2024I.V. Nwabufo, J.N. Böhm, P. Berens & D. Kobak Ifeoma Veronica Nwabufo1,2
<EMAIL_ADDRESS>
Jan Niklas Böhm1,2<EMAIL_ADDRESS>
Philipp Berens1,2<EMAIL_ADDRESS>
Dmitry Kobak1,2<EMAIL_ADDRESS>
and 1 Hertie Institute for AI in Brain Health, University of Tübingen, Germany
2 Tübingen AI Center, University of Tübingen, Germany

# Self-supervised Visualisation of Medical Image Datasets

###### Abstract

Self-supervised learning methods based on data augmentations, such as SimCLR,
BYOL, or DINO, allow obtaining semantically meaningful representations of
image datasets and are widely used prior to supervised fine-tuning. A recent
self-supervised learning method, $t$-SimCNE, uses contrastive learning to
directly train a 2D representation suitable for visualisation. When applied to
natural image datasets, $t$-SimCNE yields 2D visualisations with semantically
meaningful clusters. In this work, we used $t$-SimCNE to visualise medical
image datasets, including examples from dermatology, histology, and blood
microscopy. We found that increasing the set of data augmentations to include
arbitrary rotations improved the results in terms of class separability,
compared to data augmentations used for natural images. Our 2D representations
show medically relevant structures and can be used to aid data exploration and
annotation, improving on common approaches for data visualisation.

###### keywords:

Self-supervised learning, augmentations, contrastive learning, data
visualisation

##  1 Introduction

Medical image datasets have been quickly growing in size and complexity
[Litjens et al. (2017), Topol (2019), Zhou et al. (2021)]. Whereas medical
professionals can analyse, annotate, and classify individual images, tasks
involving large batches of images, ranging from data curation and quality
control to exploratory analysis, remain challenging.

Self-supervised learning (SSL) has recently emerged in computer vision as the
dominant paradigm for learning image representations suitable for downstream
tasks [Balestriero et al. (2023)], and it has increasingly been adopted in
medical imaging [Huang et al. (2023)]. In contrastive learning methods, such
as SimCLR [Chen et al. (2020)], BYOL (Grill et al., 2020), or DINO Caron et
al. (2021), data augmentation is used to generate different views of each
image, and a deep network is trained to keep these views close together in the
representation space. However, the learned representations are typically high-
dimensional.

Figure 1: (a) In $t$-SimCNE, the network is trained to map two random
augmentations of an input image to close locations in the 2D output space. (b)
Augmentations used for natural images in $t$-SimCNE. (c) Additional
augmentations suggested here for medical images.

Recently, Böhm et al. (2023) suggested a self-supervised contrastive method,
called $t$-SimCNE, for 2D visualisation of image datasets. Using natural image
datasets, the authors demonstrated that $t$-SimCNE obtains semantically
meaningful visualisations, representing rich cluster structure and
highlighting artefacts in the data. Their methods clearly outperformed
existing 2D embedding methods like $t$-SNE Van der Maaten and Hinton (2008)
and UMAP McInnes et al. (2020) for natural image data.

Here we apply $t$-SimCNE to several medical microscopy datasets, and
demonstrate that it yields medically relevant visualisations, outperforming
$t$-SNE visualisations of pretrained networks. Furthermore, we show that the
results improve when using rotational data augmentations (Figure 1) informed
by the rotational invariance of microscropy images. Our code is available at
https://github.com/berenslab/medical-t-simcne.

##  2 Related work

Contrastive learning methods have been widely applied to medical image
datasets (for a review, see Huang et al., 2023) but usually as pre-training
for downstream tasks such as classification or segmentation. Some recent works
visualised high-dimensional SSL representations; e.g. Cisternino et al. (2023)
used UMAP of DINO to visualise histopathology data. In contrast, our focus is
on self-supervised visualisations trained end-to-end.

Contrastive learning relies on data augmentations to create several views of
each image, and the choice of data augmentations plays a crucial role in
methods’ success Tian et al. (2020). A large number of works explored data
augmentations for medical images in a supervised setting (reviewed by Chlap et
al., 2021; Goceri, 2023). In the self-supervised context, van der Sluijs et
al. (2023) studied the effect of augmentations on the representation of X-ray
images. For histopathology images, Kang et al. (2023) advocated for using
rotations and vertical flips, as well as staining-informed color
transformations, while some other works used neighbouring patches as positive
pairs Li et al. (2021); Wang et al. (2021).

##  3 Background: SimCLR and $t$-SimCNE

SimCLR (Chen et al., 2020) produces two augmentations for each image in a
given mini-batch of size $b$, resulting in $2b$ augmented images. Each pair of
augmentations forms a so-called positive pair, whereas all other possible
pairs in the mini-batch form negative pairs. The model is trained to maximise
the similarity between the positive pair elements while simultaneously
minimising the similarity between the negative pair elements.

An augmented image $x_{i}$ is passed through a ResNet He et al. (2016)
backbone to give the latent representation $h_{i}$, which is then passed
through a fully-connected projection head with one hidden layer to yield the
final output $z_{i}$. SimCLR employs the InfoNCE loss function van den Oord et
al. (2019), which for one positive pair $(i,j)$ can be written as

$\ell_{ij}=-\log\frac{\exp\big{(}\text{sim}(z_{i},z_{j})/\tau\big{)}}{\sum_{k\neq
i}^{2b}\exp\big{(}\text{sim}(z_{i},z_{k})/\tau\big{)}}\,,$ (1)

where $\text{sim}(x,y)=x{{}^{\top}}y/\big{(}\|x\|\cdot\|y\|\big{)}$ is the
cosine similarity and $\tau$ is a hyperparameter that was set to $1/2$ in Chen
et al. (2020). Even though the loss function operates on $z_{i}$ (typically
128-dimensional), for downstream tasks, SimCLR uses the representations
$h_{i}$ Bordes et al. (2022), typically at least 512-dimensional.

The idea of $t$-SimCNE Böhm et al. (2023) is to make the network output
$z_{i}$ two-dimensional so that it is directly suitable for data
visualization. It does not make sense to apply the cosine similarity to this
representation in $\mathbb{R}^{2}$ as it would effectively normalise the
embeddings to lie on a one-dimensional circle. $t$-SimCNE replaces the
exponential of the scaled cosine similarity with the Cauchy similarity used in
$t$-SNE (Van der Maaten and Hinton, 2008): $(1+\|x-y\|^{2})^{-1}$. The
resulting loss function is

$\ell_{ij}=-\text{log}\frac{1}{1+\|z_{i}-z_{j}\|^{2}}+\text{log}\sum_{k\neq
i}^{2b}\frac{1}{1+\|z_{i}-z_{j}\|^{2}}\,.$ (2)

Böhm et al. (2023) found that directly optimizing this loss is difficult, and
suggested a three-stage process. The first stage (1000 epochs) used a
128-dimensional output which was then replaced with a 2D output and fine-tuned
in the subsequent two stages (500 epochs).

For their experiments on CIFAR datasets, the authors used a ResNet18 with a
modified first layer kernel size of $3\times 3$, and a projection head with
hidden layer size of 1024 (Figure 1a).

##  4 Experimental setup

Table 1: Used datasets. Dataset | Image dim. | Sample size | Classes | Reference
---|---|---|---|---
Leukemia | $28\times 28$ | $18\,365$ | $15$ | Matek et al. (2019a)
Bloodmnist | $28\times 28$ | $17\,092$ | $8$ | Yang et al. (2023)
Dermamnist | $28\times 28$ | $10\,015$ | $7$ | Yang et al. (2023)
Pathmnist | $28\times 28$ | $107\,180$ | $9$ | Yang et al. (2023)
PCam16 | $96\times 96$ | $327\,680$ | $2$ | Veeling et al. (2018)

### Datasets

We used five publicly available medical image datasets with sample sizes
ranging from 10 000 to over 300 000 (Table 1). Three datasets we took from the
Medmnistv2 collection (Yang et al., 2023), all consisting of $28\times 28$ RGB
images. Dermamnist is based on the HAM10000 dataset Tschandl et al. (2018), a
collection of multi-source dermatoscopic images of common pigmented skin
lesions. The images are categorised into 7 classes, which we reduced to binary
labels: melanocytic nevi and other skin conditions. Bloodmnist is based on a
dataset of microscopy images of individual blood cells from healthy donors
Acevedo et al. (2020), categorised into $8$ classes corresponding to cell
types. Pathmnist is based on a dataset of non-overlapping patches from
colorectal cancer histology slides Kather et al. (2019), categorized into $9$
classes corresponding to tissue types. The Leukemia dataset Matek et al.
(2019b) contains microscopy images of white blood cells taken from patients,
some of which were diagnosed with acute myeloid leukemia. The images were of
$224\times 224$ size, and we resized them to $28\times 28$ for consistency
with Medmnist.

The Patch Camelyon16 (PCam16) dataset Veeling et al. (2018), adapted from the
Camelyon16 challenge Bejnordi et al. (2017), consists of $96\times 96$ patches
from breast cancer histology slides. There are two classes: metastases and
non-metastases. A patch was classified as metastases if there was any amount
of tumor tissue in its central $32\times 32$ region.

Figure 2: Visualisations of the Leukemia dataset (only a subset of classes are
listed in the legend). Panel titles show $k$NN accuracy. (a) $t$-SNE of the
original images in the pixel space. (b) $t$-SNE of the 512-dimensional
representation obtained via an ImageNet-pretrained ResNet18 network. (c)
$t$-SimCNE using the same augmentations as in Böhm et al. (2023). (d)
$t$-SimCNE using augmentations including 90° rotations and flips. Note that
the EBO class is well separated here, despite only consisting of 78 images.

### Augmentations

Böhm et al. (2023) worked with natural images and used the same data
augmentations as Chen et al. (2020): cropping, horizontal flipping, color
jittering, and grayscaling (Figure 1b). Here we used all of these
augmentations with the same hyperparameters and probabilities. We reasoned
that the semantics of microscopy or pathology images should be invariant to
arbitrary rotations and arbitrary flips (Kang et al., 2023). For that reason
we considered two additional sets of augmentations: (i) vertical flips and
arbitrary 90° rotations; (ii) rotations by any arbitrary angle (Figure 1c). In
each case, all possible rotations were equally likely. When rotating an image
by an angle that is not a multiple of 90°, the corners need to be filled in
(Figure 1c, right). For this we used the average color of all border pixels
across all images in a given dataset. This color was dataset specific, but the
same for all images in a dataset.

### Architecture and training

We used the original $t$-SimCNE implementation Böhm et al. (2023) with default
parameters unless stated otherwise. For PCam16, we used the unmodified
ResNet18 He et al. (2016) without the fully-connected layer. All networks were
trained from scratch. The SimCLR models had the same architecture as
$t$-SimCNE models but had 128-dimensional output and were trained with SimCLR
loss (Eq. 1) for 1000 epochs. We then used $t$-SNE (Van der Maaten and Hinton,
2008) to reduce the 128-dimensional SimCLR representation to 2D.

We trained all $t$-SimCNE and SimCLR models on an NVIDIA RTX A6000 GPU with
the batch size of 1024, except for PCam16 where we had to reduce the batch
size to 512 to fit it into GPU memory. For comparison, we used ImageNet-
pretrained ResNet18 and ResNet152 models available in the PyTorch library
Paszke et al. (2019). To pass our images through these networks, we resized
all images to $256\times 256$, center cropped to $224\times 224$, and
normalized them (following He et al., 2016). The resulting representations had
512 and 2048 dimensions respectively. We then used $t$-SNE to reduce them to
2D. We used openTSNE $1.0.1$ Poličar et al. (2019) with default settings for
all $t$-SNE embeddings.

Figure 3: (a) $t$-SimCNE visualisation of the Leukemia dataset. Only a subset
of classes is listed in the legend. (b) $t$-SimCNE visalisation of the
Bloodmnist dataset. (c) $t$-SimCNE visualisation of the Dermamnist dataset. In
all three cases, we used augmentations including 90° rotations and vertical
flips.

### Evaluation metrics

We used two metrics to evaluate the quality of 2D embeddings. First, we used
$k$NN classification accuracy Pedregosa et al. (2011) with $k=15$ and a 9:1
training/test split. Second, we used the silhouette score (Rousseeuw, 1987).
For a single point $x$, silhouette score $s\in[-1,1]$ is defined as
$(b-w)/\max(w,b)$ where $w$ is the average distance between $x$ and points
from the same class, and $b$ is the average distance between $x$ and points
from the closest other class. The silhouette score of the entire embeddings is
the average silhouette score across all points. These two measures are
complementary: The $k$NN accuracy measures how well the classes are separated
from each other in the embedding, while the silhouette score measures how far
they are separated from each other.

##  5 Results

Table 2:  The $k$NN accuracy of 2D embeddings. Means $\pm$ standard deviations over three runs; PCam16 experiments had only one run due to its large size. Method | Dataset
---|---
Leukemia |  Bloodmnist |  Dermamnist |  Pathmnist | PCam16
$t$-SimCNE | def. augm. | $86.7\pm 0.5\%$ | $90.4\pm 0.3\%$ | $77.3\pm 0.6\%$ | $97.2\pm 0.2\%$ | $92.6\%$
\+ 90° rot. | $94.0\pm 0.0\%$ | $93.0\pm 0.3\%$ | $77.5\pm 0.3\%$ | $98.0\pm 0.0\%$ | $93.1\%$
\+ rand. rot. | $95.2\pm 0.0\%$ | $92.9\pm 0.1\%$ | $80.1\pm 0.7\%$ | $97.3\pm 0.0\%$ | $90.8\%$
$t$-SNE of SimCLR | def. augm. | $95.1\pm 0.2\%$ | $93.4\pm 0.6\%$ | $81.8\pm 0.2\%$ | $97.6\pm 0.1\%$ | $95.3\%$
\+ 90° rot. | $95.2\pm 0.2\%$ | $95.8\pm 0.3\%$ | $80.8\pm 0.5\%$ | $97.7\pm 0.1\%$ | $95.5\%$
\+ rand. rot. | $95.3\pm 0.2\%$ | $95.1\pm 0.2\%$ | $82.0\pm 0.5\%$ | $96.9\pm 0.0\%$ | $93.7\%$
$t$-SNE | pixel space | $67.4\%$ | $73.2\%$ | $78.0\%$ | $56.9\%$ | $76.9\%$
ResNet18 | $82.2\%$ | $78.1\%$ | $81.9\%$ | $87.2\%$ | $86.7\%$
ResNet152 | $72.0\%$ | $72.6\%$ | $81.0\%$ | $88.8\%$ | $86.4\%$
Table 3:  Silhouette scores (Section 4) of 2D embeddings. Same format as in Table 2. Method | Dataset
---|---
Leukemia |  Bloodmnist |  Dermamnist |  Pathmnist | PCam16
$t$-SimCNE | def. augm. | $0.02\pm 0.01$ | $0.40\pm 0.00$ | $0.13\pm 0.01$ | $0.45\pm 0.02$ | $0.04$
\+ 90° rot. | $0.13\pm 0.02$ | $0.44\pm 0.03$ | $0.11\pm 0.00$ | $0.48\pm 0.06$ | $0.05$
\+ rand. rot. | $0.24\pm 0.02$ | $0.50\pm 0.01$ | $0.13\pm 0.06$ | $0.41\pm 0.03$ | $0.05$
$t$-SNE of SimCLR | def. augm. | $-0.03\pm 0.01$ | $0.35\pm 0.00$ | $0.14\pm 0.01$ | $0.14\pm 0.01$ | $0.09$
\+ 90° rot. | $-0.00\pm 0.01$ | $0.33\pm 0.01$ | $0.14\pm 0.01$ | $0.16\pm 0.04$ | $0.10$
\+ rand. rot. | $-0.04\pm 0.01$ | $0.34\pm 0.01$ | $0.14\pm 0.00$ | $0.24\pm 0.01$ | $0.08$
$t$-SNE | pixel space | $-0.17$ | $0.07$ | $0.08$ | $-0.05$ | $0.02$
ResNet18 | $-0.22$ | $0.13$ | $0.14$ | $0.17$ | $0.04$
ResNet152 | $-0.25$ | $0.03$ | $0.14$ | $0.19$ | $0.05$

In this study, we asked (i) how the contrastive visualisation method
$t$-SimCNE (Böhm et al., 2023) could be applied to medical image datasets, and
(ii) if the set of data augmentations could be enriched compared to what is
typically used on natural images.

We considered the Leukemia dataset as our first example (Figure 2). Naive
application of $t$-SNE to the raw images in pixel space resulted in an
embedding with little class separation and low $k$NN accuracy of 67.4% (Figure
2a). Passing all images through an ImageNet-pretrained ResNet and then
embedding them with $t$-SNE improved the $k$NN accuracy to 82.2% but visually
the classes were still separated poorly (Figure 2b). Training $t$-SimCNE with
default data augmentations gave embeddings with 86.7% $k$NN accuracy (Table 2)
and much better visual class separation (Figure 2c and Table 3). This shows
that $t$-SimCNE can produce meaningful 2D visualizations of medical image
datasets.

We reasoned that the set of data augmentations could be enriched to include
90° rotations and flips because the semantics of blood microscopy images is
rotationally invariant. When training $t$-SimCNE with this set of data
augmentations, the $k$NN accuracy increased to 94.0%. Additionally including
all possible rotations by an arbitrary angle as data augmentations yielded the
highest $k$NN accuracy (95.2%) and the highest silhouette score (0.24),
indicating that domain-specific augmentations can further improve $t$-SimCNE
embeddings.

Figure 4: $t$-SimCNE visualisation of the Pathmnist dataset. Colours
correspond to classes. Images correspond to three random points close to the
tip of the annotation line.

Across the five datasets considered in this study, we saw three different
outcomes. On microscropy datasets (Leukemia and Bloodmnist), $t$-SimCNE with
random rotations performed the best: it had by far the best silhouette score
(Table 3) and visually the most separated classes (Figure 3a,b). SimCLR
followed by $t$-SNE had slightly higher $k$NN accuracies (Figure 2), but much
lower silhouette scores.

On pathology datasets (Pathmnist and PCam16), $t$-SimCNE with 90° rotations
performed the best. On Pathmnist, it had the highest $k$NN accuracy and
silhouette score (Figure 4). On PCam16, SimCLR followed by $t$-SNE showed a
slightly higher $k$NN accuracy, but $t$-SimCNE showed more well separated
structures. This difference was not captured by the silhouette scores which on
this dataset which were all close to zero (Table 3). This is because it only
had two classes, whereas $t$-SimCNE separated images not only by class but
also by tissue types (Figure 5); this led to large within-class distances and
hence misleadingly low silhouette scores.

Finally, on the dermatology dataset (Dermamnist), performance of all methods
was similarly poor: SimCLR and $t$-SimCNE resulted in embeddings not very
different from $t$-SNE in pixel space (Figure 3c).

As a control experiment, we applied $t$-SimCNE with 90° rotations and vertical
flips to the CIFAR-10 dataset Krizhevsky et al. (2009). It decreased the $k$NN
accuracy from 89% to 76%. This confirms that rotation augmentations are not
helpful for natural images because they are not invariant to rotations, unlike
microscopy and pathology images.

In the pathology datasets, $t$-SimCNE showed meaningful subclass structure.
For example, in Pathmnist, the debris class separated into three clearly
distinct subsets (Figures 4), one of which had markedly different staining
colour. In the Pcam16 dataset, the embedding clearly split patches with and
without metastasis, based on the density of chromatin and variation in the
size of the cells. The difference in visual appearance (different shades of
violet) between top-right and bottom-left likely reveals a technical artefact
resulting from different staining durations.

Figure 5: (a) $t$-SimCNE visualisation of the PCam16 dataset. (b) We
superimposed a $10\times 10$ grid over the embedding and selected one image in
each square. Frame colours show image classes. If a square had fewer than 100
images, no image was shown.

##  6 Discussion

In this paper, we showed that $t$-SimCNE (Böhm et al., 2023) can be
successfully applied to medical image datasets, yielding semantically
meaningful visualisations, and benefits from rotational data augmentations,
leveraging rotational invariance of microscropy images. In agreement with Böhm
et al. (2023), $t$-SimCNE performed better than SimCLR + $t$-SNE combination.
Even though SimCLR tended to have slightly higher $k$NN accuracy, the
silhouette score was typically much lower: $t$-SimCNE achieved visually much
stronger cluster separation, which is useful for practical visualisations.
Furthermore, parametric nature of $t$-SimCNE allows to embed new (out-of-
sample) images into an existing embedding.

We found that blood microscopy datasets benefited the most from random
rotations, while pathology datasets showed the best results with 90° rotations
and flips. We believe it is because in blood microscopy images, the
semantically meaningful part is always in the center (Figure 3a,b) and so the
corners of the image may not be important. In contrast, in histopathology
images, the edges of the image may contain relevant information, which may get
rotated out of the image and replaced by solid-color triangles (Figure 1c).
One of the datasets, Dermamnist, exhibited poor results with all analysis
methods. This may be because in this dataset the images are too small to
convey biomedically relevant information, or because the sample size was
insufficient (Table 1).

In conclusion, we argue that $t$-SimCNE is a promising tool for visualisation
of medical image datasets. It can be useful for quality control, highlighting
artefacts and problems in the data. It can also create a 2D map of cell types,
tissue types, or medical conditions, which can be useful not only for clinical
purposes but also education and research, potentially combined with an
interactive image exploration tool. In the future, it may be interesting to
extend $t$-SimCNE to learn representations invariant to technical (e.g.
staining) artefacts.

We thank Christian Schürch for discussion on histopathology data. This work
was supported by the German Science Foundation (Excellence Cluster 2064
“Machine Learning — New Perspectives for Science”, project number 390727645),
the Hertie Foundation, and the Cyber Valley Research Fund (D.30.28739). The
authors thank the International Max Planck Research School for Intelligent
Systems (IMPRS-IS) for supporting Jan Niklas Böhm. Philipp Berens is a member
of the Else Kröner Medical Scientist Kolleg “ClinbrAIn: Artificial
Intelligence for Clinical Brain Research”.

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# OGLE-2015-BLG-0845L: A low-mass M dwarf from the microlensing parallax and
xallarap effects

Zhecheng Hu (胡哲程),1 Wei Zhu (祝伟),1 Andrew Gould,2,333footnotemark: 3 Andrzej
Udalski,4 Takahiro Sumi,5 Ping Chen (陈平),6 Sebastiano Calchi
Novati,733footnotemark: 3 Jennifer C. Yee,833footnotemark: 3 Charles A.
Beichman,733footnotemark: 3 Geoffery Bryden,933footnotemark: 3 Sean
Carey,733footnotemark: 3 Michael Fausnaugh,1033footnotemark: 3 B. Scott
Gaudi,333footnotemark: 3 Calen B. Henderson,733footnotemark: 3 Yossi
Shvartzvald,633footnotemark: 3 Benjamin Wibking,1133footnotemark: 3 Przemek
Mróz,444footnotemark: 4 Jan Skowron,444footnotemark: 4 Radoslaw
Poleski,444footnotemark: 4 Michaeł K. Szymański,444footnotemark: 4 Igor
Soszynśki,444footnotemark: 4 Paweł Pietrukowicz,444footnotemark: 4 Szymon
Kozłowski,444footnotemark: 4 Krzysztof Ulaczyk,1244footnotemark: 4 Krzysztof
A. Rybicki,4,644footnotemark: 4 Patryk Iwanek,444footnotemark: 4 Marcin
Wrona,444footnotemark: 4 Mariusz Gromadzki,444footnotemark: 4 Fumio
Abe,1355footnotemark: 5 Richard Barry,1455footnotemark: 5 David P.
Bennett,14,1555footnotemark: 5 Aparna Bhattacharya,14,1555footnotemark: 5 Ian
A. Bond,1655footnotemark: 5 Hirosane Fujii,1355footnotemark: 5 Akihiko
Fukui,17,1855footnotemark: 5 Ryusei Hamada,555footnotemark: 5 Yuki
Hirao,1955footnotemark: 5 Stela Ishitani Silva,20,1455footnotemark: 5
Yoshitaka Itow,1355footnotemark: 5 Rintaro Kirikawa,555footnotemark: 5 Naoki
Koshimoto,555footnotemark: 5 Yutaka Matsubara,1355footnotemark: 5 Shota
Miyazaki,2155footnotemark: 5 Yasushi Muraki,1355footnotemark: 5 Greg
Olmschenk,1455footnotemark: 5 Clément Ranc,2255footnotemark: 5 Nicholas J.
Rattenbury,2355footnotemark: 5 Yuki Satoh,555footnotemark: 5 Daisuke
Suzuki,555footnotemark: 5 Mio Tomoyoshi,555footnotemark: 5 Paul. J.
Tristram,2455footnotemark: 5 Aikaterini Vandorou,14,1555footnotemark: 5 Hibiki
Yama,555footnotemark: 5 Kansuke Yamashita555footnotemark: 5
Affiliations are listed at the end of the paper E-mail:
hzc22<EMAIL_ADDRESS>Spitzer TeamThe
OGLE CollaborationMOA Collaboration 0009-0000-6461-5256 0000-0003-4027-4711
0000-0001-5207-5619 0000-0002-7669-1069 0000-0001-9481-7123
0000-0002-0221-6871 0000-0002-9113-7162 0000-0003-0395-9869
0000-0001-8877-9060 0000-0003-1525-5041 0000-0003-3175-2291
0000-0001-7016-1692 0000-0002-2335-1730 0000-0002-9245-6368
0000-0002-0548-8995 0000-0002-7777-0842 0000-0002-2339-5899
0000-0003-4084-880X 0000-0001-6364-408X 0000-0002-9326-9329
0000-0002-6212-7221 0000-0002-3051-274X 0000-0002-1650-1518

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the analysis of the microlensing event OGLE-2015-BLG-0845, which
was affected by both the microlensing parallax and xallarap effects. The
former was detected via the simultaneous observations from the ground and
Spitzer, and the latter was caused by the orbital motion of the source star in
a relatively close binary. The combination of these two effects led to a
direct mass measurement of the lens object, revealing a low-mass ($0.14\pm
0.05M_{\sun}$) M-dwarf at the bulge distance ($7.6\pm 1.0$ kpc). The source
binary consists of a late F-type subgiant and a K-type dwarf of $\sim
1.2M_{\sun}$ and $\sim 0.9M_{\sun}$, respectively, and the orbital period is
$70\pm 10$ days. OGLE-2015-BLG-0845 is the first single-lens event in which
the lens mass is measured via the binarity of the source. Given the abundance
of binary systems as potential microlensing sources, the xallarap effect may
not be a rare phenomenon. Our work thus highlights the application of the
xallarap effect in the mass determination of microlenses, and the same method
can be used to identify isolated dark lenses.

###### keywords:

gravitational lensing: micro – methods: data analysis – binaries: general

††pubyear: 2024††pagerange: OGLE-2015-BLG-0845L: A low-mass M dwarf from the
microlensing parallax and xallarap effects–A

##  1 Introduction

Figure 1: An illustration of the satellite parallax method and the xallarap
effect in the microlensing phenomenon. Distances and physical sizes are not to
scale. The microlensing source is a binary system, with the yellow circle
indicating the primary star (i.e., primary source) and the brown circle
indicating the secondary star (i.e., secondary source). The centre of light,
namely the photocentre, of the source system is indicated by the orange
circle. On the observer plane, Spitzer was in an Earth-trailing orbit and
observed the same source simultaneously as the ground-based observatories.

The microlensing effect (Einstein, 1936; Paczynski, 1986) is able to measure
the mass of faint or even dark objects (or stellar systems) independent from
their flux (e.g., Gould, 1992). So far, the microlensing technique has been
used to determine the masses of dozens of isolated objects, including brown
dwarfs (e.g., Gould et al., 2009; Zhu et al., 2016; Shvartzvald et al., 2019),
low-mass stars (e.g., Chung et al., 2017; Zhu et al., 2017b; Shin et al.,
2018; Zang et al., 2020a, b), and stellar remnants (Sahu et al., 2022; Lam et
al., 2022; Mroz et al., 2022), as well as another dozens of planetary systems
(e.g. Gaudi et al., 2008; Bennett et al., 2015).

Out of thousands of microlensing events that are discovered per year, only a
small fraction of them allow one to determine the lens mass directly. This is
because, in order to directly determine the lens mass, one needs to measure at
least two out of three parameters, namely the angular Einstein radius, the
microlensing parallax, and the lens flux (e.g., Yee, 2015). The angular
Einstein radius measures the characteristic angular size of the microlensing
phenomenon and is given by

$\theta_{\rm E}\equiv\sqrt{\kappa M_{\rm L}\pi_{\rm rel}}.$ (1)

Here $\kappa\equiv 4G/(c^{2}\rm AU)\approx 8.14{\rm mas}/M_{\odot}$ is a
constant, $M_{\rm L}$ is the lens mass, and $\pi_{\rm rel}\equiv{\rm
AU}(D_{\rm L}^{-1}-D_{\rm S}^{-1})$ is the lens–source relative parallax
(Gould, 2000). The microlensing parallax measures the relative parallactic
motion scaled to the angular Einstein radius (Gould, 1992)

$\pi_{\rm E}\equiv\frac{\pi_{\rm rel}}{\theta_{\rm E}}.$ (2)

The lens flux measures the observed flux of the lens at the given distance.
With a stellar mass–luminosity relation and a given extinction model, the lens
flux measurement provides a relation between the lens mass and lens distance,
similar to the other two quantities (e.g., Bennett et al., 2015). For faint or
dark lenses, the lens flux is usually not detectable, and thus the only option
towards a direct mass measurement is through the combination of angular
Einstein radius and the microlensing parallax.

Although the microlensing parallax can be measured through the orbital motion
of Earth around the Sun (Gould, 1992), such an annual parallax effect only
applies to microlensing events with relatively long timescales (e.g., Mao et
al., 2002; Poindexter et al., 2005; Wyrzykowski et al., 2016). The parameter
$\pi_{\rm E}$ of an event could also be measured with at least two well-
separated ($\sim{\rm AU}$) observatories(Refsdal, 1966; Gould, 1994b). Between
2014 and 2019, _Spitzer_ was used to observe more than five hundred
microlensing events, and so far nearly a hundred of them with parallax
solutions have been published (e.g., Calchi Novati et al., 2015a; Zhu et al.,
2017a).

Several methods are available to measure the Einstein radius, but each has its
own limitations. Observing the lens and source separately several years after
the event could obtain the lens–source relative motion $\mu_{\rm rel}$, and
then $\theta_{\rm E}$ is given by $\theta_{\rm E}=t_{\rm E}\mu_{\rm rel}$,
where $t_{\rm E}$ is the Einstein crossing time (e.g. Alcock et al., 2001;
Bennett et al., 2015; Gould, 2022). The angular Einstein radius can also be
determined through the finite source effect, which appears when the source is
close to or even crosses the caustic of the lens object/system (Gould, 1994a;
Witt & Mao, 1994; Nemiroff & Wickramasinghe, 1994), but in the case of single-
lens events, the chance is low to have the finite source effect, except for
extremely low-mass lenses such as free-floating planets (e.g., Mróz et al.,
2018; Gould et al., 2022). The astrometric microlensing has also been proposed
(Hog et al., 1995), and recently realized (Sahu et al., 2022; Lam et al.,
2022), as a method to directly measure $\theta_{\rm E}$ during the course of
the microlensing event. Additionally, interferometric observations may be able
to directly resolve the multiple microlensing images and thus determine
$\theta_{\rm E}$ (Delplancke et al., 2001; Dong et al., 2019; Cassan et al.,
2022). These latter two methods have not been applied widely because they
require high precisions and/or lucky observing conditions.

When the source is in a binary system, its orbital motion around the centre of
mass may lead to changes in the relative positions among the microlensing
objects, which also revises the light curve (Griest & Hu, 1992). This so-
called microlensing xallarap effect can also be used to measure the angular
Einstein radius (Han & Gould, 1997). Specifically, the xallarap parameter,
$\xi_{\mathrm{E}}$, is related to $\theta_{\mathrm{E}}$ via

$\xi_{\mathrm{E}}\equiv\frac{a_{\mathrm{S}}}{D_{\mathrm{S}}\theta_{\mathrm{E}}}=\frac{a_{\mathrm{S}}}{\hat{r}_{\mathrm{E}}}.$
(3)

Here $a_{\mathrm{S}}$ is the semi-major axis of the motion of the source star
(or the photocentre of the source binary) around the barycentre of the source
binary system, $D_{\mathrm{S}}$ is the distance of the source from Earth, and
$\hat{r}_{\mathrm{E}}$ is the projected Einstein radius in the source plane.
Once $\xi_{\mathrm{E}}$ and $a_{\rm S}$ are measured, the xallarap effect
provides another way to measure $\theta_{\rm E}$.

Given the abundance of stellar binaries (e.g., Duchêne & Kraus, 2013), a
substantial fraction of events might have been affected by the xallarap
effect. Poindexter et al. (2005) searched for the xallarap effect in 22
microlensing events with relatively long ($t_{\rm E}\gtrsim 70$ d) time scales
and found that about 23% of them might have been strongly affected by the
xallarap effect, although the fraction may be reduced once the shorter but
more abundant events are taken into account. The impact of xallarap effect has
also been regularly investigated in microlensing events that contain
potentially planetary signals (e.g., Furusawa et al., 2013; Miyazaki et al.,
2020; Rota et al., 2021; Satoh et al., 2023; Yang et al., 2024; Ryu et al.,
2024).

In this work, we present the analysis of the microlensing event
OGLE-2015-BLG-0845 . This event shows both xallarap and parallax effects, as
illustrated in Figure 1, which allow us to determine directly the lens mass
and distance. The observations of OGLE-2015-BLG-0845 are presented in Section
2, the detailed modelings of the event are given in Section 3, and the
physical properties of the lens and source are given in Section 4. A brief
discussion of the results is given in Section 5.

##  2 Observations

The microlensing event OGLE-2015-BLG-0845 was first identified by the OGLE-IV
collaboration on UT 15:51, 28 April 2015. With equatorial coordinates
$\rm{(RA,Dec)}_{2000}=(18^{\rm{h}}04^{\rm{m}}21\aas@@fstack{s}29,-31\degr
34\arcmin 50\aas@@fstack{\prime\prime}0)$ and Galactic coordinates
$(l,b)_{2000}=(-0\aas@@fstack{\circ}25,-4\aas@@fstack{\circ}82)$, this event
was located inside the field BLG514 of OGLE-IV survey and received three
observations per day from the 1.3 m Warsaw Telescope at the Las Campanas
Observatory in Chile (Udalski, 2003; Udalski et al., 2015a). The OGLE
observations were mostly taken in the $I$ band, but $V$ band observations were
also taken at the cadence of a few days in order to provide colour information
of the microlensing source. The OGLE data were reduced using the software
developed by Wozniak (2000) and Udalski (2003), which was based on the
Difference Image Analysis (DIA) technique of Alard & Lupton (1998).

This event was observed by the Spitzer Space Telescope as part of the 2015
microlensing campaign (Udalski et al., 2015b; Yee et al., 2015; Zhu et al.,
2017a). It was selected for Spitzer observations on June 8, 2015, as a “secret
event” and then announced as a Spitzer target on June 12, 2015. We refer to
Yee et al. (2015) for the details of the observing protocol of the Spitzer
microlensing program. The Spitzer observations started on the same date and
stopped on July 19, 2015, when the event moved out of the visibility window of
Spitzer. There were in total 148 observations taken by Spitzer. All Spitzer
observations were reduced by the customized software specifically for the
microlensing program Calchi Novati et al. (2015b).

Even OGLE-2015-BLG-0845 was also observed by the Microlensing Observations in
Astrophysics (MOA, Bond et al., 2001; Sako et al., 2008). It is called
MOA-2015-BLG-277 in the MOA database. Follow-up observations from various
small telescopes were also taken for the purpose of detecting planetary
signals. Because of the unknown systematics and/or the relatively short time
baseline, the MOA data and the follow-up data are not included in the modeling
of the parallax and xallarap signals, but they provide useful constraints on
the binary lens modeling (see Appendix A).

##  3 Modeling

The light curve of OGLE-2015-BLG-0845 was first modeled using the standard
Paczynski (1986) curve

$f(t)=f_{\mathrm{s}}A(u[t])+f_{\mathrm{b}}$ (4)

with the single-lens point-source, or 1L1S 111This notation follows the
convention that is commonly used in the microlensing literature. The numbers
in front of “L” and “S” indicates the numbers of objects in the lens and the
source systems, respectively, that directly participate in the microlensing
phenomenon in a detectable way. In other words, with “1L” (or “1S”) it does
not necessarily mean that the source (or lens) object is physically single. ,
microlensing magnification given by

$\displaystyle A(u)=\frac{u^{2}+2}{u\sqrt{u^{2}+4}}.$ (5)

Here $f_{\rm s}$ and $f_{\rm b}$, are the flux of the source star and the flux
of the blending object, respectively. The quantity, $u(t)$, is the separation
normalized to $\theta_{\mathrm{E}}$ between the lens and the source at a given
epoch $t$. For the standard Paczynski (1986) curve,

$u(t)=\sqrt{\tau^{2}+\beta^{2}},$ (6)

with

$\tau(t)\equiv\frac{t-t_{0}}{t_{\mathrm{E}}},\quad\beta(t)\equiv u_{0}.$ (7)

Here $t_{0}$ is the time when the projected separation between the lens and
the source reaches the minimum, $u_{0}$ is this minimum separation, and
$t_{\rm E}$ is the Einstein crossing time.

The standard Paczynski (1986) curve cannot fit the light curve of
OGLE-2015-BLG-0845 very well, as shown in Figure 2. While these deviations
could be modeled by the annual parallax effect reasonably well, it was soon
recognized that the parallax parameters from the annual parallax effect did
not match those from the satellite parallax effect (Section 3.1). This led to
the inclusion of additional effects, including the xallarap effect (Section
3.2). The binary lens, as well as the static binary source model, was also
considered, although it could not fit the data as well as the xallarap model
(see Appendix A).

In all the modelings, we have made use of the emcee package (Foreman-Mackey et
al., 2013) to execute a Markov Chain Monte Carlo (MCMC) analysis. The number
of walkers and burn-in steps have been adjusted so as to make sure the chain
has converged.

###  3.1 The Tension Between 1L1S Annual and Space-based Parallax Model

The accelerated motion of the Earth can lead to a distortion on the light
curve, known as the annual parallax effect (e.g., Gould, 1992; Alcock et al.,
1995). This distortion adds displacement to the rectilinear motion (Equation
7) of the Paczynski (1986) model

$(\delta\tau_{\mathrm{p}},\delta\beta_{\mathrm{p}})=\left(\boldsymbol{\pi}_{\mathrm{E}}\cdot\Delta\boldsymbol{s},\boldsymbol{\pi}_{\mathrm{E}}\times\Delta\boldsymbol{s}\right).$
(8)

Here $\Delta\boldsymbol{s}$ is the offset of the Earth-to-Sun vector in AU
(Gould, 2004). The microlensing parallax vector is defined as

$\boldsymbol{\pi}_{\mathrm{E}}=\frac{\pi_{\mathrm{rel}}}{\theta_{\mathrm{E}}}\frac{\boldsymbol{\mu}_{\mathrm{rel}}}{\mu_{\mathrm{rel}}},\quad\pi_{\mathrm{rel}}=\mathrm{AU}\left(\frac{1}{D_{\mathrm{L}}}-\frac{1}{D_{\mathrm{S}}}\right).$
(9)

Here $\pi_{\rm{rel}}$ and $\vec{\mu}_{\rm{rel}}$ are the relative parallax and
the relative proper motion between the lens and source, respectively (Gould,
2000).

The microlensing parallax can also be measured through simultaneous
observations of the same event from at least two well–separated
observatories/satellites (Refsdal, 1966; Gould, 1994b; Zhu et al., 2017b). The
two observatories—namely Spitzer and Earth in the present case—obtain light
curves that are different in both the event peak time $t_{0}$ and the impact
parameter $u_{0}$. Making use of the known projected separation between the
two observatories on the celestial plane, $D_{\perp}$, the parallax vector
$\boldsymbol{\pi}_{\mathrm{E}}$ can be estimated as

$\boldsymbol{\pi}_{\mathrm{E}}\approx\frac{\mathrm{AU}}{D_{\perp}}(\tau_{\mathrm{sat}}-\tau_{\oplus},\beta_{\mathrm{sat}}-\beta_{\oplus}).$
(10)

Here the subscripts “$\mathrm{sat}$” and “$\oplus$” represent the quantities
of the satellite and Earth, respectively.

The two approaches should yield consistent parallax parameters, provided that
the deviation from the standard Paczynski (1986) model is indeed due to the
annual parallax effect. As shown in Figure 3, for OGLE-2015-BLG-0845 , the
constraints on microlensing parallax from fitting OGLE data alone
significantly differ from the constraints from the joint fitting of OGLE and
Spitzer data, no matter which of the four degenerate solutions is considered.
Therefore, other explanations are required to explain the deviation in the
light curve from the standard Paczynski (1986) model.

Figure 2:  OGLE-IV data of OGLE-2015-BLG-0845 and the best-fit 1L1S standard
and annual parallax models. The top panel displays the OGLE $I$\- and $V$-band
data in black and yellow points, respectively. The best-fit standard model and
the parallax model are denoted by the grey dashed and the black solid curves,
respectively. The middle panel illustrates the residuals of the data and the
parallax model with respect to the standard model. The bottom panel shows the
cumulative $\Delta\chi^{2}\equiv\chi^{2}_{\rm STD}-\chi^{2}_{\rm Anu.PRLX}$
between the standard and parallax models as a function of time. Although the
best-fit parallax model improved the model $\chi^{2}$ by $>100$ over the
standard 1L1S model, there remain long-term variations in the residuals that
cannot be well fit.  Figure 3:  The tension between $\pi_{\mathrm{E}}$ as
derived by fitting OGLE-only data and joint fitting OGLE and _Spitzer_ data of
the microlensing event OGLE-2015-BLG-0845 . The 1-sigma regions are shown as
dark solid ellipses, while the 2-sigma regions are shown as lighter dashed
ellipses. For space-based parallax, all four degenerate solutions are shown
while for annual parallax, only the $u_{0}+$ solution is shown. The $u_{0}-$
solution for the OGLE-only fitting is not shown because the $\pi_{\rm{E}}$
value is close to the positive solutions, which does not change the tension.
Figure 4:  OGLE $I$ and $V$ band light curves (left panels) and _Spitzer_ 3.6
$\mu$m light curve (right panels) of the microlensing event OGLE-2015-BLG-0845
. In each panel, the best-fit 1L1S parallax model, 1L1S parallax + xallarap
model, and the 1L2S parallax + xallarap model are shown as dashed gray line,
dash-dotted line, and solid line, respectively. The OGLE $V$-band data have
been aligned to the OGLE $I$-band data based on the best-fit flux parameters,
and the _Spitzer_ flux values are shown in logarithmic scale (quasi-
magnitudes). The top panels show the observed data and the best-fit models,
and the middle panels show the residuals of data and the other models relative
to the 1L1S parallax model. In the lower panels, we show the cumulative
$\Delta\chi^{2}$ distribution of the other two models relative to the 1L1S
parallax model.

###  3.2 1L1S Xallarap + Parallax Model

The xallarap effect is the reflection of the orbital motion of the source on
the light curve, as is shown in Figure 1. In principle, one should also
consider the companion star (hereafter the secondary source) in the light
curve modeling. However, the probability is high that the secondary source
only contributes a marginal fraction of the total source flux because of the
steep relation between stellar mass and luminosity. Therefore, we begin the
xallarap modeling by including only one luminous source. For simplicity, the
orbit of the source star is assumed to be circular, and the impact of an
elliptical orbit is discussed in the section 5.1.

The xallarap effect introduces five new parameters,
$(P_{\xi},A_{\xi},B_{\xi},F_{\xi},G_{\xi})$. Here $P_{\xi}$ is the orbital
period of the source binary, and the other four parameters are variants of the
classical Thiele-Innes elements that are widely used in astrometry

$\left\\{\begin{array}[]{l}A_{\xi}=\xi_{\mathrm{E}}(\cos\phi_{\xi}\cos(-\theta_{\xi})-\sin\phi_{\xi}\sin(-\theta_{\xi})\cos
i_{\xi})\\\
B_{\xi}=\xi_{\mathrm{E}}(\cos\phi_{\xi}\sin(-\theta_{\xi})+\sin\phi_{\xi}\cos(-\theta_{\xi})\cos
i_{\xi})\\\
F_{\xi}=\xi_{\mathrm{E}}(-\sin\phi_{\xi}\cos(-\theta_{\xi})-\cos\phi_{\xi}\sin(-\theta_{\xi})\cos
i_{\xi})\\\
G_{\xi}=\xi_{\mathrm{E}}(-\sin\phi_{\xi}\sin(-\theta_{\xi})+\cos\phi_{\xi}\cos(-\theta_{\xi})\cos
i_{\xi})\end{array}\right.$ (11)

Here $i_{\xi}$ is the orbital inclination, $\phi_{\xi}$ is the phase of the
source relative to the ascending node at the reference epoch $t_{\rm ref}$,
and $\theta_{\xi}$ measures the ascending node relative to the lens–source
relative trajectory projected onto the source plane. The amplitude of the
xallarap motion, $\xi_{\rm E}$, the semi-major axis of the binary motion
scaled to the projected Einstein ring radius, as is defined in Equation 3.

The xallarap effect leads to displacement in the lens-source relative
trajectory by the amounts

$\left(\begin{array}[]{c}\delta\tau_{\mathrm{x}}\\\
\delta\beta_{\mathrm{x}}\end{array}\right)=\left(\begin{array}[]{cc}A_{\xi}&F_{\xi}\\\
B_{\xi}&G_{\xi}\end{array}\right)\cdot\Delta\boldsymbol{S}$ (12)

Following the convention in the modeling of the parallax effect (Gould, 2004),
we have introduced $\Delta\boldsymbol{S}$ to measure the displacement between
the actual position of the source under binary motion and an imaginary source
under linear motion

$\Delta\boldsymbol{S}=(t-t_{\rm ref})\boldsymbol{v_{\rm
ref}}-[\boldsymbol{S}(t)-\boldsymbol{S}(t_{\rm ref})].$ (13)

The binary motion of the source, $\boldsymbol{S}(t)$, is given by the circular
motion starting from a zero phase with an amplitude of unity, and the velocity
vector, $\boldsymbol{v_{\rm ref}}$, is the instantaneous velocity of the
source at reference epoch $t_{\rm ref}$.

The distorted lens–source relative trajectory in the presence of both parallax
and xallarap effects can be described as

$\left\\{\begin{array}[]{l}\tau=\tau_{\rm{std}}+\delta\tau_{\rm{p}}+\delta\tau_{\rm{x}}\\\
\beta=\beta_{\rm{std}}+\delta\beta_{\rm{p}}+\delta\beta_{\rm{x}}\end{array}\right..$
(14)

where $\tau_{\rm{std}}$ and $\beta_{\rm{std}}$ describe the trajectory in the
standard Paczynski (1986) model.

The best-fit 1L1S xallarap + parallax model is shown in Figure 4, and the
best-fit parameters and the associated uncertainties are given in Table 1. The
amplitude of the xallarap parameter, $\xi_{\rm E}$, derived from the xallarap
model, is also provided in the same table. Note that in this derivation we
have ignored the nonuniform prior introduced by the Thiele–Innes
parameterization of the xallarap model, and we have confirmed that the values
would only be marginally revised if a nonuniform prior were imposed.

The 1L1S Xallarap + Parallax model significantly better describes the light
curve by $\Delta\chi^{2}\simeq 340$, with respect to the 1L1S Parallax model
and removes the tension in parallax constraints between annual and space-based
parallax-only models. However, from Table 1, we can see that the
$\xi_{\mathrm{E}}$ is comparable to the amplitude of $u_{0}$. The secondary
source, which is fainter and presumably less massive, is expected to
experience a binary motion with a larger amplitude and thus produce non-
negligible features in the light curve. Therefore, it is reasonable to
introduce the secondary source as a luminous component in the microlensing
event.

Table 1: Parameter values and uncertainties of the best-fit 1L1S and 1L2S
xallarap + parallax solutions for OGLE-2015-BLG-0845 .

| 1L1S XLRP + PRLX | 1L2S XLRP + PRLX
---|---|---
Solution Type | $(+,+)$ | $(+,-)$ | $(-,+)$ | $(-,-)$ | $(+,+)$ | $(+,-)$ | $(-,+)$ | $(-,-)$
$t_{0}$ a | $7199.385\pm 0.014$ | $7199.381\pm 0.015$ | $7199.382\pm 0.016$ | $7199.388\pm 0.014$ | $7199.293\pm 0.018$ | $7199.278\pm 0.017$ | $7199.427\pm 0.014$ | $7199.432\pm 0.013$
$u_{0}$ | $0.05\pm 0.003$ | $0.0436\pm 0.0025$ | $-0.044\pm 0.003$ | $-0.05\pm 0.003$ | $0.049\pm 0.006$ | $0.046\pm 0.004$ | $-0.043\pm 0.004$ | $-0.047\pm 0.004$
$t_{\mathrm{E}}$ (days) | $42.1\pm 2.2$ | $47.9\pm 2.4$ | $47.0\pm 2.5$ | $42\pm 2$ | $46\pm 5$ | $49\pm 3$ | $46\pm 4$ | $43\pm 4$
$\pi_{\mathrm{E,N}}$ | $0.0\pm 0.0013$ | $-0.063\pm 0.004$ | $0.084\pm 0.005$ | $0.0231\pm 0.0018$ | $-0.0085\pm 0.0013$ | $-0.074\pm 0.007$ | $0.08\pm 0.007$ | $0.0125\pm 0.0015$
$\pi_{\mathrm{E,E}}$ | $0.077\pm 0.004$ | $0.077\pm 0.004$ | $0.054\pm 0.004$ | $0.074\pm 0.004$ | $0.078\pm 0.008$ | $0.071\pm 0.006$ | $0.065\pm 0.006$ | $0.07\pm 0.006$
$p_{\xi}$ (days) | $41\pm 4$ | $46\pm 4$ | $44\pm 5$ | $41\pm 5$ | $74\pm 8$ | $76\pm 7$ | $66\pm 9$ | $63\pm 9$
$A_{\xi}$ ($10^{-2}$) | $-0.01\pm 0.003$ | $-0.013\pm 0.004$ | $-0.014\pm 0.005$ | $-0.011\pm 0.003$ | $-0.065\pm 0.017$ | $-0.069\pm 0.016$ | $-0.044\pm 0.012$ | $-0.041\pm 0.012$
$B_{\xi}$ ($10^{-2}$) | $-0.011\pm 0.008$ | $-0.018\pm 0.016$ | $0.019\pm 0.018$ | $0.018\pm 0.011$ | $0.16\pm 0.04$ | $0.16\pm 0.03$ | $0.15\pm 0.04$ | $0.14\pm 0.04$
$F_{\xi}$ ($10^{-2}$) | $-0.005\pm 0.005$ | $0.003\pm 0.004$ | $-0.001\pm 0.004$ | $-0.007\pm 0.005$ | $0.029\pm 0.019$ | $0.026\pm 0.017$ | $0.02\pm 0.015$ | $0.01\pm 0.016$
$G_{\xi}$ ($10^{-2}$) | $-0.061\pm 0.019$ | $-0.062\pm 0.007$ | $0.065\pm 0.008$ | $0.063\pm 0.01$ | $0.07\pm 0.03$ | $0.09\pm 0.03$ | $0.057\pm 0.025$ | $0.06\pm 0.03$
$q_{\xi}$ | — | — | — | — | $0.33\pm 0.16$ | $0.28\pm 0.13$ | $0.3\pm 0.2$ | $0.26\pm 0.19$
$q_{\mathrm{f,OGLE}}$ | — | — | — | — | $0.14\pm 0.07$ | $0.11\pm 0.07$ | $0.18\pm 0.08$ | $0.17\pm 0.05$
$q_{\mathrm{f,OGLE,V}}$ | — | — | — | — | $0.15\pm 0.08$ | $0.14\pm 0.08$ | $0.18\pm 0.08$ | $0.17\pm 0.06$
$q_{\mathrm{f,Spitzer}}$ | — | — | — | — | $1.1\pm 0.5$ | $1.2\pm 0.3$ | $0.5\pm 0.3$ | $0.6\pm 0.2$
$\phi_{\xi}^{b}$ (deg) | $-80\pm 110$ | $-74\pm 25$ | $107\pm 12$ | $107\pm 13$ | $160\pm 6$ | $156\pm 4$ | $162\pm 5$ | $160\pm 5$
$\theta_{\xi}^{b}$ (deg) | $97\pm 5$ | $91\pm 4$ | $86\pm 4$ | $81\pm 5$ | $73.8\pm 2.5$ | $74.3\pm 2.0$ | $77.6\pm 1.4$ | $77.2\pm 1.7$
$i_{\xi}^{b}$ (deg) | $82\pm 5$ | $77.7\pm 2.5$ | $101\pm 3$ | $97\pm 4$ | $106\pm 4$ | $106\pm 3$ | $102\pm 4$ | $99\pm 4$
$\xi_{\mathrm{E}}^{b}$ | $0.063\pm 0.009$ | $0.064\pm 0.011$ | $0.068\pm 0.014$ | $0.066\pm 0.012$ | $0.18\pm 0.05$ | $0.19\pm 0.04$ | $0.17\pm 0.05$ | $0.16\pm 0.05$
Blend fraction | $0.19\pm 0.05$ | $0.24\pm 0.04$ | $0.21\pm 0.05$ | $0.13\pm 0.05$ | $0.03\pm 0.14$ | $0.03\pm 0.12$ | $0.11\pm 0.1$ | $0.13\pm 0.09$
$I_{\rm{s}}$ | $18.61\pm 0.06$ | $18.69\pm 0.06$ | $18.64\pm 0.06$ | $18.53\pm 0.06$ | $18.42\pm 0.16$ | $18.38\pm 0.13$ | $18.75\pm 0.15$ | $18.55\pm 0.12$
$\chi^{2}$/dof | 1219.38/1126 | 1193.15/1126 | 1191.66/1126 | 1219.89/1126 | 1148.25/1122 | 1145.36/1122 | 1148.09/1122 | 1143.39/1122

  * •

NOTE.
a Defined in HJD-2450000. In all fittings, we set the reference time to
$t_{0,{\rm par}}=t_{\rm ref}=2457200$.

b These are derived based on Equations (11).

###  3.3 1L2S Xallarap + Parallax Model

In order to add a luminous secondary source, at least two new parameters are
introduced into the modeling: the mass ratio and the wavelength-dependent flux
ratio of the secondary source to the primary source, denoted as $q_{\xi}$ and
$q_{\rm{f,}\lambda}$, respectively. Note that the flux ratio parameter is
different for different filters. For this modeling, we have included the
$I$-band and $V$-band data of the OGLE-IV survey and the Spitzer data, so
three flux ratio parameters are included.

The microlensing magnification from this 1L2S model is given by

$A_{2\rm{s,}\lambda}(u_{1},u_{2})=\frac{A(u_{1})+q_{\rm{f,}\lambda}A(u_{2})}{1+q_{\rm{f,}\lambda}}.$
(15)

Here $u_{1}$ ($u_{2}$) is the projected separation between the lens and the
primary (secondary) source. The position of the primary source is determined
in the same way as in Section 3.2. For the secondary source, its positional
offset relative to the rectilinear motion is given by

$\displaystyle\Delta\boldsymbol{S}_{2}=(t-t_{\rm{ref}})\boldsymbol{v}_{\rm{ref,1}}+\boldsymbol{S}_{1}(t_{\rm
ref})-\frac{1}{q_{\xi}}\boldsymbol{S}_{1}(t)$ (16)

Here the subscripts 1 and 2 denote the primary and secondary source,
respectively. Similar to that for the primary source (Equation 13), the first
two terms transform the trajectory from the barycentric frame to the frame
centred on the primary source. The last term calculates the orbit of the
secondary source, which is $1/q_{\xi}$ times larger than that of the primary
source and on the opposite side of the barycentre. The parallax effect is
included in the same way as in the 1L1S case.

As shown in Figure 4 and further detailed in Table 1, the best-fit 1L2S
Xallarap+Parallax model provides a better match to the data by
$\Delta\chi^{2}\simeq 50$, with _Spitzer_ data contributing the majority of
this improvement. Although this makes the 1L2S model suspicious at first
sight, particularly given that some fraction of the _Spitzer_ data is known to
contain systematics (e.g., Zhu et al., 2017a; Koshimoto & Bennett, 2020),
there are several pieces of evidence suggesting that the best-fit 1L2S
solution is plausible. First, the 1L2S solution is inevitable according to the
best-fit 1L1S solution, as has been explained in Section 3.2. Second, the
redder secondary source, as inferred from the binary flux ratios in different
bandpasses, is consistent with the estimated binary mass ratio directly
inferred from the light curve fitting, where more details are described in
4.1, so this binary solution is astrophysically plausible.

Some of the derived parameters in the 1L2S solution appear different from
those in the 1L1S solution. In particular, the orbital period of the source
binary, $P_{\xi}$, changes from $\sim 40\,$d to 60–80 d, and the xallarap
parameter, $\xi_{\rm{E}}$, changes from $\sim 0.06$ to $\sim 0.2$. The change
in $\xi_{\rm{E}}$ is due to the different definitions of this parameter in the
two models. In the 1L1S model, $\xi_{\rm{E}}$ is defined by the motion of the
centre of light (i.e., photocentre) relative to the centre of mass of the
binary, whereas in the 1L2S model the same parameter refers to the motion of
the primary source relative to the centre of mass of the binary. The
photocentre is closer to the barycentre than the primary source by a factor of
$(q_{\xi}-q_{\rm f})/(q_{\xi}+q_{\xi}q_{\rm f})$. Using the values in Table 1,
we find this factor to be $\sim 0.4$. Together with the statistical
uncertainties, this can basically explain the change of $\xi_{\rm E}$. The
change in the binary orbit period is related to the morphology of the xallarap
signal. As shown in the left panel of Figure 4, the xallarap signal appears to
be wave-like once the standard 1L1S model is subtracted. This wave-like
feature can be approximated by the summation of two lower-frequency (thus
longer period) terms, as illustrated in Figure 5. That figure shows the flux
variations in the lensing process while the contributions from the two source
stars are shown separately. Here the flux variations are given by

$\Delta F_{1}=\frac{f_{s}}{1+q_{f}}(A_{1}-1),\quad\Delta
F_{2}=\frac{q_{f}f_{s}}{1+q_{f}}(A_{2}-1)$ (17)

for S1 and S2, respectively.

Figure 5:  The flux variations of the two source components, S1 and S2, in the
1L2S model. The red and blue curves show the pure flux curve of the S1 and S2,
respectively, and the black curve shows the combined result. For illustration
purposes, only the $(+,+)$ solution is shown, and the flux variations of S2
have been amplified by a factor of seven. The inset shows the trajectories of
the two sources with respect to the lens object. The two sources move from the
left to the right.

Table 1 provides the best-fit parameters of the four degenerate solutions,
originating from the four-fold parallax degeneracy. These solutions have
comparable values of $\chi^{2}$ and generally consistent values of xallarap
parameters. In particular, the source binary is constrained to be nearly edge-
on, which is a preferred configuration if one is to verify the 1L2S solution
with radial velocity observations (see Section 5.2).

##  4 Physical Interpretation

###  4.1 Source Properties

The angular Einstein radius can be determined by combining the xallarap
amplitude, $\xi_{\rm E}$, and the semi-major axis of the primary source,
$a_{\rm S}$

$\theta_{\rm{E}}=\frac{a_{\rm{S}}}{D_{\rm{S}}\xi_{\rm{E}}},$ (18)

and $a_{\rm S}$ is related to the mass of the source binary via Kepler’s third
law

$\frac{a_{\rm S}}{\rm AU}=\frac{q}{(1+q)^{2/3}}\left(\frac{M_{\rm
S}}{M_{\odot}}\right)^{1/3}\left(\frac{P_{\xi}}{\rm yr}\right)^{2/3}$ (19)

Here $M_{\rm S}$ is the masses of the primary and $q_{\xi}$ is the mass ratio.
The values of these masses are key in determining $\theta_{\rm E}$, as the
xallarap amplitude and the source binary period are both measured in the
xallarap modeling.

The mass of the source binary is estimated via the isochrone fitting. To
perform this, we first derive the intrinsic colour and magnitude of both
sources, following the general method of Yoo et al. (2004). We first obtain
the observed colours and magnitudes of both sources through a linear
regression based on the 1L2S modeling of OGLE-2015-BLG-0845 . This yields

$(V-I,I)_{\rm{S1}}=(1.46\pm 0.04,18.65\pm 0.13)$ (20)

for the primary source and

$(V-I,I)_{\rm{S2}}=(1.4\pm 0.3,20.6\pm 0.5)$ (21)

for the secondary source. The above values are derived from the $(+,-)$
solution, i.e., the minimum $\chi^{2}$ solution, but the difference among the
four degenerate solutions in these numbers is statistically negligible.

The centroid of the red clumps is determined to be

$(V-I,I)_{\rm{RC}}=(1.972\pm 0.012,15.82\pm 0.07)$ (22)

following the method of Nataf et al. (2013). At the location of
OGLE-2015-BLG-0845 , the intrinsic colour and dereddened magnitude of the red
clump are

$(V-I,I)_{\rm{RC,0}}=(1.06\pm 0.06,14.46\pm 0.04)$ (23)

adapted from Nataf et al. (2013); Bensby et al. (2013), respectively. The
reddening and extinction are therefore determined to be

$(E_{\rm{S}}(V-I),A_{\rm{S}}(I))=(0.91\pm 0.06,1.37\pm 0.08).$ (24)

Together with the observed colours and magnitudes of both sources (Equations
20 and 21) and by assuming the source is at $D_{\rm S}=8.2\pm 1.4\,$kpc, we
determine the intrinsic colour and absolute magnitude of both sources to be

$(V-I,M_{\rm{I}})_{\rm{S1,0}}=(0.54\pm 0.07,2.7\pm 0.3)$ (25)

and

$(V-I,M_{\rm{I}})_{\rm{S2,0}}=(0.5\pm 0.4,4.7\pm 0.5)$ (26)

respectively. For completeness, we have included in Table 2 the measured and
derived properties of the source binary for all four degenerate solutions.

The masses of the source stars are estimated based on their positions on the
colour–magnitude diagram (CMD) with the isochrones package (Morton, 2015).
This package adopts the MIST (Dotter, 2016) isochrone and computes the log-
likelihood of five model parameters, namely the equivalent evolutionary phase
of both stars, the metallicity, age, and distance of the source system. Given
the limited observational constraints, informative priors have been adopted.
Specifically, the stellar metallicity is assumed to follow the observed
distribution of the bulge microlensing stars from Bensby et al. (2017), and
the distance is limited to the range 6.2–10.2 kpc. The posterior distribution
is sampled by emcee (Foreman-Mackey et al., 2013). The resulting constraints
on the stellar masses are shown as the black contours in Figure 6. For
illustration purposes, we have shown four typical isochrones with different
ages and metallicities that match the positions of the source stars,
especially S1, in Figure 7. These isochrones are selected in the following
way. First, four representative metallicity values, $(-1.0,-0.5,0.0,0.25)$,
are selected based on the prior metallicity distribution function (Bensby et
al., 2017). Then we determined the age of each isochrone so that it could
match the colour and magnitude of S1 well. This process yields an age range
from $2.0$ Gyr to $7.9$ Gyr, and older ages always correspond to lower values
of metallicity, which is also consistent with the observed age–metallicity
trend of bulge stars (Bensby et al., 2017).

The mass estimates from isochrone fitting also provide us with a way to verify
the light curve modeling results. Figure 6 illustrates the masses of binary
sources and the mass ratio constraints from light curve modeling (Table 1) for
all four degenerate solutions. The isochrone masses generally fall within the
1-$\sigma$ region of the mass ratio constraint, suggesting that these two sets
of estimates are broadly consistent. Given a binary consisting of $\sim 1.2$
and $\sim 0.9M_{\odot}$ stars, the flux ratio in _Spitzer_ $3.6\mu$m band is
around 0.4. This is in agreement with the flux ratio constraints for the
$(-,+)$ and $(-,-)$ solutions and slightly disfavors (by $<3\sigma$) the
$(+,+)$ and $(+,-)$ solutions.

With the mass estimates of source binary stars, the semi-major axis of the
source S1 is determined to be 0.4–0.5 au, according to Kepler’s third law.
Together with the amplitude of the xallarap parameter, which is measured from
the light curve modeling, we find the angular Einstein radius to be

$\theta_{\rm E}=0.125\left(\frac{a_{\rm S}}{0.2\,{\rm
au}}\right)\left(\frac{D_{\rm S}}{8\,\rm kpc}\right)^{-1}\left(\frac{\xi_{\rm
E}}{0.2}\right)^{-1}\rm mas.$ (27)

The exact values and uncertainties of these parameters are included in Table 2
for all four degenerate solutions.

Figure 6:  Comparison between the binary mass from the isochrone fitting and the mass ratio $q$ from light curve fitting. The red line represents the $q=1$ line. The stellar masses determined by isochrone fitting are shown in black contours, with the solid and dashed contours showing the 1 and 2 sigma regions. The light blue shadow regions mark the mass ratio 90% upper limit of each degenerate model.  Figure 7:  The Colour-Magnitude diagram (CMD) of the four solutions with 1L2S xallarap + parallax model for OGLE-2015-BLG-0845 as well as the isochrones. The four solutions are shown in different markers, i.e., the circle, square, pentagon, and diamond stand for solutions $(+,+)$, $(+,-)$, $(-,+)$, and $(-,-)$, respectively.  Table 2: Physical properties of the source binary and lens object of OGLE-2015-BLG-0845 . Parameters | $(+,+)$ | $(+,-)$ | $(-,+)$ | $(-,-)$
---|---|---|---|---
$(V-I)_{\rm{S1,0}}$ (mag) | $0.55\pm 0.08$ | $0.55\pm 0.07$ | $0.54\pm 0.07$ | $0.55\pm 0.08$
$M_{I,\rm{S1}}$ (mag) | $2.6\pm 0.3$ | $2.6\pm 0.3$ | $2.8\pm 0.3$ | $2.7\pm 0.3$
$(V-I)_{\rm{S2,0}}$ (mag) | $0.4\pm 0.4$ | $0.4\pm 0.4$ | $0.5\pm 0.3$ | $0.5\pm 0.4$
$M_{I,\rm{S2}}$ (mag) | $4.8\pm 0.6$ | $5.1\pm 0.7$ | $4.7\pm 0.5$ | $5.0\pm 0.6$
$M_{\rm{S1}}$ ($M_{\sun}$) | $1.22\pm 0.18$ | $1.23\pm 0.18$ | $1.20\pm 0.18$ | $1.20\pm 0.18$
$M_{\rm{S2}}$ ($M_{\sun}$) | $0.91\pm 0.12$ | $0.88\pm 0.13$ | $0.9\pm 0.11$ | $0.87\pm 0.12$
$a_{\rm{S}}$ (AU) | $0.19\pm 0.014$ | $0.189\pm 0.014$ | $0.181\pm 0.015$ | $0.177\pm 0.017$
$\theta_{\rm{E}}$ (mas) | $0.10^{+0.03}_{-0.02}$ | $0.091^{+0.022}_{-0.017}$ | $0.11^{+0.03}_{-0.02}$ | $0.11^{+0.03}_{-0.02}$
$\mu_{\rm{rel,geo}}$ (mas/yr) | $0.84^{+0.21}_{-0.16}$ | $0.78^{+0.18}_{-0.15}$ | $0.93^{+0.25}_{-0.19}$ | $0.90^{+0.25}_{-0.19}$
$M_{\rm L}$ | $0.15^{+0.02}_{-0.02}$ | $0.09^{+0.01}_{-0.01}$ | $0.13^{+0.02}_{-0.02}$ | $0.19^{+0.03}_{-0.03}$
$D_{\rm L}$ (kpc) | $7.68^{+0.06}_{-0.06}$ | $7.54^{+0.07}_{-0.07}$ | $7.46^{+0.10}_{-0.11}$ | $7.72^{+0.06}_{-0.08}$

###  4.2 Mass and Distance of the Lens

Given the constraints on both $\theta_{\rm E}$ and $\pi_{\rm E}$, we now
proceed to constrain the mass and distance of the lens object. The mass is
given by

$M_{\rm L}=\frac{\theta_{\rm E}}{\kappa\pi_{\rm E}}\approx
0.15\left(\frac{\theta_{\rm E}}{0.125\,\rm mas}\right)\left(\frac{\pi_{\rm
E}}{0.1}\right)^{-1}M_{\odot}.$ (28)

Similarly, the lens–source relative parallax is given by

$\pi_{\rm rel}=\pi_{\rm E}\theta_{\rm E}=0.0125\left(\frac{\theta_{\rm
E}}{0.125\,\rm mas}\right)\left(\frac{\pi_{\rm E}}{0.1}\right)\,{\rm mas}.$
(29)

This value corresponds to a typical bulge lens. Adopting a source distance of
$D_{\rm S}=8.2\,$kpc, we find the lens distance to be $D_{\rm L}\approx 7.4$
kpc. Therefore, the lens object of OGLE-2015-BLG-0845 is a low-mass M-dwarf
located in the bulge. The exact values and associated uncertainties of the
derived mass and distance for all four degenerate solutions are also included
in Table 2.

Figure 8 illustrates the constraints on the mass and distance of the lens
object for the four degenerate solutions. In addition to the constraints from
the angular Einstein radius and the microlensing parallax, we have also shown
the upper limit from lens flux. This limit is derived by taking the 90% upper
limit on the $I$-band blending flux and assuming it is due entirely to the
lens object. We have adopted the stellar mass–magnitude relations of Pecaut et
al. (2012) and Pecaut & Mamajek (2013) and the extinction relation of Bennett
et al. (2015),

$A_{\rm{L}}=(1-\exp(-D_{\rm{L}}/\tau_{\text{dust}}))/(1-\exp(-D_{\rm{S}}/\tau_{\text{dust}}))A_{\rm{S}}.$
(30)

Here $\tau_{\rm{dust}}=0.1\rm{kpc}/\sin(b)$ is the scale length of the dust
towards the galactic bulge, with $b$ the Galactic latitude of the source
system. For all four solutions, the lens mass and distance, derived from the
parallax and angular Einstein radius measurements, are well within the allowed
region by the lens flux upper limit.

Figure 8:  The constraints on the lens mass and distance for
OGLE-2015-BLG-0845 , from $\pi_{\rm{E}}$, $\theta_{\rm{E}}$, and lens flux
$f_{\rm{b}}$. The four degenerate solutions, $(+,+)$, $(+,-)$, $(-,+)$, and
$(-,-)$, are shown in different panels. The black solid line in each panel is
the upper limit from the lens flux. The coloured solid line and dashed line
with a shaded region around them are the constraints from $\pi_{\rm{E}}$ and
$\theta_{\rm{E}}$, respectively.

##  5 Discussion

###  5.1 Impact of the Eccentricity

In the 1L2S model with the xallarap effect, we have assumed a circular orbit
for the source binary. However, observations have shown that binaries with
eccentric orbits are fairly common (e.g., Duchêne & Kraus, 2013). In the
period range of 10–1000 d, which is relevant for xallarap detection, about
half of the binaries have eccentricities above $\sim$0.3 (Raghavan et al.,
2010). Therefore, it is necessary to address the impact of eccentric orbits on
the derived source and lens properties.

We adopt an eccentric orbit 1L2S model to fit the data. Together with the
orbital eccentricity $e$, the argument of periapsis $\omega$ is also
introduced to describe the eccentric motion. To avoid over-fit of the weak
signal, we fix the orbital eccentricity to values in the range 0.1–0.8 and
search the other parameters to minimize the model $\chi^{2}$. Compared to the
circular model, the best $\chi^{2}$ values of the eccentric orbits are only
marginally improved. At $e=0.3$, the improvement is $\Delta\chi^{2}\approx 5$.
More eccentric ($e\gtrsim 0.4$) orbits have even worse $\chi^{2}$ values, and
such orbits are disfavored a priori at an orbital period of $\sim 100\,$d.

With the introduction of eccentric orbits, model parameters that are relevant
to the physical properties of the source and lens objects are not changed
substantially. The changes in the flux ratios in different bands and in the
mass ratio are all within the uncertainties of the corresponding parameters
given in Table 1. With the increasing eccentricity, the xallarap parameter and
the orbital period of the source binary may both increase by up to 30–50%, but
because these two parameters have opposite effects on the angular Einstein
radius, this latter parameter is not changed as much. In the end, the mass and
distance of the lens object are only changed by $\lesssim 20\%$, which is
small compared to the difference between different degenerate solutions. It
does not change the conclusion that the lens is a low-mass M-dwarf in the
Bulge.

The inclination remains nearly unchanged with $e\leq 0.3$, but may reach up to
$128^{\circ}$ at $e=0.8$. This has some implications for the radial velocity
(RV) observations of the source binary, which will be discussed in the next
section.

###  5.2 Source Binary Confirmation

With a baseline magnitude of $I\approx 18.5$ and almost zero blending, the
source binary of OGLE-2015-BLG-0845 is potentially accessible by spectroscopic
observations from the ground. Such observations can not only confirm but also
further refine the derived properties of the source binary (and thus the lens
object).

We observed the source with the MagE spectrograph (Marshall et al., 2008) on
the 6.5 m Magellan Baade Telescope at Las Campanas Observatory on July 2,
2023. Three exposures, each with 20 min, were taken, and the resulting
spectrum has a signal-to-noise ratio (S/N) per resolution of about five. Given
the flux ratio of $\sim 0.15$, we expect to see only a single component in the
spectra. We compared the observed spectrum with synthetic spectra generated
with a stellar atmosphere grid MARCS (Gustafsson et al., 2008) and a radiative
transfer code turbospectrum (Plez, 2012) integrated together by iSpec (Blanco-
Cuaresma, 2019). The observed spectrum can be fitted with the spectra of late
F-type subgiants and dwarfs reasonably well, while it disfavors the early
F-type as well as the typical G-type stars. This is in agreement with the CMD
fitting result.

Future spectroscopic observations are able to measure the RV variations of the
source binary (e.g., Ryu et al., 2024). Given the inferred parameters, the RV
semi-amplitude is

$K=28\ {\rm{km/s}}\frac{\sin
i_{\xi}}{\sqrt{1-e^{2}}}\left(\frac{M_{\rm{S2}}}{0.9M_{\odot}}\right)\left(\frac{M_{\rm{S1}}+M_{\rm{S2}}}{2.1M_{\odot}}\right)^{-2/3}\left(\frac{P_{\xi}}{75\
\rm{day}}\right)^{-1/3}.$ (31)

This is large enough for ground-based 10-m telescopes to achieve on a late
F-type star. These RV observations will be able to directly measure the
orbital eccentricity, which is currently not constrained in the model (Section
5.1). The combination of RV and xallarap can provide useful constraints on the
angular Einstein radius, as has long been pointed out by Han & Gould (1997).

As explained in Section 3.3, the 1L2S model is favored mostly by the _Spitzer_
data. It has been argued that the _Spitzer_ data might suffer from systematics
at some level (e.g., Zhu et al., 2017a; Koshimoto & Bennett, 2020), so the
verification of the binary source via the RV method also provides an
opportunity to further investigate the issues in the _Spitzer_ data in this
particular event.

###  5.3 The Detectability of the Xallarap Effect

As demonstrated by the event OGLE-2015-BLG-0845 , the xallarap effect can
provide additional information regarding the $\theta_{\rm{E}}$ parameter and
thus has the capability to determine $\theta_{\rm{E}}$ by utilizing radial
velocity measurement or isochrone fitting. Therefore, one wonders how often
the xallarap effect appears and is detectable. In particular, under what
conditions can the xallarap effect be distinguished from the annual parallax
effect?

To answer the above question, we perform the following simplified simulations.
We adopt the survey strategy and performance of Karolinski & Zhu (2020) and
set the $t_{0}$ is at the centre of an observing season. The values of
$t_{\rm{E}}$ and $u_{0}$ were uniformly sampled in the ranges 10–150 days and
0.1–1.0, respectively, which roughly follow the observed distribution of Mróz
et al. (2019). The angular Einstein radius is assumed to be 0.55 mas,
corresponding to an M-dwarf lens located at 4 kpc. The source binary consists
of a primary star of mass $1\,M_{\odot}$ and a mass ratio of 0.3. This mass
ratio corresponds to a luminosity ratio of $\lesssim 1\%$, sufficiently small
that we can disregard the light of the secondary source in the light curve
modeling. The orbital period of the source binary is sampled from a log-flat
distribution between 0.5–10 times $t_{\rm{E}}$, and the orientation of the
binary orbit and positions of the source stars are randomized. For each of the
simulated light curves, We have performed both parallax and xallarap
modelings, and the xallarap signal is considered to be detected if the best-
fit $\Delta\chi^{2}$ between parallax and xallarap models is $>50$.

The detection efficiency of the xallarap effect is shown in Figure 9 as a
function of the binary orbital period. Source binaries with orbital periods
$P\lesssim 2t_{\rm E}$ are more likely to be detected, as their xallarap
signals are difficult to be confused with annual parallax signals. Given the
fixed sampling cadence, the detection efficiency therefore drops for shorter
timescale events, resulting in an overall frequency of around 1% once the
binary period distribution is taken into account. This fraction increases for
higher-cadence surveys and events with longer timescales. If we take roughly
the same $t_{\rm E}$ distribution with the events in Poindexter et al. (2005),
the detectability of the xallarap effect would be about 18%, consistent with
the result of Poindexter et al. (2005) that about 23% of the events with
$t_{\rm E}\gtrsim 70$ d are affected by xallarap effect. Because such long-
timescale events are of particular interest in the search for dark lenses
(e.g., Lam et al., 2020), the impact of the xallarap effect and its usage in
the lens mass determination should be evaluated seriously.

Figure 9:  The change of the xallarap effect detection efficiency with regard
to $p_{\xi}/t_{\rm E}$ ratio when $u_{0}=0.1$. The solid line, dashed line,
and dash-dotted line represent the total detection efficiency, as well as the
detection efficiency when $t_{\rm E}$ is 150 or 20 days, respectively.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
(grant No. 12133005 & 12173021) and the CASSACA grant CCJRF2105. We also
acknowledge the Tsinghua Astrophysics High-Performance Computing platform for
providing computational and data storage resources.

## Data Availability

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

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## Affiliations

1Department of Astronomy, Tsinghua University, Beijing 10084, China
2Max-Planck-Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany
3Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus,
OH 43210, USA
4Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478
Warszawa, Poland
5Department of Earth and Space Science, Graduate School of Science, Osaka
University, Toyonaka, Osaka 560-0043, Japan
6Department of Particle Physics and Astrophysics, Weizmann Institute of
Science, Rehovot 7610001, Israel.
7IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA
91125, USA
8Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge,
MA 02138, USA
9Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA
10Department of Physics and Astronomy, Texas Tech University, 1200 Memorial
Circle, Lubbock, TX 79409, USA
11Department of Physics and Astronomy, Michigan State University, 567 Wilson
Road, East Lansing, MI 48824, USA
12Department of Physics, University of Warwick, Gibbet Hill Road, Coventry,
CV4 7AL, UK
13Institute for Space-Earth Environmental Research, Nagoya University, Nagoya
464-8601, Japan
14Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
15Department of Astronomy, University of Maryland, College Park, MD 20742, USA
16Institute of Natural and Mathematical Sciences, Massey University, Auckland
0745, New Zealand
17Department of Earth and Planetary Science, Graduate School of Science, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
18Instituto de Astrofísica de Canarias, Vía Láctea s/n, E-38205 La Laguna,
Tenerife, Spain
19Institute of Astronomy, Graduate School of Science, The University of Tokyo,
2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan
20Department of Physics, The Catholic University of America, Washington, DC
20064, USA
21Institute of Space and Astronautical Science, Japan Aerospace Exploration
Agency, 3-1-1 Yoshinodai, Chuo, Sagamihara, Kanagawa 252-5210, Japan
22Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98
bis bd Arago, 75014 Paris, France
23Department of Physics, University of Auckland, Private Bag 92019, Auckland,
New Zealand
24University of Canterbury Mt. John Observatory, P.O. Box 56, Lake Tekapo
8770, New Zealand

##  Appendix A Binary Lens Interpretation

An additional companion to the lens object may also lead to distortions in the
microlensing light curve (Mao & Paczynski, 1991; Gould & Loeb, 1992). As long
as the source star stays relatively away from the caustics of the binary lens,
such distortions are not prominent features and may resemble those produced by
the xallarap effect (e.g., Rota et al., 2021; Yang et al., 2024). We therefore
carry out 2L1S modeling to make sure the detected signal is indeed caused by
the xallarap effect.

In this 2L1S modeling, we exclude the _Spitzer_ data but include the MOA data.
The latter has a higher cadence, which can be useful in constraining the
short-timescale signals arising from the binary lens. To describe the 2L1S
model, four new parameters are introduced, namely $\rho$, $\alpha$, $s$, and
$q$. Here, $\rho$ represents the angular size of the source normalized to the
angular Einstein radius, $\alpha$ is the angle between the source trajectory
and the binary axis, and $s$ and $q$ are the projected separation and mass
ratio between the two lens components, respectively. Following a thorough grid
search and refined MCMC sampling of the posterior distribution, we only
identify one binary lens solution that matches the observed data the best.
This best-fit 2L1S model is shown in Figure 10, and the parameter values are
given in Table 3 for the purpose of completeness. As shown in Figure 10, the
best-fit 2L1S model is not able to explain fully the deviations in the ground-
based data, and it is worse than the 1L2S xallarap model by $\Delta\chi^{2}$
of $\sim 130$. We therefore exclude the binary lens model as a plausible
solution for OGLE-2015-BLG-0845 .

Table 3: Parameters of 1L2S xallarap + parallax solutions for
OGLE-2015-BLG-0845

Parameters | $(+,+)$
---|---
$t_{0}$ (HJD′) | $7199.3746\pm 0.0065$
$u_{0}$ | $0.06632\pm 0.00079$
$t_{\mathrm{E}}$ (days) | $31.80\pm 0.32$
$\rho$ | $7207.56\pm 0.52$
$\alpha$ | $0.224\pm 0.015$
$s$ | $0.136\pm 0.013$
$q$ | $0.139\pm 0.013$
Blend | $-0.259\pm 0.023$
$\chi^{2}$/dof | 2761.35/2836

  * •

NOTE. HJD′=HJD-2450000.

Figure 10:  The comparison between the 2L1S model and the 1L2S xallarap +
parallax model. The figure description is similar to that of Figure 4. In the
figure, the MOA data is binned with a one-day bin size, while the fitting
process does not involve any binning. The inset in the upper panel shows the
trajectory of the source as well as the caustic curve of the binary lens and
the position of the lens.