product or market
that will help in estimating if and when a given implementation should be
contributed to the open community. [27].

##  IV Conclusions and Future Work

In this paper, we present our vision of the Requirements Analysis and
Management Model for Benefiting Openness that we designed with the specific
challenges in mind that Open Innovation brings to Requirements Engineering.
Our model recognizes the challenging aspects of stakeholder and requirements
elicitation, triage, analysis and implementation that Open Innovation brings.
We have summarized the RAMBO model in five steps and explain how each of the
steps differ from commonly accepted requirements engineering practice.

Our vision need significant further work that we plan in the near future.
Firstly, we plan to study each of the associated requirements phases in more
detail and bring more evidence of the lack of support for these activities. We
plan to use our industry network to search for empirical evidence if the
activities outlined in the RAMBO model are currently performed and in what way
or maybe not performed. Secondly, we plan to select one large company that
operated in OI and study its requirements process to create process
improvement action plan that will encourage them to adapt the parts of the
RAMBO model that are currently not present. Finally, we plan to create a
governance model for RE in OI that will contain all relevant activities and
decision points that a software-intensive firm should consider to better
utilize the benefits that openness offers.

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# An independent determination of the distance to supernova SN 1987A by means
of the light echo AT 2019xis

Aleksandar Cikota Gemini Observatory / NSF’s NOIRLab, Casilla 603, La Serena,
Chile<EMAIL_ADDRESS>Jiachen Ding Department of Atmospheric
Sciences, Texas A&M University, College Station, TX 77843, USA  Lifan Wang
George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, Texas A&M University, 4242 TAMU, College Station, TX 77843, USA
Dietrich Baade European Organisation for Astronomical Research in the Southern
Hemisphere (ESO), Karl-Schwarzschild-Str. 2, 85748 Garching b. München,
Germany  Stefan Cikota Centro Astronónomico Hispano en Andalucía, Observatorio
de Calar Alto, Sierra de los Filabres, 04550 Gérgal, Spain  Peter Höflich
Department of Physics, Florida State University, Tallahassee, FL 32306-4350,
USA  Justyn Maund Department of Physics and Astronomy, The University of
Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, UK  Ping Yang
Department of Atmospheric Sciences, Texas A&M University, College Station, TX
77843, USA

###### Abstract

Accurate distance determination to astrophysical objects is essential for the
understanding of their intrinsic brightness and size. The distance to SN 1987A
has been previously measured by the expanding photosphere method, and by using
the angular size of the circumstellar rings with absolute sizes derived from
light curves of narrow UV emission lines, with reported distances ranging from
46.77 kpc to 55 kpc. In this study, we independently determined the distance
to SN 1987A using photometry and imaging polarimetry observations of AT
2019xis, a light echo of SN 1987A, by adopting a radiative transfer model of
the light echo developed in Ding et al. (2021). We obtained distances to SN
1987A in the range from 49.09 $\pm$ 2.16 kpc to 59.39 $\pm$ 3.27 kpc,
depending on the interstellar polarization and extinction corrections, which
are consistent with the literature values. This study demonstrates the
potential of using light echoes as a tool for distance determination to
astrophysical objects in the Milky Way, up to kiloparsec level scales.

Supernovae — Large Magellanic Cloud — Distance measure — Polarimetry

##  1 Introduction

Distance determination to astrophysical objects is fundamental for improving
our knowledge about these objects. Without knowing the distance to objects,
their intrinsic brightness and size cannot be determined. Furthermore,
accurate distances to extragalactic objects are crucial for the determination
of the Hubble constant, H0, and measuring the expansion rate of the Universe
(Riess et al., 1998; Perlmutter et al., 1999). The distance to the Large
Magellanic Cloud (LMC) is important because the Cepheids in the LMC are used
as an anchor for the cosmic distance ladder (e.g. Riess et al., 2019).
Probably the most accurate distance to the LMC within 1 per cent was reported
by Pietrzyński et al. (2019), who determined the distance, d = 49.59 $\pm$
0.09 (statistical) $\pm$ 0.54 (systematic) kpc. They applied the
interferometry-based surface brightness color relation derived from single
late-type stars to fully detached eclipsing binaries with similar spectral
types.

SN 1987A, which exploded in the LMC, has also a relatively well determined
distance. Branch (1987) determined the distance to SN 1987A of 55 $\pm$ 5 kpc
by matching a distance-independent photometric angular radius of the
photosphere to a distance-dependent spectroscopic angular radius, which is
known as the Baade method (Baade, 1926) or the expanding photosphere method
(Kirshner & Kwan, 1974). Using the same method, Hoeflich (1988) obtained a
distance of 48 $\pm$ 2 kpc based on full non-LTE models by fitting a spectral
sequence of 8 spectra obtained during the first 7 months. Eastman & Kirshner
(1989) developed a model atmosphere for SN 1987A and determined the distance
to SN 1987A of 49 $\pm$ 6 kpc using the expanding photosphere method.
Hanuschik & Schmidt-Kaler (1991) also applied the Baade’s method for distance
determination and derived a distance of 52.3 $\pm$ 1.5 kpc. Panagia et al.
(1991) have compared the angular size of the circumstellar rings of SN 1987A
in the Hubble Space Telescope (HST) images with absolute sizes determined from
light curves of narrow UV emission lines and derived a distance of 51.2 $\pm$
3.1 kpc to SN 1987A. They also determined a distance of 50.1 $\pm$ 3.1 kpc to
the center of the LMC using radial velocities of the SN relative to the center
of LMC. Schmidt et al. (1992) used IR photometry of SN 1987A and determined a
distance of 49 $\pm$ 3 kpc using the expanding photosphere method. Gould
(1994) revisited the supernova-ring method and the measurements by Panagia et
al. (1991) and determined a distance to the SN of 52.7 $\pm$ 2.6 kpc and to
the LMC of 53.2 $\pm$ 2.6 kpc. Based on new measurements of the ionized-
emission light curves and the angular size of the ring around SN 1987A, Gould
(1995) derived an upper limit for the distance to the SN of D $<$ 46.77 $\pm$
0.76 kpc. Crotts et al. (1995) investigated the light echoes from the inner
region of SN 1987A to define the 3D geometry of the circumstellar nebula and
used the Panagia et al. (1991) method to determine the distances to the SN and
LMC of dSN = 51.7 $\pm$ 3.1 kpc and dLMC=51.9 $\pm$ 3.1 kpc, respectively.
Mitchell et al. (2002) conducted a detailed spectroscopic analysis of SN 1987A
and modeled the observed spectra from day 1 to 81 using hydrodynamical models.
Furthermore, they used the spectral-fitting expanding atmosphere method to
derive the distance to SN 1987A of 50 $\pm$ 5 kpc.

Distances to variable sources can also be determined by means of light echoes
(e.g. Trümper & Schönfelder, 1973; Sparks, 1994; Predehl & Schmitt, 1995;
Predehl et al., 2000; Heinz et al., 2016).

When light from a bright source, such as a supernova, reaches a cloud of dust,
some of the light is scattered in different directions and eventually reaches
an observer on Earth. Sparks (1994) proposed to measure distances to historic
supernovae by observing highly polarized circles of scattered light around
these objects. Because the maximum polarization occurs for a scattering angle
of 90 degress, the radius of the light echoes can be used as a ruler and
corresponds to the length $ct$, where $t$ is the time since the supernova
exploded and $c$ is the speed of light. Using that technique, Sparks et al.
(2008) found the geometric distance to V838 Monocerotis, an unusual variable
star with a light echo that appeared after an outburst in 2002. Even if the
scattering angle of maximum polarization is not 90 degrees, the method may
still be used if the scattering angle is known. Kervella et al. (2014)
obtained imaging polarimetry of the reflection nebula around the long-period
Galactic Cepheid RS Puppis and utilized two polarization models (one based on
Milky Way dust mixture and the other assuming Rayleigh scattering) to retrieve
the scattering angle from the degree of polarization. By taking into account
the dust distribution in the nebula, they adjusted a model of the phase lag of
photometric variations over specific nebular features to calculate the
distance to RS Puppis.

We determined the distance to SN 1987A by means of a light echo of SN 1987A,
AT 2019xis. We obtained imaging polarimetry observations of AT 2019xis, and in
combination with the publicly available photometry of AT 2019xis applied the
radiative transfer model of the SN 1987A light echo developed in Ding et al.
(2021) to determine the distance to SN 1987A.

The light echo, AT 2019xis ($\alpha$ = 05:36:13.700, $\delta$ = -69:16:24.70),
was discovered by the Optical Gravitational Lensing Experiment (OGLE) on
2019-10-15 (Gromadzki & Wyrzykowski, 2019). It was initially classified on
2019-12-28 by ePESSTO+ as a SN 1987A-like Type II supernova (Antilen et al.,
2019) at $\sim$ 70 days after the shock breakout. However, AT 2019xis was
later re-classified as a light echo of SN 1987A from the tip of a dust cloud
(Taubenberger et al., 2019), which is also visible in archival HST images,
located at an angular distance of 4.05’ from SN 1987A. Fig. 1 shows images of
AT 2019xis observed at three different epochs compared to a HST image.

In Sect. 2 we describe the observations and data reduction, in Sect. 3 the
model and method for the distance determination, and in Sect. 4 we report the
results and discuss the present findings in comparison with the literature.

Figure 1: Light echo AT 2019xis observed on January 3rd, 2020 with FORS2 in
the $V$ band (this work), April 20th, 2020 with the Multi Unit Spectroscopic
Explorer (MUSE, Prog. ID 2104.D-5041(A), PI Kuncarayakti) and April 12th, 2021
with the ESO Faint Object Spectrograph and Camera (EFOSC2) in the $V$ band (as
part of PESSTO, Smartt et al. 2015), compared to a Hubble Space Telescope
(HST) archival image (Prog. ID. 13401, PI Fransson) observed in the $F814W$,
$F606W$ and $F502N$ bands with the Wide Field Camera 3 (WFC3). The red lines
mark the position of AT 2019xis. The light echo propagated eastward from SN
1987 which is located 4.05’ west from AT 2019xis.

##  2 Data and observations

###  2.1 Imaging polarimetry

We obtained imaging linear polarimetry of the light echo AT 2019xis with the
FOcal Reducer/low dispersion Spectrograph 2 (Appenzeller et al. 1998, FORS2)
mounted on the primary focus of European Southern Observatory’s (ESO) Very
Large Telescope (VLT) Antu (Prog. ID 2104.C-5031(A), PI Cikota). The data was
taken on January 3rd, 2020, during good weather conditions. The object was at
an airmass between 1.4 to 1.5 during the observations. The DIMM seeing was
between $\sim$ 0.8-1.0”, and the Moon was set.

The observations were taken in three different passbands, the FORS2 V_HIGH
($\lambda_{0}$ = 555 nm, FWHM = 123.2 nm), R_SPECIAL ($\lambda_{0}$ = 655 nm,
FWHM = 165 nm) and I_BESS ($\lambda_{0}$ = 768 nm, FWHM = 138 nm) filters, at
four half-wave plate (HWP) angles of 0, 22.5, 45, and 67.5 degrees. The
redundancy of four HWP angles reduces the flat-fielding issue and cancels out
other instrumental effects (see Patat & Romaniello 2006). The sequence was
repeated twice in the $V$ and $I$ bands to increase the signal-to-noise ratio,
and taken only once in the $R$ band.

The brightness of 2019xis at the time of the polarimetry observations was
18.93 $\pm$ 0.03 mag in the $I$ band, as measured by the Optical Gravitational
Lensing Experiment (OGLE) project (Udalski et al., 2015). The exposure times
used were 400 sec, 250 sec and 200 sec, which led to a signal-to-noise ratio
of $\sim$80, $\sim$85 and $\sim$100 per exposure in the $V$, $R$ and $I$ band,
respectively.

The linear polarization was calculated following the standard approach (ESO,
2015; Cikota et al., 2017a). We measured the flux of the light echo in the
ordinary and extraordinary beams at all HWP angles using the aperture
photometry function from
PythonPhot111https://github.com/djones1040/PythonPhot. We used an aperture
radius of 1.3”, with the inner and outer sky radii of 1.5” and 2.5”,
respectively. Tests with slight variations of the aperture and sky radii
produced consistent results within the 1$\sigma$ uncertainties.

The normalized Stokes $q$ and $u$ parameters were calculated as described in
the FORS2 User Manual (ESO 2015, see also Cikota et al. 2017a, and Chu et al.
2022):

$\begin{array}[]{l}q=\frac{2}{N}\sum_{i=0}^{N-1}F(\theta_{i})\cos(4\theta_{i}),\\\
\\\ u=\frac{2}{N}\sum_{i=0}^{N-1}F(\theta_{i})\sin(4\theta_{i}),\end{array}$
(1)

where $N$ is the number of the half-wave retarder plate angles,
$\theta_{i}=22.5^{\circ}\times i$, with $0\leq i\leq 3$, and $F(\theta_{i})$
is the normalized flux difference between the ordinary ($f^{o}$) and
extraordinary ($f^{e}$) beams:

$F(\theta_{i})=\frac{f^{o}(\theta_{i})-f^{e}(\theta_{i})}{f^{o}(\theta_{i})+f^{e}(\theta_{i})}.$
(2)

The polarization position angles of the raw measurements have been corrected
for the half-wave plate zero angle chromatic dependence (Table 4.7 in ESO,
2015).

Finally, we calculated the degree of linear polarization, $p$, and the
polarization angle, $\theta$:

$p=\sqrt{q^{2}+u^{2}},\hskip 10.0pt\theta=\frac{1}{2}arctan(u/q),$ (3)

and apply a polarization bias correction following Wang et al. (1997). The
imaging polarimetry results are summarized in Table 1, and shown in Figure 2.

Table 1:  The imaging polarimetry measurements of the light echo AT 2019xis. The columns labeled $q_{\rm ISP}$ and $u_{\rm ISP}$ denote the Stokes $q$ and $u$ parameters of the interstellar polarization, as determined from field stars. The column labeled $p_{\rm ISPcorr}$ denotes the interstellar polarization corrected polarization of AT,2019xis. Both the observed polarization and the ISP corrected polarization have been corrected for the polarization bias. Start UT Date | Passband |  $q$ (%) |  $u$ (%) |  $\theta$ (∘) |  $p$ (%) | $q_{\rm ISP}$ | $u_{\rm ISP}$ |  $p_{\rm ISPcorr}$ (%)
---|---|---|---|---|---|---|---|---
2020-01-03 03:52:16 | $V$ |  3.30 $\pm$ 0.63 |  0.05 $\pm$ 0.67 |  0.4 $\pm$ 5.5 |  3.18 $\pm$ 0.63 |  -0.20 $\pm$ 0.07 |  0.51 $\pm$ 0.08 |  3.42 $\pm$ 0.64
2020-01-03 05:02:54 | $R$ |  3.67 $\pm$ 0.82 |  0.36 $\pm$ 0.95 |  2.8 $\pm$ 6.4 |  3.51 $\pm$ 0.82 |  -0.48 $\pm$ 0.09 |  0.50 $\pm$ 0.12 |  3.99 $\pm$ 0.83
2020-01-03 05:32:44 | $I$ |  3.97 $\pm$ 0.48 |  0.28 $\pm$ 0.38 |  2.0 $\pm$ 3.5 |  3.92 $\pm$ 0.48 |  -0.59 $\pm$ 0.09 |  0.30 $\pm$ 0.05 |  4.51 $\pm$ 0.49

###  2.2 Light curve of AT 2019xis

We used the publicly available light
curve222http://ogle.astrouw.edu.pl/ogle4/transients/transients.html of AT
2019xis from the Optical Gravitational Lensing Experiment (OGLE) project
database (Udalski et al., 2015). The light echo was observed as part of their
OGLE-IV sky survey with the 1.3 m Warsaw University Telescope located at the
Las Campanas Observatory in Chile in the $I$-band at 29 epochs between
2019-10-02 and 2020-03-13. There are also upper limit (non-detection)
measurements available before 2019-10-02. The light curve is shown in Figure
2.

Figure 2: Light curve in the $I$ band (left panel) and polarization
measurements (right panel) of the light echo AT 2019xis. Also shown is the
simultaneous best fit with the simulated model (blue squares and dashed lines
in both panels). The vertical line in the left panel marks the time of the
polarization observations, and the triangles are non-detections. The
observations shown in this plot were not corrected due to ISP or extinction
(see Case i in Sect. 3).

###  2.3 Interstellar polarization and dust extinction

The LMC itself is very dust rich and there is also some Galactic foreground
dust along the line of sight toward the LMC. Fitzpatrick & Walborn (1990)
found a total reddening along the SN 1987A sight line of $E(B-V)$ $\simeq$
0.16 mag, with the LMC component of $E(B-V)\simeq$ 0.10 mag and the Galactic
component of $E(B-V)\simeq$ 0.06 mag. This is consistent with the result by
Walker & Suntzeff (1990) who determined an $E(B-V)$ = 0.17 $\pm$ 0.02 mag from
photometry of stars near SN 1987A. Furthermore, Fitzpatrick & Walborn (1990)
measured the reddening of the star Sk-69∘ 203, $E(B-V)$ = 0.19 mag, located in
the 30 Doradus region 2.2′ north of the SN 1987A position.

There are also polarization measurements along the line of sight to SN 1987A
available in the literature. Schmidt (1976) reported the Galactic foreground
interstellar polarization (ISP) p = 0.40 $\pm$ 0.13 % and a position angle of
20° for the region where SN 1987A is located. Jeffery (1991) reported broad-
band polarimetry of SN 1987A taken at the La Plata Observatory by Mendez
(priv. comm.) from late 1988 to early 1989. Mendez determined ISP along the
sight line to SN 1987A that can be described with the following Serkowski
parameters (Serkowski et al., 1975): pmax $\simeq$ 1.05 %,
$\rm\lambda_{max}\simeq$ 5400 Å and K $\simeq$ 1.1 and a polarization angle of
$\theta\simeq$ 34∘.

The dust cloud from which the light echo propagates is, however, located 4.′05
(about $\sim$200 pc, Taubenberger et al. 2019) East from SN 1987A . In this
region the dust properties along the sight line may differ compared to the
sight line toward SN 1987A.

To investigate the dust extinction towards AT 2019xis we examined the publicly
available MUSE data of AT 2019xis (Prog. ID 2104.D-5041(A), PI Kuncarayakti)
and looked for the Na I D $\lambda$5890 Å and $\lambda$5896 Å absorption lines
which are well known tracers of dust and gas in the Milky Way (Poznanski et
al., 2012). The spectra of AT 2019xis show no obvious Na I D lines, however a
bright (14.4 G mag) star, Gaia EDR3 4657665228220710912, located 3.5″ West
from AT 2019xis, displays weak Na I D lines. The median of the geometric
distance posterior from the Earth to the star is rmedgeo $\sim$ 14.7 kpc
(Bailer-Jones et al., 2021). Therefore, this line of sight is expected to
include a large fraction of Galactic dust. The equivalent widths (EW) of the
Na I D absorption lines are EW(D2) = 0.23 $\pm$ 0.01 Å and EW(D1) = 0.22 $\pm$
0.01 Å. Following the empirical relation (valid for Milky Way dust) between
the sodium absorption and dust extinction by Poznanski et al. (2012) this
corresponds to a reddening of $E(B-V)$ = 0.05$\pm$0.01 mag (see their eq. 9).
Note that this value is comparable to the Galactic reddening component
determined by Fitzpatrick & Walborn (1990).

We also determined the ISP towards AT 2019xis following the same method as in
Chu et al. (2022, see their Fig. 2), by calculating the polarization of field
stars. The field stars polarization measurements have been corrected for
instrumental polarization, which is negligible in the center of the field but
increases up to $\sim$1% at the borders of the field (Patat & Romaniello,
2006; González-Gaitán et al., 2020). The ISP of the field was estimated by
taking the weighted mean (with inverse-variance weights) of the field stars’
Stokes $q$ and $u$, using 44, 59 and 40 field stars in the $V$, $R$ and $I$
band, respectively. The Stokes $q_{\rm ISP}$ and $u_{\rm ISP}$ are listed in
Table 1. Despite a wide range of the field stars polarization values in all
bands, ranging from $\sim$0.2 to $\sim$ 2 %, the weighted means in different
bands are relatively consistent. The mean polarization in the $V$, $R$ and $I$
bands are p = 0.53 $\pm$ 0.08 % with a polarization angle of $\theta$ = 55.9
$\pm$ 4.0 degrees; p = 0.68 $\pm$ 0.10 % with $\theta$ = 66.8 $\pm$ 4.2
degrees; and p = 0.65 $\pm$ 0.08 % with $\theta$ = 76.6 $\pm$ 3.6 degrees,
respectively. We applied the mean ISP values to correct the polarization
measurements of AT 2019xis (see Table 1) and run various simulations, with and
without the ISP correction, as explained in the next section. We also
acknowledge that the limitation of this method is that most of the stars used
to estimate the ISP are nearby compared to the large distance toward the LMC.
This means that there may be more dust along the line of sight toward the LMC
producing polarization, which is not being taken into account.

###  2.4 Geometry of the dust cloud

SN 1987’s light echoes in particular seem to come from extended dust sheets,
of course with some substructure in it, but are in general very large and
roughly perpendicular to the line of sight. Many of these light echoes are
relatively unchanged for years, i.e. the arclets do not change significantly
in shape and apparent motion (e.g. Suntzeff et al., 1988; Xu et al., 1995;
Rest et al., 2005). However, AT 2019xis is special because it is one of the
brighter SN 1987A light echoes and it appears as a point source. Figure 1
shows that the light echo coincides with a tip of a dust pillar visible in the
HST image, which is rising from a large dust structure in the east (not shown
in the cut out). The comparison of the observations taken on January 3rd, 2020
and April 20th, 2020 also shows that AT 2019xis does not change significantly
in the apparent position and shape throughout the period of the available
photometry from OGLE. Additional observations taken on April 12th, 2021 with
the ESO Faint Object Spectrograph and Camera (EFOSC2) mounted on the New
Techonlogy Telescope (NTT) show that the AT 2019xis faded away as the light
echo propagated further and is scattering from another overdensity $\sim$5
arcsec east from AT 2019xis.

AT 2019xis was observed with MUSE at two epochs, on 2020-02-19 and 2020-04-20
(Prog. ID 2104.D-5041(A), PI Kuncarayakti). A detailed analysis of the MUSE
data is out of the scope of this paper, but we used the extracted light echo
spectra to investigate the shape and morphology of the dust cloud producing AT
2019xis. We cleaned the light echo spectra using wavelet decomposition
(Holschneider et al. 1989, see also Wagers et al. 2010; Cikota et al. 2019)
and matched them to SN 1987A spectral templates. The templates have a cadance
of 0.5 days and were generated by interpolating original SN 1987A spectra
observed at more than 80 epochs by Phillips et al. (1988, 1990). To match the
spectra we followed the methods used in the Supernova Identification tool
(SNID, Tonry & Davis, 1979; Blondin & Tonry, 2007).

Fig. 3 shows the epoch-map of the light echo, derived by matching the light
echo spectra from each spaxel to SN 1987A template spectra. The light echo
propagates from right to the left in the image (eastwards), and at the
diameter of the dust cloud ($\sim$1”) is much smaller than the angular
distance to the supernova ($\sim$4.05’), so the light waves are propagating
nearly parallel. However, the points of equal epochs display some curvature.
The curvature can be explained either due to the spherical shape of the cloud
in the case of an optically thick cloud or due to distance differences of the
scattering plane in the case of optically thin dust structures. We also note
that the light echo spectra match relatively good with the SN 1987A spectra
(see bottom panel in Fig. 3), and that the best fit epochs are continuously
increasing from east to west, which implies that there is no significant
overlapping of the spectra of different epochs, suggesting that the cloud is
optically thick. Additionally, the epoch-map is consistent with the dust
pillar’s spherical appearance visible in the HST image. Thus, assuming a
spherical dust cloud seems to be a reasonable simplification for modeling this
light echo in the next section.

Figure 3: Top left panel: Light echo AT 2019xis observed with MUSE on April
20, 2020. The size of the cut out is 4x4 arcsec, and is oriented so that north
is up and east is left. We measured the Image Quality FWHM of $\sim$0.7”
($\sim$3.6 pixel) using neighbouring stars. Top right panel: Epoch-map of the
light echo determined by matching template spectra of SN 1987A to each spaxel.
The epochs are given relative to the SN explosion. Bottom panel: An example
sequence of the light echo spectra (red lines) compared to the best matched SN
1987A templates (black lines). The best fit epochs are indicated.

##  3 Model and distance determination

Ding et al. (2021) used a Monte Carlo radiative transfer model (MCRTM) to
simulate light echo observations (light curve and polarization) of SN 1987A at
the position of AT 2019xis. They used $UBVRI$ light curves of SN 1987A
obtained at the Sutherland field station of South African Astronomical
Observatory as the input (Menzies et al., 1987; Catchpole et al., 1987, 1988,
1989; Whitelock et al., 1988) and assumed a simplified dust cloud with similar
geometry to AT 2019xis to study how the simulated observations vary depending
on the dust model properties, dust cloud size, and shape. They adopted optical
properties of the dust grain models developed by Weingartner & Draine (2001)
and investigated the simulated observational differences for different dust
grain-size distributions for the Milky Way, LMC and Small Magellanic Cloud
(SMC) dust models.

We adopt the MCRTM by Ding et al. (2021) to determine the distance of the SN
1987A. The apparent brightness and shape of the light curve of the echo is
dependent on the intrinsic SN light curve, distance, dust cloud size, shape,
optical thickness and the optical properties of the dust grains. Because we
know the path length difference between the light of SN 1987A traveling
directly toward Earth and the light echo, the distance of the SN 1987A
determines the scattering angle, with which the light from the supernova is
scattered. The scattering geometry was illustrated in Ding et al. (2021),
where the distance to the SN 1987A was assumed to be 51.4 kpc (Ding, priv.
comm.). However, assuming that we do not know the distance to SN 1987A, given
the dependence of the light echo’s linear polarization on the scattering angle
(see also e.g. Sparks 1994), the scattering angle (and thus, the distance) can
be estimated by fitting the MCRTM simulations to the polarization and light
curve measurements.

The linear polarization of the light echo is also strongly dependent on the
optical thickness (Ding et al., 2021). Optically thick clouds display large
spatial variations of the polarization. For example, in an optically thick
spherical cloud, the linear polarization increases in the regions closer to
the edge of the cloud.

The dust cloud (AT 2019xis) distance to the Earth, optical thickness and size
of the dust cloud are simultaneously constrained by the light curve and
polarization measurements. The distance between the dust cloud and SN 1987A is
also constrained by the time delay and the angular distance between AT 2019xis
and SN 1987A (see Ding et al. 2021 for details). The dust cloud is assumed to
be a homogeneous sphere, whose size and optical thickness are characterized
along with the diameter. The dust particles in the cloud are assumed to be
prolate spheroids with an aspect ratio of 4. The optical properties of these
particles are computed based on the rigorous invariant imbeding T-matrix
method (Johnson, 1988; Yang et al., 2019). A size distribution defined by
Weingartner & Draine (2001) as ’LMC’ is used to obtain size-distribution
averaged dust optical properties.

We conduct the MCRTM simulations and fit the observations after applying
different extinction and interstellar polarization corrections, as follows:

(i) Without applying any ISP and extinction correction, i.e. using the
original OGLE light curve and the ISP uncorrected polarization measurements.
The best fit is shown in Figure 2.

(ii) same as (i), but assuming halved polarization measurements errors. This
test is conducted to investigate the effect of the polarization measurements
precision on the distance errors.

(iii) After applying the ISP correction to the polarization measurements
determined from field stars (determined in Sect. 2.3, see also Table 1)

(iv) After applying an ISP correction which follows a Serkowski curve with
pmax=1.05%, $\rm\lambda_{max}$=5400Å, and K=1.1, as determined in Jeffery
(1991).

(v) After applying the ISP correction to the polarization measurements
determined from field stars (determined in Sect. 2.3, see Table 1) and a
reddening correction to the light curve of $E(B-V)$=0.06 mag assuming Milky
Way dust with RV=3.1 (see Sect. 2.3). Following Cardelli et al. (1989) this
corresponds to the extinction in the $I$-band of AI = 0.035 mag. For
comparison, please note that in the case of LMC dust with RV=2.6, the
extinction is very similar, AI = 0.033 mag.

(vi) After applying the ISP correction to the polarization measurements
determined from field stars (see Sect. 2.3, and Table 1) and a reddening
correction to the light curve of $E(B-V)$=0.16 mag assuming Milky Way dust
with $R_{V}$=3.1 (see Sect. 2.3). Following Cardelli et al. (1989) this
corresponds to AI =0.094 mag. For comparison, assuming LMC dust with RV=2.6,
the extinction would be AI = 0.088.

(vii) - (xi) are same as the case (vi), but instead of using the LMC dust
model, we run the fitting for different dust models from Weingartner & Draine
(2001): LMC2, MW3.1, MW4.0, MW5.5 and SMC.

(xii) After applying the ISP correction which follows a Serkowski curve with
pmax=1.05%, $\rm\lambda_{max}$=5400 Å, and K=1.1, as determined in Jeffery
(1991), and a reddening correction to the light curve of $E(B-V)$=0.06 mag
assuming Milky Way dust with RV=3.1. Following Cardelli et al. (1989) this
corresponds to AI = 0.035 mag.

(xiii) After applying the ISP correction which follows a Serkowski curve with
pmax=1.05%, $\rm\lambda_{max}$=5400Å, and K=1.1, as determined in Jeffery
(1991), and a reddening correction to the light curve of $E(B-V)$=0.16 mag
assuming Milky Way dust with $R_{V}$=3.1 (see Sect. 2.3. Following Cardelli et
al. (1989) this corresponds to AI = 0.094 mag.

##  4 Results and discussion

Table 2:  Distance determination results with different models.

Case | Dust | ISP correction | Extinction correction | Distance | Optical | Diameter
---|---|---|---|---|---|---
| Model |  |  | (kpc) |  thicknessa | (ly)
i) | LMC | No correction | No corrections |  53.44$\pm$2.51 |  3.38$\pm$0.23 |  1.36$\pm$0.05
ii) | LMC | No correction (assuming half errors of p) | No corrections |  53.45$\pm$1.37 |  3.37$\pm$0.23 |  1.36$\pm$0.05
iii) | LMC | Field stars (this work) | No correction |  49.09$\pm$2.16 |  4.20$\pm$0.44 |  1.29$\pm$0.09
iv) | LMC | Jeffery (1991) | No correction |  58.72$\pm$3.31 |  2.63$\pm$0.23 |  1.50$\pm$0.04
v) | LMC | Field stars (this work) |  AI= 0.035 mag |  49.30$\pm$2.16 |  3.87$\pm$0.07 |  1.26$\pm$0.01
vi) | LMC | Field stars (this work) |  AI= 0.094 mag |  49.67$\pm$2.17 |  3.50$\pm$0.08 |  1.26$\pm$0.02
vii) | LMC2 | Field stars (this work) |  AI= 0.094 mag |  49.77 $\pm$2.14 |  3.68 $\pm$0.32 |  1.34 $\pm$0.07
viii) | MW3.1 | Field stars (this work) |  AI= 0.094 mag |  54.87 $\pm$2.35 |  1.22 $\pm$0.12 |  1.68 $\pm$0.02
ix) | MW4.0 | Field stars (this work) |  AI= 0.094 mag |  52.65 $\pm$2.29 |  2.03 $\pm$0.34 |  1.39 $\pm$0.04
x) | MW5.5 | Field stars (this work) |  AI= 0.094 mag |  49.23 $\pm$2.12 |  3.72 $\pm$0.12 |  1.30 $\pm$0.03
xi) | SMC | Field stars (this work) |  AI= 0.094 mag |  53.19 $\pm$2.39 |  1.89 $\pm$0.31 |  1.42 $\pm$0.03
xii) | LMC | Jeffery (1991) |  AI= 0.035 mag |  58.92$\pm$3.34 |  2.46$\pm$0.27 |  1.52$\pm$0.04
xiii) | LMC | Jeffery (1991) |  AI= 0.094 mag |  59.39$\pm$3.27 |  2.20$\pm$0.28 |  1.57$\pm$0.04

Notes. a The optical thickness is given at 0.8 $\rm\mu m$.

As shown in the right panel in Fig. 2, the observed trend in the degree of
polarization of scattered light from the dust cloud shows a possible increase
toward the red wavelengths. However, we note that the measurement errors are
relatively large and that within the errors, the degree of polarization may
also be consistent with the rise toward the blue wavelengths. A significant
polarization, however, is detected in the $I$ band.

One of the misconceptions of polarization by dust scattering is that
polarization is highest in the blue, as quoted in recent papers on the
continuum polarization observed along the lines of sight toward Type Ia
Supernovae (see e.g. Cikota et al., 2019, 2017b; Chu et al., 2022). However,
this is correct only when the cloud is optically thin at the red wavelengths.
In contrast, in the case of scattering from optically thick dust clouds, the
linear polarization is expected to show a steady increase from 0.3 $\rm\mu m$
to 1.0 $\rm\mu m$ (see White (1979); Voshchinnikov & Karjukin (1994); Kartje
(1995); Zubko & Laor (2000)). Such behavior is, for instance, observed in
reflection nebulae (Zellner, 1974).

The observed possible increase in polarization towards the red wavelengths in
the case of AT 2019xis may suggest that the scattering cloud is optically
thick in all three bands ($V$, $R$, and $I$). In optically thick clouds,
effective scattering only occurs in the outermost layer of the cloud, and the
detailed models presented in Ding et al. (2021) demonstrate that polarization
can actually peak in the red wavelengths. Additionally, Ding et al. (2021) has
demonstrated that if a scattering cloud with an optical depth close to unity
can be resolved spatially, the degree of polarization will exhibit color-
dependence at deeper layers of the cloud. Specifically, the degree of
polarization in redder filters is higher at deeper layers. Thus, multi-band
polarimetry of spatially resolved dust clouds can be utilized as a simple
diagnostic tool to determine the optical depth of clouds in general.

The distances to SN 1987A were constrained by fitting the MCRTM simulations
(Ding et al., 2021) to the AT 2019xis light curve and polarization
observations. Examples of the simulated light echo images are shown in Sect.
3.2 in Ding et al. 2021. The results are shown in Figure 4 in comparison to
literature distances, and summarized in Table 2. Figure 4 shows that all the
distances are in general comparable to the literature values. Also the
distance uncertainties of $\sim$5% are comparable to the uncertainties
achieved with other methods, the expanding photosphere method and distance
determinations using the circumstellar rings. However, our uncertainties are
highly dependent on the polarization uncertainty. When comparing the results
of cases (i) and (ii), it becomes evident that by reducing the assumed
polarization uncertainty by half, the precision of the distance estimate
approximately doubles (see Table 2). Furthermore, the degree of polarization
also has a high impact on the distance results. After correcting our initial
polarization measurements (see case i with no corrections applied) for the ISP
as determined by Jeffery (1991), we find that the estimates of distances to SN
1987A increase to $\sim$ 59 kpc (see cases iv, xii, xiii). In contrast, in the
cases iii, v, and vi, after correcting our measurements due to the ISP based
on the field stars, which effectively leads to an increase of the polarization
values (see Table 1), the distance estimates to SN 1987A decreased to $\sim$
49 kpc.

In contrast, the distance is only minimally affected by the extinction
correction of the light curve. This can be seen by comparing case iii, where
no extinction correction is applied, to case vi, where an extinction
correction of AI=0.094 mag is applied. The resulting distances are 49.09 $\pm$
2.16 kpc and 49.67 $\pm$ 2.17 kpc without and with the extinction correction,
respectively (Table 2).

The analysis of cases (vi) - (xi) highlights how different dust models impact
distance measurements. When fitting the models with LMC dust, the distance to
SN,1987A is estimated to be smaller (around 49 kpc) than when using MW and SMC
dust models, which produce distances ranging from 49 to 54 kpc. Although there
is some uncertainty introduced by variations in the optical properties of the
dust, we consider the use of LMC dust to be a reasonable assumption. This is
because the properties of the LMC dust model were derived from independent
observations of the LMC, as detailed in Weingartner & Draine 2001. The
assumption of the geometry of the scattering dust cloud may also introduce an
uncertainty in the distance estimate, but the uncertainty is less than that
resulting from the optical properties of the dust particles.

Therefore, assuming the LMC dust model is correct, the ISP correction is the
main source of uncertainty in distance estimation. Consequently, the distance
to SN 1987A can be constrained to a range between 49.30 $\pm$ 2.16 kpc (case
v) and 59.39 $\pm$ 3.27 kpc (case xiii). The weighted mean distance obtained
from cases v, vi xii and xiii is 52.4 $\pm$ 1.3(stat) $\pm$ 4.8(sys) kpc.

In summary, we have demonstrated that the use of polarization measurements of
light echoes, combined with photometric measurements, can be utilized to
estimate distances to supernovae or other transients in our Milky Way and
beyond, with a level of accuracy that is comparable to other techniques.
However, this method does have some limitations, as it relies on several
assumptions regarding dust model properties, dust grain-size distributions,
dust cloud size, and the geometric shape of the dust cloud. Despite these
limitations, one of the most significant sources of uncertainty is the
polarization uncertainty. By reducing this uncertainty, for example, by
obtaining more polarimetry observations, it may be possible to achieve greater
precision than what can be obtained with other methods.

Figure 4: Distance measurements to SN 1987A determined in this work (cases i -
xiii), compared to the values from the literature. The distances determined by
Branch (1987), Hoeflich (1988), Eastman & Kirshner (1989), Hanuschik &
Schmidt-Kaler (1991), Schmidt et al. (1992) and Mitchell et al. (2002) are
based on the expanding photosphere method, while Panagia et al. (1991), Gould
(1994), Gould (1995) and Crotts et al. (1995) derived the distance based on
the circumstellar rings. The distance determined by Gould (1995), marked with
a triangle, is an upper limit. For the description of cases i - xiii, see
Sect. 3 and Table 2.

## Acknowledgements

We would like to thank to Craig J. Wheeler, Ferdinando Patat, Yi Yang, and
Heloise Stevance for their contributions and insightful discussions that
improved the quality of the paper. The work of A.C. is supported by NOIRLab,
which is managed by the Association of Universities for Research in Astronomy
(AURA) under a cooperative agreement with the National Science Foundation.
L.W. is partially supported by the NSF through the grant AST 1813825. This
work benefited from L.A.Cosmic (van Dokkum, 2001), IRAF (Tody, 1986), PyRAF
and PyFITS. PyRAF and PyFITS are products of the Space Telescope Science
Institute, which is operated by AURA for NASA. We thank the authors for making
their tools and services publicly available. This work is based on
observations collected at the European Organisation for Astronomical Research
in the Southern Hemisphere under ESO programs 2104.C-5031(A) and
2104.D-5041(A); and on data products from observations made with ESO
Telescopes at the La Silla Paranal Observatory under programme 1106.D-0811(M):
PESSTO (the Public ESO Spectroscopic Survey for Transient Objects). The
execution in the service mode of these observations by the VLT/NTT operations
staff and PESSTO observers is gratefully acknowledged. Based on observations
made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble
Legacy Archive, which is a collaboration between the Space Telescope Science
Institute (STScI/NASA), the Space Telescope European Coordinating Facility
(ST-ECF/ESA) and the Canadian Astronomy Data Centre (CADC/NRC/CSA).

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