# Evaluation of Deep Neural Network Domain Adaptation Techniques for Image
Recognition
††thanks: *Equal contribution.

Venkata Santosh Sai Ramireddy Muthireddy* Hochschule Bonn-Rhein-Sieg
Bonn, Germany
<EMAIL_ADDRESS>Alan Preciado-Grijalva* Hochschule
Bonn-Rhein-Sieg
Bonn, Germany
<EMAIL_ADDRESS>
###### Abstract

It has been well proved that deep networks are efficient at extracting
features from a given (source) labeled dataset. However, it is not always the
case that they can generalize well to other (target) datasets which very often
have a different underlying distribution. In this report, we evaluate four
different domain adaptation techniques for image classification tasks:
DeepCORAL, DeepDomainConfusion, CDAN and CDAN+E. These techniques are
unsupervised given that the target dataset dopes not carry any labels during
training phase. We evaluate model performance on the office-31 dataset. A link
to the github repository of this report can be found here:
https://github.com/agrija9/Deep-Unsupervised-Domain-Adaptation.

###### Index Terms:

domain adaptation, unsupervised learning, convolutional neural networks,
transfer learning

## Introduction

Deep neural networks based methods have been producing state-of-the-art (SOTA)
results for many problems in machine learning and computer vision. These
methods require large amount of training and testing data to achieve the
expected result. Although the model is trained with large datasets sometimes
it will not generalize learned knowledge to new environment and datasets. This
is because deep learning algorithms assume that training and testing data is
drawn from independent and identical distributions (i.i.d.). However this
assumption rarely holds true, as there will be a shift in data distributions
across different domains this is explained in Fig. 1. This domain shift
between source and target datasets will make deep neural networks produce
wrong predictions on the target dataset. So training of a deep neural network
with source and target datasets which reduces the domain shift between
distribution of datasets is called as domain adaptation [1].

There are different types of domain adaptation (DA) techniques, a few of them
are Unsupervised DA [2][3], Semi Supervised DA[4], Weakly Supervised DA[5],
One Shot DA[6], Few Shot DA[7], Zero Shot DA[8]. In this report, we discussed
on Unsupervised DA techniques. Unsupervised DA is called so because unlabeled
target dataset is used during training to reduce the domain shift.

Further Unsupervised DA have different types based on the technique used to
reduce the domain shift between the source and target datasets. They are:

  * •

Adversarial Methods [2]

  * •

Distance-based Methods [3] [9]

  * •

Incremental Methods [10]

  * •

Optimal Transport [11]

  * •

Other Methods [12]

Figure 1: Domain adaptation in the true data space: Expectation vs.
Reality.[13]

In this report, techniques evaluated are selected from Adversarial and
Distance-based methods. In Adversarial methods, Conditional adversarial domain
adaptation (CDAN) and CDAN with Entropy conditioning (CDAN+E) [2] are
selected. In Distance-based methods, Deep domain confusion: Maximizing for
domain invariance (DDC) [9] and Deep coral: Correlation alignment for deep
domain adaptation [3] are selected. Application of domain adaptation
techniques can be used in different types of domain shift explained in Fig. 6
and application in dataset to dataset domain shift is selected for evaluation.
In this report evaluation of different domain adaptation techniques are
benchmarked on office-31 dataset [14].

Figure 2: Applications to different types of domain shift [15]

## Domain Adaptation Methods

### DeepCORAL

DeepCORAL [3] is a domain adaptation method based on a previously proposed
method called Correlation Alignment (CORAL) [16].

CORAL attempts to increase performance in domain shift by aligning the second
order statistics between both data distributions. This method consists of 1)
feature extraction, 2) apply a linear transformation and 3) implement an SVM
classifier.

DeepCORAL extends the capacities of CORAL by learning a non-linear mapping
using convolutional neural networks and a differentiable loss function that
attempts to minimize the Frobenius norm between the feature covariance
matrices of source and target distributions. This method works in an
unsupervised way where the target dataset has no labels.

Recall that the main goal is to learn representations that allow to achieve
good performance in both data distributions (image recognition, for example).
To achieve this, DeepCORAL intializes the network parameters from pre-trained
AlexNet model while at the same time minimizing the loss mentioned above
(a.k.a CORAL loss). The architecture of DeepCORAL is shown in the figure
below.

Figure 3: DeepCORAL neural network architecture.

Note that DeepCORAL takes the last fully connected layer of AlexNet (fc8) and
computes a classification and CORAL loss.

#### CORAL loss

This loss is defined for a single layer feature map. In this case for fc8,
although not mentioned, we believe the authors run a grid search to identify
the optimal layer within the network to compute the loss. This can have to do
with the task at hand.

To formalize the CORAL loss, suppose that we have source-domain training
examples $D_{s}=\\{x_{i}\\}$, where $x\in\mathbb{R}^{d}$ with labels
$L_{s}=\\{y_{i}\\}$, $i\in\\{1,...,L\\}$, and unlabeled data
$D_{T}=\\{u_{i}\\}$, $u\in\mathbb{R}^{d}$. Let $n_{s}$ and $n_{t}$ be the
number of source and target data available respectively. Recall that x and u
are the d-dimensional deep layer activations (fc8) $\phi{(I)}$ of an input
image $I$. Let $C_{S}$ and $C_{T}$ be the feature covariance matrices of these
activations for source and target domains respectively. The CORAL loss is
defined as the difference between these covariances (second order statistics):

$\textit{l}_{CORAL}=\frac{1}{4d^{2}}\|C_{S}-C_{T}\|_{F}^{2}$ (1)

where $\|\cdot\|_{F}^{2}$ refers to the squared matrix Frobenius norm. The
covariance matrices $C_{S}$, $C_{T}$ are defined as

$C_{S}=\frac{1}{n_{S}-1}\big{(}D_{S}^{T}D_{S}-\frac{1}{n_{S}}(1^{T}D_{S})^{T}(1^{T}D_{S})\big{)}$
(2)

$C_{T}=\frac{1}{n_{T}-1}\big{(}D_{T}^{T}D_{T}-\frac{1}{n_{T}}(1^{T}D_{T})^{T}(1^{T}D_{T})\big{)}$
(3)

where 1 is a column vector with all elements equal to 1.

The gradient of the CORAL loss can be calculated using the chain rule. Refer
to the paper for the analytical expression.

In practice, we do batch learning and thus the covariances are computed over
batches. Moreover, the network parameters are shared between the two networks.

In the context of multi-class classification, the total loss that is used to
train is a joint term between with classification loss (cross-entropy) and
CORAL loss. The first one trying to optimize classification accuracy and the
second trying to learn features that can work well in both data domains. The
total loss is

$\textit{l}=l_{CLASS}+\sum_{i=1}^{t}{\lambda_{i}l_{CORAL}}$ (4)

here $t$ is the number of CORAL loss computed per layers. In this case it is a
single layer. The parameter $\lambda$ is introduced as a regularizer to help
reach an equilibrium between these two losses at the end of training.

### Deep Domain Confusion (DDC)

Deep domain confusion is another domain adaptation method that uses a deep
neural network to learn a non-linear transformation. It is similar to
DeepCORAL in the sense that it introduces a transfer loss to learn features
for both data domains. [9]

In their paper, Tzeng et al. show the concept of biased datasets and how
trained classifiers don’t always transfer well to target domains. Their
approach is to maximize a loss function that minimizes classification error
and maximizes domain confusion.

Figure 4: Deep Domain Confusion approach: Maximizing domain confusion can lead
to good performance in target domains.

To minimize the distance between data distributions, DDC uses a standard
distribution distance metric called Maximum Mean Discrepancy (MMD). In this
case, this distance is computed with respect to an activation layer
$\phi(\cdot)$ acting on source data points $x_{S}$ $\in$ $X_{S}$ and target
data points $x_{T}$ $\in$ $X_{T}$. The approximation to this distance is as
follows

$MMD(X_{S},X_{T})=\left|\frac{1}{\|X_{S}\|}\sum{\phi(x_{s})}-\frac{1}{\|X_{T}\|}\sum{\phi(x_{t})}\right|$
(5)

Training a model with MMD and a classification loss will result in a strong
classifier ready to transfer accross domains. Thus, the total loss is

$l=l_{C}(X_{L},y)+\lambda MMD^{2}(X_{S},X_{T})$ (6)

Here $l_{C}$ corresponds to a classification loss on the available source
labelled data $X_{L}$ and the ground truths $y$. The hyperparameter $\lambda$
acts in a similar way a regularizer as in DeepCORAL.

The architecture proposed in DDC is similar to DeepCORAL; it intializes its
weights with a pre-trained AlexNet and weights are shared between both data
domains. The difference is that it has an extra bottleneck layer fc adapt that
is meant to regularize the training of the source classifier and prevent
overfitting to particular nuances of the source distribution. The loss is
computed after this bottleneck layer. The next figure shows the architecture
of DDC.

Figure 5: Deep Domain Confusion neural network architecture. Both data domains
share weights, a bottleneck is introduced to the AlexNet architecture and the
total loss is computed after this layer. Figure 6: Deep Domain Confusion
approach: Maximizing domain confusion can lead to good performance in target
domains.

### CDAN

Adversarial domain adaptation [17] [18] [19] consists of domain adaptation and
adversarial learning which is very similar to Generative Adversarial Networks
(GANs) [20]. In adversarial domain adaptation, a deep neural network model
learns image representations by minimizing a classification loss and
simultaneously a domain discriminator learns how to distinguish between source
and target domain. There are however two pitfalls for adversarial domain
adaptation methods. In the first place they cannot handle datasets which have
complex multi modal distributions. And secondly we cannot rely on the
uncertain information difference between source and target data to model the
domain discriminator.

These two pitfalls can be solved by conditional domain adaptation techniques.
Conditional domain adaptation is made possible due to the research carried out
in Conditional Generative Adversarial Networks (CGANs) [21]. CGANs proved a
point that domain shift between distributions of real and synthetic images can
be reduced by conditioning network using discriminative information.

CDAN [2] is an adversarial domain adaptation technique motivated by CGANs and
it makes use of the discriminative information obtained from a deep classifier
network. The key to CDAN models is a new conditional domain discriminator
conditioned to the covariance of representations of domain-specific resources
and classifier predictions. Let us consider
$D_{s}=\\{(x_{i}^{s},y_{i}^{s})\\}_{i=1}^{n_{s}}$ is a source domain data with
$n_{s}$ labeled examples and $D_{t}=\\{(x_{j}^{t})\\}_{j=1}^{n_{t}}$ is a
target domain dataset with $n_{t}$ unlabeled examples. Source and target
datasets are drawn from $P(x^{s},y^{s})$ and $Q(x^{t},y^{t})$ which are not
i.i.d. i.e., $P\neq Q$. The aim of CDAN is to design a network $G:x\to y$ that
can reduce the domain shift between source and target datasets. $f=F(x)$ and
$g=G(x)$ are feature representation and classifier predictions made from the
network G respectively.

Considering all the things mentioned above, CDAN is formalized as a minmax
optimization problem. In minmax optimization, two terms are considered as
players, namely $E(G)$ corresponding to source classifier G and minimized to
guarantee lower source risk and $E(D,G)$ on G and the domain discriminator D
on cross domains. $E(D,G)$ is minimized over D and maximized over $f$ and $g$

$E(G)=\mathbb{E}_{(x_{i}^{s},y_{i}^{s})\sim D_{s}}L(G(x_{i}^{s}),y_{i}^{s})$
(7) $E(D,G)=-\mathbb{E}_{x_{i}^{s}\sim
D_{s}}log[D(f_{i}^{s},g_{i}^{s})]-\mathbb{E}_{x_{i}^{t}\sim
D_{t}}log[D(f_{j}^{t},g_{j}^{t})]$ (8)

where L(.) is the cross entropy loss. Now CDAN is described in minmax
optimization problem as

$\stackunder{min}{G}\ E(G)-\lambda E(D,G)\ and\ \stackunder{min}{G}\ E(D,G)$
(9)

where $\lambda$ is a hyper-parameter that balances the domain adversary and
source risk.

Figure 7: Architecture of CDAN for domain adaptation [2]

In Fig. 7 D is the domain discriminator conditioned using a multilinear map
$f\otimes g$, multilinear mapping is explained in Section 3.2 of [2]. Thus
CDAN enables domain adaptation over cross domains using minmax optimization.

### CDAN+E

CDAN+E is an variant of CDAN. In CDAN+E we will have additional involvement of
Entropy Conditioning which improves the difference between source and target
domains. In CDAN, the domain discriminator allots equal weight to each and
every example irrespective of examples with uncertain predictions from G. So
uncertainity corresponding to classifier predictions are quantified using
entropy criteria explained in Eq. 10.

$H(g)=-\sum_{c=1}^{C}g_{c}log\ g_{c}$ (10)

where C is number of classes and $g_{c}$ is probability of predictions.

Now training examples are prioritized by an entropy-aware weight given by

$w(H(g))=1+e^{-H(g)}$ (11)

So CDAN+E minmax optimization is formulated as

$\begin{multlined}\stackunder{min}{G}\mathbb{E}_{(x_{i}^{s},y_{i}^{s})\sim
D_{s}}L(G(x_{i}^{s}),y_{i}^{s})\\\ +\lambda(\mathbb{E}_{x_{i}\sim
D_{s}}w(H(g_{i}^{s}))log[D(T(h_{i}^{s}))]\\\ +\mathbb{E}_{x_{j}^{t}\sim
D_{t}}w(H(g_{j}^{t}))log[1-D(T(h_{j}^{t}))])\\\
\stackunder{max}{D}\mathbb{E}_{x_{i}\sim
D_{s}}w(H(g_{i}^{s}))log[D(T(h_{i}^{s}))]\\\ +\mathbb{E}_{x_{j}^{t}\sim
D_{t}}w(H(g_{j}^{t}))log[1-D(T(h_{j}^{t}))]\end{multlined}\stackunder{min}{G}\mathbb{E}_{(x_{i}^{s},y_{i}^{s})\sim
D_{s}}L(G(x_{i}^{s}),y_{i}^{s})\\\ +\lambda(\mathbb{E}_{x_{i}\sim
D_{s}}w(H(g_{i}^{s}))log[D(T(h_{i}^{s}))]\\\ +\mathbb{E}_{x_{j}^{t}\sim
D_{t}}w(H(g_{j}^{t}))log[1-D(T(h_{j}^{t}))])\\\
\stackunder{max}{D}\mathbb{E}_{x_{i}\sim
D_{s}}w(H(g_{i}^{s}))log[D(T(h_{i}^{s}))]\\\ +\mathbb{E}_{x_{j}^{t}\sim
D_{t}}w(H(g_{j}^{t}))log[1-D(T(h_{j}^{t}))]$ (12)

where h =(f,g) is a joint variable.

Thus CDAN+E encourages the certain predictions which are achieved through
entropy minimization principle [22].

## Experiments

In this section we describe the experiments performed with the different
metods on different data domains. Our implementations are written in Pytorch
and are located in the link to the repository given above. The experiments
were carried out using Google Colab GPU support (Testla P4).

### Office-31 dataset

Office-31 dataset was first introduced in [14]. It contains a total of 4652
images collected from three different sources. The three different sources
from which images are collected are online web (amazon), digital SLR (dslr)
camera and webcam. Office-31 dataset have 31 classes 111The 31 classes are:
backpack, bike, bike helmet, bookcase, bottle, calculator, desk chair, desk
lamp, computer, file cabinet, headphones, keyboard, laptop, letter tray,
mobile phone, monitor, mouse, mug, notebook, pen, phone, printer, projector,
puncher, ring binder, ruler, scissors, speaker, stapler, tape, and trash can.
in total.

  * •

Images from the web: These are collected from amazon website. We call this
dataset as amazon dataset (A). It has around 2800 images and with an average
of 90 images per each class. The resolution of the images are each 300x300
pixels.

  * •

Images from a dslr camera: These are captured from dslr camera in real
environments. We call this dataset as dslr dataset (D). It has total of 498
images and with an average of 16 images per each class. The resolution of the
images are 1000x1000 pixels.

  * •

Images from a webcam: These are collected from a webcam with lower resolution
and will have more noise. From now this is called as webcam dataset (W). It
has around 795 images and with an average of 25 images per each class. The
resolution of the images is HxW where H and W are varying between 400 and 600.

Figure 8: Sample of keyboard category images for all the three domains [14]

### DeepCORAL

The method is evaluated with the benchmark Office dataset described above. We
have carried out 6 different experiments for all the domain shifts available
(given the tree datastes cotained in Office).

CORAL loss is applied to the last fully connected layer (fc8). The weights of
this layer are initialized following a normal distribution
$\mathcal{N}(0,0.005)$ and the weights from the other layers are initialized
with the pre-trained AlexNet. We used Stochastic gradient descent (SGD).

DeepCORAL was trained using batches of 128 images (both domains), a fixed
learning rate of $10^{-3}$, weight decay set to $5x10^{-4}$ and momentum to
0.9. The $\lambda$ factor is set in such a way that both losses are
approximately the same. In practice, given $t$ epoch iterations, $\lambda$ is
$\frac{1}{t}$. Furthermore, during training only the source dataset is
labelled, this is how we compute the classification loss.

In Table 1 we show the obtained accuracies for each method for all 6
experiments. These values correspond to the image classification accuracy on
the target dataset. From here, we see that DeepCORAL obtained the best result
in 4 out of 6 experiments. This method outputs accuracy values in the range
between 38% to 89%. The fluctuation of accuracies between domains has to do
directly with the chosen source and target distributions. It is important to
note that the method does not output accuracies greater than 50% in most of
the cases. We’ve compared our values with the original paper and although
lower by a small percentage, these follow the same trend as the original
results.

For a better visualization of the method’s performance we have added to
visualize accuracy and loss as a function of epochs for the case Amazon
$\,\to\,$ Webcam. Figure 8a) shows an increase of almost 10% in recognition
accuracy when including CORAL loss. Figure 9a) shows the behavior of the loss
functions, note that they tend to become equal due to the lambda factor.

### Deep Domain Confusion

Training for this method is very similar to DeepCORAL. The main differences
are 1) it trains with MMD loss, 2) adds the bottleneck layer and 3) it uses a
schedule learning rate which decreases as a function of epochs. Fig. 8b) shows
that accuracy with MMD (or DDC) doesn’t have a very noticeable increase
compared to training without MMD, however, it seems to be more stable and
shows an steady increase. Due to time constrains, we were only able to run one
experiment for this method. However, it is consistent with the accuracy
reported on the original paper.

### CDAN and CDAN+E

Experiments for CDAN and CDAN+E are conducted with same set of hyperparameters
and the only difference between both is the transfer loss function that is
involved. Hyperparameters used in the experiments for CDAN and CDAN+E are:

  * •

Epochs 100

  * •

Source batch size 10

  * •

Target batch size 10

  * •

Learning rate 1e-3

  * •

Momentum 0.9

  * •

Weight decay 5e-4

Dataset used for all the experiments are office-31 dataset [14] Backbone used
in this experiment is pretrained AlexNet with a small modification. There is a
bottleneck layer introduced before last fully connected layer. Bottleneck
layer has the dimension 256 (used in the original paper [2]) which is also a
user defined shape. Optimizer used is Stochastic Gradient Descent (SGD) in
which parameters of deep neural network G and domain discriminator D are added
to optimize. During the training phase each batch from source and target data
loader are passed through deep neural network G and features f from bottleneck
layer and fully connected layer output are obtained from the deep neural
network G. Thus obtained features f and final layer output of source and
target datasets are joined together. Concatenated final layer outputs are
passed through softmax layer in order to get network predictions g.

As we obtained features f and predictions g, classification loss for source
data can be calculated using $g_{s}$ and ground truth labels of source data.
Transfer loss is calculated based on domain adaptation method selected. If
CDAN is selected, joint variable h=(f,g) which have both source and target
data features and predictions are used. Thus transfer loss is calculated using
domain discriminator network whose input is h which is explained in above
Sections on CDAN. If CDAN+E is selected, along with the features and
predictions we calculated entropy from class predictions which is explained in
above Section. CDAN+E CDAN+E. This calculated entropy is used to perform
weighted operations based on uncertainty of the predictions. Total loss to do
back propagation is calculated using below equation.

$Total\ loss=classification\ loss+\lambda\ transfer\ loss$ (13)

where $\lambda$ is a hyperparameter which is described in the above Sections.
During the experiment it is observed that transfer loss obtained is very small
which resulted in very very small gradients of order $10^{-7}$. Which in turn
did not update the weights of domain discriminator and thus the transfer loss
is fluctuated with very small variance. In Table I we can observe that CDAN+E
only performed better in 2 cases out of 6. It is also observed that transfer
loss is reducing till $20^{th}$epoch in most of the cases. In Fig. 9(c) and
9(d), training and testing accuracy as well as in Fig. 10(c) and 10(d),
transfer loss and classification loss of A$\to$W domain adaptation is shown
respectively. In our opinion smaller batch size also played a key role in
results of CDAN and CDAN+E methods. Smaller batch size is chosen considering
the limitations of local machine. Although the results are not matched with
original paper we observe that there is variation in accuracy and loss when
transfer loss is included.

(a) DeepCORAL

(b) DDC

(c) CDAN

(d) CDAN+E

Figure 9: Classification and transfer loss for each of the evaluated methods
as a function of epochs.

(a) DeepCORAL

(b) DDC

(c) CDAN

(d) CDAN+E

Figure 10: Image recognition accuracies for the four evaluated methods. The domain shift in this case corresponds to Amazon to Webcam. Red and green lines correspond to the classification accuracy on the source test data. Blue and orange lines correspond to the target test data. TABLE I: Testing accuracy for different domain adaptation techniques across different datasets. |  A $\to$ W |  A $\to$ D |  W $\to$ A |  W $\to$ D |  D $\to$ A |  D $\to$ W
---|---|---|---|---|---|---
No Adaptation |  43.1$\pm$2.5 |  49.2$\pm$3.7 |  35.6$\pm$0.6 |  94.2$\pm$3.1 |  35.4$\pm$0.7 |  90.9$\pm$2.4
DeepCORAL | 49.5$\pm$2.7 |  40.0$\pm$3.3 | 38.3$\pm$0.4 |  74.4$\pm$4.3 | 38.5$\pm$1.5 | 89.1$\pm$4.4
DDC |  41.7$\pm$9.1 | — | — | — | — | —
CDAN |  44.9$\pm$3.3 |  49.5$\pm$4.6 |  34.8$\pm$2.4 |  93.3$\pm$3.4 |  32.9$\pm$2.4 |  88.3$\pm$3.8
CDAN+E |  48.7$\pm$7.5 | 53.7$\pm$4.7 |  35.3$\pm$2.7 | 93.6$\pm$3.4 |  33.9$\pm$2.2 |  87.7$\pm$4.0

## Conclusions

Based on the methods we evaluated, we can say there are three possible
scenarios for accuracy performance. Note that in the graphs presented here,
the accuracy increases when we add a domain adaptation loss (e.g. Amazon
$\,\to\,$ Webcam), however, we noticed that accuracy can decrease or stay the
same for other domain shifts. Refer to the repository to see the other plots.
Moreover, recognition accuracies don’t go higher than 70% in most experiments,
this means there is a lot of room for improvement. For future work we can
carry out trainings for 200 epochs (not 100) and understand better the
behavior of our losses, for instance why in some cases losses oscillate too
much. Finally, we can conclude that we gained hands-on experience building
end-to-end pipelines using deep neural networks for domain adaptation tasks.

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# Discovery of the Tadpole Molecular Cloud near the Galactic Nucleus

Miyuki Kaneko School of Fundamental Science and Technology, Graduate School of
Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama,
Kanagawa 223-8522, Japan  Tomoharu Oka School of Fundamental Science and
Technology, Graduate School of Science and Technology, Keio University, 3-14-1
Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan  Department of Physics,
Institute of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-
ku, Yokohama, Kanagawa 223-8522, Japan  Hiroki Yokozuka Department of Physics,
Institute of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-
ku, Yokohama, Kanagawa 223-8522, Japan  Rei Enokiya Department of Physics,
Institute of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-
ku, Yokohama, Kanagawa 223-8522, Japan  Shunya Takekawa Faculty of
Engineering, Kanagawa University 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama,
Kanagawa 221-8686. Japan  Yuhei Iwata Division of Science, National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Center for Astronomy, Ibaraki University, 2-1-1 Bunkyo, Mito, Ibaraki
310-8512, Japan  Shiho Tsujimoto School of Fundamental Science and Technology,
Graduate School of Science and Technology, Keio University, 3-14-1 Hiyoshi,
Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan

(Received 2022 August 18; Revised 2022 November 23; Accepted 2022 November 25)

###### Abstract

In this paper, we report the discovery of an isolated, peculiar compact cloud
with a steep velocity gradient at $2\farcm 6$ northwest of Sgr A*. This
“Tadpole” molecular cloud is unique owing to its characteristic head-tail
structure in the position-velocity space. By tracing the CO J=3–2 intensity
peak in each velocity channel, we noticed that the kinematics of the Tadpole
can be well reproduced by a Keplerian motion around a point-like object with a
mass of $1\\!\times\\!10^{5}\,M_{\odot}$. Changes in line intensity ratios
along the orbit are consistent with the Keplerian orbit model. The spatial
compactness of the Tadpole and absence of bright counterparts in other
wavelengths indicate that the object could be an intermediate-mass black hole.

galaxies: nuclei — Galaxy: center — ISM: clouds — ISM: molecules

††journal: ApJ

##  1 Introduction

It is widely accepted that large galaxies host a central supermassive black
hole (SMBH) with millions to billions times the mass of the Sun (e.g.,
Kormendy & Richstone 1995; Kormendy & Ho 2013). A potential scenario for SMBH
formation is based on intermediate-mass black holes (IMBHs), which have masses
of $10^{2\mbox{--}5}\,M_{\odot}$ (e.g., Mezcua et al. 2017). Thus, detecting
and studying IMBHs in detail are essential for understanding the formation and
evolution of galactic nuclei. Numerous IMBH candidates have been identified in
centers of globular clusters (e.g., Kızıltan et al. 2017), in nuclei of dwarf
galaxies (e.g., Reines et al. 2013; Baldassare et al. 2015), or as
ultraluminous X-ray sources in extragalaxies (e.g., Farrell et al. 2009).

Figure 1: Map of velocity-integrated CO J=3–2 emission. White contours are
drawn at 250 K km s-1 intervals from 250 K km s-1. The integrated intensity
was calculated by $\int_{-200}^{200}T_{\rm
MB}dV\\!+\\!5\int_{-150}^{-80}T_{\rm MB}dV$ to emphasize the Tadpole, which
appears at $(l,b)\\!=\\!(-0\fdg 090,-0\fdg 014)$. CO 0.02–0.02 (Oka et al.
1999, 2008), the C1 cloud (Oka et al. 2011; Takekawa et al. 2017), and the
negative longitude extension of the circumnuclear disk (NLE; Oka et al. 2011;
Takekawa et al. 2017) also appear in this map. The white rectangle indicates
the area presented in Figures 2–5

In the central molecular zone (CMZ) of our Galaxy, a number of compact
($d\\!<\\!5$ pc) clouds with extraordinary broad velocity width ($\Delta
V\\!>\\!50$ km s-1) have been detected (e.g., Oka et al. 1998, 1999, 2012,
2022). These peculiar clouds, namely, high velocity-dispersion compact clouds
(HVCCs), have been assumed to be accelerated by supernova explosions,
protostellar outflows, and/or cloud-to-cloud collisions (e.g., Oka et al.
2022). Subsequently, it was determined that the kinematics of CO–0.40–0.22,
which is one of the most energetic HVCCs, can be well reproduced by a cloud
being gravitationally kicked by a point-like mass of $\sim\\!10^{5}$
$M_{\odot}$ (Oka et al. 2016, 2017). The absence of any bright object near the
point-like mass suggest that it may be an IMBH. Subsequently, it was also
suggested that HVCCs HCN–0.009–0.044 (Takekawa et al. 2019a), HCN–0.085–0.094
(Takekawa et al. 2020), and CO–0.31+0.11 (Takekawa et al. 2019b) were driven
by an IMBH. These discoveries yielded a new method of finding non-luminous
massive objects, such as inactive and wandering BHs. To date, five IMBH
candidates, including IRS13E (Tsuboi et al. 2019), have been reported in the
Galactic CMZ.

When searching for gravitationally kicked gas in the CMZ, we noticed an
isolated HVCC in the CO J=3–2 data obtained with the James Clerk Maxwell
Telescope (JCMT; Parsons et al. 2018; Eden et al. 2020). It appears as an
isolated compact cloud at $(l,b)\\!\simeq\\!(-0\fdg 090,-0\fdg 014)$, which
corresponds to $\sim\\!2\farcm 6$ Galactic northwest of Sgr A* (Figure 1),
with LSR velocities between $-140$ km s-1 and $-90$ km s-1. It stands out with
its peculiar appearance and very high CO J=3–2/CO J=1–0 intensity ratio
($R_{3\mbox{--}2/1\mbox{--}0}\\!=\\!1.8$ Oka et al. 2022), which exceeds
double of the CMZ average ($R_{3\mbox{--}2/1\mbox{--}0}\\!\sim\\!0.7$ Oka et
al. 2007, 2012). In this paper, we report the discovery of the so-called
“Tadpole” molecular cloud, which is listed as id.75 in the catalog of Oka et
al. (2022). Throughout this paper, the distance to the Galactic center is
assumed to be $D\\!=\\!8.3$ kpc (Gravity Collaboration 2018).

Figure 2: (a) Map of velocity-integrated CO J=3–2 emission. The velocity range
for integration is $V_{\rm LSR}\\!=\\!-140$ to $-80$ km s-1. The intensity
unit is K km s-1. (b) Longitude-velocity map of CO J=3–2 emission at
$b\\!=\\!-0\fdg 014$ (the magenta line in panel (a)). (c) Map of velocity-
integrated 12CO J=1–0 emission. (d) Map of velocity-integrated 13CO J=1–0
emission. (e) Map of velocity-integrated CS J=2–1 emission. The integration
ranges for these lines are the same as that of the CO J=3–2 line.

##  2 Data

We first discovered the Tadpole in the JCMT CO J=3–2 data, and confirmed it in
the CO J=1–0 and CS J=2–1 line data obtained with the Nobeyama Radio
Observatory (NRO) 45 m telescope. These data sets are briefly described below.

###  2.1 CO J=3–2 Line

The 12CO J=3–2 line ($345.795990$ GHz) observations of the CMZ were performed
using the JCMT from 2013 July to 2014 July (Parsons et al. 2018). The
Heterodyne Array Receiver Program (HARP; Buckle et al. 2009) and
autocorrelation spectral imaging system (ACSIS) were used during these
observations. The half-power beam width (HPBW) of the telescope was
approximately $14\arcsec$ at 345 GHz. The ACSIS was operated in the $1$ GHz
bandwidth ($976.56$ kHz resolution) mode. During these observations, the
system noise temperature ($T_{\rm sys}$) ranged between $100\mbox{--}200$ K.
The rms noise level of the image cubes was between $0.4$ K and $0.84$ K. The
details of the CO J=3–2 data and JCMT observations are presented in Parsons et
al. (2018). We use the data after resampling onto a $7\farcs
5\\!\times\\!7\farcs 5\\!\times\\!1$ km s-1 grid.

###  2.2 CO J=1–0 Lines

The CO J=1–0 observations of the CMZ were performed using the Nobeyama Radio
Observatory (NRO) 45 m telescope (Tokuyama et al. 2019). The 12CO J=1–0
(115.27120 GHz) line data were obtained from 2011 January 19 to 29 using the
25 beam array receiver system (BEARS; Sunada et al. 2000). As the receiver
backend, the AC45 spectrometer system (Sorai et al. 2000) was employed in the
500 MHz bandwidth (0.5 MHz resolution) mode. The 13CO J=1–0 (110.20135 GHz)
line data were obtained from 2016 February to March using the four-beam
receiver system on the 45 m telescope (FOREST; Minamidani et al. 2016). The
spectral analysis machine on the 45 m telescope (SAM45; Kuno et al. 2011;
Kamazaki et al. 2012) was operated in the 1 GHz (244.14 kHz resolution) mode.
The HPBW of the telescope was approximately $15\arcsec$ at 115 and 110 GHz.
The typical $T_{\rm sys}$ was $\sim\\!800$ K and $150\mbox{--}300$ K during
the 12CO and 13CO observations, respectively. The data were resampled onto a
$7\farcs 5\\!\times\\!7\farcs 5\\!\times\\!2$ km s-1 grid. The rms noise
levels of the resultant 12CO and 13CO data cubes were 1.0 K and 0.2 K in main-
beam temperature ($T_{\rm MB}$), respectively.

###  2.3 CS J=2–1 Line

The CS J=2–1 (97.98096 GHz) line observations of the CMZ were performed during
the NRO 45 m Telescope Large Program through 2019 January–May and 2020
January–April. The mapping area was set to $-1\fdg
5\\!\leq\\!l\\!\leq\\!+1\fdg 5$ and $-0\fdg 25\\!\leq\\!b\\!\leq\\!+0\fdg 25$.
The FOREST receiver and SAM45 spectrometer were used. The SAM45 was operated
in the 1 GHz bandwidth ($244.14$ kHz resolution) mode. The HPBW of the
telescope was $\simeq\\!19\arcsec$ at 86 GHz. The $T_{\rm sys}$ ranged from
150–300 K during the CS J=2–1 line observations. The data were resampled onto
a $7\farcs 5\\!\times\\!7\farcs 5\\!\times\\!2$ km s-1 grid. The rms noise
level of the resultant CS J=2–1 data cube was $0.14$ K in $T_{\rm MB}$. The
details of the CS data and NRO 45 m observations will be presented in the
forthcoming paper (Takekawa et al. 2022, in preparation).

Figure 3: Observed line spectra at (–0 $.\\!\\!^{\circ}$091, –0
$.\\!\\!^{\circ}$016) at where the CO J=3–2 profile of the Tadpole is widest.
Black, red, blue and green lines show CO J=3–2, 12CO J=1–0, 13CO J=1–0 and CS
J=2–1 line spectra, respectively.  Figure 4: Velocity channel maps of the CO
J=3–2 line from $\hbox{$V_{\rm LSR}$}\\!=\\!-140$ to $-85$ km s-1 near the
Tadpole. The angular area is the same as that depicted in Figure 2a, c–e. The
intensity unit is K km s-1. The velocity range for integration is indicated at
the bottom right corner of each panel. The magenta cross denotes the position
of the maximum intensity pixel in each panel. The distribution of the maximum
intensity pixels is shown in the bottom right panel.

##  3 Results

###  3.1 Spatial and Velocity Structure

Figure 2 shows the spatial and velocity structure of the Tadpole in various
molecular lines. The Tadpole appears as a compact, well-defined clump in
velocity-integrated maps (Figure 2a, c–e). In the CO J=3–2 integrated map
(Figure 2a), the full width of half maximum (FWHM) angular diameter of the
Tadpole is $28\arcsec$. This angular diameter is $2$ times the angular
resolution ($14\arcsec$), corresponding to a distance of $1.1$ pc from the
Galactic center. In the 13CO J=1–0 and CS J=2–1 maps, the Tadpole has angular
sizes similar to that depicted in the CO J=3–2 map, while the 12CO J=1–0 map
depicts an appearance that is larger by a factor of 2 compared with those
depicted in the other maps.

The velocity extent of the Tadpole is from $\hbox{$V_{\rm
LSR}$}\\!\simeq\\!-135$ km s-1 to $-90$ km s-1 (Figure 2b), indicating that it
may be in the CMZ. The longitude-velocity behavior of the Tadpole is
characterized by a “head-tail” structure, from which its name is derived. It
depicts a steep velocity gradient of $\left|\Delta V/\Delta
l\right|\\!\sim\\!16$ km s-1pc-1. The tail is obscure in the 13CO J=1–0 and CS
J=2–1 data sets. We show the spectra of CO and CS lines in Figure 3.

We learned that CO J=3–2 line can trace the kinematics of the Tadpole best
because of its high intensity and well-defined appearance in the l–b–V space.
Figure 4 shows the velocity channel maps of CO J=3–2 emission. The Tadpole
appears with an elliptical shape in each velocity channel, changing its
angular size with respect to velocity. It has the largest angular size of
$36\arcsec\\!\times\\!24\arcsec$ in the $\hbox{$V_{\rm LSR}$}\\!=\\!-130$ to
$-125$ km s-1 channel, while the smallest size of $\sim\\!20\arcsec$ is
detected at both velocity ends. These angular sizes, which are comparable to
the telescope’s HPBWs, indicate that the spatial structure of the Tadpole is
not well resolved using these single-dish observations; thus, the actual
angular size must be far smaller than those observed.

The CO J=3–2 line emission shows a maximum intensity of $64$ K km s-1 in the
$-130$ to $-125$ km s-1 channel. We identified that the intensity maximum
pixel in the Tadpole at each velocity channel continuously changes with
respect to velocity. Note that these intensity maximum pixels trace an arc in
the plane of the sky (Figure 4, bottom right panel). This behavior will be
analyzed in more detail in the discussion section (§4.1).

Figure 5: Multiwavelength view near the Tadpole. White contours show the CO
J=3–2 integrated intensity with a $50$ K km s-1 interval. (a) Radio continuum
image at 1.284 GHz obtained with the MeerKAT. (b) Composite mid-infrared image
obtained with the Spitzer Space Telescope. The image depicts 8.0 $\mu$m band
flux in red, 5.8 $\mu$m in green, and 4.5 $\mu$m in blue. (c) Composite X-ray
image obtained with the Chandra X-ray Observatory. The image depicts
$0.2\mbox{--}3.3$ keV count rate in red, $3.3\mbox{--}4.7$ keV in green, and
$4.7\mbox{--}8.0$ keV in blue.

###  3.2 Physical Parameters

The physical size of the Tadpole was evaluated as the size parameter; defined
by $S\\!\equiv\\!D\tan\left(\sqrt{\sigma_{l}\sigma_{b}}\right)$ to be $2.2$
pc. The velocity dispersion was calculated to be $\sigma_{V}\\!=\\!22$ km s-1.
These give a size-linewidth coefficient ($\sigma_{V}/S^{0.5}$) of $15$ km
s-1pc-0.5 and virial theorem mass ($M_{\rm
VT}\\!\equiv\\!8.7S\sigma_{V}^{2}/G$) of $4.7\\!\times\\!10^{5}$ $M_{\odot}$.
The molecular gas mass ($M_{\rm gas}$) was estimated by summing the CO J=3–2
line integrated intensity and using the CO(3–2)-to-H2 conversion factor
($X_{\rm CO3\mbox{--}2}\\!=\\!1.4\\!\times\\!10^{20}$ [cm-2 (K km s-1)-1] Oka
et al. 2022) to be $6.6\\!\times\\!10^{2}$ $M_{\odot}$. The $X_{\rm
CO3\mbox{--}2}$ we employed is close to $1\\!\times\\!10^{20}$ that estimated
from an LVG calculation for $n({\rm H}_{2})\\!=\\!1\\!\times\\!10^{4}$ cm-3,
$T_{\rm k}\\!=\\!60$ K, and $N({\rm CO})/dV\\!=\\!1\\!\times\\!10^{17}$ cm-2
(km s-1)-1. Using the physical parameters described above, we derived the
dynamical timescale ($t_{\rm dyn}\\!\equiv\\!S/\sigma_{V}$), kinetic energy
($E_{\rm kin}\\!\equiv\\!1.5M_{\rm gas}\sigma_{V}^{2}$), and kinetic power
($P_{\rm kin}\\!\equiv\\!E_{\rm kin}/t_{\rm dyn}$) of the Tadpoe as
$2.2\\!\times\\!10^{4}$ yr, $9.4\\!\times\\!10^{48}$ erg, and
$1.4\\!\times\\!10^{38}$ erg s-1, respectively. The kinetic power of the
Tadpole is equal to $3.7\times 10^{4}\,L_{\odot}$ which is far greater than
those of molecular outflows from massive YSOs (0.01–100 $L_{\odot}$; Maud et
al. 2015).

The virial theorem mass and molecular gas mass of the Tadpole yield a
significantly high virial parameter, $M_{\rm VT}/M_{\rm gas}\\!\sim\\!700$.
This indicates that the Tadpole is not in gravitational equilibrium. Following
the method described in Stark et al. (1989), we estimated the self-gravity of
the Tadpole and tidal force by the Galactic potential. The self-gravity of the
cloud estimated here ($\sim\\!10^{-10}$ m s-2) is 2.5 orders of magnitude
weaker than the tidal force ($\sim\\!4\\!\times\\!10^{-8}$ m s-2). These
assessments clearly demonstrate that the Tadpole cannot be bound by its self-
gravity. This strongly indicates to the presence of an object with a mass
comparable to the virial mass ($4.7\\!\times\\!10^{5}$ $M_{\odot}$) inside the
Tadpole.

###  3.3 Multiwavelength View

To search for the driving source behind the Tadpole, we referred to the 1.284
GHz data obtained using MeerKAT (Heywood et al. 2022), mid-infrared data
obtained using the Spitzer Space Telescope (Ramírez et al. 2008; Churchwell et
al. 2009), and X-ray data obtained at the Chandra X-ray Observatory (Muno et
al. 2009).

No radio source brighter than 0.4 mJy beam-1 at 1.284 GHz was detected near
the angular extent of the Tadpole, while a faint filament was seen in the
Galactic south (Figure 5a). In the mid-infrared image (Figure 5b), we
identified a faint triangular rim which may be the irradiated surface of the
Tadpole. Considering the angular resolution of the Spitzer
($\sim\\!2\arcsec$), the mid-infrared rim defines the angular extent of the
Tadpole better than the CO J=3–2 appearance. We also checked the ATLASGAL
(870$\mu$m) and Hi-GAL (70, 160, 250, 350 and 500 $\mu$m) images (Contreras et
al. 2013; Molinari et al. 2016), and found no separate feature toward the
Tadpole above the sea of intense dust emission. A bright point-like source
near the Galactic southwestern edge of the Tadpole is the long-period variable
star V4872 Sgr (Matsunaga et al. 2009). In addition, we also identify dozens
of point-like, mid-infrared sources toward the Tadpole. The X-ray image also
shows numerous faint point-like sources toward the Tadpole (Figure 5c).
Although the nature of these mid-infrared/X-ray point-like sources is unclear,
we will refer to them in the discussion section (§4.5). In short, the
multiwavelength view confirms the absence of any energetic objects that can
drive the Tadpole.

##  4 Discussion

###  4.1 Tracing Molecular Gas Kinematics

The continuous change in the CO J=3–2 intensity maximum pixel along the arc
with velocity (§3.1) suggests that the bulk of warm molecular gas belongs to a
certain trajectory, such as a closed orbit. To determine the orbit trajectory
accurately, we constructed a CO J=3–2 data cube with a
$2\arcsec\\!\times\\!2\arcsec\\!\times\\!1$ km s-1 grid. Then, we calculated
the accurate intensity peak position as the CO emission center of gravity
using $5\\!\times\\!5$ pixels around the intensity maximum pixel in the
Tadpole for each $1$ km s-1 width velocity channel.

Figure 6 shows the l–b–V and l–b distributions of the intensity peak position
in the Tadpole for each velocity channel. The arc-shape apparent in the l–b
distribution became clearer than that depicted in Figure 4. The l–b–V
distribution demonstrates the striking continuity of peak positions in the
velocity direction, indicating that the bulk of warm molecular gas in the
Tadpole may predominantly follow a certain closed orbit.

Figure 6: The l–b–V distribution of CO J=3–2 intensity peak positions in the Tadpole in each $1$ km s-1 width velocity channel. The head and tail of the Tadpole are denoted by blue and green, respectively. The l–b distribution is also shown in the base plane with CO J=3–2 contours. Grey lines represent the loci of the best-fit Keplerian orbit (Table 1). The black and red filled circles denote the positions of the dynamical center and pericenter, respectively. The black arrow indicates the direction of rotation. Table 1: Parameters of the best-fit Keplerian Orbit Parameter | Value
---|---
Mass ($M_{\mathrm{dyn}}$) |  $(1.01\pm 0.05)\times 10^{5}$ $M_{\odot}$
Semimajor axis ($a$) |  $0.75\pm 0.01$ $\mathrm{pc}$
Eccentricity ($e$) | $0.35\pm 0.03$
Inclination ($i$) | $112\arcdeg\pm 1\arcdeg$
Argument of pericenter ($\omega$) | $82\arcdeg\pm 5\arcdeg$
Longitude of ascending node ($\Omega$) | $-35\arcdeg\pm 1\arcdeg$
Longitude offset ($l_{0}$) | $-0\fdg 0946\pm 0\fdg 0001$
Latitude offset ($b_{0}$) | $-0\fdg 0146\pm 0\fdg 0001$
Velocity offset ($V_{0}$) |  $-114\pm 1$ km s-1

###  4.2 Keplerian Orbit Fitting

The arc-shaped spatial structure and continuous velocity change along the arc
suggest that a rotational motion exists. Thus, we expect that the l–b–V
behavior of warm molecular gas in the Tadpole can be reproduced by a Keplerian
orbit around a point mass111Here we do not consider parabollic or hyperbolic
orbits, although the orbit could not be an elliptical.. We performed the
fitting of a Keplerian orbit to the intensity peak positions (Figure 6)
following the method described in Zhao et al. (2009). We determined nine
parameters. Five of these parameters are three-dimensional orbital parameters;
namely, the semimajor axis $(a)$, eccentricity $(e)$, longitude of ascending
node $(\Omega)$, argument of pericenter $(\omega)$, and inclination angle
$(i)$. The remaining parameters are the mass $(M_{\rm dyn})$, line-of-sight
velocity $(V_{\rm dyn})$ and l–b position $(l_{0},\,b_{0})$ of the dynamical
center. The distance to the Tadpole is assumed to be same as that to the
Galactic center. The $\chi^{2}$ minimization approach was utilized in the
l–b–V space. The $\chi^{2}$ consists of two terms, namely, the spatial
($\chi^{2}_{\rm s}$) and velocity terms ($\chi^{2}_{\rm v}$). The spatial term
is defined by the sum of the orthogonal distances between the peak position
and the modeled orbit divided by the square of the positional uncertainty,
which was set to $0.15$ pc ($3\farcs 6$). The velocity term is, similarly, the
sum of the velocity deviations between those of the peak positions and nearest
positions in the modeled orbit divided by a square of the velocity
uncertainty, which was set to $1$ km s-1.

The best-fit parameters with $1\sigma$ uncertainties are listed in Table 1.
The l–b–V locus of the orbit is presented in Figure 6. Note that the best-fit
solution is bivalent because the orbits with
$i\\!\rightarrow\\!(180\arcdeg\\!-i)$ and
$\Omega\\!\rightarrow\\!(\Omega+180\arcdeg)$ yield the same l–b–V locus. The
fitting result indicates the presence of a point mass of
$1.0\\!\times\\!10^{5}$ $M_{\odot}$ in the northwestern periphery of the
Tadpole (Figure 6).

Figure 7: Plots of the tidal force due to the point mass (solid line) and
self-gravity of the cloud (dashed lines) working on the head versus the head
diameter. A central mass of $10^{5}$ $M_{\odot}$ and head mass of $660$
$M_{\odot}$ were assumed. The thick solid line indicates self-gravity at the
pericenter distance (0.49 pc) while the thin solid line indicates that at the
apocenter (1.01 pc).

###  4.3 Tidal Stability of the Head

We examined the tidal stability of the head, assuming the Newtonian potential
of a point mass, following the method described in Stark et al. (1989). The
tidal force caused by the point mass and self-gravity doing work on the head
were calculated as functions of the head diameter (Figure 7). A central mass
of $1\\!\times\\!10^{5}$ $M_{\odot}$ and head mass of $660$ $M_{\odot}$ were
assumed. The curves of the two forces at the pericenter of the best-fit orbit
(0.5 pc) intersect at $d\\!=\\!0.14$ pc, while those at the apocenter
intersect at $d\\!=\\!0.35$ pc. Because the observed head diameter ($\sim\\!1$
pc) is larger than the critical value (0.14 pc) by a factor of seven, the
Tadpole head is tidally unstable at the pericenter. If the head includes a
self-gravitating core smaller than 0.14 pc, it may be able to survive several
turns. It is possibly that the Tadpole has been trapped by the gravitational
potential of the $10^{5}$ $M_{\odot}$ point mass, being stretched by the
strong tidal force to form the characteristic head-tail structure. This
situation is similar to the simulation of a cloud tidally disrupted by a super
massive black hole (Saitoh et al. 2014). Note that the Tadpole must have been
lost its angular momentum to be captured during the encounter with the point
mass.

Figure 8: (a) Plots of CO J=3–2 (filled circles), CO J=1–0 (crosses), and CS
J=2–1 (open circles) main-beam temperatures at the CO J=3–2 intensity peaks
versus orbital phase. (b) Plots of $R_{3\mbox{--}2/1\mbox{--}0}$ (filled
circles) and $R_{\rm CS/CO}$ (open circles) at the CO J=3–2 intensity peaks
versus orbital phase. The phase ranges of the head and tail of the Tadpole are
depicted in blue and green shades, respectively.

###  4.4 Intensity Ratios along the Orbit

Here, we refer to the CO J=1–0, CO J=3–2, and CS J=2–1 line intensities, as
well as CO J=3–2/CO J=1–0 ($R_{3\mbox{--}2/1\mbox{--}0}$) and CS J=2–1/CO
J=1–0 ($R_{\rm CS/CO}$) line intensity ratios to examine the Keplerian orbit
scenario. The critical densities of the CO J=1–0, CO J=3–2, and CS J=2–1
transitions are approximately $10^{2.5}$, $10^{4}$, and $10^{5}$ cm-3,
respectively, while their upper state energies are $5.6$, $33$, and $7.1$ K,
respectively. Thus, $R_{\rm CS/CO}$ is sensitive to the variations in density
while $R_{3\mbox{--}2/1\mbox{--}0}$ is affected by both the temperature and
density. Figure 8 shows plots of line intensities and intensity ratios along
the best-fit orbit. The motion along the orbit increases the orbital phase
($\phi$) at all times, where the pericenter of the orbit corresponds to
$\phi\\!=\\!0\arcdeg$.

Both ratios exhibit higher values at the head, indicating that both the
density and temperature are enhanced there. At the beginning of the tail,
$\phi\\!\sim\\!170\arcdeg$, $R_{3\mbox{--}2/1\mbox{--}0}$ demonstrates a
prominent peak while $R_{\rm CS/CO}$ does not. This suggest that the
temperature is enhanced at the beginning of the tail. Then, both ratios
decrease toward the tip of the tail. The higher temperature in the head may
have been caused by the shock occurring at the first encounter with the point
mass. This shock may be caused by collisions of gas clumps at the pericenter,
where adjacent orbits are very close. In a fluid mechanical treatment, the
same process is described as the “tidal compression”(Saitoh et al., 2014). The
gas accumulation near the apocenter and/or the presence on a self-gravitating
core can cause the higher density in the head. The tail may be really the low
density, tidally stretched tail of the head. The temperature increase at the
beginning of the tail could be caused by another shock at the congestion near
the apocenter. Thus, the behaviors of intensity ratios are consistent with the
Keplerian orbit scenario for the Tadpole.

Figure 9: Distribution of X-ray point-like sources (open yellow circles: Muno
et al. 2009, filled yellow circles: Zhu et al. 2018) superimposed on the
composite mid-infrared image (color; same as Fig.5b). The red cross denotes
the dynamical center of the best-fit Keplerian orbit, and the green dotted
circle has a $5\arcsec$ radius. Black contours highlight the CO J=3–2
integrated intensity with a 50 K km s-1 interval.  Table 2: Properties of
X-ray and mid-infrared sources near the point mass

|  $l$ | $b$ |  $F_{{\rm 2\mbox{--}10keV}}$${}^{\text{a}}$ | [3.6 $\mu$m]${}^{\text{b}}$ | [4.5 $\mu$m]${}^{\text{b}}$ | [5.8 $\mu$m]${}^{\text{b}}$ | [8.0 $\mu$m]${}^{\text{b}}$
---|---|---|---|---|---|---|---
ID |  |  |  |  |  |  |
| (deg) | (deg) | (10${}^{-18}\,\mathrm{W/m^{2}}$) | (mag) | (mag) | (mag) | (mag)
SSTGC 489898${}^{\text{b}}$ | –0.09513 | –0.01486 | 1.66 | 10.284 |  $\cdots$ |  $\cdots$ |  $\cdots$
SSTGC 490125${}^{\text{b}}$ | –0.09384 | –0.01444 |  $\cdots$ | 9.154 | 9.003 | 8.422 | 8.217
CXOGC 174526.9–290124${}^{\text{c}}$ | –0.09400 | –0.01346 | 16.72 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$

Note. —

a The X-ray (2–10 keV) energy flux quoted from Zhu et al. (2018).

b The source IDs and mid-infrared magnitudes are quoted from Ramírez et al.
(2008).

c The source ID defined in Muno et al. (2009).

###  4.5 What is a Point Mass?

The Keplerian orbit model requires the presence of a huge point mass at the
dynamical center. What is this point-like massive object? The mass of
$10^{5}\,M_{\odot}$ is larger than the Arches or Quintuplet clusters
($\sim\\!10^{4}\,M_{\odot}$ Figer et al. 1999a, b). According to the best-fit
model, a mass of $10^{5}\,M_{\odot}$ must be concentrated within a radius
significantly smaller than 0.5 pc (the pericenter distance), resulting in an
average mass density higher than that of the Arches cluster
($\rho\\!\sim\\!2\\!\times\\!10^{5}\,M_{\odot}\,\mathrm{pc}^{-3}$ Espinoza
2009). Anyway, the absence of a bright infrared counterpart toward the Tadpole
(Ramírez et al. 2008; Churchwell et al. 2009; Molinari et al. 2011) clearly
rules out a stellar cluster from being a candidate for the point mass.

The most promising candidate for the point-like massive object in the Tadpole
may be an IMBH. We searched for a counterpart for this point mass referring to
X-ray images (§3.3). Figure 9 shows the positions of point-like sources in the
X-ray images superimposed on the composite mid-infrared images around
$(l,b)\\!=\\!(l_{0},b_{0})$. The small statistical uncertainties in the l–b
position of the dynamical center (Table 1) should be considered with some
caution because the entire gas mass in the Tadpole is not confined within the
single closed orbit. Notice that three point-like sources in the X-ray and
mid-infrared reside within a $5\arcsec$ ($\sim\\!10$ times of the positional
ambiguity of the dynamical center) angular distance from $(l_{0},b_{0})$
(Table 2). One source, SSTGC 489898, is apparent in both X-ray and mid-
infrared images. The X-ray detected sources, SSTGC 489898 and CXOGC
174526.9–290124, exhibit hard spectra, indicating that they may be at the same
distance to the Galactic center. No time variation has been detected from
these sources (Zhu et al., 2018). If we assume the standard accretion disk
model ($L\\!=\\!\dot{M}c^{2}/12$), their X-ray luminosities correspond to
$\dot{M}\\!\sim\\!1\\!\times\\!10^{-14}\;M_{\odot}\mathrm{yr}^{-1}$. This very
low mass accretion rate could be a challenge to our interpretation. Anyway, we
suppose these three point-like sources are candidates for the luminous
counterpart of the point mass. Although we currently have very little
knowledge on these point-like sources, they should be considered as candidates
for an IMBH in future studies.

##  5 Conclusions

We have discovered the so-called Tadpole, which is an isolated, peculiar
compact cloud with an extraordinary velocity width and very high CO
J=3–2/J=1–0 intensity ratio, at $2\farcm 6$ northwest of Sgr A*. Our main
conclusions can be summarized as follows:

  1. 1.

The Tadpole molecular cloud has a size of $\sim\\!1$ pc at the distance to the
Galactic center and a velocity width of $\sim\\!50$ km s-1.

  2. 2.

It demonstrates the characteristic “head-tail” structure in position-velocity
space, having a steep velocity gradient of $16$ km s-1pc-1.

  3. 3.

Its kinematics is well reproduced by a Keplerian motion around a point-like
object with a mass of $1\\!\times\\!10^{5}\,M_{\odot}$.

  4. 4.

It is plausible that the Tadpole has been trapped by the gravitational
potential of the huge point mass, now being stretched by the strong tidal
force.

  5. 5.

The behaviors of line intensity ratios ($R_{3\mbox{--}2/1\mbox{--}0}$ and
$R_{\rm CS/CO}$) are consistent with the Keplerian orbit scenario.

  6. 6.

The absence of bright objects near the putative point-like object suggests
that the object is an inactive intermediate-mass black hole (IMBH).

These results are based on molecular line maps with
$14\arcsec\mbox{--}19\arcsec$ resolutions obtained using single-dish
telescopes. Future aperture synthesis observations of molecular lines with
millimeter/submillimeter arrays will be able to delineate the orbit stream
directly, increasing the reliability of the Keplerian orbit model for the
Tadpole.

The results presented in this paper are based on data obtained using the
Nobeyama Radio Observatory (NRO) 45-m telescope and James Clerk Maxwell
Telescope (JCMT). The NRO 45-m radio telescope is operated by the Nobeyama
Radio Observatory, a division of the National Astronomical Observatory of
Japan.

The James Clerk Maxwell Telescope is operated by the East Asian Observatory on
behalf of The National Astronomical Observatory of Japan, Academia Sinica
Institute of Astronomy and Astrophysics, Korea Astronomy and Space Science
Institute, National Astronomical Research Institute of Thailand, and Center
for Astronomical Mega-Science (as well as the National Key R& D Program of
China with No. 2017YFA0402700). Additional funding support is provided by the
Science and Technology Facilities Council of the United Kingdom, and
participating universities and organizations in the United Kingdom and Canada.

We are grateful to the NRO staff and all members of the JCMT team for
operating the telescope. T.O. acknowledges the financial support of JSPS
Grant-in-Aid for Scientific Research (A) No. 20H00178. S. Ta acknowledges
support from JSPS Grant-in-Aid for Early-Career Scientists Grant Number
JP19K14768.

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